Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational potential — where it appears

The energy per unit mass required to bring a body from infinity to a point. Its gradient is the field, and expanding it in spherical harmonics is how a planet's shape and internal density distribution are read from the motion of things orbiting it.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is.

A star is held up by its own weight

A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it.

stars · Hydrostatic equilibrium Circular and escape speed. Orbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.

The speed that does not come back, and the √2 that separates it

Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving.

spaceflight · Escape The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

spaceflight · Escape A flattened shell that pulls nothing inside it. The strength of the gravitational field of a thin homoeoid — the shell between two similar concentric ellipsoids, here with axis ratio 0.5 — along its long axis (solid) and its short axis (dashed), in units of GM/a² against distance from the centre in units of the long semi-axis a, from 20,000 equal masses placed on it. The dotted curve is a point mass. Inside the shell the field is zero along both axes to within 0.0001 — the numerical residue of a finite sampling — however flattened the shell is: this is Newton's theorem for a homoeoid, the flattened generalisation of the spherical shell's empty interior. Outside, the shell does not pull like a point: at 1.2 a along the long axis the field is 1.003 against the point's 0.694, and at 0.60 a along the short axis, just outside the shell there, 0.901. The exterior field remembers the shape; the interior does not know there is one.

The shell can be flattened if it thins at the poles

A spherical shell exerts no force anywhere inside it, and a ring exerts a force that points outward. Between them lies a shape Newton found that keeps the sphere's empty interior however far it is flattened: the homoeoid, a shell bounded by two similar ellipsoids, thick at the equator and thin at the poles in exactly the right proportion. It is why the gravity inside any flattened body can be computed at all — and why the planes of stellar streams measure the shapes of dark haloes.

gravitation · Shell theorem

Named alongside it

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

Dark matter haloEscape velocityShell theoremBlack holeEddington limitEffective temperatureEllipsoidExtrapolationHigh-velocity starHomoeoidHydrostatic equilibriumIsothermal sphere

All concepts