Homework # 6b:  Sun-Earth-Moon

Below are some exercises that will help you to deepen your understanding of Module 6b.

Please format your solutions appropriately.  Use text format, not input format, when you are typing text.  Write coherent sentences and paragraphs;  part of your grade will depend on how clearly you present your ideas.

You must work on these exercises by yourself, not in pairs;  if you get stuck and cannot work out the solution, you may consult the labbie or the instructor during their office hours.

Differential Equations

1.  What  is a differential equation?  Explain this concept in your own words.  Give examples.

2.  Using Newton's Laws explain why in the classical Newtonian framework, the laws of motion are governed by differential equations.

Your Own Law of Gravity

Here you will explore the planar three body problem using a law of gravity of your own choosing.  You may use any of the laws of gravity we studied in Module 6a or the homework for Module 6a except the inverse square law, or you may use your own law of gravity from your homework for Module 6a.

1. Using your new law of gravity, calculate the orbits of the Earth and the Moon around the Sun using NDSolve and plot the Moon's orbit about the Earth over certain periods, such as a year or a part of a year.  You should use the masses of the Sun, Earth, and Moon given in Modules 6a and 6b (see below), but you should experiment with several initial conditions for the Earth and the Moon different from the ones given below.

G= 4.981×(10^(-10));
ms=1.989×(10^30);
me=5.977×(10^24);
mm=7.349×(10^22);

sunEarthInitConds = 
{xe[0]==1.495979×(10^8),
ye[0]==0,
xe'[0]==0,
ye'[0]==2.56721625×(10^6)};

sunMoonInitConds = 
{xm[0]==356000+xe[0],
ym[0]==0,
xm'[0]==0,
ym'[0]==95398+ye'[0]};

2. Using your new law of gravity, plot the resulting motion of the Moon around the Sun.   (As in Module 6b, you may find it useful to exaggerate the relative orbital motion of the Moon with respect to the Earth ((Subscript[p, m])-(Subscript[p, e])) by a factor of 10 in order to see the Moon's orbit better.)

3. Using your new law of gravity, explore several other planar three body problems.  In particular, what if the Earth and the Moon had the same mass, equal to half the mass of the actual total of the two?  What if the Earth and the Sun had the same mass, and the Moon had mass comparable to the actual mass of the Earth?  Explore at least one other situation in addition to the two listed above.  Remember that if you change the system, you must make sure that your model (i.e. your differential equations) fits the new situation.  For example, if you make the Earth and the Sun have the same mass, then you cannot assume that the Sun stays fixed at the center of your coordinate system.