Gravitational Constant - Dimensions, Units, and Magnitude

Dimensions, Units, and Magnitude

The dimensions assigned to the gravitational constant in the equation above—length cubed, divided by mass, and by time squared (in SI units, meters cubed per kilogram per second squared)—are those needed to balance the units of measurements in gravitational equations. However, these dimensions have fundamental significance in terms of Planck units; when expressed in SI units, the gravitational constant is dimensionally and numerically equal to the cube of the Planck length divided by the product of the Planck mass and the square of Planck time.

In natural units, of which Planck units are a common example, G and other physical constants such as c (the speed of light) may be set equal to 1.

In many secondary school texts, the dimensions of G are derived from force in order to assist student comprehension:

In cgs, G can be written as:

G can also be given as:

Given the fact that the period P of an object in circular orbit around a spherical object obeys

where V is the volume inside the radius of the orbit, we see that

This way of expressing G shows the relationship between the average density of a planet and the period of a satellite orbiting just above its surface.

In some fields of astrophysics, where distances are measured in parsecs (pc), velocities in kilometers per second (km/s) and masses in solar units, it is useful to express G as:

The gravitational force is extremely weak compared with other fundamental forces. For example, the gravitational force between an electron and proton one meter apart is approximately 10−67 newtons, while the electromagnetic force between the same two particles is approximately 10−28 newtons. Both these forces are weak when compared with the forces we are able to experience directly, but the electromagnetic force in this example is some thirty nine orders of magnitude (i.e., 1039) greater than the force of gravity — roughly the same ratio as the mass of the Sun compared to a microgram mass.