Download Astrometry

Astrometry
•  Conceptually identical to radial velocity searches -- however searching for
wobble in position of host star rather than velocity
•  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, proper motion of star
•  RV technique is most sensitive to planets close to the parent star, but the
astrometry technique is more sensitive to planets far from the host star
Magnitude of effect: amplitude of stellar wobble (half peak
displacement) for an orbit in the plane of the sky is
a1 = (mp / M*) a, and hence:
⎛ m p ⎞⎛ a ⎞
Δθ = ⎜⎜
⎟⎟⎜ ⎟
⎝ M * ⎠⎝ d ⎠
for a star at distance d. Note we have again used mp << M*
Writing the mass ratio q = mp / M*, this gives:
-1
⎛ q ⎞⎛ a ⎞⎛ d ⎞
Δθ = 0.5⎜ −3 ⎟⎜
⎟ mas
⎟⎜
⎝ 10 ⎠⎝ 5AU ⎠⎝ 10pc ⎠
Note:
•  Units here are milliarcseconds - very small effect
•  Different dependence on orbital separation than radial velocity
method - astrometric planet searches are more sensitive at large
orbital distance, a
•  Explicit dependence on d (distance to the star). radial
velocity measurements not explicitely sensitive to stellar
distance, but practically more distant stars are fainter, thus the
S/N of the spectra is lower and it is harder to measure small
radial velocity variations)
•  Detection of planets at large orbital radii still requires a search
time comparable to the orbital period
Epsilon Eridani
Data obtained 1980-2006
to track the orbit
P=6.9 years, Mp=1.55 MJ
The wobble of the Sun’s projected
position due to the influence of
all the planets as seen from 10 pc
Future Astrometric Missions
•  PRIMA on VLT and GAIA (space-based) are future missions
• Planned astrometric errors at the ~10 micro-arcsecond level - good
enough to detect planets of a few Earth masses at 1 AU around
nearby stars
Radial Velocity Technique
Detection of Exoplanets through Periodic Radial Velocity
Variations in the Host Star
• Currently the most successful
method used to detect
exoplanets with 269 planets
detected with this technique
• The first planet around a normal
star, 51 Peg, was detected
through its RV signature on the
host star in 1995
• Method relies on the Doppler
shift of starlight resulting from
the star orbiting a common
center of mass with a companion
planet
a1
M*
Centre of mass
Star-Planet Systems
a2
mp
Orbital Elements:
Semi-major axis: a = a1+
Period: P
Inclination angle: i
Eccentricity: e
a2
Useful equations that apply to the system:
Kepler’s 3rd Law: G(M*+mp)P2 = 4pi2 a3
Conservation of COM: M*a1= mpa2
Conservation of Energy: M*v*2/2 + mpvp2/2 = -GM*mp/a
Force equality : M*v*2/a1 = mpvp2/a2 = GM*mp/a2
If you have a planet in orbit around a star with semimajor axis, a
a1 = distance between the center of the star and the center of mass
a2 = distance between the center of the star and the center of mass
M*
a2
a1
mp
Center of mass
The star and planet both rotate around the centre of mass with
the same angular velocity:
Ω=
G(M * + m p )
a3
Assuming mp << M*
Using Kepler’s Law and a1 M* = a2 mp (conservation of center of
mass), stellar velocity in the inertial frame is:
⎛ m p ⎞ GM *
v* ≈ ⎜⎜
⎟⎟
a
⎝ M * ⎠
Magnitude of the velocity is <= 10s of m/s
Hot Jupiter (P=5 days) orbiting a 1Msun star: 125 m/s
Jupiter orbiting the Sun: 12.5 m/s
Sun due to Earth: 0.1 m/s
This is a very small RV variation, thus specialized techniques and
spectrographs that can measure radial velocity shifts of ~10-3 of a pixel
stably over many years are required
We have calculated from basic physics the speed of the star in it’s
reference frame due to the gravitational influence of the planet. What is actually observed is the radial velocity -- the
component of the stellar velocity in the direction along the
line of sight to the observer
using the Doppler Technique
Relationship between the observed frequency, f’, and emitted freqency, fo
v = velocity of waves in the medium
vs = velocity of the source For v >> vs (electromagnetic waves where v=c) v = velocity of the source
c = speed of light This has 2 implications:
(1)  for a planet on a circular orbit, the observed velocity is a sine wave and for
a planet on an elliptical orbit, the observed velocity is more complicated, but
still varies periodically from its maximum.
Eccentric orbit:
Circular orbit: e-->0, ω--> 0
(2) The peak velocity that is observed is modified by the inclination
angle of the orbit such that:
vobs = v* sin(i )
With
⎛ m p ⎞ GM *
v* ≈ ⎜⎜
⎟⎟
⎝ M * ⎠ a
The measured quantity is:
m p sin(i )
(assuming M* is well determined)
Thus, observing vobs allows for obtaining only a lower limit on
the planetary mass if there are no other constraints on the
inclination angle
High sensitivity to small radial velocity shifts:
•  Achieved by comparing high S/N ~ 200 spectra with
template stellar spectra
•  Large number of lines in spectrum allows shifts of much
less than one pixel to be determined
Absolute wavelength calibration and stability over long timescales:
•  Achieved by passing stellar light through a cell containing
iodine, imprinting large number of additional lines of known
wavelength into the spectrum. Or simultaneous Thorium-Argon
spectrum along with the target star spectrum
•  Calibration suffers identical instrumental distortions as the data
Starlight from
telescope
Iodine cell
Spectrograph
CCD
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
• < 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
• remaining systematic errors in the observations
Currently, best observations achieve:
Best RV precision = < 1 m/s
...in a single measurement. Allowing for the detection of low
mass planets with peak Vrad amplitudes of ~ 3 m/s
HD 40307, with a radial velocity amplitude of ~ 2 m/s, has
the smallest amplitude wobble so far attributed to a planet.
Radial velocity monitoring detects massive planets (gas giants,
especially those at small a. It is now also detecting super-Earth
mass planets (< 10 ME)
Examples of radial velocity data
51 Peg, the first known exoplanet with a 4 day, circular orbit
Example of a planet with an eccentric orbit: e=0.67
Example of a planetary system with 3 gas giant planets
Planet b: 0.059 AU, 0.72 MJ e=0.04
Planet c: 0.83 AU, 1.98 MJ, e=0.23
Planet d: 2.5 AU, 4.1 MJ, e=0.36
Upsilon And: F8V star
M*=1.28 Msun
Teff=6100 K
RV Curve
Example of a multiplanet system of super Earths
•  22 planets discovered with Mp < 30 ME
•  9 super-Earths (2 ME < Mp < 10 ME)
•  Found at a range of orbital separations:
–  Microlensing detection of 5.5 ME at 2.9 AU
–  RV detections at P~ few days to few hundred
days
•  80% are found in multi-planet systems
GJ 581 M3V 0.31 Msun
581b
Bonfils, et al. 2005
Udry, et al. 2007
5.3d 15.7 ME
581c 12.9d 5.0 ME
581d 83.6d 7.7 ME
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)
log (mp sin(i))
Planet mass sensitivity as a
function of orbital separation
Detectable
Undetectable
log a
⎛ m p ⎞ GM *
v* ≈ ⎜⎜
⎟⎟
⎝ M * ⎠ a
m p sin(i ) ∝ a1 2
Selection function 2
Summary
Observables:
•  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
Current limits:
• Maximum ~ 6 AU (ie orbital period ~ 15 years)
• Minimum mass set by activity level of the star:
•  ~ 0.5 MJ at 1 AU for a typical star
•  4 ME for short period planet around low-activity star
• No strong selection bias in favour / against detecting
planets with different eccentricities