Elements of Astrodynamics

Even if one has the propulsion and power technology to send spacecraft into space this is not enough. To send a spacecraft to the moon one does not just point the rocket at the moon like a rifle pointed at a target. A spacecraft travelling anywhere in the solar system is influenced the gravity field of nearby planets and the sun. Thus the trajectory of a spacecraft is not going to be a straight line. This is most clearly the case for a satellite orbiting the Earth. So for these Newtonian velocity spacecraft some discussion of spacecraft navigation or astrodynamics is relevant.

The simplest example of this is a circular orbit of a satellite around a planet or star. Newton's second law indicates that v2 GM , N

which predicts a velocity with a magnitude v = y/GM/r tangent to a circle of radius r. A similar easy calculation can give the escape velocity. The total energy of a particle is given by the kinetic energy of that particle K = 1/2mv2 plus the potential energy in the gravity field. That potential energy is V = -GMm/r, where the force induced by this potential is F = -dV/dr. The total energy is then

If this total energy vanishes it is easy to show that v = N/2GM/ r. This velocity vanishes when r and so this velocity is the break even point for escape velocity. It is apparent that a spacecraft in a circular orbit at radius r with a velocity v must be accelerated to a velocity vesc = \/2v. A spacecraft in Earth low earth orbit has a velocity v ~ 7 km/sec and the escape velocity is v ~ 11 km/sec. A spacecraft given a higher velocity will escape as well, and if the sun were not present it would reach "infinity" at a nonzero velocity. The velocity of the Earth around the sun is 29.5 km/sec and the velocity required to escape the solar system is a/2 x 29.5 km/s = 41.7 km/sec.

To consider the motion of a body under a central 1/r2 force, such as gravity, it requires that the problem be cast in polar coordinates. The center of the coordinate plane is the center of the gravitating body, and by extension is the center of force. The radial and and transverse, or angular, components of the acceleration are

(dey i±ud8y ar-dt2 ~r\dt) ' a°-rdtV dt) ■ (5'3)

For a central force the angular acceleration is zero. For some physics problems this will not always be the case, such as with the motion of a charged particle in converging or diverging lines of magnetic force. The vanishing of the angular acceleration means that

,dd ~~dt for j a constant angular momentum per unit mass. The acceleration in the radial direction is then identified by Newton's second law with the gravitational acceleration

It is conventional to convert this problem to the variable u = 1/r. Further, since d6/dt = ju2 the problem may be converted from a time variable to the radial component of motion with d2u GM . .

Both sides of this differential equation are constant and is for an oscillating spring with a constant force. This differential equation then defines the motion of the particle by

for 90 and e constants of integration, and e defines the eccentricity of the orbit. The eccentricity is given by the angular momentum by j2/GM = ar (1 — e2), and an elliptical orbit is determined by e < 1.

Fig. 5.1. Orbital parameters used to characterize an orbit.

A real orbit around a star or planet is elliptical, as laid down by Kepler nearly 400 years ago. With the launching of Sputnik by the USSR in 1957 this was extended to astrodynamics. Orbits in astrodynamics are defined in the same way as with celestial mechanics as ellipses described by orbital elements. The Keplerian set of orbital elements [5.1] for a elliptical orbit are:

— Epoch (time): A moment in time for which celestial coordinates or orbital elements are specified. This is usually expressed in Julian seconds.

— Inclination (i): The angle subtended by the plane of the orbit of a planet or satellite and the ecliptic, which is the orbit of Earth.

— Longitude of the ascending node (I): The angle formed at the Sun from the First Point of Aries to the body's ascending node, as measured in the reference plane of the ecliptic. The ascending node is the point where the orbiting body passes through the plane of ecliptic.

— Argument of periapsis (w): The angle between the ascending node and the periapsis, or point of closest approach of an orbiting body to the central attracting body. For equatorial orbits this is not defined.

— Eccentricity (e): For elliptic orbits it is calculated from distance at periapsis (perigee or closest approach) and apoapsis (apogee or further approach) as where dp is distance at periapsis, and da is distance at apoapsis. — Mean anomaly (M) at epoch: M is a measure of time, which is a multiple of 2n radians at and only at periapsis. The mean Longitude L is defined as

— Orbital period (T): The time for a complete orbital revolution as given by Kepler's third law.

The first of these is not strictly an orbital parameter, but is how one sets the reference clock. This does become a dynamical variable if velocities are large enough to exhibit relativistic effects. However, this will be ignored.

These parameters are used to define orbits for planets and satellites in polar coordinates on a plane tilted relative to a the plane of the ecliptic. The ecliptic and periapsis define this tilting. The ascending node indicates a point of intersection of the orbit with the ecliptic, and its longitude is that angle between a line from the central body to that point and a coordinate direction of the ecliptic. The coordinate direction is the first of Aries, which is the orbital position for the spring equinox on Earth. The remaining orbital parameters give the closest approach of the planet, the structure of the elliptical orbit by its eccentricity and where at a given time the orbiting body is. These six pieces of datum are equivalent to the specification of x, y, z and vz, vy, vz in cartesian coordinates for the position and velocity of the orbiting body. However, the orbital parameters are expressed more naturally according to the geometry of elliptical orbits. This information is then used to integrate the orbit of an orbiting body according to Newton's second law. It is left to the interested reader to study this further.

For interplanetary travel a spacecraft is transferred from Earth orbit to the orbit of another planet. The transfer that performs this with the least energy is a Hohmann transfer orbit. This further assumes that the orbits of the two planets are on the same plane. This orbit is an elliptical orbit that "kisses" the orbit of the two planetary orbits. For a planet in an elliptical orbit the total energy per unit mass for this orbit is e e

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