Illustrated Physics
physics, drawn
Concept

Kepler orbit — where it appears

The path of one body round another under an inverse-square attraction: an ellipse, parabola or hyperbola with the attracting body at a focus. Its energy, angular momentum and Runge–Lenz vector are all conserved, so a bound orbit closes on itself and keeps a fixed axis.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Throws that grow into orbits. The longest throws possible from one point on the Earth, at launch speeds of 3, 5, 6.5, 7.5 km/s, ignoring air, each at its best angle, drawn to scale round the Earth. 3 km/s: launched at 42.8° above the horizontal, landing 1.0 thousand km away; 5 km/s: launched at 37.8° above the horizontal, landing 3.2 thousand km away; 6.5 km/s: launched at 29.7° above the horizontal, landing 6.8 thousand km away; 7.5 km/s: launched at 17.6° above the horizontal, landing 12.2 thousand km away. The faster the throw, the flatter the best launch and the further round the Earth it lands; at the circular-orbit speed, 7.91 km/s, the best launch is horizontal and the throw never comes down.

The parabola that is the top of an ellipse

Every textbook throw is a parabola, the best angle is 45°, and the range goes as the square of the speed. All three are statements about a flat Earth with gravity pointing the same way everywhere. On the real one a throw is a piece of an orbit — an ellipse with the Earth's centre at one focus — and the parabola is what the top of that ellipse looks like when the Earth is large. Throw faster and the best angle falls, the range runs ahead of the formula, and at eight kilometres a second the best throw is horizontal and never lands.

mechanics · Projectile An orbit, and the circle its velocities make. Left: an orbit of eccentricity 0.6 round a central mass (dot), integrated step by step, with the velocity drawn as an arrow at twelve points equally spaced in time. Right: the same twelve velocities drawn from a single point, the origin of velocity. Their tips lie on a circle — the hodograph — of radius GM/h, whose centre is displaced from the origin by e times that radius, here 0.6: the integrated velocities stay on it to within one part in ten thousand. The fastest velocity, at the closest approach, is 1.60 times GM/h and the slowest, at the furthest point, 0.40 times. The tips are bunched near the slow end, because the body spends most of its time far away.

The velocity that always lies on a circle

A planet on an elliptical orbit speeds up and slows down, its velocity swinging round as it goes. Draw all its velocities from a single point and their tips fall exactly on a circle — not an ellipse, a circle — offset from the point by the orbit's eccentricity. Hamilton noticed this in 1846; it holds for every orbit under an inverse-square force and for no other force; and it turns the ellipse into something that can be proved with a ruler and compass, as Feynman did in a lecture that was lost for thirty years.

astrophysics · Orbit stability

Named alongside it

The objects these essays reach for when they reach for this one.

Angular momentumApproximationEccentricityEllipseGravityHodographHohmann transferThe inverse-square lawOrbital speedParabolaProjectile motionRunge lenz vector

All concepts