/sci/ - Science & Math

[Anonymous No.6703398]

Given an initial position and velocity of a small object in proximity to a much larger one (i.e. a satellite and the Earth), and assuming perturbation-free motion under classical gravity, is it possible to express position of the object as a function of time (r(t) and theta(t), with origin at center of the Earth)?

I've worked on this quite a lot, and in the elliptical case (v(0) < sqrt(2mu/r(0)) I wind up with
>r = s - f*cos(a)
>s*a - f*sin(a) = √(eta)*(t-k)
where a arises in an integral substitution. So far I've been unable to render the second expression into the form a=f(t)

[Anonymous No.6703639]

>>6703398
no, in terms of elementary functions, no. But since the system is integrable in angle-action variables, this is solvable by quadratures, that is, the functions you want are at most inverses of integrals of elementary functions. This is a very well known problem with many series expansions, of course.

However, r(theta) IS elementary. If you substitute u = 1/r, you get a linear equation.

search for kepler problem.

[Anonymous No.6703648]

>>6703398
Where's the sun in this picture I always found these things to be fake.