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Assuming specific impulse is the same, in atmosphere it depends on your TWR - you don't want to go needlessly fast down low, as it pointlessly burns fuel to overcome increasing drag.
In space, a shorter burn near the periapsis is more efficient due to the Oberth effect - but not enough to outperform more efficient engine models.
Lowering throttle on a given engine in a given situation does not have any effect on the efficiency.
The times where it matters is landing and takeoff.. Suicide burn landing and rapid ascent means you spend less time fighting with gravity, thus you need less fuel. This applies mostly to non atmospheric bodies because complications and bastard calculations and all that smelly math stuff..
Essentially, at the core of it. The faster you can accelerate the better, but in practice, there are many factors that make that too impractical.
A factor that gets thrown out of the picture too often is the dead weight of the engine. That's why, disregarding thrust, Terrier is better than Poodle, and that's why the Nerv only shines on 20 t craft, or bigger.
1) a long nice slender 1,25m wide rocket with little drag or something wide and bulky with a lot of drag? If you have a lot of drag then seems better to leave amosphere as soon as possible. When the nac ball says "orbit" it means there is very little drag.
But as long as it says "surface" the rocket might t.... is sureley to tip over if you point outside the flight path marker .. the yellow little aircraft symbol. A long slender rocket might also tip over but at least compared to the rocket the available winglets are large so one can stay in atmosphere and fly fast and shallow the fins are big enough.
2) The speed you carry with you from previous stage. A decisive blast off thrust to weight 2 - 3 is nice. You then will get more speed before it burns out. If you get high speed the next stage need only maintain it and can burn at half throttle, you need more power when you near top of the arc to make an orbit and not fall down to earth again. Simply you don't wan't to go slow and fight gravity second by second. Shallow and fast is good if your rocket can do that. Otherwise you need to go to space sooner and aim for a higher orbit with more safety margin to atmosphere.
Too slow ascent means you be much more slow when you reach top of arc and then you have to burn so much fuel violently to make an orbit. And to have that power with you up there those engine need to have been lifted there.
Compare that to a blast off where much of the power is in first stage and you then ditch that stage and have so much speed you reach space without igniting next stage. Only reason for igniting next stage is to maintain speed so the difference to orbital speed is less. At the same time you can not go steep and fast cause then gravity can not turn your flight path towards horizon.
Lets say you go to steep like 50 - 60 degrees and reach 1000m/s. ( making numbers up was long time since I flew anything normal ) you might then have to have 1/3 trottle and let the speed fall again to say 700 - 800 just so that earth can turn your ship and then throttle up to 2300.
And then you can compare it to go through 30 000 at 30 degree doing 1000 m/s. And then you risk falling to earth because you are allready almost horizontal. So no need to throttle back but ignite next stage and try accelerate more to say 2200 m/s. Then ditch that stage and just have little fuel tank and a tiny light weight engine to just add 250 m/s at the top to make an orbit.
A huge blast off is always great but you need to point in the right direction. So it is try and fail repeatedly till you know how different engine work, especially SRBs burn fuel fast, and change weight very quickly / second. So with them you can fly more shallow cause you can trust that the thrust to weight ratio usually will go past 3 before they burn out and can even be 4 - 5 I have had T/W 6+ but not as much as 7. You can ofcourse create any T/W you want just buy 30 kickbacks and make a bundle of them then place a Mk1 capsule on top , ignite, and then immediately write a letter to his wife cause he will burn up before he reaches 40K altitude.
But joke aside I have had T/W 6+ on a serious rocket that I placed in orbit around kerbin. And that one flew a shallow trajectory and was a red fire ball of the worst kind I ever seen before the kickbacks burned out. I kept them on to keep the drag to reduce speed quick to save the next stage then I seprated the first stage.
To me it seems this is complicated stuff. So I say it simple as Scott Manley said, you want orbital velocity as soon as possible. But the question is the shape of your ship and the power .... can it really fly a shallow trajectory and fly fast without any negative things happening?
Because you don't move in a straight line in space.. People are usually aiming for a specific target with those burns and long ones will have you firing off prograde to compensate for the time/change in position during acceleration.
Instant acceleration would always cost exactly what the node shows, long burns cost more.
That's why this thing claims to have 20.000+ dV on paper but it doesn't amount to that in practice.
http://steamcommunity.com/sharedfiles/filedetails/?id=758254588
http://steamcommunity.com/sharedfiles/filedetails/?id=762181123&fileuploadsuccess=1
everyone has their own tricks to getting into orbit. I spend hours crunching scientific math calculations to get the right mark with a combination of brute force. I guess im a firm believer in the soviet approach to rocketry. lets come to an agreement and combine our brainpower :)
thank you
In reality you can't actualy do that as everything has a burn duration.
If you break the problem down to a number of equal burn duration impulses, we will in this example say 10 of them you would have essentially 11 ellipses, your original plus 9 tranitory ellipses and your final ellipse. At each progeassive ellipse you have to optimise your efficiency by slightly changing your burn vector. Each of these successive burns is a BiTangental maneuver.
For greater accuracy increase the number sub-impulses
In RL you use numerical analysis to attempt to find a satidfactory solution for a dynamic N body problem
There are textbook on this stuff
https://www.amazon.com/s/ref=nb_sb_noss_2/153-8419371-1714646?url=search-alias%3Daps&field-keywords=orbital+mechanics
Hence a long burn is just as efficient as fast burn, just computationally more challenging.
For the average KSP player its just way easier to do a fast burn.
If you start using ion engines everything just becomes one long contineouse burn
if your burntime is relatively low compared to your orbital period is doesnt really matter since the inefficiency is determined by the cosine of the difference in angle. plus if your burn takes too long you can always do it in multiple passes.
Too keep is short:
your gravity losses are usually a factor 10 higher than your drag losses, so it makes sense to have a as high as possible T/W. However, bigger engines are heavier engines, so at some point the extra weight will set you back more than you gained by the increased thrust.
ps: your burnprogram has no effect on your final speed difference.