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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) |
ε = |
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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