Newtonian gravitation §

The law has three ingredients, each physically load-bearing:

1. Proportionality to each mass. By Newton’s third law the force on $m_1$ from $m_2$ must equal the force on $m_2$ from $m_1$; making the force proportional to both masses is the simplest way to guarantee this symmetry. It also makes inertial and gravitational mass identical, the empirical fact later promoted to the equivalence principle in General relativity.
2. Inverse square. This is the signature of a field spreading over the surface of a sphere of area $4\pi r^2$ in three dimensions. It is what makes bound orbits closed ellipses (Bertrand’s theorem) and what makes the shell theorem work.
3. Instantaneous action at a distance. Newton’s formulation has no propagation delay. This is its deepest flaw: it is an approximation valid when the field changes slowly compared with the light travel time across the system, which is why it fails for Gravitational waves.

Example. The Earth pulls a 1 kg mass at its surface with $F = GM_\oplus m/R_\oplus^2 \approx 9.8$ N. The same law, applied to the Moon at 60 Earth radii, gives an acceleration $60^2 \approx 3600$ times smaller — about $0.0027$ m/s², which matches the Moon’s centripetal acceleration. This was Newton’s original unification: the apple and the Moon obey one law.

The field formulation is a bookkeeping upgrade with real consequences:

1. Superposition. Because the equation for $\Phi$ is linear, the potential of any mass distribution is the sum (integral) of point-mass potentials. This is why engineers can compute the gravity of an irregular asteroid, a dam’s water load, or a galaxy’s rotation curve by simple integration.
2. The shell theorem. A spherically symmetric shell attracts exterior points as if all its mass were at the center, and exerts zero force on interior points. This is why planets can be treated as point masses and why the gravity inside a uniform sphere grows linearly with radius.
3. Energy. The potential energy of two masses is $U = -Gm_1m_2/r$. Binding energy is negative: assembling a body from dispersed material releases energy, which powers accretion disks and star formation.

Example. Escape velocity: set kinetic energy equal to $|U|$: $\tfrac12 v^2 = GM/r$, so $v_{\rm esc} = \sqrt{2GM/r}$ — 11.2 km/s from Earth’s surface, independent of the projectile’s mass.

1. Conic orbits. The orbit equation is $r = \frac{h^2/GM}{1 + e\cos\varphi}$: ellipses ($e<1$), parabolas ($e=1$), hyperbolas ($e>1$). Kepler’s three laws are corollaries. This is the basis of all orbital mechanics used for satellites and spacecraft.
2. Tides. The Moon pulls the near side of Earth more strongly than the far side. The differential acceleration is $\Delta g \approx 2GMr/d^3$ — note the cube: tides are a second-derivative effect. Tidal forces raise ocean bulges, lock rotations, heat Io, and will eventually shred anything falling into a black hole.
3. Limits of the theory. Newtonian gravity predicts Mercury’s perihelion should be fixed; observation shows a residual precession of 43 arcseconds per century. General relativity explains it exactly — the first crack in the Newtonian picture.

Example problem. A satellite orbits Earth at 400 km altitude. What is its orbital period?

Surface gravity is $g = GM/R^2$. With $M$ doubled and $R$ doubled: $g' = G(2M)/(2R)^2 = \tfrac{2}{4}\,GM/R^2 = g/2$. The surface gravity is half of Earth’s — mass grows faster than the square of the radius only if density is held fixed, which is not the case here.