Wednesday, April 28, 2010

Analysis of orbital motion

Note that the following is a classical (Newtonian) analysis of orbital mechanics, which assumes the more subtle effects of general relativity (like frame dragging and gravitational time dilation) are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury's orbit about the Sun), or when extreme precision is needed (as with calculations of the orbital elements and time signal references for GPS satellites[7]).

To analyze the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the center of force. In such coordinates the radial and transverse components of the acceleration are, respectively:

a_{r}=\ddot{r}-r\dot{\theta }^{2}

and

a_{\theta }=\frac{1}{r}\frac{d}{dt}\left( r^{2}\dot{\theta } \right)

Since the force is entirely radial, and since acceleration is proportional to force, it follows that the transverse acceleration is zero. As a result,

aθ = 0

After integrating, we have

r^{2}\dot{\theta }=const

which is actually the theoretical proof of Kepler's second law (A line joining a planet and the sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass. It then follows that

\dot{\theta }=\frac{h}{r^{2}}=hu^{2}

where we have introduced the auxiliary variable

u = { 1 \over r }.

The radial force ƒ(r) per unit mass is the radial acceleration ar defined above. Solving the above differential equation with respect to time[8](See also Binet equation) yields:

\frac{d^2u}{d\theta^2} + u = -\frac{f(1 / u)}{h^2u^2}.

In the case of gravity, Newton's law of universal gravitation states that the force is proportional to the inverse square of the distance:

f(1/u) = a_r = { -GM \over r^2 } = -GM u^2

where G is the constant of universal gravitation, m is the mass of the orbiting body (planet), and M is the mass of the central body (the Sun). Substituting into the prior equation, we have

\frac{d^2u}{d\theta^2} + u = \frac{ GM }{h^2}.

So for the gravitational force — or, more generally, for any inverse square force law — the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:

u(\theta) = \frac{ GM }{h^2} + A \cos(\theta-\theta_0)

where A and θ0 are arbitrary constants.

The equation of the orbit described by the particle is thus:

r = \frac{1}{u} = \frac{ h^2 / GM }{1 + e \cos (\theta - \theta_0)},

where e is:

e \equiv \frac{h^2A}{G M}.

In general, this can be recognized as the equation of a conic section in polar coordinates (r, θ). We can make a further connection with the classic description of conic section with:

\frac{h^2}{GM} = a(1-e^2)

If parameter e is smaller than one, e is the eccentricity and a the semi-major axis of an ellipse.

Orbital planes

Main article: Orbital plane (astronomy)

The analysis so far has been two dimensional; it turns out that an unperturbed orbit is two dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two dimensional plane into the required angle relative to the poles of the planetary body involved.

The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.

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