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PH709
Extrasolar Planets
Prof Michael Smith
1
Review
We are still in the early days of a revolution in the field of planetary
sciences that was triggered by the discovery of planets around
other stars. Exoplanets now number over 200, with masses as
small as 5–7 ME (Rivera et al. 2005; Beaulieu et al. 2006).
Comparative planetology, which once included only our solar system's
planets and moons, now includes sub-Neptune to super-Jupitermass planets in other solar systems.
In April 2007, a team of 11 European scientists announced the discovery
of a planet outside our solar system that is potentially habitable, with
Earth-like temperatures.
The planet was discovered by the European Southern Observatory's
telescope in La Silla, Chile, which has a special instrument that splits
light to find wobbles in different wave lengths. Those wobbles can
reveal the existence of other worlds.
What they revealed is a planet circling the red dwarf star, Gliese 581. The
discovery of the new planet, named Gliese 581 c, is sure to fuel studies
of planets circling similar dim stars. About 80 percent of the stars near
Earth are red dwarfs.
The new planet is about five times heavier than Earth, classifying it as a
super-earth. Its discoverers aren't certain if it is rocky, like Earth, or if it
is a frozen ice ball with liquid water on the surface. If it is rocky like
Earth, which is what the prevailing theory proposes, it has a diameter
about 1 1/2 times bigger than our planet. If it is an iceball, it would be
even bigger.
Gliese 581: M star: 3480K, mass: 0.31 solar masses;
Luminosity; 0.013 solar
Hot Neptune Gl 581b 15.7 ME
0.041 AU
Super-earth Gl 581c 5.06ME 0.073 AU
Super-earth Gl 581d 8.3 ME 0.25 AU
Gl 581c: 20C (albedo = 0.5 assumed) Greenhouse? Tidal locking?
Currently the most important class of exoplanets are those that
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transit the disk of their parent stars, allowing for a determination
of planetary radii.
The 14 confirmed transiting planets observed to date are all more
massive than Saturn, have orbital periods of only a few days, and
orbit stars bright enough such that radial velocities can be
determined, allowing for a calculation of planetary masses and bulk
densities (see Charbonneau et al. 2007a). A planetary mass and
radius allows us a window into planetary composition (Guillot 2005).
The 14 transiting planets are all gas giants although one
planet, HD 149026b, appears to be 2/3 heavy elements by mass
(Sato et al. 2005; Fortney et al. 2006; Ikoma et al. 2006).
Understanding how the transiting planet mass-radius relations change
as a function of orbital distance, stellar mass, stellar metallicity, or UV
flux, will provide insight into the fundamentals of planetary formation,
migration, and evolution.
The transit method of planet detection is biased toward finding
planets that orbit relatively close to their parent stars. This means
that radial velocity follow-up will be possible for some planets as the
stellar "wobble" signal is larger for shorter period orbits.
However, for transiting planets that are low mass, or that orbit very
distant stars, stellar radial velocity measurements may not be
possible. For planets at larger orbital distances, radial velocity
observations may take years. Therefore, for the foreseeable
future a measurement of planetary radii will be our only window
into the structure of these planets.
Orbital distances may give some clues as to a likely composition, but
our experience over the past decade with Pegasi planets (or "hot
Jupiters") has shown us the danger of assuming certain types of
planets cannot exist at unexpected orbital distances.
UPCOMING SPACE MISSIONS
The French/European COROT mission, launched in 2006
December, and the American Kepler mission, set to launch in 2008
November will revolutionize the study of exoplanets. COROT will
monitor 12,000 stars in each of five different fields, each for 150
continuous days (Bordé et al. 2003).
Planets as small as 2 RE should be detectable around solar-type stars
(Moutou et al 2006). The mission lifetime is expected to be at least
2.5 yr.
The Kepler mission will continuously monitor one patch of sky,
monitoring over 100,000 main-sequence stars (Basri et al. 2005).
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Extrasolar Planets
Prof Michael Smith
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The expected mission lifetime is at least 4 yr. Detection of sub-Earth
size planets is the mission's goal, with detection of planets with radii
as small at 1 Mercury radius possible around M stars.
With these missions, perhaps hundreds of planets will be discovered
with masses ranging from sub-Mercury to many times that of
Jupiter.
Of course, while planets close to their parent stars will preferentially
be found, due to their shorter orbital periods and greater
likelihood to transit, planetary transits will be detected at all orbital
separations. In general, the detection of three successive transits
will be necessary for a confirmed detection, which will limit confirmed
planetary-radius objects to about 1.5 AU.
http://exoplanet.eu/
http://en.wikipedia.org/wiki/Extrasolar_planet
Chapter 23 of Carroll & Ostlie, Modern Astronomy,
second edition
Rapidly developing subject - first extrasolar planet around an ordinary star only
discovered in 1995 by Mayor & Queloz.
Resources. For observations, a good starting point is Berkeley extrasolar planets search
homepage
http://exoplanets.org/
Number: There are 243 planets listed — 60 in multiple planet systems, 178 in
single planet systems, 4 orbiting pulsars, 1 orbiting a brown dwarf,
2 free floating?, plenty of candidates, retractions, cluster planets,……
Detection method: 211 planets were found by radial velocity or doppler
method, 26 were found by transit method, 1 by direct imaging, and 5 by
timing method.
However: See the Interactive catalogue at the exoplanets.eu webpage:
252 planets as of 19/9/07
Mass: The planets are listed with indications of their approximate masses as multiples
of Jupiter 's mass (MJ = 1.898 × 1027 kg) or multiples of Earth's mass (ME =
5.9737 × 1024 kg). Neptune = 17.1 ME, Mercury = 0.0553 ME.
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Extrasolar Planets
Prof Michael Smith
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Orbit/Distance: approximate distances in astronomical units (1) AU = 1.496 × 108 km,
distance between Earth and Sun) from their parent stars.
Names: According to astronomical naming conventions, the official designation for a
body orbiting a star is the star's catalogue number followed by a letter. The star itself is
designated with the letter 'a', and orbiting bodies by 'b', 'c', etc
Fusing stars
There are currently 238 planets known in orbit around fusing stars.
There are currently 178 known planets in single-planet systems and 60 known planets
in 20 multiple-planet systems (14 with two planets, 4 with three and 2 with four).
"Single" here means that only one planet has been detected to date.
Detection methods are not sensitive to low-mass planets, these stars may have smaller
planets that are below the limits of detectability, or are so far from the star that they have
not yet been observed over an orbital period (Could ALL stars harbour planets?)
Pulsars
There are currently four known planets orbiting two different pulsars. The planet of PSR
B1620−26 is in a circumbinary orbit around a pulsar and a white dwarf star.
Brown dwarfs
There is currently one known planet orbiting a brown dwarf.
Free floating planets
There is currently one suspected free-floating planet, i.e. it doesn't appear to orbit a star.
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Extrasolar Planets
Prof Michael Smith
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PH709
Extrasolar Planets
Prof Michael Smith
These lectures:
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
2-component systems
Definitions, planets, disks
Detection methods
Summary of methods
Populations
Theory of formation
Theory of evolution
6
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Extrasolar Planets
Prof Michael Smith
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1 Two-Component Systems
First Law: The orbit of each planet is an ellipse with the Sun at
one focus
p
b
F
C
Q
r
S f q
a
Second Law: For any planet, the radius vector sweeps out equal
areas in equal time intervals
Kepler’s Third Law The cubes of the semi-major axes of the
planetary orbits are proportional to the squares of the planets'
periods of revolution
2
P = ka3
where P is the period and a is the average distance from the Sun. Or, if P
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is in years and a is in AU:
P 2 = a3
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
m1 v1
4 2 m1 r1


r1
P2
and
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
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Extrasolar Planets
Prof Michael Smith
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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
r1 
or, solving for r1 ,
am2
m1  m2 
Now, since we know that the mutual gravitational force is
Gm1 m2
Fgrav  F1  F2 
2
a
then substituting for r1,
2


G
m

m
P
1
2
a3 
2
4
Solving for P:
P  2
a3
G M1  M2 
Summary: Measuring the mass of a planet
• Kepler’s third law extends to: 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
Then:
ms = mass of satellite
Ps = orbital period of satellite
as = semi-major axis of satellite's orbit
about the planet.
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)
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Extrasolar Planets
Prof Michael Smith
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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:
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
Since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027 kg.
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Extrasolar Planets
Prof Michael Smith
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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:
1.1 VISUAL BINARIES:
- Resolvable, generally nearby stars (parallax likely to be available)
- Relative orbital motion detectable over a number of years
- Not possible for exoplanets!
2. ASTROMETRIC BINARY: only one component detected
3. SPECTROSCOPIC BINARIES:
- Unresolved
- Periodic oscillations of spectral lines (due to Doppler shift)
- In some cases only one spectrum seen
4. 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
• Requires angular separation ≥ 0.5 arcsec (close to Sun, long orbital periods years – remember: at 1 parsec, 1 arcsec corresponds to 1 AU)
Example: Sirius:
Also known as Alpha Canis Majoris, Sirius is the fifth closest system to the Sun
at 8.6 light-years.
Sirius is composed of a main-sequence star and a white dwarf stellar remnant.
They form a close binary, Alpha Canis Majoris A and B, that is separated "on
average" by only about 20 times the distance from the Earth to the Sun -19.8 astronomical units (AUs) of an orbital semi-major axis -- which is about
the same as the distance between Uranus and our Sun ("Sol").
The companion star, is so dim that it cannot be perceived with the naked eye.
After analyzing the motions of Sirius from 1833 to 1844, Friedrich Wilhelm
Bessel (1784-1846) concluded that it had an unseen companion.
PH709
Extrasolar Planets
Prof Michael Smith
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Hubble Space Telescope :
• Observations:
Relative positions:
 = angular separation
 = position
Absolute positions: Harder to measure orbits of more massive star A and
less massive star B about centre of mass C which has proper motion µ.
PH709
Extrasolar Planets
Prof Michael Smith
Declination
N
13
M otion of centre of mass
= proper motion µ
Secondary

E

B
Primary
Right Ascension
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".
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
Sun
For i=0°
a
a"
p
r = distance of binary star
a = a"/p" AU
A
(In general correction for i≠0 required).
PH709
Extrasolar Planets
Prof Michael Smith
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Now lets go back to Kepler’s Law …
• From Kepler's Law, the Period P is given by
2 3
4 a
P =
G (mA + mB)
2
For the Earth-Sun system P=1 year, a=1 A.U., mA+mB~msun so 4π2/G = 1
3
2
P =
a
(mA + mB)
provided P is in years, a in AU, mA, mB in solar masses.
The total system mass is determined:
a" 3 1
mA + mB = ( )
p P2
• ABSOLUTE ORBITS:
d
c
rA
*
B
rB
B
f
q
e
A
Q
A
Semi-major axes aA = (c+e)/2
aB = (d+f)/2
Maximum separation = Q = c + f
Minimum separation = q = d + e
So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a
a = aA + a B
(1)
(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
So from Kepler’s Third Law, which gives the sum of the masses, and Equation
(1) above, we get the ratio of masses, ==> mA, mB. Therefore, with both, we
can solve for the individual masses of the two stars.
1.2 Spectroscopic Binaries
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Extrasolar Planets
Prof Michael Smith
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• 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
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Extrasolar Planets
Prof Michael Smith
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See: http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm
• Data plotted as RADIAL VELOCITY CURVE:
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Astrophysics
Professor Glenn White
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orientation
Orientation:
• If the orbit is tilted to the line of sight
(i<90°), the shape is unchanged but
velocities are reduced by a factor sin i.
• Take a circular orbit with i=90°
a = rA + rB
v = v A + vB
Orbital velocities:
vA = 2π rA / P
vB = 2π rB / P
Since mA rA = mB rB
mA/mB = rB/rA = vB/vA (2)
rB
vA
r
•Shape of radial velocity curves
depends on :
v
=
rA
v
B
orbital eccentricity and
• 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
vB sin i => rB sin i
} a sin i
So can only deduce (mA + mB) sin3 i = (a sin i)3/P2
(3)
For a spectroscopic binary, only lower limits to each mass can be derived,
unless the inclination i is known independently.
PLANET MASS: DETAILED DERIVATION
********************************************************
1. Assume a planet and star, both of considerable mass, are in circular
orbits around their centre of mass. Given the period P, the star’s orbital
speed v* and mass M*, the mass of the planet, Mp is given by
Mp3/ (M* + Mp)2
=
v*3 P / (2  G)
Note that there are 9 unknowns: P, a*, ap, M*, Mp, v*, vp, a, M - 9
variables
However,
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Astrophysics
Dr. S.F. Green
18
a = a* + ap
M = M*+ Mp
Centre of mass; M* a* = mp ap
Kepler’s law relates: P, a, M
P = 2  a*/v*
P = 2  ap/vp
….so that is 6 equations.; manipulation leaves any 3 you like.
Note: we usually only know vr* = v* sin i and we assume the planet
mass is small.
However, we must independently constrain the mass of the star! How?
Depends on theoretical stellar models or model atmospheres.
In terms of a more general analysis with eccentricity e and radial speed K*
V* sin i
1.3 Eclipsing Binaries
PH507
Astrophysics
Professor Glenn White
Eclipsing Binaries
• Since stars eclipse, the orientation is
i ~ 90°
1, 1'
2, 2'
3, 3'
4, 4'
v
19
FIRST CONTACT
SECOND CONTACT
THIRD CONTACT
FOURTH CONTACT
4' 3'
2' 1'
1 2
3 4
Observer in plane
• For a circular orbit:
• Variation in brightness with time is LIGHT CURVE.
• Timing of events gives information on sizes of stars and orbital elements.
• Shape of events gives information on properties of stars and relative
temperatures. If smaller star is hotter, then:
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Astrophysics
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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 t'
4
t3 t4
t1
t'1
t2
t'2
t3
t'3
t4
t'4
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Astrophysics
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• 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.
Asymetrical 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) (4)
2RL
and
2RS + 2RL = v (t4 - t1) => 2RL =
v(t4 - t2)
(5)
2RS
RS/RL = (t2 - t1) / (t4 - t2)
• Light curves 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)
1.4 Eclipsing-Spectroscopic Binaries
• For eclipsing binaries i ≥ 70°
(sin3i > 0.9)
• If stars are spectroscopic binaries then radial velocities are known.
So: masses are derived,
radii are derived,
ratio of temperatures is derived
Examine spectra and light curve to determine which radius corresponds with which
mass and temperature:
• Densities are then derived, and
since Luminosity L = 4 R2 T4, the ratio of luminosities is derived from
(35)
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Astrophysics
LA
LB
Summary
Type
Visual
=
RA
RB
2
TA
Professor Glenn White
22
4
TB
Observed
p, motion on sky
Apparent magnitudes
Spectroscopic
velocity curves
Eclipsing
light curves
Eclipsing/
light + velocity curves
Spectroscopic distance
Derived
a, e, i, mA, mB
LA, LB
MA/MB, (MA+MB)sin3i, a sin i
e, i, RS/RL
MA, MB, RA, RB, TA/TB, a, e, i,
LA, LB, TA, TB
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Extrasolar Planets
Professor Michael Smith
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2 Extrasolar Planets or Exoplanets
Direct imaged of planets is difficult because of the enormous difference in brightness
between the star and the planet, and the small angular separation between them.. The
effects of the gravity tugging at the stars, as well as the way that gravitational
affects can influence material close to the stars, has been clearly detected.
Circumstellar dust discs. (Circumstantial evidence.) 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’
This disk around Beta Pictoris is probably connected with a planetary system. The disk
does not start at the star. Rather, its inner edge begins around 25 AUs away, farther than
the average orbital distance of Uranus in the Solar System. Its outer edge appears to
extend as far out as 550 AUs away from the star.
Analysis of earlier pictures from the Hubble Space Telescope indicated that planets were
only beginning to form around Beta Pictoris, a very young star at between 20 million
and 100 million years old. ESO ADONIS adaptive optics system at the 3.6-m telescope.
It shows (in false colours) the scattered light at wavelength 1.25 micron (J band)
Most dust grains in the disk are not agglomerating to form larger bodies; instead, they
are eroding and being moved away from the star by radiation pressure when their size
goes below about 2-10 microns. Theoretically, this disk should have lasted for only
around 10 million years. That it has persisted for the 20 to 200 million year lifetime of
Beta Pictoris may be due to the presence of large disk bodies (i.e., planets) that collide
with icy Edgeworth-Kuiper Belt type objects (dormant comets) to replenish the dust.
• How can we discover extrasolar planets?
• Characteristics of the exoplanet population
• Planet formation: theory
• Explaining the properties of exoplanets
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Extrasolar Planets
Professor Michael Smith
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Definition of a planet
Simplest definition is based solely on mass
• 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
Note 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, neptune: icy giants)...
• Substantial gaseous envelopes
• Masses of the order of Jupiter mass (Jovian planets)
• In the Solar System, NOT same composition as Sun
• Presence of gas implies formation while gas was still prevalent
Gas giants may have a rocky or metallic core—in fact, such a core is thought to
be required for a gas giant to form—but the majority of its mass is in the form of
the gaseous hydrogen and helium, with traces of water, methane, ammonia, and
other hydrogen compounds
Terrestrial planets
• Prototypes: Earth, Venus, Mars
• Primarily composed of silicate rocks (carbon/diamond planets?)
• In the Solar System (ONLY) orbital radii less than giant planets
A central metallic core, mostly iron, with a surrounding silicate mantle. The
Moon is similar, but lacks an iron core. Terrestrial planets have canyons,
craters, mountains, and volcanoes. Terrestrial planets possess secondary
atmospheres — atmospheres generated through internal vulcanism or comet
impacts, as opposed to the gas giants, which possess primary atmospheres —
atmospheres captured directly from the original solar nebula.
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Extrasolar Planets
Professor Michael Smith
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Much more massive terrestrial planets could exist (>10 Earth masses), though none are
present in the Solar System.
3 Detecting extrasolar planets
(1) Direct imaging - difficult due to enormous star / planet flux ratio
The ultimate goal of any extrasolar planet search must surely be obtaining an
image of such a planet directly. This is fraught with difficulties since planets do
not emit light, so any image would have to be captured with starlight reflected
by the planet's atmosphere or surface. This will depend of course on the albedo
of the planet, which is hard to determine unless another detection method, such
as transits, is used as well. In any case, the light from the star will swamp that of
the planet by a factor of 109 in the optical, so it seems that concentrating upon
the infrared region would have the best chance of success. In the infrared, the
difference in the emission strength between a star and a planet is 107 (Angel &
Woolf, 1997) since planets radiate strongly in the infrared and stellar emission is
weaker in this region than in the optical.
(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 early detections
(3) Astrometry
• 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: first success in 2004
• 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
s
Each method has different sensitivity to planets at various orbital radii - complete
census of planets requires use of several different techniques
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Extrasolar Planets
Professor Michael Smith
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Planet detection method : Radial velocity technique
This is also known as the "Doppler method". Variations in the speed with which
the star moves towards or away from Earth — that is, variations in the radial
velocity of the star with respect to Earth — can be deduced from the
displacement in the parent star's spectral lines due to the Doppler effect. This
has been by far the most productive technique used by planet hunters.
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….just metres per second.
Compare to previous formula:
MF = Mp3/ (M* + Mp)2 = v*3 P / (2  G)
This equation is useful because only quantities that are able to be determined
from observations are present on the right-hand side of this equation
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)
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Extrasolar Planets
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(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:
i.e. extremely small -
12.5 m/s
0.1 m/s
10 m/s is Olympic 100m running pace
The star HD 209458 was the first to have its planet detected both by
spectroscopic and photometric methods. The radial velocity of the star varies
with time over a regular period of 3.52 days.
star's radial
velocity
amplitude
period of radial
velocity
variation
star's absolute
star's
magnitude
spectral
(to get the star's
class
luminosity
and
relative to the sun, use
mass
Lstar/Lsun 2.512(4.7 - M) (solar units)
don't forget to convert
SC
to actual luminosity of
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the sun
in watts by multiplying
by Lsun
M
( Lstar/Lsun)
86.5 m/s
=
.0182 au/yr
HD209458
3.52 days
=
.00965 yr
( M/Msun)
G0 V
4.6
(1.05)
Entering the observed quantities for the symbols on the right side of equation (4)
results in a value of the mass function MF of
MF = 2.4 x 10-10 (solar masses is the unit, assuming you used the units above)
Therefore,
(5)
MF =
Mi3 sin3i / (Mi + Mv)2 = 2.4 x 10-10 Msun
Because sin i < 1,
(6)
Mi 3
/ (Mi + Mv
)2
>
2.4 x
10-10
Msun
We now have an equation in a single unknown; although it cannot be solved
analytically, it can be easily solved by trial and error (guessing values) or by using a
graphing calculator. Can you find the solution to this inequality?
(answer: approximately Mi > 0.00064 Msun or 0.67 MJupiter)
Once an upper limit to the planet's mass is known, its orbit radius (or distance from
its parent star) can be found from equation (2) above. The planet's mass is very
much smaller than its parent star's mass; therefore, the Mi term on the left-hand
side can be ignored. Similarly [because of the center of mass condition, equation
(1)], the star's orbit size around the system center of mass is much smaller than the
planet's orbit size.
Therefore, the av term on the right-hand side can be ignored, and
(8)
Mv P2 = (ai)3
Using the values of Mv and P above, we find ai = 0.046 au. This is about 9x
smaller than Mercury's orbit about the sun.
Next, let's calculate the equilibrium blackbody temperature of a planet. We assume
that thermal equilibirium (i.e., constant temperature) applies, and consequently that
the power ( = energy/time) emitted by the planet is the power absorbed from its parent
star:
(9)
Pabsorbed = Pemitted
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The left hand side is found from geometry, corrected by a coefficient that takes into
account reflected light; the right hand side is given by the Stefan-Boltzmann law:
Lstar (1 - A) ( Rp/4  dp)2 = 4 Rp2 Tp4
Lstar = luminosity (power) of the parent star
A = planet's albedo = (light reflected)/(light incident)
Rp = planet's radius
Tp = planet's temperature
dp = distance of planet from parent star
= Stefan-Boltzmann constant
Solving for Tp gives
Tp4 = Lstar(1 - A)/(16dp2)
Notice that the equilibrium temperature depends on the "guessed" albedo of the
planet; the ratio of the temperature derived with albedo = 0.95 to the temperature
derived with an albedo of 0.05 is approximately 2.
Albedos of planets in our solar system.below. The lowest albedo is around 0.05
(Earth's moon); the highest, around 0.7 (Venus).
This calculation doesn't take into account:
thermal energy released from the planet's interior, tidal energy released via a
star-planet interaction, the greenhouse effect in the atmosphere, etc.
Radial velocity measurement:
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
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Professor Michael Smith
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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.
Examples of radial velocity data
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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
Summary: observables
(1) Planet mass, up to an uncertainty from the normally unknown inclination of
the orbit. Measure mp sin(i)
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(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? Down to a fixed radial velocity:
m p sin i  a
1
2
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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.
(This implies: 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
The gravitational perturbations of a star's position by an unseen companion provides a
signature which can be detected through precision astrometry.
While very accurate wide-angle astrometry is only possible from space with a mission
like the Space Interferometry Mission (SIM), narrow-angle astrometry with an accuracy
of tens of microarcseconds is possible from the ground with an optimized instrument.
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 
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for a star at distance d. Note we have again used mp << M*
Writing the mass ratio q = mp / M*, this gives:
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
Planet detection method : Transits - Photometry
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
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Extrasolar Planets
Professor Michael Smith
35
For a planet with radius rp << R*, probability of transits is:
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)
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
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(4)
Extrasolar Planets
Professor Michael Smith
36
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:
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
HST Transit light curve from Brown et al. (2001)
A triumph of the transit method occurred in 1999 when the light curve of the
star HD 209458 was shown to indicate the presence of a large exoplanet in
transit across its surface from the perspective of Earth (1.7% dimming).
Subsequent spectroscopic studies with the Hubble Space Telescope have even
indicated that the exoplanet's atmosphere must have sodium vapor in it. The
planet of HD 209458, unofficially named Osiris, is so close to its star that its
atmosphere is literally boiling away into space.
HD 209458 b represents a number of milestones in extraplanetary research. It
was the first transiting extrasolar planet discovered, the first extrasolar planet
known to have an atmosphere, the first extrasolar planet observed to have an
evaporating hydrogen atmosphere, the first extrasolar planet found to have an
atmosphere containing oxygen and carbon, and one of the first two extrasolar
planets to be directly observed spectroscopically. Based on the application of
new, theoretical models, as of April 2007, it is alleged to be the first extrasolar
planet found to have water vapor in its atmosphere.
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Professor Michael Smith
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Star Data
Apparent Mag.: 7.65
Spectral Type: G0
Radius: 1.18 Rsolar
Mass: 1.06 Msolar
Exoplanet Data
Period: 3.52474 days
Semi-major Axis: 0.045 AU
Radius: 1.42 RJupiter
Mass: 0.69 MJupiter
Consistent with expectations - the probability of a transiting system is ~10%.
Measured planetary radius rp = 1.35 RJ:
• Proves we are dealing with a gas giant.
• Somewhat larger than models for isolated (non-irradiated) planets effect of environment on structure.
Precision of photometry with HST / STIS impressive.
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
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Extrasolar Planets
Professor Michael Smith
38
Microlensing operates by a completely different principle, based on Einstein's
General Theory of Relativity. According to Einstein, when the light emanating
from a star passes very close to another star on its way to an observer on Earth,
the gravity of the intermediary star will slightly bend the light rays from the
source star, causing the two stars to appear farther apart than they normally
would.
This effect was used by Sir Arthur Eddington in 1919 to provide the first
empirical evidence for General Relativity. In reality, even the most powerful
Earth-bound telescope cannot resolve the separate images of the source star and
the lensing star between them, seeing instead a single giant disk of light, known
as the "Einstein disk," where a star had previously been. The resulting effect is a
sudden dramatic increase in the brightness of the lensing star, by as much as
1,000 times. This typically lasts for a few weeks or months before the source star
moves out of alignment with the lensing star and the brightness subsides.
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.
Dl
Ds
Dls
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Professor Michael Smith
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The deflection: light passes by the lens at a distance DL from the
observer with impact parameter ro = tan  D L . A photon passing a
distance ro from a mass M is bent through an angle

4GM
ro c 2
radians.
ro = DL 
Two images are formed when the light from a source at distance DS
passes the gravitational lens.
The Einstein ring radius is given by:
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:
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Extrasolar Planets
Professor Michael Smith
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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:
Unlike strong lensing, in microlensing u changes significantly in a short period
of time. The relevant time scale is called the Einstein time and it's given by the
time it takes the lens to traverse an Einstein radius.
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
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Extrasolar Planets
Professor Michael Smith
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N ote: The Julian day or Julian day number (JDN) is the integer number of
days that have elapsed since the initial epoch defined as noon Universal Time
(UT) Monday, January 1, 4713 BC in the proleptic Julian calendar [1]. That noonto-noon day is counted as Julian day 0. The Heliocentric Julian Day (HJD) is
the same as the Julian day, but adjusted to the frame of reference of the Sun,
and thus can differ from the Julian day by as much as 8.3 minutes, that being
the time it takes the Sun's light to reach Earth
Lensing by a star and a planet. Model results:
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Extrasolar Planets
Professor Michael Smith
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Planet detection through microlensing
The microlensing process in stages, from right to left. The lensing star (white) moves in
front of the source star (yellow) doubling its image and creating a microlensing event. In
the fourth image from the right the planet adds its own microlensing effect, creating the
two characteristic spikes in the light curve. Credit: OGLE
Binary systems 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 real-time with dense, high precision
photometry from several sites
• Look for deviations from single star light curve due to planets
• Timescales ~ a day for Jupiter mass planets, ~ hour for Earths
• Most sensitive to planets at a ~ R0, the Einstein ring radius
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• Around 3-5 AU for typical parameters
Complementary to other methods:
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:
The microlensing event that led to the discovery of the new planet was first
observed by the Poland-based international group OGLE, the Optical
Gravitational Lensing Experiment.
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Extrasolar Planets
Professor Michael Smith
44
The microlensing light curve of planet OGLE–2005-BLG-390Lb
The general curve shows the microlensing event peaking on July 31, 2005, and
then diminishing. The disturbance around August 10 indicates the presence of a
planet.
OGLE –2005-BLG-390Lb will never be seen again. At around five times the
mass of Earth, the new planet, designated OGLE–2005-BLG-390Lb, is the
lowest-mass planet ever detected outside the solar system. And when one
considers that the vast majority of the approximately 170 extrasolar planets
detected so far have been Jupiter-like gas giants, dozens or hundreds of times
the mass of Earth, the discovery of a planet of only five Earth masses is indeed
good news.
Planet detection method: Direct detection!
Photometric :
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Extrasolar Planets
Professor Michael Smith
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Infrared image of the brown dwarf 2M1207 (blue) and its planet 2M1207b, as
viewed by the Very Large Telescope. As of September 2006 this was the first
confirmed extrasolar planet to have been directly imaged.
Direct Spectroscopic Detection? 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.
Pulsar Planets
In early 1992, the Polish astronomer Aleksander Wolszczan (with Dale Frail)
announced the discovery of planets around another pulsar, PSR 1257+12.This discovery
was quickly confirmed, and is generally considered to be the first definitive detection of
exoplanets.

Pulsar timing. Pulsars (the small, ultradense remnant of a star that has
exploded as a supernova) emit radio waves extremely regularly as they
rotate. Slight anomalies in the timing of its observed radio pulses can
be used to track changes in the pulsar's motion caused by the presence
of planets.
These pulsar planets are believed to have formed from the unusual remnants of the
supernova that produced the pulsar, in a second round of planet formation, or else to
be the remaining rocky cores of gas giants that survived the supernova and then
spiralled in to their current orbits.
PH709
Extrasolar Planets
Professor Michael Smith
4 Detecting extrasolar planets: summary
RV, Doppler technique (v = 3 m/s)
Astrometry: angular oscillation
Photometry: transits - close-in planets
Microlensing:
46