Astronomy & Celestial Mechanics
the sky, computed
Concept

Mass-to-light ratio — where it appears

The mass of a stellar population divided by its luminosity, in solar units, which rises by orders of magnitude as the population ages. It changes by a factor of thousands over a Hubble time with the mass function held fixed, purely because the luminous stars die.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it.

Most stars are small, and most of the light is not

The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch.

stars · Initial mass function The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not.

A rotation curve that refuses to fall

Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure.

galaxies · Rotation curves The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection.

The mass that is not the light

A galaxy's mass-to-light ratio is not a number, it is a curve, and it rises without turning over. What stellar populations can plausibly weigh sets a ceiling; the dynamics sit far above it, and the gap is a shape rather than a discrepancy.

galaxies · Dark matter Tully–Fisher, from two assumptions and no fitting. 44 model galaxies spanning two and a half decades in luminosity, each built from a constant disc surface brightness of 380 L☉ per square parsec with 0.13 dex of scatter and a stellar mass-to-light ratio of 1.4 with 0.1 dex. Nothing about a luminosity–speed relation is put in: the scale length follows from the surface brightness, the mass from the light, and the speed from v² = GM/R. The construction makes L ∝ v⁴ exactly, so the true slope is −10.0 magnitudes per decade of speed. Least squares of magnitude on log speed returns -8.45, and the reverse regression -9.15: the scatter here is scatter in the speed at fixed luminosity, and error in the abscissa flattens a fitted slope, so neither fit returns the value the family was built with and the distance between them is the size of the effect. Residual scatter about the drawn line is 0.54 magnitudes. Observed slopes run from about −7.5 in blue light to −10 in the near infrared, where the mass-to-light ratio is steadiest — which is the same statement as the second assumption above.

A line width that is a distance

The width of a galaxy's hydrogen line depends on how fast it rotates, which depends on its mass, which is tied to its luminosity — so a quantity no distance enters gives an absolute brightness, and the distance follows from the brightness that is seen.

galaxies · Tully–Fisher Three mass models, 2.3× apart in the disc, agreeing to 3.1 km/s everywhere. NGC 3198's rotation curve, decomposed three ways. The stellar disc has been scaled by 0.2, 0.65, 1.1 times its photometric mass, and for each scaling the halo's asymptotic speed and core radius have been fitted — not chosen — to reproduce the same total. The heavy curve and the marked points are that total: the three models agree with it to 3.08 km/s at every radius, well inside a measurement error of 4.5. The light curves below are the disc's own contribution, and at 17 kpc they differ by a factor of 2.3 — from 36 to 85 km/s — with the halo taking up exactly the slack, 126 down to 97. The curve is one function and the decomposition asks for two. The free parameter is the mass-to-light ratio of the stars, which the kinematics never measures, and it is why a "maximum disc" fit and a halo-dominated fit are both published for the same galaxy. The degeneracy does have one hard edge: scaling the disc to 1.35 times its photometric mass cannot be fitted by any halo in the family — the best leaves 5.6 km/s rms — because past the maximum-disc solution the stars alone already overshoot the curve and a halo cannot have negative mass. That is the one thing a rotation curve says about M/L on its own, and it is an upper limit. What separates the rest has to come from somewhere else: the vertical velocity dispersion of the disc, which weighs the stars alone; gas-rich dwarfs where there is scarcely a disc to argue about; or the baryonic Tully–Fisher relation, which ties the halo's speed to the baryons and would be a coincidence if the two were independent.

The same curve, two galaxies

A rotation curve is one function of radius. Decomposing it asks for two — how much of the speed is the stars and how much is the halo — and the mass-to-light ratio that trades one against the other is not measured by anything in the kinematics. A maximum disc and a halo-dominated fit run through identical points.

galaxies · Rotation curves A high end that is 2.5 in slope steeper than the one stars were born with. The mass function a star count measures against the one stars were born with, for a population that has been forming stars steadily for 10 billion years. Below about a solar mass nothing has had time to die, so the two coincide exactly. Above it the fraction still alive is the main-sequence lifetime divided by the age, and since the lifetime falls as the two-and-a-half power of the mass, the present-day function is steeper than the initial one by exactly that exponent. A count of massive stars in an old population therefore under-represents them by orders of magnitude, and reading it as an initial mass function gives a slope far too steep. The correction is large, it is calculable, and it depends on the star-formation history — which is usually the thing the mass function was going to be used to constrain.

A mass function corrected by an age

Counting stars by mass gives the stars that are alive. What every argument needs is the stars that were born, and the two differ by the fraction of each mass still on the main sequence — which is a lifetime divided by an age, and which for massive stars in an old population is a very small number.

stars · Initial mass function A virial mass inflated 23.2-fold by orbits nobody resolved. The factor by which a virial mass is overestimated when the velocity dispersion is measured from single-epoch spectra, against the system's true dispersion, for four numbers of observing epochs. Every star in a binary carries its own orbital velocity, which adds in quadrature to the system's own, and the orbital velocities of ordinary binaries are of order a kilometre a second. A system whose real dispersion is 0.3 km/s therefore measures 1.45, and since a virial mass goes as the square of the dispersion the mass comes out 23.2 times too large. Repeat epochs fix it: the orbital velocities are uncorrelated between visits while the system's own are not, so the binary variance falls as one over the number of epochs and the correction is measured rather than modelled.

A dispersion inflated by orbits nobody resolved

A velocity dispersion measured from single spectra is not the dispersion of the system's centres of mass. Every star in a binary carries its own orbital velocity of a kilometre or so, and for a dwarf galaxy whose real dispersion is smaller than that, the measured value — and the dark-matter content computed from its square — is mostly binaries.

galaxies · Velocity dispersion Three mass models, one rotation curve. Rotation curves for three decompositions of the same galaxy, from a disc contributing 30 per cent of the outer rotation to one contributing 95. Each is the quadrature sum of a stellar disc, whose shape is fixed by the light distribution and whose amplitude is an unknown mass-to-light ratio, and a dark halo with parameters of its own. All three reproduce the same flat outer rotation, because the halo's amplitude is adjusted to make up whatever the disc does not supply. They differ in the inner few kiloparsecs, by 27 kilometres a second here — which is more than the measurement error and less than the uncertainty in the disc's own contribution, since that depends on a mass-to-light ratio nobody measures directly. The maximum-disc assumption picks the largest disc consistent with the data, and it is a convention rather than a result.

Three mass models that fit the same curve

A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.

galaxies · Dark matter Five speeds from the same galaxies, and the relation each one gives. The exponent and the scatter of the baryonic Tully–Fisher relation fitted to the same 90 model galaxies with five different speeds, fitted as speed on mass. outer speed: exponent 3.98, scatter 0.080 dex in mass; peak speed: exponent 3.86, scatter 0.110 dex in mass; at 2.2 scale lengths: exponent 3.29, scatter 0.167 dex in mass; W50 ÷ 2: exponent 3.73, scatter 0.119 dex in mass; W20 ÷ 2: exponent 4.05, scatter 0.118 dex in mass. The galaxies were built with the relation in their outer halo speed, and the other four speeds each lose some of it: the ones read from the inner curve inherit how concentrated each galaxy's stars are, and the line widths add the turbulence of the gas, which matters most in the slowest galaxies.

The speed a line width stands in for

A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.

galaxies · Tully–Fisher The darkness, itemised. The extragalactic background light as a spectrum: νIᵥ against wavelength, in nanowatts per square metre per steradian, with wavelength logarithmic. The total under the two humps is not read off a plot — it is the integral (c/4π)∫ε(z)(1+z)⁻¹|dt/dz|dz of the Madau–Dickinson star formation history, taking a continuously star-forming population to radiate 10¹⁰ solar luminosities per solar mass per year, and it comes to 37.5 nW m⁻² sr⁻¹. Half of it was emitted beyond z = 0.91, which is the figure's real content: the dark sky is dominated by light released when the universe was 45 per cent of its present age. The split between the two humps is an assumption rather than a result — 50 per cent of the starlight is taken to be absorbed by dust and re-radiated near 140 µm, and the shapes are lognormals of about the observed widths — but the AREA under each is the computed quantity. Measured, the two come to about 24 and 26: the calculation falls 25 per cent short, and whether the missing light is real or a residual foreground is unsettled.

The darkness has a number in it

The night sky is not black. It carries about sixty nanowatts per square metre per steradian, and that number is the sum of every photon every star has ever emitted, redshifted and added up over thirteen billion years.

cosmology · Olbers's paradox The count theory predicts, and the inference it costs. The galaxy stellar mass function: galaxies per cubic megaparsec per dex of stellar mass, both axes logarithmic. Two Schechter components share a characteristic mass of 10^10.66 M☉ — one of slope -0.35 carrying the quenched galaxies at the knee, one of slope -1.47 carrying the star-forming ones below it — and the dashed line is the single component a luminosity function is usually fitted with. Integrated over the range drawn it gives 0.0487 galaxies per cubic megaparsec holding 2.22·10⁸ solar masses of stars, of which 51 per cent sits above the knee. This function is not measured. What is measured is a luminosity function; turning one into the other needs a mass-to-light ratio for every galaxy in the sample, and that ratio is not a constant — it runs by a factor of about five from the bluest galaxies to the reddest, so the conversion moves the red end of the distribution further than the blue end and changes the SHAPE rather than the units. A stellar mass function is a luminosity function plus a stellar population model, and the second half is where its disagreements live.

The count theory predicts, and the inference it costs

A luminosity function is measured. A stellar mass function is inferred, one galaxy at a time, through a ratio that runs by a factor of six from the bluest galaxies to the reddest — so the conversion changes the shape and not merely the units.

galaxies · Luminosity function

Named alongside it

The objects these essays reach for when they reach for this one.

Initial mass functionThe baryonic Tully–Fisher relationDark matterStellar populationRotation curveVelocity dispersionCompletenessDegeneracyThe disc–halo degeneracyEnclosed massFreeman's lawLine width

All concepts