Astronomy & Celestial Mechanics
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Galaxies

Two counts that are not the same shape

Dark matter halos are counted by a calculation that knows nothing about stars. Galaxies are counted by a survey. Laid on the same axes the two curves disagree at both ends and agree nowhere, and the knee is where the two disagreements hand over.

Assumes Luminosity function and Dark matter.

17 min read 7 figures Measuring the unreachableScales that defeat intuition

The stellar mass function is what a theory of galaxy formation predicts. It is not what a theory of galaxy formation starts from. What it starts from is a count of dark matter halos, and that count is a calculation with no stars anywhere in it: a power spectrum, the variance of the density field smoothed on a scale, and a rule for what fraction of the mass has collapsed.

Putting the two counts on one pair of axes is the most direct statement there is of what galaxy formation has to explain, and the statement is unwelcoming. They are not the same shape, and they differ at both ends.

Two counts that are not the same shape, at either end. The galaxy stellar mass function drawn against the halo mass function, both per cubic megaparsec per dex, with the halo masses multiplied by the cosmic baryon fraction 0.157 so that the upper curve is what the count would be if every halo had turned every baryon it owns into stars. The halo count comes from a linear power spectrum, a top-hat variance and the Sheth–Tormen multiplicity — nothing in it knows that stars exist. The two curves differ at both ends and agree nowhere. At 10⁸ solar masses of baryons the halos outnumber the galaxies by 83.0; at 10^11.5 by 64.2; and in between, near 10^10.5, the gap narrows to 3.4. A single mismatch would say only that the conversion is inefficient. Two mismatches with a minimum between them say that something different is happening at each end — and the knee of the luminosity function, which looked like a property of galaxies, is the place where those two somethings hand over.
Fig. 1 The galaxy stellar mass function drawn against the halo mass function, both per cubic megaparsec per dex, with the halo masses multiplied by the cosmic baryon fraction 0.157 — so the upper curve is what the count would be if every halo had turned every baryon it owns into stars. At 10810^{8} solar masses of baryons the halos outnumber the galaxies by 83; at 1011.510^{11.5} by 64; and between them, near 1010.510^{10.5}, the gap narrows to 3.4. A single mismatch would say only that the conversion is inefficient. Two, with a minimum between them, say that something different is happening at each end.

What the halo count knows

The upper curve is worth taking seriously as a prediction, because it is one of the few things in cosmology that can be computed from first principles and compared against nothing.

The chain is short. A linear power spectrum, fixed by the cosmological parameters and the transfer function that gravity and radiation pressure impose on the primordial spectrum. The variance of the density field smoothed with a top-hat enclosing a mass MM, which is one integral over that spectrum. And a multiplicity function — a rule saying what fraction of the mass sits in regions that have exceeded the collapse threshold — fitted once to simulations and used everywhere since.

Nothing in that chain refers to gas, to cooling, to stars or to anything a telescope observes. It is a statement about gravity acting on a Gaussian random field, and its accuracy is limited by the fitting function rather than by any astrophysics.

The shape it produces is a power law at low mass with a slope near −1.9-1.9, steepening into an exponential cut-off at the high end where the required overdensity is rare. That is the same qualitative form as a Schechter function — which is why the two curves can be compared at all — and the quantitative differences are the content.

The faint end, where there are too many halos

At low mass the halo count rises far more steeply than the galaxy count. The halo function’s slope is about −1.9-1.9 and the galaxy function’s low-mass component is about −1.47-1.47, and over three decades that difference compounds into a factor of eighty.

So most low-mass halos contain no galaxy worth counting, or none at all.

The standard explanation is that they could not keep their gas. A halo’s escape velocity falls with its mass, and below about 101010^{10} solar masses it is comparable to the speed of the gas driven out by a single generation of supernovae. A dwarf galaxy that forms stars blows away the material it would have made the next generation from, and the shallower the potential the more completely.

Two other mechanisms act on the same objects. Photoionisation by the ultraviolet background heats intergalactic gas to 10410^4 K after reionisation, which prevents it from falling into halos below a mass set by that temperature. And gas that cannot cool cannot collapse, which becomes a limitation for the smallest halos of all.

Two counts that are not the same shape, at either end. The galaxy stellar mass function drawn against the halo mass function, both per cubic megaparsec per dex, with the halo masses multiplied by the cosmic baryon fraction 0.157 so that the upper curve is what the count would be if every halo had turned every baryon it owns into stars. The halo count comes from a linear power spectrum, a top-hat variance and the Sheth–Tormen multiplicity — nothing in it knows that stars exist. The two curves differ at both ends and agree nowhere. At 10⁸ solar masses of baryons the halos outnumber the galaxies by 15.0; at 10^11.5 by 64.8; and in between, near 10^10.5, the gap narrows to 3.4. A single mismatch would say only that the conversion is inefficient. Two mismatches with a minimum between them say that something different is happening at each end — and the knee of the luminosity function, which looked like a property of galaxies, is the place where those two somethings hand over.
Fig. 2 The same comparison with the galaxy function’s low-mass slope steepened to −1.75-1.75 — near the upper end of what deep surveys allow. The gap at the faint end narrows and does not close, and the ratio at the bright end is untouched. The faint-end disagreement is therefore robust to the largest observational uncertainty in the measurement, which is what makes it a fact about galaxy formation rather than a fact about surveys.

The size of the faint-end gap is the reason this was once called a problem rather than a fact. Counting halos in a simulation of the Local Group and asking how many satellites the Milky Way should have gave hundreds against the eleven then known, and for fifteen years that was the missing-satellite problem. It has since been resolved from both directions — deep surveys have found dozens more, faint and diffuse, and the feedback arguments below predict that most of the rest contain no stars at all — and what is left is a quantitative question about how far down the suppression extends.

The pattern is worth noting because it recurs whenever a calculation is compared with a count. A discrepancy between a prediction that includes everything and an observation that includes only what was bright enough to see is not evidence against the prediction until the selection is modelled, and the selection here is a surface-brightness limit rather than a flux limit, which is the harder of the two to state.

The bright end, where there are too many baryons

At high mass the disagreement reverses in cause and repeats in direction. The most massive halos have far more baryons than their galaxies have stars — a halo of 101410^{14} solar masses holds 1.6×10131.6\times10^{13} of baryons and its central galaxy holds perhaps 101210^{12} of stars.

Here the gas is not escaping; the potential is far too deep for that. It is failing to cool.

Gas falling into a massive halo is shock-heated to the virial temperature, which for a cluster is 10710^{7}–10810^{8} K. At those temperatures the dominant cooling process is bremsstrahlung, whose rate rises only as the square root of the temperature while the amount of energy to be radiated rises linearly — so the cooling time rises with halo mass, and above about 101210^{12} solar masses it exceeds the age of the universe for most of the gas.

That is a threshold mass with no free parameters in it, set by the crossing of a cooling curve and an expansion rate, and it lands near the knee.

The cooling argument is not quite sufficient on its own, because gas at the centre of a cluster does have a short enough cooling time, and the star formation that should result is not observed. What prevents it is the active nucleus: the black hole at the centre accretes some of the cooling gas and returns energy to the rest, and the balance is close enough to maintain itself.

The evidence for that balance is visible rather than inferred. X-ray images of cluster centres show cavities in the hot gas, inflated by radio jets from the central galaxy, and the work done in inflating them is close to the energy the gas would otherwise have radiated away. A feedback loop that regulates itself is one of the few mechanisms in galaxy formation that can be watched operating, and the mass of the hole doing it correlates with the bulge it sits in across three decades, which is a relation the loop is usually invoked to explain.

The residual worry is one of arithmetic rather than of mechanism. A black hole of 10910^{9} solar masses accreting at a per cent of its Eddington rate produces enough power to hold a cluster’s cooling in check with a wide margin, so the question is not whether there is enough energy but why the coupling is so finely tuned that neither runaway cooling nor runaway heating occurs. Nothing in the efficiency curve answers that, and the curve’s smoothness at the high-mass end is the observational statement that something does.

The curve between them

Dividing one count by the other, rank by rank, gives the quantity the whole comparison is for.

The assumption is the crudest available: that the $n$th most numerous halo hosts the $n$th most numerous galaxy. There is no physics in it at all — only a claim that the ordering is preserved — and the curve it produces is reproduced by every more sophisticated method, which is the interesting part.

Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.6 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 21 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.
Fig. 3 Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The curve peaks at 22 per cent, at a halo mass of 1011.89 M⊙10^{11.89}\,M_\odot, and falls to 0.6 per cent at the bottom of the range and 0.3 at the top. A halo of the Milky Way’s mass sits near the peak at 21 per cent, and near the peak means near the best any halo has ever managed. Both sides are steep: the efficiency falls by more than a factor of thirty within two decades of halo mass in each direction.

The peak is the single most quoted number in galaxy formation, and its position matters more than its height. It sits at a halo mass of about 101210^{12} solar masses, which is the Milky Way’s, and the coincidence is not one — the Milky Way is a typical galaxy precisely because 101210^{12} is where galaxies are made most efficiently.

Two mechanisms with completely unrelated physics — a supernova-driven wind escaping a shallow potential, and a cooling time exceeding a Hubble time in a deep one — happen to hand over at the same mass, and the handover is what produces the knee in the luminosity function this comparison started from.

Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.4 per cent at the bottom of the range and 0.1 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 21 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.
Fig. 4 The same curve across a wider range of halo masses. The peak does not move, the falls on either side continue, and the range over which any halo converts more than a tenth of its baryons into stars is under two decades wide. Galaxy formation is a narrow business. Everything in the universe that looks like a galaxy was made in a band of halo masses that is small compared with the range of halo masses that exist.
Two counts that are not the same shape, at either end. The galaxy stellar mass function drawn against the halo mass function, both per cubic megaparsec per dex, with the halo masses multiplied by the cosmic baryon fraction 0.157 so that the upper curve is what the count would be if every halo had turned every baryon it owns into stars. The halo count comes from a linear power spectrum, a top-hat variance and the Sheth–Tormen multiplicity — nothing in it knows that stars exist. The two curves differ at both ends and agree nowhere. At 10⁸ solar masses of baryons the halos outnumber the galaxies by 64.1; at 10^11.5 by 4.8; and in between, near 10^10.5, the gap narrows to 3.1. A single mismatch would say only that the conversion is inefficient. Two mismatches with a minimum between them say that something different is happening at each end — and the knee of the luminosity function, which looked like a property of galaxies, is the place where those two somethings hand over.
Fig. 5 The same pair with the galaxy function’s knee moved up by a quarter of a dex — a factor of 1.7 in mass, which is the difference between two published initial mass functions. The bright-end ratio collapses from 64 to 4.8 and the faint-end one is barely touched. A quarter of a dex is not a large change and it very nearly removes one of the two mismatches, which is the arithmetic behind a long-running argument: the claim that galaxy formation is inefficient in the most massive halos is, in part, a claim about how much light is missed in the outskirts of a brightest cluster galaxy and about which stellar population its mass was inferred with.

The two disagreements are therefore not equally secure. The faint-end one survives every plausible change to the measurement, because it is a difference of slopes across three decades and no systematic tilts a curve that far. The bright-end one is a difference of a factor at one place, and a factor is exactly what a systematic produces.

What settles the bright end is not the count at all but the gas. A cluster’s hot intracluster medium is observed directly in X-rays, its mass measured from the emission, and it comes to several times the stellar mass of every galaxy in the cluster put together. The baryons are visibly present and visibly not in stars, which converts an inference from two counts into a direct observation of where the material went.

Why the crude assumption works

Abundance matching should not work as well as it does, and understanding why it does is worth a paragraph.

The assumption is that the relation between halo mass and stellar mass is monotonic and has no scatter. Neither is true — two halos of identical mass host galaxies differing in stellar mass by about 0.15 dex — and a relation with scatter is not the same as a relation without it, particularly at the exponential end where the count is steep.

What rescues it is that the scatter is small compared with the dynamic range. Over two decades of halo mass the stellar mass changes by three, so a 0.15 dex scatter perturbs the rank ordering only locally, and the monotone map it produces is right except in detail.

Where it fails is exactly where the scatter matters: at the massive end, where the same Eddington-bias arithmetic that inflates a mass function’s bright end also inflates the inferred efficiency. Methods that model the scatter explicitly — fitting a conditional distribution of stellar mass at fixed halo mass rather than a one-to-one map — return a peak in the same place and a high-mass tail that falls faster.

Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.6 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 2·10¹¹ M☉, sits near the peak at 10 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.
Fig. 6 The same curve with the Galaxy’s halo mass drawn at 2×10112\times10^{11} rather than 1.3×10121.3\times10^{12} solar masses — the lower end of the range different methods return. The efficiency read off for it drops by a factor of three. The Milky Way’s own halo mass is uncertain by a factor of about two, measured from satellite kinematics, from the local escape velocity, and from the timing argument against Andromeda, and the three do not agree well. So the statement that the Galaxy sits at the peak is a statement with a factor of two in it, and it survives only because the peak is broad.
Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 27 per cent, at a halo mass of 10^11.70 M☉, and falls to 7.1 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 23 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.
Fig. 7 The efficiency curve computed against a galaxy function with a steeper and more abundant low-mass component. The peak stays where it is and its height barely moves; what changes is the left-hand fall, which becomes shallower because there are more small galaxies to assign to the small halos. The peak is the robust feature and the sides are not. Every published version of this curve agrees on where galaxy formation works best and disagrees about how badly it fails away from there, which is the honest summary of two decades of work on it.

That robustness has a cause worth naming. The peak’s position is fixed by where the two counts are closest, and both counts are steep there, so a change in either moves the crossing very little. The wings are set by differences of slope between two functions whose slopes are each uncertain, and a difference of uncertain slopes is the least well determined thing a pair of curves can produce.

So the useful statements from this construction are ordinal rather than cardinal. Galaxy formation is most efficient at about 101210^{12} solar masses; it is much worse in both directions; and the efficiency at the peak is of order a fifth rather than of order one. Every one of those survives the disagreements, and no fourth statement does.

An observation against a calculation, and the lensing that checks both

The galaxy count is measured, with the caveats a mass function’s own inference carries and one more: it is the count in the local universe, and the efficiency curve is therefore the present-day one.

The halo count is not measured at all. It is computed, and what has been measured is the cosmology it is computed from — the power spectrum’s amplitude and shape, from the microwave background and from galaxy clustering.

So the comparison is between an observation and a calculation, which makes it a test of galaxy formation only if the cosmology is right. That was a live worry for two decades and is no longer one: the halo mass function’s ingredients are now measured to a per cent or better, and the factor of eighty at the faint end is not a cosmological uncertainty.

The one genuinely independent check on the halo count is gravitational lensing. The mass in and around a galaxy can be weighed by the shear it puts on the galaxies behind it, stacked over many lenses to beat down the noise, and the resulting halo masses at fixed stellar mass agree with the abundance-matched ones. That agreement is the strongest evidence the whole construction rests on.

Halos drawn as though they were isolated

The halo mass function drawn is a Sheth–Tormen fit to a BBKS transfer function, which is accurate to perhaps twenty per cent in the count at a given mass. The efficiency curve spans two orders of magnitude, so its shape survives that and its third significant figure does not.

Halos are not isolated. A large fraction of low-mass halos are subhalos inside larger ones, and their galaxies have been stripped, quenched or destroyed by the host. The count drawn treats every halo as independent, and the correction is largest exactly at the faint end where the disagreement is largest.

And the efficiency is a ratio of two present-day quantities, which is not the efficiency of anything. A halo of 101210^{12} solar masses today was smaller when its stars formed, so the number drawn is the stellar mass now over the baryon budget now, and not the fraction of gas that was available at the time which was converted.

The one number the curve is usually used for

Beyond its shape, the efficiency relation has a routine use that is worth stating because it is how most people meet it.

Given a galaxy’s stellar mass, the relation returns its halo mass. That is how the halo masses of individual galaxies are assigned in almost every catalogue that has them, and it is how a galaxy sample is split into centrals and satellites, how its clustering is predicted, and how its environment is defined.

The circularity is mild but real: the relation was derived from the counts, and using it to assign halo masses to the galaxies that produced the counts adds no information. What makes it useful is that it can then be applied to subsamples — galaxies of a given colour, or morphology, or star formation rate — and the residual variation at fixed stellar mass is what the more sophisticated versions measure.

The scatter at fixed stellar mass turns out to correlate with almost nothing. A galaxy’s halo mass is predicted by its stellar mass and barely improved by knowing anything else about it, which is either a deep fact about galaxy formation or a statement that the measurements are not good enough to see the second variable. Both readings are defended.

An argument that is really about timescales

Underneath the two mechanisms sits one comparison, made twice with opposite outcomes, and seeing it that way makes the shape of the efficiency curve inevitable.

Gas turns into stars if it can cool and collapse faster than something removes it. At the faint end the removal is fast — a supernova wind crosses a dwarf galaxy in a few million years — and the cooling is fast too, so the competition is close and the shallow potential decides it.

At the bright end the removal is slow and the cooling is slower still, so the competition is again close and the deep potential decides it, in the other direction.

In the middle neither is decisive, and the gas simply forms stars. The peak is not a place where something works especially well; it is a place where both of the things that go wrong are weak.

That is a shape worth recognising, because it recurs. The radius valley in the exoplanet population has the same structure — two processes bounding a range, with the interesting population in between — and so does the band of stellar masses that can support a stable hydrogen-burning core. A peaked distribution in nature is usually two failures rather than one success.

The reading also settles a question of emphasis that the literature does not always make clear. Feedback is usually described as the thing that regulates galaxy formation, which suggests a mechanism actively holding a rate at some value. What the two-sided curve shows is nearer to the opposite: the regulation is a boundary condition, imposed from outside the band, and inside the band the gas does what gas does. A galaxy at the peak is not being regulated. It is being left alone, and it is the only kind of galaxy that is.

Still open: whether the peak has moved

Everything here is the present-day relation. The interesting question is whether the peak’s position has changed over cosmic time, because a peak that has stayed put would mean that the physics setting it depends only on halo mass, while one that has moved would mean it depends on epoch.

Measuring it requires stellar mass functions at high redshift, which requires mass-to-light ratios for galaxies whose stellar populations are younger and dustier than anything local, and halo mass functions at the same redshifts, which the calculation supplies readily.

The current answer is that the peak sits at a slightly higher halo mass at earlier times and that the efficiency at the peak was somewhat higher, both at a significance that would not survive a change in the assumed initial mass function. The measurement is limited by the same inference a stellar mass function carries.

And there is a prior question: whether one efficiency curve is even the right object. A galaxy’s stellar mass does not depend only on its halo’s mass, and the most conspicuous second variable is not a property of the galaxy at all — it is what else is nearby.

Where this goes next

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. 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. 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.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Abundance matchingBaryon fractionCooling timeDark matter haloFeedbackHalo mass functionQuenchingSchechter functionStellar mass functionSupernova feedback