# Orbital Energy Change

1. Apr 4, 2005

### asrodan

A communications satellite is in a circular orbit around Earth at radius R and velocity v. A rocket accidently fires quite suddenly, giving the satellite an outward radial velocity v in addition to its original velocity.

a) Calculate the ratio of the new energy to the old and new angular momentum to the old.

b) Describe the subsequent motion of the satellite and plot T(r), V(r), U(r), and E(r) after the rocket fires.

I'm not sure how to do this question.

For a) E = 1/2*mu*v^2 + L^2/(2*mu*R^2) - k/R where mu is reduced mass, L is angular momentum k = G*m1*m2, and E is the total energy.

The v in the first term is for radial velocity only, so it is zero initially because there is no radial velocity in a circular orbit. The energy after the rocket fires, but before any radial position change is equal to the equation as written.

For angular momentum to change without any additional tangential velocity there has to be a radial position change. But I'm not sure how to figure out the radius of the new orbit that the satellite would attain.

For b) T(r) is the kinetic energy, I'm not sure how to plot this since the kinetic energy has a dependence on the derivative of r.

V(r) is the effective potential, consisting of the second two terms in the above energy equation.

U(r) is just the gravitational potential energy.

E(r) is the total energy.

2. Apr 4, 2005

### Andrew Mason

The new energy is:

$$E = U(R) + \frac{1}{2}m(v_t^2 + v_r^2)$$

where v_t is tangential (original) speed and v_r is the radial speed immediately after the rocket fired.

Since the force of the rocket was entirely in the radial direction, was there any torque applied? What does that tell you about angular momentum?

Using the effective potential:

$$V_{eff}(r) + T(r) = E(r)$$ where

$$V_{eff}(r) = U(r) + \frac{L^2}{2mr^2}$$

$$T(r) = \frac{1}{2}mv_r^2$$

For circular orbit, T(r) = 0. Since T(r) is non-zero, what does this tell you about the kind of orbit? Does it have a constant radius?

AM

3. Apr 4, 2005

### asrodan

Thanks Andrew.