Stability of a moon of hot jupiter in red dwarf system

[Czcibor]

I was trying to calculate possible orbit of a potential exomoon of a hot Jupiter within habitable zone of a red dwarf. Everything (heat absorbed, reasonably within Hill sphere, far from Roche limit) seamed reasonable. However, I encountered one problem - hot Jupiter should be tidally locked. It means that it rotation period would be longer than any reasonable orbit for the moon, so the moon orbit should be subject to tidal orbit decay.

Does anyone know an approximate formula to calculate speed of tidal orbit decay? (Or maybe the process would be so slow or so quick that I have a conclusive answer without calculation?)

 

[AndyW]

I appreciate your curiosity and interest in calculating the possible orbit of a potential exomoon. Tidal orbit decay is indeed an important factor to consider when studying the stability of a moon's orbit around a hot Jupiter within the habitable zone of a red dwarf.

The speed of tidal orbit decay can be approximated using the following formula:

V = -2.5 * (k/Q) * (M/m) * (R/a)^5 * (e^2)

Where:
V = orbital decay rate (meters per second)
k = tidal Love number (dimensionless)
Q = tidal dissipation factor (dimensionless)
M = mass of the planet (kg)
m = mass of the moon (kg)
R = radius of the planet (meters)
a = semi-major axis of the moon's orbit (meters)
e = eccentricity of the moon's orbit (dimensionless)

The tidal Love number and dissipation factor are properties of the planet and moon, and can be found in literature specific to the objects in question. The eccentricity of the moon's orbit can also be estimated based on the properties of the system.

It is important to note that the process of tidal orbit decay can vary greatly depending on the specific properties of the planet and moon, and can be affected by other factors such as the presence of other moons or the planet's obliquity. Therefore, it is best to use this formula as a rough estimate and to also consider other factors that may affect the stability of the moon's orbit.

I hope this information helps in your calculations and research. Keep in mind that as scientists, we are always learning and discovering new information, so it is important to constantly update our calculations and theories as we gather more data. Best of luck in your studies!