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

apap asas V P 2 = 4242 (a s + a p ) 3 G(m s + m p ) msms mpmp Newton‘s form of Kepler‘s Law

P 2 = 4242 (a s + a p ) 3 G(m s + m p ) Approximations: a p » a s m s » m p P2 ≈P2 ≈ 4242 ap3ap3 Gm s

Circular orbits: V = 2as2as P „Lever arm“: m s × a s = m p × a p a s = m p a p msms Solve Kepler‘s law for a p : a p = P 2 Gm s 4242 () 1/3 … and insert in expression for a s and then V for circular orbits

V = 22 P(4  2 ) 1/3 m p P 2/3 G 1/3 m s 1/3 V = P 1/3 m s 2/3 mpmp = 28.4 P 1/3 m s 2/3 mpmp m p in Jupiter masses m s in solar masses P in years V in m/s 28.4 P 1/3 m s 2/3 m p sin i V obs =

PlanetMass (M J )V(m s –1 ) Mercury1.74 × 10 – Venus2.56 × 10 – Earth3.15 × 10 – Mars3.38 × 10 – Jupiter Saturn Uranus Neptune Radial Velocity Amplitude of Sun due to Planets in the Solar System

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

Radial Velocity (m/s) A0 V star

M2 V star

eccentricity Elliptical Orbits = OF1/a O

P: period of orbit e: eccentricity  : orientation of periastron with viewing direction M or T: Epoch K: velocity amplitude Derived  : angle between Vernal equinox and angle of ascending node direction (orientation of orbit in sky) i: orbital inclination (unknown and cannot be determined Not derived

Observer 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 i Radial Velocity measurements only gives you one component of the full velocity:

Radial velocity shape as a function of eccentricity:

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

radialvelocitysimulator.htm

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

Important for orbital solutions: The Mass Function f(m) = (m p sin i) 3 (m p + m s ) 2 = P 2G2G K 3 (1 – e 2 ) 3/2 P = period K = Measured Velocity Amplitude m p = mass of planet (companion) m s = mass of star e = eccentricity i = orbital inclination

The orbital inclination We only measure m sin i, a lower limit to the mass. What is the average inclination? i 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 P(i) di = 2  sin i di

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

The orbital inclination P(i) di = 2  sin i di Probability i <  : P(i<  ) =  ∫ P(i) di 0  ∫ 0 2 (1 – cos  ) =  < 10 deg : P= 0.03 (sin i = 0.17)

The orbital inclination P(i) di = 2  sin i di But for the mass function sin 3 i is what is important : =  ∫ P(i) sin 3 i di 0  ∫ P(i) di 0 =  ∫ sin 4 i di  0 = 3  /16 = 0.59

Measurement of Doppler Shifts In the non-relativistic case: – 0 0 = vv c We measure  v by measuring 

The Grandfather of Radial Velocity Planet Detections Christian Doppler, Discoverer of the Doppler effect Born: , in Salzburg Died: in Venice radialvelocitydemo.htm Bild: Wikipedia

The Radial Velocity Measurement Error with Time How did we accomplish this? Fastest Military Aircraft (SR-71 Blackbird) High Speed Train World Class Sprinter Casual Walk Average Jogger

The Answer: 1. Electronic Detectors (CCDs) 2. Large wavelength coverage Spectrographs 3. Simultanous Wavelength Calibration (minimize instrumental effects)

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

Instrumentation for Doppler Measurements High Resolution Spectrographs with Large Wavelength Coverage

collimator Echelle Spectrographs slit camera detector corrector From telescope Cross disperser Grating for dispersing light

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 with disperser slit

A Modern Grating

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

As you go to higher and higher order the orders overlap spatially

3000 m= m= m= Schematic: orders separated in the vertical direction for clarity 1200 grooves/mm grating 2 1 You want to observe 1 in order m=1, but light 2 at order m=2, where 1 ≠ 2 contaminates your spectra Order blocking filters must be used

4000 m=99 m=100 m= Schematic: orders separated in the vertical direction for clarity 79 grooves/mm grating Need interference blocking filter but why throw away light? In reality:

yy ∞ 2 y m-2 m-1 m m+2 m+3   Free Spectral Range  m Grating cross-dispersed echelle spectrographs

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

Star→ Calibration source→

CCD detectors only give you x- and y- position. A Doppler shift of spectral lines will appear as  x  x →  →  v How large is  x ?

Spectral Resolution d 1 2 Consider two monochromatic beams They will just be resolved when they have a wavelength separation of d Resolving power: d = full width of half maximum of calibration lamp emission lines R = d ← 2 detector pixels

R = R >

R = →  = 0.11 Angstroms → Angstroms / pixel (2 pixel 5500 Ang. 1 CCD pixel typically 15  m (1  m = 10 –6 m) 1 pixel = Ang → x (310 8 m/s)/5500 Ang → = 3000 m/s per pixel = v  c  v = 10 m/s = 1/300 pixel = 0.05  m = 5 x 10 –6 cm  v = 1 m/s = 1/1000 pixel → 5 x 10 –7 cm

RAng/pixelVelocity per pixel (m/s)  pixel Shift in mm On detector ×10 – ×10 – – ×10 – ×10 – ×10 –4 10 – ×10 –4 2×10 –6 So, one should use high resolution spectrographs….up to a point For  v = 20 m/s How does the RV precision depend on the properties of your spectrograph?

Wavelength coverage: Each spectral line gives a measurement of the Doppler shift The more lines, the more accurate the measurement:  Nlines =  1line /√N lines → Need broad wavelength coverage Wavelength coverage is inversely proportional to R:  detector Low resolution High resolution 

Noise:  Signal to noise ratio S/N = I/  I For photon statistics:  = √I → S/N = √I I = detected photons Note: recall that if two stars have magnitudes m 1 and m 2, their brightness ratio is  B = (m 1 – m 2 )

Note: Photographic plates: S/N = CCD Detectors: S/N =

radialvelocitysimulator.htm

Velocity error  (S/N) –1 Price: S/N  t 2 exposure 14 Exposure factor Photons  t exposure

How does the radial velocity precision depend on all parameters?  (m/s) = Constant × (S/N) –1 R –3/2 (  ) –1/2  : error R: spectral resolving power S/N: signal to noise ratio  : wavelength coverage of spectrograph in Angstroms For R= , S/N=150,  =2000 Å,  = 2 m/s C ≈ 2.4 × 10 11

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?

The Effects of Stellar Rotation (v sin i) In this case i is the angle of the rotational axis

A7 star Temperature = K K0 star Temperature = 4000 K Early-type stars have few spectral lines (high effective temperatures) and high rotation rates.

A0 A5 F0 F5 RV Error (m/s) G0G5 K0 K5 M0 Spectral Type Main Sequence Stars Ideal for 3m class tel. Too faint (8m class tel.). Poor precision 98% of known exoplanets are found around stars with spectral types later than F6

Including dependence on stellar parameters 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)  (m/s) ≈ Constant ×(S/N) –1 R –3/2 v sin i ( 2 ) f(T eff ) (  ) –1/2

Eliminate Instrumental Shifts Recall that on a spectrograph we only measure a Doppler shift in  x (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

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

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

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

Advantages of simultaneous Th-Ar calibration: Large wavelength coverage (2000 – 3000 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

One Problem: Th-Ar lamps change with time!

Absorption lines of the star Absorption lines of cellAbsorption lines of star + cell Solution #2: Use a gas absorption cell

Campbell & Walker: Hydrogen Fluoride cell: Demonstrated radial velocity precision of 13 m s –1 in 1980!

Drawbacks: Limited wavelength range (≈ 100 Ang.) Temperature stablized at 100 C Long path length (1m) Has to be refilled after every observing run Dangerous

A better idea: Iodine cell (first proposed by Beckers in 1979 for solar studies) Advantages over HF: 1000 Angstroms of coverage Stablized at 50–75 C Short path length (≈ 10 cm) Cell is always sealed and used for >10 years If cell breaks you will not die! Spectrum of iodine

Spectrum of star through Iodine cell:

The iodine cell used at the CES spectrograph at La Silla

HARPS To improve RV precision you also need to stabilize the spectrograph

Detecting Doppler Shifts with Cross Correlation  (x-u) a1a1 a2a2 g(x) a3a3 a2a2 a3a3 a1a1 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 CCF

ObservationStandard CCF S/N=100 S/N=10 S/N=5

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“ Additional information on RV method:

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

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

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

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

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.

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(  t +  ) + Constant Where  = 2  /P,  = phase shift Best fit minimizes the  2 :  2 =  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

Period Analysis 2. Discrete Fourier Transform: Any function can be fit as a sum of sine and cosines FT(  ) =  X j (T) e –i  t N0N0 j=1 A DFT gives you as a function of frequency the amplitude (power = amplitude 2 ) of each sine wave that is in the data Power: P x (  ) = | FT X (  )| 2 1 N0N0 P x (  ) = 1 N0N0 N 0 = number of points [(  X j cos  t j +  X j sin  t j ) ( ) ] 2 2 Recall e i  t = cos  t + i sin  t X(t) is the time series

A pure sine wave is a delta function in Fourier space t P AoAo FT  AoAo 1/P

Period Analysis 3. Lomb-Scargle Periodogram: Power is a measure of the statistical significance of that frequency (1/period): 1 2 P x (  ) = [  X j sin  t j –  ] 2 j  X j sin 2  t j –  [  X j cos  t j –  ] 2 j  X j cos 2  t j –  j 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 tan(2  ) =  sin 2  t j )/  cos 2  t j ) j j

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

Bootstrap Randomization: 1. Take your data and randomly shuffle the values (velocity, brightness, etc.) keeping the times fixed 2. Calculate Lomb-Scargle Periodogram 3. Find the peak with the highest power 4. Reshuffle data, repeat times 5.The fraction of runs where the highest peak in your shuffled data exceeds the maximum power of your real data is your false alarm probabilty

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?

Try 16.3 minutes:

Lomb-Scargle Periodogram of the data:

The first Tautenburg Planet: HD 13189

Least squares sine fitting: The best fit period (frequency) has the lowest  2 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 Amplitude (m/s)

Noise level Alias Peak False alarm probability ≈ 10 –14

Alias periods: Undersampled periods appearing as another period

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 (fs =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.

Raw data After removal of dominant period

Given enough measurements you can find a signal in your data that has an amplitude much less than your measurement error.

Scargle Periodogram: The larger the Scargle power, the more significant the signal is:

Rule of thunb: If a peak has an amplitude 3.6 times the surrounding frequencies it has a false alarm probabilty of approximately 1%

Confidence (1 – False Alarm Probability) of a peak in the DFT as height above mean amplitude around peak

To summarize the period search techniques: 1. Sine fitting gives you the  2 as a function of period.  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 Period04: multi-sine fitting with Fourier analysis. Tutorials available plus versions in Mac OS, Windows, and Linux Generalized Periodogram: