Where a body can no longer be any shape it likes

At its defaults it draws where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

self-gravity-limit is one function in lib/figures/extremes.js — radiation, cross-sections and self-gravity. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does.

The gravitational, kinetic and total energies of a uniform self-gravitating sphere of 1 solar mass, against its radius, in units of the total energy it has at one solar radius. The kinetic energy is exactly minus half the gravitational one at every radius, which is the virial theorem, and the total is exactly minus the kinetic. So the curve that says how much energy the body has and the curve that says how hot it is are the same curve upside down: the mean temperature at one solar radius is 2.78 million K, and shrinking the body raises it.

The maximum height σ/ρg for four body sizes, at a crushing strength of 200 MPa and a density of 3000 kg/m³. The number falls as 1/R, because a larger body's own gravity is stronger at its surface in proportion to its radius. On the smallest, the limit exceeds the body — which is why small objects are shaped like anything at all, and large ones are shaped like spheres.

The energy a self-gravitating body has given up in reaching a given radius, divided by the rate it is radiating — the Kelvin–Helmholtz time — for 1 solar mass at 1 solar luminosity. At the Sun's radius it is 9.4 million years. Half of the gravitational energy released on the way down was radiated and the other half is the heat the body is now holding, which is the virial theorem doing the accounting. The Earth's rocks are 477 times older than this number, and that discrepancy is what a source of energy nobody knew about had to resolve.

How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all.

physicscheck asserts something about self-gravity-limit that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.