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PH507
Astrophysics
Professor Michael Smith
1
Week 6: Lectures 13-14
For instance, the UBV system has about 100 standard stars measured to about ±
0.01 magnitude. Then if we can calibrate the flux of just one of these stars, we
have calibrated the system. The calibration is usually given for zero magnitude
at each filter; all fluxes are then derived from this base level. The star usually
chosen as the calibration star is Vega.
Colour index in the BV system. Blackbody curves for 20,000 K and 3000 K, along with
their intensities at B and V wavelengths. Note that B - V is negative for the hotter star,
positive for the cooler one.
Nearby objects………
PH507
Astrophysics
Professor Michael Smith
2
PH507
Astrophysics
Professor Michael Smith
3
STARS
The Hertzsprung-Russell Diagram
In 1911, Ejnar Hertzsprung plotted the first such two-dimensional diagram
(absolute magnitude versus spectral type) for observed stars, followed
(independently) in 1913 by Henry Norris Russell.
PH507
Astrophysics
Professor Michael Smith
4
The simple HR diagram represents one of the great observational syntheses in
astrophysics. Note that any two of luminosity, magnitude, temperature, and
radius could be used, but visual magnitude and temperature are universally
obtained quantities.
Important stars: no obvious pattern…Sirius B, Betelgeus in opposite corners:
Nearby stars: main-sequence appears. Most stars are less luminous and
cooler than the Sun (alpha Centauri, nearest to us and a triple system, is
similar).
Note the hot small stars: the white dwarfs.
PH507
Astrophysics
Professor Michael Smith
5
Most stars have properties within the shaded region known as the main
sequence. The points plotted here are for stars lying within about 5 pc of the
Sun. The diagonal lines correspond to constant stellar radius, so that stellar
size can be represented on the same diagram as luminosity and temperature.
The first H-R diagrams considered stars in the solar neighbourhood and plotted
absolute visual magnitude, M, versus spectral type, which is equivalent to luminosity versus spectral type or luminosity versus temperature. Note (a) the welldefined main sequence (class V) with ever-increasing numbers of stars toward
later spectral types and an absence of spectral classes earlier than A1 (Sirius),
(b) the absence of giants and supergiants (class III and I), and (c) the few white
dwarfs at the lower left.
The brightest stars:
PH507
Astrophysics
Professor Michael Smith
6
An H-R diagram for the 100 brightest stars in the sky. Such a plot is biased in favour of
the most luminous stars--which appear toward the upper left--because we can see
them more easily than we can the faintest stars. These are the GIANTS and
SUPERGIANTS
In contrast, the H-R diagram for the brightest stars includes a significant number of
giants and supergiants as well as several early-type main-sequence stars. Here we
have made a selection that emphasises very luminous stars at distances far from the
Sun. Note that the H-R diagram of the nearest stars is most representative of those
throughout the Galaxy: the most common stars are low-luminosity spectral type M.
The most prominent feature of the H-R diagram is the Main Sequence:

Strong correlation between Luminosity and Temperature.

Hotter stars are Brighter than cooler stars along the M-S.

About 85% of nearby stars, including the Sun, are on the M-S.
All other stars differ in size:
Giants & Supergiants:
PH507

Astrophysics
Professor Michael Smith
7
Very large radius, but same masses as M-S stars
White Dwarfs:

Very compact stars: ~Rearth but with ~0.6 Msun!

Example: Betelgeuse: M2 Iab (supergiant)
o

L ~ 40,000 Lsun, T ~ 3,500 K
Sun: G2 V (main-sequence)
o
T ~ 5,000 K
Stellar luminosity classes:

Ia : Brightest Supergiants

Ib : Less luminous supergiants

II : Bright giants

III : Giants

IV : Subgiants

V : Main-sequence stars
Luminosity Classes
Stellar luminosity classes in the H-R diagram. Note that a star's location could be
specified by its spectral type and luminosity class instead of by its temperature and
luminosity. Giants possess cool low-density photospheres, hence absorption lines
identify them (e.g. narrower lines). After spectral classification, their distance can be
estimated according to their luminosity class. This is their spectroscopic parallax.
PH507
Astrophysics
Professor Michael Smith
8
Magnitude versus Colour
Because stellar colours and spectral types are roughly correlated, we may construct a
plot of absolute magnitude versus colour - called a colour-magnitude diagram. The
relative ease and convenience with which colour indices (such as B - V) may be
determined for vast numbers of stars dictates the popularity of colour-magnitude plots.
The resulting diagrams are very similar to the magnitude-spectral type H-R diagrams
considered above.
The Mass-Luminosity Relationship
Just as the determination of the period and size of the Earth’s orbit (by Kepler’s third
law) leads to the Sun’s mass, so also have we deduced binary stellar masses.
Because it is necessary to know the distance to the binary system in order to establish
these masses, we need only observe the radiant flux of each star to find its luminosity.
When the observed masses and luminosities for stars in binary systems are plotted,
we obtain the correlation called the mass-luminosity relationship.
PH507
Astrophysics
Professor Michael Smith
9
In 1924, Arthur S. Eddington calculated that the mass and luminosity of normal stars
like the Sun are related by
L  M 


L  M  

His first crude theoretical models indicated that α ≈ 3. On a log-log plot, this gives a
straight line with a slope of 3. Main sequence stars do seem to conform to this
α ≈ 3 for luminous and massive
-type stars to α
dim red stars of low mass.
From a sample of 126 well-studied binary systems, we find that the break in slope
below this value is 2.26; above it, 3.99. Or :
n,
value of exponent n
3.9
3.0
2.7
Lifetime
Mass range M
M<7M
7M
< M < 25 M
25 M < M
Mass-3
Mass/Luminosity
If we use the mass-luminosity relation for stars of 0.4MSun and greater,
or
so a star with 10x the mass of the Sun will have a main sequence lifetime of only 10
million yrs!
So we know that O stars, the most massive stars, have main sequence lifetimes of
only a million years so the fact that we see some O stars now means that star
formation is still occurring in the Milky Way.
See:
http://www.shef.ac.uk/physics/teaching/phy111/
Stellar (Main Sequence) Properties With Mass
Mass
40 MSun
17
Temp
35,000 K
21,000
Radius
18 RSun
8
Luminosity
320,000 LSun
13,000
tMS
106
yrs
7
10
habitable zone
350-600 AU
PH507
Astrophysics
Professor Michael Smith
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7
13,500
4
630
8x107
2
8,100
2
20
2x109
1
5,800
1
1
1010
1-2
0.2
2,600
0.32
0.0079
5x1011
0.1-0.2
The more massive stars burn their fuel very rapidly, leading to short lifetimes:
Spectral
Class
Mass
(Msun)
L
(Lsun)
Temp.
(K)
Radius
(Rsun)
O5
40
400,000
40,000
13
B0
15
13,000
28,000
4.9
A0
3.5
80
10,000
3.0
F0
1.7
6.4
7,500
1.5
G0
1.1
1.4
6,000
1.1
K0
.08
.46
5,000
0.9
M0
0.5
0.08
3,500
0.8
M*/Msun
60
30
10
3
1.5
1
0.1
time (years)
3 million
11 million
32 million
370 million
3 billion
10 billion
1000's billions
Spectral type
O3
O7
B4
A5
F5
G2 (Sun)
M7
Today, astrophysical theories of stellar structure explain these results in terms of the
different internal structures of stars of different mass and the opacities of stellar
atmospheres at different temperatures. Note that the M-L law does not apply to highly
evolved stars, such as red giants (with extended atmospheres) and white dwarfs (with
degenerate matter. While most stellar masses lie in the narrow range from
0.085Msun
to 100Msun , stellar luminosities cover the vast span 10-4 ≤ L/L ≤ 106.
A useful relationship to give a rule of thumb estimate of a stars surface temperature is;
0.5
 M 
T  5870  
M* 
Mean Stellar Density: Mean Density = Mass / Volume
Main Sequence: small range of mean densities:

Sun (G2v): ~1.6

O5v Star:
g/cc
~0.005 g/cc
PH507

Astrophysics
M0v Star:
~5
Professor Michael Smith
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g/cc
Giants: Low-density stars: ~10-7 g/cc (e.g., K5III)
Supergiants: Very low-density: ~10-9 g/cc (e.g., M2I)
White Dwarfs: High-density stars: ~105 g/cc
For reference, at sea level on Earth, water has a density of 1 g/cc, and air has a
density of ~0.001 g/cc.
Stellar Evolution:
In this section, we explain the HR tracks qualitatively in terms of:
1. The energy source…..chemical, gravitational and nuclear reactions. We
exclude chemical energy (e.g. forest fires) for stars.
2. Transport from the source to the surface…..conduction, convection or radiation.
We exclude conduction as ineffective.
3. Radiative transfer through the photosphere, as discussed above. Hydrogen
ions can provide the opacity in stars like the Sun.
The internal structure of stars will be quantified in later lectures.
PH507
Astrophysics
Professor Michael Smith
12
The end of Hydrogen burning:

During main sequence lifetime hydrogen burning is confined to the core
Hydrogen burning converts hydrogen into helium in the core.

Eventually the core hydrogen is exhausted . Energy then comes from a
hydrogen shell

With no energy production in the core, it contracts to maintain thermal
hydrostatic equilibrium. The collapse of the core will cause it to heat up.

The hydrogen burning shell dumps further helium onto the core. Hydrogen
burning moves outward.

Over a million years, the core of a Sun sized star decreases to about 1/10
original size.

The temperature rises from 15 to about 100 million K.

The core is composed of helium ‘ash’.

The outer layers of the star become heated by their proximity to the energy
source. The inert hydrogen outside the shell hinders the movement of the
photons.

When photons have trouble moving through a medium, they end up pushing
outwards on the matter. This is called radiation pressure.

The outer layers are not so tightly bound by gravity and will expand enormously
forming a red giant
Why Helium won’t burn yet

Hydrogen, a single proton, has a single electrostatic charge

Helium has two.
PH507

Astrophysics
Professor Michael Smith
13
Helium nuclei must have a much higher kinetic energy (speed) to get close
enough to bind
Helium burning begins

When the central temperature reaches 100 million K, helium burning starts.

Two helium nuclei fuse to form an isotope of beryllium.

This is very unstable.

If it is hit by another helium they fuse into a stable isotope of carbon.

This is known as the triple alpha process.

A high energy gamma ray is released by each reaction
A Star’s Safety-valve

Gravity tries to compress a star

When a perfect gas is compressed its density and temperature increase.

If a gas heats up its pressure increases.

The pressure tries to expand the star.

If a reaction starts to run away, the temperature rises and the star expands.

This drops the temperature and the reaction is slowed.
Perfect and degenerate

In a low-mass red giant (< 3 Msol), the core must undergo considerable
compression to drive the temperature high enough to start helium burning.

No two identical particles may occupy the same quantum state.

The electrons obey the Pauli exclusion principle (Wolfgang Pauli, 1925) and will
not be compressed any further.

The gas is said to be degenerate and is supported by degenerate-electron
pressure.

In the highly compressed core, free electrons are so crowded together that
quantum effects must be considered.
Helium flash:

When the temperature in the core reaches that required for helium fusion,
energy begins to be released.

Because the star is supported by electron degenerate pressure, it does not
expand.

(Remember degeneracy is a quantum effect and not influenced by temperature
in the same way.)

Without its safety valve the temperature soars and the fusion process runs
away.

This runaway takes only a few seconds and is called a Helium Flash

It releases a vast quantity of energy which drives the temperature so high that
the gas behaves in an ideal way again.
PH507

Astrophysics
Professor Michael Smith
14
The Helium Flash is not observable, since the photons produced in the
explosion are trapped in the Hydrogen layers.
Low mass stars:

After the helium flash, substantial carbon and oxygen ‘ash’ is dumped at the
core.

The core contracts until electron degeneracy again supports the star.

The temperature reached is enough to start shell helium burning around the
core

Helium shell burning, like the hydrogen shell before it, heats the outer layers of
the star and it expands again to form a red supergiant.
Low mass planetary nebulae

The helium shell is much thinner than the hydrogen one and is unable to swell
the star to relieve the temperature build up.

The process runs away until the helium layer is thick enough to expand the star
thus cooling it.

These helium flashes raise the luminosity from 100 to 100,000 times that of the
Sun.

The flashes can also re-start the hydrogen burning.

Can be so energetic that the outer layers of the star are blown clean off. The
escape velocity from the surface of a star is vesc = (2GM/R)1/2 .

The expanding shell of ejected gasses, ionized by ultraviolet light from the hot
core left behind. The White Dwarf core has a surface temperature over
100,000 K. Wein's law for a hot body with this temperature gives a peak
wavelength of 2.9 x 10-8m, corresponding to ultraviolet light.

When the electrons recombine with the surrounding ions, they often enter an
excited state and then jump down to the ground state emitting visible photons.
This process is known as fluorescence.
HST images of
Planetary Nebulae
Henize 1357
NGC 6543
PH507
Astrophysics
Professor Michael Smith
The Helix
15
MyCn18
Planetary nebulae

Last for around 50,000 years after which it has dispersed and faded from view.

Accounts for 15% of matter returned to the Inter-Stellar Medium (ISM) by stars.

The planetary nebula takes ~ 60% of the star with it leaving only the core.
White dwarfs

< 4 Msol, never produce temperature high enough to ignite carbon and oxygen.

During this phase, the star moves to the left on the H-R diagram.

The track will sometimes loop corresponding to thermal pulses.

As the ejected nebula fades and the core cools, the stars track turns sharply
downward.

The core becomes more and more compressed as the temperature drops.

Most of the matter becomes degenerate again and the contraction halts.

The star is now called a white dwarf - about the same size as the Earth.

Its density is typically 109 kg/m3.

One teaspoon weighs as much as an elephant (5.5 tons)

Remember that electron degeneracy is a quantum effect. This means that the
more massive a white dwarf, the smaller smaller it becomes.
The end of the road

The Chandrasekhar mass is the largest mass that a white dwarf can possibly
have.

Highly ionized atoms floating in a sea of degenerate electrons.

As the star cools, the random motions of the particles slow and the electric
forces between ions line them up in a crystalline lattice.

From this point on the star is ‘solid’
PH507
Astrophysics
Professor Michael Smith
16

the electrons, though degenerate, may move around the lattice.

The core is similar to copper or silver.

As it cools further it evolves into a cold dark diamond sphere of carbon and
oxygen, about the size of the Earth.
Higher Mass Stars: How far can it go?

For an element to serve as fuel energy must be given off when its nuclei collide
and fuse.

This energy comes from packing together more tightly the neutrons and
protons in the ash nuclei than in the fuel nuclei.

Once iron is reached with 56 protons and neutrons, no further energy can be
extracted by the addition of more.

Iron does not burn.

The fuel layers burn outward dumping more and more iron onto the core which
is supported by degeneracy pressure alone.

Eventually this fails, catastrophically and violently.
Supernova

The photons released at this temperature are so powerful they can smash iron
nuclei.

Another 1/10 of a second and the electrons merge with the protons to form
neutrons.

At about 1/4 of a second the density in the core reaches nuclear density; 4 x
1017 kg/m3.

This is virtually uncompressible and so the rapid collapse is very suddenly
stopped dead.

Electron degeneracy fails at about 1 trillion (1012) kg/m3.

The core then collapses

in less than 1/10 second, the central temperature exceed 5 billion K.

Infalling matter essentially bounces and sends a violent shock wave back up to
the surface.

The star explodes; its luminosity going up by a factor of 100 million.

For a few days a star may shine as brightly as the whole galaxy.

And then its gone. Apart from the small super dense core.
Neutron stars

After the supernova has exploded, it leaves a small super dense core.

This consists of neutrons at nuclear density, 4 x 1017 kg/m3.

One thimble full would weigh 100 million tons

It is now only ~ 30 km across, escape velocity = 1/2 the speed of light
PH507
Astrophysics
Professor Michael Smith
17
Pulsars: Little Green Men?

In 1967 Jocelyn Bell detected regular pulses of radio noise.

these pulses were incredibly regular between 0.25 - 1.25 seconds.

For a while some wondered whether these were artificial in origin.

Spinning magnetospheres with intense magnetic fields.
Black holes

If the mass of a burned out core exceeds about 3 Msol its gravity will exceed
both:
o
Electron degenerate pressure
o
Neutron degeneracy pressure

It ‘falls through’ white dwarf and neutron star.

It becomes a black hole.

All the material has collapsed to a central point.

The escape velocity exceeds the speed of light.
Protostar – Young Star evolution is not so well known:
PH507
Astrophysics
Professor Michael Smith
18
From Protostar to Young Star

Protostars are cool when they begin to shine in the visible so start to the right of
the diagram.

Continued gravitational contraction of the protostar.

Decreasing surface area means a reduction of luminosity

Decreasing radius (higher pressure and therefore higher temperature)

Different masses of star will follow different paths to their main destination on
the main sequence.
Protostars shine because they are hotter than their surroundings:

Need an energy source to stay hot, but

Central temperature is too cool for nuclear fusion to ignite
Initial energy source is Gravitational Contraction (aka, the Kelvin-Helmholz
Mechanism):

The Protostar shrinks slowly, releasing gravitational energy

50% goes into photons, and is radiated away as starlight

other 50% goes into heating the Protostar interior
How long can does this last?
Kelvin-Helmholz Timescale
To understand how long a Protostar can shine by Gravitational Contraction, we need to
compare two numbers

The Energy Source: (M2/R)

The Energy Loss Rate: Luminosity (L)
The ratio is the Kelvin-Helmholz Timescale:
The Kelvin-Helmholz timescale is ~30 Myr for a 1 solar mass protostar.
Consequences:

Shorter K-H time for high-mass protostars

Longer K-H time for low-mass protostars
H-R Diagram of pre-Main Sequence evolution for stars of various masses:
Brown Dwarfs: Failed Stars
• Stars between 1/100 and 1/12 the mass of the Sun may be able to burn
deuterium into helium for a short time, but cannot sustain nuclear reactions.
Such “failed” stars are called brown dwarfs. They are similar in size to Jupiter
with masses of 10-80 times that of Jupiter
• At a temperature of 2 million K, a lithium atom can combine with a proton to form
two He atoms. In a star that can sustain the P-P chain, the core is hot enough to have
burned all the Li to He. If Li does appear in the spectrum, the center of the star must
be cooler than 2 million K. In addition to Li, brown dwarfs show methane and water
absorption in their spectra.