Understanding Kepler's Three Laws of Planetary Motion

Several years ago, Sky and Telescope published my work on "Pittendreigh's Law of Planetary Motion."  It was a joke, but there is truth to it.  Pittendreigh's Law states that the earth is moving faster and faster around the sun every year you age - and those of us who are getting older can certainly attest to this fact!  The older you get, the faster the years go by. We can also attest to Pittendreigh's Law of Gravity - because the earth is moving faster around the sun with every year we age, gravity is getting stronger.  Take my word for it - that 10 inch reflector telescope used to be so easy to move into the backyard.  But with gravity getting stronger every year, I struggle with it more and more.  By the time I'm 80, that thing will weigh a ton!

On a serious side, Kepler produced three laws of planetary motion.  I'm amazed that he was able to develop these at the dawn of modern astronomy.  He came up with two in 1609 and the other in 1619.  Keep in mind Galileo applied the telescope to astronomy for the first time in 1609. 

Kepler's laws are:

1.  The orbit of every planet is an ellipse with the Sun at one of the two foci.
2.  A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3.  The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

So, what does all this mean?

Let's take Kepler's First Law.  In Kepler's mind, God created a perfect and orderly universe.  That meant that the planets orbiting the Sun must do so in a perfect circle.  Observations and calculations of Mars and other planets did not fit that theory of orbits being perfect circles.  God, it turns out, may have created a perfect and orderly universe, but as the Creator is more complex than Kepler expected, God's perfect universe was more complex.  And more interesting.

The planets do not orbit in a perfect circle, but rather in an ellipse.

Now a circle is an ellipse, but it is one specific type of an ellipse.  An ellipse can be thought of as "sort of a round thing."  Or more accurately, an ellipse is a circle or oval.  Or to be exact, an ellipse is a  plane curve, especially a conic section whose plane is not parallel to the axis, base, or generatrix of the intersected cone, or the locus of points for which the sum of the distances from each point to two fixed points is equal.

If the plane is parallel to the base, the ellipse is a circle.

If the plane cuts into the cone at an angle, the ellipse is not a circle but an oval.

In other words, an ellipse is a sort of round thing.

But more than that, an ellipse has two focal points, or two focuses -- or since "focuses" is not the grammatically correct plural of a focal point, we use the term "foci."

What are these foci?  Where do they come from? 

Keep in mind, Kepler's First Law says, "The orbit of every planet is an ellipse with the Sun at one of the two foci."  So if the Sun is at one foci, what is at the other foci and how do you know which of the two foci is the Sun?

The answer is simple, but first you have to understand a little more about the nature of an ellipse, which is about to get a little more complicated than just "a sort of round thing."  After all, the borders of the states of Colorado or Texas or "sort of round things" and no decent planetary orbit would ever look like Colorado or Texas (no disrespect to the residents of those states)

For a moment, let's start over in thinking about how to define an ellipse. Let's start with the two foci. 



Now let's connect the two foci with a string.

 Now let's tighten the string so that it does not hang lose:

 Now let's move the string around, keeping it tight, so that it moves from one place to another. 

Let's say I keep track of the extreme point of the string as I move it from place to place.  And then let's say I connect those dots where the string had been extended as I moved it around.

Hey!  I have just formed an ellipse.

Not only that, but I have two focal points in that ellipse.  One focal point has one end of the string, and the other focal point has the other end. 

The Sun goes on one of those two foci.

What goes on the other?

Nothing.  It is empty space.

How do you know which of the two foci is the Sun's position?  Doesn't matter.  Either one will do.

Now back to Kepler's First Law of Planetary Motion:  "The orbit of every planet is an ellipse with the Sun at one of the two foci."


 
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Now for the Kepler's Second Law of Planetary Motion: "A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."

Let's look at the image below.  You have the Sun, represented in yellow.  A planet, represented as green, and the Sun is at one foci of the ellipse and the planet is orbiting along that elipse.  The planet is at a specific point in time, which we could call January 1st, but for our purposes, we will call "Point A."

Let's say a week goes by.  Now the planet is at "Point B."

Now let's say an EQUAL amount of time goes by - one more week.,

and now the planet is at "Point C."

Kepler's Second Law says that the area of the triangle formed by A, B and the Sun,

 is equal to the area of the triangle formed by B, C and the Sun. 

Let's give our imaginary planet a year of 8 months.  Take any of the two shaded areas, each representing a single month, would have equal areas.

One of the implications for this is that the planet moves slower when it is far from the Sun, but faster when the orbit takes that planet closer to the sun.

Now let's go to Kepler's Third Law of Planetary Motion:  "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

Huh?

What's a "semi-major axis" of a planet's orbit?

If we are going to understand what a semi-major axis is, it sounds logical to first understand what a major axis is.

The major axis of an ellipse is its longest diameter, a line that runs through the center and both foci of the ellipse.  In other words, the two ends of the line would each be at the widest point of the shape of the ellipse.

The semi-major axis is one half of the major axis, and thus runs from the center of the ellipse, through one of the two foci of the ellipse and to the edge of the ellipse.

Essentially, the semi-major axis is the measure of the radius of an orbit taken at the orbit's two most distant points.

Now that we cleared that up, what was Kepler's Third Law again?  "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

Huh?

Maybe it would help if we expressed it as a mathematical formula:

Huh?

OK, each of these letters means something, so let's take a moment to interpret them.

The letter "T" represents the orbital period.  It can be a year, days, seconds, whatever.  But I have to have a way of telling the "T" or the orbital periods of the two planets apart, so I'll give each a little subscription.

The little subscriptions on both sides of the equations designate Planet A or Planet B, so that on the first side of the equation, you have the orbital period of Planet A divided by the orbital period of Planet B.

On the other side of the equation you have this lower case "r."  That equals the average orbital radius of the planets, or that "semi-major axis."

There is a little more to this equation.  If you look at the upper right corner of each equation, you will see that one side of the equation will be squared and the other side will be cubed.

So let's give it a try.

Since we are currently waiting for the Transit of Venus, let's play with Venus and see how far away it is from the Sun.  Since Kepler's Third Law of Planetary Motion compares two orbital bodies around a central object, we don't really need to know anything about that central body, the Sun, but we do need to know something of another planet with which we can compare Venus.  Let's take Earth.

We know that Earth has an orbital period of 365.245 days and that it's semi-major axis is 149,598,261 km.  How do I know these things? I looked it up online.  That's OK, because in order to use this formula, you have to have some known elements of the equation. 

As for Venus, we know its orbital period is 224.7 days.

So let's plug these figures into our equation.

Our next step is to have both sides speak the same language.  One side speaks the language of "days" and the other side speaks the language of "kilometers." We want the answer in terms of distance, so the "days" on the top and bottom of the left side of the equation cancel out, so let's take them out of our equation and just leave the numbers.

Next step, divide 224.7 by 365.245.

Next, let's square the result of that division problem.  To "square" a number is to multiply it by itself, so .6152 times .6152 gives us .3785.

Now we can make this equation look a lot simpler if we take the cube root of both sides.  The cube root of .3785 is .7233.  By taking the cube root of the other side, we just remove the parenthesis and that cube number in the upper right corner. 

Now the equation is looking much simpler.  Let's get the unknown factor, the radius of Venus' orbit, or the semi-maximum axis, all by itself.  To do that, let's multiply both sides by 149,598,261 km.

So, there is our answer 108 million kilometers.