Interplanetary Circularization

[funkcanna]

Hi all,

When I do an interplanetary transfer, I always fine tune my final approach mid-trip so I'm in the desired plane and altitude when I switch to the destination planets SOI. This saves massive amounts of fuel for plane correction etc. My question is, what is the best altitude to do a Circularization burn? Obviously I know to do it at Pe but is a Pe of 100k or 1000km more efficient, and why?

Edited May 15, 2015 by funkcanna

[Vanamonde]

It depend on what you want to do once you get there. For one thing, if it's a round trip, the lower you orbit the more fuel you'll have to burn to climb back up on the return leg.

[allmhuran]

The lower the better, since changes of velocity are most efficient at higher velocity. Why? Well, it's maths, but it's fairly simple maths.

Remember that kinetic energy is equal to 1/2 times the mass times the square of the velocity. Ek = 1/2 mv^2

OK, let's plug this into a few scenarios. We'll arbitrarily set m to 1 to make things simple.

First, let's change our velocity from 0 m/s to 10 m/s. Starting energy is 1/2 * 0^2 = 0. Final energy is 1/2 * 10^2 = 50. So our energy changes by 50. (50 - 0).

Now, let's change from 10 m/s to 20 m/s. Starting energy is 50. Final energy is 1/2 * 20^2 = 200. So our energy has changed by 150 (200 - 50).

But that was the same velocity change of 10 m/s both times! When we were going faster the 10 m/s velocity change resulted in a much larger energy change.

The closer your periapsis is to the target, the higher your periapsis velocity will be. Let's assume we have a high periapsis. Then we burn some amount of fuel (F), reducing our velocity by some amount (this is our delta-V) which results is some change in energy (delta E).

If we burn exactly the same amount of fuel (F) with a closer periapsis, we will get exactly the same amount of velocity change (delta V), but our delta-E will be much higher. So the same amount of fuel cost has resulted in a much larger change in energy. This is called the Oberth effect (wiki that).

If, however, you actually wanted a very high orbit for some reason, then you have to factor in the cost of pushing your periapsis back up to that altitude (unless, of course, you can gravity slingshot there off a moon, which you can actually do on the way in if you want).

Edited May 15, 2015 by allmhuran

[funkcanna]

The lower the better, since changes of velocity are most efficient at higher velocity. Why? Well, it's maths, but it's fairly simple maths.

Remember that kinetic energy is equal to 1/2 times the mass times the square of the velocity. Ek = 1/2 mv^2

OK, let's plug this into a few scenarios. We'll arbitrarily set m to 1 to make things simple.

First, let's change our velocity from 0 m/s to 10 m/s. Starting energy is 1/2 * 0^2 = 0/ Final energy is 1/2 * 10^2 = 50. So our energy changes by 50. (50 - 0).

Now, let's change from 10 m/s to 20 m/s. Starting energy is 50. Final energy is 1/2 * 20^2 = 200. So our energy has changed by 150 (200 - 50).

But that was the same velocity change of 10 m/s both times! When we were going faster the 10 m/s velocity change resulted in a much larger energy change.

The closer your periapsis is to the target, the higher your periapsis velocity will be. Let's assume we have a high periapsis. Then we burn some amount of fuel (F), reducing our velocity by some amount (this is our delta-V) which results is some change in energy (delta E).

If we burn exactly the same amount of fuel (F) with a closer periapsis, we will get exactly the same amount of velocity change (delta V), but our delta-E will be much higher. So the same amount of fuel cost has resulted in a much larger change in energy. This is called the Oberth effect (wiki that).

If, however, you actually wanted a very high orbit for some reason, then you have to factor in the cost of pushing your periapsis back up to that altitude (unless, of course, you can gravity slingshot there off a moon, which you can actually do on the way in if you want).

Perfect, thank you!

[Snark]

Depends what you mean by "most efficient" and what your constraints are.

If the goal is "get into a circular orbit using the least amount of dV", then the optimal approach is this:

Come in from interplanetary so that your periapsis is as surface-grazingly low as possible. Right at periapsis, burn retrograde until your apoapsis is at the desired height. Coast to apoapsis, then burn prograde to circularize.

The bigger your final circular radius, the less dV you need.

- - - Updated - - -

If, however, you actually wanted a very high orbit for some reason, then you have to factor in the cost of pushing your periapsis back up to that altitude (unless, of course, you can gravity slingshot there off a moon, which you can actually do on the way in if you want).

Actually, you can have your cake and eat it too. Do two burns: the first one with a really low periapsis to take advantage of the Oberth effect as you described, but don't circularize-- leave your apoapsis at the desired circular radius. Then coast up to apoapsis and circularize there.

The goal being "do as much of your burn at as low an altitude as possible," as you say.

[ajburges]

Also keep in mind your mission. While keeping the mother ship in high orbit means your landers need more dV those landers typically need less fuel to realize that dV. For a low number of landings it may be more efficient to park the mother ship in high orbit. It also allows more efficient inclination changes of the lander.

The orbit is all about where you want your energy. Higher orbits require less change in the ships kinetic energy, but landed is the lowest kinetic energy state you work with. Oberth effect just describes the most efficient way to convert mechanical energy to kinetic.

I tend to prefer mid orbits (a buffer in the 100x accel zone) to shorten lander round trips, but an elliptical initial orbit allows for more fuel efficient visitation of the pole areas while maximizing Oberth effect utilization on insertion. Once the poles are explored, I can hohmann to a lower orbit for near equitorial exploration and eventual interplanetary transfer burn.