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Chapter 2- Radial Velocities

Explore the principles and accuracies of measuring radial velocities in celestial bodies, such as planets and stars. Learn about orbital properties and multiple planet systems. Understand Kepler’s laws and orbital parameters for exoplanets.

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Chapter 2- Radial Velocities

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  1. PHY6795O – Chapitres Choisis en Astrophysique • Naines Brunes et Exoplanètes Chapter2- Radial Velocities

  2. Contents 2.1 Description of orbits 2.2 Measurementprinciples and accuracies 2.3 Instrument programmes 2.4 Results to date 2.5 Properties of radial velocityplanets 2.6 Multiple planetsystems 2.7 Planetsaroundbinary and multiple stars PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  3. 2.1 Description of orbits (1) • Principle: the motion of a single planet in orbitaround a star causes the star to undergo a reflex motion around the barycenter (center of mass) defined as whereM and arefers to the mass and semi-major axis, respectively. (Notation: subscriptp for planet and ★ for the host star; arelisused for a planetorbit relative to the star) • Reflex motion results in periodic perturbation of: radial velocity, angular position on the sky, and in the time arrival of someperiodic signal associatedwith the host star. • The orbitis an ellipse described in polar coordinates by The semi-major axis a, semi-major axis b are related to the eccentricitye by (2.1) (2.3) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  4. 2.1 Description of orbits (2) • ν: trueanomaly, angle referred to the elliptical (true) orbit. • E: eccentricanomaly, angle referred to auxiliarycircle. • M: meananomaly, angle refering to a fictitiousmean motion around the orbitrelated to E and ν. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  5. 2.1 Description of orbits (3) • The position of an objectalong the orbitatanychosen time t isobtained first by calculating the meananomalyM, thensolving for E (transcendentalequation 2.10) and thenusing Equation 2.6 to obtainν(t) and r(t)from Equation 2.1. • ν, E are related by • Let Pbe the orbital period. The meananomalyat time afterpericenter passage isdefined as • and related to E by (2.6) (2.9) (2.10) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  6. 2.1 Description of orbits (4) • Orbitspecification. A Keplerianorbit in three dimension isdescribed by the followingsevenparameters: • : orbit inclination (i=0; face on) • : longitude of the ascendingnode • : argument of pericenter (measured • in true orbital plane PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  7. 2.1 Description of orbits (5) • Kepler’sthreelaws of planetary motion: (1) the orbit of planetis an ellipse with the Sun at one focus; (2) the line joining the planet and the Sun sweeps out equal areas in equalintervals of time; (3) the squares of the orbital periods of the planets are proportional to the cubes of theirsemi-major axes. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  8. 2.1 Description of orbits (6) (b) Absoluteorbits: the motion of the planetis relative the barycenter. We have The sizes of the threeorbits are in proportion a★ : ap : arel = Mp: M★ : (M★ + Mp), witharel = a★ + ap. • (a) Relative orbits: the motion of the planetis relative to the star ratherthan the barycenter. The 3rd Kepler’slawis • SinceM★ >> Mp (2.15) (2.16) (2.17) (2.18) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  9. 2.1 Description of orbits (7) From Figure 2.2, we have and leading to where K is the radial velocitysemi- amplitude • The radial velocitysemi-amplitude (2.19) (2.20) (2.21) (2.22) Figure 2.2 PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  10. 2.1 Description of orbits (8) • The shape of the radial velocitycurveisdetermined by e and ω(see Figure 2.4). TogetherwithP, theircombinationconstrains the value of a★sini.Neither a★noricanbedeterminedseperately. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  11. 2.1 Description of orbits (9) Substituting 2.17 and 2.18 into 2.22 yields Alternatively, For a circularorbit and M★ >> Mp For Jupiter around the Sun (a=5.2 AU, P=11.9 yr), KJ= 12.5 m s-1. For Earth, For an Earthwithin the HZ of a anM5 (P~10 days; M★~0.1 M), K~1 m/s. • Other expressions for K (2.23) (2.26) (2.28) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  12. 2.1 Description of orbits (10) • Fitting a single planet • Radial velocitycanconstrain the following 5 observables: e, P, tp, ω and the combinationK=f(a,e,P,i). • Twoadditionalterms are usuallytakenintoaccount: (1) the systemicvelocityΥdescribing the constant component of the radial velocity of the system’s centre of mass relative to the solar system barycentre and (2) a linear trend parameterd,whichmayaccommodate instrumental drifts as well as unidentified contributions from massive, long-periodcompanions. • The radial velocity signal of a star with an orbitingplanetisthus, from Equation 2.21, (2.29) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  13. 2.1 Description of orbits (11) • Data analysis • A χ2 minimizationisused for determining the orbital parameters. Complications due to the non-linear nature of the problem. • Algorithms • Lomb-Scargleperiodigram • Levenberg-Marquardt (MPFIT in IDL) • Linearisation techniques • Bayesianmethods • Markov Chain Monte Carlo (ex: EXOFIT) • For npplanets, there are 5np + 1 • parameters to fit, Υincluded. • Correction for dynamical interaction • oftenneeded for multiple systems. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  14. Contents 2.1 Description of orbits 2.2 Measurementprinciples and accuracies 2.3 Instrument programmes 2.4 Results to date 2.5 Properties of radial velocityplanets 2.6 Multiple planetsystems 2.7 Planetsaroundbinary and multiple stars PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  15. 2.2 Measurementprinciples and accuracies (1) Doppler shift An instantaneousmeasurement of the stellar radial velocity about the star-planetbarycenterisgiven by the small, systematic Doppler shift in wavelength of the many (thousands) absorption linesthatmake the star’sspectrum. In the observer’sreference frame, the source isreceedingwithvelocityvat an angle θ relative to the direction from the observer source, the change in wavelengthΔλ=λobs-λemisgiven by the relativistic Doppler shift equation whereλobs, λem are observed and emittedwavelengths, β=v/c. For v<<c and θ<< π/2, the expression reduces to the classicalform (2.38) (2.39) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  16. 2.2 Measurementprinciples and accuracies (2) • Specialrelativisticterms are significantcorresponding to changes in vr of several m/s. • Index of refraction of the air at the spectrograph, nair, and itsdependance in wavelengthsintroduceserrors ≤ 1 m/s. • Nair=1.000277 at standard temperature and pressure • Measuringrequireslong-term (months to years) radial velocityaccuraciesat the level of 1 m/s, i.e., one part in 108. Not easy! • Precision radial velocitiesusuallyachievedwithdedicatedcross-dispersed échelle spectrographswithhighresolving power (λ/Δλ ~50 000 – 100 000) operating in the optical (450 – 700 nm). • High instrumental stability and accuratewavelength calibration isrequired. • Relatively large (4m-10m) telescopesrequired to achievehighsignal-to-noise. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  17. 2.2 Measurementprinciples and accuracies (3) Echelle spectrumof the Sun PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  18. 2.2 Measurementprinciples and accuracies (4) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  19. 2.2 Measurementprinciples and accuracies (5) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  20. 2.2 Measurementprinciples and accuracies (6) Cross-correlationspectroscopy • The Doppler shift information iscontained in many absorption lines. • Cross-correlation techniques withmasks are used to extract the information • Initiallywithphysicalmasks but limited to match various spectral types • Nowadays, maskisimplementednumerically Let εbe the Doppler shift, S(v) the spectrum and M(v) the mask, bothexpressed in velocityspacev, the cross-correlationfunctionisdefined as • The Doppler shift εisobtainedisobtained by minimizingC(ε). • The preciseshape of C(ε) depends on the intrinsic line shapes and on the template line width. • Width of C(ε) yields the stellarrotationalvelocityv sin i. • Equivalenthwitdhprovides a metallicityestimate if Teffisknown. (2.40) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  21. 2.2 Measurementprinciples and accuracies (7) Cross-correlationspectroscopy – Principle Eggenberger & Udry (2009) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  22. 2.2 Measurementprinciples and accuracies (8) Cross-correlationspectroscopy - Example PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  23. 2.2 Measurementprinciples and accuracies (9) Deriving radial velocitiesfrom Doppler shifts The followingeffects must betakenintoaccount for deriving the stellar Doppler shift. • Earth motion • Frame of referenceis the Solar System barycenter. • Earthmovementdeterminedfromephemeridesprovided by JPL. • Line shift fromgravitationalredshift • 636 m/s for the Sun, ~500 m/s for an M5V. • Stellarspace motion • Intrinsicspace motion of the star, accelerationincluded. Data obtainedfromHipparcos • Zero point • Verydifficult to establish an absolutereference for radial motion < 50 m/s PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  24. 2.2 Measurementprinciples and accuracies (10) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  25. 2.2 Measurementprinciples and accuracies (11) Wavelength calibration Wavelength calibration iskey for achievingaccurate Doppler measurements. It iscertainly one of the most important design drivers of modern precision radial velocity (PRV) instruments. • Spectrographslitwidtheffect • In velocityspace, the slitwithcanbeseveral km/s wide. This meansthat the illumination within the slit must bekeptuniformat the level of 10-3 to maintain a radial velocityaccuracylessthan 1 m/s. Variousobservational and instrumental strategiescanbeused for PRV calibration. • Use telluric (atmospheric) lines • Advantage of commonopticalpathwith the target but limitedwavelength range. Lines are intrinsically variables due to winds. Accuracy possible at the level of ~ 20 m/s. • Gascell in the opticalpath of the spectrograph • Provides a dense and accuratewavelengthreference, superimposed on the stellarlines. HF used in the past but toxic; iodine (I2) nowcommonlyused. • Pros: large number (1000s) of lines, sameopticalpath as target, provides a simultaneoustracking of the instrumental point spreadfunction. • Cons: 20-30% transmission loss, lines not uniformlydistributed and clusteredbetween 500 and 620 nm (exclude M dwarfs). Data analysis not trivial. • In the infrared: NH3 cell for VLT-CRIRES (R=100 000; 0.96-5.2 μm) yields a precision of 3-5 m/s over weeks of months. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  26. 2.2 Measurementprinciples and accuracies (12) Gascellspectroscopyprinciple PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  27. NH3cellwith CRIRES • Non-optimalexperimentdonewith CRIRES with a NH3gascellsuggeststhat ~3-5 m/s is possible. • CRIRES not designed for PRV • Veryencouragingresult • SPIROU willcover a wavelength range 70 times that of CRIRES! Bean et al, 2010 PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  28. 2.2 Measurementprinciples and accuracies (13) Wavelength calibration • Fiber-fedspectrograph • The entrance spectrographslitisreplaced by a fiber and the instrument features a decidatedfiberused for calibration. • Allow the spectrograph to beinstalledswayfrom the telescope in a separatethermally-controlled room to minimize instrumental wavelength shifts. • Optical scrambling • Minimizes variable light illumination due to multiple reflectionwith the fibers. • Octogonal fibers are particularly effective atscrambling light. • Calibration lamps • Thorium-Argonlamps (e.g. HARPS and predecessor: ELODIE) • Offer a large number of strongemissionlines over a wideoptical to infrared range • No throughputloss • Laser frequencycombs • Ideal calibration source, wide and uniformwavelengthcoverage, • Based on single laser pulse maintained on a repetitivepath, circulating in a cavity • Frequency of the comb: with , and T is the round-triptravel time withn~105-106. is the carrier envelopefrequency. • Both and syncronized by reference to atomicclocks • Enablehighstabilityat the level of ~0.01 m/s PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  29. 2.2 Measurementprinciples and accuracies (14) Example of frequencycombfrom HARPS Spectrum of a star obtainedusing the HARPS instrument on the ESO 3.6-metre telescopeat the La Silla Observatory in Chile. The lines are the light from the star spread out throughvariousorders. The dark gaps in the lines are absorption featuresfromdifferentelements in the star. The regularlyspacedbright spots justabove the lines are the spectrum of the laser frequencycombthatisused for comparison. The very stable nature and regularspacing of the frequencycombmakeit an idealcomparison, allowing the detection of minute shifts in the star’sspectrumthat are induced by the motion of orbitingplanets. Source: http://www.eso.org/public/images/ann12037a/ PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  30. 2.2 Measurementprinciples and accuracies (15) • Exposuremetering • A means to take a small fraction (a few %) of the starlight to monitor its flux during the exposure. Twopurposes: • Calculateprecisely the photon-weightedmid-point of the exposure for barycentric correction. • Optimizeexposure time to reachrequiredsignal-to-noise • Accuracylimits • Currentsensitivity (instrumental only): 0.3-0.5 m/s, a record held by the Swiss (Geneva team) • Corresponds to displacement of a few nm on the detector ! • Accuracy must bemaintained over severalyears. • RV amplitude independant of distance but SNR considerationslimit observations to stars brighterthan V < 8-10 • Easiesttargets for RV: massive planets, smallP (a) and large e. (e.g. 51 Pegasus) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  31. 2.2 Measurementprinciples and accuracies (16) • Accuracylimits • Error sources imposinglimits in on RV accuracy • Instrumental (mechanical/thermal stability, wavelength calibration) • Photon noise (fundamentallimit) • HARPS on 3.6m ESO. ~1 m/s on V~7.5 with SNR~100 on G star in 30s. • Stellar noise (so-called « jitter » noise) • Jitter noise • Activity in the stellaratmosphere (spots, plages) • Oftenverysignificant. Spots with a filling factor of a few % caninducejitter of a few m/s. • Varies on timescales comparable to stellar rotation period. • Correlatedwithchromospheric (magnetic) activity. Core of CaII H/K linesis a good proxy (S-index). • Stellar oscillations • 15-min exposuresufficient to dampresidual RV down to 0.2 m/s (HARPS) • Surface granulation • Of order 1 m/s on solar-type star. Hour-longexposureneeded to damp (seenextslide). • Unrecognized (long period) planetarycompanions PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  32. 2.2 Measurementprinciples and accuracies (17) Simulation of oscillation jitter PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  33. 2.2 Measurementprinciples and accuracies (18) • Jitter noise ischaracterized as an excess in the radial velocity standard error is the total RV standard error. includes all sources of jitter noise. • Astrophysical contributions are dependent of several variables: rotationalvelocity, age, activity (2.44) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  34. 2.2 Measurementprinciples and accuracies (19) • Jitter noise iscorrelatedwithrotationalvelocity (v sin i). PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  35. 2.2 Measurementprinciples and accuracies (20) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  36. Contents 2.1 Description of orbits 2.2 Measurementprinciples and accuracies 2.3 Instrument programmes 2.4 Results to date 2.5 Properties of radial velocityplanets 2.6 Multiple planetsystems 2.7 Planetsaroundbinary and multiple stars PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  37. 2.3 Instrument Programmes (1) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  38. 2.3 Instrument Programmes (2) Two good examples of productive instruments • HARPS (High Accuracy Radial VelocityPlanetSearcher) • 3.6m on LaSilla (Chile) • Fiber-fed échelle, R=115 000 • Twofibers: one science, one for wavelength calibration • TwoCCDs 4kx4k, 15 um pixels) • Wavelength range: 378-691 nm • In operationsince 2003 • Best instrument in the world (~1 m/s) • 2ndmost productive (# of discoveries) • HARPS-North • Copy of HARPS for the 3.6m TNG on LaPalma. In operationsince 2012 PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  39. 2.3 Instrument Programmes (3) • KECK HIRES • 10m Keck I telescope (MaunaKea; Hawaii) • échelle slitspectrograph, R=80 000 • Iodinecell for wavelength calibration • Wavelength range: 390-620 nm • First light in 1993 • Not designed for exoplanetdetection ! • Sensitivity: 1-2 m/s • Most productive in the world • # of discoveries PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  40. 2.3 Instrument Programmes (3) Future developments (smallselection) • ESPESSO • VLT 8m. • Could combine the 4 VLT together. • HARPS-likefiber-fed échelle spectrograph; R=140 000 • Wavelength range: 350-720 nm • RV accuracyrequirement: 10 cm/s • First light: 2016 • Available to the community in 2017. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  41. 2.3 Instrument Programmes (4) Future developments (smallselection) • SPIRou (SPectropolarimètreInfraROUge) • CFHT 3.6m • HARPS-likefiber-fed échelle spectropolarimeter • R=75 000 • Spectral resolution • Wavelength range: 0.95-2.35 μm • RV accuracyrequirement: 1 m/s • First light: 2017 PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  42. An 8m-class instrument on a 4m class telescope PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  43. SPIrou data simulator 4Kx4k K H J Y PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  44. SPIROU vs CRIRES (VLT) PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  45. A spectrumisworth a thousandpictures! M3V spectrum Telluriclines MaunaKea Calar Alto Animation credit: E. Artigau PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  46. A roadmap to habitable exoplanets • Act #1: detection • Build up a catalog (> 100) of rockyplanetcandidates in the habitable zone. • Transit & RV required • Lensingveryuseful for statistic (planetfrequency) • Act #2: characterization • Constraindensity of the planet • Requiresboth transit & RV data. • Atmosphericcharacterization. Seek spectral signature of H2O, CO2, CH4 and O2/O3. • Large space-based IR telescoperequired: JWST. • Best targets: M dwarfs. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  47. The M dwarfopportunity • There are lots of them! The mosttypical star in the Galaxyis a M3V (M~0.3 M). • Theyspan a factor of 5 in mass. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  48. Orbital period in the HZ ismeasured in weeks, not years PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  49. The RV signal isrelativelystrong • Higher transit probability: Pt=Rp/a • Sun: Pt=0.5%; M3V: Pt=1.5%; M6V: Pt=2.3% PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

  50. But… • M dwarfs are faint. • Observations in the IR absolutelyrequiredespecially for late Ms. • M dwarfs are active. They are fully convective and show significantmagneticactivity (stellar spots) • Source of jitter noise for the RV signal. Expected to be 4-5 smaller in the IR compared to the visible. • RV at IR wavelengthsis more complicated • Lots of telluriclines to deal with. • Instrumentation is more complex (cryogenic) i.e. expensive. PHY6795O – Naines brunes et Exoplanètes 2. Radial Velocities

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