Kepler’s First Law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

Figure 1: (a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci (f1f1 size 12{f rSub { size 8{1} } } {} and f2f2 size 12{f rSub { size 8{2} } } {}) is a constant. You can draw an ellipse as shown by putting a pin at each focus, and then placing a string around a pencil and the pins and tracing a line on paper. A circle is a special case of an ellipse in which the two foci coincide (thus any point on the circle is the same distance from the center). (b) For any closed gravitational orbit, mm size 12{m} {} follows an elliptical path with MM size 12{M} {} at one focus. Kepler’s first law states this fact for planets orbiting the Sun.

Kepler’s Second Law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see Figure 2).

Kepler’s Third Law

The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is

where TT size 12{T} {} is the period (time for one orbit) and rr size 12{r} {} is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality.

Figure 2: The shaded regions have equal areas. It takes equal times for mm size 12{m} {} to go from A to B, from C to D, and from E to F. The mass mm size 12{m} {} moves fastest when it is closest to MM size 12{M} {}. Kepler’s second law was originally devised for planets orbiting the Sun, but it has broader validity.

Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions.

People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause.

We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one).

Let us consider a circular orbit of a small mass mm size 12{m} {} around a large mass MM size 12{m} {}, satisfying the two conditions stated at the beginning of this section. Gravity supplies the centripetal force to mass mm size 12{m} {}. Starting with Newton’s second law applied to circular motion,

The net external force on mass mm size 12{m} {} is gravity, and so we substitute the force of gravity for FnetFnet size 12{F rSub { size 8{ ital "net"} } } {}:

The mass mm size 12{m} {} cancels, yielding

The fact that mm size 12{m} {} cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius rr size 12{r} {}, all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period TT size 12{T} {} into the equation. By definition, period TT size 12{T} {} is the time for one complete orbit. Now the average speed vv size 12{v} {} is the circumference divided by the period—that is,

Substituting this into the previous equation gives

Solving for T2T2 size 12{T rSup { size 8{2} } } {} yields

Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields

This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body MM size 12{M} {} cancel.

Now consider what we get if we solve T2=4π2GMr3T2=4π2GMr3 for the ratio r3/T2r3/T2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } } {}. We obtain a relationship that can be used to determine the mass MM size 12{M} {} of a parent body from the orbits of its satellites:

If rr size 12{r} {} and TT size 12{T} {} are known for a satellite, then the mass MM size 12{M} {} of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio r3/T2r3/T2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } } {} should be a constant for all satellites of the same parent body (because r3/T2=GM/4π2r3/T2=GM/4π2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } = ital "GM"/4π rSup { size 8{2} } } {}). (See Table 1).

It is clear from Table 1 that the ratio of r3/T2r3/T2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } } {} is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the rr size 12{r} {} and TT size 12{T} {} data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation.

The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:

A non-satellite body fulfilling only the first two of the above criteria is classified as “dwarf planet.”

In 2006, Pluto was demoted to a ‘dwarf planet’ after scientists revised their definition of what constitutes a “true” planet.

The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics.

Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure 3(a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity.

Figure 3(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident.

Figure 3: (a) The Ptolemaic model of the universe has Earth at the center with the Moon, the planets, the Sun, and the stars revolving about it in complex superpositions of circular paths. This geocentric model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints as to what are the causes of these motions. (b) The Copernican model has the Sun at the center of the solar system. It is fully explained by a small number of laws of physics, including Newton’s universal law of gravitation.