Delta-V, liftoff and the math...

[Eurobertics]

Hello @all,

today I have just a mathematical question.

I play around with calculating mission parameters by hand. I have a crucial question about that.

Actually I'm good wood with Hohmann-transfers, TWR, DeltaV and orbital velocity as well. Most of this was taken from the advanced rocket design page from the KSP wiki in association with Google.

But one questions still remains:

What do I need to calculate the Delta-V for liftoff to low orbit injection like starting from Kerbin and raise to a orbit let's say 100km?
Or on the other hand, calculation from the surface of Mun to low orbit.
Most Delta-V maps show that this would be about 4000 to 4700 m/s (Kerbin).
How is this calculated?

Looked around the internet with many good calculators. But none of them shows Me the math behind that.

Thanks in advance
Eurobertics.

[RoofCat]

not sure about math, but 4000..4700 for Kerbin orbit is a really bad number. Your rocket must be terrible to need as much.

Good rockets should have it closer to 3400..3600. Best can go below 3200 (I have reached 3150m/s in short tests), but that requires really high TWR and that in turn requires low total delta v onboard (low mass->not enough fuel for far flights). So the quickest ships aren't actually the best considering the distant targets you have. So lets say have it ~3600m/s for LKO and fly safe.

I would stay away from exact math in this case. Practice your launch profiles until you find good ones and that's it. In general most beginners fear drag, while invisible gravity does more evil to properly built rockets over time. Of course, if your rocket has uncovered flat frontal surfaces the drag may be as bad. Sharp nose cones everywhere - rather simple rule.

The reason for me not recommending you to do full math is you have to keep in mind drag and gravity at once. It is optimizing multiple equations. Can be done, but it is still tricky. Since gravity force impact changes with altitude and your trajectory and velocity. And drag changes with your altitude, flight angle vs.rocket angle (gliding drag), ship profile, length, velocity.
And since you will most likely launch it manually, small inaccuracy will invalidate all your math you have probably spent 20 minutes on calculating. Good launch is good enough in the long run and a small reserve hasn't hurt anybody. Take it easy and have fun I guess.

[TinfoilChef]

I use a general rule of thumb based on KER data. To launch most ships from Kerbin I design the craft to have a 1.5 TWR on the pad and have around 4000 dv to reach an 80-100km parking orbit. (any fuel on payload isolated by either locking tanks or disabling crossfeed)

I know there's charts that say you can reach orbit with 3200 dv but I'd rather have more fuel than not enough.

[Eurobertics]

Thanks for your explanations.
I know there is a lot of trial and error which is the fun part too.

But is there somewhere explained how to calculate the needed dv from surface to LKO?
I find it just interesting how KSP can teach orbital mechanics and mathematics at the same time.

If I do some math, I only come to about 2300m/s dv for surface launch to 80km orbit.
Even with some reserves, this is far away from the mentioned 3600m/s to get into a parking orbit.

[Technicalfool]

https://wiki.kerbalspaceprogram.com/wiki/Cheat_sheet
You'll find some useful Delta V equations here. Warning: Lots of math.

[George Kerman]

Originally posted by Eurobertics:

Thanks for your explanations.
I know there is a lot of trial and error which is the fun part too.

But is there somewhere explained how to calculate the needed dv from surface to LKO?
I find it just interesting how KSP can teach orbital mechanics and mathematics at the same time.

If I do some math, I only come to about 2300m/s dv for surface launch to 80km orbit.
Even with some reserves, this is far away from the mentioned 3600m/s to get into a parking orbit.

That kind of calculation would be decent for non-atmospheric bodies like the Mun. But regardless, your calculation is way off. Why would you need JUST 2300m/s to get from 0m (aprox) to 80 000m AND have at 80 km almost 2300m/s horizontal velocity? What about the vertical velocity spent to go from zero to 80 km? What about those 3-5 mins spent combating almost 10m/s negative acceleration from gravity (only at launch 9.8 something m/s, afterwards slowly decreasing as your horizontal velocity increases) ? But for Kerbin you need plenty of things to be taken into account like:
a) all the atmospheric characteristics found here for Kerbin: https://wiki.kerbalspaceprogram.com/wiki/Kerbin
b) The aerodynamic properties of your craft
c) TWR/ acceleration ultimately tells you how much time you'll spend combating gravity before reaching orbital velocity. Low TWR = more time spent combating gravity => more gravity losses.

For bodies like the Mun, the absolute best deltaV for orbit is obtained by launching and immediately going almost fully horizontal if the terrain allows you and keeping the vertical velocity close to zero (much safer to be + rather than minus), then circularizing at the desired apo height. It's not gonna be just 550m/s. I haven't done these test trials, but I'm pretty sure it's gotta be around 600 or a bit more. There's still some gravity to combat here as well.

It ain't a fixed normal from SL/GL to orbit, it depends on the craft and your piloting. Even in space it ain't a fixed number when you play with maneuver nodes (well there is an optimal number, hard to accomplish with manual piloting and manual maneuver nodes; we can get very close though). You can do the maneuver imperfectly, then you gotta corrections. It is safe to add 10% more dV than you think you'll need.

[Eurobertics]

Thank you for your answer and suggestion. I know that 2300 is way to low.
Acutally I'm on about 3100 to 3300m/s, taking the drag with the gravity turn into account. I know that's still not enough, but I think I'm on the right track (well, I hope ^^ )

Thank you all for your great explainations.

[Semaj]

For me personally, I always go for a 4000m/s dv rocket however I'm always aiming for the orbit of least darkness (128,745m) above Kerbin if it's a capsule or manned craft.

[dnrob7]

I'd say break it down into it's individual components and give yourself some data sets to work with.

Build a "standard test rocket" and have a "standard launch plan".. Figure out what angles you are at during what altitudes, writing down dV spent and semi major axis on each launch. Once you have that repeatable within a smallish margin, launch again with gravity at 0, do this a few times and compare results. After that, turn gravity back on and turn drag off, launch launch launch, scribble scribble scribble.. then launch with both gravity and drag off and see if the numbers add up..
I think this will reveal whats under that particular skirt.

With atmospherics it gets a bit more test-heavy with multiple designs needed for multiple rocket sizes but if you like math, you won't mind.. I do. I don't like math, that's why I haven't done this myself.. Though I'm kinda hoping you will and then share the results ;)

[RoofCat]

with the gravity equal 9.8m/s², going quite vertical for just 60 seconds (largest Kickback burns for ~62 sec at 100% power), you will lose to gravity 60*9.8=588m/s delta v. And it is just the start.

Going with TWR 1.5 you accelerate ~5m/s. After 60 sec you will have just 300m/s. Which is great for ~60° trajectory at that point. Still high loses to gravity. Your altitude will be ~60*150=9ooo m. Considering you fly at an angle, actually closer to 8ooo and velocity slightly higher since it's easier to gain velocity not flying at 90° and also your ship only gets lighter on average. The thing is - almost all of that were pure loses because you are going up, not orbital. Because you need to escape thick atmosphere first.

It is rather safe to add the same amount of delta v again. Which would take you closer to
600m/s at 40°, which means largest part of your acceleration already goes for orbital velocity at this point. And your altitude 8000+60*450=35ooo. Again, considering flight angle it is rather slightly below 30km and in fact noticably higher than 600m/s already. In theory you target ~900m/s at 30° and 30ooo m. While in case your craft is powerful and sharp enough, it is even better to have it lower and faster. As gravity is your main enemy.

At this point you still waste some delta v on ascent and gravity, but is should be roughly similar to what you gained in orbital velocity during first 2 stages. So it almost evens out.

Orbital velocity is 2290m/s at ~72km. Adding 1200m/s (wasted on ascent and gravity) to that you have 3500m/s.



For best ascent attach a lot of Hammer boosters to your ship. They have incredible TWR. So you can gain ~300m/s quickly (they have ~26s burn time iirc) and thus can escape thick atmosphere much quicker and thus go much faster at lower angles etc., etc. much sooner.
Here is simple test example for that. It also has extreme TWR and low launch angle. It falls to ~40° almost immediately while also gaining 300+m/s. Then another 600m/s almost as quickly while getting closer to 30°. With so much TWR it flies almost straight without losing much angle to gravity turn. Slightly later Hammer is done and small Spark finishes the job.
http://steamcommunity.com/sharedfiles/filedetails/?id=927437439
You may consider it (almost) ballistic launch, for which ~30° are the best as we know since middle ages with a circularization engine.

The trouble with Hammer (over)boosted solution though - radial decouplers in KSP are as expensive as engines for some reason (like 600$ each). Nose cones half that. So in the end you just chill and waste a bit more fuel and delta v with less impressive ascent, while saving probably 1/3 of the launch price.



Fun part - I know how to build stock manned orbiter with MK1 command pod for less than 3ooo$ launch price. Which is even less than that unmanned beast in the picture. Since probe core and some other parts aren't cheap. It really isn't worth the trouble to go for least delta v spent records trying to reach LKO.

Max delta v left after reaching LKO with least amount of money spent for launch is what space travel is about.

[Eurobertics]

Thank you RoofCat for this detailed explanation.
I just have a few questions about that:

Originally posted by RoofCat:

with the gravity equal 9.8m/s², going quite vertical for just 60 seconds (largest Kickback burns for ~62 sec at 100% power), you will lose to gravity 60*9.8=588m/s delta v. And it is just the start.

Going with TWR 1.5 you accelerate ~5m/s. After 60 sec you will have just 300m/s. Which is great for ~60° trajectory at that point. Still high loses to gravity. Your altitude will be ~60*150=9ooo m. Considering you fly at an angle, actually closer to 8ooo and velocity slightly higher since it's easier to gain velocity not flying at 90° and also your ship only gets lighter on average. The thing is - almost all of that were pure loses because you are going up, not orbital. Because you need to escape thick atmosphere first.

It is rather safe to add the same amount of delta v again. Which would take you closer to
600m/s at 40°, which means largest part of your acceleration already goes for orbital velocity at this point. And your altitude 8000+60*450=35ooo. Again, considering flight angle it is rather slightly below 30km and in fact noticably higher than 600m/s already. In theory you target ~900m/s at 30° and 30ooo m. While in case your craft is powerful and sharp enough, it is even better to have it lower and faster. As gravity is your main enemy.

The first part is clear to Me and helps a lot what I did wrong.

On the second part, where do you take the 150 (60*150=9000) from?
If this is the TWR why did you multiply this with 100?

On the last, third part the same. Where do the 450 come from (8000+60*450=35000)?
Is that the new TWR on about 8000m bases on your calculation? If so, why is this *100 again?

Please correct Me if I'm wrong.

Thanks a lot in advance
Eurobertics

[RoofCat]

I recon it is about 150 and 450 I haven't explained anywhere. Those are average speeds.
(0+300)/2 and (300+600)/2 for each "stage" mentioned.
60 are seconds. Why 60? Just as an example, because 1 minute and it is also close to the largest SRB burn time. Easy to relate.

Since you fly with variable speed due to TWR>1, you have to use average speed to caculate the distance flown over that time.

Of course, since your weight changes all the time (=>TWR increases) and your trajectory becomes shallower with gravity turn, this was really very "aproximate" math. You would have to use the real start-end velocities (like "(324+691)/2") and then also adjust the altitude reached with the parabola function, because you aren't flying at 90°.
You can do that with exact math. I just did it based on a very rough aproximation of what I remember from sin&cos values (more like wild feeling about triangles) and my empirical KSP experience with altitudes, velocities and angles.

[Eurobertics]

Ah I understand. I think I get close to this.
That your calculations are just for example is absolutely perfect for Me.
Now I understand all this much clearer.

Many thanks for patience and explanations.

Eurobertics

[RoofCat]

if you are interested in really exact math, keep in mind I used simplified average speed which would be true for fixed acceleration. TWR changes are rather modest on most rockets (except pure SRB) so for all purposes here it was close enough. While in fact acceleration in KSP is changing all the time (rather slightly exponential) and that would require more complicated average speed over time formula to calculate distance. More like median or weighted average speed.
It may include logarithmic function in a simplified case where just the fuel burns out at fixed rate per sec. As soon as you start dropping empty tanks or stages, it becomes more and more complicated.

Or you could do the rather simple math I did with 1 sec intervals and then use that fine ladder to get really close to exact one without making your head explode.

When I'm trying to estimate average Isp during launch for a particular engine, I actually read it at 75% and 25% tanks and then have the average number of those two. It is still quite inaccurate, though much closer to the real Isp than (0%+100%)/2 or the one measured at 50% tanks. The same would be true for speed. The more steps you use, the better the result.
Comparing two "ladders" you can see how close you are, since the accuracy increase will diminish consideraby the more steps you have. At some point eventual accuracy increase won't be worth the trouble anymore. Good is good enough. Like how many decimals do you really need? :)

[MAD]

http://steamcommunity.com/sharedfiles/filedetails/?id=578501464

liftoff is highly dependand on assent trajectory and drag.
I would just do some trials in sandbox and see what is optimal for each craft.

[Aristizabal95]

I know I'm late for this discussion, but I have found an amazing course on space mission design, and this video talks about the calculations for ascent and reentry. Not easy to grasp, but unveils all the mistery behind the numbers

https://www.youtube.com/watch?v=xCKQryLgCrk