Earth's Keplerian system & towards universal gravitation

Updated 31 May 2026

Having investigated the Solar, Jovian and Saturnian Keplerian Systems, let's take a closer look at home and start with trying to determine the value of its Kepler 'constant' (should it exist) by looking at the Moon's orbit about the Earth.

It's at a distance (measured with reasonable accuracy by Ptolemy nearly 2000 years ago) of about 60 Earth radii and takes about 27 days to complete one orbit.

From this we can say the value of the Kepler 'constant' for the Earth is ~ 60³/27² ~ 296.

In the case of the other systems, there are many orbiting bodies and we can calculate an average, but in this case, if we go back many decades at least, there is only one such body orbiting the Earth.

However, let's take inspiration from Newton, who realized that an apple falling to the ground also experiences an acceleration towards the centre of the Earth. This acceleration is commonly referred to as g and can be measured accurately by means of a pendulum, length l, the period of which is 2π√(l/g).

We can generalize the formula from the previous post and say that the acceleration at distance d, regardless of whether the object is orbiting, is:

4π²k/d².

At the Earth's surface, we state in this case that d = 1 (the same as if all the Earth's mass were concentrated at its centre) and g = 4π²k, where k is the Kepler 'constant' for the Earth. If we know g, we can therefore determine that:

k = g/(4π²).

In metric units g ~ 9.8 m/s² and we need to convert this to the units used when calculating the value from observation of the Moon, i.e. Earth radii/day²:

• the radius of the Earth (also measured, albeit in different units, with reasonable accuracy by the Ancient Greeks) is ~ 6.4 x 10⁶ m and we divide our value of g by this number
• there are 86400 seconds in a day, squared is ~ 7.5 x 10⁹ and we multiply the above result by this number.

In these new units g ~ 11719 and dividing by 4π² gives ~ 297, which is very close indeed to the value we derived earlier.

We can conclude therefore that the Earth, as with the Sun, Jupiter and Saturn also has its own Keplerian system, where the central body exerts a gravitational force that follows an inverse-square law.

From earlier considerations, that force appears to be related to the mass of the central object (Sun, Jupiter etc), possibly simply proportional to it, in which case the Kepler 'constant' for the system is also proportional to the mass of the central object.

We can surmise therefore that all massive bodies, including those bodies without satellites of their own, exert such a gravitational force on surrounding matter. (Surface features on the Moon, for example, and the near spherical shape of all planets, including those without moons, provide strong evidence for this).

This was to be confirmed experimentally by Henry Cavendish in the laboratory at the end of the 18th century - an incredible achievement for the time given the weakness of the gravitational force.