Trajectory of object around the moon as function of time

[Henriklm]

Hello :-)

I'am trying to figure out how to describe a trajectory of an object as a function of time. The object is a spacecraft and it sets of from Earth, travels around the moon, and then heads back to Earth.

http://courses.ncssm.edu/math/NCSSM%20Student%20Materials/InvestigationsTrimester%203/Moon.pdf

I found a solution to this in this pdf (p. 5-6), but I have some questions.

I want to find r(t)= ( x(t) y(t) ). This pdf points me in the right direction (i think), but I how can I find the acceleration in the x and y directions? I´ve tried different things, but I always end up a very frustrating loop, where everything is dependent of each other.

I´ve also look at Keple's laws, but i can't find a solution where i get r(t)= ( x(t) y(t) ). Instead I get r("angle") = ( x("angle") y("angle") ).

Thank you :-D

Henrik

 

[D H]

This is a three body problem and it is does not involve one of the five Lagrange points. There is no solution in the elementary functions.

 

[Henriklm]

Thanks for the answer :-)

Hmm okay, I've read that somewhere before, just didn't understand what it meant.

Can it be solved numerically? Or is there another convenient way to descripe the trajectory?

 

[D H]

Yes, it can be solved numerically. If you want an accurate answer that is the way to go.But not so much back in the 1960s. Computers then were downright pathetic by modern standards. The CPU in a smart phone is about twice as powerful as top-of-the-line 1960s era mainframe, and that computer was housed in a huge special purpose room about the size of a high school gymnasium. The Apollo engineers employed a variety of approximations to arrive at answers that were close enough. Google the term "patched conic".

 

[Henriklm]

Nice! :D Must have been quite a task for NASA then.

How can I get a computer to solve it numerically? I can calculate the acceleration in the x and y direction at a given position, an then calculate the new velocity and position, and then start over, but I need to define an equation to do this in a computer, right? How would these equations look like?

I'm using Mathcad 2001i Professional.

 

[Nik_2213]

One gotcha if you were planning close approach: Moon & Mars have mascons, which mean that the local gravity field is *lumpy* and cannot be modeled as a uniform sphere or spheroid beyond first approximation. IIRC, Apollo circum-lunar orbits were subject to several scary seconds of 'jitter' on each pass, amounting to many kilometres of positioning error and much hair-tearing, until NASA devised corrections...

http://en.wikipedia.org/wiki/Mass_concentration_(astronomy)

 

[Henriklm]

Thank you, i didn't know that.

But I think i'll just calculate it as uniform sphere, just to keep it simple ;-)

 

[D H]

An interesting article on lunar mascons: http://science.nasa.gov/science-news/science-at-nasa/2006/06nov_loworbit/

Instead, something bizarre happened.

The orbit of PFS-2 rapidly changed shape and distance from the Moon. In 2-1/2 weeks the satellite was swooping to within a hair-raising 6 miles (10 km) of the lunar surface at closest approach. As the orbit kept changing, PFS-2 backed off again, until it seemed to be a safe 30 miles away. But not for long: inexorably, the subsatellite's orbit carried it back toward the Moon. And on May 29, 1972—only 35 days and 425 orbits after its release—PFS-2 crashed.​