LKO and then hohmann to final orbit vs. straight to final orbit

[davidpsummers]

I think the OP question is: If I intend to get to a 250x250 orbit, should I:

• set my AP to 250 and when I reach it, I circularize?
  
• set my AP to 71, circularize and THEN do Hohman transfer to 250x250?
  

Unfortunately, I don't know the answer either... I'd very like to learn it from you guys...

Addressing this question (is it better to establish a low orbit and raise it or go right for the higher orbit), I would say that, absent the Oberth effect, it doesn't matter.

Each orbit has an energy associated with the sum of the kinetic energy (dV) and potential energy (altitude). As long as the efficiency you convert fuel to kinetic energy (and hence also to potential energy) is the same, it shouldn't matter?

Now the Oberth effect can mean the efficiency of converting fuel to kinetic energy varies. I think the main difference is that when you circularize your higher orbit (where your speed will be the lowest, and you have the least Oberth effect), you will have less dV to get if you periapsis is higher. That will be if you have already achieved a lower orbit. But for the orbits we are talking about, I don't think that is a big difference?

[OhioBob]

but we aren't 'essentially in orbit'... we're still suborbital. And the longer burn is coming at a higher AoA, or you'd be orbital. You're essentially slowing the gravity turn drastically, and continuing to burn more vertically. You still need to burn horizontally to circularize, and now raise the Periapsis to the much higher orbit all in one burn. The more I think about it, the harder it is to believe that this is as efficient as a small orbit, raised by burning horizontally (relative to planet)...

I don't think you really mean AoA (angle of attack), I think what you are referring to is "flight path angle". Flight path angle is the angle that the velocity vector makes with the horizontal plane.

In the example I gave, the flight path angle at the time of the extended burn is 3.086^o. That is so close to horizontal that the losses are negligible. Also note that after I complete the burn I am effectively orbital, 37km x 250km, and I have nearly a half orbit to complete before I reach apoapsis (155^o). What I'm doing is almost a Hohmann transfer from the burnout point up to the 250 km apoapsis.

Maybe you are use to more highly lofted ascent trajectories?

Edited October 6, 2015 by OhioBob

[ajburges]

"Gravity Losses" is a deceptively broad term.

Orbital mechanics can approximate any suborbital trajectory as an orbital trajectory that goes below the surface of the orbited body. Orbital insertion then becomes a problem of how to grow am elliptic orbit efficiently starting at Pe with limited TWR (and atmosphere). There is no "gravity losses" from suborbital trajectories alone. What gravity losses actually are is radial thrust to keep you from the section of the trajectory that is subterranean; most craft don't do well underground much less several hundred km below the surface. Once you complete a gravity turn, you can eliminate that radial thrust completely.

From an impulse perspective, applying thrust at any point other than Ap and Pe is a suboptimal way to alter the orbital energy. Thrusting of prograde is a different form of inefficiency.

Which approach to orbital insertion is better is evaluating which inefficiency is greater. Thrusting off Ap/Pe or raising Pe from the lower Ap (and thus raising Ap with less velocity). Typically, the low difference in phase from PE means that insertion to high orbit is more efficient than raising a low orbit with the velocity loss. High Ap insertion spends less time in atmo.

Note: a naive reader could take this as support for the notorious up and turn right approach. This is not so. Remember, we are trying to grow an extremely small, elliptic orbit without going subterranean. Going "up" is radial thrust starting with 0 component velocity. Better to use some of the velocity we get from planetary rotation. Going "up" is just to prevent us from passing Ap and needing even less efficient thrust application.

Inclination changes are another interesting aspect. The role velocity plays in plane adjustment would imply that plane correction for arrival should occur at the Ap of a minimal capture (since the highly elliptical orbit is a prerequisite to circular capture). In practice, for small plane changes, I often find that a normal component in my capture burn is cheaper because of trigometric ratios of an already large capture burn make normal components incredibly cheap. The sum of sine and cosine is greater than 1 between 0 and 90 degrees! That means you can get net component thrust greater than their combined vector!

[OhioBob]

I often find that a normal component in my capture burn is cheaper because of trigometric ratios of an already large capture burn make normal components incredibly cheap.

I frequently do the same thing. For instance, if I want an equatorial orbit around the target planet, I adjust my approach trajectory so that the periapsis lies in the planet's equatorial plane. I can, therefore, combine a plane change with the capture burn to end up in an equatorial orbit. Much cheaper ÃŽâ€v-wise than performing two burns.

[Jason Patterson]

but we aren't 'essentially in orbit'... we're still suborbital. And the longer burn is coming at a higher AoA, or you'd be orbital. You're essentially slowing the gravity turn drastically, and continuing to burn more vertically. You still need to burn horizontally to circularize, and now raise the Periapsis to the much higher orbit all in one burn. The more I think about it, the harder it is to believe that this is as efficient as a small orbit, raised by burning horizontally (relative to planet)...

The only reason I was suborbital in my example is because Kerbin has an atmosphere. The periapsis was right at Kerbin's surface. I suspect that some of the confusion in general (though perhaps not in your case) comes from thinking of an orbit of this type as 71x0 instead of the far more accurate 671x600km that it really is. It's only 11% out of circular at the time we were deciding either to coast to a 71km apoapsis or continue burning to a 250km apoapsis.

Another way to think of this: If I burnt at periapsis to raise my apoapsis by 1km, then circularized, then repeated until I reached my destination orbit, that would be substantially less economical than a few well-timed burns to reach the same goal due to increased use of the Oberth effect. The limit of how few short duration burns you can make in this case is two - one to raise the apoapsis and one to circularize, and that should be most efficient.

[Nothalogh]

I think the OP question is: If I intend to get to a 250x250 orbit, should I:

• set my AP to 250 and when I reach it, I circularize?
  
• set my AP to 71, circularize and THEN do Hohman transfer to 250x250?
  

Unfortunately, I don't know the answer either... I'd very like to learn it from you guys...

My approach would be to gravity turn to 55km, immediately hohmann transfer to 250km, raise PE at AP, and circularize from there

[cantab]

I frequently do the same thing. For instance, if I want an equatorial orbit around the target planet, I adjust my approach trajectory so that the periapsis lies in the planet's equatorial plane. I can, therefore, combine a plane change with the capture burn to end up in an equatorial orbit. Much cheaper ÃŽâ€v-wise than performing two burns.

I've started to get vaguely competent at approaching in the correct inclination to start with. Key is to make a mid-course correction and play around with the timing of said correction. There are probably still situations when I'll need to make a plane change after initial capture though.

[Red Iron Crown]

I frequently do the same thing. For instance, if I want an equatorial orbit around the target planet, I adjust my approach trajectory so that the periapsis lies in the planet's equatorial plane. I can, therefore, combine a plane change with the capture burn to end up in an equatorial orbit. Much cheaper ÃŽâ€v-wise than performing two burns.

Would it not be cheaper to just barely capture in a highly elliptical orbit, plane change at Ap, then circularize at Pe?

[FyunchClick]

Edit: ninja'd.

I frequently do the same thing. For instance, if I want an equatorial orbit around the target planet, I adjust my approach trajectory so that the periapsis lies in the planet's equatorial plane. I can, therefore, combine a plane change with the capture burn to end up in an equatorial orbit. Much cheaper ÃŽâ€v-wise than performing two burns.

With periapsis being at the same place as ascending/descending node, wouldn't the cheaper option be to burn just enough retrograde at periapsis for a highly eccentric capture , and then burn for a plane change at apoapsis? Since your speed is so much lower at apoapsis when very eccentric, any change to the speed vector(lit. dV) should be that much lower too. Of course, you need to combine normal/anti-normal with retrograde to prevent lifting periapsis). You can then finalize your burn for a circular orbit once you swing back to periapsis.

Or in other words, since periapsis is where you're V vector is biggest, it also requires a lot of dV to modify the direction that vector; thus it would be the worst place to do a plane change.

Or am I missing some arcane aspect of orbital maths here?

[ajburges]

Would it not be cheaper to just barely capture in a highly elliptical orbit, plane change at Ap, then circularize at Pe?

Yes but no.

The normal dV component is indeed smaller with the low V burn vs the high V burn. The savings comes from the non-linear behavior of trigometric functions and the large dV cost of a capture burn.

For an off prograde burn of angle a the prograde component uses the coefficient cos(a) and the orthogonal component has a coefficient of sin(a). The sun of those coefficients is greater than 1 for the range (0°, 90°). That means that the sun of the burn components is greater than the burn. As a result, while the normal component of a high V burn may be greater, adding it to a large burn is paradoxically cheaper than the small burn at a low dV.

Note the primary savings is due to the trigometric sum. The trigometric sun is greatest at 45°. Past that, the sum decreases. A 45° off retrograde capture burn will not even produce 45° of inclination change. The math may indicate a lower thrust angle being the break even point. Someone would need to derive the equations to evaluate the optimal solution. Even then, there will be no easy answer since it depends on SoI, initial V, and initial Pe.

The cherry on top of this is this manuver is the one of the few times that the optimal manuver is also the fast one.

- - - Updated - - -

My phone does not like it when I speak math. Please excuse the excessive auto complete errors.

[Laie]

Would it not be cheaper to just barely capture in a highly elliptical orbit, plane change at Ap, then circularize at Pe?

Depends on the magnitude of the plane change.

You can "hide" a normal component in your capture burn at very little cost. For my typical capture burn, this doesn't amount to much: on the order of two or five degrees, ten tops. Any more than that and it becomes cheaper to schedule another burn at high apoapsis. (Cheaper than doing it all at once I mean)

Edited October 7, 2015 by Laie

[ajburges]

I've started to get vaguely competent at approaching in the correct inclination to start with. Key is to make a mid-course correction and play around with the timing of said correction. There are probably still situations when I'll need to make a plane change after initial capture though.

I've come to the line of thinking that inclination tuning is the last concern for incoming trajectory.

In order:

1) intercept

2) pro/retrograde equatorial/polar orbit (avoid orbit reversals)

3) establish planar intercept with target orbit at Pe (I want to do as much plane change work as possible for "free" with the capture burn)

4) if possible use minor radial adjustment and follow up to decrease arrival time and/or inclination difference.

The added advantage of this approach is I can get a 1 orbit, 1 (rarely 2) burn intercept with any target in the SoI upon arrival. Most my satilite contracts also complete with the capture burn.

[FyunchClick]

while the normal component of a high V burn may be greater, adding it to a large burn is paradoxically cheaper than the small burn at a low dV.

Interesting. Until now I've saved my plane changes for a point high in my orbital swing on the assumption that would always be cheapest there. I'll go and try how it works out with my next few captures.

[OhioBob]

Interesting. Until now I've saved my plane changes for a point high in my orbital swing on the assumption that would always be cheapest there. I'll go and try how it works out with my next few captures.

The equation to compute the ÃŽâ€v of a combined altitude and plane change is,

where V_i is initial velocity, V_f is final velocity, and θ is the plane change angle.

You can use this equation to compute the ÃŽâ€v with the plane change and without it. You can then compare the difference it to what the ÃŽâ€v would be if you performed the plane change at apoapsis of a highly eccentric orbit. This way you can decide which is cheaper ÃŽâ€v-wise. However, you will find that small plane changes, up to a few degrees, can be made almost for free when you combine them with the capture burn. Even if you plan to do the bulk of the plane change at high apoapsis, you should always try to do at least a few degrees at capture because it comes so cheap.*

(ETA) * This technique is used in real life whenever a geostationary satellite is launched out of Cape Canaveral. Because of the latitude of the launch site, the satellite is initially launched into a low Earth orbit with an inclination of about 29 degrees. When they perform the burn to inject the satellite into the geostationary transfer orbit, they’ll take a few degree of inclination out of the orbit because they can do so for so little additional ÃŽâ€v, even though the velocity is very high at that point. Then, when they reach geostationary distance and the velocity is slow, they perform the bulk of the plane change in combination with the orbit circularization burn.

Edited October 8, 2015 by OhioBob