The redshift that weighs a dead star

The clock that runs slow lower down derived the gravitational redshift from a photon climbing out of a potential well: light that leaves a surface where the potential is lower arrives with its frequency reduced by the difference in potential over $c^2$. The radius a mass adds that no circumference shows found that clocks read only the time part of the geometry around a mass, and that the space part needs light. Both stayed close to home, where the shift is a few parts in ten thousand million and has to be dug out of the noise of the best clocks ever built.

At the surface of a white dwarf the shift is ten thousand times larger than at the Sun’s and visible in an ordinary spectrograph. The interesting thing about it is not its size, though. It is that for this one kind of star the redshift is not merely a measure of gravity. It is a measure of mass, because a white dwarf is not allowed to choose its own radius.

A shift that needs a second number #

For a body of mass $M$ and radius $R$, light leaving the surface and arriving far away is shifted by

$$z = \left(1 - \frac{2GM}{Rc^2}\right)^{-1/2} - 1 \;\approx\; \frac{GM}{Rc^2},$$

the approximation holding to a part in ten thousand for anything less compact than a neutron star. Astronomers quote it as the velocity $cz$ that would give the same Doppler shift. For the Sun it is 636 metres a second.

The shift depends on the ratio $M/R$ and on nothing else. That is its limitation for an ordinary star. The Sun’s radius is set by the balance between its pressure and its weight, and its pressure depends on its temperature, which depends on how fast it is burning hydrogen and how easily the heat leaks out. Two stars of the same mass can have very different radii — a dwarf and a giant — and two stars of different masses can have the same radius. A redshift measured from a main-sequence star tells the ratio of its mass to its radius and leaves both unknown. It is also small, comparable to the turbulent motions of the star’s own atmosphere, and on the Sun the convective upwellings shift the lines blueward by an amount of the same order, which is why the solar redshift took until the 1960s to be measured cleanly.

A star with no say in its size #

A white dwarf is the exhausted core of a star like the Sun, a mass comparable to the Sun’s compressed to the size of the Earth. The mass no cold matter can hold up worked out why such a thing gets smaller as it gets heavier. Its pressure comes not from heat but from the exclusion principle: the electrons are packed so closely that, as the pressure that is not a temperature found for the electrons in a metal, they fill every momentum state up to a ceiling, and the ceiling rises with the density. The pressure depends on density and on nothing else. Cool the star and it does not shrink.

That removes the radius from the list of free parameters. Given the mass, and the number of nucleons per electron (two, for helium, carbon or oxygen), hydrostatic balance determines the whole structure, the radius included.

The curve in the figure is computed, not quoted. The pressure of a degenerate electron gas whose highest momentum is $x$ times $m_e c$ is a known function — Chandrasekhar wrote it down in 1931 — that goes as the five-thirds power of the density while the electrons are slow and as the four-thirds power once they are relativistic. Integrating the balance between that pressure and the weight of the layers above it, outward from a chosen central density until the pressure falls to zero, gives one star. Repeating for a range of central densities gives the sequence.

At low mass the computed curve follows the slow-electron law, $R \propto M^{-1/3}$, and a 0.6 solar-mass star has a radius of 1.27 per cent of the Sun’s, about 8,800 kilometres. Heavier stars have faster electrons, whose pressure rises less steeply with density, so they must be compressed further to hold themselves up, and the radius falls faster than the slow law allows. At 1.459 solar masses the radius reaches zero. Above that no cold configuration exists.

One shift, one mass #

Put the radius the star is forced to have into the redshift and the result is a function of mass alone.

In the slow-electron regime the redshift goes as $M/R \propto M^{4/3}$, so doubling the mass multiplies the shift by two and a half. Near the limit the radius collapses and the redshift climbs without bound on the scale of this figure. A typical white dwarf, at 0.6 solar masses, shifts its light by thirty kilometres a second — fifty times the Sun’s shift — and a massive one by hundreds.

The curve is single-valued. Every mass has one redshift and every redshift one mass. That is what makes the measurement a scale. A spectrum of the star, a line identified, its wavelength compared with the laboratory’s, and the mass comes out without any knowledge of the star’s distance, brightness, temperature or companion — provided the speed of the star itself along the line of sight can be separated from the shift, a provision that turns out to be the whole practical difficulty.

There is something stranger in it too. The radius of a degenerate star is set by the balance between gravity and a pressure whose size is fixed by Planck’s constant, and in the slow-electron regime it comes out as

$$R \propto \frac{\hbar^2}{G\,m_e\,(\mu_e m_u)^{5/3}}\,M^{-1/3},$$

so the redshift $GM/Rc^2$ goes as $G^2 m_e (\mu_e m_u)^{5/3} M^{4/3}/\hbar^2 c^2$. A gravitational redshift — the cleanest prediction of general relativity, a statement about the geometry of time — measured on a white dwarf weighs the star in units set by the quantum mechanics of its electrons. The two theories that have never been joined meet, without any trouble at all, in the position of a spectral line.

How sharp the scale is #

A scale’s usefulness depends on how quickly its reading moves when the load does. For the redshift that is the logarithmic slope $d\ln z/d\ln M$, which starts near four-thirds for light stars and rises without limit as the radius collapses: 2.29 at one solar mass, 5.1 at 1.3. A fixed uncertainty in the velocity — five kilometres a second is a good measurement — then translates into a mass uncertainty that falls steeply with mass, for two reasons acting together: the shift is larger, so the same error is a smaller fraction of it, and each fraction of shift is worth a smaller fraction of mass.

At 0.3 solar masses, five kilometres a second leaves the mass uncertain by a third. At 1.3 the same measurement fixes it to half a per cent. The gauge is most precise exactly where the star is closest to collapse, which is also where the mass matters most, since a white dwarf near the limit is the candidate for the thermonuclear explosions that astronomers use to measure the expansion of the universe. Near the limit, though, the steepness cuts both ways. The predicted radius is so sensitive to the mass that small corrections to the model — the star’s heat, its composition, its rotation — move it by more than they would anywhere else, and the mass read from the scale is only as good as the model behind it.

The forecast and the confirmation #

The first white dwarf known, the faint companion of Sirius, was found to be white rather than red by Walter Adams in 1915, which meant that it was hot, and therefore, being faint, that it was small. Arthur Eddington put the numbers together in 1924. From the star’s brightness and an assumed surface temperature of 8,000 kelvin he inferred a radius of 19,600 kilometres; from the orbit of the pair the mass was then put at about 0.85 solar masses; the density was some fifty thousand times water’s, which was absurd by any standard of the day; and the redshift, he pointed out, would be about twenty kilometres a second — a test of general relativity, and of the absurd density, in one.

Adams measured it the next year and reported nineteen kilometres a second. Joseph Moore at Lick found twenty-one in 1928. The agreement was celebrated as the third of the classical confirmations of Einstein’s theory, beside the perihelion of Mercury and the bending of starlight, and it stood in the textbooks for forty years.

Both numbers were wrong, and by the same factor. Sirius B’s surface is not at 8,000 kelvin but about 25,000; most of its light comes out in the ultraviolet, where nobody in 1924 could see it, and a star that hot and that faint must be much smaller than Eddington’s estimate — under six thousand kilometres. With its modern mass of 1.018 solar masses, the redshift should be four times what he forecast, and when Jesse Greenstein and his collaborators measured it in 1971, at a moment when the white dwarf was well separated from its companion in the sky, they got 89 ± 16. Spectra taken with the Hubble Space Telescope, which can isolate the white dwarf completely, gave 80.4 ± 4.8 kilometres a second in 2005.

So what did Adams measure? Sirius A is nearly ten thousand times brighter than its companion and, in 1925, only about eight seconds of arc away from it. Its light, scattered in the atmosphere and the telescope, lay across the faint star’s spectrum, and Sirius A’s own lines, with no significant gravitational shift, dragged the measured positions towards zero. Historians have argued about how much an expectation of twenty kilometres a second shaped which lines were measured and how; nobody now doubts that the number was contaminated. The lesson the episode leaves is plain enough without deciding that. A measurement that agrees with a prediction confirms only that the two are consistent, and if the prediction has an unexamined input — here a temperature — the agreement can be between two wrong numbers.

Two further things make the episode worth retelling. The theory that would have given the right radius did not exist in 1925. Ralph Fowler showed in 1926 that the electrons in such a star were degenerate, and Chandrasekhar’s relativistic treatment came in 1931; Eddington’s radius came from photometry, not physics. And the modern prediction in the figure, made from the measured mass and the cold-star model with no reference to the star’s temperature or brightness at all, falls within a kilometre a second of the measured shift. The star is about as hot as a white dwarf gets while staying close to the cold model, and its hydrogen envelope adds only a little to the radius. The scale works.

A speed that has to be taken out #

A spectral line from a star is shifted by the star’s motion along the line of sight as well as by its gravity, and the two shifts are indistinguishable in the spectrum. A single white dwarf drifting through the Galaxy at thirty kilometres a second towards the Earth would show no net shift at all at 0.6 solar masses. The gravitational shift can only be isolated when the star’s velocity is known by some other route, and in practice that means a companion.

Sirius B has one, Sirius A, whose own lines carry the pair’s velocity with a negligible gravitational shift, and the orbit’s contribution at any date is known from a century of observation. Other white dwarfs are measured in wide pairs with ordinary stars, which move together through space, or in star clusters, whose members share a common velocity; the Hyades cluster supplied the first sample of white dwarfs whose redshifts could be read off against their neighbours. The single most precise measurements made since have been of white dwarfs in binaries where all three quantities — orbit, radius and redshift — can be had at once, and where the three can be checked against one another.

There is a second difficulty in the line itself. The hydrogen lines of a white dwarf are not sharp. The atmosphere is so dense that every atom sits in the electric field of its neighbours, and the levels are shifted and split by an amount that varies from atom to atom, just as the field an atom calls strong found that a field of the right size reshapes a level’s structure entirely. The resulting lines are tens of ångströms wide, while a shift of eighty kilometres a second moves a line at 656 nanometres by less than two ångströms. Only the narrow core at the bottom of the strongest line, formed high in the atmosphere where the density is lower, is sharp enough to measure, and modelling its shape is the main systematic error in every measurement.

Weighing as a consistency check #

Three routes to a white dwarf’s mass exist where all three can be followed: from the orbit of a binary by Kepler’s law, from the radius — measured from the star’s brightness, distance and temperature — through the mass–radius relation, and from the redshift through the same relation. The first is pure gravity. The second and third both rest on the model, but differently: the radius route uses the relation directly, the redshift route uses it through the ratio of mass to radius. When all three agree, as they do for Sirius B and for 40 Eridani B, the agreement tests the degenerate-electron model and general relativity at once.

The checks have repeatedly found the model right. The most interesting failures have pointed to composition. A star whose core was mostly iron, with more nucleons per electron, would be smaller at the same mass and redder, and a few white dwarfs have at times seemed too small for carbon and oxygen; most of those cases have dissolved under better distances. That the scale can distinguish an iron core from a carbon one at all is a measure of how sharp it is.

The ceiling on any surface #

The same formula holds for anything that holds itself up, and white dwarfs sit in the middle of a range that runs over ten decades.

The weak-field rule, redshift equal to half the compactness, holds up to about a tenth, which takes in every white dwarf with a large margin. Neutron stars are past it. A 1.4 solar-mass neutron star twelve kilometres in radius reddens its surface light by about a quarter — by z = 0.24 — so a spectral line from it would arrive at four-fifths of its emitted frequency. Measuring that would weigh a neutron star the way Sirius B has been weighed, and it would fix the ratio of mass to radius that the pressure that weighs down what it holds up found is the one thing the dense matter’s equation of state decides. A claimed detection of redshifted absorption lines from an X-ray burster in 2002, at z = 0.35, was not confirmed by later observations, and the star was afterwards found to spin fast enough to smear such lines into invisibility. A neutron star’s surface redshift has not been measured from a line.

The curve does not run to the horizon, where the redshift becomes infinite. Hans Buchdahl showed in 1959 that a static ball whose density does not increase outwards cannot be held up inside nine-eighths of its Schwarzschild radius, whatever it is made of, because the pressure needed rises without limit before that point is reached. At nine-eighths the surface redshift is exactly two. Light leaving the surface of anything that is standing still can therefore arrive with no less than a third of the frequency it left with. A larger redshift from a surface — a thermal spectrum, say, reddened by a factor of five — would mean something collapsing or something not made of matter in equilibrium at all.

What the curve leaves out #

Every figure here except the record is drawn from the cold model: a non-rotating, non-magnetic ball of carbon and oxygen at absolute zero, with no atmosphere, in which the electrons supply all the pressure and the nuclei only the weight. Real white dwarfs depart from it in four ways, and each moves the radius by a few per cent at most for the stars whose redshifts have been measured.

Heat adds a little pressure, and a young, hot white dwarf is slightly larger than the cold curve says; the effect is largest for light stars and fades as the star cools over billions of years. A hydrogen and helium envelope, a hundredth of the mass or less, sits on top and adds a few per cent to the radius. Rotation flattens the star and lowers the effective gravity at the equator. Strong magnetic fields, up to tens of thousands of tesla on some white dwarfs, shift and split the lines in ways that can mimic or swamp the gravitational shift. And at the high-mass end, the nuclei begin to capture electrons and the electrons’ interactions with the ions reduce the pressure, so the real limiting mass and the radii near it differ from the ideal gas’s by amounts that matter precisely where the scale is sharpest.

The figures also draw a smooth curve where a measurement gives a point with an error bar, and the curve hides that the radius near the limit is the least certain part of the model, not the most. The redshift’s steepness there is real; the precision it seems to promise is precision on the model’s terms.

The domain of the argument is a star held up by cold degenerate electrons, between a few tenths of a solar mass and a little under the limit, with its own velocity known from a companion, and a line narrow enough to measure. Inside that domain the redshift is a weight. Outside it, it is again what it is for every other star: one number about two unknowns.

Still open: what a neutron star’s line would say #

The neutron-star analogue of the Sirius B measurement is the measurement that dense-matter physics most wants and has not got. A surface redshift would fix the ratio of mass to radius directly, and with a mass known from an orbit it would fix the radius, which is what decides between the candidate equations of state for matter beyond nuclear density. Radii are now being inferred instead from the way a hot spot on a spinning neutron star bends its light, and from the tidal deformation of neutron stars in merging binaries, each with its own model behind it. Whether a sharp line will ever be seen from a neutron star’s surface — from a slowly spinning, weakly magnetised star with a cool enough atmosphere to hold heavy atoms — is not known, and neither is whether the redshift it showed would agree with the radius the other methods find.

The redshift of a degenerate star is a scale because the star has no freedom left in its size. A white dwarf’s radius is fixed by its mass through the pressure of its electrons, so its surface redshift — 30 km/s at 0.6 solar masses, 78 at 1.0, 223 at 1.35 — is a single-valued function of mass, steepening towards the limit at 1.459, and a shift measured to five kilometres a second weighs a heavy one to half a per cent. The first time the scale was read, the reading agreed with a forecast built on the wrong temperature, and it took the separated light of the star itself, half a century later, to show that the true shift was four times larger and exactly where the electrons said it should be.