Final answer:
The ratio of orbital speeds at perihelion and aphelion is derived from the conservation of angular momentum and is given by vp/va = (1+e)/(1-e), where 'e' is the orbital eccentricity.
Step-by-step explanation:
From the conservation of angular momentum, we can show that the ratio of orbital speeds at perihelion (vp) and aphelion (va) for an object in an elliptical orbit is given by vp/va = (1+e)/(1-e), where e is the eccentricity of the orbit. The angular momentum, L, is conserved at all points in the orbit, meaning Lp = La, and because angular momentum is the product of linear speed, mass, and the radius of orbit (L = mvr), the speeds at perihelion and aphelion can be related through their respective distances from the center of mass.
At perihelion, the object is at its closest approach, rp, and moves with speed vp. At aphelion, it is at its furthest point, ra, moving with speed va. The conservation of angular momentum implies that m * vp * rp = m * va * ra. Since the mass of the orbiting object remains constant, it cancels out, simplifying the equation to vp/va = ra/rp.
Knowing that ra/rp = (1+e)/(1-e) from the geometry of an ellipse (where rp = a(1-e) and ra = a(1+e), with 'a' being the semi-major axis), we can combine these equations to get vp/va = (1+e)/(1-e).