2.6 Entering a Phasing Orbit

Explanation

At this point we have matched our orbit to the space station orbit. We are at the correct speed and altitude, and we are in the same orbital plane. However, one task remains. In our Orbit and Orbit Sync MFDs, we can see that although we are flying in the same path as the space station, we are leading by 927 seconds, or about 15 minutes. 

Note that even small differences in time will result in large differences in physical distances between the spaceship and the target. Since the space station is traveling at nearly 1.5 km/s, a difference of just 5 minutes would translate to a physical distance of nearly 500 km. 

To synchronize, we will temporarily adjust our circular orbit into an elliptical phasing orbit that has a higher apoapsis. This will cause us to require more time per orbit relative to the space station, allowing the space station to catch up to us. 

In the code below we calculate the desired period of the temporary orbit by adding the time lag to our current orbital period. Then we work backwards to compute how much speed we should add to achieve this temporary orbit. 

Computation

Here is the computation used to estimate the fuel burn in our flight plan. Note that in our flight plan we used a ballpark estimate for the time lag, and assumed our ship would be ahead of the space station. The time lag was estimated as follows:

• we computed the time required from launch to apex

• we computed the time required for the first orbit (i.e., from apex and then continuing until we passed overhead the lunar base)

• we knew that we launched when the space station was directly south of us, so we can subtract the space station period from our spaceship's first orbit period

Here are the printouts:

Once we reach the rendezvous point, we can exit our phasing orbit at its periapsis by firing the engines in retrograde to re-circularize the orbit. The delta-V and fuel required are identical to the amounts we used to enter the phasing orbit. 

This maneuver is a good example of how orbital mechanics can be counterintuitive. In effect, in order slow down to let the space station catch up,  we sped up!

Flight Results

In Orbiter we have an MFD called Orbit Sync which can help visualize this synchronization orbit. In the images below, we have performed prograde burn to enter the phasing orbit, and the Orbit Sync MFD shows that it will take our ship and the target station about 10,250 seconds (2.8 hours) to reach the rendezvous point. 

Once we reach the rendezvous point, we fire the engines in retrograde to re-circularize our ship's orbit, and then begin maneuvering the ship to dock with the target. At the rendezvous point, the ships were about 2 km from each other. 

On the map MFD, we can see our positions are now matched:

In the next section we will show the docking process.