Parachute formulas for 1.2

[Gaarst]

This thread is copied straight from another thread I wrote in KSP Discussion. I'm opening this one here to make it easier to find for people who would need it (thank Slashy for this suggestion).

Quote

AFAIK the parachute calculator thingy is not updated since a long time so I've bothered doing this stuff. If you wonder how many chutes you need to slow down a craft to a given speed then take a look: I've done a few measurements and have derived a few formulas for parachutes mechanics.

 

First, let's start with a bit of physics before the maths. The terminal velocity of an object in air is obtained by equating drag and gravitational pull and is:

, v is the terminal velocity of the object, m its mass, g the gravitational acceleration, ρ is the density of the air, C_x is the drag coefficient and A is the effective area.

In practice, C_x and A are always grouped together since dissociating one from the other is hard, especially in KSP. From now on, g will be taken to be 9.81 m.s^-2 and ρ to be the air density ASL on Kerbin (about 1.116 kg.m^-3), so the results will only be valid at the surface of Kerbin, but adapting them to other bodies would be simple as you would just need to change these two parameters.

 

For a single chute we can then write the terminal velocity as: , defining the constant B as  (the greater B is, the faster the body will fall).

B is a constant that is valid for a given parachute part and on the surface of Kerbin, it has a different value for each parachute in the game that can be found by reversing the equation above:

• Mk16 chute: B = 0.029
• Mk2-R radial chute: B = 0.039
• Mk16-XL chute: B = 0.020
• Mk25 drogue chute: B = 1.099
• Mk12-R radial drogue chute: B = 1.065

 

The above is valid for a single chute. When you consider multiple chutes, the only part that changed in B is the parameter C_xA which depends on the surface of parachutes: you can consider several chutes to be a single larger chute. When trying stuff in game, you find that the stack chutes behave nicely when adding chutes: 2 chutes means C_xA is twice greater, B is twice smaller and your craft falls √2 times slower.

Radial chutes are annoying. As is turns out, when you place several radial chutes in symmetry their area does not scale linearly, ie: 2 in chutes symmetry doesn't mean B becomes 2*B, instead you find that B becomes about 2^1.5*B. The total area of several radial chutes placed in symmetry scales with their number to the power 1.5. This means that 2 radial chutes placed in symmetry are more efficient than 2 identical parachutes placed separately (same thing for more chutes).
Note that if you place several radial chutes not in symmetry (placing them separately) they add up linearly, like stack chutes.

For several chutes, the equation to find the terminal velocity of a vessel becomes: .

B is the "parachute constant" from earlier, unchanged even for several parachutes, m is the mass of the vessel, n is the number of parachutes and α is a number that depends on the type of chute used: it is 1 for stack chutes and radial chutes placed independently, and it is 1.5 for radial chutes placed in symmetry (to account for the non-linear increase).

If you have two different types of chutes, your terminal velocity is then:  (the summation is done this way because math), and so on for more types of chutes: .

By rearranging the equation, you can solve other parachute related problems:

• The maximal mass that can be slowed to under a given speed by a given number and type of parachutes is: , or if you have several types of parachutes: 
• The number of parachutes of a given type needed to slow a given mass to a given speed is: 

 

I am not physically (and mentally) capable or writing web pages or making mods so if anyone wants to take these stupid equations and make something useful out of them, feel free to.

 

EDIT:

I did a few measurements for the other bodies of the Kerbal system, to help people for interplanetary parachute landings. Use the constant B for another body with the same equations above, nothing else changes.

	Kerbin	Duna	Eve	Laythe
Mk16	0.029	0.079	0.009	0.036
Mk2-R (radial)	0.039	0.105	0.012	0.047
Mk16-XL	0.020	0.054	0.006	0.024
Mk25 (drogue)	1.099	2.967	0.340	1.331
Mk12-R (radial drogue)	1.065	2.877	0.330	1.291

All measurements were taken ASL (~370m for Duna) so your craft will fall a little faster than predicted if you're higher up.

If someone feels like getting values for the Sun or Jool go ahead.

 

[neil470]

This is exactly what I was looking for when designing my Eve lander. Thank you for putting the time into looking into it. Using the info you provided, I made a spreadsheet that runs through the terminal velocity calculations for your chosen craft mass, body, and parachute setup. While it's fairly simple and there's not much to go wrong, I take no responsibility for Jebediah's safety during descent.