Question

6. In studying the Moon and the Earth, we saw that the tidal forces are causing the Moon to be pushed away from the Earth. In our solar system, we have another curious case. Triton, the main moon of Neptune, is in a retrograde orbit, which means it is orbiting in the opposite direction that Neptune is rotating. It is likely a captured object. Because of tidal forces and this odd orbit, Triton is slowly spiraling in toward Neptune. This means that in the future, Triton will be ripped apart once it gets close enough to Neptune due to the tidal forces acting on it. At what distance will Triton be when Neptune begins to rip it apart? Use the following steps to determine the distance. (a) Determine the force of gravity if you were on the surface of Triton in terms of m (the generalized mass of an object on the surface). (b) Use the results from Part (a) and the equation for the tidal force to determine the distance between Neptune and Triton when Triton will be ripped apart. Note: For consistency, please use d as the distance between the main gravitational source and the mass m experiencing the tidal force and do as the distance between the centers of the two objects. [You may need to use a program like Maple or Wolfram Alpha to solve the equation for what you are looking for. If you do this, please include either a copy of what you did or completely describe how you used that program.]

Answer 1

Gravitational force acting on a moon as it orbits its planet:

On subtracting the center of mass force, we see the differential force acting on it:

Therefore Gravitational Force:

$F_g = -G\frac{Mm}{r^2}$

Tidal force is the approximately the differential force

$F_t \approx \left ( \frac{dF}{dr} \right ) \Delta r = 2G\frac{Mm}{r^3} \Delta r$

The Roche limit, sometimes referred to as the Roche radius, is the distance within which a celestial body held together only by its own gravity will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction. To calculate the rigid body Roche limit for a spherical satellite, the cause of the rigidity is neglected but the body is assumed to maintain its spherical shape while being held together only by its own self-gravity.

The gravitational pull F_G on the mass u towards the satellite with mass m and radius r can be expressed according to Newton's law of gravitation

$F_g = \frac{GMu}{r^2}$

The tidal force F_t on the mass u towards the planet with radius R and a distance d between the centers of the two bodies can be expressed as:

$F_t = \frac{2GMur}{d^3}$

The Roche limit is reached when the gravitational pull and the tidal force cancel each other out.

$\therefore F_t = F_g$

$\implies \frac{2GMur}{d^3} = \frac{Gmu}{r^2}$

$\therefore d = r \left ( \frac{2M}{m} \right )^{\frac{1}{3}}$

Here is the radius of the satellite (Triton)

I have a few things which isn't clear in this question. What is meant by main gravitational force? Is the Sun needs to be considered? Then the physics will be a lot complicated and cannot be solved analytically. What are actually and $d_0$ mentioned in the question? Please follow up so that I can give you a precise answer.