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PH709
Extrasolar Planets - 3
Professor Michael Smith
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Extrasolar Planets or Exoplanets
GENERAL REVISION OUTLINE FOR FINAL EXAM
• How can we discover extrasolar planets? L3
• Characteristics of the exoplanet population L4
• Planet formation: theory L5
• Explaining the properties of exoplanets L6
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
Types of planet
A. Giant planets (gas giants, `massive’ planets)
• Solar System prototypes: Jupiter, Saturn,
(Uranus, Neptune: ice 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
Cores: 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.
H and He: 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.
B. Terrestrial planets
Prototypes: Earth, Venus, Mars
Primarily composed of silicate rocks (carbon/diamond planets?)
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In the Solar System (ONLY) orbital radii less than giant planets
Core: A central metallic core, mostly iron with a surrounding silicate
mantle. The Moon is similar, but lacks an iron core.
Atmosphere of Planet
Gas giants possess primary atmospheres — atmospheres
captured directly from the originalsolar nebula.
Terrestrial planets possess secondary atmospheres —
atmospheres generated through internal vulcanism and comet
impacts.
Temperature of Planet
Estimate the temperature of an exoplanet based on the intensity of
the light it receives from its parent star.
We 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:
Pabsorbed = Pemitted
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 
p
2
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
p
2)
Albedo? Notice that the equilibrium temperature depends on
the "guessed" albedo of the planet;
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Extrasolar Planets - 3
Professor Michael Smith
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This calculation doesn't take into account the thermal energy
released from the planet's interior, tidal energy released via a starplanet interaction, the greenhouse effect in the atmosphere, etc.
3.2 Detecting extrasolar planets
(1)
Direct imaging - difficult due to enormous star / planet flux ratio.
Any optical 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,
Infrared. 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.
Detection may be possible when the planet is especially large
(considerably larger than Jupiter), widely separated from its parent
star, and young (so that it is hot and emits intense infrared
radiation).
(2) Radial velocity
• Observable: line of sight velocity of star orbiting centre of
mass of star - planet binary system.
(3) Astrometry
• Observable: stellar motion in plane of sky
(4) Transits: photometry
• 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.
• Rare - requires monitoring millions of background stars, and
also unrepeatable
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3.3 Direct Imaging
Direct imaging of planets is difficult because of the enormous
difference in brightness between the star and the planet, and the
small angular separation between them.
Direct detection: must be large and distant from star
Circumstellar dust discs. (Circumstantial evidence.) Disc of
material around the star Beta Pictoris bably connected with a
planetary system. The disk does not start at the star. Rather, its
inner edge begins around 25 AU away, farther than the average
orbital distance of Uranus in the Solar System.
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.
Young stars are preferred because young planets are expected to
be more luminous than older planets. In addition, direct imaging is
based on detection of planet luminosity, which must be related to
planet mass or size through uncertain theoretical models.
Some stunning individual systems have been reported (Marois et al.
2010, Lagrange et al. 2010), but the surveys indicate that fewer
planets are found than would be predicted by extrapolating the
power-law (of Eqn. (1) – see next lecture) out to 10-100 AU
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).
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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
3.4 Planet detection method : Radial velocity
technique
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.
We observe the star. So what can we say about the exoplanet?
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:
G(M * + m p )
W=
a3
Using a1 M* = a2 mp and a = a1 + a2, then the stellar speed
(v* = a ) in an inertial frame is:
V* =
mp
G(M * + m p )
M*
a
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(assuming mp << M*). i.e. the stellar orbital speed is small …. just
metres per second.
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:
Vobs = V* sin(i)
Hence, measurement of the radial velocity amplitude produces a
constraint on:
mp sin(i)
This assumes 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 yields 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 running pace
12.5 m/s
0.09 m/s
10 m/s is Olympic 100m
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.
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star's radial
velocity
amplitude
HD209458
period
of radial
velocity
variation
86.5 m/s
=
.0182 au/yr
Professor Michael Smith
star's absolute
magnitude
3.52 days
=
.00965 yr
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star's
spectral
class
and
mass
(solar
units)
G0 V
4.6
1.05 M/Msun
Entering the observed quantities for the symbols on the right side of
equation (4) results in a value of the mass function f(M) of
f(M) = 2.4 x 10-10 (solar masses is the unit, assuming you used the
units above)
Therefore,
(5)
f(M) =
Mi3 sin3i / (Mi + Mv)2 = 2.4 x 10-10 Msun
Because sin i < 1,
(6)
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Mi3 / (Mi + Mv)2 >
8
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)
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 centre of mass condition, the star's orbit size around
the system centre of mass is much smaller than the planet's orbit size.
Therefore we return to:
(8)
Mv
P2
=
ai 3
Using the values of Mv and P above, we find ai = 0.046 au.
This is about 9 x smaller than Mercury's orbit about the sun.
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 over many years are required
For circular orbit:
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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 where
e = 1 – b2/a2 periastron = a (1-e)
apastron = a (1+e)
a = semi-major axis, b = semi-minor axis
Summary: three parameters derived from observables
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(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? Down to a fixed radial velocity:
m p sin i µ a
1
2
.
3.5 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.
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
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æ mp ö
÷÷ ´ a
a1 = çç
è M* ø
In terms of the angle:
æ m p öæ a ö
÷÷ç ÷
Dq = çç
M
è * øè d ø
for a star at distance d. Note we have again used mp << M*
æ m p öæ a ö
÷÷ç ÷
Dq = çç
radians
è M * øè d ø
Detection threshold as function of semi-major axis
3.6 Planet detection method : Transits Photometry
TRANSITS
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.
SELECTION: 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.
CONFIRMATION: 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.
DENSITIES: The first confirmed transiting planets observed were all
more massive than Saturn, have orbital periods of only a few days,
and orbit stars bright enough such that radial velocities can also
be determined, allowing for a calculation of planetary masses and
bulk densities. A planetary mass and radius allows us a window
into planetary composition (Guillot 2005).
The first transiting planets were mainly 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.
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
The photometric transit technique can determine the radius of a
planet, but generally not the mass and hence does not immediately
indicate if a transit signal is due to a planet or a binary star system.
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Probability. For a planet with radius rp << R*, probability of a transit-
æR ö
Ptransit = sin(q ) » ç * ÷
èaø
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
DF æ rp ö
=ç ÷
Fo è R* ø
2
This is a measure of planetary radius! No dependence on
distance from star.
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%.
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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 limb-darkening
Deviations from profile expected from a perfectly opaque disc
could provide evidence for satellites, rings etc
Transit depth for an Earth-like planet is:
Photometric precision of ~ 10-5 seems achievable from space
.
Transit timing variation method (TTV) and transit
duration variation method (TDV)
If a planet has been detected by the transit method, then variations
in the timing of the transit provide an extremely sensitive method
which is capable of detecting additional planets in the system with
sizes potentially as small as Earth-sized planets.
Duration variations may be caused by an exomoon.
Orbital phase reflected light variations
Short period giant planets in close orbits around their stars will
undergo reflected light variations changes because, like the
Moon, they will go through phases from full to new and back again.
Since telescopes cannot resolve the planet from the star, they see
only the combined light, and the brightness of the host star seems
to change over each orbit in a periodic manner. Although the effect
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is small — the photometric precision required is about the same as
to detect an Earth-sized planet in transit across a solar-type star —
such Jupiter-sized planets are detectable by space telescopes such
as the Kepler Space Observatory.
3.7 Method : Gravitational microlensing
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
Dl
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• Observer-source distance
• Lens-source distance
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Ds
Dls
Azimuthal symmetry -> light from the source appears as a ring
...with radius R0 - the Einstein ring radius - in the lens plane
relevant time scale is called the Einstein time and it's given by the
time it takes the lens to traverse an Einstein radius.
Timescales for sources in the Galactic bulge, lenses ~ halfway
along the line of sight:
• Solar mass star ~ 1 month (Einstein radius of order a
few AU)
• Jupiter mass planet ~ 1 day (0.1 AU)
• 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.
3.8 Timing: Pulsar Planets
In early 1992, the Polish astronomer Aleksander Wolszczan (with
Dale Frail) announced the discovery of planets around another
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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.
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