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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|
|E = 1/2m + -|
|= - -|
|= (1 + εcosθ)|
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 =|
|r max =|
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|
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|
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