Astronomy & Celestial Mechanics
the sky, computed
Generator

The mass-function generator

One population, three integrals, two ends
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.

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.

5 essays call mass-function. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Stars

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.

 One star in 380 makes essentially all the ionising light. Three cumulative fractions against stellar mass, for a broken power-law initial mass function with slopes 1.3 and 2.3 breaking at 0.5 solar masses. Each curve says what share of one quantity is produced by stars heavier than the mass on the axis, and the three do not resemble one another. Only 0.41 per cent of the hydrogen-ionising photons come from stars below 15 solar masses, because the ionising output of a star climbs by five orders of magnitude between eight and twenty. One star in 380 is above that mass, and between them those stars hold 14 per cent of the mass. Those two numbers are the leverage in every star-formation rate quoted from an Hα line. What is measured is the light of a handful of very massive stars; what is reported is the mass of a whole population; and the number in between is an integral over a part of the mass function that no extragalactic observation reaches. The medians are marked but should be read with care, and the reason is visible in the curves: the mass-weighted median at 1.27 solar masses is a property of the population, while the light-weighted one at 58 is a property of where the plot stops — halving the upper mass limit moves it to 36. An integrand that rises with mass has its median wherever the axis ends. Galaxies

A birth rate measured from light nothing young emitted

A star-formation rate is quoted in solar masses per year, and nothing in the measurement counts a star or weighs anything. What is measured is a luminosity produced by about one star in four hundred, and the conversion to a mass is an integral over a part of the mass function that no extragalactic observation reaches.

 Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took. Cosmology

It ends when the walls meet

Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

 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. Stars

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.

 A bound on the baryon density from an abundance nobody can extrapolate. Primordial deuterium against the baryon density, with the upper bounds that the solar system's own deuterium and helium-3 place on it. The inequality is D_p ≤ D_obs + ³He_obs/g₃, and it holds for any star-formation history whatever: a deuteron entering a star becomes ³He, so the pair can only be moved from one member to the other and then destroyed, never increased. With pre-solar values of D/H = 2.0e-5 and ³He/H = 1.5e-5, the bounds are 3.50e-5 at g₃ = 1, 5.00e-5 at g₃ = 0.5, 8.00e-5 at g₃ = 0.25 — and because deuterium falls with density, each upper bound on the abundance is a lower bound on η: η₁₀ > 4.98, η₁₀ > 3.99, η₁₀ > 2.97. The weakest of them, at a survival fraction of 0.25, still excludes 52 per cent of the density range below the answer. That was the state of the measurement for most of the 1980s, and the shape of it is the thing worth carrying: an abundance too poorly understood to be extrapolated at all still constrained the quantity, because its direction of change under processing was known even though its magnitude was not. The microwave background's 6.13 sits above every bound drawn, which is not a coincidence and is not evidence — a bound that excluded the answer would have been an error, and a bound that admits it is only a bound. Cosmology

An abundance with no direction to correct in

Every primordial abundance is measured today and extrapolated backwards, and the extrapolation works because processing moves each species one way. Helium-3 is made by small stars and destroyed by large ones, so the sign of its correction is not merely uncertain — it is unknown.

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