3 The Detection and Properties of Planetary Systems: Wed hHörsaal 2, Physik, Helmholz 5Prof. Dr. Artie HatzesThe Formation and Evolution of Planetary Systems:Thurs hHörsaal 2, Physik, Helmholz 5Prof. Dr. Alexander Krivov
4 Detection and Properties of Planetary Systems 14. April Introduction/The Doppler Method21. April The Doppler Method/Results from Doppler Surveys I28. April Results from Doppler Surveys II05. May Transit Search Techniques: (Ground )06. May Transit Search Techniques (Space: CoRoT and Kepler)12. May Characterization: Internal Structure19. May Characterization: Atmospheres26. May Exoplanets in Different Environments (Eike Guenther)02 June The Rossiter McClaughlin Effect and “Doppler Imaging” of Planets09. June Astrometric Detections16. June Direct Imaging23. June Microlensing30. June Habitable Planets07. July Planets off the Main Sequence
5 Literature Planet Quest, Ken Croswell (popular) Extrasolar Planets, Stuart Clark (popular)Extasolar Planets, eds. P. Cassen. T. Guillot, A. Quirrenbach (advanced)Planetary Systems: Formation, Evolution, and Detection, F. Burke, J. Rahe, and E. Roettger (eds) (1992: Pre-51 Peg)
6 Resources: The Nebraska Astronomy Applet Project (NAAP) This is the coolest astronomical website for learning basic astronomy that you will find. In it you can find:Solar System ModelsBasic Coordinates and SeasonsThe Rotating SkyMotions of the SunPlanetary Orbit SimulatorLunar Phase SimulatorBlackbody Curves & UBV FiltersHydrogen Energy LevelsHertzsprung-Russel DiagramEclipsing Binary StarsAtmospheric RetentionExtrasolar PlanetsVariable Star Photometry
7 The Nebraska Astronomy Applet Project (NAAP) On the Exoplanet page you can find:Descriptions of the Doppler effectCenter of massDetectionAnd two nice simulators where you can interactively change parameters:Radial Velocity simulator (can even add data with noise)Transit simulator (even includes some real transiting planet data)
9 Resources: The Extrasolar Planets Encyclopaedia In 7 languagesGrouped according to techniqueCan download data, make plots, correlation plots, etc.
10 This Week:Brief Introduction to ExoplanetsThe Doppler Method: Technique
11 Why Search for Extrasolar Planets? How do planetary systems form?Is this a common or an infrequent event?How unique are the properties of our own solar system?Are these qualities important for life to form?Up until now we have had only one laboratory to test planet formation theories. We need more!
12 The Concept of Extrasolar Planets Democritus ( B.C.):"There are innumerable worlds which differ in size. In some worlds there is no sun and moon, in others they are larger than in our world, and in others more numerous. They are destroyed by colliding with each other. There are some worlds without any living creatures, plants, or moisture."
13 Giordano Bruno ( )Believed that the Universe was infinite and that other worlds exists. He was burned at the stake for his beliefs.
14 What kinds of explanetary systems do we expect to find? The standard model of the formation of the sun is that it collapses under gravity from a proto-cloudBecause of rotation it collapses into a disk.Jets and other mechanisms provide a means to remove angular momentum
15 The net result is you have a protoplanetary disk out of which planets form.
16 Expectations of Exoplanetary Systems from our Solar System Solar proto-planetary disk was viscous. Any eccentric orbits would rapidly be damped outExoplanets should be in circular orbitsOrbital axes should be aligned and progradeGiant planets need a lot of solid core to build up sufficient mass to accrete an envelope. This core should form beyond a so-called ice line at 3-5 AUGiant Planets should be found at distances > 3 AUOur solar system is dominated by JupiterExoplanetary systems should have one Jovian planetOnly Terrestrial planets are found in inner regions
17 So how do we define an extrasolar Planet? We can simply use mass:Star: Has sufficient mass to fuse hydrogen to helium.M > 80 MJupiterBrown Dwarf: Insufficient mass to ignite hydrogen, but can undergo a period of Deuterium burning.13 MJupiter < M < 80 MJupiterPlanet: Formation mechanism unknown, but insufficient mass to ignite hydrogen or deuterium.M < 13 MJupiter
18 IAU Working Definition of Exoplanet Objects with true masses below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity) that orbit stars or stellar remnants are "planets" (no matter how they formed). The minimum mass/size required for an extrasolar object to be considered a planet should be the same as that used in our Solar System.Substellar objects with true masses above the limiting mass for thermonuclear fusion of deuterium are "brown dwarfs", no matter how they formed nor where they are located.Free-floating objects in young star clusters with masses below the limiting mass for thermonuclear fusion of deuterium are not "planets", but are "sub-brown dwarfs" (or whatever name is most appropriate).In other words „A non-fusor in orbit around a fusor“
19 How to search for Exoplanets Radial VelocityAstrometryTransitsMicrolensingImagingTiming Variations
20 Radial velocity measurements using the Doppler Wobble radialvelocitydemo.htm8
21 Radial Velocity measurements Requirements:Accuracy of better than 10 m/sStability for at least 10 YearsJupiter: 12 m/s, 11 yearsSaturn: 3 m/s, 30 years
22 Astrometric Measurements of Spatial Wobble q =Center of massMD2q2q = 8 mas at a Cen2q = 1 mas at 10 pcsCurrent limits:1-2 mas (ground)0.1 mas (HST)Since D ~ 1/D can only look at nearby stars
23 Jupiter only1 milliarc-seconds for a Star at 10 parsecs
28 Timing VariationsIf you have a stable clock on the star (e.g. pulsations, pulsar) as the star moves around the barycenter the time of „maximum“ as observed from the earth will vary due to the light travel time from the changing distance to the earthThe Pulsar Planets were discovered in this way
33 ( ) 2pas V = Circular orbits: P Conservation of momentum: ms × as = mp × apas =mp apmsSolve Kepler‘s law for ap:ap =P2Gms4p2()1/3… and insert in expression for as and then V for circular orbits
34 V =2pP(4p2)1/3mp P2/3 G1/3ms1/3mp in Jupiter massesms in solar massesP in yearsV in m/sV =0.0075P1/3ms2/3mp=28.4P1/3ms2/3mp28.4P1/3ms2/3mp sin iVobs =
35 Radial Velocity Amplitude of Planets in the Solar System Mass (MJ)V(m s–1)Mercury1.74 × 10–40.008Venus2.56 × 10–30.086Earth3.15 × 10–30.089Mars3.38 × 10–4Jupiter1.012.4Saturn0.2992.75Uranus0.0460.297Neptune0.0540.281Pluto3×10–5
36 Radial Velocity Amplitude of Planets at Different a
40 Not 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 determinedP: period of orbitw: orientation of periastrone: eccentricityM or T: EpochK: velocity amplitude
41 Radial velocity shape as a function of eccentricity:
42 Radial velocity shape as a function of w, e = 0.7 :
43 Eccentric orbit can sometimes escape detection: With poor sampling this star would be considered constant
45 Important for orbital solutions: The Mass Function f(m) =(mp sin i)3(mp + ms)2=P2pGK3(1–e2)3/2P = periodK = Amplitudemp = mass of planet (companion)ms = mass of stare = eccentricity
46 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 planeWe only measure MPlanet x sin iiObserver
47 The orbital inclination We only measure m sin i, a lower limit to the mass.What is the average inclination?iP(i) di = 2p sin i diThe 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
48 The orbital inclination P(i) di = 2p sin i diMean inclination:p∫P(i) sin i diP(i) di<sin i> == p/4 = 0.79Mean inclination is 52 degrees and you measure 80% of the true mass
49 The orbital inclination P(i) di = 2p sin i diBut for the mass function sin3i is what is important :p∫P(i) sin 3 i diP(i) di= 0.5∫sin 4 i dip<sin3 i> == 3p/16 = 0.59
50 The orbital inclination P(i) di = 2p sin i diProbability i < q :q∫P(i) dipP(i) di2P(i<q) =(1 – cos q )=q < 10 deg : P= 0.03(sin i = 0.17)
51 Measurement of Doppler Shifts In the non-relativistic case:l – l0Dvc=l0We measure Dv by measuring Dl
52 The Radial Velocity Measurement Error with Time How did we accomplish this?
53 The Answer: Electronic Detectors (CCDs) Large wavelength Coverage SpectrographsSimultanous Wavelength Calibration (minimize instrumental effects)
54 Instrumentation for Doppler Measurements High Resolution Spectrographs with Large Wavelength Coverage
59 yxOn a detector we only measure x- and y- positions, there is no information about wavelength. For this we need a calibration source
60 CCD detectors only give you x- and y- position CCD detectors only give you x- and y- position. A doppler shift of spectral lines will appear as DxDx → Dl → DvHow large is Dx ?
61 Consider two monochromatic beams dl Spectral Resolution← 2 detector pixelsConsider two monochromatic beamsdlThey will just be resolved when they have a wavelength separation of dlResolving power:R =ldldl = full width of half maximum of calibration lamp emission linesl1l2
62 R = → Dl = 0.11 Angstroms→ Angstroms / pixel (2 pixel 5500 Ang.1 pixel typically 15 mm=vDl cl1 pixel = Ang → x (3•108 m/s)/5500 Ang →= 3000 m/s per pixelDv = 10 m/s= 1/300 pixel= 0.05 mm = 5 x 10–6 cmDv = 1 m/s = 1/1000 pixel → 5 x 10–7 cm
63 Velocity per pixel (m/s) For Dv = 20 m/sRAng/pixelVelocity per pixel (m/s)DpixelShift in mm0.0053000.060.0010.1257500.0274×10–40.02515000.01332×10–450 0000.05030000.006710–425 0000.1060000.0335×10–510 0000.25150002×10–55 0000.5300006.6×10–410–51 0002.51500001.3×10–42×10–6So, one should use high resolution spectrographs….up to a pointHow does the RV precision depend on the properties of your spectrograph?
64 Each spectral line gives a measurement of the Doppler shift Wavelength coverage:Each spectral line gives a measurement of the Doppler shiftThe more lines, the more accurate the measurement:sNlines = s1line/√Nlines→ Need broad wavelength coverageWavelength coverage is inversely proportional to R:DldetectorLow resolutionHigh resolutionDl
65 Noise:sII = detected photonsSignal to noise ratio S/N = I/sFor photon statistics: s = √I → S/N = √I
66 s (S/N)–1Price: S/N t2exposure14Exposure factor1636144400
67 How does the radial velocity precision depend on all parameters? s (m/s) = Constant × (S/N)–1 R–3/2 (Dl)–1/2: errorR: spectral resolving powerS/N: signal to noise ratioDl : wavelength coverage of spectrograph in AngstromsFor R= , S/N=150, Dl=2000 Å, s = 2 m/sC ≈ 2.4 × 1011
68 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 F6Poor RV precision → cool stars of spectral type earlier than F6Why?
69 A7 starK0 starEarly-type stars have few spectral lines (high effective temperatures) and high rotation rates.
70 ( ) Including dependence on stellar parameters v sin i 2 s (m/s) ≈ Constant ×(S/N)–1 R–3/2(Dl)–1/2f(Teff)v sin i : projected rotational velocity of star in km/sf(Teff) = factor taking into account line densityf(Teff) ≈ 1 for solar type starf(Teff) ≈ 3 for A-type starf(Teff) ≈ 0.5 for M-type star
71 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 motionz.B. for TLS spectrograph with R= our best RV precision is 1.8 m/s → 1.2 x 10–6 cm
72 Traditional method:Observe your star→Then your calibration source→
73 Problem: these are not taken at the same time… ... Short term shifts of the spectrograph can limit precision to several hunrdreds of m/s
75 Spectrographs: CORALIE, ELODIE, HARPS Solution 1: Observe your calibration source (Th-Ar) simultaneously to your data:Stellar spectrumThorium-ArgoncalibrationSpectrographs: CORALIE, ELODIE, HARPS
76 Advantages of simultaneous Th-Ar calibration: Large wavelength coverage (2000 – 3000 Å)Computationally simple: cross correlationDisadvantages of simultaneous Th-Ar calibration:Th-Ar are active devices (need to apply a voltage)Lamps change with timeTh-Ar calibration not on the same region of the detector as the stellar spectrumSome contamination that is difficult to modelCannot model the instrumental profile, therefore you have to stablize the spectrograph
77 RVs calculated with the Cross Correlation Function Take your spectrum, f(l) :2. Take a digital mask of line locations, g(l) :
78 RVs calculated with the Cross Correlation Function 3. Compute Cross Correlation Function (related to the convolution)4. Location of Peak in CCF is the radial velocity:l
83 Example: The companion to HD 114762 using the telluric method Example: The companion to HD using the telluric method. Best precision is 15–30 m/sFilled circles are data taken at McDonald Observatory using the telluric lines at 6300 Ang.
84 Limitations of the telluric technique: Limited wavelength range (≈ 10s Angstroms)Pressure, temperature variations in the Earth‘s atmosphereWindsLine depths of telluric lines vary with air massCannot observe a star without telluric lines which is needed in the reduction process.
85 b) Use a „controlled“ absorption cell Absorption lines of star + cellAbsorption lines of the starAbsorption lines of cell
86 Campbell & Walker: Hydrogen Fluoride cell: Demonstrated radial velocity precision of 13 m s–1 in 1980!
87 Drawbacks:Limited wavelength range (≈ 100 Ang.)Temperature stablized at 100 CLong path length (1m)Has to be refilled after every observing runDangerous
88 A better idea: Iodine cell (first proposed by Beckers in 1979 for solar studies) Spectrum of iodineAdvantages over HF:1000 Angstroms of coverageStablized at 50–75 CShort path length (≈ 10 cm)Can model instrumental profileCell is always sealed and used for >10 yearsIf cell breaks you will not die!
90 The iodine cell used at the CES spectrograph at La Silla
91 Modelling the Instrumental Profile: The Advantage of Iodine What is an instrumental profile (IP):Consider a monochromatic beam of light (delta function)Perfect spectrograph
92 Modelling the Instrumental Profile (IP) We do not live in a perfect world:A real spectrographIP is usually a Gaussian that has a width of 2 detector pixelsThe IP is a „smoothing“ or „blurring“ function caused by your instrument
94 Convolutionf(x-u)a1a2a3g(x)a3a2a1Convolution is a smoothing function. In our case f(x) is the stellar spectrum and f(x) is our instrumental profile.
95 The IP is not so much the problem as changes in the IP No problem with this IPOr this IPShift of centroid will appear as a velocity shiftUnless it turns into this
96 Use a high resolution spectrum of iodine to model IP Iodine observed with RV instrumentIodine Observed with a Fourier Transform Spectrometer
97 Observed I2FTS spectrum rebinned to sampling of RV instrumentFTS spectrum convolved with calculated IP
98 Gaussians contributing to IP IP = Sgi Model IP Sampling in Data spaceSampling in IP space = 5×Data space sampling
99 Instrumental Profile Changes in ESO‘s CES spectrograph 2 March 199414 Jan 1995
100 Instrumental Profile Changes in ESO‘s CES spectrograph over 5 years:
101 Mathematically you solve this equation: Iobs(l) = k[TI2(l)Is(l + Dl)]*PSFWhere:Iobs(l) : observed spectrumTI2 : iodine transmission functionk: normalizing factorIs(l): Deconvolved stellar spectrum without iodine and with PSF removedPSF: Instrumental Point Spread FunctionProblem: PSF varies across the detector.Solve this equation iteratively
102 Modeling the Instrumental Profile In each chunk:Remove continuum slope in data : 2 parametersCalculate dispersion (Å/pixel): 3 parameters (2 order polynomial: a0, a1, a2)Calculate IP with 5 Gaussians: 9 parameters: 5 widths, 4 amplitudes (position and widths of satellite Gaussians fixed)Calculate Radial Velocity: 1 parametersCombine with high resolution iodine spectrum and stellar spectrum without iodineIterate until model spectrum fits the observed spectrum
106 Additional information on RV modeling: 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“
107 Barycentric Correction Earth’s orbital motion can contribute ± 30 km/s (maximum)Earth’s rotation can contribute ± 460 m/s (maximum)
108 Needed for Correct Barycentric Corrections: Accurate coordinates of observatoryDistance of observatory to Earth‘s center (altitude)Accurate position of stars, including proper motion:a′, d′a, dWorst case Scenario: Barnard‘s starMost programs use the JPL Ephemeris which provides barycentric corrections to a few cm/s
109 For highest precision an exposure meter is required No cloudstimePhotons from starMid-point of exposuretimePhotons from starCentroid of intensity w/cloudsClouds
110 Differential Earth Velocity: Causes „smearing“ of spectral linesKeep exposure times < min