Apapsis and revolution synchonization

[Warzouz]

I'm returning from Jool to Kerbin. I might come too fast no to blow into Kerbin atmosphere.

If I want to do a gravity assist at Kerbon to slow down, what Sun apoapsis I need to target to encounter Kerbin again at my Periaps on my next orbit ?

I assume this should be an table of apoapsis for each Kerbin orbit (x2, x3, x4, x5...)

 

[Snark]

Can you be a bit more specific about your return trajectory?

Are you coming in with a solar Pe set to Kerbin's orbit, and you want to lower your Ap to get a Kerbin encounter when you're at solar Pe?

Or are you coming in with a solar Pe much lower than Kerbin, and you want to lower your Ap to get a Kerbin encounter when you're at solar Ap?

Or something else?

[Warzouz]

16 minutes ago, Snark said:

Can you be a bit more specific about your return trajectory?

Are you coming in with a solar Pe set to Kerbin's orbit, and you want to lower your Ap to get a Kerbin encounter when you're at solar Pe?

Or are you coming in with a solar Pe much lower than Kerbin, and you want to lower your Ap to get a Kerbin encounter when you're at solar Ap?

Or something else?

I'm coming on a near solar PE at Kerbin orbit. I want to gravity assist to meet Kerbin again around the same place on next orbit (with probably a small adjustment on apoapis.

I suppose there is some sun apoapsis altitude which should allow me to encounter Kerbin at nearly the same place (this the small adjustment).

Is there a way to calculate those apoapsis ?

[MaxL_1023]

I think he wants to encounter Kerbin once at solar PE (with AP at Jool from the ejection burn), slow down due to a reverse gravity assist, then encounter Kerbin again on the next solar PE.

It looks like you need to capture into a resonant orbit - you want either 2 Kerbin orbits for one of yours, or 3 Kerbin orbits for 2 of yours (depends on how long you are willing to wait). Either one would have a lower entry velocity than a direct transfer from Jool.

For a 2:1 resonance, you need an AP of about 1.5 times Kerbin's.

 

[Warzouz]

22 minutes ago, MaxL_1023 said:

I think he wants to encounter Kerbin once at solar PE (with AP at Jool from the ejection burn), slow down due to a reverse gravity assist, then encounter Kerbin again on the next solar PE.

It looks like you need to capture into a resonant orbit - you want either 2 Kerbin orbits for one of yours, or 3 Kerbin orbits for 2 of yours (depends on how long you are willing to wait). Either one would have a lower entry velocity than a direct transfer from Jool.

For a 2:1 resonance, you need an AP of about 1.5 times Kerbin's.

 

Yes RESONANT Orbit, that's it.

How to calculate the apoapsis of those orbit ?

Why I want to do that : if I find myself going too fast, I'll change my encounter to a slowdown gravity assist, and I would like to set my apopsis to the correct orbit so I'll get Kerbin again easily.

Edited April 10, 2016 by Warzouz

[OhioBob]

Assuming your periapsis is the orbit of Kerbin, then

For a 2:1 resonance, Ap = 29,315,361,202 m

For a 3:1 resonance, Ap = 42,716,175,169 m

For a 4:1 resonance, Ap = 54,677,459,799 m

These numbers are altitudes, not radii.  It should be the number you read when hovering your mouse pointer over the Ap.

Rather than worrying too much abut getting Pe and Ap just right, it might be easier to go by semimajor axis.  If your semimajor axis is correct, you should come back to the same spot regardless of whether your Pe and Ap are just right.  Here is what the semimajor axis should be:

For a 2:1 resonance, a = 21,588,400,729 m

For a 3:1 resonance, a = 28,288,807,713 m

For a 4:1 resonance, a = 34,269,450,027 m

[MaxL_1023]

How much delta-v do you have left once you get to Kerbin's PE? You might be able to capture into a highly eccentric orbit then aerobrake down (PE near 55km). It will take a fair bit to get into a resonant orbit - even with the oberth effect likely several hundred m/s.

[OhioBob]

31 minutes ago, MaxL_1023 said:

For a 2:1 resonance, you need an AP of about 1.5 times Kerbin's.

Not quite, you need a semimajor axis about 1.5 times Kerbin.

It really is rather simple, the square of the period is proportional to the cube of the semimajor axis (Kepler's third law).

(p₂ / p₁)² = (a₂ / a₁)³

Therefore, if we double the period then the ratio of the semimajor axes becomes

a₂ / a₁ = (2²)^1/3 = 1.587401052

Tripling the period we get,

a₂ / a₁ = (3²)^1/3 = 2.080083823

and so on.

FYI, Kerbin's semimajor axis is 13,599,840,256 meters.

[Blaarkies]

TL;DR -> Solar Ap = 29 576 962 461.7 m should give an orbital period twice that of Kerbin(you will encounter Kerbin again 2 years after your first approach, at the same location)...that is an Ap a bit past Duna's orbit?

"sidereal period" is the term you are looking for(an orbit position relative to the stars):

where:

•  is the orbit's semi-major axis in meters
•  is the standard gravitational parameter in 
  •  is the gravitational constant,
  •  is the mass of the more massive body.

Kerbol:
Orbital Characteristics
Standard gravitational parameter: 1.1723328×10^18 m3/s2

Kerbin:
Orbital Characteristics
Semi-major axis: 13 599 840 256 m
Apoapsis: 13 599 840 256 m
Periapsis: 13 599 840 256 m
Sidereal orbital period: 9 203 545 s -> 426 d 0 h 32 m 24.6 s

So I am assuming you dropped your solar Pe down to 13 599 840 256m, got an encounter with Kerbin and you need to know what solar Ap(after gravity assist) you should aim for(so that your orbital period is 2x, 3x, 4x,..etc that of Kerbin). So turn the above equation around(you want alpha):


a = sqrt3( (T^2 * u) / (4 * pi^2) )


Ap = 2( a - Pe/2 )

2x kerbin's orbital period:
T=(9 203 545 s) * 2
=  18407090 s
alpha = 21 588 401 358.8451 m
Solar Ap = 29 576 962 461.7 m

3x
alpha = 28 288 808 537.9473
Solar Ap = 42 977 776 819.9 m

4x
alpha = 34 269 451 027.3265 m
Solar Ap = 54 939 061 798.7 m

1/2x
alpha = 8 567 362 756.83162 m
Solar Ap = 3 534 885 257.7 m

1/3x
alpha = 6 538 121 445.91076
Solar Ap = -523 597 364.2m
*Note: Negative apoapsis could be difficult.

Remember that these are basic ratios for YOUR first orbit...you could get a lot more different encounters by using a 1.53789x ratio and waiting for 10 orbits to go by(that random number is not tested, this is just for arguments sake).

Shamelessly lifted a lot of info from these:
https://en.wikipedia.org/wiki/Orbital_period
http://wiki.kerbalspaceprogram.com/wiki/Kerbin
http://wiki.kerbalspaceprogram.com/wiki/Kerbol
http://www.sciweavers.org/free-online-latex-equation-editor -> Pasted .png does not work on forums

Edited April 11, 2016 by Blaarkies
png equation pictures do not show

[Warzouz]

OK, I did some math

• T : Kerbin orbital periode
• G : Gravitationnal constant
• M : Kerbol mass
• r : resonance

I get : Target semi major axis^3 = (r * T/2pi)^2 * GM

EDIT (multi ninja'd)

Edited April 10, 2016 by Warzouz

[Blaarkies]

@OhioBob Damn this forum is fast, well done that was a quick answer. I just want to add that, there exists some resonant periods between 1:1 and 2:1
-> 1.333:1 - This should give a low solar Ap, but the next encounter will only happen in 6 years

But the difference in speed when swinging by Kerbin will be in the order of a few hundred m/s or so(since 940m/s burn at LKO gets you out of Kerbin SOI, but a 1200m/s burn will get you to Duna)