# Shoot the moon - differential equation for motion in earth - moon syst

1. May 24, 2014

### visio

1. The problem statement, all variables and given/known data
We have given coordinates on the Earth from where we are shooting to the Moon (bullet has really small mass). The Moon orbit and therefore Moon position in time t is known. The task is to compute the initial velocity vector (the angle and velocity of the bullet), so the bullet will reach the Moon.

2. Relevant equations
Maybe we can use this equations?

$\frac{d^2x}{dt^2} = 2\Omega \frac{dy}{dt} +\Omega^2x-\frac{GM_e(x-x_e)}{r^3_e}-\frac{GM_m(x-x_m)}{r^3_m}$

$\frac{d^2y}{dt^2} = -2\Omega \frac{dx}{dt} +\Omega^2y-\frac{GM_e(y_e)}{r^3_e}-\frac{GM_m(y_m)}{r^3_m}$

Ω is angular system velocity
G is gravitational constant
$M_e, M_m$ is mass of the Earth, Moon
$r_e, r_m$ is distance between Earth, Moon and the bullet
$x_e, x_m$ coordinates of the Earth, Moon centre of mass

3. The attempt at a solution
I know that I need to solve it numerically with shooting method, but the problem is, how the differential equation describing bullet trajectory looks like. I found the ones above, but I am not physicist (the main problem is to find the numeric solution of that equation), I do not know, if I can use them or not.

If anyone can give me some reference to the literature about this problem or something (the equations can be simple - no need to include all the influences, just the main ones as gravity field and rotation), I would be very happy. Thank you

Last edited: May 25, 2014
2. May 24, 2014

### dauto

You mean you're not a physicist. A physician is another name for a medical doctor. The equations are fine. The origin is placed at the center of mass of the earth-moon system.There are probably easier ways to solve that problem. Hard to tell since you didn't really give enough details about the problem

3. May 25, 2014

### visio

Thanks for language note. Unfortunately I do not have more details, it is up to me to find out the differential equations to solve. But if you can tell me what details are missing, I can probably add them. Maybe I can add that as the initial conditions for given system I am using coordinates of the place on Earth (at time = 0) and coordinates of the Moon (at time = end time).

Last edited: May 25, 2014
4. May 25, 2014

### D H

Staff Emeritus
The equations you found are for the synodic frame in the circular restricted three body problem. The Earth and Moon don't move in that frame. Having a stationary target should make your shooting algorithm a bit easier.

5. May 25, 2014

### visio

Thanks for an advice. But still, I do not know what equation should I use then. Can you be a little bit more specific?