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Properties of Circular Orbits
Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth.

The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M.

For all parts of this problem, where appropriate, use G for the universal gravitational constant.

Part A  
Find the orbital speed v for a satellite in a circular orbit of radius R.
Part A.1 Find the force

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Part A.2 A basic kinematic relation

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Hint A.3 Newton's 2nd law

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Express the orbital speed in terms of G, M, and R.
  v =  sqrt(G*M/R) 
Part B  
Find the kinetic energy K of a satellite with mass m in a circular orbit with radius R.
Express your answer in terms of m, M, G, and R.
  K =  G*M*m/(2*R) 
Part C  
Express the kinetic energy K in terms of the potential energy U.
Part C.1 Potential energy
What is the potential energy U of the satellite in this orbit?
Express your answer in terms of m, M, G, and R.
  U =  -G*M*m/R 
  K =  -1/2*U 
This is an example of a powerful theorem, known as the Virial Theorem. For any system whose motion is periodic or remains forever bounded, and whose potential behaves as

U \propto R^n,

Rudolf Clausius proved that


where the brackets denote the temporal (time) average.
Part D  
Find the orbital period T.
Hint D.1  

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Express your answer in terms of G, M, R, and pi.
  T =  2*pi*R^(3/2)*(G*M)^(-1/2) 
Part E  
Find an expression for the square of the orbital period.
Express your answer in terms of G, M, R, and pi.
  T^2 =  (2*pi)^2/(G*M)* R^3 
This shows that the square of the period is proportional to the cube of the semi-major axis. This is Kepler's Third Law, in the case of a circular orbit where the semi-major axis is equal to the radius, R.
Part F  
Find L, the magnitude of the angular momentum of the satellite with respect to the center of the planet.
Hint F.1 Definition of angular momentum
Recall that \vec{L} = \vec{R} \times \vec{p}, where p_vec is the momentum of the object and R_vec is the vector from the pivot point. Here the pivot point is the center of the planet, and since the object is moving in a circular orbit, p_vec is perpendicular to R_vec.
Express your answer in terms of m, M, G, and R.
  L =  m*(G*M*R)^0.5 
Part G  
The quantities v, K, U, and L all represent physical quantities characterizing the orbit that depend on radius R. Indicate the exponent (power) of the radial dependence of the absolute value of each.
Hint G.1 Example of a power law
The potential energy behaves as U=GMm/R, so U depends inversely on R. Therefore, the appropriate power for this is {-1} (i.e., U \propto R^{-1}).
Express your answer as a comma-separated list of exponents corresponding to v, K, U, and L, in that order. For example, -1,-1/2,-0.5,-3/2 would mean v \propto R^{-1}, K \propto R^{-1/2}, and so forth.
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