The Sun-Earth-Moon System

Comp110 Course Module 6b

Three body problems

In Module 6a we studied two body problems (Sun-Earth);  here we are going to study three body problems (Sun-Earth-Moon).  While two body problems can always be solved analytically (the orbits are always conic section), three body problems typically can only be solved numerically.  Since we have already solved two body problems numerically, we can apply a similar approach to solve three body problems.  As you will see, the resulting trajectories can become extremely complicated even using the normal (1/(r^2)) law.

Setting up the differential equations

As with the case for the Earth and the Sun, we must settle on a coordinate system for the Earth, Sun and Moon in order to model their motion. The simplest approach is to extend the model for the Earth and the Sun by adding a third quantity pm[t] that represents the position of the moon over time.

pe[t_]={xe[t],ye[t]};
ps[t_]={xs[t],ys[t]};
pm[t_]={xm[t],ym[t]};

Adding the moon to our diagram yields three bodies.  The vectors  ps[t]-pe[t] and pm[t]-pe[t] point from the Earth to the Sun and from the Earth to the Moon.

The gravitational force on the Earth is the sum of the gravitational force due to the Sun plus the gravitational force due to the Moon.  We already have a formula from Module 6a for the gravitational force on the Earth due to the Sun in terms of ps[t] and pe[t].  We can build an analogous formula for the effect of the Moon on the Earth and by addition arrive at the total gravitational force fe[t] on the Earth:

fe[t]==((G ms me (ps[t]-pe[t]))/((leng[ps[t]-pe[t]])^3))+((G mm me (pm[t]-pe[t]))/((leng[pm[t]-pe[t]])^3))

Remember the function leng computes the length of a vector.

leng[{x_,y_}]:=(Sqrt[(x^2)+(y^2)])

Recalling Newton's third law that force is equal to mass time acceleration, we have

fe[t]=me pe''[t].

Substituting the definition of fe[t] into this equation and canceling out the factor of me on both sides, we arrive at the fundamental differential equation that describes the motion of the Earth based on the positions of the Earth, the Sun and the Moon.

pe''[t]== ((G ms(ps[t]-pe[t]))/((leng[ps[t]-pe[t]])^3))+((G mm (pm[t]-pe[t]))/((leng[pm[t]-pe[t]])^3)).

Note that the acceleration of the Earth does not depend on its mass!  (Remember Galileo and the leaning tower of Pisa.)  A similar approach can be used to derive analogous equations for the acceleration of the Moon and the Sun.

Solving the equations

To solve the three body problem, we will need the masses for the three bodies and gravitational constant G,

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

In the case where one body is very massive in comparison to all the others, we can consider the massive body to be fixed at the center of mass of the n-body system, and take this location to be the origin of our coordinate system.  We will use this approach for our initial analysis of the Earth-Sun-Moon system by fixing the Sun at the origin of the coordinate system.

ps[t_]={0,0};

As we shall see, the force of the moon on the Earth is negligible, so the motion of the Earth is still an ellipse with the Sun as a focus.  The semi-major axis of this ellipse is 1.495979× (10^(8 ))km.  As you will discover, the motion of the Moon around the Sun is more complicated.  The distance between the Earth and the Moon varies between 356,000 kilometers and 406,700 kilometers, with a mean distance of 384,400  kilometers.

You may have thought that since the Moon is so close to the Earth, the gravitational force of the Sun on the Moon would be negligible in comparison to the force of the Earth on the Moon.  You will discover, however, that the Sun's force on the Moon is a little more than twice the force of the Earth on the Moon.  Thus we have a legitimate three body problem, and we can expect the motion of the Moon to be less uniform than it looks to the casual, Earthbound observer.

Exercises

Your assignment is to develop computational routines to compute and plot the orbits of the Moon and Earth.  You can assume that the orbits all lie in the same plane.  This assumption is not in fact true, but it is a reasonable assumption for these exercises. (Actually the orbit of the Moon around the Earth is inclined at an angle of about 5° to the orbit of the Earth around the Sun.)  Copy the problems below and enter your solutions in a separate notebook.  Your results should enable you to address the following problems:

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]};

pe[t]+10(pm[t]-pe[t]).