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Exoplanets
Edge on
Face on
Transit Exoplanet: Light Curve Analysis and Limb Darkening Effect
Exoplanets
Let’s look into how we analyze the planet transit light curve …
First, let’s assume the following:
o The planet is in a circular orbit. This is often true for planets
close to their star as tidal interaction with the star acts to
circularise the orbit of the planet.
o The stellar intensity is uniform across the stellar disk, i.e., the
stellar limb darkening is negligible. This is true at long
wavelengths, e.g. the I band (806±149 nm). ( We will come
back to this assumption later.)
o The planet is dark compared to the central star.
o The light comes from a single star, i.e., the light from the
planet host star is not blended with the light from another star.
Transit light curve analysis
Transit light curve analysisAstronomers’ (silly) confusion in the
use of the impact parameter b:
b = a cos i
to represent the projected distance
between the center of the star and that
of the planet.
However, often it is also used as
b = (a cos i) /R*
to be dimensionless. This normalized
definition gives simpler mathematical
terms in related equations. So we need
to pay attention to which b is used
under the circumstances.
Transit light curve analysis
The schematic of a planet transiting its host star (middle) with the corresponding
variation in brightness during the transit (top) and during the occultation (bottom). The
impact parameters b and the transit parameters (δ, tF, tT) used in the equations here
after are indicated on this figure.
Another view of the transit planet light curve.
Transit light curve analysis
Transit light curve analysis
Another view of the transit planet light curve.
b: impact parameter
a: distance between the star and planet
i: inclination angle
Transit light curve
analysis
Transit light curve
analysis
transit shape: the ratio of the
duration of the flat part of the
transit (tF) to the total transit
duration (tT), so this is purely
geometrical parameter
Transit shape and
Pythagorean theorem
Transit light curve
analysis
Transit light curve
analysis
Transit light curve
analysis
Transit light curve
analysis
Transit light curve
analysis
Transit light curve
analysis
Transit light curve analysis
But there is another step ……
Transit light curve analysis
Limb Darkening Effect
Limb Darkening Effect
The solar disk is (gradually) dimmer at the
edge than the center. Why?
(http
://spiff.rit.e
du/c
lasses/p
hys440/le
ctu
res/lim
b/lim
b.h
tml)
Parameterization of the Limb Darkening Effect
Center
Limb Limb
Limb darkening effect equation should be
able to produce profiles like this one.
Observed intensity profile
across the Solar disc.
Limb Darkening Effect
(http://spiff.rit.edu/classes/phys440/lectures/limb/limb.html)
(https://sites.ualberta.ca/~pogosyan/teaching/ASTRO_122/lect9/lecture9.html)
We see photons from the photosphere where the optical depth is not greater than 1.
(The optical depth is proportional to the product of the number density, cross section
and the physical length. It’s dimensionless.)
Limb Darkening Effect
o In reality, the stellar luminosity is not constant across
the stellar disk.
o The stellar disk is brighter at its centre than at its
edge.
o The photons received from the centre of the stellar
disk come from deeper into the stellar atmosphere
than those received from the edge of the disk.
o A photon coming from deeper into the stellar
atmosphere has a higher temperature and thus
appears brighter at the associated wavelength.
o Thus, at the corresponding wavelength, the stellar
centre appears brighter than the stellar limb, hence
the expression "limb darkening".
Limb Darkening Effect
(http://abyss.uoregon.edu/~js/ast121/lectures/lec23.html)
Transit light curve analysis
Planet transit illustration with limb darkening
Transit light curve analysis
Limb darkening in real
transit light curves
Without limb
darkening
Transit
Depth
Limb
Darkening
Ingress Egress
Limb Darkening Effect in Transit Light curve
Transit light curve & Limb Darkening Effect
Using a realistic model of stellar limb darkening is important when
fitting transit light curves, as the shape of the limb darkening will
influence the derived planet parameters (mainly the planet radius
and impact parameter on the stellar disk).
This intensity variation across the stellar disk is calculated from
stellar atmosphere models (e.g. ATLAS9, PHOENIX) where the
emergent intensity with regard to the line of sight is known. This
intensity is then passed through different instrumental filters (e.g.
the standard filters in Claret 2000 and Claret 2004, and the
CoRoT and Kepler filters in Sing 2010), and fitted with different
limb darkening laws to derive the associated limb darkening
coefficients.
http://kurucz.harvard.edu/grids.html (ATLAS9)
http://www.hs.unihamburg.de/EN/For/ThA/phoenix/index.htm (PHOENIX)
Parameterization of the Limb Darkening Effect
Let’s consider this geometry
and define the limb darkening
coefficient (u) as
which is a normalized
differential intensity between
the center and the limb.
Edge
The simplest known way to parameterize the limb darkening
effect is
and this is known as “linear case” of limb darkening effect.
r : projected distance to the
surface (= radial distance from
the observed center of the disc)
( = cos)
Parameterization of the Limb Darkening Effect
Comparisons of the expected limb darkening profiles from the linear case
with u = 1.0, 0.8, 0.6, 0.4 and 0.2 (from lowest curve upwards). The curve
for u = 1 is a circle. The radius of the disc is taken to be 1, r = 0 is the
centre of the disc and r = ± 1 is the limb.
( = cos)
Parameterization of the Limb Darkening Effect
Summary of the most commonly used limb darkening effect equations.
In practice, the choice of which limb-darkening law to use depends on the
signal-to-noise ratio (S/N) of the transit, the observational bandpass and
the stellar type. High S/N observations allow the shape of the limb
darkening to be more accurately constrained, and so can justify the usage
of a limb-darkening law with more coefficients.
Transit light curve analysis
