1. The Detection and Properties of Planetary Systems
Prof. Dr. Artie Hatzes

2. Artie Hatzes
Tel:

3. The Detection and Properties of Planetary Systems:
Wed h
Hörsaal 2, Physik, Helmholz 5
Prof. Dr. Artie Hatzes
The Formation and Evolution of Planetary Systems:
Thurs h
Hörsaal 2, Physik, Helmholz 5
Prof. Dr. Alexander Krivov

4. Detection and Properties of Planetary Systems
14. April Introduction/The Doppler Method
21. April The Doppler Method/Results from Doppler Surveys I
28. April Results from Doppler Surveys II
05. May Transit Search Techniques: (Ground )
06. May Transit Search Techniques (Space: CoRoT and Kepler)
12. May Characterization: Internal Structure
19. May Characterization: Atmospheres
26. May Exoplanets in Different Environments (Eike Guenther)
02 June The Rossiter McClaughlin Effect and “Doppler Imaging” of Planets
09. June Astrometric Detections
16. June Direct Imaging
23. June Microlensing
30. June Habitable Planets
07. 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 Models
Basic Coordinates and Seasons
The Rotating Sky
Motions of the Sun
Planetary Orbit Simulator
Lunar Phase Simulator
Blackbody Curves & UBV Filters
Hydrogen Energy Levels
Hertzsprung-Russel Diagram
Eclipsing Binary Stars
Atmospheric Retention
Extrasolar Planets
Variable Star Photometry

7. The Nebraska Astronomy Applet Project (NAAP)
On the Exoplanet page you can find:
Descriptions of the Doppler effect
Center of mass
Detection
And 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)

8. And where is Nebraska?

9. Resources: The Extrasolar Planets Encyclopaedia
In 7 languages
Grouped according to technique
Can download data, make plots, correlation plots, etc.

10. This Week:
Brief Introduction to Exoplanets
The 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-cloud
Because 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 out
Exoplanets should be in circular orbits
Orbital axes should be aligned and prograde
Giant 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 AU
Giant Planets should be found at distances > 3 AU
Our solar system is dominated by Jupiter
Exoplanetary systems should have one Jovian planet
Only 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 MJupiter
Brown Dwarf: Insufficient mass to ignite hydrogen, but can undergo a period of Deuterium burning.
13 MJupiter < M < 80 MJupiter
Planet: 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 Velocity
Astrometry
Transits
Microlensing
Imaging
Timing Variations

20. Radial velocity measurements using the Doppler Wobble
radialvelocitydemo.htm 8

21. Radial Velocity measurements
Requirements:
Accuracy of better than 10 m/s
Stability for at least 10 Years
Jupiter: 12 m/s, 11 years
Saturn: 3 m/s, 30 years

22. Astrometric Measurements of Spatial Wobble
q =
Center of mass M D 2q
2q = 8 mas at a Cen
2q = 1 mas at 10 pcs
Current limits:
1-2 mas (ground)
0.1 mas (HST)
Since D ~ 1/D can only look at nearby stars

23. Jupiter only
1 milliarc-seconds for a Star at 10 parsecs

24. Microlensing

25. Direct Imaging: This is hard!
times fainter planet
4 Arcseconds
Separation = width of your hair at arms length

26. For large orbital radii it is easier

27. Transit Searches: Techniques

28. Timing Variations
If 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 earth
The Pulsar Planets were discovered in this way

29. The Discovery Space

30. Radial Velocity Detection of Planets: I. Techniques
Keplerian Orbits
Spectrographs/Doppler shifts
Precise Radial Velocity measurements

31. Newton‘s form of Kepler‘s LawV mp ms ap as
P2 =
4p2
(as + ap)3
G(ms + mp)

32. P2 =
4p2
(as + ap)3
G(ms + mp)
ap » as
Approximations:
ms » mp
P2 ≈
4p2
ap3
Gms

33. ( ) 2pas V = Circular orbits: P Conservation of momentum:
ms × as = mp × ap
as =
mp ap ms
Solve Kepler‘s law for ap:
ap =
P2Gms
4p2 ( )
1/3
… and insert in expression for as and then V for circular orbits

34. V = 2p
P(4p2)1/3
mp P2/3 G1/3ms1/3
mp in Jupiter masses
ms in solar masses
P in years
V in m/s
V =
0.0075
P1/3ms2/3 mp =
28.4
P1/3ms2/3 mp
28.4
P1/3ms2/3
mp sin i
Vobs =

35. Radial Velocity Amplitude of Planets in the Solar System
Mass (MJ)
V(m s–1)
Mercury
1.74 × 10–4
0.008
Venus
2.56 × 10–3
0.086
Earth
3.15 × 10–3
0.089
Mars
3.38 × 10–4
Jupiter
1.0
12.4
Saturn
0.299
2.75
Uranus
0.046
0.297
Neptune
0.054
0.281
Pluto
3×10–5

36. Radial Velocity Amplitude of Planets at Different a

39. Elliptical Orbits O
eccentricity
= OF1/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 determined
P: period of orbit
w: orientation of periastron
e: eccentricity
M or T: Epoch
K: 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

44. Eccentricities of bodies in the Solar System

45. Important for orbital solutions: The Mass Function
f(m) =
(mp sin i)3
(mp + ms)2 = P
2pG
K3(1–e2)3/2
P = period
K = Amplitude
mp = mass of planet (companion)
ms = mass of star
e = 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 plane
We only measure MPlanet x sin i i
Observer

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

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

49. The orbital inclination
P(i) di = 2p sin i di
But for the mass function sin3i is what is important : p ∫
P(i) sin 3 i di
P(i) di
= 0.5 ∫
sin 4 i di p
<sin3 i> =
= 3p/16 = 0.59

50. The orbital inclination
P(i) di = 2p sin i di
Probability i < q : q ∫
P(i) di p
P(i) di 2
P(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 – l0 Dv c = l0
We 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 Spectrographs
Simultanous Wavelength Calibration (minimize instrumental effects)

54. Instrumentation for Doppler Measurements
High Resolution Spectrographs with Large Wavelength Coverage

55. Echelle Spectrographs
camera
detector
corrector
Cross disperser
From telescope
slit
collimator

57. Most of light is in n=0 5000 A n = –2 4000 A 5000 A n = –1 4000 A

58. Free Spectral Range Dl = l/mDy ∞ l2 y
m+3
Grating cross-dispersed echelle spectrographs

59. 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

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 Dx
Dx → Dl → Dv
How large is Dx ?

61. Consider two monochromatic beams dl
Spectral Resolution
← 2 detector pixels
Consider two monochromatic beams dl
They will just be resolved when they have a wavelength separation of dl
Resolving power:
R = l dl
dl = full width of half maximum of calibration lamp emission lines l1 l2

62. R = → Dl = 0.11 Angstroms
→ Angstroms / pixel (2 pixel 5500 Ang.
1 pixel typically 15 mm = v
Dl c l
1 pixel = Ang → x (3•108 m/s)/5500 Ang →
= 3000 m/s per pixel
Dv = 10 m/s
= 1/300 pixel
= 0.05 mm = 5 x 10–6 cm
Dv = 1 m/s = 1/1000 pixel → 5 x 10–7 cm

63. Velocity per pixel (m/s)
For Dv = 20 m/s R
Ang/pixel
Velocity per pixel (m/s)
Dpixel
Shift in mm
0.005
300
0.06
0.001
0.125
750
0.027
4×10–4
0.025
1500
0.0133
2×10–4
50 000
0.050
3000
0.0067
10–4
25 000
0.10
6000
0.033
5×10–5
10 000
0.25
15000
2×10–5
5 000
0.5
30000
6.6×10–4
10–5
1 000
2.5
150000
1.3×10–4
2×10–6
So, one should use high resolution spectrographs….up to a point
How 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 shift
The more lines, the more accurate the measurement:
sNlines = s1line/√Nlines
→ Need broad wavelength coverage
Wavelength coverage is inversely proportional to R: Dl
detector
Low resolution
High resolution Dl

65. Noise: s I
I = detected photons
Signal to noise ratio S/N = I/s
For photon statistics: s = √I → S/N = √I

66. s  (S/N)–1
Price: S/N  t2exposure 1 4
Exposure factor 16 36
144
400

67. How does the radial velocity precision depend on all parameters?
s (m/s) = Constant × (S/N)–1 R–3/2 (Dl)–1/2
: error
R: spectral resolving power
S/N: signal to noise ratio
Dl : wavelength coverage of spectrograph in Angstroms
For R= , S/N=150, Dl=2000 Å, s = 2 m/s
C ≈ 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 F6
Poor RV precision → cool stars of spectral type earlier than F6
Why?

69. A7 star
K0 star
Early-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)–1/2
f(Teff)
v sin i : projected rotational velocity of star in km/s
f(Teff) = factor taking into account line density
f(Teff) ≈ 1 for solar type star
f(Teff) ≈ 3 for A-type star
f(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 motion
z.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 spectrum
Thorium-Argon
calibration
Spectrographs: CORALIE, ELODIE, HARPS

76. Advantages of simultaneous Th-Ar calibration:
Large wavelength coverage (2000 – 3000 Å)
Computationally simple: cross correlation
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
Cannot 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

79. One Problem: Th-Ar lamps change with time!

80. HARPS

81. Solution 2: Absorption cell
a) Griffin and Griffin: Use the Earth‘s atmosphere:

82. O2
6300 Angstroms

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/s
Filled 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 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.

85. b) Use a „controlled“ absorption cell
Absorption lines of star + cell
Absorption lines of the star
Absorption 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 C
Long path length (1m)
Has to be refilled after every observing run
Dangerous

88. 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 50–75 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!

89. Spectrum of star through Iodine cell:

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 spectrograph
IP is usually a Gaussian that has a width of 2 detector pixels
The IP is a „smoothing“ or „blurring“ function caused by your instrument

93. Convolution
 f(u)f(x–u)du = f * f
f(x):
f(x):

94. Convolution
f(x-u) a1 a2 a3
g(x) a3 a2 a1
Convolution 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 IP
Or this IP
Shift of centroid will appear as a velocity shift
Unless it turns into this

96. Use a high resolution spectrum of iodine to model IP
Iodine observed with RV instrument
Iodine Observed with a Fourier Transform Spectrometer

97. Observed I2
FTS spectrum rebinned to sampling of RV instrument
FTS spectrum convolved with calculated IP

98. Gaussians contributing to IP IP = Sgi Model IP
Sampling in Data space
Sampling in IP space = 5×Data space sampling

99. Instrumental Profile Changes in ESO‘s CES spectrograph
2 March 1994
14 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)]*PSF
Where:
Iobs(l) : observed spectrum
TI2 : iodine transmission function
k: normalizing factor
Is(l): Deconvolved stellar spectrum without iodine and with PSF removed
PSF: Instrumental Point Spread Function
Problem: PSF varies across the detector.
Solve this equation iteratively

102. Modeling the Instrumental Profile
In each chunk:
Remove continuum slope in data : 2 parameters
Calculate 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 parameters
Combine with high resolution iodine spectrum and stellar spectrum without iodine
Iterate until model spectrum fits the observed spectrum

103. Sample fit to an observed chunk of data

104. Sample IP from one order of a spectrum taken at TLS

105. WITH TREATMENT OF IP-ASYMMETRIES

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 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

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

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