Transcript Document

Transit Searches: Techniques
Why are Transits Important?
R*
DI
The drop in intensity is give by the ratio of the cross-section areas:
DI = (Rp /R*)2 = (0.1Rsun/1 Rsun)2 = 0.01 for Jupiter
Radial Velocity measurements => Mp (we know sin i !)
=> density of planet
→ Transits allows us to measure the physical properties of the
planets
Also in transit spectroscopy.
As starlight passes through the atmosphere of the planet, part of it is absorbed.
Transit Probability
i = 90o+q
q
a
sin q = R*/a = |cos i|
90+q
Porb =  2p sin i di / 4p =
90-q
–0.5 cos (90+q) + 0.5 cos(90–q) = sin q
= R*/a for small angles
R*
Transit Duration
t = 2(R* +Rp)/v
where v is the orbital velocity and i = 90 (transit across disk center)
For circular orbits
From Keplers Law’s:
2R* P (4p2)1/3
t
2p P2/3 M*1/3G1/3
v = 2pa/P
a = (P2 M*G/4p2)1/3
t 1.82 P1/3 R* /M*1/3 (hours)
In solar units, P in days
Note t3 ~ (rmean)–1 i.e. it is related to the mean density of the star
Transit Duration
Note: The transit duration gives you an estimate of the stellar radius
0.55 t M1/3
R=
P1/3
R in solar radii
M in solar masses
P in days
t in hours
For more accurate times need to take into account the
orbital inclination and the fact that you have a finite radius planet
for i  90o need to replace R* with R:
R2 + d2cos2i = R*2
d cos i
R*
R = (R*2 – d2 cos2i)1/2
R
The Full Duration of the transit is including inclination is
given by:
R
{ a sin
i √(
*
a is the semi-major axis
2
(
arcsin
Rp
1+ R
*
–
(
a cos i 2
R*
{
D=
P
p
Making contact:
1.
2.
3.
4.
First contact with star
Planet fully on star
Planet starts to exit
Last contact with star
Note: for grazing transits there is
no 2nd and 3rd contact
1
4
2
3
DI/I
Prob.
N
t (hrs)
forbit
Mercury
1.2 x 10-5
0.012
83
8
0.0038
Venus
7.5 x 10-5
0.0065
154
11
0.002
Earth
8.3 x 10-5
0.0047
212
13
0.0015
Mars
2.3 x 10-5
0.0031
322
16
9.6 x 10-4
Jupiter
0.01
0.0009
1100
29
2.8 x 10-4
Saturn
0.007
0.00049
2027
40
1.5 x 10-4
Uranus
0.0012
0.000245
4080
57
7.7 x 10-5
Neptune
0.0013
0.000156
6400
71
4.9 x 10-4
51 Peg b
0.01
0.094
11
3
0.03
Planet
Moon
6.2 x10-6
Ganymede
1.3 x 10-5
Titan
1.2 x 10-5
N is the number of stars you would have to observe to see a transit, if all stars had
such a planet
Note the closer a planet is to the star:
1. The more likely that you have a favorable orbit
for a transit
2. The shorter the transit duration
3.
Higher frequency of transits
→ The transit method is best suited for short period planets.
Prior to 51 Peg it was not really considered a viable detection
method.
Shape of Transit Curves
2
tflat
tflat
ttotal
=
[R* – Rp]2 – d2 cos2i
[R* + Rp]2 – d2 cos2i
ttotal
Note that when i = 90o tflat/ttotal = (R* – Rp)/( R* + Rp)
For main sequence stars we can use the mass radius relationship to
eliminate either the mass or the radius from expression for transit
time.
Shape of Transit Curves
HST light curve of HD 209458b
A real transit light curve is not flat
Shape of Transit Curves
Effects of Limb Darkening (or why the curve is not flat).
Bottom of photosphere
q2
q1
dz
Temperature
Temperature profile
of photosphere
10000
8000
6000
4000
z=0
tn =1 surface
Top of photosphere
z
z increases going
into the star
Limb darkening in
other stars
Use transiting planets
No limb darkening
transit shape
At the limb the star has less flux than is expected, thus the planet blocks less light
To model the transit light curve and derive the true radius of
the planet you have to have an accurate limb darkening law.
Problem: Limb darkening is only known very well for one
star – the Sun!
Why Worry about Limb Darkening?
Suppose someone observes a
transit in the optical. The
„diameter“ of the stellar disk
is determined by the limb
darkening
Years later you observe the
transit at 10000 Ang. The star
has less limb darkening, it
thus has a larger „apparent
diameter. You calculate a
longer duration transit
because you do not take into
account the different limb
darkening
And your wrong conclusion:
More limb darkening →
short transit duration
Less limb darkening in red
→ longer transit duration
→ orbital inclination has
changed!
Shape of Transit Curves
Grazing eclipses/transits
These produce a „V-shaped“
transit curve that are more
shallow
Planet hunters like to see a flat part on the bottom of the transit
Probability of detecting a transit Ptran:
Ptran = Porb x fplanets x fstars x DT/P
Porb = probability that orbit has correct orientation
fplanets = fraction of stars with planets
fstars = fraction of suitable stars (Spectral Type later than F5)
DT/P = fraction of orbital period spent in transit
Estimating the Parameters for 51 Peg systems
Porb
Period ≈ 4 days → a = 0.05 AU = 10 R‫סּ‬
Porb  0.1
fplanets
Although the fraction of giant planet hosting stars is
5-10%, the fraction of short period planets is
smaller, or about 0.5–1%
Estimating the Parameters for 51 Peg systems
fstars
This depends on where you look (galactic plane,
clusters, etc.) but typically about 30-40% of the stars
in the field will have radii (spectral type) suitable for
transit searches.
Radius as a function of Spectral Type for Main Sequence Stars
A planet has a maximum radius ~ 0.15 Rsun. This means that a star can
have a maximum radius of 1.5 Rsun to produce a transit depth consistent
with a planet.
Take 1% as the limiting depth that you can detect a transit from
the ground and assume you have a planet with 1 RJ = 0.1 Rsun
Example:
B8 Star: R=3.8 RSun
DI = (0.1/3.8)2 = 0.0007
Suppose you detect a transit event with a depth of 0.01. This
corresponds to a radius of 50 RJupiter = 0.5 Rsun
Additional problem: It is difficult to get radial velocity
confirmation on transits around early-type stars
Transit searches on Early type, hot stars is not effective
You also have to worry about late-type giant stars
Example:
A KIII Star can have R ~ 10 RSun
DI = 0.01 = (Rp/10)2
→ Rp = 1 RSun!
Unfortunately, background giant stars are
everywhere. In the CoRoT fields, 25% of the stars
are giant stars
Giant stars are relatively few, but they are bright and can be seen to
large distances. In a brightness limited sample you will see many
distant giant stars.
Estimating the Parameters for 51 Peg systems
Fraction of the time in transit
Porbit ≈ 4 days
Transit duration ≈ 3 hours
DT/P  0.08
Thus the probability of detecting a transit of a planet in a single
night is 0.00004.
For each test orbital period you have to observe enough
to get the probability that you would have observed the
transit (Pvis) close to unity.
E.g. a field of 10.000 Stars the number of expected transits is:
Ntransits = (10.000)(0.1)(0.01)(0.3) = 3
Probability of a transiting Hot
Jupiter
Frequency of Hot Jupiters
Fraction of stars with suitable radii
So roughly 1 out of 3000 stars will show a transit event due to a
planet. And that is if you have full phase coverage!
CoRoT: looks at 10,000-12,000 stars per field and is finding on
average 3 Hot Jupiters per field. Similar results for Kepler
Note: Ground-based transit searches are finding hot Jupiters 1 out
of 30,000 – 50,000 stars.
Catching a transiting planet is thus like playing
Lotto. To win in LOTTO you have to
1. Buy lots of tickets → Look at lots of stars
2. Play often → observe as often as you can
The obvious method is to use CCD photometry
(two dimensional detectors) that cover a large
field.
CCD Photometry
CCD Imaging photometry is at the heart of any transit
search program
1. Aperture photometry
2. PSF photometry
3. Difference imaging
Aperture Photometry
Get data (star) counts
Get sky counts
Magnitude = constant –2.5 x log [Σ(data – sky)/(exposure time)]
Instrumental magnitude can be converted to real magnitude by
looking at standard stars
And to remind you what a magnitude is. If two stars
have brightness B1 and B2, their brightness ratio is:
B1/B2 = 2.512Dm
5 Magnitudes is a factor of 100 in brightness,
larger values of m means fainter stars.
Aperture photometry is useless for crowded fields
Term: Point Spread Function
PSF: Image produced by the instrument + atmosphere = point
spread function
Atmosphere
Camera
Most photometric reduction
programs require modeling of
the PSF
Note: PSF is the two dimensional version of the Instrumental Profile (IP)
Crowded field Photometry: DAOPHOT
Computer program developed to obtain accurate photometry of blended
images (Stetson 1987, Publications of the Astronomical Society of the
Pacific, 99, 191)
DAOPHOT software is part of the IRAF (Image Reduction and Analysis
Facility)
IRAF can be dowloaded from http://iraf.net (Windows, Mac)
or
http://star-www.rl.ac.uk/iraf/web/iraf-homepage.html (mostly Linux)
In iraf: load packages: noao -> digiphot -> daophot
Users manuals: http://www.iac.es/galeria/ncaon/IRAFSoporte/Iraf-Manuals.html
In DAOPHOT modeling of the PSF is done
through an iterative process:
1. Choose several stars as „psf“ stars
2. Fit psf
3. Subtract neighbors
4. Refit PSF
5. Iterate
6. Stop after 2-3 iterations
Original Data
Data minus stars found in first
star list
Data minus stars found in
second determination of star
list
Image Subtraction
If you are only interested in changes in the brightness (differential
photometry) of an object one can use image subtraction (Alard,
Astronomy and Astrophysics Suppl. Ser. 144, 363, 2000)
Image subtraction: Basic Technique
• Get a reference image R. This is either a synthetic image (point sources)
or a real data frame taken under good seeing conditions (usually your best
frame).
• Find a convolution Kernal, K, that will transform R to fit your observed
image, I. Your fit image is R * I where * is the convolution (i.e. smoothing)
• Solve in a least squares manner the Kernal that will minimize the sum:
S ([R * K](xi,yi) – I(xi,yi))2
i
Kernal is usually taken to be a Gaussian whose
width can vary across the frame.
In pictures:
Observation
Reference profile: e.g.
Observation taken under excellent
conditions
Smooth your reference profile
with your Kernel function. This
should look like your observation
In a perfect world if you subtract
the two you get zero, except for
differences due to star variabiltiy
These techniques are fine, but what happens when some light
clouds pass by covering some stars, but not others, or the
atmospheric transparency changes across the CCD?
You need to find a reference star with which you divide the flux
from your target star. But what if this star is variable?
In practice each star is divided by the sum of all the other stars
in the field, i.e. each star is referenced to all other stars in the
field.
T: Target, Red:
Reference Stars
T
A
C
B
T/A = Constant
T/B = Constant
T/C = variations
C is a variable star
Sources of Errors
Sources of photometric noise:
1. Photon noise:
error = √Ns (Ns = photons from source)
Signal to noise ratio = Ns/ √ Ns = √Ns
Root mean scatter (rms) in brightness = 1/(S/N)
Sources of Errors
2. Sky:
Sky is bright, adds noise, best not to observe
under full moon or in downtown Jena.
Ndata = counts from star
Error = (Ndata + Nsky)1/2
Nsky = background
S/N = (Ndata)/(Ndata + Nsky)1/2
rms scatter = 1/(S/N)
Nsky = 1000
Nsky = 100
Nsky = 10
rms
Nsky = 0
Ndata
Sources of Errors
3. Dark Counts and Readout Noise:
Dark: Electrons dislodged by thermal noise,
typically a few per hour.
This can be neglected unless you are looking at
very faint sources
Readout Noise: Noise introduced in reading out the CCD:
Typical CCDs have readout noise counts of 3–11 e–1
(photons)
Sources of Errors
4. Scintillation Noise:
Amplitude variations due to Earth‘s atmosphere
s ~ [1 + 1.07(kD2/4L)7/6]–1
D is the telescope diameter
L is the length scale of the atmospheric turbulence
Star looks fainter
Star looks brighter
For larger telescopes the diameter of the telescope is much
larger than the length scale of the turbulence. This reduces the
scintillation noise.
Sources of Errors
5. Atmospheric Extinction
Atmospheric Extinction can affect colors of stars and photometric
precision of differential photometry since observations are done at
different air masses
Major sources of extinction:
• Rayleigh scattering: cross section s per molecule ∝
l–4
• Aerosol Extinction
• Absorption by gases
Intensity
Atmospheric extinction can also affect differential photometry because
reference stars are not always the same spectral type.
A-star
K-star
Wavelength (Å)
Atmospheric extinction (e.g. Rayleigh scattering) will affect the A star more
than the K star because it has more flux at shorter wavelength where the
extinction is greater
Intensity
Atmospheric extinction could produce false detections:
Drop due to atmospheric
extinction
Intensity
Time
Transit detection algorithms
would detect these as a
transit
Time
Sources of Errors
6. Stellar Variability: Signal that is noise for our purposes
Stellar activity, oscillations, and other forms of variability can hinder
one‘s ability to detect transit events due to planets.
e.g. sunspots can cause a variations of about 0.1-1%
Fortunately, most of these phenomena have time scales
different from the transit periods.
Light Curves from Tautenburg taken with BEST
Step 1: Observe a field of stars, plot up the photometry and make
sure that it is consistent with what you expect from noise sources
A not-so-nice
looking curve
from an open
cluster
Saturated
bright stars
CCD Counts
Saturation
CCD Counts
t
Saturation + nonlinearity
t
Finding Transits in the Data
1.
Produce a time series light curve of your observations
using your favorite technique (aperture, psf, or difference
imaging photometry)
Finding Transits in the Data
2. Remove the bumps and wiggles due to instrumental effects
and stellar variability using low pass filters
Finding Transits in the Data
3. Phase fold the data using a trial period
Finding Transits in the Data
3. Perform a least squares fit using a box (BLS = box least squares)
w
d
Find the best fit box of width, w, and depth d.
Define a frequency spectrum of residuals (parameter of best
fit) as a function of trial periods. Peaks occur at most likely
values of transit periods. The BLS is the most commonly used
transit algorithm
Kovaks, Zucker, & Mazeh, A&A, 391, 369, 2002
The quantity SR is defined as such that when one has low
residuals (i.e. good fit), SR has a maximum value.
Confirming Transit Candidates
A transit candidate found by photometry is only a candidate
until confirmed by spectroscopic measurement (radial
velocity)
Any 10–30 cm telescope can find transits. To confirm these
requires a 2–10 m diameter telescope with a high resolution
spectrograph. This is the bottleneck.
Current programs are finding transit candidates faster than
they can be confirmed.
Light curve for HD 209458
Radial Velocity Curve for HD 209458
Transit
phase = 0
Period = 3.5 days
Msini = 0.63 MJup
Confirming Transit Candidates
Radial Velocity measurements are essential for confirming the
nature (i.e. get the mass) of the companion, and to exclude socalled false postives.
False Positives
It looks like a planet, it smells like a planet, but it is not a planet
1. Grazing eclipse by a main sequence star:
One should be able to distinguish
these from the light curve shape and
secondary eclipses, but this is often
difficult with low signal to noise
These are easy to exclude with Radial
Velocity measurements as the
amplitudes should be tens km/s
(2–3 observations)
2. Giant Star eclipsed by main sequence star:
G star
Giant stars have radii of 10–100 R‫ סּ‬which translates
into photometric depths of 0.0001 – 0.01 for a
companion like the sun
These can easily be excluded using one spectrum to
establish spectral and luminosity class. In principle no
radial velocity measurements are required.
Often a giant star can be known from the transit time.
These are typically several days long!
e.g. giant with R = 10 Rsun and M = Msun and
a transit period of 10 days:
t ≈ 1.3 days!
Probably not detectable from ground-based observations
A transiting planet around a solar-type star with a 4 day
period should have a transit duration of ~ 3 hours. If the
transit time is significantly longer then this it is a giant or
an early type star.
3. Eclipsing Binary as a background (foreground) star:
Fainter binary
system in
background or
foreground
Total = 17% depth
Light from bright
star
Light curve of
eclipsing
system. 50%
depth
Difficult case. This results in no radial velocity variations as the fainter
binary probably has too little flux to be measured by high resolution
spectrographs. Large amounts of telescope time can be wasted with
no conclusion. High resolution imaging may help to see faint
background star.
4. Eclipsing binary in orbit around a bright star (hierarchical
triple systems)
Another difficult case. Radial Velocity Measurements of the
bright star will show either long term linear trend no variations
if the orbital period of the eclipsing system around the primary
is long. This is essentialy the same as case 3) but with a
bound system
5. Unsuitable transits for Radial Velocity measurements
Transiting planet orbits an early type star with rapid rotation
which makes it impossible to measure the RV variations or
you need lots and lots of measurements.
Depending on the rotational velocity RV measurements are
only possible for stars later than about F3
Results from the CoRoT Initial Run Field
26 Transit candidates:
Grazing Eclipsing Binaries: 9
Background Eclipsing Binaries: 8
Unsuitable Host Star: 3
Unclear (no result): 4
Planets: 2
→ for every transiting planet found there are 10 false
positive detections. These still must be followed-up with
spectral observations
Search Strategies
Look at fields where there is a high density of stars.
Strategy 1:
Look in galactic plane with a small (10-20 cm) wide field (> 1 deg2)
telescope
Pros: stars with 6 < V < 15
Cons: Not as many stars
WASP
• WASP: Wide Angle Search For Planets (http://www.superwasp.org). Also
known as SuperWASP
• Array of 8 Wide Field Cameras
• Field of View: 7.8o x 7.8o
• 13.7 arcseconds/pixel
• Typical magnitude: V = 9-13
Search Strategies
Strategy 2:
Look at the galactic bulge with a large (1-2m) telescope
Pros: Potentially many stars
Cons: V-mag > 14 faint!
Image in
galactic
bulge
OGLE
• OGLE: Optical Gravitational Lens Experiment
(http://www.astrouw.edu.pl/~ogle/)
• 1.3m telescope looking into the galactic bulge
• Mosaic of 8 CCDs: 35‘ x 35‘ field
• Typical magnitude: V = 15-19
• Designed for Gravitational Microlensing
• First planet discovered with the transit method
Search Strategies
Strategy 3:
Look at a clusters with a large (1-2m) telescope
Pros: Potentially many stars (depending on cluster)
Cons: V-mag > 14 faint! Often not enough stars, most open
clusters do not have 3000-10000 stars
A dense open cluster: M 67
Stars of interest have
magnitudes of 14 or
greater
A not so dense open cluster:
Pleiades
h and c Persei double cluster
A dense globular cluster: M 92
Stars of interest have
magnitudes of 17 or
greater
Search Strategies
Strategy 4:
One star at a time!
The MEarth project
(http://www.cfa.harvard.edu/~zberta/mearth/)
uses 8 identical 40 cm telescopes to search
for terrestrial planets around M dwarfs one
after the other
A transiting planet candidate is
only a candidate until it is
confirmed with Radial Velocity
measurements!
Radial Velocity Follow-up for a Hot Jupiter
The problem is not in finding the transits, the problem
(bottleneck) is in confirming these with RVs which requires
high resolution spectrographs.
Telescope
Easy
Challenging
Impossible
2m
V<9
V=10-12
V >13
4m
V < 10–11 V=12-14
V >15
8–10m
V< 12–14
V >17
V=14–16
It takes approximately 8-10 hours of telescope time on a
large telescope to confirm one transit candidate
Summary
1. The Transit Method is an efficient way to find
short period planets.
2. Combined with radial velocity measurements it
gives you the mass, radius and thus density of
planets
3. Roughly 1 in 3000 stars will have a transiting hot
Jupiter → need to look at lots of stars (in galactic
plane or clusters)
4. Radial Velocity measurements are essential to
confirm planetary nature
5. Anyone with a small telescope can do transit work
(i.e even amateurs)
Tautenburg Exoplanet Search Telescope: a 30 cm
“Amateur” Telescope
A proud amateur and his light curve of the
new hot transiting Saturn