The candle that has to be standardised

A supernova is not a star exploding in the ordinary sense of a release of stored energy. In a Type Ia the light seen is not the energy of the explosion at all — that went almost entirely into kinetic energy of the ejecta within seconds and is invisible. What is seen, weeks later, is the decay of a radioactive isotope made during the burning.

The isotope is ⁵⁶Ni, produced by fusion at densities high enough to reach nuclear statistical equilibrium, and its half-life is 6.1 days. Producing it at all requires temperatures at which nuclear statistical equilibrium is reached, which is not a condition any main-sequence star’s core comes near. It decays to ⁵⁶Co, which decays with a half-life of 77 days to ⁵⁶Fe, which is stable. A Type Ia light curve is a plot of that chain, delayed and smeared by the time it takes photons to escape an expanding cloud.

Why the peak is a mass #

Arnett’s rule looks like a coincidence and is a theorem. In the one-zone model the luminosity satisfies

$$L(t) = \int_0^{x}2z\,e^{z^2-x^2}\,Q(z\tau_m)\,dz,\qquad x = t/\tau_m,$$

with $Q$ the heating rate and $\tau_m$ the diffusion time. Differentiate and the result is $L' = 2x\,[Q - L]$, so $L = Q$ exactly at the maximum, whatever $Q$ is. The peak luminosity is the instantaneous radioactive power at the moment of the peak, and since the power at a given time is proportional to the nickel mass, the peak brightness measures the nickel mass and nothing else.

That is a strong statement and it is worth noting how easy it is to break in a figure. A first version of the generator behind these curves paired the published $A(z)$ and $B(z)$ integrands with the physical Bateman heating rate — two formulations with different implied source terms — and the luminosity at maximum came out at 0.68 of the deposited power. Every curve looked plausible; the one property the whole subject rests on had quietly failed.

What explodes #

The progenitor is a white dwarf, and the reason a white dwarf can explode at all is that it is held up by a pressure that does not depend on temperature.

The Chandrasekhar limit is also what makes the explosions similar to each other. If ignition occurs when the star approaches 1.4 solar masses, then every Type Ia burns about the same amount of material under about the same conditions, and the spread in outcomes is the spread in how the burning propagates rather than in how much fuel there is.

How the white dwarf gets there #

A white dwarf that formed at 0.6 solar masses is not going to reach 1.4 by itself. It has to be fed, and the feeding requires a companion. The two channels — accretion from a normal companion, or the merger of two white dwarfs — predict different things and neither has been ruled out. What is worth noticing is that this is a genuine gap: the objects whose brightnesses calibrate the expansion of the universe are of unknown origin, and the calibration works anyway because it is empirical.

How the burning propagates #

The similarity of Type Ia supernovae to one another is not automatic even at fixed mass, and the reason is that the outcome depends on how fast the burning front moves.

A carbon–oxygen white dwarf ignited at its centre can burn in two regimes. A deflagration is subsonic: the front moves by thermal conduction, the star has time to expand ahead of it, and the burning at lower density produces intermediate-mass elements — silicon, sulphur, calcium — rather than iron-group ones. A detonation is supersonic: a shock compresses material before burning it, densities stay high, and almost everything becomes nickel.

Pure detonation would turn the whole star into iron-group elements, and the observed spectra show large quantities of silicon — the defining feature of the class. Pure deflagration burns too little and leaves the star underluminous and mixed. The favoured picture is a transition between the two, with the front starting subsonic and going supersonic partway out, and where that transition happens sets the nickel mass.

That is the physical origin of the one parameter behind the width–luminosity relation, and it is worth being clear that it is not derived from first principles. The transition density is an input to the models, not an output, so the family of light curves is generated by tuning the same quantity the observations are being used to measure.

They are not standard #

Type Ia peak absolute magnitudes span about 0.6 magnitudes — nearly a factor of two in luminosity. At that spread they would be useless: a distance good to forty per cent is not a distance.

Mark Phillips found the way out in 1993. The supernovae that are brighter at maximum are also the ones that decline more slowly afterwards, and the correlation is tight enough to correct on.

The physical link between nickel mass and diffusion time is stated in that model rather than derived — a power law with an exponent chosen to reproduce the observed range of decline rates. That is exactly the role the real relation’s empirical calibration plays, and it is why the Phillips relation is honestly described as a correction rather than as a theory. What the model does establish is that a one-parameter family ought to produce a correlation of about this slope, which is worth knowing.

The spectrum, which is how one is recognised #

A light curve alone does not identify a supernova, and the classification is spectroscopic. The scheme is old and partly awkward, having been built up before anybody knew what any of the classes were.

Type I means no hydrogen lines. Type II means hydrogen lines. Within Type I, Ia has a strong silicon absorption near 615 nm and the others do not. Everything else about the classification followed from that one line.

The physical division turns out to cut across it. Types Ib, Ic and II are all core collapses of massive stars, differing only in how much of the envelope was stripped before the explosion; Type Ia is the only thermonuclear class, and it is grouped with the core collapses because it happens to lack hydrogen. A classification by what is absent produced one useful category and one accidental one, and the useful one is the accident: silicon is present because the burning front spent time at intermediate density, which is precisely the property that makes the class uniform.

The absence of hydrogen is itself a constraint on the progenitor. If a Type Ia comes from accretion onto a white dwarf from a normal companion, some of that companion’s hydrogen ought to be swept up in the ejecta and visible in a late-time spectrum. Deep searches have not found it in any well-observed case, which is one of the stronger arguments for the merger channel and one of the more awkward facts for the accretion one.

What the tail says #

Past about fifty days the light curve becomes a nearly straight line on a logarithmic axis, and the slope is a laboratory number rather than an astrophysical one.

The heating at that stage is entirely ⁵⁶Co decay, whose mean life is 111.3 days, so full trapping of the decay products would give a decline of $2.5/(111.3\ln 10) = 0.0098$ magnitudes per day. Observed tails decline faster, because the ejecta expand until they are transparent to the gamma rays that do the heating, and the escaping fraction rises as the column falls.

That difference is not a nuisance — it is a measurement. The rate at which the tail steepens gives the ejecta’s column density and hence its mass, independently of anything else, and it is one of the few direct constraints on whether the exploding object really did have 1.4 solar masses.

What was actually measured, and when #

The chain of inference here is unusually long and worth setting out, because the conclusion drawn from it in 1998 was large.

A supernova gives an apparent magnitude and a light-curve shape. The shape gives a correction, and the corrected apparent magnitude gives a relative distance — how much further one supernova is than another — with no absolute scale at all. That is all that is needed to measure the expansion history’s shape.

Getting an absolute distance, and hence the expansion rate itself, is a different and harder problem: it requires the supernovae to be calibrated against something whose distance is known geometrically, which means finding Cepheids in the same galaxies as recent Type Ia supernovae.

The progenitor nobody has seen #

For a core-collapse supernova the progenitor is sometimes identifiable: the star was bright, it was there in an archival image of the same galaxy, and after the explosion faded it was gone. That identification has been made a couple of dozen times and it is how the mapping from stellar mass to explosion type is calibrated.

For a Type Ia it has never been made. A white dwarf is far too faint to appear in any pre-explosion image of another galaxy, so what the archival images constrain is not the dwarf but its companion.

Those constraints are real and they are one-sided. A red-giant companion would be luminous enough to detect in a nearby galaxy, and deep pre-explosion imaging of several well-placed Type Ia sites has found nothing — excluding the brightest companion models for those particular events. A main-sequence companion of a solar mass or so would be below the limit and is not excluded.

There is a second test that works after the explosion rather than before it. A companion sitting a few solar radii away when the ejecta arrive is struck by them, and the collision produces a brief excess of ultraviolet light in the first day or two — before the nickel-powered light curve has risen. Surveys that catch supernovae early enough now search for that excess routinely.

Most events show none, and a few show something. The interpretation is contested, because a nickel distribution reaching close to the surface of the ejecta produces an early excess too, and distinguishing the two requires colours in the first hours.

A third test looks for what the companion should have left behind. If a normal star donated the material, it survives the explosion, is kicked to a high velocity, and should still be sitting in the remnant centuries later. Searches inside the historical galactic remnants have produced candidates and no consensus.

Three independent tests, all consistent with the single-degenerate channel being a minority contributor, and none of them decisive — which is the state of a question the entire extragalactic distance scale does not depend on and probably should.

The delay between forming and exploding #

There is one observable that discriminates between the channels without needing to see any individual progenitor, and it is a population statistic.

A white dwarf takes time to form and then more time to be pushed over the limit. The distribution of that total delay — from the birth of the stars to the explosion — differs between the two routes. Accretion from a normal companion requires the companion to evolve and fill its Roche lobe, which sets a delay determined by that star’s lifetime and therefore concentrated in a range. A merger of two white dwarfs requires their orbit to shrink by gravitational radiation, which takes a time depending steeply on the initial separation — and since separations are distributed roughly uniformly in the logarithm, the resulting delay distribution falls as one over the time.

That is a sharp prediction and it is measurable. Compare the rate of Type Ia supernovae per unit stellar mass in galaxies of different ages: an old elliptical, whose stars formed ten billion years ago, against a star-forming spiral. The ratio of the rates measures the delay distribution’s shape.

The measurement has been made several ways — from supernova rates in galaxy samples binned by colour, from rates as a function of redshift, and from the rates in galaxy clusters where the star-formation history is known — and the answer is consistent: the rate falls roughly as the inverse of the delay time, over two orders of magnitude in time.

That is the merger channel’s prediction and not the accretion channel’s. It does not exclude the second, because a mixture with a minority of prompt events fits too, and the majority of Type Ia supernovae appear to come from systems whose stars formed billions of years earlier.

A question about individual objects has been partly answered by counting a population, which is the usual resolution in this subject when the objects themselves are unobservable.

Two of the essay’s three claims about the light curve can be checked by moving a parameter, and both of them are claims about a family rather than about one object.

The other supernova that measures a distance #

Type Ia supernovae dominate the distance scale, and they are not the only supernovae that can be used as one. The alternative is worth knowing because it shares no calibration with them at all.

A core-collapse supernova whose progenitor kept its hydrogen envelope produces a light curve with a plateau: for two or three months the luminosity is nearly constant, because a recombination front is receding through the expanding envelope at just the rate that keeps the emitting surface’s temperature fixed.

During that plateau the photosphere is a blackbody at a known temperature, and its velocity is measurable from the Doppler width of the absorption lines. A velocity times a time is a radius; a radius and a temperature give a luminosity; and a luminosity with an observed flux gives a distance.

Nothing in that chain is a standard candle. It is a geometric measurement of an object’s physical size, and the distance follows without any calibration against a nearer object of the same kind.

The difficulties are in the details rather than in the principle. The photosphere is not a blackbody — it is a scattering-dominated atmosphere whose emergent flux is diluted relative to a blackbody by a factor that has to be computed from a model. The velocity measured from a line is the velocity at the line-forming region rather than at the photosphere. And the plateau is not perfectly flat.

The resulting distances are good to about ten or fifteen per cent per object, which is worse than a Type Ia and is obtained through an entirely different chain. Applied to the same host galaxies, the two methods agree — which is a check on the Type Ia calibration that owes nothing to Cepheids, to parallaxes, or to anything below them on the ladder.

What the model leaves out #

Three things, stated because the figures above look more definitive than the subject is.

The one-zone model has one zone: the nickel is assumed to sit where the photons have to diffuse through everything, and in reality it is distributed through the ejecta in a way that affects both the rise time and the colour evolution. Multi-dimensional models with resolved burning fronts do considerably better and do not have a closed form.

The gamma-ray escape is a one-parameter description of a geometry-dependent process, and the parameter is fitted rather than computed.

And the light curves above are bolometric, which is not what anybody observes. Real photometry is in bands, and converting between them requires a bolometric correction that evolves as the supernova cools and its spectrum changes — which is where a good deal of the residual scatter in the corrected magnitudes comes from.

And the other kind of standard candle drawn beside it, since the two rungs of the ladder overlap in exactly one class of galaxy.

Where the ladder goes next #

The rung above is the second correction, on colour: Type Ia supernovae that are redder are also fainter, and separating intrinsic colour variation from dust reddening in the host galaxy is currently the largest systematic in the whole method — the same separation problem that dust makes everything look further away is about, arriving in a place where a per cent matters. Beyond that is the question of whether the relation evolves — whether a supernova at redshift 1, in a younger and more metal-poor galaxy, obeys the same width–luminosity relation as one nearby. Nothing observable settles it directly, and every cosmological result quoted from these objects carries the assumption.