Equations of motion
We can write expressions for both the angular momentum and the total energy. If p θ is the magnitude of the momentum in the tangential direction, then since this perpendicular to , L = rp θ . But p θ = mv θ = m = mr = mr . Hence L = r(mr ) = mr 2 . Hence:
|L = mr 2|
We can also write an expression for the total energy as a sum of the radial kinetic energy term, the angular kinetic energy term and the potential term:
|E = 1/2m + -|
Rearranging and dividing through the left side by m 2 r 4 and the right by L 2 , and canceling factors of dt 2 we find:
|= - -|
To find the equations of motion we want to find r in terms of θ . In principle we could take the square root of both sides of the above equation, separate the variables, integrate to find θ(r) , and then invert to find r(θ) . This involves a lot of messy algebra which is not very enlightening, so we will just state the result:
|= (1 + εcosθ)|
where ε is the eccentricity that we saw in Ellipses and Foci and is now given by:
This equation determines the motion of all orbital systems in the universe.
We can also find the maximum and minimum values of r . The minimum occurs where the expression for 1/r is a maximum. This is when cosθ = 1 and the maximum is therefore . Thus:
|r min =|
The maximum occurs for the minimum of 1/r . There are two cases: first, when ε < 1 , the minimum of 1/r is . Thus:
|r max =|
When ε≥1 , the expression for 1/r can take on the value zero when cosθ = - 1/ε . Hence the maximum value r can take on is infinite in this case.
We can also take the equation and using r 2 = x 2 + y 2 , and cosθ = x/r , we can write:
|x 2 + y 2 = - xε|
Solving the orbits
The orbits are determined by the various values that ε can take.
When ε = 0 , the expression for ε tells us that E = - . The negative value of the energy just means that the potential energy is more negative than the kinetic energy is positive. In this case we have r min = r max = . The particle is trapped at the very bottom of a potential well, and the radius does not change as it goes around the orbit, hence forming a circle. Substituting this value for r into the energy we have E = - . Note that we could have derived this directly by summing the potential energy we found for a circular orbit with the kinetic energy (Gravitational Potential Energy).
|E = 1/2mv 2 + U = - = -|
Elliptical orbits occur when 0 < ε < 1 . This means that - < E < 0 . Again the particle is trapped in a potential well, oscillating now between r min and r max .
|+ = 1|
where a = and b = . This is an ellipse with its center at (- L 2 ε/GMm 2(1 - ε 2), 0) , and with semimajor and semiminor axis length a and b respectively. It can also be shown that one focus of this ellipse is at the origin.
Parabolic and Hyperbolic Orbits
Parabolic orbits occur when ε = 1 , which means that E = 0 .