Rendezvous

In spaceflight, rendezvous refers to the event in which two spacecraft meet. This does not neccessarily occur in space: a rendezvous can also happen on the surface of a celestial body.

In orbit

A rendezvous usually takes place in orbit, e.g. when spacecrafts are travelling to a space station. If two spacecraft are close enough to each other (< 300m) and travel in similar orbits they are said to rendezvous. In such a situation, both spacecraft can stay close to the space station with minimal corrections (linear RCS).

There are two ways to bring about rendezvous:

• direct insertion into the space station's orbit, or
• insertion into a catch-up orbit.

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

Catch-up orbit

The best launch window for a catch-up orbit is one that places the ship into the same orbital plane as the target but with a significantly lower orbit. The important measure is the difference between the large semiaxes, since the large semiaxis defines the time needed for one revolution around Earth.

The point at which the spacecraft should leave the (almost circular) catch-up orbit can be calculated using this formula:

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

where

• $\Delta \varphi$ - 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).

When this angle is reached, the chasing spacecraft performs a prograde burn 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 orbital parameters of both craft should be equal. At close distances, its enough to just neutralize any velocity difference.

On the surface

Rendezvous on the surface of a planet is not nearly as difficult as in orbit since the speeds involved are much smaller. The presence of an atmosphere also simplifies maneuvers for aerodynamic vessels. The simplest way to perform a surface rendezvous is to fly directly towards the other spacecraft.

When the other spacecraft is far away (such as on the other side of the planet), a more fuel-efficient way to get there may be via a ballistic trajectory. This can be thought of as "throwing" the spacecraft towards the target, and letting it fall close to it. On planets with no atmosphere the Map MFD can be used to estimate and adjust the ballistic trajectory and the landing point.