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Landing Delta-V Calculation


Kerano
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I've been working on an program to calculate combined takeoff and landing delta-v from a wide range of planet sizes (comets through superearths) and atmospheric thicknesses (vacuum through supervenuses).

I'm reasonably happy with the takeoff delta-v calculation - a two-burn Hohmann transfer from surface to orbit assuming a vacuum, plus a term to approximate atmospheric drag. It's not perfect - it makes several assumptions including unlimited TWR on the rocket - but it's a decent first approximation.

Takeoff%20equations.png

The landing delta-v calculation involves a deorbit burn and then a braking burn. Deorbit is easy enough - just reverse the circularization burn to bring the periapsis back to the surface. But the braking burn is more involved, because I'm looking to land a rocket capable of taking off back to orbit (not just a capsule).

We can set certain limits. Braking delta-v can be as low as 0 m/s (super-thick atmosphere and/or tiny comet where descent to the surface is very slow) or as high as 110% of the takeoff delta-v (vacuum descent with unlimited TWR, allowing 10% safety margin). Between these two values - where the atmosphere is thick enough to slow descent but not to a safe landing speed - is where I could use some ideas on how to proceed.

The rocket we're landing will vary greatly in mass depending on the surface gravity and thickness of the atmosphere we're dealing with. My initial thinking is to find the terminal velocity at the surface and use that to deduce the braking delta-v. This won't be the same as the terminal velocity on ascent though, because on descent there'll be more drag (rocket travelling rear-end first). Also, any parachutes will have much more of a drag effect on low-mass rockets than heavy ones.

Clearly there's a lot going on here. I'm not looking for an exact solution, but a decent approximation. How do we estimate landing delta-v for a rocket - across a range of planet sizes - when there's not enough atmosphere to land safely without a braking burn? Any thoughts are welcome! :)

Edited by Kerano
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If your atmosphere is enough dense to take into account its drag force, probably you should first calculate the terminal velocity limited by the drag force.
(I.e. solve Mg = ρCSv2/2 and find v). Usually ~150 m/s for Kerbin + Mk1 pod without chutes.
Of course, C and S depend on chutes or airbrakes state.
That's the remain of velocity you must eliminate if your lander is in equilibrium with air drag force.
If this is greater than value for nonatmospheric body, then the atmosphere is not too dense, just forget about her.

Given the max acceleration appropriate for your lander (say, 4g for crew), find the duration of braking. t = v/(a-g).

Calculate the required thrust F = Ma.
Decrease it by, say, 0.75. F" = 0.75 F.
(Because you can't be absolutely accurate, while remaining several m/s at already zero height or remaining several meters at already zero speed will crash your lander.
So, indeed you need a reserve of thrust and must begin rocketbraking above the minimally required altitude).

Recalculate acceleration from this thrust. a" = F"/M.
Recalculate duration t" = v/(a"-g) + tyour reaction time if ignition is manual, say 1..2 s + tengine reaction time, say 0.5..1 s + tduration of hovering before landing, 3..5 s because you should get v=0 at several meters above the ground to avoid scratching

Calculate the fuel rate: q = F"/(ISP*g)  if your ISP is in seconds
Calculate the spent fuel mass dm = qt"
Calculate dV = ISP*g*ln(M/(M-dm))

Round this value up because non-vertical trajectory, inaccuracy, etc.

And you get a rough estimated value.

Then calculate the estimated altitude of rocketbraking.
hbraking = v * (tyour reaction time if ignition is manual, say 1..2 s + tengine reaction time, say 0.5..1 s ) + v2/(2(a"-g)) + hseveral meters to hover at last.

Round it up.

Correct your terminal velocity value v takng into account that density is lower as the altitude is higher. 
ρ" = ρ exp(-hbraking/H) where H is the atmosphere scale (you can get it, say, from Kerbal wiki, where you get density, about 7990 m for Earth and Kerbin).

Repeat all calculations with corrected v.

Edited by kerbiloid
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