CheersKevin has some nice videos where he writes various KOS scripts that cope with this problem:
The video #10 - Space Rescue! is the one about the rendezvous, but some of the scripts used (like to calculate orbit times etc) are developed in previous videos. IIRC he solves it like Meithan described vith the variation that he first burns for a transfer orbit, THEN burns for a higher "wait/sync" orbit with a period of
+phase time. So a bit less elegant and less efficient but this works for launch to rendezvous without waiting on the ground and you are guaranteed to have a sync orbit outside the atmo (for its always has higher APo than the target ones), making it a nice KOS base script.
There are less math solutions to the problem. Given: radius central body = 600k, a rescue target at 100k alt on equitorial orbit (90'), atm psuedo limit at 70k, launch site on equator, surface gravity = 9.8
Method 1. Direct intercept. Also known as "watch your target zip by on close approach as you are struggling to get your 1g*mass thruster to throw out 1000 dV in desired 2 seconds."
0. Nu is the sqrt of (mu). Nu is our cheat, I keep it close to my heart. mu = 3.5316E12 (obtain by multiplying accel(=9.81)*r(=0.6M)^2: Nu = sqrt(9.81) * r(=0.6M) = 1.88M
1. Calculate target orbit
a. nu/sqrt® = velocity = 2246 m/s. The circumferance is what pi*rad(=700k)*2 = 4398229m. The period circ/vel = 1959 seconds.
b. angular velocity(omega) = 2*pi()/period = 0.0033 radians/sec - done
2. Emperically a vessel takes 5 minutes to reach near orbital velocity. During the first minute of flight its horizontal velocity is near zero, but climbs from near zero to orbital over 4 minutes.
a. thus it is traveling about 2/5ths orbital velocity (discounting ground movement) for 5 minutes, and orbital velocity thereafter.
b. this means if a target is directly overhead it need to warp 3 minutes forward to catch up after reaching target orbit. But since we know angular velocity we can calculate what? lets see
c. 3min(=60sec)*0.0032 = 0.57 radians = 33 degrees.
3. So if you know the longditudinal coordinates of the launch site (in mech jeb this will be given as angle to prograde) and you know the angle to prograde of the target, just wait until the launch site preceeds the anlge to prograde of the target by 33, launch and set the target altitude at 100. The only trick here is to make the orbit realtively flat over 42k. With reference to the targets humor, if you are not experienced in making your gravity turn you might have to thrust in the -radial direction to get orbital velocity were it will intercept. If you are not, then allow your self a few extra degrees. When you reach the target your horizontal velocity relative to the orbital velocity (target velocity also)/2*accel(=thrustMax/mass) is the lead time you need over the target at your apo. If you are one of those who likes to do most of the circularization at target altitude, then you need to factor this time into your lead over the target at launch. For example, if you are 1000 m/s slower and your thrust is 1 g, then it takes 100 seconds, you will need half this time. so 50 seconds = 0.16rad (=10 degrees).
Oh, and how much dV do you need, its the same as orbital insertion from launch. IOW practice making orbit at 75k and then use that method and extend your apo from 75k to 100k at an altitude of 60k or so.
In any case if you know your insertion methods (wipe that grin off your face) and you know how to guestimate targets position relative to launch site even goofing this method and sill having a 100k equitorial then you are going to get much closer to your target than you might imagine, 5k is ideal, more than that and you might have to increase or decrease your orbit a tad to close the gap.
Method 2.
Overkill phasing. This method is less risky than the first, but really overkills on fuel
1. So we know what our Nu is, is that cute, so how do we get an orbit going half the velocity. r (=600K+alt) needs to be 4 fold. So to get an orbit half as fast we need an orbit 4 times the radius or 2800K, that means we need an alt of 2200K. But then our target is 1/4th the radius, so its radians per second are v/r and ours is (v/2)/4r = 1/8th its making 8 oribt for our 1 and basically we have a window every 37 minutes. Thats not to clever, but on the bright side we are long 400km from a KSO.
2. So we are going to have to close the gap, but do we actually need to think about this too much just match the planes. Actually this does not need much math either. Just burn retro until the orbit is nearly the targets orbit alt or even the targets orbit. It is recommended that you do this at your targets theta(=apo) this way your intercept is at the targets pa. Now we might get lucky and get close to the target, but actually chances are no and don't waste the effort trying. When the ship hits apo, just retro, as you retro your ships ellipse will intercept the target maybe six times as you close your orbit, you are interested in the smallest circle. Each of these you can calculate with relative ease if you know the period. Mech jeb tells you the periods of your orbit and your target will give its period. so basically you can use this to estimate the intercept if you know the angle to prograde.
Method 3
Smart phasing. So basically we have done the direct, and we have overkilled on the phasing. How about a better phasing. Once again target is at 100 k
1. This time you want to be about 90 degrees ahead of your target.
2. Launch to about 125 to 150 km alt.
3. Once at target apo, correct the pe to 100 km alt, (125 alt apo = ~ 0.0031 rad/sec, 150 alt apo = ~0.00305 rad per second)
4. At pe squeeze the apo in just as in method 2
5. time the intercept.
Note this can be done at any launch position, but who actually wants to wait
So the question is what makes method 1 and 3 work. The short answer is we are not lingering around waiting for a transfer window, in method 1 we are essentially allowing our non-inertial reference frame (=surface of Kerbin see Einstien's theory of relativity) to phase us with a target in its inertial reference frame. We only need to wait 32 minutes at most to launch, and we are trying to get within a 5 km window where our docking manuevers are more important than transfer dynamics.
In method 3 we know a higher orbit has a slower angular velocity, so we allow ourselves to get about 60 degrees ahead of the target, and then use that lead to close our orbit with the target successively such that when we are ready to make the final closure we are on top of the targets. In method three we retro at pe which is targets alt, so no fuel is wasted in making multiple retros at the same kick position. The better you are at approximating your launch with target intercept, the lower altitude that is convienient (avoiding wasted time in orbit).
And the lesson we learn is that if we had to rescue a naut in crisis its almost always best to plan a launch window that will intercept a target in a more or less direct manner. Simple math on the ground is alot cheaper on the brain than complex math in space.
