Astrometric Detection of Planets

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)

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


So why did Galileo fail?

d = 1 parsec

= 1 arcsec

F f = F/DD

d = 1/d in parsecs, in arcseconds

1 parsec = 3.08 ×1018 cm


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 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

×

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


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)


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

A s in ia b s

=P K 1√ ( 1 - e 2 )

2

= semi major axis = parallax

K1 = Radial Velocity amplitudeP = periode = eccentricity

× 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 (mas)

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 scale

2. Monitor changes in the plate scale (instrumental effects)

3. Give additional measures of your target

Focal „plane“

Detector

Example

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?

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

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?

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 13 years these have not been confirmed.

Real Astrometric Detections with the Hubble Telescope Fine Guidance Sensors

HST uses Interferometry!

The first space interferometer for astrometric measurements:The Fine Guidance Sensors of the Hubble Space Telescope

Interferometry with Koesters Prisms

Transmitted beam has phase shift of /4

Reflected beam has no phase shift

Interferometry with Koesters Prisms

CONSTRUCTIVE INTERFERENCE

0

HST generates a so-called S-curve

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

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

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

245300024520002451000245000024490002448000

Julian Date

The RV orbit of the quadruple planetary system 55 Cancri e

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!!

The Planet around Eridani

What the eye does not see a periodogram finds:

Radial Velocities

Activity indicator

FAP ≈ 10–9

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 (HIP)M A ~ 0.8 Msun , M B ~ 0.0017 Msun = 2.2 mas, i = 30°

Orbital inclination of 30 degrees is consistent with inclination of dust ring

Space: The Final Frontier

1. Hipparcos

• 3.5 year mission ending in 1993

• ~100.000 Stars to an accuracy of 7 mas

2. Gaia

• 1.000.000.000 stars

• V-mag 15: 24 as

• V-mag 20: 200 as

• Launch 2011

3. Space Interferometry mission

• 60 solar-type stars

• precision of 4 as

Space Interferometry Mission

9 m baseline

Expected astrometric precision : 4 as

Programs:

1. Detecting Earth-like Planets

2. Young Planets and Star Formation

3. Multiple Planet Systems

SIMLiteZero?

Its 5 year mission is to boldly go where no planet hunter has gone before:

• Demonstrated precision of 1 as and noise floor of 0.3 as amplitude.

• Multiple measurements of nearest 60 F-, G-, and K- stars.

• Directly test rocky planet formation

„This paucity of low mass planets is almost certainly an artfact of sensitivity, as the Doppler technique struggles to detect lower-mass planets. Thus, we have reached a roadblock in planetary science and astrobiology.“

Detecting Earth-like Planets

Jupiter only

1 milliarc-seconds for a Star at 10 parsecs

Multiple Planet Systems

Goal is to get three-dimensional orbit information

• Relationship between terrestrial and giant planets

• Mean motion resonances

• Long term eccentricty evolution (co-planarity)


Detectabilty of Earth-mass planets investigated through blind tests

• Team A: generated 150 planetary systems with the best guess of mass and period distributions

• Team B: rotated systems, set up realistic observing schedules, created synthetic RV and astrometric signals and added noise.

• Team C: detection groups (4)


Reliability = ratio of true detections to all detections (true + false)

Results:

1 group: 100% reliability

2 groups: over 80%

1 group over 40%

Based on 1% false alarm probability

The SIM Reference Grid

• Must be a large number of objects

• K giant stars V = 10–12 mag

• Distance ≈ several kilopcs

• Over 4000 K giants were observed with precise radial velocity measurements to assure that they were constant

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).

Similar effect in radial velocity:


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


Gravitational deflection of light:

defl =2 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) 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 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 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

Astrometric signal as a function of spot filling factor

Aast (as) = 7.1f 0.92 (10/D)

D = distance in pc, f is filling factor in percent

For a star like the sun, sunspots can cause an astrometric displacement of ~1 as

1 milliarcsecond

Our solar system from 32 light years (10 pcs)

40 as

In spite of all these „problems“ SIM and possibly 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 sucessful 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

Documents

Astrometric Detection of Planets