Help with my delta-V map

[Rusty6899]

I am trying to make a delta-V map of the Jool system. I am fairly sure that the pentagram design is the most effective for this purpose, but I have a few questions for anyone who has decent grasp of orbital mechanics.

Firstly I will explain the numbers on my map. The lines between Jool and each moon should show the delta-V required to reach that moon. In red are the delta-V values for an encounter from LJO (150km), on the other side of the line (in black) are the dV requirements for an encounter from an elliptical orbit with a 150km periapsis and a apoapsis at the edge of Jool's SOI. This is to give a realistic delta-V requirement for someone reaching a moon from Kerbin, whereas the map I use shows the delta-V requirement for LJO then another for transfer to e.g. Laythe, meaning there would be a huge wastage of delta-V if the map was followed (however, I realise that there are maps available that take this into account). The numbers on the same line closer to the moons are the Delta-V requirements to circularise to a low orbit. So for example, from a LJO it takes 1600 m/s to encounter Laythe and another 780m/s to reach a 65km orbit about Laythe. Whereas from Jool orbit from a 150km periapsis and apoapsis at the edge of Jool's SMA (2.46Gm) it takes 920 m/s to encounter Laythe and 780 to reach a 65km orbit (the orbits would be equivalent prior to the circularisation burn so the same dv values have been used). The numbers shown on each moon are the delta-V requirements of a landing/launch.

There is an arc joining each moon. Starting at one moon and following the arch to your target, the first value (next to the moon of origin) denotes the delta-V requirement of the ejection burn to reach the destination. The second value (on the same line, next to the target moon) denotes the delta-V requirement of the circularisation burn. So to get from low Laythe orbit to low Vall orbit, you need to perform a 590 m/s ejection burn and 300 m/s circularisation burn.

Here are (a few of) the assumptions that the map makes.

The ejection burn from moon A to moon B is equal to the circularisation burn if you were travelling from moon B to moon A.

Pol and Bop have circular Joolian orbits with the characteristics of an orbit with their in-game semi-major axis.

Low orbit on Laythe, Vall, Tylo, Bop and Pol is 65km, 15km, 20km, 10km and 10km respectively.

My questions are as follows...

Is my first assumption correct? If not, how do I calculate circularisation burns?

Is there any way of knowing at which point it would be most efficient to transfer from Bop/Pol to other moons? I assume that the transfer from either to Vall, Tylo or Laythe is most efficient at either Pe or Ap (although I'm not sure which one). Between Bop and Pol it seems that it would be very complicated to calculate the most efficient transfer possible.

How are my numbers? I have a feeling that they might be a little bit off (particularly for transfers involving Bop and Pol). This is my first attempt at calculating delta-V requirements (they are all rounded to the nearest 5 m/s by the way).

I would like to put the ejection angles on the map, but http://ksp.olex.biz doesn't give values for every transfer. Does anyone know the best way to calculate these? (my own calculation attempts broke down due to mathematical absurdities e.g. theta = cos^-1(1/e)... where e<1)

Any help with this would be greatly appreciated.

Edited March 13, 2014 by Rusty6899

[Tom Asz]

Whoa! This is great I can't help you, cause I'm not that advanced as a player, but this is plain and simple. Just what I needed

[Supernovy]

Your assumption is correct, Orbital mechanics are generally symmetrical. I'm not sure about the numbers, but you could cross reference with other Δv maps.

On ejection angles, I'd find the angle between the asymptotes of the hyperbolas given by each escape trajectory for each transfer, then use that to find the difference between moon-prograde and jool-prograde.

[Yasmy]

For some low delta-v transfers from some moons, the ejection orbit is not hyperbolic but elliptical.

Essentially, the SOI is too small compared to the size of a slightly sub-escape elliptical orbit.

For example, a Minmus to Mun transfer burn is still an elliptical orbit around Minmus with apoapsis outside the SOI.

Thus you can't use the hyperbolic orbit escape angle calculation.

Here's an snippet from the source for http://ksp.olex.biz/js/kspcalc.js for elliptical escape burns.

                if (e < 1) {
                        // maltesh's solution for elliptical transfers
                        var a = -o.mu/(2*eta);
                        var l = a*(1-e*e);
                        var nu = Math.acos((l-o.soi)/(e*o.soi));
                        var phi = Math.atan2((e*Math.sin(nu)), (1+e*Math.cos(nu)));
                        //eject = (270 - (phi*180/Math.PI)) % 360;

                        // Kosmo-nots fix to maltesh's solution
                        eject = (90 - (phi*180/Math.PI) + (nu*180/Math.PI)) % 360;
                }

It's the same answer I got when I did this problem, so I use the code as a quick reference.

Edited March 10, 2014 by Yasmy

[Rusty6899]

Your assumption is correct, Orbital mechanics are generally symmetrical. I'm not sure about the numbers, but you could cross reference with other ÃŽâ€v maps.

I actually found someone who had calculated the delta-V requirements between each moon, so I have access to updated numbers now. They are fairly similar to the ones I had calculated, though. I can't really use the numbers from most delta-V maps because they aren't showing the same information. Most just give the delta-V requirements for travel to/from Kerbin, rather from travel between other bodies.

For some low delta-v transfers from some moons, the ejection orbit is not hyperbolic but elliptical.

Essentially, the SOI is too small compared to the size of a slightly sub-escape elliptical orbit.

For example, a Minmus to Mun transfer burn is still an elliptical orbit around Minmus with apoapsis outside the SOI.

Thus you can't use the hyperbolic orbit escape angle calculation.

Here's an snippet from the source for http://ksp.olex.biz/js/kspcalc.js for elliptical escape burns.

                if (e < 1) {
                        // maltesh's solution for elliptical transfers
                        var a = -o.mu/(2*eta);
                        var l = a*(1-e*e);
                        var nu = Math.acos((l-o.soi)/(e*o.soi));
                        var phi = Math.atan2((e*Math.sin(nu)), (1+e*Math.cos(nu)));
                        //eject = (270 - (phi*180/Math.PI)) % 360;

                        // Kosmo-nots fix to maltesh's solution
                        eject = (90 - (phi*180/Math.PI) + (nu*180/Math.PI)) % 360;
                }

It's the same answer I got when I did this problem, so I use the code as a quick reference.

Thanks, I was unaware that the calculations were only used for hyperbolic orbits (although it makes sense from the equations). I better get back to work on it.