Illustrated Physics
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Series

Self-gravity — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    The size at which a body becomes round

    A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

    part 1 · astrophysics
  2. The line that slopes the wrong way. 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 ball of gas that heats up as it cools

    Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

    part 2 · astrophysics
  3. Cold matter, and the mass above which nothing holds it up. The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it.

    The mass no cold matter can hold up

    A white dwarf gets smaller as it gets heavier, which no ordinary object does. Follow that curve upward and the radius reaches zero at 1.46 solar masses — because once the electrons are relativistic the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball does not depend on its radius at all.

    part 3 · astrophysics
  4. Which wavelengths grow instead of oscillating. The dispersion relation of a self-gravitating isothermal gas, ω² = c²k² − 4πGρ, with each axis measured in the scale the gas sets for itself. Short wavelengths oscillate: pressure wins, and the disturbance is a sound wave. Long wavelengths do not: ω² is negative, so the disturbance grows exponentially instead of travelling, and the region collapses. The changeover is at λ_J = c√(π/Gρ), and it happens because pressure support acts on a sound-crossing time that grows with the region while gravity's collapse time does not depend on the size at all. Four densities spanning 6 decades are drawn and they lie exactly on top of one another, to 6e-16, because the criterion has no scale of its own: it is the same curve for a diffuse cloud and for a protostellar core, with different numbers written on the axes. In this gas at 10 K those numbers are 2.12 pc and 28.72 solar masses at 10² cm⁻³, 0.21 pc and 2.87 solar masses at 10⁴ cm⁻³, 4372 AU and 0.29 solar masses at 10⁶ cm⁻³, 437 AU and 0.03 solar masses at 10⁸ cm⁻³.

    The disturbance that grows instead of travelling

    A sound wave in a gas oscillates because pressure restores what the disturbance displaced. Add the gravity the gas exerts on itself and the restoring force acquires a competitor that does not weaken with size — so above one wavelength the sum changes sign, the frequency becomes imaginary, and the disturbance stops travelling and starts growing. It is the same wave equation with one term subtracted.

    part 4 · astrophysics
  5. The distance that does not know how big the moon is. 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.

    The distance that forgets the moon

    A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

    part 5 · astrophysics
  6. The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.

    The moon kept molten by its neighbours

    Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

    part 6 · astrophysics
  7. The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up.

    The pressure that weighs down what it holds up

    In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

    part 7 · astrophysics
  8. Two ways to spin, against angular momentum. The spin Ω²/πGρ of the Maclaurin spheroids (blue) and the Jacobi ellipsoids (red) against their angular momentum, in units of √(GM³R) with R the radius of a sphere of the same volume. The two families meet at the bifurcation, angular momentum 0.304. Below it only spheroids exist. Above it a body of given angular momentum could be either, and the two differ: the ellipsoid spins more slowly, its spin falling as its angular momentum grows, because it gets longer and its moment of inertia climbs faster than its angular momentum. Every Maclaurin spheroid past the bifurcation is a shape a slightly viscous body will leave.

    The spinning fluid that becomes a cigar

    A ball of fluid held together by its own gravity flattens when it spins, as the Earth bulges at the equator. Spin it faster and it flattens further, into a lens — but only up to a point, and long before that point something unexpected happens. In 1834 Jacobi found that a fast-spinning fluid can also take a shape that is not symmetric about its axis at all: an ellipsoid with three different axes, stretched along its equator, tumbling end over end. Past a definite spin the stretched shape has less energy than the flattened one, and a body that can shed energy will choose it.

    part 8 · astrophysics

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