I've been working on this for a while with little success. I'm trying to model a continuous burn orbit maneuver for say 5 minutes. I've tried to split the burn into multiple sub-burns that are calculated instantaneously, but this doesn't seem to work.

My end game here is to model 2 continuous burns both around apogee (so first burn, orbit around to apogee again, second burn). The error I'm currently getting is that if I compare the results for a single burn of length N to the results of two burns of length N/2, I use more propellant to get to my target perigee altitude. I've found references to Gauss planetary equations, and even a book they are in:

Battin RH (1999) Introduction to the Mathematics and Methods of Astrodynamics AIAA Education Series. American Institute of Aeronautics and Astronautics, Reston, Virginia

Apparently page 489 has the sorts of equations I'm looking for. But I can't get access to this book. Any online source of this book, or the appropriate method/equations would be a perfect answer!

  • $\begingroup$ You just use an integrator. As for converting position and velocity to orbital elements, that question has already been answered here. $\endgroup$
    – Mark Adler
    Mar 11 '15 at 15:25
  • $\begingroup$ @Mark Adler I've already modelled this use many instantaneous burns (say 100 1s instant burns), but my results don't make much sense $\endgroup$
    – ThePlanMan
    Mar 12 '15 at 8:14

It isn't entirely clear to me whether you are doing a series of impulsive (zero time duration) maneuvers with coasts between, or an actual "finite burn", a more realistic maneuver that takes some non-zero time to accomplish. In either case, however, you might be carrying some of the propellant to a higher potential energy before burning it, relative to an instantaneous burn. This "gravity loss" is noticeable. For an extreme example, this effect was used by the onboard "Gamma Guidance" algorithm to adjust the timing of the third of three solid rocket motor burns by the IUS/PAM-S upper stage to counter dispersions in the first two burns to place the ESA spacecraft Ulysses more accurately onto its initial hyperbolic trajectory away from Earth in 1990.

The Ph.D. dissertation of Greg A. Dukeman, Georgia Institute of Technology, 2005, "Closed-Loop Nominal and Abort Atmospheric Ascent Guidance for Rocket-Powered Launch Vehicles" in part (Chapter 3) discusses finite burns giving equations for "Thrust Integrals" (analytic solutions for the change in radius and velocity vectors over the course of a finite burn), for constant thrust or constant thrust acceleration for simplified equations of motion (burn direction changes linearly with time, and gravity is taken as constant). See the thesis for why this is useful even with these simplifications. (Beware the typo in equation 3-3. I believe f11*T*A should instead be f11*A in the first term for delta R sub T.) Unless your maneuvers are very long (and perhaps even then) this technique could be used to target your burns.

There is much more of interest in this very fine dissertation, available online. http://hdl.handle.net/1853/6820


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