Limits on moons to transiting planets

Limits on the Orbits and Masses of Moons around Currently-Known Transiting Exoplanets
http://arxiv.org/abs/1007.4500

Abstract wrote:

Aims. Current and upcoming space missions may be able to detect moons of transiting extra-solar planets. In this context it is important to understand if exomoons are expected to exist and what their possible properties are. Methods. Using estimates for the stability of exomoon orbits from numerical studies, a list of 87 known transiting exoplanets is tested for the potential to host large exomoons. Results. For 92% of the sample, moons larger than Luna can be excluded on prograde orbits, unless the parent exoplanet's internal structure is very different from the gas-giants of the solar system. Only WASP-24b, OGLE2-TR-L9, CoRoT-3b and CoRoT-9b could have moons above 0.4 m\oplus, which is within the likely detection capabilities of current observational facilities. Additionally, the range of possible orbital radii of exomoons of the known transiting exoplanets, with two exceptions, is below 8 Jupiter-radii and therefore rather small.

Briefly looking at it, it seems that the probability for OGLE2-TR-L9b and Corot-13b is low, but a great interest is directed towards Corot-9b with a very high probability for at least one moon.
But I don't know why the authors did not figure out that the presence of a moon around WASP-12b is simply impossible. If there was a moon (say the size of Apophis), it would have been destroyed while passing trough the flow of gas falling onto the star.

Gonna digg this for more results...

The case of HD 80606b is quite impressive: essentially all moons are ruled out thanks to the highly eccentric orbit.

As for WASP-12b, it is for a start clear that they are dealing with a theoretical planet-star model not taking into account environmental effects (while you're going on about WASP-12b, what about the comet-like atmospheric evaporation of HD 209458b). It is also not entirely clear to me how much effect the gas would have on the orbit of a moon passing through it and whether it would be a significant factor in its orbital evolution.

Lazarus wrote:

As for WASP-12b, it is for a start clear that they are dealing with a theoretical planet-star model not taking into account environmental effects (while you're going on about WASP-12b, what about the comet-like atmospheric evaporation of HD 209458b). It is also not entirely clear to me how much effect the gas would have on the orbit of a moon passing through it and whether it would be a significant factor in its orbital evolution.

For the case of WASP-12b, it's quite hard to imagine. The gas itself would do nothing on the possible moon, the destruction (or if the moon is not destroyed, it would be pulled apart) would be caused by the speed of the same gas (which, I think, must be strong). A practical example would be someone trying to walk outside during a strong hurricane.

About HD 209458b, the case is somewhat different. Despite the 35,000 km/h of gas' speed, the possible moon could resist if it's an Enceladus-like moon, but if the moon has the mass of an asteroid it would be pulled apart too.

But this is just what I think.

From this graphic of WASP-12, I get the impression the stream is outside the Hill sphere, where any moons would be anyway.

Yes, but what about the space between the L1 point and the surface of the planet ? There must be some stream there.

I interpret the purple areas on the XY plane in that graph to be the wind. The size and direction of the arrows being the speed and direction of the wind respectively. If this is right, then the wind between L₁ and the planet surface isn't as bad as jut outside the L₁ point.

The other question is the density of the gas - fast speed is not the only consideration. You need to have enough material there to do the damage.

Lazarus wrote:

The other question is the density of the gas - fast speed is not the only consideration. You need to have enough material there to do the damage.

Some calculation will do the trick. Assumptions first. See the space between the dotted lines on the XY plane, at the projection of the L1 point ? Let's assume that it's the bottleneck in which the gas leaves the planet. We assume that this bottleneck is a perfect cylinder (for simpler calculations).
Looking at the graphic and knowing that the possible moon will pass between the L1 point and the surface of the planet, we have 85,000 km for the length (h) of the cylinder (roughly one third of the planet's diameter + its envelope). For the radius (r) of the cylinder we take the space between the dotted lines at the L1 point's projection, making a radius of 70,000 km.
The volume of the cylinder is given by V = pi*r²*h. Thus we have V = 3.14159265*(70,000,000)²*85,000,000, making 1.3x10^24 m³ as the answer.
We also know that the planet is loosing 6 billion tons of material per second.
Thus, the density at a given second is roughly 4.615x10^-12 kg/m³.

Looking at this, the possible moon would have a certain chance to survive and see the last breath of its parent planet. I expected the density to be much higher.

The main question is about the wind, which can only be solved by observations.

Edit: I could measure the speed of the gas falling onto the star by taking a piece of it and make the calculations with the formula of free fall, but the likelihood to be wrong would be huge.

I have some problems in verifying the Equation 5 in the paper. Authors state a maximum mass of 27 Earth masses for a satellite around Corot-3 b. But I find totally different and unreal values.

I don't get how the quantities are specified.

M_max= 2/13(fR_Hill)^3/12 x Q_p/3k_2PTR_p⁵ x (m_p/G)^1/2

T= gyrs
Rhill= in AUs, Rj or Kms?
R and M in Rj and Mj?
G= is for gravitational constant, 6.6748x10¹¹ N (m/kg)^-2

so Masses and radii are to be expressed in Kg and meters.

Trying each options I don't find results provided from authors

Help!

The formula would be more readable if you add some parenthesis (remember the precedence rules, this can avoid many mistakes), as follows:

M_max= 2/13(fR_Hill)^13/2 x (Q_p/(3k_2PTR_p⁵)) x (m_p/G)^1/2

I took a look here and I found this:

Barnes & O'Brien wrote:

Recently, Holman & Wiegert(1999) investigated the stabilityof planets in binarystar systems, and theirresults are applicable tothe planet-satellite situation aswell. Through numerical integrationsof a test particleorbiting one component ofa binary star system,Holman & Wiegert (1999)found that for highmass ratio binaries, thecritical semimajor axis forobjects orbiting the secondaryin its orbital planeis equal to aconstant fraction (f) ofthe secondary's Hill radiusor



We treat a starorbited by a muchless massive planet asa high mass ratiobinary system and deducethat the critical semimajoraxis for a satelliteorbiting the planet is0.36R_H (f = 0.36)for prograde satellites (fromHolman & Wiegert 1999,Fig. 1). This agrees closelywith Burns (1986). Infact, none of theprograde moons of oursolar system orbit outsidethis radius (see Table 1).Holman & Wiegert (1999)did not treat objectsin retrograde orbits (whichare expected to bemore stable than progradeones), so to treatpossible captured satellites wetake f_retrograde = 0.50,based on the solarsystem values for a_m/R_Hin Table 1.

It tells that fR_Hill is not simply the Hill Sphere but a dimensionless fraction of it.
For the units, the authors did not mention it so they are in SI.

I did the calculations with it but there are still problems. I think that their formula is not clear enough. I suggest that you send a mail to the authors about it.

Edasich you have to keep your units consistent.

If you take your time units as fortnights, your mass units as Planck masses and your distance units as furlongs, you must obtain the value of G in furlongs³ m_Planck^-1 fortnights^-2

Quote :

The formula would be more readable if you add some parenthesis (remember the precedence rules, this can avoid many mistakes), as follows

It's not easy to type mathematical formulae without Equation editor. I try my best. In any case there is URL to referred paper to check reference.

Quote :

It tells that fR_Hill is not simply the Hill Sphere but a dimensionless fraction of it.

Yes, I have noticed it, both for prograde and retrograde orbits. But situation doesn't change.

Indeed I have used SI units (masses in Kg, distance, planetary radius and Hill radius in meters, since gravitational constant G includes both Kg both meters, not kilometers or AUs), but I keep on getting quite unlikely values. And also Sedna seems having trouble with.

Concluding,the research paper looks incorrect or at least unclear.

f is the dimensionless fraction, R_Hill does have dimensions (of length, obviously).

In fact reading the paper carefully shows the prediction of maximum prograde mass of 27 M_E is for COROT-9b not COROT-3b, for which they predict 1.7 M_E (and 111 M_E for a retrograde orbit!!!)

Nevertheless I am not managing to reproduce the values in the table, including the maximum orbital radii. I suspect something's gone wrong somewhere.

Edasich wrote:

It's not easy to type mathematical formulae without Equation editor. I try my best. In any case there is URL to referred paper to check reference.

I know, don't feel bad, I wasn't yelling at you. Maybe there's some LaTeX for websites and forums.

Plus, with such values the moon would have to be big and we would have already seen it.
I planned to use these formulae for my own database but since there's something wrong I won't use them.

Could it be that their numbers came from dynamical simulations of the systems (as opposed to direct calculation from formulae)? How closely do these formulae reflect reality?