Radial Velocity Detection of Planets: I. Techniques and Tools
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1 Radial Velocity Detection of Planets: I. Techniques and Tools 1. Keplerian Orbits 2. Spectrographs/Doppler shifts 3. Precise Radial Velocity measurements 4. Searching for periodic signals
2 Detection and Properties of Planetary Systems Schedule of Lectures 06. Apr: Introduction and Background: 13. Apr: The Radial Velocity Method: Techniques and Keplerian Orbits 20. Apr: The Radial Velocity Method: Results 27. Apr: Astrometric Detections 04. May: Transiting Exoplanets: Method and Ground-based Results 11. May: No class 18. May: Transiting Exoplanets: Space Missions 25. May: Transiting Exopanets : Atmospheres 01. Jun: Host Stars 08. Jun: No Class 15. Jun: Gravitational Microlensing 22. Jun: Planets in Extreme Envirnonments 29. Jun: Planets in other environments (Dr. Eike Guenther) 06. Jul: In Search of Earth 2.0
3 Newton s form of Kepler s Law V m p m s a p a s P 2 = 4π 2 (a s + a p ) 3 G(m s + m p )
4 P 2 = 4π 2 (a s + a p ) 3 G(m s + m p ) Approximations: a p» a s m s» m p P 2 4π 2 a p 3 Gm s
5 Circular orbits: V = 2πa s P Lever arm : m s a s = m p a p a s = m p a p m s Solve Kepler s law for a p : a p = ( P 2 Gm s ) 1/3 4π 2 and insert in expression for a s and then V for circular orbits
6 V = 2π V = m p P 1/3 m 2/3 = s m p P 2/3 G 1/3 m 1/3 s P(4π 2 ) 1/ m p P 1/3 m 2/3 s m p in Jupiter masses m s in solar masses P in years V in m/s V obs = 28.4 m p sin i P 1/3 m 2/3 s
7 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
8 Radial Velocity Amplitude of Sun due to Planets in the Solar System
9 Up the main sequence: 1. Higher Mass 2. Higher temperature 3. Shorter life 2 M סּ 1 M סּ (s.m.) 0.1 M סּ
10 Radial Velocity Amplitude of Planets at Different a G2 V star Radial Velocity (m/s)
11 Radial Velocity (m/s) A0 V star
12 M2 V star
13 Elliptical Orbits O eccentricity = OF1/a
14 Derived P: period of orbit e: eccentricity ω: orientation of periastron with viewing direction M or T: Epoch K: velocity amplitude Not derived Ω: angle between Vernal equinox and angle of ascending node direction (orientation of orbit in sky) i: orbital inclination (unknown and cannot be determined
15 Radial Velocity measurements only gives you one component of the full velocity: Observer 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
16 Radial velocity shape as a function of eccentricity:
17 Radial velocity shape as a function of ω, e = 0.7 :
18 Eccentric orbit can sometimes escape detection: With poor sampling this star would be considered constant
19 Important for orbital solutions: The Mass Function f(m) = (m p sin i) 3 (m p + m s ) 2 = P K 3 (1 e 2 ) 3/2 2πG P = period K = Measured Velocity Amplitude m p = mass of planet (companion) m s = mass of star e = eccentricity i = orbital inclination
20 The orbital inclination We only measure m sin i, a lower limit to the mass. What is the average inclination? i P(i) di = 2π 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
21 The orbital inclination P(i) di = 2π sin i di Mean inclination: <sin i> = π 0 π 0 P(i) sin i di P(i) di = π/4 = 0.79 Mean inclination is 52 degrees and you measure 80% of the true mass
22 The orbital inclination P(i) di = 2π sin i di Probability i < θ : P(i<θ) = 2 θ 0 π 0 P(i) di P(i) di = (1 cos θ ) θ < 10 deg : P= 0.03 (sin i = 0.17)
23 Measurement of Doppler Shifts In the non-relativistic case: λ λ 0 λ 0 = Δv c We measure Δv by measuring Δλ
24 The Grandfather of Radial Velocity Planet Detections Christian Doppler, Discoverer of the Doppler effect Born: , in Salzburg Died: in Venice First radial velocity measurement for a star made on Sirius by Sir William Higgins in 1868 Bild: Wikipedia
25 The Radial Velocity Measurement Error with Time Fastest Military Aircraft (SR-71 Blackbird) Average Jogger High Speed Train Casual Walk World Class Sprinter How did we accomplish this?
26 The Answer: 1. Electronic Detectors (CCDs) 2. Large wavelength coverage Spectrographs 3. Simultanous Wavelength Calibration (minimize instrumental effects)
27 Charge Coupled Devices Up until the 1970s astronomers used photographic plates that had a quantum efficiency of 1-3% CCD detectors have a quantum efficiency of 80-90%. Plus the data is in digital form
28 Instrumentation for Doppler Measurements High Resolution Spectrographs with Large Wavelength Coverage
29 Echelle Spectrographs camera detector corrector From telescope slit Cross disperser Grating for dispersing light collimator
30 A Modern Grating
31 5000 A 4000 A n = A 4000 A 4000 A 5000 A 4000 A 5000 A n = 1 n = 1 n = 2 Most of light is in n=0
32 As you go to higher and higher order the orders overlap spatially
33 1200 grooves/mm grating Schematic: orders separated in the vertical direction for clarity m= λ m= m=3 λ You want to observe λ 1 in order m=1, but light λ 2 at order m=2, where λ 1 λ 2 contaminates your spectra Order blocking filters must be used
34 79 grooves/mm grating Schematic: orders separated in the vertical direction for clarity In reality: m=99 m= m= Need interference blocking filter but why throw away light?
35 δλ Free Spectral Range Δλ = λ/m m-2 m-1 m m+2 y m+3 Δy λ 2 Grating cross-dispersed echelle spectrographs
36 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
37 Star Calibration source
38 CCD detectors only give you x- and y- position. A Doppler shift of spectral lines will appear as Δx Δx Δλ Δv How large is Δx?
39 Spectral Resolution 2 detector pixels dλ Consider two monochromatic beams They will just be resolved when they have a wavelength separation of dλ Resolving power: λ 1 λ 2 R = λ dλ dλ = full width of half maximum of calibration lamp emission lines
40 R = R >
41 R = Δλ = 0.11 Angstroms Angstroms / pixel (2 pixel 5500 Ang. 1 CCD pixel typically 15 µm (1 µm = 10 6 m) 1 pixel = Ang x ( m/s)/5500 Ang = 3000 m/s per pixel v = Δλ c λ Δv = 10 m/s = 1/300 pixel = 0.05 µm = 5 x 10 6 cm Δv = 1 m/s = 1/1000 pixel 5 x 10 7 cm
42 R Ang/pixel Velocity per pixel (m/s) For Δv = 20 m/s Δpixel Shift in mm On detector So, one should use high resolution spectrographs.up to a point How does the RV precision depend on the properties of your spectrograph?
43 Wavelength coverage: Each spectral line gives a measurement of the Doppler shift The more lines, the more accurate the measurement: σ Nlines = σ 1line / N lines Need broad wavelength coverage Wavelength coverage is inversely proportional to R: Low resolution Δλ High resolution Δλ detector
44 Noise: I σ I = detected photons Signal to noise ratio S/N = I/σ For photon statistics: σ = I S/N = I Note: recall that if two stars have magnitudes m 1 and m 2, their brightness ratio is Β 2 /Β 1 = (m 1 m 2 )
45 Note: Photographic plates: S/N = CCD Detectors: S/N =
46 Exposure factor Velocity error (S/N) 1 Price: S/N t 2 exposure Photons t exposure
47 How does the radial velocity precision depend on all parameters? σ (m/s) = Constant (S/N) 1 R 3/2 (Δλ) 1/2 σ : error R: spectral resolving power S/N: signal to noise ratio Δλ : wavelength coverage of spectrograph in Angstroms For R= , S/N=150, Δλ=2000 Å, σ = 2 m/s C
48 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 (~1.2 solar masses, ~6000 K) Poor RV precision hot stars of spectral type earlier than F6 Why?
49 The Effects of Stellar Rotation (v sin i) In this case i is the angle of the rotational axis
50 A7 star Temperature = 8000 K K0 star Temperature = 4000 K Early-type stars have few spectral lines (high effective temperatures) and high rotation rates.
51 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
52 G0 K5 M0 A0 B9
53 Including dependence on stellar parameters σ (m/s) Constant (S/N) 1 R 3/2 v sin i (Δλ) 1/2 ( 2 ) f(t eff ) v sin i : projected rotational velocity of star in km/s f(t eff ) = factor taking into account line density f(t eff ) 1 for solar type star f(t eff ) 3 for A-type star (T = K, 2 solar masses) f(t eff ) 0.5 for M-type star (T = 3500, 0.1 solar masses)
54 Eliminate Instrumental Shifts Recall that on a spectrograph we only measure a Doppler shift in Δx (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= our best RV precision is 1.8 m/s 1.2 x 10 6 cm
55 Traditional method: Observe your star Then your calibration source
56 Problem: these are not taken at the same time... Short term shifts of the spectrograph can limit precision to several hunrdreds of m/s
57
58 Solution 1: Observe your calibration source (Th-Ar) simultaneously to your data: Stellar spectrum Thorium-Argon calibration Spectrographs: CORALIE, ELODIE, HARPS
59 Advantages of simultaneous Th-Ar calibration: Large wavelength coverage ( Å) Computationally simple Disadvantages of simultaneous Th-Ar calibration: Th-Ar are active devices (need to apply a voltage) Lamps change with time Th-Ar calibration not on the same region of the detector as the stellar spectrum Some contamination that is difficult to model
60 One Problem: Th-Ar lamps change with time!
61 Solution 2: telluric lines a) Griffin and Griffin: Use the Earth s atmosphere:
62 O Angstroms
63 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.
64 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.
65 Solution #3: Use a gas absorption cell Absorption lines of star + cell Absorption lines of the star Absorption lines of cell
66 Campbell & Walker: Hydrogen Fluoride cell: Demonstrated radial velocity precision of 13 m s 1 in 1980!
67 Drawbacks: Limited wavelength range ( 100 Ang.) Temperature stablized at 100 C Long path length (1m) Has to be refilled after every observing run Dangerous
68 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) Cell is always sealed and used for >10 years If cell breaks you will not die!
69 Spectrum of star through Iodine cell:
70 The iodine cell used at the CES spectrograph at La Silla
71 HARPS To improve RV precision you also need to stabilize the spectrograph
72 Detecting Doppler Shifts with Cross Correlation φ(x-u) a 1 a 2 g(x) CCF a 3 a 2 a 3 a 1 A standard method to computer the Doppler shift is to take a spectrum of a standard star, compute the cross correlation function (CCF) with your data and look for the shift in the peak
73 Observation Standard CCF S/N=100 S/N=10 S/N=5
74 Additional information on RV method: Valenti, Butler, and Marcy, 1995, PASP, 107, 966, Determining Spectrometer Instrumental Profiles Using FTS Reference Spectra Butler, R. P.; Marcy, G. W.; Williams, E.; McCarthy, C.; Dosanjh, P.; Vogt, S. S., 1996, PASP, 108, 500, Attaining Doppler Precision of 3 m/s Endl, Kürster, Els, 2000, Astronomy and Astrophysics, The planet search program at the ESO Coudé Echelle spectrometer. I. Data modeling technique and radial velocity precision tests
75 Barycentric Correction Earth s orbital motion can contribute ± 30 km/s (maximum) Need to know: Position of star Earth s orbit Exact time Earth s rotation can contribute ± 460 m/s (maximum) Need to know: Latitude and longitude of observatory Height above sea level
76 Needed for Correct Barycentric Corrections: Accurate coordinates of observatory Distance of observatory to Earth s center (altitude) Accurate position of stars, including proper motion: α, δ α, δ Worst case Scenario: Barnard s star Most programs use the JPL Ephemeris which provides barycentric corrections to a few cm/s
77 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
78 Differential Earth Velocity: Causes smearing of spectral lines Keep exposure times < min
79 Finding a Planet in your Radial Velocity Data 1. Determine if there is a periodic signal in your data. 2. Determine if this is a real signal and not due to noise. 3. Determine the nature of the signal, it might not be a planet! (More on this next week) 4. Derive all orbital elements The first step is to find the period of the planet, otherwise you will never be able to fit an orbit.
80 Period Analysis: How do you find a periodic signal in your data 1. Least squares sine fitting: Fit a sine wave of the form: V(t) = A sin(ωt + φ) + Constant Where ω = 2π/P, φ = phase shift Best fit minimizes the χ 2 : χ 2 = Σ (d i g i ) 2 /N d i = data, g i = fit Note: Orbits are not always sine waves, a better approach would be to use Keplerian Orbits, but these have too many parameters
81 Period Analysis 2. Discrete Fourier Transform: Any function can be fit as a sum of sine and cosines N 0 FT(ω) = X j (T) e iωt j=1 1 Power: P x (ω) = FT X (ω) 2 P x (ω) = 1 N 0 N 0 Recall e iωt = cos ωt + i sinωt X(t) is the time series N 0 = number of points 2 2 [( Σ X j cos ωt) j + ( Σ X j sin ωt )] j A DFT gives you as a function of frequency the amplitude (power = amplitude 2 ) of each sine wave that is in the data
82 P FT A o A o t 1/P ω A pure sine wave is a delta function in Fourier space
83 Period Analysis 3. Lomb-Scargle Periodogram: P x (ω) = 1 2 [ Σ X j cos ω(t j τ) ] 2 j Σ X j cos2 ω(t j τ) j [ Σ X j sin ω(t j τ) ] 2 j Σ X j sin2 ω(t j τ) tan(2ωτ) = (Σsin 2ωt j )/ (Σcos 2ωt j ) j j Power is a measure of the statistical significance of that frequency (1/ period): False alarm probability (FAP) 1 (1 e P ) N Ne P = probability that noise can create the signal. The FAP is the probability that random noise would produce your signal. N = number of indepedent frequencies number of data points
84 As a good rule of thumb for interpreting Lomb-Scargle Power, P: P < 6 : Most likely not real 6 < P < 10: May be real but probably not 10 < P < 14: Might be real, worth investigating more 14 < P < 20: Most likely real, but you can still be fooled P > : Definitely real Caveat: Depends on noise level and the sampling. Always best to do simulations
85 Period Analysis How do you know if you have a periodic signal in your data? Here are RV data from a pulsating star What is the period?
86 Lomb-Scargle Periodogram of the data:
87 16.3 minutes:
88 The first Tautenburg Planet: HD 13189
89 Least squares sine fitting: The best fit period (frequency) has the lowest χ 2 Amplitude (m/s) Discrete Fourier Transform: Gives the power of each frequency that is present in the data. Power is in (m/ s) 2 or (m/s) for amplitude Lomb-Scargle Periodogram: Gives the power of each frequency that is present in the data. Power is a measure of statistical signficance
90 False alarm probability Alias Peak Noise level
91 Alias periods: Undersampled periods appearing as another period
92 The Nyquist frequency tells you the sampling rate for which you can find periodic signals in your data. If f s is the sampling frequency, then the Nyquist frequency is 0.5 f s. Example: If you observe a star once a day (f s =1 d 1 ), your Nyquist frequency is 0.5 d 1. This means you cannot reliably detect periods shorter than 2 day in your data due to alias effects.
93 Raw data After removal of dominant period
94 Given enough measurements you can find a signal in your data that has an amplitude much less than your measurement error.
95 Scargle Periodogram: The larger the Scargle power, the more significant the signal is:
96 Rule of thunb: If an amplitude peak in the DFT has an amplitude 3.6 times the surrounding frequencies it has a false alarm probabilty of approximately 1%
97 Confidence (1 False Alarm Probability) of a peak in the DFT as height above mean amplitude around peak
98 To summarize the period search techniques: 1. Sine fitting gives you the χ 2 as a function of period. χ 2 is minimized for the correct period. 2. Fourier transform gives you the amplitude (m/s in our case) for a periodic signal in the data. 3. Lomb-Scargle gives an amplitude related to the statistical significance of the signal in the data. Most algorithms (fortran and c language) can be found in Numerical Recipes Generalized Periodogram: Period04: multi-sine fitting with Fourier analysis. Tutorials available plus versions in Mac OS, Windows, and Linux
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