To follow or not to follow the node...

[Warzouz]

[quote name='OhioBob'][FONT=Calibri][SIZE=3][COLOR=#000000]I just tested the two scenarios using a simulation.[/COLOR][/SIZE][/FONT][SIZE=3][COLOR=#000000][FONT=Calibri]
[/FONT][/COLOR][/SIZE][FONT=Calibri][SIZE=3][COLOR=#000000]
The test was for a hypothetical trip to Jool. [/COLOR][/SIZE][SIZE=3][COLOR=#000000]I figured an 80 km parking orbit and a theoretical instantaneous Δv of 1990.4 m/s.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] This Δv gives a hyperbolic excess velocity of 2800 m/s, which is typical for a Jool mission. [/COLOR][/SIZE][SIZE=3][COLOR=#000000]I equipped the vehicle with a Poodle engine and gave it an initial mass of 51721 kg.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] This mass was chosen because it yields a theoretical burn time of 312.47 s, which is 1/6[/COLOR][/SIZE][SUP][SIZE=2][COLOR=#000000]th[/COLOR][/SIZE][/SUP][SIZE=3][COLOR=#000000] of an orbit.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] I started the burn 30[/COLOR][/SIZE][SUP][SIZE=2][COLOR=#000000]o[/COLOR][/SIZE][/SUP][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000] before the maneuver node, i.e. half before the node and half after.
[/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000]
In scenario #1 the thrust vector was maintained in a prograde direction throughout the burn. [/COLOR][/SIZE][SIZE=3][COLOR=#000000]In scenario #2 the thrust vector remained locked in a fixed attitude in relation to the stars.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] The fixed attitude was that of the orbit tangent through the maneuver node. [/COLOR][/SIZE][SIZE=3][COLOR=#000000]Scenario #2 should approximate keeping the vehicle aligned to the maneuver node marker on the Navball.[/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000] In both cases I terminated the burn the instant the hyperbolic exess velocity reached 2800 m/s.
[/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000]
In both cases the actual Δv ended up greater than the theoretical, which is what we would expect.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] However, the difference between the two scenarios was small.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] 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. [/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000]In the second scenario, the vehicle altitude dipped as low as 75.26 km.
[/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000]
Although both scenarios resulted in the same hyperbolic excess velocity, the resulting orbits were different.[/COLOR][/SIZE][SIZE=3][COLOR=#000000] 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.[/COLOR][/SIZE][/FONT][FONT=Calibri][SIZE=3][COLOR=#000000] As you can see, scenario #2, i.e. alignment to the maneuver node, is a closer match to the theoretical orbit. [FONT=Calibri]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.

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

Nice data, as I suspect the test #2 is nearer the intended trajectory, the test #1 would need a more expensive correction burn, isn't it ? Do you have an estimation of that ?

OR

It should be possible to change the start burn time to achieve the same orbit as node targeting using prograde burn. Is it possible ?

[OhioBob]

I think that any of the scenarios can be made to work well. It's just a matter of timing the burn so that the resulting departure is at the correct angle. However, I don't think we want a method that involves a bunch of complex math to determine when to start the burn. The common practice is to split the burn time 50/50 around the maneuver node, which is easy to do. In that case, following the maneuver node appears to get us closer to the desired trajectory and, therefore, should require a smaller course correction (though I have not done any calculations to determine the magnitude of the correction). This concurs with my in game experience as well, as little as there is.

The idea of following the maneuver node for half the burn and then switching to the prograde marker I think requires more testing. It appears to work very nicely for this particular example, but I need to make sure that is not a fluke. I'd like to test the idea under different sets of circumstances to see if there is any consistency to the result.

Edited November 30, 2015 by OhioBob
fixed formatting

[GarrisonChisholm]

[quote name='OhioBob'][COLOR=#222222][FONT=Verdana]

[/FONT][/COLOR][COLOR=#222222][FONT=Verdana]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?[/FONT][/COLOR] I think some in game testing is warranted.

[TABLE="width: 500"]
[TR]
[TD="width: 222, bgcolor: transparent"][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]Scenario #3 aligned to Node/1[SUP]st[/SUP] half, Prograde/2[SUP]nd[/SUP] half[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]Theoretical Ideal Orbit[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=black][FONT=Verdana]Semi-major axis, km[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]-450454[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]-450459[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=black][FONT=Verdana]Eccentricity[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]2.54487[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]2.50957[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=black][FONT=Verdana]Periapsis altitude, km[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]95892[/FONT][/COLOR]
[/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]80000[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=black][FONT=Verdana]Longitude of periapsis, deg.[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]0.170[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]0.000[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=black][FONT=Verdana]Longitude at infinity, deg.[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]113.31[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=black][FONT=Verdana]113.48[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[/TR]
[TR]
[TD="width: 222, bgcolor: transparent"][COLOR=#222222][FONT=Verdana]Δv, m/s[/FONT][/COLOR][/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=#222222][FONT=Verdana]2058.1[/FONT][/COLOR][/CENTER]
[/CENTER]
[/TD]
[TD="width: 116, bgcolor: transparent"][CENTER][CENTER][COLOR=#222222][FONT=Verdana]1990.4[/FONT][/COLOR]
[/CENTER]
[/CENTER]
[/TD]
[/TR]
[/TABLE]
[/QUOTE]

As it happens, by pure observation and a "general feel", I had struck upon this method to execute all my burns, based on the logic that the maneuver mode's movement during the second half of the burn seemed to be much more unstable. I have no data to support it, only my impression that it is preferable to me.

However, for what it is worth, consider this a "character witness" for your discovery; I've been doing it this way for several months. Edited November 24, 2015 by GarrisonChisholm

[OhioBob]

Sorry, guys. I just discovered a typo in my formula that computed the longitude. This changes the numbers. In the comparison between "align to prograde" versus "align to maneuver node," the latter still wins out as being the closest to the theoretical orbit, however the Δv has gone up slightly. In regard to the hybrid method, it is now appears to be the worst option in terms of accuracy. Please forget everything I said about it. Below are the revised numbers.
 

	Scenario #1, aligned to Prograde	Scenario #2, aligned to Maneuver Node	Scenario #3, aligned to Node then Prograde	Theoretical Ideal Orbit
Semi-major axis, km	-450459	-450459	-450459	-450459
Eccentricity	2.58601	2.50234	2.54686	2.50957
Periapsis altitude, km	114432	76742	96799	80000
Longitude of periapsis, deg.	-2.006	0.924	5.083	0.000
Longitude at infinity, deg.	110.74	114.48	118.20	113.48
Δv, m/s	2054.4	2076.4	2056.9	1990.4

In regard to Yasmy's method, using his equation to estimate the burn start time and aligning to the maneuver node now comes out very close in terms of accuracy. The method appears to warrant further study.
 

	Yasmy method, aligned to Maneuver Node
Semi-major axis, km	-450459
Eccentricity	2.50326
Periapsis altitude, km	77158
Longitude of periapsis, deg.	-0.228
Longitude at infinity, deg.	113.32
Δv, m/s	2077.1

I'd like to stress again that when I say "aligned to maneuver node," what I'm really doing is maintaining a fixed attitude throughout the burn in the direction of a tangent drawn through the maneuver node. This may not be exactly the same as keeping aligned to the maneuver node on the Navball. When performing maneuvers in the game, the location of the maneuver node on the Navball seems to float around a bit, particularly toward the end of a burn. Since I don't know how the game computes the direction of the maneuver node marker, I can't exactly replicate it in my simulation. Therefore, in game results might be different than the results of my simulations.

I'd like to also note that, in reporting my results, I've arbitrarily set zero longitude to be in the direction of the maneuver node. This is probably apparent but I should have mentioned it earlier.

Sorry again about the error.

Edited November 30, 2015 by OhioBob
fixed formatting

[EdFred]

I can't believe the thread went this long without the blatantly obvious solution:

MOAR BOOSTERS!

If my TWR isn't sufficient enough that I am going to end up in the atmosphere, I burn in a higher orbit - minimum required to complete the node in one burn.

[Yasmy]

This is an interesting theoretical question. For practical purposes, it usually doesn't matter, because delta-v budgets are often not incredibly tight. None the less, I'd love to see more simulation. Last time I was involved in this question I wrote a numerical integrator (because that's more fun than downloading one) to test these ideas, and I've been working on the physics for an approximate answer. After about 5 pages of latex, it is starting to get hairy, and, well, life does interrupt frequently. Hopefully someday soon I'll be able to present some formulae demonstrating the effect of each method on the final orbit in a way that is easy to compare. At the moment it is a bit too ugly to be usable. Even first order approximations get pretty hairy when estimating the change in the pe, ap, e and argument of periapsis due to a small arbitrary burn near pe. I've never seen a uglier set of first order approximations.

But like OhioBob, I wouldn't advocate using a complex formula to calculate when to start the burn. The method I brought up (splitting the delta-v) is an idea multiple people on these forums have suggested. I had just never seen anyone do the simple algebra for start time or do the simulation to evaluate its value.

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

[quote name='EdFred']I can't believe the thread went this long without the blatantly obvious solution:

MOAR BOOSTERS!

If my TWR isn't sufficient enough that I am going to end up in the atmosphere, I burn in a higher orbit - minimum required to complete the node in one burn.[/QUOTE]

Yes, yes. Of course. But this question equally applies to short duration burns, like 5 degrees instead of 40 to 90 degrees, where there is a small difference between following prograde and following the maneuver node. One method is theoretically better under some situations. I wouldn't advocate using a method with no practical difference, but I would always advocate learning the difference. Until we have a corpus of tests, or a set of well founded equations, this remains an open question which gets asked again and again here.

[EdFred]

[quote name='Yasmy']
Yes, yes. Of course. But this question equally applies to short duration burns, like 5 degrees instead of 40 to 90 degrees, where there is a small difference between following prograde and following the maneuver node. One method is theoretically better under some situations. I wouldn't advocate using a method with no practical difference, but I would always advocate learning the difference. Until we have a corpus of tests, or a set of well founded equations, this remains an open question which gets asked again and again here.[/QUOTE]

I would assume the multiple burn technique would be the most efficient. Anything not exactly at the node is wasted on orbital. But like was said, I don't think anyone really wants to develop the formula. Probably involves some pretty deep calculus.

[OhioBob]

Quote

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.

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 November 30, 2015 by OhioBob
fixed formatting and image

[g00bd0g]

I would love to see ohiobobs analysis but WITH correction burns to make orbits match the target orbit exactly.

[Yasmy]

Matching the target orbit exactly would be way more expensive than ensuring reasonably matching arrival conditions. You can see that in the divergence of trajectories in OhioBob's plots. There's no good reason to force your periapsis back to your initial parking orbit for example. This leaves a looser set of constraints on what is an acceptable final result. If you had very specific mission goals, like arrival at a moon in a precise location in its orbit at a precise velocity and radius, that is something you could optimize for. Otherwise you would have to define exactly what the goal of your correction burn is.

[OhioBob]

If everything with the ejection is spot on other than the flight path angle, then we just need to make a slight change to the direction of the velocity vector. If we wait and make the correction immediately after leaving Kerbin's SOI, then, for the example given, I estimate that the delta-v would be about 49 m/s for the 'align to maneuver node' trajectory and about 134 m/s for the 'align to prograde' trajectory. In practice, however, it's not quite that simple. I usually just create a new maneuver node and figure out what burn I need to make to reestablish a good intercept with the target. As Yasmy said, there's no need to get on the originally planned trajectory, we just need to get on a trajectory that ends with the correct arrival conditions.

Edited November 29, 2015 by OhioBob
fixed formatting

[Streetwind]

Wow, a lot has happened while I was asleep! :O

Thanks OhioBob, the graphical representation really does help. And it makes me wonder as to how the game actually draws your maneuver node marker. Isn't it a tangent as well, when you think about it? If you are 90 degrees off the node, and you point at the node marker on the navball, then you will point directly at Kerbin. You won't point at the location of where on the orbit the maneuver node is set. So that feels like your 'burn at tangent through the node' is in fact an accurate representation of what happens ingame... before you start the burn, at least. Because the game auto-adjusts the vector through the node to guide the player towards compensating for errors he makes during the burn.

The graphical view also explains why burning prograde during the second half actually becomes the worst possible result, not the best as we initially thought. Looking at the graph, you can see that the blue trajectory is already slightly too curved inwards, and the second part of the burn is where you would burn outwards in that case. If you burned prograde instead, you would greatly increase the inwards curve, moving you further away from the ideal orbit instead of closer. A theoretical reversal, where you would burn prograde during the first half and then burn along the (unmodified) tangent, would similarly result in an orbit that's curved outward even worse than the red one.

If you burned prograde during the first half and then followed the modified position of the marker on the navball... well, I'm not exactly sure what would happen. The marker would auto-adjust itself to make you burn radial-in to cancel out the outwards error you have introduced, so the resulting orbit would be better than the red one. But there's probably no simple way for OhioBob to simulate that in his own application.

Still, the graphs suggest that there is a variant somewhere in there where you burn just a little bit off node towards prograde near the start the burn, then stick to the node marker for the majority of it. That should push the blue line a little bit more outwards, to match the ideal.

[OhioBob]

Quote

Thanks OhioBob, the graphical representation really does help. And it makes me wonder as to how the game actually draws your maneuver node marker. Isn't it a tangent as well, when you think about it? If you are 90 degrees off the node, and you point at the node marker on the navball, then you will point directly at Kerbin. You won't point at the location of where on the orbit the maneuver node is set. So that feels like your 'burn at tangent through the node' is in fact an accurate representation of what happens ingame... before you start the burn, at least. Because the game auto-adjusts the vector through the node to guide the player towards compensating for errors he makes during the burn.

From my observations, I agree with this. I think that maintaining a fixed heading along the maneuver node tangent line is a reasonable representation of what happens in the game. The thing I don't fully understand, and therefore cannot simulate, is how the vector auto-adjusts. My simulation is comparable to aligning to the original maneuver node marker and then switching to attitude hold for the duration of the burn. This would surely result in the maneuver node marker drifting away from the ship's heading, most noticeably toward the end of the burn.
 

Quote

The graphical view also explains why burning prograde during the second half actually becomes the worst possible result, not the best as we initially thought. Looking at the graph, you can see that the blue trajectory is already slightly too curved inwards, and the second part of the burn is where you would burn outwards in that case. If you burned prograde instead, you would greatly increase the inwards curve, moving you further away from the ideal orbit instead of closer. A theoretical reversal, where you would burn prograde during the first half and then burn along the (unmodified) tangent, would similarly result in an orbit that's curved outward even worse than the red one.

That's correct.
 

Quote

If you burned prograde during the first half and then followed the modified position of the marker on the navball... well, I'm not exactly sure what would happen. The marker would auto-adjust itself to make you burn radial-in to cancel out the outwards error you have introduced, so the resulting orbit would be better than the red one. But there's probably no simple way for OhioBob to simulate that in his own application.

Since I don't know how the game computes the adjustment, there's no way I can simulate it.
 

Quote

Still, the graphs suggest that there is a variant somewhere in there where you burn just a little bit off node towards prograde near the start the burn, then stick to the node marker for the majority of it. That should push the blue line a little bit more outwards, to match the ideal.

That is what Red Iron Crown suggested a couple days ago when he posted the following. My experiments simply show that he had it right.
 

Quote

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.

 

Edited November 30, 2015 by OhioBob
fixed formatting

[Padishar]

The manoeuvre node marker shows the direction of the resulting vector when you subtract the current velocity vector from the post-burn velocity vector of the node. E.g. It shows what direction you need to burn in to get your final velocity vector correct but does nothing about your position. This positional error is the reason for the resulting orbit difference.