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
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.
Brief History
• Hooke, Flamsteed, Picard, Cassini, Horrebrow, Halley also tried and failed
• Galileo was the first to try measure distance to stars using a 2.5 cm telescope. He of course failed.
• 1887-1889 Pritchard used photography for astrometric measurements
• Modern astrometry was founded by Friedrich Bessel with his Fundamenta astronomiae, which gave the mean position of 3222 stars.
• 1838 first stellar parallax (distance) was measured independently by Bessel (heliometer), Struve (filar micrometer), and Henderson (meridian circle).
• Astrometry is also fundamental for fields like celestial mechanics, stellar dynamics and galactic astronomy. Astrometric applications led to the development of spherical geometry.
• Mitchell at McCormick Observatory (66 cm) telescope started systematic parallax work using photography
• 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?
d = 1 parsec
= 1 arcsecond
F f = F/DD
d = 1/d in parsecs, in arcseconds
1 parsec = 3.08 ×1018 cm
Astrometry: Parallax
So why did Galileo fail?
D = 2.5cm, f ~ 20 (a guess)
Plate scale = 360o · 60´ ·60´´
2 F=
206369 arcsecs
F
F = 500 mm Scale = 412 arcsecs / mm
Displacement of Cen = 0.002 mm
Astrometry benefits from high magnification, long focal length telescopes
Astrometry: Proper motionDiscovered 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
×
a1
a2
a1m1 = a2m2
a1 = a2m2 /m1
D
To convert to an angular displacement you have to divide by the distance, D
mM
aD
=
The astrometric signal is given by:
m = mass of planet
M = mass of star
a = orbital radius
D = distance of star
= mM2/3
P2/3
D
Astrometry: Orbital Motion
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
This is in radians. More useful units are arcseconds (1 radian = 206369 arcseconds) or milliarcseconds (0.001 arcseconds) = mas
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
- longitude of periastron passage
e -eccentricity
• Solve for these with astrometry - semiaxis major
i - orbital inclination
- position angle of ascending node - proper motion - parallax
• Solve for these with radial velocity - offset
K - semi-amplitude
All parameters are simultaneously solved using non-linear least squares fitting and the Pourbaix & Jorrisen (2000) constraint
= semi major axis = parallax
K1 = Radial Velocity amplitudeP = periode = eccentricity
A s in ia b s
=P K 1√ ( 1 - e 2 )
2 × 4.705
-9.45
-9.40
-9.35
-9.30
-9.25
53.5053.4553.4053.3553.30
53.50
53.45
53.40
53.35
53.30
2.4500x106
2.44902.4480
Julian Date
So we find our astrometric orbit
-9.45
-9.40
-9.35
-9.30
-9.25
53.5053.4553.4053.3553.30
53.50
53.45
53.40
53.35
53.30
2.4500x106
2.44902.4480
Julian Date
But the parallax can disguise it
-9.5
-9.4
-9.3
-9.2
-9.1
-9.0
-8.9
-8.8
53.853.653.4
53.8
53.6
53.4
2.4500x106
2.44902.4480
Julian Date
And the proper motion can slinky it
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 Displacment (as)
Jupiter at 1 AU 100
Jupiter at 5 AU 500
Jupiter at 0.05 AU 5
Neptune at 1 AU 6
Earth at 1 AU 0.33
Parallax 100000
Proper motion (/yr) 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
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Astrometry, a simple example5 "plates"different scalesdifferent orientations
Result of OverlapSolution toPlate #1
Precision = standard deviation of thedistribution of residuals ( ) from themodel-derived positions (*)
1 2 3
4
5
I
0.002 arcsec
The Importance of Reference stars
Perfect instrument Perfect instrument at a later time
Reference stars:1. Define the plate scale2. Monitor changes in the plate scale (instrumental effects)3. Give additional measures of your target
Focal „plane“
Detector
Example
Typical plate scale on a 4m telescope (Focal ratio = 13) = 3.82 arcsecs/mm = 0.05 arcsec/pixel (15 m) = 57 mas/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!
A signal is present, but what is it due to?
Frequency (cycles/year)
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 19 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
0.2
0.1
0.0
-0.1
Dec
lina
tion
(ar
csec
)
-0.2 -0.1 0.0 0.1RA (arcsec)
0° (N)
90°(E)
12
3
4 56
8
9
1011
1214
15
1617
W 1062 AB
HST astrometry on a Binary star
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 b
P = 2173 d
Msini = 10.2 MJup
B
i = 4 deg → m = 142 MJup
= 0.142 Msun
Bean et al. 2007AJ....134..749B
GL 876
M- dwarf host star
Period = 60.8 days
Gl 876
• 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 mass of Gl876b
55Cnc d
Perturbation due to component d,
P = 4517 days = 1.9 ± 0.4 mas
i = 53° ± 7°
Mdsin i = 3.9 ± 0.5 MJ
Md = 4.9 ± 1.1 MJ
McArthur et al. 2004 ApJL, 614, L81
Combining HST astrometry and ground-based RV
The 55 Cnc (= 1 Cnc) planetary system, from outer- to inner-most ID r(AU) M (MJup)d 5.26 4.9 ± 1.1c 0.24 0.27 ± 0.07b 0.12 0.98 ±0.19e 0.04 0.06 ± 0.02
Where we have invoked coplanarity for c, b, and e
= (17.8 ± 5.6 Mearth) a Neptune!!
Distance = 3.22 pcs = 10 light years
Period = 6.9 yrs
The Planet around Eridani
-4
-2
0
2
4
mas
-4 -2 0 2 4 mas
Eri = 0.3107 arcsec (HIP)M A ~ 0.8 M sun , M B ~ 0.0017 M sun = 2.2 mas, i = 30°
2000.52001
2001.5
2002
2002.5
2003
2003.5
2004
HST Astrometry of the extrasolar planet of Eridani
-3
-2
-1
0
1
2
3
mas
-3 -2 -1 0 1 2 3 mas
N
EM A ~ 4.0 M O, M B ~ 0.45 M O = 1.9 mas, i = 133°
K1 = 2.8 km s-1
HD 213307 = 3.63 mas
Mass (true) = 1.53 ± 0.29 MJupiter
Eri
= 0.3107 arcsec (parallax)
a = 2.2 mas (semi-major axis) i = 30° (inclination)
X-d
ispl
acem
ent (
arc-
seco
nds)
Y-d
ispl
acem
ent (
arc-
seco
nds)
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
A s in ia b s
=P K 1√ ( 1 - e 2 )
2 × 4.705
Brown Dwarf
The Planetary System of 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
The astrometric perturbation of Vb 10
Mass = 6.4 MJup
A possible problem: The RV measurements show no variability, but these are at low precision
It is unlikely that it is a more massive companion in an eccentric orbit
Red lines: A high amplitude radial velocity model showing that the measurements would have missed the periastron passage
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?
Taken with a different slit width!
“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
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
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 (Th-Ar 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!
AstrometryRadial Velocity
Space: The Final Frontier
GAIA from the Thüringer Landessternwarte Tautenburg
detectionParameters determined
GAIA Detection limits
Red: G-stars Blue: M Dwarfs
Casertano et al. 2008
Number of Expected Planets from GAIA
8000 Giant planet detections4000 Giant planets with orbital parameters determined1000 Multiple planet detections500 Multiple planets with orbital parameters determined
Sources of „Noise“
Secular changes in proper motion:
Small proper motion
Large proper motion
Perspective effect
ddt = –
2vr
AU
ddt = –
vr
AU
In arcsecs/yr2 and arcsecs/yr if radial velocity vr in km/s, in arcsec, 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:
No observer motion observer motion
aber ≈vc sin –
14
v2
c2 sin2 +16
v3
c3 sin2 (1 + 2 sin2 )
=
20-30 arcsecs
=
1-3 mas
= angle between direction to target and direction of motion
=~ as
Sources of „Noise“
Gravitational deflection of light:
defl =4 GM
Ro c2cot
2
M = mass of perturbing body
Ro = distance between solar system body and source
c, G = speed of light, gravitational constant
= angular distance between body and source
Source (as) @limb dmin (1 as)
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 as
is for a limb-grazing light ray
Spots :
y
x
Brightness centroid
Astrometric signal of starspots
2 spots radius 5o and 7o, longitude separation = 180o
T=1200 K, distance to star = 5 pc, solar radius for star
Latitude = 10o,60o Latitude = 10o,0o
Horizontal bar is nominal precision of SIM
1 milliarcsecond
Our solar system from 32 light years (10 pcs)
40 as
In spite of all these „problems“GAIA has the potential to find planetary systems
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
Summary