Modern mathematical techniques
Conic orbits
For simple things like computing the delta-v for coplanar transfer ellipses,
traditional approaches work pretty well. But time-of-flight is harder,
especially for near-circular and hyperbolic orbits.
The patched conic approximation
The transfer orbit alone is not a good approximation for interplanetary
trajectories because it neglects the planets' own gravity. Planetary gravity
dominates the behaviour of the spacecraft in the vicinity of a planet, so it
severely underestimates delta-v, and produces highly inaccurate prescriptions
for burn timings.
One relatively simple way to get a
first-order approximation of delta-v is based on the patched conic
approximation technique. The idea is to choose the one dominant gravitating
body in each region of space through which the trajectory will pass, and to
model only that body's effects in that region. For instance, on a trajectory
from the Earth to
Mars, one would begin by considering only the Earth's gravity until the
trajectory reaches a distance where the Earth's gravity no longer dominates that
of the Sun. The
spacecraft would be given
escape velocity to send it on its way to interplanetary space. Next, one
would consider only the Sun's gravity until the trajectory reaches the
neighbourhood of Mars. During this stage, the transfer orbit model is
appropriate. Finally, only Mars's gravity is considered during the final portion
of the trajectory where Mars's gravity dominates the spacecraft's behaviour. The
spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde
burn would slow the spacecraft enough to be captured by Mars.
The size of the "neighborhoods" (or
spheres of influence) vary with radius rSOI:
-
where ap is the
semimajor axis of the planet's orbit relative to the
Sun;
mp and
ms are the
masses of the
planet and Sun, respectively.
This simplification is sufficient to compute rough estimates of fuel
requirements, and rough time-of-flight estimates, but it is not generally
accurate enough to guide a spacecraft to its destination. For that, numerical
methods are required.
The universal variable formulation
To address the shortcomings of the traditional approaches, the universal
variable approach was developed. It works equally well on circular,
elliptical, parabolic, and hyperbolic orbits; and also works well with
perturbation theory. The differential equations converge nicely when integrated
for any orbit.
PerturbationsThe universal variable formulation works well with the variation of
parameters technique, except now, instead of the six Keplerian orbital elements,
we use a different set of orbital elements: namely, the satellite's initial
position and velocity vectors x0
and v0 at a given epoch
t = 0. In a two-body simulation, these
elements are sufficient to compute the satellite's position and velocity at any
time in the future, using the universal variable formulation. Conversely, at any
moment in the satellite's orbit, we can measure its position and velocity, and
then use the universal variable approach to determine what its initial position
and velocity would have been at the epoch. In perfect two-body motion,
these orbital elements would be invariant (just like the Keplerian elements
would be).
However, perturbations cause the orbital elements to change over time. Hence,
we write the position element as x0(t)
and the velocity element as v0(t),
indicating that they vary with time. The technique to compute the effect of
perturbations becomes one of finding expressions, either exact or approximate,
for the functions x0(t)
and v0(t).
Non-ideal orbitsThe following are some effects which make real orbits differ from the simple
models based on a spherical earth. Most of them can be handled on short
timescales (perhaps less than a few thousand orbits) by perturbation theory
because they are small relative to the corresponding two-body effects.
Equatorial bulges cause precession of the node and the perigee Tesseral harmonics [1] of the gravity field introduce additional perturbations lunar and solar gravity perturbations alter the orbits Atmospheric drag reduces the semi-major axis unless make-up thrust is
used
Over very long timescales (perhaps millions of orbits), even small
perturbations can dominate, and the behaviour can become
chaotic. On the
other hand, the various perturbations can be orchestrated by clever
astrodynamicists to assist with orbit maintenance tasks, such as
station-keeping,
ground
track maintenance or adjustment, or phasing of perigee to cover selected
targets at low altitude.
|
