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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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