Solving the orbits

# Solving the orbits - Solving the orbits The orbits are...

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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 = 1/2 mv 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 .

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Figure %: Potential well for a elliptical orbit. We can also solve the equation for
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## This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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Solving the orbits - Solving the orbits The orbits are...

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