Intercepting satellites in orbit

[Benredder]

I have a few contracts that I want to finish where I’m required to rescue some kerbals stuck in satellites in orbit. Currently I’ve figured out how to replicate said orbit, but the satellite-to-be-rescued is going the exact same speed but is a quarter of the planets length behind me. My question is this: Is there any way to increase the speed of your orbit while, at the same time, not changing the shape of your orbit?

[OHara]

No.  

The shape of the orbit determines how long it takes to complete an orbit, assuming that by 'orbit' you mean the free motion under gravity; to take the path faster of slower would require continuous thrusting.

You will need to thrust forward into a higher orbit, which takes longer to complete, to let the target catch up to you ---figuring your much higher by trial-and-error, using the manuever nodes intercept marks in the game, or pre-computing with Kepler's law if you like ---  and then slow back down when you meet it.

[Blaarkies]

22 minutes ago, Benredder said:

I have a few contracts that I want to finish where I’m required to rescue some kerbals stuck in satellites in orbit. Currently I’ve figured out how to replicate said orbit, but the satellite-to-be-rescued is going the exact same speed but is a quarter of the planets length behind me. My question is this: Is there any way to increase the speed of your orbit while, at the same time, not changing the shape of your orbit?

No
Orbits with exactly the same shape will have exactly the same orbital period, orbital velocities at the same points of the orbit and...they are the same after all

It sounds like you are ahead of the satellite? What you need to do is, thrust towards prograde(forward). This will increase your speed, which will stretch out your orbit on the other side(making it bigger). This will cause your orbital period to increase(you take longer to finish on lap). This way that satellite will catch up to you, just wait until the point where your orbital line intersects the satellite's orbital line. From there, re-evaluate whether or not you are ahead or behind the satellite. 

[DrLicor]

Looks like you need this  

 

 

[Zhetaan]

9 hours ago, Benredder said:

My question is this: Is there any way to increase the speed of your orbit while, at the same time, not changing the shape of your orbit?

Technically, yes, you could have a higher speed while you forced your orbit down by burning radial in--but to accomplish this kind of forced orbit would require effectively infinite fuel.

What you probably want to do instead is have something called a phasing orbit.  Saying that the satellite is a quarter of the planet's length behind you doesn't actually tell much--different orbits are different sizes and shapes, so a planet's-length means different things in terms of orbital proximity depending on the actual shape and size of the orbit and your current location on it--so instead, I will assume that your orbits are nearly circular and that your target is a quarter-orbit behind you.

The important parameter to know is how far away the target is in terms of time.  If you're on the same orbit but at different points on the orbit, then the target is going to be some number of seconds ahead of or behind you.  If you know that number and you know also your orbital period, then you can burn to an orbit that has a period equal to your current orbit plus the difference (or, if you're behind, you burn to an orbit that is the current period minus the difference).  The equation for it is this:

a³ = μT² / 4π²

where:

a = semi-major axis of the orbit (in metres)
μ = gravitational parameter (this is constant for each celestial body, so for anything in Kerbin orbit, you need Kerbin's parameter, which is equal to 3.5315984×10¹² m³/s²)
T = orbital period (in seconds)

Let's say for example that you are chasing a satellite in a perfectly circular 100-kilometre orbit.  Your current orbital speed is irrelevant, but the orbital period is not.  You can fairly easily find the orbital period (subtract the values of the-time-to-apoapsis and time-to-periapsis markers to get half of an orbit, and multiply that time by 2 for the full orbit), but let's run through it once just for the sake of knowing what we're doing.  To get orbital period, the equation rearranges to:

T² = 4π²a³ / μ

The semi-major axis of a 100 km orbit is not (100_Pe + 100_Ap) / 2 = 100 km:  the apsis values in KSP are surface altitudes, so they don't take into account the radius of the planet.  Kerbin is 600 km in radius, so finding the total semi-major axis of the orbit requires you to add 1200 km to the apsides before you divide by 2.  That makes the total semi-major axis 700 km, or 700,000 metres.

T² = 4π² (700000)³ / (3.5315984×10¹²)
T² = 1.3541097×10¹⁹ / 3.5315984×10¹²
T² = 3834268.7
T = 1958.1 seconds, or about 32 minutes, 38 seconds

Let's assume that your satellite is a quarter-orbit behind you.  You can find this value by setting up a manoeuvre node and watching the time-to-node at the moment the satellite passes it.  Since the node is ahead of you, you'll have to subtract that time from the period to see how far behind you the satellite is, but that shouldn't be a problem.  Alternatively, set up a node, let it pass you, and see the time when the satellite passes it.  Since we're assuming one quarter, that puts the satellite 489.5 seconds behind you, or a little over 8 minutes.  Therefore, you want an orbit with a period equal to your current orbit plus 8 minutes, so that in the time it takes you to go around once and return to your starting position, the satellite goes around once and then catches up to you.

That means you want an orbit whose period is equal to 2447.6 seconds:

a³ = μT² / 4π²
a³ = (3.5315984×10¹²)*(2447.6)² / 4π²
a³ = 5.3591074×10¹⁷
a = 812264.5 metres, or 812.2645 km

Multiply this by 2 to get the new major axis (not semi-major!) of about 1624.5 km.

This is not much:  remember that 1200 km of that is the planet, and another 100 km is the altitude to your periapsis (which is your burn point and will not change because you want to return to it).  That leaves 324.5 km, which is the altitude of your new apoapsis.  Burn for that altitude and when you return to your periapsis, you'll be on top of the satellite.  Burn to re-circularise and intercept.

Alternatively, you can do this entirely by eye by setting up a manoeuvre node and watching the intercept markers change as you alter the burn.

Edited October 20, 2017 by Zhetaan