By Joshua Winn, Princeton University
Imagine, the year is 1660. The law of gravity is unknown. We’ve just read Johannes Kepler’s books and puzzled over those three patterns he observed in the motion of the planets. Can we use those patterns to figure out the law of gravity? This is a tough problem; it took Newton years to solve it. But we have an advantage: we know calculus. Newton had to invent calculus, which naturally, took some time.
Three Laws, Three Equations
First, let’s write Kepler’s laws in equation form. The first law says the orbits are ellipses with the Sun at one focus, so if we use a polar coordinate system with the Sun at the origin, the path of the planet, r of theta is equal to a times one minus e-squared over one plus e cos-theta. That’s the equation for an ellipse.
Kepler’s second law says the line from the Sun to the planet sweeps out area at a steady rate. This implies 1/2 r-squared d-theta/dt is a constant, a certain area per unit time, that is specific to each planet. For the Earth, the numerical value is pi AU-squared per year, since the Earth’s orbit is approximately a circle of radius one, which has a total area of pi.
More generally, 1/2 r-squared d-theta/dt is equal to the area of the ellipse, pi a-squared times the square root of one minus e-squared divided by the orbital period, P. That’s Kepler’s second law. And, Kepler’s third law says that P in proportional to a to the 3/2 power. So, that’s our trio of equations. Now, let’s get to work.
Rewriting the Equations
Pretend we already know from laboratory experiments that force equals mass times acceleration. But, we don’t yet know the equation for the force of gravity. To obtain a clue, we need to calculate the acceleration of a planet that obeys Kepler’s laws. To calculate acceleration, first, we need to know the planet’s position as a function of time. Then we’ll take the time derivative to get the velocity.
Well, Kepler’s first law tells us the position, but not as a function of time; it’s a function of angle, theta. All the time information is the second and the third laws. So, we need to combine the equations, somehow.
Let’s convert to Cartesian coordinates. In general, when the polar coordinates are r and theta, the x coordinate is r times cosine of theta, and y equals r sine-theta. So, for our planet, x is a time one minus e-squared times cos-theta over one plus e cos-theta. And we get a similar equation for y. We can do the same thing with unit vectors.
We Still Need to Simplify
Now, let’s calculate the velocity by taking the time derivatives of x and y. Since they’re written as functions of theta, and not time, we need to use the chain rule: vx, the x-component of velocity, is dx/dt, which we can write as dx/d–theta times d-theta/dt. Since x has functions of theta in the top and bottom of the expression, we use the quotient rule.
What about d-theta/dt? For that we need Kepler’s second and third laws, the ones relating to time. Let’s consolidate them, by writing the P in the second law in terms of a, using the third law. The third law says P equals some constant times ato the 3/2 power. We can label that constant however we want. But instead, let’s be clever.
The second law has a 1/2 on the left side, and a pi and a square root on the right side. So, to make the result as simple as possible, let’s write the third law as P equals 2pi over root-K times ato the 3/2, where K is a constant. That way, the 1/2 and the pi cancel out, and the K will fit nicely under the square root, so what we’re left with is r-squared d-theta/dt is equal to the square root of Ka times one minus e-squared.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
We plug in the expressions we just derived, which leads to an equation in terms of r and theta. To put everything in terms of just one variable, theta, we insert the ellipse equation for r-theta, and simplify.
That gives v_x is equal to minus the square root of K over a times one minus e-squared times sine-theta. That factor in front of sine-theta is a constant. It doesn’t depend on r or theta or time and it has units of velocity. To make the equation look even simpler, let’s name that constant v_naught. That way, v_x is simply minus v_naught times sine-theta.
That leaves the other component of velocity, v_ y which we calculate as dy/d-theta times d-theta/dt. Let’s just jump to the answer: v_ y is equal to v_naught cos-theta plus v_naught times e.
An Ironic Coincidence
What does all this mean? Let’s find out, by tracking the planet’s velocity vector over a full orbit. We’ll plot v_x on the horizontal axis, and v_ y on the vertical axis. That kind of chart is called velocity space; each point in the chart specifies a velocity, rather than a position. As theta increases, the equations tell us that v_x starts at zero and v_ y starts at v_naught plus e.
Then as theta increases, v_x goes negative and v_ y shrinks. When we keep going, what we find is amazing. The tip of the velocity vector moves in a circle! You can prove it algebraically, too, by showing that our equations imply v_x-squared plus v_ y minus–e v_ naught quantity squared equals v_naught-squared. That’s the equation for a circle in velocity space, with radius v_naught, centered at the point zero e times v_naught.
Now, this is ironic. Ancient astronomers were sure the planets moved in circles, because the circle was just such a perfect shape. And when the data got good enough to rule out uniform circular motion, they added more circles, to make epicycles. And it took astronomers a long time to ditch the circles and arrive at the truth that the planets move on ellipses.
The ancient astronomers were right, after all. Planetary motion does involve perfect circles; it’s just that the circles are in velocity space. While the planet moves in an ellipse, its velocity vector traces out a circle. Well, that was an unexpected treat.
Common Questions about Kepler’s Laws of Planetary Motion and Newton’s Laws
The first of Kepler’s laws can be written as such: r of theta is equal to a time one minus e-squared over one plus e cos-theta.
After combining the equation forms of Kepler’s laws we’re left with r-squared d-theta/dt is equal to the square root of Ka times one minus e-squared.
Ancient astronomers thought that planets moved in perfect circles. But after some time they gave up on that idea. Now, using Kepler’s laws we have proven that they indeed do move in circles.