Minimum velocity for GEO orbit transfer

[orbitsnerd]

Homework Statement

A GEO spacecraft crosses the earth’s equatorial
plane when its true anomaly is 30 deg. The
eccentricity of the orbit is 0.1 and its initial
inclination is 5 deg with respect to the equator.
What minimum velocity increment is required to
transfer this GEO to an equatorial orbit?

Homework Equations

e=.1
theta=30 degrees
i=5 degrees
cos E=e+cos(theta)/1+e*cos(theta)
deltav=2*v*sin(i/2)

The Attempt at a Solution

Not sure where to start. I am thinking I begin with the equations above. I think it is a plane transfer but not sure how to do it.

 

[gabbagabbahey]

Use conservation of energy.

What is the energy of the spacecraft in is original (tilted) orbit? What would be its energy if it were in an equatorial orbit?

 

[D H]

Use conservation of energy.

What is the energy of the spacecraft in is original (tilted) orbit? What would be its energy if it were in an equatorial orbit?

Conservation of energy is not going to help here. Even a small plane change maneuver can be fairly expensive delta-v wise, and ideally there will be no change in orbital energy. For example, the International Space Station's orbit has a 51.6 degree inclination because plane change maneuvers are expensive. A 51.6 degree inclination is the lowest inclination orbit into which vehicles launched from Baikonur can be placed. (Baikonur is at 45.6 degrees latitude, so vehicles could launch into a 45.6 degree orbit from there. However, that would result in spent first stages dropping on China. A 45.6 degree inclination is the lower limit imposed by physics. Politics adds a 6 degree surtax.)

Homework Equations

e=.1
theta=30 degrees
i=5 degrees
cos E=e+cos(theta)/1+e*cos(theta)
deltav=2*v*sin(i/2)

When can you perform a plane change? For an elliptical orbit, which of the choices requires the least delta-v?

From the last of the relevant equations, you obviously you need to compute the velocity. How do you compute that?

BTW, that last equation is not quite right. That is the equation for a pure plane change from a circular orbit. What is the equation for a pure plane change from an elliptical orbit?


Note: I am assuming you are to do a pure plane change in this problem -- that is, you are to keep the satellite in geosynchronous orbit. It is a tad cheaper to simply cancel the z component of velocity at the right time, but that will result in a non-geosynchronous orbit.


These lecture slides might be of use. Start at slide 34.
http://web.nps.navy.mil/~ssweb/AA4362/week8/wk8_lecture2.htm