Requesting cycler calculations

[Superluminaut]

To anyone who enjoys orbital math, I would really appreciate it if you could calculate some numbers for me.

Because in ksp, kerbin and duna have the same resonance frequency as earth and mars, I’m sourcing the numbers I have from this earth-mars example.

First I would like to know when to place the cycler into solar orbit, this should be the earliest occurrence of a kerbin-duna phase angle of 55.38 degrees. When does this event take place?

The cyclers orbit has a synodic period of 2.145x kerbin years. If the cyclers periapsis is 13,683,999,542m, what is the apoapsis?

I believe with these numbers I can get a cycler running… at least for the first cycle.

[Tommygun]

Is this what you need to know?

http://www.reddit.com/r/KerbalAcademy/comments/1x06wa/how_to_make_a_kerbinduna_aldrin_cycler/

and here:

http://forum.kerbalspaceprogram.com/threads/24393-Proof-of-concept-Duna-Cycler-trajectory

[Superluminaut]

Is this what you need to know?

http://www.reddit.com/r/KerbalAcademy/comments/1x06wa/how_to_make_a_kerbinduna_aldrin_cycler/

and here:

http://forum.kerbalspaceprogram.com/threads/24393-Proof-of-concept-Duna-Cycler-trajectory

Thank you but this its not the information I'm looking for. The reddit guy got his numbers confused, and the other is not a cycler.

I know the numbers can be derived very quickly from the info I have given if you know the equations needed.

[K^2]

T = 2*pi * sqrt(a³/μ)

Substitute in period in seconds for T and Kerbol's gravitational parameter for μ. That will give you the semi-major axis a in meters, from which you can get your apsides.

[Bill Phil]

If the orbital periods coincide, and they start in the right places, then maybe...

[PakledHostage]

I believe with these numbers I can get a cycler running… at least for the first cycle.

I did a free return flyby of Duna back before maneuver nodes were added in version 0.18 of e game, so it should be possible to set up a true cycler with all the bells and whistles that are currently available. I don't have the specifics of my trajectory at hand but I'll see what I can dig up.

[Superluminaut]

T = 2*pi * sqrt(a³/μ)

Substitute in period in seconds for T and Kerbol's gravitational parameter for μ. That will give you the semi-major axis a in meters, from which you can get your apsides.

Hey thanks,

my math is not so strong, I get semi-major axis = μ^1/3(T/2pi)^2/3

How do I get apsides? I think I just subtract the periapsis from the semi-major axis...

Also how can the time point of a certain phase angle be calculated?

Edited March 19, 2015 by Superluminaut

[K^2]

How do I get apsides? I think I just subtract the periapsis from the semi-major axis...

Almost. It's twice the semi-major axis. (Also known as the major axis.)

r1 + r2 = 2a

However, keep in mind that in KSP, the apsides are reported as altitudes, and the r1 and r2 above are from center of the planet. So:

h1 + h2 = 2a - 2R

Here, h1 and h2 are the apsides as reported by KSP.

Also how can the time point of a certain phase angle be calculated?

Now that's tricky. It's easier than going from time to angle, but it's still tricky. First, you need to understand the concepts of Mean Anomaly and Eccentric Anomaly.

Mean anomaly is the angle (in radians) you'd make in this fraction of your orbit if it was circular. In other words, after time t, your mean anomaly is M = 2pi * t/T. So if you know M and T, you know t.

Now, the Mean anomaly is related to Eccentric Anomaly via Kepler's equation.

M = E - e sin(E)

Here, e is the eccentricity of the orbit, and it's equal to (r1-r2)/(r1+r2). e = 0 corresponds to circular orbit. e = 1 corresponds to escape trajectory. Elliptical orbits will fall somewhere in between.

Now, Eccentric anomaly is the angle the body makes from the center of the ellipse. And that is related to the true anomaly θ (which is the phase you want) by the following.

tan(θ/2) = (sqrt(1+e) / sqrt(1-e)) * tan(E/2)

Once you know tan(E/2) from formula above, you can find E/2 by taking arctan of it. (It's also called tan^-1 on some calculators). Make sure that your θ is in radians and is between 0 and 2pi.

Edited March 19, 2015 by K^2