How do I calculate tidal forces from gravitational potential derivatives?

[Allday]

Hey people,

Im doing some analysis of some N-body simulation data. I'm trying to calculate the tidal forces exerted on the smaller groups of particles by the other mass. I have a model for the distribution of matter causing the tidal field so I can analytically calculate the gravitational potential and the directional second derivatives, but how do I translate that into the forces. Anybody have a reference for some good reading on the subject.

 

[pervect]

Hey people,

Im doing some analysis of some N-body simulation data. I'm trying to calculate the tidal forces exerted on the smaller groups of particles by the other mass. I have a model for the distribution of matter causing the tidal field so I can analytically calculate the gravitational potential and the directional second derivatives, but how do I translate that into the forces. Anybody have a reference for some good reading on the subject.

Well if you have a potential V(x,y,z) then the force in cartesian coordinates (x,y,z) is given by
$$F_x = \frac{\partial{V}}{\partial{x}} \hspace{.25 in} F_y = \frac{\partial{V}}{\partial{y}} \hspace{.25 in} F_z = \frac{\partial{V}}{\partial{z}}$$

and if you have a unit vector U, the tidal force T is another vector, the gradient of the force F in the direction of vector U, given by

$$T_x = \frac{\partial^2{V}}{\partial x \partial x}} U_x + <br /> \frac{\partial^2{V}}{\partial x \partial y}} U_y + <br /> \frac{\partial^2{V}}{\partial x \partial z}} U_z$$
$$T_y = \frac{\partial^2{V}}{\partial y\partial x}} U_x + <br /> \frac{\partial^2{V}}{\partial y\partial y}} U_y + <br /> \frac{\partial^2{V}}{\partial y\partial z}} U_z$$
$$T_z = \frac{\partial^2{V}}{\partial z\partial x}} U_x + <br /> \frac{\partial^2{V}}{\partial z\partial y}} U_y + <br /> \frac{\partial^2{V}}{\partial z\partial z}} U_z$$

You can write this in tensor notation

$$T^i = K^i{}_j U^j$$

where $$K^i{}_j = \frac{\partial^2{V}}{\partial x^i \partial x^j}$$


It gets more complicated if you want to use general (non-cartesian) coordinates

But you can always say that the tidal forces at a point are given by a second rank tensor, one that takes in a vector (the displacement) and spits out a vector (the tidal force).

I *think* that the partial derivates should normally alll commute, so $$\frac{\partial^2 V}{\partial x \partial y} = \frac{\partial^2 V}{\partial y \partial x}$$

Google finds "Clairaut's theorem"

http://planetmath.org/encyclopedia/ClairautsTheorem.html

 

[R0man]

Hi there, it sounds like you are on the right track with your analysis of the N-body simulation data. To calculate the tidal forces, you will need to use the gravitational potential and the directional second derivatives. These can be calculated using mathematical equations, as you mentioned.

To translate this into forces, you will need to use the equation F = -m∇Φ, where F is the force, m is the mass of the smaller group of particles, and ∇Φ is the gradient of the gravitational potential. This will give you the magnitude and direction of the tidal force exerted on the particles.

As for references, I would recommend looking into textbooks or articles on celestial mechanics or astrophysics. You can also find some helpful resources online, such as lectures or tutorials on tidal forces and N-body simulations. Good luck with your analysis!