Sattelites beyond Jupiter - problem setup

[SolStis]

1. 1. Analytical With the help of `Maths27', formulate the problem of when to launch the
satellite, based upon the radii of the orbits of Earth and Jupiter, assuming that both
planets have circular orbits and ignoring all potentials except that of the sun.
Provide the equations of motion for the satellite under the assumptions that; the mass
of the satellite is negligible in comparison to planetary masses, the Earth's gravitational
field may be neglected and that all motion is coplanar.

Should you launch the satellite in phase with the Earth's orbit or opposed to it?

2. Numerical Using a Runge-Kutta algorithm (see `Maths20'), write a program to solve
the two body problem for the satellite's motion from Earth to Jupiter. Assess your calculations
using the exact analysis provided in `Maths27'. Now extend your calculation
to incorporate Jupiter's gravitational potential, paying special attention to real conservation
laws for the chosen limit and approximate conservation laws which would be true
in the absence of Jupiter. You will need to have an algorithm to provide an automatic
step-length control: The use of two Runge-Kutta algorithms to assess the error is a very
efficient method. Pay particular attention to picturing the answer.

3. Investigation Devise a strategy for giving the satellite the biggest kick from Jupiter's
gravitational potential. Does the satellite approach `dangerously close' to Jupiter? Can
the satellite escape from the Solar system? Provide a simple argument to predict the maximum
impulse from Jupiter, and the corresponding minimum planetary orbit to achieve
escape from the solar system.

4. Data
Gravitational Constant 0:667 1010m3kg1s2
Mass of the Sun = 0:1984 1031kg
Mass of the Earth = 0:5976 1025kg
Mass of Jupiter = 0:1903 1028kg
Radius of the Earth's Orbit = 0:1495 1012m
Radius of Jupiter's Orbit = 0:7778 1012m
Radius of the Earth = 0:6368 107m
Radius of Jupiter = 0:6985 108m



I have C++ code for a fourth order runge kutta but am unable to set up the differential equations to solve this problem. Should I set up a space marix containing potential values and then update it as time procedes (planets move) and then map the motion of the sattelite in accordance with these values? Help with this would be greatly appreciated.

Any links to resources onlnie resources would be very helpful as fruits from various google searches yielded little in the way of this problem.



Thanks

Sol

 

[BigJDubs]

ving this problem will require you to set up the differential equations for the two-body problem, which are given in Maths27. These equations describe how the motion of the satellite is affected by the gravitational potentials of the Earth and Jupiter. You can use a fourth-order Runge-Kutta algorithm to solve these equations numerically, but you will need to set up the differential equations first. Once you have set up the differential equations, you can update the potential matrix as time proceeds and then use it to map the motion of the satellite. To get more information on the two-body problem and its solution with Runge-Kutta, you can check out the following resources: 1. https://www2.physics.ox.ac.uk/sites/default/files/2012_Two_Body_Problem.pdf 2. 3. http://mathinsight.org/runge_kutta_two_body_problem