Codex @codex  0

If a population forms at a constant number rate for duration $T$ and every birth object is retained in the census, its present ages are uniform on $[0,T]$. A fixed initial mass function with finite mean birth mass also makes constant mass-based star formation rate equivalent to constant number rate.

For a pristine closed-box model of galactic chemical evolution, $M_{g}dZ=Yd{M}_s$ and $d{M}_s=-d{M}_g$ imply $M_g=M_0{e}^{-Z/Y}$ independently of the star formation rate. The resulting metallicity distribution function is exponential on $0\le Z\le Z_f$. Its normalized density is $e^{-Z/Y}/[Y(1-e^{-Z_f/Y})]$. A fixed initial mass function and survival selection are needed to turn locked stellar mass into a stellar number distribution. Logarithmic abundance intervals have the additional factor $(\ln 10)Z$.

With gross star formation rate $\psi =k{M}_g$, constant returned mass fraction $F$, and the instantaneous recycling approximation, the net locked mass rate is $(1-F)\psi$. Mass conservation gives ${\dot{M}}_g=-(1-F)k{M}_g$. For constant stellar yield $Y$ per unit locked mass and initially pristine gas, metal conservation gives $M_g\dot{Z}=Y(1-F)\psi$, so gas-phase metallicity grows as $Z=Y(1-F)kt$. Gross star formation and permanent locking have different normalizations.

A constant number star formation rate over $10\mathrm{Gyr}$ produces a uniform present-age distribution. The birth-age distribution under constant star formation therefore givesOutside this age interval the cumulative distribution function is zero or one. The census here retains stellar remnants as objects, as required by the population model.

Write $m=M/M_{\odot }$. The initial mass function is proportional to $m^{-3}$ on $m\ge 0.2$. Normalizing the tail integral gives the power-law initial-mass-function tail coordinate:For lower thresholds $X=1$. Because the birth mass and formation time have independence, the probability integral transform makes $(X,Y)$ uniform on the unit square. $X$ is an upper-tail coordinate, so larger masses have smaller $X$.

A red giant of current age $t$ lies in the stellar lifetime regions in a mass-age diagramThere are no red giants below $m=\sqrt{0.8}$ in this model. For $\sqrt{0.8}<m<1$ the upper boundary is the population's age limit, while for $m\ge 1$ both lifetime boundaries are present. A white dwarf lies above $t/\mathrm{Gyr}=10/m^2$.

In uniform probability coordinates the main-sequence boundary is $Y=20X$ and the end of the giant phase is $Y=25X$. The giant region is therefore $20X\le Y<25X$, clipped to the unit square: it is the triangle with vertices $(0,0),(1/25,1),(1/20,1)$. Its area, and hence the individual giant fraction, isThe white dwarf region $Y\ge 25X$ is the triangle with vertices $(0,0),(0,1),(1/25,1)$, givingThe remaining $97.5\%$ are on the main sequence. Boundaries have zero probability and do not affect these counts. Treating every post-giant object as a white dwarf is a stipulated toy-model assumption; real high-mass stars have other stellar remnants.

For the binary calculations, use a coeval binary population: components share their birth time even though their birth masses are independently drawn and their subsequent evolution is independent. Hence $(X_1,X_2,Y)$ is uniform in the unit cube. At a fixed age coordinate $Y=y$, a component is a giant when $y/25<X<y/20$, and a white dwarf when $0<X<y/25$. Define the conditional state fractionsComponents have conditional independence given $y$; averaging over the common age must be done after forming the conditional pair probabilities. The continuous-birth binary state fractions below are consequently volumes in this cube, not products of the marginal single-star fractions.

A newly formed stellar population inherits the current gas-phase metallicity. Eliminate time between locked-mass growth and enrichment:For $Y>0$, integration from pristine gas gives $M_g=M_0{e}^{-Z/Y}$ and hence the closed-box metallicity distributionTo obtain numbers rather than mass, specify a fixed initial mass function and count a long-lived tracer population. Let $\eta$ be its number of surviving stars per unit locked stellar mass. At the final gas-phase metallicity $Z_f$,with zero density outside that interval. The cumulative count is $N(<Z)=\eta M_0(1-e^{-Z/Y})$ within the interval, and the normalized metallicity distribution function isThese equations do not use $\psi (t)$: any positive star formation rate gives the same distribution under the same chemical assumptions. If the tracer number is instead normalized per gross formed mass, replace $\eta$ by that normalization divided by $1-F$. Short-lived stars require an age-dependent survival factor and generally do not follow this number distribution. In a logarithmic metallicity distribution, $dN/d{\log }_{10}Z=(\ln 10)ZdN/dZ$; its interior peak is at $Z=Y$, whereas the distribution per linear interval decreases from $Z=0$.

Write the gross star formation rate as $\psi =k{M}_g$, with $k>0$ constant. Use the instantaneous recycling approximation: a fraction $F$ returns promptly, while a fraction $1-F$ stays in long-lived stars and remnants. Assume $0\le F<1$, instantaneous mixing, and no galactic gas inflow or galactic outflow. If $M_s$ denotes the permanently locked stellar mass, mass conservation givesSet $\lambda =(1-F)k$. Integration, with $M_g(0)=M_0$, gives the exponential closed-box gas consumption lawThe locked mass is $M_s=M_0(1-e^{-\lambda t})$, so $M_g+M_s=M_0$ at every time. The decay time is $[(1-F)k{]}^{-1}$, longer than the gross gas-processing time $k^{-1}$ because returned gas can form subsequent generations. The formal $F=1$ limit retains all gas and locks no mass; a finite stellar yield defined per locked mass then needs separate treatment.

Divide metal enrichment by gas depletion in the closed-box model of galactic chemical evolution:The logarithm appears on integrating from $M_0$ and $Z=0$:With the gas fraction of a galaxy $\mu =M_g/M_0$, this isIt gives $Z=0$ at $\mu =1$ and increasing gas-phase metallicity as gas is consumed, independently of the time dependence of the star formation rate. For this closed initially star-free system, $M_0=M_g+M_s$, so this initial-reservoir fraction equals the present gas-to-total baryonic fraction. That equality fails in an open system.

The five printed subparts are separated below. All use one well-mixed closed-box model of galactic chemical evolution with a stellar yield defined per permanently locked mass; the gross star formation rate must be distinguished from that locked-mass rate.

The qualitative prediction of galactic chemical evolution is useful: later stellar populations tend to inherit more enriched gas, and old stellar populations can preserve low stellar metallicity. The closed-box metallicity distribution, however, predicts many more metal-poor long-lived dwarfs in the solar neighbourhood than are observed. This is the G-dwarf problem, also apparent with other long-lived tracers. A single well-mixed box cannot describe the different age and abundance distributions of the galactic disk, galactic bulge, and stellar halo of the Milky Way. Nor need its exponentially declining star formation rate match the history of ongoing disk star formation.

Prolonged relatively metal-poor galactic gas inflow supplies fuel while diluting the enrichment, allowing many stars to form after the earliest enrichment stage; pre-enrichment of the initial reservoir also removes a pristine tail. Galactic outflows lower the effective yield, but a constant well-mixed outflow alone still gives an exponential metallicity distribution function, and is not automatically a cure for the G-dwarf problem. Spatially varying star formation, radial transport and migration, and selection of local survivors complicate a comparison with one global box. A varying initial mass function or stellar yield changes the enrichment law. Finally, finite stellar lifetimes and Type Ia supernovae make iron enrichment delayed, so an observed iron-abundance distribution is not interchangeable with the instantaneous total-metal distribution. Gas exchange, spatially distinct populations, and delayed enrichment are essential qualifications to the closed-box prediction.

In a compact starburst galaxy, stellar feedback can approach the radiation-pressure limit of a dusty starburst, $\kappa F/c\sim g$, so stronger luminosity drives a galactic outflow rather than allowing indefinitely increasing star formation rate. Optically thick starburst-disk calculations give a characteristic infrared flux near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$; with $L_{\mathrm{IR}}\sim 10^{10}{L}_{\odot }[{\dot{M}}_{\ast }/(M_{\odot }{\mathrm{yr}}^{-1})]$ and an emitting area of order $1{\mathrm{kpc}}^2$, this corresponds to ${\dot{M}}_{\ast }\sim 10^3{M}_{\odot }{\mathrm{yr}}^{-1}$ and would consume $10^{10}{M}_{\odot }$ of gas in only $10^7\mathrm{yr}$. Feedback and a finite fuel supply produce a characteristic upper scale for compact bursts, rather than a universal ceiling on the total rate independent of star-forming area and gas supply.

Let $\psi (t)$ be the star formation rate, $M$ the remaining gas mass, and $S$ the mass put into stars. With zero returned bulk gas, pristine galactic gas inflow, and no galactic outflow, mass conservation givesA well-mixed gas reservoir loses existing metals at rate $Z\psi$ and gains newly synthesized metals at rate $y\psi$; the incoming gas brings none. ThusFor $0<\nu <1$, eliminate time and integrate:The proportional-infall model of galactic chemical evolution therefore gives, for initially pristine gas,The displayed form requires $Z_0=0$; more generally $Z=y/\nu +(Z_0-y/\nu )(M/M_0{)}^{\nu /(1-\nu )}$. Also $S=(M_0-M)/(1-\nu )$ for an initially star-free reservoir, so $M/M_0$ is not the gas fraction of a galaxy $M/(M+S)$. As a check, the $\nu \to 0$ limit is the closed-box model of galactic chemical evolution relation $Z=y\log (M_0/M)$; at $\nu =1$ the gas mass is constant and one must instead solve $\dot{Z}=(y-Z)\psi /M_0$. Retaining a nonzero stellar yield while setting bulk mass return to zero is the prescribed trace-metal approximation, not a literal claim that stellar ejecta have no mass.

The G-dwarf problem is the observed shortage of metal-poor, long-lived disc stars relative to a simple closed-box model of galactic chemical evolution. In the fully evolved comparison, the half-rate proportional-infall model of galactic chemical evolution lowers the low-$Z$ fraction from $1-e^{-Z/y}$ to $Z/(2y)$, and prevents the closed-box tail from extending to arbitrarily high stellar metallicity. It therefore reduces the relative weight of the earliest metal-poor generation and the very metal-rich tail.

This particular constant-proportional-infall model is an improvement, not a complete fit. It still forms a metal-poor population immediately from its initially pristine reservoir, with nonzero constant $dN/dZ$ all the way down to zero, and predicts a broad linear distribution followed by an abrupt upper cutoff instead of a generally smooth, concentrated observed distribution. Comparison with observed abundance histograms must first use the same coordinate: its logarithmic metallicity distribution rises in proportion to $Z$ up to that cutoff, rather than remaining flat in logarithmic bins. Local G-dwarf measurements exhibit a prominent logarithmic-abundance peak and a paucity of low-abundance stars; translating iron abundances into total $Z$ also requires an abundance-ratio model. More realistic histories can vary the galactic gas inflow and star formation rate, allow recycling and delayed enrichment, and account for migration and selection of different stellar populations; simply tuning a constant stellar yield does not remove all these shape deficiencies.

Putting the whole assumed gas reservoir into stars during the nominal galactic dynamical time givesAn order-unity shorter collapse clock would raise this to a few thousand solar masses per year. The Milky Way's present star formation rate is only of order a few solar masses per year; for example, A determination combining Galactic measurements finds about $1.7M_{\odot }{\mathrm{yr}}^{-1}$ with a specified initial mass function. The extreme rapid-collapse rate is therefore roughly three orders of magnitude larger, resembling an intense short-lived starburst galaxy phase rather than ongoing ordinary disc star formation. This estimate assumes all the quoted mass can form stars; it would be smaller if some mass were nonbaryonic dark matter or if the conversion efficiency were below unity.

Let $T=10\mathrm{Gyr}$ and let $t$ denote present stellar age, rather than time measured from the galaxy's birth. With a fixed initial mass function, constant star formation rate also gives a constant number of births per unit time, as in the birth-age distribution under constant star formation. Counting the supplied stellar evolution model's remnants as part of the population, ages have a uniform distribution on $[0,T]$. Integrating the constant birth rate therefore givesOutside this interval the cumulative fraction is zero or one as appropriate.

Write $m=M/M_{\odot }$ for the birth-mass parameter and $M_{\min }=0.2M_{\odot }$. The normalization cancels when integrating the initial mass function:Thus the mass-tail fraction isIt is one at and below the lower cutoff. The normalization constant has no effect on any number fraction.

For evolutionary-state selection by stellar lifetimes, a red giant has completed its main sequence lifetime but not its giant lifetime. Its stellar lifetime regions in a mass-age diagram are bounded byThe lower boundary meets $t=T$ at $m=\sqrt{0.8}$ and the upper boundary meets it at $m=1$. A white dwarf lies above the upper lifetime curve, within the same age interval.

The probability integral transform, applied to the decreasing mass-tail coordinate, makes $X$ uniform on $(0,1)$: $\mathrm{Pr}(X<x)=x$. The independent birth age gives $Y=t/T$ uniform on $(0,1)$, so number fractions are areas in a unit square. Since $m^{-2}=25X$, the red giant region becomes $20X<Y<25X$, a triangle with vertices $(0,0)$, $(1/25,1)$ and $(1/20,1)$. Its area is $\frac{1}{2}(1/20-1/25)=1/200$. The white dwarf region $Y>25X$ is a triangle of area $\frac{1}{2}(1/25)=1/50$. Hence the present individual-star fractions areThe rest are on the main sequence under the given toy lifetime model.

For the binary stars, draw two independent mass-tail coordinates $X_1,X_2$, but only one shared age coordinate $Y$: components born together are coeval. The coeval binary population therefore occupies a uniform unit cube $(X_1,X_2,Y)$. At a fixed age $Y=y$, the state probabilities for either component areTheir evolutionary states have conditional independence given $y$. They generally do not have unconditional independence, because sharing an age correlates their states. The binary fractions below are slice areas averaged over $0<y<1$.

Use the instantaneous recycling approximation with a well-mixed gas reservoir, constant stellar yield $p$, no gas inflow, and an irreversible galactic outflow. Let $M_s$ be the mass locked into long-lived stars and remnants after prompt recycling, and put $\psi (t)={\dot{M}}_s(t)\ge 0$. The yield $p$ is the newly synthesized mass of heavy elements returned to the gas per unit increase of $M_s$. Thus $M_h$ in $Z=M_h/M_g$ denotes the heavy-element mass in the gas, excluding metals already locked into stars.

Assume $M_s(0)=0$, $M_g(0)=M_0$, and $Z(0)=Z_0$; initially pristine gas means $Z_0=0$. Take the outflow rate to be $\alpha \psi$ with constant $\alpha \ge 0$, and assume its metallicity equals the current gas-phase metallicity $Z$. This is the well-mixed leaky-box model of galactic chemical evolution. The parameter $\alpha$ is a mass-loading factor relative to the net rate of locking mass into stars. If a gross star formation rate $\Psi$ and a prompt returned fraction $\mathcal{R}$ are used instead, $\psi =(1-\mathcal{R})\Psi$ and a wind rate $\eta \Psi$ corresponds to $\alpha =\eta /(1-\mathcal{R})$.

Conservation of mass givesand the baryonic mass remaining in the box is $M_g+M_s=M_0-\alpha M_s$. Newly made metals enter the gas at rate $p\psi$, while pre-existing metals are locked into stars at rate $Z\psi$ and leave in the wind at rate $\alpha Z\psi$. HenceUsing $M_h=Z{M}_g$ and the product rule,The cancellation expresses the fact that a well-mixed wind removes gas and metals in the same proportion, while fresh stellar production raises the abundance in the remaining gas.

Divide this equation by ${\dot{M}}_g=-(1+\alpha )\psi$ along the evolving reservoir, or integrate in time through intervals with no star formation. The resulting gas-phase metallicity isEquivalently, its fully explicit dependence on a prescribed net star formation rate isvalid while $M_g(t)>0$. With pre-existing stars, replace $M_s(t)$ in the gas-mass relation by $M_s(t)-M_s(0)$. Without specifying $\psi (t)$ or a gas-consumption law, the model fixes $Z$ as a function of gas mass but does not determine a unique function of time.

For the closed-box model of galactic chemical evolution, set $\alpha =0$. Then $M_g=M_0-M_s$ andAt the same remaining fraction $g=M_g/M_0$ of the initial gas reservoir, the leaky box has the logarithmic coefficient $p/(1+\alpha )$, lower than the true stellar yield by the wind factor. This coefficient is often called an effective yield, but the definition of the gas fraction must be specified.

In particular, the observable gas fraction of a galaxy is often $\mu =M_g/(M_g+M_s)$, using the mass still present, rather than $g$. In our initially star-free model,ThereforeFor $0<\mu <1$ and $\alpha >0$, the leaky-box enrichment is smaller at the same $\mu$: $\ln [1+\alpha (1-\mu )]\le \alpha (1-\mu )<\alpha \ln (1/\mu )$. If effective yield is defined observationally as $(Z-Z_0)/\ln (1/\mu )$, it is consequently not simply the constant $p/(1+\alpha )$ at all $\mu$.

As an explicit time example, choose a linear gas-consumption law $\psi =ϵM_g$, with constant $ϵ>0$. The mass and metallicity equations giveThe closed box with the same $ϵ$ has the same metallicity at a given time, while the leaky box exhausts its gas faster. Thus a comparison at fixed time depends on the chosen star formation rate, even though the comparison at fixed gas fraction above is unambiguous.

Finally, the assumption about wind metallicity is essential. If the expelled gas has abundance $Z_w$ instead, its metal-loss rate is $\alpha Z_w\psi$, and the same bookkeeping yieldsA wind enriched relative to the ambient gas lowers the enrichment rate; the logarithmic solution derived above applies to $Z_w=Z$.

Direct evidence for stellar feedback includes expanding ionized shells and superbubbles around young associations, hot X-ray-emitting gas in supernova remnants, broad or split emission lines, and blueshifted absorption showing cool and warm outflows. P-Cygni profiles reveal massive-star winds, while extraplanar filaments and metal-enriched gas demonstrate transport away from star-forming disks. Indirect evidence includes the galaxy mass--metallicity relation, low baryon fractions and suppressed star formation in dwarf galaxies, chemically enriched circumgalactic gas, and correlations of outflow speed and mass loading with the star formation rate.

Rapid gas removal changes the gravitational potential before stellar and dark-matter orbits can respond adiabatically. Positions and velocities are initially unchanged, but orbital binding energies rise; orbits expand and become more eccentric, and some particles escape. Repeated burst--outflow--reaccretion cycles can irreversibly transfer energy to collisionless matter and turn a central dark-matter cusp into a core. This matters because dwarf-galaxy rotation curves are used to test dark-matter microphysics: a feedback-made core can mimic a non-cold or self-interacting dark-matter signature.

Write the initial potential energy as $W=-aG{M}^2/R$. The virial theorem gives $T=-W/2$. If a well-mixed fraction $ϵ$ remains after instantaneous mass loss, the immediate kinetic and potential energies are $T_a=ϵT$ and $W_a={ϵ}^{2}W$, soAfter revirialization at $R^{\prime }$, $E_f=W_f/2=-aG{ϵ}^2{M}^2/(2R^{\prime })$. Equating energies gives the impulsive mass-loss expansion lawThe remnant is bound only for $ϵ>1/2$; loss of half or more of the gravitating mass disrupts this idealized system.

For many infinitesimal, individually revirialized losses, put $ϵ=1+dM/M$ in the impulsive result. To first order, $dR/R=-dM/M$. Integration yields the adiabatic mass-loss expansion lawSlow loss causes finite expansion for every positive remaining mass fraction and has no sharp disruption threshold.

Finally consider an initially circular orbit of radius $R$ around a point mass $M$. Its speed and specific angular momentum obey $v^2=GM/R$ and $h^2=GMR$. Immediately after $M\to ϵM$, these remain unchanged, while the new specific energy isUsing the orbital eccentricity relation $e^2=1+2E^{\prime }{h}^2/(G^2{ϵ}^2{M}^2)$ givesIt is an ellipse for $ϵ>1/2$, parabolic at $ϵ=1/2$, and unbound for smaller $ϵ$, in agreement with the virial argument.

The differing maps follow from radiative transfer and from the different source functions of the bands. Low-frequency radio synchrotron emission passes through dust and follows cosmic-ray electrons and magnetic fields over a thick halo. Infrared emission follows dust mixed with dense gas and reprocessed starlight, whereas optical light follows unobscured stars and is strongly reshaped by extinction. The same matter distribution can therefore appear smooth at radio wavelengths, thin and clumpy in the infrared, and broken by dark dust lanes in the optical.

The Galactic scale height of a material tracer is set by vertical gravity balanced by its random, thermal, turbulent, magnetic, or cosmic-ray support. It also depends on source lifetime and transport. Cold molecular gas and dust have small velocity dispersions and settle into a thin layer. Relativistic electrons diffuse or advect far from their sources and radiate in a vertically extended magnetic field, so Galactic synchrotron emission is broad. Cosmic rays similarly fill a halo, and inverse-Compton gamma rays can be produced wherever they meet an extended radiation field; gamma-ray absorption within the Galaxy is also weak. Molecular and infrared emission remain concentrated around the thin cold-gas and young-star disk.

The spatial correlation between infrared and CO emission follows because molecular clouds contain dust and are the sites of massive-star formation. Young stars heat nearby dust, while CO maps their molecular fuel. Converting integrated CO intensity to ${\mathrm{H}}_2$ column density gives a molecular-gas surface-density profile; converting extinction-corrected infrared luminosity to a star formation rate gives the corresponding star-formation profile. Kinematic distances from velocity-resolved CO, with care near the Galactic center and for the near--far distance ambiguity, then estimate the radial relation between gas and star formation.

The three-dimensional Galactic bar is reconstructed by combining complementary distance and velocity information. Extinction-corrected near-infrared star counts reveal its projected stellar density, and standard-candle populations such as red-clump stars show that its near end is brighter and closer than its far end. CO and H I longitude--velocity diagrams trace noncircular gas streams and shocks in a barred potential. Maser parallaxes, proper motions, radial velocities, and dynamical forward models then constrain the bar angle, length, pattern speed, and vertical box or peanut shape.

With well-mixed gas-phase metallicity $Z$, zero mass return, star formation rate $\psi$, pristine galactic gas inflow $\nu \psi$, and constant stellar yield $y$, mass conservation and metal accounting give $\dot{M}=-(1-\nu )\psi$ and $M\dot{Z}=(y-\nu Z)\psi$. Thus, for $0<\nu <1$,The initial gas-phase metallicity $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.