Download Sun, Stars and Planets Part 1: The Sun: its structure and energy

Sun, Stars and Planets
J C Pickering 2014
Part 1: The Sun: its structure and energy generation
Lecture 1. Basic properties of the Sun – an introduction
1.1
1.2
1.3
1.4
Setting the scene…
Definition of a star
Properties of the Sun
How are these properties quantified?
1.1 Setting the scene
Earth- Sun distance = 1 AU (Astronomical Unit), or 8 minutes for light to reach us
(in light years)
Distances:
Nearest star Proxima Centauri
4.2
Sirius
8.6
Orion Nebula
1500
Galactic Centre
28,000
Galactic Diameter
100,000
Andromeda galaxy
2,300,000 (biggest galaxy in our Local Group of 30 galaxies
6 million lys across)
Virgo Cluster 50,000,000 (2000 galaxies, nearest other galactic cluster, and centre of our
Local Supercluster – diameter 100,000,000)
Most Distant Observed Object ~13,000,000,000
Number of stars and galaxies…
Galaxies contain anything from between 100,000 up to 3,000,000,000,000 (3 thousand billion)
stars!
Our Galaxy contains 100 billion stars.
Number of galaxies estimated in observable Universe – 200 - 300 billion…
So, it is vital we understand the physics of stars!
How do we go about describing the internal structure of stars and their evolution?
Evolution of stars is very slow; for a typical ordinary star like our Sun a small change in the Sun’s
properties would make the Earth uninhabitable for us, but we have been on Earth for hundreds of
thousands of years. Geologists believe the Earth’s crust has been solid for several thousand
million years, and that the Sun’s luminosity cannot have changed significantly during this time
span. So how can we make progress in our understanding of stars and their evolution… ?
What are the fundamental things we know about the Sun? and what is the Sun?
1.2 Definition of a star
A star is a self-gravitating mass of gas that radiates energy.
1.3 Properties of a star we need to know about
Before we can talk about evolution and internal structure we need to know the basic properties of a
star: it has a Mass, which means there is a Pressure, and a gas pressure means that we want to
know about the temperature of the gas. The star radiates heat which can be quantified by
Luminosity.
Mass pressure temperature heat luminosity
The Sun – is our closest star and has global properties:
mass
M = 1.99 x 1030 kg (one solar Mass)
(solar radius)
radius
R = 6.96 x 108 m
26
luminosity L = 3.83 x 10 W
(solar luminosity)
11
Sun-Earth mean distance = 1 Astronomical Unit (AU) = 1.50 x 10 m
1
1.4 How are these quantities determined?
1.4.1 Distance:
2
3
P µ D
P is planet’s orbital period
D is the planet’s orbital semimajor axis)
- gives relative scale of the solar system but not the
absolute scale;
then use accurate measurement of distance to a
planet to set the absolute scale e.g. radar-ranging to
Venus.
(Earlier methods of distance measurement: transit
observations; Greek astronomy)
Kepler’s 3rd law
1.4.2 Radius of the Sun:
Measure the angular size of the Sun, and knowing the Earth-Sun distance ⇒ radius
Using the small angle approximation, in radians, θ/2 = r/d
where θ = angular size of the Sun, r = Sun’s radius, d = Earth-Sun distance
1.4.3 Mass of the Sun:
Knowing the orbital motions of planets and their distance ⇒ G M to high precision.
[G = the gravitational constant, Ms mass of star, mp mass of planet, d star-planet
separation, ω angular velocity, P orbital period]
1.4.4 Luminosity and flux:
A star’s Luminosity, L, is the power (W) emitted by the entire surface of a star.
At a distance d from the star, its luminosity is found by measuring flux density F (Wm-2):
At distance d from a star, its luminosity is spread over a sphere of area 4 π d2. We assume that the
star radiates uniformly in all directions. So at any point on the sphere the flux density is
given by:
inverse-square law: F = L / (4π
πd2 )
( d = 1 A.U. )
For the Sun we can measure flux (energy incident from the Sun
per unit time per unit area) at the Sun-Earth distance,
known as the solar “constant” = 1368 W m-2
and from this we can calculate solar luminosity.
d
F
2
TO DO: calculate the solar luminosity using the solar constant 1368 Wm-2.
Also, we can compare
the solar flux at the Earth with
the solar flux at other planets:
for example to estimate the
surface temperature of other
planets.
1.4.5
TO DO: what would be the “solar
constant” as measured on:
(i) Mercury
(ii) Pluto ?
d2
d1
F1
F2
Surface temperature and radiation:
Reminder: Black-body Radiation.
A Black body:
- is in thermodynamic equilibrium: photons and matter have the same temperature.
Reach this through collisions/interactions, mean free path (mfp) is short, ie are opaque.
- The light within the source is more likely to interact with the material of the source than to
escape, it will only escape after considerable interaction with material within the source. So a
common feature of BB sources is that they are opaque.
At a given T, any body in thermodynamic equilibrium will show a black-body (≈ continuum)
spectrum.
For a star, the BB spectrum is a good approximation up to a point only – where the stellar
atmosphere’s mfp increases, a line spectrum is formed (and this can tell us about the star’s
composition, etc).
Many astronomical sources produce continuous spectra with reasonably good approximation
to BB form.
Planck’s law :
Wien’s law:
Stefan’s law:
3
Radiation from the surface of stars can be approximated to that of black-body radiation:
For black body, flux F per unit area of emitter
where σ = Stefan-Boltzmann constant
F = σ T4
(Stefan’s law)
The Effective temperature is defined such that a black body of temperature Teff with the same
radius as the star would radiate the same total amount of energy.
Define effective temperature of surface of star by:
L = 4 π R2 σ (Teff)4
where R is the radius of the star.
TO DO: For the Sun, using values of L and R from 1.3 calculate the effective temperature of
the Sun
L = 4 π R2 σ (Teff)4 Teff =
K
1.4.6 Chemical composition of the Sun:
We believe this to be similar to typical composition in the universe. The composition of a sample of
material at any depth within the Sun can be defined using 3 simple parameters:
hydrogen mass fraction X = mass of hydrogen in sample/mass of sample, and similarly
helium mass fraction Y, and metallicity Z = mass of other elements in sample/mass of sample,
X
Hydrogen
~73% by mass
Helium
~25%
Y
Heavier elements ~2%
Z (O, C, N, Ne, Fe, … in order of abundance)
How do we know this? : Observational data: solar spectrum, meteorites
1.4.7 Age of the Sun:
Only known indirectly: radioactive dating of rocks;
computed from evolutionary models of the Sun.
~ 4.6 x 109 years
LECTURE KEY POINTS:
• Definition of a star
• Sun’s global properties: mass, radius, luminosity
• Astronomical unit AU, solar units
• Luminosity, inverse square law, flux density
• Definition of effective temperature; Stefan’s law
• Sun’s chemical composition and age
Learning outcomes include:
• be able to give definitions of a star, astronomical unit, effective temperature
• be able to explain what is meant by luminosity, flux, and a black body source.
• have an awareness of how the Sun’s properties are measured, more details to follow later in the
course as indicated during the lecture.
• know Stefan’s law, and the definition of effective temperature such that you are able to write
down the equations and use them
To Do: -
Problem Sheet 1 - Question 1
Check you understand what a black body is
calculate the solar luminosity using the measured solar constant
Quick calculations: solar flux at Mercury and Pluto
Calculate the effective temperature of the Sun
4
Sun Stars and Planets
J C Pickering 2014
Part 1 continued: The Sun: its structure and energy generation
Lecture 2: Modelling the Stellar interior
Aims: derive equations governing the physical properties of a star and examine their validity
2.1 Hydrostatic equilibrium
2.2 Equation of mass continuity
2.3 How good an approximation is hydrostatic equilibrium?
2.4 How long would it take a star to collapse if pressure forces were negligible?
2.5 Mean density of the Sun
2.6 Very simple estimate of the mean pressure of the Sun
2.7 Estimate of minimum central pressure of the Sun
2.1 Hydrostatic equilibrium
A star is held together by the force of gravitation, the attraction exerted on each part of the star by
all other parts. If this force was the only important one then the star would rapidly shrink – but this
attractive gravitational force is resisted by the pressure of the stellar material in the same way that
the kinetic energy of the molecules, or equivalently the pressure of the Earth’s atmosphere,
prevents the atmosphere from collapsing to the surface of the Earth. These two forces,
gravitational attraction and thermal pressure, play key roles in determining stellar structure.
2.1.1. Assumptions
(a) assume stars are spherical and symmetrical about their centres
(b) The stellar properties change so slowly with time - neglect the rate of change with time of
these properties.
With these assumptions a star’s structure is governed by a set of equations in which all the
physical quantities depend only on the distance from the centre of the star.
2.1.2 Balance between pressure and gravitational forces:
Consider the forces acting on a small cylinder of matter
(infinitesimal element) in a spherical star, gravitational and
pressure forces.
The lower face of cylindrical element of matter is distance r
from the centre, and upper face is distance r + δr, both faces
have equal area δA.
Volume of the elemental cylinder of matter is δA δr,
Mass of the infinitesimal element m = ρ(r) δA δr
where ρ (r) = density of stellar material at r.
1) The gravitational force acting on the elemental mass at r in a spherical body whose density
depends only on distance from the centre is the same as if all the mass interior to the element
were concentrated at the centre of the body and all the remainder of the body were neglected.
Force pulling cylinder towards centre
‫=ܨ‬
ீ ெሺ௥ሻ ௠
௥మ
=
ீ ெሺ௥ሻ ఘሺ௥ሻ ఋ஺ ఋ௥
௥మ
Where G is the Newtonian gravitational constant, M(r) is the mass contained within the sphere of
radius r.
1
(2) Pressure force: the forces due to the pressures balance exactly except for the forces of the
stellar material on the inner and outer face of the elemental cylinder at r and r + δr
Net pressure force =
p(r+δr) δA - p(r) δA
For equilibrium, the forces must balance:
మ
giving:
Equation of hydrostatic equilibrium (also known as hydrostatic support):
[2.1]
Applicable in stellar interiors, planetary atmospheres, etc.
2.2 Equation of mass continuity
The quantities M, ρ and r are not independent as the
mass M(r) contained within a sphere of radius r is determined
by the density of the material at all points within radius r.
We can obtain a relation between M, ρ and r
r
Stellar mass
M(r) = 4 π 0∫ r 2 ρ(r) δr
Consider the mass of a spherical shell between radii r and r+δr.
Mass of thin shell = 4 π r2 ρ δr
provided δr is small.
The mass of the shell can be written = M (r + δr) – M(r)
=
δr
for a thin shell
Hence
[2.2]
So we have two differential equations [2.1] and [2.2] describing the structure of the stellar interior –
but we have three functions M(r), p(r), ρ (r).
Clearly we need a further relation between them if we are to determine all the parameters. We
also need an equation for temperature (see later lecture).
2.3 How good an approximation is hydrostatic equilibrium?
In deriving equation [2.1], the equation of hydrostatic equilibrium,
equilibrium, we assumed that the forces
acting on any element of material in a star are exactly in balance. If these are not in balance, for
example if the star was expanding or contracting, then there would be a net force on an element of
matter equal to the product of its mass and acceleration (F=ma).
If forces are not in balance, a fluid element would accelerate:
ρ a= - ρ g +
ௗ௣
ௗ௥
= λ ρ g say
2
From rest, in time t the fluid element travels distance thus
[2.3]
-1
7
So, for the sun, if we consider a 10% decrease in radius, s = R /10 = 7 x 10 m, and g=300 ms ,
ଷ
time 10 ଵൗ sec.
ଶ
3
10
Solar eclipse records show that there has been no appreciable radius change in 10 yrs ~ 10 sec.
therefore
-14
λ < 10
Since geological evidence concerning the ages of radioactive elements in the Earth’s crust and of
9
fossils suggests that the properties of the Sun have not changed significantly for at least 10 years
(3 x 1016 seconds), the present λ < 10-27.
So, hydrostatic equilibrium is a very good approximation.
2.4 How long would a star take to collapse if pressure forces were negligible (dynamical
timescale) ?
మ
and
Putting s=R, the stellar radius, gives
మ
2 ~
2.4
To do: calculate the dynamical timescale for the Sun.
2.5 Mean density of the Sun
Simply the mass per unit volume:
[2.5]
What is this density similar to?
2.6 Very simple estimate of the Sun’s mean pressure
Using
Make approximation for mean value, mass = M☼/2
density [2.5], approximating
2
2
and take radius R☼/2 and take mean
2 2.6
To do: calculate a value for the mean pressure for the Sun using [2.6]
3
2.7 Estimate of the Sun’s minimum central pressure
Use the equations of hydrostatic equilibrium [2.1] and mass continuity [2.2] to estimate a minimum
value of the central P of a star of known mass and radius:
⁄ ()
− () ()
−()
≡
= ଶ
=
⁄
()
4 ଶ ()
4 ସ
()
−()
=
()
4 ସ
= ௖ − ௦ = − 4 ସ
ெೞ
ெೞ
଴
଴
Where c= centre of star, s surface, MS total mass, PC central pressure, PS surface pressure.
But in a star r < rS
and so
ଵ
௥ర
>
ଵ
௥ೞర
So we can estimate the minimum value of the integral:
௦ଶ
> =
4 ௦ସ
4 ସ
8 ௦ସ
ெೞ
ெೞ
଴
଴
Rearranging, and taking PS << PC gives:
௦ଶ
௦ଶ
௖ > ௦ +
>
8 ௦ସ
8 ௦ସ
[2.7]
For the sun MS = M and rS = R so PC > 4.5 x 1013 N m-2 . This result requires no
knowledge of the chemical composition or physical state of the solar material.
For stars other than the Sun, we can write:
ଶ
௦
⨀ ସ
ۨ
௖ > 8 ௦ସ
ۨ
௦
ଶ
[2.8]
LECTURE KEY POINTS:
• Derivation of equation of hydrostatic equilibrium – including assumptions: spherical
symmetry, static - some assessment of their validity
• Equation for dM/dr mass continuity
• Dynamical timescale tdyn
• Mean density of Sun
• Estimates of mean pressure of Sun, and minimum central pressure
To Do: calculate the dynamical timescale for the Sun, and the mean pressure.
Problem sheet 1 Question 2
Learning outcomes include:
• be able to derive the equations of hydrostatic equilibrium and mass continuity, giving
assumptions and reasoning, and a comment on their validity.
• be able to derive, calculate and explain what is meant by the dynamical timescale,
• be able to estimate the mean density and minimum central pressure of the Sun
4
Sun Stars Planets
Part I continued :
3.
J C Pickering 2014
The Sun: its structure and energy generation
Lecture 3. Modelling the Sun’s interior: pressure, density, temperature
3.1
3.2
3.3
3.4
3.5
3.6
Local thermodynamic equilibrium, and stellar plasma
Pressure in solar interior, and the equation of state
Mean molecular weight
Estimate of central temperature of Sun
The virial theorem
Contraction of a star
3.1 Local thermodynamic equilibrium, and stellar plasma
The interiors of stars are in thermodynamic equilibrium to high degree of accuracy: well-defined
temperature T(r), pressure p(r), etc
3.1.1 What do we mean by thermodynamic equilibrium?
• If a physical system is isolated and left alone for a sufficiently long time it settles down into a
state of thermodynamic equilibrium
• In thermodynamic equilibrium the overall properties of a physical system do not vary from point
to point and do not change with time
• But individual particles of the system are in motion and do have changing properties, eg
electrons being removed from or attached to atoms
• But there is a statistically steady state in which any process and its inverse occur equally
frequently
• Because the properties are not varying from point to point when in thermodynamic equilibrium,
all parts have the same temperature
• Key point – in thermodynamic equilibrium all physical properties of the system (eg P, internal
energy, specific heat) can be calculated in terms of its density, temperature and chemical
composition alone.
3.1.2 Stellar plasma: an ideal gas and local thermodynamic equilibrium, radiation
The stellar material is an ionised gas or plasma, and is assumed to be an ideal gas.
Because of the high T in stars all but the most tightly bound electrons are separated from the
atoms. This allows a greater compression of stellar material without deviation from the perfect gas
law because a nuclear dimension is 10 -15 m compared with a typical atomic dimension of 10-10 m,
and so the plasma has a higher density than you might intuitively expect is possible for an ideal
gas. Plasma also differs from an ordinary gas because the forces between electrons and ions
have a much longer range than the forces between neutral atoms.
In contrast to many space plasmas that are low density, most of the plasma in a star is very dense
(as mentioned above), i.e., it has a short mean free path and there are many collisions and
interactions between the electrons, ions and photons. As the timescale between collisions is much
shorter than the changes in pressure, temperature and composition, one might expect the plasma
to be in thermodynamic equilibrium. As temperature and pressure are functions of the stellar
radius, we do not have global, but local thermodynamic equilibrium (LTE). This is a huge
simplification and one of the reasons that we can easily define the temperature in these regions: In
LTE the electrons, photons and ions all have the same temperature and this is equivalent to the
1
kinetic temperature of the gas (in contrast, in many space plasmas, the electron and ion
temperatures are not the same!).
•
•
In the deep interior of a star it is very close to being in thermodynamic equilibrium
Near the surface of a star there start to be departures from thermodynamic equilibrium and
there is a net outflow of energy. However if there are enough collisions then we have a kinetic
temperature and a state of local thermodynamic equilibrium.
Unlike in typical laboratory conditions, in a stellar interior the radiation
radiation is in thermodynamic
equilibrium with matter – and in thermodynamic equilibrium the intensity of radiation is given by the
Planck function:
3.2 Pressure in the solar interior
3.2.1 Radiation Pressure
Just as the particles in a gas exert a pressure which can be calculated from the kinetic theory of
gases, by considering collisions of particles with an imaginary surface in the gas, the photons in a
Planck distribution exert a pressure known as radiation pressure.
1
3
(where a is the radiation constant= 7.55 x 10 -16 J m -3 K -4 )
3.2.2 Gas pressure
Pressure contribution from ions and electrons. Particles in the gas exert P calculated from kinetic
theory of gases, and we will see that:
( where R is the gas constant = 8.26 x 10 3 J K-1 kg -1 and µ is the mean molecular weight)
3.2.3
Pressure in the Solar interior
In general
P P୥ୟୱ P୰ୟୢ
But for a star like the Sun it has been found that ௚௔௦ ௥௔ௗ and so we can neglect Prad for the
Sun.
2
3.2.4 Equation of state
Rather than using the usual expression for the ideal gas equation, P = n kB T, it is customary to
express the number density n in terms of the mass density and introduce the mean molecular
weight, µ. The mean molecular weight is the mean mass of the gas particles in units of the
hydrogen mass, mH. The mean particle mass is thus µmH.
The mass density of the gas, ρ, can be expressed in terms of its number density n and mean
particle mass:
ߩ = ߤ ݉ு ݊ ⟹
ߤ =
The ideal gas equation then becomes:
ܲ = ݊݇஻ ܶ =
Where specific gas constant for hydrogen ℛ =
ఘ
௡ ௠ಹ
ߩℛܶ
ߩ ݇஻ ܶ
=
ߤ ݉ு
ߤ
݇஻ൗ
ିଵ ିଵ
݉ு ≃ 8300 ‫ܭ ݃݇ ܬ‬
3.3 Mean Molecular Weight µ
Now we need an expression for the mean molecular weight as a function of ρ, T and chemical
composition. Calculating µ for completely general values of ρ and T is very complicated because
to find n (number density of each species), the fractional ionization of all the elements has to be
computed. But this can be simplified as follows:
Recall that in order to find µ we will need to calculate the total number density n for all the species
in the gas. Let us define X, Y and Z as the mass fractions for hydrogen, helium and other
elements heavier than He (astronomers conventionally call these the ‘metals’).
X is thus the ratio of the mass of the gas that is in the form of hydrogen to the total mass of the
gas, or ‫)ܪ(ܯ‬/‫ܯ‬௧௢௧ . The total mass of hydrogen is the number of hydrogen atoms (or, for the fully
ionised gas, the total number of protons, N(H+)) multiplied by the mass of a hydrogen atom, mH.
We can neglect the electron masses here, and we also do not have to distinguish between proton,
hydrogen or atomic mass. We thus have:
ܰ(H ା )݉ு
≃
‫୲ܯ‬୭୲
݊ (H ା )
=
݉ு
ߩ
ܺ = H mass fraction
‫(ܯ‬H)
=
‫୲ܯ‬୭୲
ܻ = He mass fraction
‫(ܯ‬He)
ܰ(Heଶା )݉ு௘
݊ (Heଶା )
=
≃
=
4݉ு
‫୲ܯ‬୭୲
‫୲ܯ‬୭୲
ߩ
ܼ = mass fraction for metals =
‫(ܯ‬metal)
‫୲ܯ‬୭୲
Here ρ is the mass density and n denotes number density. As we have absorbed all the metals in
Z, we have X + Y + Z = 1.
We now need to consider how each element contributes to the mass and the number density.
3
When ionised, each hydrogen atom contributes an electron and a proton to the number density; it
contributes mH to the mass density.
Each He atom contributes two electrons and its nucleus (an α particle, made up of 2 neutrons and
2 protons), i.e., 3 particles, to the number density and adds 4 mH to the mass density.
Finally, consider the species heavier than helium. For a species with electron number l, a fully
ionised ion will contribute l electrons along with one nucleus. The mass due to this species is
approximately 2 l mH (this assumes that the number of neutrons and protons is equal in a nucleus,
which is not a bad assumption).
The total number density is thus given by:
݊ = 2 ݊ு శ + 3 ݊ு௘ మశ + (݈ + 1)݊metal೗శ
We can now express the number densities for the different species in terms of their mass densities
and mass fractions, for example according to ݊ு శ = ߩܺ/݉ு , so
݊=2
ܺߩ
ܻߩ
ܼߩ
3
1
ߩ
+ 3
+ ሺ݈ + 1ሻ
≃ ൬2 ܺ + ܻ + ܼ൰
݉ு
4݉ு
2 ݈ ݉ு
4
2
݉ு
X + Y + Z = 1. We can now simplify, and say Z≈0, and so:
ߤ =
ߩ
1
4
=
=
3
݉ு ݊
5ܺ + 3
ቀ2ܺ + 4 ܻቁ
TO Do: Problem sheet 1. Add your more accurate calculation of µ, where Z≠0, to your notes here.
For the Sun, the surface values for X and Y are about 74% and 24%, so µ ≈ 0.6. In the solar core,
the value of Y increases to more than 60% and µ ≈ 0.8.
…We can now use the equation of state as giving an expression for P…
3.4 Estimate of Sun’s central temperature
Take
ܲ=
and simplify for rough estimate, using ߤ
ఘℛ்
ఓ
recall from lecture 2:
ܲ௖ ≃
మ
ீ ெ⨀
ర
ଶ ோ⨀
[2.6]
≈ 1, ߩ ≈ ߩҧ , ܲ = ܲ௖
and ߩҧ =
‫⨀ܯ‬
4ൗ ߨܴ3⨀
3
[2.5]
so
ீ ெమ
൬ ଶ ோర⨀ ൰ =
⨀
ℛ
ఓ
൬ସ
ெ⨀
య
ൗଷ గோ⨀
൰ ܶ஼
⟹ ܶ஼ =
ఓ ீ ெ⨀
ℛ ோ⨀
To do: calculate Tc Compare this with the value from a more accurate solar model = 1.56 x 107 K.
భ
Note: how does TC vary with stellar mass? Since ‫= ܯ‬
ସ
ଷ
ߨ ܴ ߩ, ܴ =
ଷ
ଷெ య
ቀସగఘቁ ,
if we assume that the
1/3
mean ρ ≈ constant, if M increases by factor of 2, radius increases by 2 ≈1.26, ie mass increases
more rapidly than R. So M/R is larger for stars of larger mass, and hence Tc is larger for greater M.
Our calculation has made no assumptions about the method of energy release, but is not a bad
estimate. So, with modest densities and high temperature we can see that the stellar material is
gaseous and ionised. With the high central pressure (as found in lecture 2) it might initially be
surprising that the stellar material is an ionised gas, or plasma, but as it is ionised there can be
much greater compression without deviation from the perfect gas law as the nuclear dimension is
10-15 m compared to typical atomic dimension of 10-10 m.
4
3.5 The virial theorem
A further consequence of the equations:
and
can be found by integrating the equations over the entire volume of the star: …
Start with:
Multiply by 4πr3 and integrate over the interior of the star:
Integrate LHS by parts:
and since
we have
First term: negligible. If the star were surrounded by vacuum, its surface pressure
p(R) would be zero and first term = 0. In fact, surface pressure will not be zero, but will be many
orders of magnitude smaller than the central P, or mean P, so this term is so small compared to
the other two terms that it may be neglected.
Third term: gravitational potential energy Ω of the star (negative). It is the energy released in
forming the star from its component parts dispersed to infinity.
So we have:
3
Ω 0
for a perfect gas, the internal energy per unit mass u is given by
3
3 1 3 1tot 2 tot
Where Utot is the total thermal energy, γ = ratio of the specific heat at constant P to the specific
heat at constant V, and since a fully ionized gas is a monoatomic gas, γ = 5/3.
Putting it all together gives the virial theorem:
Thus the negative gravitational energy is just equal to twice the thermal energy.
5
3.6 Contraction of a star (aside)
An important result of using
is as follows:
The total energy of a star can be defined by E = U + Ω
energy. If the star radiates into space, E must decrease.
Combining
2U + Ω = 0 and
E=
so
U – 2U
=
E = U + Ω,
provided there are no other sources of
gives:
-U = Ω / 2
E= –U = Ω/2
Thus, the total energy of the star is negative and is equal to half the gravitational energy, or minus
the thermal energy. So, a decrease in E leads to a decrease in Ω, but to an increase in U. Thus a
star composed of a perfect gas, with no hidden energy
energy supplies, contracts and heats up as it
radiates energy. (Ω is always –ve for a self gravitating body and becomes more negative as a
body of fixed mass contracts.) This may seem rather paradoxical, such a star finds it difficult to
cool down; any attempt to lose energy causes the star to contract and to release energy at a rate
that not only supplies the energy loss from the surface, but also heats up the material of the star.
We had assumed a fully ionised gas, with γ = 5/3, but this is also true as long as γ > 4/3.
Contrast the behaviour of a star with that of a cooling ember – the ember, hotter than its
surroundings, radiates into the surroundings. As it radiates, it cools, eventually coming to
thermodynamic equilibrium with the surroundings (same temperature). This is not so for a
classical self gravitating mass of gas, a star, that radiates into cooler surroundings (universe) – it
becomes hotter and hotter, and increases the disparity between its temperature and the
surrounding temperature. However the laws of thermodynamics are not broken – the zeroth law
requires heat to flow from hot (star) to cold (universe), and for T to become uniform when the
system has reached thermodynamic equilibrium, and of course the star has not yet reached
thermodynamic equilibrium with the universe!
Lecture 3:
Key points
• Gas pressure and radiation pressure, thermodynamic equilibrium
• Mean molecular weight
• Estimate of central temperature of the Sun
• Solar interior is gaseous and essentially fully ionized
• Virial theorem, and contraction of a star
After this lecture you should:
- Be able to explain what is meant by local thermodynamic equilibrium;
- Understand that the Planck function describes the radiation of a star;
- Understand that the Sun is composed of a plasma, and understand why this may be
approximated as an ideal gas;
- Understand what is meant by gas and radiation pressure, know the expressions for these;
- Be able to estimate the mean molecular weight;
- Know what the equation of state is, and be able to use it where necessary;
- Be able to estimate the central temperature of the Sun;
- Virial theorem: be able to derive this and use it as demonstrated in notes and lecture;
To Do:
Calculate TC for the Sun
Problem sheet 1, Q3 and Q4
6
Sun Stars Planets
Part 1 :
4.
J C Pickering 2014
The Sun: its structure and energy generation, continued.
Lecture 4. Energy production in the Sun
4.1 Estimate of the minimum mean temperature of the Sun
4.2 How does the Sun shine?
4.3 The source of the Sun’s energy
4.4 Nuclear fusion
4.5 Stability
4.6 Solar neutrino problem
4.1 Estimate of the minimum mean temperature of the Sun using the virial theorem.
In lecture 3 we derived the virial theorem: 2 Utot + Ω = 0 and used this to explain the phenomenon of
a star contracting and heating up as it radiates heat. Here we will use the virial theorem to estimate
the minimum mean temperature of the Sun. The virial theorem can be written as:
ܲ
݀݉ + Ω = 0
ߩ
3න
But we know that
ெೄ
Ω = −න
଴
(1)
‫݉ܩ‬
݀݉
‫ݎ‬
But r < rS (surface radius) everywhere, so 1/r > 1/rS is true. Therefore:
‫ܯ ܩ‬ௌଶ
‫݉ܩ‬
݀݉ =
2 ‫ݎ‬௦
‫ݎ‬௦
ெೄ
−Ω > න
଴
If we have an ideal gas
ெೄ
3න
଴
ܲ=
(2)
ℜఘ்
ఓ
,
so
ெೄ
ܲ
ℜܶ
3 ℜ ܶത‫ܯ‬ௌ
݀݉ = 3 න
݀݉ =
ߩ
ߤ
ߤ
଴
(3)
ഥ = ‫݉݀ ܶ ܵܯ׬‬
Where ₸ is the mean temperature, defined by ‫ܶ ܵܯ‬
0
Combining (1) (2) and (3):
3 ℜ ܶഥ ‫ܯ‬ௌ
‫ܯܩ‬ௌଶ
>
ߤ
2 ‫ݎ‬ௌ
ܶത >
‫ܯ ܩ‬ௌ ߤ
6 ‫ݎ‬ௌ ℜ
We assume Sun is composed of ionised H, so µ = ½, take MS = M , rS = R , and so
₸
> 2 x 106 K
1
4.2 How does the Sun shine ?
4.2.1 Could the Sun’s energy source be gravitational energy?
-----No, since…
Total available gravitational energy = G M
2
/ R
This could sustain the Sun’s present luminosity for time = gravitational energy/luminosity
= (G M
2
/R
)/L
~ 107 yrs
4.2.2 Is the Sun were shining by cooling down?
If it were then by the virial theorem,
the thermal time is half of the time that the Sun could shine through gravitational energy (above),
since thermal energy ~ ½ gravitational energy.
Neither can explain how Sun has shone for > 109 yrs
This means that, if the Sun’s radiation were supplied by either contraction (gravitational) or
cooling it would have changed substantially in the last 10 million years, but geologists estimate it
can hardly have altered in a time a hundred times longer > 109 yrs
The thermal timescale (also known as the Kelvin-Helmholtz timescale) is:
So, there has to be another source for the Sun’s radiant energy.
4.3 The source of the Sun’s energy
So, if the sun radiates energy at rate = 4 x 1026 W, using E = m c2 the Sun is losing mass at a rate
of 4 x 109 kg s-1. From geologists, we know L has not changed significantly over last few thousand
million years, so over this time mass loss ≈ 2 x 10 - 4 M .
So if neither gravitational energy nor thermal energy can account for the sun’s energy, then the
source of the Sun’s energy must be released by the conversion of matter from one form to another,
and be capable of releasing at least 2 x 10-4 of the rest mass energy of the Sun. This rules out
chemical reactions such as combustion of coal, gas and oil which only release up to 5 x 10-10 of the
rest mass energy. The only way known in which quantities of energy as large as this can be
released, by the change of matter from one form to another, is through nuclear reactions.
- either fission reactions of heavy nuclei, like in an atomic bomb and nuclear reactions which can
release 5 x 10 -4 of the rest mass energy
- or fusion reactions of light nuclei that occur in the hydrogen bomb and can release almost 1 %
of the rest mass energy.
It is believed that nuclear fusion reactions are the source of the energy radiated during most phases
of a star’s evolution.
2
4.4 Nuclear fusion
Hydrogen
4 1H
Mass:
E = m c2
4 mH
Helium
4He
⇒
3.97 mH
∴ energy production = (0.03 mH) c2
This could power the Sun for
i.e. fraction 0.007 of mass converted to energy
tnuc ~ 0.007 M c2 / L
~ 1011 yr
Note:
tdyn <<
Main energy source in Sun:
tK-H <<
tnuc
Proton-proton reaction
Branch I
85%
Branch II
15%
Branch III
<1%
[In main-sequence stars more massive than the Sun, the main energy source is also H burning but
by the CNO cycle.
Requires higher temperature.]
Nuclear fusion is a process in which nuclei of relatively low mass are fused together to form nuclei
of somewhat greater mass. Fusion is brought about by a sequence of nuclear reactions in which
colliding nuclei combine and fragment, to produce new nuclei together with other particles. No
energy is actually created in these reactions: energy is liberated from the reactants and is
redistributed amongst the products so that some of it replaces the energy radiated by the Sun, thus
maintaining the high core T and sustaining the nuclear reaction rates.
The most important series of nuclear reactions occurring in main sequence stars are those
converting hydrogen into helium – this is termed hydrogen burning. There are several routes by
which H can be converted to He, and they must obey conservation laws: conservation of charge and
energy, and conservation of baryon number.
There are two basic reaction chains for conversion of H to He: the Proton-Proton (PP) chain
and the carbon-nitrogen cycle in which nuclei of C and N are used as catalysts in the conversion of
H to He. The particular temperature in the Sun’s core dictates that the PP I chain dominates. But
in stars with progressively higher T than the Sun, the other 2 chains PP II and PP III become
important. For stars with even higher central temperatures the CNO cycle dominates.
3
The net effect for each of the PP chains is production of a helium nucleus from four protons.
4
PP I overall
1
→
Step [1]
1
1H
+
Step [2]
2
1H
+ 11H →
Step [3]
3
2He
1H
1
1H
→
2
1H
3
2He
4
2He
+
+
2e+ + 2νe + 2γ
e+ + ν e
e+ = positron
νe = neutrino
γ = gamma ray
X2
+ γ
+
+
3
2He
→
4
2He
+
1
1H
+
1
1H
[The Appendix of this handout gives a schematic diagram of the PP I,
cycle.]
II, and III reactions and the CNO
Stellar central temperature and relative rate of energy release for PP and CNO reactions.
Reaction rates = rate of energy generation
per unit volume of gas.
n = number density of reactant nuclei
T = temperature
ܴ௣௣ ∝ ݊ଶ ܶ ସ
ܴ஼ேை ∝ ݊ଶ ܶ ଵ଻
Stellar properties are very sensitive to T.
L very dependent on T and hence M.
4
2e+ + 2νe + 2γ produces neutrinos. The theories of
The overall PPI 4 11H →
2He +
thermonuclear reactions can be tested by calculating the rate of neutrinos expected at Earth, and
checking these with experiments. The reaction rates are very sensitive to temperature, so this will
allow an estimate of the core temperature.
To Do: Estimate the rate of solar neutrinos passing through the top of your head: [2005 Q1]
The overall nuclear PPI reaction chain is: 4
1
1H
→
4
2He
+
2e+ + 2νe + 2γ
Αssuming the neutrino (νe) energy to be negligible, calculate approximately the energy
liberated from just one completion of this PPI chain.
Assuming this PPI chain were the only kind of nuclear reaction taking place, derive an expression
for and estimate the number of solar neutrinos passing through the top of your head per second.
[Take helium nucleus mass = 3.97 mH, proton mass mH =1.673 x 10-27 kg. 1 AU = 1.5 x 1011m.
Solar luminosity=3.83x1026 W]
4
We know that stars on the main sequence have a stable “hydrogen burning” lifetime that is over 3
billion years. The thermonuclear energy source is however not infinite, and gravity is relentless.
Every star must confront the paradox by which it gets hotter and hotter while losing more and more
energy to the cold dark universe. This confrontation and its ultimate resolution is the central plot to
the life stories of the stars.
4.5 Stability
The hydrogen bomb works on the same principle as the thermonuclear reactions which power the
sun. Why does the Sun “burn” stably for billions of years, why does it not just explode like a bomb?
It has a built in “safety valve”, and “burns” by “controlled thermonuclear fusion”.
Make notes in lecture.
“Missing” neutrinos due to neutrino oscillations…
4.6 Solar neutrino problem
TO DO: Additional reading: - read and make notes!
John N. Bahcall, How the Sun Shines
available at: http://www.nobelprize.org/nobel_prizes/themes/physics/fusion/
Although EM radiation has a hard time escaping from the Sun’s core, the neutrinos produced there
have no such difficulty. Observing these solar neutrinos gives a direct test of our solar models and
solar nuclear reactions. Although huge numbers are produced (100 billion pass through your
thumbnail every second) their low interaction rate makes them hard to detect when they reach and
pass through the Earth.
The higher the energy of the neutrino the greater the probability of detection (generally
probability of detection µ (neutrino energy)2 ). The neutrino emitted by β decay of 85B (PP III
chain) is much more energetic than the other neutrinos emitted by H burning, and the number of
reactions going through PP III at the estimated solar central T is µ T18. Because of this high
dependence on T, if these neutrinos could be detected, the central T of the Sun could be
determined very accurately and it would be compared with the central T predicted by theoretical
calculations.
How to measure the neutrinos? Davis and Bahcall, experimentalist and astronomer, proposed
an experiment :– The only way to detect neutrinos, is to cause a stable atomic nucleus to be
transformed into an unstable nucleus by capturing a neutrino and then to observe the β decay of the
unstable nucleus. Neutrinos must be captured in a place where no other particles could produce
the same unstable nucleus – eg in a deep mine shielded from cosmic rays. The process studied
was:
37
Cl + νe →
37
Ar
→
37
18Ar
+ e-
Cl + e + + νe
37
Chlorine was in perchloroethylene (cleaning fluid) C2Cl4 400,000 Litres in a tank deep in a mine.
Argon was removed before it decayed. At the end of a typical 80 day run, the tank was emptied,
and the contents were analysed, and the 3718Ar nuclei were counted – no mean feat, as there were
usually only about 50 3718Ar nuclei among the 1031 nuclei in the tank! The result however, was
only about 1/3 of the neutrinos expected! This result was also confirmed by a Japanese group.
This became known as the solar neutrino problem. Initially people thought the solar models might
be flawed in some important aspect – and attempts were made to construct models giving lower
central T – but this was never achieved. Helioseismology experiments started to show that solar
models were valid.
5
Physicists know of 3 kinds of neutrino: Electron neutrino νe , Muon neutrino νµ ,Tauon neutrino ντ.
Neutrinos created in the Sun’s core are electron neutrinos, and this is the only kind that can be
detected by the C2Cl4 tank experiment. There was a suggestion that a particular kind of interaction,
between the electron neutrino leaving the core and the solar material through which they pass,
causes the neutrinos to change type – so that only 1/3 of the neutrinos leaving the Sun are still
electron neutrinos.
Experiments were constructed, Sudbury Neutrino Observatory, Canada, which can detect all 3 kinds
of neutrino – big tank of heavy water. The flux of all 3 types of neutrino can be measured –
confirming the hypothesis that neutrinos can change their type.
Lecture 4: SUMMARY outcomes and TO DO:
Summary:
• estimate of minimum mean temperature of the Sun
• Insufficiency of gravitational and thermal energy to power Sun
• Nuclear power – fusion : be able to describe the H burning chains of nuclear reactions and
discuss with respect to the Sun and hotter stars.
• p-p reaction chain as source of solar energy – be able to calculate resultant energy
• Solar neutrino problem: be able to discuss the solar neutrino problem, and how investigated, and
current explanation
• be able to discuss the contraction and heating up of a star, and stability of nuclear fusion
TO DO: • Calculate the number of neutrinos from the Sun passing through your head per second
• Read web site article on the solar neutrino problem – add to notes in this handout on this topic
• Problem sheet 1, Q 5 and Q6
6
APPENDIX.
7
Copy Right The Open University
8
Sun Stars Planets
Section 1:
5.
J C Pickering 2014
The Sun: its structure and energy generation, continued.
Lecture 5: Energy production and heat transport
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Luminosity, the energy generation equation
Heat transport
Radiative heat transport
Opacity
Photon energy during diffusion process
Heat transport: conduction
Convective heat transport
5.1 Luminosity
Derive an equation relating the rate of energy release and the rate of energy transport.
Assume: star is spherically symmetrical and energy is transported in a radial direction.
Suppose energy flows across a sphere of radius r, and at a rate L(r) (in units W).
Define luminosity L(r) = energy per unit time crossing sphere of radius r, centred on centre of Sun,
Let ε = energy per unit time produced per unit mass at any given location in Sun (W kg -1).
Consider the energy release in a spherical shell. L (r +  r) exceeds L(r) by the energy released in
the shell.
Equate the difference between energy crossing a sphere of radius r +  r, and a sphere of radius r,
to the energy released in the spherical shell:
ADD NOTES DURING LECTURE
Where ε is function of ρ, T and chemical abundances.
1
Only in the inner 25% of the Sun are temperatures high enough for significant energy production.
We have neglected the change with time of the stellar properties. We can neglect the time
dependence if tnuc >> t th , that is if the energy sources being used are capable of supplying the
star’s radiation for a long time compared to the thermal time.
We have also not considered the possibility that some of the energy released in the shell is used to
heat up or change the volume of the shell.
We now have one further equation for the structure of a star, but only by introducing 2 more
unknown quantities: and L – so we will still need several more equations…
5.2 Heat transport in the Sun
Consider the way in which energy is transported
outwards in a star:
3 ways of transporting heat:
radiation, conduction, convection
Radiation and conduction both depend on the
collision of energetic particles with less energetic particles resulting in an exchange of energy. In
the case of radiation the energy is carried by photons.
5.3 Radiative heat transport (heat transport by photons)
If photons could escape freely from centre of Sun, they would take R/ c = 2 seconds to reach the
surface.
But we know it must take at least more
like the Kelvin-Helmholtz time (lecture 4)
The energy released at the centre of the Sun slowly diffuses outwards. We have seen that the total
thermal energy of the Sun could supply its rate of radiation for about 3 x 107 years – so this gives us
an estimate of how long it takes for a photon to diffuse from the solar centre to the surface.
Photons escape by random walk, scattered many times by ionized matter in the solar interior.
5.3.1 Random walk
With a central T  107 K photons associated with black-body radiation have wavelength in the X ray
range. But light from the stellar surface is typically in the visible spectral range – so the photon
4
energy is about 10 times smaller than the average energy per photon in the stellar core. The
source of this degradation of photon energy is the coupling between radiation and matter. Photons
diffuse through most stellar matter, a process in which a given photon travels, on average, a mean
free path l before being scattered or absorbed and re-emitted in a random direction. The
description of this process is similar to the statistical problem of a random walk, where N random
choices for the next step (here – re-emission in arbitrary direction) result in net displacement N1/2 l
from starting point.
2
Steps: all steps of equal length l for simplicity; directions random
r4
r3
O
r1
r2
Total net displacement after N steps
r = r1 + r2 + … + rN
And root mean square radial displacement after N scatterings:
< rN2 >1/2 = N1/2 l
So, the net distance travelled in N steps
|r| = N1/2 l
The number of steps to travel from the star’s centre to the star’s surface is N = ( R / l ) 2
Time taken for one step = l / c
On average in the Sun,
l ≈ 10 -3 m = 1 mm (l is the mean free path – m.f.p.)
Hence the total time for photon to escape (the diffusion time) is:
[5.2]
Note
- N ~ 10 24
- It takes 1012 times longer to escape than if photons had free flight
- The mean free path is the distance particles travel between collisions – if
m.f.p.is large then particles can get from a point where the T is high to one where it is
significantly lower before colliding and transferring energy, and a large transport of
energy results.
3
5.3.2 Heat transport by radiation
Derive an expression for dT/dr:
Assume: l , the mean free path, is much much less than the scale on which temperature T and
density  vary, and that energy transport by photons is a diffusive process.
Radiative energy density in photons
u=aT4
r+l
u(r+l)
Flux
(1/6) v u(r+l)
r
Only 1/6 of photons travelling
in any one principal direction.
(1/6) v u(r–l)
u (r–l)
r–l
Net flux (in +r direction) across surface r
is
(1/6) v u(r-l) - (1/6) v u(r+l) = - (1/3) v l du
dr
So
We have v = c and u = a T 4
so:
and
Define opacity κ by
Also,
F = L / 4π r 2
So finally,
[5.3]
Opacity к depends on density, temperature, chemical abundances.
4
5.4 Opacity
The flow of energy by conduction and radiation is essentially similar in nature, and the rate at which
energy flows by these processes is determined by the opacity  . The radiative transport equation
(derived in 5.3) relates the rate of energy transport to the temperature gradient and the opacity. The
opacity of stellar material is a measure of the resistance of the material to the passage of radiation.
[The probability that a photon will be absorbed in travelling distance δx = к ρ δx . So if l= δx the
probability of absorption is 1, and so 1 = к ρ l.]
The calculation of stellar opacity is a very complicated process as all atoms and ions must be
considered. Looking at sources of opacity all the microscopic processes contributing to the
absorption of radiation at different frequencies need to be considered. There are 4 basic types of
process involved:
5.4.1 Bound-bound absorption
An e- is moved from one bound orbit
in an atom or ion to an orbit of higher
energy with the absorption of a
photon
5.4.2 Bound-free absorption
(Most important in Sun)
An e- in a bound state around a nucleus is moved into a free hyperbolic orbit by the absorption of a
photon.
5.4.3 Free-free absorption
An e- initially in a free state absorbs a photon and moves to a state of higher energy.
5.4.4 Electron scattering
(Most important at higher
temperatures)
It is possible for a photon to be scattered by an e- or an atom. Although this process doesn’t lead to
true absorption of radiation it does slow down the rate at which energy escapes from a star because
it continually changes the direction of the photons.
5
5.5 Photon energy
Photons are generated in the Sun’s core through thermonuclear reactions
as rays, typical wavelength 10 -12 m, at high temperature ~15 x 10 6 K. But photons at the Sun’s
surface have typical wavelength 10-6 m and the temperature is 5800 K. How does the average
energy of photons degrade from high energy from the PP chain reactions, to the relative low energy
of visible light ? It is a 2 stage process:
(i) rays from PP (proton-proton thermonuclear reactions) chains undergo multiple scattering with
electrons and ions in the star’s core. Each scattering redistributes the energy between photon and
particles. So, some of the energy of a very high energy photon is transferred to the surrounding
plasma. Eventually the distribution of photon energies changes from that of PP ray to a blackbody
spectrum characterised by the temperature of the core. We can say thermalisation takes place.
(ii) electrons and ions in the solar material interact with photons (scattering or absorption and reemission) as they make their way out of the Sun by diffusion (random walk). At each point in the
star the spectrum is characteristic of a black body source at the temperature of that point.
Overall, photons have energies distributed according to the characteristic black body curve
dependent on the temperature at the point in the star – and we see that an initially relatively small
number of high energy photons produced in the core of the star, eventually becomes at the star’s
surface a large number of lower energy photons.
5.6 Heat transport by Conduction Is heat transport by particles (ions and electrons) as
opposed to photons in radiative heat transport. Is heat transport by particles important in the Sun?
Consider the flux of energy:


For a gas: Energy density ug = 3/2 n kB T (monoatomic gas) and Pgas = n kB T, so
For radiation: Energy density urad for radiation urad = a T4 , and Prad = 1/3 a T4 , so
Therefore
(1)
104
Find n and T:
(2)
<< 1
(3)
<< 1
If T = ₸ (from 4.1) then T≈ 2 x 106 K
(
̅
)

Where we have taken μ≈1/2 as we are assuming fully ionised H. So:
(
)
So (1)
.
However, this is more than offset by (2) and (3) being <<1, so that overall conduction is
unimportant in the solar interior. But this is not the case in dense stars, like white dwarfs, where
conduction is very important.
6
5.7 Convective heat transport (heat transport by bulk motion of material)
We have seen that the energy generated in the core by nuclear reactions must escape from the
star. If the outer parts of the star are in radiative equilibrium, a certain T gradient must be set up to
maintain the requisite energy flow L (see equation of radiative heat transport [5.3]).
If the opacity к becomes high, or if L is high, then a steep T gradient must be set up to maintain
energy flow – a steep T gradient is unstable in a star just as it is unstable in the Earth’s atmosphere.
A star will be unstable against convection if the actual T gradient > T gradient if an element of mass
rose adiabatically. Under what conditions does convection occur?
Schwarzschild stability criterion Energy transport by conduction and radiation occurs whenever
there is a temperature gradient maintained in a body. However, convection, which occurs only in
liquids and gases, only occurs when the temperature gradient exceeds some critical value. We
must ask whether a given run of T(r), p(r), ρ(r) is stable.
Consider an element of fluid displaced upwards, slowly so it remains
in pressure balance but quickly enough that no heat
is exchanged with its surroundings (adiabatic).
Change is adiabatic means
And so:
ρf P f
surroundings
δr
Fluid
element
Pressure
ρ1 P1
ρo P o
Density
Original
New position
(surroundings)
New position
(fluid element)
If fluid element denser than new surroundings, sinks back: stratification stable
If fluid element lighter than new surroundings, keeps rising: stratification unstable
Stable if
i.e.
i.e.
Unstable if:
[5.4]
This is the Schwarzschild stability criterion. If unstable this means bulk motion (convection) will
transport heat
7
Note: Mixing length
Phenomenological picture: fluid elements rise distance lmix and then merge
with surroundings ( lmix is called the mixing length):
It is assumed that convective elements of a characteristic size rise or fall through a distance,
comparable with their size before exchanging their excess heat with their surroundings.
Estimate lmix - pressure scale height (~R/10 in solar interior)
Find an expression for dT/dr for convective heat transport
Convection is very efficient at transporting heat. Stratification has only to be marginally unstable to
transport the solar energy output. So in convectively unstable regions:
is an excellent approximation
Equation of state
, hence
P  T
so:
ln  = ln P - ln T + const
Differentiating:
Substituting in:
Gives
[5.5]
in convectively unstable regions
Convection will thus occur when the magnitude of dT/dr is greater than the adiabatic temperature
gradient. This will be the case in cooler regions with high opacity and the hot core.
Key Points
Equation for dL/dr – be able to derive this
Heat transport by radiation – diffusive process – be able to discuss
Photons escape by random walk – very slow – be able to estimate diffusion time, and discuss
Equation for dT/dr if radiation is heat-transport mechanism – be able to derive this
Sources of opacity – be able to list, and explain these
Photon energy – understand and be able to explain
Conduction is not important inside the Sun: be able to estimate importance of conduction
Schwarzschild criterion for convective stability: be able to derive this and explain
Convection very efficient at transporting heat: be able to explain why
Equation for dT/dr if convection is the mechanism of heat-transport: be able to derive this
Heat transport will be by:
convection if stratification unstable or
radiation if stratification stable
To do: Problem Sheet 1 – finish questions 1-6
8
Sun, Stars, Planets
J C Pickering
2014
Lecture 6: Overview of the Sun’s Structure
6.1 Summary of equations of stellar structure
6.2 The solar interior
6.3 Overview of the structure of the Sun: interior and atmosphere
6.4 The atmosphere of the Sun
6.5 Sunspots and magnetic activity
Lectures 1-5 have completed the derivations of all the equations of stellar structure:
6.1 Summary of equations of stellar structure
equation of hydrostatic equilibrium
equation of mass continuity
equations of heat transport
radiative
and
convective
Energy generation equation
Also need an equation of state (ideal gas equation)
[where the following symbols are: T temperature, P pressure,  density, r radius, L luminosity, m
mass,  energy per second generated per unit mass,  mean molecular weight,  opacity, G
gravitational constant, c speed of light, a radiation constant,  gas constant]
Appropriate boundary conditions for the differential equations need to be considered:
Boundary conditions:
m = 0,
L= 0
at r = 0
m=M
at r = R
p=T=0
at r = R



(zero boundary conditions: can do better, but this is a good enough approximation for simple use of
equations)
1
Also we have expressions for P (Pressure),  (energy per unit time produced per unit
mass at any given location in Sun (W kg-1)), and  opacity :
(
)
(
)
(
)
Given the chemical composition of a star, we now have 7 equations with 7 unknowns:
P,  T, m, L,  and as functions of r.
Calculations of the structure of a star now involve obtaining expressions for P,  and  and
then the solution of the 4 differential equations.
Comment on boundary conditions:
(i) at centre m = 0, L = 0
when r = 0
(ii) at surface: use approximations – if the stars were truly isolated bodies then we would expect
 and P = 0 at the surface. But, there is no sharp edge to a star: the density  of the Sun near
to the visible surface is estimated to be  10-4 kg m -3 which is much less than the Sun’s mean
density (see Lecture notes section 2.5).
Also, surface temperature is much less than mean temperature, for sun surface T = 5780 K,
compared to typical mean T ~ 2 x 10 6 K (see lecture notes section 4.1).
So it is possible that the solution of the equations of stellar structure throughout the interior of a star
will not be seriously affected if the true surface boundary conditions were replaced by the
assumption  = 0, T = 0 at r = R

6.2 The solar interior
6.2.1 Composition. Variation of the mass fractions X, Y and Z with the fractional radius r / R for

the Sun: add diagram during lecture
Throughout most of its interior the Sun is by mass ~ 73% hydrogen, 25% helium and ~ 2% of
everything else. But in the central 30% of its radius the percentage of hydrogen is increasingly
depleted (and He increased), with about half the hydrogen initially present in the core of the Sun
having been converted into He.
2
6.2.2 Variation of pressure, density and temperature with fractional radius in the Sun
Lecture 7
Structure of Sun
according to a
Standard solar model
Density
ρ (103 kg m-3)
150
Temperature
radiative
0
0
convective
0.5
1.0
r / R
(Fractional Radius)
p (1016 Nm-2)
T (106 K)
15
0
0
1.0
r / R
Pressure
2
0
0.5
0
0.5
1.0
r / R
The rise of T, P and  with increasing depth is as expected. It is not until a fractional radius of about
0.5 that the density is equal to that of water on Earth (1000 kg m -3). Even at the centre of the Sun,
where the T ~ 15.6 x 106 K, the density is predicted to be only fourteen times that of lead, though
the pressure is more than 1010 times that of the Earth’s atmosphere at sea-level.
Lecture 7
6.2.3 Variation of luminosity L and
energy generation rate  with
fractional radius in the Sun
4
L (1026 W )
Hydrogen abundance
0.7
X
0
0
2
0
0.5
r / R
Fractional radius
1.0
0.5
r / R
1.0
Energy generation rate
ε (10-3 J s-1kg-1)
0
Luminosity
0
0.5
r / R
1.0
3
6.3 Overview of the Structure of the Sun: interior and atmosphere
6.4 The atmosphere of the Sun
Photosphere
Chromosphere
Transition Region
Corona
Solar wind
The surface of the Sun is not really a surface at all, but a thin, semi-transparent shell of gaseous
material:
6.4.1 The Photosphere:
The light that we see from the sun comes from the photosphere (= sphere of light), a layer of
atmosphere about 500 km thick (very thin compared to Sun ~1.4 million kms across!). In physical
terms the photosphere is more like an atmospheric layer and is the closest thing to a surface. The
air you are breathing now is over 1000 times denser than the material of the photosphere! The
photosphere emits light as it is hot: 9000 K at its lower boundary, and ~4500 K at the top. Most
photospheric light comes from region 5800 – 6000 K, and this represents the surface temperature of
the sun. What wavelength does this correspond to?
If we take the emitted light as being that of a blackbody at 5800K,
Wien’s law: maxT = 2.8979 x 10-3 m K gives max ~ 500 nm – visible light
(We will discuss the photosphere in greater depth later in the course…)
4
6.4.2. The Chromosphere
During a total eclipse of the Sun, the moon blocks the light from the photosphere for a few minutes
(it is an amazing coincidence that, although the Sun and Moon differ greatly in size and distance
from the Earth, their diameters and distances are just right to allow such a precise blockage to
occur). The solar “atmosphere” can then be seen – mostly white, but close to the photosphere is a
narrow region with pink/red tinge, the chromosphere…
The gases of the Sun extend far beyond the photosphere, which may be considered the lowest level
of the solar atmosphere. The region immediately above the photosphere is called the
chromosphere. The chromosphere is 2000-3000 km thick. It glows faintly relative to the photosphere
and can only be seen easily in a total solar eclipse. The faint glow of the chromosphere is due to an
emission spectrum from hot, low density gases emitting at discrete wavelengths. In the
chromosphere energy continues to be transported by radiation. Hydrogen atoms absorb energy
from the photosphere and most of the energy is then emitted as red light (because of strong Balmer
H-alpha emission). This colour is the origin of its name (chromos meaning “colour''). The
chromosphere is most easily viewed by filtering out all other wavelengths of light from the Sun and
only letting the red light from the chromosphere through. (The discovery of helium was from
emission lines seen in the chromosphere during an eclipse in 1868. Helium was found on the Earth
in 1895.)
6.4.3. Transition region
Above the chromosphere is a very thin layer of the Sun's atmosphere about 100 km thick over which
the temperature rises drastically from 20,000 degrees Kelvin in the upper chromosphere to over 2
million degrees Kelvin in the corona. This region is called the transition region. The large
temperature rise was unexpected these outermost layers of the Sun which are far from the heat
producing core. The complicated structure of the Sun's magnetic field provides clues to the dramatic
increase in temperature over such a small change in radius.
6.4.4. The Corona
The outermost layer of the Sun’s atmosphere is called the corona. It gets its name from the crown
like appearance evident during a total solar eclipse. The corona stretches far out into space and, in
fact, particles from the corona reach the Earth's orbit. The corona is very extensive, very tenuous
and has an extremely high temperature (~106 K). It does however appear faint compared to the
photosphere, and therefore can only be seen from Earth during a total solar eclipse or by using a
coronagraph telescope which simulates an eclipse by covering the bright solar disk.
What is the source of heating in the Corona ?
The Corona cannot be heated simply by energy radiated by the photosphere or by heat conducted
through the chromosphere; 2nd law of thermodynamics says no transfer of energy from a cooler
body to a hotter one is possible in this situation by either of these methods… But, the energy must
come from the lower regions of the Sun somehow – how?
The behaviour of the magnetic field in
the lower parts of the Corona plays an important role in heating the coronal gas…
The shape of the corona is mostly determined by the magnetic field of the Sun.
5
Closed field lines
- helmet streamers
Open field lines
Charged particles can stream away from Sun along open field lines - fast solar wind several
hundred km/s.
The free electrons in the corona move along magnetic field lines and form many different structures
including helmet streamers which can be seen as long, spiked cones in solar eclipse images.
Sometimes the magnetic field emerges from the lower regions and loops back down into the Sun.
These magnetic structures can be seen extending up into the corona. As particles follow the path
created by the magnetic field they form dynamic loops and arches that are most readily visible with
special telescopes. These structures are known as solar prominences.
As the density is so low, the corona is a poor emitter compared to the photosphere. The Corona is
not a black-body because it has such a very low density and is not opaque. Although not a blackbody source of radiation, we can still estimate the wavelength at which the Corona predominantly
radiates: Photon energy  kT where k = Boltzmann const.
Temperature of corona  2 x 106 K
E = h = hc/ 
so
 = hc/E
E  kT
1.38 x 10-23 x 2 x 106 = 2.76 x 10-17 J
= 6.63 x 10-34 x 3 x 108 / (2.76 x 10-17 )
= 7.21 x 10-9 m  7 nm
(X ray)
Some phenomena arise when the magnetic fields undergo very rapid changes:
Solar flares: rapid energetic outbursts of EM radiation. Duration 100-1000s, and may release up
to 1025 J in radiation in this time! And the material may have a temperature of over 107 K! Incidence
of flares follows solar activity cycle. Radiation emitted by flares is dominant in X ray and EUV.
Coronal mass ejection (CME): is a violent outflow of coronal material. The mass of ejected
material is typically 5 x 1012 – 5 x 1013 kg, and frequency of CMEs is typically 1 per day, or 3 per day
at solar maximum of the Sun’s activity cycle. It appears that the magnetic field is moving and the
plasma is forced to move with the field. Origin of these CME events is probably in a rapid
reconfiguration of the magnetic field in the lower parts of the solar corona – this allows energy
stored in the magnetic field to be released suddenly, causing a violent outflow of coronal material.
Magnetic reconnection occurs: – which converts energy stored in magnetic field into kinetic
energy of particles. In the case of solar flares and CMEs the reconnection event occurs high in a
coronal loop – particles are accelerated down to the foot points of the loops to give solar flares.
Much research into this area is still going on…
6
The incredibly hot temperature of the corona, however, requires a permanent heating mechanism,
or the plasma of the corona would cool down rapidly. There are many mechanisms which could heat
some gas above the surface of the Sun, but none of those mechanisms could account for the large
rate of heating necessary to heat the corona. This phenomenon remained a mystery for more than
50 years. However, although not all details are clear there is a likely solution to this mystery:
Using data from instruments onboard the SOlar and Heliospheric Observatory (SOHO) and from the
Transition Region And Coronal Explorer (TRACE), solar physicists have identified small patches of
magnetic field covering the entire surface of the Sun. Contrary to the bright, large magnetic field
loops which are linked to the "active regions" during periods of solar maxima, these patches seem
to appear and disappear randomly in time scales on the order of 40 hours. This “magnetic carpet”
is probably a source of the corona's heat. It is now believed that the heating of the corona is linked
to the interaction of the magnetic field lines radiating out of the small patches mentioned above.
Because the laws of electromagnetism prohibit the intersection of two magnetic field lines, every
time magnetic field lines come close to crossing they are "rearranged," and this magnetic
reconnection continuously heats the solar corona. It's a fairly inefficient source of energy, but the
vast number of these small magnetic patches on the Sun’s surface makes the process a viable
solution to the 50 year old problem of what heats the solar corona.
6.4.5. Solar Wind Particles from the corona also stream out along the magnetic field lines of the
Sun that extend into interstellar space. This "solar wind" transports particles through space at 400
kilometers per second. (At the Earth the number density of protons in the Solar wind  7 x 10 6 m-3 ;
very low). When the solar wind reaches the Earth the magnetic field of the Earth will sometimes
trap these electrons and protons and pull them into the Earth's atmosphere. Atoms in the Earth's
atmosphere interact with these high energy particles by accepting energy from them and then
releasing that energy in the form of coloured light – aurora.
6.5 Sunspots and magnetic activity
Sunspots are indicators of solar activity.
Measurements of magnetic fields reveal that all sunspots are characterized by magnetic field
strengths that are much higher than elsewhere in the photosphere. Possibly the intense magnetic
field suppresses convection, which reduces the rate of energy transport to the photosphere resulting
in the relatively cool regions that characterise a sun spot (hence they appear darker than their
surroundings). Sunspots tend to occur in groups or pairs with opposite polarities.
Magnetic field lines in a sunspot Add diagram during lecture
Magnetic field lines and a pair of sunspots
The field lines emanate from one sunspot and form a loop which arches above the photosphere and
returns to enter the Sun at the other member of the pair.
7
The sun has an 11 year cycle of
sunspot activity, which can be
seen in the varying number of
sunspots observed.
The Approximately 11 year solar
sunspot cycle.
Why does the sun have a magnetic field? We believe that the Sun’s magnetic fields are
generated by electrical currents acting as a magnetic dynamo inside the sun. Magnetic fields can
be formed when electrically conducting fluids flow according to certain patterns, and we think such
electrical currents are generated by the flow of hot ionised gases in the Sun’s convection zone.
The Sun’s magnetic dynamo has a 22 year cycle. During the first half of this cycle, the Sun’s
magnetic north pole is in its northern hemisphere and the magnetic south pole is in the Sun’s
southern hemisphere. At the time of the peak in sunspot cycle (solar maximum) the magnetic poles
flip, exchanging places, and the magnetic north is now situated in the Sun’s southern hemisphere.
This “flip” happens every 11 years at solar maximum.
Lecture 6: Summary
• Summary of eqs. of stellar structure; boundary conditions –be able to discuss, and derive
these equations (see previous lectures)
• Solar atmosphere structure: photosphere, chromosphere, transition region, corona, and
solar wind – be able to discuss, describe, comment on.
• Coronal heating problem – be able to discuss
• Importance of magnetic fields in the corona - be able to describe & discuss related phenomena
• Solar magnetic activity cycle , 11-year sunspot cycle, 22 year magnetic dynamo cycle – be
able to comment on
TO DO: look through the first 6 lectures of the course which have covered the derivations of the
stellar structure equations. Make sure you understand this material, and attempt past exam
questions of relevance.
8
Part 1 continued:
Sun, Stars and Planets
7.
J C Pickering
The Sun: its structure and energy generation
2014
Lecture 7: The solar spectrum and stellar spectra
7.1 Thermal radiation spectrum
7.2 Absorption lines
7.3 Non thermal radiation
7.4 Common uses of stellar spectrum analysis
7.5 Spectral Classification of stars
7.6 Estimating luminosity and luminosity Classification of Stars
To Do: re-read sections covered in previous lectures of particular relevance:
3.1 Thermodynamic equilibrium
5.5 Photon Energy
7.1 Thermal radiation spectrum
7.1.1 Black body radiation
We see a thermal radiation spectrum from the Sun’s photosphere (the Sun’s visible bright
surface). In the photosphere the photons are in thermodynamic equilibrium with matter
and so have a black-body spectrum with intensity distribution B or Bλ :
Black body [Reminder]:
There are 2 key features of sources that produce black-body spectra:
i) The energy that is emitted as light has its origin in the internal, or thermal, energy of the
material making up the source – such sources are thermal sources of radiation.
ii) In addition to being a thermal source of light, the condition is also needed that the light
within the source is more likely to interact with the material of the source than to escape, it
will only escape after considerable interaction with material within the source. So a
common feature of BB sources is that they are opaque. Many astronomical sources
produce continuous spectra with reasonably good approximation to BB form.
The spectral distribution of this intensity is dependent solely on the temperature of the
material T, and is given by the Planck function:
Planck Function:
(
)
(
)
(
⁄
(
⁄
)
)
h = 6.6 x10-34 Js-1
k =1.4 x10-23 JK-1
c= speed of light, λ wavelength, υ
frequency, T temperature
B has dimensions of: energy / unit area / unit time / unit frequency / unit solid angle
B has dimensions of: energy/unit area/ unit time/ unit wavelength/unit solid angle
1
Planck distribution for black body objects at a range of temperatures
Notice that:
The area under the Planck curve increases
dramatically as T increases.
This is not surprising as for a blackbody flux (W/m2)
per unit area of emitter is given by Stefan’s law:
Also, since
luminosity:
where R is stellar radius,
So as temperature of the star increases the
luminosity increases dramatically.
7.1.2 Wien’s displacement law: maxT = 2.8979 x 10
-3
mK
gives maximum of Planck distribution. So provided an object is a black-body source, by studying its
spectrum one can determine its temperature. Temperatures of stars can be found using this.
For solar surface temperature (T=5800K), Bλ(T) peaks in the visible range of the spectrum, at a
wavelength of about 500nm, in yellow part of the visible spectrum:
violet 400nm (4000 Å)
red 700nm (7000Å)
(Å is Angstroms)
Cooler stars - redder (longer wavelength) radiation
Hotter stars - for hotter stars than the Sun the peak of the Planck distribution of flux density
moves to shorter wavelengths and these stars will appear white, or bluer than the Sun.
So a star’s colour will give an indication of its surface temperature (more on this in lecture 9)
Question: A star has luminosity 5 times the solar luminosity, and from its spectrum its
effective temperature is 30,000K.
What is the stellar radius? At what wavelength does
the star’s radiation peak?
[ The solar max= 500nm and Teff = 5800K ]
7.2 Absorption lines
7.2.1 Formation of absorption lines in stellar spectra
Dark lines are seen in the
continuous blackbody spectrum.
These lines are formed in the
cooler overlying material in the
Sun’s atmosphere.
2
7.2.2 Energy levels and transitions
A photon can be absorbed if its energy matches the difference between two energy levels of an
atom. An electron is excited to a higher lying energy level or atom is ionized:
Photon of energy h is absorbed:
---------------------------------- En’
h = ΔE
------------------------------------ En
ΔE = hc/, =hc/ΔE, h = Planck Constant (Js), = wavelength, ħ =h/2, me(e- mass)= 9.11 x 10-31kg
o = electric permittivity of free space.
For hydrogen:
Or
En= -h c R / n2
(J)
= -13.6/ n2 (eV)
where the Rydberg constant, R = m e4 /(8 o2 h3 c), and 1eV=1.6 x 10-19 J.
In fact, about 25,000 lines have been identified in the Sun’s visible spectral region, most originating
in the photosphere, which is responsible for nearly all the light between the lines. Each element,
each atom and ion has its own unique spectrum – ranging from simple hydrogen spectrum, to the
complex spectrum of iron with 1000s of transitions.
7.2.3 Occurrence and strength of lines depends on:
i) amount of element present (relative abundance)
ii) probability that electron is in appropriate energy level (depends on T; the higher the T the more
likely that energy levels corresponding to higher energies will be occupied)
iii) probability that photon of given frequency will be absorbed
So, solar absorption lines give information on: Chemical abundance and temperature, and hence
density and pressure.
The number of lines observed depends on the complexity of an atom or ion’s energy level structure.
The energy level structure of hydrogen is very simple, Helium is more complicated, and the level
structure of the iron group of elements (3d transition elements) is complex with hundreds of energy
levels and so these have thousands of lines in their spectra.
In fact astronomers use vast databases of atomic and molecular data in order to identify transitions
seen in stellar spectra, and to interpret what these tell them about a star. The data bases contain
both laboratory observed experimental spectra for each atom and ion, as well as calculated values –
data on millions of lines!
3
7.2.4 The spectrum of the Sun (see Slides file for lecture 7, on course web site)
An image of a spectrum can be represented graphically in terms of flux m-2 nm-1 against  and the
absorption lines have different relative depths, or strength (corresponding to different levels of
blackening of photographic plate or nowadays, detection by detector.) The spectrum of the Sun
follows the curve for a blackbody at 5800K, with absorption and emission lines also.
Solar spectrum overview.
On this scale the emission and
absorption lines cannot be seen.
Much higher resolution would be
needed to see the spectral line
features.
[Interesting web site: allows you to plot a segment of solar spectrum in detail at:
http://bass2000.obspm.fr/solar_spect.php - select “tools – solar spectrum”]
7.3 Also non-thermal radiation is observed
7.3.1 Bremstrahlung: (free-free transitions – see opacity, Lecture 5)
Transitions are possible that involve only the continuum of the H atoms. These are called free-free
transitions or Bremsstrahlung (braking radiation) – and the corresponding absorption is free-free
opacity. (In stellar interiors, where H is completely ionized free-free absorption is one of the most
important contributors to opacity.)
7.3.2 Synchrotron radiation: electrons spiral around B-lines
Synchrotron radiation is seen in the X-ray and radio emission from hot coronal plasma and from hot
plasma in flares. Any charged particle moving in a magnetic field is forced to move in a spiral
around magnetic field lines. The spiraling motion of electrons in a magnetic field gives rise to the
emission of electromagnetic radiation.
7.4 Some common uses of stellar spectral analysis
a) classification of spectral type of star (7.5 below)
b) classification of luminosity class
(7.6 below)
c) measurement of photospheric chemical abundances
d) measurement of radial velocity of star from Doppler shift of centre of spectral line
e) measurement of stellar rotation from additional broadening of spectral line by the
rotational velocity
f) measurement of the mass inflow or outflow from asymmetries in the line profile
g) measurement of photospheric magnetic fields by the Zeeman effect
7.5 Spectral Classification of stars
In order to be able to systematically discuss and understand the behaviour of stars we need a
system of classification of stars. We can use the temperature and colour, as well as the main
spectral characteristics of the star for such as system:
4
We have looked at the solar spectrum, and how absorption lines are seen in stellar spectra – these
absorption lines can be used as an indicator of the star’s temperature, as they show us what
species (atoms/ions or molecules) are a dominant or noticeable source of opacity, and thus roughly
what the temperature is – in simple terms: cool stars will show lines from molecules and atoms, hot
stars will show H and He lines relatively more strongly.
Spectral Classification
Type
Colour
Teff (K)
Main characteristics
Examples
O
Blue
> 25000
He+ lines; strong UV
Mintaka
B
Blue-white
11000–25000
Neutral He lines
Rigel, Spica
A
White
7500 – 11000
Strong H lines
Sirius, Vega
F
Yellow-white
6000 – 7500
Weak metal lines
Procyon
G
Yellow
5000 – 6000
Solar-like spectrum
K
Orange
3500 – 5000
M
Red
< 3500
Metal lines dominate
Molecular bands
noticable
Sun,Capella
Arcturus,
Aldebaran
Betelgeuse,
Antares
The spectroscopic method of obtaining photospheric temperatures was well established by the
1920’s, led by astronomer Anne Jump Canon at Harvard University. On the basis of the strengths
of their spectral lines, stellar spectra are classified by letter in a scheme called the Harvard Spectral
Classification. In order of descending temperature the spectra classes are labeled O B A F G K M
Each class also divided into 10 subdivisions, from 0 to 9, with 0 being the hottest. So F9 and G0
stars are very similar.
so e.g. …, B8, B9, A0, A1, A2, … A9, F0, F1, …
Example: the Sun is a G2 star, and Rigel is classified as a B8 star. If we can establish a star’s
spectral type, we can determine its average photospheric temperature with an uncertainty of only
about 5 %. [Note: OBAFGKM phrases can be used to remember order… ]
The strengths of
various absorption lines
versus photospheric
temperature
(From: S.Green & M Jones, An
Introduction to the Sun and
Stars (Cambridge University
Press))
7.6 Estimating luminosity and luminosity Classification of Stars
In addition to the spectral classification scheme in 7.5 which assigns a letter to a star of particular
spectral characteristics and temperature, we also have a luminosity classification based upon the
widths of the absorption lines in the star's spectrum. (An absorption line is not infinitely thin. It has
a strength and a width – the strength is the area within the line profile.) A luminosity classification is
necessary as stars may have a similar temperature and spectral characteristics, but have very
different luminosity – they may be a different kind of star (will be discussed later in the course…).
5
If one looks at spectra of stars of the same temperature, but different luminosity (where L has been
determined non-spectroscopically), then at a given temperature, the more luminous the star, the
narrower are its spectral absorption lines, and the stronger are the absorption lines due to
certain ionised atoms.
Luminosity has an effect on the line width – because of the conditions in stellar atmospheres: A
large star has lower density in its outer layers than a small star of the same temperature, because
the mass of these layers is spread over a larger volume. In small stars, density and pressure are
higher, atoms and ions collide and interact more, causing distortions in the energy levels of the
atom, and thus a wider range of wavelength for a particular photon. So, we get broader absorption
lines for smaller stars (lower luminosity). This process is called pressure broadening.
Also, because of more collisions and recombinations between particles in the atmosphere, there are
less ions present in a smaller star, and so the spectral lines from ions are weaker for a lower
luminosity star than for a larger star of the same temperature.
This correlation of luminosity and spectral features is used to determine luminosity for stars of
unknown L, it is not precise but gives luminosity classes which can distinguish different groups of
stars.
So: Absorption lines are Pressure-sensitive:


Spectral lines get broader as the pressure increases.
Big stars have lower pressure in their atmospheres compared to smaller stars.
Implications:


Larger stars have narrower absorption lines.
Larger stars are brighter at the same temperature.
Since larger stars are brighter at a given stellar temperature (more surface area to radiate),
measuring differences in the line widths for stars of the same temperature gives an estimate of the
stellar luminosity. This gives us a way to assign a relative luminosity to stars based upon their
spectral line properties!
Luminosity Classes:
Ia: the most luminous supergiants
Ib: less luminous supergiants
II: luminous giants
III: normal giants
IV: subgiants
V: dwarfs (main-sequence stars)
The full designation of a star’s spectral type also
includes its luminosity class.
The Sun is spectral type G2 V,
Betelgeuse is a spectral type M2 Ia.
←Diagram showing approximate positions of
luminosity classes I to V. (From: S.Green & M Jones, An
Introduction to the Sun and Stars (Cambridge Univ. Press))
Lecture 7: Summary
• Thermal radiation – form given by Planck’s function: be able to describe, and explain
• know Wien’s law - know and be able to use maxT=constant.
• know (be able to write down) and be able to use Stefan’s law
• Absorption and emission lines at frequencies determined by energy levels of electrons in
atom: be able to understand and explain origin of lines in stellar spectra and what information
may be obtained about the star from stellar spectra
• Balmer and Lyman series for hydrogen: be able to explain and describe these
• Stellar spectral types – be able to understand and explain these
• Luminosity and line widths – understand and be able to discuss the link between these
• Luminosity classes – know what this is, more details later in the course.
TO DO: Problem sheet 2 - Questions 1, 2, 8
6
Sun Stars Planets
Section
8.
J C Pickering
2014
Stars, putting the sun in context.
Lecture 8: Understanding the Main Sequence
8.1 The main-sequence
8.2 Equations of stellar structure, state, and boundary conditions
8.3 Homology transformation, and relationships between parameters
8.4 Main sequence lifetime
8.1 The main-sequence.
90% of the stars lie on the main-sequence, the longest phase in their life, during which the energy
generation is hydrogen burning. We can use the stellar structure equations to understand
observed stellar properties.
These diagrams (known as HertzsprungRussell diagrams, more on this in later
lectures) plot luminosity and temperature
of each star: in the solar neighbourhood
and in the Pleiades star cluster.
8.2 Equations of stellar structure, equations of state and boundary conditions
8.2.1 Given here is the summary of the equations covered so far during the course – see your
lecture notes for the derivations, etc
Four differential equations governing stellar structure: In the stellar interior we have:
Hydrostatic equilibrium
[1]
Mass conservation
[2]
[3a]
Radiative heat transport
Convective heat transport
Radiative equilibrium
(
)
[3b]
[4]
1
We need expressions for P, and . If a star is in a steady state, and in a state close to
thermodynamic equilibrium, then these three quantities depend on , T and C (chemical
composition), and these will generally be functions of radius r. So we have a problem of basic
physics, to determine the pressure, opacity and rate of energy generation of a medium for given
conditions of density and temperature:
=  (, T, C)
Opacity
We could assume opacity of stellar material is given by Kramer’s opacity formula:
[5]
 
 = constant
Energy generation rates
=  (, T, C)
For many thermonuclear burning stages, the energy-generation rate, may, within a limited T range be
approximated by:
[6]

where: = constant, and  are usually slowing varying functions of and T, and may be taken
as constants for simple calculations. For most fuels  1, and   4 for hydrogen burning, and
  30 or more for carbon burning.
Chemical composition this enters through parameters μ (mean molecular weight,

к (opacity)
and
(energy generation rate).
Equations of state
P = P (, T, C)
Assume either gas pressure or radiation pressure operate, but not both. Possible expressions for
pressure:
[7a]
[7b]
Where  is the gas constant and a the radiation constant.
8.2.2 Boundary conditions and calculations
(i)
Inner boundary conditions
R
O
At
r =0
(ii)
At r = R
r
m(r) = 0
L(r) = 0
surface boundary conditions
T=0
and P=0
(since TS << Tc and PS <<Pc)
So given the chemical composition of a star, we have 7 equations for 7 unknowns P, , T, M, L,
and  as functions of r.
Model calculations then involve: obtaining expressions for P , and  and then solving the four
differential equations [1] – [4] with boundary conditions.
This yields P, ,T, (C), as functions of r, L, Teff
2
8.3 Homology transformation
The equations (listed in 8.2) are subject to a homology transformation: that is, given a solution to the
equations, with the quantities (P, T, L, ) stated as functions of r, for a given total mass M and given
chemical composition C, then we can find a new solution for a new total mass, simply by multiplying
the other physical variables by appropriate scaling factors.
So since the input physics is the same, we expect that in terms of dimensionless coordinate x = r/R
the internal structure of stars will be identical except for a scaling:
So for two stars, say star 1 and star 2:
O
r
R
and so for example:
Add notes
This expresses the fact that each star contains the same fraction of its mass within the same fraction
x of its radius.
And (add notes →):
We have the same curves for all stars with the same physics and composition.
So, for example, if we know the P in one star at a particular fraction of its radius, then we would know
there is a rule to scale this to determine the P in another star at another fraction of its radius.
So we can write:
Where Po
, To , mo , o , and
Lo
functions are the same for all the
stars.
We thus have a set of answers, which
are the same for all stars.
We can find the relations, for how ρc,
Pc, Tc and L scale with M and R:
3
8.3.1 Central density, M and R
- make notes during lecture
3
ρc  M / R
[8]
shows how the central density scales from star to star.
8.3.2 Central P, M and R
- make notes during lecture
2
4
Pc  M / R
[9]
Shows how the central pressure scales from star to star
8.3.3 Central T, M and R
TO DO
You work through this one
And similarly using equation of state for pressure:
substitute in for
…gives
Pc  ρc Tc
P
ρ
T
= P c Po
= ρc ρo
= Tc To
But from [8] ρc  M / R
2
3
4
2
4
and from [9] Pc  M / R
3
M / R  ( M / R ) Tc
so:
giving:
Tc  M / R
[10]
8.3.4 Luminosity-mass
and also, similarly, using the radiative heat transport equation:
[where of course T, L, and ρ, are functions of r]
ρ
substitute in for:
= ρc ρo
=r/R,
L(r) = L Lo
T
= Tc To
d /dr=1/R and  = o   T-
giving:
[-------------------------------------------------]
Same for all stars
3
But ρc  M / R [8]
and
Tc  M / R [10]
so this gives, after substituting in for Tc and ρc :
L M
3+ β –α
R
3α –β
[11]
4
L M
3+ β –α
3α –β
R
[11]
For low-mass stars ( M
1M )

 : Kramers opacity  = 1,  = 7/2

 


: p-p chain
 4
- so [11] becomes:
L M
L M
5.5
R
-0.5
5.5
 M
approximately because R dependence is so weak
5.5
[12]
(for low mass stars)
For higher mass stars
 : electron scattering = 0,  = 0

: CNO cycle
  17
L M
so [11] becomes:
3
[13]
(for higher mass stars)
So a reasonable compromise across the entire mass range is:
L M
4
[14]
[14] is an important relation – you should know this
TO DO:
PLEASE work through the above calculations yourselves
8.3.5 Radius- mass
A similar method, as used in the derivations 8.3.1 – 8.3.4, is used here.
Starting with
[4]
where L, ρ and  are functions of r
Substituting in [4] for:
ρ
= ρc ρo
(you work this through):
 (r) = o ρ(r) T(r) 
T(r) = Tc To(x)
=
( L/R ) dLo
but comparing with [4] we can say that:
L
/d
dLo
= [4 
/d
2
= L Lo
ρo
=4
o ρo
2
ρo
….gives
To

o ρo
To
] R2 ρc2 Tc 

will be true for all stars (it is one of the 4 equations describing stellar structure) in terms of our new
variable , fractional radius.
5
So we can therefore state that :
L/R  R2 ρc2 Tc [15]
Because energy generation [6]  (r) = o ρ(r) T(r) 
does vary with the mass of the
stars, we consider now the two cases for low and higher mass stars :
(i) low mass stars
(  4)
Using equation [15] L/R  R2 ρc2 Tc 
And substituting in for L, Tc, and ρc into [15] using:
5.5
-0.5
R
[12] (for low mass stars – more precise version before simplification of
L M
neglecting of R)
… and (  4)
gives
-0.5
R3 ρc2 Tc   M5.5 R
3
then substitute in for ρc  M / R
gives
[8] and for Tc  M / R [10]
R3 (M2 / R6) (M4 / R4)  M5.5 R
-0.5
giving M 0.5  R 6.5
so:
M R
13
or
R M
1/13
[16]
for low mass stars
(ii) higher mass stars
(  16)
You Do this:
follow the same procedure as in (i) above, with   16 but use [13]
higher mass stars.
You should get the result:
15 / 19
R M
[17]
For higher mass stars
3
L  M which is for
These relations [16] and [17] between R and M, allow you to revisit relations [8], [9] and [10]
for low and higher mass stars to now give the dependence of central density ρc, central
pressure Pc and central temperature Tc on stellar mass M.
For example:
ρc  M / R
3
[8]
would become:
(i) for low mass stars, using [16],  ρc  M /M3/13
so approximately
ρc  M
for low mass stars
[18]
6
(ii) but for higher mass stars, using [17],  ρc  M / M45/19
ρc  1 / M
so approximately
for higher mass stars
[19]
These are strikingly different relations for ρc !
8.3.6 The slope of the main-sequence
The H-R diagram is a plot of log L
against log Teff . So we are looking
for the relationship between these two
physical parameters, for both low and
higher mass stars.
–8
Log L
–4
Log Teff
We already know L = 4  R2  Teff4 from previous lectures, so
we can write Teff4  L/ R2
(i) low mass stars:
Use
[12]
L M
5.5
gives: Teff4  M5.5 / M2/13
L M
but
[16] R  M
and
1/13
and substitute into Teff4  L/ R2
approximately Teff4  M5.5
or
5.5
so approximately,
L  Teff4
for the lower main sequence
[20]
(ii) higher mass stars:
use
L M
[13]
3
and
gives: Teff4  M3 / M30/19 ,
but
L M
→
15/19
and substitute into Teff4  L/ R2
or approximately Teff4  M 3/2
(Teff4) 2/3
M 
so
[17] R  M
3
so approximately,
L  Teff8
L  [ (Teff4 )
2/3
for the upper main sequence
]3
[21]
These relations of L and Teff for upper and lower MS are qualitatively correct, but not as steep in
observation.
7
We have seen that for stars of the same chemical composition, chemically homogeneous
stars, the equations governing the stellar structure need only to be solved once, and the
properties of stars of all masses could then be obtained. Such a set of models of stars in
which the dependence of the physical quantities on fractional mass mo and x is independent
of the total mass of the star is known as a homologous sequence of stellar models.
8.3.7 Departures from homology scaling:
Any departures from homology scaling are due to:
(i)
Radiation pressure (important in high-mass stars)
(ii)
Convection in (a) the core and/or (b) the outer part of the star
8.4 Lifetime of a star on the main sequence
Amount of fuel available during hydrogen burning phase
But
So
Time t for star to exhaust fuel in its H burning phase is, since
Main sequence lifetime
[14],
[22]
Massive stars are quickest to exhaust their hydrogen fuel.
LECTURE 8: SUMMARY – learning outcomes




Can qualitatively understand form of MS
On Main Sequence (MS), L increases with M
More massive stars have shorter main sequence lifetimes
Homology scaling, a useful tool: be able to quickly show useful relationships between
parameters starting from equations of stellar structure, and equations of state
TO DO: Problem Sheet 2 Q 7.
8
Sun, Stars, Planets
9.
JC Pickering
2014
Lecture 9: The Sun in context, stellar magnitudes and measurement of distance
9.1 Basic parameter ranges for stars
9.2 Stellar magnitudes
9.3 Stellar colours and temperatures
9.4 Measuring stellar distances
9.5 Proper motion
9.6 Summary of methods of distance determination
9.1 Basic stellar parameter ranges: (mass M, luminosity L, radius R, effective temperature Teff)
0.1 M
< M
<
0.01 R
< R
< 1000 R
10
-4


L

< L
2000 K < Teff
50 M
6
< 10


L

< 100 000 K
How are these measured? We have already described briefly how M R
L and Teff are




determined. We can now turn to the determination of these parameters for more distant objects.
We can use theoretical models (for example as discussed in lecture 8) then in order to understand
the values we find for a large number of stars.
9.2 Stellar magnitude
How bright are the stars?
When we observe stars to determine their brightness, we are actually measuring the amount of
light from the star reaching us – its apparent brightness. To study and compare stars and their
properties we have to know their intrinsic brightness. By intrinsic brightness we are talking about
the total amount of power a star radiates into space over all wavelengths. This is known as
luminosity, (measured in Watts).
Hipparchus, the Greek astronomer, catalogued and classified stars visible to the naked eye
according to their visual brightness - into six magnitudes: the brightest being first magnitude and
the faintest 6th magnitude. A 1st magnitude star is approx 100 times brighter than a 6th magnitude
star. The human perception of brightness as measured by the eye is logarithmic in nature (equal
steps of perceived brightness correspond to equal ratios of flux density). So the magnitude scale is
a logarithmic scale. The modern quantification of brightness was then designed to agree with the
old Greek measures. The modern system: quantifies magnitudes as:
5 steps of magnitude = factor of 100 in Flux.
So for example:
 a 10th magnitude star is 100 times fainter than a 5th magnitude star.
 a 20th magnitude star is 10,000 times fainter than a 10th magnitude star.
If you look up tables of stellar properties, commonly you would find values given for the stars’
apparent or visual magnitudes rather than values of their luminosity L.
9.2.1. Definition of stellar magnitude
is the apparent magnitude (definition)
is the flux of light (energy per unit time per unit area) received from star at the observing point.
The value of the constant K is chosen to match to the scale of Hipparchus.
Note: the larger the value of (i.e. brighter star), the smaller the magnitude
The apparent magnitude scale can be applied to any object in the sky:
Sun:
-26.73
Full moon: -12.7
Venus at its brightest: -4.1
1
Brightest star: -1.46
Naked eye dimmest stars (urban): +3
Naked eye, faintest visible, dark skies: +6
Faintest objects detectable by HST ≈ +31
Faintest objects detectable through the largest telescopes (eg ELT) ≈ +36
The Sun and the dimmest objects detectable through telescopes are 62 magnitudes apart, this
24
corresponds to an actual difference in brightness of over 6 x 10 .
The difference in magnitudes of two stars can be expressed:
(
)
( )
Where
where d = distance of star to the Earth.
range from 10 -8 W m -2 for Sirius, the brightest star in the sky, to 10-20 W
Values of flux density
m -2 for the faintest detectable objects, ranges from –1.5 to 36.
So apparent magnitude depends on the star’s luminosity L and its distance d from the observer.
is what can be observed
To compare intrinsic properties of stars, we need to remove dependence on distance.
So we want to define absolute magnitude M of a star to be what its apparent magnitude would be
if it were at some chosen standard distance away. What distance?
9.2.2 Parsec: unit of distance
Astronomers measure distances in units of a parsec.
1 parsec (pc) is defined as the distance at which a length of 1 A.U. would subtend an angle of one
arc-second.
Add diagram
By definition (using small angle approximation):
Aside:
16
15
1 pc = 206265 AU = 3.09 x 10 m.
(Just for reference, 1 light-year = 9.5 x 10 m)
1 AU = 1.5 x 1011 m = average distance from Earth to Sun.
2
9.2.3. Definition of absolute magnitude
The absolute magnitude M of a star is the apparent magnitude the star would have if it were placed
at a standard distance of 10 pc away.
Thus we can calculate the absolute magnitude M of a star as follows:
Where is the flux of the star observed at Earth with the star at its actual distance , and
flux that would be received if the star were at a standard distance =10pc from Earth.
We know
is the
and L is an intrinsic property of the star. So
(
(
)
But d2 = 10pc
( )
(
)
)
Where d=distance to star in parsecs.
Most stars have absolute visual magnitudes within the range -6 < Mv < 16
Sun has absolute visual magnitude Mv = 4.8, and it is a very average star.
Example: what is the luminosity of a star whose distance is 60pc and with an
apparent magnitude of 3.61 ?
9.2.4 U B V passbands
So absolute magnitudes of stars can be used as a comparison of the stars’ brightness.
But to determine the magnitude we need to measure the stars’ flux. In practice the flux density is
measured using detectors and filters which are sensitive to different wavelength ranges.
We can measure magnitudes in different wavelength bands. The most well known of these is the
“UBV “ system.
Three standard filters are:
U “Ultraviolet”
B “ blue”
V “ visual” passband
Each of U, B, V filters about 100 nm wide, centred on following wavelengths:
λU ~ 365 nm
λB ~ 440 nm
λV ~ 550 nm
So e.g. we talk about: apparent visible magnitude mV
And
absolute visible magnitude MV
These magnitudes measured in the different bands can be used to assign a “colour” to the star and
give an indicator of T.
3
9.2.5 Bolometric magnitude
The total apparent or absolute magnitude of the star, taken over all wavelengths, is usually called
the bolometric magnitude. The difference between the bolometric magnitude of a star and its
magnitude in, for example the V band, is called the bolometric correction, BC.
Theorists are interested in the total energy output of star. The magnitude corresponding to total
energy output is called the bolometric magnitude: mbol , Mbol.
[Where, as before, Mbol and
mbol - M denotes “absolute”, and m denotes “apparent” bolometric magnitudes.]
A star’s energy output is approximately black-body in distribution, so its effective temperature can
be calculated from measuring two of mU, mV, mB.
Then from say mV, the bolometric magnitude mbol is calculated by adding the bolometric
mbol = mV + BC
correction BC:
BC values are tabulated for each type of star (OBAFGKM and Luminosity class), and are generally
negative, since there is more energy in the whole spectrum than in a limited part of it.
Exercise:
Mbol = 0 for a main-sequence star with L = 3 x 10
28
W.
Show that Mbol = -2.5 log ( L /(3 x 1028))

and calculate Mbol and mbol


Recall that stars radiate approximately as black bodies,
9.3 Stellar colours and temperatures
so their intensity is given by the Planck function:
V
log B
B
U
T
Note: x axis is
frequency in this plot,
unlike the Planck
distribution plots
shown in lecture 7,
which are against
wavelength.
log

Measuring a star's brightness in (say) U, B, V bands gives a measure of its effective (surface)
temperature.
and
B – V = mB – mV
Define colour indices:
U – B = mU – mB
Teff
U–B
B–V
40 000 K
-1.15
-0.35
9 900 K
0.0
0.0
Cooler stars are redder ;
U-B, B-V positive
4 900 K
0.47
0.89
hotter stars are bluer
U-B, B-V negative
We can therefore determine a star’s temperature using colour indexes, known as the photometric
method of temperature determination. (Note: this is independent of distance.)
However, a more accurate measurement of photospheric temperature can be obtained for
individual stars by the spectroscopic method (explained in lecture 7) based on analysis of the
spectral absorption lines in the observed stellar spectra.
4
9.4 Measuring stellar distances: Method of trigonometric parallax
Distance is measured directly geometrically by this method. It is important to know the distances in
order to: be able to estimate the luminosity of an object; to find masses of objects from their orbital
motions (using Kepler’s 3rd law); and for estimating physical sizes of objects.
Nearby stars
have small periodic apparent motions (with respect to more distant background stars) due to
Earth’s motion in its orbit around the Sun. This is not a “true” motion and is called Stellar Parallax.
d
In the diagram the line of sight to
the star in June differs from that in
December, when the Earth is on the
other side of its orbit around the
Sun. The nearby star, as seen from
Earth, appears to sweep through
the angle shown. Half of this angle,
is the parallax, p.
d=1/p
Naturally as we observe the parallax of more distant stars the parallax decreases with increasing
distance to the star. As the nearest stars are still far from Earth, the largest measured parallaxes
are of the order of less than an arcsecond. Our nearest star, alpha Centauri, has a parallax angle of
0.76 arcsec. Parallaxes are measured by both photography and digital imaging, and preferably
from space to avoid blurring and problems observing through the atmosphere of the Earth.
d=1/p
Where: p = parallax angle in arcseconds , d = distance from Earth to star in parsecs
[ Reminder: Parallax Second= Parsec (pc) Fundamental unit of distance : "A star with a parallax of
1 arcsecond has a distance of 1 Parsec." ]
1 parsec (pc) is equivalent to: 206,265 AU, 3.26 Light Years, 3.086x1013 km ]
eg: a star has a parallax of 0.02 arcsec – what is its distance? [Ans:50pc]
If p = 1 arc-sec, d = 1 parsec and
If p = N arc-secs,
d = (1/N) parsecs
Limitations: if the stars are too far away, the parallax is too small for accurate measurement.
Smallest measurable parallax from the ground is ~ 0.01 arcsec, so this method of trigonometric
parallaxes is limited: although it is a good method to distances up to 100 pc, there are not many
stars that are this close. However, satellite measurements (Hipparcos) measure parallaxes to
accuracy of 0.001 arcsec. Hipparcos measured parallaxes for about 120,000 stars, out as far as
1000pc for brighter stars. Gaia space mission will measure parallax accurate to a few  arcsecs !
9.5 Proper Motion
Stars not fixed in space: they move relative to the Sun. The motions
however are very tiny seen from the Earth.
Two components of the motion:
Motion along line of sight does not change star’s
position on sky.
Motion perpendicular to it does change star’s
position in the sky (by tiny amount).
Such a change in position of a star is called its
proper motion, so called because it is intrinsic
to the star, and not a result of the motion of the
observer or a moving reference point. Proper
μ
motion is usually expresses in seconds of arc
d
per year (3600 arc sec = 1 degree).
5
We can find the traverse speed of the star using the proper motion from
vt = d sin μ
(μ = proper motion, d = distance to star, which could be obtained by measuring its parallax) small
angle, sin μ ≈ μ
Example: Barnard’s star has the largest known proper motion: 10.3 arc-sec per year and also has
parallax 0.55 arc-sec (hence lies at 1.8 pc from Earth).
The radial velocity (vr), the component of the star’s motion relative to us (toward or directly away
from us) can be obtained by a method relying on the Doppler effect. This gives a change in the
). Knowing this wavelength shift (by
wavelength of a spectral line emitted by this object (
measuring it in a stellar spectrum), and comparing it to the known wavelength of the transition line,
(measured in the laboratory) the rest wavelength() , and using:
(
)
gives us the radial velocity . [ redshift: increase in wavelength, object moving away from us,
blueshift: decrease in wavelength, object moving towards us].
Note: since, in general, the proper motion of a star would be expected to decrease with distance
from Earth, the proper motion of a star can be used as a rough guide to the star’s distance.
The radial velocity and transverse velocity specify the overall motion of the star through space
with respect to Earth.
True Space velocity of star
√
These overall motions of the stars are not random, they are in part related to the large-scale
motions in our Galaxy, and by the groupings of many stars in clusters. In fact, whether a star
belongs to a cluster can often be decided by comparing its motion through space with that of the
other cluster members.
9.6 Brief summary: Some important distance measurement methods:
1] Parallax - method possible for nearer stars
2] Proper motion – greater for nearer stars in general, measure many stars’ proper motion and
we get a rough guide to distance
3] Variable stars – Period-Luminosity relation, measure period, calculate L, then measure
apparent magnitude (flux), and so get absolute magnitude – and then get distance (not covered as
part of this lecture course, but it is a very important method)
4] Colour and spectral type: gives Teff, but L=4R2  T4eff, assume stars of same type have
approximately the same size, so use R of similar spectral type star whose radius is known, then
with T and R, get L. Measure apparent magnitude with detectors from Earth, knowing L, then
calculate absolute magnitude – which then gives you the distance.
LECTURE 9: SUMMARY and outcomes
 Ranges of stellar parameters M, R, L, Teff – be aware of, and how the sun compares
 Definition of apparent and absolute magnitude – be able to define, know and use
 Apparent magnitude– relation between apparent magnitude, distance, luminosity, be able to
write down relevant equations and use.
 Definition of parsec – be able to define a parsec
 U B V photometry – be able to explain what this is
 Bolometric magnitude, bolometric correction – be able to define and explain
 Colour indices – be able to explain, and describe
 Stellar temperature measurement from colour indices – be able to explain and discuss
 Measuring stellar distances – by trigonometric parallax – be able to explain and calculate
parallax or distance
 Proper motion – be able to explain & describe this, and comment on use as distance estimate
 TO DO Questions now do-able: Problem sheet 2, Q 4,5,6,8,9, Problem sheet 3: Q 1,2,7
6
Sun Stars Planets
10.
J C Pickering 2014
Stars - Putting the Sun in context
Lecture 10: The Hertzsprung-Russell Diagram
10.1 Observers’ H-R diagram
10.2 Theorists’ H-R diagram
10.3 H-R diagrams: 100 brightest stars, 100 nearest stars
10.4 Distribution of stars on the H-R diagram
10.5 Mass and the H-R diagram
10.6 H-R diagram and clues to stellar evolution
10.7 Mass and rate of stellar evolution – clusters: open and globular
To understand more about stars and obtain insight into their evolution we need to look at the
overall distribution of stellar properties rather than individual properties of individual stars.
We can make use of Hertzsprung-Russell Diagrams to systemize our observations of
stars. Suitable properties for comparing stars are Teff, L and R – however we don’t need all
three – because stars emit like black bodies, Teff L and R are related (L = 4 π R2 σ Teff4) - so
if we know two of Teff , L and R, then we can get the third property.
10.1 Observers’ H-R diagram: Colour-magnitude diagram
- add during lecture
Note: magnitude is always plotted increasing downwards (larger magnitude=lower luminosity)
Absolute magnitude is related to luminosity, (see lecture 9)
B-V colour index, spectral type and effective temperature are related (see lectures 7 and 9)
So possible HR diagrams include: Colour-magnitude, and log L – log Teff, etc
10.2 Theorists’ H-R diagram: log L – log Teff
Since L = 4 π R2 σ Teff4 , R= [L/(4 π σ Teff4) 1/2 so for each point on the HR diagram there is a
unique stellar radius. So for stars of the same radius :
d log L = 4 and stars of the same radius lie on lines of slope –4 in the H-R diagram.
d log T
Annotate
during lecture
[ NB: temperature always
plotted increasing to the left]
1
10.3
HR diagrams – brightest and nearest stars
10.3.1 HR diagram of the 100 brightest stars
-8
Luminosity
Class I
Mv
Luminosity
Class II
absolute
magnitude
Luminosity
class V
4
B0
G0
M0
10.3.2 H-R diagram of the nearest 100 stars
2
Annotate
during lecture
Mv
16
A0
M5
10.4 Distribution of stars on the H-R diagram
Assume:
• any particular star is luminous for only a finite time
• there are distinct stages between star’s birth and death – each stage characterised by
some range of Teff and L; thus the star moves around the H-R diagram as it evolves
• stars we see today are not all at the same stage of evolution
So:
• if a large population of stars is observed today, then the longer a particular evolutionary
stage lasts, the greater will be the number of stars that we observe in that stage. Only very
few stars will be observed going through a short lived stage.
• the concentrations of points on the H-R diagram are those regions where stars spend a
comparatively large fraction of their lives. On this basis a star must spend most of its life
on the main sequence, because this is where 90% of the stars lie. Based on the above
2
assumptions we can say that the red giant, supergiant and white dwarf regions are where
we might expect some stars to be for a significant time.
• the concentration of stars on the H-R diagram depends not only on how quickly a star
passes through a region, but also on what fraction of stars pass through the region at all.
• some regions of the H-R diagram might appear to have no stars, or scarcely any, simply
because they correspond to stages in a stellar lifetime when stars tend to be shrouded in
cooler material and are therefore not observable directly.
Hipparcos
The Hipparcos satellite carried out precise measurements of all stars down to approximately
magnitude +9 of: position, parallax, proper motion and photometry (colour index etc) to
unprecedented accuracy. http://www.rssd.esa.int/Hipparcos
An H-R diagram of the stars measured using Hipparcos is shown below:
3
10.5 Mass and the H-R diagram
So we have our observational data: absolute magnitude and colour index, which gives us L against
Teff – and hence R, but one further property of a star is enormously important – mass.
Diagram:
Stellar mass and the
H-R diagram
Masses are given in
multiples of M
(From: S.Green & M Jones,
An Introduction to the Sun
and Stars (Cambridge
University Press)
• Stellar masses are measured from 0.08 M to about 50 M . The Sun is a very average star.
• The lower the mass, the greater the number of stars – huge stars are rare.
If we have a first look at stellar evolution: (see diagram above) – looking at the different masses on
an H-R diagram – notice:
(i)
supergiants tend to have greater masses than red giants which have a greater mass than white
dwarfs.
(ii) Within the class of supergiants, red giants or white dwarfs, there is no correlation of M with L or
photospheric Teff – masses appear “jumbled”.
(iii) Along the main sequence (MS) stars, mass does correlate to L, and hence Teff. As mass
increases, L and Teff increase. In fact we see a 500 times increase in mass giving a 1010
increase in L along the MS.
(iv) In the lower MS, masses are comparable with red giants, in the upper part of the MS the
masses are comparable with supergiants.
Do stars change their mass during their evolution?
We now have information on masses of stars on the H-R diagram, but before we use our L, Teff and
M information to look at stellar evolution we need to know whether a star’s mass varies during its
lifetime.
From observational evidence:
• one observes MS stars, red giants and super giants losing mass in terms of stellar winds. But this
mass loss is very much less than the star’s initial mass.
• In images of planetary nebulae we can see impressive mass loss, with shells of material having
been flung off by a central star. Some stars end their live by shedding a planetary nebula or more
violently a supernova – great mass loss…
• So for most of the life of a star, severe mass loss occurs only when a planetary nebula is shed,
and the stellar remnant becomes a white dwarf, or when a massive star ends its life as a
supernova.
10.6 Using H-R diagram features to devise a model for some of the stages of stellar evolution
At this stage, with the features we see in the H-R diagrams, and knowledge of stellar masses, and
mass loss, we can deduce the following…
1] Main sequence phase: during the MS phase a star does not change L or photospheric Teff very
much (otherwise it would move along the MS and this doesn’t fit with the range of masses seen).
4
However stars do drift very slightly above the MS during the MS phase, hence it appears as a band
in the H-R diagram rather than a narrow line.
2] After the MS phase – less massive stars become red giants, and more massive stars become
supergiants. This is consistent with stellar masses seen in these regions in the HR diagram. Also
consistent with the rarity of supergiants – there are very few very massive stars.
3] red giants evolve to a point where they shed planetary nebulae and the stellar remnant evolves
to become a white dwarf
4] super giants become star destroying type II supernovae.
10.7 Do stars of different masses evolve at different rates?
To answer this question we can look at star cluster HR diagrams. Make 2 assumptions:
1) That star clusters occur because the stars in them form at about the same time.
2) The compositions of stars in a cluster are similar as they formed from the same cloud of gas.
These two points are vital – if the stars form at the same time and have the same composition then
the stars in the cluster only differ in their mass, they are a homologous group. So, looking at an HR
diagram of stars from a single cluster will show if stars with different masses evolve at different rates.
10.7.1 Open Clusters (galactic clusters) These are comparatively young systems (age 100
million to a few billion years),with 102–105 stars, egs: the Pleiades (M45) and NGC 188.
Example HR diagrams for open clusters:
The Pleiades open cluster
Open cluster NGC 188
In the H-R diagram of stars in the Pleiades cluster notice:
• Almost all the stars are on the main sequence. This cluster is not old enough for many stars to
have reached the end of their MS phase.
• The most luminous stars visible on diagram appear to be moving away from the MS.
• The upper end of the MS, where the most massive stars are expected to be, is unpopulated.
In the H-R diagram of NGC 188, notice:
• The absence from the MS of all but the low-mass stars.
• The presence of considerable numbers of stars between the main sequence and the red giant
region, which could represent the higher masses missing from the main sequence. This suggests
the more massive a star, the sooner it leaves the main sequence, and that most stars that have left
the MS go on to become red giants.
• The most massive stars have left the MS, and must therefore have shorter MS lifetimes.
• The point at which this depopulation of the MS occurs is called the Main sequence turn-off, and
can be used as an indicator of the ages of clusters.
• Supergiants are absent in this cluster, which could be because massive MS stars, which are their
precursors, are rare. Also, if massive stars evolve rapidly, then any supergiants could have
become Type II supernovae, and have thus vanished from the H-R diagram.
5
• The absence of white dwarfs is presumably because they are too faint to detect.
• This is an older cluster than the Pleiades, because in the Pleiades the MS is populated to higher
stellar masses than the MS in NGC 188. This occurs because the more massive the star the
sooner it leaves the MS. In NGC 188 there has been enough time for all but the low mass stars to
leave the MS, whereas the Pleiades is too young (100 million years) for this to have happened.
10.7.2 Globular clusters are far older than open clusters, billions of years old, typically 106 stars.
In the H-R diagram of globular cluster
M13 the more massive stars have
evolved to become red giants, and
more details of the further evolution
will be given in the next lectures. The
MS Turn-off point can be used to
estimate the cluster’s age, in this
case 14 billion years!
10.7.3 Estimating cluster age from MS Turn-off point
At the Turn-off point the star leaves the main sequence as it ends H burning in its core.
Cluster age ߬ =
Total energy released on MS(ாౣ౩)
Luminosity
‫ܧ‬ெௌ = mass H "burnt" × Nuclear energy released/Kg (‫ܧ‬ே௄௚)
But
= ݂ ܺு ‫ܧ × ܯ‬ே௄௚
Where ݂ = fraction of mass depleted of H before the star leaves the MS ≈ 20%
ܺு = mass fraction of hydrogen
But ‫ܯ ∝ ܮ‬ସ and so ‫ܮ ∝ ܯ‬ଵ/ସ and we can write
‫ ⨀ܯ = ܯ‬ቀ௅ ቁ
௅
⨀
‫ܧ‬ெௌ
݂ ܺு ‫ܮ ⨀ܯ‬
߬=
=
ቆ ቇ
‫ܮ‬
‫ܮ‬
‫⨀ܮ‬
ଵ/ସ
∴
‫ܮ ⨀ܯ‬
߬ = ݂ ܺு
ቆ ቇ
‫⨀ܮ ⨀ܮ‬
ଵ/ସ
ିଷ/ସ
‫ܧ‬ே௄௚
‫ܧ‬ே௄௚
௅
݈‫ ݃݋‬௅ can be read from the HR diagram at the MS turn-off point, ݂ ≈ 20%, ܺு estimated (eg 0.6),
⨀
‫ܧ‬ே௄௚ can be calculated (eg ≈ 6.575 x 1014 J/kg). Putting these into the above expression allows the
age of the cluster to be calculated.
LECTURE 10: SUMMARY
• Hertzsprung-Russell diagram: be able to understand, sketch, explain, discuss
• Main sequence, Red giants, White dwarfs: introduction to these stars
• Typical HR diagrams: be able to discuss and describe, understand various regions, and what
we can deduce about stellar evolution. Nearby stars – Brightest stars – be able to describe,
explain and comment on the difference in the HR diagrams for these
• Open and Globular clusters - be able to comment, explain and describe the HR diagrams.
• Estimate of cluster age using MS Turn off point - be able to estimate using HR diagram
6
Sun Stars Planets
Lecture 11
J C Pickering
2014
Life of a star: part 1. The main sequence and post - main sequence evolution
11.1 Main sequence life
11.2 Life beyond the main sequence: post main sequence evolution
Consider the three phases of stellar evolution:
- The first phase follows the formation of a star from a cloud of interstellar material through its
gravitational contraction and subsequent heating to the main sequence, defined as the point at which
hydrogen burning begins.
- The second phase follows the development of a star on through the main sequence (MS): a star
spends most of its lifetime on the MS so this is an important, though least spectacular place.
- The final phase, post main sequence life, has more rapid evolution that begins when H burning
has been exhausted in the stellar core, through to the star’s death.
11.1 Main sequence life
11.1.1. The zero age main sequence ZAMS
The curve defined by values of L and Teff corresponding to static stars that are homogeneous and
have just commenced hydrogen burning forms the zero age main sequence (ZAMS).
The star is essentially static, supplying energy losses by nuclear burning.
Equations of stellar structure hold well (see earlier lectures).
As we saw for the Sun, nuclear processes take place in the core. Nuclear reaction rates vary with T,
so moving out from the centre of a star, the boundary defining the limit of nuclear reactions is sharp.
The size of the core varies with stellar mass, as do the reactions and the mechanisms of energy
transport to the outer layers of a star.
On the main sequence:
11.1.2. Nuclear reactions and energy transport:
Lower main sequence stars ( < 1.5 M )

Core H burning. Radiative transport from core, further out there is a convective envelope.
Upper main sequence stars ( > 1.5 M )
Core T high enough for CNO H burning to take place,

and energy release in the core is sufficiently concentrated to trigger convective instability – and the
centre of the star is convective.
Diagram: Structure of 1 M MS star

5 M MS star

1
11.1.3 Why do we have a main sequence?
Consider L and T and why they remain constant for so long:
(i) stability
A stable star is in hydrostatic equilibrium: the inward force of gravity on each layer of a star is
balanced by the net outward pressure forces.
- If there is a small change what happens? – if a star’s radius decreases a little at constant T, then
(a) the density of gas in each layer of the star would rise as the volume has decreased.
As P = k T/m for each layer of the star, the P will also rise, and
(b) the gravitational force will also rise a small amount if the radius is decreased slightly – but this will
be accompanied by a rise in T due to conversion of the gravitational potential energy into kinetic
energy of the gas particles, and hence the P also rises.
(c) because of these P rises the net outward force due to the variation of P with depth will also rise
and there will no longer be a balance with the inward gravitational force. The star will thus expand
back to its original dimensions.
A star on the MS is very stable against any dimensional changes.
(ii) The Russell-Vogt theorem states
“The equilibrium structure of an ordinary star is
determined uniquely by its mass and chemical composition”.
A certain mass of stellar material of fixed composition can only reach one stable configuration. This
stable configuration corresponds to one point on the H-R diagram.
A star of different mass occupies a different point on the H-R diagram.
We have a MS because the stars on it are stable, with similar chemical composition but with different
masses. ( Stars away from the MS have a different chemical composition.)
11.1.4 Main sequence lifetimes
We have already seen in previous lectures that the more massive the star the shorter its MS lifetime.
Lifetime on MS:
So for a star double the mass of another, its MS lifetime is 1/8 as long.
Note: to calibrate with lifetime scale we could use the Sun, with
 1010 years and M= 1 M

So massive stars, eg M = 15 M are predicted to have relatively short MS lifetimes, even as brief as

10 million years. So, many of the massive upper MS stars currently observed must have formed
fairly recently on the astronomical timescale.
[NB age of Sun 4.5 x 109 years]
11.1.5 range of stellar masses
There are far more low mass stars than high mass stars.
Are we seeing all the low mass stars? The less massive a star is, the less luminous it is, and the
harder it is to observe – the true number of low mass stars is easy to underestimate.
Lower limit of the mass of a star:
The lower the mass of a star, the lower is the core T. At lower T, energy generation from nuclear
processes is reduced and becomes insignificant. Objects, more massive than planets, but < 0.08 M ,

which is not sufficient mass to run nuclear reactions at a rate high enough to match the surface
radiation rate, are called brown dwarfs. These brown dwarfs have a low T and don’t emit much
visible radiation, they radiate in the IR (infrared). Brown dwarfs can be distinguished from gas giant
planets as: they form from the ISM (interstellar medium) and so have a similar starting composition
as other stars; whilst planets form by accretion of dust in nebulae surrounding protostars.
2
Upper limit of the mass of a star:
- limited initially obviously by the mass of the original contracting cloud (not a severe limit as typical
cloud masses are several thousand M s).

- and also, in the most massive stars there is an important process which is not important for lower
mass stars:
radiation pressure, for photons within a black-body source, is given by:
Prad = 1/3 aT4
whereas gas pressure Pgas= n k T
and so
(a=radiation density constant)
(n=number density of particles)
Prad / Pgas = aT3 / 3 nk
Note: For the Sun, radiation pressure is negligible compared to the gas pressure.
But as T increases with higher mass stars, the effect of Prad will increase rapidly compared to the
effect of Pgas .
 The stability of lower/intermediate mass MS stars results from the balance of gravitational force
and force due to gas pressure.
 For most massive stars stability needs a balance between gravitational force and the force due to
radiation P.
 Modelling has shown that radiation P increases so rapidly with T that a very massive star would
easily be blown apart by the radiation P.
 Upper limit to mass of a star is 100 M but its lifetime would of course be very short.

11.2 Life beyond the Main sequence
There are two main principles important for understanding stellar evolution
1. a battle between pressure and gravity
2. fusion of heavier elements takes place at successively higher temperatures
H-R diagram showing the evolutionary track of a moderate/low mass star – eg the Sun
Add labels A – H during lecture
3
Recapping, for the main sequence (MS):
[A] Main sequence
 the distribution of stars on the HR diagram shows that most stars lie on the MS
 stellar models show that the stability of MS stars is the result of a balance between inward force of
gravity and outward force of gas pressure (or radiation P for higher M stars)
 outward pressure is sustained by the energy provided from fusion of H to He
 the rate of energy production and lifetime of a star on the MS is highly dependent on the T and
hence the M of the star.
 as H burning continues, H is converted to He in the star’s core, so its composition is changing a
little. the mean molecular weight, increases and thus central Pc drops. As P drops the core
contracts a little, so central density c increases, and hence Tc increases to restore the P balance,
resulting in a slight increase in L during a star’s MS life. This explains the finite width of the MS on
the H-R diagram, with stars of the same mass but different ages, and hence different compositions,
being in slightly different locations.
 The end of the star’s MS lifetime is marked by exhaustion of hydrogen in the star’s core.
H – burning
core
Structure of a: (a) MS star and
Add in lecture
(b) star leaving the MS because of core H exhaustion.
11.2.1 Low/moderate mass star post main sequence evolution – eg Sun
[B]
 A critical point will come when hydrogen in the star’s core is all gone.
 Nuclear reactions in the core stop and the star leaves the main sequence.
 The core slowly starts to contract as it is no longer releasing energy at a sufficient rate to generate
a pressure gradient sufficient to support the surrounding layers.
 Because of the contraction of the core, gravitational potential energy is converted into thermal
energy and hence the T will begin to rise.
[C]
 Eventually the shell of unprocessed hydrogen surrounding the original core will be heated
sufficiently for hydrogen burning to start in a clearly defined shell – shell hydrogen burning.
 The star is now ascending the red giant branch (RGB) with energy production from
hydrogen shell burning.
 Because of the shell source of H burning thermonuclear reactions (no reactions yet in core of He),
the envelope of the star expands as the core continues to contract, – during this process the L
remains nearly constant.
 L = 4  R2  Teff4 so surface T (Teff) actually drops (as L  constant and R is increasing).
 Because of dropping surface T, the star moves horizontally to the right on the HR diagram.
 Eventually L increases as convection carries energy to the star’s surface and the star continues
ascending the RGB.
4
For a star of 1 M : the core has been compressed to 1/50 th of its original size,

the core T has risen from 15 x 106 K  100 x 106 K,
and surface T has dropped to 3500 K.
radius has increased from 1 R  10 R


The star now shines orange-red: it is a red giant:
- add - Structure of a red giant star
[D]
Once the core T has risen sufficiently, at this point helium burning starts in the star’s core.
3 4He 
12
C
“Triple-alpha thermonuclear reaction”
How does He burning start?
 In the red giant star core contraction had continued, driving the envelope out rapidly.
 In low mass stars electron degeneracy sets in before core He ignition is reached.
 On He ignition, the energy release raises the core T.
 When He ignition does occur, because matter in the core is degenerate, there is no significant
change in P; so the increased core temperature Tc leads to even further energy release (nuclear
fusion at a faster rate), and a thermal runaway develops – known as a helium flash – and
evolution is dynamic.
 The huge energy release rate at the peak of the He flash (~ 1011 L ) lasts very briefly

 and the large energy release rapidly lifts the electron degeneracy and allows the core to expand
(because of the high Tc)
 as this happens the core T drop means that He burning ceases so the envelope contracts slightly.
 Most of the increased luminosity of the core goes into expansions so star’s luminosity decreases.
 Finally the core re-adjusts its structure so that He burning can occur under non degenerate
conditions. The star is now on the horizontal branch of the HR diagram
[E]
The star is now in the region of the HR diagram called the horizontal branch (HB)
 On the HB the star is now in a state of core He burning and shell H burning.
Structure of a horizontal branch giant star.
Enlarged view of the core
-ADD
5
 Eventually the He in the core of the HB star is exhausted (has burnt  C and O).
 As before, the core contracts, and so core P and T increase until He ignites in a shell just outside
the core, with H burning in the next shell outside of that.
 The star is in a double shell burning stage.
 The mass of the now inert core, consisting of C and O, continues to increase as the He shell
produces more C and O,
 and the core continues to contract further raising the core T.
 As the T continues to rise, the energy generation of the two shell sources goes even faster, and
the increasing L distends the outer envelope of the star.
 The star moves up to the asymptotic giant branch.
HR diagram showing evolutionary track of a
solar mass star from MS to AGB.
HR diagram of a globular cluster
showing the:
Main Sequence (MS),
Red giant Branch (RGB),
Horizontal Branch (HB) and
Asymptotic giant branch (AGB)
[F]
On the asymptotic giant branch (AGB) – the asymptotic giant star is in a state of double shell
burning (with an inert core, and H and He burning shells). Add diagram below during lecture:
Structure of an asymptotic giant star.
Enlarged view of the core
6
Eventually:
 more contraction of the core causes free electrons there to become degenerate.
 As the shell sources burning H and He generate higher luminosities the star may become a large
red giant or supergiant, depending on its mass, with high energy expenditure, and so is not able
to live much longer. This is not a stable situation!
 The He-burning shell is thin causing a thermal runaway:
- as He shell burning gets underway, the shell (because it is so thin and insubstantial), cannot lift
the material above it – and so cannot expand. So the temperature rise resulting from He burning is
not moderated. With this temperature increase, the He burning rate increases yet more, and so
increasing the T even further.
- This leads to rapid energy release, a thermal pulse, (which can last a few 100 years), which is
approximately periodic with intervals of 104 – 105 years, and considerable expansion. After the
thermal pulse the T has dropped and so the cycle of thermal runaway then repeats itself.
This thermal runaway, and thermal pulse means that the star is variable and undergoes mass loss
from its outer envelopes.
 The ejected material may form a circumstellar shell. High mass loss is characteristic of the
very late stages of evolution of a giant. These extended gaseous and dust envelopes were
historically called planetary nebulae.
 The central hot star is still contracting, and the envelope continues to expand and is thrown off as
a planetary nebula
[G] what is now left behind after mass loss as a planetary nebula is the naked core of the star – a
white dwarf.
 In a white dwarf degeneracy pressure and gravity are in balance.
[H]
The only source of energy remaining for the white dwarf is thermal energy, and the star
gradually cools getting dimmer and dimmer.
H-R diagram with evolutionary track for a star from red giant, through planetary nebula stage,
and cooling to white dwarf stage (plotted are dots showing central stars of planetary nebulae, and
open circles cooling white dwarfs).
7
11.2.2 high mass star post main sequence evolution (M >10 M )

what we already know:
 Massive stars have a shorter lifetime on the MS than less massive stars
 Due to their higher mass they have higher central T, so H burning reactions proceed faster
 Because of the faster nuclear reactions, energy is released more quickly and so the stars are
brighter – more luminous
 For a short while these stars are hot and bright on the MS – O and B type stars.
Evolution for high mass stars:
[A] hot and bright O and B type stars on the Main sequence.
 H burning through CNO cycle
 Towards the end of the MS life, H becomes depleted in the core – end of MS life
[B]
 H depleted in core – H burning in core stops
 Same process as for lower mass stars now happens: core contracts and so core T rises.
[C] Red giant – H burning shell
 H shell burning stars. The core becomes choked with “ash” = nuclei that are a products of the
nuclear reaction.
 As nuclear reactions diminish (moves to right in HR diagram), there is no longer the pressure
gradient from the escaping radiation to balance the gravitational force, and so the core contracts
under gravity
 Contracting core raises the density and T of the core, so this “ash” (Helium) begins to burn
[D] He burning begins in core.
 The ignition of He begins without a helium flash (as M > 2.25 M )

 With He burning, the T in the core rises, increasing the rate of energy release (nuclear reaction
rate depends on T). (star moves to left in HR diagram)
 If stellar material is fairly opaque, energy finds it hard to escape. So the T rises, and then the gas
expands and cools, the drop in T leading to a drop in the rate of nuclear reaction
 Star is stable – lies on the horizontal branch and is a He burning supergiant
 the colour of the star depends on its surface T – which depends on its mass.
Structure of a He burning supergiant
8





As the He in the core diminishes noticeably, the He burning nuclear reaction wanes.
This means that the P gradient due to escaping energy diminishes,
so the core of the star once again contracts under gravity
and so the core T goes up again.
When the core T reaches 3 x 108 K carbon burning can start in the core.
12
C + 4He  16O + 
 Outside the core (C burning) there are shells burning HeC and HHe.
Schematic of supergiant star structure:
C burning stage, with He shell burning
and H shell burning
Schematic diagram of structure of a
supergiant star, last day!
H-rich outer envelope limits are not shown.
The supergiant star goes through this pattern of steps a number of times as successively heavier
elements become scarce in the core. But each new fuel releases less energy than the previous one
on burning. There is a process of diminishing returns…
 This cycle continues and the star evolves by successively using up one round of fuel in the core,
undergoing core contraction to raise the T sufficiently to ignite the previous ash into a new fuel.
 Finally iron is produced in the core – and the story reaches a final climax
 Iron is the ultimate slag heap of the universe – no further nuclear extraction of energy is possible
by burning of heavier elements at this stage.
 Once the core is iron, the core has no alternative but to contract because of the halt in nuclear
reactions. The core contracts and heats up catastrophically – no new source of energy can be
found to balance the gravitational force. Every channel of heat loss only results in the star
contracting even further…
 Very high temperatures are reached, several billion K. Intense temperatures lead to iron photodisintegration, taking the core’s heat energy to overcome the particles’ binding energy.
 Now there is no balance for the gravitational force – catastrophic core collapse results, and it
falls inwards freely under its own self gravity.
 The rapidly rising density squeezes the free electrons into the protons to form a huge hot mass of
neutrons.
 A supernova explosion results leaving either a neutron star or a black hole
11.2.3 Diminishing returns – nuclear reactions in core
 Each new thermonuclear reaction is less efficient than previous one at releasing energy
 So the reactions need to proceed at a higher rate to yield radiation to balance gravity
 Faster reaction rates mean more neutrinos are produced – these carry away a growing proportion
of the energy generated
9
 The star goes through its life cycle at an ever increasing pace – squandering its reserves at a
faster and faster rate:
Duration:
Burning reaction stage:
H
7 x 106 years
He
5 x 105
(~10% of MS lifetime)
C
600 years
Ne
1 year
O
6 months
Si
1 day !
HR diagram showing evolutionary tracks for low mass (1 M ) and high mass (5M ) stars.


All extra diagrams
in this handout are
taken from “Sun and
Stars”, Green and
Jones.
Predicted paths of stars on the H-R
diagram as they evolve off the MS to the
red (or supergiant phase): Note- detail for
1 and 2 M tracks is not included (see

earlier diagram).
A onset of hydrogen core fusion – start
of MS life
B dashed line B marks end of H core
fusion – end of MS life – and start of
H shell fusion
C hydrogen shell fusion continues
D helium core fusion starts
E helium core fusion continues
F helium shell fusion starts
LECTURE 11: SUMMARY
•Hertzsprung-Russell diagram: evolutionary tracks of low mass and high mass stars: understand,
and be able to sketch, explain, and compare and contrast
• Post Main sequence evolution: know different stages, be able to describe and explain what
happening in star in terms of nuclear reactions, physical changes and processes – and position at
each stage of evolution on H-R diagram. Be able to sketch star’s structure at each evolutionary
stage.
10
Sun Stars Planets
12.
J C Pickering 2014
Lecture 12: The Life of a star, part 2: STAR FORMATION
12.1 Where do stars form?
12.2 Formation of a protostar.
12.3 H-R diagram, Hayashi tracks, evolution onto the main sequence (MS)
12.4 Observations of protostars, bipolar outflows, and protoplanetary disks
Consider the three phases of stellar evolution:
• The first phase follows the formation of a star from a cloud of interstellar material through its
gravitational contraction and subsequent heating to the main sequence (MS), defined as the
point at which hydrogen burning begins.
• The mid phase follows the development of a star on through the main sequence: a star spends
most of its lifetime on the MS so this is an important, though least spectacular place
• The final phase of stellar evolution, has more rapid evolution that begins when H burning has been
exhausted in the stellar core, through to the star’s death.
This lecture covers star birth. (Lecture 11 covered main sequence and post MS phases.)
Stars are born from more widely dispersed gas which condenses because of the gravitational
attraction of the gas on itself.
12.1 Where would we expect to find star birth?
Interstellar space is not empty – it contains dust and gas; the Interstellar Medium (ISM).
Observational evidence seems to point to the dense clouds as being interstellar nurseries:
 we observe young star clusters which seem to be surrounded by the remnants of the original
cloud from which they are formed (dust and gas – eg Orion nebula)
 some dense clouds have a large number of compact infra red (IR) light sources – this is due to
either light being absorbed by dust and re-emitted at the temperature of the dust (in the IR:
from Wien’s law), or the source is cool and emits only in the IR.
The IR radiation comes not from the star itself but from the cocoon of dust still surrounding it.
12.2. Formation of a protostar
12.2.1 Contraction of a dense cloud: (i) spontaneous and (ii) triggered cloud collapse
(i) spontaneous collapse – Jean’s instability
 All atoms, molecules and particles in a cloud are attracted to each other by gravitational forces.
Each particle is affected by the gravitational attraction of the combined mass of all the others.
 However observations show many clouds appear to be in a state of equilibrium – ie not
contracting. So each gas particle is in continuous motion (average translational KE = 3/2 kT).
This motion produces a gas pressure that provides an outward force to counteract the
tendency of the gas to contract. So there is a balance of gas pressure against gravitational
force: if the force due to gravity is greater, then gravitational contraction occurs.
 For a uniform spherical cloud “Jean’s mass”, MJ , is the mass above which the gravitational
force will overcome the opposing pressure due to particle motion:
(
)
(
)
Where n = particle number density, m=mass of “average” gas particle, T=gas temperature
So at lower T, lower mass clouds will contract more easily than at higher T.
1
(ii) collapse triggered externally
Trigger mechanisms are probably needed to cause a cloud to change from an equilibrium state to
one in which contraction has been initiated. A slight increase in density is required to trigger Jean’s
collapse. Possible triggers include:
 a shock front: from a supernova has a compressed gas region just inside the expanding shell.
 regions where several O and B stars form (OB associations). These, forming close together,
produce large amounts of visible and UV light causing a shock in the material in the surrounding
ISM (due to force of radiation pressure).
 Spiral density waves in a galaxy maintain a galaxy’s spiral structure and compress all material
they pass. Star formation regions are more concentrated in spiral arms.
 Close approach or collision of another cloud or star: produces a local gravitational disturbance
which could trigger gravitational contraction.
 Star formation can be triggered throughout an entire galaxy (called a starburst galaxy) by the
interaction with another nearby galaxy.
12.2.2 Fragmentation
Stars are often found in clusters – they appear to form from fragments of a cloud, each collapsing on
its own. They form open clusters of typically a few hundred stars.
12.2.3 From fragment to protostar
A star is formed in a region of large radius and low temperature in the HR diagram. If the collapse is
quasi-static according to the virial theorem (see earlier lecture), half the energy of collapse goes to
heating the material, the other half is radiated away. If the collapse is not quasi-static, some of the
energy of the collapse goes into rotational energy of the whole star (this is what happens in practice).
Initially - the gas is transparent.
- Collapse occurs at nearly uniform T – most of change in gravitational PE goes into the
moment of inertia of the star – collapse is rapid.
Then - as contraction occurs, R (fragment radius) decreases so gravitational PE decreases. Energy
is conserved, so gravitational PE converts to KE of the molecules - the KE increases, and is
converted by mutual collisions of molecules into thermal energy of the gas, and T goes up. The
excited molecules emit radio waves, microwaves and IR radiation, which escape the cloud as it is
initially transparent, so the T rise remains minimal (only 10-20 K).
As contraction progresses further:  the number density of molecules increases, so it is harder for the radiation to escape – it becomes
trapped by the surrounding layers.
 The gas becomes opaque to radiation; so now some of the gravitational PE goes into heating, and
the fragment’s internal T rises more rapidly.
 This heating reduces the rate of collapse- a state of quasi-static equilibrium is approached.
 The interior of the star may be hot enough for H burning to begin, at which point the collapse is
halted and the star adjusts itself to equilibrium on the zero age main sequence.
Note: for this H burning to begin there must be enough gravitational PE available to raise the T to the
hydrogen burning ignition point. For a gas cloud of very low mass, this will not happen as the core T
will be insufficient for H burning, and in this case the collapse is halted by electron degeneracy
pressure. The lowest mass star than can burn H is  0.08 M . Objects of lower mass than this

(13-80 MJ, Jupiter masses), substellar objects, brown dwarfs, have core deuterium burning if they are
sufficiently massive for core T to exceed 106 K. Brown dwarfs may shine for a hundred million years
at most before their deuterium supply is burned out.
2
12.3 HR diagram and evolution onto the Main sequence
On an H-R diagram a fragment of dense ISM cloud falls in the low T region. The track of the
contracting cloud fragment across the H-R diagram is far from certain, because the process takes
place behind gas and dust, screening the fragment from view and takes place over a short timescale
- so we are less likely to observe stars in this phase. We therefore rely on models, that predict:
 after a few thousand years of gravitational contraction, surface T rises to 2000 – 3000 K.
 the contracting fragment is still quite large and so L is high (L = 4  R2  T4eff), approximately 10100 times its eventual L as a star on the MS.
 the fragment eventually becomes a protostar.
The H-R diagram below shows the predicted tracks for protostars of various masses as they evolve
towards the MS region – called Hayashi tracks.
Diagram: theoretical tracks
(Hayashi tracks) of
protostars of various masses
across the H-R diagram as
they evolve towards the MS.
(Times for the protostars to
reach different stages of their
evolution are shown.).
[Diagram from Sun and Stars, Green
and Jones]
Notice: the timescale for this early stage of evolution – the more massive the protostar, the quicker it
reaches the MS. A 15 M protostar takes 105 years to reach the MS, this is less than 1% of the time

for a 1 M protostar to reach the MS. Virtually all protostars that then become MS stars take less

than 108 years to pass through the protostar phase, this is very quick compared to other phases of
their life.
The HR diagram track shapes depend on changes in internal structure and the way energy is
transported through protostars as they collapse. Recall L = 4  R2  T4eff , and notice:
 intermediate/low mass protostars – an early drop in L is due to the effect of the increase in surface
T being more than off-set by the effect of the decrease in R
 > 2 M higher mass prototstars – effects of increasing surface T and decreasing surface area just

about balance, so L changes little as T increases (tracks are ≈ horizontal on H-R diagram).
 shortly before joining the MS, tracks generally show a drop in L as the effect of the contraction of
the protostar tends to dominate over T effects.
3
12.4. Observations of protostars:
Observation is difficult: the number of stars observed in a given phase of evolution should be roughly
proportion to the time an individual star spends in that phase. When evolution is fast, as in this phase
of evolution, then the statistics are against our finding any stars in that place. However, with radio
and IR telescopes we are learning much more about star formation: for example, many protostars
show a phenomenon called bipolar outflow.
Diagram: Gas flowing away from protostar at high speed
(50 km s-1 ) in opposite directions. Phase lasts ≈ 104 years.
As the cloud fragment contracts it spins faster and
flattens into a circumstellar disc, or torus, with the
protostar at the centre. Material in a disk rotating
around the star moves radially and is gradually
accreted. At the same time mass loss from the star
is probably channelled in a bipolar outflow by the
star’s magnetic field and the disk. Mass loss is a
strong stellar wind - radiation pressure contributes to
propelling the stellar wind (pressure exerted by
photons on any object that absorbs or reflects them).
As the surrounding envelope of dust around the protostar disperses, the accretion process stops, the
central globe of gas is no longer a protostar – it becomes a pre-main sequence star. Eventually
when the core has contracted further, the core T will be sufficient for H burning and it become a main
sequence star.
12.5 Why do the planets orbit the Sun close to the ecliptic plane? Why are there bi-polar
mass outflows observed in protostars ?
12.5.1 conservation of angular momentum, accretion disk formation
Typically a real cloud from which a star would form would have a small amount of net angular
momentum (for example due to galactic rotational shear, or other bulk motions or interactions). This
angular momentum is conserved as the cloud collapses.
Consider angular momentum L, particle of mass m, at distance from cloud centre r, with angular
velocity , with centripetal acceleration ac and gravitational acceleration ag.
Since angular momentum l = m r2 ω, and l and m are constant:
But as r decreases during cloud collapse, ω must increase. What does this imply for a collapsing
cloud?
The radial acceleration of a particle of mass m at radius r has 2 components:
(1) Gravity directed towards the collapsing cloud centre:
where M(r)is the mass contained within sphere of radius r.
4
(2) Centripetal term directed away from the axis of rotation
With the radial component thus being
← Annotate diagram during lecture
Net radial acceleration
Note: a(r) for a rotating cloud is less than for a non-rotating cloud.
At the pole
but at the equator
In the equatorial plane, accelerations are both radial (and assume the mass inside the contracting
mass point is constant):
We have
and
,
, but from earlier we know,
so
As the contraction proceeds, and r decreases:
Initially r large, ω small:
increases faster than
r decreases further, and eventually
=
to the centre. (At points with even lower r,
, and at this point the gas doesn’t contract any closer
>
).
So contraction continues only in regions outside of the equatorial disk, but not in the plane of the
disk. It is the balance between the centripetal acceleration and gravitational acceleration that
causes a collapsing cloud core to form an accretion disk: ac is acting perpendicular to the axis of
rotation of the gas cloud, and ag is acting radially. Flow of mass parallel to the axis of rotation is
unimpeded by rotation, whilst flow towards the centre near the equatorial plane is prevented by the
rotation.
5
12.5.2 Why are forming stars seen to have bipolar jets and outflows?
Whilst the physics of precisely what is going on is still not clear, these outflows can in part be
understood by considering the conservation of angular momentum. We have just seen that because
of conservation of angular momentum:
r 2  1/, so for a collapsing cloud, with initial radius ri
and angular velocity and i, and final stellar radius rf and angular velocity f
if
ri  1015 m
say,
then
 
( )


The resulting star might be expected to have a rotation period of under a second if angular
momentum were strictly conserved as the cloud contracts, and an equatorial rotation velocity greater
than the speed of light… The Sun however has a rotation period of about a month… So a protostar
must be losing angular momentum somehow. It does this by throwing matter outward – we believe
this because of observations of jets of particles outward along a forming star’s axis of rotation.
So the star is formed by accretion of matter from an accretion disk, but it is simultaneously throwing
matter outwards at high speed along its axis of rotation. As this mass is thrown out, the rotation
decreases, and then mass can once more flow inwards in the accretion disk to the forming star. The
flow outwards at the star’s poles prevents accretion from the pole direction, so accretion is primarily
from matter in the accretion disk. This accretion disk, and the matter remaining after the star has
formed, is involved in planet formation, and hence accounts for the plane of the orbits of the planets,
the prograde rotation and revolution of planets.
LECTURE 12: SUMMARY of stellar formation

Protostars and evolution onto the Main sequence: be able to describe the different stages of
evolution, physical processes, changes. Be able to sketch diagram of protostar with bipolar outflow.

ISM – be able to comment on the ISM briefly with reference to star formation.

Description of cloud collapse, and triggers: be able to list and comment on triggers, and
describe processes in cloud collapse.

HR diagram, Hayashi tracks for protostars: be able to sketch and comment on for 1 M star.


Protostars and bipolar outflow: be able to sketch and describe
6
Sun, Stars, Planets
13.
J C Pickering
2014
Lecture 13: Part I: Binary stars: measuring stellar masses
Part II : The Sun in the Galaxy
13.1 Binary stars
13.2 Binary star orbits
13.3 Measuring stellar masses using binary star observations
13.4 The Sun in the Galaxy
13.1 Binary stars and measuring stellar masses
Approximately 2/3 of stars are in binary systems. (Also even triple systems.) This is however
not very surprising given that young stars are found in clusters. A “binary star” is in fact
actually two stars – but they are so far away that they appear as one star in the sky – their
angular separations are so small that the binary star usually cannot be resolved.
Binary star systems provide one of the few ways of measuring stellar masses.
The basis of the method of measuring the mass of a star is to observe how it moves when a
force is applied to it: Newton’s 2nd law of motion F=ma. We must observe the star being
accelerated. Any star is accelerated by any other mass in the Universe through gravitational
attraction, but the effect is only large enough to measure if the other mass is large and
relatively close to the star – a binary system.
13.2 Binary orbits
The stars of a binary orbit a common centre of mass, COM, on elliptical orbits.
Annotate during lecture:
Define the angle i to be the angle between the line of sight to the binary system and the
normal of the orbital plane of the binary system:
Annotate during lecture:
1
We will assume that the stars have circular orbits about the centre of mass of the binary
system
(Annotate the diagram below during lecture:)
Balancing the gravitational and centrifugal
forces:
For Star 1:
[13.1]
For Star 2:
[13.2]
[13.3]
(
)
(
)
[13.4]
These results are also true for elliptical orbits (radius of orbit r can be replaced by semi-major axis a).
Notice: as M1 or M2 increase, P decreases, and as r (or a) increases, P increases.
13.3 Measuring the stellar masses
Stellar masses for the two components of a binary system can be measured in three cases:
(i) visual and astrometric binaries
(ii) spectroscopic binaries
(iii) eclipsing binaries
(i) Visual binaries: here we can see both stars separately as 2 distinct points of light.
We can measure each star’s orbit on the sky.
We can measure the ratio of the orbit sizes and use [13.3] 
The angular separation of the stars, α, is measured.
Distance, d, from the Earth to the binary star system is found from the measured parallax.
Then: a = sinα /d
α
a
d
and knowing the stars’ separation, a , and measuring
Using [13.4]
the period P, gives M (=M1 + M2 ).
...and thus, having the ratio of the stellar masses and
their sum we can find M1 and M2 individually.
2
(ii) Spectroscopic binaries: these are known only from the periodic variation of the Doppler shift of
the spectral lines observed.It is not possible to resolve these as two objects when seen visually alone.
We know the velocity components along the line of sight (measured from the stellar spectra)
but we do not know r.
Period P is measured from the
spectroscopic observations.
Annotate diagram during lecture
Projected velocities v1 and v2 are found,
where:
[13.5]
Therefore [13.3] can be written:
And so we can also write:
[13.6]
)
(
We want to use [13.4] and express mass in terms of what can be measured, v1 or v2 . To do
this we need to replace r with v. Start by multiplying [13.4] by (sin i)3 :
(
)
(
)
(
(
(
)
(
)
)
)
[13.7]
“mass function”
And we can also get a similar expression to [13.7] containing v2 and M1 .
If one of the two stars is very faint (a “single-lined binary”), we can measure P and v1 , and derive only
the mass function from these. For a “double-lined binary”, we can measure P, v1 and v2 , allowing us
to get the stellar mass provided that we know inclination i. For an unknown inclination we therefore
find a minimum mass by setting sin i =1.
3
If the orbit of the stars is seen nearly edge-on, we have …
(iii) Eclipsing binaries: a bound pair of stars is deduced from periodic changes of the total
light from the system arising from eclipses of one star by the other - variations in light curves
due to eclipses
 i  90o
Get an accurate measure of m1 and m2
(Also the lengths of the eclipses can be used to estimate the radii of the stars,)
Example: Light curve of an eclipsing binary.
Brightness
Time
[An analysis of the light curve can often yield an estimate of the inclination of the orbit of the
system relative to the line of sight.]
Further comments about binary systems
Stars’ radii tend to increase with age (e.g. red giants!)
If one star of binary system expands, it may reach stage where its outermost material is more
attracted to the companion star. Mass transfer from one star to the other follows. Also can
transfer angular momentum. These are called interacting binaries.
This interaction may affect the evolution of both stars.
Examples include:
e.g. Algol (β Per) – first eclipsing binary discovered (P=2.87 days)
high-mass MS star and a low-mass red giant
e.g. dwarf novae
brighten by 2-5 mag for a few days each month (probably an
interacting binary with WD accreting matter onto a disk)
e.g. binary pulsar
neutron star rotating hundreds of times a second, spun up by mass
4
13.4 The Sun in the Galaxy
Galaxy classifcation
The best known and most often used general scheme, in which galaxies are grouped
according to their appearance was devised by Edwin Hubble. It splits galaxies into ellipticals,
spirals (normal and barred), and irregulars, and is represented by the “tuning-fork” diagram
below.
Elliptical galaxies are graded from E0 (spherical) to E7 (very elongated) in terms of
increasing eccentricity).
Spirals have a planar disk – and are divided into those without (S) and with (SB) central bar.
Normal spirals range from Sa (arms tightly wound around the nucleus) to Sc (arms widely
spread from the nucleus), and,
similarly, barred spirals from SBa (arms tightly wound) to SBc (arms widely spaced).
Irregulars are designated Ir.
To this original scheme, another category, S0, was added to describe lenticular systems with
a nucleus surrounded by a disk-like structure that lacks spiral arms.
Hubble’s tuning-fork diagram of galaxy morphologies
Comments on galaxy types
Spirals have obvious rotation – and gas and young stars can be seen
In ellipticals – stellar motions appear to be mainly random – so no rotation. There is little gas
or dust, and little current star formation
Origin of different galaxy types – may depend on initial conditions, whether there was
significant rotation or not. The type will also depend on the history of the galaxy – it may have
experienced collisions and mergers.
5
Components of our Galaxy
Schematic diagram of our Galaxy – ADD DURING LECTURE
(i)
(ii)
(iii)
thin disk of stars, gas and dust
central bulge
halo of old stars. Globular clusters are found in a roughly spherical halo, distances
to these can be measured using variable stars.
The Sun is 8.5 kpc from the Galactic centre in a spiral arm.
Our Galaxy rotates with a period of 2 x 108 years – so the Sun’s velocity around the Galactic
centre is 220 km s-1 .
Of course the stars also have random motions (for example due to gravitational interactions.)
Our neighbours…and beyond
Our Galaxy has satellite galaxies: well known are the Large Magellanic Cloud (LMC) and
Small Magellanic Cloud (SMC).
The Galaxy is part of the Local Group of galaxies
(includes M31).
Galaxies tend to be grouped into clusters and superclusters.
Galaxies are found on large scale to be moving away from one another:
Hubble expansion of the universe.
The recession velocity:
v=Hd
Where H ~ 70 km s-1 Mpc-1
On very large scales we find galaxies tend to be confined to 2-D structures in space (a bit like
walls of a bubble-foam.)
LECTURE 13: SUMMARY
 Derivation of relations between binary masses, separations and orbital period – be able to
do this from basics
 Using binaries to estimate stellar masses – be able to calculate
 Visual binaries – get masses if distance known – be able to calculate
 Spectroscopic binaries – be able to calculate msin3i (inclination i)
 Eclipsing binaries: most useful – understand and be able to use light curve information
 Mass transfer between binary companions leads to interesting evolution –be aware of this
 Galaxy types: elliptical, spiral, irregular – be aware of the different types, especially with
respect of star formation
 Main components of our spiral Galaxy: bulge, disk - be able to sketch the components of our
Galaxy, and comment on
 Sun’s place in Galaxy; Galactic rotation - be able to comment on this
TO DO: questions 4 and 5, problem sheet 3.
6
Imperial College London
Physics Department
Sun, Stars and Planets
An introductory course in astrophysics
Dr Juliet Pickering, Dr David Clements
2013-14
Comments and corrections to [email protected]
Dave Clements Office 1011 Blackett
Lecture notes may be found on Blackboard (http://blackboard.ic.ac.uk)
ii
Contents
1 The Planets Half of the Course
1
2 Planets: Textbooks
3
3 Outline Syllabus
3.0.1 The Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
4 An
4.1
4.2
4.3
4.4
4.5
4.6
Overview of the Solar System
Aims of this Lecture . . . . . . . . . . . . . . . . . . . .
Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overall Inventory of the Solar System . . . . . . . . . .
Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . .
Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 The Moon . . . . . . . . . . . . . . . . . . . . . .
4.7 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Phobos and Deimos: The Moons of Mars . . . .
4.8 The Asteroid Belt . . . . . . . . . . . . . . . . . . . . .
4.9 Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9.1 The Moons of Jupiter . . . . . . . . . . . . . . .
4.10 Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.1 The Rings . . . . . . . . . . . . . . . . . . . . . .
4.10.2 The Moons of Saturn . . . . . . . . . . . . . . .
4.11 Uranus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Neptune . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 Pluto, Trans-Neptunian Objects (TNOs) and the Kuiper
4.14 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 The Oort Cloud . . . . . . . . . . . . . . . . . . . . . . .
4.16 Formation of the Solar System . . . . . . . . . . . . . .
4.17 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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15
5 Planetary Orbits: Kepler’s Laws
5.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Kepler’s Three Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . .
5.3 Derivation of Kepler’s Three Laws . . . . . . . . . . . . . . . . . . . . . . . .
17
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Contents
6 Terrestrial Planets: Heating, Cooling Processes and Interiors
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The Unquiet Earth . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Primordial Heating . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 The Structure of the Earth . . . . . . . . . . . . . . . . . . . . .
6.5 Long Duration Heat Sources . . . . . . . . . . . . . . . . . . . . .
6.6 The decay of long term heating sources . . . . . . . . . . . . . .
6.7 Heat Loss from Planets . . . . . . . . . . . . . . . . . . . . . . .
6.8 Cooling Processes . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Volcanism and Tectonics on Other Terrestrial Planets . . . . . .
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7 Terrestrial Planet Surfaces and Temperatures
7.1 Introduction . . . . . . . . . . . . . . . . . . . .
7.2 Major Factors in Shaping Planetary Surfaces .
7.3 Impact Cratering . . . . . . . . . . . . . . . . .
7.4 Volcanism and Tectonics . . . . . . . . . . . . .
7.5 Erosion . . . . . . . . . . . . . . . . . . . . . .
7.6 The Surface Temperatures of Planets . . . . . .
7.7 The Greenhouse Effect . . . . . . . . . . . . . .
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8 Terrestrial Planet Atmospheres
8.1 Introduction . . . . . . . . . . . . . . .
8.2 Why do we have an atmosphere at all?
8.3 Atmospheric Density and Pressure . .
8.4 Temperature Variations with Height .
8.5 Thermal Escape . . . . . . . . . . . .
8.6 Current Atmospheric Composition . .
8.7 Origin of Atmospheres . . . . . . . . .
9 Gas
9.1
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9.6
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39
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40
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43
44
Giants: Structure and Atmospheres
Introduction . . . . . . . . . . . . . . . . . . . .
Basic Properties of Gas Giants . . . . . . . . .
The Internal Structure of Jupiter and Saturn .
Excess Heat in Jupiter and Saturn . . . . . . .
The Internal Structure of Uranus and Neptune
Gas Giant Atmospheres . . . . . . . . . . . . .
Ring Systems . . . . . . . . . . . . . . . . . . .
9.7.1 Derivation of the Roche Limit . . . . . .
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10 Moons: Formation and Properties
10.1 Introduction . . . . . . . . . . . . . . . . . . .
10.2 Orbits and Masses . . . . . . . . . . . . . . .
10.3 Formation . . . . . . . . . . . . . . . . . . . .
10.3.1 The Moon . . . . . . . . . . . . . . . .
10.4 Tidal Forces and Tidal Heating . . . . . . . .
10.5 Tidal Locking, Libration and Circularisation .
10.6 Orbital Resonances . . . . . . . . . . . . . . .
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Contents
11 Small Bodies: Comets, Asteroids and the
11.1 Introduction . . . . . . . . . . . . . . . . .
11.2 Asteroids . . . . . . . . . . . . . . . . . .
11.3 Kuiper Belt and Trans-Neptunian Objects
11.4 Comets . . . . . . . . . . . . . . . . . . .
12 Detecting Exoplanets
12.1 Introduction . . . . . . . . . . . . . . . . .
12.2 Units . . . . . . . . . . . . . . . . . . . . .
12.3 What is a Planet Anyway? . . . . . . . .
12.4 Direct Detection: How Hard Can it Be? .
12.5 Reflex Motion and Doppler Measurements
12.6 Planetary Transit Searches . . . . . . . .
12.7 Other ways to detect planets . . . . . . .
12.7.1 Pulsar Planets . . . . . . . . . . .
12.7.2 Gravitational Lensing . . . . . . .
13 The
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
System
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69
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77
Exoplanet Population
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
The Current State of Planet Searches . . . . . . . . . . .
Selection Effects . . . . . . . . . . . . . . . . . . . . . . .
Exoplanet Masses . . . . . . . . . . . . . . . . . . . . . . .
Exoplanet Composition . . . . . . . . . . . . . . . . . . .
Exoplanet Orbits: Hot Jupiters and Planetary Migration .
Host Star Metalicity . . . . . . . . . . . . . . . . . . . . .
Exoplanets: A young Science . . . . . . . . . . . . . . . .
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14 Astrobiology: Life on Other Planets
14.1 Introduction . . . . . . . . . . . . . . . . .
14.2 Life on Earth: History . . . . . . . . . . .
14.3 Lessons from the History of Life on Earth
14.4 Life Elsewhere in the Solar System . . . .
14.4.1 Mars . . . . . . . . . . . . . . . . .
14.4.2 Europa . . . . . . . . . . . . . . .
14.4.3 Enceladus . . . . . . . . . . . . . .
14.5 Life Outside the Solar System . . . . . . .
14.5.1 Host Star . . . . . . . . . . . . . .
14.5.2 Gas Giant Moons . . . . . . . . . .
14.6 The Galactic Habitable Zone . . . . . . .
14.7 How to Find Life on Other Planets . . . .
15 The
15.1
15.2
15.3
15.4
15.5
Outer Solar
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87
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95
Search for Extraterrestrial Intelligence
Introduction . . . . . . . . . . . . . . . . . . .
The Drake Equation . . . . . . . . . . . . . .
The Fermi Paradox . . . . . . . . . . . . . . .
SETI and CETI . . . . . . . . . . . . . . . . .
The Future . . . . . . . . . . . . . . . . . . .
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Contents
vi
Chapter 1
The Planets Half of the Course
The second half of the Suns Stars and Planets lecture course concerns Planets, both within
our solar system and those outside it that are now being discovered by both ground and space
based satellites. This section of the course will comprise 12 lectures, one guest lecture (TBD)
and three problems sheets. As with the first part of the course, your tutors will not cover
the course, so you will need to find solutions for the problem sheets on Blackboard. Lecture
notes, problem sheets, as well as their solutions, and other materials as necessary, will also
be available on Blackboard.
Dr Clements’ office is 1011 Blackett, and his office hours for this course are 12pm-1pm on
Tuesdays.
If necessary you can also contact him via email at: [email protected]
He can be found online on Twitter as @davecl42 and on Wordpress blogs as davecl.wordpress.com.
Some of the items on these sites may occasionally be of interest to those doing this course.
1
2
Sun Stars and Planets 2012-13
Chapter 2
Planets: Textbooks
There is no single textbook for this part of the course, but much useful material can be found
in the following books:
• An Introduction to the Solar System, edited by David A. Rothery, Neil McBride and
Iain Gilmour, published by Cambridge University Press
ISBN: 978 1 107 60092 8
As with Introduction to the Sun and Stars, this is a large, colourful and very well
illustrated introductory text to the Solar System side of this course. It has lots of facts,
descriptions, diagrams and figures, but is rather light on mathematics. It covers basic
ideas well, but without the rigour that is usual for any Imperial College course.
• Exploring the Solar System, by Peter Bond, published by Wiley-Blackwell
ISBN: 978 1 4051 3499 6
This is another well presented and illustrated introductory text much like the Rothery
book above. It covers a number of topics somewhat more deeply, and in addition includes
chapters on the Sun and on Explanets. It is also rather lightweight on mathematics.
• Planets & Planetary Systems, by Stephen Eales, published by Wiley-Blackwell
ISBN: 978 0 470 01693 0
This is a shorter text book than many of the others listed, and lacks many of the colourful
illustrations. However, it makes up for this in taking a much more mathematical point
of view of the subject material. It also includes sections on exoplanets and on life in
the universe. It is thus a very useful textbook for this part of the course. If you only
get one textbook, this should probably be it.
• An Introduction to Astrobiology, edited by David A. Rothery, Iain Giilmour and Mark
Sephton, published by Cambridge University Press
ISBN: 978 1 107 60093 5
3
4
Sun Stars and Planets 2012-13
Astrobiology is a relatively young subject that brings together astronomy, physics, biology and much else besides. This book provides an excellent introduction to these
disparate fields and their contributions to astrobiology. As such, it goes rather further
than this course will in many areas, but will provide a lot of extra material for those
who are interested in this new and growing field.
• Transiting Exoplanets, by Carole Haswell, published by Cambridge University Press
ISBN: 978 0 521 13938 0
This book provides in depth coverage of the increasingly important study of transiting
exoplanets, as well as good coverage of the overall field of exoplanet searches. The
coverage includes mathematical discussion of a wide range of topics concerned with
transiting exoplanets, including their atmospheres, the structure of giant planets and the
overall exoplanet population. This textbook goes well beyond the material this course
will cover on exoplanets, but it provides a lot of interesting and rigorous material on
exoplanets in general, and the application of transit methods to the search for exoplanets
and exo-planetary systems.
There are, of course, many other textbooks, popular books and well illustrated coffee table
books available that cover these areas as well, and the library is well stocked with such texts.
In addition to academic texts there are a number of popular non-fiction and fiction books
which deal with the planets, moons and other objects found in the solar system. These can
provide a more intuitive feel for what the landscapes of the solar system might be like, whether
dealing with the giant volcanic landscape of Olympus Mons on Mars, or the icy surfaces of
the moons of Saturn. Of the many such books available I would recommend Kim Stanley
Robinson’s Mars series (Red Mars, Green Mars and Blue Mars) for their coverage of the
landscapes, geography and geology of Mars, Paul McAuley’s The Quiet War and Gardens
of the Sun for their coverage of the moons of the outer planets, and Alastair Reynold’s Blue
Remembered Earth for a tour of the solar system. Kim Stanley Robinson’s recent 2312 also
adds some interesting coverage of Mercury and Venus.
Chapter 3
Outline Syllabus
The lectures dealing with the planets in our own and other solar systems will be divided up
as follows:
1. An Overview of the Solar System and its formation
2. The orbits of planets and Kepler’s Laws
3. Terrestrial Planets: Heating, Cooling and Interiors
4. Terrestrial Planets: Surfaces and Surface Temperatures
5. Terrestrial Planet Atmospheres
6. Gas Giants: Structure and Atmospheres
7. Moons: Formation and Properties
8. Small Bodies: Comets, asteroids and the Outer Solar System
9. Exoplanets: Detection
10. Exoplanets: Properties and Characterisation
11. Astrobiology: Life on Other Planets
12. The Search for Extraterrestrial Intelligence
3.0.1
The Examination
In principle, all the material in these lecture notes, and the lectures, is examinable. In
practice, less than that is easily examined. At the end of each chapter I will highlight things
to remember, which are the central issues in each chapter that may appear in the examination.
5
6
Sun Stars and Planets 2012-13
This course contains a lot of information. You will need not only to recall that for the
examination, but also be able to use it. An exam question might ask you to compare and
contrast the properties of two different types of planetary body covered in separate parts of
the course. To achieve a high mark you will need not only to recall the basic facts but use
them to draw such contrasts and make conclusions.
Chapter 4
An Overview of the Solar System
4.1
Aims of this Lecture
Much of this half of the course will deal with the objects in our own Solar System. This first
section is meant to provide an overview of those objects, planets and otherwise, where they
are, what their key properties are and how they differ, and what the key physical drivers were
behind their formation. There will also be some historical discussion about how we arrived
at our present knowledge of the Solar System, both theoretical and observational.
When looking at the properties of extrasolar planets and planetary systems it is also useful
to see how these compare and contrast with the local example of the Solar System. A broad
idea of what our solar system contains is thus a necessary first step in our study of planets.
4.2
Units
In many circumstances, astronomers do not use standard SI units since the numbers involved
are, literally, astronomically large. We thus use, in addition to SI, units that might be called
‘astronomer’s units’ which are based on scalings from known objects or places. We may thus
talk about numbers of solar masses or solar luminosities, scaling to the mass and luminosity
of the Sun. Such quantities are denoted with or Sun eg. M or LSun . Similarly we also
scale to the Earth using ⊕ or E and Jupiter using Jup .
We also have the special unit of distance called the Astronomical Unit, or AU. This is the
distance between the Earth and the Sun, which is 149.6×106 km.
4.3
Overall Inventory of the Solar System
The Solar System consists of the following ingredients:
7
8
Sun Stars and Planets 2012-13
• The Sun - a star with a surface temperature of ∼5780K, mass of 2 × 1030 kg., and radius
7 × 108 km, with a rotational period of ∼ 27 days and magnetic activity. The first part
of this course will have told you much more about the Sun.
• 8 planets. Four of these are terrestrial planets, like the Earth, four are gas giants like
Jupiter. Many of these planets have moons.
• Asteroids
• Kuiper-Belt objects and Trans-Neptunian objects
• All of the above are in the same orbital plane, known as the ecliptic, and are on mostly
circular, prograde (ie. in the same direction as the Sun) orbits. A small number of
KBOs and comets are exceptions to this.
• The Oort cloud - which is roughly spherical, contains about 1011 nascent comets, and
possibly extends out to 10000 AU.
The age of the Solar System is roughly equal to the age of the Sun and the age of the Earth,
which are found to be ∼ 4.6 × 109 years.
We shall now look at each of these objects in turn.
4.4
Mercury
Mercury is the closest planet to the Sun. It has no atmosphere and images reveal that its
surface is heavily cratered. Surface temperatures range from 740K in the powerful glare of
the Sun, to 80K on the far side of Mercury from the Sun. It is clearly a very harsh place.
The NASA satellite Messenger is currently in orbit around Mercury, and the ESA mission
BepiColombo will be launched in 2015, due to arrive in 2022.
Mercury has a weak magnetic field about 1.1% as strong as Earth’s. This has implications,
as we shall see later, about the internal structure of the planet.
The heavily cratered surface implies that the surface is very old. Some regions are less
cratered, suggesting that they have been resurfaced at some point in the distant past by
geological activity. Other surface features suggestive of tectonic activity exist (eg. faults),
but there are no indications of recent geological activity.
4.5
Venus
Venus is the next planet as we travel outwards from the Sun. Unlike Mercury, its surface
features cannot easily be studied since it has a thick, opaque atmosphere, mostly made up
of carbon dioxide. Clouds can be seen in the atmosphere - they are made from tiny droplets
9
of sulphuric acid. Venus is actually a more hostile environment than Mercury. Apart from
the sulphuric acid rain. the atmospheric pressure is 100 times that of Earth, and the surface
temperature is 670K, hot enough to melt lead. Carl Sagan frequently described Venus as
Hell.
Despite the challenges of these conditions, several Russian probes have managed to land on
the surface of Venus and beam back images during their brief lifetimes, while the NASA
Magellan satellite used radar to see through the obscuring atmosphere and map the surface.
These missions have revealed a surface with few impact craters and, instead, signs of lava
planes and volcanoes.
Venus lacks a magnetic field, making it different from the Earth and Mercury, and implying
that its internal structure may be rather different to the Earth, which, given that they have
similar mass, is surprising. The young surface, with an absence of cratering, and the absence
of plate tectonics suggest the interesting possibility that Venus goes through periodic total
resurfacing events, where the entire surface is covered by layers of lava. Counting impact
craters suggests that the most recent resurfacing event could have occurred 300-500 Myr ago.
4.6
Earth
The Earth is the planet we are most familiar with. In the context of this survey its most
important aspects are that it has an atmosphere that is ∼80% Nitrogen and 20% Oxygen
and has a surface temperature of 288K. The presence of oxygen in the atmosphere is unique
in the Solar System and is something we will discuss later on.
Why do you think oxygen is so unusual as an atmospheric constituent?
Earth has few visible impact craters, indicating that the surface is young. It has a strong
magnetic field, and active volcanoes and tectonic plates. These combined, as we shall see,
provide information on the internal structure of the planet. Water is common, with oceans
covering 70% of the surface.
4.6.1
The Moon
The Earth also has an unusually large moon, the Moon, which has no atmosphere. It shows
many impact craters but also signs of historic lava flows, the mare or seas. The Moon is in
a synchronous orbit with the Earth, so that the same face of the Moon always points to the
Earth. The Moon does not have a significant global magnetic field. The Earth-Moon system
is thought to have been formed through a huge impact between the proto-Earth and a Mars
sized body about 100-150Myr after the formation of the Solar System.
10
4.7
Sun Stars and Planets 2012-13
Mars
Mars is smaller than the Earth or Venus, but larger than Mercury. It has a very thin atmosphere, with a pressure only about 0.6% of Earth’s and mostly made up of carbon dioxide.
The mean surface temperature is 233K. The surface of Mars has a distinct orangey-red colour
thanks to the colour of the rocks and dust on its surface. There are many huge geological
features on Mars, including the largest volcano in the Solar System, Olympus Mons, which
is 24km high, and a huge canyon system, Valis Marineris, that extends 400km across the
surface.
Despite the thin atmosphere Mars has strong weather systems with seasons, and dust storms
that can last for weeks and that can cover a significant fraction of the planet’s surface.
The current central question about Mars is whether it was once hospitable for life, and whether
life ever formed there. Mounting evidence for the existence of substantial quantities of water
ice adds credence to these ideas, and the new generation of Mars rovers and orbiting satellites
are gathering large volumes of data about the role of water on the surface of Mars in its
distant past. New results from the Curiosity rover will continue to emerge during the course
of these lectures.
4.7.1
Phobos and Deimos: The Moons of Mars
Mars also has two small moons, Phobos and Deimos. They are likely asteroids that have been
captured by Mars’s gravitational field.
4.8
The Asteroid Belt
The asteroid belt lies between the orbits of Mars and Jupiter and is made up of a large
number of rocky and metallic bodies ranging in size from Ceres, with a diameter of 950 km,
downwards, with many, many more small bodies than large. Those that have been studied
in detail have plentiful impact craters. The NASA Dawn mission is currently in the asteroid
belt, studying various asteroids in detail, so we will soon learn much more about these objects.
4.9
Jupiter
Jupiter is the largest planet in the Solar System and the first ‘gas giant’. It is composed
mostly of hydrogen (90%) and helium(10%) with traces of methane, ammonia and water
vapour. The features we see on Jupiter and not those of a solid ‘surface’ but are in fact
ever-changing cloudscapes that lie at the top of its deep atmosphere. The different colours of
the cloud bands represent detailed differences in content, chemistry and depth. An example
of one of these weather systems is the Great Red Spot, a storm system larger than the Earth,
11
that has persisted for several hundred years. The temperature of the cloud tops that we see
is ∼120K, but the temperature and pressure will rise as you go deeper into the atmosphere.
At a depth of 10000km, the temperature should be ∼6000K with a pressure 106 times that
on the surface of the Earth. Jupiter also has a large and powerful magnetic field, 20000 times
stronger than that of Earth.
4.9.1
The Moons of Jupiter
Jupiter has a large number of moons, dominated by the four large ‘Galilean’ satellites, socalled because they were first observed by Galileo, called Io, Europa, Ganymede and Callisto,
in order outward from Jupiter. Io is the most volcanically active body in the Solar System,
with a large number of active volcanoes and a surface covered by sulphur deposits from the
eruptions. Because of this, it’s surface visibly resembles a giant pizza. The heating necessary
to maintain this level of volcanic activity comes from ‘tidal heating’, something we will discuss
later. Io is largely made up of rocky material, not dissimilar from the Earth. Europa, the
next moon outwards from Jupiter, is very different from Io, with a surface that is made up
of ice. Despite this, Europa is a largely rocky body, but the icy layer is expected to be about
100km deep. The icy surface shows few impact craters, and instead appears to be made up
of broken ice packs and fractured plains. This suggests that liquid water may at times reach
the surface through ‘cryovolcanoes’. Water vapour escaping form these may have recently
been detected by the Hubble Space Telescope. This in turn suggests the possibility that a
subsurface ocean of liquid water might lie beneath the surface, kept liquid through similar
tidal heating processes to those on Io, but operating at lower temperatures. The remaining
Galilean moons, Ganymede and Callisto are broadly similar, made predominantly of ice and
with heavily cratered, old surfaces. Ganymede is the largest moon in the Solar System, with
a diameter larger than that of Mercury, but, since it is largely made of ice rather than rock,
it has a substantially smaller mass.
All four of Jupiter’s Galilean moons lie within its magnetosphere, and are thus bombarded
by charged particles. Io is especially strongly affected, and suffers from an especially harsh
radiation environment as a result. Sulphur and Oxygen atoms released by Io’s volcanism are
heated by the charged particles in Jupiter’s magnetosphere and escape the moon’s gravity
to eventually form a ring of plasma around Jupiter. Ions streaming from this ‘plasma torus’
are picked up by the magnetic field and accelerated into Jupiter’s ionosphere, producing an
electrical current of several million amps and leading to spectacular aurorae around Jupiter’s
poles.
4.10
Saturn
Saturn is the second biggest gas giant in our Solar System, having a radius about 15% smaller
than that of Jupiter. Its atmosphere also has a banded appearance similar to that of Jupiter.
Storm systems have been observed in Saturn’s atmosphere by the Cassini spacecraft, but
nothing on the scale of Jupiter’s Great Red Spot. Saturn has such a rapid rotation speed,
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Sun Stars and Planets 2012-13
with a day only 10.7 hours long, that there is significant atmospheric bulging at Saturn’s
equator. Similar to Jupiter, Saturn also has a large magnetic field.
4.10.1
The Rings
The most distinctive feature of Saturn, of course, is the ring system. While all the gas giants
have ring systems of some kind, Saturn’s is the most visible. It is not solid, but is made up
of many small icy and rocky particles, ranging in size from 1cm to a few metres. The rings
are most likely the result of the break up of a moon following a catastrophic impact. The
ring particles all orbit in the equatorial plane of Saturn, creating the disk we see. The rings
themselves are surprisingly thin, only 100m thick, but the reflectivity of the particles makes
the ring system highly visible in reflected sunlight.
The rings are structured into many smaller subrings as a result of the gravitational influence
of small ‘shepherd’ moons that are among Saturn’s 61 satellites.
4.10.2
The Moons of Saturn
Saturn’s moons are generally small, with only seven having radii greater than 200km. However, the largest of these, Titan, is one of the most interesting objects in the entire Solar
System. Titan is one of the largest moons in the Solar System, and is roughly half the size
of the Earth. It has a thick atmosphere that is predominantly nitrogen and methane. The
remainder of the atmosphere, <1%, is made up of complex hydrocarbons. These make Titan’s atmosphere opaque, but also indicate that complex hydrocarbon chemistry is taking
place. The Cassini spacecraft and the Huygens lander have examined Titan in detail, and
have revealed a surface of ice beneath the smoggy atmosphere, together with lakes and seas
of liquid hydrocarbons, filled by a rain of ethane and methane.
Another moon of Saturn that has aroused considerable interest of late is Enceladus. Its surface
has few impact craters, but is instead covered by many cracks, suggesting that it has been
resurfaced through cryovolcanic activity at some time in its past. Direct evidence for this was
found by Michele Dougherty of Imperial College, who discovered geysers of water vapour,
mixed with other compounds, being vented into space from cracks in Enceladus’ surface.
4.11
Uranus
Uranus, like Neptune, is smaller than Jupiter or Saturn, but still has a mass 15 times that of
Earth. It appears as a rather featureless blue-green planet. It is unusual in the Solar System
in that its axis of rotation is tipped ∼98 degrees away from being ‘vertical’ to the plane of the
ecliptic. It is suspected that this is due to a major impact in its earlier history that knocked
the planet onto its side. One pole of Uranus thus always points towards the Sun, while the
other always points away. This results in unusual atmospheric flows with one side of the
13
planet always warmer than the other. Uranus has a ring system, the second most prominent
in the Solar System, but its ring particles are much darker than those found in Saturn’s rings.
Like all the other gas giants, Uranus has a large magnetic field.
Uranus has at least 27 moons, but only five are larger than 200km in radius. Some of these
show evidence for cryovolcanism in their past.
4.12
Neptune
Neptune is the last planet in our Solar System, and is the last of the four gas giants. It has
a mass about 17 times that of Earth. Its atmosphere is a distinct blue colour resulting from
the small amount of methane it contains absorbing light at the red end of the spectrum while
the rest is reflected. More features are observable in Neptune’s atmosphere than in Uranus’,
with banding and pale clouds being detectable by both flypast missions such as Voyager and
by remote observation from the Hubble Space Telescope. Giant storm systems also occur,
although the ‘Great Dark Spot’ detected by Voyager 2 in 1989 had gone away by the time
HST observed the planet in 1994.
Neptune has at least 13 moons, but only three have a radius larger than 200 km. The largest
of these, Triton, is an unusual object that orbits in the opposite direction to all the other
Neptunian moons. This suggests that it did not form at the same time as Neptune and the
rest of its moons, but was instead captured by the planet at a later date. Such a capture
would have been associated with impacts and other activity that would leave their mark on
the surface of Triton, and indeed we find that Triton has a strange divided surface, with
geyser-like plumes evident in one area, and a rough, resurfaced, geography elsewhere. The
geysers are likely responsible for Triton’s tenuous nitrogen atmosphere.
4.13
Pluto, Trans-Neptunian Objects (TNOs) and the Kuiper
Belt
Beyond Neptune, there are no single dominant mass planets. Instead, there is a plethora of
small bodies that form a belt of objects, known as the Kuiper Belt, extending outwards from
the orbit of Neptune. The first of these to be discovered was Pluto, long regarded as a planet
in its own right, but the discovery of other, similarly sized, if not larger, TNOs starting in
the 1990s has led to a re-evaluation of Pluto’s status. The discovery of Eris, which is larger
than Pluto, tipped the balance, and Pluto was demoted to being a ‘minor planet’ by the
International Astronomical Union in 2006.
Kuiper belt objects are left overs from the formation of the Solar System and, since they
represent relatively pristine material from the formation epoch, they are of great interest.
Their distance from the Sun makes them difficult to study, but the NASA New Horizons
mission will fly past Pluto in July 2015, and will tell us much more about these objects.
14
4.14
Sun Stars and Planets 2012-13
Comets
Comets are small bodies from the outer Solar System whose orbits take them close to the
Sun. When this happens, their surface heats up, and volatiles boil off, forming the distinctive
tail that, in the case of bright comets, can even be visible during daylight. Comets come in
two different types, defined by whether they are short or long period. Short period comets
are thought to come from the Kuiper Belt, while long period comets, with periods greater
than about 100 years, come from further away. They come from the last, and most distant,
part of the Solar System to be discussed here: The Oort Cloud.
The ESA Rosetta mission will rendezvous with the comet 67P/ChuryumovGerasimenko in
August 2014, drop[ping a lander on the comet, and then following it as its orbit takes the
comet on its closest approach past the Sun. If all goes well, we will know much more about
comets by the time this course is given in 2015.
4.15
The Oort Cloud
The Oort Cloud is named after Jan Oort, the astronomer who first suggested its existence.
Oort’s idea was that a large population of cometry bodies, that formed in the inner Solar
System at the same time as the rest of the planets, would be thrown out of the Solar System
by gravitational interactions with giant planets like Jupiter and Saturn. The resulting cloud
of comets could include as many as 1011 objects and extend to tens of thousands of AU
in distance, forming a roughly spherical cloud surrounding the Solar System. Long period
comets, with high inclinations relative to the ecliptic, many of which have retrograde orbits,
fall into the inner Solar System from the Oort Cloud.
4.16
Formation of the Solar System
The physics of star and planetary formation is a large and complex topic, the theory of which
will be covered in the 3rd Year Astrophysics course. The core concepts, though, emerge from
several observational facets of our Solar System, which allow us to get some basic idea for
these processes without going into details. The first key observation is that the orbits of most
bodies in the Solar System are roughly circular, and they are all prograde ie. orbiting in the
same direction as the Sun’s rotation. This suggests that the Sun and planets all formed as
part of the collapse of a solar nebula. This broad picture was first developed by Laplace in
the late 18th century.
The starting point is a slowly rotating molecular cloud, which starts to collapse under the
force of its own gravity. As this happens, thanks to conservation of angular momentum, the
cloud contracts and spins faster. The collapse then continues, perpendicular to the rotation
axis. The pre-stellar material, made up of dust and gas, settles into the rotation plane and
the collapse proceeds fastest at the centre, where the Sun will eventually condense. Away
15
from the centre, clumps of dust and gas start to coalesce as the diffuse disk material breaks
up. Once these planetesimals reach 10km in size they begin to accrete material themselves
through runaway gravitational attraction, eventually forming the planets. At the same time
the infant Sun ignites and the resultant stellar wind clears away gas and dust that is not
already gravitationally bound into condensed objects. The eventual composition of a planet
will depend on where in the solar nebula the planet formed.
This is the broad picture of star and planet formation that astronomers work with, but the
details are still uncertain. Observations of planets in other star systems, ie. exoplanets, are
prompting rapid development in this field. For example, it seems that gas giant planets often
migrate to the inner regions of a star system, even though they have to form at distances
from their parent star comparable to those of Jupiter and Saturn.
4.17
Summary
This part of the course has provided a rapid tour of the Solar System, showing both the
variety of objects in it, and looking at some of the features common to all of them. Solar
System science is a very rapidly moving field, with active research going on from the ground
and in space. The Curiosity rover is busy on Mars, the Cassini spacecraft is continuing its
survey of the Saturnian system, and the Dawn satellite is providing us with our first view
of the largest asteroids. Meanwhile, the New Horizons mission is getting closer to Pluto and
Rosetta is nearing its target comet.
Things to Remember
• The names of the planets
• Their order going out from the Sun and that they lie in the same plane - the
ecliptic
• The basic geography of the Solar System including the asteroid belt, the most
famous moons/rings, the Kuiper Belt and the Oort Cloud
• The basic principles behind our model of the formation of the Solar System
• The natural consequences of this model with respect to prograde orbits, and
variation of planetary composition with distance from the Sun
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Sun Stars and Planets 2012-13
Chapter 5
Planetary Orbits: Kepler’s Laws
5.1
Historical Introduction
The predominant view of the structure of the Solar System until the late 16th century was
established in Ancient Greece by Aristotle. This model of the Solar System was geocentric
- the Earth was at the centre and all other bodies, including the Sun, orbited around it,
against a backdrop of ‘fixed’ stars. This was not the only model of the Solar System during
that period, and in fact Artstarchus suggested, in 200BC, that the Sun rather than the Earth
was at the centre of the Solar System, but the geocentric view prevailed.
The geocentric model explained the usual motion of the planets against the fixed stars as a
result of the planets’ motion around the Earth in their respective orbits. However, one aspect
of planetary motion that was difficult to explain in the geocentric system was retrograde
motion. This is the stage in the orbit of planets like Mars and Uranus where their direction
of motion in their orbit, as seen from Earth, reverses, and they appear to move backwards
against the frame of reference of the fixed stars. The solution to this in the geocentric model
was add an epicycle to the planet’s orbit, and epicycle being an additional circular motion of
the planet about its circular orbital track around the earth.
In 1543 Copernicus began a revolution in our understanding of the Solar System and the rest
of the Universe. In his book De revolutionibus orbium coelestium (On the Revolutions of the
Celestial Spheres), published just before his death1 he describes how a heliocentric model of
the Solar System, with all the planets including Earth orbiting the Sun in circular orbits, can
explain retrograde motion without the need for epicycles. In the heliocentric view, epicycles
arise when the Earth catches up and overtakes a planet, like Mars, that is further from the Sun
and thus orbiting more slowly. Some tweaks and contrivances are still necessary to explain
anomalies that arise because the actual orbits of the planets are somewhat elliptical, but this
work represented a huge advance in our understanding.
1
Legend has it that the very first copy of De Devolutionibus was placed in his hands as he lay on his
deathbed in a stroke induced coma. He is said to have woken, glanced at the book, and then died peacefully
17
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Sun Stars and Planets 2012-13
The publication of De Revolutionibus did not change things immediately, and did not cause
any great controversies. At that stage the geo- and heliocentric models were simply that
- competing ideas that had yet to be fully tested by observations. Beginning in 1576, the
astronomer Tycho Brahe began observations at his observatory at Uraniborg, near Copenhagen, that would provide the data necessary for a far more precise understanding of the
orbits of the planets. He and his staff and students produced highly accurate measurements
of planet positions, without the aid of telescopes, between 1576 and 1597. He also discovered
a supernova in our own galaxy which still bears his name.
Johanes Kepler worked as an assistant to Tycho at the end of Tycho’s life, when he had
moved to Poland. On Tycho’s death in 1601, Kepler was appointed his successor as imperial
mathematician to Emperor Rudolph II. While the main part of his work for the Emperor was
to provide advice of an astrological nature (astrology paid better than astronomy, even then!),
Kepler continued his work analysing the detailed observations that Tycho had obtained of the
motions of the planets. Over the next decades, Kepler derived three laws for planetary motion
from Tycho’s observations, and also attempted to provide a physically-based explanation for
these laws, arguably making him the first astrophysicist (not that we would recognise his
physics today). These results were published in 1609 in Astronomia Nova. He also wrote a
book, Somnium published posthumously, discussing the possibility of astronomy from another
planet, which has been described as the first work of science fiction.
At the same time, Galileo was working with the first optical telescopes in Italy. Among many
other things, in 1610 he discovered the four large moons of Jupiter, known to this day as
the Galilean moons, and found that they orbited Jupiter in just the same way as the planets
orbitted the Sun in the heliocentric model.
Between Kepler and Galileo, the stage was set for a revolution in our understanding of the
Solar System.
5.2
Kepler’s Three Laws of Planetary Motion
Kepler derived three laws of planetary motion from Tycho’s data:
1. Planets follow an elliptical orbit with the Sun at one focus
2. The line joining the planet and the Sun sweeps out area at a constant rate
3. The square of the time a planet takes to go round the Sun, P, is proportional to the
cube of the semi-major axis of its orbit ie.
P 2 ∝ a3
Kepler’s explanation for these laws involved Platonic solids and the harmony of the celestial
spheres. It wasn’t until Newton turned his attention to planetary orbits that we arrived at
something we would recognise today as a full physical explanation of Kepler’s laws, using
Newton’s laws of motion and gravitation.
19
r
θ
Planet in orbit
Origin
r
Sun
Figure 5.1
5.3
Derivation of Kepler’s Three Laws
The natural coordinate system for studying the orbits of the planets is a plane polar system,
with the origin at the centre of mass of the massive object, in this case the Sun, around which
the planets are orbiting.
Basis vectors in a plane polar system are b
r, along the direction joining, in this case, a planet
b
to the Sun’s centre of mass, and θ, along the tangent to this direction. Expressing these in
cartesian coordinates, in terms of the angle θ we find that:
−sinθ
cosθ
b
b
(5.1)
,
θ=
r=
cosθ
sinθ
From this you can show that:
d
b
b
r = θ̇θ,
dt
db
θ = −θ̇b
r
dt
(5.2)
Exercise Use the definition of plane polar coordinates in terms of cartesian coordinates given
in equation 5.1 to show that the definition of the differentials in equation 5.2 are correct.
Given these definitions, and using the general result for vector differentiation that:
d
da dφ
(φa) = φ
+
a
du
du du
(5.3)
we can obtain the velocity and acceleration:
r = rb
r
(5.4)
ṙ = ṙb
r + rθ̇θb
r̈ = r̈ − rθ̇2 b
r + 2ṙθ̇ + rθ̈ θb
(5.5)
(5.6)
The gravitational force of the Sun on an orbiting planet will be given by Newton’s inverse
square law of gravity. Combining this with Newton’s first law of motion F = ma we then get:
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Sun Stars and Planets 2012-13
r θ̇∆t
r
Figure 5.2: Area traced out by an orbiting body in time ∆t.
mr̈ = f (r)b
r
(5.7)
GM m
(5.8)
r2
where f (r) is the force due gravity, M is the mass of the Sun, and m is the mass of the planet.
Using the formula for r̈ given in equation 5.6, and examining the b
r and θb directions separately
we find:
GM
(5.9)
r̈ − rθ̇2 = − 2
r
Since there is no force in the θb direction, as gravity only acts along b
r, the vector joining the
planet to the Sun, the term in θb must vanish:
f (r) =
2ṙθ̇ + rθ̈ = 0
This last term can be written more usefully, and more compactly as:
1 d 2
r θ̇ = 0
r dt
(5.10)
(5.11)
Exercise: Use the chain rule for differentiation on equation 5.11 to show that this is equal
to equation 5.10.
Equation 5.11 is equivalent to saying that r2 θ̇ is a constant. r2 θ̇ is the specific angular
momentum, ie. the angular momentum per unit mass, and we will define this as equal to the
constant h.
r2 θ̇ = h
(5.12)
What does this last result mean in terms of Kepler’s Laws? Figure 5.2 shows how area is
traced out by a body as it orbits around the Sun. The area of a triangle is simply
1
1
1 base × height = r rθ̇∆t = h∆t
(5.13)
2
2
2
21
But we know that h, the specific angular momentum, is constant, so the area covered in equal
times ∆t, is also constant. Kepler’s Second Law is proven.
Before going further in the mathematics, let us also look at the combination of equations 5.9
and 5.11.
Circular motion will have a constant distance form the Sun, r, which we will set to the value
a, and a constant rate of change of θ, ie. θ̇ = ω, the angular velocity. This is constant so
equation 5.11 is satisfied. In equation 5.9 we then have r̈ = 0 which means:
rθ̇2 = aω 2 =
GM
a2
(5.14)
Both sides of this equation are constant, so as long as they are equal we have solved both 5.9
and 5.11, indicating that circular motion is an acceptable solution for the orbit of a planet.
Furthermore, since the time taken for a planet to orbit the Sun, p = 2π
ω by definition, we thus
have:
2
GM
a3
GM
2π
2
= 2 =⇒ 2 =
= const
(5.15)
aω = a
p
a
p
4π 2
Which is Kepler’s Third Law proven, at least in the case of circular orbits.
For a more general solution to 5.9 and 5.11 we define a new variable u such that r = 1/u and
differentiate to obtain u̇ and ü.
ṙ = −
1
du dθ
du
du
u̇ = −r2
= −r2 θ̇
= −h
2
u
dθ dt
dθ
dθ
and
d
r̈ = −h
dt
du
dθ
dθ d
= −h
dt dθ
du
dθ
= −hθ̇
d2 u
dθ2
(5.16)
(5.17)
Substituting this into equation 5.9 gives:
hθ̇
d2 u r2 θ̇
GM
+
θ̇ =
θ̇
dθ2
r
r2 θ̇
(5.18)
Given that r2 θ̇ = h this can be rewritten:
hθ̇
d2 u
GM
+ huθ̇ =
θ̇
2
dθ
h
(5.19)
Dividing through by h and θ̇ then yields:
d2 u
GM
+u= 2
dθ2
h
(5.20)
This is the equation of simple harmonic motion with constant forcing, the general solution to
which is:
GM
(5.21)
u = Acos(θ − θ0 ) + 2
h
22
Sun Stars and Planets 2012-13
We can choose a coordinate system where θ0 = 0 and rewrite this as:
1
1
(1 + ecosθ)
=
r
r0
(5.22)
where r0 = h2 /(GM ) and where we can set e ≥ 0 without loss of generality. There are then
four cases, depending on the value of e.
• For e = 0 we simply get a circle, with r = r0 at all times.
• For 0 < e < 1 we have an ellipse with eccentricity e - this is in fact the situation for the
orbits of all the planets, proving Kepler’s first law.
• For e = 1 we get a parabola.
• For e > 1 we get a hyperbola.
The first two of these options are bound orbits, so is what we see for the planets. The last
two options are unbound orbits, and are what is seen for objects that achieve escape velocity.
In the case of the Solar System, the Pioneer and Voyager satellites have managed this, and
will travel forever between the stars of our Galaxy.
Things to Remember
• Kepler’s 3 laws as stated in section 5.2
• How to explain all three, and the consequences with respect to circular, elliptical, parabolic and hyperbolic orbits
• Be able to show the 3rd law in the context of a circular orbit
Chapter 6
Terrestrial Planets: Heating,
Cooling Processes and Interiors
6.1
Introduction
The Earth is a terrestrial planet, along with Mercury, Venus and Mars. There are commonalities between them, but also substantial differences. In this chapter we will look at the
internal structure of terrestrial planets and the factors that drive that structure. This will
provide some key insights into how and why the four terrestrial planets differ. Along the way
we’ll also uncover some of the forces that have shaped the Earth and its geography over its
4.5 billion year history.
6.2
The Unquiet Earth
In London, it is easy to think of the Earth as fixed and unchanging, but we know that this
isn’t in fact the case. Earthquakes and volcanic eruptions are just two reminders that our
planet is a dynamic system, even if much of that dynamism operates on timescales far longer
than that of a human life.
The Earth beneath us is in fact a lot more dynamic even than that, as can be seen when
material from deeper beneath the surface bursts out during a volcanic eruption. Where did
the energy for that heat come from and how has this driven the large scale structure of the
Earth and the surface features we see today?
23
24
6.3
Sun Stars and Planets 2012-13
Primordial Heating
The formation of the Earth was a violent process, with large impactors peppering the forming
planet and with smaller bodies accreting at a high rate. The kinetic energy of these impactors
is largely turned to heat during the collision, and the accretion of smaller bodies also leads
to heating of the young earth. In general, the potential energy of all the mass that falls onto
a planet during its formation, which is converted to heat, is given by:
PE ∼
GM 2
R
(6.1)
where M is the mass of the planet, G has its usual value, and R is the radius of the planet.
The immediate consequence of all this energy being deposited into the young planet as it is
forming, is to make much of the material molten, and to keep it that way for many thousands
of years.
6.4
The Structure of the Earth
The material that made up the forming Earth includes substances with a range of densities. If
a mixture of liquids with different densities is allowed to settle, the densest material will end
up at the bottom, in this case at the centre of the Earth. The constituents of the young Earth
can be determined by looking at the constituents of meteorites. These include Al, Si, Ti, Fe,
Ni, Mg, Ca, with Fe and Ni being quite abundant. The densest of these materials are Fe and
Ni, so these constituents separated out, and fell towards the core of the Earth, leading to the
formation of a solid, largely iron core, surrounded by a liquid nickel-iron outer core. Above
this is the mantle, made of a material called peridotite which includes minerals containing
Mg, Ca, Fe, Al, Si, Na, O, Cr, but which is essentially 40-60% SiO2 . The minerals that make
up peridotite include feldspar, olivine, pyroxene, spinel, garnet and others. Above the mantle
lies the crust, made of basalt and ∼75% SiO2 . On top of this crust are the sedimentary rocks
produced by erosion processes which make up most of the landscapes that we can see on the
surface (see Fig. 6.1).
The separation of the Earth into core, mantle and crust is based on the results of seismology
- essentially looking at how the speed of sound changes as seismic waves travel through the
Earth. Figure 6.2 shows how the physical properties that affect the transmission of seismic
waves change with depth, and their effect on the passage of seismic waves.
An alternative way of thinking about the structure of the Earth is based not on the constituents of the material but on its physical state. This leads to a different classification that
we will find useful later on. In this approach the core is the same, but the mantle is then divided not into the upper and lower mantle, which is based on composition, but into the region
where the rock is molten or under sufficient pressure that it can flow — the asthenosphere —
and the region where the rock is rigid — the lithosphere — where flow is not possible. The
upper parts of the mantle and the crust make up the lithosphere. This is also shown in Fig.
6.1.
25
Figure 6.1: Structure of the Earth from Fig. 2.1 of Rothery, McBride & Gilmour.
26
Sun Stars and Planets 2012-13
ut: Structure and Atmospheres of Planets
Interiors of the Earth and terrestrial planets
arth’s structure
onal:
h)
% SiO2)
iO2)
outer shell
onvecting part of the mantle
usgs.gov/gip/dynamic/inside.html Interior structure of the Earth, taken from
Karttunen et al Fundamental Astronomy
Left: Comparison of the composition
Figure 6.2: Internal
structure
of the Earth
andand
how the
this changes the results of seismology eg. speed
of the
terrestrial
planets
of seismic waves vs. depth. Taken from Karttunen et al. Fundamental Astronomy.
Moon (McBride & Gilmour)
Below: relative sizes & core masses
(in %) taken from Karttunen et al
27
Isotope
235 U
238 U
232 Th
40 K
Half-life (109 ) yrs
0.71
4.5
13.9
1.3
Present Rate of Heat Generation (10−12 W kg−1 )
0.04
0.96
1.04
2.8
Table 6.1: Half-lives of the most important radiogenic heat sources in the Earth’s crust and mantle
today.
6.5
Long Duration Heat Sources
As well as the initial heat input from the formation of the planet, there are two long duration
sources of heat that help to keep the Earth’s interior hot. The most important of these is
radiogenic heating from long lived unstable isotopes. The radiation given off by their decay is
absorbed by their surroundings, leading to an increase in temperature. Table 6.1 summaries
the most important long lived isotopes in the Earth for long term radiogenic heating. The
Earth’s age is 4.5 billion years, which is comparable to the half lives of these isotopes.
Tidal heating is the other potential source of long term heating. Tidal heating comes from
the effects of a nearby orbiting massive body - in the case of the Earth, the Moon produces
tidal effects. The most noticeable are the tides in the Earth’s oceans, but there is also a ∼1m
maximum rise and fall of the Earth’s rocky surface due to the Moon. This deformation of
the Earth imparts energy which appears as heating. It is thought that thsi heating is largely
deposited in the crust and mantle. The amount of energy imparted to the Earth from the
Moon by this tidal interaction is small, nearly two orders of magnitude less than the energy
input from radiogenic heating, but tidal heating is very important for other bodies in the
Solar System, as we will see later.
6.6
The decay of long term heating sources
While radiogenic heating and tidal heating persist today, they are not an infinite resource.
With time the radioactive species responsible for heating will decay away. The tidal heating
rate will also fall as angular momentum is transferred from the rotation of the Earth to the
orbit of the Moon, and as the Moon moves away from the Earth. Ultimately, therefore, the
interiors of all terrestrial planets cool.
6.7
Heat Loss from Planets
The Earth cools by radiating heat away into space. Volcanic eruptions are the most obvious
example of this, but there are many other, less dramatic ways in which the heat from the
upper layers of the asthenosphere travels through the lithosphere. The amount of heat that the
Earth, or any other planet, can thus expel is determined by its surface area. In contrast, the
parison between the different
terrestrial planets
28
Sun Stars and Planets 2012-13
sizes and core
planet and terrestrial moon structure
%
12%
ume
McBride & Gilmour Introduction to the Solar System
Figure 6.3: Comparison of the interiors of the 4 terrestrial planets and the Moon. Note that the
smaller the object, the thicker the crust. The Moon, for example, has a crust that is 1000km deep.
From McBride & Gilmour.
amount of heat, and, indeed, the amount of active radiogenic heating underway, is dependent
on the volume of the Earth, or other object.
Heat loss rate ∝
Surface area
4πR2
1
=
∝
Volume
4/3πR3
R
(6.2)
Thus smaller planets cool more rapidly than larger planets. The long term result of cooling
is that the lithosphere thickens and the asthenosphere becomes thinner. Mars, for example,
has a much thicker crust than the Earth, and Mercury has an even thicker crust.
6.8
Cooling Processes
How does the heat travel from the core of the Earth to the surface?
29
Figure 6.4: Convection transferring heat from the core to the surface. Fig 2.14 of Rothery, McBride
& Gilmour.
There are four key processes that allow the Earth to cool:
• Conduction
This is the most familiar process, whereby heat transfers from a hotter to a cooler region
through thermal conduction. In the lithosphere, where rocks are rigid and cannot flow,
this is the main method of heat transference.
• Convection
In the asthenosphere, where material is able to flow, convection operates, and is the
most efficient way that heat is transferred. Hotter material expands, and is thus less
dense, so rises, while cooler material contracts, becomes more dense, and falls. The
cooler material then warms up and the process continues. Large scale convection cells
exist in the asthenosphere, where this process can operate. Solid state convention, in
which rocks flow by a few cm/year, drives this process. See Fig 6.4 for a diagram of
how this operates.
• Eruption/Advection
The lithosphere is too rigid to allow convection, so the last stage of the process of heat
transference from the core to the surface takes place when molten rock, or magma,
spreads over the surface and cools, or as it is injected into the lithosphere and cools
beneath the surface, and the heat is conducted away to the surrounding crust.
30
Sun Stars and Planets 2012-13
Figure 6.5: The key features of plate tectonics: sea floor spreading, continental plates, subduction
zones, and arcs of volcanoes around the edges of continental plates. From USGS
• Plate techtonics
The surface of the Earth is made up of a series of plates that essentially float on the
surface of the asthenosphere. Some of these plates are thicker, and form the continents,
while others are thinner, and form the floor of the oceans. The plates move relative
to each other, with new material being produced by hot magma emerging from the
asthenosphere at mid-ocean ridges, leading to sea floor spreading, and with old, cold,
material sinking into the asthenosphere at the edges of continents in subduction zones.
6.9
Volcanism and Tectonics on Other Terrestrial Planets
Given that the cooling rate of a planet is set by its surface area to volume ratio, we would
expect smaller planets, and similar objects like the Moon, to have thicker lithospheres and thus
be less tectonically active. The smallest terrestrial planet in our Solar System is Mercury.
Studies show that its surface is old, as evidenced by heavily cratering, but that there are
regions where some resurfacing has taken place. Our best estimate is that this resurfacing
took place roughly a billion (109 ) years ago (1 Gyr). The Moon, while not a planet, shares
many of the properties of a terrestrial planet, so is another useful check of our ideas about
planetary volcanism. Like Mercury, there are regions of the Moon that are old and heavily
cratered, but others, the maria, or seas, that are younger and appear to have been resurfaced
31
by more recent lava flows. These are also old, about 1 Gyr old, and are thought to be related
to impact events that punched holes through thinner portions of the Moon’s lithosphere,
allowing lava to flow over the surface.
Venus, with a similar size and mass to Earth, would be expected to have a similar lithosphere
thickness and thus similar tectonic activity. However, observations, conducted using the radar
mapping instruments of the Magellan spacecraft, have found no evidence for tectonic activity.
The surface of Venus, though, is young, showing none of the extensive cratering that is seen
on the older surfaces of the Moon or Mercury. If there are no tectonic plates on Venus, how
does its interior cool?
One idea is that the lithosphere of Venus acts like the lid on a huge pressure cooker. Instead of
the continuous leaking of internal heat that we see on Earth, the idea is that Venus occasionally
blows its top, with the entire planet being resurfaced through periodic volcanic catastrophes.
The surface of Venus appears to be between 700Myr and 500Myr old, which would set the
date of the most recent catastrophic resurfacing. Volcanoes are seen on Venus, often showing
a strange, flat topped appearance suggesting slow growth. Historical lava flows up to 2000km
have been found, but there is no evidence of ongoing volcanic activity. The volcanoes of Venus
are presumably awaiting the next epoch of catastrophic volcanism.
Mars is intermediate in mass between Mercury and the Earth, so might be expected to have
an intermediate level of tectonic activity. There is indeed some evidence of tectonics on Mars,
with significant differences between the northern and southern hemispheres. The southern
highlands are similar to the thick crust of Earth’s continents, while the northern lowlands
are similar to the thinner crust of the oceans. However, the lithosphere of Mars is now much
thicker than that of Earth, so any ancient tectonic activity is likely to have stopped long
ago. Age estimations using cratering statistics suggest that the southern highlands are older,
at about 4.5 Gyr, while the northern lowlands and the Tharsis Bulge, home to Mars’ giant
volcanoes, are younger at 3.7Gyr.
Mars has the largest volcanoes in the Solar System, including the giant Olympus Mons. These
are all found in the Tharsis Bulge region, which appears to be similar to the ‘hot spots’ found
in several locations on the Earth. These hot spots seem to be the result of upwellings in the
asthenosphere at certain positions in the mantle. These mantle plumes bring heat into the
lithosphere in a way that is largely separate from tectonic activity. The Hawaiian Islands
on Earth are a result of a hot spot located in the middle of the Pacific Ocean plate, well
away from any region of sea floor spreading. Since the Pacific plate is moving, the volcanic
islands, that grow around the location of the hot spot, are gradually dragged away from the
hot spot, leading to the production of a chain of volcanic islands and, further away, a series of
sub-surface sea mounts. On Mars, there is no tectonic activity, so the large ‘shield’ volcanoes
that grow from them just continue growing. This is why Olympus Mons is the largest volcano
in the Solar System.
32
Sun Stars and Planets 2012-13
Things to Remember
• The structure of the Earth (Fig 6.1) including the names, constituents and properties
of different layers
• Heat sources for terrestrial planets, long and short duration including primordial
heating
• Heat loss processes, including conduction, convection, advection/eruption & plate
techtonics
• Dependence of heat sources and heat losses on the size of body
• Consequences of this for the internal structure and surface volcanism on other
planets
Chapter 7
Terrestrial Planet Surfaces and
Temperatures
7.1
Introduction
In the previous chapter we found that the surface properties of planets are far from typical of
their interiors. The vast majority of the Earth’s volume is made up of molten rock, flowing,
albeit slowly, in giant convection currents. The continental plates that make up the surface of
our planet are just low density material floating on this ocean of rock. However, the surface
of the Earth is something that we are intimately concerned with, and the surfaces of other
planets are, by and large, all that we can see of them. Understanding the processes responsible
for the planetary surfaces that we see, and that determine the basic properties of planetary
environments, including surface temperature, are thus a key ingredient to understanding the
nature of planets in our own and other solar systems.
7.2
Major Factors in Shaping Planetary Surfaces
While the four terrestrial planets are all very different in appearance, they are all shaped by
similar physical process. The importance of these different processes, though, varies between
planets, and this will apply just as much in other solar systems as it does in our own. There
are four central processes that determine the surface geography of planets:
• Impact cratering
The majority of impact events occurred during the earliest stages of the Solar System,
but impacts continue to happen today, albeit at a much lower rate. Examples on Earth
include Meteor Crater in Arizona, a crater 550 feet deep and about 1 mile across that
was produced by an impact about 50000 years ago, and the Tunguska event, likely
an airbursting small meteor, that levelled 2150 sq. km of forrest in Siberia in 1908. In
33
34
Sun Stars and Planets 2012-13
Planet
Cratering
Tectonics
Volcanism
Erosion
Comments
Mercury
Heavily cratered,
very old surface (∼4.5 Gyr)
No (some in past)
No
No
Most geologically inactive
terrestrial planet
Venus
Few craters, surface
only ∼500Myr old
No tectonics seen
Past volcanoes seen
Yes
Catastrophic
resurfacing possibility
Earth
Few craters, geologically
active, erosion effects
Yes
Yes
Yes
Interior not yet solidified
Mars
Heavily cratered in parts
No current tectonics
Past giant volcanoes seen
Yes
Probably almost solidified
Table 7.1: Summary of role of shaping processes in terrestrial planets
February 2013, a somewhat smaller meteor passed over the Russian city of Chelyabinsk.
The shockwave as it passed through the atmosphere causing moderate damage over a
large area and injuring roughly 1500 people. With an estimated mass of 12000-13000
tonnes and a size of 20m, the Chelyabinsk meteor is probably the largest natural object
to enter the Earth’s atmosphere since Tunguska.
• Volcanism
As discussed in the previous lecture, volcanoes are where the heat contained within
a planet can escape to the surface and, over geological timescales, allow the planet’s
interior to cool. The effects of lava flows can be seen in many places in the Solar System,
and there are giant volcanoes on Mars.
• Tectonics
The surfaces of some terrestrial planets are made up of tectonic plates that float on top
of the hot, molten, interior. The interactions of these plates, and their movements on
the surface, are a driving force for shaping the surface of planets.
• Erosion
In the presence of a fluid, whether gas or liquid, surface features are eroded and modified
over time.
The importance, or otherwise, of each of these factors varies form planet to planet. Those
planets that have cooled rapidly, for example, so that their lithospheres are thick, are less likely
to experience tectonic or volcanic activity, while those with no atmosphere will not experience
extensive erosion. The importance of these factors for each of the terrestrial planets is listed
in Table 7.1.
7.3
Impact Cratering
Cratering is ubiquitous throughout the Solar System, caused by the impact of small bodies
with larger objects. Impacts can be thought of as the process of accretion of planetary
material that continuing to this day, though at a much lower rate than during the epoch of
planet formation. Younger surfaces on planets and moons have fewer craters, and this can be
used to date the surfaces that are seen. Where large scale resurfacing has occurred, through
35
volcanic activity, for example, signs of cratering are erased. Erosion, also, can erase the signs
of cratering given sufficient time.
During an impact, rocks are heated and subjected to very high pressures. Rocks melt and
fracture as a result. Material will be ejected from the impact crater, in both solid and
molten form, and a cavity is excavated leading to the familiar shape. A sufficiently powerful
impact will expel large quantities of material, leading to large area effects. Ejected rocks
can sometimes be expelled from the atmosphere and even given sufficient kinetic energy to
achieve escape velocity. In fact some meteorites found on Earth were ejected from Mars by
past impacts.
Impacts on water, which will be common in the case of Earth since it is 70% covered by water,
would produce massive tsunamis.
A large enough impact will produce significant environmental damage. The extinction of the
dinosaurs has been linked to the Chixulub crater, found beneath the Yucatan Penunsula in
Mexico. Still bigger impacts can change the nature of the objects involved. The Moon, for
example, is thought to have been formed as a result of an impact between the young Earth
and a body roughly the size of Mars.
The impact rate in the Solar System has been in decline for at least the past ∼4 Gyr, but
there are suggestions, based on crater counting studies on the Moon, that there was a brief
increase in the impact rate about 3.8 - 4 Gyr ago. This phase in the development of the
Solar System has been termed the Late Heavy Bombardment, and may be linked to broader
aspects of the evolution of the Solar System.
7.4
Volcanism and Tectonics
The physical background to volcanism and plate tectonics were discussed in the previous
chapter. Both can have a considerable effect on shaping planetary surfaces. Evidence for
historical large scale resurfacing events involving volcanic activity can be seen on the Moon
and Mercury, but these most ended 3 Gyr ago. Volcanoes are clearly present on Mars,
including the massive Olympus Mons. The surface of Venus appears to be geologically young,
less than 0.5 Gyr, and volcanoes have been seen on its surface in radar mapping observations
by the Magellan satellite, but the lack of tectonic activity on Venus has led to the idea that
its surface is periodically subject to catastrophic volcanism, where it is completely resurfaced
from time to time.
On Earth, there is evidence for large scale resurfacing during events known as flood basalts.
Examples of these include the Deccan Traps in India, about 65 million years old, the Columbia
River flood basalts, about 15 million years old, and the Siberian Traps, about 248 million
years old. These flood basalts are made up of tens to hundreds of separate lava flows stacked
on top of each other, reaching thicknesses of 1 to 3 km and covering thousands of square
kilometres of the surface. They are the result of the production of lava volumes up to 2
million cubic kilometres in size that erupted over timescales of 1 to 5 million years. This
36
Sun Stars and Planets 2012-13
Planet
Venus
Earth
Mars
Mercury
Albedo
0.77
0.3
0.25
0.10
Mean Surface Temperature (K)
733
288
223
443
Surface Atmospheric Pressure (bar)
92
1.0
6 × 10−3
10−15
Table 7.2: Albedos of terrestrial planets. From Rothery McBride & Gilmour
represents an annual eruption rate over twenty times greater than those observed for present
day hot spots such as Hawaii. Many of the historical flood basalts are associated with mass
extinctions, and the volume of lava, ash and gas they can produce is certainly enough to cause
major environmental effects.
Plate tectonics are responsible for the shape and distribution of continents and oceans across
the surface of the Earth, driven by the convection currents in the upper parts of the mantle.
There is evidence for historical tectonic activity on Mars. As with volcanism, tectonic activity
is expected to decline with time as a planet cools, and the lithosphere extends to greater
depths. Smaller planets, such as Mars or Mercury, will cool much faster than the Earth,
leading to the cessation of tectonic activity, and the geologically quiescent state we see today
on these planets.
7.5
Erosion
Where fluids are able to flow on a planet’s surface, erosion can take place. The flowing
fluids may be gas or liquid, leading to two different types of erosion: fluvial erosion where
liquids are involved - this can be seen as water erosion on Earth and, possibly, on Mars; and
aeolian erosion where the flowing fluid is the gas of an atmosphere - this can be seen in dry
environments on Earth, on the surface of Mars and, to some extent, on Venus. These different
erosive processes leave different signatures on the environment, allowing us to get some idea
of the presence, or absence, of water on the surface of Mars in the past. Both fluvial and
aeolian erosion leads to the formation of stratified sedimentary rocks, like sandstone.
7.6
The Surface Temperatures of Planets
The presence, or absence, of liquid water on the surface of a planet is of great interest in
the context of exobiology - the search for life on other planets. In the case of Mars we are
interested in whether water flowed on its surface in the past, and in the case of planets around
other stars - exoplanets - we are interested in determining whether life, or the conditions for
life, might persist today.
The temperature of a planet can be estimated, under certain assumptions, quite simply by
looking at the energy balance of incoming to outgoing radiation.
37
The energy input to a planet is the radiation received from its parent star, minus the fraction
of that radiation that is reflected away. The reflection fraction is given by the planet’s albedo,
a, where a = 1 means total reflection and a = 0 means total absorption. The total energy
received by a planet can then be calculated as follows:
Flux density in Wm−2 received = F =
L
4πd2
(7.1)
where L is the total luminosity of the star and d is the distance from the star to the planet.
The planet intercepts a total of:
L
πR2 F = πR2
(7.2)
4πd2
where R is the radius of the planet. Some of this energy is reflected by the planet’s albedo
so the total power absorbed by the planet is then:
Total Power Received = PR = πR2
L
(1 − a)
4πd2
(7.3)
If we then assume that the planet radiates this heat away as a perfect black body we can
find its no-atmosphere temperature TN A . The reason why we have to specify that this is
a no-atmosphere temperature will become apparent shortly. The total power emitted by a
black body at temperature TN A and radius R, PE is:
PE = 4πR2 .σTN4 A
(7.4)
from the Stefan-Boltzman equation, and where σ is the Stefan-Boltzman constant, σ =
5.670373 × 10−8 Wm−2 K−4 . By setting the power received equal to the power radiated away
we can determine the temperature at which these balance, and find TN A .
L
(1 − a)
4πd2
L (1 − a) 1/4
=
16 π σ d2
4πR2 .σTN4 A = πR2
⇒ TN A
(7.5)
(7.6)
Note that this is independent of R, the radius of the planet.
How well does this equation work?
Given the albedo values for the various planets found in table 7.2 we can calculate the noatmosphere temperatures of a variety of planets in our Solar System.
Exercise: Calculate TN A for the planets listed in Table 7.2.
When you do this, you will find that the TN A values for Mars and Mercury are a good match
to the results of the energy balance equation, but the Earth and Venus are anything but, with
Venus having a surface temperature 500K higher than that estimated by TN A . Why is this?
38
7.7
Sun Stars and Planets 2012-13
The Greenhouse Effect
The answer to this is that Venus and the Earth both have significant atmospheres, and that
those atmospheres contain gases that allow more heat to be retained than is assumed by the
no-atmosphere approximation. Solar radiation peaks in the optical part of the electromagnetic
spectrum. The atmosphere is (largely) transparent at these wavelengths, so the light of
the Sun passes straight through, allowing us to see, and allowing its radiation to heat the
planet. The re-emitted thermal radiation, however, peaks at longer wavelengths, in the midinfrared (you can determine this using the Wien Displacement law). The atmosphere is not
as transparent at these wavelengths as in the optical, thanks to the presence of CO2 , methane
and other so-called greenhouse gases. These gases in the atmosphere absorb some of the midIR radiation from the surface, warm up a little, and reradiate it, again as thermal emission,
in all directions. A reduced fraction of the radiation emitted from the surface thus reaches
space, and the planet therfore retains more of the energy received form the Sun than it would
without an atmosphere. The with-atmosphere temperatures, as observed for Venus and the
Earth, are thus higher than the calculated no-atmosphere temperatures, TN A .
The overall climate system on the Earth is of course more complex than this simple analysis
suggests, with the presence of clouds leading to local increases in albedo, the detailed content
of the atmosphere changing the albedo further, modifying the degree of the greenhouse effect,
and other factors such as the condensation of water vapour into rain providing other inputs
of heat. The details of these and other factors are studied in the Atmospheric Physics course.
One thing, though, is clear - if the fraction of greenhouse gases, such as CO2 and methane,
in the atmosphere increases, there will be more mid-IR absorption, and greater retention of
heat.
Things to Remember
• The major factors shaping planetary surfaces: impacts, volcanism, tectonics and erosion
• The dependence of impact rate on time
• The presence or absence of volcanism on other Solar System bodies and the reasons for this
• How to calculate the no-atmosphere surface temperature of planets
• How the Greenhouse Effect can change these no-atmosphere temperatures
Chapter 8
Terrestrial Planet Atmospheres
8.1
Introduction
In the last chapter we saw how important the presence, or absence, of an atmosphere is for the
surface temperatures of terrestrial planets. Atmospheres are also important for many other
processes, including erosion and, of course, anything biological. In this chapter we will look
at the origin of terrestrial planet atmospheres in the Solar System, how atmospheres escape
from a planet, how atmospheres are structured, and what they contain.
8.2
Why do we have an atmosphere at all?
Venus and the Earth have significant atmospheres. Mars has only a thin atmosphere, while
Mercury, the Moon and smaller rocky bodies in the asteroid belt and elsewhere usually have
little or no atmosphere at all. What makes Venus and the Earth so different? The answer to
this is that Venus and the Earth are more massive bodies than the others, and the atmosphere
is held onto their surface by the effects of gravity. We can examine the effects of gravity on the
structure of the atmosphere of a planet using the principle of hydrostatic equilibrium, whereby
the downward force due to gravity on each part of the atmosphere, must be balanced by the
pressure gradient in the atmosphere.
8.3
Atmospheric Density and Pressure
If the pressure gradient is to balance the force of gravity then:
dP
= − ρg
dz
39
(8.1)
40
Sun Stars and Planets 2012-13
where P is the atmospheric pressure, z is height above the surface, ρ is the atmospheric
density and g is the gravitational acceleration. Pressure and density are connected by the gas
law:
P V = N kB T
(8.2)
where P is pressure, V volume, kB is Boltzmann’s constant, and T is temperature. We can
rewrite this in terms of the density of a gas of mean molecular weight hµA i and the atomic
mass unit mamu as follows:
P hµA i mamu
= kT
(8.3)
ρ
Since V = M
ρ and the mass M of a given mole of material, M = N hµA i mamu . Combining
equations 8.1 and 8.3 we get:
dP
P hµA i mamu g
= −
dz
kT
(8.4)
To be able to solve this equation we need to make two simplifying assumptions. The first is
that g doesn’t vary with height. This is a good assumption since the atmospheres of terrestrial
planets are thin compared with the sizes of the planets. The second is that T doesn’t vary
with height. This isn’t as good, but is adequate since pressure changes more rapidly with
height than temperature. Equation 8.4 can then be solved by separation of variables, and we
get:
P = P0 e −
hµA imamu
gz
kT
(8.5)
and there is a similar exponential fall off for density as a function of height. The quantity
(kT / hµA i mamu g) is known as the scale height and gives height at which pressure and density
fall to 1/e times their surface values. The scale height is actually quite small on Earth, with
pressure falling to about 60% of the sea level value at the summit of Mauna Kea, a height
of 4200m, and to just 33% of the sea level value at the height of Everest, about 6000m.
This is why mountaineers on Everest often develop serious breathing and respiration related
problems. This also shows that our approximation that g doesn’t vary with height is a good
one.
8.4
Temperature Variations with Height
The atmosphere is largely transparent to solar radiation, so does not absorb energy from the
Sun. Instead, solar radiation is absorbed by the surface of the Earth, which heats up, and
this heats the atmosphere around it. The temperature of the Earth’s atmosphere, and that
of other terrestrial planets, thus decreases with height. The behaviour of temperature with
height divides the atmosphere into different regions:
• Troposphere
This is the lowest layer of the atmosphere, in contact with the surface of the planet.
Temperature drops rapidly with height in the troposphere
41
Atmospheric structure
Earth
temperature [K]
temperature [K]
Mars
Altitude [km]
Venus
temperature [K]
Troposphere: T drops dramatically with height
Mesosphere: T drops, but more gently
Figure 8.1: Temperature vs. height for Venus, Earth and Mars, showing the separate regions of the
Thermosphere: T rises; large day-night variations
atmosphere. Only Earth has a stratosphere as only Earth has Ozone in its atmosphere to absorb UV
Stratosphere:
T rises
due to atmospheric absorption (O3mostly); unique to Earth
radiation.
From McBride
& Gilmour.
• Stratosphere
Only the Earth has a stratosphere, characterised by temperature rising with height. This
is a result of ozone molecules absorbing ultra-violet radiation from the Sun, leading to
an injection of energy, and thus heat, in this layer. Other planets, such as Mars or
Venus, do not have any ozone in their atmospheres, and thus lack a stratosphere
• Mesosphere
Temperature falls with height in the mesosphere, though at a slower rate than in the
troposphere
• Thermosphere
Temperature rises with height in the thermosphere as a result of several energy injection processes, including the absorption of extreme-UV photons from the Sun, and
interactions with the Solar Wind.
8.5
Thermal Escape
If gravity keeps an atmosphere around a planet, how do low mass planets like Mercury and
Mars, lose their atmospheres?
42
Sun Stars and Planets 2012-13
Figure 8.2: The shape of the Maxwellian velocity distribution for three different temperatures. Note
that it has a tail to high velocities. These high velocity particles are those most likely to escape from
a planetary atmosphere. From Tanner McCarron and Weston McCarron.
For an atom or molecule of a gas to be able to escape from a planet, it must have a velocity
greater than the escape velocity of the planet. Escape velocity is such that the kinetic energy
of the particle is equal to the change in potential energy required to climb out of the planet’s
potential well ie. ∆P E = KE. This is given by:
r
1
2GM
GM m
2
mvesc =
⇒ vesc =
(8.6)
2
R
R
where M is the mass of the planet and R is its radius. Note that escape velocity is independent
of the mass of the object trying to escape the planet’s gravity well.
The velocity distribution of particles in a gas at a temperature T is given by the Maxwell
distribution:
4 m 32 2 − mv2
P (v)dv = √
v e 2kT dv
(8.7)
π 2kT
where m is the mass of the particle, T the temperature, v the velocity and P (v) is the
probability of that velocity. This distribution has a high velocity tail, shown in Fig. 8.2.
These are the particles most likely to escape from a planetary atmosphere.
The most probable and rms velocities in a Maxwellian are:
vp =
r
2kT
;
m
vrms =
r
3kT
m
(8.8)
e velocity and thermal velocity
43
eep its
e:
ity has to
y ~ factor 6)
cape
can sustain
spheres;
on can keep
eir
es
rs can
O2 and N2 in
pheres
Figure 8.3: Escape velocity plotted against surface temperature for a number of planets and other
Solar System bodies.
We can set a rough requirement for atmospheric retention, for example, that the most probable
velocity should be less than a tenth the escape velocity, ie. vp < (1/10)vesc . A more accurate
formula, called the Jeans Escape Fraction, gives the rate of particles escaping by thermal
evaporation:
n √
vesc 2
−λ
φJ = 2 π(λ + 1)e
; where λ =
(8.9)
vp
vp
where n is the number density of gas particles.
One common factor in all these equations is that, at a given Temperature, lighter particles
will have a higher velocity. We would thus expect that lighter species, such as Hydrogen and
Helium, will be more likely to escape, and this is in fact what we find in Figure 8.3. The gas
giants, Jupiter, Saturn, Neptune and Uranus, are all massive enough to retain hydrogen and
helium, the most abundant elements in the universe. Earth and Triton can keep water in their
atmospheres, while Mars, Venus and Titan can retain carbon dioxide. Mercury and the Moon
cannot even retain carbon dioxide, and thus have absent or extremely thin atmospheres.
8.6
Current Atmospheric Composition
The current atmospheric composition of the terrestrial planets with appreciable atmosphere
is shown in Fig 8.4. The atmospheres of Venus and Mars can be classed as ‘oxidised’ in
that there is no free oxygen. Instead, oxygen is bound into compounds, mostly CO2 but also
44
Sun Stars and Planets 2012-13
water, H2 O and sulphur dioxide, SO2 . The gas giants, as we will see in more detail later, have
atmospheres rich in hydrogen, since they are massive enough to retain this light species, and
are dominated by hydrogen, helium and compounds like methane (CH4 ), ammonia (NH3 .)
water and hydrogen sulphide (H2 S). They thus have atmospheres classed as ‘reducing’. Earth
is the only planet to have an oxidising atmosphere, containing free oxygen (and also the
compound ozone, O3 , which shields the surface from UV light and, thanks to the heating
from this absorption, a stratosphere).
The presence of free oxygen in the Earth’s atmosphere is rather special since Oxygen is a
reactive element that would usually be bound into compounds, as seen on Venus and Mars.
The lack of carbon dioxide in Earth’s atmosphere, a major constituent in the atmospheres of
Mars and Venus, is also rather odd. Oxygen was not always a major constituent of Earth’s
atmosphere. In fact, the latest estimates suggest that it was only a major constituent for the
past billion years. Oxygen, of course, is produced by photosynthesis, so the presence of life
on the Earth is the reason that we have oxygen in the atmosphere. If life were to suddenly
disappear, the oxygen levels would gradually decrease as it combines chemically with other
elements, like carbon.
The relative absence of CO2 in the Earth’s atmosphere is another issue. This is the result of
the Urey weathering process, whereby carbon dioxide dissolved in water reacts with silicates
in rocks, leading to the deposition of calcium carbonate (CaCO3 ). An example of this reaction
is:
CaSiO3 + 2CO2 + H2 O → Ca2+ + SiO2 + 2HCO−
3
→ CaCO3 + CO2 + H2 O
Ca2+ + 2HCO−
3
(8.10)
The CaCO3 produced by this reaction is dissolved in the water and eventually precipitates
out to form sedimentary rocks such as limestone and chalk. These same rocks, on Earth at
least, can also be produced by biological processes, that also serve to remove CO2 from the
atmosphere.
8.7
Origin of Atmospheres
Where did planetary atmospheres come from originally?
Any primordial atmosphere is likely to have been lost, since the planets were all very hot
after formation, the solar wind would have been quite powerful in the early Sun, and the
bulk of the content would have been the most common gases, hydrogen and helium, which
the terrestrial planets are too small to retain. Instead, what we seen now are likely to be
secondary atmospheres, the content of which will be controlled by the balance between gas
sources and gas sinks:
• Sources
45
Composition of atmospheres:
Venus
Earth
Mars
Venus & Mars: oxidised atmospheres
Figure 8.4: Current composition of terrestrial planet atmospheres (Mercury essentially has no atmoEarth:sphere).
ratherNote
special!
Most
of CO
fixed
rocks and in oceans. O2
that both
on Venus
and2Mars
COin
2 dominates.
consequence of photosynthesis at later stage: oxidising
atmosphere
◦ Outgassing from planetary interiors eg. volcanoes
◦ Evaporation and sublimation of material on the surface (eg. water, solid CO2 )
◦ Bombardment by bodies rich in volatiles
• Sinks
◦ Condensation and chemical reactions (temporary)
◦ Stripping by the solar wind
◦ Impacts
◦ Thermal escape
The key difference between Earth and Venus, which are otherwise quite similar planets, may
be that Venus never acquired a significant amount of water - forming from planetesimals that
lacked it, missing out on bombardment by material rich in water ice, or because it lost its
water through photoionisation through being closer to the Sun, with the resulting hydrogen
being lost to thermal escape. A lack of water would have meant that the Urey weathering
process was never able to leach CO2 out of the atmosphere. Outgassing from the core would
then not have been counteracted by chemical reactions, and so the thick atmosphere could
build up and make Venus the unpleasant place it is today. Mars, conversely, had too little
outgassing to counteract the increased rate of thermal escape due to its lower mass, at least
in recent times, and thus lost its atmosphere, becoming the cold, low pressure environment
currently being explored by the Mars rovers
46
Sun Stars and Planets 2012-13
Things to Remember
• That atmospheric pressure drops exponentially with height, and the definition of
atmospheric scale height
• The structure of Earth’s atmosphere, with troposphere, stratosphere, mesosphere
and thermosphere, and the way temperature varies with height in these four regions
• The atmospheric structures of Venus and Mars, and why they differ from the
Earth
• Know what escape velocity is and how to derive it
• Be able to derive a simple formula (using a constant times vp rather than the
Jeans Escape Fraction) to show whether a given molecular species can be retained by a
planet of given mass and temperature
• The consequences of this for the atmospheres of planets in the Solar System
• The compositions of the atmospheres of the terrestrial planets and how they relate to other properties & the Urey weathering process
• The sources and sinks of terrestrial planet atmospheres
Chapter 9
Gas Giants: Structure and
Atmospheres
9.1
Introduction
The planets Jupiter, Saturn, Uranus and Neptune are collectively known as the Gas Giants.
They are the four largest planets in the Solar System and have properties that are very different from the terrestrial planets. In this chapter we will examine their properties, including
their structure and atmospheres, and look at the differences between Jupiter and Saturn,
which share a number of properties, and Uranus and Neptune, which are similar to each
other but subtly different from the other gas giants. We will also examine the origins of a
feature that is common to all the gas giants, but which is most distinctive in Saturn - ring
systems.
9.2
Basic Properties of Gas Giants
The Gas Giants are much lower density than the terrestrial planets, and have much deeper
atmospheres. Saturn, for example, has a sufficiently low density that it would float on water
- if you could find a big enough bucket. This means that they cannot be dominated by the
kind of rocky and metallic material that makes up most of the structure of the terrestrial
planets. Instead, the bulk of their mass comes from light elements, and they have much
higher Hydrogen and Helium abundancies. Consideration of the mass and temperature of the
gas giants in the context of the thermal escape of gases from an atmosphere (see Fig. 8.3)
shows that Hydrogen and Helium can be retained by them.
Since the gas giants are effectively large balls of gas, it is difficult to define a ‘surface’ for
them, or to probe very far beneath the cloud layers that we see from outside, to determine
what their internal structure might be. The ‘surface’ issue is solved by arbitrarily defining
the surface of these planets to be where their atmospheric pressure equals that of Earth i.e.
47
48
Sun Stars and Planets 2012-13
the radius of these planets is defined as the radius at which P = 1 bar. What we know of
their internal structure has largely been determined by examinations of their effect on the
passage of spacecraft that fly past or orbit them, measurements of their magnetic fields, and
the combination of these results with models of their deep interior. These results indicate
that the gas giants have rocky and icy cores that are at high temperature and pressure - in
this case we use the term ice to refer to volatile materials such as H2 O, CH4 and NH3 rather
than something that is actually frozen. Observations and flypasts also reveal that the gas
giants are all somewhat flattened in shape - they are prolate, bulging a little at their equators
as a result of their high rotation speeds. All of the gas giants have magnetic fields.
The gas giant planets formed further out in the Solar System than the terrestrial planets,
and the formation process took longer, especially for the outermost planets. The increased
distance from the Sun allowed them to accrete more Hydrogen and Helium, leading to their
greater mass. Models of the early Solar System suggest that the gas giants formed somewhat
closer to the Sun than we see them now, and then proceeded to migrate outwards, clearing
the Solar System of debris as they did so. The asteroid belt and the Kuiper belt are all that
they left behind.
Examination of the temperatures of the Gas Giants compared to the energy they receive
from the Sun reveals that all except Uranus are emitting excess heat ie. they are warmer
than the energy they receive from the Sun would suggest. The origin of this excess heat will
be discussed below.
9.3
The Internal Structure of Jupiter and Saturn
The internal structure of the two largest gas giants is shown in Fig 9.1. Their cores are
thought to be a mixture of rock and ices (ie. volatiles), surrounded by a layer of ices. This
material is at temperatures and pressures of up to ∼16000K and 50 Mbar (ie. 50 million Earth
atmospheres) in Jupiter, and ∼10000K and 18 Mbar in Saturn. The rocky/icy cores in both
planets have masses of about 10 Earth masses in total. There may be further differentiation
in the cores, leading to a metallic iron centre, as in Earth, but we do not have sufficient data
to be sure of this.
Outside the rocky/icy cores is a region made up of helium and metallic hydrogen. The latter
is a form of hydrogen that only arises under intense pressure, where the hydrogen nuclei are
pressed together so hard that their electrons become delocalised from their parent nuclei,
and form a fermi gas of free electrons that can flow throughout the volume of the metallic
hydrogen. This material is conductive and liquid at the temperatures and pressures prevalent
in the cores of these planets. You can think of this material as being somewhat like mercury
at room temperature and pressure.
Further out form the centre, the pressure subsides to values below 2Mbar, where hydrogen
returns to its more familiar molecular form as H2 . This is a gradual process so there is
no sharp boundary between metallic and molecular Hydrogen layers. Further out still, a
similar smooth transition occurs between liquid and gas. While hydrogen and helium are the
49
Figure 9.1: The internal structure of Jupiter and Saturn (from Rothery, McBride & Gilmour).
dominant materials in these outer laters, they are also mixed with other icy material and
some small amount of rocky material.
9.4
Excess Heat in Jupiter and Saturn
Jupiter and Saturn are both warmer than the energy they receive from the Sun would suggest.
This is known as having excess heat. The origin for this excess is probably different for each
planet.
• Jupiter
There are three likely explanations for excess heat in the case of Jupiter. Firstly, as
the largest planet in the Solar System, it may still be radiating away the residual heat
from its formation - the cooling rate for this primordial heating, as for all heat loss in
planets, goes as the inverse of the radius. Secondly, there is the possibility that Jupiter
is still slowly contracting, converting potential energy to thermal energy as it does so.
Finally, and uniquely for Jupiter in the Solar System, there is the possibility that there
may be a low rate of deuterium fusion in the hottest densest regions.
• Saturn
Saturn is too small to have significant residual heat from its formation. Instead, the best
idea for how it generates its excess heat is that it comes form the separation of Helium
50
Sun Stars and Planets 2012-13
Figure 9.2: The internal structure of Uranus and Neptune (from Rothery, McBride & Gilmour).
from Hydrogen in the metallic Hydrogen layer. The Helium then rains downwards,
releasing potential energy as it does so. This process cannot work in Jupiter as the
metallic hydrogen layer there is hotter, allowing helium to be dissolved in it, and stirred
by convection, keeping the materials well mixed.
9.5
The Internal Structure of Uranus and Neptune
The internal structure of the smaller two gas giants, Uranus and Neptune, is shown in Fig.
9.2. There are some similarities between them and Saturn and Jupiter, but also some striking
differences. The first difference is that Uranus and Neptune have less Hydrogen and Helium
than the larger gas giants. This leads to the second key difference, which is that they are too
small to be able to produce the conditions necessary for the formation of metallic hydrogen.
The two outer gas giants have more volatiles in them than hydrogen or helium - roughly 20%
of their mass comes from these gases while they account for about 90% of the mass of Jupiter
and Saturn. Some people therefore classify them as ‘ice giants’, rather than the more generic
gas giants.
Apart from the absence of metallic hydrogen and the reduced amount of H and He, their
structure is broadly similar to that of the other two gas giants, with a rocky and icy core,
and inner region of icy material, and then an outer region of Hydrogen and Helium.
51
Figure 9.3: The atmospheric structure of Jupiter and Saturn (from Rothery, McBride & Gilmour).
Neptune is found to produce excess heat. This is likely a result of continuing differentiation in
its internal structure, with denser material falling towards the centre and releasing potential
energy as heat. Uranus, in contrast, releases no excess heat. This is a puzzle since the two
planets are very similar. Where one has an internal heat source, one would thus expect the
other to have one as well. The only significant difference between the two is that the orbital
axis of Uranus is pointed towards the Sun. This leads to a very different distribution of heat
within its atmosphere and could, in principle, lead to the disruption of the convection flows
that would otherwise allow internally generated heat to reach the surface and be radiated
away. Further research is needed to determine whether this explanation is correct.
9.6
Gas Giant Atmospheres
When we observe a gas giant planet, we are looking at their atmospheres not their surfaces.
These have many colours and structures in them, arising from the various processes, chemical,
physical and meteorological that drive them. The temperature profile of the atmospheres show
similarities to some aspects of the terrestrial planets, with a troposphere, a convective layer,
where the temperature falls with height, and then a thermosphere where the temperature
rises with height. Jupiter and Saturn (see Fig. 9.3) have highly reflective clouds in their
topmost layers, with the constituents of lower layers not yet fully identified. The colouring in
52
Sun Stars and Planets 2012-13
Figure 9.4: The atmospheric structure of Uranus and Neptune (from Rothery, McBride & Gilmour).
the atmospheres is not yet fully understood, but is likely due to trace elements, including, in
the case of Jupiter, sulphur.
A similar division between troposphere and thermosphere is seen in the atmospheres of Uranus
and Neptune, but the temperature increase in the atmosphere of Uranus with height in the
thermosphere is very slow, indicating that there is little or no convection and that energy
transport is very inefficient. This relates to the issue of there being no excess heat detected
in Uranus, as discussed above. Methane in the upper layers of both these planets give them
their bluish colour.
All the gas giants have banded structures in their atmospheres which are related to variations
in wind speeds, with different bands traveling at different speeds, and with turbulence occurring at the interfaces between different bands. The differing colours are related to different
materials, with dark bands, called belts, coming from rising material and light bands, called
zones, from sinking material. This can be explained in two ways - either the planets can be
seen as a series of coaxial rotating cylinders, or as a number of convection cells.
Long duration weather systems also appear within this banded structure, the most obvious
of which is the Great Red Spot on Jupiter, a 14000 x 26000 km storm that has been raging
for at least 170 years. Smaller storms have been seen on the other gas giant planets, but none
as persistent as this.
53
2R
r
M
m
m
d
Figure 9.5: Diagram showing two self gravitating bodies of mass m close to a larger body of mass
M.
9.7
Ring Systems
All of the gas giant planets have ring systems. Saturn has the most obvious, partly because its
ring material has a high ice content and is thus highly reflective, but rings have been detected
for all of the others. The rings are composed of orbiting debris, with particle sizes ranging
from 1cm to 5m. Ring systems have a wide range of structures. In Saturn we see gaps and
divisions in the rings due to small moons clearing their orbit, moons that shepherd material,
and to the effects of orbital resonance (see later). Formation scenarios for rings include the
idea of satellites shattered by an impact, that they are made up of material left over from the
formation of the solar system that never coalesced to form a planet, and the suggestion that
they are the remains of a moon that migrated towards the plant and was then disrupted.
Central to all these ideas is the result that the rings of all the gas giants are within their
Roche Limit. This is the radius around a planet within which an object that would otherwise
be held together by its self-gravity, will be torn apart by tidal forces.
9.7.1
Derivation of the Roche Limit
We have two bodies of mass m a distance r apart, lying a distance d from a body of larger
mass M and radius R, where M m and d r.
The tidal force from the body of mass M that is working to pull the mass m bodies apart is
the difference in gravitational attraction on them. Thus:
Ft = ∆F (d, d + r) =
GM m
GM m
−
2
d
(d + r)2
Multiplying out the term on the right we get:
2
(d + r)2 − d2
d + 2dr + r2 − d2
Ft = GM m
=
GM
m
d2 (d + r)2
d2 (d2 + 2dr + r2 )
(9.1)
(9.2)
54
Sun Stars and Planets 2012-13
we know that r is small compared to d, so we can eliminate terms in r2 and, in the denominator, terms in r as well, giving:
r
Ft = 2GM m 3
(9.3)
d
The Roche Limit is then defined as the distance at which this tidal force from the large mass
M is balanced by the gravitational attraction between the two smaller masses m. Thus:
Gmm
GM m
=2
r ⇒ dR =
2
r
d3
M
2
m
1/3
.r
(9.4)
The Roche Limit is usually expressed in terms of densities, with ρP for the planet and ρs for
the satellite, and using R for the radius of the planet and rs for the radius of the satellite.
Looking at things this way we find that:
ρP =
M
4
3
3 πRP
;
ρs =
m
(9.5)
4
3
3 πrs
Substituting this into equation 9.4 we get:
1/3
dR = 2
Rp3 ρP
rs3 ρs
!1/3
1/3
rs ⇒ dR = 2
ρp
ρs
1/3
R
(9.6)
A more detailed calculation by Edouard Roche in 1848 leads to the actual value of Roche
Limit:
1/3
ρp
dR = 2.456
R
(9.7)
ρs
Things to Remember
• The internal structures of the gas giants and the reasons for differences between them
• The atmospheric structures of gas giant planets and their constituents
• The properties of gas giant ring systems
• The simple derivation of the Roche Limit that leads to equation 9.6
• How to apply the Roche Limit to moons and ring systems
Chapter 10
Moons: Formation and Properties
10.1
Introduction
Nearly all of the planets in the Solar System have moons, with some of these moons, such as
Titan or Ganymede, being larger than the planet Mercury. Even minor planets, such as Pluto
and Eris, have their own moons. Eris has one known moon, called Dysnomia, while Pluto has
five detected moons - Charon, Hydra, Nix and the recently discovered, and recently named
Styx and Kerberos. The presence of these moons around Pluto, and the likelihood that there
are other, smaller, and thus harder to detect, companions, is causing difficulties in plans for
the NASA New Horizons flyby mission to Pluto.
The number and range of properties of moons around planets is summarised in Table 10.1.
The history and formation of moons can provide extra information about the formation of
the solar system and about the planets around which they orbit.
10.2
Orbits and Masses
The mass of a planet can be determined if you know the details of the orbit of a moon around
it. If we make the simplifying assumption that the orbit of a moon around a planet is circular,
and that the planet’s mass is much greater than that of the moon, then we can use equation
5.14 to derive the planet’s mass. All we need to know is the distance between the planet and
the moon, and the moon’s orbital period. These can all be determined observationally. Thus:
aω 2 =
GM
ω 2 a3
⇒M =
2
a
G
(10.1)
For the Earth-Moon system the parameters are: a = 386 × 106 m while the orbital period
of the Moon is 27.3 days, which converts to ω = 2π/2.36 × 106 s. Put these numbers into
equation 10.1 and we get the mass of the Earth = 6 × 1024 kg.
55
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Sun Stars and Planets 2012-13
Planet
Mercury
Venus
Earth
Mars
Moons
0
0
1
2
Jupiter
8 + >50 captured
Saturn
23 +>30 captured
Uranus
Neptune
27+
13+
Comments
The Moon, 1700km radius
Phobos and Deimos, both irregular and small
(22 and 15km diameter respectively)
Ganymede (2600 km), Callisto (2400 km)
Io (1800 km), Europa (1600 km)
Titan (2600 km) earthlike atmosphere!
Rhea (800 km), Iapetus (700 km)
Enceladus (250 km; water)
7 large + other Trojans, co-orbiting moons, rings...
Titania (800 km), Oberon (800 km)
Triton (1400 km)
Table 10.1: Summary of the number of moons around each planet in the Solar System, with comments
on some of their properties.
Exercise: Calculate the mass of Mars and Jupiter given that Phobos has an orbital diameter
of 9.4 × 103 km and an orbital period of 0.32 days, and that Callisto has an orbital diameter
of 1883 × 103 km and an orbital period of 16.7 days.
10.3
Formation
One of the reasons why the moons of the Solar System have such a wide range of properties is
that they have formed in a variety of ways. The three principle ways that moons are formed
are:
• Condensation
In a process similar to that which formed the plants around the Sun, smaller bodies are
thought to be able to form in orbit around a larger planet. Moons formed through this
process will be prograde ie. orbiting the planet in the same direction that the planet
rotates, and, since they formed in relatively dense material around a forming planet,
from a proto-satellite condensation disk, they will be relatively higher mass objects.
The four Galilean moons of Jupiter, Io, Europa, Ganymede and Callisto, are likely to
have formed through condensation.
• Capture
Gravitational interactions between planets and smaller, free floating, bodies can lead
to the smaller bodies becoming gravitationally bound to planets. Such captured moons
may have retrograde orbits. Examples of these include the moons of Mars, Phobos and
Deimos. Larger retrograde moons, such as Triton, are likely to be protoplanetary cores
that were captured by a larger planet during the later stages of planet formation.
57
• Collision and fragmentation
Small prograde moons, especially those close to a planet, are likely to have been formed
by collisions between moons that were once larger. These larger bodies are then split
into smaller, separate moons. This process takes place predominantly near to a planet
since the orbital velocities will be higher and thus the collisions more energetic, leading
to increased chances of fragmentation during a collision. Saturn’s moon Hyperion, for
example, is likely to have formed this way.
Smaller moons in general, whether formed through fragmentation or capture, are likely
to have irregular shapes since their gravitational fields are too weak to produce a spherical surface.
10.3.1
The Moon
Our own Moon is somewhat of an exception among this range of formation methods, since it
appears to have formed as the result of a giant impact between the young Earth and a Mars
size body roughly 50-100 Myrs after the formation of the proto-Earth. Denser material in the
impactor would have remained with the young Earth, while the Moon subsequently formed
from the lighter ejecta that resulted from the collision. This explains why the Moon has a
smaller nickel-iron core, relative to its size, than other terrestrial-type bodies. The energy of
this collision would have re-melted the surfaces of both bodies.
10.4
Tidal Forces and Tidal Heating
We have already looked at one aspect of tidal forces when we examined the Roche limit, but
tidal effects also apply that are less dramatic than the production of ring systems, through
objects being broken apart. Where bodies orbit each other there will be a difference in forces
from one side of the object to the other. As discussed above (equation 9.3), the tidal force
on a body of mass m a distance r from the centre of mass due to its orbit a distance d from
a larger body of mass M is:
r
Ft = 2GM m 3
(10.2)
d
If some part of the object is liquid, then this fluid will flow in response to the tidal force and
you get what we see in the seas of Earth - tides.
Exercise: Using the orbital parameters of the Earth, Sun and Moon, compare the tidal
forces on a kg of water on the surface of the Earth due to the Moon and due to the Sun. You
will find they are of comparable magnitude. This is what gives rise to ‘spring tides’ when
the Moon is full or new, and thus aligned with the Sun. You might also want to compare
the magnitude of the tidal forces on Earth with the tidal forces experienced by Io as it orbits
around Jupiter.
These forces will also result in the deformation of the moon or planet subject to the tidal
forces. This leads to heating in just the same way that repeatedly squashing a tennis ball
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Sun Stars and Planets 2012-13
produces heating. In the case of the Earth and Moon there isn’t very much tidal heating, but
in other systems around massive planets, like Jupiter, there can be a considerable amount of
heat generated. This is what powers the volcanoes of Io, leads to the water geysers found
on Saturn’s moon Enceladus, and which may maintain a liquid ocean beneath the surface
of Europa. Indeed water geysers may have recently been found on Europa, similar to those
already found on Enceladus.
10.5
Tidal Locking, Libration and Circularisation
A rotating moon (or planet) will, in general, tend to pull the bulge produced by tidal forces
away from perfect alignment with the centre of mass of the body around which it is orbiting,
producing a force that acts to bring the tidal bulge back into alignment. Over time, this force
will act to synchronise the orbital period and rotational period of the objects. The end point
of this effect, known as tidal locking, can be seen in the Earth-Moon system, where the Moon
always points the same face at the Earth. A one-to-one ratio in this kind of tidal locking
is not always the end point. Mercury, for example, has 1.5 rotation periods for each orbital
period.
Our discussion of tidal effects to this point has assumed that the orbits are circular, but this
is not, in general, the case. Instead, most orbits are elliptical to some extent, with the orbital
speed varying in accordance with Kepler’s Laws. The result of this is that the face of the
moon that points to the planet at the closest (pericentre) and furthest (apocentre) points of
the orbit, wobbles from side to side during the rest of the orbit. (see Fig 10.1). Over time,
the forces that result from this libration will tend to circularise the orbit.
10.6
Orbital Resonances
One object orbiting around another in isolation from everything else is a simple physical system, where the orbital parameters can be calculated analytically. However, in the real world,
there are always other bodies involved which can add complexity to the orbital mechanics. In
the case of a planet with many moons, the orbit of one moon can be affected by contributions
from other moons. In the Solar System more broadly, as we will see in the next lecture,
planets like Jupiter or the Earth can influence the orbits of other objects around the Sun.
In many circumstances, the gravitational interactions between orbiting bodies will occur at
random intervals and will average out over time. However, if an orbital configuration repeats regularly and with a period that is a small integer number of orbits, then the small
perturbations from these interactions will not average out. This is a process known as orbital
resonance and it occurs quite often in complex systems of orbiting bodies. Such resonances
are described in terms of the number of orbits of the inner body to the number of orbits of
the outer body (or bodies, for more complex interactions) eg. the Galilean moon Io is in a
4:2:1 resonance with the moons Europa and Ganymede. So Io orbits Jupiter 4 times for every
2 orbits that Europa makes and for every one orbit that Ganymede makes.
Libration and circularisation
59
On elliptical orbit, object moves
faster near pericentre
But rotation is constant, so point on
object appears to wander back
and forth over a rotation (see
red dot shown here)
Example: Moon (shown here) only
roughly faces Earth, can see
more than 50% of surface as
the spin lags and leads the
orbital motion
Non-balancing forces will act to
circularise the orbit
Exception: Io… it is in 4:2:1 resonance with Europa and
http://www.astrosurf.com/cidadao/moon_obs_04.htm
Ganymede tidal heating
and http://antwrp.gsfc.nasa.gov/apod/ap070902.html
volcanism
Figure 10.1: How the face of a moon on an elliptical orbit changes
during the course of that orbit,
producing the effect known as libration. Taken from www.astrosurf.com
There are two possible effects from such an orbital resonance:
• The body (or bodies) are locked into its orbit, and cannot, for example, move outwards
as a response to tidal forces. This effect happens typically when the orbits of the objects
concerned never approach each other very closely, and are called stable resonances.
This is the situation for Io in its orbit around Jupiter. In isolation, the tidal effects that
squeeze Io as it orbits around Jupiter would have led it to move outwards, in the same
way that the Earth’s Moon has moved away over millions of years. Orbital angular
momentum is thus transferred to the Moon from the Earth, and the Moon climbs out
of the Earth’s gravity well. This cannot happen to Io because of the orbital resonance
it is in with Ganymede and Europa. The energy that would otherwise move Io up
the gravity well from Jupiter instead goes into tidal heating of Io’s interior, leading
to the rampant volcanism that we can see on its surface. Europa, too, is involved
with this orbital resonance and, like Io, is also locked into its orbit. It is further from
Jupiter, so there is less tidal heating as a result, but this is still enough to melt some
of Europa’s icy interior, leading to a layer of liquid water beneath its surface, and
producing cryovolcanism. Saturn’s moon Enceladus is in a 2:1 orbital resonance with
the moon Dione. Tidal heating here is likely to be responsible for the cryovolcanism
that produces the geysers seen on this moon.
• The body gets accelerated or decelerated in its orbit until it is no longer in resonance.
In effect this means that the orbital configuration affected is cleared of objects subject
to the resonance. These are called unstable resonances.
Observation of the rings of Saturn show a variety of gaps. The most obvious of these,
observable from Earth and discovered originally in 1675 by Giovanni Cassini and named
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Sun Stars and Planets 2012-13
after him, is the Cassini Division. This gap in the rings is produced by a 2:1 orbital
resonance between any particles in this ring and the moon Mimas.
Orbital resonances and other effects in fact make the structure of Saturn’s rings very
complex, with a wide variety of different gaps, sub-rings, and sub-structures such as
spokes, braids and shepherd moons.
Orbital resonances are not restricted to the orbits of moons around planets, but also apply
in the broader Solar System. They produce the Kirkwood Gaps seen in the distribution of
orbits in the asteroid belt, and, in the outer Solar System, lead to the class of objects called
Plutinos (see the next chapter for more details).
Orbital interactions and resonances can also lead to quite bizarre non-circular and nonelliptical orbits. The Saturnian moons Janus and Epimetheus, for example, have ‘bean’
shaped orbits around Saturn. This kind of thing is not restricted to the outer Solar System.
A near Earth asteroid, Cruithne, lies in a 1:1 orbital resonance between the Earth and the
Sun, leading to a strange ‘bean’ shaped orbit that reaches outwards to the orbit of Mars and
inwards beyond the orbit of Venus.
Things to Remember
• Use of the orbital period of an orbiting body to calculate the mass of its parent planet
• The names and general properties of the most famous moons in the Solar System (the Moon, Mars’ moons, the Galilean moons of juliter, Titan, Enceladus, Triton)
• The main formation mechanisms for moons
• Tidal forces, tidal heating, tidal locking and libration & circularisation
• Orbital resonances in moon systems and how this can lead to tidal heating, especially in the example of Io
Chapter 11
Small Bodies: Comets, Asteroids
and the Outer Solar System
11.1
Introduction
A full analysis of the orbital dynamics of the Solar System shows that the regions between
most of the planets lack stable orbits because of gravitational resonance effects. The only
regions in the Solar System where this is not the case are between Mars and Jupiter, and
outside the orbit of Neptune. Unsurprisingly, we find plentiful small objects orbiting in these
regions, forming the Asteroid Belt, between Mars and Jupiter, and the Kuiper Belt, beyond
the orbit of Neptune. Further out, there is also the Oort cloud, where proto-comets kicked
out of the young Solar System, live. There are also other, shorter lived, populations of
objects, such as the Centaurs, and comets, which occasionally visit the inner Solar System.
All together, these objects form the small bodies and minor planets of the Solar System.
While they are not a significant constituent of the Solar System by mass, small bodies have
not been through the reprocessing involved in planet formation that the rocks and gases in
larger bodies have endured. The small bodies can thus provide us with clues about what
material in the early Solar System might have been like. Asteroids also occasionally collide
with planets, including the Earth, so keeping an eye on them is not only useful scientifically,
it may provide early warning for major disasters.
11.2
Asteroids
There are approximately 100,000 asteroids currently known in the main asteroid belt between
Mars and Jupiter, and about 30,000 of these have well determined orbits. The total mass
of these asteroids only amounts to about 0.001 Earth masses. The largest asteroid in the
main belt is Ceres, with a diameter of 900km. The vast majority are much smaller than this,
61
Kirkwood gaps
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Plot of the number of asteroids vs period
 Kirkwood gaps
(discovered 1857)
These correspond to orbital
resonances with Jupiter.
Most prominent at
2:1, 3:1, 5:2, 7:3
[Remember
# orbits (inner body) : # orbits (outer body)]
Nature (2009) 457
but additional gaps, not explained
by obvious orbital resonances.
New studies (e.g., Minton &
Malhotra 2009, Nature 457,
1109) suggest it is a result of
orbital resonances + planetary
migration.
Figure 11.1: The distribution of main belt asteroids as a function of their orbital radius. The gaps in
the distribution are known as the Kirkwood Gaps. Also noted are the orbital resonances with Jupiter,
which coincide with most, but not all, of the Kirkwood Gaps. From Minton & Malhotra, Nature
(2009), 457, 1109.
with their size distribution mirroring the size distribution of impact craters on objects like
the Moon and Mercury. This isn’t surprising since the impacts were produced by asteroids.
While most asteroids lie between Mars and Jupiter, some exist in the inner Solar System.
Near Earth Asteroids are bodies that come close to the Earth. About 7000 of these are
currently known. Potentially Hazardous Asteroids (PHAs) are those that come very close to
Earth and might collide with it at some point in the future. About 1000 of these are currently
known.
Most asteroids are too small to have gone through any surface differentiation themselves.
Rather then being solid bodies, like Earth or Mercury, they are thought to be ‘rubble piles’
made up of lots of separate sub-fragments bound together by mutual gravity. The different
fragments can move relative to one another, leading to a somewhat ‘molten’ appearance, with
finer, dusty regolith material settling to lower points in the local gravitational potential, and
larger fragments moving upwards in a manner similar to the motion of brazil nuts in museli
when it is shaken. Data taken by the Hyabusa spacecraft on its mission to the near-Earth
asteroid Itokawa are consistent with this idea.
The distribution of Main belt asteroids as a function of their semi-major axis (in AU) is
shown in Fig. 11.1. As you can see, the distribution is not uniform, but is characterised by
63
several distinct gaps. These were discovered in 1857 and are still known as Kirkwood Gaps in
honour of the astronomer who found them. Most of the Kirkwood Gaps are easily explained
by orbital resonances with Jupiter. The 2:1, 3:1, 5:2 and 7:3 resonances are clearly seen.
However, there are other gaps that do not coincide with resonances with the current orbit of
Jupiter.
The explanation for this, as proposed by Minton and Malhotra (Nature (2009), 457, 1109,
M&M) is that the giant planets went through a period of migration in the early Solar System.
M&M’s model suggests that Jupiter moved inwards by about 0.2 AU, while Saturn, Uranus
and Neptune moved outwards by 0.8, 3 and 7 AU respectively. This migration is the result
of the exchange of angular momentum between planetesimals, left over from the main phase
of planet formation, and the giant planets. They tested this idea by modelling the effects of
the known planetary orbits on an initially uniform distribution of main belt asteroids over
a period of 4 Gyr and comparing this to the same simulation, but with the added assumed
period of planetary orbit evolution. These simulations were then compared with the observed
asteroid distribution. The simulation with the planet orbit evolution was by far the better fit.
The planetary orbit evolution would also give rise to the observed late heavy bombardment
of the inner Solar System, and fits with other results.
Asteroids come in a range of classes, largely determined by their reflectance spectrum which
allows an estimation of the material on their surface. Classes include:
• C class: surface dominated by carbon (carbonaceous), with reflectance ∼5%. These are
the most common type, representing 40% at 2AU and 80% at 4AU.
• S class: surfaces dominated by silicates (stony material), with reflectance ∼16%, and
with a distinct spectral absorption signature at ∼1µm. These are the second most
common class.
• M class: these asteroids are almost entirely metal, containing Ni and Fe. They are rarer,
but have reflectance ∼15%.
• D-class: these asteroids are very dark, with reflectance only ∼3%. They are increasingly
common at greater distances from the Sun. Their surfaces may include organic material.
There are also other classes including E and P. The overall numbers of different classes,
especially the low reflectance ones, are difficult to judge since different classes are detected
with differing efficiencies, so the statistics are dominated by selection effects.
The size distribution of asteroids is a power law, with roughly equal amounts of mass in each
logarithmic mass bin, so there are many small asteroids, but only a few very large ones.
One dynamically interesting subclass of asteroid are the Trojan asteroids. They share the
same orbit as Jupiter, but lie 60 degrees ahead and 60 degrees behind the planet’s orbital
position. These points are the so-called L4 and L5 Lagrange points, where the gravitational
and centrifugal forces of two orbiting masses cancel out for a third, smaller, orbiting body.
The L4 and L5 points are stable saddle points in the gravitational potential, so that objects
that arrive there will stay there. The Trojan asteroids might have arrived at Jupiter’s L4 and
L5 points during the period in which Jupiter migrated closer to the Sun.
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Sun Stars and Planets 2012-13
Figure 11.2: The orbits of a selection of Kuiper Belt Objects compared to the orbits of Jupiter,
Saturn, Uranus, Neptune and Pluto (J, S, U, N, P). From Rothery, McBride & Gilmour.
11.3
Kuiper Belt and Trans-Neptunian Objects
The migration of the giant planets, as described above, has largely cleared small bodies from
the orbit of Jupiter to the orbit of Neptune. Beyond the orbit of Neptune, though, small
bodies can persist undisturbed. As early as the 1950s, Gerard Kuiper proposed the existence
of a belt of small bodies beyond the orbit of Neptune. Pluto, and its largest moon Charon,
were already known at this point, discovered in 1930, but it wasn’t until 1992 that any
further trans-neptunion objects (TNOs) were discovered. We now know of at least 70,000
such objects, with diameters >100km, forming what has been called the Kuiper belt, which
lies 30-50 AUs or more from the Sun (see Fig. 11.2). The total mass of objects in the Kuiper
Belt has been estimated to be ∼0.1 Earth mass, meaning that the Kuiper Belt actually
includes more mass than the ‘main’ asteroid belt between Mars and Jupiter.
Kuiper Belt objects (KBOs) are thought to be the left overs from earlier stages of the formation of the Solar System, made up of material with a high fraction of ices and volatiles.
Reflectance spectra of KBOs have a wide range of properties, which may be the result of long
term changes in their surface properties resulting from exposure to UV light from the Sun,
but also resulting from more abrupt surface changes coming from impacts between KBOs.
Our knowledge of the outer solar system is still very incomplete. The NASA New Horizons
mission will help with this when it encounters Pluto in 2015 and then moves on to study
other KBOs in 2016-2020.
65
Figure 11.3: The orbital eccentricity plotted against semi-major axis for KBOs. From Rothery,
McBride & Gilmour.
While we await the results from New Horizons, and other detailed studies using ground
and space based telescopes, the classifications of KBOs are largely based on their orbital
properties, and in particular on their orbital eccentricity and semi-major axis. When these
are plotted together you can see three separate groups of orbital characteristics which leads
to the division of KBOs into separate classes. The majority are classified as Classical KBOs,
and have low eccentricity orbits with radii of ∼44 AU. The second largest group have a range
of eccentricities and all lie at a radius close to 39.4 AU. Pluto is one of these objects, leading
them to be termed Plutinos. If the orbital period of the Plutions is compared with that of
Neptune, you find that Neptune orbits the Sun three times for every two orbits that a Plutino
makes: they are in a 3:2 orbital resonance with Neptune, keeping them in this orbital position.
Pluto is in many ways indistinguishable from the other Plutinos, a result which eventually
led to the reclassification of Pluto as a dwarf planet. The final group of KBOs have high
eccentricities and large semi-major axis. They are classified as Scattered Disk Objects and
are likely the source of short period comets.
An additional class of small solar system body that is likely associated with KBOs are the socalled Centaurs. These are objects whose orbits cross the orbits of one or more major planet.
Such orbits will not be long lasting because they will eventually encounter the gravitational
field of a major planet and be captured or scattered into a different orbit. They may even
hit one of the giant planets. An example of this was the impact of Comet Showmakey-Levy
9 with Jupiter in 1994. The discovery of Centaurs predates that of KBOs other than Pluto
by a number of years. The first Centaur, called Chiron, was found in 1977. The fact that it
was in an orbit that was not stable in the long term hinted at the existence of a larger body
of similar objects in more stable orbits that would be able to feed Centaurs into the Solar
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Sun Stars and Planets 2012-13
System. The reservoir for the Centaurs, and for short period comets, is the Kuiper Belt,
which was discovered 15 years later.
11.4
Comets
The final class of small body in the Solar System that we will discuss are perhaps the most
spectacular: comets.
Comets are objects on highly eccentric orbits that come from the outer to the inner parts of
the Solar System. They are objects rich in ices and volatiles which are released when they
heat up in the inner parts of the system, giving rise to the spectacular tails (one of dust one
of ionised material, which interact differently with the solar wind). They have very short
lives, only a few 104 years before mass loss and dynamical interactions with planets lead to
their destruction. Their internal structure is thought to resemble a ‘dirty snowball’, where
dust and rock is mixed with ices, including water ice and other frozen volatiles. The dirty
smowball forms the comet’s nucleus, which is small (10-20km) and very porous, When they
near the Sun they heat up and volatiles boil off, leading to the familiar shape of these objects.
There are two classes of comet based on their orbital period - short period (<200 years) and
long period. The short period comets, like Halley, have low orbital inclination and are usually
prograde. These comets are thought to originate in the Kuiper Belt. Long period comets have
much higher orbital inclination and are as often retrograde as prograde. They are thought
to come from the Oort cloud, a spherical reservoir of comets believed to lie at much greater
distances from the Sun than the Kuiper Belt, out to as far as 50000 AU, about a quarter of
the way to the nearest star. So far, no definitive detection of an object in the Oort Cloud has
been made. Instead, its existence is currently inferred from the presence of the long period
comets, in much the same way that the existence of the Kuiper Belt was once inferred from
the discovery of Centaurs. Sedna, a TNO with a very eccentric orbit with an aphelion of
∼1000 AU is our best current candidate for a member of the Oort cloud.
Towards the end of 2014 the Rosetta mission will rendezvous with the periodic comet 67P/ChuryumovGerasim
and send a lander down to the surface in November 2014. Over the following year, the spacecraft will follow the comet as it falls into the inner Solar System. Depending on the results
of this mission, by the time this lecture course is given next year the section on comets might
be very different.
67
Things to Remember
• The basic parameters of the asteroid belt and the classes & constituents of asteroids
• The Kirkwood gaps and their origin in orbital resonances.
oids.
The Trojan aster-
• Properties of Kuiper Belt objects. The Centaurs.
• The internal structure and origin of comets, including the orbital properties of
long & short period comets
• How these relate to the Kuiper Belt and Oort Cloud
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Sun Stars and Planets 2012-13
Chapter 12
Detecting Exoplanets
12.1
Introduction
We have spent the last 8 chapters, and 8 lectures, looking at the properties of objects in our
own Solar System. Not so long ago, that would be where things stopped, since we knew of
no other planetary bodies in the Universe. Whole theories of planet and planetary system
formation were developed on the basis of the 8 planets and many minor bodies of our own
Solar System, but there was lack of other places where these models could be tested.
Over the last 15 years, though, there has been a revolution in our understanding of planetary
systems, resulting from a series of technological breakthroughs that have allowed planets
in other solar systems - exoplanets - to be discovered in ever greater numbers. Specific
observatories on the ground and in space are completely dedicated to planet searches, and
over a thousand planets in other systems are now know 1 iscoveries continue at a fast pace last years lecture notes said ’nearly a thousand’. More planet discoveries are announced every
day, so some of the raw numbers in these notes will already be out of date. You can keep
track of the latest results through dedicated websites such as exoplanet.eu.
12.2
Units
For the rest of this course we will be looking at objects far away from the Solar System,
and will thus have to use astronomer’s units. Many of these are based on scalings to known
objects eg. the mass of the Sun (M ), the astronomical unit, the Solar luminosity (L ), but
there are other more specialised units that we will use.
The first, and most important, of these is the parsec. This is a distance of 3.26 light years,
3.08×1016 m, or 2.1 ×105 AU. The parsec is derived from the distance at which an object
must be for it to have a proper motion on the sky of 1 arcsecond when the Earth moves by 1
1
D
69
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Sun Stars and Planets 2012-13
Figure 12.1: The number of exoplanets discovered each year. Plot produced using the tools and
database available at exoplanet.eu
71
Figure 12.2: Diagram showing how the parsec is derived
AU. Essentially this means that a parsec is the length of the adjacent side of a triangle which
has an angle of 1 arcsecond and an opposite length of 1 AU. See Fig. 12.2 for a diagram.
The other thing that we will be using that could be termed an astronomer’s unit are magnitudes. These will be used to express the brightness of stars and changes in the flux received
from them. Magnitudes where introduced to you in the first (Stars) part of this course, but
as a reminder, the difference in magnitude between two objects is given by:
F1
m1 − m2 = −2.5 log
(12.1)
F2
where m1 and m2 are the magnitudes of the two objects, and F1 and F2 are the fluxes of the
two objects. This just deals with how to compare two objects. To place magnitudes on an
absolute scale, one also has to define the flux that corresponds to zero magnitude. There are
two ways of doing this in astronomy. The traditional way has been to define the magnitude
of the star Vega (a bright A-type star) to be identically zero for all observations. These
are called Vega magnitudes. The second, is to define a specific flux density, of 3631 Jy (a
unit which will be explained shortly) to be zero magnitude for all observations. These are
called AB magnitudes. Since the star Vega has (roughly)2 ega is a good, single temperature,
black body at optical wavelengths, which led to its adoption as a standard star. However,
it was later found that it is surrounded by a dust disk, left over from its formation, which
contributes significantly in the mid- and near-IR. This causes some confusion when Vega is
used as a standard star at these wavelengths, though these considerations are not important
here. a black body spectrum, its flux varies with wavelength, so the flux corresponding to
2
V
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Sun Stars and Planets 2012-13
zero magnitudes in the vega system varies with wavelength. This is not the case for AB
magnitudes.
The Jansky (Jy) is another astronomer’s unit, but is more closely related to standard SI units.
1Jy = 10−26 Wm−2 Hz−1
12.3
(12.2)
What is a Planet Anyway?
Within our own Solar System, there is a formal definition of a planet. It is a body orbiting
the Sun, whose gravity is strong enough to make it spherical, and which has cleared its
neighbourhood of smaller bodies. The latter part of this definition, agreed by the International
Astronomical Union (IAU) in 2005, is what demoted Pluto to minor planet status.
Outside our Solar System the definition of what is a planet is less clear. The IAU definition
states that to qualify as an exoplanet a body must be orbiting a star and have a mass below the
threshold at which thermonuclear fusion of deuterium can take place. This sets the maximum
mass for a planet at ∼ 13 MJ . No consideration is given to how these bodies formed, and
the minimum mass should match the minimum mass to qualify as a planet in our own Solar
System.
Things that are not considered exoplanets in this scheme include objects above the deuterium
burning mass limit, which are defined as brown dwarfs (these are essentially failed stars) and
free floating bodies that are low enough mass to qualify as an exoplanet, but which do not
orbit around a star. These are termed sub-brown dwarfs, and recent results suggest that they
may be more numerous than stars in our galaxy.
A physical definition of a planet based on formation history and/or composition, which might
be a more scientific approach, is still lacking. The discovery of objects like Cha 110913-773444,
which appears to be a sub-brown-dwarf (ie. with mass <13 MJ ) but which has a dust disk
in which planets (moons?) might be forming, just makes the issue more complicated.
12.4
Direct Detection: How Hard Can it Be?
In the current era of space telescopes and large, 8-10m, telescopes on the ground, one might
think that directly detecting an exoplanet orbiting around another star would be easy. Unfortunately, this is far from true, mainly because of the huge contrast between the light that
comes from the star and that which is reflected from the planet, and because of the small
angular separation between any planet and its parent star.
Consider a planet with an albedo of 1, visible only because of light reflected from its parent
star, radius Rp , orbiting a distance d from a star of luminosity L∗ . The stellar flux received
by the planet will be:
L∗
F =
(12.3)
4πd2
73
The planet’s luminosity, Lp then comes from the total power it intercepts, assuming that it
has an albedo of 1:
L∗ Rp 2
2
2
πRp ⇒ Lp = F πRp =
(12.4)
4
d
If we put in numbers appropriate for Jupiter into this equation - Rp ∼ 7 × 107 m and d ∼ 5
AU - and calculate its relative luminosity to the Sun, we find:
L
LJ
=
L
4
RJ
d
2
1
1
=
×
L
4
7 × 107
5 × 1.5 × 1011
2
⇒
LJ
∼ 2 × 10−9
L
(12.5)
So, to directly detect a planet like Jupiter orbiting another star you will have to remove the
light of that star to an accuracy of about 1 part in a billion - the star outshines the planet by
that much. This is very difficult to achieve. The situation is somewhat better in the infrared,
where the Black Body spectrum of the hot star is declining, but where that of the cooler
planet is peaking, but this still requires better than 1 part in a million exclusion of stellar
light. There are ways that this can be achieved, using techniques such as choronography and
nulling interferometry, but this all means that direct detection is not an efficient way to search
for planets. However, it can be, and has been, used to follow up planets that have already
been detected by indirect methods, so as to better characterise the objects. For example,
the first direct spectrum of an exoplanet was obtained in 2010 by Bowler et al using such an
approach.
If we cannot search for exoplanets directly, how can they be found?
Fortunately there are a range of indirect methods that look for the effects of any planets that
may be present on their parent star.
12.5
Reflex Motion and Doppler Measurements
A family of detection methods are based on studying the dynamical effects of an orbiting
planet on the parent star.
Just as with binary stars, a star and planet actually orbit around a common centre of mass,
but with the planet mass much smaller than the stellar mass. Viewing this in the centre of
mass frame it looks like Fig. 12.3. From the stars part of the course we know that for binary
stars:
M1
r2
4π 2 r3
=
and M1 + M2 =
(12.6)
M2
r1
G P2
where P is the orbital period. So, for a planet of mass mp orbiting a star of much larger mass
Ms :
mp
as
4π 2 a3p
=
and Ms =
(12.7)
Ms
ap
G P2
where as and ap are the distances from the star and planet to the common centre of mass
respectively.
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Sun Stars and Planets 2012-13
a_p
x
a_s
Figure 12.3: A star orbits around the common centre of mass with radius a s, while the planet orbits
around it with a radius a p.
The first result from this analysis, as can be seen from Fig. 12.3 is that the position of a
star being orbited by a planet will appear to wobble around on the sky. While the planet
cannot be directly seen, its existence can be inferred if we can measure the star’s regular
displacement as . In fact the angular displacement β = as /d, where d is the distance to the
star from the observer, is what is usually measured. From the above we can find that
β=
mp
mp ap
as
as
but
=
⇒β=
d
Ms
ap
Ms d
(12.8)
So you get a larger, and thus more detectable, angular displacement β for large mass planets
in wide orbits around low mass stars. For our own Solar System viewed form a distance of
10 pc, you would see a displacement lower than 0.4 milliarcsec per year from the effect of
Jupiter orbiting the Sun. This is a very small angular shift, corresponding to the width of a
finger at 5000 km, so it is not a particularly viable method of planet detection.
EXERCISE: Calculate the displacement you get from the Earth’s orbit around the Sun,
when viewed from 10 pc.
However, an alternative method based on this same idea comes from looking at the motion
of the star along the line of sight, rather than in the plane of the sky. What can be measured
here are changes in the velocity of the star in the line of sight, which can be done by looking
at Doppler shifts from spectral lines.
Recalling the results for binary stars:
P
M23 sin3 (i)
= v13
2
M
2πG
(12.9)
75
where i is the inclination angle between the orbital plane and the line of sight, and P is the
orbital period.
For planets M2 = mp , M = Ms + mp = Ms and v1 = vs , so:
mp sin(i) =
Ms2 P
2πG
1/3
vs
(12.10)
Without knowing the inclination angle i, this method only allows you to calculate the minimum mass for a planet.
The largest values of vs come for large mass planets orbiting low mass stars with short orbital
periods.
If you were observing along the plane of the ecliptic, the velocity shifts from the Sun-Jupiter
system amount to 12 m/s, and from the Sun-Earth system amount to 0.1 m/s.Velocity shifts
as low as 0.5 m/s have been measured, and the first clear detections of exoplanets around
main sequence stars were obtained using this method.
EXERCISE: Calculate the velocity shift you would get if Jupiter was orbiting the Sun with
an orbital period of 88 days (ie. in the orbit of Mercury) rather than its actual orbital period
of 11.9 years.
12.6
Planetary Transit Searches
If a planetary system’s orbital plane lies along our line of sight, planets will from time to time
pass in front of their star, absorbing some of the light from the star that would otherwise
reach us. This kind of thing can be seen in our own Solar System where Venus or Mercury
can be seen to pass in front of the Sun. The last transit of Venus was in June 2012. Planetary
transits will cause a small, but potentially measurable, dip in the brightness of a distant star
observed from Earth (see Fig. 12.4).
What flux decrease will a planetary transit produce?
If the uneclipsed flux of the star is Fs , the eclipsed flux Ft , the flux of the planet is Fp , the
radius of the star is Rs and of the planet is Rp then:
Ft = Fs −
Rp
Rs
2
Fs + Fp
(12.11)
Since Fp << Fs we get:
Ft = Fs
1−
Rp
Rs
2 !
and the flux decrease ∆F = Fs − Tt =
Rp
Rs
2
Fs
(12.12)
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parameters:
* orbital period
* planetary radius RP/RS
* planetary mass (need MS)
* inclination of orbit
Sun Stars and Planets 2012-13
flux decrease ≈ (RP/RS)2
Sensitivity/bias:
• easier to detect larger
planetary radii and
• small semi-major axes
→ short periods
→ `hot Jupiters’
∆F
Figure 12.4: A diagram showing schematically what happens to the light received form a star as a
planet transits along our line of sight.
So:
∆F
=
F
Rp
Rs
2
(12.13)
For the Solar System, Jupiter would cause a 1% drop in the light seen from the Sun, which
is large enough to be measurable from the ground, while the Earth would produce a 0.01%
drop, which can be measured from space.
If you know that the planet is transiting then doppler measurements can determine the
planet’s mass. The star’s radius can be determined from the duration of the transit, leading
to the radius of the planet. That, combined with the mass, allows the density to be calculated,
which is the first step towards understanding what the planet is made of.
For a transit to be detected the planet’s orbital plane must be quite closely aligned with the
line of sight to the star. Assuming a random orientation of orbital inclinations for planetary
systems, and considering the diameter of the Sun, it can be shown that there is a chance of
about 1 in 200 for the transit of an Earth-like planet around a Sun-like star to be visible.
Such transits would happen only once a year for the Earth, and you would need to observe
at least two such transits to be sure that it was detected, and to measure the orbital period.
The Kepler satellite was thus designed to monitor a total of 105 stars for a period of 3 to 5
years in search of, among other things, Earth like planets. If they are common, it should be
able to detect several such systems. Sadly, a technical failure on the satellite has led to the
possible end of its planet hunting mission after only about 3 years, but efforts are under way
to restore its planet hunting capabilities.
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12.7
Other ways to detect planets
While the transit and doppler methods are responsible for the detection of most of the exoplanets that we know, there are a couple of other methods that have proven useful.
12.7.1
Pulsar Planets
The first of these is the timing of pulsars, which led to the detection of the very first exoplanets.
Pulsars are rotating neutron stars, remnants of supernova explosions, which emit beams
of electromagnetic radiation from their magnetic poles. These beams act like lighthouses,
producing regular pulses which can be timed to picosecond accuracies. Regular deviations in
these pulses, produced by the same centre-of-mass shifts seen above for the transit method,
can be measured to accuracies better than 1m/s in velocity and 1000 km in distance. This
has allowed a small number of planets, with masses down to 0.0004 Earth masses, to be
discovered. None of them are likely to be particularly nice places to live, though, since these
systems have survived a supernova and are now bathed in hard radiation from the pulsar.
12.7.2
Gravitational Lensing
Gravitational lensing is the process by which light is bent and focussed as it passes close to a
large mass. Stars and planets are both large enough to produce a measurable magnification
of the light of a background star if they pass close enough to our line of sight. Large scale
monitoring projects like OGLE, originally intended to search for Baryonic Dark Matter, can
in principle detect the lensing amplification produced by a planet orbiting a star responsible
for lensing, and there are a small number of cases where this has been found. The advantage
of this approach is that it is sensitive to essentially all possible planetary masses, but the
disadvantage is that the lensing signal is not repeatable, so one can never be absolutely
certain what has produced it, or determine the full characteristics of any planetary system
the lensing has revealed.
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Things to Remember
• The definitions of parsec, AU, magnitude and Jy
• The definition of a planet, and how to distinguish it from a brown dwarf and a
minor planet
• The problems of direct detection of exoplanets; derivation of luminosity ratios
• The derivation of reflex motion
• The derivation of the change in the flux from a star due to a planetary transit
• The use of the above methods in detecting planets and other planet detection
methods
Chapter 13
The Exoplanet Population
13.1
Introduction
At the time of writing, over a thousand planets are now known outside our own Solar System,
including at least 129 in multi-planet systems. More are being discovered all the time thanks
to ongoing survey programmes such as the Kepler satellite and SuperWASP survey, looking
for planetary transits, and the HARPS radial velocity survey. We have reached the point
where we can draw conclusions about some aspects of the exoplanet population. However,
the methods of exoplanet detection all have limitations, so our view of the population as a
whole is necessarily incomplete, and biased by what are known as ‘selection effects’. In this
chapter we will look at what is known about the exoplanet population, try to deconvolve some
of the selection effects, and draw some conclusions about the overall population of planets in
our galaxy.
13.2
The Current State of Planet Searches
New planet discoveries are announced all the time, so any attempt to describe the current
state of planet searches is doomed to become rapidly out of date. However, the broad picture
is that we now (Feb 2014) have discovered over 1000 planets. Most of these were discovered
using the radial velocity technique, but the transit method is rapidly producing new candidate
and confirmed planets thanks to the Kepler mission. About 15% of discovered planets are
found in multiple planet systems. The parent stars of the planets discovered so far are largely
F, G and K type main sequence stars. This does not, however, mean that other stellar types
do not have stars, since the majority of planet searches have been targeted at F, G and K type
stars. This is because these stars are sufficiently long lived that life might have developed
on planets around them, they are relatively bright, compared to the more common low mass
M stars, and they are well suited to the radial velocity method, since they have many well
defined spectral lines, and have stable stellar atmospheres. The nature of planets around
other stellar spectral types is thus largely unconstrained.
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Sun Stars and Planets 2012-13
On the basis of current results we can say that at least 20-50% of F, G and K-type stars have
at least one giant planet, comparable in mass to Jupiter, in an orbit whose semi-major axis
is <20 AU.
13.3
Selection Effects
The issue of host stellar type is the first in a series of ‘selection effects’ that constrain and
bias what we are able to say about the exoplanet population. Selection effects arise in a
wide variety of sciences, especially observational ones like astrophysics, where one does not
have full control over what you find. Selection effects are often quite subtle and can require
careful consideration, but they can also be quite obvious once the observational problem
is understood. The preponderance of F, G and K-type stars in the radial velocity method
searches is a case in point. Another is the sensitivity, or lack of it, of various methods to
various types of planets. The radial velocity method, for example, is not sensitive enough
to discover an Earth mass planet in an Earth-like orbit around any other stars. The radial
velocity changes that the Earth produces on the Sun have an amplitude of ∼ 0.1 m/s. The
most accurate radial velocity measurements so far achieved are five times bigger than this, at
0.5 m/s, and that was only possible after a long monitoring programme and heroic efforts to
exclude the effects of other motions on the surface of the star being observed (Alpha Centauri
- Udry et al., 2012, Nature, 491, 207).
While some selection effects will exclude some classes of planet from what we can detect, other
selection effects will lead to other classes of planet being much easier to detect. High mass
planets that are in orbits very close to their parent stars produce the largest radial velocity
shifts. Such ‘Hot Jupiters’ are thus very well represented in the current results of exoplanet
searches.
These and other selection effects must be carefully considered when using the current set of
known exoplanets to derive conclusions about the overall population of exoplanets. Nevertheless, this is what we are about to do.
13.4
Exoplanet Masses
Figure 13.1 shows a histogram of currently known planet masses, measured in terms of Jupiter
masses. As can be seen, the vast majority of currently known exoplanets have masses that
would class them as gas giants if they were in our own Solar System, with masses > 0.05
MJ , the mass of Uranus. In fact, most of these masses are not actual masses but are lower
limits to the mass of planets detected using the radial velocity method, and thus are in
fact measurements of M sin i where i is the angle of inclination of the orbit to the line of
sight. To go from such a radial velocity minimum mass to an actual mass measurement, a
determination of the inclination angle is needed. If a transit observation is available then
we known the the inclination angle i is high. An example of this is the planet HD209458b,
where a transit observation and a radial velocity shift is seen. Estimates for the inclination
81
Figure 13.1: Histogram of planet masses. The x-axis shows log(planet mass/Jupiter mass). Earth’s
mass is 0.003 times that of Jupiter. As can be seen, nearly all currently known planets have masses ion
the range of gas giants, with hardly any planets known that have masses comparable to that of Earth.
This is because current detection techniques make it very difficult to detect and confirm a planet with
mass comparable to that of Earth. Generated using data from exoplanet.eu
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Sun Stars and Planets 2012-13
angle can also be obtained by other observations. In the case of Epsilon Eridani, a dust ring
around the star is observed. This appears elliptical in the observations, but such rings are
expected to be circular. From the dust ring ellipticity an inclination angle of 46 degrees can
be estimated. The orbital plane of the planets in this system should match that of the dust
ring, so we can then estimate the true mass of the planet Epsilon Eridani b. From radial
velocity measurements we have M sin i = 0.86MJ , thus:
M sin i = 0.86MJ ; i = 46◦ ⇒ M =
0.86MJ
= 1.2MJ
sin 46◦
(13.1)
The fact that most of the planets known have masses in the gas giant range is not surprising.
Our current detection techniques are largely insensitive to lower mass planets. We thus would
not expect to have found many, or in fact any, Earth mass planets in our studies to date. The
few Earth and lower mass planets that appear in Fig. 13.1 are the result of pulsar timing
or gravitational lensing detections which are not subject to the selection biases in favour of
large mass planets that apply to the radial velocity and, to some extent, transit methods that
are responsible for the majority of planet detections. Nevertheless, a statistical analysis of
Kepler planet candidates, including many that are potentially Earth-mass but which cannot
be unconfirmed by radial velocity measurements (Petigura et al., 2013), has concluded that
22% of Sun-like stars harbour planets in orbits such that liquid water is possible on their
surfaces (a region known as the habitable zone - see next chapter). The nearest star with
such a planet could be as close as 12 light years away.
13.5
Exoplanet Composition
Detailed analysis of the composition of an exoplanet is not something we can yet achieve.
However, simply being able to measure the density of an exoplanet would be a big step
towards understanding what it might be made of, especially bearing in mind the range of
densities of planets in our own Solar System, with the terrestrial planets being much denser
than the gas giants. Masses and planetary radii are available for about a few hundred planets
so far. The vast majority of these turn out to have low densities, comparable to those of
our own gas giants. There are a handful of exoplanets that have higher densities, though,
and these can be considered candidate terrestrial planets. In most of these cases their mass
estimates are currently rather uncertain, so there are large uncertainties on their derived
densities.
The best way to determine the composition of an exoplanet, or at least its atmosphere, is
to obtain spectroscopy, but this is an even more difficult job than direct detection of the
continuum light of a planet. However, for transiting exoplanets, a number of tricks are
possible. Time resolved spectroscopy allows us to look at the effect of the exoplanet on the
light of the star as it passes through the planet’s atmosphere. Comparison of stellar spectra
before, during and after the transit allow the size and some aspects of the planet’s composition
to be measured. This was first achieved on the planet HD209458b, a gas giant orbiting 0.045
AU from its parent star. Absorption in the Lyman α line coming from hydrogen in the
planet’s atmosphere was found to cover 15% of the stellar disk, rather than the 1.5% covered
83
Figure 13.2: Mass and orbital semi-major axis for non-pulsar planets. Many more gas giants close
to their parent star are found than expected. From Rothery, Gilmour & Sephton.
by the opaque core of the planet, implying that the atmosphere of this planet is very extended,
resulting from the fact that it is being heated to high temperatures by the star. Since these
initial observations, similar spectroscopic transit studies have revealed water vapour, carbon
dioxide and methane in HD209458b’s atmosphere, as well as hydrogen. These are all things
you would expect to find in a gas giant’s atmosphere.
13.6
Exoplanet Orbits: Hot Jupiters and Planetary Migration
One of the big surprises when exoplanets started to be discovered was that there are a large
number of ‘hot Jupiters’ - gas giant planets that orbit very close to their parent stars (se
Fig. 13.2). These are in fact the easiest objects for both radial velocity and transit studies
to detect, but there was no expectation at all, before their detection, that such things would
exist. The reason for this is that gas giants are expected to form much further out in their
solar systems since the young star will, on first ignition, heat and boil off all the volatiles in
the inner regions of the protoplanetary disk. This is why we see terrestrial planets close to
our own Sun and gas giants further out.
Hot Jupiters must therefore migrate inwards, from their formation location, to where they
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Figure 13.3: Orbital eccentricities vs. semi-major axis for exoplanets compare to those of Jupiter
and Saturn. From Rothery, Gilmour & Sephton.
are seen by our exoplanet observations. The best current idea for how this occurs is that
there is an interaction between the forming gas giant and the protoplanetary disk during the
process of formation that causes it to move inwards. The infall cannot proceed too far or
the gas giant will end up in the star, so some other process has to terminate the migration,
possibly as a result of the young star boiling away the protoplanetary disk. As a gas giant
moves inwards in its system, smaller terrestrial planets will be scattered out of their systems
or dumped into their stars.
If this is common, how did our own solar system and planet stay as they are? There is some
evidence from our asteroid belt that Jupiter moved inwards by about 0.2 AU, but then this
motion stopped. It turns out that interactions between gas giants when there is more than
one in a system can slow or halt any inward migration. Perhaps our existence on Earth is a
result of such an interaction between Jupiter and Saturn.
The orbital eccentricities of exoplanets are often much larger than those seen in our own Solar
System (see Fig. 13.3). This may come about through orbital resonances between two gas
giants, or through close encounters between gas giants, which would result in one gas giant
being expelled from the system and the other acquiring a high eccentricity orbit, passing close
to its parent star.
85
13.7
Host Star Metalicity
One other result that has emerged from studies of exoplanets and their host stars is that it
appears that planets are more likely to be found orbiting stars with higher metallicities - ie.
that contain more enriched material. The origin of this effect is currently unclear, and it may
be that this is actually the result of a subtle selection effect and not a genuine signal. If real,
two possible explanations are:
• That an inherently more metal rich star will have more metals in its protoplanetary
disk, possibly enhancing the condensation of dust into planetessimals and increasing
the likelihood of planet formation.
• Alternatively, it might be that inner, rocky terrestrial planets often have their orbits
disrupted and end up falling into their parent star, enriching its atmosphere
13.8
Exoplanets: A young Science
The study of exoplanets is still very young. We have seen many surprises so far, including
the discovery of hot jupiters and more broadly that solar systems cover a much wider range
of properties, such as orbital eccentricities, than was once expected. It seems possible that
our own Solar System is rather more stable gravitationally than many of the other systems
uncovered so far. Whether we are lucky in living in a solar system where the young Earth
could survive or not is unclear. However, we still cannot detect terrestrial plants in other
systems, and the observations we do have are potentially subject to a wide range of selection
effects and biases. There is much work to be done in this field in exploring the properties of
exoplanets and how they are related. You can do some of this yourselves with exoplanet.eu,
which collects data on all exoplanets as they are discovered and provides tools for analysing
their properties.
Things to Remember
• The current state of exoplanet searches
• The results of selection effects in exoplanet searches
• The calculation & observed distribution of exoplanet masses
• The determination of exoplanet composition
• Exoplanet orbits, hot jupiters and planetary migration
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Chapter 14
Astrobiology: Life on Other Planets
14.1
Introduction
Not so long ago, the quest for life elsewhere in the universe could be regarded as speculation
that would remain impossible to test. A range of discoveries of the last 20 years, however, have
drawn this topic into the scientific mainstream, and there are now many people working in the
general area of astrobiology. This includes astronomers and physicists, but also biologists and
geologists, since studies of the history and diversity of life on Earth can inform our searches
for life elsewhere.
14.2
Life on Earth: History
The Earth formed roughly 4.5 Gyr ago, and the late heavy bombardment ended about 4Gy
ago. The earliest clear signs of life on Earth are structures called stromatolites, which are
built up by the action of a thin later of photosynthesising blue green algae. The oldest known
stromatolites are 3.46 Gyr old, and were discovered in Australia. Evidence for life arising
even earlier than this is provided by carbon isotope ratios in Earth’s oldest sediments. These
suggest that autotrophic organisms that fixed atmospheric carbon were well established 3.8
Gyr ago, though it has also been suggested that simple chemical processes might be mimicking
this signature of life. If life was genuinely well established 3.8 Gyr ago that is only a very
short time, compared to the age of the Earth, after the planet became inhabitable at all,
following the late heavy bombardment.
How life arose is a subject of great debate. One well established scenario suggests that
life started with simple, self-sustaining chemical reactions which gradually increased in complexity. These chemical reactions likely took place in places where there was a rich mix of
chemicals and plentiful available energy. Hydrothermal vents in the deep ocean are one possible site for the first emergence of these processes. The self sustaining chemical networks
require catalysts to operate. These may originally have been mineral catalysts, such as the
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Sun Stars and Planets 2012-13
iron oxides available in hydrothermal vents, but other organic materials, proteins and RNA
(Ribonucleic acid), are also capable of such catalysis. In addition to catalysis, RNA is also
capable of self-reproduction, which would have given it such an advantage over the other
processes operating at the time, that it likely took over and the earliest biology on Earth was
based on RNA. Later, DNA (Deoxyribonucleic acid) came to dominate since, as long as the
proteins necessary for its reproduction are around, since it is more stable and less subject to
reproduction errors.
The early history of life on Earth may thus move from mineral catalysed chemistry, to a
simple RNA-world, which then suffered a genetic takeover as the more stable and efficient
DNA came to dominate.
While photosynthesis is the dominant energy generation mechanism on Earth today, this is
dependent on the availability of sunlight. Deep ocean hydrothermal vents, while possessing a
rich chemistry, are well away from sunlight, and thus feed themselves not through photosynthesis but through chemosynthesis, deriving energy from chemical processes rather than from
light. These organisms would have lived in what we consider extreme environments, and we
can see their descendants today, a class of single cell organism known as archaea, in similarly
extreme environments like hydrothermal vents and hot springs.
As life developed and spread, photosynthesis started, and began producing oxygen as a by
product. This was not a significant constituent of the atmosphere until very recently in
geological terms. It was only about 500 Myr ago that oxygen levels approached those of today
(see Fig 14.1) and were high enough to allow the ozone layer to form. Up until this point most
of life on Earth was anaerobic - ie. operated in the absence of oxygen. In fact, oxygen is toxic
to anaerobic life, so the first mass extinction we know about was the result of photosynthetic
organisms polluting the Earth with the deadly poison that is oxygen. Anaerobic life is, of
course, still with us in the oxygen free slime at the bottom of oceans and in stagnant pools
of water. Many of the waste products of anaerobic life are in fact toxic to us, which is why
pond gas and the results of other anaerobic processes smell so bad.
For most of the history of life on Earth, life was made up of single celled organisms. Multicellular life, like us, only emerged about 1 Gyr ago, first as multicellular algae, then as the
first attempts at more complex multicellular life, the still poorly understood ediacarans. It
was not until about 550 Myr ago that what we would regard as modern multicellular life
emerged during the Cambrian explosion, a sudden flowering of great diversity in forms of life,
the results of which we can still see today.
14.3
Lessons from the History of Life on Earth
What lessons can be drawn for the search for life elsewhere from this rapid overview of the
history of life on Earth?
Firstly, we can look at what appear to be the essentials for life:
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Figure 14.1: The oxygen content of the Earth’s atmosphere over time. The Berker-Marshall Point is
the stage at which there is enough oxygen in the atmosphere for the ozone layer to form. From Paumann
et al., Biochimica et Biophysica Acta (BBA) - Bioenergetics, Volume 1707, Issues 23, AprilMay 2005,
Pages 231253
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Figure 14.2: Key dates in the history of life on Earth. From Rothery, Gilmour & Sephton.
91
• A supply of energy of some kind (photosynthesis dominates currently, chemosynthesis,
in the absence of light, probably dominated the early stages of life on Earth).
• The presence of liquid water. This is necessary to allow chemical reactions to take place
at all.
All other things that we might think are essential, such as DNA or the presence of oxygen,
are likely to be beneficial to specific types of life, but not to the existence of life in general.
Secondly, the world as we know it today is in fact a relatively recent occurrence. For much of
the history of life on Earth there were only unicellular species existing in a largely anaerobic
environment.
14.4
Life Elsewhere in the Solar System
Using the lessons gained from examining the history of life on Earth, what can we say about
the potential for life elsewhere in our own Solar System?
14.4.1
Mars
The place in the Solar System most likely to have once had an environment fairly similar
to the Earth is the planet Mars. While it is currently a cold, dry place with a very thin
atmosphere, there is now a growing body of evidence that suggests that liquid water once
flowed on the surface of Mars during a warm wet phase as recently as 3 Gyr ago. Other
observations suggest that small amounts of water may have flowed on the surface much more
recently.
If liquid water existed, or exists today, on Mars, is there any evidence for life? As yet, there
is nothing unambiguous - if there was you would have heard about it - but there are some
interesting hints. The presence of methane in the atmosphere of Mars suggests that there
is something on the planet producing this gas. It is one of the byproducts of anaerobic
life on Earth, but it can also be produced by geological processes. Examination of isotope
abundances in any methane detected by the Curiosity Rover should be able to determine the
origin of this gas, since biological processes operate differently for different isotopes. So far,
however, Curiosity has found no methane on Mars, despite its detection by orbiting satellites
elsewhere on the planet. Meanwhile, possible evidence for historical life on Mars may have
emerged in meteorites from Mars that have landed on Earth. In 1996 the discovery of martian
microfossils was claimed in the meteorite ALH84001, which originated on Mars. This result
is far from agreed, but the possibility of finding fossil martian lifeforms, whether in martian
meteorites or in situ on Mars using rovers such as Curiosity, is one way in which the presence
of ancient life on Mars could be confirmed.
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14.4.2
Europa
As discussed under the section on the Solar System’s moons, there is evidence for a liquid
ocean beneath the icy surface of the Galilean moon Europa. The conditions in such a subsurface ocean are very uncertain, but it is possible that the tidal heating of the moon by its
orbit around Jupiter, could lead to the presence of hydrothermal vents in this ocean, similar
to those thought to have been the cradle for life on Earth. Similar processes within Europa
could lead to the same kind of primitive life that emerged on the young Earth. Future missions to the Jovian moons such as ESA’s JUICE project will be looking for signs of this ocean
and any biological processes that might be taking place within it. The recent discovery of a
water plume on Europa, similar to that found en Enceladus, means that we may be able to
get an idea of subsurface conditions on this moon through observations from JUICE or even
from observatories closer to home. There may also be a similar subsurface ocean on the moon
Ganymede as well, though this would be buried under an even thicker layer of ice since it is
subject to less tidal heating than Europa.
14.4.3
Enceladus
The one other place in the solar system where there is clear evidence for the presence of liquid
water is Enceladus, the moon of Saturn, where jets of water vapour emerge from cracks in
parts of its surface. Enceladus, like Europa, is tidally heated, so here too there may be a
subsurface ocean and hydrothermal vents that could host biological systems. There is also
some evidence of a subsurface water layer in Saturn’s moon Titan.
14.5
Life Outside the Solar System
Having looked at possible homes for life in our own Solar system it is now time to look for
it elsewhere. The requirements for a habitable planet outside our own Solar system will be
broadly similar to what we have found locally, with the presence of liquid water being of
paramount importance. For an Earthlike planet to be capable of supporting life the following
conditions would have to hold:
• Large enough mass so that the atmosphere can provide sufficient pressure for water to
be a liquid on its surface. Atmospheric pressure is given by:
P = mc
GMp
R2
(14.1)
where P is the pressure, mc is the mass of a column through the atmosphere, Mp is the
mass of the planet and R is the radius of the planet.
• The planet must be large enough to have geological activity so that volatiles can be
incorporated into the crust, as seen in the carbon cycle on Earth. It should be noted
that Venus is an interesting exception to this rule, since it is similar in mass to Earth,
We thus want planetary masses between 0.5 and 10 Earth masses; these are sometimes
called ‘Earth-mass planets’.
2. Planet positions: the habitable zone
For carbon-based life, (and carbon cycle) need liquid water, and thus temperatures be93
tween 273 and 373 K. For naive calculation of habitable zone, see PS 4 where we found
a distance of 0.6 to 1.1 AU for the habitable zone.
from http://www.geosc.psu.edu/ kasting/PersonalPage/Kasting.htm
Figure 14.3: The location of the Continuously Habitable Zone for a range of stellar types compared
to the position of the planets in our own Solar System. From www.geosc.psu.edu.
but lacks a carbon cycle, so while this is a necessary condition it is not sufficient for
there to be an active geological cycle.
• The planet must be large enough to retain an atmosphere: remembering the thermal
escape of atoms from atmospheres discussed in Section 8.5. This implies that the planet
must have a mass ≥ 0.5 M⊕ .
• The planet must be small enough not to have accreted an extended hydrogen rich
atmosphere, and to have become a gas giant. This implies a mass ≤ 10 M⊕
For liquid water to exist the planet must also be at an appropriate temperature, between
273 and 373K. These surface temperature limits define what is called the Habitable Zone
for planets in any given system. A simple calculation of the width of the Sun’s habitable
zone, using the considerations discussed in section 7.6, would estimate it to lie from 0.6
to 1.1 AU in distance form the Sun. Lower mass stars will have smaller habitable zones
closer to them, while higher mass stars will have them further out.
The power output of stars changes over the course of geological time. For life to have
time to evolve, we are actually interested in a narrower region where liquid water can
persist for the entire history of the planet. For our own Solar System this extends from
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0.95 to 1.15 AU. Figure 14.3 shows where the continuously habitable zone lies for a
range of stellar types.
14.5.1
Host Star
The host star has influence beyond just keeping a planet’s temperature at the right level for
liquid water to exist. High mass spectral types, such as O, B and A stars, evolve too quickly
for life to have time to evolve - recall that stellar life time decreases as M −3 on the main
sequence. High mass stars also have high surface temperatures and would thus emit copious
amounts of UV light which might be harmful to life.
Very low mass stars such as M stars have their CHZ closer than the tidal locking radius. This
would mean that one side of the planet has permanent day and the other permanent night.
This might be a problem, with the two sides being respectively very hot and very cold, so the
cold side could act as a trap, freezing out the atmosphere over time. Recent modelling work,
though, suggests that a sufficiently dense atmosphere can circulate heat from one side of a
tidally locked planet to the other, avoiding this problem.
Many stars lie in binary systems which may lead to instability in planetary orbits. Orbits
may also be affected by other bodies in the same solar system as we have seen with the inward
evolution of hot jupiters in Chapter 13. Conversely it may be that a gas giant in a stable
orbit further from the star than a terrestrial planet, as is the case for Jupiter in the Solar
System, might limit the number of impacts in the inner system.
14.5.2
Gas Giant Moons
The considerations given above apply to life on the surface of a terrestrial planet. As discussed
in section 14.4, life might also exist beneath the icy surfaces of gas giant moons, like Europa
and Enceladus, in our own Solar System. Gas giant moons elsewhere might also be capable of
harbouring life in this way, or, if they are warm enough and have a dense enough atmosphere
for the presence of surface water, they might also harbour life on their surface.
14.6
The Galactic Habitable Zone
The large scale geography of our galaxy influences where it is most likely to find life bearing
planets. The formation of terrestrial planets requires high-metallicity stars. These are most
likely to be found in the thin disk or bulge of the galaxy. The outer regions of the disk have
low metallicity so would have fewer terrestrial planets, while regions closer to the centre of
the galaxy would suffer from two disadvantages: firstly the stars are closer together, leading
to overcrowding and the possible gravitational disruption of a stellar system by a passing
star; secondly, a given star will be more likely to be close to an energetic event, such as a
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supernova, which could wipe out life in nearby star systems. The core of our galaxy is thus
not thought to be hospitable for life.
The orbit of a star around the galaxy is also important. The Sun’s orbit is nearly circular, so
it is less likely to stray into crowded regions in the core. The Sun’s orbit also avoids crossing
the spiral arms of the galaxy, which are also regions of high stellar density and thus hazardous
to life.
14.7
How to Find Life on Other Planets
We have discussed the considerations for an extrasolar planet to be able to support life, but
would we be able to detect such life if it were present?
We have already seen how exoplanet studies are beginning to be able to determine various
parameters for the atmospheres of hot jupiters. Our observational capabilities are improving and there are now plans for instruments that will eventually be able to take spectra of
terrestrial exoplanet atmospheres. There are a number of ‘biomarkers’ that could appear in
these spectra if life is present. Chief among these is ozone, which has a prominent absorption
feature in the infrared at about 10µm. This would be a clear sign of the presence of life
since oxygen is a highly reactive molecule, which, unless constantly replenished, would soon
be locked up in other compounds like CO2 . The only process we are aware of that can keep
oxygen levels high enough for an ozone absorption layer to exist is photosynthesis in plants.
Other possible biomarkers include methane and spectral features associated with chlorophyll
in plants.
However, as we have seen, the ozone layer in the Earth’s atmosphere is relatively recent
in geological terms, and chlorophyll, while common on Earth, will not necessarily be the
molecule of choice for photosynthesis on other planets. A more general signature of life will
be signs of any chemistry which is out of equilibrium - the abundance of oxygen in the Earth’s
atmosphere is an example of this - since the action of biological processes are the only way
we know that can maintain such a disequilibrium over time. Quite what we might find in the
atmospheres of biologically active exoplanets remains to be seen.
Things to Remember
• The history of life on Earth
• The requirements for life and the likely sites elsewhere in the Solar System
• The requirements for and potential sites of life around other stars
• The galactic habitable zone
• How life might be found on exoplanets
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Chapter 15
The Search for Extraterrestrial
Intelligence
15.1
Introduction
Assuming that life does exist on other planets, the next great question is whether intelligent
life exists elsewhere. We do not yet have any evidence for extraterrestrial intelligence (ETI),
but absence of evidence is not evidence of absence. There are many issues surrounding the
search for extraterrestrial intelligence - how it should be conducted, whether we should try to
make contact ourselves, what to do if we ever do find evidence for it - but few hard and fast
results. The lecture for this section of the course will thus largely take the form of a discussion
about how we can guesstimate the number of extraterrestrial intelligences in the galaxy, of
what uncertainties there are in such a prediction, and about broader issues concerning the
search and possible discovery of ETI.
The two key results in this area are the Drake Equation and the Fermi Paradox, which will
be described here in turn.
15.2
The Drake Equation
The Drake equation (devised by Frank Drake in 1961) encompasses the terms needed to predict the number of intelligent civilisations in the galaxy that at any given time are interested
in communication with other civilisations. At the time the equation was devised, there were
few constraints on any of the terms, but an extra 50 years of astrophysics has begun to tie
some of them down.
N = R × fp × nE × fL × fi × fc × Lc
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(15.1)
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where:
N = the number of technological civilisations in the galaxy that are interested in communication
R = the average rate of star formation in the galaxy (in stars per year)
fp = the fraction of those stars with planetary systems
nE = the average number of habitable planets in each system
fL = the fraction of those habitable planets on which life develops
fi = the fraction of those planets on which intelligent life develops
fc = the fraction of intelligent species interested and able to communicate with other species
Lc = the lifetime of a communicating civilisation
At this point the only terms in the Drake Equation where we have accurate values are R
and fp , which have values of roughly 8 to 20 for R and about 0.5 for fp . nE is a term that we
should have good constraints on fairly soon, from long term transit studies using instruments
like Kepler. Our best guess at it so far is that it is likely to be close to 1, and the best
current results suggest a value of 0.2 (which, for our purposes here, is very close to 1). fL is,
essentially, the goal of the whole field of exobiology, but that study is currently in its infancy
and any guess that can be made at this stage will be highly uncertain.
That leaves fi , fc & Lc , which are not easily determined, and which are controlled by factors
that are biological and sociological.
What values do you think are reasonable for fi , fc & Lc , and what are your justifications for
these estimations?
15.3
The Fermi Paradox
The next important consideration in this field is known as the Fermi Paradox. This arises
from a question that the famous physicist Enrico Fermi ask in an informal discussion in 1950.
Fermi made the observation that, given the great age of the universe (about 13.5 Gyr) and the
very large number of stars in the galaxy (about 1011 ), then we should be able to see evidence
of intelligent life through interstellar probes or spacecraft, unless intelligent life capable of
interstellar travel and/or communication is very rare. His key question was ‘Where are they?’
since it can be shown quite easily (I have run a 3rd year UG project on this) that a civilisation
capable of interstellar travel and colonisation can spread throughout the galaxy in what is,
in geological and cosmological terms, a relatively short time - 1-100 Myr.
This issue is also known as the ‘great silence’ since it applies just as much to communication
as it does to physical contact with alien intelligences.
What possible resolutions can you come up with for the Fermi paradox, and what is your
justification for these solutions?
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15.4
SETI and CETI
The Search for Extraterrestrial Intelligence (SETI) and attempts at Communication with
Extraterrestrial Intelligence (CETI - confusingly pronounced the same way) are terms for
various attempts to observationally test the idea that extraterrestrial intelligences exist and
might be contacted by us. Much of the work has focussed on radio observations looking for
narrow band, artificial signals coming from Sun-like stars, but there are many other possible
ways in which extraterrestrials might communicate with each other.
Can you think of alternatives to radio waves for interstellar communication?
What kind of problems exist when trying to detect a narrow band radio signal at an unknown
frequency?
Within CETI, a small number of attempts have been made to transmit powerful radio signals
towards certain locations in the galaxy. The most significant of these was the use of the
Areceibo interplanetary radar to send a message coded as an image towards the globular
cluster M31. The message will take 25000 years to reach the cluster and any reply would
take 25000 years to come back, so we don’t expect a snappy conversation. Meanwhile, normal
radio and TV broadcasts from Earth are propagating through the nearby galaxy. While cable
and satellite TV mean that less power has been expended on such transmissions over the last
few decades, powerful transmissions of previous years are still heading out into space. One of
the most powerful and furthest travelling signals so far is that of the 1936 Olympics in Berlin,
presided over by Adolf Hitler.
Given the experience on Earth of what happens when two cultures with very different levels
of technological development interact, is it a good idea to be advertising our presence on the
galactic stage?
15.5
The Future
Developments in radio astronomy currently underway will allow us to have far greater sensitivity to narrow band signals in the next decade. The Square Kilometer Array project (SKA)
in particular will be able to detect airport radars 50 to 60 light years away, while more powerful early-warning type radars could be detected at even greater ranges. At the same time,
large ground based telescopes such as the E-ELT, and space-based projects such as SPICA,
JWST and, eventually, Darwin, will be able to detect terrestrial planets and search for signs
of life in their atmospheric spectra.
In the next few decades we will thus be able to not only start filling in some more of the
astrophysical and astrobiological terms in the Drake Equation, but might also be able to
conduct the first studies in SETI that could detect nearby civilisations like our own.
SETI, and CETI, might soon stop being the preserve of scientific speculation, and become
actual observational and practical studies in their own right.
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Things to Remember
• The Drake Equation
• The Fermi Paradox
• The current status and potential for SETI observations