Monday, 4 April 2011

Mekanika Orbit: Rules of Thumb

"Ilmu Pengetahuan dan Teknologi Antariksa di Indonesia Mesti Dikembangkan Secepat Mungkin"
*Arip Nurahman*

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.
  • Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating bodies, in the absence of non-gravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects:
    • Orbits are either circular, with the planet at the center of the circle, or elliptical, with the planet at one focus of the ellipse.
    • A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
    • The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
  • Without firing a rocket engine (generating thrust), the height and shape of the satellite's orbit won't change, and it will maintain the same orientation with respect to the fixed stars.
  • A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
  • If a brief rocket firing is made at only one point in the satellite's orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus to move from one circular orbit to another, at least two brief firings are needed.
  • From a circular orbit, a brief firing of a rocket in the direction which slows the satellite down, will create an elliptical orbit with a lower perigee (lowest orbital point) at 180 degrees away from the firing point, which will be the apogee (highest orbital point). If the rocket is fired to speed the rocket, it will create an elliptical orbit with a higher apogee 180 degrees away from the firing point (which will become the perigee).
The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, unless they are very close, the trailing craft cannot simply fire its engines to go faster.

This will change the shape of its orbit, causing it to gain altitude and miss its target. One approach is to actually fire a reverse thrust to slow down, and then fire again to re-circularize the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up.

A third firing at the right time will put the trailing craft in an elliptical orbit which will intersect the path of the leading craft, approaching from below.
To the degree that the standard assumptions of astrodynamics do not hold, actual trajectories will vary from those calculated. For example, simple atmospheric drag is another complicating factor for objects in Earth orbit.

These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system. (Celestial mechanics uses more general rules applicable to a wider variety of situations.) The differences between classical mechanics and general relativity can also become important for large objects like planets.