Astronomy & Celestial Mechanics
the sky, computed
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Polytropes — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface. The dimensionless density θ against the dimensionless radius ξ, for polytropic indices 0, 1, 1.5, 3, 4.5, 5. Each curve is the whole structure of a star whose pressure is K times its density to the power 1 + 1/n: the equation of state and hydrostatic equilibrium leave one second-order differential equation, and this is its solution. Every curve starts at θ = 1 with zero slope, because the density is greatest at the centre and has no cusp there. What separates them is where they end. At n = 0 the density is uniform and the surface is at ξ₁ = 2.4495; by n = 3 it has moved out to 6.8968 and the central density is 54.2 times the mean. At n = 5 the curve reaches zero only at infinity — a configuration of infinite radius and, remarkably, finite mass — and every index above it has neither. The three curves that have closed forms, n = 0, 1 and 5, are drawn from the same numerical integration as the rest and agree with those forms to better than two parts in a million, which is what licenses reading the others off the picture. What the figure cannot show is the scale: ξ is radius divided by a length that depends on the central density and on K, so two stars of the same index and wildly different sizes have the same curve here.

    An equation of state is already a star

    Write down how a gas's pressure depends on its density, insist that the pressure hold the weight up, and everything else follows — the run of density, the fraction of the mass inside each radius, and, at one particular index, a mass that does not care what the radius is.

    part 1 · stars
  2. The most pressure a core can supply, and what its envelope asks for. The pressure at the boundary of a star's isothermal core, on a logarithmic scale, against the fraction of the star's mass the core holds. The coloured curves are the most pressure a core of that mass can supply at its edge — the maximum of the isothermal sphere, 1.40 c⁸/(G³M²) for a core of mass M — for cores of mean molecular weight 0.6, 1.0, 4/3; the flat grey line is the pressure the envelope, of molecular weight 0.6, exerts on the core at the same temperature. The core's ceiling falls as the square of its mass, and it can hold the envelope up only while it lies above the line. The curves cross it at core fractions of 0.370, 0.133, 0.075: a core of the same molecular weight as its envelope could hold 37 per cent of the star, and a fully ionised helium core, μ = 4/3, under this envelope only 7.5 per cent. The limit exists because the core is heavier per particle than the gas it has to hold up.

    A core too heavy for the gas around it

    When a star's centre runs out of hydrogen, the helium left behind stops making energy, settles to one temperature, and becomes the one member of the polytrope family with no surface of its own. It can hold up the envelope around it only while it is less than about a tenth of the star. Past that there is no balance to be had, and the star crosses to the giant branch in a few per cent of its lifetime — which is why that part of the diagram is empty.

    part 2 · stars
  3. Where relativity overtakes the margin: 306 Schwarzschild radii for 10⁵ solar masses. Two things against the size of a contracting star, measured in its own Schwarzschild radius, on logarithmic axes. The flat lines are the margin by which the star's pressure-weighted adiabatic exponent exceeds 4/3 — the Newtonian condition for stability — for stars of 1000, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸ solar masses with mean molecular weight 0.61; it does not change as the star shrinks, because it is fixed by the gas fraction of the pressure, and it falls with mass from 0.036 to 1.2·10⁻⁴. The diagonal is what general relativity adds to the requirement, 1.1245 × 2GM/(Rc²), which in these units is the same line for every mass. Contraction moves each star leftward along its flat line, and where the line meets the diagonal the star becomes dynamically unstable: at 31, 97, 306, 966, 3,052, 9,649 Schwarzschild radii respectively. Every one of those crossings lies between 31 and 9,649 Schwarzschild radii out, where the star is not remotely compact — the relativistic correction is tiny, and it decides the matter only because the margin it is compared with is tinier.

    A star tipped over at a thousand times its horizon

    A star of a million suns is held up almost entirely by its own light, which leaves it stable by about a thousandth. General relativity raises the bar by an amount that is tiny until the star is compact, and a thousandth is tiny too — so the star collapses at nearly a thousand times the radius of the black hole it becomes, at a mean density below that of air, and with no fuel yet burning.

    part 3 · stars
  4. The centre goes soft at 19 solar masses; the whole core only at 27. Two criteria for the pair instability, against the mass of the oxygen core, each the deepest value reached as the core contracts through the pair-producing range. The orange curve is local: the lowest Γ₁ − 4/3 at the core's centre. It goes negative at 19.1 solar masses — the centre of every such core passes through gas that, taken by itself, cannot hold itself up. The blue curve is the criterion that actually decides the matter, the pressure-weighted mean over the whole core, and it stays positive until 27.5 solar masses of oxygen. In the shaded band between the two the centre goes soft and the core does not go unstable, because the stiffer gas around the centre outvotes it. The threshold is computed for a core built as an n = 3 polytrope with the pairs ignored in its structure; detailed evolutionary models find the first pair-instability pulses in stars whose helium cores exceed roughly thirty solar masses, so the construction lands in the right neighbourhood rather than on a precise number.

    A soft centre the rest of the core outvotes

    Above a billion kelvin, light in a star's core starts turning into electron–positron pairs, and gas that spends its heat on making mass cannot push back hard enough to hold itself up. But stability belongs to the whole core, not to its centre. An oxygen core goes soft in the middle from about nineteen solar masses and holds; it gives way only near twenty-eight, when the soft gas carries two-thirds of the pressure-weighted volume.

    part 4 · stars

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