Isaac Newton founded modern physics with his brilliant analysis of planetary motion. Both the greatest mathematician and the greatest physicist of his age, he began by inventing a new branch of mathematics — the infinitessimal calculus — in order to think about motion in a completely novel way. Our story begins with a full account of how Newton the mathematician used his new tool to convert Kepler’s laws of planetary motion into an equivalent inverse square law of acceleration. This mathematical translation led Newton the physicist to perhaps the greatest “Aha!” moment in the history of science — the realization that the force of gravity we all feel on Earth extends up to the Moon and beyond, binding the whole solar system together.
Let’s follow along as Newton tackles one of the most difficult and important math problems in history — then recognizes the implications of his solution to learn something entirely new about our universe.
“A planet moves in an ellipse with the Sun at one focus, such that the line between the Sun and planet sweeps out equal areas in equal times.” Kepler’s first two laws are a very natural way to think about motion: First you describe the trajectory of a body, then you find a rule for how the body progresses along that trajectory over time.
The figure shows the planet P and the ellipse of its trajectory with the Sun S at one focus. The gray area represents the sector of the ellipse the radius SP will sweep out in some fixed but arbitrary time. The arbitrary choice here is one twentieth the period of the planet; the planet P will advamce to the leading edge of the gray sector in one twentieth as long as a full orbit. According to Kepler, the speed of the planet varies in such a way that the area of the gray sector remains constant. Therefore, the area of the gray sector in this figure is always one twentieth the area of the whole ellipse.
Newton will complicate this picture, because he wants to fit the way a planet moves into his framework for how bodies in general move. The payoff is that his laws of motion will apply to objects here on Earth as well as to planets — a radically different approach than Kepler or anyone else before Newton.
Newton introduced a completely new way to think about motion. He imagined motion as a series of tiny steps or snapshots very slightly separated in time. This is exactly how animation works — a rapidly displayed sequence of slightly changing snapshots looks like a continuous motion when the frame rate is high enough. Chopping a trajectory into a series of small steps is essentially differentiation in calculus jargon, while assembling steps back into a trajectory is integration.
Here we begin with ten frames per period. The gray sector is initially one tenth the total ellipse area, so the radius vector SP jumps to the end of the gray sector at each step. The orbit looks very jerky, so after a half period we double the frame rate to twenty per period, halving the area of the gray sector. After a second half period we again double frame rate and halve the gray area, and so on. Soon the animated radius appears to move continously as far as your eye can tell, and the gray area showing the size of each step is invisibly narrow. Newton described the animation step duration as infinitessimal, meaning “as brief as necessary for present purposes.”
You can represent the velocity vector — the rate of change of position — by an arrow pointing along the instantaneous direction of motion, with length proportional to speed. That is, if the tail of the arrow is at the body, the tip of the arrow is at the place the body would be after some fixed but arbitrary time if its velocity did not change. Here, the fixed time is one twentieth the period of the planet. As before, the leading edge of the shaded sector shows where the planet, with its changing velocity, will actually be after that same time interval.
Kepler's first two laws specify exactly how the planet moves, so by taking very short time steps, you can compute the instantaneous velocity as accurately as you please. If you imagine infinitessimal steps, you understand that the velocity vector must always be tangent to the ellipse, and the equal area law means it will be longer or shorter as the planet P approaches or recedes from the Sun S.
When you keep track of the tip of the velocity vector — a velocity trajectory — something interesting happens: It seems to trace a circle around P. Of course, the center of this circle is displaced upward from P, since the length of the vector has to change. We will show that the velocity trajectory is not just approximately a displaced circle, but precisely so for any body orbiting according to Kepler's first two laws.
“A body at rest tends to stay at rest; a body in motion tends to stay in motion in a straight line at a constant speed. A force changes the velocity of a body; the body accelerates at a rate directly proportional to the force and inversely proportional to the body's mass.” Newton's first two laws of motion serve the same purpose as Kepler's first two laws, but apply to material bodies in general, on Earth as well as in the heavens.
We've already used Kepler's laws to numerically work out the velocity trajectory of a planet. Newton's laws suggest we also look at its acceleration. Like velocity, acceleration is a vector; in fact, acceleration is to the velocity trajectory what velocity is to the ordinary position trajectory. Just as the velocity must be tangent to the trajectory of the planet, its acceleration must be tangent to its velocity trajectory. The length of the acceleration vector, like the velocity vector, is smaller when the planet is far from the Sun, larger when near the Sun.
The line from the center of the velocity circle to the tip of the velocity vector appears to always remain exactly perpendicular to the position vector SP. Since the acceleration is tangent to the velocity circle, this means the acceleration vector of the planet always points exactly in the direction from P to S. That is, the planet always accelerates directly toward the Sun. Newton's second law says there must be a force on each planet pulling it directly toward the Sun!
"The planet accelerates directly toward the Sun," means exactly the same thing as, "the Sun-planet line sweeps out equal areas in equal times." To see why, begin with a body P moving in a straight line at a constant speed past a fixed point S. Taking snapshots at equal time intervals, P sweeps over a series of equal area gray triangles. The areas are equal because they share a common height h, the distance from the linear trajectory of P to the point S, and have equal bases b, the distance P moves in each time step.
Next, suppose that the velocity suddenly changes at one of the time steps, so the body switches to a new trajectory line. If SP continues to sweep out area at the same rate, then the areas swept out along the new line in each time step must equal the areas stepped off along the original line. But the last old triangle and the first new triangle share a common side SP. In order to have equal areas, their altitudes perpendicular to SP must be equal. Therefore, area will continue to be swept out at the same rate along the new trajectory line if, and only if, the component of the velocity perpendicular to SP remains unchanged. In other words, any sudden change in velocity of the body at P which keeps the rate SP is sweeping out area constant must be parallel to SP.
Imagine breaking down a continously changing velocity into a huge number of infinitessimal steps. You've just proven that each infintessimal change in velocity must be directly toward or away from S. In other words, the body P must accelerate directly toward or away from S in order for SP to sweep out equal areas in equal times and vice versa.
The rate area is swept out is the angular momentum of P around S (divided by twice the mass of the body P). Kepler's equal area law is an example of conservation of angular momentum.
An ellipse is the set of points for which the sum of the distances from two fixed points — its foci — is constant. The major axis, or largest diameter, is on the line connecting the foci. (The minor axis, or smallest diameter, is on the perpendicular bisector of the major axis and passes through the center of the ellipse.) For a Keplerian ellipse, the Sun S is at one of focus, the planet P is a point on the ellipse, and the other focus O is a point in empty space. The planet is closest to the Sun (its perihelion) at the endpoint of the major axis near S, and farthest from the Sun (its aphelion) at the endpoint near O.
Since the distance from S to perihelion is the same as from O to apehelion, the constant sum of distances SP plus PO equals the major axis of the ellipse. Extend SP to Q such that PQ equals PO. Then SQ remains constant as P moves around the ellipse; in other words, SQ sweeps out a circle with center S and radius equal to the major axis of the ellipse.
By construction, OPQ is always an isoceles triangle, which means that P lies on the perpendicular bisector MP of OQ, where M is the midpoint of OQ. In fact, MP must be tangent to the ellipse at P.
To understand why, move P to any other point on the perpendicular bisector of OQ. For any point on MP you choose, PQ always equals PO. When P is not on SQ, SPQ is not straight, so the sum of SP and PQ, hence the sum of SP and PO, must be greater than SQ. Because the ellipse is all the points P with SP plus PO equal SQ, the points on MP are all outside the ellipse, except for the original P that lies on SQ, which of course also lies on the ellipse. Thus, line MP touches the ellipse only at one point (the original P on SQ), so MP must be tangent to the ellipse at P.
Since MP is tangent to the ellipse, and OQ is perpendicular to MP, if you rotate the circle (along with points O and Q) by ninety degrees about S, OQ will now be parallel to the ellipse tangent at P. Also, SQ will now be perpendicular to SP. The diagram looks very much like a scaled and translated version of the velocity trajectory we computed numerically from Kepler's laws — OQ is proportional to the velocity, OS to the displacement of the center of the velocity circle from the zero velocity point O, and SQ is the constant velocity radius.
In fact, the vector OQ you have constructed has exactly the properties of the velocity vector of the planet: First, it is parallel to the tangent of the ellipse at P. Second, it moves around a circle centered on S with SQ always exactly ninety degrees ahead of SP, so its rate of change is always exactly in the PS direction. Thus, if the planet moves around its ellipse with velocity OQ, its acceleration vector will point directly toward S, which means SP will sweep out equal areas in equal times. In other words, you now know how to geometrically construct the velocity vector corresponding to Kepler's first two laws.
Kepler describes planetary motion as an elliptical trajectory plus an equal area law. Instead of the equal area law, you could say that the planet moves around its ellipse at a speed proportional to OQ in your velocity trajectory construction. In other words, specifying both the position and velocity trajectories of a planet — both its ellipse and the associated eccentric circle —is an alternative to Kepler's description.
Newton realized that if you can find a rule for the acceleration, you need neither the position nor the velocity trajectory. If you simply know where the planet is, how fast and in what direction it's moving at one instant of time, you can use your acceleration rule to figure out exactly where it will be at any future time (or where it was at any past time): The velocity tells you where it will move after a short time, and the acceleration tells you its direction and speed at that new time for the next time step.
Just as the equal area law and elliptical trajectory determine the velocity trajectory, the acceleration rule is also built into Kepler's laws. In fact, you already found the rule for the direction of the acceleration — directly toward the Sun. You only need to find its magnitude - exactly how fast is the velocity vector of the planet changing as it moves around the Sun?
The trajectories in position and velocity space are coupled by the fact that the two radius vectors SP and SQ are always exactly ninety degrees apart. Thus, even though P and Q speed up and slow down as they move around S, the instantaneous angular speeds of SP and SQ are always equal. Call that varying but common angular speed ω.
Consider first the motion of the planet around the Sun in ordinary space. The planet’s velocity has both a radial component, directly toward or away from the Sun, and a transverse component, perpendicular to that. The angular speed of the planet around the Sun is independent of its radial velocity, and directly proportional to its transverse velocity. In fact, if you measure ω in radians per unit time, the transverse velocity is simply the product ωr, where r is SP, the instantaneous distance of the planet from the Sun. Transverse velocity is the rate of change of the height of the triangle with the radius vector r as its base. Thus, the rate of change of area swept out by the radius is half of the product of r and the transverse velocity (ωr), or half of ωr².
Calling twice the rate the planet’s radius sweeps out area L (a symbol used for angular momentum), you have L = ωr². But Kepler’s second law says that (half of) L remains constant as the planet orbits. Therefore, the angular speed of the planet around the Sun, ω, varies inversely as the square of its distance from the Sun, ω = L/r², where the constant L is twice the rate the planet sweeps out area according to Kepler’s second law.
Recall that the angular speed of the planet around the Sun ω is also the angular speed of the velocity radius SQ around its circular trajectory in velocity space. Let u denote the constant length of SQ. Since the acceleration is entirely transverse to SQ, its magnitude g is simply ωu, the velocity space analog of the formula for transverse velocity ωr. Since the velocity radius u is constant, and ω varies inversely as the square of the distance r, the acceleration g must also vary inversely as the square of the distance. In fact, g = uL/r², so the constant of proportionality is uL.
You have proven that if a planet accelerates directly toward the Sun at a rate which varies inversely as the square of its distance from the Sun, then it will automatically obey Kepler’s first two laws of planetary motion. The trajectories in both velocity and ordinary space emerge by integrating this inverse square law, that is, by taking a large number of very small steps in time starting from the current position and velocity of the planet. Again, in Newton’s reformulation, the trajectories are secondary; what you need to study are the relationships between instantaneous position, velocity, and acceleration.
So far, all you know is that each planet accelerates toward the Sun at a rate inversely proportional to the square of its distance, g = uL/r². The constant of proportionality in this inverse square law might still differ from one planet to the next. Since L is an area per unit time, uL has dimensions of length cubed divided by time squared — exactly the same dimensions as the constant of proportionality in Kepler’s third law: the cube of the major axis of a planet’s ellipse is proportional to the square of its orbital period. How is uL related to a³/T²?
Since the ellipse is just a circle elongated on one axis, its area is πab (the stretched πr²), where a and b are its major and minor semi-axes (that is, measured from the center like the radius of a circle). Since the constant L is twice the area swept out per unit time, L = 2πab/T, where T is the period of the planet.
You can derive an expression for the velocity circle radius u from the rate area is being swept out at two specific points of its orbit: At perihelion, when the planet is closest to the Sun, L = (a-c)(u+w), where c is the distance from the center of the ellipse to one focus, and w is the displacement of the center of the velocity circle from zero velocity (so w is the length of OS). Similarly, at aphelion, when the planet is farthest from the Sun, L = (a+c)(u-w). Eliminating w from these two formulas results in u = La/(a²-c²). The denominator is b², so u = La/b².
Combining these expressions for L and u, you find uL = 4π²a³/T². Kepler’s third law states that a³/T², and hence the constant of proportionality uL, is the same for all the planets. (To be clear, u and L individually differ from planet to planet, but their product uL is the same for all planets.) Therefore, in fact, a single inverse square law for the acceleration, g = uL/r², applies to every planet.
Thus, with Newton’s description, you don’t really need a separate rule analogous to Kepler’s third law — a single constant automatically connects the orbits of all the planets: The Sun causes planets to accelerate directly toward it at a rate inversely proportional to their distance from it. In fact, any object at the same distance from the Sun accelerates toward the Sun at that same rate. By Newton’s time, astronomers knew that comets travel around very eccentric Keplerian orbits, often falling from beyond Jupiter’s orbit to within Earth’s orbit. There is only a single puzzle - why this inverse square law? Newton acknowledged he had no clue about this final mystery with the quip, "Hypotheses non fingo."
The same year Kepler published his first two laws, 1609, Galileo invented the telescope. He made several huge discoveries immediately: He saw the disks of the planets — planets are not simply points of light! — a sight no one had seen before. Furthermore, the phases of Venus confirmed that planets are globes lit by sunlight, like the Moon. And most importantly for this story about Newton, Galileo discovered the four largest moons of Jupiter. By Newton’s time over half a century later, astronomers had studied Jupiter’s moons in considerable detail. They learned that the four moons orbited Jupiter according to Kepler’s laws, except with a value of a³/T² about a thousand times smaller than for the planets or comets orbiting the Sun.
Newton devised a brilliant indirect test of Kepler’s cube-square law for the case of the Earth: A cannonball accelerates downward at a rate of 9.8 m/s² (as Galileo measured), independent of how fast it is moving horizontally. Newton realized that if its horizontal speed were high enough, the cannonball would follow the curve of Earth’s surface, and (absent air resistance) it would continue to fall all the way around the Earth in a circular orbit. He could use this hypothetical (to him) orbiting cannonball and the Moon to check Kepler’s cube-square law for objects orbiting the Earth.
As you have seen, if objects accelerate toward a
center at a rate inversely proportional to the square of their
distance, they obey Kepler’s laws. The ancient Greeks had
measured the radius of the Earth (6380 km) and the distance to
the Moon (60 Earth radii). Since the Moon goes around with respect
to the stars once more each year than with respect to the Sun, its
sidereal period is about 27.3 days. For a roughly circular
orbit, you have already worked out the acceleration
g = ωv = ω²r, so the downward
acceleration of the Moon works out to be 0.0027 m/s²
((2π/(27.3×
Newton’s reasoning here is arguably the greatest breakthrough in the history of science: The familiar force of gravity that pulls cannonballs downward near the surface of the Earth is the same force that holds the Moon in orbit around us. And furthermore, the connection between the trajectories of cannonballs and of the Moon is none other than Kepler’s laws for the motion of planets around the Sun, or of the Galilean moons around Jupiter. The millenia of labor poured into understanding the motion of the planets has brought Newton right back to the one force we all feel from birth — gravity.
Newton’s great insight — far more important than Archimedes’s “Eureka” moment — is that his inverse square law of acceleration relates the gravity we all feel at the surface of the Earth to the acceleration of the Moon in its orbit around the Earth. Gravity may be the familiar force you’ve always known, but it does far more than just pull you down to Earth. Gravity extends upward throughout the solar system and beyond, binding the moons to their planets and the planets to the Sun. We sense the solar system by touch as well as by sight, or at least we all feel the force holding the whole thing together.
Notice that you never used Newton’s laws of motion in proving the equivalence of the inverse square law of planetary acceleration to Kepler’s laws. We mentioned his laws of motion merely to motivate why instantaneous position, velocity, and acceleration is an interesting way to describe motion. The inverse square law of acceleration is simply a description of how planets and moons and earthbound projectiles do, in fact, move. It is not a theoretical prediction, but an experimental observation, just like Kepler’s laws.
However, Newton also developed a general theory of mechanics — how things move — including his iconic formula F = ma. When he applied this mechanics to the case of planetary motion, Newton discovered his theory of universal gravitation. Universal gravitation offers an explanation for the differences in the constant of proportionality in the inverse square acceleration laws observed for the Sun, Jupiter, and Earth — it is no longer simply a description of how things move, but a theory explaining how any object can serve as a center of attraction. Universal gravitation also explains why the Moon does not accurately move in a Keplerian ellipse, and very accurately predicts its observed orbit. But all this belongs to a separate story, “Binding the Solar System.”