Rendezvous

Rendezvous is in spaceflight the event of two spacecraft meeting each other. This does not neccessarily have to be in space, the event of two spacecraft meeting on the surface of a celestial body is also called rendezvous.

In orbit

The most often appearing rendezvous takes place in orbit, eg spacecrafts travelling to a space station. A rendezvous takes place, if both spacecraft are close enough to each other (< 300m) and travel in similar orbits. In that case, both spacecraft can keep station with minimal corrections (linear RCS).

There are two possible ways to get to rendezvous:

• Direct insertion into the space stations orbit
• insertion into a catch-up orbit.

The direct insertion needs more propellant and allows only few windows for insertion. If launching from the surface, the spacecraft has to get not only inserted into the orbit plane of the other spacecraft, but also have to do its final orbit insertion burn close to the space station. Because of the very rare launch windows and high propellant demands, the alternative way of inserting the spacecraft into a catch-up orbit is prefered.

Catch-up orbit

The right launch window for a catch-up orbit is, a launch, which places the ship into the same orbital plane as the target, but with a significantly lower orbit. The important measure is the difference in large semiaxis, which defines the time needed for one revolution around earth.

The moment for leaving the (almost circular) catch-up orbit can be calculated by this formula:

$\Delta \varphi =\pi \left[1-(2m-1)\left({\frac {a}{r_{z}}}\right)^{\frac {3}{2}}\right]$

With:

• $\Delta \varphi$ - the distance angle between the chasing spacecraft and the target spacecraft.
• $m$ - the number of apogee passes on the transfer ellipse.
• $a$ - the large semiaxis of the transfer ellipse.
• $r_{z}$ - the radius of the target orbit (assumed circular).

If this angle is reached, the chasing spacecraft does a propulsion maneuver to enter the planned transfer ellipse. This is called the intercept burn.

Rendezvous burn

When reaching the final apogee pass of the transfer ellipse, all possible errors will usually bring the spacecraft away from its desired course (including errors because the initial orbits have not been perfectly circular). Because of that influences, the next orbit is usually not the final target orbit, but again a less eccentric chasing orbit. The parameters of this orbit can be calculated using the following formula:

• The spacecraft arrives at the target point before the target:

$a=r_{z}\cdot {}^{3}{\sqrt {\left(1+{\frac {1}{n}}\cdot {\frac {(2\pi -\Delta \varphi )}{2\pi }}\right)^{2}}}$

• The target arrives at the target point before the chasing spacecraft:

$a=r_{z}\cdot {}^{3}{\sqrt {\left(1-{\frac {1}{n}}\cdot {\frac {\Delta \varphi }{2\pi }}\right)^{2}}}$

Rendezvous

When the target is reached, the orbit parameters of both craft should be equal. At close distances, its enough to just neutralize any velocity difference.