Kepler's Laws of Planetary Motion

Three empirical laws describing planetary orbits: elliptical paths, equal areas in equal times, and the harmonic law relating period to distance.

Orbital Mechanics Equation - Mathematical formulations
Observable Universe - Planetary systems in cosmos
Dark Matter - Galactic rotation curves

1. Kepler's Third Law

We might wonder how Kepler derived his famous 3rd law? If we take the logarithm of both sides of the relationship:

\log (T^{2}) = \log (k a^{3})

where $k$ is an arbitrary dimensioning constant, we get:

2 \log (T) = 3 \log (a) + \log (k)

When plotted on a $\log$ v. $\log$ graph, this gives a straight line:

\log (T) = \frac{3}{2} \log (a) + C

where $C$ is the ordinate intercept. The important parameter here is $\frac{3}{2}$ , which directly leads to his third law. The $\frac{3}{2}$ coefficient is independent of the base of the logarithms used. There were natural (base- $e$ ), base-2, and Briggs base-10 log tables in use at the time.

Kepler used this logarithmic technique on known planetary data and published it in Kepler’s Ephemerides (1620), explicitly dedicating his work to Napier (the inventor of logarithms). This is what is called an 'empirical' law, as it was obtained from observational data.

Interestingly, Kepler's Third Law can be derived theoretically from Newton's Law of Universal Gravitation (the Inverse Square Law). However, it's important to note that Newton used Kepler's Laws to formulate his theories later on—so it's a bit of a "cart and horse" situation!

Newton's version of Kepler's Third Law incorporates his Law of Universal Gravitation, providing a theoretical basis for the empirical relationship Kepler discovered.

2. Newton's Version of Kepler's Third Law

Kepler's Third Law, in its original empirical form, states:

T^{2} \propto a^{3}

where $T$ is the orbital period of a planet, and $a$ is the semi-major axis of its orbit.

Newton, using his Law of Universal Gravitation, showed that this relationship could be derived theoretically. According to Newton's Law of Universal Gravitation, the gravitational force $F$ between two masses $M$ (the mass of the Sun) and $m$ (the mass of the planet) is:

F = \frac{G M m}{r^{2}}

where $G$ is the gravitational constant, and $r$ is the distance between the two masses.

For a planet in a circular orbit, this gravitational force provides the necessary centripetal force to keep the planet in orbit:

F = \frac{m v^{2}}{r}

where $v$ is the orbital velocity of the planet. Equating the two expressions for $F$ , we get:

\frac{G M m}{r^{2}} = \frac{m v^{2}}{r}

Simplifying, we find:

v^{2} = \frac{G M}{r}

The orbital velocity $v$ is related to the orbital period $T$ by:

v = \frac{2 π r}{T}

Substituting this into the equation for $v^{2}$ , we get:

{(\frac{2 π r}{T})}^{2} = \frac{G M}{r}

Simplifying further:

\frac{4 π^{2} r^{2}}{T^{2}} = \frac{G M}{r}

T^{2} = \frac{4 π^{2} r^{3}}{G M}

This is Newton's version of Kepler's Third Law. It shows that $T^{2} \propto r^{3}$ , with the proportionality constant being $\frac{4 π^{2}}{G M}$ . This equation not only confirms Kepler's empirical law but also provides the means to calculate the mass of the central body ( $M$ ) if the orbital period and distance are known.

3. Kepler and Aristarchus

Aristarchus of Samos (c. 310 – c. 230 BC) was a pioneering Greek astronomer who proposed a heliocentric model, placing the Sun at the center of the universe. However, his ideas did not gain much traction because they conflicted with the prevailing Aristotelian and Ptolemaic views that supported a geocentric universe, where Earth was the center.

The Ptolemaic model was more accurate for predicting planetary positions at the time, largely because it employed epicycles (small circles along which planets moved as they orbited Earth). This accuracy was achieved through increasingly complex adjustments rather than a fundamentally correct understanding of planetary motion. The preference for the geocentric model was not just due to its predictive power but also due to the philosophical and religious views of the era, which saw Earth as the center of the universe.

Nicolaus Copernicus (1473–1543) revived the heliocentric model, inspired in part by Aristarchus. However, Copernicus still used circular orbits and epicycles; he did not anticipate Kepler’s later discovery of elliptical orbits. Kepler’s work, based on accurate observational data, finally established the elliptical nature of planetary orbits, which eliminated the need for epicycles.

4. Galileo's Trial Incident

Galileo Galilei (1564–1642) was an Italian astronomer and physicist who championed the Copernican model of the solar system. This was in direct conflict with the Catholic Church’s teachings at the time, which were based on the geocentric model.

Galileo's Dialogue Concerning the Two Chief World Systems (1632) was a major factor in his trial. In this work, Galileo defended the heliocentric model and presented it as superior. However, the book was seen as mocking Pope Urban VIII, who had previously been a supporter of Galileo. This perceived mockery contributed to the Church’s decision to put Galileo on trial. The trial was not solely because of personal offense but also due to the broader implications of Galileo's support for the Copernican theory, which challenged the established religious and scientific views of the time. The Church saw this as heretical because it contradicted the Scriptures as they were interpreted then. Galileo was ultimately found "vehemently suspect of heresy," forced to recant his views, and spent the rest of his life under house arrest.

Next Action

Derive Kepler's laws from Newtonian gravity with examples