# 19.4 Planetary Motion

The gravitational interaction between a planet and the sun is described by the inverse square central force law.

For convenience we place the sun at the origin of our coordinates, and start our planet at the point \((1,0,0)\), with the initial first derivative of its position given by \((a,b,0)\).

We will assume that the planet is much lighter than the sun, (as the earth is compared to our sun) so that the sun does not move. (Actually, what is fixed in planetary motion is the center of mass of the system. Jupiter and Saturn are sufficiently large that when they are in the same part of our sky the center of mass of all the planets does not lie inside the sun, so that the sun moves around, but not very much.)

With these coordinates and this assumption the equations of motion for \(\vec{r}\), the position vector of the planet obeys the equation

\[\frac{d\vec{r}}{dt} = -c\frac{\vec{r}}{r^3}\]

Since the force on the planet points toward the sun, and we are starting the planet in the \((x,y)\) plane, our \(z\) coordinate will always be \(0\), and we can ignore it.

This is a second order differential equation with two dependent variables, \(x(t)\) and \(y(t)\). We can set this up on a spreadsheet devoting a column each to \(t, x, y\) and the derivatives of \(x\) and \(y\). In terms of coordinates, the equations of motion are

\[\frac{d^2x}{dt^2} = -c\frac{x}{r^3} ~ \text{and} ~ \frac{d^2y}{dt^2} = -c\frac{y}{r^3}\]

Since \(r\) occurs in both of these equations and \(r\) is \((x^2 +y^2)^\frac{1}{2}\), it is convenient to devote a column to \(r\) as well. Setting \(r=1\) defines a scale for \(r\), but not for \(t\). This means we can choose our unit of time so that \(c\) is \(1\).

With that choice, we can set up our spreadsheet as follows:

We put the time variable \(t\) in column A, and start at row 7 with A7 set to 0. We must choose an increment for \(d\), and you can determine the one you like best. It must be small enough so that \(\frac{d\vec{r}}{dt}\) is small, but large enough that you can plot orbits. You might start with \(d=10^{-2}\) and change that if it does not work well. We can put the letter d in A2 and its value in B2. The other parameters we will need to specify are the initial values of the derivatives of \(x\) and \(y\). So enter "initial x speed" in A3 and its value (say 0) in B3, with "initial y speed" in A4 and its value (say 1) in B4.

Put t in A6, x in B6, y in C6, r in D6, x' in E6, and y' in F6. We put \(x\) and \(y\) in columns B and C and so put 1 in B7 and 0 in C7. We put \(r\) in column D, setting D7 to =(B7^2+C7^2)^0.5. We put \(\frac{dx}{dt}\) (call it \(x'\)) in column E, setting E7 to =B3 and put \(\frac{dy}{dt}\) (call it \(y'\)) in column F setting F7 to =B4.

We next set A8 to =A7+$B$2

B8 to =B7+$B$2*E7

C8 to =C7+B$2*F7 (you can copy B8 into C8)

Copy D7 into D8

Set E8 to =E7-$B$2*B7/$D7^3

Copy E8 to F8

Now copy A8 through F8 down the columns

This will give the crudest approximation to the solution for your values of the parameters.

**When you are done, an \(x,y\) plot of columns B and C will give the orbits in space. Adjust your
parameters as needed.**

**Exercise 19.5: Set this up. What values of \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) give circular motion
in these coordinates?**

In the past, dealing with equations like these numerically was excruciatingly horrible. Instead, physicists from Newton on solved the equations by introducing quantities, namely energy and angular momentum, which do not change with this motion, and deduced orbits by reasoning rather than numerical computation.

The actual behavior of planets was carefully observed by astronomers over centuries and was crisply summarized in Kepler's three laws, which are as follows:

**1. The motion of planets and other bodies subject to the same force is in orbits that are "conic
sections": ellipses or hyperbolae or in very special circumstances parabolas (all with the sun as a focus), or
straight lines.**

**2. The area swept out per unit time in any orbit is constant.**

3. There is a certain specific relation between the period of an elliptical orbit and a measure of its radius, which relation we will not discuss further.

**Final Note:** The last few chapters contain lots of material that is not contained in any normal
single variable calculus course. The purpose of this material is for your enjoyment and not to intimidate you.
The problem is that the applets and the approaches here allow you to learn calculus much faster than you can be
expected to do with a regular calculus course. But what you learn and retain is heavily dependent on how much
time you spend doing it. If the end result was that you spent much less time learning calculus, that would bad
for you. So you might as well spend the same amount of time, and just learn more!