Using polar coordinates, equations that describe the orbital trajectory of a satellite or planet can be derived.

If a particle moves under the influence of a force that is directed toward a fixed point, the particle is said to be in central force motion (CFM). CFM is commonly caused by electrostatic or gravitational forces.

A satellite orbiting a planet, for example, exhibits CFM. As the satellite orbits the planet, the magnitude and direction of both the velocity and force change, but the force is always in the direction of the planet. Basically, the satellite is always accelerating towards the planet, but due to its sideways velocity, it stays in orbit.

The general force equilibrium equation is extremely simple since the only force acting on the satellite is gravity,

     F = GMm/r²

where G is the Gravitation Constant (not 'g'), M is the mass of the central body, and m is the mass of the satellite. For earth, G_e = 66.73×10^-12 m³/(kg-s²) and M_e = 5.9×10³⁰ kg.

Before presenting the basic equation of motion for orbits, it is important to understand Kepler's 2nd Law. In the θ direction of orbits (polar coordinates), there is no force, giving,

This requires r v_θ to be a constant, or

Another way to derive this is with conservation of angular momentum. Note, the velocity in the θ direction is not always tangent to the orbit except for circular orbits (see diagram below).

The general equilibrium equation above is actually a second order differential equation that can be solved. This section will concentrate on the basic motion of a satellite orbiting the earth when the satellite mass is negligible when compared to the earth.

To set the boundary condition, consider a satellite at time t = 0, at a distance r_o from the center of the earth, with a velocity v_o parallel to the tangent of the earth's surface as shown. Then the complete general solution is (derivation omitted),

But, 'G' is not a convenient constant for engineers. The equation can be convert to 'g' using the GM = gR²,

where R_e is the radius of the earth, 6,370 km or 3,960 mi. This form can be used for other planets, but both R and g will have different constants. There is a more convenient format when the eccentricity definition of conic sections (see section below) is used,

where g_e = 9.81 m/s² or 32.2 ft/s². After some algebraic rearranging, the central force equation can be written as,

Conic Section Parameters

Four Possible Conic Sections

From analytical geometry, a conic section is a curve that has the property that the ratio of the radius r to the perpendicular distance to a straight line d (called the directrix) is constant.

The ratio r/d = r_o/d_o is called the eccentricity. From the figure one can see that

     r cosθ + d = r_o + d_o

This can be written in terms of the eccentricity as

Comparing this with the first equation above, it is seen that the satellite's free-flight trajectory describes a conic section with eccentricity ε.

The type of trajectory taken by the satellite is determined by the value of ε. There are four types of trajectories to consider.

If v_o is chosen such that ε = 0, then r must equal r_o, and the trajectory is circular.

Setting ε = 0, the velocity for a circular trajectory can be determined as

Apogee and Perigee for
Elliptical Orbits

If 0 < ε < 1, the trajectory is elliptical.

The minimum radius, or radius at perigee r_p, occurs at θ = 0°. Thus, the initial radius is the radius at perigee,

     r_o = r_p

The maximum radius, or radius at apogee r_a, occurs at θ = 180°. Setting θ = 180° gives

Notice that the maximum radius increases without limit as e approaches 1.