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Astrometric Detection of Exoplanets
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 other physical phenomena
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
1 AU projects to 1 arcsecond at a
distance of 1 pc = 3.26 light years
Astrometry: Parallax
So why did Galileo fail?
q= 1 arcsecond
d = 1/q, d in parsecs, q
in arcseconds
d = 1 parsec
F
1 parsec = 3.08 ×1018 cm
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
Hipparchus 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
a1m1 = a2m2
a1 = a2m2 /m1
a2
×
a1
D
To convert to an angular displacement
you have to divide by the distance, D
Astrometry: Orbital Motion
The astrometric signal is given by:
q= m
M
a
D
This is in radians. More useful units are
arcseconds (1 radian = 206369 arcseconds) or
milliarcseconds (0.001 arcseconds) = mas
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 - true semi-major axis
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
53.40
53.45
53.50
53.30
53.35
53.40
53.45
53.50
53.50
53.50
53.45
53.45
53.8
53.40
53.40
53.6
53.35
53.35
53.30
53.30
53.4
53.6
2.4480
2.4490
53.8
53.4
2.4480
2.4490
2.4500x10
Julian Date
6
2.4480
2.4490
2.4500x10
Julian Date
6
2.4500x10
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
*
*
*
*
*
*
*
*
2
*
*
*
*
*
*
*
*
3
*
*
*
*
1
*
*
*
4
*
*
*
*
*
*
*
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
Example
Focal „plane“
Detector
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
Typical plate scale on a 4m telescope (Focal ratio = 13) = 3.82 arcsecs/mm =
0.05 arcsec/pixel (15 mm) = 57mas/pixel. The displacement of a star at 10
parsecs with a Jupiter-like planet would make a displacement of 1/100 of a
pixel (0.00015 mm)
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?
In search of a perfect reference.
You want reference objects that move little with respect
to your target stars and are evenly distributed in the sky.
Possible references:
K giant stars V-mag > 10.
Quasars V-mag >13
Problem: the best reference objects are much fainter
than your targets. To get enough signal on your target
means low signal on your reference. Good signal on
your reference means a saturated signal on your
target → forced to use nearby stars
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!
Frequency (cycles/year)
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 < 1 Mjupiter at 3 AU. After 16 years these have not been
confirmed.
Real Astrometric Detections with the Hubble Telescope Fine
Guidance Sensors
HST uses Narrow Angle Interferometry!
The first space interferometer for astrometric measurements:
The Fine Guidance Sensors of the Hubble Space Telescope
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
HST astrometry
on a Binary star
8
0.1
9
90°(E)
17
11
16
10
15
0.2
2
0° (N)
-0.2
1
14
-0.1
12
0.0
RA (arcsec)
0.1
Image size at best sites from ground
HST is achieving astrometric precision of 0.1–1 mas
One of our planets is missing: sometimes you need the true mass!
HD 33636 bB
Bean et al. 2007AJ....134..749B
P = 2173 d
msini = 10.2 MJup
i = 4 deg → m = 142 MJup
= 0.142 Msun
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 ± 0.34 MJup and an orbital inclination i =
84° ± 6°.
• Assuming coplanarity, the inner companion (Gl 876c) has a
mass Mc = 0.56 MJup
The Planet around e Eridani
Cambel and Walker : e Eri was a
„probable variable“
The Planet around e Eridani
What the eye does not see a periodogram finds:
Period = 6.9 years
Radial
Velocities
Activity
indicator
FAP ≈ 10–9
X-displacement (arc-seconds)
Y-displacement (arc-seconds)
e Eri
p = 0.3107 arcsec (parallax)
a = 2.2 mas (semi-major axis)
i = 30° (inclination)
Mass (true) = 1.53 ± 0.29 MJupiter
Orbital inclination of 30 degrees is consistent with inclination of
dust ring
One worrisome point: The latest radial velocities do not
fit the orbit:
Astrometric
measurements of
HD 38529
2
a A s i n i P K 1√ ( 1 e )
=
wa b s
2 p × 4.705
Brown Dwarf
The Planetary System of u And
Note: the planets do
not have the same
inclination!
The Purported Planet around Vb10
Up until now astrometric measurements have only detected known exoplanets.
Vb10 was purported to be the first astrometric detection of a planet. Prada and
Shalkan 2009 claimed to have found a planet using the STEPS: A CCD camera
mounted on the Palomar 5m. 9 years of data were obtained.
Vb 10
Control star
Control star
Control star
The Periodograms show a
significant signal at 0.74 years
The astrometric perturbation of Vb 10
Mass = 6.4 MJup
Looks like a confirmation with radial velocity measurements,
but it is only driven by one point
The RV data does not support the
previous RV model. The only way
is to have eccentric orbits which is
ruled out by the astrometric
measurements.
Is there something different about the first point?
“Science is a way of trying not to
fool yourself. The first principle is
that you must not fool yourself, and
you are the easiest person to fool.”
– Richard Feynman
Taken with a
different slit width!
Comparison between Radial Velocity Measurements
and Astrometry.
Astrometry and radial velocity measurements are fundamentally
the same: you are trying to measure a displacement on a detector
Radial Velocity
Astrometry
1. Measure a displacement of a
spectral line on a detector
1. Measure a displacement of a
stellar image on a detector
2. Thousands of spectral lines
(decrease error by √Nlines)
2. One stellar image
3. Hundreds of reference lines (ThAr or Iodine) to define „plate
solution“ (wavelength solution)
3. 1-10 reference stars to define
plate solution
4. Reference lines are stable
4. Reference stars move!
Space: The Final Frontier
1.
2.
Hipparcos
•
3.5 year mission ending in 1993
•
~100.000 Stars to an accuracy of 7 mas
Gaia
•
1.000.000.000 stars
•
V-mag 15: 24 mas
•
V-mag 20: 200 mas
•
Launch 2011 2012 2013
GAIA Detection limits
Casertano et al. 2008
detection
Parameters
determined
Red: G-stars Blue: M Dwarfs
Number of Expected Planets from GAIA
8000 Giant planet detections
4000 Giant planets with orbital parameters determined
1000 Multiple planet detections
500 Multiple planets with orbital parameters determined
Our solar system from 32 light years (10 pcs)
1 milliarcsecond
40 mas
In spite of all these „problems“GAIA has the potential to find
planetary systems
Summary
1. Astrometry is the oldest branch of Astronomy
2. It is sensitive to planets at large orbital distances
→ complimentary to radial velocity
3. Gives you the true mass
4. Least successful of all search techniques because
the precision is about a factor of 1000 to large.
5. Will have to await space based missions to have a
real impact
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)
The Secular Acceleration of
Barnard‘s Star (Kürster et
al. 2003).
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 =
4 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) @limb
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
16260
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
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