Table of Contents

Orbital Mechanics

"To Infinity and Beyond" (B.L. 1995)

Gravity's Big Reach!

Newton's Law of Universal Gravitation can be written in the form Fg = GMm/r² (1) where Fg is the gravitational force of attraction (in newtons) between two masses M and m (in kilograms), separated from each other by a distance r ( in metres). G is the universal gravitational constant of
6,67 x 10-11Nm²/kg².

The acceleration g that a mass m would experience under the influence of a gravitational force Fg is

g = F_g/m (Newton's 2nd Law of Motion)

rearranging we get Fg = g m (2)

setting the right side of equation (2) equal to the right side of equation (1) the m's drop out and we get

g = GM/r² (3)

The quantity g is called the local acceleration due to gravity (m/s²) attributed to mass M as a distance r(from its centre of mass).

The quantity g is also called the gravitational field strength (N/kg) attributed to mass M as a distance r (from its centre of mass).

Graphically Newton's equation (3) (using M as the mass of the Earth) can be plotted as shown in the graph below. This graph plots the gravitational field strength surrounding the Earth at various distances from the Earth's centre. Of course the portion of the graph to the left of the Earth's radius line would only apply if the Earth were compressed so that all of the Earth's mass were inside the radius selected.

A gravitational field can be characterized in one of two identical ways.

1. The local acceleration (due to gravity). On the Earth's surface this is approximately 9.8m/s².
   
   
2. The local gravitational field strength. This quantity is numerically the same as the local acceleration due to gravity and on the Earth's surface is approximately 9.8 N/kg.

By dimensional analysis one can easily show that the units of N/kg and m/s² are equivalent.

One of the most interesting things about the graph above is that it predicts black holes. If the Earth were to be compressed so that all of its mass were in an extremely compact form, the local gravitational field would become enormous as seen in the upper left of the graph.

One can extrapolate the field strength to a distance approaching zero and see that the gravitational field strength approaches infinity.

As r approaches zero, the gravitational field strength approaches infinity. This is just a graphic representation of the fact that the ratio GMm/r² an infinitely large value as r gets smaller and smaller.

Similarly one can see from extrapolating the line on the graph that in order for the gravitational field strength to approach zero, the distance from the Earth must approach infinity. An examination of the ratio GMm/r² shows why this is true. In order to make the ratio exceedingly small, the distance r must get larger and larger.