# The Doppler Method, or Radial Velocity Detection of Planets: I. Technique

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1 ASTs309L The Doppler Method, or Radial Velocity Detection of Planets: I. Technique 1. Keplerian Orbits 2. Spectrographs/Doppler shifts 3. Precise Radial Velocity measurements

2

3 ASTs309L The Doppler Effect:

4 ASTs309L The Radial Velocity Technique:

5 ASTs309L

6 Johannes Kepler ( ) Three laws of planetary motion: 1. Planets move in ellipses with the Sun in on focus 2. The radius vector describes equal areas in equal times 3. The squares of the periods are to each other as the cubes of the mean distances

7 ASTs309L Newton s form of Kepler s Law V m p m s a p a s V obs = 28.4 m p sin i P 1/3 m 2/3 s Approximations: m s» m p

8 ASTs309L Radial Velocity Amplitude of Sun due to Planets in the Solar System Planet Mass (M J ) V(m s 1 ) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

9 ASTs309L Radial Velocity Amplitude of Planets at Different a G2 V star Radial Velocity (m/s)

10 ASTs309L i 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 M Planet x sin i

11 ASTs309L

12 ASTs309L Radial Velocity measurements Requirements: Precision of better than 10 m/s Stability for at least 10 Years V obs = 28.4 m p sin i P 1/3 m 2/3 s Jupiter: 12 m/s, 11 years Saturn: 3 m/s, 30 years

13 Radial velocity shape as a function of eccentricity:

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

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

16 Measurement of Doppler Shifts In the non-relativistic case: l l 0 l 0 = Dv c We measure Dv by measuring Dl

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

18 The Answer: 1. Electronic Detectors (CCDs) 2. Large wavelength Coverage Spectrographs 3. Simultaneous Wavelength Calibration (minimize instrumental effects) 4. Fast Computers...

19 Instrumentation for Doppler Measurements High Resolution Spectrographs with Large Wavelength Coverage

20 Echelle Spectrographs camera detector corrector From telescope slit Cross disperser collimator

21 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 slit without disperser slit with disperser

22 A Spectrum Of A Star:

23 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

24 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?

25 R Ang/pixel Velocity per pixel (m/s) For Dv = 20 m/s Dpixel Shift in mm So, one should use high resolution spectrographs.up to a point How does the RV precision depend on the properties of your spectrograph?

26 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?

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

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

29 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

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

32 Solution 1: Observe your calibration source (Th-Ar) simultaneously to your data: Stellar spectrum Thorium-Argon calibration Spectrographs: CORALIE, ELODIE, HARPS

33 Solution 2: Absorption cell a) Griffin and Griffin: Use the Earth s atmosphere:

34 O Angstroms

35 Example: The companion to HD using the telluric method. Best precision is m/s Filled circles are data taken at McDonald Observatory using the telluric lines at 6300 Ang.

36 Limitations of the telluric technique: Limited wavelength range ( 10s Angstroms) Pressure, temperature variations in the Earth s atmosphere Winds Line depths of telluric lines vary with air mass Cannot observe a star without telluric lines which is needed in the reduction process.

37 b) Use a controlled absorption cell Absorption lines of star + cell Absorption lines of the star Absorption lines of cell

38 Campbell & Walker: Hydrogen Fluoride (HF) cell: Drawbacks: Limited wavelength range ( 100 A) Temperature stablized at 100 C Long path length (1m) Has to be refilled every observing run Dangerous Demonstrated radial velocity precision of 13 m s 1 in 1980!

39 A better idea: Iodine cell (first proposed by Beckers in 1979 for solar studies) Spectrum of iodine Advantages over HF: 1000 Angstroms of coverage Stablized at C Short path length ( 10 cm) Can model instrumental profile Cell is always sealed and used for >10 years If cell breaks you will not die!

40 HARPS Simultaneous ThAr cannot model the IP. One has to stabilize the entire spectrograph

41 Barycentric Correction Earth s orbital motion can contribute ± 30 km/s (maximum) Earth s rotation can contribute ± 460 m/s (maximum)

42 Needed for Correct Barycentric Corrections: Accurate coordinates of observatory Distance of observatory to Earth s center (altitude) Accurate position of stars, including proper motion: a, d a, d Worst case Scenario: Barnard s star Most programs use the JPL Ephemeris which provides barycentric corrections to a few cm/s

43 For highest precision an exposure meter is required Photons from star No clouds time Clouds Mid-point of exposure Photons from star Centroid of intensity w/clouds time

44 Differential Earth Velocity: Causes smearing of spectral lines Keep exposure times < min

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