Elisabeth Telescope (SAAO), South Africa
L11: Finding exoplanets
the Transit technique
credit: Waldmann
UCL Certificate of astronomy
Dr. Ingo Waldmann
Detecting exoplanets
Detecting exoplanets
*
•
•
•
•
•
•
Astrometry
Radial Velocity
Occultations (Transit photometry)
Direct Imaging
Gravitational Lensing
Timing (pulsars)
Photometry & Exoplanets
Kinematic
Photometric
11/2
Transit Photometry
Transit Photometry
Photometry measures the brightness of the celestial
objects.
NB: fluxes, not luminosities !!
Flux: Energy per unit time and area
Luminosity: Energy per unit time
If a planet passes in front of a star, the star will be
partially eclipsed and its light will be dimmed.
Photometry & Exoplanets
11/3
Transit of an exoplanet
Photometry measures the brightness
of a celestial object.
!
NB: It’s measuring the stars flux
(not luminosity)
FLUX: energy per unit time and area
LUMINOSITY: energy per unit time
If planet passes infront of parent star
the flux of the star will temporarily drop
drawing out a lightcurve
Lightcurve
Transit of an exoplanet
Even a Jupiter size planet will
only obscure ~ 1% of the
stellar light.
out-oftransit
The transit depth is
proportional to the surface
ratios of the planet and the
star:
ingress
in-transit
egress
F
F=
!
✓ !
Rplanet
Rstar
◆2
Lost in the glare (NASA)
Transit Photometry
Transit Photometry
• A planet around a star on an orbit viewed close to edge-on
towards us will produce a periodic dimming (eclipse) which can
be detected.
• Even the largest planets only block the light of the star by a
tiny amount.
• For instance, Jupiter’s radius is RJ~RꙨ/10: At best, at the
position of the Sun it would block ~1/100 of its disk area
• Earth-like planets will produce much smaller effects
• In order to see a transit, the orbit must be aligned edge-on (or
close to edge-on). That means ~1-2% of all possible orbits
• Only the flux is measured ! The planet is not resolved
Photometry & Exoplanets
11/4
Transit of Venus
Photometry & Exoplanets
11/5
Estimating the loss of flux
Estimating the loss of flux
Consider angular distances:
θS
D = Distance to star-planet system
RS
D
Radius of star: θS = RS/D (in radians)
Radius of planet: θP=RP/D (assume distance Star-Planet<<D)
Hence, out of πθS2, the planet blocks πθP2 (let us neglect the
brightness of the dark side of the planet)
The flux will fall by a similar amount, namely:
ΔΦ/Φ0=[θP2/θS]2 = [RP/RS]2 2
F = (✓p /✓⇤ ) = (Rp /R⇤ )
Sun-Jupiter:
Sun-Earth:
1% change in flux
10-4 change in flux
Photometry & Exoplanets
11/7
Transit duration
Transit duration
We can easily compute the duration of the eclipse (τ) taking into
account the parameters that describe the system:
Radius of star (from luminosity and temperature, also M s)
– Rs
Radius of planet (Attenuation ΔΦ/Φ0)
– Rp
–δ
stellar latitude of trajectory (angle)
–a
orbital radius (assumed circular; Kepler’s 3rd law)
–P
orbital period (interval between transits)
Photometry & Exoplanets
11/8
Transit probability
Transit probability
The probability of having an eclipsing planetary system can be
computed from the inclination of the orbit with respect to the
observer. The inclination angle (i) and latitude of trajetory (δ) are
equivalent:
δ
δ
Line-of-sight view
i
P
h
a
Edge-on view
Rs
Transit probability
Transit probability
In order to have a transit we need:
h ≤ Rs + Rp
δ
i
P
h
a
A random distribution will have equally
probable values over the range of the
angle i. The geometric transit probability
is:
Edge-on view
For example, for the Sun-Earth system
the probability is ~0.005, i.e. we will need
to monitor at least 200 stars to have a
chance to detect one transit.
Photometry & Exoplanets
11/10
Solving the system
Solving the system
In fact, a combination of transit data and radial velocities allows
us to fully understand the orbit of the exoplanet and break the
degeneracy that one gets with radial velocity measurements
between the mass of the planet and the inclination angle (one can
only constrain Msin i with the radial velocities)
The condition for the presence of a transit implies that the orbit
must be close to edge-on. The transit timing and dimming of the
star allows us to uniquely determine the orbit inclination, the ratio
Rp/Rs and in combination with the radial velocity, the mass of the
planet.
Photometry & Exoplanets
11/12
ration,
tT ¼
0
P
R$
arcsin@
!
a
(&
)1=2 1
1 þ Rp =R$ "½ða=R$ Þ cos i(
A;
2
1 " cos i
$
%'2
2
ð3Þ
Solving the star-planet system
First to fourth contact points
R*
Rp
1 2
3
bR*= a cosi
1
2
3
4
4
tF =
Duration of full eclipse
(contact points 2-3)
tT = Duration of total eclipse
(contact points 1-4)
Lightcurve
ΔF
tF
tT
Transit depth
Seager & Mallén-Ornelas (2003)
R# ¼ kM# ;
$
%' 2
#&
(1=2
1 " Rp =R$ "½ða=R$ Þ cos i(2
sinðtF !=PÞ
¼ #&
$
%' 2
(1=2 ;
sinðtT !=PÞ
1 þ Rp =R$ "½ða=R$ Þ cos i(2
ach stellar sequence (main sequence,
giants, etc.) and x describes the power law
the total transit duration,
0 (&
) 1
$
%'
equence stars; Cox 2000).
P
R$ 1 þ R =R$ "½ða=R$ Þ cos i(
Solving the star-planet system
2
tT ¼
!
arcsin@
E SOLUTION OF PLANET AND STAR PARAMETERS
3.2. Analytical Solution
2
p
a
1=2
A;
1 " cos2 i
1041
From
Kepler’s
and radial
velocity
we
have: and M the planet mass,
3.2.1.
Four
Parameters
Derivable
from
Observables
ular
orbit,
where
G isthird
the law
universal
gravitational
constant
p
five unknown parameters
4!2M
a3 *, R*, a, i, and Rp from the five equations above. It
2
P ¼
;
ð4Þ
R(DF,
*
,
tF, a
of physical parameters
be
found
directly
from
the
observables
t
T
GðMcan
þ
M
Þ
#
p
3.1 (the three transit geometry equations and Kepler’s
1 2third law with M3p 5
4 M#
SEAGER
&
MALLÉN-ORNELAS
Vol.
58
,mass-radius relation.
From the transit depth measurement we have the
bR*= a cosi
x
rs
are
as
follows:
the
planet-star
radius
ratio,
which
trivially
follows
from equation
R# ¼ third
kM#ratio:
;law with Mp 5 M! ,
ð5Þ
planet/star
om M* and from Kepler’s
1
2
3
4
p
ffiffiffiffiffiffiffi
!
"
R
1=3
r each stellar sequencep (main
sequence,
giants, etc.) and x describes the power law of the
P
¼2 GM
DF
! ;
in-sequence stars; Cox
a¼
;
ð1
R#2000).
2
4!
Rp
projected
distance
between
the
planet
3.2.
Analytical
Solution
We
can
now
the
orbital
inclination:
rameter
(eq.
[7]),
thecalculate
orbital
inclination
is and star centers during midtransit in units
directly
from
transit Derivable
shape
(2), together with equation (6),
! equation
"
3.2.1.
Fourthe
Parameters
from Observables
ΔF
R
!
("five unknown
)
pffiffiffiffiffiffiffi#2 parameters
pffiffiffiffiffiffiffi
$ cos2 %1 M
i¼
b *, R*,; a,2i, and Rp %"
the
from the
five#equations
2 1=2 t F above. It is firstð1
aÞ=
t T t , t , and P)
1 'physical
DF parameters
' sin ðtcan
sin ðdirectly
tT !=PÞfrom
1 þthe observables
DF
F !=P
ons of
be
found
(DF,
T F
;
$
%
Fig.
of transit
light-curve observables.
Twolaw
schematic
light curves
on the bottom
(solid and
2 1.—Definition
5M
in
x 3.1 is
(the three
transit
and
Kepler’s
third
with
Marepshown
# ); this
radius
geometry
ofT
the!=P
star and planet
is shown on the top. Indicated on the solid light curve are the transit depth DF, the tota
1'
sin2geometry
ðtF !=PÞ=equations
sin
ð
t
Þ
duration between ingress and egress t (i.e., the ‘‘ flat part ’’ of the transit light curve when the planet is fully superimpos
llar mass-radius relation.
third, and fourth contacts are noted for a planet moving from left to right. Also defined are R , R , and impact parameter b
!
"
i. Different impact parameters b (or different i) will result in different transit shapes, as shown by the transits correspondin
x=ð1%3xÞ
eters are
as follows:
the planet-star
radius ratio,
which trivially follows
equation(2003)
(1),
ffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
Rthe
Seager from
& Mallén-Ornelas
p
! ptransit
rectlyRfrom
duration
1=x "! equation (3),
¼
DF ¼ k
DF :
ð1
F
*
p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
$
%'
#&
(
"
#2law"(eq. [4]),#and
i(
1 "relation
R =R$ "½ða=R
sinðt !=PÞ
$ Þ cos
sit depth (eq. [1]), Kepler’s
third
the mass-radius
(eq.
[5])
rema
2
¼ #&
;
$
%'
(
Rp
PR"
a
1=2
sinðt !=PÞ
1 þ R =R$ "½ða=R$ Þ cos i(
ote that
by
substituting
ða=Rlaw
icos
and
ðDF
" Þ cos
ip 5
: Þ! , ¼ Rp =R" the above equations take
ð
$
tT ¼
1bþ¼third
rom
M* and
from Kepler’s
with
M
M
the total transit duration,
R
R"
!a
"
0 (&
) 1
$
%'
! 2
"1=3
P
R$ 1 þ R =R$ "½ða=R$ Þ cos i(
A;
P GM!
t ¼ arcsin@
Solving
the
star-planet
system
!
a
1
"
cos
i
nsit depth (eq. [1]),a Kepler’s
third
law
(eq.
[4]),
and
the
mass-radius
relation
(eq.
[5])
rem
¼
;
ð1
2
1=2
3.3.2.
The
Simplified
Analytical
Solution
Note that by substituting b4!
¼ ða=R" Þ cos i and ðDF Þ ¼ Rp =R" the above equations tak
uations is(eq.
more
useful
than the
exact solution
for considering the general properties of lig
arameter
[7]),
the
orbital
inclination
is
can now
thefactor
orbitalthat
inclination:
of the We
solution
or iscalculate
a simple
can cancel out in parameter ratios. The impa
!
"
R!
1, becomes
roximation3.3.2.
tT !=P5
The
Simplified
%1 Analytical
i ¼ cos
b
; Solution
ð1
R
*
a$ solution
%2 3for
1=2considering the general properties of li
quations2is$morepuseful
ffiffiffiffiffiffiffi%2 than the exact
p
ffiffiffiffiffiffiffi
2
R
1 2parameter ratios. The
3 4 imp
1
$
DF
$
ð
t
=t
Þ
1
þ
DF
F
T
ut
of
the
solution
or
is
a
simple
factor
that
can
cancel
out
in
6
7
y radius
is
- it’s called
the impact
parameter:
b What
¼ 4 is b??
ð1
5 ;
1, becomes
proximation
tT !=P5
2
bR*= a cosi
1 $!ðtF =tT Þ "x=ð1%3xÞ pffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffi
Rp2$ R!
"!
%2 3:1=2
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiDF
¼
DF %¼2 k1=x 2 $
ð1
1
2
3
4
R& 1R$
$ðtF =tT"Þ& 1 þ DF 7
& DF
6
b¼4
;
ð
%0:57 5 1=2
2
x ' 0:8, in which case R
¼1=4
ðDF Þ .
1 p$=R
ðt&DF
F =t
TðÞ"! ="& Þ
a
2P
ð1
¼
&2
'1=2 ;
2
R
!
" Set of Equations
tT $ tF and Their Solution
3.3. The Simplified
F
p
T
p
2
2 1=2
2
2 1=2
2
2
p
T
1=2
2
p
ΔF
rcomes
solution take on a simpler form1=4
under the assumption R! 5 a. This assumption
is equiv
We can even
calculate
the
stellar
density:
a
2P i5DF
t F in generally ha
has as its consequence
cos
1 (from eq.
[13]). Systems we are interested
;
ð
¼
&
'
tT
1
1=2
3=4
2
2
R
!
1 (or R! =ae8). Mathematically
assumption allows arcsin x ' x and sin x ' x. Und
" 32
tTDF
$ tthis
F
P
"
¼
: 1.—Definition
Fig.
of transit light-curve observables. Two schematic light curves are shown on the bottom (solidð1
and
&2
'
3=2
nðtT !=PÞ ' tF =t"T . AG!
comparison
of
these
two
terms
is
shown
in
Figure
2a;
for
cases
geometry
of
the
star
and
planet
is
shown
on
the
top.
Indicated
on
the
solid
light
curve
are
the
transit
depth
DF,
the
tota
between ingress and egress t (i.e., the ‘‘ flat part ’’ of the transit light curve when the planet is fully superimpos
tT $ t2F duration
impact parameter
and fourth contacts
noted for a planet moving from left totright.
Also defined are
ecomes
1,R , aR , and
second
terb
an
4% and much better in most cases.i.third,
Under
theare approximation
T !=P5
Different impact parameters b (or different i) will result in different transit shapes, as shown by the transits correspondin
Seager
& Mallén-Ornelas
(2003)
'of
1. "A' comparison
term with
as3=4
a function
of tT!/P
(Fig.
2b)
shows
agreement
to bet
for P, tF, andoftTthis
in days,
the first factor
on the
right-hand
side
of equation
(1
F
*
p
Gaudi & Winn (2007)
Transits and Radial Velocity
... more information from transit data
The trajectory of the planet can be further constrained if we have the
radial velocity as a function of time (Rossiter-McLaughlin effect)
Photometry & Exoplanets
11/11
First planet ‘seen’ with the
First transiting exoplanet detected
photometry method
In 1999, the first extrasolar planet to
show transits across the disc of its
star (HD209458) was detected
HD209458 is a G0V star (like the
Sun)
The planet was originally detected via
radial velocities
Photometry & Exoplanets
11/13
HD 209458b
HD 209458b
• HST light curve
• Planet: 0.69 MJ
P=3.524 d
τ~3h
a=0.045 AU
R=1.35 RJ
T>1700K
Brown et al. (2001)
–
–
–
–
–
Data wrapped over many transits
• Spectral transit
– Sodium
– Oxygen
– Carbon
Charbonneau et al. (2000)
Photometry & Exoplanets
11/14
HD 209458b
HD 209458b
• HST light curve
• Planet: 0.69 MJ
Data wrapped over many transits
Brown et al. (2001)
– P=3.524 d
– τ~3h
– a=0.045 AU
– R=1.35 RJ
– T>1700K
• Spectral transit
– Sodium
– Oxygen
– Carbon
Photometry & Exoplanets
11/15
HD209458b: exercise
Exercise
Let us estimate the radius of the planet as a fraction of the star’s
radius
Flux loss =
= [RP/RS]2 = 0.016
RP = 0.13 RS
0.016 (1.6%)
HD209458 is a G0V
1.1MꙨ star with
RS=1.15RꙨ
Planet radius
Rp = 0.16RꙨ
Photometry & Exoplanets
11/16
Three currently active search networks
MEarth
Hat-NET
Super-WASP
The start of a dedicated search
Kepler mission
CoRoT mission
Ground Based Survey Era
HD209458b
51 Peg b
Pulsar timing
Exoplanet detection at ULO
Exoplanet detection at ULO
The transit of HD 80606b in front
of its parent star was dectected
at ULO (Fossey, Waldmann and
Kipping, MNRAS) on Feb. 13,
2009
Jupiter-sized planet, e=0.93
Photometry & Exoplanets
11/19
HD 80606b discovered at ULO
Transits and planet
composition
more on that later...
Gillon et al. (2007)
The transit method in combination
with the radial velocity method
gives information both about
planet size and mass. The density
is the zeroth order approach to
study the composition.
Photometry & Exoplanets
11/20
Space Missions
•
COROT (ESA, France)
Transit
–
–
–
–
•
–
Imaging
30cm telescope
Launched Dec. 2006, in operation
One-half of 2.8o x 2.8o field of view (the other half used for
astroseismology studies)
10-40 rocky planets expected
Kepler (NASA)
–
–
–
–
–
•
Space Missions
1.4m telescope
Launched March 2009,in operation.
105 sq deg field of view (42 2kx1k CCDs)
Monitors brightness of 100,000 stars over 4 years
Space-based photometry eliminates the noise from background
atmosphere
Expected: about 50 planets if similar to Earth (640 planets if size
R~2.2RE)
Terrestrial Planet Finder-Coronograph (NASA)
–
–
–
No launch date yet
Light from central star is blocked
4m x 6m telescope
Photometry & Exoplanets
11/21
Region explored by the Kepler mission
Region explored by Kepler
Photometry & Exoplanets
11/22
Secondary eclipse
Primary eclipse
Aims & Objectives
• Understand how transit observations are performed
!
• How to measure the planet/star ratio and other parameters
from the lightcurve
!
• Understand how transits break the radial velocity
degeneracy