1: Use both a leapfrog and Runge Kutta solver for the following assignment. Numerically
integrate the equations for two bodies orbiting each other. You can do the Sun and the Earth and
look up initial conditions online.Essentially you need to solve F~ = GM1M2
r2 r for both objects. Do
this in two dimensions (no z coordinate) so that F~1,2 = GM1M2
[(x1x2)2+(y1y2)2]3/2 (x1x2)i+(y1y2)j If you
are clever, recall that F~1,2 = F~2,1 and dont do twice the calculations needed.Plot your solution
for 10-20 orbits and plot then total energy. Do this using the RK4 method and the leapfrog method
and comment on the dierences in the results between solvers. Google around for dierent initial
conditions but recall that the angular momentum is conserved and take your initial condition to be
one where the initial velocity i purely in the y direction.As no one in this course is required to
know programming you may just use the built in solvers which default to RK4 methods. As such,
you may just solve the equations of motion using maple or python with the RK4 solvers. Solving
with the leap frog method as well will net you a plus 40 extra credit for this homework assignment.
2: (Medium) Integrate equation 8.38 to obtain 8.41. Take the constant to be /2. You may
manipulate the integral into a form and then use a table, you may not use an integrator. Hint, let u
=1/r also look at appendix E.
3: (Longer) For a force F(r) = k/r3 set up the equation of motion 8.20. There are three
possible cases l
2 = k, l
2 > k, and l
2 < k. Each of these three cases results in a dierent solution
to the dierential equation youll derive using equation 8.20.Solve it in terms of u then convert
it to r. As this is a second order equation each solution will have two unknown constants. Pick a
value for each case, write it out explicitly and produce a plot of the solution for me.Describe what is physically happening in each case is the motion unbounded, bounded, or
inspiralling? One trick to explore the behavior is to plot the eective potential (equation 8.34,
depends on your form for r) and examine behavior based on your eective potential. Specifically,
either plot the eective potential using your solution for r in each of the three cases or plot your
solution.Hint combinations of A cos () + B sin () = C cos ( + ). Another Hint for the case of
sinusoidal motion the eective potential is the easiest way to determine the motion. Similarly linear
combinations of exponentials of the form Aex + Bex = C cosh (x + ).
4: (easy) Taking the center of the sun as the origin, calculate the center of mass of the solar
system 8 times. Once with The Sun and Mercury, then the Sun and Venus, then the Sun and Earth,
then the .. keep doing it as if each of the eight planets were the only planets in the solar system.
Look up the planet masses and semi-major axis on the internet. Finally, calculate the center of
mass of the solar system with all planets assuming they are all aligned in a perfect line on the same
side of the sun. The best way to do this problem is with an excel spreadsheet where you can write
1.5 1030 as 5E30.Graph this and label the x axis by the planets with the last entry being the combined center of
mass. You will need to plot the log of the center of mass to see dierences. Write a comment as
to whether or not this is a reasonable estimate for the average center of mass and if not why not.
NOTE You must arrange the planets in order from closest to farthest. Order them this way for the
next problem as well.
5: (Tedious only due to looking things up, combine this with 4 on the same spreadsheet.) The
spin angular momentum of a solid body about its own center is Lspin = I!rotation where you can
take the moment of inertia to be for a solid ball and the radius to be the radius of the planet. The
orbital angular momentum of a planet around the sun is Lorbit = md2!orbit where d is the planets
distance from the sun.A) Recalling the definition or orbital and rotational ! and using values from the internet, create
an excel spreadsheet which calculates the individual orbital and spin angular momentum of each
planet about its own axis and about the sun. Also, calculate the spin orbital angular momentum
of the sun.Add each planets spin and orbital momentum together and give each planets TOTAL
angular momentum. Add a tenth entry that has the total spin, orbital, and spin+orbital angular
momentum of all planets and the sun.Your 1st column should read SUN. Mercury, Venus, Earth,
Mars, Jupiter, Saturn, Uranus, Neptune, TotalIn three separate plots plot the spin angular momentum versus planet, the orbital angular momentum of each planets, and the total angular momentum of each planet. Comment on where most
of the angular momentum lies. How much of the total angular momentum is in the sun?Give me the plots as bar plots with axis titles with the appropriate units. Plot the log of the
quantity, this can be done by clicking on the y axis and selecting scale. The only plot that you
should present as a scatter plot is the last one in the question above. You will be graded on the
neatness of the plot, if you need to rest minimum values to neaten up plots then do so. Set the graph
of total angular momentum with a minimum y value of 1e36.B) The binding energy of a spherical mass is given by Ebinding = 3GM2
5r . The kinetic energy of a
spinning sphere is L2/(2I). Determine the angular momentum which would cause the kinetic energy
in spin to equal the binding energy. Is the sun in danger of spinning itself out? Is there enough
angular momentum in the solar system, that, if it was all put into the sun, would be problematic?
6: 8-6 Hint use conservation of energy and conservation of momentum.
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