Astronomy & Celestial Mechanics
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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.

Assumes Hydrostatic equilibrium and Degeneracy.

17 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

A star is a great deal of gas that has not fallen in. The condition for that is one line — the pressure gradient at every radius has to carry the weight of what is above it — and by itself it settles nothing, because it is one equation in two unknowns. Pressure and density are both free, and the balance holds for infinitely many pairs of them.

Supply a second relation between those two, and the underdetermination vanishes completely. Not approximately, and not in the sense of narrowing a range: the structure of the star becomes the solution of a single ordinary differential equation, with no freedom left in it except two scale factors that fix how big the star is and how dense.

The relation usually supplied is a power law,

P=Kρ1+1/n,P = K\rho^{1 + 1/n},

and the number nn is called the polytropic index. It is not a fitted parameter. A non-relativistic degenerate gas has n=1.5n = 1.5 exactly; a fully relativistic one has n=3n = 3; a star whose pressure is a fixed fraction gas and the rest radiation has n=3n = 3 as well; a convective envelope stirred to a uniform entropy has n=1.5n = 1.5; incompressible matter has n=0n = 0. Each of those is a statement of physics, and each of them, put into hydrostatic equilibrium, produces a complete star.

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.
Fig. 1 The Lane–Emden equation solved for six indices. The vertical axis is θ, whose n-th power is the density in units of the central density, and the horizontal axis is radius in units of a length that depends on the central density and on K. Every curve leaves the centre flat, because the density is greatest there and has no cusp. What separates them is where they end: n = 0 reaches zero at ξ₁ = 2.449 and n = 3 at 6.897, and n = 5 never reaches it at all. The three curves with closed forms — n = 0, 1 and 5 — are drawn from the same numerical integration as the others and agree with those forms to better than two parts in a million.

One equation, and what it costs to write it down

Substituting the power law into hydrostatic equilibrium and eliminating the pressure gives

1ξ2ddξ(ξ2dθdξ)=−θn,\frac{1}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}\frac{d\theta}{d\xi}\right) = -\theta^{n},

with ρ=ρcθn\rho = \rho_c\theta^{n} and r=aξr = a\xi. Both scale factors have been divided out. The equation carries no mass, no radius, no temperature and no composition — only the index. Two stars of the same index and utterly different sizes have the same curve on the figure above, and differ only in what the axes are worth.

That is the trade the polytrope makes, and it is worth being explicit about which half is bought and which is given up. What is bought is that the whole interior comes out of an integration a person can do by hand in an afternoon. What is given up is any account of where the energy comes from: nothing here mentions nuclear reactions, and nothing here mentions the flow of heat outward. A polytrope is a star in mechanical equilibrium and nothing more.

There is a second reading of the same fact, and it is the one that makes the family worth learning. Because the equation contains only nn, the shape of a star is a one-parameter object. Everything a structure can be — how sharply the mass is piled toward the middle, how much of the volume is nearly empty, how the pressure falls away — has been reduced to a single dial. That is a very strong claim about a class of objects that includes red dwarfs, white dwarfs, the convective envelopes of giants and the radiative interiors of the Sun, and its strength is exactly what makes the cases where it fails informative.

The boundary conditions at the centre are θ=1\theta = 1 and θ′=0\theta' = 0, and the second of those is the one that does the work: a nonzero slope at the origin would mean a density cusp, which would mean an infinite force. The origin is a removable singularity, which is why the integration starts a little way out from it on a series expansion rather than at zero.

What comes out, for one index

Solving the equation once gives everything at once. The density is θn\theta^{n}, the pressure is θn+1\theta^{n+1}, and if the gas is ideal the temperature is θ\theta — three profiles from one curve, differing only in an exponent. The mass inside a radius is −ξ2θ′-\xi^{2}\theta', read straight off the same solution rather than integrated afterwards.

The inside of an n = 3 polytrope: density, pressure, temperature and enclosed mass. One polytrope opened out. The horizontal axis is fractional radius and the four curves are everything a structure is: density over central density, pressure over central pressure, temperature over central temperature — all three of them powers of the same θ — and the mass enclosed within each radius. For n = 3 the centre is 54.18 times denser than the average, and half the star's mass lies inside 28.3 per cent of its radius, which is the sense in which a star of this index is centrally concentrated. The pressure falls faster than the density and the temperature more slowly, because P ∝ θ^4 and ρ ∝ θ^3 while T ∝ θ, and those three exponents are the whole content of the equation of state. The enclosed-mass curve is not an integral performed afterwards: it is ξ²θ′ read straight off the same solution, which is why it reaches one exactly at the surface. What the figure does not show is any actual quantity — no grams, no kelvin, no metres. Two scale factors were divided out to get here, and putting them back requires two numbers this equation cannot supply.
Fig. 2 The n = 3 polytrope opened out. Density over central density, pressure over central pressure, temperature over central temperature and enclosed mass over total mass, all against fractional radius. The centre is 54.2 times denser than the average and half the star’s mass lies inside 25 per cent of its radius, which is the sense in which a star of this index is centrally concentrated. The pressure curve falls faster than the density and the temperature more slowly, because the three exponents are n + 1, n and 1 — and those three exponents are the entire content of the equation of state.

The most useful single number to come out of it is the central condensation, the ratio of the central density to the mean. For n=0n = 0 it is 1 by construction. For n=1.5n = 1.5 it is 5.99, for n=3n = 3 it is 54.2, and for n=4n = 4 it is 622. That number decides a great deal downstream: how sharply the nuclear burning is concentrated, how much of the star the convection can reach, and — as the next rung of this ladder turns on — how stiffly the body resists being deformed from outside.

The inside of an n = 1.5 polytrope: density, pressure, temperature and enclosed mass. One polytrope opened out. The horizontal axis is fractional radius and the four curves are everything a structure is: density over central density, pressure over central pressure, temperature over central temperature — all three of them powers of the same θ — and the mass enclosed within each radius. For n = 1.5 the centre is 5.99 times denser than the average, and half the star's mass lies inside 52.1 per cent of its radius, which is the sense in which a star of this index is centrally concentrated. The pressure falls faster than the density and the temperature more slowly, because P ∝ θ^2.5 and ρ ∝ θ^1.5 while T ∝ θ, and those three exponents are the whole content of the equation of state. The enclosed-mass curve is not an integral performed afterwards: it is ξ²θ′ read straight off the same solution, which is why it reaches one exactly at the surface. What the figure does not show is any actual quantity — no grams, no kelvin, no metres. Two scale factors were divided out to get here, and putting them back requires two numbers this equation cannot supply.
Fig. 3 The same figure at n = 1.5, the index of a non-relativistic degenerate gas and of a fully convective star. The centre is only 5.99 times denser than the mean and half the mass sits inside 43 per cent of the radius: a far more even body than the one above. The two figures are the same integration at two values of one number, which is the whole point of drawing them together — everything that distinguishes a white dwarf’s interior from a main-sequence star’s, at this level of description, is the exponent in the pressure law.

The same structure can be drawn against the physical quantities it stands for rather than against the dimensionless ones, which is what the older figure in this collection does.

The inside of an n = 4.5 polytrope: density, pressure, temperature and enclosed mass. One polytrope opened out. The horizontal axis is fractional radius and the four curves are everything a structure is: density over central density, pressure over central pressure, temperature over central temperature — all three of them powers of the same θ — and the mass enclosed within each radius. For n = 4.5 the centre is 6189.47 times denser than the average, and half the star's mass lies inside 6.7 per cent of its radius, which is the sense in which a star of this index is centrally concentrated. The pressure falls faster than the density and the temperature more slowly, because P ∝ θ^5.5 and ρ ∝ θ^4.5 while T ∝ θ, and those three exponents are the whole content of the equation of state. The enclosed-mass curve is not an integral performed afterwards: it is ξ²θ′ read straight off the same solution, which is why it reaches one exactly at the surface. What the figure does not show is any actual quantity — no grams, no kelvin, no metres. Two scale factors were divided out to get here, and putting them back requires two numbers this equation cannot supply.
Fig. 4 A stiffer polytrope than the standard model, and the direction the family is heading. At n=4.5n = 4.5 the star is far more centrally concentrated than at n=3n = 3: the density falls by a factor of 88 across the inner tenth of the radius, and the outer half of the star holds almost none of the mass. Push the index to 5 and the radius becomes infinite — the solution never reaches zero density, so there is no surface and no star. The whole family lives between that limit and n=0n = 0, the incompressible sphere at the other end, and every real star’s structure is a statement about where between them its equation of state puts it.

Where the radius goes

Two scale factors were divided out to get the dimensionless equation, and putting them back is where the physics returns. If KK is fixed by the material rather than by the star — which is what it means for the pressure law to be an equation of state rather than a fitting formula — then the mass and the radius are no longer independent. Eliminating the central density between them leaves

R∝M(1−n)/(3−n),R \propto M^{(1-n)/(3-n)},

and the exponent is the whole story.

Radius against mass for each polytropic index, and the vertical line at n = 3. What survives when both of a polytrope's scale factors are eliminated. Requiring P = Kρ^(1+1/n) with K fixed by the physics rather than by the star gives R ∝ M^((1−n)/(3−n)), and the exponent is drawn here for each index on logarithmic axes through the Sun. The three regimes are qualitatively different rather than quantitatively so. Below n = 1 the star grows with mass, as ordinary intuition expects. At n = 1 the exponent is exactly zero and every mass has the same radius. Above it the exponent turns negative — at n = 1.5, the index of a non-relativistic degenerate gas, R ∝ M^(−1/3), and a heavier white dwarf is a smaller one. And at n = 3 the denominator vanishes: the radius drops out of the relation altogether and what is left is a statement about the mass alone, drawn here as the vertical line. That is not a curiosity of the algebra. An index of 3 is what a fully relativistic degenerate gas has and what a radiation-dominated star has, and the vertical line is the reason a limiting mass exists at all — a mass that is the same whatever size the object is. The figure cannot show which index a real star has, and no real star has one index throughout.
Fig. 5 Radius against mass for each index, on logarithmic axes through the Sun, so that each relation is a line whose slope is its exponent. Below n = 1 a heavier star is a larger one. At n = 1 the exponent is exactly zero and every mass has the same radius. At n = 1.5 it is −1/3 and a heavier body is a smaller one, which is the mass–radius relation of a white dwarf. And at n = 3 the denominator vanishes: the radius drops out of the relation entirely, leaving the vertical line — a statement about the mass alone, valid at any size.

Three regimes, and they are different in kind rather than in degree. The everyday one is on the left: pile on more material and the object gets bigger. At n=1n = 1 the growth stops, and the radius becomes a property of the material rather than of the object. Past that the sign flips, and this is where a degenerate gas lives: more mass makes a smaller star, because the extra weight compresses the existing material faster than the new material adds volume.

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.
Fig. 6 The real relation for a cold star, computed from the full degenerate equation of state rather than from a single index. The low-mass end has slope −1/3, which is the n = 1.5 polytrope’s answer, and the curve peels away from it as the electrons become relativistic and the effective index climbs toward 3. The radius then falls to zero at a finite mass. The polytrope’s two limiting indices are the two ends of this one curve, and the interesting part is the transition between them, which no single index describes.

At n=3n = 3 the exponent is infinite, and the honest way to read that is not that the radius is undefined but that it has left the equation. What remains is M=M = constant: a mass fixed by KK and by the constants of nature, the same however large or small the object is. That is not an accident of the algebra. It is the reason a limiting mass exists at all, and it is why the Chandrasekhar mass can be written down without ever solving for a radius.

The same index from a different direction

Eddington reached n=3n = 3 by a route with no degeneracy in it whatever. Suppose the pressure in a star is partly gas and partly radiation, and suppose the fraction β\beta carried by the gas is the same at every radius. That supposition is not obviously true, and Eddington’s justification for it was that it makes the problem tractable — but it turns out to force the structure to be an n=3n = 3 polytrope automatically, and to leave one algebraic relation between β\beta and the star’s mass.

Eddington's standard model: the pressure split between gas and radiation, 1 to 150 M☉. The same index reached from the other end. Requiring that the ratio of gas pressure to total pressure be the same everywhere in a star makes the structure an n = 3 polytrope automatically, and it leaves one algebraic relation between that ratio β and the star's mass: (1 − β)/β⁴ = 3.02×10⁻³ (M/M☉)² μ⁴, drawn here for a mean molecular weight of 0.61. The two curves are β and 1 − β against mass on a logarithmic axis, and the crossing is the mass at which radiation carries half the pressure. At one solar mass the gas carries 99.958 per cent of it and radiation is a rounding error; at 150 solar masses the gas is down to 48.4 per cent. The relation is quartic in β and quadratic in the mass, which is why the transition is gradual over two decades rather than sharp: nothing switches on. What the curve is really saying is that a star cannot be made arbitrarily massive and stay a star — as β falls the structure approaches the n = 3 case whose mass is fixed and whose radius has dropped out, and the effective adiabatic index approaches 4/3, where a star is neutrally stable against its own pulsation. The figure shows the pressure split and not the instability; the instability is what the split implies.
Fig. 7 Eddington’s quartic, (1 − β)/β⁴ = 3.02×10⁻³ (M/M☉)²μ⁴, drawn for a mean molecular weight of 0.61. The two curves are the fraction of the central pressure carried by gas and by radiation, and they cross where radiation carries half. At one solar mass the gas carries 99.979 per cent and radiation is a rounding error; at 150 solar masses the gas is down to 48.4. The transition is gradual over two decades because the relation is quartic in β and quadratic in the mass, so nothing switches on — the star becomes a radiation-supported object by degrees.

The relation deserves to be read slowly, because it says something a modern reader may take for granted and Eddington did not. The mass enters squared. The molecular weight enters to the fourth power. And β\beta enters to the fourth power on the other side, so a small departure from β=1\beta = 1 corresponds to a large change in mass. The consequence is that the amount of radiation pressure in a star is not a free parameter to be adjusted: it is fixed by the mass, and above a few tens of solar masses it is not small.

This is where the standard model connects to the brightness that would blow a star apart. As β\beta falls the effective adiabatic index of the mixture falls toward 4/34/3, which is exactly the value at which a self-gravitating body becomes neutrally stable against its own pulsation — neither restoring nor collapsing. A star made too massive is not merely bright; it is structurally on the edge, and the same index that produced a limiting mass for a cold star produces a stability limit for a hot one.

Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end.
Fig. 8 What the family is used for once its limitations are admitted. The main sequence’s luminosity–mass relation can be derived by homology over a polytropic family — assume the structure is self-similar, put in radiative transport and an ideal gas, and the exponent comes out near 3.5 without solving anything. The prefactor is wrong by tens of per cent and the power is right, which is the general pattern for what a polytrope is good for.

What the model refuses to say

It is worth listing what has been assumed away, because polytropes are so obliging that it is easy to forget how much they do not know.

There is no energy generation. Nothing in the Lane–Emden equation says that the centre is hot enough to burn, and a polytrope will sit there indefinitely with no source at all. The temperature profile is a consequence of the pressure profile, not a cause of it.

There is no transport. A real star’s structure is set jointly by hydrostatic balance and by the requirement that the heat produced can actually get out, and it is the second condition that decides whether a region is convective or radiative and how transparent the material is. A single index cannot represent a star with a convective core and a radiative envelope, because those are two different indices.

And there is no surface. A polytrope’s density and temperature both go to zero at ξ1\xi_1, which is a fair description of nothing: a real photosphere is where the optical depth falls to about two thirds, at a finite density and a temperature of thousands of kelvin. The outer few per cent of the radius is exactly where the approximation is worst, and unfortunately it is also all anyone can see.

The index that is not a number

The whole construction assumes that one exponent describes the material everywhere, and there is a class of phenomena that exists precisely because that is false.

The quantity that matters dynamically is not the exponent in the pressure–density relation of a static configuration but the adiabatic exponent — how the pressure responds when a parcel of gas is compressed quickly, with no time to exchange heat. For a fully ionised or fully neutral ideal monatomic gas that exponent is 5/3. In a region where the gas is partly ionised it is not, and it can fall well below 4/3.

The reason is that compression there does not go entirely into raising the temperature. Some of the energy goes into ionising more of the gas, which is a store that absorbs work without producing pressure. The material becomes soft, locally, in a shell.

That softness is not a curiosity. A star with such a shell at the right depth is unstable to radial pulsation: the shell absorbs heat on compression and releases it on expansion, which is a heat engine driving the oscillation rather than damping it. That is why Cepheids and RR Lyrae stars pulsate, why they do so only within a narrow strip of the temperature–luminosity plane, and why the strip has sharp edges — it is where the partial-ionisation zone of helium sits at the depth that makes the engine work.

None of this can be represented by a polytrope, because a polytrope has one exponent by construction and this is an effect of the exponent varying with radius. What the polytrope supplies is the background the shell sits in, and the calculation that decides whether a star pulsates is a perturbation about a structure the polytrope approximates.

A single index is a statement that the material is the same everywhere, and the places where stars do their most conspicuous things are the places where it is not.

Where the family came from

The equation carries two names and neither belonged to an astrophysicist in the modern sense.

Jonathan Homer Lane was an American patent examiner who published in 1870 what is generally taken to be the first attempt to compute the interior of the Sun. His question was not what the Sun is made of — nobody knew — but whether a gaseous body held up by its own pressure could have a surface temperature like the Sun’s and an interior consistent with the gas laws. He wrote down the balance, assumed the relation between pressure and density, and integrated numerically by hand.

Robert Emden’s Gaskugeln of 1907 systematised it into the family described here, computed the solutions for a range of indices, and tabulated them. Those tables were how stellar structure was done for the next fifty years: an astronomer wanting a model looked up an index, took the tabulated dimensionless solution, and multiplied by two scale factors.

Chandrasekhar’s Introduction to the Study of Stellar Structure of 1939 is the book that made the family a tool rather than a subject, and it is also where the mass limit is derived — from the n = 3 solution, by exactly the route sketched above.

What is worth taking from that history is how much was extracted before anything was known about the energy source. Lane and Emden had no nuclear physics, and the mechanical part of a star’s structure turned out not to need any: hydrostatic balance plus an equation of state is enough to fix the run of density, and the burning arranges itself to fit. The structure came first and the physics of the source came sixty years later, which is the reverse of the order the subject is usually taught in.

Why the family is still used

A model with no nuclear physics and no radiative transfer might be expected to have been retired once computers could integrate the real equations, and it has not been. Four reasons.

The first is that the limiting cases are exact. A fully convective star really is an n=1.5n = 1.5 polytrope, to the accuracy with which the mixing-length description of convection holds, because a convective region is stirred to uniform entropy and uniform entropy for an ideal monatomic gas is exactly P∝ρ5/3P \propto \rho^{5/3}. The same is true of a non-relativistic degenerate gas, for a completely different reason that produces the same exponent.

The second is that polytropes give the right dependences even where they give the wrong numbers. The scalings that make mass the dominant variable in a star’s life and set its lifetime can all be derived from homology arguments over a polytropic family, and they come out with the correct exponents. The prefactors are wrong by tens of per cent; the powers are right.

The third is that some of the family’s members are not stars at all and behave the same way. An n=5n = 5 polytrope has infinite radius and finite mass, which sounds like a pathology and is in fact the Plummer sphere, the standard analytic model of a star cluster. Above n=5n = 5 the mass diverges too, and there is no self-gravitating configuration of that index — a fact about a differential equation that turns into a fact about which objects the universe is allowed to contain.

The fourth is that a polytrope is what a numerical model is checked against. An integration of the full structure equations is a program with several thousand lines in it, and the only way to know whether it works is to feed it a problem whose answer is known independently. Setting the opacity and the energy generation to their polytropic equivalents and demanding that the code reproduce ξ1=6.89685\xi_1 = 6.89685 is the standard test, and it has caught a great many errors.

Where the ladder goes

The polytrope is a one-parameter family, and the next rungs of this ladder are all about what happens when the single parameter is not enough. The star that swells because its centre shrank has two indices at once, an inner one and an outer one, and the interesting physics is at the boundary between them. The tidal response of a body depends on its central condensation and therefore on its index, which is how a number measured from outside becomes a statement about the run of density within.

And the whole edifice rests on a relation that a laboratory can, in some cases, be asked about directly. KK for a degenerate electron gas is a combination of fundamental constants and nothing else. KK for the mixture in a real stellar interior is not, and the disagreement between the two ways of knowing the Sun that this collection takes up under opacity is at bottom a disagreement about how the pressure and the density are related in a place nobody can reach.

Where this goes next

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. 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. From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at. The mass a cold star cannot exceed A degenerate star's radius falls as its mass rises, and nothing in that relation suggests a limit. Making the electrons relativistic softens the pressure law until pressure and gravity scale the same way with radius — and the radius drops out of the balance, leaving one mass and no room to argue. Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end. Mass decides everything, by a power of three and a half Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

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Central condensationChandrasekhar limitDegeneracy pressureEddington limitEquation of stateHydrostatic equilibriumLane emden equationMass radius relationPolytropeRadiation pressureStellar density