TRANSIT DEPTH, TRANSIT LENGTH, ORBITAL PERIOD AND STAR COLOR
Bruce L. Gary; 2007.03.04, updated 2010.10.22

This 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.
INTRODUCTION

This web page describes a simple model that I developed for converting a transit light curve (LC) to size of the secondary, which is then used to discriminate between the secondary being an exoplanet versus an eclipsing binary (EB) star. A "concept description" section uses actual R-band measurements of an exoplanet to illustrate how a LC can be interpreted. The "model" employs limb darkening relationships for each filter band. The primary star's B-V color (closely associated with spectral type) is used to derive the star's mass and radius, on the assumption that it's a main sequence star (like ~90% of stars). Oribtal period is used to calculate orbital velocity (assuming a circular orbit). The planet's radius and "central miss distance" (related to inclination) are adjusted to match the LC depth and duration.
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.

Section 1 is a case study that is used to illustrate the concepts employed. Far more steps are shown than would be used in practice. The goal for this section is to determine the size of the secondary (exoplanet or EB binary star).

Section 2 shows how to use information about the LC's shape to assess whether the LC is compatible with an exoplanet or an EB whose light is blended with a nearby star to produce what merely appears to be a small transit depth.

Section 3 is a summary of only those things that need to be done, after the underlying concepts are understood, to convert the basic properties of a LC and star color to a solution for secondary size and liklihood of the transit belonging to an exoplanet versus an EB.

Section 4 describes an Excel Spreadsheet that can be downloaded and run to do just about everything described on this web page. The user enters transit depth, transit length period and star color (B-V) in cells corresponding to the LC's filter band and a cell displays a 3-iteration solution for Rp/Rj (if a solution exists). It also can be used to assess the liklihood of the LC shape belonging to the exoplanet domain, based on the user's input of a shape parameter, S.

1.0 CONCEPTS OF LC INTERPRETATION - A CASE STUDY

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:

GIVEN:
B-V = +0.66 ± 0.05
orbital period, P = 3.9415 days,
R-band observations
transit depth, D = 23.7 ± 0.4 mmag,
transit length, L = 2.97 ± 0.03 hours (contact 1 to 4).

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.

SOLUTION:

Rstr = 2.23 - 2.84 * (B-V) + 1.644 * (B-V)^2 - 0.285 * (B-V)^3

Planet radius, Rp/Rj = 1.41 (1st iteration)
Rp/Rj = 9.73 * Rs * SQRT [1 - 10 ^(-D/2500)], which assumes central transit and no limb darkening

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)
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 (also called "impact parameter"), b = 0.42 (ratio of closest approach to center divided by star's radius)
b = SQRT [1 - (L / Lx)^2]

Limb darkening effect, LDe = 1.16 (divide D by this)
I(b)/I(av) = [1 - 0.98 + 0.15 + 0.98*c - 0.15*c^2] / 0.746, for B-band
"       = [1 - 0.92 + 0.19 + 0.92*c - 0.19*c^2] / 0.787, for V-band
"       = [1 - 0.85 + 0.23 + 0.85*c - 0.23*c^2] / 0.828, for R-band
"       = [1 - 0.78 + 0.27 + 0.78*c - 0.27*c^2] / 0.869, for I-band

where c = SQRT(1 - b^2) (=  cosine(theta), and theta is angle between line-of-sight and star radius line to location of nearest approach of exoplanet to star center)
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 b = 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, b = 0.405
(Same eqn as above)

Limb darkening correction, LDe = 1.165 (divide D by this)
(Same eqn as above)
No more iterations are needed since the two miss distance results (& limb darkening corrections) are the same. We have a stable solution:
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.".

EVALUATING THE SOLUTION

How good is this result? Let's compare it with a detailed model-fitting analysis by professional astronomers. The "mystery" exoplanet is no mystery. It's XO-1, whose discovery was announced May 18, 2006 and published in the September issue of the Astrophysical Journal (McCullough, P.R., Stys, J. E., Valenti, J. A., Johns-Krull, C. M., Janes, K. A., Heasley, J. N., Bye, B. A., Dodd, C., Fleming, S. W., Pinnick, A., Bissinger, R., Gary, B. L., Howell, P. J., Vanmunster, T., "A Transiting Planet of a Sun-like Star", Astrophys. J., 648, 2, 1228-1238, September 2006; http://xxx.lanl.gov/abs/astro-ph/0605414 (abstract) and http://arxiv.org/PS_cache/astro-ph/pdf/0605/0605414.pdf (complete article).

This article reports that Rp/Rj = 1.30 +/- 0.11, which compares well with the simple model result calculated here, of Rp/Rj = 1.31 ± 0.07. (The larger SE for the professional result reflects a  realistic assessment of such systematic uncertainties as converting B-V to stellar radius and mass.)

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 ("b") 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 b < 0.73 (coutesy of Cindy Foote, 2007.03.04).

Figure 1.06. Transit depth is greatest for R-band, consistent with miss distance b > 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.

As an additional check the shape of the measured transit light curve can be compared with a model calculation. First, let's consider LC shapes for various sized secondaries (either an exoplanet or EB star) transiting through the center of the star they orbit. The following figure was derived from a model that used sun-like R-band limb darkening.

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

The following figure shows the dependence of "threshold secondary" Rp/Rstr versus B-V.

Figure 2.06. Relationship of "threshold secondary" Rp/Rstr versus B-V.

In order to use the above figure to distinguish between exoplanet versus EB shapes we need to take into account the primary star's color. For example, if B-V is sun-like, we can draw a vertical line at Rp/Rstr = 0.15 and consider everything leftward to be exoplanets and everything rightward to be EBs. Similarly, for any other B-V a vertical line can be placed upon this figure to show the domains where exoplanets and EBs are to be found, as the following figure illustrates.

Figure 2.07. Domans for distinguishing exoplanets from EBs based on B-V, shape parameter S, and miss distance b, for two examples of B-V. The blue circle in the left panel is located at the measured shape and center miss distance for XO-1.

Since XO-1 has B-V = 0.66 ± 0.05 we can use the left panel to determine that it must be an exoplanet. This determination is based on the shape parameter, S, and the miss distance that was determined from Section 1 (plus the B-V color for XO-1). Even if we hadn't performed a solution for miss distance we could say that's it was likely that the B-V and S information was in the exoplanet domain. If S were slightly smaller, say 0.27, then there would be no dispute about the light curve belonging to an exoplanet. (Well, all this is subject to my model assumtions, such as the "main sequence" one.)

There's another graph that can be used for the same purpose as the previous ones, and I think it's much more useful than the graphs in the previous figure because it doesn't require knowledge about miss distance. Instead, it requires knowledge about transit depth, D, which is easily measured.

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.

<>This figure allows a quick assessment of a LC's association with an exoplanet versus an EB. If the LC is an EB blend, such as the triplet case described by Mandushev et al (2005), there may not be a "solution" using either the above figure or the analysis of Section 1. To assist in evaluating this it is helpful to have transit light curves for more than one filter band.

Again, this procedure is only as good a guide as the underlying assumptions, the principal one bring that the star undergoing transit is on the main sequence.

3.0 SUMMARY OF TRANSIT LIGHT CURVE ANALYSIS

Much of the preceding was meant to show the underlying concepts for quickly evaluating a transit LC. It may have given an unfair impression of the complications involved. This section will skip the explanations for "why" and just present a sequence of what to do, like a cookbook. I'll repeat the figures that are needed below the list of steps.

1) Determine the candidate star's B-V
2) Use the measured LC to determine transit Depth and Shape parameter, D and S.
2) Using D and P determine if the LC is likely to be an exoplanet, or EB, or neither
(cf. Fig. 3.01)
3) If the LC is for an EB, no more analysis is needed. If it's an exoplanet, then proceed.

4) Use the Excel spreadsheet (link below) to convert B-V, D, L and filter band to Rp/Rj and miss distance,

OR, do it manually by following the steps below...

4) Determine the star's radius, Rstr, and mass, Mstr, from B-V (cf. Fig. 3.02)

5) Calculate 1st iteration of Rp/Rj, using following equation:

Secondary size, Rp/Rj = 9.73 * Rstr * SQRT [1 - 10 ^(-D/2500)]

6) Calculate the secondary's orbital velocity, central transit length and miss distance using these equations:
Planet orbital radius, a = 1.496e8 * [Mstr^1/3 * (P / 365.25)^2/3], where P[days], Mstr[sun's mass] & a[km]
Transit length maximum, Lx = (Rstr * Rsun + Rp/Rj * Rj) / ( pi a / 24 * P) where Rsun = 6.955e5 km, Rj = 7.1492e4 km
Miss distance,  b = SQRT [1 - (L / Lx)^2]

7) Using the miss distance and filter band, determine limb darkening effect, LDe (cf. Fig. 3.03)

8) Convert the measured transit depth D to a value that would have been measured if there were no limb darkening, using the following eq:

D' = D / LDe

9) Repeat steps 5, 6 and 7 using D' instead of D.

10) If step's 7 LDe is the same as the 1st time, then there's no need for additional iterations. The last calculated Rp/Rj is the answer. Otherwise, repeat steps 5, 6, 7 and 8 until a stable solution emerges.

Figure 3.01. If the D/S location for the LC is above the red line corresponding to the star's B-V, then it's probably an EB. If D/S is below then it's probably an exoplanet. If it is to the left of the upward sloping trace (central transit), then there's no solution, and you may be dealing with an EB blending or triple star system.

Figure 3.02. Star's radius and mass from B-V.

Figure 3.03. Limb darkening effect, LDe, versus transit miss distance and filter band.

This completes the summary of what is done to assess a transit LC to determine if it's due to an exoplanet or EB, and if it's an exoplanet to determine its size. This web page's purpose has been to demonstrate that a simple procedure can be used to guide the choice of survey candidates for a night's observing in order to avoid spending time on unlikley (EB blend) candidates.