Gravitation — Kepler and Newton

Newton's law states that every pair of point masses attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = Gm_1m_2/r^2$.

Key properties:

• Always attractive: gravity never repels.
• Infinite range: the force never reaches exactly zero, no matter how far the bodies are.
• Inverse-square law: doubling the distance reduces the force by a factor of 4; tripling reduces it by 9. This is a geometric consequence of the force "spreading" over a sphere of area $4\pi r^2$.
• Acts on point masses: for spherically symmetric bodies (like planets) the law holds exactly with $r$ measured between centres (Shell Theorem).

Surface gravity: setting $F = mg$ and $F = GmM_T/R_T^2$ gives $g = GM_T/R_T^2 \approx 9.81\,\mathrm{m/s^2}$.

The gravitational field $\vec{g}(r) = -GM\hat{r}/r^2$ is defined as the gravitational force per unit mass at a point. It points toward the source mass and falls off as $1/r^2$.

The gravitational potential $V(r) = -GM/r$ is negative everywhere (by convention $V(\infty) = 0$). The potential energy of a mass $m$ in the field is $U = mV = -GMm/r$. The field is the negative gradient of the potential: $\vec{g} = -\nabla V$.

For a body in orbit around a central mass the total mechanical energy $E = KE + U$ determines the orbit type:

• $E < 0$: bound orbit — ellipse (circle is the special case $e = 0$).
• $E = 0$: body just barely escapes — parabolic trajectory; this defines escape velocity.
• $E > 0$: body escapes with kinetic energy to spare — hyperbolic trajectory.

The virial theorem tells us that for a circular orbit $KE = -E$ and $U = 2E$, so the kinetic energy is half the magnitude of the (negative) potential energy.

Kepler's three empirical laws, derived from precise astronomical observations, are exact consequences of Newton's law of gravitation:

First Law (Law of Orbits): every planet moves in an ellipse with the Sun at one focus. The shape is described by eccentricity $e$: for a circle $e = 0$, for a highly elongated ellipse $e \to 1$.

Second Law (Law of Areas): the radius vector from Sun to planet sweeps equal areas in equal times. This is a direct consequence of the conservation of angular momentum: because gravity is a central force (no torque about the Sun), $L = mr^2\dot{\theta}$ is constant, hence $dA/dt = L/(2m) = \mathrm{const}$. A planet moves fastest at perihelion and slowest at aphelion.

Third Law (Harmonic Law): the square of the orbital period is proportional to the cube of the semi-major axis: $T^2 \propto a^3$. The proportionality constant $4\pi^2/(GM)$ is the same for all bodies orbiting the same central mass, so comparing two planets gives $T_1^2/a_1^3 = T_2^2/a_2^3$.

Escape velocity is the minimum launch speed needed to escape a planet's gravity without further propulsion. It is derived by setting the total mechanical energy to zero ($KE + U = 0$): $\tfrac{1}{2}mv_{esc}^2 - GMm/R = 0$, giving $v_{esc} = \sqrt{2GM/R}$.

Note that escape velocity is independent of the mass of the escaping object and of the direction of launch — it only depends on the mass and radius of the planet.

For a circular orbit at radius $r$, gravity provides the centripetal force: $GMm/r^2 = mv_{orb}^2/r$, giving $v_{orb} = \sqrt{GM/r}$. Comparing with $v_{esc}$: $v_{esc} = \sqrt{2}\,v_{orb}$, so escape speed is always $\sqrt{2} \approx 1.41$ times the circular orbital speed at the same altitude.

The total energy of an elliptical orbit depends only on the semi-major axis $a$: $E = -GMm/(2a)$. A lower orbit has more negative energy (more tightly bound) and — counterintuitively — a higher speed, since $v_{orb} \propto 1/\sqrt{r}$.