# Burn time to raise to an elliptical orbit from an elliptical orbit

I am trying to simulate orbit raising like India's MOM mission did in order to get escape velocity. But I can't determine the best/most-suitable point/time on the current orbit where the probe should start burning the propellant in order to raise to a new orbit, whether the point the burn starts should be at perigee of the current orbit, or it should start somewhere before the perigee. Please keep in mind that all orbits of MOM probe were elliptical.

• Ok, the question is actually not about MOM and has nothing to do with ellipses. the actual question is that given a desired delta-v change and point in time assuming infinte TWR, the correct start of the burn is calculated for the actual TWR. – Polygnome Aug 10 '16 at 10:43
• Yes you are right, but keep MOM orbits in mind. As the probe being in one elliptic orbit, has to go in an other elliptic orbit with higher apoapsis. – Nadeem Jamali Aug 10 '16 at 10:46
• Yes, but thats irrelevant... the eccentricity only has an effect on the question in how many burns you split the desired delta-v change, it doesn't influence how you distribute delta-v for the same burn. Those are two completely different (and unrelated) questions (although both are relevant when designing a trajectory). – Polygnome Aug 10 '16 at 10:50
• The first question (into how many burns do you split a burn for maximum efficiency) is simple: infinitely many. that way the most change is in the actual, desired direction, and the fewest m7s are expended to correct for the directional change. But its not feasible. how to spit the burns then becomes a matter of feasibility, which is different for every spacecraft and engine. The question how to distribute the burn (when to start, when to stop) on the other hand has a stright-forward, optimal mathematical answer. – Polygnome Aug 10 '16 at 10:54
• So can you please post/suggest such a mathematical equation. That can give a hint for the most suitable time to start the burn? – Nadeem Jamali Aug 10 '16 at 10:56

tl;drCenter the burn around periapsis.

Your objective is to increase the specific energy of the orbit by the desired amount, while minimizing the burn time. The specific energy is:

$$\mathcal{E}={v^2\over 2}-{\mu\over r}$$

$\mathcal{E}$ is a constant of motion of an unperturbed orbit, and can be calculated from the velocity $v$ and distance from the center of the body $r$ at any time in the orbit (with those two measured at the same time of course). $\mu$ is the $GM$ of the body.

A change to the energy at any instant in the orbit is then:

$$d\mathcal{E}=v\,dv+{\mu\over r^2}dr$$

To get the most change, we want $v$ to be large and $r$ to be small. Those are both achieved at periapsis. Hence why we do these burns around periapsis.

For small excursions from periapsis, $dr$, the change in $r$ due to either the orbit itself or the burn, is very small. So we will drop that, leaving:

$$d\mathcal{E}=v\,dv$$

So for a given instantaneous $\Delta V$ of $dv$, we get the most $d\mathcal{E}$ when $v$ is maximized. That is right at periapsis. However we have finite thrust, so we can't do the whole burn there. $v$ is lower on each side of periapsis, by approximately the same amount for the same time before or after periapsis. To maximize the average $v$ during the burn, we would center the burn on periapsis, so it starts and stops at the same $v$.

I said "approximately" since the burn itself is changing $v$. So while centering the burn is a good starting point, the optimal start and end time is little different. What's more, the $dv$ as a function of time, $dv/dt$ or the acceleration, is increasing over the burn assuming constant thrust, due to the fact that the mass of the vehicle is decreasing as it expends propellant. You can simulate all of that and just change one variable, the start time, and integrate each start time until the desired change in $\mathcal{E}$ is achieved. Then the minimum of that curve is the optimal start time for the burn.

We haven't talked about the optimal direction for the burn to maximize $d\mathcal{E}$. That would be in the current instantaneous velocity direction, with the derivation left to the reader. Turning during the burn adds complexity and perhaps some risk to the maneuver, so the burn is sometimes done at a fixed inertial attitude, accepting some loss of efficiency for simplicity. The optimal attitude is very close to the velocity direction at periapsis, adjusting it slightly for the realities outlined above.

• And note that if you really care about saving fuel and your engines are capable of many restarts you do multiple short burns at periapsis (one per orbit) so Oberth provides as much of your delta-v as possible. – Loren Pechtel Oct 10 '16 at 19:23
• Sure, if you have enough time left before your Earth departure window expires. – Mark Adler Oct 10 '16 at 19:37
• If you're going to do it with multiple burns you plan that in advance and start raising your apoapsis before the window. The only burn that has to be done in one piece is the one that puts you into an escape orbit. – Loren Pechtel Oct 10 '16 at 20:43