Astronomy & Celestial Mechanics
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Galaxies

A gradient the old stars have walked away from

The gas in a disc galaxy is richer in metals near the centre than at the edge, by about six-hundredths of a dex per kiloparsec in the Milky Way. Two ingredients of disc growth make that slope, a disc that grows from the inside out makes it flatten with time — and the old stars that should carry the steeper history have moved several kiloparsecs from where they were born.

Assumes Chemical evolution, Galactic structure and Star formation.

15 min read 6 figures Small perturbations, large consequencesMeasuring the unreachable

A galaxy’s gas has one metallicity only in the equilibrium that makes the mass–metallicity relation a law. Inside any disc galaxy the gas is richer towards the centre. In the Milky Way, measured from the young Cepheid variables and ionised nebulae that trace the present-day gas, the oxygen and iron abundances fall by about 0.06 dex for every kiloparsec out from the centre — a factor of four between the inner disc at 4 kpc and the outer disc at 14.

A gradient that steep is not an accident of where stars happened to form. It is a record of how the disc was assembled, and it is the first place in a galaxy’s chemistry where time and space are tangled: the gradient seen today is a snapshot of a pattern that has been changing for ten billion years, and the stars that could tell how it changed have been moving the whole time.

A disc built from the inside out, with a gradient of −0.066 dex per kiloparsec at 12 Gyr. The metallicity of the gas, in solar units on a logarithmic scale, against galactocentric radius, at ages of 2, 6, 12 Gyr, for a disc in which every ring is its own box with infall: pristine gas arrives on a timescale that grows with radius, from 1 Gyr in the centre to 7 Gyr at 8 kpc, turns into stars on the depletion time of a Kennicutt law, which is shorter where the gas is denser and much longer below a threshold of 7 M☉ pc⁻², and keeps everything it makes. The slopes fitted between 4 and 14 kpc: −0.259 dex/kpc at 2 Gyr, −0.124 dex/kpc at 6 Gyr, −0.066 dex/kpc at 12 Gyr. The inner disc has had its gas early and turned it over many times, so it is near the yield; the outer disc is still accreting and forming stars slowly, so its gas is diluted and young in the chemical sense. At 8 kpc the model's present abundance is 1.16 of the yield. Every ring is independent here: no gas flows between them and no star moves, which are the two processes that real discs add and which both act to flatten what is drawn.
Fig. 1 The metallicity of the gas against galactocentric radius at 2, 6 and 12 Gyr, for a disc in which each ring accretes pristine gas on a timescale that grows with radius — 1 Gyr at the centre, 7 Gyr at 8 kpc — and forms stars on a density-dependent depletion time with a threshold at 7 M☉ pc⁻². The gradients fitted between 4 and 14 kpc are −0.259, −0.124 and −0.066 dex/kpc. At the Sun’s radius the present abundance is 1.16 of the yield. The rings are independent; no gas flows between them and no star moves.

What the slope is measured with

Measuring a gradient from inside the disc it belongs to is a problem of its own. Every tracer needs two numbers — a composition and a distance from the Galactic centre — and each tracer is good at one and poor at the other.

The cleanest present-day tracer is the Cepheid variables. They are young, a few tens to a couple of hundred million years old, so they carry the composition of the gas they formed from with little time to have moved; they are luminous enough to be seen across much of the disc; and a Cepheid gives its own distance from how slowly it blinks, which is what turns an abundance into a point on a gradient. Their spectra give iron, oxygen and a dozen other elements directly, and their gradient between about 5 and 15 kpc is close to −0.06 dex/kpc in iron.

Ionised nebulae round young massive stars trace the gas itself and give oxygen, but through the same temperature-sensitive lines whose calibration divides the mass–metallicity relation, and their distances come from kinematics, which near the Sun’s line of sight to the centre and anticentre are poorly defined. Young B stars give abundances with different systematics again. Where the three overlap they agree on a slope of 0.04 to 0.07 dex/kpc, which is the band every model of the disc has to hit.

The dust in the plane is the limiting factor at both ends. The inner disc inside about 4 kpc, where a bar dominates the dynamics, is hidden at optical wavelengths behind tens of magnitudes of extinction, and infrared spectroscopy of giants there is only recently reaching the precision of the optical work nearer the Sun. The fitted range of 4 to 14 kpc used in every figure here is the range in which the gradient is actually measured.

A ring at a time

The model behind every figure here is the box with infall that the solar neighbourhood’s metallicity distribution demanded, applied separately to each ring of the disc. Each ring receives pristine gas at a rate that decays exponentially; turns gas into stars at a rate set by the local gas surface density; returns forty per cent of each generation’s mass promptly, with new metals; and keeps the rest. The total mass each ring eventually receives follows the disc’s exponential profile, with a scale length of 2.6 kiloparsecs.

Two things in that description can depend on radius, and each is observed to.

The infall timescale. Discs are thought to form from the inside out: gas with low angular momentum settles near the centre first, gas with more settles further out and later. In chemical-evolution models of the Milky Way the infall timescale has for decades been taken to grow roughly linearly with radius, from about a billion years in the inner disc to seven at the Sun and longer beyond. The evidence is indirect but consistent — the inner disc’s stars are older, the outer disc is still gas-rich, and galaxies at earlier times are more compact.

The efficiency. The rate of star formation per unit area rises faster than the gas surface density does, roughly as its 1.4 power, so the depletion time is shorter where gas is denser. And below a surface density of a few solar masses per square parsec, star formation becomes much less efficient again: a disc that thin is stable against the collapse that forms clouds, and outer discs are observed to hold neutral hydrogen they are barely turning into stars.

Neither ingredient alone

Two ingredients make the gradient: −0.066 with both, 0.000 with neither. The gas metallicity against radius at 12 Gyr for the same disc built four ways, separating the two things that can make its outer parts poorer. Infall that is inside-out delivers gas to the inner disc first, so the outer disc is younger in the sense that matters. A depletion time that is shorter where the gas is denser — and far longer below the threshold density at which discs stop forming stars efficiently — processes the inner gas faster and leaves the outer gas unprocessed. With both, the gradient fitted between 4 and 14 kpc is −0.066 dex/kpc; with inside-out infall alone it is −0.020; with the efficiency law alone −0.049; with neither — the same infall timescale and the same depletion time everywhere — 0.000, because every ring is then the same box scaled in mass and a box's abundance does not depend on its mass. A gradient is not evidence of either ingredient by itself: the present-day gradient of the gas is reproduced by different combinations, and what separates them is how the gradient has changed with time, which only old stars and distant galaxies record.
Fig. 2 The same disc at 12 Gyr built four ways. With inside-out infall and the density-dependent efficiency together, the gradient is −0.066 dex/kpc; with inside-out infall alone, −0.020; with the efficiency law and threshold alone, −0.049; with neither — one infall timescale and one depletion time everywhere — 0.000. A disc whose rings differ only in how much gas they receive has no gradient at all, because a box’s abundance does not depend on its mass.

The last case is the instructive one. A disc in which every ring runs the same history, scaled up or down in mass, has exactly the same metallicity at every radius. The exponential profile — more gas in the centre than at the edge — does nothing by itself, because metallicity is a ratio and a ratio does not care how much of everything there is. A gradient needs the rings to run different histories.

Inside-out infall supplies one difference: at any given time, an outer ring has received a smaller fraction of its eventual gas and is still being diluted. The efficiency law supplies another: an outer ring with less gas per unit area turns it into stars more slowly, so it has cycled its gas fewer times. The threshold adds a strong third effect at large radii, where gas can sit for billions of years without forming the stars that would enrich it.

Each ingredient alone produces a gradient, and neither alone produces the observed one in this model. That is the kind of result a model should be read for. It does not say that these are the only ingredients — radial gas flows and radially varying winds are both real and both move the answer — but it says that the present gradient is a joint product, and that any single observation of it constrains a combination.

A slope that changes with age

The gradient of an inside-out disc flattens from −0.345 to −0.065 dex per kiloparsec. The slope of the gas metallicity gradient, fitted between 4 and 14 kpc, against the age of the disc, for a disc whose infall is inside-out and for one whose infall timescale is the same at every radius; both have a depletion time that depends on the local gas density. The inside-out disc starts steep — −0.345 dex/kpc at 1 Gyr, when the outer rings have barely begun — and flattens to −0.065 as they catch up. The uniform disc goes from −0.297 to −0.048. The two present-day values are within reach of one another, and the histories are not: a gradient measured today cannot say how the disc was built, but the gradient that planetary nebulae, open clusters and old stars carry from earlier epochs can, if their birth radii are known. That last condition is the difficulty, because stars move.
Fig. 3 The fitted gradient between 4 and 14 kpc against the age of the disc, for inside-out infall and for infall with the same timescale at every radius; both have the density-dependent efficiency. The inside-out disc starts at −0.345 dex/kpc at 1 Gyr, when the outer rings have barely begun, and flattens to −0.065 as they catch up. The uniform disc goes from −0.297 to −0.048. Today’s values are close to one another; their histories are not.

An inside-out disc is steepest when it is youngest. In its first billion years the inner rings have received most of their gas and enriched it, while the outer rings have received almost none; the abundance contrast is enormous. As the outer rings fill, they process their gas and approach the yield too, and the gradient relaxes. By twelve billion years it has flattened by a factor of five.

That prediction has two ways of being tested, and both are hard. The first is to look back: measure abundance gradients in disc galaxies at high redshift, whose light left when the universe was a few billion years old. Such measurements need spatially resolved spectroscopy of distant galaxies, possible for strongly lensed systems and with adaptive optics, and the results are a mixture — some galaxies at redshift two have steep negative gradients, many have flat ones, and a few have inverted gradients with metal-poor centres, which is what a galaxy looks like when fresh gas has just been funnelled into its middle. The average is flatter than the simple inside-out model predicts, and the spread is larger than any smooth model produces.

The model’s own numbers say what such a survey should find for a galaxy like this one. Six billion years into its history — roughly the epoch of redshift one for a disc that began forming at redshift three — its gradient would be −0.124 dex/kpc, about twice today’s; at two billion years, −0.259. Those are large slopes, easily measurable in principle, and they are steeper than most measured high-redshift gradients. Either discs at that epoch were not assembling from the inside out as cleanly as the model assumes, or their gas was being stirred radially by the turbulence and mergers that high-redshift discs visibly have, or the measurements — made at a resolution of a kiloparsec or more on galaxies a few kiloparsecs across — are smoothing real gradients flat. Beam smearing of that kind biases every gradient towards zero, and correcting for it needs the very profile being measured.

The second test is to look inside the Milky Way at objects whose ages are known: open star clusters, planetary nebulae whose progenitors had different masses and therefore different ages, and individual stars with ages from asteroseismology. Each carries the abundance of the gas it formed from at the place it formed. Plotted against radius by age group, they should draw the history in the figure.

Where the old stars have gone

They do not quite, and the reason is that the place a star formed is not the place it is found.

A star’s orbit in a disc changes over time. Encounters with molecular clouds and spiral arms gradually heat an orbit, making it more eccentric and more inclined, and that heating is observed: older stars have larger velocity dispersions. But there is a second process that changes a star’s orbit without heating it. A star near the corotation radius of a spiral pattern — where the pattern and the star move at the same angular speed — can exchange angular momentum with the arm and be moved inward or outward by kiloparsecs while keeping a nearly circular orbit. A spiral pattern is allowed to turn only where the resonances permit, and its corotation radius is where this happens. Because transient spirals come and go with different pattern speeds, every part of the disc is at some corotation at some time.

A star that has migrated this way looks entirely at home. Its orbit is circular, its velocity dispersion unremarkable; only its chemistry — carried from a birth radius several kiloparsecs away — is out of place.

Stars 8–10 Gyr old were born on a gradient of −0.195 and now show −0.045. The mean metallicity of stars against the radius at which they are found today, for two age groups — 0–1 Gyr, 8–10 Gyr — beside the gradient of the gas they formed from at the radii where they formed. Each star inherits its birth ring's gas abundance at its birth time and then moves in radius, with a spread that grows as the square root of its age: 0.8 kpc for the 0–1 Gyr stars, 3.3 kpc for the 8–10 Gyr stars. Migration of this kind happens without heating the orbits, when a star is carried along by a spiral arm at corotation, so the migrants remain on nearly circular orbits and look entirely at home where they are found. The youngest stars' gradient barely changes, from −0.069 to −0.064 dex/kpc. The oldest stars' is flattened from −0.195 to −0.045. An old population's gradient is therefore a lower limit on the gradient it was born with, and the solar neighbourhood's oldest stars include many born several kiloparsecs inside the Sun's orbit — which is part of why stars of the same age near the Sun span a wide range of metallicity. The migration spread is a representative value from simulations and from the scatter in the age–metallicity relation, not a measurement for this disc.
Fig. 4 The mean metallicity of stars against the radius where they are found today, for stars 0–1 Gyr and 8–10 Gyr old, beside the gradient of the gas at the radii and times where they formed. Each star moves in radius with a spread growing as the square root of its age: 0.8 kpc for the youngest, 3.3 kpc for the oldest. The young stars’ gradient barely changes, from −0.069 to −0.064 dex/kpc. The old stars were born on a gradient of −0.195 and are now found on one of −0.045 — flatter than the gas today.

The figure’s result is stark. The old stars were born when the disc’s gradient was steep, three times steeper than the present gas. After nine billion years of migration with a spread of three kiloparsecs, the gradient they show is flatter than the gas’s present gradient. A population that should record the disc’s steeper past records, instead, a gradient shallower than anything the gas has ever had.

Stars 8–10 Gyr old were born on a gradient of −0.195 and now show −0.045. The mean metallicity of stars against the radius at which they are found today, for three age groups — 0–1 Gyr, 4–6 Gyr, 8–10 Gyr — beside the gradient of the gas they formed from at the radii where they formed. Each star inherits its birth ring's gas abundance at its birth time and then moves in radius, with a spread that grows as the square root of its age: 0.8 kpc for the 0–1 Gyr stars, 2.5 kpc for the 4–6 Gyr stars, 3.3 kpc for the 8–10 Gyr stars. Migration of this kind happens without heating the orbits, when a star is carried along by a spiral arm at corotation, so the migrants remain on nearly circular orbits and look entirely at home where they are found. The youngest stars' gradient barely changes, from −0.069 to −0.064 dex/kpc. The 4–6 Gyr stars' goes from −0.110 to −0.052. The oldest stars' is flattened from −0.195 to −0.045. An old population's gradient is therefore a lower limit on the gradient it was born with, and the solar neighbourhood's oldest stars include many born several kiloparsecs inside the Sun's orbit — which is part of why stars of the same age near the Sun span a wide range of metallicity. The migration spread is a representative value from simulations and from the scatter in the age–metallicity relation, not a measurement for this disc.
Fig. 5 The same calculation with an intermediate group, 4–6 Gyr old, which migrated with a spread of 2.5 kpc. Its gradient goes from −0.110 at birth to −0.052 today; the oldest stars go from −0.195 to −0.045. Birth gradients steepen with age and observed gradients flatten, so the observed gradient’s change with age runs the opposite way to the birth gradient’s — and the migration spread assumed here is representative of simulations, not a measurement for the Milky Way.

That inversion matters for what the observations have been taken to show. Open clusters and planetary nebulae of different ages in the Milky Way have gradients that are similar, or somewhat flatter for older objects. Read naively, that says the gradient has flattened slowly or not at all. Read with migration, it is consistent with a gradient that was once much steeper and has been blurred in the stars that carried it — and the naive and the migrating readings cannot be separated without knowing each star’s birth radius.

Birth radii from chemistry

Birth radii cannot be observed, but they can be inferred, and the inference turns the problem inside out. If the gas at any radius and time had a definite metallicity — a smooth gradient at every epoch — then a star’s age and metallicity together pick out where it formed. A star of 8 billion years and solar metallicity, found at the Sun, cannot have formed at the Sun’s radius if the gas there 8 billion years ago was a third of solar; it must have formed further in. Surveys with precise ages and abundances for hundreds of thousands of stars now reconstruct birth radii this way, and they find that a large fraction of the solar neighbourhood’s old stars formed kiloparsecs inside the Sun’s orbit.

This also explains a long-standing puzzle. Stars of the same age near the Sun have a wide range of metallicities — nearly a dex — which a single ring enriching steadily with time cannot produce. Mixing stars from rings with different histories produces it naturally. The age–metallicity relation near the Sun is flat and broad not because the local gas never enriched but because the local stars are a sample of the whole disc.

The circularity is obvious and it is not fatal. The inferred birth radii depend on the assumed gradient history, and the gradient history is what the birth radii were supposed to test. The way out is consistency: the gradient history must reproduce the observed distribution of ages and abundances everywhere in the disc at once, including the metallicity distributions at other radii, which the solar neighbourhood’s histogram is one slice of.

What winds do to a gradient

A disc built from the inside out, with a gradient of −0.031 dex per kiloparsec at 12 Gyr. The metallicity of the gas, in solar units on a logarithmic scale, against galactocentric radius, at an age of 12 Gyr, for a disc in which every ring is its own box with infall: pristine gas arrives on a timescale that grows with radius, from 1 Gyr in the centre to 7 Gyr at 8 kpc, turns into stars on the depletion time of a Kennicutt law, which is shorter where the gas is denser and much longer below a threshold of 7 M☉ pc⁻², and loses as much gas in a wind as it locks into stars. The slope fitted between 4 and 14 kpc: −0.031 dex/kpc at 12 Gyr. The inner disc has had its gas early and turned it over many times, so it is near the yield; the outer disc is still accreting and forming stars slowly, so its gas is diluted and young in the chemical sense. At 8 kpc the model's present abundance is 0.42 of the yield. Every ring is independent here: no gas flows between them and no star moves, which are the two processes that real discs add and which both act to flatten what is drawn.
Fig. 6 The inside-out disc with a wind that removes as much gas from each ring as it locks into stars. The present gradient flattens from −0.066 to −0.031 dex/kpc, and the abundance at the Sun’s radius falls to 0.42 of the yield. A uniform wind lowers the inner disc more than the outer in logarithm, because the inner disc was closest to saturation; winds that are stronger where the potential is shallower — at large radii — would steepen the gradient instead.

The wind the mass–metallicity relation requires is small for a galaxy as massive as the Milky Way, and a wind that is uniform across a disc flattens its gradient by lowering the inner, saturated part most. But winds need not be uniform. Supernova-driven outflows escape more easily from the thin outer disc, where both the gravitational restraint and the column of gas above the plane are smaller. Such a wind acts on the outer disc preferentially and steepens the gradient. The sign of a wind’s effect on a gradient therefore depends on where the wind is launched, and that is not something the global relation can say.

The same is true of gas flowing radially within the disc. A slow inward flow of a kilometre per second, driven by the viscosity of the gas or by the torques of bars and spirals, carries metal-poor outer gas inwards and dilutes the inner disc, and depending on its radial profile it can flatten or steepen the present gradient. Every ingredient that moves material in radius moves the gradient, and the rings of the model drawn here, independent of one another, are the limiting case in which nothing does.

Beyond the edge the model assumes

Past about twelve to fifteen kiloparsecs the Milky Way’s gradient does not continue falling. Open clusters and Cepheids in the far outer disc have abundances that level off near a third of solar, and the neutral hydrogen there extends well beyond the stars, in a disc that is warped and flared.

The ring model has a partial explanation in its threshold: below 7 M☉ pc⁻² star formation is slow, so outer gas neither enriches nor dilutes much, and its abundance is set largely by whatever it arrived with. That points at the assumption the model makes most confidently and should make least — that infalling gas is pristine. Gas accreted onto the outer disc today has passed through the halo, where it has mixed with metals blown out of the inner disc over billions of years. An outer disc fed with gas at a tenth of solar abundance has a floor at a tenth of solar, and a gradient that flattens into it.

The outer disc is also where the figures’ distances become least certain and the surface brightness of the stars becomes something distance cannot help with: an outer disc is faint per unit area however near it is, and its stellar populations are measured star by star or not at all. In external galaxies the same flattening appears in the outer parts of discs where only the gas can be measured, which suggests it is general rather than a property of the Milky Way’s own history.

A common slope in the right units

A result from integral-field surveys of hundreds of nearby disc galaxies puts the Milky Way in context. Measured in kiloparsecs, gradients vary widely from galaxy to galaxy. Measured in units of each galaxy’s own effective radius, they are strikingly similar: about −0.1 dex per effective radius, with small scatter, for galaxies of very different masses. That is what inside-out growth predicts if the infall timescale scales with the disc’s own size rather than with an absolute distance. The Milky Way’s gradient, converted to its own effective radius, is somewhat steeper than the common value — which may be telling or may be an artefact of measuring a gradient from inside the disc, through dust, over a limited range of radii.

Still open: when the white dwarfs explode

Every model here assumes that metals are returned promptly. Oxygen is, because it comes from massive stars that live a few million years. Iron is not: most of it comes from Type Ia supernovae, which explode long after the stars that made them formed, with delays that range over three orders of magnitude. A gradient in iron and a gradient in oxygen are therefore gradients measured with two different clocks, and the ratio of the two elements — the α-knee that the first model of this subject used as a single delay — depends on the whole distribution of those delays, which has been measured and is not a single number.

Where this goes next

A slope of 0.57 at the low-mass end, against the 0.59 and 0.24 two winds give there. Gas-phase oxygen abundance against stellar mass for star-forming galaxies. The solid measured curve is the relation found from electron-temperature abundances in stacked spectra, which rises as a power of mass below a turnover near 10^8.9 solar masses and saturates above it at 12 + log(O/H) = 8.798. The two model curves are a galaxy in equilibrium with its gas supply, whose metallicity is the yield divided by one plus the mass it ejects per unit mass of stars and one plus the dilution by the gas it must keep accreting. Only the ejection changes with mass. A wind driven by the energy of supernovae must lift gas out of a potential whose depth goes as v², so its loading goes as v⁻² and, with v ∝ M^(1/3), as M^(−2/3); a wind driven by momentum goes as v⁻¹ and M^(−1/3). Those set the limiting slopes far below the turnover, 0.67 and 0.33; at log M = 7.5, where dilution still matters, the drawn curves have slopes of 0.59 and 0.24. The measured slope there is 0.57. The dashed measured curve uses a strong-line calibration of the same galaxies' spectra; it sits higher, flattens sooner and has a slope of only 0.40 at log M = 9 — so which wind the data prefer is decided as much by the choice of abundance calibration as by the galaxies. Above the turnover every curve saturates, because a galaxy that ejects almost nothing keeps what it makes and its abundance is the yield, diluted. The metals a galaxy keeps measure what it threw away Small star-forming galaxies are metal-poor and large ones are not, in a relation tight enough to be a law. Read as an equilibrium between inflow, star formation and wind, its slope says how the wind is driven, its turnover says where galaxies stop losing what they make — and its redshift evolution says galaxies were poorer because they were still being filled. A one-parameter model, and 9 times too many metal-poor stars. The metallicity distribution of long-lived stars near the Sun — the bars — against two models of how a galaxy enriches itself. The closed box is as simple as a model of a galaxy can be: gas turns into stars, stars make metals and return them, nothing enters and nothing leaves. It has exactly one parameter, the yield, and it predicts the whole curve. It gets the peak roughly right and the tail catastrophically wrong: 13.7 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 9. This is the G-dwarf problem, and the reason it is an argument rather than a discrepancy is that the failure cannot be fixed by changing the yield: the yield sets where the peak is, and moving the peak to fix the tail moves it away from the data. What is wrong is a boundary condition. Let gas keep arriving — pristine, at roughly the rate it is being consumed — and the gas is never both abundant and metal-poor for long, so few stars form while it is. That is the second curve, with 3.2 per cent below -1, and it needed no new nucleosynthesis and no new parameter beyond the fact of accretion. The picture cannot show the thing that would settle it directly, which is the infall itself: the gas arriving on the disc now is a few solar masses a year spread over twenty kiloparsecs, and it has never been securely detected. A histogram that says the box was not closed The simplest model of a galaxy enriching itself has exactly one parameter and predicts the whole metallicity distribution of its surviving stars. The solar neighbourhood's disagrees, in a specific direction — and the failure is not in the nucleosynthesis but in a boundary condition. Where a 2-armed pattern is allowed to exist in the Milky Way. Angular speed against radius, with the two combinations that matter for a pattern of 2 arms. A wave turning at 25 km/s per kpc corotates with the stars at 8.7 kpc, has an inner Lindblad resonance at 1.9 kpc and an outer one at 13.8 kpc. Between the Lindblad radii the wave propagates; at them it is absorbed. Every one of these radii is found by bisection on the curves as drawn, not quoted — and none of them is a property of the pattern alone, because κ comes from the rotation curve, which comes from the mass. Where a pattern is allowed to turn A spiral wave has one pattern speed, and that single number decides where in the disc it can exist at all. The boundaries are set by the frequency at which a star rocks radially about its circular orbit — a frequency the rotation curve already contains.

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Abundance gradientAge metallicity relationChemical evolutionCorotationDepletion timeGalactic outflowGas infallInside out formationThe Kennicutt–Schmidt lawRadial migrationStar formation threshold