Calculating Work for Orbital Transfer of a Satellite

[seiferseph]

How would i go about doing a problem like this?

A 1500 kg satellite is orbiting the Earth at a height of 250 km. How much work would it take to move it into an 800 km orbit?

 

[marlon]

Use the gravitational potential energy. If the radius increases, this energy will become less negative, ergo, the energy rises...

marlon

 

[marlon]

The formula is $$- \frac{GMm}{R}$$

G is the universal gravitational constant
M is the mass of the first object
m is the mass of the second object
r is the radius

be sure to treat the Earth as a point mass with all mass centered in that point. Thus, you will also need the Earth's radius...

marlon

 

[seiferseph]

that is just the equation for gravitational potential energy at one point. wouldn't it have kinetic? and it asks for work in the question. also, its worth 10 marks, so i don't think it can be that simple.

i've done some where you take it off the surface and throw it into orbit, and i used

Etotal(orbit) = Epotential(surface) + W

 

[dextercioby]

The total mechanical energy (KE + gravitational potential) that a body has is found,when solving the Kepler problem in CM,

$$\mbox{Tot \ E}=-G\frac{m_{body}M_{Earth}}{2a}$$

,where "a" is the big semiaxis of the elliptical orbit.However,it can be proven really easliy that "a" goes to R (radius of orbit) for a circular orbit.

It's all u need to know.The work done is simply the variation in total energy of the body.

Daniel.

 

[seiferseph]

The total mechanical energy (KE + gravitational potential) that a body has is found,when solving the Kepler problem in CM,

$$\mbox{Tot \ E}=-G\frac{m_{body}M_{Earth}}{2a}$$

,where "a" is the big semiaxis of the elliptical orbit.However,it can be proven really easliy that "a" goes to R (radius of orbit) for a circular orbit.

It's all u need to know.The work done is simply the variation in total energy of the body.

Daniel.

isn't that just its total energy at that point? what is the work used to bring it up into a higher orbit? this is the equation my professor gave me for moving something from the surface up into an orbit: Etotal(orbit) = Epotential(surface) + W

 

[dextercioby]

Mind that there's a trick here.The height is given wrt Earth's surface,while R in the formulas is the distance between the Earth's center & the orbit (the circle's radius).

So u need to add the mean Earth's radius

$$\bar{R}_{Earth}\simeq 6371 \ \mbox{Km}$$

Daniel.


P.S.Of course,for consistency of units,u need to transform every length from Km to m...

 

[dextercioby]

The sattelite is already in orbit...W is just the diff.between total energies...

Daniel.

 

[seiferseph]

The sattelite is already in orbit...W is just the diff.between total energies...

Daniel.

so its just Etotal(2nd orbit) - Etotal(1st orbit)? using the equation you said, except using the larger value 800km for R for the 2nd orbit and smaller 250km for 1st orbit?

 

[dextercioby]

Yeah,that's right...

Daniel.