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To follow or not to follow the node...


Streetwind
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...that is a question I have! :P

Assume I have low-thrust propulsion and wish to make an ejection burn in low Kerbin orbit. Assume I have set up a maneuver node to do so, and that maneuver node tells me I need to burn, in total, for a full 90 degree arc of the planet. Or in other words, 45 degrees before the node, all the way to 45 beyond it. Finally, assume that my vessel's mass and TWR do not change enough during the burn to be relevant.

How should I execute this burn? Should I set my vessel to follow the maneuver node, resulting in a strong radial-in component at the start of the burn and a strong radial-out component at the end? Or should I set my vessel to follow prograde, resulting in burning off-node in one direction at the start and off-node in the other direction at the end? Or is there perhaps no difference between the two approaches? Does one of the two perhaps result in the maneuver node counting the expended dV wrong?

And, does it make a difference if we're talking about a one-time ejection burn or about periapsis kicking?


(And please don't tell me to split the burn, this is a contrived example to aid the discussion, not an actual burn I need to make ingame right now. ;))
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Fantastic analysis, OhioBob. Although I still don't understand why it works out that way (I am a potato at orbital mechanics, hence this thread)


As to the why, perhaps the following image will help to illustrate it.

DepartureTrajectories.png

This is just a graph that I made using Excel, so I couldn't get real fancy with labels, etc. Imagine that the planet Kerbin is centered on the origin of the axes (0, 0) with a radius of 600 km. The colored curved lines represent the trajectories that the spacecraft will follow as it leaves low Kerbin orbit and heads out into space. The dashed green line represents the theoretical orbit that the spacecraft would follow if the dV were applied instantly at the maneuver node. The maneuver node is located at the coordinates 0, -680. The red line represents the trajectory we follow by burning prograde. The blue line represents the trajectory we follow by burning to the maneuver node or, more correctly, to the right as the image is oriented. Both the red and blue trajectories start 30 degrees before reaching the maneuver node, with the burn time centered on the maneuver node.

Not only do we want to exit Kerbin space with the correct velocity, but also in the correct direction. The correct direction is along the path of the dashed green line. You can see that when we burn prograde, we immediately start to push the vehicle farther and farther away from Kerbin, ending in a significant deviation from where we want to go. When burning in the direction of the maneuver node, we stay much truer to the desired trajectory. In fact, it looks to me like the location of the maneuver node on the Navball auto-corrects as the burn progresses, which likely keeps us even closer on course than this image represents.

Of course, either trajectory can be made to approximately match the green line by adjusting the start time of the burn. In the case of the blue line, starting the burn a little sooner would move the outgoing path a little closer to the green line (this is what Yasmy suggests). In the case of the red line, we would want to start the burn later, though I'm not sure there's a simple formula to compute the start time.

Since the dV loss that comes with burning to the maneuver node is relatively small, there seems to be little advantage to using the prograde method. Burning to the maneuver node appears to be the easier and better solution.

Edited by OhioBob
fixed formatting and image
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At least burning prograde is not wasting energy. Neither of those approaches will get you to the orbit you expect to get while looking at patched conics. Splitting the node to consecutive burns, each about 1 minute-long - is the best way to approximate the result. I don't know about any mods, that actually perform numerical integration to show you precise resulting orbit.
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Buring prograde is not loosing fuel, but your PE won't be at the same location. Thus your AP won't be where you want it to be. You may burn less dV BUT you'll be off course. With the speed you gained, you correction burn may be expensive.

Personally I always stick to Node. If your burn is long, you'll waste fuel (some kind of anti-radial, then radial), but you'll be much near you target course.

The ideal way to do it is to do multiple short burns at PE. The issue become the timing, but it's more accurate and fuel efficient. You should target specific orbit AP with know orbital revolution duration, so you can manage timing correctly. I find that uneasy for my skills.

The other solution is to change interplanetary stage to a higher TWR, which is easier to manage, but usually more expensive (more engines, more fuel, more LKO stage.
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As Warzouz said, raise your AP over a few orbits & then do the final burn. The timing isn't really that big a deal, for the initial burn set the node up for your complete transfer & use that to raise AP the first time, then just raise AP as normal for a few times. You can try and hit the absolute ideal transfer window if you like, but it's not that big a deal if you're a couple of days off ( which is a *lot* of orbits ).

Only annoying thing about doing that is moons getting in the way.
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First off, I'm assuming you're aligning your burn such that the middle of the burn coincides with the maneuver node. I highly suspect that you are too, but I just wanted to clarify.

Setting SAS to point prograde and just burning prograde for the length of the maneuver should be the most fuel-efficient way to complete your maneuver, but for a burn that long your final trajectory will be different than your planned trajectory. Make sure you have enough deltaV budget for correction burns to tweak your flight path.

Following the maneuver through the whole length of the burn will land you on a trajectory that's closer to your planned one, but will be less efficient. The real question is whether the correction burns from burning straight prograde make up the amount that gets wasted by following the maneuver. I suspect that they're about the same, but I have no evidence of this beyond a handful of personal experiences.

Hope this helps.

Note: I deliberately ignored multiple burns (aside from the correction burns) and TWR because, hey, it's your thread. ;^)
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Long burns have also another side effect, except burning more fuel : you could cross the atmosphere, or hit the planet.

I remembered a slowdown at Moho. I set a PE at 50km, the burn was so long the the PE reduced a lot. I don't remember precisely, but I think It ended at less than 20km, the I had to continue burning after PE and the actual orbit crossed Moho. I had to fix it at AP, the redo another PE burn to circularize.

Now, I keep a larger distance with Moho...

Anyway, capturing is an easy manover, you only lose fuel. Departure is more delicate because you have to be on he correct course. I remembered messing with a departure and being off course by 10°. I ended burning a lot of fuel to fix it. Bur again, that was a long time ago, in the beta.

As for burn angles I'm happy if the whole burn is under 30° (2.6min for 80km orbit, up to 9.4min for 1000km)
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[quote name='BenCushwa']Setting SAS to point prograde and just burning prograde for the length of the maneuver should be the most fuel-efficient way to complete your maneuver, but for a burn that long your final trajectory will be different than your planned trajectory. Make sure you have enough deltaV budget for correction burns to tweak your flight path.

Following the maneuver through the whole length of the burn will land you on a trajectory that's closer to your planned one, but will be less efficient. The real question is whether the correction burns from burning straight prograde make up the amount that gets wasted by following the maneuver. I suspect that they're about the same, but I have no evidence of this beyond a handful of personal experiences.[/QUOTE]

Yep, this is precisely the thing I was looking for. Thanks BenCushwa, for being the one reply out of five that actually read the OP and responded to the [I]question[/I], not to the artificial situation :)

Interestingly enough, I have heard prominent KSP figures - Scott Manley, to name one example - mention that they're getting better results by following the prograde marker instead of the node, especially as the arc traversed (and thus the radial component) gets larger. Which means I now have some votes for one approach and some votes for the other. This makes me even more curious - what is the exact difference between the two approaches in the sense of orbital mechanics?

By following the prograde marker, you begin to raise your apoapsis at the wrong location, pushing it off to one side of where you want to be going. But as you travel around the planet and past the node, you counter this by pushing the apoapsis off into the [I]other[/I] direction from where you want to be going, twisting your orbit. Technically, as long as your ship's acceleration is reasonably close to constant over the burn, these errors should largely cancel each other out, though there will be a small error introduced by the difference in time spent on each arc due to your speed... which may in turn get canceled by a growing Oberth effect... and there's also the fact that by burning prograde before arriving at the node, you slightly [I]raised[/I] the orbit where the node itself is sitting.

By following the node marker, you begin to twist your orbit anticlockwise with a radial-in component in the burn, making your apoapsis start to rise off to the side of where you want to be going. But as you travel around the planet and past the node, you counter this with a radial-out component on the second leg of the burn, twisting your orbit clockwise and pushing your apoapsis into the other direction. Technically, as long as your ship's acceleration is reasonably close to constant over the burn, these errors should largely cancel each other out, though there will be a small error introduced by the difference in time spent on each arc due to your speed... which may in turn get canceled by a growing Oberth effect... and there's also the fact that by burning radial-in before arriving at the node, you slightly [I]lowered[/I] the orbit where the node itself is sitting.

It's striking how similar that sounds, no? In one approach, you raise your orbit a bit above the node as you pass it, while twisting your orbit first clockwise, then anticlockwise. In the other approach, you lower your orbit a bit below the node as you pass it, while twisting your orbit first anticlockwise, then clockwise. At first glance, it's almost like they're polar opposites that should lead to a very similar end result with a very similar deviation, if perhaps in the opposite direction.

How can it be that one method generates less of an error than the other?
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I don't have much experience with this, but I did once perform an experiment comparing the two methods. I didn't bother to compare the dV because I was more interested in finding out which method got me closer to the desired trajectory. (I figured that if the less accurate method were cheaper in terms of dV, the difference would likely be negated by having to perform a larger course correction.) My conclusion was that following the maneuver node was better, so that's what I've done ever since.

Edited by OhioBob
fixed formatting
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[quote name='Streetwind']How can it be that one method generates less of an error than the other?[/QUOTE]

Again, this is a total SWAG, but I think it has something to do with the fact that if you burn prograde, the direction of prograde changes over the duration of a long burn, whereas maintaining a maneuver assumes making a burn in a fixed direction. (This is ignoring the fact that the maneuver marker can change position over time because I have no idea how that change is calculated. I could be dead wrong here.) It's like burning normal/anti-normal to change your inclination: as your orbital inclination changes, so do your N/AN vectors, so you have to change your heading otherwise you'll start adding some prograde or retrograde.

Keep in mind, the whole notion of patched conics is based on the idea of instantaneous (i.e. zero burn length) deltaV being applied at the maneuver node. Putting the maneuver node at the middle of a burn with an actual duration provides a close approximation, but the longer the burn the less accurate the approximation.

EDIT: Take the notion of a long-burn maneuver node to the extreme for a moment. Assume that you have a simple prograde maneuver node that requires a burn so long that you have to start it on the direct opposite side of whatever body you're orbiting. If you burn according to the maneuver vector, you'll be burning the [I]exact wrong direction[/I] for the maneuver and you'll end up on a trajectory nothing like what you planned, and you'll likely de-orbit to boot. If you burn prograde and maintain prograde until you generate enough deltaV, you'll likely get much closer to the intended results.

I <3 orbital mechanics. B^) Edited by BenCushwa
Obvious edit is obvious
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[quote name='OhioBob'][FONT=Calibri][SIZE=3][COLOR=#000000]I don’t have much experience with this, but I did once perform an experiment comparing the two methods. [/COLOR][/SIZE][SIZE=3][COLOR=#000000]I didn’t bother to compare the Δv because I was more interested in finding out which method got me closer to the desired trajectory. [/COLOR][/SIZE][SIZE=3][COLOR=#000000](I figured that if the less accurate method were cheaper in terms of Δv, the difference would likely be negated by having to perform a larger course correction.) [/COLOR][/SIZE][SIZE=3][COLOR=#000000]My conclusion was that following the maneuver node was better, so that’s what I’ve done ever since.[/COLOR][/SIZE][/FONT][/QUOTE]

Roughly how big was the error in your test? Were the two approaches pretty close together, or noticably different from each other?



[quote name='BenCushwa']Again, this is a total SWAG, but I think it has something to do with the fact that if you burn prograde, the direction of prograde changes over the duration of a long burn, whereas maintaining a maneuver assumes making a burn in a fixed direction. (This is ignoring the fact that the maneuver marker can change position over time because I have no idea how that change is calculated. I could be dead wrong here.) It's like burning normal/anti-normal to change your inclination: as your orbital inclination changes, so do your N/AN vectors, so you have to change your heading otherwise you'll start adding some prograde or retrograde.

Keep in mind, the whole notion of patched conics is based on the idea of instantaneous (i.e. zero burn length) deltaV being applied at the maneuver node. Putting the maneuver node at the middle of a burn with an actual duration provides a close approximation, but the longer the burn the less accurate the approximation.

EDIT: Take the notion of a long-burn maneuver node to the extreme for a moment. Assume that you have a simple prograde maneuver node that requires a burn so long that you have to start it on the direct opposite side of whatever body you're orbiting. If you burn according to the maneuver vector, you'll be burning the [I]exact wrong direction[/I] for the maneuver and you'll end up on a trajectory nothing like what you planned, and you'll likely de-orbit to boot. If you burn prograde and maintain prograde until you generate enough deltaV, you'll likely get much closer to the intended results.

I <3 orbital mechanics. B^)[/QUOTE]

Not sure the example is a good one, since burning more than 90 degrees from the node is counterproductive in every case and heading... though maybe the 90 degree variant might be a good extreme-case test scenario, I suppose.

And about change in direction versus fixed direction, hmm. I'm not sure. You're twisting your orbit about a whole lot with that radial component, which should shift the apoapsis around... maybe it's a result of radial burns being so inefficient when deep in a gravity well, that the deviation is less than what you get from shifting your ap around with fully prograde burns? I can only speculate...
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It's a bit of a tradeoff. Point at the node and you risk pushing Pe down into atmo if your starting orbit is low. Point at prograde and you raise Pe before you get there, squandering precious Oberth effect. My usual technique for longer burns is to point between the two and watch periapsis carefully, if it starts rising point closer to the node, if it starts falling point closer to prograde. Once past Pe (and assuming it's an ejection and not a periapsis kick), just point at the node.
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[quote name='Red Iron Crown']My usual technique for longer burns is to point between the two and watch periapsis carefully, if it starts rising point closer to the node, if it starts falling point closer to prograde. Once past Pe (and assuming it's an ejection and not a periapsis kick), just point at the node.[/QUOTE]

So you treat periapsis kicking differently?

And, wouldn't that introduce errors, since you burned off-node during the first half but on-node during the second? You would need a little bit of radial-out to cancel out the anticlockwise orbit twist you reated during the initial radial-in burn, no?



[quote name='Brainlord Mesomorph']How many times to I have to post this:

Long Burns? pls see the tutorial in my sig.[/QUOTE]

About as many times as I need to point out that this thread has [I]nothing whatsoever[/I] to do with needing a better way to handle low-TWR burns... :rolleyes:

You'd think that literally spelling out "please don't tell me to split the burn" in the OP would help. Edited by Streetwind
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[quote name='Brainlord Mesomorph']How many times to I have to post this:

Long Burns? pls see the tutorial in my sig.

(RIC's method requires mods. Mine works in a stock game :D)[/QUOTE]
My method works with a protractor on the screen too, no mods required. My method also doesn't take a significant part of a year to complete. :P

[quote name='Streetwind']So you treat periapsis kicking differently?[/quote]
On a kick I continue to maintain periapsis altitude after Pe, burning until I've reached the desired SMA. For an ejection once you pass Pe it doesn't matter anymore.

[quote]And, wouldn't that introduce errors, since you burned off-node during the first half but on-node during the second? You would need a little bit of radial-out to cancel out the anticlockwise orbit twist you reated during the initial radial-in burn, no?[/QUOTE]
The node automatically adjusts itself for off-node burning, so pointing at the node will get fairly close to the desired trajectory.
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[quote name='Red Iron Crown']On a kick I continue to maintain periapsis altitude after Pe, burning until I've reached the desired SMA. For an ejection once you pass Pe it doesn't matter anymore.[/QUOTE]

Interesting technique. Whenever I do that, I rely on KER to keep the argument of periapsis as static as possible... I assume maintaining the periapsis height has a comparable effect, as long as your margin of error does not increase too far through burns taking too long?
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[quote name='Red Iron Crown']My method works with a protractor on the screen too, no mods required. My method also doesn't take a significant part of a year to complete. :P

[/QUOTE]

A protractor?! hadn't thought of that.
Do i still HAVE one?... yes, its in my collection of antique drafting tools(with my slide-rule! Literally!)

oh, and yes, my method can take tree quarters of a Kerbal year. Actually it works BETTER on Eve and Moho. ... and not very well at al; at Jool and Eloo. Where Alignment can be years before departure. :P Edited by Brainlord Mesomorph
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*** NOTE THAT THESE RESULTS CONTAIN A MATHEMATICAL ERROR. PLEASE SEE REPLY #30 FOR CORRECTION. ***

I just tested the two scenarios using a simulation.

The test was for a hypothetical trip to Jool. I figured an 80 km parking orbit and a theoretical instantaneous Δv of 1990.4 m/s. This Δv gives a hyperbolic excess velocity of 2800 m/s, which is typical for a Jool mission. I equipped the vehicle with a Poodle engine and gave it an initial mass of 51721 kg. This mass was chosen because it yields a theoretical burn time of 312.47 s, which is 1/6th of an orbit. I started the burn 30o before the maneuver node, i.e. half before the node and half after.

In scenario #1 the thrust vector was maintained in a prograde direction throughout the burn. In scenario #2 the thrust vector remained locked in a fixed attitude in relation to the stars. The fixed attitude was that of the orbit tangent through the maneuver node. Scenario #2 should approximate keeping the vehicle aligned to the maneuver node marker on the Navball. In both cases I terminated the burn the instant the hyperbolic exess velocity reached 2800 m/s.

In both cases the actual Δv ended up greater than the theoretical, which is what we would expect. However, the difference between the two scenarios was small. Keeping the vehicle aligned to the prograde marker was slightly better at 2054.4 m/s, versus 2067.4 m/s for the fixed thrust vector alignment. In the second scenario, the vehicle altitude dipped as low as 75.26 km.

Although both scenarios resulted in the same hyperbolic excess velocity, the resulting orbits were different. Below is a comparison of the orbital parameters of the two scenarios and the theoretical ideal orbit based on the Δv being applied instantly at the maneuver node. As you can see, scenario #2, i.e. alignment to the maneuver node, is a closer match to the theoretical orbit. Of course, it is possible to adjust the start time of either scenario to improve the longitudes; however, if we stay with the practice of centering the burn on the maneuver node, then scenario #2 is better.
 

 

Scenario #1,
aligned to
Prograde

 

Scenario #2,
aligned to
Maneuver Node

 

Theoretical
Ideal
Orbit

 

Semi-major axis, km

-450472

-450473

-450459

Eccentricity

2.58596

2.51543

2.50957

Periapsis altitude, km

114430

82660

80000

Longitude of periapsis, deg.

-7.506

-2.466

0.000

Longitude at infinity, deg.

105.24

110.96

113.48

 

Edited by OhioBob
fixed formatting
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[I]Edit: Oops: I just saw the second page of responses. Missed it before writing this. Nice tests OhioBob![/I]

<Removed a bunch of junk>

[COLOR="silver"][SIZE=1]- - - Updated - - -[/SIZE][/COLOR]

OhioBob, if you are up for some more tests, how about splitting the burn [url=http://forum.kerbalspaceprogram.com/threads/125413-Precision-Burns-Before-the-Node-After-the-Node?p=2020522&viewfull=1#post2020522]like this instead[/url]:

t = T (m/mf) (1 - sqrt(1 - mf/m)), where T is the total burn time, mf is the fuel consumed, and m in the initial total mass. Use the theoretical values for an instantaneous impulse burn.

This should roughly split the delta-v in half before and after the node. I don't think it is the best split, but I suspect it is better than T/2. I expect that it will reduce the error in the longitude of periapsis. On the other hand, it would increase the periapsis. I'm interested in if it puts you closer to your intended orbit.

Edit: Here is an equivalent, but perhaps more usable form: t = T (1-exp(-dv/(2 Isp g0))) / (1-exp(-dv/(Isp g0)).
Then from your numbers, t = 0.571 T = 178.7 seconds, instead of T/2 = 156.2 seconds. Edited by Yasmy
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[quote name='Streetwind']Not sure the example is a good one, since burning more than 90 degrees from the node is counterproductive in every case and heading... though maybe the 90 degree variant might be a good extreme-case test scenario, I suppose.

And about change in direction versus fixed direction, hmm. I'm not sure. You're twisting your orbit about a whole lot with that radial component, which should shift the apoapsis around... maybe it's a result of radial burns being so inefficient when deep in a gravity well, that the deviation is less than what you get from shifting your ap around with fully prograde burns? I can only speculate...[/QUOTE]
It was intended as an extreme example to show that for burn long enough you cannot simply burn to the maneuver node and expect to land up in the target orbit. And I totally agree re: radial component and shifting apoapsis; this is why I included provisions for a correction burn in the "burn straight prograde" scenario.

[quote name='OhioBob'][FONT=Calibri][SIZE=3][COLOR=#000000]
[TABLE="width: 575"]
[TR]
[TD="width: 147, bgcolor: transparent"][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]Scenario #1 aligned to Prograde

[/FONT][/COLOR]
[/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]Scenario #2 aligned to Maneuver Node[/FONT][/COLOR]
[/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]Theoretical Ideal Orbit[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"][COLOR=black][FONT=Calibri]Semi-major axis, km[/FONT][/COLOR][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]-450472[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]-450473[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]-450459[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"][COLOR=black][FONT=Calibri]Eccentricity[/FONT][/COLOR][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]2.58596[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]2.51543[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]2.50957[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"][COLOR=black][FONT=Calibri]Periapsis altitude, km[/FONT][/COLOR][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]114430[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]82660[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]80000[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"][COLOR=black][FONT=Calibri]Longitude of periapsis, deg.[/FONT][/COLOR][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]-7.506[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]-2.466[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]0.000[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"][COLOR=black][FONT=Calibri]Longitude at infinity, deg.[/FONT][/COLOR][/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]105.24[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]110.96[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 77, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Calibri]113.48[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 147, bgcolor: transparent"]Δv, m/s[/TD]
[TD="width: 77, bgcolor: transparent, align: center"]2054.4[/TD]
[TD="width: 77, bgcolor: transparent, align: center"]2067.4[/TD]
[TD="width: 77, bgcolor: transparent, align: center"]1990.4[/TD]
[/TR]
[/TABLE]

[/COLOR][/SIZE][/FONT][/QUOTE]

Excellent work. It seems I was reasonably correct: burning to a maneuver node will cost you more deltaV and be closer to your desired orbit, whereas burning prograde will save you some deltaV but require correction. In your case, I'd wager that the 13 deltaV saved won't be enough for a correction burn, so burning to the maneuver node is likely the better option.

Cheers!
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Quote

OhioBob, if you are up for some more tests, how about splitting the burn [URL=http://forum.kerbalspaceprogram.com/threads/125413-Precision-Burns-Before-the-Node-After-the-Node?p=2020522&viewfull=1#post2020522] like this instead:

t = T (m/mf) (1 - sqrt(1 - mf/m)), where T is the total burn time, mf is the fuel consumed, and m in the initial total mass. Use the theoretical values for an instantaneous impulse burn.

This should roughly split the delta-v in half before and after the node. I don't think it is the best split, but I suspect it is better than T/2. I expect that it will reduce the error in the longitude of periapsis. On the other hand, it would increase the periapsis. I'm interested in if it puts you closer to your intended orbit.


*** NOTE THAT THESE RESULTS CONTAIN A MATHEMATICAL ERROR. PLEASE SEE REPLY #30 FOR CORRECTION. ***

Using your equation I compute t = 178.5 s. This means is that I have to start the burn 34.3o prior to the maneuver node rather than 30o. Making this change modifies the numbers to the following:
 

 

Scenario #1,
aligned to
Prograde

 

Scenario #2,
aligned to
Maneuver Node

 

Theoretical
Ideal
Orbit

Semi-major axis, km

-450472

-450467

-450459

Eccentricity

2.58596

2.51420

2.50957

Periapsis altitude, km

114430

82098

80000

Longitude of periapsis, deg.

-11.806

-3.616

0.000

Longitude at infinity, deg.

100.94

109.82

113.48

Δv, m/s

2054.4

2071.8

1990.4

 

- - - Updated - - -

I just discovered something interesting. On a whim I decided to see what happened if I aligned the thrust vector with the maneuver node for the first half of the burn, and then with the prograde marker for the second half of the burn. The resulting longitude of periapsis was a very close match to the theoretical orbit. I think this is the best result of the options tested so far. Perhaps we've found a better way to execute our burns? I think some in game testing is warranted.
 

 

Scenario #3,
aligned to Node
then Prograde

 

Theoretical
Ideal
Orbit

Semi-major axis, km

-450454

-450459

Eccentricity

2.54487

2.50326

Periapsis altitude, km

95892

77158

Longitude of periapsis, deg.

0.170

-0.228

Longitude at infinity, deg.

113.31

113.32

Δv, m/s

2058.1

2077.1

 

Edited by OhioBob
fixed formatting
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Though I haven't tracked it explicitly, I find my emperical experience matches OhioBob's post (the previous one). I tend to stick with burning to the maneuver node, because I found that burning on prograde seemed to require a lot more adjustment after the burn.

If anything for a long burn (like burning 6+ minutes for Jool), I found that I saved way more by working to split the burn, rather than trying to really split hairs on the prograde/maneuver node portion.


Though OhioBob's most recent post (burn at the maneuver node, then prograde) sounds very interesting. Efficient burn AND ends up at the right target.

Cheers,
~Claw
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Cool. Thanks for the second set of tests. Splitting the delta-v around the node appears to be worse than splitting the burn time for either method. I haven't seen someone do that test before.

I like your method of node, then prograde. Looks like a winner.
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Fantastic analysis, OhioBob. Although I still don't understand [I]why[/I] it works out that way (I am a potato at orbital mechanics, hence this thread), it appears you've discovered something new! :) I especially like how much that method preserves the longitudes, meaning it should be suitable both for direct burns and for periapsis kicking.
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