11.2. Classical models of galactic chemical evolution¶
Simple models of galactic chemical evolution consider at any given time for a given region in a galaxy (i) how much gas is accreted from the environment, (ii) what fraction of the gas in the region turns into stars (the star formation efficiency), (iii) how many metals are returned to the interstellar medium from previous generations of stars, and (iv) how much gas is lost from the system. Alternatively, we can prescribe the number of stars formed at any given time—the star formation rate—which in the above is a combination of points (i) and (ii), instead of the gas accretion. In both cases, these four basic processes can be modeled using simple prescriptions. In the classical models that we consider in this section, the gas accretion history is specified.
The classical models simplify the chemical enrichment treatment by considering that enrichment by the different enrichment processes discussed above happens directly after a stellar population forms. This is called the instantaneous recycling approximation. We also assume that any enriched material is immediately mixed into the ISM, which could technically be considered to be part of the instantaneous recycling approximation as well, but we list it explicitly. We further assume that all yields are absolutely constant, that is, they do not depend on time, stellar abundance, or on any other property of the stellar population or galaxy. Then the distinction between the different enrichment processes vanishes and the evolution of the ISM’s elemental abundance becomes the same for all elements. Thus, this approach cannot be used to investigate how abundance ratios of different elements vary over time as observed, but it is useful to get an overall sense of the build-up of heavy elements in galaxies.
While the approximations in the classical models may appear to be strong, they in fact produce realistic trends for a few elements, notably oxygen and magnesium. As we saw above, these elements are produced primarily in type II supernovae, which, because they originate from massive stars, happen so soon after star formation that they can be considered to be instantaneous to a good approximation. Because their abundance in the ISM receives very small contributions from the other enrichment processes, they are therefore well captured by the approximations of the classical models. However, the classical models were, classically, applied to the iron abundance as well. Because the present-day iron abundance in the ISM receives significant contributions from both type II and type Ia SNe enrichment and the latter occurs typically 1 Gyr after star formation, the instantaneous recycling and single-enrichment-process approximations fail to capture significant aspects of the evolution of the iron abundance.
11.2.1. The closed box model¶
The simplest model for galactic chemical evolution considers a small region (or annulus) in a galaxy and removes any inflow or outflow of gas from consideration (e.g., van den Bergh 1958; Salpeter 1959; Schmidt 1959). Thus, all of the gas that turns into stars is assumed to be present initially, this gas turns into stars over time with some efficiency, metals are returned to the ISM through enrichment and mixed into the ISM, and this enriched gas forms the next generations of stars without losing any gas to the environment. Because of this setup, this is called the closed box model (Talbot & Arnett 1971).
Because it is one of the simplest possible galactic chemical evolution models, the closed box model is often used as a baseline to compare more complex models to. It is therefore instructive to derive the evolution of the heavy-metal content in the ISM in the closed box model. For this we consider the evolution of three quantities: the gas mass \(M_g\), the stellar mass \(M_*\), and the gas mass in metals \(M_Z\); the gas mass includes the gas mass in heavy metals and we denote the metallicity as \begin{equation} Z = M_Z / M_g\,. \end{equation} Because enrichment happens instantaneously, the following sequence of events happens at every time: (i) some gas turns into stars at a star-formation rate \(\dot{M}_*\), (ii) all mass that would ever be returned due to stellar evolution and stellar explosions is returned to the ISM, some enriched in heavy-metal content, (iii) all other stellar mass formed is locked up into stars forever. Because of this, at the end of a given time step, any difference in the gas mass from the previous step is mass that has been turned into stars, or \begin{equation}\label{eq-closedbox-dotmg} \dot{M}_g = -\dot{M}_*\,. \end{equation} Note that here and in everything that follows in this chapter, we ignore the fact that the metals returned to the ISM slightly increase its mass; this increase at most a few percent at the highest ISM abundances observed and it therefore produces a negligible effect on the overall ISM mass and its abundances. It is also important to note that in the instantaneous recycling approximation, it can be the case that \(\dot{M}_* \neq \mathrm{d} M_* / \mathrm{d} t\), that is, the instantaneous star-formation rate \(\dot{M}_*\) does not equal the change in stellar mass in an infinitesimal time step. This is because \(\dot{M}_*\) equals \(\mathrm{d} M_* / \mathrm{d} t\) from step (i) in the sequence above, but in the instantaneous recycling approximation, step (ii) happens within the same \(\mathrm{d}t\) and it also modifies \(M_*\). This is particularly important in the presence of recycling of unenriched material (see Section 11.2.3 below), where \(\mathrm{d} M_* / \mathrm{d} t = \dot{M}_*\,(1-r)\) with \(r\approx 0.4\).
The evolution of \(M_Z\) is slightly more complicated. At any given time, \(M_Z\) decreases as some of the metals in the ISM are incorporated into stars, through \(\dot{M}_{Z,*-} = Z\,\dot{M}_g=-Z\,\dot{M}_*\), while \(M_Z\) increases as metals are yielded through enrichment, as \(\dot{M}_{Z,*+} = p\,\dot{M}_*\). The parameter \(p\) is the population-level yield of the stellar population, the mass in metals returned to the ISM relative to the mass in stars formed for an entire population of stars; above we discussed that \(p \approx0.035\) for the yield of metals of any kind. Note that winds from massive stars also return (“recycle”) mass to the ISM that has the star’s birth metallicity \(Z\), but we will ignore this for now (we will see below that we can incorporate this by a simple redefinition of the yield \(p\)). The full evolution equation of \(M_Z\) is therefore \begin{align} \dot{M}_Z & = \dot{M}_{Z,*-} + \dot{M}_{Z,*+}= (p-Z)\,\dot{M}_*\,.\label{eq-closedbox-dotmz} \end{align} Combining Equations (11.2) and (11.3), we have that \begin{align}\label{eq-closedbox-dotz} \dot{M}_Z = -(p-Z)\,\dot{M_g}\,, \end{align} or in terms of \(Z\) \begin{equation}\label{eq-closedbox-dotz-2} \dot{Z} = -p\,\frac{\dot{M}_g}{M_g}\,, \end{equation} using that \(\dot{Z} = \dot{M}_Z/M_g - Z\,\dot{M}_g/M_g\). The solution of Equation (11.5) is \begin{equation}\label{eq-closedbox-zt} Z(t) = -p\,\ln\left[\frac{M_g(t)}{M_g(0)}\right]\,. \end{equation} Thus, we see that the metallicity \(Z\) increases monotonically as the gas mass decreases due to star formation.
To go further and obtain a specific expression for \(Z(t)\) and \(M_g(t)\), we require the fourth ingredient mentioned at the start of this section, a prescription for the fraction of gas turned into stars at each time step, which we have not done so far. Before doing that, however, we can already obtain a few interesting results. The first is an approximate determination of the population-level yield. This we can obtain using the fact that the current gas fraction in the Milky Way is about 10% (see Section 1.2.2), so within the assumptions of the closed box model that means that \(M_g(\mathrm{now})/M_g(0) \approx 0.1\). The present-day abundance of the ISM is approximately solar (Nieva & Przybilla 2012), and \(Z_\odot \approx 0.014\) (Asplund et al. 2009). Then we can determine the yield \(p\) using Equation (11.6) and we find that \(p \approx 0.006\). Comparing this to the value that we quoted above, \(p \approx0.035\), we see that \(p\) is substantially lower. As we will discuss below, this is an indication that the closed box model is incorrect and, in particular, that the yield is low due to outflows of gas that are ignored in the closed box framework.
Secondly, we can derive the metallicity distribution in the closed box model without specifying the star formation efficiency. To do this, we note that because the ISM’s metallicity monotonously increases with time, the number of stars with a given metallicity \(Z'\) is given by the number of stars formed at the time that \(Z(t) = Z'\) and, equivalently, the cumulative number of stars up to a given metallicity \(Z'\) is proportional to the amount of mass in stars formed up to this same time. The latter in the closed box model is simply \(M_g(0) - M_g(t)\) and putting this all together we have that \begin{align} N(< Z') & \propto M_*(t) = M_g(0) - M_g(t) = M_g(0)\,\left[1-\exp\left(-Z'/p\right)\right]\,, \end{align} where we have used Equation (11.6) in the last step. The metallicity distribution is then \begin{align} \frac{\mathrm{d}N}{\mathrm{d}Z} & = M_g(0)\,\frac{\mathrm{d}}{\mathrm{dZ}}\left[1-\exp\left(-Z/p\right)\right] \propto \exp\left(-Z/p\right)\,.\label{eq-closedbox-dndz} \end{align}
Observationally, stellar metallicities and their distribution are typically represented as \(\log_{10}\left(Z/Z_\odot\right)\), which we will refer to as \([\mathrm{M/H}]\) to make it clear that this refers to the full metallicity rather than the abundance of a specific element. The metallicity distribution in bins of \([\mathrm{M/H}]\) is \begin{align} \frac{\mathrm{d}N}{\mathrm{d}[\mathrm{M/H}]} & \propto Z\,\exp\left(-Z/p\right)\,, \end{align} where \([\mathrm{M/H}]=\log_{10}\left(Z/Z_\odot\right)\). Note that when we look at the metallicity distribution after a finite time \(t\), we should cut off this distribution at \(Z(t)\) or its \([\mathrm{M/H}]\) equivalent, but to understand the full range of metallicities that could form in the closed box model, we will simply plot the entire distribution for \(t \rightarrow \infty\) below.
Using the yield \(p = 0.006\) obtained above from the present-day mass and abundance of the ISM, we plot the closed box metallicity distribution in Figure 11.4.
[6]:
mhs= numpy.linspace(-3.,1.,201)
Zsolar= 0.014
def mhdist_closedbox(mhs,p=0.006):
Zs= Zsolar*10**mhs
mhdist= Zs*numpy.exp(-Zs/p)
# Normalize such that int d mh mhdist = 1
mhdist/= numpy.sum(mhdist)*(mhs[1]-mhs[0])
return mhdist
figure(figsize=(7,5))
plot(mhs,mhdist_closedbox(mhs,p=0.006),label=r'$p = 0.006$')
plot(mhs,mhdist_closedbox(mhs,p=0.035),label=r'$p = 0.035$')
plot(mhs,mhdist_closedbox(mhs,p=0.003),label=r'$p = 0.003$')
xlabel(r'$[\mathrm{M/H}]$')
ylabel(r'$\mathrm{relative\ fraction}$')
legend(frameon=False,fontsize=18.);
Figure 11.4: The metallicity distribution in the closed-box model.
The resulting metallicity distribution is broad with a large number of stars below one-tenth solar metallicity and a peak at \([\mathrm{M/H}] = \log_{10}\left(p/Z_\odot \right) \approx -0.368\). We compare this distribution to observations of the metallicity distribution in the solar neighborhood in the section below. If we change the yield \(p\), for example to the value \(p \approx 0.035\) for the enrichment expected from stellar evolution (which would lead to a present-day ISM abundance of \(Z \approx 5.75 Z_\odot\) or \([\mathrm{M/H}] \approx 0.76\)!) or to half of the value that we derived, the metallicity distribution shifts while largely keeping the same shape, as shown in Figure 11.4. Physically, the reason that many stars formed have low metallicity in the closed box model is as follows. With all of the gas in place at the start of star formation in the closed box model, early star formation and enrichment produces an amount of heavy metals that is easily diluted in the large gas reservoir necessary to form all of the stars that will ever form. Thus, the metallicity of the ISM at early times grows slowly and many stars therefore form from gas that remains low metallicity for a long while. As the gas supply starts to get exhausted at later times, star formation and subsequent enrichment produces the same yield of heavy elements, but they start to be mixed into a smaller and smaller gas supply, such that the metallicity of the gas rapidly increases and fewer and fewer stars form in a given metallicity range. Thus, \(\mathrm{d}N/\mathrm{d}Z\) steadily decreases with \(Z\) (that there is a maximum in \(\mathrm{d}N/\mathrm{d}[\mathrm{M/H}]\) is solely due to the transformation \(Z \rightarrow [\mathrm{M/H}]\)).
That we have been able to determine the metallicity distribution in the closed box model without specifying how much gas turns into stars at each time is a demonstration of a general rule that the metallicity distribution is largely determined by interplay of the gas supply and the enrichment yield, with only a minor dependence on the star-formation history.
To fully solve chemical evolution in the closed box model, we need to specify how efficiently star formation happens. A simple model is that the star formation efficiency time scale \(\tau_*\) is constant over time, where this quantity expresses how fast the available gas is consumed by star formation and it is defined as \begin{equation}\label{eq-closedbox-taustardef} \tau_* = \frac{M_g}{\dot{M}_*}\,. \end{equation} (Note that a larger value of \(\tau_*\) corresponds to a lower efficiency of star formation, because it means that it takes longer to consume the available gas). Combining this with Equation (11.2), we have that \(\tau_* = -M_g/\dot{M}_g\) or \(M_g(t) = M_g(0)\exp\left(-t/\tau_*\right)\). Thus, the gas supply dwindles exponentially for a constant star formation efficiency in the closed box model. The star formation rate \(\dot{M}_*\) similarly declines, using Equation (11.10) \begin{equation}\label{eq-closedbox-taustarconst-sfr} \dot{M}_* = \frac{M_g(0)}{\tau_*}\,\exp\left(-t/\tau_*\right)\,. \end{equation} The evolution of the metallicity \(Z(t)\) is then given by plugging this into Equation (11.6) \begin{align} Z(t) & = -p\,\ln\left[\frac{M_g(t)}{M_g(0)}\right] = p\,\frac{t}{\tau_*}\,.\label{eq-chemev-Zt-closedbox-consttauSFE} \end{align} Thus, the metallicity \(Z\) increases linearly, reaching the peak \(Z=p\) of the \([\mathrm{M/H}]\) distribution after \(t= \tau_*\) and solar metallicity at \(t = \tau_*\,Z_\odot/p\) or \(t \approx 2.33\tau_*\) for the \(p=0.006\) yield that we derive. If we require that the metallicity of the ISM in the solar neighborhood has solar metallicity today at \(t \approx 10\,\mathrm{Gyr}\), we have that \(\tau_* \approx 4.3\,\mathrm{Gyr}\).
For the constant star-formation efficiency in Equation (11.10), we can explicitly derive \(\mathrm{d} N / \mathrm{d} Z\) as \begin{align} \frac{\mathrm{d} N }{\mathrm{d} Z} & \propto \dot{M}_*\,\left(\frac{\mathrm{d} Z }{\mathrm{d} t}\right)^{-1} = \frac{M_g(0)}{\tau_*}\,\exp\left(-t/\tau_*\right)\,\left(\frac{p}{\tau_*}\right)^{-1} \propto \exp\left(-Z/p\right)\,, \end{align} in agreement with Equation (11.8). Alternatively, we could assume an alternative constant star-formation efficiency \(\tilde{\tau}_*\) defined as \begin{equation}\label{eq-closedbox-tildetaustardef} \tilde{\tau}_* = \frac{M_g^2}{M_g(0)\,\dot{M}_*}\,, \end{equation} which is the previous star formation efficiency \(\tau_*\) multiplied by the ratio of the gas mass to the initial gas mass. Constant \(\tilde{\tau}_*\) therefore corresponds to an increasing \(\tau_*\) with time, or a decreasing star formation efficiency with decreasing gas mass, which is more consistent with the non-linear form of the Kennicutt-Schmidt law; see Equation (18.40) and surrounding discussion. Then using Equation (11.2), we have that \(\dot{M}_g/M_g^2= -1/(\tilde{\tau}_*\,M_g(0))\) or \begin{equation} M_g(t) = \frac{M_g(0)}{t/\tilde{\tau}_*+1}\,, \end{equation} Thus, the gas mass declines as \(\approx 1/t\) rather than exponentially, as we expected from the decreasing efficiency of star formation with time in this model. The star formation rate in this case is \begin{equation} \dot{M}_* = \frac{M_g(0)}{\tilde{\tau}_*\,(t/\tilde{\tau}_*+1)^2} \end{equation} and the metallicity increases with time as \begin{equation} Z(t) = p\,\ln \left[ t/\tilde{\tau}_*+1\right] \end{equation}
To obtain a solar-metallicity ISM after 10 Gyr now, we require that \(\tilde{\tau}_* \approx 1.07\,\mathrm{Gyr}\). While the time dependence of the star-formation rate and the ISM’s metallicity is different than in the example above, we still have that \begin{align} \frac{\mathrm{d} N }{\mathrm{d} Z} & \propto \dot{M}_*\,\left(\frac{\mathrm{d} Z }{\mathrm{d} t}\right)^{-1} \propto \frac{M_g(0)}{\tilde{\tau}_*\,(t/\tilde{\tau}_*+1)^2}\,\left(\frac{p}{t/\tilde{\tau}_*+1}\right)^{-1} = \exp\left(-Z/p\right)\,, \end{align} Thus, we explicitly see that for two different assumptions of how gas turns into stars at each time, the metallicity distribution of the formed stars is the same in the closed box model. Figure 11.5 shows the different time evolution in these two models.
[7]:
p= 0.006
tau_star_1= 4.3
tau_star_2= 1.07
ts= numpy.linspace(0.,10.,101)
figure(figsize=(13,4))
subplot(1,3,1)
plot(ts,numpy.exp(-ts/tau_star_1))
plot(ts,1./(ts/tau_star_2+1))
xlabel(r'$t$')
ylabel(r'$M_g$')
subplot(1,3,2)
plot(ts,1./tau_star_1*numpy.exp(-ts/tau_star_1),label=r'$\tau_* = M_g/\dot{M}_*$')
plot(ts,1./(tau_star_2*(ts/tau_star_2+1)**2.),label=r'$\tau_* = M_g^2/[M_g(0)\,\dot{M}_*]$')
xlabel(r'$t$')
ylabel(r'$\dot{M}_*$')
legend(frameon=False,fontsize=18.)
subplot(1,3,3)
plot(ts,p*ts/tau_star_1)
plot(ts,p*numpy.log(ts/tau_star_2+1))
xlabel(r'$t$')
ylabel(r'$Z$')
tight_layout();
Figure 11.5: The time evolution of the closed-box model for two different star formation efficiency time scales.
We see that the metallicity increases faster in the second model, because the gas mass declines faster as well. This creates a steeply dropping star-formation rate, such that the resulting metallicity distribution comes out the same.
11.2.2. The G dwarf problem¶
The closed box model was the first proposed model for galactic chemical evolution and one of the first orders of business was to compare its prediction for the metallicity distribution function to the observed distribution for stars near the Sun. This was done in the early 1960s, when stellar spectroscopy was available for only small samples of stars, which were moreover biased in the way they sampled the metallicity distribution (in that stars were often selected for spectroscopy because they seemed to have interesting metal abundances; this remained an issue until the early 2010s!). However, pioneering work by Wallerstein (1962) demonstrated a tight correlation between the metal content of a G-type dwarf (a star similar to the Sun) and its UV excess. The UV excess of a star in this context is the photometric difference between a UV color (e.g., \(U-B\) in the Johnson \(UBV\) system) and that of a typical star with the same optical color (e.g., \(B-V\); Wallerstein 1962 used stars in the Hyades open cluster to define the typical star colors). This correlation is the result of different levels of metal-line blanketing in the UV (e.g., Schwarzschild et al. 1955), with metal-rich stars having so many metal lines in the UV that their UV flux is significantly suppressed with respect to that of more metal-poor stars. Crucially, this relation meant that an unbiased view of the metallicity distribution could be obtained by applying the UV-excess–metallicity correlation to an unbiased photometric sample of stars. Unfortunately, there are no similar relations between photometric observables and abundance ratios (such as [O/Fe]) and obtaining an unbiased determination of the distribution of abundance ratios therefore remained an observational challenge for a long time.
van den Bergh (1962) and Schmidt (1963b) were the first to compare the observed distribution of metallicities obtained using the UV-excess method to the predictions from the closed box model. We can re-create what they did by using a more modern sample, the sample of 1,111 FGK stars obtained from the HARPS exoplanet program (Mayor et al. 2003) and analyzed by Adibekyan et al. (2012). To download this catalog, we use the same function that we defined in Chapter 9.2.2 to download a catalog from the Vizier service:
[8]:
# vizier.py: download catalogs from Vizier
import sys
from pathlib import Path
import shutil
import tempfile
from ftplib import FTP
import subprocess
_ERASESTR= " "
def vizier(cat,filePath,ReadMePath,
catalogname='catalog.dat',readmename='ReadMe'):
"""
NAME:
vizier
PURPOSE:
download a catalog and its associated ReadMe from Vizier
INPUT:
cat - name of the catalog (e.g., 'III/272' for RAVE, or J/A+A/... for journal-specific catalogs)
filePath - path of the file where you want to store the catalog (note: you need to keep the name of the file the same as the catalogname to be able to read the file with astropy.io.ascii)
ReadMePath - path of the file where you want to store the ReadMe file
catalogname= (catalog.dat) name of the catalog on the Vizier server
readmename= (ReadMe) name of the ReadMe file on the Vizier server
OUTPUT:
(nothing, just downloads)
HISTORY:
2016-09-12 - Written as part of gaia_tools - Bovy (UofT)
2017-09-12 - Copied to galdyncourse (yes, same day!) - Bovy (UofT)
2019-12-20 - Copied directly into notes to avoid additional dependency - Bovy (UofT)
"""
_download_file_vizier(cat,filePath,catalogname=catalogname)
_download_file_vizier(cat,ReadMePath,catalogname=readmename)
catfilename= Path(catalogname).name
with open(ReadMePath,'r') as readmefile:
fullreadme= ''.join(readmefile.readlines())
if catfilename.endswith('.gz') and not catfilename in fullreadme:
# Need to gunzip the catalog
try:
subprocess.check_call(['gunzip',filePath])
except subprocess.CalledProcessError as e:
print("Could not unzip catalog %s" % filePath)
raise
return None
def _download_file_vizier(cat,filePath,catalogname='catalog.dat'):
sys.stdout.write('\r'+"Downloading file %s from Vizier ...\r" \
% (Path(filePath).name))
sys.stdout.flush()
try:
# make all intermediate directories
os.makedirs(Path(filePath).parent)
except OSError: pass
# Safe way of downloading
downloading= True
interrupted= False
file, tmp_savefilename= tempfile.mkstemp()
os.close(file) #Easier this way
ntries= 1
while downloading:
try:
ftp= FTP('cdsarc.u-strasbg.fr')
ftp.login()
ftp.cwd(str(Path('pub') / 'cats' / cat))
with open(tmp_savefilename,'wb') as savefile:
ftp.retrbinary('RETR %s' % catalogname,savefile.write)
shutil.move(tmp_savefilename,filePath)
downloading= False
if interrupted:
raise KeyboardInterrupt
except:
raise
if not downloading: #Assume KeyboardInterrupt
raise
elif ntries > _MAX_NTRIES:
raise IOError('File %s does not appear to exist on the server ...' % (Path(filePath).name))
finally:
if Path(tmp_savefilename).exists():
os.remove(tmp_savefilename)
ntries+= 1
sys.stdout.write('\r'+_ERASESTR+'\r')
sys.stdout.flush()
return None
We then write a function to read the catalog:
[9]:
import os
from pathlib import Path
from astropy.io import ascii
_CACHE_BASEDIR= Path(os.getenv('HOME')) / '.galaxiesbook' / 'cache'
_CACHE_VIZIER_DIR= _CACHE_BASEDIR / 'vizier'
_ADIBEKYAN_VIZIER_NAME= 'J/A+A/545/A32'
def read_sn_abu(verbose=False):
"""
NAME:
read_sn_abu
PURPOSE:
read the Adibekyan et al. catalog of abundances of
stars in the solar neighborhood
INPUT:
verbose= (False) if True, be verbose
OUTPUT:
pandas dataframe
HISTORY:
2020-10-14 - Written - Bovy (UofT)
"""
# Generate file path and name
tPath= _CACHE_VIZIER_DIR/ 'cats'
for directory in _ADIBEKYAN_VIZIER_NAME.split('/'):
tPath = tPath / directory
filePath= tPath / 'table4.dat'
readmePath= tPath / 'ReadMe'
# download the file
if not filePath.exists():
vizier(_ADIBEKYAN_VIZIER_NAME,filePath,readmePath,
catalogname='table4.dat',readmename='ReadMe')
# Read with astropy, w/o the .gz
table= ascii.read(str(filePath),readme=str(readmePath),format='cds')
return table.to_pandas()
and a function to plot the abundance distribution in differential \(\mathrm{d}N([\mathrm{X/H}]) / \mathrm{d} [\mathrm{X/H}]\) or cumulative \(N(<[\mathrm{X/H}])\) form:
[10]:
def plot_sn_xhdist(element,plotrange=[-3,1],bins=31):
sndata= read_sn_abu()
hist(sndata[f'[{element.capitalize()}/H]'],
range=plotrange,bins=bins,histtype='step',
color='k',lw=1.5,
density=True,label=r'$\mathrm{data}$')
xlabel(rf'$[\mathrm{{{element.capitalize()}/H}}]$')
return None
def plot_sn_xhcumuldist(element,plotrange=[-3,1]):
sndata= read_sn_abu()
sort_indx= numpy.argsort(sndata[f'[{element.capitalize()}/H]'])
plot(sndata[f'[{element.capitalize()}/H]'][sort_indx],
numpy.arange(1,len(sndata)+1)/len(sndata),
'k-',lw=1.5,label=r'$\mathrm{data}$')
xlabel(rf'$[\mathrm{{{element.capitalize()}/H}}]$')
ylabel(rf'$N(<[\mathrm{{{element.capitalize()}/H}}])$')
return None
Following in van den Bergh (1962) and Schmidt (1963b)’s footsteps, we can then compare the distribution of \([\mathrm{Mg/H}]\) to the prediction of the closed box model in Figure 11.6, using the yield \(p = 0.006\) that we obtained from matching the present-day abundance of the ISM. Note that while the evolution equations that we have derived consider the abundance of all metals \(Z\) and an element like Mg only makes up a small fraction of the total metal content of a star or the ISM, the predicted distribution \(\mathrm{d}N/\mathrm{d}[\mathrm{Mg/H}]\) is equal to \(\mathrm{d}N/\mathrm{d}[\mathrm{M/H}]\). This is because the fact that the abundance ratio \([\mathrm{Mg/H}]\) is normalized to the equivalent ratio in the Sun causes the factor that scales the overall metallicity to the Mg abundance to drop out (this is the case as long as we assume that all elements move in lockstep, as we do in the instantaneous recycling approximation, but it will cease to be true when we consider different enrichment channels; see Section 11.3).
[11]:
figure(figsize=(7,5))
plot_sn_xhdist('Mg')
plot(mhs,mhdist_closedbox(mhs,p=0.006),
label=r'$\mathrm{Closed\ box},\ p = 0.006$')
legend(frameon=False,loc='upper left',fontsize=18.);
Figure 11.6: The G dwarf problem.
It is clear that the closed box model predicts a much too wide distribution of metallicities, with the closed box prediction having a significant number of stars with \([\mathrm{Mg/H}] < -1\), while such stars are essentially absent in the data. This was already known to van den Bergh (1962) and Schmidt (1963b), was confirmed many times over the ensuing decades (e.g., Pagel & Patchett 1975), and it remains the case even using the larger, unbiased samples of stellar abundances that are now available (e.g., Hayden et al. 2015; Buder et al. 2019).
This excess of metal-poor G-type dwarf stars predicted in the closed box model is known as the G dwarf problem. Because G-type dwarfs are long-lived, with typical lifetimes of 10 Gyr, it is difficult to explain this problem by assuming that old, metal-poor stars have evolved to stellar remnants and subsequent work furthermore demonstrated that the metallicity distribution of K- and M-type dwarfs has the same lack of metal-poor stars (e.g., Mould 1982; Favata et al. 1997; Schlesinger et al. 2012). Thus, the G dwarf problem is a true problem with the closed box model, not an observational artifact.
The G dwarf problem is one of those classical problems in astrophysics that has been solved numerous times, but remains presented as a problem (other examples include the missing-satellites problem and the missing baryons problem). This is both because until we have a definitive theory of galaxy formation, the exact resolution of the problem is difficult to isolate, but also because it is a highly educational problem in learning to think about galactic chemical evolution. Many solutions to the G dwarf problem have been proposed over the years.
The reason that the closed box model overpredicts the number of metal-poor stars in the solar neighborhood is because the abundance of metals in the ISM does not rise quickly enough at early times. Therefore, many long-lived stars are formed while the ISM is still metal-poor and these should be present in the present-day sample of G dwarfs. Any solution of the G dwarf problem therefore has to increase the rate of metal production in the past relative to what it is today significantly above the value of this ratio in the closed box model. Early resolutions of the G dwarf problem focused on changing the initial mass function with time, such that more massive stars were produced relative to low-mass stars in the past than there are today (e.g., Schmidt 1963b). A larger number of early massive stars increases the metal-production rate in the past, because these massive stars explode as supernovae and enrich the ISM, while the long-lived, low-mass stars that are more commonly formed at later times in this scenario lock up metals from the ISM without enriching the ISM much. However, models of this type that successfully reproduce the observed metallicity distribution are always about to exhaust their gas supply at the present time and combining this with the relatively constant rate of star formation observed in the solar neighborhood, this means that we would have to be living at a very special time in the life of our Galaxy (Thuan et al. 1975). We also now know that the initial mass function cannot change as significantly over time as required by these models.
Another easy way to resolve the G dwarf problem is to propose that the initial abundance of the ISM at the start of star formation was not the primordial \(Z=0\) value, but that instead the ISM was pre-enriched to some value \(Z_0\) (Truran & Cameron 1971). In this scenario, the equations describing the time evolution of the closed box model remain the same, but the solution of Equation (11.5) is now \begin{equation}\label{eq-closedbox-zt-with-zinit} Z(t) = Z_0-p\,\ln\left[\frac{M_g(t)}{M_g(0)}\right]\,, \end{equation} and following the same steps as those leading up to Equation (11.8), the cumulative metallicity distribution is now \begin{align} N(< Z') & \propto M_g(0)\,\left[1-\exp\left({Z_0-Z' \over p}\right)\right]\,, \end{align} and zero for \(Z < Z_0\). Comparing this model with \(Z_0 = Z_\odot/10\) with the cumulative metallicity distribution of the data and with the original closed box model, we find the result shown in Figure 11.7 (accounting for \(Z_0\) in the estimate of \(p\) based on the present-day gas fraction and the ISM’s present-day solar abundance, only lowers it to \(p = 0.0055\), so we keep using \(p=0.006\) for simplicity).
[12]:
mhs= numpy.linspace(-3.,1.,201)
Zsolar= 0.014
def mhcumuldist_closedbox(mhs,p=0.006,Z0=Zsolar/10.):
Zs= Zsolar*10**mhs
mhcumuldist= 1.-numpy.exp((Z0-Zs)/p)
mhcumuldist[Zs<Z0]= 0.
# Normalize such that int d mh mhdist = 1
mhcumuldist/= mhcumuldist[-1]
return mhcumuldist
figure(figsize=(7,5))
plot(mhs,mhcumuldist_closedbox(mhs,p=0.006,Z0=0.),
label=r"$\mathrm{Closed\ box},\ p = 0.006,\ Z_0 = 0$")
plot(mhs,mhcumuldist_closedbox(mhs,p=0.006,Z0=Zsolar/10.),
label=r"$\mathrm{Closed\ box},\ p = 0.006,\ Z_0 = Z_\odot/10$")
plot(mhs,mhcumuldist_closedbox(mhs,p=0.01,Z0=Zsolar/3.),
label=r"$\mathrm{Closed\ box},\ p = 0.010,\ Z_0 = Z_\odot/3$")
plot_sn_xhcumuldist('Mg')
ylim(0.,1.52)
legend(frameon=False,loc='upper left',fontsize=18.);
Figure 11.7: Pre-enrichment and the G dwarf problem.
We see that abundances now only start at \([\mathrm{Mg/H}] = -1\), but once star formation commences, the abundance of metals increases as fast as it does in the non-pre-enriched closed box model and, therefore, we still end up with far too many metal-poor stars. The reason that the pre-enriched model follows almost the same evolution as the non-pre-enriched model is that in the context of the closed box model, the initial pre-enriched state is almost the same as the state of the non-pre-enriched model once it reaches \(Z_0\). This is because any stars that have formed at \(Z < Z_0\) in the non-pre-enriched model do not affect the evolution of the gas metallicity (because enrichment is instantaneous), so the only difference is in the gas mass at \(Z = Z_0\), which is \(M_g(0)\) in the pre-enriched model and \(M_g(0)\,e^{-Z_0/p}\) in the non-pre-enriched model. However, the total gas mass does not affect the rate at which metals are produced (see Equation 11.5).
Pre-enrichment therefore does not entirely solve the problem of the over-production of metal-poor stars in the closed box model, but if we are willing to vary the parameters of the model beyond what we have considered so far, it is possible to find a reasonable match to the data. Figure 11.7 includes a line that uses a much higher level of initial enrichment \(Z_0 = Z_\odot/3\) (\([\mathrm{Mg/H}] \approx -0.5\)) and a higher yield \(p = 0.01\). As you can see, this provides a good match to the observed cumulative metallicity distribution. In this model, the present-day abundance of the ISM would be \(\approx 2Z_\odot\) or \([\mathrm{Mg/H}] \approx 0.3\), which is uncomfortably high given observational determinations of the abundances of young stars (which should reflect the ISM’s abundance well; Nieva & Przybilla 2012), but not impossible. Thus, if the gas that forms the stars in the solar neighborhood starts out with \(Z \approx Z_\odot/3\), the pre-enriched closed box model matches the observed data quite well.
Historically, the pre-enrichment model was considered a good model, because it was believed that the stellar halo was formed through star formation during the initial collapse of the gas in our Galaxy (Eggen et al. 1962) before this gas settled in a disk and commenced star formation in the disk. In this model, it is then plausible that the gas that settles in the disk is pre-enriched through the star-formation and enrichment cycle that occurred during the formation of the stellar halo. Nowadays, we have much evidence that the stellar halo largely formed through the accretion of small satellite systems, such as dwarf galaxies and globular clusters (Searle & Zinn 1978), with an additional contribution from stars perturbed out of the disk at early times, but no star formation occurring in the stellar halo itself. Pre-enrichment is therefore not a viable model when considering the entire disk. But a more modern version of the pre-enrichment scenario is that there may be different epochs of star formation in the disk, with in particular an early epoch that enriches the gas to \(Z \approx Z_\odot/3\), but whose stars are rare in stellar samples close to the Sun because these old stars are typically closer to the center or far from the mid-plane where the Sun lies.
The pre-enrichment scenario is a solution of the G dwarf problem that does not qualitatively change the basic assumptions of the model. Below, we consider possible solutions that do change the assumptions, in particular stepping away from the closed nature of the system.
11.2.3. The leaky box model¶
The basic assumption of the closed box model which gives it its name—that the system is closed—has long been known to be manifestly untrue. Galaxies as a whole, and galactic neighborhoods within galaxies, typically accrete gas from their environment throughout cosmic history and gas gets removed by winds driven by massive stars, supernova explosions, and active galactic nuclei. To resolve the G dwarf problem, it therefore makes sense to consider the effect of inflows and outflows of gas. We start by looking at the effect of outflows of gas, in what is known as the leaky box model.
That galaxies expel a significant amount of gas to their environment is observed through the ubiquitous existence of superwinds and outflows at low (Heckman et al. 1990) and high redshift (Shapley et al. 2003; Weiner et al. 2009) and from the fact that the gas in the halos of nearby disk galaxies is rich in metals and must therefore consist at least partly of enriched gas driven from the disk into the halo (e.g., Peeples et al. 2014). Simulations of galaxy formation in the expanding Universe also consistently show that an amount of gas similar to that consumed by star formation is expelled from disk galaxies through outflows (e.g., Oppenheimer & Davé 2006; Finlator & Davé 2008).
To open up the closed box model to its leaky cousin, the amount of gas lost to outflows is typically parameterized as a fraction \(\eta\), called the outflow efficiency or the mass loading factor, of the gas turned into stars \begin{equation}\label{eq-chemev-leakybox-etadef} \dot{M}_{\mathrm{outflow}} = \eta\,\dot{M}_*\,. \end{equation} This parameterization makes sense for all but the largest galaxies, because outflows are driven by the winds or supernova explosions of massive stars, the latter of which occur in the gas-rich environments of star-forming regions. Type Ia supernovae cause less gas to be driven out of the galaxy, as they typically occur long after a star has separated from its birth cloud and therefore have less gas in their surroundings to drive out. In large galaxies (with dark-matter halos well above the Milky Way’s mass of \(10^{12}\,M_\odot\)), significant outflows are driven by active galactic nuclei and these outflows are not directly related to star formation in the galaxy.
To determine the effect of gas outflows on the predictions of the simple box model that we have been considering, we can assume that the outflow efficiency \(\eta\) is constant. Then Equation (11.2) becomes \begin{align}\label{eq-leakybox-dotmg} \dot{M}_g & = -\dot{M}_* - \dot{M}_{\mathrm{outflow}} = -(1+\eta)\,\dot{M}_*\,. \end{align} The evolution of the amount of mass in metals \(M_Z\) is also affected, because now some amount of gas with metallicity \(Z\) is lost to outflows and Equation (11.3) becomes \begin{align}\label{eq-leakybox-dotmz-1} \dot{M}_Z & = (p-Z)\,\dot{M}_*-Z\,\dot{M}_{\mathrm{outflow}} = (p-Z-\eta\,Z)\,\dot{M}_*\,. \end{align} Combining Equations (11.22) and (11.23), we now have that \begin{align}\label{eq-leakybox-dotz} \dot{M}_Z = -{(p-Z-\eta\,Z)\over (1+\eta)}\,\dot{M_g}\,, \end{align} or in terms of \(Z\) \begin{equation}\label{eq-leakybox-dotz-2} \dot{Z} = -p'\,\frac{\dot{M}_g}{M_g}\,, \end{equation} where \(p' = p/(1+\eta)\). The solution of Equation (11.25) is \begin{align}\label{eq-leakybox-zt} Z(t) & = -p'\,\ln\left[\frac{M_g(t)}{M_g(0)}\right] = -{p \over 1+\eta}\,\ln\left[\frac{M_g(t)}{M_g(0)}\right]\,. \end{align} Thus, as in the closed box model, the metallicity \(Z\) increases monotonically as the gas mass decreases due to star formation and outflows. In fact, comparing Equation (11.26) to the equivalent Equation (11.6), we see that the evolution of \(Z\) in the leaky box is the same as that in the closed box if we replace the yield \(p\) with the effective yield \(p' = p/(1+\eta)\). The effective yield is lower than the pure population-level yield of heavy metals returned to the ISM by stars because of the existence of outflows, which remove enriched gas from the galaxy.
As we discussed in the context of the closed box model, winds from massive stars also return mass to the ISM at the stars’ birth abundance \(Z\). Assuming that this return also happens instantaneously and that the return is a recycling fraction \(r\) of the amount of mass turned into stars \(\dot{M}_{\mathrm{recycle}} = -r\,\dot{M}_*\) (with a minus sign to indicate that this is mass returned to the ISM), it is clear that this recycling leads to exactly the same evolution as the leaky box with \(\eta = -r\). A closed-box with recycling fraction \(r\) therefore has an effective yield \(p' = p/(1-r)\), while if we want to take into account both recycling and outflows, we obtain \(p' = p/(1+\eta-r)\). The recycling fraction is not negligible, but in fact around 40% of the mass of the mass of a stellar population is returned to the ISM at its birth metallicity (Weinberg et al. 2017). Note that as discussed in Section 11.2.1, in the presence of recycling, we have that \(\mathrm{d} M_* / \mathrm{d} t = \dot{M}_*\,(1-r)\), which is important to keep in mind for calculations that include recycling.
Because the leaky-box model is mathematically equivalent to the closed box model for \(p \rightarrow p'\), the predicted metallicity distribution is the same as that of the closed box model. The only difference is in the interpretation of the yield parameter. In the closed box model, the yield parameter only takes into account enrichment from stars and from using the fact that the present-day ISM has approximately solar abundances, we determined that \(p \approx 0.006\). This was a surprisingly low value compared to the expected yield of \(p \approx 0.035\) from our theoretical understanding of stellar and explosive nucleosynthesis in high- and low-mass stars. Outflows provide one way of resolving this discrepancy: the determined yield in the context of the leaky-box model is \(p' = 0.006\), while the nucleosynthetic yield is \(p = 0.035\) and they are related through the outflow parameter as \(p/p' = 1+\eta\) (or as \(p/p' = 1+\eta-r\) in the presence of recycling). We can therefore reconcile these two yields by having \(\eta \approx 5\), that is, if at all times we expel from the solar neighborhood a significantly larger amount of gas than the gas consumed by star formation, then the ISM’s present-day solar abundance is natural given the nucleosynthetic yields. As we will see below, this combination of the nucleosynthetic yield and the outflow efficiency are what sets the abundance of the ISM at late times even in more complex models of galactic chemical evolution and the outflow efficiency therefore sets the metallicity of the ISM in galaxies in the local Universe: lower outflow efficiencies in higher mass galaxies lead to an increasing metallicity as a function of galaxy mass, a relation known as the mass–metallicity relation (Tremonti et al. 2004).
11.2.4. The accreting box model¶
Beside expelling gas in outflows, galaxies also accrete pristine gas throughout their lives from their intergalactic environment. Such inflows are difficult to detect, but can be seen, e.g., as high-velocity gas clouds falling into the Milky Way (Oort 1970). Cosmological simulations demonstrate that much of a galaxy’s gas is accreted slowly in the form of cold streams (Katz et al. 1996; Keres et al. 2005; Dekel et al. 2009) and that this is the main driver of star formation (Schaye et al. 2010). Models of galactic evolution have therefore included infalling gas for a long time (e.g., Larson 1972; Tinsley 1974). Because in the closed and leaky box models, the metallicity of the ISM is directly determined by the ratio of the current amount of gas to the initial amount, it is clear that we should expect the evolution of the metal content of the ISM to be significantly different in models where new gas is (semi-)continuously supplied to the system.
As a simple example of adding accretion to our closed box model and turning it into the accreting box model, let’s assume that the accretion of gas is such that the total amount of gas in the box is constant in time. In this case, Equation (11.3) for the evolution of the mass of metals \(M_Z\) in the ISM is still valid, but we can no longer use Equation (11.2) to write this in terms of the evolution of the gas mass \(M_g\). Instead, we convert Equation (11.3) to an equation for the time evolution of \(Z\): \begin{equation} \dot{Z} = (p-Z)\,{\dot{M}_* \over M_g}\,, \end{equation} from \(\dot{Z} = \dot{M}_Z/M_g - Z\,\dot{M}_g/M_g\) where the second term on the right-hand side is now zero, because the gas mass is constant (\(\dot{M}_g = 0\)). We cannot solve this equation without an assumption about how stars turn into gas (through an assumed star-formation efficiency), but we can generally understand how the metallicity evolves by replacing time with the total mass \(M = M_g + M_*\) in the system as the independent variable. Because \(\dot{M} = \dot{M}_*\), we have that \(\mathrm{d} Z / \mathrm{d} M = \dot{Z} / \dot{M}_*\), such that \begin{equation}\label{eq-accretingbox-dzdm} {\mathrm{d} Z \over \mathrm{d} M} = {(p-Z) \over M_g}\,. \end{equation} This is a linear inhomogeneous differential equation, which we can solved by combining the general solution for the homogeneous version (obtained by setting \(p=0\) in this case and solved by \(Z = C\,\exp\left(-M/M_g\right)\) with \(C\) an integration constant) with a particular solution (\(Z = p\) in this case), such that \(Z = p + C\,\exp\left(-M/M_g\right)\). To determine the integration constant, we note that initially \(M = M_g\) and \(Z=0\), such that \(C = -p\,e\) and therefore the solution of Equation (11.28) is \begin{equation}\label{eq-accretingbox-zM} Z = p\,\left[1-\exp\left(1-{M \over M_g}\right)\right]\,. \end{equation} After star formation commences, the total mass quickly becomes larger than the gas mass (e.g., in the Milky Way \(M / M_g \approx 10\) at the present time) and when \(M \gg M_g\), we see that \(Z \approx p\). Therefore, the abundance of the ISM reaches a constant value \(Z_{\mathrm{eq}}\) that represents an equilibrium between enrichment from supernovae and infall of pristine gas and the value of \(Z_{\mathrm{eq}}\) is directly set by the stellar yield. In more general models of accretion, this latter statement continues to be true, with \(Z_{\mathrm{eq}}\) being independent of the rate of infall (a conclusion first reached by Larson 1972). An easy way to see why the equilibrium point has \(Z = p\) is to consider what happens to a volume of gas consumed by star formation: when a volume with abundance \(Z\) and mass \(m_g\) is turned into stars, two things happen: (a) metals produced by enrichment processes are returned with a total mass of \(m_Z = p\,m_g\) and (b) the gas supply is replenished by adding \(m_g\) of pristine \(Z=0\) gas. The net result of this is that a gas mass with metallicity \(Z\) is replaced by one of the same mass and metallicity \(Z'= (p\,m_g)/m_g = p\). The metallicity of the ISM therefore increases until it reaches a steady state at \(Z=p\). Remember that, as always in this chapter, we are working under the assumption that \(Z \ll 1\), such that the mass in metals can be ignored in the total gas mass.
To determine the yield \(p\) in the accreting box model for the solar neighborhood, we can again calibrate the model to the Milky-Way’s present-day gas fraction, \(M/M_g \approx 10\), and the ISM’s solar abundance, \(Z \approx Z_\odot = 0.014\). From Equation (11.29), we then derive \(p \approx 0.014\) (\(p \approx Z_\odot\), because \(M/M_g \gg 1\)). As in the closed box model, this value is smaller than the value expected from stellar evolution, \(p \approx 0.035\), although not as small as the \(p\approx 0.006\) that we obtained in the closed box model.
To derive the metallicity distribution in the accreting box model, we can again start by noting that the number of stars \(N(<Z)\) with metallicity less than a given value \(Z'\) is proportional to the total mass in stars \(M_*\) when the ISM has reached metallicity \(Z'\), but now we use Equation (11.29) to express that mass in terms of \(Z\) and constants: \begin{align} N(< Z') & \propto M_*(Z')= M(Z')-M_g = -M_g\,\ln \left(1-{Z\over p}\right)\,. \end{align} The metallicity distribution is then \begin{equation} {\mathrm{d} N \over \mathrm{d} Z} \propto {1 \over p-Z}\,,\quad \mathrm{or}\quad {\mathrm{d} N \over \mathrm{d} [\mathrm{M/H}]} \propto {Z \over p-Z}\,. \end{equation} We see that the metallicity distribution is sharply peaked at \(Z \approx p\), although in this model, the singularity at \(Z=p\) is never reached, because that would require \(M = \infty\).
By adding outflows to the accreting box of the same form \(\dot{M}_{\mathrm{outflow}} = \eta\,\dot{M}_*\) as in the leaky box model, it is straightforward to show that Equation (11.29) becomes \begin{equation}\label{eq-accretingbox-zM-outflow} Z = {p \over 1+\eta}\,\left\{1-\exp\left(\left[1+\eta\right]\,\left[1-{M \over M_g}\right]\right)\right\}\,, \end{equation} and the metallicity distribution is \begin{equation} {\mathrm{d} N \over \mathrm{d} Z} \propto {1 \over p'-Z}\,, \end{equation} where as before \(p' = p/(1+\eta)\) is the effective yield. To match the present-day solar abundance of the ISM with \(M/M_g = 10\) and \(p=0.035\) we then require \(\eta = 1.5\) (for \(M/M_g =10\gg 1\), essentially \(Z = p/[1+\eta]\), so \(\eta \approx p/Z_\odot-1\)). Thus, compared to the \(\eta \approx 5\) value that we required in the leaky box model without accretion, less gas needs to be driven from the solar neighborhood for the ISM’s current solar abundance to make sense. Taking into account recycling as well is trickier than before, because recycling affects the total mass \(M\) in a different way than outflows. It is left as an exercise to show that with both outflows and recycling \begin{equation}\label{eq-accretingbox-zM-outflow-recycling} Z = {p \over 1+\eta-r}\,\left\{1-\exp\left(\left[{1+\eta-r \over 1-r}\right]\,\left[1-{M \over M_g}\right]\right)\right\}\,. \end{equation} For the equilibrium, this expression does simplify to again replacing \(1+\eta\) with \(1+\eta-r\) and, because the solar neighborhood is close to equilibrium in this model, for \(r=0.4\), the outflow efficiency therefore rises to \(\eta \approx 2\) to match the solar neighborhood data. As discussed in Section 11.2.3, the combination \(p/(1+\eta-r)\) sets the abundance of the ISM at late times, which continues to be the case even if we let the gas mass increase or decrease slightly over time.
We can compare these accreting box models (without recycling) to the solar-neighborhood metallicity distribution in Figure 11.8. Because the model is so sharply peaked, measurement uncertainties in the abundances are important in the comparison, because any non-zero uncertainties will smooth out the intrinsic sharp peak in the metallicity distribution. Therefore, we smooth the predicted accreting-box distributions with a Gaussian kernel of \(0.1\,\mathrm{dex}\) to account for the typical uncertainty in the stellar abundance measurements.
[13]:
from scipy.ndimage import gaussian_filter1d
def mhdist_accretingbox(mhs,p=0.006,eta=0.,e_mh=0.1):
Zs= Zsolar*10**mhs
mhdist= Zs/(p/(1+eta)-Zs)
mhdist[Zs > p/(1+eta)]= 0.
mhdist= gaussian_filter1d(mhdist,e_mh/(mhs[1]-mhs[0]))
# Normalize such that int d mh mhdist = 1
mhdist/= numpy.sum(mhdist)*(mhs[1]-mhs[0])
return mhdist
figure(figsize=(7,5))
plot_sn_xhdist('Mg',bins=51)
# Make sure to avoid [M/H]=0 where there is a singularity
mhs= numpy.linspace(-3.,1.,200)
plot(mhs,mhdist_closedbox(mhs,p=0.006),
label=r'$\mathrm{Closed\ box},\ p = 0.006$')
plot(mhs,mhdist_accretingbox(mhs,p=0.014,e_mh=0.1),
label=r'$\mathrm{Accreting\ box},\ p = 0.014$')
plot(mhs,mhdist_accretingbox(mhs,p=0.035,eta=1.5,e_mh=0.1),
label=r'$\!\!\!\!\!\!\mathrm{Accreting\ leaky\ box},$'
+r'\\\phantom{hah}'+r'$p = 0.035,\ \eta=1.5$')
legend(frameon=False,loc='upper left',fontsize=18.);
Figure 11.8: Continuous gas accretion and the G dwarf problem.
The prediction from the non-leaky and leaky versions of the accreting box model are of course the same, because they have the same effective yield \(p'\). We see that the tail of metal-poor stars that was present in the closed and leaky box models is now absent and that the support of the metallicity distribution—the range over which it is non-zero—in the data and the model is much more similar. The shape of the accreting-box metallicity distributions is is similar to that of the data, although it is narrower to a degree that cannot be easily explained by the measurement uncertainties. From our considerations of the closed box model, it should be clear that the low-metallicity tail in the accreting box model could be populated by having a higher gas mass in the past than today, for example, by letting accretion be lower today than it was in the past. Because the Universe was more dense in the past than it is now, this is in fact what we would expect to happen. Populating the high-metallicity tail with more stars is more difficult, because the yield sets an upper limit on the metallicity in the accreting box model. These stars were most likely not formed near the solar neighborhood, but they were born in a region of the Galaxy where the effective yield (once we take into account the effect of outflows as well) is higher than it is locally.
Thus, we see that by considering the effects of outflows and accretion, we can solve two of the major issues with the closed box model as applied to the solar neighborhood:
The lack of observed metal-poor stars (the G dwarf problem) can be explained through slow accretion of pristine gas: because the initial gas reservoir is small in the accreting box scenario compared that in the closed box, the metallicity of the ISM increases much more rapidly as the same number of enriched ejecta get mixed into a small gas reservoir and few metal-poor stars are formed;
The low abundance of the present-day ISM (\(Z \approx Z_\odot=0.014\)) compared to that expected in the closed box model (\(Z \approx 2.3\,p\) where \(p \approx 0.035\)), is explained by a combination of inflows and outflows: outflows lower the effective yield, with stronger outflows leading to bigger suppression (\(p \rightarrow p'=p/[1+\eta]\)) and inflows cause the ISM’s abundance to quickly reach an equilibrium value at \(Z \approx p'\) at which point enrichment by stellar evolution is balanced by inflow of pristine gas from the galactic environment.
A relatively continuous inflow of gas that keeps the star formation rate in the disk of a galaxy steady combined with outflows that are such that approximately the same amount of gas is expelled at any given time as is consumed by star formation therefore suffice to understand the overall chemical evolution of disk galaxies.
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