By Joshua N. Winn, Princeton University
Kepler’s laws are the three patterns pointed out by Johannes Kepler, a 17th century astronomer, in the motion of the planets. While Kepler’s second law is about how fast the planets move; when they’re close to the Sun, they move faster, in a specific way, his third law is about total time required to go all the way around: the orbital period.
Area of the Movement of Planets
As the planet moves, the line joining the planet and the star, the planet’s radius vector, sweeps out area at a steady rate. If we would color the area swept out by the radius vector, and switch the color every second, we would notice that when the planet is near the star, the radius is shorter, but the planet moves faster, so the colored sector has a large angular width, and when the planet is far away, the radius is longer but the planet is slower, making a tall skinny sector.
What Kepler found is that all the sectors have the same area: equal areas in equal times. However, this can mathematically be put it another way.
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Kepler’s Second Law
Let’s imagine and put the planet at an arbitrary position and watch it move for an infinitesimal time interval dt. It sweeps out a thin sector spanning an angle d-theta, with an area dA. And what’s the area of that sector? Well, the planet moves in both the radial direction, the r-direction, and the perpendicular direction, which we’ll call the theta-direction—the direction of increasing theta. It’s the motion in the theta-direction that sweeps out area. Purely radial motion doesn’t sweep any area.
In time dt, the planet moves in the theta-direction by an amount r times d-theta, using the usual small angle approximation. So, the swept-out sector is basically a skinny right triangle with sides of r and rd-theta. The area of that triangle is 1/2 r times rd-theta. That does leave out a tiny little corner piece of the sector, its dimensions of dr and rd-theta, that’s the product of 2 tiny numbers. So, in the limit of infinitesimal dt, that little piece is vanishingly small compared to the rest of the triangle. All of which means we can write dA equals 1/2 r times rd-theta. And dA/dt, the rate at which area is swept out, is 1/2 times r-squared d-theta/dt.
If that rate is a constant, as Kepler observed, then dtheta/dt must be proportional to one over r-squared. That is the modern way to state Kepler’s second law. Theta advances at a rate that varies as one over r-squared.
Kepler’s Third Law
Then there’s Kepler’s third law. This one is about total time required to go all the way around, the orbital period – the bigger the orbit, the longer the period.
To be quantitative, let’s think of a logarithmic chart, where the horizontal axis shows the semimajor axis, in AU, and the vertical axis shows the period, in Earth years. So, the point representing the Earth is at 1 AU and one year, and the other points are for the other planets. Strikingly, they all fall on a single straight line! It has a slope of 3/2; if we move 2 units to the right, the line goes up 3 units. Since this is a log plot, it means the log of P is 3/2 times the log of a, plus some constant. Which in turn means that P in proportion to a to the3/2 power. That’s Kepler’s third law.
Kepler’s third law is the most reliable way we have to measure the mass of just about anything in astrophysics. The basic idea is that to measure an object’s mass, we need to watch other things moving in response to its gravity. It works for stars, planets, black holes, neutron stars, entire galaxies, and even, in a sense, for measuring the mass of the entire universe.
Kepler’s Third Law in a Scaling Relation
Let’s write Kepler’s third law in a scaling relation. We know that when M equals one solar mass, and a equals 1 AU, the period will come out to be a year; that’s the situation for the Earth. And we also know that P is proportional to M to the minus-1/2 power, and a to the 3/2 power. We can write Kepler’s third law as P equals one year times M over M-dot to the minus 1/2 times a over 1 AU to the 3/2 power.
What we’ve done is to evaluate the equation for a benchmark case, the Earth, and then specified how the answer scales with the inputs, M and a. The scaling relation is handy because as long as we use units of years, AU, and solar masses, we won’t have to look up the numerical values of G and the mass of the Sun , or worry about the 2pi. We can just plug the numbers directly into the scaling relation.
We can make a scaling relation of out just about any equation because it helps give us a sense of the numbers. It keeps us from getting lost in the orders of magnitude. Let’s illustrate with Jupiter. We know that Europa has a P of 0.01 years, and the value of a is about 0.0045 AU. If we plug those numbers into our scaling relation, and solve for M over M-dot, we get 0.0009; we just learned that Jupiter is a bit less than a thousandth the mass of the Sun.
Common Questions about Kepler’s Second and Third Laws
Kepler’s second law is about how fast the planets move; when they’re close to the Sun, they move faster, in a specific way.
Kepler’s third law is about total time required to go all the way around, the orbital period – the bigger the orbit, the longer the period.
Kepler’s third law is the most reliable way we have to measure the mass of just about anything in astrophysics. The basic idea is that to measure an object’s mass, we need to watch other things moving in response to its gravity. It works for stars, planets, black holes, neutron stars, entire galaxies, and even, in a sense, for measuring the mass of the entire universe.
