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6.1.2 Constants of Motion

Using the two-body equation of motion, we can derive several constants of motion of a satellite orbit The first is

where e is the total specific mechanical energy, or mechanical energy per unit mass, for the system and is the sum of the kinetic energy per unit mass and potential energy per unit mass. We refer to Eq. (6-4) as the energy equation. Because the forces in the system are conservative, the energy is a constant The term for potential energy, -p/r, defines the potential energy to be zero at infinity and negative at any radius less than infinity. Using this definition, the specific mechanical energy of elliptical orbits will always be negative. As the energy increases (approaches zero), the ellipse gets larger, and the elliptical trajectory approaches a parabolic trajectory. From the energy Eq. (6-4), we find that a satellite moves fastest at perigee of the orbit and slowest at apogee.

We also know that for a circle the semimajor axis equals the radius, which is constant. Rearranging the energy equation, we find the velocity of a satellite in a circular orbit.

where Vdr is the circular velocity in km/s, RE is the Earth's radius in km, and r is the orbit's radius in km.

From Table 6-1, the energy of a parabolic trajectory is zero. A parabolic trajectory is one with the minimum energy needed to escape the gravitational attraction of Earth. Thus, we can calculate the velocity required to escape from the Earth at any distance, r, by setting energy equal to zero in Eq. (6-4) and solving for velocity.

= 11.179 88 (RE/ryn = 892.8611 r~m where V^ is the escape velocity in km/s, and r is in km.

Another constant of motion associated with a satellite orbit is the specific angular momentum, h, which is the satellite's total angular momentum divided by its mass. We can find it from the cross product of the position and velocity vectors.

We find that from Kepler's second law, the angular momentum is constant in magnitude and direction for the two-body problem. Therefore, the orbital plane defined by the position and velocity vectors must remain fixed in inertial space.

When solving the two-body equations of motion, we need six constants of integration (initial conditions) for the solution. Theoretically, we could find the three components of position and velocity at any time in terms of the position and velocity at any other time. Alternatively, we can completely describe the orbit with five constants and one quantity which varies with time. These quantities, called classical orbital elements, are defined below and are shown in Fig. 6-3. The coordinate frame shown in the figure js the geocentric inertial frame,* or GCI, defined in Chap. 5 (see Table 5-1). Its origin is at the center of the Earth, with the X-axis in the equatorial plane and pointing to the vernal equinox. Also, the Z-axis is parallel to the Earth's spin axis (the North Pole), and the Y-axis completes the right-hand set in the equatorial plane. The classical orbital elements are:

a: semimajor axis: describes the size of the ellipse (see Fig. 6-1).

e: eccentricity: describes the shape of the ellipse (see Fig. 6-1).

°A sufficiently inertial coordinate frame is a coordinate ñame that we can consider to be non-accelerating for the particular application. The GCI frame is sufficiently inertial when considering Earth-orbiting satellites, but is inadequate for interplanetary travel because of its rotational acceleration around the Sun.

i: inclination: the angle between the angular momentum vector and the unit vector in the Z-direction.

£2: right ascension of the ascending node: the angle from the vernal equinox to the ascending node. The ascending node is the point where the satellite passes through the equatorial plane moving from south to north. Right ascension is measured as a right-handed rotation about the pole, Z.

to: argument of perigee: the angle from the ascending node to the eccentricity vector measured in the direction of the satellite's motion. Hie eccentricity vector points from the center of the Earth to perigee with a magnitude equal to the eccentricity of the orbit v: true anomaly: the angle from the eccentricity vector to the satellite position vector, measured in the direction of satellite motion. Alternately, we could use time since perigee passage, T.

Fig. 6-3. Definition of the Keplerlan Orbital Elements of a Satellite In an Elliptic Orbit We define elements relative to the GCI coordinate frame.

Fig. 6-3. Definition of the Keplerlan Orbital Elements of a Satellite In an Elliptic Orbit We define elements relative to the GCI coordinate frame.

Given these definitions, we can solve for the elements if we know the satellite's position and velocity vectors. Equations (6-4) and (6-7) allow us to solve for the energy and the angular momentum vector. An equation for the nodal vector, n, in the direction of the ascending node is n = Z x h (6-8)

We can calculate the eccentricity vector from the following equation:

Table 6-2 lists equations to derive the classical orbital elements and related parameters for an elliptical oibit.

Equatorial (i = 0) and circular (e = 0) orbits demand alternate orbital elements to solve the equations in Table 6-2. These are shown in Fig. 6-3. For equatorial orbits, a single angle, 77, can replace the right ascension of the ascending node and argument of perigee. Called the longitude ofperigee, this angle is the algebraic sum of Si and (0. As i approaches 0, 77 approaches the angle from the X-axis to perigee. For circular

Symbol |
Name |
Equation |
Quadrant Check |

a e |
semimajor axis eccentricity |
a = -p/(2e)=(rA + rP)/2 e = |e| = 1 - (rP/a)= fe/a)-1 |
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