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Radial Velocity Detection of Planets: I. Techniques

Radial Velocity Detection of Planets: I. Techniques. Keplerian Orbits Spectrographs/Doppler shifts Precise Radial Velocity measurements. 4 p 2. (a s + a p ) 3. P 2 =. G(m s + m p ). Newton‘s form of Kepler‘s Law. V. m p. m s. a p. a s. 4 p 2. a p 3. P 2 ≈. Gm s. 4 p 2.

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Radial Velocity Detection of Planets: I. Techniques

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  1. Radial Velocity Detection of Planets:I. Techniques • Keplerian Orbits • Spectrographs/Doppler shifts • Precise Radial Velocity measurements

  2. 4p2 (as + ap)3 P2 = G(ms + mp) Newton‘s form of Kepler‘s Law V mp ms ap as

  3. 4p2 ap3 P2≈ Gms 4p2 (as + ap)3 P2 = G(ms + mp) ap» as Approximations: ms» mp

  4. Solve Kepler‘s law for ap: 2pas V = Circular orbits: P 1/3 ( P2Gms ) ap = ms× as = mp× ap Conservation of momentum: mp ap 4p2 as = ms … and insert in expression for as and then V for circular orbits

  5. 2p mp P2/3 G1/3ms1/3 V = P(4p2)1/3 mp 0.0075 mp V = 28.4 = P1/3ms2/3 P1/3ms2/3 mp sin i 28.4 P1/3ms2/3 Vobs = mp in Jupiter masses ms in solar masses P in years V in m/s

  6. Radial Velocity Amplitude of Sun due to Planets in the Solar System

  7. Radial Velocity Amplitude of Planets at Different a G2 V star Radial Velocity (m/s)

  8. A0 V star Radial Velocity (m/s)

  9. M2 V star

  10. Elliptical Orbits O eccentricity = OF1/a

  11. Not important for radial velocities Important for radial velocities W: angle between Vernal equinox and angle of ascending node direction (orientation of orbit in sky) i: orbital inclination (unknown and cannot be determined P: period of orbit w: orientation of periastron e: eccentricity M or T: Epoch K: velocity amplitude

  12. Radial velocity shape as a function of eccentricity:

  13. Radial velocity shape as a function of w, e = 0.7 :

  14. Eccentric orbit can sometimes escape detection: With poor sampling this star would be considered constant

  15. Eccentricities of bodies in the Solar System

  16. (mp sin i)3 (mp + ms)2 P K3(1 – e2)3/2 = f(m) = 2pG Important for orbital solutions: The Mass Function P = period K = Amplitude mp = mass of planet (companion) ms = mass of star e = eccentricity

  17. Because you measure the radial component of the velocity you cannot be sure you are detecting a low mass object viewed almost in the orbital plane, or a high mass object viewed perpendicular to the orbital plane We only measure MPlanet xsin i i Observer

  18. The orbital inclination We only measure m sin i, a lower limit to the mass. What is the average inclination? i P(i) di = 2p sin i di The probability that a given axial orientation is proportional to the fraction of a celestrial sphere the axis can point to while maintaining the same inclination

  19. p p ∫ ∫ P(i) sin i di P(i) di 0 0 The orbital inclination P(i) di = 2p sin i di Mean inclination: <sin i> = = p/4 = 0.79 Mean inclination is 52 degrees and you measure 80% of the true mass

  20. p p p ∫ ∫ P(i) sin 3i di P(i) di ∫ sin 4i di = 0.5 0 0 0 The orbital inclination P(i) di = 2p sin i di But for the mass function sin3i is what is important : <sin3 i> = = 3p/16 = 0.59

  21. q 2 ∫ P(i) di (1 – cos q ) = 0 p ∫ P(i) di 0 q < 10 deg : P= 0.03 (sin i = 0.17) The orbital inclination P(i) di = 2p sin i di Probability i < q : P(i<q) =

  22. Dv c Measurement of Doppler Shifts In the non-relativistic case: l – l0 = l0 We measure Dv by measuring Dl

  23. The Radial Velocity Measurement Error with Time How did we accomplish this?

  24. The Answer: • Electronic Detectors (CCDs) • Large wavelength Coverage Spectrographs • Simultanous Wavelength Calibration (minimize instrumental effects)

  25. Instrumentation for Doppler Measurements High Resolution Spectrographs with Large Wavelength Coverage

  26. Cross disperser slit collimator Echelle Spectrographs camera detector corrector From telescope

  27. with disperser A spectrograph is just a camera which produces an image of the slit at the detector. The dispersing element produces images as a function of wavelength without disperser slit

  28. 5000 A n = –2 4000 A 5000 A n = –1 4000 A 4000 A n = 1 5000 A 4000 A n = 2 5000 A Most of light is in n=0

  29. These are the instensity distribution of the first 3 orders of a reflection grating

  30. 6000 14000 m=1 m=2 4000 9000 m=3 3000 5000 1200 gr/mm grating l1 Schematic: orders separated in the vertical direction for clarity l2 You want to observe l1 in order m=1, but light l2 at order m=2, where l1 ≠ l2 contaminates your spectra Order blocking filters must be used

  31. In reality: Need interference blocking filter but why throw away light? 79 gr/mm grating Schematic: orders separated in the vertical direction for clarity 9000 14000 m=99 m=100 5000 9000 m=101 4000 5000 2000 3000

  32. ∞ y l2 Dy dl Free Spectral Range Dl = l/m m-2 m-1 m m+2 m+3 Grating cross-dispersed echelle spectrographs

  33. y x On a detector we only measure x- and y- positions, there is no information about wavelength. For this we need a calibration source

  34. CCD detectors only give you x- and y- position. A doppler shift of spectral lines will appear as Dx Dx → Dl → Dv How large is Dx ?

  35. ← 2 detector pixels dl l R = dl Spectral Resolution Consider two monochromatic beams They will just be resolved when they have a wavelength separation of dl Resolving power: dl = full width of half maximum of calibration lamp emission lines l1 l2

  36. Dl c v = l = 1/300 pixel Dv = 10 m/s = 0.05 mm = 5 x 10–6 cm R = 50.000 → Dl = 0.11 Angstroms → 0.055 Angstroms / pixel (2 pixel sampling) @ 5500 Ang. 1 pixel typically 15 mm 1 pixel = 0.055 Ang → 0.055 x (3•108 m/s)/5500 Ang → = 3000 m/s per pixel Dv = 1 m/s = 1/1000 pixel → 5 x 10–7 cm

  37. For Dv = 20 m/s So, one should use high resolution spectrographs….up to a point How does the RV precision depend on the properties of your spectrograph?

  38. sNlines = s1line/√Nlines → Need broad wavelength coverage Wavelength coverage is inversely proportional to R: High resolution Low resolution Dl Dl detector Wavelength coverage: • Each spectral line gives a measurement of the Doppler shift • The more lines, the more accurate the measurement:

  39. Noise: s I I = detected photons Signal to noise ratio S/N = I/s For photon statistics: s = √I → S/N = √I Note: recall that if two stars have magnitudes m1 and m2, their brightness ratio is DB = 2.512(m1–m2)

  40. Exposure factor 1 4 16 36 144 400 Price: S/N t2exposure s  (S/N)–1 S/N t2exposure

  41. How does the radial velocity precision depend on all parameters? s (m/s) = Constant × (S/N)–1 R–3/2 (Dl)–1/2 • : error R: spectral resolving power S/N: signal to noise ratio Dl : wavelength coverage of spectrograph in Angstroms For R=110.000, S/N=150, Dl=2000 Å, s = 2 m/s C ≈ 2.4 × 1011

  42. The Radial Velocity precision depends not only on the properties of the spectrograph but also on the properties of the star. Good RV precision → cool stars of spectral type later than F6 Poor RV precision → hot stars of spectral type earlier than F6 Why?

  43. A7 star K0 star Early-type stars have few spectral lines (high effective temperatures) and high rotation rates.

  44. Too faint (8m class tel.). Poor precision Ideal for 3m class tel. Main Sequence Stars RV Error (m/s) M0 K5 F0 K0 A5 G0 G5 A0 F5 Spectral Type 98% of known exoplanets are found around stars with spectral types later than F6

  45. v sin i ( ) 2 Including dependence on stellar parameters s (m/s) ≈ Constant ×(S/N)–1 R–3/2 (Dl)–1/2 f(Teff) v sin i : projected rotational velocity of star in km/s f(Teff) = factor taking into account line density f(Teff) ≈ 1 for solar type star f(Teff) ≈ 3 for A-type star f(Teff) ≈ 0.5 for M-type star

  46. Eliminate Instrumental Shifts Recall that on a spectrograph we only measure a Doppler shift in Dx (pixels). This has to be converted into a wavelength to get the radial velocity shift. Instrumental shifts (shifts of the detector and/or optics) can introduce „Doppler shifts“ larger than the ones due to the stellar motion z.B. for TLS spectrograph with R=67.000 our best RV precision is 1.8 m/s → 1.2 x 10–6 cm

  47. Traditional method: Observe your star→ Then your calibration source→

  48. Problem: these are not taken at the same time… ... Short term shifts of the spectrograph can limit precision to several hunrdreds of m/s

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