Suicide-burn handy equation

[Oan]

v₀ is the initial vertical speed [m/s]

μ is the planet (moon) gravitational parameter [m³/s²]

y_S is the altitude of the surface [m]

R is the planet (moon) radius [m]

y_B is the burn-start altitude [m]

y₀ is the initial altitude [m]

m is the mass of the ship (with fuel) [Kg]

F_T is the max. thrust [N]

all the altitudes are refered to the "sea" level

the actual ÃŽâ€v needed is roughly 10% more

kill the horizontal velocity,

wait until the ship's altitude is equal to y_B,

then burn up at maximum throttle

TIPS:

shut down the engines and set the throttle to maximum, to start burning hit space;

if during the falling the ship is slowly moving horizontally is safer to start the burn when the radar altimeter reads y_B-y_S.

Edited November 19, 2013 by Oan

[SanderB]

Hi Oan nice formula, more complicated then I would be willing to create . I think it isn't simple enough though because I and most people cant use your formula to calculate suicide burn altitude fast enough with your formula.

I've written this little thing in Word which I think is more simple and thus easier for laypeople to use so long as they are willing to do relatively easy mental arithmetic that some could conceivably calculate during vertical descent without pausing the game (Not recommended for safety purposes.) I only recommend its use on low gravity moons and planets because at least there the velocity of descent doesn't vary very much as you descend to the surface at a relatively constant rate and so pre calculated suicide burn altitudes don't change very much.

[rkman]

Suicide burn is fuel efficient, but first killing horizontal speed makes the landing inefficient.

It's probably a lot more complicated to calculate suicide burn time if the burn is more retrograde-like. Even mechjeb's suicide burn countdown is barely useful for such a trajectory.

[wingnut]

SanderB and Oan, could you please post examples for both your formulas?

In Oan's formula I am not clear what y0 is supposed to be.

In SanderB's formular I don't know where to get the velocity of descent Vd from and likewise Ga.

Could you please post an example for a suicide burn of a vessel in a 20x20 km Mun orbit with a weight of 5 tons and thrust of 20 kN?

[Oan]

I forgot to mention it, y0 is the initial altitude

example:

m=5x10^3 kg, y0=20x10^3 m, Ft=20x10^3 N

μ=6.514x10^10 m^3/s^2, R=200x10^3 m

ys = 0 m, v0 = 0m/s

ÃŽâ€V=sqrt(2*6.514x10^10 * ( 1/200x10^3 - 1/220x10^3 ) ) = 243,35 m/s

yb=(5x10^3 * 243,35^2)/(2 *20x10^3) = 7402 m

this is only an example, you need to use the actual value of ys that is the height of the surface under the ship

[wingnut]

Thanks Oan, now the calculation in my spreadsheet has a result after all and it is equal to yours.

[Oan]

you're welcome

[Hetsin]

I hope this is not necro-ing, but I have some clarifications regarding this, due to testing and maths I did. This assumes you have access to an information mod and are landing on a body with no atmosphere.

I'd recommend having the landing engine(s) on a toggle action group instead of staging like OP suggests.

Assuming any given orbit:

The vertical dv is ~ sqrt(2*g*h + v0^2).

Where:

g is gravity at the surface. (This will be obviously less at higher altitudes, but to be conservative aka not make kerbal paste, I figure it best to use surface g. Conversely, you can probably do some maths to figure out the average g, but this is the quick and dirty version)

h is the craft's radar height.

v0 is the craft's vertical velocity at a given moment. (If you are in a circular orbit this is effectively 0)

The burn altitude comes to the OP's calculation, restructured a bit.

Altitude fraction = (Vertical dv ^2) / (2 * 1000 * Thrust (kN))

Note: I put the thrust in kN, mostly for ease of reference

You may ask, where'd the mass go? The reason I didn't multiply the mass is because, at this point you should kill your horizontal velocity, which uses fuel, which decreases mass. After completing the horizontal burn :

Suicide burn altitude ~ Altitude fraction * mass (kg)

This should, when your vertical velocity gets to 0, leave you a few hundred m above the surface. If you were skilled enough, or had the automated aid, to keep your Thrust to Mass ratio (TMR) constant it -should- leave you a few m above surface (because of the conservative g value).

That's how I figured out how to calculate suicide burns, now on to clarifications.

Suicide burns are not particularly efficient. We all know that hovering is as inefficient as you can get (using the highest possible dv to go nowhere) and can be defined as having a TMR equal to g. A suicide burn is TMR - g (a) for t seconds to equal the vertical dv, so your 'gravity waste' dv is equal to g*t.

That isn't even the biggest offender of dv usage, which is the suicide burn itself. The higher from the surface you are, the more vertical dv you need to burn.

Therefore minimize the dv needed, you need to minimize the vertical dv (reduce height) and decrease burn time (increase TMR and reduce vertical dv). The increase to TMR has to be tempered with mass and fuel usage which is beyond my scope of understanding to figure out.

Example (Mun suicide burns, total values include the dv used to bring it down after the suicide burn thus breakdown values are approximate, using a ~ 3.8t lander with 50kN engine):

28.8k x 28.6k : 900 dv (530 h dv, 285 v dv, 75 gw dv)

10.2k x 8.8k : 760 dv (540 h dv, 166 v dv, 60 gw dv)

The transfer orbit between them: ~23 dv

So the lower orbit saved about 115 total dv on the suicide burn.

The main points are suicide burns are not as efficient as most people think, however they do allow a certain amount of ease for determining landing sites. Also, the burn should be done from as low an orbit as possible (mainly within limits of maneuverability and terrain). And as always make sure to buffer your dv budget, in case fun things happen.

ADDENDUM (theory crafting aka untested):

That was the quick and dirty way, now to increase accuracy!

The biggest way to increase the accuracy of the burn (lower the height when vertical velocity is 0), it to keep a constant TMR. You can adjust the throttle (or presumably thrust in 0.23) to match the mass lost with the burn, but this is ... difficult to do manually. You can also plan around average mass of the craft while burning, but to do this you need to know how much fuel is used. This gets us:

The quick and dirty method:

First we need to know the fuel flow rate (m-dot) for the stage (thanks Advanced Rocket Design tutorial).

m-dot = thrust / (Isp * 9.82) // This is a constant for the stage (unless you turn off/on engine(s))

then

m-avg = m0 - (dv / (2 * m-dot * (TMR - g)))

The accurate method (based off of the Tsiolkovsky Rocket Equation):

m-avg = (m0 + (m0 / e ^ (dv / (Isp * 9.82)))) / 2

Where:

m-avg is the average mass to plug in to the altitude fraction

m0 is the starting mass

TMR is the starting Thrust Mass ratio // Hey, I did say quick and dirty

e is, of course, Napier's constant. ~2.71828

So now you're finishing your burn closer to the surface. Which may or may not be a good thing. Next step, how do we use less dv on the burn? Easy! Basic trigonometry. Now I nearly failed my trig class all those years ago, so there is undoubtedly a more efficient way, but I figure this will do as a start. This also requires a relatively flat stretch of area, so kilometerage may vary.

So you've calculated your vertical dv. Good. Now look at your horizontal speed. Figure out the dv needed to reduce your horizonal velocity to your vertical dv. You see where I'm going with this, right?

So to calculate your mass after the horizonal burn. This will be:

Quick and Dirty:

m1 = m0 - (hdv / (m-dot * (TMR - g)))

Accurate:

m1 = m0 / e ^ (hdv / (Isp * 9.82))

Where:

m1 is the final mass after the burn

m0 is the starting mass

hdv is the horizontal delta v

TMR is the starting Thrust Mass ratio

e is, of course, Napier's constant. ~2.71828

You plug m1 into the m-avg formula as m0, and multiply the dv in that formula by sqrt(2) (~1.4142). Instead of rad+, you put the pitch to 45 deg retrograde, wait for the altitude it spits out and burn. This should save about 1/4 of the vertical dv that you initially calculated, which may or may not be worth the effort.

Edited December 19, 2013 by Hetsin
Correction