Planetary Motion: Orbit Transfers, Hohmann transfer

[Lost1ne]

The thought of increasing a satellite's (for example) speed to allow it to transfer from a "higher energy" elliptical orbit to a "lower energy" circular orbit (in reference to the effective potential energy plot that arises after introducing the concept of an effective mass to simplify the two-body problem into a one-body problem) seems a bit counterintuitive. It's easy to find this illogical with a circular orbit simply occurring at a lower energy level than an elliptical orbit in the effective potential energy plot.

Examining other equations such as ΔE = Δ(-C/A) (where C = GMm) clearly depicts mathematically why, in the final stage of the Hohmann transfer process, transferring to a circular orbit with a larger major axis requires an in-take of energy, gained through the rocket engine. However, how should this be viewed with respect to the effective potential energy plot (equal to [L^(2)[/[2μr^(2)] - (GMm)/r, with the remaining term being the kinetic energy from the radial velocity; μ is the effective mass, Mm/(M+m))? Would the best way to think of this be a shift in this graph due to the work done by the propellant force on the satellite (thus a new graph that allows for the new, final energy to take a circular orbit)?

 

[LURCH]

Don’t yet have a conclusive answer for you, but would like to see if we can work it out together. Just off the top of my head, have you included the energy of the reaction mass (the expelled propellant) in your thinking? Also, since an elliptical orbit continuously exchanges kinetic energy for potential, the total of the two remains nearly constant, and that may be the key to understanding this. Will spend some time trying to flesh out these ideas and come back to see this thread again later.

 

[Lost1ne]

Don’t yet have a conclusive answer for you, but would like to see if we can work it out together. Just off the top of my head, have you included the energy of the reaction mass (the expelled propellant) in your thinking?

Oooh. I don't think I thought too much about that actually. I guess I'm then encouraged to think of the "flow mass" portion of this problem? If I choose my system to be the satellite and any propellant that is exerted, I'm assuming that the energy in that system would be conserved. However, our concern should be with the energy of the satellite after it loses a bit of propellant over some time duration and then experiences some change of velocity as a result of it. The net external force on this system would be the gravity exerted on the system by the massive mass that the system is orbiting. Am I on the right track? Is it encouraged to think of the problem in this manner to answer my question more effectively? I'll return to this later.

 

[LURCH]

I do believe that is the right track to follow. Was able to think about it a little, and I noticed that the Transfer burn for rounding out an orbit at the top of the ellipse requires thrust in the opposite direction from the burn required for keeping the satellite at the bottom of the ellipse. The propellant gets thrown into higher orbit when the satellite settles into a lower one, and vice versa. That seems like an important clue.

My only hesitation comes when considering that, even when settling into a round orbit with the exact same energy as the original ellipse, an acceleration is still required. That one has got me puzzled, because no orbital energy has been gained or lost, yet we had to expend energy to get there. Will continue to ponder, and return later (I’m at work, browsing the Forums during my breaks).

 

[A.T.]

My only hesitation comes when considering that, even when settling into a round orbit with the exact same energy as the original ellipse, an acceleration is still required. That one has got me puzzled, because no orbital energy has been gained or lost, yet we had to expend energy to get there.

If the craft (not counting the propellant expelled in the whole maneuver) gains no energy, and you lose chemical energy, then the expelled propellant must have gained it (the propellant has more KE+PE than it had originally in the tank).

 

[LURCH]

Ah yes, that’s the piece I was missing. Thanks A.T.

 

[Lost1ne]

I've created a diagram.

https://imgur.com/a/JmM230i

After discussing this with my professor, this is what we concluded the effective potential energy curve should look like (assuming it's correct). This would apply to the Hohmann transfer orbit where our craft transfers from an elliptical orbit to a circular orbit where the craft is accelerated tangentially (thus, angular momentum increases) into a circular orbit when at the apogee of the elliptical orbit. The apogee of the elliptical orbit is equal to the radius of our circular orbit, all measured from the more massive, "stationary" object that our craft is orbiting about. My professor didn't mention the flow mass aspect of this problem, but I believe that the system analyzed in our curve will be the craft, excluding the propellant released.