Transit timing analysis of the exoplanets TrES-1 and TrES-2

Transit timing analysis of the exoplanets TrES-1 and TrES-2

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The aim of this work is a detailed analysis of transit light curves from TrES-1 and TrES-2, obtained over a period of three to four years, in order to search for variabilities in observed mid-transit times and to set limits for the presence of additional third bodies. Using the IAC 80cm telescope, we observed transits of TrES-1 and TrES-2 over several years. Based on these new data and previously published work, we studied the observed light curves and searched for variations in the difference between observed and calculated (based on a fixed ephemeris) transit times. To model possible transit timing variations, we used polynomials of different orders, simulated O-C diagrams corresponding to a perturbing third mass and sinusoidal fits. For each model we calculated the $\chi^2$ residuals and the False Alarm Probability (FAP). For TrES-1 we can exclude planetary companions ($>$ 1 M$_{\oplus}$) in the 3:2 and 2:1 MMRs with high FAPs based on our transit observations from ground. Additionally, the presence of a light time effect caused by e. g. a 0.09 M_Sun mass star at a distance of 7.8 AU is possible. As for TrES-2, we found a better ephemeris of T$_c = 2 453 957.63512(28) + 2.4706101(18) \times$Epoch and a good fit for a sine function with a period of 0.2 days, compatible with a moon around TrES-2 and an amplitude of 57 s, but it was not a uniquely low $\chi^2$ value that would indicate a clear signal. In both cases, TrES-1 and TrES-2, we were able to put upper limits on the presence of additional perturbers masses. We also conclude that any sinusoidal variations that might be indicative of exomoons need to be confirmed with higher statistical significance by further observations, noting that TrES-2 is in the field-of-view of the Kepler Space Telescope.

Using a common procedure to study orbital variations in CV variables, W-UMa type, Algol etc. eclipsing binary, they would have detected hints of a low mass stellar companion, above enough hydrogen-deuterium limit (0.09 Solar masses or 90 Jupiter masses) at 7.8 AUs.
If real, interesting new questions about planet formation in close binary systems.

Moreover there would also hints of an exomoon orbiting TrES-2 b, but minimum mass limit seems oddly great for a satellite, 52 M_Earth with possible orbital period of 0.2 days (if I have read correctly, of course. On the contrary feel free to correct me).

The wording of the abstract is kind of confusing, especially their use of the word "possible", which could be interpreted as "slightly suggested by the data", or "not ruled out."

According to the paper, they determined that the maximum moon mass that could exist for TrES-1 is ~10^-6 Earth-masses, and ~10^-7 Earth-masses for TrES-2. The 52-Earth mass moon detection thus seems spurrious.

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This upper mass limit is clearly way below our detection limit for masses causing transit timing variations in the system TrES-1 and TrES-2. The closest moon at this upper mass limit would cause a timing amplitude of the order of 10^-6 s, which is not detectable even from space by several orders of magnitude.

The very low mass limits there are those suggested by predictions of tidal theory, which generally use a Q value for the planet of 10⁵. Whether this is appropriate for hot Jupiter satellites is another matter.

The 52 Earth mass is derived from a low-significance signal from the data. Seems rather large for a satellite though.

Unless it may be a trojan exoplanet or even a binary planet?? Just weird

Remember that both TrES planets transit across their host star. If it were a binary planet or 52-Earth-mass trojan exoplanet, it would have been detected.

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Remember that both TrES planets transit across their host star. If it were a binary planet or 52-Earth-mass trojan exoplanet, it would have been detected.

By a more complex light curve.

I made some calculations with the data given in the paper and I got these tables.

TRES-1
Upper mass limit	10-6 M Earth	10-6 M Earth
Period (range)	4.2 hours	15 hours
Semi-major axis	77,568.28 km	181,236.43 km
Minimum density to avoid tidal destruction	1.222	0.096
Maximum diameter	210.545 km	491.933 km

TRES-2
Upper mass limit	10-7 M Earth	10-7 M Earth	10-7 M Earth
Period (range)	3.9 hours	"best peak": 4.8 hours	12 hours
Semi-major axis	92,482.06 km	106,212.4 km	195,644.93 km
Minimum density to avoid tidal destruction	1.426	0.94	0.15
Maximum diameter	92.809 km	106.647 km	196.336 km

All this indicates that these moons are very small, the largest diameter possibility is near the size of Vesta. The moons would be not spherical, but would have the shape of an asteroid. More particularly, these moons would represent the larger versions of the Pluto's moons Nix and Hydra.
Plus, if it's confirmed, it shows that we are able to find moons with that size and, more generally, we are able to detect Earth-like moons. But the range of semi-major axis is quite big, it's not impossible that the effect is caused by two or three moons instead of one. Returning to the case of TRES-1 and TRES-2, these planets have no rings (I don't think that rings can exist with a semi-major axis less than 70,000 km).
The paper shows that a similar event happened in 2002 with the planet HD 209458 b.

About the burning question of the tidal lock, some of you says that the moons of planets very close to their stars would be tidally locked to the host stars. But in the Hill Sphere of the planet, the gravity of the planet is higher than the gravity of the star, so the moons would tend to be tidally locked to the planet instead of the star.

At this time, all the things I put here are my espectations about the possible moons of TRES-1 and TRES-2.

Bye

Sedna

PS: Does somebody know how to put the HTML code option ON ?

I do not succeed in understanding. Were these moons discovered or are them only hypothetical? And how one they discovered? Can you give me a paper concerning the moon of HD 209458 b?

Sedna wrote:

Plus, if it's confirmed, it shows that we are able to find moons with that size and, more generally, we are able to detect Earth-like moons.

Correct me if I am wrong, but there isn't anything to be confirmed regarding the existence of a moon to either TrES-1 b or TrES-2 b. The moon masses you and the paper describe (10^-7, 10^-6) are theoretical upper mass limits to hypothetical moons, as opposed to detected moons. The paper makes it quite clear that it's more of a "If any moons exist, they have no more than this mass." They do not report the detection of any moons compatable with that mass.

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This upper mass limit is clearly way below our detection limit for masses causing transit timing variations in the system TrES-1 and TrES-2. The closest moon at this upper mass limit would cause a timing amplitude of the order of 10^-6 s, which is not detectable even from space by several orders of magnitude.

(Emphasis mine)

In summary, the paper says that there are no detected timing variations compatable with a moon of any realistic mass. Thus, no detected moons.

Stalker wrote:

I do not succeed in understanding. Were these moons discovered or are them only hypothetical? And how one they discovered? Can you give me a paper concerning the moon of HD 209458 b?

The moons are purely hypothetical, and there is no paper reporting the discovery of any moon, of any extrasolar planet (hot or cold).

Sedna wrote:

About the burning question of the tidal lock, some of you says that the moons of planets very close to their stars would be tidally locked to the host stars. But in the Hill Sphere of the planet, the gravity of the planet is higher than the gravity of the star, so the moons would tend to be tidally locked to the planet instead of the star.

I agree.

I also agree with the idea that either of these planets have no rings. The paper tells of objects of low mass (for which ring particles certainly below to) get evaporated away pretty quickly.

Sedna wrote:

PS: Does somebody know how to put the HTML code option ON ?

Perhaps this post will clear up some questions. If not, that thread would be the place to discuss it.

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The wording of the abstract is kind of confusing, especially their use of the word "possible", which could be interpreted as "slightly suggested by the data", or "not ruled out."

Okay, I've fallen into the trap, I read the paper too quickly. After reading your comments, I read the paper slowly and saw that it's only simulations. But my post is not useless, I just made a more detailed portrait.

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Can you give me a paper concerning the moon of HD 209458 b?

There's no paper about that moon, it was just indicated in a sentence in the paper.

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which is not detectable even from space by several orders of magnitude.

Yes, but why people said that Kepler will or would be able to detect exomoons ?

About the HTML code, I understand why I cannot put HTML lines directly in the answer box. Thank you for your help.

Bye

Sedna

Well going to the formula in the paper referenced for tidal limits, for TrES-2 I get a moon mass limit of 10^-6 Earth masses for Q_p=10⁵. On the other hand the paper I referred to above suggests that for hot Jupiter satellites, Q_p values of ~10¹² might be more appropriate, which boosts the limit to about 10 Earth masses for this planet.

Sedna wrote:

Yes, but why people said that Kepler will or would be able to detect exomoons ?

Possibility of a photometric detection of "exomoons"
http://arxiv.org/abs/astro-ph/0601186

On the detectability of habitable exomoons with Kepler-class photometry
http://arxiv.org/abs/0907.3909

Abstract wrote:

We find that habitable-zone exomoons down to 0.2 Earth masses may be detected and ~25,000 stars could be surveyed for habitable-zone exomoons within Kepler's field-of-view. A Galactic Plane survey with Kepler-class photometry could potentially survey over one million stars for habitable-zone exomoons. In conclusion, we propose that habitable exomoons will be detectable should they exist in the local part of the galaxy.

I find myself able to produce HTML tables in the fast-reply box.

Lazarus wrote:

Well going to the formula in the paper referenced for tidal limits, for TrES-2 I get a moon mass limit of 10^-6 Earth masses for Q_p=10⁵. On the other hand the paper I referred to above suggests that for hot Jupiter satellites, Q_p values of ~10¹² might be more appropriate, which boosts the limit to about 10 Earth masses for this planet.

A super-Earth moon around such planets, isn't that too big ? I don't think that such moons may be stable with a semi-major axis less than 350,000 km.

About Kepler, I already saw the two paper, but the news came out in french only a few days ago (why such a difference ?), so I didn't pay attention before. But, if Kepler is able to detect exomoons, that's good news.

For HTML, I just need to adapt the code with the rules of this forum and it will work.

Sedna wrote:

A super-Earth moon around such planets, isn't that too big ?

It's too big under the assumption that Q_p = 10⁵.
It's okay under the assumption that Q_p = 10¹².
(where Q_p is the tidal dissipation parameter, and is a quantization of how efficient the planet is at dissipating tidal energy).

Ultimately, it comes down to what value of Q_p better fits these planets, and this isn't a value that is easy to measure. It's commonly assumed that Q_p = 10⁵, but if it isn't, then that changes the limit as to what the maximum moon mass can be.

Of course there's also the issue of whether such a moon could form in the first place... theoretical predictions suggest that the maximum moon mass is a few ×10^-4 of the mass of the gas giant, but then again there is no observational evidence for scaling ratios of exomoons around exoplanets. There's also the possibility of capturing a terrestrial planet.

Sirius_Alpha wrote:

It's too big under the assumption that Q_p = 10⁵.
It's okay under the assumption that Q_p = 10¹².
(where Q_p is the tidal dissipation parameter, and is a quantization of how efficient the planet is at dissipating tidal energy).

Ultimately, it comes down to what value of Q_p better fits these planets, and this isn't a value that is easy to measure. It's commonly assumed that Q_p = 10⁵, but if it isn't, then that changes the limit as to what the maximum moon mass can be.

Sorry if you think that I am stupid after reading this, but I cannot understand the term of tidal dissipation, even though I try to get it.

Lazarus wrote:

Of course there's also the issue of whether such a moon could form in the first place... theoretical predictions suggest that the maximum moon mass is a few ×10^-4 of the mass of the gas giant, but then again there is no observational evidence for scaling ratios of exomoons around exoplanets. There's also the possibility of capturing a terrestrial planet.

Moons can form when the planet is already on a close orbit, but it's possible if the accretion disk is not completely wiped out by the stellar wind.
About the capture of a rocky planet, I think there are six possible scenarii:
1: The gas giant crosses the orbit of a rocky planet but that planet is too far to be captured.
2: The rocky planet enters the Hill Sphere of the gas giant but it orbits too quicky and it escapes like NEOs.
3: The rocky planet enters the Hill Sphere but crashes on the gas giant.
4: The rocky planet is succesfully captured but when the gas giant comes closer to its star, the captured planet escapes because the Hill Sphere of the gas giant is shrinking.
5: The rocky planet succesfully captured and cannot escape but when the gas giant is close to its star the orbit of the moon is unstable and crashes on the planet.
6: The gas giant has captured the rocky planet, close enough so that it cannot escape and the orbit is stable.

Looking at this, the likeliness for a gas giant on a close orbit to have a "big" moon is quite small.

Bye

Sedna

Sedna wrote:

Sorry if you think that I am stupid after reading this, but I cannot understand the term of tidal dissipation, even though I try to get it.

No not at all. I found it difficult to grasp tidal forces too. Tidal dissipation is how easy it is for a planet to get rid of tidal energy without it affecting the orbit of the moon.

As a moon orbits around a planet, it raises a tide on that planet like the Moon does on Earth. Earth's rotation will carry this bulge ahead of the sub-lunar point (the place on Earth directly below the moon).


Since that bulge on the primary is ahead of the sub-lunar point, the bulge's gravity will pull on the moon ever so slightly. Since the bulge is ahead of the moon, the moon will be gravitationally pulled in the direction it is already orbiting around Earth. This causes it to speed up a little, and an increase in speed translates to an increase in altitude (orbital dynamics).

Earth is pretty solid being a terrestrial planet. The tide raised on it by the moon won't go away too easily. What really helps dissipate this tidal bulge on Earth is the oceans, because they're easily movable and can flow. The more efficiently a planet can get rid of this bulge, the less the bulge will be able to affect the moon.

Considering a terrestrial planet without oceans, such a planet will have more difficulty getting rid of this tidal bulge because the solid nature of the planet doesn't allow the tidal bulge to be dissipated as easily as it would if the planet were made of gas. As such, gas planets are much more efficient at getting rid of the tidal bulges caused by their moons than terrestrial planets.

In the case of a hot Jupiter, which will certainly be tidally locked to the star, any stable satellite orbital periods will be shorter than the rotation period of the planet. This will cause the tidal bulge to drag behind the moon, lowering its orbit instead. Now tidal dissipation here determines how quickly the moon will fall into the planet, instead of how long it will take to escape (like at the Earth/moon system).

The precise reason for why, (and by extension, how to measure Q_p) is a topic of continuing research. Quoting from the paper Lazarus linked to earlier,

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The physical origin of the tidal Q_p in gas giant planets has been an outstanding question since at least the 1970's. The Q_p for Jupiter is constrained to be Q_p ~ 10⁵ - 10⁶ in order that Io's orbit expanded into the Laplace resonance. Early theoretical work by Hubbard (1974) showed that the turbulent viscosity generated by convective eddies required for the outward transport of heat would give rise to Q_p ~ 10⁵, perhaps explaining the observed value. Goldreich & Nicholson (1979) then pointed out that turbulent eddies in the planet have long turnover rates, compared to the forcing periods of interest, severely decreasing the turbulent viscosity used in Hubbard's calculation. Goldreich & Nicolson (1977) estimated Q_p ~ 10¹³ for "equilibrium-tide" flow in Jupiter, underpredicting the observed tidal dissipation rate by a factor of 10⁷ - 10⁸. Wu (2005) revisited Goldreich and Nicholson's calculation, revising Q_p downward to ~ 10¹²

Tidal quality factor Q is the ratio of the energy stored in the tidal bulges to the energy dissipated per tidal cycle (from here. So higher Q means the planet doesn't dissipate the energy so quickly, hence longer survival times around planets with higher Q.

Does a stable zone exist? The entirety of the Hill sphere is not stable. Using the result from here that puts the outermost stable orbit of a prograde satellite at 0.4895 times the Hill sphere (for zero eccentricity planetary orbits). For TrES-2, the outermost stable prograde orbit is at 2.09 planetary radii. Retrograde satellites are stable out to 0.9309 times the Hill sphere, or almost 4 planetary radii, slightly smaller than the orbit of our moon and comparable to Triton's orbit around Neptune. The Roche limit for an Earth-density satellite is at about 1.23 planetary radii, so the stable zone is not zero.

Wait, it seems I'm confused. Higher Q_p means the planet is worse at dissipating tidal energy? Wouldn't that cause the bulge to persist and drag the moon in quicker?

For the orbit to decay, the energy has to go somewhere... your choices are basically rotation of the planet, rotation of the satellite, heat...

I assumed that in getting rid of the tidal bulge, the energy would go into heating of the planet (with a rocky planet having lower Q and thus gets heated more than a gas planet with higher Q). Is that incorrect?
And what do I need to correct in my post up there?

Thank you very much Sirius_Alpha for your comment, that helps me out. So, if I'm right: the greater the tidal dissipation parameter is, the bigger the potential moons are (feel free to correct if it's wrong).
About Earth and Moon and going back to the time when the Moon was still a "baby". At that time, Earth had no oceans and the Moon was very close to it (approx. 30,000 km). The Q must have been very high to avoid the Moon to crash on it (or maybe the lava oceans replaced the water oceans). If it's true, I think that rocky planets in close orbits could host moons so that the ratio could be small, like the confusing-I-think Corot-7 b.

Bye

Sedna