Is there any mathematical formula for the optimal ascent to orbit?

[hydropos]

For the last 3 days I've been trying to develop an equation for the lowest ∆V launch trajectory to LKO. Specifically, I'm looking for the optimal radial and circumferential velocity as a function of time or altitude. I've filled pages with differential equations and assorted calculus, but I can't seem to put all the pieces together. Has anyone calculated this already, and if so, where can I find that?

[UmbralRaptor]

There is no analytic solution; you'll need to go for numeric approximations. Beyond that, we can talk about rules of thumb for pitching over and speed...

[hydropos]

Can you direct me to any plots or tables of the numerical solutions? I've tried searching, but the only thing that comes up are a lot of "well I usually do this" posts. The closest thing to a rigorous approach were some folks who used mechjeb and different turn parameters on a given ship. However, my math has already shown that any circumferential velocity prior to 34km up is actually bad for your ∆V, so those are unlikely to be good approximations.

Edited February 10, 2014 by hydropos

[purpletarget]

What I found in the approximations was the vertical dV required to get to orbital altitude is ludicrous below 10km, because the air drag for such high velocities is prohibitive. So it never gets within a significant fraction of the horizontal dV to reach orbital velocity. So it makes most sense to put all the effort of the flight into the vertical.

After 10km, two things generally happen.

1) The terminal velocity starts increasing at a rate that any rocket is quickly unable to accelerate to match.

2) The additional dV "Upwards" requirement starts being small enough (and quickly as the rocket ascends) to be within the orders of magnitude to compare properly with the Orbital Velocity requirement on the horizontal. When the shortest path between two points is a straight line, the tangent of these two requirements starts generating an angle that can be used to guide a gravity turn.

There may be some links at The Drawing Board: A library of tutorials and other useful information which may have some further information close to what you're looking for...and/or you can also check the link in my sig for some vids, some of which touch on the series of calculations and how one could approach the problem.

[Pecan]

http://forum.kerbalspaceprogram.com/threads/46194-I-need-someone-help-me-do-some-math-for-launch-optimization/page2

NB: the optimal ascent path is heavily dependent on the vehicle design (especially TWR at each stage). mhoram has written a calculator for optimal ascent given a specific ship.

[Mmmmyum]

Wasn't this the Goddard Problem?

http://en.wikipedia.org/wiki/Goddard_problem

[hydropos]

When the shortest path between two points is a straight line, the tangent of these two requirements starts generating an angle that can be used to guide a gravity turn.

Can you clarify this?

There may be some links at The Drawing Board: A library of tutorials and other useful information which may have some further information close to what you're looking for.

I just combed through that. There was a lot of good stuff, but nothing quite like I was hoping for.

http://forum.kerbalspaceprogram.com/threads/46194-I-need-someone-help-me-do-some-math-for-launch-optimization/page2

This, OTOH, is almost exactly what I was looking for. Imma be honest, I TLDR'd at page 5, but it looks like they got stuck where I did. I think I'm going to continue my efforts in that thread.

NB: the optimal ascent path is heavily dependent on the vehicle design (especially TWR at each stage). mhoram has written a calculator for optimal ascent given a specific ship.

This is very true. I am prepared to work under the assumption that I have infinite TWR at my disposal. I suspect you really don't need more than about 3g's of thrust for a "perfect" kerbin launch.

Wasn't this the Goddard Problem?

http://en.wikipedia.org/wiki/Goddard_problem

Sort of. It's much more complicated with realistic aerodynamics.

[Kasuha]

You might try checking this thread, too.

[impyre]

<snip> my math has already shown that any circumferential velocity prior to 34km up is actually bad for your ∆V, so those are unlikely to be good approximations.

Everything I've read and understand about the math agrees with the concept that it cannot be analytically tackled, what math suggests that you should wait until 34km to turn? Also, does this account for KSP's lacking aerodynamics? As far as I can tell, the whole issue can be summarized by looking at the amount of potential work done against the vessel by different components, namely atmospheric drag, gravity drag, and inefficient fuel usage. Since I use NEAR to provide enhanced aerodynamic simulation, atmospheric drag also has a component related to the vehicle's AoA (which adds another level of complexity)... but even without it one can see that it takes a lot of work to change the velocity vector from pointing nearly at the horizon to straight up, so it seems like you should point as prograde as possible as early as possible... at least after a balance point (where the savings in fuel efficiency are offset by additional costs incurred from staying in the atmosphere longer), although good aerodynamics can allow lift to offset the penalty for turning earlier and may in fact provide an incentive to do so. I've been working on this problem for awhile, and what I'm seeing is that the tipping point is actually about 24km... and if you assume a smooth turn is better than an impulsive turn (which it usually is [at least with decent aerodynamics]) then depending on velocity and rate of turn, it could be completely reasonable to begin a slow steady turn at about 10-12km, be at the 45 degree point at about 24km, and then horizontal by 34-36km. This seems to be in line with what i've noticed in-game; however, I must admit that I doubt a truly constant turn rate results in the absolute best result... based on the exponential nature of the atmosphere, I imagine it's probably something that looks more like a turn which begins gradually, but picks up speed as you gain altitude and then slams on brakes when you are horizontal... though I have nothing that mathematically supports this.

[Pecan]

Both the threads I and Kasuha referenced (posts 5 and 8 respectively) have the actual workings in them and more recent posts. Since the only answers in this one are to read those two you're probably better off posting in one of those.

However, since real-life and FAR both recommend starting the gravity turn immediately, or very shortly, after liftoff I think you'll find the 'rapid turn at 10km' is very sub-optimal because it's too high not too low.

[hydropos]

As far as I can tell, the whole issue can be summarized by looking at the amount of potential work done against the vessel by different components, namely atmospheric drag, gravity drag, and inefficient fuel usage.

That's more or less what I did, except I set everything up in terms of wasted ∆V instead of wasted energy.

Everything I've read and understand about the math agrees with the concept that it cannot be analytically tackled, what math suggests that you should wait until 34km to turn? Also, does this account for KSP's lacking aerodynamics?

Let's say your craft is ascending straight vertically. Every second, you waste 9.8 m/s per second plus drag. If you have some circumferential velocity, say 1/2 orbital speed, then you only waste 0.5²*9.8 m/s per second, but now you waste more ∆V counteracting drag. The altitude (34km) was for a craft with average "drag value" of 0.3, ie something made largely from solid rockets. At that altitude, if you added any circumferential velocity, you would lose less ∆V from gravity than you would gain from drag.

That value was based on an embarrassingly bad omission. My mathematical framework was such that radial and circumferential velocity were two completely independent vectors. That is to say, I didn't account for the fact that if you expend a given amount of ∆V at an angle, you add sin(θ)∆V to radial velocity AND cos(θ)∆V to circumferential.

[impyre]

Yep. Since I'm using NEAR the drag is a little different (it probably costs me more to stray too far from current airspeed vector), which causes me to favor gradual turns. But negating the effects of increased drag due to AoA, simple angular turns are a fairly good approximation if executed at the proper altitudes/speeds.

[Friman]

Hi,

I wanted to mathematically calculate the best ascending path but I also thing that it is impossible to solve to equations for all cases. Therefore I tried so solve it numerically with computer simulations... It also turned out to be extremely difficult if you don't know the kind of ascending path you want. Finally I decided to figure out if the "typical" ascending trajectory that most users follow could be improved.

I started with a typical ascend orbit and then I run a simulation to calculate the final results (delta-v spent, final velocity, h...). Then I did a program that would change some parameters and then re-run the simulation until the best values for them were found.

The trajectory had the following layout:

1. Ascending at terminal velocity until h1 (h1 is a parameter that the program had to find)

2. Turn 45º and burn at constant acceleration (a2) until h2 (more parameters).

3. Engines off and free ascending until h3.

4. Engines on and 90º burn at constant acceleration (a4) until gets to orbit (I set orbit at 75km although I know space starts at 70km because of drag eff.)

The program gave some results so the final values for those parameters where:

h1 = 9500 m (+-50)

h2 = 26500 m (+-50)

h3 = 64250 m (+-50)

a2 = 5.47g (+-0.01g)

a3 = 1.7g (+-0.01g)

Delta-V = 4676 m/s

That's interesting because it's mostly what we do but at least now we know that considering this kind of ascending path these are the best values for those parameters...

I hope this helps,

Friman

PS: in the future I'll try to include de angles in the parameters and I'm open to your suggestions

[impyre]

You didn't really include much detail about the program itself, but by what you've said... it seems extremely limited. I understand the difficulties associated with this type of calculation, but your post suggests that some things are optimal which are not optimal (by quite a wide margin in some cases). For example, you could use a lower angle than 45 degrees and less thrust on the second part (think 30 degrees and 2.5 g's), and just burn a little longer. Also, breaking this up into many more segments would increase its accuracy.

I'm not trying to discourage you from doing further research, in fact I encourage it, I just want to also encourage you to be careful how you present things in this particular forum... as there may well be people looking for answers here that don't have the requisite knowledge to judge the efficacy of the method presented from an analytical standpoint.

[rkman]

my math has already shown that any circumferential velocity prior to 34km up is actually bad for your ∆V

How much delta-v does it take you to get into 75km LKO?

The record stands at 4343m/s http://forum.kerbalspaceprogram.com/threads/39196-Launch-Efficiency-Exercise-Updated-for-0-21-1?p=502762&viewfull=1#post502762

Starting the turn at ~10km i regularly get there with a little under 4400m/s.

(that is without FAR or NEAR)

[GoSlash27]

How much delta-v does it take you to get into 75km LKO?

The record stands at 4343m/s http://forum.kerbalspaceprogram.com/threads/39196-Launch-Efficiency-Exercise-Updated-for-0-21-1?p=502762&viewfull=1#post502762

Starting the turn at ~10km i regularly get there with a little under 4400m/s.

(that is without FAR or NEAR)

I don't know exactly how much DV it takes for me since I don't have any add-ons to give me an exact measurement, but it's got to be under 4,400 m/sec according to the calculated DV at launch and fuel reserves remaining in orbit.

I launch vertical to 7.5 KM at 1.5-2G, then follow a very close approximation of a gravity turn with an approximation of the sine of pitch for t/w the rest of the way. I use benchmarks as I go; 68* and 1.9G at 15 KM, 45* and 1.4G at 25KM, etc.

I can't guarantee that this is the most efficient path there is, but it is very efficient for every vehicle that uses a ballistic launch profile.

Best,

-Slashy

[rkman]

I launch vertical to 7.5 KM at 1.5-2G, then follow a very close approximation of a gravity turn

Thanks, but I am curious specifically about the efficiency of hydropos' ascent, who starts the turn at 34km.

[hydropos]

Thanks, but I am curious specifically about the efficiency of hydropos' ascent, who starts the turn at 34km.

As I said before, that number was based on a bad assumption in my math. That would only apply if you had an engine pointing downward and another pointing sideways and had no way to turn your ship.

[rkman]

As I said before, that number was based on a bad assumption in my math.

I overlooked that, thanks for clarifying.

There have been some developments in another discussion about launch optimization here: http://forum.kerbalspaceprogram.com/threads/46194-I-need-someone-help-me-do-some-math-for-launch-optimization/page15 , with talk of theoretical minimum d-v in the range of 4200~4300m/s, and apparently a new record at 4322~4326m/s.

A comment about gravity turn:

afaik the point of doing an actual gravity turn (letting gravity pull the rocket into the desired curve - which most of us don't really do), is to minimize steering loss (which according to mechjeb for most ascents is in the range of several 10's of m/s). It seems logical to me that just as optimal ascent involves a compromise between drag loss and gravity loss, simply minimizing steering loss is not optimal but rather steering loss to is a component of the compromise.

I think a real gravity turn is generally gradual and slow, meaning the rocket spends more time being closer to vertical than it would in case of a more aggressive turn, thus causing more gravity loss.

[GoSlash27]

Hydropos,

There is simply no way to create an equation that can answer this question. It has interdependent functions. velocity is a function of drag, which is a function of velocity. Likewise, drag is also a function of atmospheric density, which is a function of altitude, which is a function of time/acceleration, which is a function of... drag.

Even very simple problems of this sort are unsolvable by mathematical analysis. You have to approximate an optimal solution through repeated simulation with small adjustments to input variables.

Thankfully, we have an outstanding simulation of KSP physics at our disposal; KSP itself. We also have a large distributed processor to run the simulation (all the players) and we've come up with a pretty good approximation of the ideal solution.

Best,

-Slashy