Elementary introduction to the physics of Newtonian gravity

According to Newton's Law of Universal Gravitation - stated in Proposition VII, Book III, of his Philosophiae Naturalis Principia Mathematica, 1687 - two point-like bodies attract each other with a force whose strength is given by the formula

$$F = G\frac{m_1m_2}{r^2}$$

where $$m_1$$ and $$m_2$$ are the masses of the bodies and r is the distance between them.

In stellar dynamics, star sizes are in most cases much less than their mutual distances and thus it is justified to consider them as point-like bodies. Newton proved also that the same formula applies to spherical bodies, provided r is the distance between their centers - Proposition VIII, Book III.

G is a proportionality constant, whose value in the International System of units ( SI ) is about $$6.6743 \times 10^{-11}~\mathrm{m}^3~\mathrm{kg}^{-1}~\mathrm{s}^{-2}$$.

Often the International System is not the most convenient choice for stellar dynamics. In astronomy it is more convenient to measure distances in parsecs ( pc ), masses in units of one solar mass ( $$1 \mathcal{M}_{\odot} = 1.9884 \times 10^{30}$$ kg ), and speeds in kilometers per second ( km/s ). Then the time unit is 0.9778 My and the gravitational constant is given as

$$G = 4.301 \times 10^{-3} \frac{(\mathrm{km/s})^2~\mathrm{pc}}{\mathcal{M}_{\odot}}$$.

Another system of units ( sometimes named N-body units ) is used in numerical calculations, that is what we are mostly interested in. Masses are given in units of M, the total mass of the physical system under consideration ; G is taken to be one, and length units are adapted so that E, the total energy of the ( supposedly bound ) system, is -1/4. In this system then the length unit ( called virial radius ) is $$\frac{G M^2}{4 |E|}$$ ; the mass unit is M ; the time unit is $$\frac{G}{8}\sqrt{\frac{M^5}{|E|^3}}$$ ; and the speed unit is $$2\sqrt{\frac{|E|}{M}}$$.