How do I calculate dV for ARMs?

[BoilingCold]

Hi, I'm looking for some advice regarding asteroid capture missions.

How can I go about calculating (or even estimating) the delta-V I'll need to do them? I've got to the point where I'm comfortable calculating interplanetary maneuvers using things like the Transfer Window Planner, but I have no idea how to do this for asteroids. Pre-0.25 I would just use trial and error and launch reverting, now however I'm attempting a hard mode career game so I can't revert!

I've tried simple calculations but they don't seem to help. For example, here's an asteroid that looks like I should be able to easily capture it:

There's only a 126m/s velocity difference between Kerbin & EQU-850. Kerbin's escape velocity (according to the Wiki) is 3431m/s, so if I eject from Kerbin prograde (relative to the sun) will that mean I need 9284 (Kerbin orital velocity) + 3431 (escape velocity) - 9410 (asteroid orbital velocity) = 3305m/s dV to match velocities with the asteroid?

How do I then calculate/estimate the dV needed to get it into orbit with the Mun? There is no info that I can see about how much a Class C asteroid weighs, nor have I come across any tools (in-game or out-of-game) that would let me calculate such a maneuver.

If ARMs are effectively impossible in hard mode career games then fair enough, lesson learned, but it would be great to work out how to do them

Edited October 15, 2014 by BoilingCold

[Empiro]

You can't really add speeds like that, because you're not taking into account things like the direction of travel (imagine the asteroid going 9284 m/s retrograde) and the Oberth effect (which will reduce your cost by a bit, though in this case very little at all).

You'll need to establish orbit first (~4500 m/s in stock aerodynamics), then you need about 1000 m/s to escape Kerbin and head toward the asteroid. However, it'll take more than 126 m/s to match up with the asteroid. Assuming that you want to intercept it a good ways before it reaches Kerbin's SOI, you'll be heading toward it with a decent amount of speed. Plus, you'll need some delta-V to correct your course along the way.

You shouldn't need too much d-V to get it around the Mun (I would guess <500 m/s assuming you do it reasonably efficiently). You might even be able to use very little if you manage to get a good gravity assist from the Mun to slow you down. I'm not sure how much a class C asteroid masses, however, so I think that part will have some guesswork involved.

[BoilingCold]

Yeah I guessed my back-of-an-envelope calculations would be horribly simplistic, but I didn't know where else to start . Thanks for the estimates, I'll consider having a go if my career funds can stand the risk! I'd still love a way to actually calculate this though, or at least make more than an educated guess

[mhoram]

It is not easy to make estimations about the ∆v needed for catching asteroids.

You will basically need to perform the following steps:

- Launch craft to Low Kerbin Orbit

- Get to an intercept course with the asteroid

- Change course to match the orbit of the asteroid

- Dock with the asteroid

- Reduction of ∆v for the remaining maneuvers

- Perform a course correction so that the asteroid can get into an intercept course with the intended final orbit

- Change course to amtch the final orbit

Launch craft to Low Kerbin Orbit

you will need around 4500 m/s

Get to an intercept course with the asteroid

In order to intercept the asteroid outside Kerbins SOI you need around 950 m/s to leave the SOI and an additional amount of ∆v. This additional amount depends on the question "how far away and how fast do you want to intercept the asteroid?"

The farer away the more ∆v you need and the faster the more ∆v you need.

Change course to match the orbit of the asteroid

The ∆v you need for this depends on the vector-difference between the asteroids velocity and your ships velocity at the interception point.

As a general rule of thumb: the more ∆v you spent at LKO departure-burn the more ∆v you need for this burn.

Dock with the asteroid

Depending on the distance you managed to intercept the asteroid you will need a couple of dozen of m/s for docking.

Reduction of ∆v for the remaining maneuvers

The remaining ∆v after docking is much less than before docking. The reason is that the asteroid adds mass to the ship while the fuel remains the same.

The difference can be calculated using the Tsiolkovsky rocket equation.

dv₁ = ∆v before docking

m₀ = ships mass without fuel

m_a = asteroids mass

g = 9.82 m/s² (http://wiki.kerbalspaceprogram.com/wiki/Specific_impulse#Conversion_factor)

I_sp = ships I_sp before docking

From this you can calculate the ships mass before docking m₁:

m₁ = m₀ * exp(dv₁ / (I_sp * g))

And the ∆v after docking dv₂ is:

dv₂ = ln((m₁ + m_a) / (m₀ + m_a)) * I_sp * g

Depending of the ratio of ships mass to asteroid mass the difference might be quite large, in one example my ∆v was reduced by a factor of 10.

Perform a course correction so that the asteroid can get into an intercept course with the intended final orbit

This amount depends on the distance to Kerbins SOI. The farther away you are the less ∆v you need. However since you now have much less ∆v available (because now you are docked to the asteroid), it can be beneficial to intercept the asteroid as early as possible. In this case you can change the orbit to intercept the Mun with less than 100 m/s.

Change course to match the final orbit

This value depends on the difference between the velocities of Kerbin and the Asteroid.

But since the orbits of the two are usually not that far apart 100 m/s should be enough to capture the asteroid within Kerbins SOI.

Why 100? You need according to this ∆v-map at most 130 m/s to intercept Duna or Eve. And since the asteroids are usually not that far away, less ∆v is needed.

An additional 50-100 m/s to get from the captured orbit to the Mun-orbit should bring you to your goal.

A final note on the question "When should I intercept the asteroid to require the least amount of fuel?"

Unfortunately this question is not easy to answer, because the ∆v needed to intercept the asteroid decreases with time to intercept while the ∆v needed to bring the asteroid to the target orbit increases with time to intercept.

Edited October 15, 2014 by mhoram
corrected value for g

[BoilingCold]

Fantastic, that's great thank you very much, gives me a lot more to go on than my previous trial & error method

Now... how much do asteroids weigh?

Edit: found this -

where M = mass in tons and x = class (A=1, E=5).

/me starts a new page in the KSP notebook!

Edit2: Another question, the asteroid I'm currently tracking actually has a Kerbin encounter (just) in about 300 days, am I right in thinking that if I wait until it's within Kerbin's SOI that the dV needed to capture it will be slightly less due to help from Kerbin's gravity well? Or would the difference be too little to worry about?

Edited October 15, 2014 by BoilingCold

[mhoram]

Edit2: Another question, the asteroid I'm currently tracking actually has a Kerbin encounter (just) in about 300 days, am I right in thinking that if I wait until it's within Kerbin's SOI that the dV needed to capture it will be slightly less due to help from Kerbin's gravity well? Or would the difference be too little to worry about?

I don't know if I understand your question right, so I try to guess what you mean.

If you just want to decellerate the asteroid so that is stays within Kerbins SOI, the only place you can do that is within Kerbins SOI - so under ideal circumstances you don't need to leave Kerbins SOI to do that, if the Asteroid is within Kerbins SOI in 300 days.

If you want to capture the asteroid on a final target orbit that it would not cross on its current course, then you will need less ∆V if you encounter the asteroid before it reaches Kerbins SOI and change it's course as early as possible.

You ask for "slightly less" ... compared to what alternative?

Edit: and about the asteroids mass: some people sent probes to attach to the asteroid. That way they were able to see the exact mass.

Edited October 15, 2014 by mhoram

[BoilingCold]

Ahh OK, thanks, that makes sense. I'm guessing that capturing it into Kerbin orbit first and then transferring it to the Mun (my intended destination) is going to be the best way to do this.

And the probe idea just occurred to me too, while I was in the shower

I've also just discovered that you can add a body on the Transfer Window Planner. \o/ \o/

Edited October 15, 2014 by BoilingCold