INTRODUCTIONBruce L. Gary; 2007.03.04This web page is a place where I summarize a procedure I use to estimate the size of an exoplanet candidate from the transiting star's spectral class, transit depth, transit length, orbital period and filter band. A link is included for downloading an Excel spreadsheet that performs the calculations.
A proper solution for planet radius from a LC will involve more information about the LC's shape than is used by the simple solution on this page. A crude method is presented for determining if the shape is similar to what an exoplanet can produce, versus what a blend of an eclipsing binary (EB) with another nearby star would produce. My shortcuts reduce accuracy, of course, but if an approximate answer is acceptable then the procedures described here may be useful.
I like explaining things through the use of a specific example. The
reader's job is to "generalize" from the specific. I'm going to
treat real observations of a "mystery" star's transit light curve; this
way we can grade the results of my crude analysis procedure using a
rigorous treatment by professionals.
Let's assume the following:
which were derived from the following transit light curve (measured with an amateur 14-inch telescope).
Figure 1.01 Transit light curve for a mystery star whose LC we shall try to "solve" using the procedures described on this web page.
After a transiting candidate has been observed, and before radial
velocity measurements have been made to assess the mass of the secondary,
this is all the information we have to work with. Using this limited
information there are many steps for interpreting the LC to estimate secondary size, Rp/Rj.
At this point we have an approximate planet size. It's a first iteration since limb darkening has been neglected.
The next group of operations is a 2nd iteration.
Star's mass, Mstr = 0.97 (times sun's mass)No more iterations are needed since the two miss distance results (& limb darkening corrections) are the same. We have a stable solution:
Mstr = 2.57 - 3.782 * (B-V) + 2.356 * (B-V)^2 - 0.461 * (B-V)^3
Planet orbital radius, a = 7.22e6 [km]
a = 1.496e8 * [Mstr^1/3 * (P / 365.25)^2/3], where P[days], Mstr[sun's mass] & a[km]
Transit length maximum, Lx = 3.28 [hr] (corresponds to central transit)
Lx = 2 (Rstr * Rsun + Rp/Rj * Rj) / (2 pi a / 24 * P) where Rsun = 6.955e5 km, Rj = 7.1492e4 km
Miss distance, m = 0.42 (ratio of closest approach to center to star's radius)
m = SQRT [1 - (L / Lx)^2]
Limb darkening effect, LDe = 1.16 (divide D by this)
I(m)/I(av) = [1 - 0.98 + 0.15 + 0.98*m - 0.15*m^2] / 0.746, for B-band
" = [1 - 0.92 + 0.19 + 0.92*m - 0.19*m^2] / 0.787, for V-band
" = [1 - 0.85 + 0.23 + 0.85*m - 0.23*m^2] / 0.828, for R-band
" = [1 - 0.78 + 0.27 + 0.78*m - 0.27*m^2] / 0.869, for I-band
- Corrected transit depth, D = 20.4 mmag (1st iteration for D)
- D = D / LDe (this is the D that would have been measured if the star were uniformly bright)
- Planet radius, Rp/Rj = 1.31 (2nd iteration for Rp/Rj)
- (Same eqn as above but now assumes m = 0.42 and appropriate limb darkening)
- Transit length maximum, Lx = 3.25 [hr] (2nd iteration for central transit length)
- (Same eqn as above)
- Miss distance, m = 0.405
- (Same eqn as above)
- Limb darkening correction, LDe = 1.165 (divide D by this)
- (Same eqn as above)
Rp/Rj = 1.306
To assign a SE to this solution it is necessary to repeat the above
procedure using a range of values for the measured transit depth and
length. When this is done (using the Excel spreadsheet, below) we get
Rp/Rj = 1.306 +/- 0.063 (with B-V uncertainty contributing the greatest
component of SE). Note that the stated SE doesn't include the
uncertainties associated with the equations converting B-V to stellar
radius and mass, nor does it allow for the possibility that the star is "off the main sequence.".
The B-band light curve for XO-1 is measured to have D = 24.8
± 0.5 and L = 2.95 ± 0.03. For these inputs the procedure
described above gives Rp/Rj = 1.29 +/- 0.06.
GRAPHICAL REPRESENTATION OF EQUATIONS
The following graphs can be used instead of the equations for
deriving star radius, mass and limb darkening correction (derived from
Allen's Astrophysical Quantities, Fourth Edition, 2000):
Figure 1.2. Converting star color B-V to stellar radius (assuming main sequence).
Figure 1.03. Converting star color B-V to stellar mass (assuming main sequence).
Figure 1.04. Converting miss distance and filter band to intensity at that location, normalized by disk-average intensity (assuming a sun-like star).
The following two figures show how transit shape depths chould
behave when the miss distance changes from near-center to near-edge.
These are real measurements (graciosly provided by Cindy Foote) that
were categorized as EB based on the depth values. The concept is the
same, whether it's an exoplanet or small EB, because in both cases a
central transit produces a greater loss of light in B-band than R-band,
and for a near-edge transit the reverse is true.
Figure 1.05. Transit depth is greatest for B-band, consistent with miss distance <0.73 (coutesy of Cindy Foote, 2007.03.04).
Figure 1.06. Transit depth is greatest for R-band, consistent with miss distance >0.73 (coutesy of Cindy Foote, 2007.03.04).
2.0 VERIFYING THAT LIGHT CURVE SHAPE IS NOT AN E.B. BLENDING
Wide-field survey telescopes provide an efficient means for
detecting stars that are undergoing periodic fadings with depths small
enough to be caused by an exoplanet transit. A fundamental limitation of
such a survey is that in order to achieve a wide field of view the
telescope's resolution is poor, and this leads to many "false alarms"
due to the "blending" of light from stars within the resolution circle.
If, for example, the resolution circle has a radius of 1 'arc and the
star flux within this circle corresponds to ~11th magnitude, it is
common for several stars to be present within the resolution circle
that are fainter than 11th magnitude but with fluxes that add up to
11th magnitude. If this situation occurs, and when one of those stars is an
eclipsing binary (EB) with a large transit depth, the transit depth
measured by the survey telescope may be small enough to resemble one
produced by a planet. This is a common occurence.
There are two blending situations
that need to be considered: 1) the EB is part of a triple star system,
so the blending star is too close to the EB to be resolved by even
large telescopes, and 2) the EB and the blending star are
far enough apart (usually gravitationally unrelated but close together
in our line-of-sight) that their angular separation are within the
resolution limits of
ground-based telescopes. The second of these blending situations
is probably more common than the first.
When a survey telescope system produces many candidates per month it
is not feasible to rule out an EB explanation for each one by measuring
radial velocities during the course of a few nights with a telescope
large enough to produce spectrograms that have the required accuracy. Although radial velocity measurements would allow the
determination of the secondary's mass, and thus distinguish between EB
and planet transits, large telescope time is too costly for such an
approach unless there is good evidence ruling out an EB explanation and leaving open the exoplanet possibility.
A better alternative is to perform follow-up observations of the
survey candidates using telescope's with apertures sufficient to
identify the most common blending situation. Amateurs with 14-inch
aperture telescopes, for example, have sufficient resolution to determine which star
within the survey's resolution area is undergoing transit, thus easily
identifying most cases of non-gravitationally related blending. These
telescopes also have sufficient SNR for an 11th magnitude star, for
example, to allow the transit light curve to be determined with good
enough quality to sometimes identify the presence of a triple star
system EB. There may be many more cases of triple star EBs that
resemble exoplanet transits than there are actual exoplanet transits.
Therefore, it is important to be able to interpret a transit light
curve to distinguish between a triple star EB and an exoplanet.
This section demonstrates how amateurs can
distinguish between exoplanet light curves and "EB triple star blended
light curves" of similar depth, so that additional amateur observing time is not
wasted on non-exoplanet candidates.
Figure 2.01. Model light curves for central transits by different sized secondaries. An R-band sun-like limb darkening function was used.
First contact occurs when the intensity begins to drop, and second
contact can be identified by the inflection where the slope changes
from steep to shallow. A "shape" parameter is defined as the ratio of
time the secondary is partially covering the star to the entire length
of
the transit (e.g., contact 1 to contact 2 divided by contact 1 to
mid-transit). For example, in the above figure consider Rp/Rstr = 0.12:
contact 1 and 2 occur at -0.55 and -0.44, and contact 1 to mid-transit
is
0.55. For this transit the shape parameter is S = (0.55-0.44) / 0.55 = 0.20.
Let's estimate the shape parameter for a real transit.
Figure 2.02. Measured light curve with the contact times indicated.
My readings of contact 1 and 2 are
-1.48 and -1.05 hour. The shape parameter is therefore
0.29 (0.43 / 1.48). Assigning SE uncertainties and propogating them
yields: S = 0.29 ± 0.01.
The following figure shows how the shape parameter varies with secondary size for central transits.
Figure 2.03. Shape parameter, S, versus planet size for central transits.
We next consider how the LC shapes vary with
miss distance. We'll adopt one secondary size and vary the miss distance.
Figure 2.04. Shape of LCs for various miss distances (b) and a fixed secondary size of Rp/Rstr = 0.08.
Note the change of terminology for "center miss distance" from m to b. Sorry, but I use both symbols for this parameter throughout this web page.
The following figure summarizes the dependence of S on many choices for planet size and miss distance.
Figure 2.05. Shape parameter for a selection of secondary sizes and center miss distances, b.
Recall that for this LC we determined that S = 0.29 ± 0.01. The shape alone tells us that Rp/Rstr < 0.17. From the previous section we derived m = b
= 0.40 (the thick black trace in the above figure), so this means
Rp/Rstr ~ 0.13. It's not our purpose here to re-derive Rp/Rj, but let's
do it to verify consistency. Rp/Rj = 9.73 * Rp/Rstr * Rstr/Rsun = 9.73
* 0.13 * 0.99 = 1.25. This is smaller than 1.31 derived from the
transit depth, but notice that the 1.25 estimate came from the light
curve shape, S, and extra information about miss distance. This consistency check is successful.
Our goal in this section is merely to distinguish between exoplanet
light curve shapes and EB shapes. It will be instructive to consider
secondaries at the threshold of being a star versus a planet. This is
generally taken to be Rp/Rj ~1.5. For such "threshold secondaries" the
Rp/Rstr will depend on the size of the star, which in turn depends on
its B-V (spectral type). Let's list some examples, going from blue to
red stars.
Blue star (B-V ~ 0.30, spectral type F1V), Rstr/Rsun ~1.5, Rp/Rstr ~0.10
Sun-like (B-V ~ 0.65, spectral type G2V), Rstr/Rsun ~1.0, Rp/Rstr ~0.15
Red star (B-V ~ 1.20, spectral type K6V), Rstr/Rsun ~0.7, Rp/Rstr ~0.22
Figure 2.08. Domains for exoplanets and EBs, using parameters S and D as input (yielding Rp/Rstr and miss distance as answers).
This figure requires knowledge of transit depth, D, instead of miss distance. This is better since D
is easily determined by casual inspection of a LC. The shape
parameter S is also easily determined by visual inspection. Therefore,
without any attempts to "solve" the LC this plot can be used
to estimate Rp/Rstr and miss distance. Then, by knowing B-V we can
specify a Rp/Rstr "threshold secondary" boundary in the figure that separates the exoplanet and the EB domains.
Consider the previous example, where XO-1 was determined to have S
= 0.29 and D ~ 24 mmag. Given that B-V = 0.66 we know that a
"threshold secondary" will have Rp/Rstr = 0.156 (cf. Fig. 2.6). Now,
using the above figure draw a trace at this Rp/Rstr value, as in the
following figure.
Figure 2.09. Domains for exoplanets and EBs for independent variables S and D
with a "threshold secondary" Rp/Rstr domain separater (thick red trace) at Rp/Rstr = 0.16,
corresponding to B-V = 0.66. The blue circle corresponds to the S and D location for XO-1.
From this graph it is immediately apparent that, subject to the
assumptions of the model, XO-1 is an exoplanet instead of an EB. This
conclusion does not require solving the LC for Rp/Rstr, as described
in Section 1. Indeed, this graph gives an
approximate solution for miss distance, m = 0.5 (not as accurate as the
solution in Section 1, but somewhat useful).
Here's a handy plot showing "threshold secondary" boundaries for other B-V values.
Figure 2.10. The thick red traces are "secondary
threshold" boundaries, labeled with the B-V color of the star, above
which is the EB realm and below which is the exoplanet realm.
Now
that you understand the concepts I can save you time by offereing an
Excel spreadsheet that does most of what's described on this web page.
The user simply enters LC depth D, LC length L, and star color B-V in the appropiate cells and the
spreadsheet calculates a 3-iteration solution for Rp/Rj (provided a
solution exists). Here's the link for the Excel spreadsheet that does
everything described in Section 1: RpIterationExcelSpreadsheet
Figure 4.01. Example of the Excel spreadsheet with XO-1
entries for several filter bands (B5:C8 for B-band, etc) and the Rp/Rj
solution (B10:B11 for B, etc).
The line for SE of "Rp/Rj solution" is based on changes in D, L and
B-V using their respective SE. In this example note that the Rp/Rj
solutions for all bands are about the same, 1.30. This provides a good
"reality check" on data quality as well as the limb darkening model.
Rows 13-15 show the SE on Rp/Rj due to the SE on B-V, D and L
separately. The largest component of uncertainty comes from B-V. Even
if B-V were known exactly there's an uncertainty in converting it to
star radius and mass, given that the "main sequence" of the HR diagram
consists of a spread of star locations and there's a corresponding
spread in the relationship bewteen radius versus B-V and mass versus
B-V.
A future version of this spreadsheet will include a section for the user to enter a transit shape parameter value, S,
and an answer cell will show the likelihood of S/D being associated
with an exoplanet versus an EB. I also plan on expanding the limb
darkening model to take into account limb darkening dependence on star
color.
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This site opened: March 3, 2007 Last Update: March 4, 2007