Calculating Custom Delta-V

[Atubara]

I've been constructing a custom solar system lately, and being the math geek I am, I decided to try and create a Dv map for my system. The only problem being that I have no clue what I'm doing. I can't even seem to recreate the numbers I see on Dv maps for the stock game. I know that I should consider the Vis-Viva equation and assume everything's co-planar and perfectly circular but is there something I'm missing?

 

I would also like to discuss about any other formulas, such as which would yield the most effective results.

 

I would really appreciate it if somebody could enlighten me on this subject. It would also help other system-builders with their work. 

[YNM]

dV maps are unrepresentative of how the calculation is done - it's there only to look at the minimum (or average minimum) budget you should provide.

For interplanetary transfers, assume you're making a hohmann transfer between the first planet's orbit to the next one. The dV will be

1. Between 1st planet orbit to transfer

2. Between transfer orbit to 2nd planet.

If you can calculate the usual hohmann transfer then things will be easy. Or you can be lazy and model them in KSP and bring MechJeb out.

[ElWanderer]

OhioBob's website is pretty handy: http://www.braeunig.us/space/orbmech.htm

Try calculating a Kerbin-Duna transfer.

1. Start by assuming Kerbin isn't there. What velocity do you need to be going at Kerbin's orbit, to reach Duna's orbit (pick Duna's periapsis, apoapsis or something in between, it shouldn't make much difference)?

2. You can calculate the difference between this velocity and Kerbin's orbital velocity (which is about 9.3km/s). That gives you the velocity you need to be going, relative to Kerbin, at the edge of Kerbin's sphere of influence.

3. Based on this velocity and the size of Kerbin's sphere of influence, you can calculate the hyperbolic excess velocity. This may be in the link I gave. If not, try Wikipedia. I got it from one of the two!

4. Then, by reversing the calculation, based on the hyperbolic excess velocity from 3, you can determine the velocity at, say, a periapsis of 100km above Kerbin.

5. Finally, find the difference between the velocity calculated in step 4 and the velocity you need to be going at in your circular 100km orbit of Kerbin. That gives you the magnitude of your ejection burn.

You can perform a similar set of calculations at Duna's sphere of influence to find out how fast you would be going at periapsis, and thus how much delta-v you would need to go from a fly-by into orbit.

The answers should be pretty close to those given by delta-v maps.

It's the same steps for other bodies, but as eccentricities and inclinations get further from 0, you may not be able to assume ideal orbits. Then again, it's in those conditions that delta-v maps get further from reality, as anyone who has naïvely tried to get to Moho discovers.

[Atubara]

I'm really sorry if I'm not getting something here unless I'm missing an important equation or if I'm simply doing it wrong.

I'm still not getting anything close to the numbers I see. I would use MechJeb, but I'm still messing around with fidgety Kopernicus to generate my system.

[cubinator]

Note that realistically sized bodies will require about sqrt(10) times bigger delta-V budgets than what you'd expect in stock KSP.

[Atubara]

Ah, don't worry. I just wanted to check my answers first with KSP's parameters before I used the maths for a realistically sized system.

[YNM]

Slight clarification of my previous post : Basically in interplanetary transfer you're making a hohmann transfer around the Sun. But it's sliightly complicated by having to escape the planet's gravitational force (other than Sun's).

[ElWanderer]

7 hours ago, Atubara said:

I'm really sorry if I'm not getting something here unless I'm missing an important equation or if I'm simply doing it wrong.

I'm still not getting anything close to the numbers I see. I would use MechJeb, but I'm still messing around with fidgety Kopernicus to generate my system.

Can you show your working. I'm typing this on a phone, so this'll be a bit brief (no formulae, sorry), but here's my calculation of Kerbin-Duna. Hopefully any differences will jump out.

1. An eccentric orbit around Kerbol (GM = 1.7233*10^18) with a periapsis (altitude) of 1.3338*10^10m and an apoapsis (altitude) of 2.0726*10^10m (Duna's semimajoraxis, less the radius of Kerbol) has a velocity at periapsis of 10228m/s.

2. A circular orbit around Kerbol (GM = 1.7233*10^18) with a constant radius of 1.36*10^10m has a velocity of 9285m/s. The difference between the two velocities is 943m/s.

3. Now switch to caring about Kerbin (GM = 3.5316*10^12). To be going 943m/s at the edge of Kerbin's sphere of influence (84159286m from centre of planet) gives a hyperbolic excess velocity of 897m/s.

4. To get a hyperbolic excess velocity of 897m/s, at an altitude of 100km (radius 700000m) you must be going about 3300m/s

5. A circular orbit of Kerbin with an altitude of 100km has a velocity of 2246m/s. Therefore the ejection burn would be about 1050m/s.

That's a middle of the road figure - you can get a lower value if you aim for Duna's periapsis. From memory, delta-v maps suggest 1030m/s to get to Duna.

Note - I've used the old values from a spreadsheet I set up a while ago. I'm aware that the masses and orbital velocities have changed slightly in v1.2 to accommodate the more precise values for G and g0.

[Atubara]

Oh dear, now that I'm looking over your work, I realise that I didn't use seperate GM products! What a naïve mistake on my part.

My home has really bad wifi, so I would show you my work, but it wouldn't have uploaded anyways. I would type it, but I have no motivation to type it after focusing on my derivatives/integrals homework. 

Just to clarify, this can be used for destinations at both apoapsis and periapsis, right? Would aiming for the periapsis be more efficient? I did end up with a ballpark number of around 1600, and a test flight to Duna from a 100km orbit from Kerbin took about 1800 Dv due to my inefficient piloting skills.

[ElWanderer]

21 minutes ago, Atubara said:

Would aiming for the periapsis be more efficient? I did end up with a ballpark number of around 1600, and a test flight to Duna from a 100km orbit from Kerbin took about 1800 Dv due to my inefficient piloting skills.

Using circular 100km parking orbits at Kerbin and Duna, I get a total delta-v of 1690m/s to transfer if arriving at Duna periapsis, and 1620m/s if arriving at Duna apoapsis. That doesn't take into account any plane changes or corrections burns.

I wonder if it's always going to be cheaper to transfer to an outer planet when you arrive at apoapsis. The ejection burn is larger, but in the case of Duna the insertion burn is much smaller. It may not always be the case, though.

I've had to fiddle a lot with my spreadsheet to calculate these numbers, so errors may have crept in.