Logs make an Impact: Kepler’s Third Law

Johannes Kepler

Johannes Kepler. Discoverer of the laws of planetary motion, and early logarithm enthusiast.

It was only a few short years after Napier’s discovery of logarithms that they made their first, and lasting, impact on the sciences. They were instrumental in uncovering the most important of Kepler’s Laws of planetary motion.

The German astronomer Johannes Kepler had published his first two laws of planetary motion in 1609. These were

  • I — The planets orbit the sun in ellipses, with the sun at one focus.
  • II — The line joining a planet to the sun sweeps out equal areas in equal times.

But for nearly a decade afterwards Kepler, a prodigious calculator, laboured in vain to find some further relationship hidden in the masses of observational data that the late Tycho Brahe has gathered. In particular, he had yet to find a law linking the orbits of all the planets in some way.

Enter the logarithm. Kepler was an instant convert on discovering Napier’s logs in 1616, and would eventually publish tables of his own. But before this, he seems to have used logs to finally discover the elusive third law he’d been looking for.

As in the previous post, we’ll ‘cheat’ a little here by using modern notation (and data). Kepler had data on the orbital parameters of the first six planets, among them the orbital period $T$, and the semi-major axis $R$. Below is a table of suitably normalised values of $T$, $R$ for each of the planets in the solar system, and their logarithms $\log_{10}(T)$ and $\log_{10}(R)$. To make things clearer here, two planets which were as yet undiscovered in Kepler’s time — Uranus and Neptune — have both been included in the data.
\text{Planet} & \text{T (/yrs)}& \text{R (/$10^6$ km)} & \log_{10}(T) & \log_{10}(R)\\
\text{Mercury (m)} & 0.24 & 57.91 &-0.618 & 1.763\\
\text{Venus (V)} & 0.62 & 108.21 &-0.211 & 2.034\\
\text{Earth (E)} & 1.0 & 149.6 &0.0 & 2.175\\
\text{Mars (M)} & 1.88 & 227.94 & 0.274 & 2.358\\
\text{Jupiter (J)} & 11.86 & 778.34 &1.074 & 2.891\\
\text{Saturn (S)} & 29.45 & 1426.71 & 1.469 & 3.154\\
\text{Uranus* (U)} & 84.02 & 2870.63 &1.924 & 3.458\\
\text{Neptune* (N)} & 164.79 & 4498.39 &2.217 & 3.653\\

Plotting these logarithm pairs gives what is called a log-log plot of the data, and this is also shown below.

Solar system planets T,R, Log-Log plot

Log-Log plot of the orbital period $T$ and semi-major axis $R$ of eight planets in the solar system. The graph is a perfect straight line, revealing a power law relationship between the variables.

The plot shows immediately that there is a linear relationship between the logs. In fact, the slope of the line in the plot can be measured to be $\cong 2/3$. This means that
\[\log_{10}(R) = \frac{2}{3} \log_{10}(T) +C \]
for some constant $C$. Solving this log equation therefore gives
\[ T^2 = \alpha R^3\]
where $\alpha$ is a constant. This power law is the planetary relationship being sought. Logarithms will always reveal any such power law relationships as straight lines on a log-log plot.

It seems likely that a log enthusiast like Kepler would have tabulated logarithms of his data values at some point. Upon doing so, he would have been one step away from discovering this last law. In 1619, three years after coming across logarithms, he published his third and final law of planetary motion

  • III — The square of a planet’s orbital period is proportional to the cube of its semi-major axis.

This law completed the set by relating the orbits of all the planets around the sun together. In fact, the correct constant $\alpha = 4\pi^2/G M_{\odot}$, where $M_{\odot}$ is the mass of the sun, revealing that it is the sun which is the common cause of planetary motion in the solar system. This was the law which later allowed Newton in 1687 to show that it was in fact the inverse square gravitational pull of the sun that caused the planets to orbit it.

So, from the very start, logs were more than just a simple arithmetical tool. Their discovery opened up gateways to whole new avenues of research in the sciences. Logs have since come to be recognised as an important element of the scientific revolution in general. By now have become a ubiquitous part of both mathematics and the sciences. Not bad for a something Napier had originally intended as simply an aid to pen and paper arithmetic.

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