# Calculating Secondary Eclipse Depth

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WBY
Calculating Secondary Eclipse Depth

I may have captured a secondary transit (eclipse of exoplanet by host star) of a known exoplanet. I am tying to determine if that is a reasonable hypothesis. Perhaps one of the followers of this forum can advise how one might go about determining if the secondary transit hypothesis is within reason, and how one would make such an estimate, if possible, given the known information.

The midpoint of the transit event I observed at 12/31/2019 06:46:21 UT was within 1-segma error of the predicted 0.5 phase given by the NASA  Exoplanet archive transit tool (21/31/2019 03:12 UT +/- 3h 37m 08s). The transit depth I measured was 0.23% if I include limb darkening. If I set limb darkening to zero, which I think would be the case for a secondary transit, the depth changes to 0.20% and the goodness of fit parameters remain almost exactly the same as those with limb darkening included.  In both cases I used the orbital period for the known exoplanet (~3.337d). These transit depths are somewhat less than half the observed primary transit depth of 0.58%. The known exoplanet orbits very near the host star which has been classified as M2.0Ve. in Simbad. The orbit of the known exoplanet is close to circular. I have no idea how to go about estimating a secondary transit depth from the known information for this star-planet system or whether it is possible to do so since I have no idea of the planet's albedo. Someone with much more complete knowledge of planetary science may have know what a reasonable range of albedo may be.

Summary info for the star and known exoplanet are attached. .

Thanks,

bpablo
Not a secondary eclipse (probably)

The short answer is that it is theoretically possible, given how strange this particular system is, but I don't think it's likely . Let's assume that the planet is completely reflective so it has exactly the same temperature. That means the luminosity (4*pi*R^2*T^4) difference will be dependent on the ratio of the two radii.

Rp = 0.37 Rj or 0.037 Rs

Rs = 0.48

(Rp/Rs)^2 = 0.006

This means that the total contribution of the secondary.

0.006/(1.+0.006)  or 0.0059

meaning that even if the planet was just as hot as the star  it would contribute less than 0.6 percent of the system light and that's all that will be gone when it is eclipsed. If you put a reasonable temperature for a hot Jupiter which on the high end appears to be around 1400K, then

fracL = (Rp/Rs)^2*(Ts/Tp)^4 = 0.00012 or 0.01 %

so the secondary would contribute

0.00012/(1+0.00012) or 0.000119

so the planet likely contributes less than 0.01 % to the luminosity of the system and that is what would be missing when it is eclipsed. Secondary transits are almost never seen, even in Kepler data, and would be virtually impossible to see from the ground except in the most extreme of circumstances. However GJ 3470 is a weird system with a hot neptune that is so hot it's evaporating mass at a rate never before seen. Based on the transit depth though of 0.23 %  the temperature would need to be around 2859 K, which is 1000 degrees hotter than normal. I would say it's more likely to be a second planet or an artifact, than a secondary eclipse, but it's definitely worth follow up observations.

Thanks,

Bert Pablo

Staff Astronomer, AAVSO

WBY
Not a secondary eclipse (probably)

Thanks, Bert.

I follow the logic. I was not sure if there could be additional contribution from albedo.I had a vague recollection from the Princeton University Coursera exoplanet course several years ago that when all the incident radiation from the star was not absorbed by the planet and its atmosphere, the reflected light added to the thermal radiation from the planet.

There are several things that lead me to think this could be a secondary transit

1. the mid transit timing and transit duration are right (but the duration is strongly affected by the assumed orbital period.

2. The transit depth assuming limb darkening coefficients are zero, which seems to be a reasonable assumption for a secondary transit, is about 0.2%one-third of the primary transit depth for this exoplanet or about 0.2%, which I thought might be possible at least it meant the planet is cooler than the star

3. Because this planet is so close to the host star with an orbital radius of only 0.036 au - about 1/10th of Mercury's it seemed possible that it might be close to the temperature of the star and/or there could be hot gas around the star contributing to radiation from the "planet."and if the gas is hot enough, it would not have a commensurate dimming effect during primary transit. I also thought their might be some additional contribution from reflected light.  Without those two additional effects, the surface of the planet facing the star would have to e about 76 % of the temperature of the star. -(1/3)^0.25. That would put the temperature of the planet's radiating surface at about 2670 degrees. I didn't know if that made sense or not.

Am I wrong about an additional contribution from reflected light when all of the incident radiation from the star is not absorbed? Of course, in equilibrium that means the planet should be cooler than the star. If I am not mistaken about reflection,  Is the addition just the intensity of the star's radiation at the planet's orbital radius times the surface area of the planet's disk times the albedo?

WBY
Additional secondary eclipse depth from reflected light

If I am correct that one should ad the energy reflected by the planet to the thermal energy radiated by it, and assuming geometric albedo of 0.2, I arrive at the reflected light adding 0.046 x primary transit depth to the secondary eclipse depth one derives from the planet's thermal radiation alone. That isn't a huge difference but neglecting other factors contributing to flux from the planet that I can't identify, it would reduce the required planet radiating surface temperature to about 2550 K. As you pointed out this planet is hotter than the normal hot Jupiter.

My calculation is attached. It is a bit complicated since the planet is close to the star and one can't use the simple inverse square law you apply to point sources.

bpablo
More correct but..

My assumption was that the planet itself radiated like a star and if so that is the amount of light it would put off. It however, would never do that on it's own. That is a gross over-simplification. In reality the light that came from the planet would be as you described, reflected light from the star (there is some heating as well but we tend to blend them into the same effect). In that case the answer would be closer to what you derived, but then the actual planet source heat would be negligible as it will likely never reach the temperatures I mentioned. The estimates for this particular planet (which as far as I can tell is the hottest known) is ~1700 F or 1400 K. Like I said, I was giving the simplest calculation possible which gave the planet the best possible case and it is still barely possible. I still don't think a secondary transit is very likely, but if you can reproduce it in further observations then I may become less skeptical.  Part of my skepticism lies in the fact that I don't believe a secondary exoplanet transit has ever been observed from the ground.  If it does turn out to be real, it would almost certainly be a huge deal. If you can confirm this, would you like me to help you find an exoplanet person to work with?  I am positive someone would be interested in this.

Bert Pablo
Staff Astronomer, AAVSO

WBY
Secondary Eclipse of Hat-P-11 b as Comparison

Hat-P-11 b has very similar characteristics of orbit and planet size to GJ 3470 b. The planet is extremely close in size. orbital semi-major axis is about 1.5x Gj 3470 and its host star is about 1000 K hotter. The attached Nov. 2016 A&A Manuscript article about the discovery of  the Hat-P-11 b secondary eclipse puts the possible event I observed at roughly 300x the secondary eclipse depth one would expect to observe.

That doesn't seem very probable. Unforntunately we are expecting bad weather for the rest of the week and I won't be able to observe the star at the next couple of orbital periods after my observation but I will look as soon as weather allows.

pablotwa
Depth Variations in Different Passbands

Hi Brad. Have you read the "Exoplanet False Positive Detection with Sub-meter Telescopes" by
Dennis M. Conti, PhD?

There's a section there on Depth variations on different passbands that may applicable to your situation (or not)...anyway I attached it here just in case.

Pablo

WBY
Depth variations in different passbands

Thanks for the suggestion Pablo,

I have read it, and even more remarkably, I remember a lot of it. The nearest star in Gaia Dr2 is approx 30 arcsec away with g mag 20.9. The host star is at g mag 11. 35 There are no nearby stars to contaminate my measurement. That doesn't mean some other "artifact" (such a pleasant, innocuous word for "screw-up") didn't cause a false positive.  This is pretty faint stuff for amateur observing, ~2.5 millimags. I am a journeyman photometrist in the trade rankings from apprentice to master but I am fortunate to have access to an observatory with outstanding equipment - a 24"  custom RC on a massive equatorial fork mount with a Princeton Instruments PIXIS 2048B eXcelon camera located out in the boonies west of Waco, TX. When the weather cooperates we can do some high precision photometry that tends to make club members using the observatory seem much more accomplished  than we are. Some nice pix of the observatory here:    https://www.centexastronomy.org/ .

NASA exoplanet archive transit depth for GJ 3470 b is  given as 0.58% from the Awiphan, et al paper https://arxiv.org/pdf/1606.02962.pdf. That appears to be the transit depth in z'. The transit depth in i' appears to be a bit deeper and a bit shallower in Rc. My observations were using a Bessel formula I filter (not an Ic).. Therefore, I expect that my observed transit depth would be similar if it were an observation of the transit of the same planet.  The midpoint of the observed (potential) event agreed within 1-sigma error to the secondary eclipse time predicted on my observation date  which suggested this might be an occultation of the planet by the star. However, unless their is some very large effect due to the rapid boiling off of the star's atmosphere, which I cannot assess, the depth of the possible event is much too large. Comparisons are secondary eclipse depth of Hat-P-11 b mined from Kepler data https://arxiv.org/pdf/1611.00153.pdf and ground observations of Wasp-19 b https://arxiv.org/pdf/1303.0973.pdf suggest the secondary eclipse depth would be a couple percent of the depth I observed.

I need to make additional observations at times spanning the midpoint of my observation plus multiples of the orbital period of GJ 3470 b. . Weather is not cooperating, however. This is our worst time of year  for observing based on cloud cover.