Orbital Dynamics

Orbital dynamics is the study of the motion of objects in the presence of gravitational forces of other bodies, due to gravity. All bodies provide a gravitational pull on other bodies surrounding it.

The motion of one body about another body due to gravity, is known as an Orbit. Orbits are described by a body of theory called "Orbital Dynamics". Kepler discovered that in Newtonian Physics, which ignore Einstein's theories of relativity, orbits are elliptical in shape, at least for the simple case of one body orbiting another body without any influence from any third body. Kepler's Laws are three equations which describe elliptical orbits, and still hold true today.

Newton's Law of Gravity

The force exerted by two bodies on each other is given by

<math>F = \frac{Gm_{1}m_{2}}{r^{2}}</math>

where G is the Universal Gravitational Constant whose value is 6.67300 × 10^-11 m³ kg^-1 s^-2, <math>m_1</math> is the mass of the first body, <math>m_2</math> is the mass of the second body, and r is the distance between them.

The centripetal force of an object that is held in rotation around a fixed point is expressed as

F = (m₂ * V²) / r

where V is the linear velocity of the object, which in our case is the satellite. Notice that the velocity (and later the period) are independent of the mass of the satellite. Combining these two equations together and solving for the velocity of the satellite gives an orbital velocity of a circular orbit as

<math>V = \sqrt{\frac{m_{1}G}{r}} = \sqrt{\frac{\mu G}{r}} </math>

where <math>\mu = m_{1}G </math> is referred to as a gravitational parameter, for which tables can be found for the Sun and each of the planets.

The circumference of a circular orbit is <math>C = 2 \pi r </math>. The period is then equal to the circumference divided by the velocity, or

<math>P = \frac{C}{V} = 2 \pi \sqrt{\frac{r^3}{\mu}} </math>

Keplerian Orbits

A Keplerian orbit is a simplified orbit with the following assumptions

1. There are only two bodies exerting a gravitational force.
2. The bodies are homogeneous and spherical.
3. No other forces act on the system.

These assumptions lead to an elliptical orbit. In fact, both bodies will orbit around a point that is located along the line connecting the center of mass of the two bodies. The location on this line is the center of mass <math>m_2/(m_1+m_2)</math>. For a man-made satellite orbiting the Earth, the center of mass of the combined system is essentially at the center of mass of the earth. For the Moon orbiting the Earth, the center of mass of the combined system is displaced 2949 miles from the center of mass of the Earth. It is 1014 miles below the surface of the Earth.

Simple Celestial Mechanics for Simple Problems

In the case in which the tabulated data available is surface gravity and radius, as is the case for some amatures dealing with moons and planets, then surface orbital velocity can be found as the square root of the product of the radius and the surface gravity.

V_os = (g_s * r) ^ 0.5

To find circular orbital velocity at some altitude, first find the gravity at that altitude. Then the orbital velocity at that altitude is the square root of the product of gravity at altitude the radius of the orbit.

g_a = g_s * (r_s² / r_a²)

V_oa = (g_a * r_a) ^ 0.5

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