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PH507
Multi-wavelength
Professor Michael Smith
1
Lecture 4
Mass can be measured in two ways. We could count up the atoms, or count up
the molecules and grains of dust and infer the number of atoms. This method
can be used if the object is optically thin and we have good tracer: a radiation
or scattering mechanism in which the number of photons is related to the
number of particles.
Otherwise, measuring the mass of an object relies upon its gravitational
influence….on nearby bodies or on itself (self-gravity).
Newton’s second law states: F = m a, while the first law relates the acceleration
to a change is speed or direction.
Kepler’s empirical laws for orbital motion thus describe the nature of the
acceleration from which masses can be derived.
Kepler's Laws
First Law: The orbit of each planet is an ellipse with the Sun at one focus
p
b
F
r
S f q
a
C
Q
S = Sun, F = other focus, p = planet.
r = HELIOCENTRIC DISTANCE.
f = TRUE ANOMALY
a = SEMI-MAJOR AXIS = mean heliocentric distance),
size of the orbit.
b = SEMI-MINOR AXIS.
e = ECCENTRICITY,
defines shape of orbit.
Ellipse:
SP + PF = 2a
(1)
e = CS / a
(2)
Therefore
b2 = a2(1-e2)
(3)
which defines the
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Multi-wavelength
•When CS = 0,
When CS = ,
e = 0,
e = 1,
Professor Michael Smith
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b = a, the orbit is a circle.
the orbit is a parabola.
• q = PERIHELION DISTANCE = a - CS = a – ae
q = a(1-e)
(4)
• Q = APHELION DISTANCE = a + CS = a + ae
Q = a(1+e)
(5)
Second Law: For any planet, the radius vector sweeps out equal areas in
equal times
• Time interval t for planet to travel from p to p1 is the same as time taken for
planet to get from p2 to p3. Shaded areas are equal.
• Let the time interval t be very small. Then the arc from p to p1 can be
regarded as a straight line and the area swept out is the area of the triangle
S p p1. If f1 is the angle to p1, and f is the angle to p:
p1
p2
p3
r1
p
r
f
S
i.e
Area = 1/2 r r1 Sin (f1-f).
Since t is very small, r ~ r1 and Sin (f1-f) ~ (f1-f) = f
Area = 1/2 r2 f
The rate this area is swept out is constant according to Kepler's second
law, so
r2 df/dt = h
(6)
where h, a constant, is twice the rate of description of area by the radius
vector. It is the orbital angular momentum (per unit mass.
The total area of the ellipse is πab which is swept out in the orbital period
P, so using eqn (7)
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2ab/P = h.
The average angular rate of motion is n = 2/P, so
n a2(1-e2)1/2 = h
(7)
Kepler’s Third Law
Kepler's third law took another ten years to develop after the first two. This law
relates the period a planet takes to travel around the sun to its average distance and
the Sun. This is sometimes called the semi major axis of an elliptical orbit.
P 2 = ka3
where P is the period and a is the average distance from the Sun.
Kepler’s Third Law follows from the central inverse square nature of the law of
gravitation. First look at Newton's law of gravitation - stated
mathematically this is
F
Gm1 m2
r2
Newton actually found that the focus of the elliptical orbits for two bodies of masses m1 and
m2 is at the centre of mass. The centripetal forces of a circular orbit are
F1
r1
v2
X
Centre of M ass
m1
m2
v1
F2
r2
2
F1
and
m1 v1
4 2 m1 r1


r1
P2
2
F2
m2 v2
4 2 m2 r2


r2
P2
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where
v
2r
P
and since they are orbiting each other (Newton’s 2nd law)
r1
m2

r2
m1
Let's call the separation a = r1 + r2. Then;
a  r1 
m1 r1
 m1

 r1 
 1 and multiplying both sides by m2 , am2  m1 r1  m2 r1
m2
m2
r1 
or, solving for r1 ,
am2
m1  m2 
Now, since we know that the mutual gravitational force;
Fgrav  F1  F2 
then substituting for r1,
3
a 
Gm1 m2
2
a
G m1  m2  P2
42
Solving for P:
P  2
a3
G M1  M2 
Third Law is therefore: The cubes of the semi-major axes of the planetary
orbits are proportional to the squares of the planets' periods of
revolution.
Example
Europa, one of the Jovian moons, orbits at a distance of 671,000 km from the centre
of Jupiter, and has an orbital period of 3.55 days. Assuming that the mass of Jupiter
is very much greater than that of Europa, use Kepler's third law to estimate the
mass of Jupiter.
Using Kepler's third law:
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
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the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds
Thus:
m jupiter  meuropa 
c
4 2 6.71  108
c
h
3
hc
6.67  10 11 3.07  105
 19
.  1027 kg
h
2
and since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027 kg.
Summary of Kepler’s Laws
Summary: Measuring the mass of a planet
• Kepler’s third law gives G(M+m) =  a3/P2
Since M >> m for all planets, it isn't possible to make precise enough
determinations of P and a to determine the masses m of the planets.
However, if satellites of planets are observed, then Kepler's law can be used.
• Let mp = mass of planet
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
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mp = 42 as3/(G Ps2)
• All the major planets have satellites except Mercury and Venus. Their
masses were determined from orbital perturbations on other bodies and
later, more accurately from changes in the orbits of spacecraft.
So: we can determine the masses of massive objects if we can detect and follow the
motion of very low mass satellites. That doesn’t lead very far. How can we determine
the masses of distant stars and exoplanets?
BASIC STELLAR PROPERTIES - BINARY STARS
• For solar type stars, single:double:triple:quadruple system ratios are
45:46:8:1.
• Binary nature of stars deduced in a number of ways:
VISUAL BINARIES:
- Resolvable, generally nearby stars (parallax likely to be available)
- Relative orbital motion detectable over a number of years
ASTROMETRIC BINARY: only one component detected
SPECTROSCOPIC BINARIES:
- Unresolved
- Periodic oscillations of spectral lines (due to Doppler shift)
- In some cases only one spectrum seen
SPECTRUM BINARY: 2 sets of lines but no apparent orbital motion but
spectrum is clearly combined from stars of differing spectral class.
ECLIPSING BINARY:
- Unresolved
- Stars are orbiting in plane close to line of sight giving eclipses
observable as a change in the combined brightness with time (‘’light
curves).
Some stars may be a combination of these.
Visual Binaries
• Angular separation ≥ 0.5 arcsec (close to Sun, long orbital periods - years) –
example Sirius:
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• Observations:
Relative positions:
 = angular
 = position
Absolute positions: Harder to measure orbits of more massive star A and
separation less massive star B about centre
of mass C which has proper motion µ.
Declination
N
M otion of centre of mass
= proper motion µ
Secondary

E

Right Ascension
B
Primary
C
A
NB parallax and aberration must also be accounted for.
• RELATIVE ORBITS:
- TRUE orbit:
q = peri-astron distance (arcsec or km)
Q = apo-astron distance (arcsec or km)
a = semi-major axis (arcsec or km)
a = (q + Q)/2
- APPARENT orbits are projected on the celestial sphere
Inclination i to plane of sky defines relation between true orbit and apparent
orbit. If i≠0° then the centre of mass (e.g. primary) is not at the focus of the
elliptical orbit.
Measurement of the displacement of the primary gives inclination and true
semi-major axis in arcseconds a".
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i
i
Incline by 45°
Apparent orbit
True orbit
• If the parallax p in arcseconds is observable then a can be derived from a".
Earth
B
radius
of Earth's
orbit
a
a"
Sun
For i=0°
p
r = distance of binary star
a = 1 AU . a"/p"
A
(In general correction for i≠0 required).
Now lets go back to Kepler’s Law …
• From Kepler's Law, the Period P is given by
2 3
2
P =
4 a
G (mA + mB)
(26)
For the Earth-Sun system P=1 year, a=1 A.U., mA+mB~msun so 4π2/G = 1
3
a
P =
(mA + mB)
2
P in years, a in AU, mA,mB in solar masses.
From (25) and (26),
a" 3 1
mA + mB = ( )
p P2
Sum of masses is determined
• ABSOLUTE ORBITS:
d
c
rA
A
*
B
rB
e
f
(27)
B
q
A
Q
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Semi-major axes aA = (c+e)/2 Minimum separation = q = d + e
aB = (d+f)/2
Maximum separation = Q = c + f
So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a
a = aA + a B
(28)
(and clearly r =rA + rB)
From the definition of centre of mass, mA rA = mB rB ( mA aA = mB aB)
mA/mB = aB/aA = rB/rA
(29)
So from Kepler’s Third Law, which gives the sum of the masses, and Equation
(29) above, we get the ratio of masses, ==> mA, mB. Therefore, with both, we
can solve for the individual masses of the two stars.
Spectroscopic Binaries
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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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• Data plotted as RADIAL VELOCITY CURVE:
recession
+
v
(km s-1)
recession
+
0
time
approach
approach
3
2
4
1
3
4
1
3
Observer
4
2
time
-
ABSOLUTE
Relative Orbit
2
v
(km s-1) 0
RELATIVE
Relative •radial
velocity
If the
orbit iscurve
tilted to the line of
2sight (i<90°), the shape is unchanged
but velocities are reduced by a factor
1
sin i.
v
1 • Take3 a circular orbit with i=90°
a = rA + rB
v = v A + vB
4
Orbital velocities:
2
vA = 2π rA / P
1
v
v
=
2π
r
/
P
B
B
3
1
Since mA rA = mB rB
4
2
mA/mB = rB/rA = vB/vA (31)
1
rB
v
1
3
vA
r4
1
v
=
rA
v
B
• Shape of radial velocity curves
depends on orbital eccentricity
and orientation.
• In general, measured velocities are vB sin i and vA sin i, so sin i terms cancel.
• From Kepler's law
mA + mB = a3/P2 (in solar units).
Observed quantities: vA sin i => rA sin i
} a sin i
vB sin i => rB sin i
So can only deduce (mA + mB) sin3 i = (a sin i)3/P2
(32)
For a spectroscopic binary, only lower limits to each mass can be derived,
unless i is known independently.
Eclipsing Binaries
• Since stars eclipse i ~ 90°
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• For a circular orbit:
1, 1' FIRST CONTACT
2, 2' SECOND CONTACT
3, 3' THIRD CONTACT
4, 4' FOURTH CONTACT
4' 3'
2' 1'
v
1 2
3 4
Observer in plane
• Variation in brightness with time is LIGHTCURVE.
• Timing of events gives information on sizes of stars and orbital elements.
• Shape of events gives information on properties of stars and relative
temperatures.
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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
t3
t'
4
t4
t1
t'1
t2
t'2
t3
t'3
t4
t'4
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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.
Assymetrical and/or unevenly spaced minima indicate eccentricity and
orientation of orbit.
• For a circular orbit:
t1 t2
t3 t 4
Distance = velocity x time
2RS = v (t2 - t1) (33)
2RL
and
2RS + 2RL = v (t4 - t1) => 2RL =
v(t4 - t2)
(34)
2RS
and ratio of radii
RS/RL = (t2 - t1) / (t4 - t2)
• Lightcurves are also affected by:
Non-total eclises
No flat minimum
Limb darkening
(non-uniform
brightness)
"rounds off"
eclipses
Ellipsoidal stars
(due to
proximity)
"rounds off"
maxima
Reflection effect
(if one star is
very bright)
(35)
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Astrophysics
Professor Glenn White
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Eclipsing-Spectroscopic Binaries
• For eclipsing binaries i ≥ 70°
(sin3i > 0.9)
• If stars are spectroscopic binaries then radial velocities are known.
So from eqns (31) and (32) masses are derived,
from eqns (33) and (34) radii are derived,
from (36) ratio of temperatures is derived
Examine spectra and lightcurve to determine which radius corresponds with which
mass and temperature:
+
v
-
B
From radial velocity curve
star A is more massive
A
Initially A is approaching (blue shift)
so first eclipse is A in front of B
Since first eclipse is primary eclipse
B is hotter than A
F
If 2 sets of lines are seen then B is larger
If 1 set of lines is seen then A is larger
time
• Since Luminosity L = 4 R2 T4, the ratio of Luminosities is derived from
(TO BE DISCUSSED LATER IN COURSE!)
LA
LB
=
Summary
Type
Visual
Spectroscopic
Eclipsing
Eclipsing/
LA/LB
spectroscopic
RA
RB
2
TA
4
TB
Observed
p, motion on sky
Apparent magnitudes
Derived
a, e, i, mA, mB
LA, LB
velocity curves
lightcurves
light + velocity curves
MA/MB, (MA+MB)sin3i, a sin i
e, i, RS/RL
MA, MB, RA, RB, TA/TB, a, e, i,
distance
LA, LB, TA, T
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Professor Glenn White
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Lecture 6: Extrasolar Planets
134 other stars are now known to possess planetary systems. 157 planets have
been discovered. Although none of the stars has been directly imaged, the
effects of the gravity tugging at the stars, as well as the way that gravitation
affects can affect material close to the stars, has been clearly seen.
Disc of material around the star Beta Pictoris – the image of the bright central
star has been artificially blocked out by astronomers using a ‘Coronograph’
• How can we discover extrasolar planets?
• Characteristics of the exoplanet population
• Planet formation
• Explaining the properties of exoplanets
Rapidly developing subject - first extrasolar planet around an ordinary star only
discovered in 1995 by Mayor & Queloz.
Observations thought to be secure, but theory still preliminary...
Resources. For observations, a good starting point is Berkeley extrasolar planets search
homepage
http://exoplanets.org/
Theory: Annual Reviews article by Lissauer (1993) is a good summary of the state of
theory prior to the discovery of extrasolar planets
Definition of a planet
Simplest definition is based solely on mass
• Stars: burn hydrogen (M > 0.075 Msun)
• Brown dwarfs: burn deuterium
• Planets: do not burn deuterium (M < 0.013 Msun)
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Professor Glenn White
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Deuterium burning limit occurs at around 13 Jupiter masses (1 MJ = 1.9 x 1027 kg ~
0.001 Msun It is important to realise that for young objects, there is no large change in
properties at the deuterium burning limit. ALL young stars / brown dwarfs / planets
liberate gravitational potential energy as they contract
Types of planet
Giant planets (gas giants, `massive’ planets)
• Solar System prototypes: Jupiter, Saturn, Uranus...
• Substantial gaseous envelopes
• Masses of the order of Jupiter mass
• In the Solar System, NOT same composition as Sun
• Presence of gas implies formation while gas was still prevelant
Terrestrial planets
• Prototypes: Earth, Venus, Mars
• Primarily composed of rocks
• In the Solar System (ONLY) orbital radii less than giant planets
Much more massive terrestrial planets could exist (>10 Earth masses), though none are
present in the Solar System. The Solar system also has asteroids, comets, planetary
satellites and rings - we won’t discuss those in this course.
Detecting extrasolar planets
(1) Direct imaging - difficult due to enormous star / planet flux ratio
(2) Radial velocity
• Observable: line of sight velocity of star orbiting centre of mass of star - planet
binary system
• Most successful method so far - all detections to date
(3) Astrometry
• 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)
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Astrophysics
Professor Glenn White
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(5) Gravitational lensing
• Observable: light curve of a background star lensed by the gravitational
influence of a foreground star. The light curve shape is sensitive to whether the
lensing star is a single star or a binary (star + planet is a special case of the
binary)
• Rare - requires monitoring millions of background stars, and also unrepeatable
• Some sensitivity to Earth mass planets
Each method has different sensitivity to planets at various orbital radii - complete
census of planets requires use of several different techniques
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Professor Glenn White
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Planet detection method : Radial velocity technique
A planet in a circular orbit around star with semi-major axis a
Assume that the star and planet both rotate around the centre of mass with an angular
velocity:
Using a1 M* = a2 mp and a = a1 + a2, then the stellar speed (v* = a ) in an inertial
frame is:
(assuming mp << M*). i.e. the stellar orbital speed is small.
For a circular orbit, observe a sin-wave variation of the stellar radial velocity, with an
amplitude that depends upon the inclination of the orbit to the line of sight:
Hence, measurement of the radial velocity amplitude produces a constraint on:
mp sin(i)
(assuming stellar mass is well-known, as it will be since to measure radial velocity we
need exceptionally high S/N spectra of the star).
Observable is a measure of mp sin(i).
-> given vobs, we can obtain a lower limit to the planetary mass
In the absence of other constraints on the inclination, radial velocity searches
provide lower limits on planetary masses
Magnitude of radial velocity:
Sun due to Jupiter:
Sun due to Earth:
12.5 m/s
0.1 m/s
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Professor Glenn White
i.e. extremely small -
20
10 m/s is Olympic 100m running pace
Spectrograph with a resolving power of 105 will have a pixel scale ~ 10-5 c ~ few km/s
Therefore, specialized techniques that can measure radial velocity shifts of ~10-3 of
a pixel stably over many years are required
High sensitivity to small radial velocity shifts is achieved by:
• comparing high S/N = 200 - 500 spectra with template stellar spectra
• using a large number of lines in the spectrum to allow shifts of much less than
one pixel to be determined.
Absolute wavelength calibration and stability over long timescales is achieved by:
• passing stellar light through a cell containing iodine, imprinting large number
of additional lines of known wavelength into the spectrum
• with the calibrating data suffering identical instrumental distortions as the data
Error sources:
(1) Theoretical: photon noise limit
• flux in a pixel that receives N photons uncertain by ~ N1/2
• implies absolute limit to measurement of radial velocity
• depends upon spectral type - more lines improve signal
• around 1 m/s for a G-type main sequence star with spectrum recorded at
S/N=200
• practically, S/N=200 can be achieved for V=8 stars on a 3m class
telescope in survey mode
(2) Practical:
• stellar activity - young or otherwise active stars are not stable at the m/s
level and cannot be monitored with this technique
• remaining systematic errors in the observations
Currently, the best observations achieve:
 ~ 3 m/s
...in a single measurement. Thought that this error can be reduced to around 1 m/s with
further refinements, but not substantially further. The very highest Doppler precisions
of 1 m/s are capable\of detecting planets down to about 5 earth masses.
Radial velocity monitoring detects massive planets, especially those at small a, but
is not sensitive enough to detect Earth-like planets at ~ 1 AU.
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Examples of radial velocity data
51 Peg b was the first known exoplanet with a 4 day, circular orbit: a hot Jupiter, \lying
close to the central star.
Example of a planet with an eccentric orbit: e=0.67
Summary: 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?
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.
Planet formation is a contest, where a growing planet must fight for survival lest it be
swallowed by the star that initially nurtured it.
Planet detection method : Astrometry
Conceptually identical to radial velocity searches. Light from a planet-star binary is
dominated by star. Measure stellar motion in the plane of the sky due to presence
of orbiting planet. Must account for parallax and proper motion of star.
Magnitude of effect: amplitude of stellar wobble (half peak displacement) for an orbit in
the plane of the sky is
 mp 
  a
a1  
M
 *
In terms of the angle:
 m p  a 
 
  
 M *  d 
for a star at distance d. Note we have again used mp << M*
Writing the mass ratio q = mp / M*, this gives:
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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
END OF NOTES