Astronomy & Celestial Mechanics
the sky, computed
Concept

Luminosity function — where it appears

The distribution of a population's intrinsic brightnesses, giving how many objects there are per unit luminosity. Its width decides how strongly a flux-limited survey is biased, and it is usually measured from the same biased survey it is needed to correct.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The luminosity function has a knee, and the knee is the point. A Schechter function with a faint-end slope of -1.25, a characteristic magnitude of -20.9 and a normalisation of 0.0093 per cubic megaparsec per magnitude, plotted logarithmically. Fainter than the knee the curve is a straight line — a power law — and brighter than it the count falls off exponentially, which is why there is no such thing as a galaxy ten times brighter than the brightest. Integrated across the range drawn, galaxies brighter than the knee are 1.3 per cent of the number and 26 per cent of the light: almost every galaxy is a dwarf, and almost all the light is not in one.

A count with a knee in it

Count galaxies by luminosity and the answer is a power law at the faint end and an exponential cut-off at the bright one. Almost every galaxy is a dwarf; almost none of the light is in one; and the bend between those two statements is where galaxy formation stops being efficient.

galaxies · Luminosity function A sample that gets brighter with distance because the faint ones drop out. The mean absolute magnitude of a magnitude-limited sample, relative to the population it is drawn from, against distance. The population has a spread of 0.5 magnitudes about a mean of -4, and the survey stops at apparent magnitude 20. Nearby, everything is detected and the sample is unbiased. Beyond about 316228 parsecs the faint end of the distribution starts falling below the limit and the survivors are brighter than average; further out the bias deepens without limit, because eventually only the extreme tail is detectable. The horizontal line is the classical Malmquist value of 1.382 times the square of the spread, which is what the bias averages to over a magnitude-limited sample as a whole — it is a property of the sample rather than of any one object, and using it as a correction for an individual star is a common and specific mistake.

A sample brighter than the population it came from

Every survey stops at some apparent brightness. At any distance it therefore contains only the objects luminous enough to make the cut, so the average object in it is brighter than the average object in the universe — by an amount that grows with distance and that has been shortening every distance in astronomy since 1920.

starlight · Distance ladder A 1.1 solar-mass white dwarf held up for 2.8 extra billion years. The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. The lower band is bare Mestel cooling — thermal energy of the ions leaking out through an envelope whose opacity is Kramers'. On top of it sits the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; and on top of that the gravitational energy of ²²Ne settling through what is left, taken here as 0.6 kT per ion. Neither is fuel: both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time. The delay rises from 1.88 billion years at 0.5 solar masses to 2.81 at 1.1 — 58 per cent of the cooling already done. A white dwarf age computed from the bare law is too young, and it is too young by more the heavier the star is, which is exactly the direction that matters, because the massive white dwarfs are the ones used to date the oldest populations.

A clock that stops while its interior freezes

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling — until the interior crystallises. The latent heat of that phase change, and the settling of a heavy isotope through what is left, hold a massive white dwarf up for nearly three extra billion years.

stars · White dwarf cooling 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 The same count, taken in two places. The ratio of a cluster's luminosity function to the field's, per galaxy at the knee, against absolute magnitude. Both are Schechter functions — the field at a faint-end slope of -1.25 and a characteristic magnitude of -20.9, the cluster at -1.05 and -21.4 — and they are normalised to agree at -21 so that what is drawn is a difference of SHAPE rather than of density, a cluster being some 240 times denser than the field by construction. Two things differ. The cluster's faint end is shallower: at -15 it holds 0.22 of the field's dwarfs per bright galaxy. And its knee is 0.5 magnitudes brighter, which is a factor of 1.6 in luminosity. Neither difference can be read as a cause. A cluster's galaxies are also redder, and the same photometry measures both — so a shallower faint end could mean that dwarfs were destroyed, or that they were never made, or that they are still there and have faded below the survey's limit because their star formation was stopped. The count says the populations differ; it does not say which of a galaxy's life stages the difference happened in.

The same census, taken in two places

Fit a Schechter function to a rich cluster and to the field around it and the two come back with different slopes and different knees. Both differences are real, and neither can be read as a cause — a cluster's galaxies are also redder, and the same photometry measures both.

galaxies · Luminosity function

Named alongside it

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

CompletenessFaint end slopeSchechter functionCharacteristic luminosityEddington biasMalmquist biasNumber densityQuenchingVolume-limited sampleCooling ageCoulomb couplingCrystallisation

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