Survey
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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