maximum orbital frequency formula derivation

[toribonidie]

Hello! I am writing to you to ask for some clarification in your documentation on how you derive your expression for maximum orbital frequency. I am using this code to search for giant planets around red clump stars, and the minimum period I calculate is very different than what TLS algorithm is using to search for transits.

In your paper, you say that the most short- period (high-frequency) circumstellar orbit is given by

$$f_{max} = \frac{\sqrt{GM_{s}/(3R_{s}})^{3}}{2\pi}$$

where the $3R_{s}$ term is calculated from the Roche limit, assuming a pessimistic density of $\rho_{p} = 1 g/cm^{3}$. I do not see a derivation for this formula in your paper, but I assume you're using Kepler's 3rd law

$$f_{max} = 1/P = \frac{\sqrt{GM_{s}/(a})^{3}}{2\pi}$$

And you are using the roche limit to calculate the semi major axis. For a rigid transiting body, the roche limit is given by,

$$a = R_{s} (2 \frac{\rho_{s}}{\rho_{p}})^{1/3}.$$

Or, for a fluid transiting body, the roche limit is given by,

$$a \approx 2.44R_{s} (\frac{\rho_{s}}{\rho_{p}})^{1/3}.$$

It says that you are assuming $\rho_{p} = 1 g/cm^{3}$, but I do not see any assumption about $\rho_{s}$. Now, if use this assumption of planet density and I plug in my value for a typical density of a RC star (0.002 $g/cm^{3}$), I get

$$a^{3} = 0.004 R_{s}^{3}$$

for a rigid body and

$$a^{3} = 0.03 R_{s}^{3}$$

These are both extremely different from the value than the $a^{3} = (3R_{s})^3$ from your paper. Are you inputing a typical density for a solar mass star (~1.4 $g/cm^{3}$ for the sun)? Because that would produce an $a^{3}$ term that is roughly $(3R_{s})^3$.

I don't see anything in the paper that makes this assumption, so I could be misinterpreting something, but this formula is restricting the period range that TLS searches and I believe shorter periods are physically possible.

Please let me know if this makes sense. Thank you so much!

[hippke]

Hi @toribonidie, thank you for your question. This seems less like a code question, assuming the code replicates the paper correctly. Instead, I believe your question is better answered with physical reasoning. I'm a little tight on time at the moment, but I'll try to ping my co-author René for an answer.

[toribonidie]

Hi @hippke ! Thank you for getting back to me. Please take your time to respond. I am able to override the default period grid that TLS provides while performing my searches, but I just wanted to make sure I'm interpreting this correctly so that I am not searching for periods that are physically impossible.

So, no rush! But I look forward to hearing back from you or René for clarification.

[reneheller]

Hello @toribonidie! Your calculations look alright to me. Our formulae have been adopted from Ofir (2014), see his Sect. 3.1 therein for a few more details. In brief, yes, f_max is derived from Kepler's 3rd Law as you illustrate. The factor of 3 R_s comes from the division of the stellar density by the planetary density and then taking the third root. For a fluid-like Sun-like star and very low-density planet, we have 2.44 × (ρ_s/ρ_p)^1/3 ~ 2.44 × (1.4 g cm^-3 / 1 g cm^-3)^1/3 ~ 2.44 × 1.12 ~ 2.73 ~ 3.

I checked our paper and you're right, we should have explicitly mentioned that we assumed a solar-type star. In fact, for low-mass stars the density is higher than for solar-type stars and the Roche limit can move out to ~ 10 R_s. Michael (@hippke), maybe this is something that we should note in the TLS documentation for users to be aware that it might make sense to adjust the Roche limit and therefore f_max?

Tori, for your particular case of an extremely low-density star you end up with a Roche radius inside the star, did you notice? a³ between 0.004 R_s³ and 0.03 R_s³ is equivalent to a between 0.16 R_s and 0.3 R_s. I would expect any planet at that "distance" to the star to lose angular momentum and fall into the stellar core within < 1 million years. What do you think?

[toribonidie]

Hi René!

Thank you so much for getting back to me and clarifying. I think adding that to the documentation might be useful in the future since I know of other people using this algorithm to look for planets around hot subdwarfs and m dwarfs, which would both have very different Roche radii.

And yes, for my case, since the Roche radius is inside the star, I am just setting the minimum semi major axis to be $a = R_{star}$, so my period grid now spans from 6-32 days.

Thank you again!