By Joshua N. Winn, Princeton University
Kepler’s first law: the orbits are ellipses, with the Sun at one focus. Kepler’s second law: d-theta/dt in proportion to one over r-squared. And Kepler’s third law: the orbital period P in proportion to a to the3/2. However, these are not really ‘laws’ in the sense of ‘fundamental laws of physics’; Kepler didn’t understand why these patterns hold. That task fell to Isaac Newton.
Newton’s law of motion is that the force acting on a body equals the mass of that body times its acceleration, F equals ma, or m dv/dt. And Newton’s law of gravity is F equals minus-G times big-M times little-m over r-squared times r-hat, where in this case the big-M is the Sun’s mass and little-m is the planet’s mass. So, now, let’s roll up our sleeves and show how these laws relate to Kepler’s laws.
One might expect that we’d go in order, starting with Kepler’s first law. But it turns out Kepler’s second law is the most fundamental. So, let’s start there. The key concept is the conservation of angular momentum. Before getting to angular momentum, though, let’s think a moment about momentum and velocity.
Momentum is mass times velocity, p equals mv. And the velocity has 2 components. In a time dt, the velocity takes the planet from one position to another, changing both r and theta. That means the velocity has a radial component, toward or away from the origin, and an angular component, in the perpendicular direction.
The radial component, v-sub-r is equal to dr/dt. And the angular component is rd-theta, the distance moved in the direction of increasing theta, divided by dt. So, v_theta equals rd-theta/dt.
Angular momentum, which we’ll consider as L, is defined as r times mv_theta. Only the angular component, the ‘sideways component’, of the velocity matters. And since v_theta is rd-theta/dt, we can write L as mr–squared dtheta/dt.
In vector language, L equals r-cross mv, the cross product being the way to pick out only the perpendicular component of a vector. It has a magnitude of r times the component of mv that’s perpendicular to r.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Why Do We Define Angular Momentum?
Angular momentum, in some circumstances, is conserved. It doesn’t change with time, even if the body is changing in lots of other ways. The classic example is the twirling figure skater who pulls in her arms, effectively reducing her r, which means her d-theta/dt must increase, to compensate. That’s why she twirls faster. Angular momentum is conserved whenever there’s no net torque; no force in the theta direction. This is certainly true for the planets; the only force is gravity, which is in the radial direction, toward the Sun.
So, as a planet goes around, even though r and v are always changing, r–squared d-theta/dt is a constant, and therefore, d-theta/dt varies as one over r-squared. That’s Kepler’s second law! So, we now see that Kepler’s second law is a consequence of the conservation of angular momentum. Just as the ice skater twirls faster when she pulls in her arms, the planets twirl faster when they approach the Sun.
This is an important result, with implications beyond planetary motion. It helps explain why material speeds up as it spirals into a black hole, why a star spins faster when it contracts in size, and why a young star is surrounded by a spinning disk of material, within which the planets are formed.
Explaining Kepler’s Third Law
Why is the orbital period proportional to the 3-halves power of the semimajor axis? We’ll prove it for a circular orbit, where we can see the physics very clearly.
Imagine a planet moving in a circle of radius a, with some constant speed v. Over a full orbital period, P, the planet travels all the way around the circle. Therefore, v must equal the circumference of the circle, 2pi a divided by P. Or equivalently, P equals 2pi a over v. So, we already see one reason why P increases with a: the circumference of the circle gets bigger. There’s a longer way to go. In addition, when a is larger, v is lower; the planet moves more slowly because the gravitational attraction is weaker. This increases P even more, so that at the end of the day P goes like a to the 3/2.
In a time dt, the planet advances by a small angle d-theta, which corresponds to an arc length of a d-theta. So, v equals a d-theta/dt. During that same time interval, the velocity vector rotates by the same angle d-theta. The change in the velocity vector is v times d-theta. And so, the magnitude of the acceleration, the rate of change of velocity, must be v d-theta/dt. We can combine the equations, by solving the first one for d-theta/dt, and then inserting the answer, v over a, into the second equation. This gives an acceleration of v-squared over a.
Rewriting Kepler’s Third Law
We have just proven something you might have already known: to keep a body moving at speed v in a circle of radius a, you need to supply an inward acceleration, a centripetal acceleration, of v-squared over a. In the case of a planet, that acceleration is provided by the Sun’s gravitational force, G time big-M over a-squared. We set that equal to v-squared over a, and solve for v, finding v equals the square root of GM over a. Which we can then insert into our earlier expression for the period and, presto, we see that P is proportional to a to the 3/2 power. We also see that the proportionality constant is 2pi over the square root of GM. It goes down with the mass of the attracting body.
Let’s rewrite Kepler’s third law in a more convenient form, a so-called 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. So, we can rewrite 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.
Newton’s law of motion is that the force acting on a body equals the mass of that body times its acceleration.
Momentum is mass times velocity, p equals mv.
Angular momentum is conserved whenever there’s no net torque. Kepler’s second law is a consequence of the conservation of angular momentum.