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PH709
Extrasolar Planets - 3
Professor Michael Smith
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3.1 Extrasolar Planets or Exoplanets
Approach…..
• How can we discover extrasolar planets?
• Characteristics of the exoplanet population
• Planet formation: theory?
• Explaining the properties of exoplanets
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.
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.
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)
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 AU 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.
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Professor Michael Smith
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Analysis of Hubble Space Telescope data indicated that planets
were only beginning to form around Beta Pictoris, a very young star
at between 20 million and 100 million years old.
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.
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Using high-contrast, near-infrared adaptive optics observations with the Keck
and Gemini telescopes, the team of researchers were able to see three orbiting
planetary companions to HR8799
3.2 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
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Extrasolar Planets - 3
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Types of planet
A. 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
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?)
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.
Terrestrial planets have canyons, craters, mountains, and
volcanoes.
Atmospheres?
Gas giants possess primary atmospheres — atmospheres
captured directly from the originalsolar nebula.
Terrestrial planets possess secondary atmospheres —
atmospheres generated through internal vulcanism or comet
impacts.
Much more massive terrestrial planets could exist (>10 Earth
masses), though none are present in the Solar System.
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Extrasolar Planets - 3
Professor Michael Smith
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3.3 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.
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.
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.
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
• 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: 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: first success in 2004
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• 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.
3.4 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:
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(assuming mp << M*). i.e. the stellar orbital speed is small …. just
metres per second.
Compare to previous formula:
f(M) = 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)
(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 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
86.5 m/s
=
.0182 au/yr
period
of radial
velocity
variation
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
[To calculate the mass from the magnitude M of a star:
Lstar/Lsun 2.512(4.7 - M)
Since the Sun has a magnitude of 4.7. ]
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)
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Therefore,
(5)
f(M) =
Mi3 sin3i / (Mi + Mv)2 = 2.4 x 10-10 Msun
Because sin i < 1,
(6)
3
2
Mi / (Mi + Mv) >
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
=
ai3
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.
Temperature of Planet
It is possible to 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:
(9)
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:
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Extrasolar Planets - 3
Professor Michael Smith
Lstar (1 - A) ( Rp/4  dp)2 = 4 
p
2
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 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; 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. The lowest albedo is around 0.05
(Earth's moon); the highest, around 0.7 (Venus).
This calculation doesn't take into account the 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 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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Extrasolar Planets - 3
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:
.in a single measurement.
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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Extrasolar Planets - 3
Professor Michael Smith
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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
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Extrasolar Planets - 3
Professor Michael Smith
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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? Down to a fixed radial velocity:
m p sin i  a
1
2
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 semi-major axis
• No strong selection bias in favour / against detecting planets
with different eccentricities
Of the first 100 stars found to harbour 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.)
.
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Extrasolar Planets - 3
Professor Michael Smith
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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.
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 
for a star at distance d. Note we have again used mp << M*
Writing the mass ratio q = mp / M*, this gives (in milliarcseconds):
Note:
• 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
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Extrasolar Planets - 3
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• 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
3.6 Planet detection method : Transits Photometry
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
Probability. For a planet with radius rp << R*, probability of a transit-
Close-in planets are more likely to be detected. P = 0.5 % at
1AU, P = 0.1 % at the orbital radius of Jupiter
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What can we measure from the light curve?
(1) Depth of transit = fraction of stellar light blocked
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%.
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
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:
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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 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. It is
also found to have water vapor in its atmosphere.
Star Data
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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: 1.35 RJupiter
Mass: 0.69 MJupiter
Measured planetary radius rp = 1.35 RJ:
• Proves we are dealing with a gas giant.
• Somewhat larger than models for isolated (nonirradiated) planets - effect of environment on structure.
A reflected light signature must also exist, modulated on the orbital
period, even for non-transiting planets. No detections yet.
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.
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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
Dl
Ds
Dls
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.
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Extrasolar Planets - 3
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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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:
Einstein time. 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 (Einstein radius of order a few AU)
• Jupiter mass planet ~ 1 day (0.1 AU)
• Earth mass planet ~ 1 hour
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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
Note: 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 .
That noon-to-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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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
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
• Around 3-5 AU for typical parameters
Complementary to other methods:
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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 a 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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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 .
3.8 Planet detection method: Direct detection!
Photometric :
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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.
3.9 Timing: 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

(1) a second round of planet formation, or else to be
(2) the remaining rocky cores of gas giants that survived the
supernova and then spiralled in to their current orbits.
3.10 Detecting extrasolar planets: summary
RV, Doppler technique (v = 3 m/s)
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Astrometry: angular oscillation
Photometry: transits - close-in planets
Microlensing:
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