[Tutorial] Timed-Visual Sattellite Injection Timings

[Frederf]

I thought I would share my method for a Kerbin-Minimus injection which relied only on a stopwatch and looking out the window of the IVA spacecraft. I found it both satisfying and effective.

Initial conditions: Space craft in Kerbin circular orbit, approx 1000km in this example. Orbital plane matches destination object, in this case Minimus. Orbits are same direction for simplicity but not required.

Concept: Using simple orbital math a time can be derived between seeing an object at the periphery of Kerbin's disk and when a rendezvous burn is initiated.

Step 1. Phase angle and transit time

In this example assume the selected transfer orbit has a transit time of 48 hours. A successful arrival requires that our burn takes place on the opposite side of Kerbin from the position of Minimus 48 hours in the future. Minimus will travel 48/T of its orbital path in this time. Let's say the orbital period is 320 hours. 48/320 is 0.15 or 15% or 54 degrees. If you burn when you are located opposite of this future position of Minimus then you will arrive at 54 degrees in the future of Minimus's orbit and so will Minimus.

Step 2. Time from opposition

Assume you knew the exact moment you were directly opposed to Minimus's current position. Given that information how long would you wait until you began your burn? That is the goal of this step. The spacecraft in the parking orbit (and the opposition direction) is constantly changing its orbit angle. Minimus is also orbiting and changing its angle but more slowly as it is distant. If a space craft is orbiting 360 degrees in 18 minutes, then it is orbiting at a rate of 20 degrees per minute. If Minimus is orbiting at 360 degrees per 320 hours then that is 0.01875 degrees per minute. The result is that the space craft is generating 19.98125 degrees of relative orbit angle per minute. Going back to Step 1, we want a 54 degree angular difference so we must wait 2.7025 minutes after the spacecraft is directly opposed (opposite of Kerbin) to Minimus before beginning our burn.

Step 3. Time from sighting

However we don't know the exact moment of opposition, yet. What we can know, by looking out the window, is when Minimus appears out from behind Kerbin. From this we can learn how far in the past (before the sighting) Minimus was in opposition. When Minimus is sighted, it must be some distance along its orbit behind the current opposition point. At an altitude of 1000km I am 1600km from the center of Kerbin along the axis which passes through the spacecraft, Kerbin's center, and the current point of opposition of my orbit (the last one by definition). Minimus at an altitude of 46,400km is 47,000km from Kerbin's center approximately along this same axis. Assuming the angle to be small, Minimus is 48600km away from me. The slice of Kerbin which Minimus emerging from behind is 600km lateral from the axis. Minimus is an unknown lateral distance from this axis and is what we currently wish to find. Using the properties of similar triangles 600/1600 = X/48600 where X is the lateral distance (or in another approximation the orbital path distance) of Minimus.

The tangent of (X/47000) is the number of degrees orbit angle between the current position of Minimus and the current position of the opposition point at sighting. In this case the angle is 21.195 degrees. At 19.98125 degrees per minute this means 1.061 minutes ago, Minimus and the spaceship were on opposite sides of Kerbin.

Step 4. Putting it all together

The last step is straightforward. If we want to burn at 2.7025 minutes after opposition, but by the time we actually see Minimus it will be 1.061 minutes past that, we must burn 2.7025-1.061 minutes after we see Minimus. 1:38 after Minimus is visible will be 2:42 after opposition and will be the required 54 degrees ahead such that when you arrive in the spot two days later, Minimus will be there as well.

Edited October 12, 2012 by sal_vager