By Joshua N. Winn, Princeton University
The 17th century astronomer Johannes Kepler pointed out three patterns in the motion of the planets. They’re called Kepler’s laws. Kepler’s first law is that the planets trace out ellipses as they go around the Sun. The orbits look like circles, but they’re not. They’re slightly flattened into ovals, and even the Sun is off-center.
Circles and Ellipses
Mathematically, a circle is defined as the set of all points that are the same distance from some chosen center. Put a pin in a corkboard, then take a piece of string and tie both ends to the pin, making a loop. Then, put a pen in the loop, stretch the string tight, and sweep around the board, drawing as you go. You’ve just made a circle, centered on the pin.
For an ellipse, you need 2 pins, and you tie the ends of the string to the pins, with some slack. And stretch the string tight with the pen, and sweep around the board, drawing as you go. What you get in that case is an ellipse. The 2 pins are the focus points, or foci, of the ellipse. For all the points on the ellipse, the distance to the first focus, plus the distance to the second focus, is a constant, that’s the length of the string, in our construction.
The length of that string is also equal to the length of the long axis of the ellipse, the major axis. If we unpinned the string and straightened it out, it would reach exactly across the major axis. We conclude that for any point on an ellipse, the distance to one focus plus the distance to the other focus equals the length of the major axis.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Radii and Areas
To make a circle, we only needed to make one choice: the radius, which we’ll call a. For an ellipse, the equivalent is the radius along the major axis, which we’ll also label a, and we’ll call that the semi-major axis. But with an ellipse, we also have to choose the distance between the center and either focus.
We can choose whatever distance we want, as long as it’s smaller than a. Tradition dictates that we express that distance as a times e, where e is a number smaller than one, which is called the eccentricity. When e is zero, the foci coincide at the center, and we go back to a circle of radius a; as e gets larger, and closer to one, the foci separate and we get a more elongated ellipse.
For a circle, we know area equals pi a-squared. For an ellipse, it turns out to be pi a-squared times the square root of one minus-e-squared.
A Coordinate System
To understand the mathematical equation for an ellipse, in polar coordinates, let’s start by introducing a coordinate system. We’ll put the origin at one of the foci, and we’ll lay down x and y axes along the major and minor axes. To specify the points on the ellipse, we would use polar coordinates: r is the distance from the origin, and theta is the angle measured counter-clockwise from the x axis.
First, let’s think about what we expect. It’s going to be a rising and falling function, for theta equals zero, r has its minimum value of a minus ae, that’s a times one minus-e. As we dial theta up to higher values, r increases, and at theta equals pi, 180°, r achieves its maximum value of a times one plus-e. Then it shrinks back down as theta wraps around to 2pi.
To find the equation, we start with the fact that at any point on the ellipsis, the sum of distances to the foci is equal to the major-axis length, 2a. We can write that as r plus r-prime equals 2a, where r is the distance to the focus at the origin, and r-prime is the distance to the other focus. But that’s not such a convenient equation. We want it purely in terms of r and theta, not r-prime, so how do we get rid of the r-prime? We use the Law of Cosines.
The Law of Cosines
The Pythagorean theorem says c-squared is equal to a-squared plus b-squared, where a, b, and c are the lengths of the sides of a right triangle. The Law of Cosines is the generalization to any triangle. It says that for any triangle, c-squared equals a-squared plus b-squared minus 2ab times the cosine of gamma, where gamma is the angle across from the c-side.
We’ll apply it to our triangle, with r-prime as our c-side. So, we have r-prime squared equals r-squared plus 2ae -squared minus 2r times 2ae times the cosine of angle opposite r-prime, which is pi minus theta. And the cosine of pi minus theta is minus the cosine of theta. At the end, we find r equals a times one minus e-squared divided by one plus e cos theta. That’s our equation!
Semi Major Axis and Eccentricity
So, does it make sense? When the eccentricity is zero, the equation reduces to r equals a times one over one, that’s just r equals a; that’s a circle. And when e is not zero, and we dial theta around the clock, cos theta goes from one to minus one and back to one, the denominator starts big and gets small, then big again. That, too, makes sense; it says r oscillates between a minimum value at theta equals zero and a maximum at pi.
In particular, if we plug in theta equals 0, we get a times one minus e-squared over one plus e. And since one minus e-squared is one plus e times one minus e, the one plus e’s cancel out, and we’re left with a times one minus-e, which is what we expected. That’s the minimum distance from the focus. Likewise, when theta equals pi, we get the expected maximum distance of a times one plus-e.
So, what Kepler noticed, his ‘first law’, is that all the planets move on ellipses, with the Sun not at the center but rather at one of the foci. Each planet has its own value for the semi major axis and eccentricity.
Common Questions about Kepler’s First Law
Kepler’s first law is that the planets trace out ellipses as they go around the Sun.
Mathematically, a circle is defined as the set of all points that are the same distance from some chosen center.
The Pythagorean theorem says c-squared is equal to a-squared plus b-squared, where a, b, and c are the lengths of the sides of a right triangle.