What is the Orbital Period of Positronium in a Circular Orbit?

[PinkFlamingo]

Hi there! I hope someone can help me with this problem. I've been working on this for over 5 hours and I've gotten nowhere!

A positron is a particle with the same mass as an electron but with a positive charge. A positron and an electron can briefly form an unusual atom known as positronium. Imagine a situation where the two particles are in a circular orbit about their center of mass. Since the particles have equal mass, the center of mass is midway between them. Let r be the separation of the particles (so that the orbits are each of radius r/2).

(a) Show that the orbital period T is related to the separation distance r by:

T^2 = (16)(pi^3)(E0)(me)(mp) (r^3)
---------------------
(e^2)[(me) + (mp)]

This is a consequence of Kepler's third law for electrical orbits.

(b) Show that if an electron and a proton are in circular orbits about their center of mass (which is not at the midway point between them but much closer to the proton), then the same expression results.

* * * * *

OK, so so far, I'm guessing that I somehow use the formulae:

q = ne

F = 1 |Q||q|
-------- x ---------
4(pi)(E0) (r^2)

But I'm not really sure where the rest of it comes from

If someone could help me out, I would really appreciate it!

Thanks!

Mandy

 

[swansont]

The bad idea of Keplerian orbits being used aside, if the orbit is indeed a circle, then the force each particle feels must obey the centripetal force equation, F=mv²/r. In this case the centripetal force is the electrostatic force, so they two equations can be set equal to each other.