Look up on a clear night and almost everything you can see is falling. The Moon is falling around the Earth; the Earth and the other planets are falling around the Sun; a comet swings in from the cold dark, whips once around our star, and is flung back out never to return. A spacecraft on a gravity-assist flyby steals a little speed from a giant planet and slingshots on to the outer Solar System. All of these journeys — the tidy circle of a communications satellite, the long thin ellipse of a comet, the open hyperbola of an interstellar visitor like ʻOumuamua — are governed by one force law and one idea.

The force is gravity, and near a single dominant mass M it always points straight at that mass and weakens with the square of the distance:

A force that always points along the line to a fixed centre — depending only on how far out you are, never on which direction — is called a central force. This page is about what such a force does to a moving body. The astonishing punchline, which we will build up to piece by piece, is that every possible orbit under this inverse-square law is a conic section: a circle, an ellipse, a parabola, or a hyperbola. The type you get is fixed by nothing more than the body's energy.

First gift of a central force: angular momentum is conserved

Because the force always points along \hat{\mathbf r} — straight toward the centre — it exerts no torque about that centre. Torque is \vec{\tau} = \vec{r}\times\vec{F}, and the cross product of two parallel vectors is zero. No torque means the angular momentum

This single fact does two enormous things. First, since \vec{L} never changes direction, both \vec{r} and \vec{v} must forever stay perpendicular to it: the motion is trapped in a single plane. A three-dimensional problem collapses to a flat, two-dimensional one before we have done any real work.

Second, the magnitude L controls how fast the body sweeps out area. In a small time dt the radius vector sweeps a thin triangle of area dA = \tfrac12\,r^2\,d\theta, and a short calculation gives

The rate at which area is swept never changes. This is exactly Kepler's second law: a planet sweeps out equal areas in equal times. It is not a special property of gravity's 1/r^2 shape — it holds for any central force — and it means a planet must race along when it is close to the Sun (a short, fat triangle) and dawdle when it is far away (a long, thin one). Speed is highest at perihelion (closest approach) and lowest at aphelion (farthest).

Second gift: the two-dimensional problem becomes one-dimensional

With the motion pinned to a plane, use polar coordinates (r,\theta) centred on the mass. The body's kinetic energy splits into a radial part and a sideways (tangential) part:

Here is the trick that unlocks everything. The angular momentum L = m r^2\dot\theta is a constant we already know, so we can use it to eliminate \dot\theta entirely: \dot\theta = L/(m r^2), and the tangential energy becomes \tfrac12 m r^2\dot\theta^2 = L^2/(2m r^2). Substituting,

Look at what has happened. The energy now has exactly the form of a particle moving in one dimension — the single coordinate r — inside a potential we call the effective potential:

It is a tug-of-war between two terms. The first, -GMm/r, is the real gravitational well: it pulls the body inward and plunges toward -\infty as r\to 0. The second, L^2/(2m r^2), is the centrifugal barrier: it is not a "real" force but the bookkeeping cost of the sideways motion, and because it grows like 1/r^2 it beats gravity's 1/r at small radius, throwing up a wall that (for any L\neq 0) stops the body from ever reaching the centre. The whole story of orbits is the story of this one curve.

Reading the orbit off the curve

Since the radial kinetic energy \tfrac12 m\dot r^2 can never be negative, the body is confined to the radii where U_{\text{eff}}(r) \le E. Draw the total energy as a flat horizontal line at height E; the body lives only where the curve dips below that line, and it turns around (\dot r = 0) exactly where the line crosses the curve. Drag the two sliders and watch the orbit change character.

Four regimes, all on one picture:

• At the bottom of the well — if E equals the minimum value of U_{\text{eff}}, the two crossing points merge into one. r can't change, so \dot r = 0 forever: this is a circular orbit. The minimum of the effective potential is the circular orbit.
• A dip below zero (E < 0) — the energy line cuts the curve at two radii, an inner r_{\min} (perihelion) and an outer r_{\max} (aphelion). The body oscillates between them: a bound elliptical orbit.
• Exactly zero (E = 0) — there is an inner turning point but no outer one; the body can just barely escape to infinity, arriving with zero speed. This is a parabolic trajectory, the knife-edge escape.
• Above zero (E > 0) — one turning point, and the body flies off to infinity with speed to spare: an unbound hyperbolic flyby, the path of an interstellar comet or a spacecraft slingshot.

The orbits are conic sections — Kepler's three laws

Solving the radial equation for the actual shape r(\theta) (a standard but longer calculation) gives a beautifully compact answer, r(\theta) = \dfrac{p}{1 + e\cos\theta}, which is precisely the polar equation of a conic section with eccentricity e. The energy sign we just read off the effective-potential picture maps straight onto the shape:

• e = 0 — circle (E at the well minimum);
• 0 < e < 1 — ellipse (E < 0, bound);
• e = 1 — parabola (E = 0);
• e > 1 — hyperbola (E > 0, unbound).

Worked examples

Example 1 — speed in a circular orbit. For a circular orbit gravity supplies exactly the centripetal force, \dfrac{GMm}{r^2} = \dfrac{m v^2}{r}. Cancel and solve:

A low Earth satellite orbits at r \approx 6.7\times 10^6\ \text{m} with GM_\oplus \approx 3.99\times 10^{14}\ \text{m}^3\text{s}^{-2}, giving v \approx \sqrt{5.95\times 10^{7}} \approx 7.7\ \text{km/s} — about 28{,}000\ \text{km/h}. Notice the mass m of the satellite has dropped out: a bolt and a bus orbit at the same speed.

Example 2 — Kepler's third law in friendly units. Measure time in years and distance in astronomical units (AU) for bodies orbiting the Sun. Then 4\pi^2/GM_\odot = 1 and the third law becomes the clean T^2 = a^3. Mars sits at a = 1.52\ \text{AU}, so

A Martian year really is about 687 days. Kepler nailed it with naked-eye data.

Example 3 — escape speed. To just barely escape (reach infinity with zero speed) the total energy must be exactly zero: \tfrac12 m v_{\text{esc}}^2 - \dfrac{GMm}{r} = 0. Solving,

Escape speed is always \sqrt2 \approx 1.41 times the circular-orbit speed at the same radius. From low Earth orbit that is about 11\ \text{km/s} — the figure every rocket to the planets must beat.