# Light corpuscle derivation

1. Jan 16, 2017

### Liska

Hi all, this is not a homework question but a derivation in lecture notes that I am trying to understand... It says:

any light corpuscle should experience the acceleration d^2r/ dt^2 = − Gmr/ r^3 , where ~r define the position of the corpuscle in the gravitational field of the body whose mass is m; • the solutions of this equation of motion are conic sections. They can describe bound or unbound orbits. However, the speed of light is so large that it exceeds the escape velocity. Thus, the resulting orbit will be an hyperbolic orbit, which can be parametrically written as r = R(1 + e) 1 + e cos φ , r^2 dφ/dt = [GmR(1 + e)]^1/2 . In the previous equations R is the radius of the point of closest approach between the corpuscle and the body of mass m, chosen to lie of the x axis, e is the eccentricity of the orbit and φ is an angle, counted from the x axis, called true anomaly. r and φ define the position of the corpuscle with respect to the mass m in polar coordinates. • the vector ~r is written as ~r = r(~ex cos φ + ~ey sin φ) in terms of the two components along the x and the y axes. Thus, the velocity ~v is:

~v = d~r/ dt = Gm R(1 + e) 1/2 [−~ex sin φ + ~ey(cos φ + e)]
(It's all much better written here: http://www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf)

But I just don't understand how the author gets to the expression of the velocity from the previous equations. I tried getting dr/dt and then getting dφ/dt and plugging it into the equation with dφ/dt but I'm getting nowhere... Please help. Thank you!