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Astrometric Detection of Planets
Stellar Motion
There are 4 types of stellar „motion“ that astrometry can
measure:
1. Parallax (distance): the motion of stars caused by
viewing them from different parts of the Earth‘s
orbit
2. Proper motion: the true motion of stars through
space
3. Motion due to the presence of companion
4. „Fake“ motion due to physical phenomena (noise)
Brief History
Astrometry - the branch of astronomy that deals with the
measurement of the position and motion of celestial bodies
• It is one of the oldest subfields of the astronomy dating back at
least to Hipparchus (130 B.C.), who combined the arithmetical
astronomy of the Babylonians with the geometrical approach of the
Greeks to develop a model for solar and lunar motions. He also
invented the brightness scale used to this day.
• Galileo was the first to try measure
distance to stars using a 2.5 cm telescope.
He of course failed.
• Hooke, Flamsteed, Picard, Cassini, Horrebrow, Halley also tried
and failed
• 1838 first stellar parallax (distance) was measured
independently by Bessel (heliometer), Struve (filar micrometer),
and Henderson (meridian circle).
• Modern astrometry was founded by
Friedrich Bessel with his Fundamenta
astronomiae, which gave the mean position
of 3222 stars.
• 1887-1889 Pritchard used photography for astrometric
measurements
• Mitchell at McCormick Observatory (66 cm)
telescope started systematic parallax work
using photography
• Astrometry is also fundamental for fields like celestial
mechanics, stellar dynamics and galactic astronomy. Astrometric
applications led to the development of spherical geometry.
• Astrometry is also fundamental for cosmology. The
cosmological distance scale is based on the measurements of
nearby stars.
Astrometry: Parallax
Distant stars
Astrometry: Parallax
So why did Galileo fail?
q= 1 arcsec
d = 1/q, d in parsecs, q
in arcseconds
d = 1 parsec
F
D
f = F/D
Astrometry: Parallax
So why did Galileo fail?
D = 2.5cm, f ~ 20 (a guess)
Plate scale =
360o · 60´ ·60´´
=
2pF
F = 500 mm
Scale = 412 arcsecs / mm
206369 arcsecs
F
Displacement of a Cen = 0.002 mm
Astrometry benefits from high magnification, long focal length
telescopes
Astrometry: Proper motion
Discovered by Halley who noticed that Sirius, Arcturus, and
Aldebaran were over ½ degree away from the positions
Hipparcos measured 1850 years earlier
Astrometry: Proper motion
Barnard is the star with the highest proper motion (~10
arcseconds per year)
Barnard‘s star in 1950
Barnard‘s star in 1997
Astrometry: Orbital Motion
The astrometric signal is given by:
q= m
M
a
D
m = mass of planet
M = mass of star
a = orbital radius
D = distance of star
m P2/3
q = 2/3
M D
Note: astrometry is sensitive to companions of
nearby stars with large orbital distances
Radial velocity measurements are distance independent, but
sensitive to companions with small orbital distances
Astrometry: Orbital Motion
With radial velocity measurements and astrometry one can
solve for all orbital elements
• Orbital elements solved with astrometry and RV:
P - period
T - epoch of periastron
w - longitude of periastron passage
e -eccentricity
• Solve for these with astrometry
a - semiaxis major
i - orbital inclination
W - position angle of ascending node
m - proper motion
p - parallax
• Solve for these with radial velocity
g - offset
K - semi-amplitude
All parameters are simultaneously solved using non-linear least
squares fitting and the Pourbaix & Jorrisen (2000) constraint
a A s i n i P K 1√ ( 1 - e 2 )
=
wa b s
2 p × 4.705
a = semi major axis
w= parallax
K1 = Radial Velocity amplitude
P = period
e = eccentricity
So we find our astrometric orbit
But the parallax can disguise it
And the proper motion can slinky it
-9.25
-8.8
-9.25
-8.9
-9.30
-9.30
-9.35
-9.35
-9.0
-9.1
-9.2
-9.40
-9.3
-9.40
-9.4
-9.45
-9.5
-9.45
53.30
53.35
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53.45
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53.30
53.35
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53.45
53.50
53.50
53.50
53.45
53.45
53.8
53.40
53.40
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53.35
53.35
53.30
53.30
53.4
53.6
2.4480
2.4490
53.8
53.4
2.4480
2.4490
2.4500 x1 0
Julia n Date
6
2.4480
2.4490 2.4500x10
Julian Date
6
2.4500 x1 0
Julian Date
6
The Space motion of Sirius A and B
Astrometric Detections of Exoplanets
The Challenge:
for a star at a distance of 10 parsecs (=32.6 light years):
Source
Jupiter at 1 AU
Jupiter at 5 AU
Jupiter at 0.05 AU
Neptune at 1 AU
Earth at 1 AU
Parallax
Proper motion (/yr)
Displacment (mas)
100
500
5
6
0.33
100000
500000
The Observable Model
Must take into account:
1. Location and motion of target
2.
Instrumental motion and changes
3.
Orbital parameters
4. Physical effects that modify the
position of the stars
Astrometry, a simple example
5 "plates"
different scales
different orientations
*
*
*
*
*
*
*
*
1
2
*
*
*
*
*
*
*
*
*
*
3
*
*
*
*
QuickTime™ and a Vi deo decompre sso r are need ed to se e th is p icture.
*
4
*
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*
*
*
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*
5
Result of Overlap
Solution to
Plate #1
*
*
Precision = standard deviation of the
distribution of residuals ( ) from the
model-derived positions ( * )
*
*
*
*
I
0.002 arcsec
The Importance of Reference stars
Perfect instrument
Perfect instrument at a later time
Reference stars:
1. Define the plate scale
2. Monitor changes in the plate scale (instrumental effects)
3. Give additional measures of your target
Good Reference stars can be difficult to find:
1. They can have their own (and different) parallax
2. They can have their own (and different) proper motion
3. They can have their own companions (stellar and planetary)
4. They can have starspots, pulsations, etc (as well as the target)
Where are your reference stars?
Fossil Astronomy at its Finest - 1.5% Masses
MTot =0.568 ± 0.008MO
MA =0.381 ± 0.006MO
MB =0.187 ± 0.003MO
πabs = 98.1 ± 0.4 mas
W 1062 AB
-0.1
4
5
6
Declination (arcsec)
3
0.0
8
0.1
9
90°(E)
17
10
11
16
15
0.2
2
0° (N)
-0.2
1
14
-0.1
12
0.0
RA (arcsec)
0.1
Astrometric detections: attempts and failures
To date no extrasolar planet has been discovered with the
astrometric method, although there have been several false
detections
Barnard´s star
Scargle Periodogram of Van de Kamp data
False alarm
probability = 0.0015!
A signal is present, but what is it due to?
New cell in lens
installed
Lens re-aligned
Hershey 1973
Van de Kamp detection was
most likely an instrumental
effect
Lalande 21185
Lalande 21185
Gatewood 1973
Gatewood 1996:
At a meeting of the American Astronomical Society Gatewood claimed
Lalande 21185 did have a 2 Mjupiter planet in an 8 yr period plus a second
one with M < 1Mjupiter at 3 AU. After 10 years these have not been
confirmed.
Real Astrometric Detections with the Hubble Telescope Fine
Guidance Sensors
GL 876
M- dwarf host star
Period = 60.8 days
Gl 876
The mass of Gl876B
• The more massive companion to Gl 876 (Gl 876b) has a
mass Mb = 1.89 ± 0.34MJup and an orbital inclination i = 84° ±
6°.
• Assuming coplanarity, the inner companion (Gl 876c) has a
mass Mc = 0.56MJup
55 Cnc d
Combining HST astrometry and
ground-based RV
McArthur et al. 2004
ApJL, 614, L81
Perturbation due to
component d,
P = 4517 days
a = 1.9 ± 0.4 mas
i = 53° ± 7°
Mdsin i = 3.9 ± 0.5 MJ
Md = 4.9 ± 1.1 MJ
The RV orbit of the
quadruple planetary system
55 Cancri e
The 55 Cnc (= r1 Cnc)
planetary system, from outerto inner-most
ID
r(AU) M (MJup)
d
5.26 4.9 ± 1.1
c
0.24 0.27 ± 0.07
b
0.12 0.98 ±0.19
= (17.8 ± 5.6 Mearth) a Neptune!!
e
0.04 0.06 ± 0.02
Where we have invoked
coplanarity for c, b, and e
Distance = 3.22 pcs = 10 light years
Period = 6.9 yrs
HST Astrometry of the extrasolar
planet of  Eridani
4
3
MA ~ 4.0 MO, MB ~ 0.45 MO
a = 1.9 mas, i = 133°
-1
K1 = 2.8 km s
E
 Eri
p = 0.3107 arcsec (HIP)
MA ~ 0.8 Ms un , MB ~ 0.0017 Ms un
a = 2.2 mas, i = 30°
2
2
N
2000.5
2001
0
mas
mas
1
-1
2001.5
0
2002
-2
2002.5
HD 213307
p = 3.63 mas
-3
2003
-2
2003.5
-3
-2
-1
0
mas
1
2
3
2004
-4
-4
-2
0
mas
2
4
 Eri
p = 0.3107 arcsec (HIP)
MA ~ 0.8 Msun , MB ~ 0.0017 Msun
a = 2.2 mas, i = 30°
Mass (true) = 1.53 ± 0.29 MJupiter
Orbital inclination of 30 degrees is consistent with inclination of
dust ring
The Future: Space Interferometry Mission
9 m baseline
Expected astrometric precision : 4 mas
The SIM Reference Grid
• Must be a large number of objects
• K giant stars V = 10–12 mag
• Distance ≈ several kilopcs
• Currently 4000 K giants are being observed with
precise radial velocity measurements to assure that
they are constant
Sources of „Noise“
Secular changes in proper motion:
Small proper
motion
Perspective effect
Large proper
motion
2vr
dm
mp
dt = – AU
vr
dp
2
p
–
=
dt
AU
In arcsecs/yr2 and
arcsecs/yr if radial
velocity vr in km/s, p in
arcsec, m in arcsec/yr
(proper motion and
parallax)
Similar effect in radial velocity:
The Secular Acceleration of
Barnard‘s Star (Kürster et
al. 2003).
2. Sources of „Noise“
Relativistic correction to stellar aberration:
20-30 arcsecs
v3
sin2 q (1 + 2 sin2 q)
3
c
1
v2
2
sin q + 6
2
c
1-3 mas
=
=
aaber ≈
v
1
–
c sin q
4
q = angle between direction to
target and direction of motion
observer motion
=
No observer motion
~ mas
Sources of „Noise“
Gravitational deflection of light:
adefl =
2 GM
y
cot
2
2
Ro c
M = mass of perturbing body
Ro = distance between solar system body and source
c, G = speed of light, gravitational constant
y = angular distance between body and source
Source
a(mas)
dmin (1 mas)
Sun
1.75×106 180o
Mercury
83
9´
Venus
493
4o.5
Earth
574
123o (@106 km)
Moon
26
5o (@106 km)
Mars
116
25´
Jupiter
16.27
90o
Saturn
5780
17o
Uranus
2080
71´
Neptune
2533
51´
Ganymede
35
32´´
Titan
32
14´´
Io
31
19´´
Callisto
28
23´´
Europa
19
11´´
Triton
10
0.7´´
Pluto
7
0.4´´
dmin is the angular
distance for which the
effect is still 1 mas
a is for a limb-grazing
light ray
Spots :
y
x
Brightness
centroid
…and pulsations:
Astrometric signal of starspots
Latitude = 10o,60o
2 spots radius 5o and 7o,
longitude separation = 180o
DT=1200 K, distance to star
= 5 pc, solar radius for star
Latitude = 10o,0o
Horizontal bar is nominal
precision of SIM
Astrometric signal as a function of spot filling factor
Aast (mas) = 7.1f 0.92 (10/D)
D = distance in pc, f is filling factor in percent
Our solar system from 32 light years (10 pcs)
1 milliarcsecond
40 mas