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,

Fnet=mac=mv2r.Fnet=mac=mv2r. size 12{F rSub { size 8{ ital "net"} } = ital "ma" rSub { size 8{c} } =m { {v rSup { size 8{2} } } over {r} } } {}

(6)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"} } } {}:

GmMr2=mv2r.GmMr2=mv2r. size 12{G { { ital "mM"} over {r rSup { size 8{2} } } } =m { {v rSup { size 8{2} } } over {r} } } {}

(7)The mass mm size 12{m} {} cancels, yielding

GMr=v2.GMr=v2. size 12{G { {M} over {r} } =v rSup { size 8{2} } } {}

(8)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,

v=2πrT.v=2πrT. size 12{v= { {2π`r} over {T} } } {}

(9)Substituting this into the previous equation gives

GMr=4π2r2T2.GMr=4π2r2T2. size 12{G { { ital "mM"} over {r rSup { size 8{2} } } } =m { {v rSup { size 8{2} } } over {r} } } {}

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

T2=4π2GMr3.T2=4π2GMr3. size 12{T rSup { size 8{2} } = { {4π rSup { size 8{2} } } over { ital "GM"} } r rSup { size 8{3} } } {}

(11)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

T
1
2
T
2
2
=
r
1
3
r
2
3
.
T
1
2
T
2
2
=
r
1
3
r
2
3
.
size 12{ { {T rSub { size 8{1} } rSup { size 8{2} } } over {T rSub { size 8{2} } rSup { size 8{2} } } } = { {r rSub { size 8{1} } rSup { size 8{3} } } over {r rSub { size 8{2} } rSup { size 8{3} } } } } {}

(12)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:

r3T2=G4π2M.r3T2=G4π2M. size 12{ { {r rSup { size 8{3} } } over {T rSup { size 8{2} } } } = { {G} over {4π rSup { size 8{2} } } } M} {}

(13)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.

Newton’s universal law of gravitation is modified by Einstein’s general theory of relativity, as we shall see in Particle Physics. Newton’s gravity is not seriously in error—it was and still is an extremely good approximation for most situations. Einstein’s modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions.