The Detection and Properties of
Planetary Systems
Prof. Dr. Artie Hatzes
Artie Hatzes
Tel:036427-863-51
Email: [email protected]
www.tls-tautenburg.de→Lehre→Vorlesungen→Jena
The Detection and Properties of Planetary Systems:
Wed. 14-16 h
Hörsaal 2, Physik, Helmholz 5
Prof. Dr. Artie Hatzes
The Formation and Evolution of Planetary Systems:
Thurs. 14-16 h
Hörsaal 2, Physik, Helmholz 5
Prof. Dr. Alexander Krivov
Detection and Properties of Planetary Systems
14. AprilIntroduction/The Doppler Method
21. AprilThe Doppler Method/Results from Doppler Surveys I
28. AprilResults 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
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)
Resources: The Nebraska Astronomy Applet Project (NAAP)
http://astro.unl.edu/naap/
This is the coolest astronomical website for learning basic
astronomy that you will find. In it you can find:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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
The Nebraska Astronomy Applet Project (NAAP)
On the Exoplanet page you can find:
1.
2.
3.
Descriptions of the Doppler effect
Center of mass
Detection
And two nice simulators where you can interactively change
parameters:
1. Radial Velocity simulator (can even
add data with noise)
2. Transit simulator (even includes
some real transiting planet data)
And where is Nebraska?
Resources: The Extrasolar Planets Encyclopaedia
http://exoplanet.eu/
• In 7 languages
• Grouped according to technique
• Can download data, make plots, correlation
plots, etc.
This Week:
1. Brief Introduction to Exoplanets
2. The Doppler Method: Technique
Extrasolar Planets
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!
The Concept of Extrasolar Planets
Democritus (460-370 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."
Giordano Bruno (1548-1600)
Believed that the Universe was infinite and that other
worlds exists. He was burned at the stake for his
beliefs.
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
The net result is you have a protoplanetary disk out
of which planets form.
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
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
IAU Working Definition of Exoplanet
1.
2.
3.
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“
How to search for Exoplanets
1.
Radial Velocity
2. Astrometry
3. Transits
4.
Microlensing
5.
Imaging
6. Timing Variations
Radial velocity measurements using the Doppler Wobble
radialvelocitydemo.htm
8
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
Astrometric Measurements of Spatial Wobble
Center of mass
q= m
M
a
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
Jupiter only
1 milliarc-seconds for a Star
at 10 parsecs
Microlensing
Direct Imaging: This is hard!
1.000.000.000 times
fainter planet
4 Arcseconds
Separation = width of your hair at arms
length
For large orbital radii it is easier
Transit Searches: Techniques
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
The Discovery Space
Radial Velocity Detection of Planets:
I. Techniques
1.
Keplerian Orbits
2. Spectrographs/Doppler shifts
3.
Precise Radial Velocity measurements
Newton‘s form of Kepler‘s Law
V
mp
ms
ap
as
P2 =
4p2 (as + ap)3
G(ms + mp)
4p2 (as + ap)3
G(ms + mp)
P2 =
Approximations:
ap » as
ms » mp
P2 ≈
4p2 ap3
Gms
Circular orbits:
2pas
V=
P
ms × as = mp × ap
Conservation of momentum:
mp ap
as =
ms
Solve Kepler‘s law for ap:
1/3
2Gm
P
s)
ap = (
2
4p
… and insert in expression for as and then V for circular
orbits
V=
2p mp P2/3 G1/3ms1/3
P(4p2)1/3
0.0075 mp
V = 1/3 2/3
P ms
=
28.4 mp
P1/3ms2/3
mp in Jupiter
masses
ms in solar masses
P in years
V in m/s
Vobs =
28.4 mp sin i
P1/3ms2/3
Radial Velocity Amplitude of Planets in the Solar System
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Mass (MJ)
1.74 × 10–4
2.56 × 10–3
3.15 × 10–3
3.38 × 10–4
1.0
0.299
0.046
0.054
1.74 × 10–4
V(m s–1)
0.008
0.086
0.089
0.008
12.4
2.75
0.297
0.281
3×10–5
Radial Velocity Amplitude of Planets at Different a
Elliptical Orbits
O
eccentricity
= OF1/a
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
Important for radial velocities
P: period of orbit
w: orientation of periastron
e: eccentricity
M or T: Epoch
K: velocity amplitude
Radial velocity shape as a function of eccentricity:
Radial velocity shape as a function of w, e = 0.7 :
Eccentric orbit can sometimes escape detection:
With poor sampling this star would be considered constant
Eccentricities of bodies in the Solar System
Important for orbital solutions: The Mass Function
f(m) =
i)3
(mp sin
(mp + ms)2
P K3(1–e2)3/2
=
P = period
K = Amplitude
mp = mass of planet (companion)
ms = mass of star
e = eccentricity
2pG
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 MPlanet x sin i
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
The orbital inclination
P(i) di = 2p sin i di
Mean inclination:
<sin i> =
p
∫0
P(i) sin i di
∫0
P(i) di
p
= p/4 = 0.79
Mean inclination is 52 degrees and you measure 80% of
the true mass
The orbital inclination
P(i) di = 2p sin i di
But for the mass function sin3i is what is important :
p
<sin3 i> =
∫0
P(i) sin 3 i di
∫0
P(i) di
p
= 3p/16 = 0.59
p
= 0.5 ∫ sin 4 i di
0
The orbital inclination
P(i) di = 2p sin i di
Probability i < q :
P(i<q) =
q
2 ∫0 P(i) di
p
∫0
P(i) di
q < 10 deg : P= 0.03
=
(1 – cos q )
(sin i = 0.17)
Measurement of Doppler Shifts
In the non-relativistic case:
l – l0
l0
=
We measure Dv by measuring Dl
Dv
c
The Radial Velocity Measurement Error with Time
How did we accomplish this?
The Answer:
1.
Electronic Detectors (CCDs)
2. Large wavelength Coverage Spectrographs
3.
Simultanous Wavelength Calibration
(minimize instrumental effects)
Instrumentation for Doppler
Measurements
High Resolution Spectrographs with Large
Wavelength Coverage
Echelle Spectrographs
camera
detector
corrector
Cross disperser
From telescope
slit
collimator
5000 A
n = –2
4000 A
5000 A
4000 A
4000 A
5000 A
n = –1
Most of
light is in
n=0
n=1
4000 A
n=2
5000 A
dl
Free Spectral Range Dl = l/m
m-2
m-1
m
m+2
m+3
y
Dy ∞ l2
Grating cross-dispersed echelle spectrographs
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
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 ?
Spectral Resolution
← 2 detector pixels
dl
Consider two monochromatic
beams
They will just be resolved when
they have a wavelength
separation of dl
Resolving power:
l1
l2
l
R=
dl
dl = full width of half
maximum of calibration
lamp emission lines
R = 50.000 → Dl = 0.11 Angstroms
→ 0.055 Angstroms / pixel (2 pixel sampling) @ 5500 Ang.
1 pixel typically 15 mm
v =
Dl c
l
1 pixel = 0.055 Ang → 0.055 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
For Dv = 20 m/s
R
Ang/pixel
Velocity per
pixel (m/s)
Dpixel
Shift in mm
500 000
0.005
300
0.06
0.001
200 000
0.125
750
0.027
4×10–4
100 000
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
0.00133
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?
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:
Low resolution
Dl
detector
High resolution
Dl
Noise:
s
I
I = detected photons
Signal to noise ratio S/N = I/s
For photon statistics: s = √I → S/N = √I
Exposure
factor
1 4 16 36
Price: S/N  t2exposure
144
400
s  (S/N)–1
How does the radial velocity precision depend on all
parameters?
s (m/s) = Constant × (S/N)–1 R–3/2 (Dl)–1/2
s: error
R: spectral resolving power
S/N: signal to noise ratio
Dl : wavelength coverage of spectrograph in Angstroms
For R=110.000, S/N=150, Dl=2000 Å, s = 2 m/s
C ≈ 2.4 × 1011
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?
A7 star
K0 star
Early-type stars have few spectral lines (high effective
temperatures) and high rotation rates.
Including dependence on stellar parameters
s (m/s) ≈ Constant ×(S/N)–1 R–3/2 (Dl)–1/2 (
v sin i
f(Teff)
)
2
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
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=67.000 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:
Stellar
spectrum
Thorium-Argon
calibration
Spectrographs: CORALIE, ELODIE, HARPS
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
RVs calculated with the Cross Correlation
Function
1. Take your spectrum, f(l) :
2. Take a digital mask of line locations, g(l) :
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
One Problem: Th-Ar lamps change with time!
HARPS
Solution 2: Absorption cell
a) Griffin and Griffin: Use the Earth‘s atmosphere:
O2
6300 Angstroms
Example: The companion to HD 114762 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.
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.
b) Use a „controlled“ absorption cell
Absorption
lines of star +
cell
Absorption lines of the
star
Absorption lines of 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)
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!
Spectrum of star through Iodine cell:
The iodine cell used at the CES spectrograph at La Silla
Modelling the Instrumental Profile: The Advantage of
Iodine
What is an instrumental profile (IP):
Consider a monochromatic beam of light (delta function)
Perfect
spectrograph
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
Convolution
 f(u)f(x–u)du = f * f
f(x):
f(x):
Convolution
f(x-u)
a2
a1
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.
The IP is not so much the problem as changes in the IP
No problem with this IP
Or this IP
Unless it turns into this
Shift of centroid will appear as a velocity shift
Use a high resolution spectrum of iodine to model IP
Iodine
observed
with RV
instrument
Iodine
Observed
with a
Fourier
Transform
Spectrometer
Observed I2
FTS spectrum rebinned to
sampling of RV instrument
FTS spectrum convolved
with calculated IP
Gaussians
contributing to IP
Model IP
IP = Sgi
Sampling in Data space
Sampling in IP space =
5×Data space sampling
Instrumental Profile Changes in ESO‘s CES spectrograph
2 March 1994
14 Jan 1995
Instrumental Profile Changes in ESO‘s CES spectrograph
over 5 years:
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
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
Sample fit to an observed chunk of data
Sample IP from one order of a
spectrum taken at TLS
WITH TREATMENT OF IP-ASYMMETRIES
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“
Barycentric Correction
Earth’s orbital motion can
contribute ± 30 km/s (maximum)
Earth’s rotation can contribute
± 460 m/s (maximum)
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
For highest precision an exposure meter is required
No clouds
Photons from star
time
Clouds
Mid-point of exposure
Photons from star
Centroid of intensity
w/clouds
time
Differential Earth Velocity:
Causes „smearing“ of
spectral lines
Keep exposure
times < 20-30 min
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