Kepler-138 (KOI-314)
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Shellface
So recently, I've been looking back through literature I skipped over some during the past year. I expect you folks will recall the confirmation via TTVs of two of the planets orbiting KOI-314 (now Kepler-138) in this paper from the HEK project. I've been looking over the result, but feel that it isn't fully adequate.
The lightcurve of the system contains a third transit signal, interior to the other two. It is far shallower than the others, corresponding to a 0.45 R⊕ planet compared to 1.61 R⊕ for the others. This makes it comparable in size to Mars. Though it was not possible to verify the planetary nature of the candidate in the relevant paper, the statistical work of Rowe et al. validated it as a real planet in the same system.
The period ratio of the small planet (b) and the inner large planet (c) is ~1.3355, which places it about 0.1% away from a perfect 4:3 resonance. The period ratio for the two large planets (c and d) is ~1.6754, about 0.5% from than a perfect 5:3 resonance. The period ratio for the inner and outer planets is ~2.2389, close to 20:9 (2.222…), but this happens to be similarly close to the lower order 9:4 (2.25) ratio. This would make the expectation that all three planets are interacting with each other to a moderate degree, with interactions being more dominated by the two larger planets.
In Kipping et al., the contribution of the innermost planet's mass to their solution is ignored because it must be small; sensible densities for the planet yield m ≈ 0.1 M⊕, which is about an order of magnitude smaller than the expected masses of the other two. However, they do briefly study its TTV diagram, with its representation in figure 7 apparently modelling two sinusoids at the expected TTV periods for its period ratios with c and d. The fit seems decent, but is hampered by the irregular precision of the transit times, presumably due to the small radius of the planet.
Now, I can't claim to understand how TTV modelling works very well. But, I understand that the TTV amplitude contains information about the mass of the relevant peturber. Without knowledge of the reciprocal TTV signal in the other component this, if I understand correctly, does not give the mass of the object through which TTVs are observed, but for thise case, given the displayed precision, I expect that meaningful upper limits on b's TTV signal on d and perhaps c can be obtained, if not marginal detections. This would provide either a mass limit or a tentative mass for b - which is, again, Mars-sized - and such a measurement could be very valuable considering the difficulty of measuring the masses of sub-Earths at the current epoch.
If somebody understands the equations better than I do - is this plausible for this system? It sure seems like a potential opportunity.
I suppose, regardless of the outcome of the above, it would be useful to study the extended lightcurve of the star for TTVs, as this would result in more up-to-date values for the masses of c and d.
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ajamyajax
in response to Shellface's comment.
Coincidentally, I asked Joey for some tips about this after his recent paper:
"Planet Hunters VII. Discovery of a New Low-Mass, Low-Density Planet (PH3 c) Orbiting Kepler-289 with Mass Measurements of Two Additional Planets (PH3 b and d)"
http://arxiv.org/abs/1410.8114
He told me they (Eric Agol and Kat Deck) used TTVFast to derive the masses, and also kindly provided me with these links:
http://arxiv.org/abs/1207.4192 "provides an analytic way to calculate a 'nominal' mass, which in most cases, should be a decent estimate of the mass."
Section 2 of this paper http://arxiv.org/abs/1309.2329 "does it in a more condensed and somewhat clearer way."
"TTVFast: An efficient and accurate code for transit timing inversion problems" http://arxiv.org/abs/1403.1895
Hope this helps. (And Joey, hope you don't mind a few quotes here.)
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Shellface
Yeah, that helps some, thanks. Using the equations for the TTV amplitudes I recover the approximate values for c - d (0.003, 0.02d). Using an estimate for b's mass (0.08 M⊕, which is towards the highest sensible value), I find that b - d interactions are negligible for their respective errors, but b - c interactions are larger (0.02, 0.0007d). c's signal on b should be the trend observed in Kipping et al., which can therefore give a second value for c's mass. b's signal on c is comparable to the errors on its individual transit times; importantly, the TTV period for both planets' interactions with c have similarly long periods, so only modelling one as has been done will give an innaccurate mass for d by perhaps ~30%. This doesn't reconcile d's anomalously low density, but it should make it less severe.
Because short cadence observations were started relatively late, the data precision is somewhat imperfect for the first ~half of the dataset. This is particularly poor for c, as the minimum that occurred during LC observations can be complemented by the minimum that occurs at the end of observations, which could detect b's signal - but with such high scatter, this may not be possible. Nevertheless, an analysis of the full lightcurve is greatly desirable. aja, would you mind taking a look at some point?
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ajamyajax
in response to Shellface's comment.
I'll try SF, but like the BEER algorithm stuff it could be a while until I can make it work. And I'm sure you will figure this out in the meantime.
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Shellface
in response to ajamyajax's comment.
Hey now, I can only ask you to study the transit times - you've shown me well enough that you know how to derive them. I don't know if it would be too much to apply that to a full lightcurve with so many transits, but it would be fantastic if you could. But please don't feel obligated.
I can hopefully deal with the maths on my own, even if it's at the end of my abilities. (how can there be an imaginary component of eccentricity oh my)
(And hey, I wasn't trying to ask you to model the phase curve, only look for an eclipse! Turns out I can do that quick enough, anyway)
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ajamyajax
in response to Shellface's comment.
Aw Man, maybe you didn't realize this, but it takes hours to create those TTV transit times.. The reason why is this: I need to inspect every single transit closely (and yes I started writing a program for that once, but too many exceptions to make it practical). So I must ask is this project that important to you? Tell you what, if you post the TTV times here either from your own inspection or as data from someone else's paper, I will run them through my O-C or other plots that might be of interest. And also by the way, that p=10.3x transit is really blended in at times, so estimates would need to be derived for those as well. Anyway, you asked. And as I've said before on this board, you sometimes get what you pay for. ; )
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Shellface
in response to ajamyajax's comment.
You do it all manually? @_@
Though I was unable to access the star's lightcurve previously (for some reason), looking at it now the precision isn't as good as I hoped (though the binned SC data is rather better), so I could see this being… difficult. But if I understand your methodology correctly, I could probably do this - it'd take a long time because there's something like 400 transits in all, but I'll be damned if a computer can out-pattern-recognise me!
Anyway, interesting as this is, I'll probably leave it for a while. I've been planning on a moderate-scale study of phase variations of Hot Jupiters in K1 data, to expand on what I did with those two K2 binaries down to actual planets. But after that, I'll see about this peculiar little system.
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Shellface
Before I could even get around to doing what I was planning, someone takes the rug out from under me! Jontof-Hutter et al. have done… basically everything I could have suggested, and measured the mass of b, like I suggested; though not to very high precision, it is confidently non-zero. Excellent to see that all carried out!
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