Astronomy & Celestial Mechanics
the sky, computed
Generator

The lane-emden generator

Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface
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.

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.

1 essay calls lane-emden. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Stars

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.

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