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Gravitation: Orbits


Terms (orbits)

Orbits, page 2

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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:

ε =    

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 = -    

Solving the orbits

The orbits are determined by the various values that ε can take.

Circular orbits

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 = - = -    

Figure %: Potential well for a circular orbit.
In the case ε = 0 we can see that this equation simplifies to x 2 + y 2 = . This describes a circle with radius .

Elliptical Orbits

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 .

Figure %: Potential well for a elliptical orbit.
We can also solve the equation for ε in this range. The algebra turns out to be complicated, but we find that:

+ = 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.