Download ph507weeks1

PH507
Multi-wavelength
Professor Michael Smith
THE MULTIWAVELENGTH UNIVERSE AND EXOPLANETS
School of Physical Sciences Convenor Prof. Michael Smith
Taught in Term 2
Teaching Provision:
1
PH507
ECTS Credits 7.5
Kent Credits 15 at Level I
30 lectures + 4 workshops + 2 class tests
Prerequisites:
PH300, PH301, PH304
Aims: To provide a basic but rigorous grounding in observational, computational and
theoretical aspects of astrophysics to build on the descriptive course in Part I, and to
consider evidence for the existence of exoplanets in other Solar Systems.
Learning Outcomes:
1. An understanding of the fundamentals of making astronomical observations across the
whole electromagnetic spectrum, including discussion of photometry and spectroscopy,
and the physics of the astrophysical radiation mechanisms.
2. An understanding of the motions of objects in extrasolar systems and the basic techniques
required to solve the 2-body problem to measure their properties.
3. An understanding of observational characteristics of stars, and how their physical
structures are derived from observation and using simple physical models.
4. To be able to discuss coherently the origin and evolution of Solar Systems and be able to
evaluate claims for evidence of Solar Systems other than our own.
SYLLABUS:
•
•
•
•
•
•
Part 1: measurements
Part 2: radiation
Part 3: dynamics
Part 4: star and planet formation
Part 5: telescopes/instruments
Part 6: stars and stellar structure
Assessment Methods:
Examination 70%, Homework 10%, 1st class test 10%, 2nd class test 10%.
Recommended Texts:
Carroll & Ostlie, An Introduction to Modern Astrophysics, Addison-Wesley, [QB461]
Stuart Clark. Extrasolar Planets, Wiley Press
C.R.Kitchin. Astrophysical Techniques, Adam Hilger Press.
[Note: Changes may occur to the syllabus during the year]







Prof. Michael Smith:
101 Ingram, x7654, [email protected]
Office hours: 10-12am Wed
Bad weather
Numbers, names
Locations, times of lectures
Lecturers
PH507
Multi-wavelength
Professor Michael Smith
2
PART 1: Measurement
LECTURE 1
Distance: Distance is an easy concept to understand: it is just a length in some
units such as in feet, km, light years, parsecs etc. It has been excrutiatingly
difficult to measure astronomical distances until this century.
Unfortunately most stars are so far away that it is impossible to directly
measure the distance using the classic technique of triangulation.
Trignometric parallax: based on triangulation – need three parameters to fully
define any triangle e.g. two angles and one baseline.
To triangulate to even the closest stars we would need to use a very large
baseline. In fact we do have a long baseline, because every 6 months the earth
is on opposite sides of the sun. So we can use as a baseline the major axis of the
earth's orbit around the sun.
BASELINE: 2 x earth-sun distance = 2 Astronomical Units (AU)
(The average distance from the earth to the sun is called the Astronomical Unit.)
The parallactic displacement of a star on the sky as a result of the Earth’s orbital
motion permits us to determine the distance from the Sun to the star by the
method of trigonometric (heliocentric) parallax. We define the trigonometric
parallax of the star as the angle  subtended, as seen from the star, by the
Earth’s orbit of radius 1 AU. If the star is at rest with respect to the Sun, the
parallax is half the maximum apparent annual angular displacement of the star
as seen from the Earth.
PH507
Multi-wavelength
Professor Michael Smith
3
PH507
Multi-wavelength
Professor Michael Smith
4
PH507
Multi-wavelength
Professor Michael Smith
5
1 radian is defined as:
360
 57.3 degrees = 206265 arc seconds, approximately. There are 2
2
rad in a circle (360˚), so that 1 radian equals 57˚17´44.81” (206, 264.81”).
1 radian =
Independent distance unit is the light year:
c  t ( year )  9.47 1015 m
The light year is not used much by professional astronomers, who work instead
with a unit of similar size called the parsec, where 1 parsec = 1 pc = 206265
AU = 3.086 x 1016 m = 3.26 light years.

The measurement and interpretation of stellar parallaxes are a branch of
astrometry, and the work is exacting and time-consuming. Consider that
the nearest star, Proxima/Alpha Centauri (Rigil Kent), at a distance of 1.3
pc, has a parallax of only 0.76”; all other stars have smaller parallaxes.
PH507
Multi-wavelength
Professor Michael Smith
6
Formula:
tan p 
1AU
d
or
d 
1
AU
p
where p is in radians
for small angles.
To convert to arcseconds:
2.063 105
d 
AU
p ''
or
d 
1
pc .
p"
Technological advances (including the Hubble Space Telescope) have improved
parallax accuracy to 0.001” within a few years. Before 1990, fewer than 10,000
stellar parallaxes had been measured (and only 500 known well), but there are
about 1012 stars in our Galaxy. Space observations made by the European Space
Agency with the Hipparcos mission (1989-1993) accurately determined the
parallaxes of many more stars. Though a poor orbit limited its usefulness,
Hipparcos was expected to achieve a precision of about 0.002”. It actually
achieved 0.001” for 118,000 stars. The method of trigonometric parallax is
important because it is our only direct distance technique for stars.
The ground-based trigonometric parallax of a star is determined by
photographing a given star field from a number (about 20) of selected points in
the Earth’s orbit. The comparison stars selected are distant background stars of
nearly the same apparent brightness as the star whose parallax is being
measured. Corrections are made for atmospheric refraction and dispersion and
for detectable motions of the background stars; any motion of the star relative
to the Sun is then extracted. What remains is the smaller annual parallactic
motion; it is recognised because it cycles annually.
Because a seeing resolution of 0.25” is considered exceptional (more typical it is
1”), it may seem strange that a stellar position can be determined to ±0.01” in
one measurement; this accuracy is possible because we are determining the
centre of the fuzzy stellar image.
PH507
Multi-wavelength
Professor Michael Smith
7
In 2011 – 2013, Gaia will be set into orbit with a Soyuz rocket (and SIM Space
Interferometric Mission from the US). It will be able to measure parallaxes
of 10 micro-arcseconds. It consists of a rotating frame holding three
telescopes. Some aims:
…….Accurate distances even to the Galactic centre, 8000 parsecs away.
……..Photometry: accurate magnitudes.
……..Planet quest
……..Reference frame from distant quasars (3C273 is 800 Mpc away)
In the meantime, to go further, we construct the COSMIC LADDER.
If we can estimate the luminosity of a star from other properties, they can be
used as STANDARD CANDLES.
2 LUMINOSITY.
We can actually only measure the radiant flux of a flame and need to make a
few assumptions to find the true luminosity. Luminosity depends on the
distance and extinction (as well as relativistic effects).
The measured flux f is in units of W/m2 , the flow of energy per unit area. The
radiated power L, ignoring extinction, is given by:
f 
d2 
L
4d 2
L
4f
’
showing that a standard candle can yield the distance.
The Stellar Magnitude Scale
The first stellar brightness scale - the magnitude scale - was defined by
Hipparchus of Nicea and refined by Ptolemy almost 2000 years ago. In this
qualitative scheme, naked-eye stars fall into six categories: the brightest are of
first magnitude, and the faintest of sixth magnitude. Note that the brighter the star,
the smaller the value of the magnitude. In 1856, N. R. Pogson verified William
Herschel’s finding that a first-magnitude star is 100 times brighter than a sixthmagnitude star and the scale was quantified. Because an interval of five
magnitudes corresponds to a factor of 100 in brightness, a one-magnitude
difference corresponds to a factor of 1001/5 = 2.512. (This definition reflects the
PH507
Multi-wavelength
Professor Michael Smith
8
operation of human vision, which converts equal ratios of actual intensity to
equal intervals of perceived intensity. In other words, the eye is a logarithmic
detector). The magnitude scale has been extended to positive magnitudes
larger than +6.0 to include faint stars (the 5-m telescope on Mount Palomar can
reach to magnitude +23.5) and to negative magnitudes for very bright objects
(the star Sirius is magnitude -1.4). The limiting magnitude of the Hubble Space
Telescope is about +30.
Astronomers find it convenient to work with logarithms to base 10 rather than
with exponents in making the conversions from brightness ratios to magnitudes
and vice versa. Consider two stars of magnitude m and n with respective
apparent brightnesses (fluxes) lm and ln. The ratio of their fluxes fn / fm
corresponds to the magnitude difference m - n. Because a one-magnitude
difference means a brightness ratio of 1001/5, (m - n) magnitudes refer to a ratio
of (1001/5)m-n = 100(m-n)/5, or
fn / fm = 100(m-n)/5
Taking the log10 of both sides (because log xa = a log x and log 10a = a log 10 =
a),
log (fn / fm) = [(m - n)/5] log 100 = 0.4(m - n)
or
m - n = 2.5 log (fn / fm)
This last equation defines the apparent magnitude; note that m > n when fn >
fm, that is: brighter objects have numerically smaller magnitudes. Also note that
when the brightnesses are those observed at the Earth, physically they are
fluxes. Apparent magnitude is the astronomically peculiar way of talking
about fluxes.
Here are a few worked examples:
PH507
Multi-wavelength
Professor Michael Smith
9
(a) The apparent magnitude of the variable star RR Lyrae ranges from 7.1 to 7.8
- a magnitude amplitude of 0.7. To find the relative increase in brightness from
mini-mum to maximum, we use
log (fmax / fmin) = 0.4 x 0.7 = 0.28
so that
fmax / fmin = 100.28 = 1.91
This star is almost twice as bright at maximum light than at minimum.
(b) A binary system consists of two stars a and b, with a brightness ratio of 2;
however, we see them unresolved as a point of magnitude +5.0. We would like
to find the magnitude of each star. The magnitude difference is
mb - ma = 2.5 log (fa / fb) = 2.5 log 2 = 0.75
Since we are dealing with brightness ratios, it is not right to put ma + mb = +5.0.
The sum of the luminosities (fa + fb) corresponds to a fifth-magnitude star.
Compare this to a 100-fold brighter star, of magnitude 0.0 and luminosity l0:
PH507
Multi-wavelength
Professor Michael Smith
10
ma+ b - m0 = 2.5 log [l0 / (fa + fb)]
or
5.0 - 0.0 = 2.5 log 100 = 5.
But
fa = 2 fb, so that fb = (fa + fb)/3.
Therefore
(mb - m0) = 2.5 log (f0 / fb) = 2.5 log 300 = 2.5 x 2.477 = 6.19.
The magnitude of the fainter star is 6.19, and from our earlier result on the
magnitude difference, that of the brighter star is 5.44.
What units are used in astronomical photometry?
The well-known magnitude scale of course, which has been calibrated using standard stars whic
vary in brightness.
But how does the astronomical magnitude scale relate to other photometric units? Here we assum
unless otherwise noted, which are at least approximately convertible to lumes, candelas, and lux
1 mv=0 star outside Earth's atmosphere = 2.54 10-6 lux = 2.54 10-10 phot
Luminance: ( 1 nit =1 candela per square metre)
1 mv=0 star per sq degree outside Earth's atmosphere = 0.84E-2 nit
= 8.4 10-7 stilb
1 mv=0 star per sq degree inside clear unit airmass
= 6.9 10-7 stilb
= 0.69E-2 nit
(1 clear unit airmass transmits 82% in the visual, i.e. it dims 0.2 magnitudes)
One star, Mv=0 outside Earth's atmosphere = 2.451029 cd
Apparent magnitude is thus an irradiance or illuminance, i.e. incident flux per unit
area, from all directions. Of course a star is a point light source, and the incident
light is only from one direction.
Apparent magnitude per square degree is a radiance, luminance, intensity, or
"specific intensity". This is sometimes also called "surface brightness".
Still another unit for intensity is magnitudes per square arcsec, which is the
magnitude at which each square arcsec of an extended light source shines.
Only visual magnitudes can be converted to photometric
units. U, B, R or I magnitudes are not easily convertible
to luxes, lumens and friends, because of the different
wavelengths intervals used. The conversion factors would be
strongly dependent on e.g. the temperature of the blackbody
PH507
Multi-wavelength
Professor Michael Smith
11
radiation or, more generally, the spectral distribution of
the radiation. The conversion factors between V magnitudes
and photometric units are only slightly dependent on the
spectral distribution of the radiation.
What units are used in radiometry/infrared astronomy?
Here we're not interested in the photometric response of some detector with a
well-known passband (e.g. the human eye, or some astronomical photometer).
Instead we want to know the strength of the radiation in absolute units: watts
etc. Thus we have:
Radiance, intensity or specific intensity:
W m-2 ster-1 [Å-1]
SI unit
erg cm-2 s-1 ster-1 [Å-1]
CGS unit
photons cm-2 s-1 ster-1 [Å-1] Photon flux, CGS units
Irradiance/emittance, or flux:
W m-2 [Å-1]
SI unit
-1
erg cm-2 s-1 [Å ]
CGS unit
photons cm-2 s-1 ster-1 [Å-1] Photon flux, CGS units
Note the [A-1] within brackets. Fluxes and intensities can
be total (summed over all wavelengths) or monochromatic
("per Angstrom Å" or "per nanometer").
In Radio/Infrared Astronomy, the unit Jansky is often used as a measure of
irradiance at a specific wavelength, and is the radio astronomer's equivalence to
stellar magnitudes. The Jansky is defined as: 1 Jansky = 10-26 W m-2 Hz-1
Absolute magnitude represents a total flux, expressed in e.g. candela, or lumens.
Absolute Magnitude and Distance Modulus
So far we have dealt with stars as we see them, that is, their fluxes or apparent
magnitudes, but we want to know the luminosity of a star. A very luminous
star will appear dim if it is far enough away, and a low-luminosity star may
look bright if it is close enough. Our Sun is a case in point: if it were at the
distance of the closest star (Alpha Centauri), the Sun would appear slightly
fainter to us than Alpha Centauri does. Hence, distance links fluxes and
luminosities.
PH507
Multi-wavelength
Professor Michael Smith
12
The luminosity of a star relates to its absolute magnitude, which is the
magnitude that would be observed if the star were placed at a distance of 10 pc
from the Sun. (Note that absolute magnitude is the way of talking about
luminosity peculiar to astronomy). By convention, absolute magnitude is
capitalised (M) and apparent magnitude is written lowercase (m). The inversesquare law of radiative flux links the flux f of a star at a distance d to the
luminosity F it would have it if were at a distance D = 10 pc:
F / f = (d / D)2 = (d / 10) 2.
If M corresponds to L and m corresponds to luminosity l, then
m - M = 2.5 log (F / f ) = 2.5 log (d/10)2 = 5 log (d / 10)
Expanding this expression, we have useful alternative forms: since
m1  m2  2.5 log
d1
 5 log d1  5 log d2 ,
d2
defining the absolute magnitude m2 = M at d2 = 10 pc, so m1 = m and d2 = d,
m - M = 5 log d - 5
M = m + 5 - 5 log d
In terms of the parallax,
M = m + 5 + 5 log p”
Here d is in parsecs and p” is the parallax angle in arc seconds.
PH507
Multi-wavelength
Professor Michael Smith
13
The quantity m - M is called the distance modulus, for it is directly related to
the star’s distance. In many applications, we refer only to the distance moduli
of different objects rather than converting back to distances in parsecs or lightyears.
Magnitudes at Different Wavelengths
The kind of magnitude that we measure depends on how the light is filtered
anywhere along the path of the detector and on the response function of the
detector itself. So that problem comes down to how to define standard
magnitude systems.
PH507
Multi-wavelength
Professor Michael Smith
14
Magnitude Systems
Detectors of electromagnetic radiation (such as the photographic plate, the
photoelectric photometer, and the human eye) are sensitive only over given
wavelength bands. So a given measurement samples but part of the radiation
arriving from a star.
Four images of the Sun, made using (a) visible light, (b) ultraviolet light, (c) X rays,
and (d) radio waves. By studying the similarities and differences among these views of
the same object, important clues to its structure and composition can be found.
Because the flux of starlight varies with wavelength, the magnitude of a star
depends upon the wavelength at which we observe. Originally, photographic
plates were sensitive only to blue light, and the term photographic magnitude
(mpg) still refers to magnitudes centred around 420 nm (in the blue region of
the spectrum). Similarly, because the human eye is most sensitive to green and
yellow, visual magnitude (mv) or the photographic equivalent photo visual
magnitude (mpv) pertains to the wave-length region around 540 nm.
Today we can measure magnitudes in the infrared, as well as in the ultraviolet,
by using filters in conjunction with the wide spectral sensitivity of photoelectric
photometers. So systems of many different magnitudes (colour combinations)
are possible. In general, a photometric system requires a detector, filters, and a
PH507
Multi-wavelength
Professor Michael Smith
15
calibration (in energy units). The properties of the filters are typified by their
effective wavelength, 0, and bandpass, ∆ which is defined as the full width at
half maximum in the transmission profile. The three main filter types are wide
(∆≈ 100 nm), intermediate (∆≈ 10 nm), and narrow (∆≈1 nm). There is a
trade-off for the bandwidth choice: a smaller ∆ provides more spectral
information but admits less flux into the detector, resulting in longer
integration times. For a given range of the spectrum, the design of the filters
makes the greatest difference in photometric magnitude systems.
A commonly used wide-band magnitude system is the UBV system: a
combination of ultraviolet (U), blue (B), and visual (V) magnitudes, developed
by H. L. Johnson. These three bands are centred at 365, 440, and 550 nm; each
wavelength band is roughly 100 nm wide. In this system, apparent magnitudes are
denoted by B or V and the corresponding absolute magnitudes are sub-scripted:
MB or MV.
To be useful in measuring fluxes, the photometric system must be calibrated in
energy units for each of its bandpasses. This calibration turns out to be the
hardest part of the job. In general, it relies first on a set of standard stars that
define the magnitudes, for a particular filter set and detector; that is, these stars
define the standard magnitudes for the photometric system to the precision with
which they can be measured.
Infrared Windows
The UBV system has been extended into the red and infrared (in part because of
the development of new detectors, such as CCDs, sensitive to this region of the
spectrum). The extensions are not as well standardised as that for the Johnson
UBV system, but they tend to include R and I in the far red and J, H, K, L, and
PH507
Multi-wavelength
Professor Michael Smith
16
M in the infrared.
As well as measuring the properties of individual stars at different
wavelengths, observing at loner wavelengths, particularly in the infrared,
allows us to probe through clouds of small solid dust particles, as seen below
A visible-light (left) vs. 2MASS infrared-light (right) view of the central regions of the
Milky Way galaxy graphically illustrating the ability of infrared light to penetrate
the obscuring dust. The field-of-view is 10x10 degrees
Infrared passbands which allow transmission (low absorption):
J Band: 1.3 microns
H Band: 1.6 microns
K band: 2.2 microns
L band 3.4 microns
M band 5 microns
N band 10.2 microns
Q band 21 microns
Bolometric magnitudes can be converted to total radiant energy flux: One star
of Mbol = 0 radiates 2.97 1028 Watts.
System is defined by Vega at 7.76 parsecs from the Sun with an apparent
magnitude defined as zero.
With Lbol = 50.1 Lsolar and Mbol = 0.58.
Sun: mbol = -26.8
Full moon: -12.6
Venus: -4.4
Sirius: -1.55
Brightest quasar: 12.8
For Vega: mb = mv = 0. mk = +0.02
Sun: Mb = 5.48, Mv = 4.83, Mk = 3.28
Colour Index: B-V, J-H, H-K are differences in magnitude….flux ratios.
PH507
Multi-wavelength
Professor Michael Smith
17
But cooler, redder objects possess higher values.
Extinction
Interstellar Medium modifies the radiation. Dust particles with size of order of
the wavelength of the radiation.
Blue radiation is strongly scattered compared to red: blue reflection nebulae
and reddened stars.
Colour Excess:
measures the reddening.
E(B-V) = B-V - (B-V)o
Modified distance modulus:
m() = M() + 5 log d – 5 + A()
where A () is the extinction due to both scattering and absorption, strongly
wavelength dependent. The optical depth is given by
exp(  ) 
I
Io
.
Therefore A() = 1.086 

The optical depth is

where N is the total column density of dust (m-2) between the star and the
observer and is the scattering/absorption cross-section (m2).
ISM Law related extinction to reddening:
Av / E(B-V) = 3.2 + - 0.2
Spectroscopic Parallax
Hertzsprung-Russell deduced the main-sequence stars for nearby objects,
relating their luminosity to their colour. Groups of distant stars should
also\line along the same main-sequence strip. However they appear very
dim,\of course due to their distance. On comparison of fluxes, we determine
the distance. This works out to about 100,000 pc, beyond which main-sequence
stars are too \dim.
Cepheids as Standard Candles: The Period-Luminosity
Relationship
Cepheids show an important connection between period and luminosity: the
pulsation period of a Cepheid variable is directly related to its median
PH507
Multi-wavelength
Professor Michael Smith
18
luminosity. This relationship was first discovered from a study of the variables
in the Magellanic Clouds, two small nearby companion galaxies to our Galaxy
that are visible in the night sky of the southern hemisphere. To a good
approximation, you can consider all stars in each Magellanic Cloud to be at the
same distance. Henrietta Leavitt, working at Harvard in 1912, found that the
brighter the median apparent magnitude (and so the luminosity, since the stars
are the same distance), the longer the period of the Cepheid variable. A linear
relationship was found.
Harlow Shapley recognised the importance of this period-luminosity (P-L)
relation-ship and attempted to find the zero point, for then a knowledge of the
period of Cepheid would immediately indicate its luminosity (absolute
magnitude).
This calibration was difficult to perform because of the relative scarcity of
Cepheids and their large distances. None are sufficiently near to allow a
trigonometric parallax to be determined, so Shapley had to depend upon the
relatively inaccurate method of statistical parallaxes. His zero point was then
used to find the distances to many other galaxies. These distances are revised
as new and accurate data become available. Right now, some 20 stars whose
distances are known reasonably well (because they are in open clusters) serve
as the calibrators for the P-L relationship.
Further work showed that there are two types of Cepheids, each with its own
separate, almost parallel P-L relationship.
PH507
Multi-wavelength
Professor Michael Smith
19
The classical Cepheids are the more luminous, of Population I, and found in
spiral arms. Population II Cepheids, also known as W Virginis stars after their
prototype, are found in globular clusters and other Population II systems.
Classical Cepheids have periods ranging from one to 50 days (typically five to
ten days) and range from F6 to K2 in spectral class.
Population II Cepheids vary in period from two to 45 days (typically 12 to 20
days) and range from F2 to G6 in spectral class.
Population I and II Cepheids are both regular, or periodic, variables; their
change in luminosity with time follows a regular cycle.
Cepheids are bright and distinct. They can be used to determine distances to
quite distant galaxies, to about 5 Mpc. HST stretched this to 18 Mpc (Virgo
cluster).
Tully-Fisher Relation
In a spiral galaxy, the centripetal force of gas and stars balances the
gravitational force:
PH507
Multi-wavelength
Professor Michael Smith
20
mV2/R = GmM/R2.
If they have the same surface brightness ( L/R2 is constant) and the same massto-light ratio (M/L is constant), then
L ~ V4. So, provided we can measure V, certain galaxies can be used as standard
candles. (determine V through the 21 cm line of atomic hydrogen in the galaxy).
Type 1a Supernovae.
The peak light output from these supernovae is always about M b = -19.33 +0.25. Therefore we can infer the distance from the inverse square law. Being so
bright , they act as standard cadles to large distances: to 1000 Mpc.
Why are they standard candles? White dwarfs I binary systems. Material from\
a companion red giant is dumped on the white dwarf surface until the WD
reaches a critical mass (Chandrasekhar mass) of 1.4 solar masses. Explosion
occurs with fixed rise and fall of luminosity.
Other methods: time delay of light rays due to gravitational lensing, cluster size
influences Compton scattering of CMB radiation and bremsstrahlung emission
(X-rays). Combining, yields the size estimate (Sunyaev-Zeldovich effect). Or,
rotational properties of stars with starspots…….
New Method?
Reverse argument: knowing the Hubble constant is 72 km/s/Mpc, (WMAP
result), distances can be found directly from the redshift!
PH507
Multi-wavelength
Professor Michael Smith
Questions
How do we scale the solar system?
How do we find the distance to gas clouds?
PLANET REVIEW
The Terrestrial Solar System
21
PH507
Multi-wavelength
Professor Michael Smith
22
In the picture above we see the positions of the asteroid belt (green) and other nearearth objects
The material in the plane of the Solar System is known as the Kuiper Belt. Surrounding
PH507
Multi-wavelength
Professor Michael Smith
23
this is a much larger region known as the Oort Cloud, that contains material that
occasionally falls in, under the influence of gravity, towards the Sun as comets.
The Sun
At over 1.4 million kilometers (869,919 miles) wide, the Sun contains 99.86
percent of the mass of the entire solar system: well over a million Earths
could fit inside its bulk. The total energy radiated by the Sun averages 383
billion trillion kilowatts, the equivalent of the energy generated by 100 billion
tons of TNT exploding each and every second.
Planetary configurations
• Some of the definitions below make the assumption of coplanar circular orbits. True
planetary orbits are ellipses with low eccentricity and inclinations are small so the
concepts are applicable in real cases.
• Copernicus correctly stated that the farther a planet lies from the Sun, the slower it
moves around the Sun. When the Earth and another planet pass each other on the
same side of the Sun, the apparent retrograde loop occurs from the relative motions
of the other planet and the Earth.
PH507
Multi-wavelength
Professor Michael Smith
24

As we view the planet from the moving Earth, our line of sight reverses its
angular motion twice, and the three-dimensional aspect of the loop comes about
because the orbits of the two planets are not coplanar. This passing situation is
the same for inferior or superior planets.

A Retrograde loop occurs when a superior planet moves through opposition, and
occurs as the earth's motion about its orbit causes it to overtake the slower
moving superior planet. Thus close to opposition, the planet's motion relative to
PH507
Multi-wavelength
Professor Michael Smith
fixed background stars, follows a small loop.
25
PH507
Multi-wavelength
Professor Michael Smith
26
Lecture 4
Mass can be measured in two ways. We could count up the atoms, or count up
the molecules and grains of dust and infer the number of atoms. This method
can be used if the object is optically thin and we have good tracer: a radiation
or scattering mechanism in which the number of photons is related to the
number of particles.
Otherwise, measuring the mass of an object relies upon its gravitational
influence….on nearby bodies or on itself (self-gravity).
Newton’s second law states: F = m a, while the first law relates the acceleration
to a change is speed or direction.
Kepler’s empirical laws for orbital motion thus describe the nature of the
acceleration from which masses can be derived.
Kepler's Laws
First Law: The orbit of each planet is an ellipse with the Sun at one focus
p
b
F
r
S f q
a
C
Q
S = Sun, F = other focus, p = planet.
r = HELIOCENTRIC DISTANCE.
f = TRUE ANOMALY
a = SEMI-MAJOR AXIS = mean heliocentric distance),
size of the orbit.
b = SEMI-MINOR AXIS.
e = ECCENTRICITY,
defines shape of orbit.
Ellipse:
SP + PF = 2a
(1)
e = CS / a
(2)
Therefore
b2 = a2(1-e2)
(3)
which defines the
PH507
Multi-wavelength
•When CS = 0,
When CS = ,
e = 0,
e = 1,
Professor Michael Smith
27
b = a, the orbit is a circle.
the orbit is a parabola.
• q = PERIHELION DISTANCE = a - CS = a – ae
q = a(1-e)
(4)
• Q = APHELION DISTANCE = a + CS = a + ae
Q = a(1+e)
(5)
Second Law: For any planet, the radius vector sweeps out equal areas in
equal times
• Time interval t for planet to travel from p to p1 is the same as time taken for
planet to get from p2 to p3. Shaded areas are equal.
• Let the time interval t be very small. Then the arc from p to p1 can be
regarded as a straight line and the area swept out is the area of the triangle
S p p1. If f1 is the angle to p1, and f is the angle to p:
p1
p2
p3
r1
p
r
f
S
i.e
Area = 1/2 r r1 Sin (f1-f).
Since t is very small, r ~ r1 and Sin (f1-f) ~ (f1-f) = f
Area = 1/2 r2 f
The rate this area is swept out is constant according to Kepler's second
law, so
r2 df/dt = h
(6)
where h, a constant, is twice the rate of description of area by the radius
vector. It is the orbital angular momentum (per unit mass.
The total area of the ellipse is πab which is swept out in the orbital period
PH507
Multiwavelength
28
P, so using eqn (7)
2ab/P = h.
The average angular rate of motion is n = 2/P, so
n a2(1-e2)1/2 = h
(7)
Kepler’s Third Law
Kepler's third law took another ten years to develop after the first two. This law
relates the period a planet takes to travel around the sun to its average distance and
the Sun. This is sometimes called the semi major axis of an elliptical orbit.
P 2 = ka3
where P is the period and a is the average distance from the Sun.
Kepler’s Third Law follows from the central inverse square nature of the law of
gravitation. First look at Newton's law of gravitation - stated
mathematically this is
F
Gm1 m2
r2
Newton actually found that the focus of the elliptical orbits for two bodies of masses m1 and
m2 is at the centre of mass. The centripetal forces of a circular orbit are
r1
F1
v2
X
Centre of M ass
m1
m2
v1
F2
r2
2
F1
and
m1 v1
4 2 m1 r1


r1
P2
PH507
Multiwavelength
29
2
F2
m2 v2
4 2 m2 r2


r2
P2
where
v
2r
P
and since they are orbiting each other (Newton’s 2nd law)
r1
m2

r2
m1
Let's call the separation a = r1 + r2. Then;
a  r1 
m1 r1
 m1

 r1 
 1 and multiplying both sides by m2 , am2  m1 r1  m2 r1
m2
m2
or, solving for r1 ,
r1 
am2
m1  m2 
Now, since we know that the mutual gravitational force;
Fgrav  F1  F2 
then substituting for r1,
3
a 
Gm1 m2
2
a
G m1  m2  P2
42
Solving for P:
P  2
a3
G M1  M2 
Third Law is therefore: The cubes of the semi-major axes of the planetary
orbits are proportional to the squares of the planets' periods of
revolution.
Example
Europa, one of the Jovian moons, orbits at a distance of 671,000 km from the centre
of Jupiter, and has an orbital period of 3.55 days. Assuming that the mass of Jupiter
is very much greater than that of Europa, use Kepler's third law to estimate the
mass of Jupiter.
Using Kepler's third law:
PH507
Multiwavelength
30
m jupiter  meuropa
4 2 a 3

GP 2
The semi-major axis, a = 6.71 x 105 km = 6.71 x 108 m, and
the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds
Thus:
m jupiter  meuropa 
c
4 2 6.71  108
h
3
 19
.  1027 kg
c6.67  10 hc3.07  10 h
 11
5 2
and since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027 kg.
Summary of Kepler’s Laws
Summary: Measuring the mass of a planet
• Kepler’s third law gives G(M+m) =  a3/P2
Since M >> m for all planets, it isn't possible to make precise enough
determinations of P and a to determine the masses m of the planets.
However, if satellites of planets are observed, then Kepler's law can be used.
• Let mp = mass of planet
ms = mass of satellite
Ps = orbital period of satellite
as = semi-major axis of satellite's orbit
about the planet.
PH507
Then:
Multiwavelength
31
G(mp+ms) = 42 as3/Ps2
If the mass of the satellite is small compared with the mass of the planet then
mp = 42 as3/(G Ps2)
• All the major planets have satellites except Mercury and Venus. Their
masses were determined from orbital perturbations on other bodies and
later, more accurately from changes in the orbits of spacecraft.
So: we can determine the masses of massive objects if we can detect and follow the
motion of very low mass satellites. That doesn’t lead very far. How can we determine
the masses of distant stars and exoplanets?
BASIC STELLAR PROPERTIES - BINARY STARS
• For solar type stars, single:double:triple:quadruple system ratios are
45:46:8:1.
• Binary nature of stars deduced in a number of ways:
VISUAL BINARIES:
- Resolvable, generally nearby stars (parallax likely to be available)
- Relative orbital motion detectable over a number of years
ASTROMETRIC BINARY: only one component detected
SPECTROSCOPIC BINARIES:
- Unresolved
- Periodic oscillations of spectral lines (due to Doppler shift)
- In some cases only one spectrum seen
SPECTRUM BINARY: 2 sets of lines but no apparent orbital motion but
spectrum is clearly combined from stars of differing spectral class.
ECLIPSING BINARY:
- Unresolved
- Stars are orbiting in plane close to line of sight giving eclipses
observable as a change in the combined brightness with time (‘’light
curves).
Some stars may be a combination of these.
Visual Binaries
• Angular separation ≥ 0.5 arcsec (close to Sun, long orbital periods - years) –
example Sirius:
PH507
Multiwavelength
32
• Observations:
Relative positions:
 = angular
 = position
Absolute positions: Harder to measure orbits of more massive star A and
separation less massive star B about centre
of mass C which has proper motion µ.
Declination
N
M otion of centre of mass
= proper motion µ
Secondary

E

Right Ascension
B
Primary
C
A
NB parallax and aberration must also be accounted for.
• RELATIVE ORBITS:
- TRUE orbit:
q = peri-astron distance (arcsec or km)
Q = apo-astron distance (arcsec or km)
a = semi-major axis (arcsec or km)
a = (q + Q)/2
- APPARENT orbits are projected on the celestial sphere
Inclination i to plane of sky defines relation between true orbit and apparent
orbit. If i≠0° then the centre of mass (e.g. primary) is not at the focus of the
elliptical orbit.
Measurement of the displacement of the primary gives inclination and true
semi-major axis in arcseconds a".
PH507
Multiwavelength
33
i
i
Incline by 45°
Apparent orbit
True orbit
• If the parallax p in arcseconds is observable then a can be derived from a".
Earth
B
radius
of Earth's
orbit
a
a"
Sun
For i=0°
p
r = distance of binary star
a = 1 AU . a"/p"
A
(In general correction for i≠0 required).
Now lets go back to Kepler’s Law …
• From Kepler's Law, the Period P is given by
2 3
2
P =
4 a
G (mA + mB)
(26)
For the Earth-Sun system P=1 year, a=1 A.U., mA+mB~msun so 4π2/G = 1
3
a
P =
(mA + mB)
2
P in years, a in AU, mA,mB in solar masses.
From (25) and (26),
a" 3 1
mA + mB = ( )
p P2
Sum of masses is determined
• ABSOLUTE ORBITS:
d
c
rA
A
*
B
rB
e
f
(27)
B
q
A
Q
PH507
Multiwavelength
34
Semi-major axes aA = (c+e)/2 Minimum separation = q = d + e
aB = (d+f)/2
Maximum separation = Q = c + f
So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a
a = aA + a B
(28)
(and clearly r =rA + rB)
From the definition of centre of mass, mA rA = mB rB ( mA aA = mB aB)
mA/mB = aB/aA = rB/rA
(29)
So from Kepler’s Third Law, which gives the sum of the masses, and Equation
(29) above, we get the ratio of masses, ==> mA, mB. Therefore, with both, we
can solve for the individual masses of the two stars.
Spectroscopic Binaries
PH507
Multiwavelength
35
• Orbital period relatively short (hours - months) and i≠0°.
• Doppler shift of spectral lines by component of orbital velocity in line of sight
(nominal position is radial velocity of system):
wavelength
wavelength
Time
Time
2 Stars observable
1 Star observable
PH507
Multiwavelength
36
• Data plotted as RADIAL VELOCITY CURVE:
recession
+
v
(km s-1)
recession
+
0
time
approach
approach
3
2
4
1
3
4
1
3
Observer
4
2
time
-
ABSOLUTE
Relative Orbit
2
v
(km s-1) 0
RELATIVE
Relative •radial
velocity
If the
orbit iscurve
tilted to the line of
2sight (i<90°), the shape is unchanged
but velocities are reduced by a factor
1
sin i.
v
1 • Take3 a circular orbit with i=90°
a = rA + rB
v = v A + vB
4
Orbital velocities:
2
vA = 2π rA / P
1
v
v
=
2π
r
/
P
B
B
3
1
Since mA rA = mB rB
4
2
mA/mB = rB/rA = vB/vA (31)
1
rB
v
1
3
vA
r4
1
v
=
rA
v
B
• Shape of radial velocity curves
depends on orbital eccentricity
and orientation.
• In general, measured velocities are vB sin i and vA sin i, so sin i terms cancel.
• From Kepler's law
mA + mB = a3/P2 (in solar units).
Observed quantities: vA sin i => rA sin i
} a sin i
vB sin i => rB sin i
So can only deduce (mA + mB) sin3 i = (a sin i)3/P2
(32)
For a spectroscopic binary, only lower limits to each mass can be derived,
unless i is known independently.
Eclipsing Binaries
• Since stars eclipse i ~ 90°
PH507
Astrophysics
Professor Glenn White
37
• For a circular orbit:
1, 1' FIRST CONTACT
2, 2' SECOND CONTACT
3, 3' THIRD CONTACT
4, 4' FOURTH CONTACT
4' 3'
2' 1'
v
1 2
3 4
Observer in plane
• Variation in brightness with time is LIGHTCURVE.
• Timing of events gives information on sizes of stars and orbital elements.
• Shape of events gives information on properties of stars and relative
temperatures.
PH507
Multiwavelength
38
Case 1
Smaller star is hotter
Case 2
Larger star is hotter
F
or
magnitude
Secondary minimum
Primary minimum
time
Case 1 t'1
Case 2 t 1
t'2
t2
t'3
t3
t'
4
t4
t1
t'1
t2
t'2
t3
t'3
t4
t'4
PH507
Multiwavelength
39
• If orbits are circular: minima are symmetrical ie t2-t1 = t4-t3 = t2'-t1' = t4'-t3';
minima are half a period apart; eclipses are of same duration.
Assymetrical and/or unevenly spaced minima indicate eccentricity and
orientation of orbit.
• For a circular orbit:
t1 t2
t3 t 4
Distance = velocity x time
2RS = v (t2 - t1) (33)
2RL
and
2RS + 2RL = v (t4 - t1) => 2RL =
v(t4 - t2)
(34)
2RS
and ratio of radii
RS/RL = (t2 - t1) / (t4 - t2)
• Lightcurves are also affected by:
Non-total eclises
No flat minimum
Limb darkening
(non-uniform
brightness)
"rounds off"
eclipses
Ellipsoidal stars
(due to
proximity)
"rounds off"
maxima
Reflection effect
(if one star is
very bright)
(35)
PH507
Multiwavelength
40
Eclipsing-Spectroscopic Binaries
• For eclipsing binaries i ≥ 70°
(sin3i > 0.9)
• If stars are spectroscopic binaries then radial velocities are known.
So from eqns (31) and (32) masses are derived,
from eqns (33) and (34) radii are derived,
from (36) ratio of temperatures is derived
Examine spectra and lightcurve to determine which radius corresponds with which
mass and temperature:
+
v
-
B
From radial velocity curve
star A is more massive
A
Initially A is approaching (blue shift)
so first eclipse is A in front of B
Since first eclipse is primary eclipse
B is hotter than A
F
If 2 sets of lines are seen then B is larger
If 1 set of lines is seen then A is larger
time
• Since Luminosity L = 4 R2 T4, the ratio of Luminosities is derived from
(TO BE DISCUSSED LATER IN COURSE!)
LA
LB
=
Summary
Type
Visual
Spectroscopic
Eclipsing
Eclipsing/
LA/LB
spectroscopic
RA
RB
2
TA
4
TB
Observed
p, motion on sky
Apparent magnitudes
Derived
a, e, i, mA, mB
LA, LB
velocity curves
lightcurves
light + velocity curves
MA/MB, (MA+MB)sin3i, a sin i
e, i, RS/RL
MA, MB, RA, RB, TA/TB, a, e, i,
distance
LA, LB, TA, T
164PH507
Multiwavelength
41
Lecture 6: Extrasolar Planets
134 other stars are now known to possess planetary systems. 157 planets have
been discovered. Although none of the stars has been directly imaged, the
effects of the gravity tugging at the stars, as well as the way that gravitation
affects can affect material close to the stars, has been clearly seen.
Disc of material around the star Beta Pictoris – the image of the bright central
star has been artificially blocked out by astronomers using a ‘Coronograph’
• How can we discover extrasolar planets?
• Characteristics of the exoplanet population
• Planet formation
• Explaining the properties of exoplanets
Rapidly developing subject - first extrasolar planet around an ordinary star only
discovered in 1995 by Mayor & Queloz.
Observations thought to be secure, but theory still preliminary...
Resources. For observations, a good starting point is Berkeley extrasolar planets search
homepage
http://exoplanets.org/
Theory: Annual Reviews article by Lissauer (1993) is a good summary of the state of
theory prior to the discovery of extrasolar planets
Definition of a planet
Simplest definition is based solely on mass
164PH507
Multiwavelength
42
• Stars: burn hydrogen (M > 0.075 Msun)
• Brown dwarfs: burn deuterium
• Planets: do not burn deuterium (M < 0.013 Msun)
Deuterium burning limit occurs at around 13 Jupiter masses (1 MJ = 1.9 x 1027 kg ~
0.001 Msun It is important to realise that for young objects, there is no large change in
properties at the deuterium burning limit. ALL young stars / brown dwarfs / planets
liberate gravitational potential energy as they contract
Types of planet
Giant planets (gas giants, `massive’ planets)
• Solar System prototypes: Jupiter, Saturn, Uranus...
• Substantial gaseous envelopes
• Masses of the order of Jupiter mass
• In the Solar System, NOT same composition as Sun
• Presence of gas implies formation while gas was still prevelant
Terrestrial planets
• Prototypes: Earth, Venus, Mars
• Primarily composed of rocks
• In the Solar System (ONLY) orbital radii less than giant planets
Much more massive terrestrial planets could exist (>10 Earth masses), though none are
present in the Solar System. The Solar system also has asteroids, comets, planetary
satellites and rings - we won’t discuss those in this course.
Detecting extrasolar planets
(1) Direct imaging - difficult due to enormous star / planet flux ratio
(2) Radial velocity
• Observable: line of sight velocity of star orbiting centre of mass of star - planet
binary system
• Most successful method so far - all detections to date
(3) Astrometry
164PH507
Multiwavelength
43
• Observable: stellar motion in plane of sky
• Very promising future method: Keck interferometer, GAIA, SIM
(4) Transits
• Observable: tiny drop in stellar flux as planet transits stellar disc
• Requires favourable orbital inclination
• Jupiter mass exoplanet observed from ground HD209458b
• Earth mass planets detectable from space (Kepler (2007 launch. NASA
Discovery mission), Eddington)
(5) Gravitational lensing
• Observable: light curve of a background star lensed by the gravitational
influence of a foreground star. The light curve shape is sensitive to whether the
lensing star is a single star or a binary (star + planet is a special case of the
binary)
• Rare - requires monitoring millions of background stars, and also unrepeatable
• Some sensitivity to Earth mass planets
Each method has different sensitivity to planets at various orbital radii - complete
census of planets requires use of several different techniques
164PH507
Multiwavelength
44
Planet detection method : Radial velocity technique
A planet in a circular orbit around star with semi-major axis a
Assume that the star and planet both rotate around the centre of mass with an angular
velocity:
Using a1 M* = a2 mp and a = a1 + a2, then the stellar speed (v* = a ) in an inertial
frame is:
(assuming mp << M*). i.e. the stellar orbital speed is small.
For a circular orbit, observe a sin-wave variation of the stellar radial velocity, with an
amplitude that depends upon the inclination of the orbit to the line of sight:
Hence, measurement of the radial velocity amplitude produces a constraint on:
mp sin(i)
(assuming stellar mass is well-known, as it will be since to measure radial velocity we
need exceptionally high S/N spectra of the star).
Observable is a measure of mp sin(i).
-> given vobs, we can obtain a lower limit to the planetary mass
In the absence of other constraints on the inclination, radial velocity searches
provide lower limits on planetary masses
Magnitude of radial velocity:
Sun due to Jupiter:
Sun due to Earth:
12.5 m/s
0.1 m/s
164PH507
Multiwavelength
45
i.e. extremely small -
10 m/s is Olympic 100m running pace
Spectrograph with a resolving power of 105 will have a pixel scale ~ 10-5 c ~ few km/s
Therefore, specialized techniques that can measure radial velocity shifts of ~10-3 of
a pixel stably over many years are required
High sensitivity to small radial velocity shifts is achieved by:
• comparing high S/N = 200 - 500 spectra with template stellar spectra
• using a large number of lines in the spectrum to allow shifts of much less than
one pixel to be determined.
Absolute wavelength calibration and stability over long timescales is achieved by:
• passing stellar light through a cell containing iodine, imprinting large number
of additional lines of known wavelength into the spectrum
• with the calibrating data suffering identical instrumental distortions as the data
Error sources:
(1) Theoretical: photon noise limit
• flux in a pixel that receives N photons uncertain by ~ N1/2
• implies absolute limit to measurement of radial velocity
• depends upon spectral type - more lines improve signal
• around 1 m/s for a G-type main sequence star with spectrum recorded at
S/N=200
• practically, S/N=200 can be achieved for V=8 stars on a 3m class
telescope in survey mode
(2) Practical:
• stellar activity - young or otherwise active stars are not stable at the m/s
level and cannot be monitored with this technique
• remaining systematic errors in the observations
Currently, the best observations achieve:
 ~ 3 m/s
...in a single measurement. Thought that this error can be reduced to around 1 m/s with
further refinements, but not substantially further. The very highest Doppler precisions
of 1 m/s are capable\of detecting planets down to about 5 earth masses.
Radial velocity monitoring detects massive planets, especially those at small a, but
is not sensitive enough to detect Earth-like planets at ~ 1 AU.
164PH507
Multiwavelength
46
Examples of radial velocity data
51 Peg b was the first known exoplanet with a 4 day, circular orbit: a hot Jupiter, \lying
close to the central star.
Example of a planet with an eccentric orbit: e=0.67
164PH507
Multiwavelength
47
Summary: observables
(1) Planet mass, up to an uncertainty from the normally unknown inclination of
the orbit. Measure mp sin(i)
(2) Orbital period -> radius of the orbit given the stellar mass
(3) Eccentricity of the orbit
Summary: selection function
Need to observe full orbit of the planet: zero sensitivity to planets with P > Psurvey
For P < Psurvey, minimum mass planet detectable is one that produces a radial velocity
signature of a few times the sensitivity of the experiment (this is a practical detection
threshold)
Which planets are detectable?
m p sin i  a
1
2
164PH507
Multiwavelength
48
Current limits:
• Maximum a ~ 3.5 AU (ie orbital period ~ 7 years)
• Minimum mass ~ 0.5 Jupiter masses at 1 AU, scaling with square root of semimajor axis
• No strong selection bias in favour / against detecting planets with different
eccentricities
Of the first 100 stars found to harbor planets, more than 30 stars host a Jupiter-sized
world in an orbit smaller than Mercury's, whizzing around its star in a matter of days.
Planet formation is a contest, where a growing planet must fight for survival lest it be
swallowed by the star that initially nurtured it.
Planet detection method : Astrometry
Conceptually identical to radial velocity searches. Light from a planet-star binary is
dominated by star. Measure stellar motion in the plane of the sky due to presence
of orbiting planet. Must account for parallax and proper motion of star.
Magnitude of effect: amplitude of stellar wobble (half peak displacement) for an orbit in
the plane of the sky is
 mp 
  a
a1  
 M* 
In terms of the angle:
 m p  a 
 
  
M
 *  d 
for a star at distance d. Note we have again used mp << M*
Writing the mass ratio q = mp / M*, this gives:
164PH507
Multiwavelength
49
Note:
• Units here are milliarcseconds - very small effect
• Different dependence on a than radial velocity method - astrometric planet
searches are more sensitive at large a
• Explicit dependence on d (radial velocity measurements also less sensitive for
distant stars due to lower S/N spectra)
• Detection of planets at large orbital radii still requires a search time comparable
to the orbital period
Detection threshold as function of semi-major axis
• Lack of units deliberate! Astrometric detection not yet achieved
• As with radial velocity, dependence on orbital inclination, eccentricity
• Very promising future: Keck interferometer, Space Interferometry Mission
(SIM), ESA mission GAIA, and others
• Planned astrometric errors at the ~10 microarcsecond level – good enough to
detect planets of a few Earth masses at 1 AU around nearby stars
Lecture 7:
Planet detection method : Transits
Simplest method: look for drop in stellar flux due to a planet
transiting across the stellar disc
Needs luck or wide-area surveys - transits only occur if the orbit is almost edge-on
For a planet with radius rp << R*, probability of transits is:
164PH507
Multiwavelength
50
Close-in planets are more likely to be detected. P = 0.5 % at 1AU, P = 0.1 % at the
orbital radius of Jupiter
What can we measure from the light curve?
(1)
Depth of transit = fraction of stellar light blocked
This is a measure of planetary radius!
In practice, isolated planets with masses between ~ 0.1 MJ and 10 MJ, where MJ
is the mass of Jupiter, should have almost the same radii (i.e. a flat mass-radius
relation).
-> Giant extrasolar planets transiting solar-type stars produce transits
with a depth of around 1%.
Close-in planets are strongly irradiated, so their radii can be (detectably) larger.
But this heating-expansion effect is not generally observed for short-period
planets.
(2)
(3)
(4)
Duration of transit plus duration of ingress, gives measure of the orbital radius
and inclination
Bottom of light curve is not actually flat, providing a measure of stellar limbdarkening
Deviations from profile expected from a perfectly opaque disc could provide
evidence for satellites, rings etc
Photometry at better than 1% precision is possible (not easy!) from the ground.
HST reached a photometric precision of 0.0001.
Potential for efficient searches for close-in giant planets
Transit depth for an Earth-like planet is:
164PH507
Multiwavelength
51
Photometric precision of ~ 10-5 seems achievable from space
May provide first detection of habitable Earth-like planets
NASA’s Kepler mission, ESA version Eddington
A reflected light signature must also exist, modulated on the orbital period,
even for non-transiting planets. No detections yet.
Planet detection method : Gravitational microlensing
Light is deflected by gravitational field of stars, compact objects, clusters of galaxies,
large-scale structure etc
Simplest case to consider: a point mass M (the lens) lies along the line of sight to a
more distant source
Define:
• Observer-lens distance
• Observer-source distance
• Lens-source distance
Azimuthal symmetry -> light from the source appears as a ring
...with radius R0 - the Einstein ring radius - in the lens plane
Gravitational lensing conserves surface brightness, so the distortion of the image of
the source across a larger area of sky implies magnification.
The Einstein ring radius is given by:
Dl
Ds
Dls
164PH507
Multiwavelength
52
Suppose now that the lens is moving with a velocity v. At time t, the apparent distance
(in the absence of lensing) in the lens plane between the source and lens is r0.
Defining u = r0 / R0, the amplification is:
Note: for u > 0, there is no symmetry, so the pattern of images is not a ring and is
generally complicated. In microlensing we normally only observe the magnification A,
so we ignore this complication...
Notes:
(1) The peak amplification depends upon the impact parameter, small impact
parameter implies a large amplification of the flux from the source star
(2) For u = 0, apparently infinite magnification! In reality, finite size of source
limits the peak amplification
(3) Geometric effect: affects all wavelengths equally
(4) Rule of thumb: significant magnification requires an impact parameter
smaller than the Einstein ring radius
(5) Characteristic timescale is the time required to cross the Einstein ring radius:
Optical depth to microlensing
Define the optical depth to microlensing as:
164PH507
Multiwavelength
53
This is just the integral of the area of the Einstein ring along the line of sight to the
source. For a uniform density of lenses, can easily show that the maximum contribution
comes from lenses halfway to the source.
Several groups have monitored stars in the Galactic bulge and the Magellanic clouds to
detect lensing of these stars by foreground objects (MACHO, Eros, OGLE projects).
Original motivation for these projects was to search for dark matter in the form of
compact objects in the halo.
Timescales for sources in the Galactic bulge, lenses ~ halfway along the line of sight:
• Solar mass star ~ 1 month
• Jupiter mass planet ~ 1 day
• Earth mass planet ~ 1 hour
The dependence on M1/2 means that all these timescales are observationally feasible.
However, lensing is a very rare event, all of the projects monitor millions of source
stars to detect a handful of lensing events.
Lensing by a single star
Lensing by a star and a planet
164PH507
Multiwavelength
54
What has this to do with planets?
Binaries can also act as lenses
Light curve for a binary lens is more complicated, but a characteristic is the presence of
sharp spikes or caustics. With good enough monitoring, the parameters of the binary
doing the lensing can be recovered.
Orbiting planet is just a binary with mass ratio q << 1
Planet search strategy:
• Monitor known lensing events in realtime with dense, high precision
photometry from several sites
• Look for deviations from single star lightcurve due to planets
• Timescales ~ a day for Jupiter mass planets, ~ hour for Earths
• Most sensitive to planets at a ~ R0, the Einstein ring radius
• Around 3-5 AU for typical parameters
Sensitivity to planets
Complementary to other methods:
164PH507
Multiwavelength
55
Actual sensitivity is hard to evaluate: depends upon frequency of photometric
monitoring (high frequency needed for lower masses), accuracy of photometry (planets
produce weak deviations more often than strong ones)
Very roughly: observations with percent level accuracy, several times per night, detect
Jupiter, if present, with 10% efficiency
Many complicated light curves observed:
...but no strong evidence for planets seen yet
RV, Doppler technique (v = 3m/s)
164PH507
Multiwavelength
Astrometry: angular oscillation
Photometry: transits - close-in planets
Microlensing:
56
164PH507
Multiwavelength
57
Direct detection! Photometric :
2005 image of 2M1207 (blue) and its planetary companion, 2M1207b, one of the
first exoplanets to be directly imaged, in this case from the Very Large
Telescope array in Chile
Spectroscopic? The starlight scattered from the planet can be distinguished
from the direct starlight because the scattered light is Doppler shifted by virtue
of the close-in planet's relatively fast orbital velocity (~ 150 km/sec).
Superimposed on the pattern given by the planet's albedo changing slowly with
wavelength, the spectrum of the planet's light will retain the same pattern of
photospheric absorption lines as in the direct starlight.
Lecture 8 The extrasolar planet population
Current status of exoplanet searches:
Radial Velocity Method (Doppler technique, gravitational wobble)
• 156 exoplanets hosted by134 stars discovered, with masses M.sin(i) as low as 6
Earth masses. Generally:~ 0.06 MJ and 10MJ…
164PH507
Multiwavelength
58
164PH507
Multiwavelength
59
and orbital radii from 0.02 AU to 6 AU.
• Planet fraction among ~ solar-type stars exceeds 7%
• Most are beyond 1 AU
• Around 1% of stars have hot Jupiters - massive planets at orbital radii a < 0.1
AU
• Four very low mass planets have been detected ….20 earth masses.
• Planet occurrence rises rapidly with stellar metallicity
• Multiple planets are common, often in resonant orbits
Microlensing: two strong detections, low detection rate imply upper limit of ~1/3 on the
fraction of lensing stars (~ 0.3 Msun) with Jupiter mass planets at radii to which lensing
is most sensitive (1.5 - 4 AU)
Transits: 7 known planets (5 found with OGLE photometrically – dimming).
Interesting upper limit from non-detection of transits in globular cluster 47 Tuc
Transits + Doppler yields mass and size, hence the density of the planet: 0.2 – 1.4
gm/cm3 : mainly gaseous. In addition, sodium and nitrogen found in their atmospheres.
Direct Imaging: reports of detections with HST and VLT.
Eccentricity: • Except at very small radii, typical planet orbit has significant
eccentricity
164PH507
Multiwavelength
60
164PH507
Multiwavelength
61
Eccentricity:
Eccentricity vs planet mass
164PH507
Multiwavelength
62
164PH507
Multiwavelength
Nothing very striking in these plots:
63
164PH507
Multiwavelength
64
• Accessible region of mp - a space is fully occupied by detected planets
• Ignoring the hot Jupiters, no obvious correlation between planet mass and
eccentricity...
Results from radial velocity searches
(1) Massive planets exist at small orbital radii. Closest in planet is at a = 0.035 AU,
cf Mercury at ~ 0.4 AU. Less than 10 Solar radii.
(2) Hot Jupiters have close to circular orbits. All detected planets with semi-major
axis < 0.07 AU have low e. This is similar to binary stars, and is likely due to
tidal circularization.
(3) Remaining planets have a wide scatter in e, including some planets with large e.
Note that the distance of closest approach is a(1-e), and that the effect of
tidal torques scales as separation d-6. The very eccentric planet around
HD80606 (a = 0.438 AU, e = 0.93, a(1-e) = 0.03 AU) may pose some
problems for tidal circularization theory.
164PH507
Multiwavelength
65
Account for this by considering only planets with masses large enough to be detectable
at any a < 2.7 AU.
-> dN / dlog(a) rises steeply with orbital radius
Implies that the currently detected planet fraction ~7% is likely to be a substantial
underestimate of the actual fraction of stars with massive planets.
Models suggest 15-25% of solar-type stars may have planets with masses 0.2 MJ < mp <
10 MJ.
Strong selection effect in favour of detecting planets at small orbital radii, arising from:
- lower mass planets can be detected there
- mass function rises to smaller masses
Observed mass function increases to smaller Mp:
164PH507
Multiwavelength
Note: the brown dwarf desert!
66
164PH507
Multiwavelength
67
Constraint from monitoring of 43 microlensing events. Typically, the lenses are
low mass stars.
At most 1/3 of 0.3 Solar mass stars have Jupiter mass planets between 1.5
AU and 4 AU.
Currently consistent with the numbers seen in radial velocity searches - not yet
known whether there is a difference in the planet fraction between 0.3 - 1 Solar
mass stars.
Transit lightcurve from Brown et al. (2001)
Consistent with expectations - the probability of a transiting system is ~10%.
Measured planetary radius rp = 1.35 J:
• Proves we are dealing with a gas giant.
• Somewhat larger than models for isolated (non-irradiated) planets effect of environment on structure.
• In detail, suggests planet reached its current orbit within a few x 10 Myr
after its formation.
Precision of photometry with HST / STIS impressive...
Metallicity distribution of stars with and without planets
164PH507
Multiwavelength
68
Left plot: metallicity of stars with planets (shaded histogram) compared to a sample of
stars with no evidence for planets (open histogram) (data from Santos, Israelian &
Mayor, 2001)
Host star metallicity
Planets are preferentially found around stars with enhanced metal abundance.
Cause or effect? High metal abundance could:
(a) Reflect a higher abundance in the material which formed the star +
protoplanetary disc, making planet formation more likely.
(b) Result from the star swallowing planets or planetesimals subsequent to
planets forming. If the convection zone is fairly shallow, this can apparently
enrich the star with metals even if the primordial material had Solar abundance.
Detailed pattern of abundances can distinguish these possibilities, but results currently
still controversial.
Lack of transits in 47 Tuc
A long HST observation monitored ~34,000 stars in the globular cluster 47 Tuc looking
for planetary transits.
Locally: 1% of stars have hot Jupiters
~ 10% of those show transits
 Expect 10 -3 x 34,000 ~ few tens of planets
None were detected. Possible explanations:
164PH507
Multiwavelength
69
• Low metallicity in cluster prevented planet formation
• Cluster environment destroyed discs before planets formed
• Stellar fly-bys ejected planets from bound orbits
All of these seem plausible - make different predictions for other clusters.
164PH507
Multiwavelength
70
Lecture 9: Radiation processes
Almost all astronomical information from beyond the Solar
System comes to us from some form of electromagnetic
radiation (EMR). We can now detect and study EMR over a
range of wavelength or, equivalently, photon energy,
covering a range of at least 1016- from short wavelength,
high photon energy gamma rays to long wavelength low
energy radio photons. Out of all this vast range of
wavelengths, our eyes are sensitive to a tiny slice of
wavelengths- roughly from 4500 to 6500 Å. The range of
wavelengths our eyes are sensitive to is called the visible
wavelength range. We will define a wavelength region
reaching somewhat shorter (to about 3200 Å) to somewhat
longer (about 10,000 Å) than the visible as the optical part of
the spectrum. (Note: Physicists measure optical
wavelengths in nanometers (nm). Astronomers tend to use
_Angstroms. 1 Å = 10-10 m = 0.1 nm. Thus, a physicist would
say the optical region extends from 320 to 1000 nm.)
All EMR comes in discrete lumps called photons. A photon
has a definite energy and frequency or wavelength. The
relation between photon energy (Eph) and photon frequency
 is given by:
Eph = h
164PH507
Multiwavelength
71
or, since c = 
E ph 
hc

where h is Planck’s constant and  is the wavelength, and c
is the speed of light. The energy of visible photons is
around a few eV (electron volts). (An electron volt is a nonmetric unit of energy that is a good size for measuring
energies associated with changes of electron levels in atoms,
and also for measuring energy of visible light photons. 1 eV
= 1.602 x 10-19 Joules.)
In purely astronomical terms, the optical portion of the
spectrum is important because most stars and galaxies emit
a significant fraction of their energy in this part of the
spectrum. (This is not true for objects significantly colder
than stars - e.g. planets, interstellar dust and molecular
clouds, which emit in the infrared or at longer wavelengths
- or significantly hotter- e.g. ionised gas clouds, neutron
stars, which emit in the ultraviolet and x-ray regions of the
spectrum. Another reason the optical region is important is
that many molecules and atoms have electronic transitions
in the optical wavelength region.
Blackbody Radiation
Where then does a thermal continuous spectrum come from?
Such a continuous spectrum comes from a blackbody whose
spectrum depends only upon the absolute temperature. A
blackbody is so named because it absorbs all electromagnetic
energy incident upon it - it is completely black. To be in perfect
thermal equilibrium, however, such a body must radiate energy
at exactly the same rate that it absorbs energy; otherwise, the
164PH507
Multiwavelength
72
body will heat up or cool down (its temperature will change).
Ideally, a blackbody is a perfectly insulated enclosure within
which radiation has come into thermal equilibrium with the
walls of the enclosure. Practically, blackbody radiation may be
sampled by observing the enclosure through a tiny
pinhole in one of the walls. The gases in the interior of a star are
opaque
(highly
absorbent)
to all radiation (otherwise, we would see the stellar interior at
some wavelength!); hence, the radiation there is blackbody in
character. We sample this radiation as it slowly leaks from the
surface of the star - to a rough approximation, the continuum
radiation from some stars is blackbody in nature.
We will define the regions of the Electromagnetic Spectrum to
have wavelengthds as follows:
 Gamma-rays: < 0.1Å, highest frequency, shortest
wavelength, highest energy.

X-Rays: 0.1Å -- 100Å

Ultraviolet light: 100Å -- 3000Å

Visible light: 3000Å -- 10000Å = 1µm (micrometer or
micron)

Infrared Light: 1µm -- 1mm

Radio waves: >1mm, lowest frequency, longest
wavelength, lowest energy.
Planck’s Radiation Law
After Maxwell's theory of electromagnetism appeared in 1864,
many attempts were made to understand blackbody radiation
theoretically. None succeeded until, in 1900, Max K. E. L. Planck
(1858-1947) postulated that electromagnetic energy can propagate only in discrete quanta, or photons, each of energy E = hv.
164PH507
Multiwavelength
73
He then derived the spectral intensity relationship, or Planck
blackbody radiation law:



2h 3  1
I( )d   2  h 

 c   kT



 e  1 


where I(v)dv is the intensity (J/m2 . s . sr) of radiation from a
blackbody at temperature T in the frequency range between v
and v + dv, h is Planck's constant, c is the speed of light, and k
is Boltzmann's constant. Note the exponential in the denominator.
Because the frequency v and wavelength of electromagnetic
radiation are related by v = c, we may also express Planck's
formula in terms of the intensity emitted per unit wavelength
interval:
This is illustrated for several values of T:
164PH507
Multiwavelength
74
Note
that
both
I() and I(v)
increase as the
blackbody temperature increases - the blackbody becomes
brighter. This effect is easily interpreted when we note that
I(v)∆v is directly proportional to the number of photons emitted
per second near the energy hv. The Planck function is special
enough so that its given its own symbol, B() or B(v), for
intensity.
Wien’s Law
A blackbody emits at a peak intensity that shifts to shorter
wavelengths as its temperature increases.
164PH507
Multiwavelength
75
 Wilhelm Wien (1864-1928) expressed the wavelength
at which the maximum intensity of blackbody
radiation is emitted - the peak (that wavelength for
which dI()/d = 0) of the Planck curve (found from
taking the first derivative of Planck's law) - by Wien's
displacement law:
max = 2.898 x 10-3 / T
where max is in metres when T is in Kelvin. Note that
because maxT = constant, increasing one proportionally
decreases the other.
164PH507
Multiwavelength
76
For example, the continuum spectrum from our Sun is
approximately blackbody, peaking at max ≈ 500 nm.
Therefore, the surface temperature is near 5800 K.
The Law of Stefan and Boltzmann
The area under the Planck curve (integrating the Planck
function) represents the total energy flux, F (W/m2),
emitted by a blackbody when we sum over all wavelengths
and solid angles:
164PH507
Multiwavelength
77
where = 5.669 x 10-8 W/m2 . K4. The strong temperature
dependence of this formula was first deduced from
thermodynamics in 1879 by Josef Stefan (1835-1893) and
was derived from statistical mechanics in 1884 by
Boltzmann. Therefore we call the expression the StefanBoltzmann law. The brightness of a blackbody increases as
the fourth power of its temperature. If we approximate a
star by a blackbody, the total energy output per unit time of
the star (its power or luminosity in watts) is just L =
4R2T4 since the surface area of a sphere of radius R is
4R2
To summarise: A blackbody radiator has a number of
special characteristics. One, a blackbody emits some energy
at all wavelengths. Two, a hotter blackbody emits more
energy per unit area and time at all wavelengths than does a
cooler one. Three, a hotter blackbody emits a greater
proportion of its radiation at shorter wavelengths than does a
cooler one. Four, the amount of radiation emitted per
second by a unit surface area of a blackbody depends on the
fourth power of its temperature.
Stellar Material
Our Sun is the only star for which I( has been accurately
observed. Indeed, Ibol is related to the solar constant: the total
164PH507
Multiwavelength
78
solar radiative flux received at the Earth’s orbit outside our
atmosphere (1370 W/m2). The solar luminosity L (3.86 x 1026
W) is calculated from the solar constant in the following manner.
Using the inverse-square law, we find the radiative flux at the
Sun’s surface R. Then Lis just  times this flux. The solar
energy distribution curve may be approximated by a Planck
blackbody curve at the effective temperature Teff, defined as the
temperature of a blackbody that would emit the same total
energy as an emitting body, such as the Sun or a star. Then the
Stefan-Boltzmann law implies
L = 4π R2 T4eff
J s-1
where is the Stefan-Boltzmann constant.
Stellar Atmospheres
The spectral energy distribution of starlight is determined in a
star’s atmosphere, the region from which radiation can freely
escape. To understand stellar spectra, we first discuss a model
stellar atmosphere and investigate the characteristics that
determine the spectral features.
Physical Characteristics
The stellar photosphere, a thin, gaseous layer, shields the stellar
interior from view. The photosphere is thin relative to the stellar
radius, and so we regard it as a uniform shell of gas. The physical
properties of this shell may be approximately specified by the
average values of its pressure P, temperature T, and chemical
composition µ (chemical abundances).
164PH507
Multiwavelength
79
We also assume that the gas obeys the perfect-gas law:
P =nkT
where k is Boltzmann’s constant. This relationship is also known
as Boyle’s Law.
An important result that follows from it is that the kinetic energy
of a particle, or assemblage of particles, is given by the
relationship;
KE 
3
kT
2
Thus temperature is just a measure of the kinetic energy of a gas,
or an assemblage of particles. This equation applies equally well
to a star as a whole, as to a single particle, and later we will look
at the comparison between a star’s kinetic and gravitational
(potential) energies.
The kinetic energy is also a measure of the velocity that atoms or
molecules are moving about at - the hotter they are, the faster
they move. Thus, for a cloud of gas surrounding a hot star of
temperature T = 15,000 K, which consists of hydrogen atoms
(mass = 1.67 10-27 kg);
3
1
kT  KE  mv2
2
2
v
3kT
 19 km s 1  50,000 mph
m
The particle number density is related to both the mass density
(kg/m3) and the composition (or mean molecular weight) µ by
the following definition of µ:
164PH507
Multiwavelength
80
1


mH n

where mH = 1.67 x 10-27 kg is the mass of a hydrogen atom. For
a star of pure atomic hydrogen, µ = 1. If the hydrogen is
completely ionised, µ = 1/2 because electrons and protons
(hydrogen nuclei) are equal in number and electrons are far less
massive than protons. In general, stellar interior gases are
ionised and

1
3
1
2X  Y  Z
4
2
where X is the mass fraction of hydrogen, Y is that of helium,
and Z is that of all heavier elements. The mass fraction is the
percentage by mass of one species relative to the total. Thus, for a
pure hydrogen star (X=1.0, Y = 0.0, Z = 0.0),  ~ 0.5, and for a
white dwarf star (X = 0.0, Y = 1.0, Z = 0.0)  ~ 1.33.
Temperatures
The continuous spectrum, or continuum, from a star may be
approximated by the Planck blackbody spectral-energy
distribution. For a given star, the continuum defines a colour
temperature by fitting the appropriate Planck curve. We can
also define the temperature from Wien’s displacement law:
maxT = 2.898 x 10-3 m . K which states that the peak intensity of
the Planck curve occurs at a wavelength max that varies
inversely with the Planck temperature T. The value of max
then defines a temperature. Also note here that the hotter a star
is, the greater will be its luminous flux (in W/m2), in accordance
with the Stefan-Boltzmann law: F = T4 where = 5.67 x 10-8
W/m2 . K4. Then the relation
164PH507
Multiwavelength
81
L = 4πR2T4eff
defines the effective temperature of the photosphere.
A word of caution: the effective temperature of a star is usually
not identical to its excitation (Boltzmann eqn) or ionisation
temperature (Saha eqn) because spectral-line formation
redistributes radiation from the continuum. This effect is called
line blanketing and becomes important when the numbers and
strengths of spectral lines are large.
When spectral features are not numerous, we can detect the
continuum between them and obtain a reasonably accurate value
for the star’s effective surface temperature. The line blanketing
alters the atmosphere’s blackbody character.
Spectrophotometry
164PH507
Multiwavelength
82
The goal of the observational astronomer to to make
measurements of the EMR from celestial objects with as
much detail, or the finest resolution, possible. There are of
course different types of detail that we want to observe.
These include angular detail, wavelength detail, and time
detail. The perfect astronomical observing system would
tell us the amount of radiation, as a function of wavelength,
from the entire sky in arbitrarily small angular slices. Such a
system does not exist!
We are always limited in angular and wavelength coverage,
and limited in resolution in angle and wavelength. If we
want good information about the wavelength distribution
of EMR from an object (spectroscopy or spectrophotometry)
we have to give up angular detail. If we want good angular
resolution over a wide area of sky (imaging) we usually
have to give up wavelength resolution or coverage.
The ideal goal of spectrophotometry is to obtain the
spectral energy distribution (SED) of celestial objects, or
how the energy from the object is distributed in
wavelength. We want to measure the amount of energy
received by an observer outside the Earth's atmosphere, per
second, per unit area, per unit wavelength or frequency
interval. Units of spectral flux (in cgs) look like:
f  = ergs s-1 cm-2 Å -1
if we measure per unit wavelength interval, or
f = ergs s-1 cm-2 Hz -1
(pronounced f nu if we measure per unit frequency interval.
164PH507
Multiwavelength
83
Classifying Stellar Spectra
Observations
A single stellar spectrum is produced when starlight is focused
by a telescope onto a spectrometer or spectrograph, where it is
dispersed (spread out) in wavelength and recorded
photographically or electronically. If the star is bright, we may
obtain a high-dispersion spectrum, that is, a few mÅ per
millimetre on the spectrogram, because there is enough radiation
to be spread broadly and thinly. At high dispersion, a wealth of
detail appears in the spectrum, but the method is slow (only one
stellar spectrum at a time) and limited to fairly bright stars.
Dispersion is the key to unlocking the information in starlight.
The Spectral-Line Sequence
At first glance, the spectra of different stars seem to bear no
relationship to one another. In 1863, however, Angelo Secchi
found that he could crudely order the spectra and define
different spectral types. Alternative ordering schemes appeared
in the ensuing years, but the system developed at the Harvard
Observatory by Annie J. Cannon and her colleagues was
internationally adopted in 1910. This sequence, the Harvard
spectral classification system, is still used today. (About 400,000
stars were classified by Cannon and published in various
volumes of the Henry Draper Catalogue, 1910-1924, and its
Extension, 1949.
At first, the Harvard scheme was based upon the strengths of the
hydrogen Balmer absorption lines in stellar spectra, and the
spectral ordering was alphabetical (A through to P). Some letters
were eventually dropped, and the ordering was rearranged to
correspond to a sequence of decreasing temperatures (see the
effects of the Boltzmann and Saha equations): OBAFGKMRNS.
164PH507
Multiwavelength
84
Stars nearer the beginning of the spectral sequence (closer to O)
are sometimes called early-type stars, and those closer to the M
end are referred to as late-type. Each spectral type is divided
into ten parts from 0 (early) to 9 (late); for example, . . . F8 F9 G0
G1 G2 . . . G9 K0 . . . . In this scheme, our Sun is spectral type
G2. In 1922, the International Astronomical Union (IAU)
adopted the Harvard system (with some modifications) as the
international standard.
Many mnemonics have been devised to help students retain the
spectral sequence. A variation of the traditional one is “Oh, Be a
Fine Girl, Kiss Me Right Now, Smack.”
The next Figure shows exemplary stellar spectra arranged in
order; note how the conspicuous spectral features strengthen and
diminish in a characteristic way through the spectral types.
164PH507
Multiwavelength
85
Comparison of spectra observed for seven different stars having a
range of surface temperatures. The hottest stars, at the top, show
lines of helium and multiply-ionised heavy elements. In the
coolest stars, at the bottom, helium lines are not seen, but lines of
neutral atoms and molecules are plentiful. At intermediate
temperatures, hydrogen lines are strongest. The actual
compositions of all seven stars are about the same.
164PH507
Multiwavelength
86
The Temperature Sequence
The spectral sequence is a temperature sequence, but we must
carefully qualify this statement. There are many different kinds
of temperatures and many ways to deter-mine them.
Theoretically, the temperature should correlate with spectral
type and so with the star’s colour. From the spectra of
intermediate-type stars (A to K), we find that the (continuum)
164PH507
Multiwavelength
87
colour temperature does so, but difficulties occur at both ends of
the sequence. For O and B stars, the continuum peaks in the far
ultraviolet, where it is undetectable by ground-based
observations.
Through satellite observations in the far
ultraviolet, we are beginning to understand the ultraviolet
spectra of O and B stars. For the cool M stars, not only does the
Planck curve peak in the infrared, but numerous molecular
bands also blanket the spectra of these low-temperature stars.
164PH507
Multiwavelength
88
When the strengths of various spectral features are plotted
against excitation-ionisation (or Boltzmann-Saha) temperature;
the spectral sequence does correlate with this temperature as
seen below;
164PH507
Multiwavelength
89
In practice, we measure a star’s colour index, CI = B - V, to
determine the effective stellar temperature.
If the stellar
continuum is Planckian and contains no spectral lines, this
procedure clearly gives a unique temperature, but observational
uncertainties and physical effects do lead to problems: (a) for the
very hot O and B stars, CI varies slowly with Teff and small
uncertainties in its value lead to very large uncertainties in T; (b)
for the very cool M stars, CI is large and positive, but these faint
stars have not been adequately observed and so CI is not well
determined for them; (c) any instrumental deficiencies,
calibration errors, or unknown blanketing in the B or V bands
affect the value of CI - and thus the deduced T. Hence, it is best
to define the CI versus T relation observationally.
SPECTROSCOPY
• Last year discussed stellar spectra and classification on an
empirical basis:
Spectral sequence
O B A F G K M
Temperature
~40,000 K
---->
2500 K
Classification based on relative line strengths of He, H,
Ca, metal, molecular lines.
• We will now look a little deeper at stellar spectra and
what they tell us about stellar atmospheres.
Radiative Transfer Equation
• Imagine a beam of radiation of intensity I passing
through a layer of gas:
164PH507
Multiwavelength
Power passing into volume
90
Area
dA
E = I d dA d

Power passing out
of volume
E  + dE 
where I = intensity into
solid angle element d 
path length ds
NB in all these equations subscripts  can be replaced by 
In the volume of gas there is:
ABSORPTION - Power is reduced by amount
dE = -   E ds = -   I d dA d ds
where 
is the ABSORPTION COEFFICIENT or
OPACITY
= the cross-section for absorption of radiation of
wavelength  (frequency ) per unit mass of gas. Units of
are m2 kg-1
The quantity  is the fraction of power in a beam of radiation of wav
 absorbed by unit depth of gas. It has units of m-1. (NB in many texts
simply given the symbol  in the equations given here beware!)
EMISSION - Power is increased by amount
dE = j  d dA d ds
(1)
where j = EMISSION COEFFICIENT = amount of energy
emitted per second per unit mass per unit wavelength
into unit solid angle.
Units of j (j) are W kg-1 µm-1 sr-1 (W kg-1 Hz-1 sr-1) or m s-3 sr-1
(NB power production per unit volume per unit wavelength into unit so
164PH507
Multiwavelength
91
angle is  =j More confusion is possible here, since is also the
symbol used for total power output of a gas, units are W kg-1, - Bewar
So total change in power is
dE = dI d dA d = -   I d dA d ds + j  d dA d ds
which reduces to
dI = -   I ds + j  ds
dI
ds
= - I + j 
(2)
(3)
This is a form of the radiative transfer equation in the
plane parallel case.
Optical depth
• Take a volume of gas which only absorbs radiation (j =
0) at  :
dI = -   I ds
For a depth of gas s, the fractional change in intensity is
given by
I (s)

dI
 I
I (0)

s

=

0 -  ds

ln (
Integrating ==>
I (s)

I (0)

s
) = -
  ds
0

s
-
==>
I (s) = I (0) e

 ds
0

We define Optical depth 
s

   ds



(4)
164PH507
Multiwavelength
92
-

So
(5)
I (s) = I (0) e


• Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its
original value if optical depth = 1.
• Optical depth is not a physical depth. A large optical
depth can occur in a short physical distance if the
absorption coefficient  is large, or a large physical
distance if  is small.
Full Radiative Transfer Equation again
dI
= - I + j 
ds
divide by  
dI

 ds
= -I +


j



dI

d
= -I + S


As ds --> 0,  is constant over ds. 
This is the RADIATIVE TRANSFER EQUATION in the
plane parallel case.
Define:

S =
where

j


or
j =  S




and S is the SOURCE FUNCTION.
Radiative transfer in a blackbody
• Remember definition of a blackbody as a perfect absorber
and emitter of radiation. Matter and radiation are in
THERMODYNAMIC EQUILIBRIUM, ie gross properties
do not change with time. Therefore a beam of radiation
in a blackbody is constant:
(6)
164PH507
Multiwavelength
dI
ds
93
= 0 = - I + j 
from definition of source function, j =  S
==> 0 =  (I - S),
i.e. I = S.
but for a blackbody I = B
the PLANCK FUNCTION
2
B =

2hc

3
1
hc/kT
(e
- 1)
B =

2h
2
c
1
hkT
(e
- 1)
Summary: in complete thermodynamic equilibrium the source
function equals the Planck function,
i.e.
j =  B
• In studies of stellar atmospheres we make the assumption
of LOCAL THERMODYNAMIC EQUILIBRIUM (LTE),
i.e. thermodynamic equilibrium for each particular layer
of a star.
• Note that if incoming radiation at a particular
wavelength (e.g. in a spectral line) enters a blackbody gas
it is absorbed, but emission is distributed over all
wavelengths according to the Planck function. All
information about the original energy distribution of the
radiation is lost. This is what happens in interior layers of
a star where the density is high and photons of any
wavelength are absorbed in a very short distance. Such a
gas is said to be optically thick (see below).
Emission and Absorption lines
•  the absorption coefficient describes the efficiency of
absorption of material in the volume of gas. In a low
density gas, photons can generally pass through without
interaction with atoms unless they have an energy
corresponding to a particular transition (electron energy
level transition, or vibrational/rotational state transition
in
molecules).
At
this
particular
(7)
(K
164PH507
Multiwavelength
94
energy/frequency/wavelength the absorption coefficient
 is large.
• Let's imagine the volume of gas shown earlier with both
absorption and emission:
I
I  (0)
path length s
dI

d
= S - I



Multiply both sides by e and re-arrange
dI

==>
d



e + I e = S e



d
==>
d


(I e ) = S e 



integrate over whole volume, i.e. from 0 to s, or 0 to 


I e

==>


=
0

S e


0
assuming S = constant along path
I e - I(0) = S e - S
==>
==>
I
I(0) e- +
radiation left
over from light
entering box.
=
S (1 - e- )
(8)
light from radiation
emitted in the
box.
 >> 1: OPTICALLY THICK CASE
If  >> 1, then e- --> 0, and eqn (8) becomes
• Case 1
I =
164PH507
Multiwavelength
95
S
In LTE
S  = B ,
the Planck function.
So for an optically thick gas, the emergent spectrum is
the Planck function, independent of composition or
input intensity distribution. True for stellar photosphere
(the visible "surface" of a star).
• Case 2
(9)
 << 1 OPTICALLY THIN CASE
If  << 1, then e- ≈ 1 - 
(first two terms of Taylor series expansion)
eqn (8) becomes I = I(0) (1 - ) + S (1 - 1 + )
==>I = I(0) +  ( S - I(0) )
(10)
• If I(0) = 0 : no radiation entering the box (from
direction of interest):
From eqn (8)
I =  S
(= 
B in LTE)
Since  = ∫  , then
I =   s S
If  is large (true at wavelength of spectral lines) then I is large,
we see EMISSION LINES. This happens for example in gaseous
nebulae or the solar corona when the sun is eclipsed.
• If I(0) ≠ 0 , let's examine eqn (8)
I = I(0) +  ( S - I(0) )
If S > I(0) then right hand term is +ve
when  is large (ie  is large) we see higher intensity
than I(0)
==> EMISSION LINES ON BACKGROUND INTENSITY.
If S < I(0) then right hand term is -ve
when  is large (ie  is large) we see lower intensity
than I(0)
==>ABSORPTION LINES ON BACKGROUND INTENSITY.
For stars we see absorption lines. This means I(0) > S,
164PH507
Multiwavelength
96
i.e. (intensity from deeper layers) > (source function for the top layers
Assuming LTE (S = B) the source function increases as
temperature increases:I(0) = B(Tdeep layer) > S = B(Touter layer)
Therefore temperature must be increasing as we go into
the star for absorption lines to be observed.
• To summarise
- We see CONTINUUM RADIATION for an optically
thick gas
(= PLANCK FUNCTION assuming LTE).
- We see EMISSION LINES for an optically thin gas.
- We see ABSORPTION LINES + CONTINUUM for an
optically thick gas
overlaid by optically thin gas with temperature decreasing outwards.
- We see EMISSION LINES + CONTINUUM for an
optically thick gas overlaid by an optically thin gas with
temperature increasing outwards.
Atomic Spectra - Absorption & Emission line series and
continua
• Bohr theory (last year's physics unit) adequately describes
electron energy levels in Hydrogen. Quantum mechanics
is required for more massive atoms to describe the
dynamics of electrons. However, we are interested here
only in the energy levels of electron states rather than a
detailed model or description of atomic structure. We can
therefore use ENERGY LEVEL DIAGRAMS without
164PH507
Multiwavelength
97
worrying too much about the theory behind them.
• There are 3 basic photon absorption mechanisms related
to electrons. Using Hydrogen as the example, the electron
energy levels, n, are described by
E(n) = - 2 2 me e4 Z2 / n2 h2
from Bohr Theory
Bound-Bound Transitions
• BOUND - BOUND transitions give rise to spectral lines.
• ABSORPTION LINE if a photon is absorbed, causing
increase in energy of electron.
Energy of absorbed
photon
h = E(nu) - E(nl)
(1)
where E(nu) and E(nl) are energies of upper and lower
energy levels respectively.
This is RADIATIVE
EXCITATION.
• Note energy can also be absorbed from collision of a free
particle (COLLISIONAL EXCITATION) - no absorption
line is seen in this case.
• Atom remains in excited state until
SPONTANEOUS EMISSION (photon is emitted typically
after ~10-8 s)
or INDUCED EMISSION (Photon emitted at same energy
and coherently
with incoming photon - as in lasers).
Both produce EMISSION LINES.
• Narrow lines are seen since () is high, otherwise gas
transitions can only occur if is transparent and  ()
photon
has
energy is low.
(frequency/wavelength)
• Energy
level
diagram
corresponding to difference shows electron energy
in energy levels. If this is level
changes
for
the
case
then
the absorption of a photon.
absorption coefficient  Lowest energy level set to
PH507
Astrophysics
Dr. S.F. Green
zero energy. 1eV = 1.6 x
10-19 J.
98
n=•
n=4
n=3
13.6 eV
12.73 eV
12.07 eV
n=2
10.19 eV
n=1
Lyman
Series
Balmer Paschen
Series Series
0 eV
• Series of lines seen
-LYMAN SERIES transitions to/from n=1 lines seen in
ultraviolet
-BALMER SERIES ""
n=2
""
visual
-PASCHEN SERIES""
n=3
""
infrared ...
• Energy of absorbed photon
is
h = ( - E(nl)) + 1/2 me
v2
Bound-free transitions
• If photon has energy
1/2 m ev 2
13.6 eV
greater than that required n=•
12.73 eV
to move an electron in an n=4
12.07 eV
atom from its current n=3
energy level to level n=∞, n=2
10.19 eV
the
electron
will
be
released, ionizing the atom.
• Ionization potential for n=1
0 eV
Hydrogen is
13.6 e
• Since one of the states (free electron) can have any energy,
the transition can have any energy and the photon any
frequency (above a certain value determined by  and
E(nl)).
Thus BOUND-FREE transitions give an ABSORPTION
CONTINUUM.
• RE-COMBINATION is a FREE-BOUND transition and
PH507
Astrophysics
Dr. S.F. Green
99
results in an EMISSION CONTINUUM.
• The spectrum produced by
absorption from a single continuum • 
energy level will therefore
appear as a series of lines


of
increasing
energy
(increasing
frequency, decreasing wavelength) up to a limit defined by
-E(nl), with an absorption continuum shortward of this
limit.
• For nl=1 the Lyman series (Lyman-, Lyman- etc) is
observed together with the Lyman continuum shortward
of =91.2 nm. (Since interstellar space is populated by
very low density and low temperature hydrogen (ie with
n=1) photons with <91.2nm are easily absorbed so it is
opaque in the near-UV).
For nl=2 the Balmer series (H, H etc) is observed
together with the Balmer continuum shortward of
=364.7 nm.
Free-free transitions
• Absorption of a photon by a free electron in the vicinity of
an ion.
Electron changes from free energy state with velocity v1
to one with velocity v2
i.e. h = 1/2 me v22 - 1/2 me v12
Determination of 
• The actual spectrum of a star depends on the physical
conditions (notably temperature) and composition of the
stellar atmosphere. The intensity is produced at a
physical level in the star where  ~ 2/3. In order to
determine the total spectrum, the value of  needs to be
determined at all wavelengths. The overall  is the sum
PH507
Astrophysics
Professor Glenn White
100
of the contributions from each atomic/molecular species
in the atmosphere. Each component of  depends on the
number of atoms/molecules with a given energy state
capable of absorbing radiation at that frequency and the
absorption efficiency. We will deal with the energy state
populations first:
Boltzmann's equation (Excitation equilibrium)
• Boltzmann's
equation
describes
the
population
distribution of energy states for a particular atom in a gas.
The ratio of number of atoms per m3 in energy state B to
energy state A:
NB
NA
gB (EA - EB)/kT
e
gA
=
(50)
where gA and gB are STATISTICAL WEIGHTS (number
of different quantum states of the same energy), k =
Boltzmann const and T = temperature of gas.
NB EB > EA so exponential power is -ve.
• The probability of finding an atom in an excited state
decreases exponentially with the energy of the excited
state, but increases with increasing temperature.
Saha Equation (Ionization Equilibrium)
• The Boltzmann eqn does not describe all the possible
atomic states. Excitation may cause electrons to be lost
completely. There are therefore a number of different
ionization states for a given atom, each of which has one
or more energy states.
• The ratio of the number of atoms of ionization state i+1 to
those of ionization state i (i=I is neutral, i=II is singly
ionized, etc) is given by
3/2
Ni+1
Ni
=
Ui+1 2
Ui Ne
2 me k T
2
h
-i /kT
e
where Ne is the electron density (number of electrons per
PH507
Astrophysics
Professor Glenn White
101
m3),
i is the ionization potential of the ith ionization state,
Ui+1 and Ui are PARTITION FUNCTIONS obtained from
•
weights:
Ui = gi1 +
-Ein /kT
e
in
g
n=2
the statistical
• The higher the Ionization potential, i, the lower the
fraction of atoms in the upper ionization state,
The higher the Temperature, , the higher the fraction of
atoms in the upper ionization state, (Collisional excitation
is more likely to ionize atom),
The higher the electron density, the lower the fraction of
atoms in the upper ionization state (due to recombination).
• The Boltzmann and Saha Equations give the fraction of
atoms in a given ionization state and energy level
allowing (when combined with absorption/emission
probabilities)  and hence the line strengths to be related
to abundances.
Example - Abundances in the Sun
• In line forming regions in the Sun:T ~ 6000 K, Ne ~ 7x1019 m3.
Gas
I
II UII/UIUIII/UII g1
Hydrogen 13.6 eV
2
2
Calcium
6.1 eV 11.9 eV ~2
~0.5
1
g2
2
6
From Saha Equation for Hydrogen, the ratio of ionized to un-ionized H
NII/NI ≈ 6x10-5
i.e. most of Hydrogen is un-ionized.
From Boltzmann equation, ratio of number of atoms with electrons in l
n=2 to those in level n=1 (E1-E2 = -10.19 eV) is
N2/N1 ≈ 3x10-9
i.e almost all H atoms in ground state.
The H Balmer lines which originate from level n=2 are
PH507
Astrophysics
Professor Glenn White
102
strong only because the H abundance is so high.
From Saha Equation for Calcium,NII/NI ≈ 600 and NIII/NII ≈ 2x10
i.e. most of Calcium is in singly ionized state.
From Boltzmann equation, ratio of number of atoms with electrons in
energy states which contribute to the H and K lines to those in the
ground state (E1-E2 = -3.15, -3.13 eV) is (NB/NA)II ≈ 10-2
i.e most
Ca atoms in ground state.
The H and K lines of Calcium are therefore strong
because most Ca atoms in the Sun are in an energy state
capable of producing the lines.
• For stars cooler than the Sun more H is in the ground
state so Balmer lines will be weaker, for stars hotter than
the Sun more H is in n=2 state so Balmer lines will be
stronger. (T ~ 85000 K needed for N2/N1 =1). But at this
temperature NII/NI = 105 so little remains un-ionized.
• Balmer line strength depends on excitation (function of T)
and ionization (function of T and Ne). Balance of effects
occurs at T ~ 10,000 K so Balmer lines are strongest in A0
stars.
• A similar effect occurs for other species but at different
temperatures.
Transition probabilities
• Once we know the population of all energy states for a
given gaseous species we need to know the transition
probabilities for each energy state change before the
absorption coefficient can be determined.
• The transition probabilities must be calculated from
atomic theory or determined by experiment - much time
has been invested in this major problem in astrophysics.
• The EINSTEIN TRANSITION PROBABILITY (inverse of
lifetime):
PH507
Astrophysics
Professor Glenn White
103
for spontaneous emission, A21  2
for stimulated emission
B21  -1
for absorption
A12  -1
Total 
• We can now calculate  for a given gaseous species. For
Hydrogen (removing spectral line opacities for clarity):
Lyman continuum
absorption
falls off with decreasing 

due to  -1 dependence
Log 
T~25000K (B star)
Balmer
continuum
absorption
Paschen
continuum
absorption
T~5000K (G star)
(nm)
• Similar diagrams exist for other species. The total  will
be the sum for all species in the star.
• The region of a star for which optical depth ~2/3
determines where observed radiation originates. So if 
is large, then = 2/3 at a high level in the atmosphere
and if  is low, = 2/3 deep in the atmosphere.
Solar photospheric opacity
• The solar atmosphere is dominated by hydrogen. The
visible surface, the photosphere, has a temperature ~5800
K. However, as can be seen from the diagram above, 
for hydrogen at low temperatures is low in the visible
region (~400-700nm). This is because the continuum
absorption in the visible is due to Paschen absorption
PH507
Astrophysics
Professor Glenn White
104
(electrons originating in level n=3) and most hydrogen is
in ground state or n=2 level. We would therefore expect
the continuum to come from much deeper in the sun
where temperatures are higher. So what causes the high
solar photospheric opacity?
500.
•The solar opacity comes from the
H- ion. The ionization potential for
H- -->
Log 
T~25000K (B star)
is 0.75 eV (=1650nm).
H - bound-free H - free-free
From Boltzmann eqn
N3/N1 =
T~5000K (G star)
But from Saha eqn
(nm)
N(H)/N(
Therefore N(H-)/N3 ≈
i.e. number of H- ions is greater than number of H atoms
in level n=3, so absorption of photons to dissociate H- to
H dominates the continuum absorption in the optical.
Limb darkening
• The Sun is less bright near the limb than at the centre of
the disk.
 The continuum spectrum of the entire solar disk defines a
Stefan-Boltzmann effective temperature of 5800 K for the
photosphere, but how does the temperature vary in the
photosphere?
A clue is evident in a white-light
photograph of the Sun.
PH507
Astrophysics
Professor Glenn White
105

 We see that the brightness of the solar disk decreases
from the centre to the limb - this effect is termed limb
darkening.
Limb darkening arises because we see deeper, hotter gas
layers when we look directly at the centre of the disk and
PH507
Astrophysics
Professor Glenn White
106
higher, cooler layers when we look near the limb.
Assume that we can see only a fixed distance d through the
solar atmosphere. The limb appears darkened as the
temperature decreases from the lower to the upper
photosphere because, according to the Stefan-Boltzmann
PH507
Astrophysics
Professor Glenn White
107
law (Section 8-6), a cool gas radiates less energy per unit
area than does a hot gas. The top of the photosphere, or
bottom of the chromosphere, is defined as height = 0 km.
Outward through the photosphere, the temperature drops
rapidly then again starts to rise at about 500 km into the
chromosphere, reaching very high temperatures in the
corona.
Formation of solar absorption lines. Photons with energies well
away from any atomic transition can escape from relatively deep
in the photosphere, but those with energies close to a transition
are more likely to be reabsorbed before escaping, so the ones we see
on Earth tend to come from higher, cooler levels in the solar
atmosphere. The inset shows a close-up tracing of two of the
thousands of solar absorption lines, those produced by calcium at
about 395 nm.
PH507
Astrophysics
Professor Glenn White
108
At this point, you may have discerned an apparent paradox:
how can the solar limb appear darkened when the
temperature rises rapidly through the chromosphere?
Answering this question requires an understanding of the
concepts of opacity and optical depth. Simply put, the
chromosphere is almost optically transparent relative to the
photosphere. Hence, the Sun appears to end sharply at its
photospheric surface - within the outer 300 km of its 700,000
km radius.
Our line of sight penetrates the solar atmosphere only to the
depth from which radiation can escape unhindered
(where the optical depth is small). Interior to this point,
solar radiation is constantly absorbed and re-emitted (and
so scattered) by atoms and ions.
PH507
Astrophysics
Professor Glenn White
109
Y
Length of each solid bar is
approximately the same,
i.e. depth for which =2/3
R
y
Observer
X
Rx
Since R y > R x, radiation from the edge
of the disk, Y, originates from a higher
(cooler) region than at the centre of the
disk, X.
Assuming LTE, the continuum radiation
is described by the Planck function since
Y is at lower temperature, radiation is of
lower intensity
Spectral line formation
• Lines form higher in atmosphere than continuum. For
optical lines this corresponds to lower temperature than
continuum and therefore lower intensity (absorption
lines) (see p21 where S < I).
small
~2/3 low in
atmosphere
6500
T (K)
high
~2/3 high in
atmosphere
4500
0
200
400 km
Height above photosphere
F

PH507
Astrophysics
Professor Glenn White
110
Spectral line strength
Spectral lines are never perfectly monochromatic.
Quantum mechanical considerations govern minimum
line width, and many other processes cause line
broadening .
• For abundance calculations we want to know the total
line strength. Total line strength is characterised by
EQUIVALENT WIDTH.
F

Normal line
F

Fully saturated line

Shaded areas are equal
 = equivalent width


Stellar composition
• Derived from spectral line strengths in stellar
atmospheres.
In the solar neighbourhood, the
composition of stellar atmospheres is:
Element
H
He C,N,O,Ne,Na,Mg,Al,Si,Ca,Fe, others
% mass
70
28
~2.
Spectral line structure
• NATURAL WIDTH:
Due to uncertainty principle,
E=h/t, applied to lifetime of excited state.
For
"normal" lines the atom is excited (by a photon or
collision) to an excited atate which has a short lifetime t
~ 10-8 s. The upper energy level therefore has uncertai
energy E and the resultant spectral line (absorption or
emission) has an uncertain energy (wavelength). Line has
a Lorentz profile,  ~ 10-5 nm for visible light.
• COLLISIONAL/PRESSURE BROADENING:
Outer energy levels of atoms affected by presence of
neighbouring charged particles (ions and electrons).
random effects lead to line broadening since the energy of
PH507
Astrophysics
Professor Glenn White
111
upper energy level changes relative to the unexcited state
energy level.
This is the basis of the Luminosity
classification for A,B stars. Gaussian profile.  ~ 0.02 - 2
nm.
• DOPPLER BROADENING:
Due to motions in gas producing the line. Doppler shift
occurs for each each photon emitted (or absorbed) since
the gas producing the line is moving relative to the
observer (or gas producing the photon).
Thermal Doppler broadening due to motions of
individual atoms in the gas. ~0.01 - 0.02 nm for Balmer
lines in the Sun. Gaussian profile.
Bulk motions of gas in convection cells. Gaussian profile.
• ROTATION:
If there is no limb darkening, then lines have
hemispherical profile due to combination of radiation
from surface elements with different radial velocities.
Effect depends on rotation rate, size of star and angle of
polar tilt.
In general, V sin i is derived from the profile.
_
V -1
(km s )
200
Receding
+V
A
Approaching
-V
B
C
F

A
B
o
C
100

0
O B A F G K
• ATMOSPHERIC OUTFLOW:
Many different types.
Star with expanding gas shell (result of outburst) gives PCYGNI PROFILE.
Continuum (+ absorption lines) from star, emission or
PH507
Astrophysics
Professor Glenn White
112
absorption lines from shell:
F
Expanding
gas
shell
B

D
D
C
Star
D
A

o
Observer
B
C
A
C
B
Radiation from star, A, passes through cooler cloud
giving absorption line due to shell material which is blue
shifted relative to star. Elsewhere, emission lines are
seen.
Be STARS: Very rapid rotators with material lost from
the equator:
Radiation from star, A, passes through cooler cloud
giving absorption line.
Overall line structure is
hemispherical rotation line (B,D). Emission lines seen due
to shell material (C,E).
C
F
Rotating
gas
shell
E
Star
B
A
o
Observer
C
B
A
D
E
D

PH507
Astrophysics
Professor Glenn White
113
Forbidden lines
• Only certain transitions are generally seen for two
reasons:
1) Outer energy levels are far from the nucleus so in dense
gases, levels are distorted or destroyed by interactions.
2) Selection rules for change of quantum numbers restrict
possible transitions.
• In fact forbidden transitions are not actually forbidden.
However, the probability of a forbidden transition is very
low, so an allowed transition will generally occur. The
lifetimes in an excited state for which there are no allowed
downward transitions are ~10-3 - 109 seconds (ie very
low transition probability).
These are called
METASTABLE STATES.
• De-excitation from a metastable state can be by:
1) Collisional excitation, or absorption of another photon
to higher energy state allowing another downward
transition to the equilibrium state,
2) FORBIDDEN TRANSITION producing a FORBIDDEN
LINE. Usually denoted with [], e.g. [OII 731.99].
• Forbidden lines are usually much fainter than those from
allowed transitions due to low probability.
• In interstellar nebulae excited by UV from nearby hot
stars, some elements' excited states have no allowed
downward transitions to the ground state. In the absence
of frequent collisions (due to low density) or high photon
flux, a forbidden transition is the only way to the ground
state.
• These lines were not understood for a long while. A new
element Nebulium was invented to account for them.
PH507
Astrophysics
Professor Glenn White
114
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.
Lectures 13-14 : Nearby objects
PH507
Astrophysics
Professor Glenn White
115
PH507
Astrophysics
Professor Glenn White
116
PH507
Astrophysics
Professor Glenn White
117
PH507
Astrophysics
Professor Glenn White
118
PH507
Astrophysics
Professor Glenn White
119
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; today, this plot is called a Hertzsprung-Russell (H-R) diagram.
We infer the properties of stars from their light - in bulk for fluxes and spread out for
spectra. This chapter deals with the wealth of information that can be discerned by
studying stellar spectra. First we consider stellar atmospheres, for here the stellar
spectra originate. Then we tell the story of spectral observations - how they have been
made, correlated, and interpreted. Finally, we present that famous and crucial synthesis
- the Hertzsprung-Russell diagram - and some of its implications. This discussion will
lead to an understanding of the stars themselves.
PH507
Astrophysics
Professor Glenn White
120
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. (Recall that
stands for the Sun.)
PH507
Astrophysics
Professor Glenn White
121
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.
PH507
Astrophysics
Professor Glenn White
122
PH507
Astrophysics
Professor Glenn White
123
Stellar (Main Sequence) Properties With Mass
Mass
Temp Radius Luminosity
40 MSun 35,000 K 18 RSun 320,000 LSun
tMS
habitable zone
106 yrs
350-600 AU
17
21,000
8
13,000
107
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
As will soon become clear to you, this simple 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 for stars.
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.
PH507
Astrophysics
Professor Glenn White
124
PH507
Astrophysics
Professor Glenn White
125
Magnitude versus Spectral Type
The first H-R diagrams considered stars in the solar neighbourhood and plotted absolute
visual magnitude, M, versus spectral type, Sp, which is equivalent to luminosity
versus spectral type or luminosity versus temperature. The Figure below shows this
type of plot for the stars with well-determined distances within about 5 pc of the Sun.
Note (a) the well-defined 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.
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.
We can also trace the birth of young stars on this plot:
PH507
Astrophysics
Professor Glenn White
126
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. Let’s see what information we can glean from them.
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 Glenn White
127
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 
≈ 3. On a log-log plot, this gives
a straight line with a slope of . Main sequence stars do seem to conform to this
relationship, although the exponent varies from 
≈ 3 for luminous and massive
stars through 
-type stars to 
2 for 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.
3
L
~M
The value of the exponent varies for different kinds of stars, and in normally between ~
2.7 and 4. The value of 3 is appropriate to stars more massive than the sun.
PH507
Astrophysics
Professor Glenn White
128
The more massive stars burn their fuel very rapidly, leading to short lifetimes:
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
For L Mn ,
value of exponent n
3.9
3.0
2.7
Mass range M
M < 7 M
7 M < M < 25 M
25 M < M
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 0.085M
to 100M , 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* 
PH507
Astrophysics
Professor Glenn White
129
PH507
Astrophysics
Professor Glenn White
130
PH507
Astrophysics
Professor Glenn White
131
PH507
Astrophysics
Professor Glenn White
132
PH507
Astrophysics
Temperature inside the Sun
Professor Glenn White
133
PH507
Astrophysics
Professor Glenn White
134
PH507
Astrophysics
Professor Glenn White
135
Photospheric Temperatures

The continuum spectrum of the entire solar disk defines a Stefan-Boltzmann
effective temperature of 5800 K for the photosphere, but how does the
temperature vary in the photosphere? A clue is evident in a white-light
photograph of the Sun.

We see that the brightness of the solar disk decreases from the centre to the
limb - this effect is termed limb darkening.
Limb darkening arises because we see deeper, hotter gas layers when we look
directly at the centre of the disk and higher, cooler layers when we look near
the limb.
PH507
Astrophysics
Professor Glenn White
136
Assume that we can see only a fixed distance d through the solar atmosphere. The limb
appears darkened as the temperature decreases from the lower to the upper photosphere
because, according to the Stefan-Boltzmann law, a cool gas radiates less energy per unit
area than does a hot gas. The top of the photosphere, or bottom of the chromosphere, is
defined as height = 0 km. Outward through the photosphere, the temperature drops
rapidly then again starts to rise at about 500 km into the chromosphere, reaching very
high temperatures in the corona.
Formation of solar absorption lines. Photons with energies well away from any atomic transition can
escape from relatively deep in the photosphere, but those with energies close to a transition are more likely
to be reabsorbed before escaping, so the ones we see on Earth tend to come from higher, cooler levels in the
solar atmosphere. The inset shows a close-up tracing of two of the thousands of solar absorption lines,
those produced by calcium at about 395 nm.
PH507
Astrophysics
Professor Glenn White
137
At this point, you may have discerned an apparent paradox: how can the solar limb
appear darkened when the temperature rises rapidly through the chromosphere?
Answering this question requires an understanding of the concepts of opacity and
optical depth. Simply put, the chromosphere is almost optically transparent relative to
the photosphere. Hence, the Sun appears to end sharply at its photospheric surface within the outer 300 km of its 700,000 km radius.
Our line of sight penetrates the solar atmosphere only to the depth from which radiation
can escape unhindered (where the optical depth is small). Interior to this point, solar
radiation is constantly absorbed and re-emitted (and so scattered) by atoms and ions.
PH507
Astrophysics
Professor Glenn White
138
The Chromosphere

The solar chromosphere extends about 10,000 km above the photosphere, and its
gas density is far less than that of the photosphere. This thin layer has a reddish hue
- as a result of the Balmer (H) emission of hydrogen - visible during a total solar
eclipse.

At the limb of the Sun, tenuous jets of glowing gas 500 to 1500 km across extend to
a distance of 10,000 km upward from the chromosphere.
PH507
Astrophysics
Professor Glenn White
139
In the spicules, which are best observed in H, gas is rising at about 20 to 25 km/s.
Although spicules occupy less than 1% of the Sun’s surface area and have lifetimes of
15 min or less, they probably play a significant role in the mass balance of the
chromosphere, corona, and solar wind, and occur in regions of enhanced magnetic fields
PH507
Astrophysics
Professor Glenn White
140
Solar spicules, short-lived narrow jets of gas that typically last mere minutes, can be seen sprouting up
from the solar chromosphere in this H alpha image of the Sun. The spicules are the thin, dark, spikelike
regions. They appear dark against the face of the Sun because they are cooler than the solar photosphere
The Corona

At solar eclipses, the corona appears as a pearly white halo extending far from the
Sun’s limb. A brighter inner halo hugs the solar limb, and coronal streamers extend
far into space.
PH507
Astrophysics
Professor Glenn White
141
The change of gas temperature in the lower solar atmosphere is dramatic. The minimum temperature
marks the outer edge of the chromosphere. Beyond that, the temperature rises sharply in the transition
zone, finally levelling off at over 1,000,000 K in the corona.
Why the Sun's atmosphere is shockingly hot
Nobody would have guessed it: the atmosphere of the Sun is much, much
hotter than its surface. By more than one million degrees Centigrade in fact. Since 1939,
when scientists first determined the temperature of the solar atmosphere - known as the
corona - they were unable to come up with a convincing theory of why it greatly
exceeded the "mere" 6000° of the visible surface.
Nearly 60 years later, SOHO solved the mystery. Once again the MDI acronym (short
for Michelson Doppler Imager) is the code needed to decipher the secrets of our nearest
star. With MDI, scientists gathered data showing that huge numbers of small, closely
intertwined magnetic loops continuously emerge from the Sun's visible surface, clash
with one another and dissolve within 40 hours.
PH507
Astrophysics
Professor Glenn White
142
The loops seem to form a tight pattern that scientists call a magnetic carpet. Their
interaction generates electrical and magnetic short-circuits and releases enough energy
to heat the corona to temperatures hundreds of times higher than those of the solar
surface.
The solar atmosphere is permeated with magnetic fields, generated by electrified gas, or
plasma, churning violently beneath the visible surface. Solar astronomers have long
observed loops of plasma, called coronal loops, which appear to trace out the corona's
complex magnetic-field structure, much as iron filings reveal the invisible magnetic
field surrounding a magnet. Coronal loops come in various sizes, but most are
enormous, capable of spanning several Earths.
Solar astronomers know the particles comprising plasma are electrically charged and
feel magnetic forces. Thus, scientists thought coronal loops were tubes of plasma
trapped by and enclosed in the arch-shaped magnetic fields of the corona. The coronal
loops have puzzling features, however. The strong pull of solar gravity led astronomers
to believe that the plasma should be dense at the bases of the loop and thin at the top,
just as the Earth's gravity pulls our atmosphere close to the surface, causing it to thin
with increasing altitude. In fact, coronal loops seem to be about the same density
throughout their height, even though some of them extend several hundred thousand
miles (over a million kilometers) above the solar surface.
PH507
Astrophysics
Professor Glenn White
143
Coronal loops come in a variety of shapes and sizes, but most are enormous, capable of spanning several
Earth's. Photo: NASA and the TRACE team
The Visible Corona
Coronal continuum radiation has a temperature of 1 to 2 x 106 K.
PH507
Astrophysics
Professor Glenn White
144
Photograph of the solar corona during the July, 1991 eclipse, at the peak of the sunspot cycle. At
these times, the corona is much less regular and much more extended than at sunspot
minimum. Astronomers believe that coronal heating is caused by surface activity on the Sun.
The changing shape and size of the corona are the direct result of variations in prominence and
flare activity over the course of the solar cycle.
PH507
Astrophysics
Professor Glenn White
145
PH507
Astrophysics
Professor Glenn White
146
Line Emission
Forbidden Lines
Superimposed on the visible coronal continuum are some emission lines that
were unidentified until about 1942, when W. Grotrian of Germany and B. Edlén
of Sweden interpreted them. (They were long called “coronium” lines, for they
did not fit any known atomic transition). The two strongest lines are the green
line of Fe XIV (530.3 nm) and the red line of Fe X (637.4 nm); both are forbidden
lines.
Two significant obstacles hindered the identification of the coronal emission
lines: (1) the responsible transitions are forbidden, and (2) the temperatures of
the corona are unexpectedly high. In quantum mechanics, certain energy levels
of an atom are metastable because downward transitions from such levels are
strongly prohibited. While an ordinary permitted transition takes place in about
10-8 s, these metastable levels may persist for seconds or even days before a
forbidden transition occurs. In most laboratory and astrophysical situations,
gas densities are so high that collisional de-excitation empties metastable levels
very rapidly - there is just not enough time for a forbidden transition to take
place. In the near vacuum of the corona, however, metastable levels populated
by either photospheric radiation or collisions can decay and forbidden emission
features are formed.
PH507
Astrophysics
Professor Glenn White
147
Solar Activity
The Solar Cycle
Sunspots




Sunspots are photospheric phenomena that appear darker than the
surrounding photo-sphere (at about 5800 K) because they are cooler
(sunspot continuum temperatures are about 3800 K, and sunspot excitation
temperatures are about 3900 K). The darkest, central part (with the
temperatures just mentioned) is termed the umbra; the umbra is usually
surrounded by the lighter penumbra with its radial filamentary structure.
The most important characteristic of a sunspot is its magnetic field. Typical
field strengths are near 0.1 T, but fields as strong as 0.4 T have been
measured. Related to the magnetic field is a horizontal flow of gas in the
sunspot penumbra: gas moves out along the lower filaments and inward
along the higher filaments (at speeds up to 6 km/s).
A given sunspot has an associated magnetic polarity. Lines of magnetic
force diverge from a north magnetic pole and converge at a south pole; you
are familiar with this characteristic of bar magnets and our Earth. Two
sunspots of complementary polarity are generally found together in a
bipolar spot group.
The magnetic field in a typical sunspot is about 1000 times greater than the
field in neighbouring, undisturbed photospheric regions (which is itself
several times stronger than the Earth's field).
PH507
Astrophysics
Professor Glenn White
148
PH507
Astrophysics
Professor Glenn White
149
(a) The looplike structure of this prominence clearly reveals the magnetic field lines connecting the two
members of a sunspot pair. (b) This image of a particularly large solar prominence was observed by
ultraviolet detectors aboard the Skylab space station in 1979
Sunspot Numbers
In 1610, shortly after viewing the sun with his new telescope, Galileo Galilei made the
first European observations of Sunspots. Daily observations were started at the Zurich
Observatory in 1749 and with the addition of other observatories continuous
observations were obtained starting in 1849. The sunspot number is calculated by first
counting the number of sunspot groups and then the number of individual sunspots. The
"sunspot number" is then given by the sum of the number of individual sunspots and ten
times the number of groups. Since most sunspot groups have, on average, about ten
spots, this formula for counting sunspots gives reliable numbers even when the
observing conditions are less than ideal and small spots are hard to see. Monthly
averages (updated monthly) of the sunspot numbers (25 kb GIF image), (37 kb
postscript file), (62 kb text file) show that the number of sunspots visible on the sun
waxes and wanes with an approximate 11-year cycle.
PH507
Astrophysics
Professor Glenn White
150
PH507
Astrophysics
Professor Glenn White
151
Positional Variation
Detailed observations of sunspots have been obtained by the Royal Greenwich
Observatory since 1874. These observations include information on the sizes and
positions of sunspots as well as their numbers. These data show that sunspots do not
appear at random over the surface of the sun but are concentrated in two latitude bands
on either side of the equator. A butterfly diagram (142 kb GIF image) (610 kb postscript
file) (updated monthly) showing the positions of the spots for each rotation of the sun
since May 1874 shows that these bands first form at mid-latitudes, widen, and then
move toward the equator as each cycle progresses. The cycles overlap at the time of
sunspot cycle minimum with old cycle spots near the equator and new cycle spots at
high latitudes. An alternate version of this diagram with different colors for even and
odd numbered cycles is available as a 610kb postscript file.
The distribution of sunspots in solar latitude varies in a characteristic way
during the 11-year sunspot-number cycle. Sunspots tend to reside at high
latitudes (±35˚) at the start of a cycle.
PH507
Astrophysics
Professor Glenn White
152

Most spots are near ±15 at maximum, and the few spots at the end of the
cycle cluster near ±8. Very few sunspots are ever found at latitudes greater
than ±40. The lifetime of a sunspot ranges from a few days (small spots) to
months (large spots). In fact, a sunspot dies at the same latitude where it
was born (a characteristic that permits us to determine the solar rotation).
What takes place is this: as the cycle progresses, new spots appear at ever
lower latitudes. The first high-latitude spots of a cycle appear even before
the last low-latitude spots of the previous cycle have vanished.

So the sunspots follow active latitude belts during the course of a cycle. Less
clear, but just as tantalising, large active regions and spot groups seem to fall
into preferred longitudes during a cycle - active longitude belts.
Concentrations of magnetic fields appear to persist below the photosphere,
so that new spot groups arise from about the same locations as previous
ones. The active longitude belts sometimes appear about 180 apart, though
not all the time; frequently they are not symmetrical across the equator.
PH507
Astrophysics
Professor Glenn White
153
They reveal a different persistent pattern in the magnetic field structure in
the convective zone.

Predictions of sunspot numbers are now routinely made – an excellent web
site is www.spaceweather.com
The Babcock Model
PH507
Astrophysics
Professor Glenn White
154
Solar Rotation

By observing sunspots with his telescope, Galileo determined that the Sun’s
surface rotates eastward (synodically) in about one month. Today, the same
method is used in the sunspot zone (other methods, such as Doppler shifts,
are necessary above latitude ±40), and we know that the Sun rotates
differentially. That is, the rotation period is shorter at the solar equator
(about 25d) than at higher latitudes (about 27d at 40 and 30d at 70).

To sum up, sunspots reveal on a small scale the complexity and variability
of solar magnetic phenomena. The parts of a sunspot are all transient
magnetic structures - basically a sheaf of magnetic flux tubes filling the
umbra and penumbra and fanning out above them.
PH507
Astrophysics
Professor Glenn White
155
The Sun: A Model Star

Our Sun is the nearest star. The fascinating properties and phenomena of
the solar surface layers are easily observed and have been studied intensely.
Unfortunately, models for understanding solar phenomena have not kept
pace with such detailed data. Because the Sun is a fairly typical star and
because it is the only star that spans a large angular diameter as seen from
the Earth, the discussion here serves as the physical basis to investigate the
other stars.
Sun
Mass (1024 kg)
1,989,100.
6
3
2
GM (x 10 km /s )
132,712.
Volume (1012 km3)
1,412,000.
Volumetric mean radius (km) 696,000.
Mean density (kg/m3)
1408.
Surface gravity (eq.) (m/s2) 274.0
Escape velocity (km/s)
617.7
Ellipticity
0.00005
Moment of inertia (I/MR2) 0.059
Visual magnitude V(1,0) -26.74
Absolute magnitude
Luminosity (1024 J/s)
Mass conversion rate (106 kg/s)
Mean energy production (10-3 J/kg)
Surface emission (106 J/m2s)
Spectral type
Model values at center of Sun:
Central pressure:
Central temperature:
Central density:
Earth
5.9736
0.3986
1.083
6371.
5515.
9.78
11.2
0.0034
0.3308
-3.86
+4.83
384.6
4300.
0.1937
63.29
G2 V
(Sun/Earth)
333,000.
333,000.
1,304,000.
109.2
0.255
28.0
55.2
0.015
0.178
-
2.477 x 1011 bar
1.571 x 107 K
1.622 x 105 kg/m3
The Structure of the Sun

1 AU from the Earth with a radius of 6.96 x 105 km (109Ro) and a mass of 1.99 x
1030 kg (333,000M), and luminosity, (rate of total radiative energy output) of 3.86
x 1026 W.

The average density of the Sun is only 1400 kg/m3 - consistent with a composition
of mostly gaseous hydrogen and helium.

From its angular size of about 0.5° and its distance of almost 150 million kilometres,
we determine that its diameter is 1,392,000 kilometres (109 Earth diameters and
almost 10 times the size of the largest planet, Jupiter).

All of the planets orbit the Sun because of its enormous gravity. It has about 333,000
times the Earth's mass and is over 1,000 times as massive as Jupiter.
PH507

Astrophysics
Professor Glenn White
156
The Sun is made of 94% Hydrogen, 6% Helium, - the other elements make up just
0.13% (the three most abundant ‘metals’ Oxygen, Carbon, and Nitrogen make up
0.11%).
PH507
Astrophysics
Professor Glenn White
157
This spectacular image of the Sun was made by capturing X rays emitted by our star's most active
regions. It was taken by a camera on a rocket lofted shortly before the total solar eclipse of July 1991.
(Note the shadow of the Moon approaching from the west, at top.) The brightest regions in all these
images have temperatures of about 3 million Kelvin.
Lectures Week 7: Star Formation
Lectures Week 8: Theory of exoplanets
PH507
Astrophysics
Professor Glenn White
158
Debris Disks
Debris disks are remnant accretion disks with little or no gas left (just dust &
rocks), outflow has stopped, the star is visible.
Theory: Gas disperses, “planetesimals” form (100 km diameter rocks), collide &
stick together due to gravity forming protoplanets (Wetherill & Inaba 2000).
Protoplanets interact with dust disks: tidal torques cause planets to migrate inward
toward their host stars. Estimated migration time ~ 2 x 105 yrs for Earth-size
planet at 5 AU (Hayashi et al. 1985).
Perturbations caused by gas giants may spawn smaller planets (Armitage 2000):
Start with a stable disk
around central star.
Jupiter-sized planet forms
& clears gap in gas disk.
Planet accretes along spiral Disk fragments into more
arms, arms become unstable. planetary mass objects.
Debris Disks – Outer Disk
AB Aurigae outer
debris disk nearly
face on – see
structure &
condensations
(possible protoplanet formation
sites? Very far
from star) .
(Grady et al. 1999)
Two obvious differences between the exoplanets and the giant planets in the Solar
System:
• Existence of planets at small orbital radii, where our previous theory suggested
formation was very difficult.
• Substantial eccentricity of many of the orbits. No clear answers to either of these
surprises, but lots of ideas...
PH507
Astrophysics
Professor Glenn White
159
Most conservative possibility:
• Planet formation in these extrasolar systems was via the core accretion model ie same as dominant theory for the Solar System
• Subsequent orbital evolution modified the planet orbits to make them closer to
the star and / or more eccentric
We will focus on this option. However, more radical options in which exoplanets form
from gravitational instability are also possible.
Orbital evolution – migration
Need a migration mechanism that can move giant planets from formation at ~5 AU to a
range of radii from 0.04 AU upwards.
Three theories have been proposed:
• Gas disc migration: planet forms within a protoplanetary disc and is swept
inwards with the gas as the disc evolves and material accretes onto the star. The
most popular theory, as by definition gas must have been present when gas giants
form.
• Planetesimal disc migration: as above, but planet interacts with a disc of rocks
rather than gas. Planet ejects the rocks, loses energy, and moves inwards.
• Planet scattering: several massive planets form – subsequent chaotic orbital
interactions lead to some (most) being ejected with the survivors moving inwards
as above.
Gas disc migration
Planet interacts with gas in the disc via gravitational force
Strong interactions at resonances, eg where disc = nplanet, with n an integer.
For example the 2:1 resonance, where n = 2, which lies at 2-2/3 rp = 0.63 rp
Resonances at r < rp: Disc gas has greater angular velocity than planet. Loses
angular momentum to planet -> moves inwards
Resonances at r > rp: Disc gas has smaller angular velocity than planet. Gains
angular momentum from planet -> moves outwards
PH507
Astrophysics
Professor Glenn White
160
Interaction tends to clear gas away from location of planet, Result:
planet orbits in a gap largely cleared of gas and dust
This process occurs for massive planets (~ Jupiter mass) only - Earth mass planets
remain embedded in the gas though gravitational torques can be very important source
of orbital evolution for them too.
How does this lead to migration?
PH507
Astrophysics
Professor Glenn White
161
Angular momentum transport in the gas (viscosity) tries to close the gap (recall,
diffusive evolution of an accretion disc).
Gravitational torques from planet try to open gap wider.
Gap edge set by a balance:
-> Internal viscous torque = planetary torque
Planet acts as a angular momentum ‘bridge’:
• Inside gap, outward angular momentum flux transported by viscosity within disc
• At gap edge, flux transferred to planet via gravitational torques, then outward
again to outer disc
• Outside gap, viscosity again operative
Typically, gap extends to around the 2:1 resonances interior and exterior to the planet’s
orbit.
As disc evolves, planet moves within gap like a fluid element in the disc - ie usually
inwards.
Inward migration time ~ tn = R2 / n ~ few x 105 yr from 5 AU.
Mechanism can bring planets in to the hot Jupiter regime.
Believe that this mechanism is quantitatively consistent with the distribution of
exoplanets at different orbital radii – though the error bars are still very large!
PH507
Astrophysics
Professor Glenn White
162
Points: data Curve: theoretical migration model
Eccentricity
Substantial eccentricities of many exoplanets orbits do not have
completely satisfactory explanation. Possibilities:
(1) Scattering among several massive planets
Assumption: planet formation often produces a multiple system
which is unstable over long timescales:
• Chaotic evolution of a, e (especially e)
• Orbit crossing
• Eventual close encounters -> ejections
• Eccentricity for survivors
Advantages:
• Given enough planets, close together, definitely works
• Can produce very eccentric planets cf e=0.92 example discovered
• Some (stable) multiple systems are already known
Disadvantages:
• Requires planets to form very close together.
For two planets, with mass ratios to star q1 = m1 / M*, q2,
approximate stability boundary for separations:
> 2.4 (q1 + q2)1/3
System is unstable on short timescale if: a2 < a1 (1 + )
For Jupiter mass planets, (1 + ) ~ 1.3
Is it plausible that unstable systems formed in a large fraction
of extrasolar planetary systems?
• Collisions may produce too many low e systems
(2) Disc interactions
Assumption: gravitational interaction with disc generates eccentricity
PH507
Astrophysics
Professor Glenn White
163
Advantages:
• Same mechanism as invoked for migration
• Works for just one planet
• Theoretically, interaction is expected to increase eccentricity
if dominated by 3:1 resonance
Disadvantages:
• Gap is only expected to reach the 3:1 resonance for brown dwarf type
masses, not massive planets. Smaller gaps definitely tend to circularize the
orbit instead.
• Seems unlikely to give very large eccentricities
(3) Protoplanetary disc itself is eccentric
Assumption: why should discs have circular orbits anyway?
Eccentric disc -> eccentric planet?
Not yet explored in much depth. A possibility, though again seems unlikely to lead to
extreme eccentricities.
Scattering theory is currently most popular, possibly augmented
by interactions with other planets in resonant orbits.
PH507
Astrophysics
Professor Glenn White
164