Astronomy & Celestial Mechanics
the sky, computed
Generator

The hydrostatic-balance generator

Inside a star that is holding itself up (polytrope n = 3)
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.

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.

7 essays call hydrostatic-balance. 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.

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. Stars

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.

 Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has. Stars

The part of a star that boils

Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite.

 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here. Stars

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

 Gas outweighs stars 6:1, and the baryons come to 15%. Above: enclosed mass against radius for a 7-keV cluster with a beta-model gas profile, β = 0.65 and a 250-kpc core. The total is from hydrostatic equilibrium — the same equation that holds up a star, with the mass following from the density and temperature gradients and from nothing else — and comes to 9.98·10¹⁴ solar masses inside 2 megaparsecs. The gas, integrated from the same profile, is 1.6·10¹⁴; the stars in all the galaxies are 2.5·10¹³, 6 times less. Most of a cluster's ordinary matter is not in anything anybody would call an object. The two total-mass curves differ by the 15 per cent hydrostatic bias: some of the pressure holding the gas up is turbulence left from the last merger rather than heat, and a mass computed from the thermal pressure alone is low by about that — the assumption that makes the measurement possible is the one that biases it. With the correction, the baryons come to 15 per cent of the total, against the 15.7 per cent the microwave background gives for the universe as a whole. A cluster is large enough to have kept everything it started with, so its own accounting is a cosmological measurement. Below: the same gas seen two ways. X-ray surface brightness falls as (1+z)⁻⁴, so a cluster at redshift one is 16 times fainter per unit sky than the same cluster nearby; the Sunyaev–Zel'dovich distortion of the microwave background does not fall at all, because it is a fraction of a background whose own brightness rises by exactly the same factor. That is why a millimetre survey finds clusters at any distance and an X-ray survey finds the near ones. Galaxies

The baryons that are not in the galaxies

The stars in a cluster are about a seventh of its ordinary matter. The rest is ten-million-kelvin gas that is invisible optically and dominant in X-rays — and weighing it with hydrostatic equilibrium biases the answer low by exactly the amount the assumption is wrong by.

 Central pressure bracketed without a model: 6 bodies, 23 decades apart. What can be said about the middle of a body from its mass and its radius alone. The lower end of each bar is GM²/8πR⁴, which follows from hydrostatic equilibrium and nothing else — no equation of state, no composition, no temperature, no assumption whatever about how the density is arranged inside. The upper end costs one more assumption, that the density does not increase outward, and it needs a central density, which is a model output rather than an observation and is why that edge is drawn as the softer one. The dot is what a full structural model gives. For the first five bodies every dot lies inside its bar, and what is worth noticing is how wide the bar is: Sun's rigorous floor is 4.48e+13 pascals against a modelled 2.34e+16, a factor of 522. The bound is true and nearly useless there, because most of a centrally condensed body's pressure comes from the concentration and the derivation deliberately knows nothing about it. The relativistic entry is the exception, and the reason to draw the figure at all. neutron star's modelled central pressure is 8.5 times the Newtonian ceiling — a body no Newtonian arrangement of matter with density falling outward can produce. The floor still holds, and holds for a statable reason: relativity makes the pressure gradient steeper than Newtonian gravity does, so the true central pressure can only exceed what the Newtonian derivation demands. The bracket therefore does more than constrain an interior. Applied at a small enough radius it breaks, and where it breaks is where Newtonian hydrostatics has stopped being the right equation. Stars

A floor under the centre that assumes nothing

There is a lower bound on the pressure at the centre of any body in hydrostatic equilibrium, and it needs no equation of state, no composition and no temperature — only a mass and a radius. For the Sun it is nearly useless. For a neutron star it says which theory of gravity the interior needs.

 The least a star can average, whatever it is made of. The lower bound on the mass-averaged temperature of a main-sequence star of ideal gas, μmₕGM/6kR, against its mass, both logarithmic, with the radius taken from the main-sequence mass–radius relation, for mean molecular weights of 0.5 and 0.62 — 0.5 is fully ionised hydrogen, the lightest possible gas and so the lowest floor; 0.62 is the solar mixture. For a star of the Sun's mass and radius the floor is 1.9 million kelvin whatever the star is made of. The dashed curve is the average a centrally condensed star actually has, for the index-three polytrope that approximates the Sun: three times the floor, 7.2 million kelvin for the solar mixture. The floor rises slowly with mass, because radius grows almost as fast. Nothing in it refers to nuclear reactions: mechanics alone requires a star to be hot enough inside that hydrogen is ionised throughout and fusion is within reach. Stars

Mechanics sets a star's temperature before fusion does

A ball of gas with the Sun's mass and radius must average at least two million kelvin inside, whatever it is made of and whatever powers it, because nothing else can hold it up. The bound comes from the virial theorem and one inequality about gravitational binding. Where it is met, a body can be a star; where it demands temperatures a body plainly does not have — the Earth, Jupiter, a white dwarf — it proves that something other than hot gas is holding that body up.

 The middle of a cluster cools faster than the universe ages. The time the gas of a cool-core cluster, of central electron density 0.05 per cubic centimetre, would take to radiate its heat at constant pressure, against radius, both logarithmic. Inside 76 kiloparsecs it is less than 7.7 billion years, the time since a redshift of about one when the cluster's core is thought to have settled, and at 10 kiloparsecs it is 0.8 billion years. Beyond a few hundred kiloparsecs it is longer than the age of the universe, and the gas is simply hot. Galaxies

A cooling flow that never arrives

The gas at the centre of a rich cluster radiates its heat in under a billion years, so it should be sinking inward and turning into stars at hundreds of solar masses a year. For two decades that was the accepted picture and it was treated as a measurement. Then X-ray spectra looked for the gas on its way down and found that it stops at a third of the temperature it began with, while the galaxy in the middle forms stars a hundred times too slowly. Something pays the radiative bill, and bubbles of the right size are visible in the X-ray images.

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