| Rules of thumb |
Rules of thumb
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 in the absence of thrust:
Orbits are either circular, with the planet at the center of the
circle, or form an
ellipse,
with the planet at one
focus. 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 assumptions do not hold, actual trajectories will vary
from those calculated.
Atmospheric drag is one major complicating factor for objects in
Earth orbit. The differences between
classical mechanics and
general relativity can become important for large objects like planets.
These rules of thumb are decidedly inaccurate when describing two or more bodies
of similar mass, such as a
binary star system.
|
