An exponent that is a slope, not a law

The first rung of this anchor established the relation and the consequence: double the mass and the output multiplies by eleven, which is why a modest range of masses produces a colossal range of stars. Eleven is two to the three and a half.

Three and a half is not a law. It is the average slope of a curve that has three straight pieces and two corners, and the corners are where the physics changes hands, at 0.43 and 2 solar masses.

That distinction matters in a specific way rather than as a general caution. An exponent is used to extrapolate — to infer a luminosity from a mass or a mass from a luminosity, over a range wider than the one it was fitted in — and an exponent that is really three exponents makes an extrapolation wrong in a direction that depends on which end it started from. Using 3.5 to convert the luminosity of a red dwarf into a mass overestimates the mass by a third; using it at the top of the main sequence underestimates it.

What homology predicts #

The prediction comes from a scaling argument that assumes stars of different masses are the same star at different sizes — homologous, in the sense that the run of pressure, temperature and density is the same function of the fractional radius with different scale factors.

Under that assumption, radiative energy transport gives

$$L \;\propto\; \frac{\mu^4 M^3}{\bar\kappa}$$

with μ the mean molecular weight and κ̄ a suitably averaged opacity. Everything about the exponent is in that last factor.

If the opacity is a constant, which it is when electron scattering dominates, the exponent is 3 exactly. Electron scattering — the same process whose opacity sets the ceiling on how bright a star can be — has no frequency dependence, no temperature dependence and no density dependence — a free electron scatters a photon and that is all — so κ is a pure number, 0.2(1 + X) square centimetres per gram, and the mass dependence is untouched.

If the opacity is Kramers’, which is bound-free and free-free absorption by partly ionised heavy elements, then $\kappa \propto \rho T^{-7/2}$. Working the scalings through — the central temperature goes as M/R and the density as M/R³ — the exponent comes out at 5.5.

If radiation pressure holds the star up, the luminosity is pinned near the Eddington value, which is proportional to the mass, and the exponent is 1. That regime is not reached by ordinary stars, but it is approached, and the approach is what bends the top of the relation.

Three regimes, three numbers, and no fitting anywhere in the derivation.

It is worth seeing where the homology exponent comes from, because it is three lines and it explains why the opacity is the only thing that moves it.

Hydrostatic equilibrium gives a central pressure of order $GM^2/R^4$, and the ideal gas law converts that into a central temperature of order $\mu GM/R$. Radiative diffusion carries a flux proportional to the temperature gradient over the opacity times the density, which for a star of mass $M$ and radius $R$ gives

$$L \;\sim\; \frac{R\,T_c^4}{\bar\kappa \rho} \;\sim\; \frac{\mu^4 G^4 M^3}{\bar\kappa}$$

and the radius has cancelled. That cancellation is the reason the relation is a relation: the luminosity of a radiative star depends on its mass and its opacity and not on how big it happens to be, which is why a star can swell into a giant without its luminosity collapsing.

Why the measured steps are not the predicted ones #

The gap between 4 and 5.5 in the middle of the main sequence, and between 3.5 and 3 above it, is not an error in either the fit or the derivation. It is what homology leaves out, and there are three things.

Convection. A star below about 0.35 solar masses is convective from centre to surface; one around the Sun’s mass has a convective envelope over a radiative interior; one above about 1.3 has a convective core and a radiative envelope. The homology argument assumes radiative transport everywhere. Where convection carries the flux the temperature gradient is set by the adiabat rather than by the opacity, and the opacity’s exponent stops mattering.

The composition gradient. The mean molecular weight enters to the fourth power and is not the same in a star’s core as at its surface, and it changes as the star burns. A star is homologous to another star only at the same evolutionary stage, and a fit to a sample of real binaries averages over whatever stages they happen to be at.

The energy source. Below about 1.3 solar masses hydrogen burns through the proton–proton chain, whose rate goes as roughly the fourth power of temperature; above it the CNO cycle takes over, at something like the eighteenth power. That changes how concentrated the energy generation is, which changes the structure, which feeds back into the relation. It is also the origin of the convective core: a power of eighteen concentrates the burning into a small central region, the flux there exceeds what radiation can carry, and convection starts.

The three omissions do not push the same way. Convection flattens the relation, the composition gradient steepens it, and the change of energy source does both at different masses. That the net effect is a step from 2.3 to 4 to 3.5 rather than something less orderly is a fact about stars rather than a consequence of the argument.

There is a fourth omission, and it is the one that makes the low-mass end genuinely hard rather than merely approximate. Below about 0.35 solar masses the gas is dense enough for electron degeneracy pressure to contribute, and cool enough for the equation of state to be non-ideal — the electrons are partly degenerate and the ions are strongly coupled. The ideal-gas step in the derivation above simply fails, and the low-mass relation is not a scaling result at all. It is the output of a numerical model, checked against the few dozen low-mass eclipsing binaries that exist, and those models are known to under-predict the radii by about five per cent for reasons that are still argued about.

The ceiling #

The fall of the exponent towards 1 at the top is not a gradual softening. It is a collision with a limit.

That intersection is an upper limit on the mass of a star, derived from an opacity and two constants of nature, and it lands within a factor of two or three of where the observed main sequence ends. It is also the reason the exponent falls to one: a star cannot exceed its own ceiling, so the relation has nowhere to go but along it.

There is a subtlety about which luminosity is limited, and it matters for how sharp the limit is. The Eddington argument compares radiation pressure with gravity at a point, and in a real star the two are compared at every radius with the local opacity — which near the surface of a hot star is not electron scattering but a forest of iron lines whose opacity is several times larger. The effective ceiling there is correspondingly lower, and the outer layers of a very massive star are unstable long before the star as a whole reaches the classical limit.

That is why the most massive stars are not merely near their ceiling but visibly losing mass: the instability drives a wind, the wind removes the envelope, the envelope’s removal lowers the opacity, and the star settles at a mass its own radiation can tolerate. A brightness that would blow the star apart is the mechanism, and the observed upper end of the main sequence is where it has finished acting.

What the varying exponent changes #

Two consequences follow from the exponent not being a constant, and both are larger than they look.

The main-sequence lifetime. A star’s lifetime goes as its fuel over its consumption, which is $M/L$, so the lifetime exponent is $1 - \alpha$ — a lifetime falling as $M^{-1.3}$ at the bottom of the main sequence and as $M^{-3}$ in the middle. That is the difference between a red dwarf outliving the universe by a factor of ten and by a factor of a thousand, and it is entirely a statement about which opacity dominates in an object nobody can see inside.

The initial mass function’s light. A population’s light comes overwhelmingly from its heaviest members, and how overwhelmingly depends on the exponent. With a mass function falling as $M^{-2.35}$ and a luminosity rising as $M^{\alpha}$, the light per logarithmic mass interval goes as $M^{\alpha - 1.35}$ — rising steeply if α is 4 and barely at all if α is 2.3. So the same population’s light is dominated by different stars in different mass ranges, and most stars being small while most of the light is not is a statement whose strength depends on which piece of this curve is being integrated over.

What is actually measured #

The relation is not a theoretical curve. It is a fit, and it is a fit to a small and peculiar sample.

Masses of stars are measured directly in exactly one way: two stars orbiting each other, where Kepler’s third law gives the sum and the ratio of the reflex motions gives the individual values. To get a luminosity as well, the system must also eclipse, so that the radii follow from the light curve and the distance from the radius and the flux. The only stars whose masses are known are the eclipsing double-lined spectroscopic binaries, and there are a few hundred with parameters good enough to use.

The chain from a light curve to a point on this diagram is worth setting out, because every step is a measurement rather than a model. The spectroscopic orbit gives $M_1 \sin^3 i$ and $M_2 \sin^3 i$; the eclipse gives the inclination, so the sines come out; the eclipse durations give the two radii in units of the semi-major axis, which the orbit supplies in kilometres; and the surface brightnesses from the depths, with the radii, give the luminosities. No stellar model enters anywhere. That is why these few hundred systems carry the whole relation.

Those stars are not a random sample. They are close binaries, which means they may have interacted; they are biased towards short periods, which is where eclipses are likely; and they are biased towards nearly equal masses, which is where both spectra can be seen. The relation fitted to them is then applied to single stars.

There is one independent check, and it is a good one. Asteroseismology measures a star’s mean density from its oscillation frequencies, with no binary and no eclipse, and combining that with an interferometric radius gives a mass. For the handful of stars where both exist the masses agree with the relation to a few per cent. That is a small sample and it is entirely single stars, so it tests the one thing the binary sample cannot: whether being in a close binary has changed the star.

The honest summary is that the exponent’s variation is real and well established at the level of the steps, and that the second decimal place of any of the three numbers is a property of the sample rather than of stars.

Where the numbers came from historically #

The relation is older than the theory, which is worth knowing because it is the reverse of the usual order and it explains why 3.5 became the quoted number.

Eddington had it by 1924, from a handful of visual binaries with measured masses and estimated luminosities, and he had the theory at the same time — the radiative-transport argument above is essentially his. What he did not have was the opacity: the atomic physics that gives Kramers’ law and the electron-scattering constant came later, and Eddington’s own opacity estimates were wrong by a factor of several in a direction that happened to cancel against another error.

The exponent of 3.5 dates from mid-century fits to a sample of a few dozen. The corners were visible in those fits and were treated as scatter, because the sample was too small to establish them. They became established when the eclipsing-binary sample reached a few hundred with good photometry, which is recent enough that a textbook of the 1980s still quotes one exponent and one of the 2010s quotes three.

One more thing the history explains. The 3.5 is quoted because it fits the range where the sample is densest, and the sample is densest around a solar mass for the ordinary reason that solar-type stars are common and bright enough to have their spectra taken. So the canonical exponent is a statement about the observability of stars as much as about their structure, and it happens to be very nearly the value the middle regime and the upper regime average to — which is a coincidence of where the sample sits rather than a compromise anybody negotiated.

The generalisation #

The structure worth extracting is that a power law fitted over a wide range is almost always a chord across several power laws, and that the exponent it returns is a weighted average of the real ones with weights set by where the data are.

Here the fitted 3.5 is a compromise between 2.3, 4 and a fall towards 1, and it is nearer 4 than anything because the sample is densest around a solar mass. A sample of only red dwarfs would return 2.3 and a sample of only O stars something near 2. The fitted exponent is therefore a joint statement about stars and about who was observed, and the second half is not usually reported. All three would be correct and none would be the law.

The same shape recurs constantly in this collection. The crater chronology’s exponential branch and linear branch are two regimes of one curve, and a single power law fitted across them would be a chord. A debris size distribution has one slope for the fragments and another for the intact objects. And an initial mass function corrected by an age is a power law whose fitted slope depends on which masses the sample could reach.

There is a second reading, about what an exponent is evidence of. A power law with a constant exponent over many decades is a strong hint of a single mechanism with no scale in it; a power law whose exponent changes is a hint of several mechanisms handing over. So the interesting thing about the mass–luminosity relation is not that it is approximately a power law but that it is approximately three of them, and each corner is a place where the star’s interior reorganises itself.

The corollary is a habit rather than a rule. Before quoting an exponent, draw its derivative. A power law’s signature is a flat derivative, and the derivative is the only picture in which a corner is obvious rather than a matter of judgement.

Where the ladder goes next #

The next rung takes the fourth power of the mean molecular weight seriously. Two stars of the same mass and different composition have different exponents and different luminosities: a metal-poor star has a lower opacity and a lower molecular weight, so it is hotter, smaller and brighter than a solar-composition star of the same mass — which is the whole reason a globular cluster’s main sequence sits below the field’s in a colour–magnitude diagram, and the reason an age fitted from one requires a composition first.

Further rungs on this anchor: the mass–radius relation, which has its own two regimes and its own corner at the same place; the exponent at the very bottom, below the hydrogen-burning limit, where the object is degenerate and the luminosity falls with time rather than depending on mass at all; the relation for stars that are not on the main sequence, which is not a relation; and the empirical scatter’s decomposition into age, composition and binarity, which is what a modern fit reports instead of a single exponent.