# Fermiology of two-dimensional titanium carbide and nitride MXenes

Mohammad Bagheri1 Rina Ibragimova2 Hannu-Pekka Komsa1,2 1Microelectronics
Research Unit, University of Oulu, Oulu, Finland
2Department of Applied Physics, Aalto University, Aalto, Finland.

###### Abstract

MXenes are a family two-dimensional transition metal carbide and nitride
materials, which often exhibit very good metallic conductivity and are thus of
great interest for applications in, e.g., flexible electronics,
electrocatalysis, and electromagnetic interference shielding. However,
surprisingly little is known about the fermiology of MXenes, i.e, the shape
and size of their Fermi-surfaces, and its effect on the material properties.
One reason for this may be that MXene surfaces are almost always covered by a
mixture of functional groups, and studying Fermi-surfaces of disordered
systems is cumbersome. Here, we study fermiology of four common Ti-based
MXenes as a function of the surface functional group composition. We first
calculate the effective band structures of systems with explicit mixed
surfaces and observe gradual evolution in the filling of the Ti-d band and
resulting shift of Fermi-level. We then demonstrate that these band structures
can be closely approximated by using pseudohydrogenated surfaces, and also
compare favorably to the experimental angle-resolved photoemission
spectroscopy results. By modifying the pseudohydrogen charge we then proceed
to plot Fermi-surfaces for all systems and extract their properties, such as
the Fermi-surface area and average Fermi-velocity. These are in turn used to
evaluate the electrical conductivity with the relaxation time fitted to
experimentally measured conductivities.

##  I Introduction

MXenes are a large class of two-dimensional (2D) materials of transition metal
(M) carbides and nitrides (X) Naguib et al. (2012); Gogotsi and Anasori
(2019). Distinct from other 2D materials, MXenes are synthesized by etching
out layers of atoms from a layered bulk precursor phase. As a result of the
etching process, the dangling bonds on the surface are passivated by a mixture
of organic groups from the etching solution, such as -O, -OH, and -F Naguib et
al. (2011). MXenes are metallic and exhibit very high conductivity in addition
to good mechanical properties and easy solution processing. As a result, they
have shown great promise for applications in conductive inks, battery
electrodes, electromagnetic interference shielding, and various types of
sensors Shahzad et al. (2016); Anasori et al. (2017); Fu et al. (2019); Lee
and Kim (2020).

Given the large interest on these materials, surprisingly little effort has
been devoted to understanding the fundamental properties of their Fermi-
surfaces, i.e., fermiology, and its effect on the material properties Matsuda
and Hasegawa (2007). Experimentally, Fermi-surfaces are commonly investigated
using angle-resolved photoemission spectroscopy (ARPES) Matsuda and Hasegawa
(2007); Lasek et al. (2021). ARPES results for delaminated Ti3C2, the
prototypical MXene, were reported by Schultz et al. Schultz et al. (2019), but
since the measured film consists of randomly oriented flakes, the ARPES data
was azimuthally averaged, which prevented the extraction of the Fermi-surface
shape. Quasiparticle interference within scanning tunneling microscope (STM-
QPI) is also often used to gain insight on the Fermi-surface, but requires
high quality surface with small density of scattering centers Chen et al.
(2017) — a luxury not afforded in the case of MXenes, where the surfaces are
covered by a random mixture of organic groups and other residues from the
synthesis. In fact, to the best of our knowledge, clean STM images of MXene
surfaces have not been reported in the literature. Finally, Fermi-surfaces
could be probed via de Haas van Alphen effect Ouisse and Barsoum (2017), but
again no reports exist for MXenes.

Also computational studies of Fermi-surfaces are very rare, even when the band
structures have been reported in numerous papers and are relatively well
understood Xie and Kent (2013); Khazaei et al. (2013, 2017). Hu et al.
calculated the Fermi-surface of monolayer and bulk Ti2C2(OH)2 for two
different layer stackings Hu et al. (2015). In addition, some insight can be
gained from the Fermi-surfaces of the precursor MAX phases, which were
presented in Ref. Ouisse and Barsoum, 2017. While the Fermi-surfaces for pure
O or OH terminated structures are straightforward to evaluate, since the
surfaces are passivated by a mixture of groups (stabilized by strong nearest
neighbor interactions between the groups Ibragimova et al. (2019, 2021)) and
the electronic structure of O and OH-terminated surfaces differ markedly, it
is not clear how the Fermi-surface evolves with changing functional group
composition. Calculations for the mixed group surfaces are cumbersome due to
requiring the use of supercells and the resulting band folding. The band
structures can be unfolded with the effective band structure method, where
each state from the supercell Brillouin zone is projected to each of the
folded planewaves from the primitive cell Brillouin zone Ku et al. (2010);
Popescu and Zunger (2010), but smearing of the bands still complicates the
analysis.

In this paper, we show how to overcome these problems and report results for
the Fermi-surfaces of Ti3C2, Ti2C, Ti4N3, and Ti2N monolayers with varying
O/OH composition. We calculate the effective band structures for mixed group
surfaces and show that they can be well approximated by pseudohydrogenated
surfaces. We analyze the Fermi-surface properties as a function of the surface
group composition and compare the evaluated electrical conductivities to the
experiments. We also compare the Ti3C2 band structure to those obtained from
recent ARPES experiments.

##  II Methods

All density-functional theory calculations are carried out using VASP software
Kresse and Furthmüller (1996a, b), together with projector augmented plane
wave method Blöchl (1994). We adopt the Perdew-Burke-Ernzerhof exchange-
correlation functional for solids (PBEsol)Perdew et al. (2008). The plane wave
cutoff is 550 eV and the Brillouin zone of the primitive cell is sampled with
a 16$\times$16 k-point mesh.

The effective band structures (EBS) are calculated using BandUP software
Medeiros et al. (2014). The atomic structures for the mixed surfaces were
determined via combination of cluster expansion and Monte Carlo simulations in
our previous publication Ibragimova et al. (2021), and we use the same
4$\times$4 supercell special quasi-ordered structures as constructed therein.

Electronic band structures are interpolated using BoltzTrap2 software, which
is also used for the integration of the transport properties Madsen et al.
(2018). We use k-point mesh interpolation factor 200 and temperature 300 K.
Fermi-surfaces are analyzed and plotted using the ifermi software Ganose et
al. (2021), with interpolation factor 20.

##  III Results

###  III.1 Band structures of mixed surface MXenes

Figure 1:  (a) Atomic structure for Ti3C2 with a surface functionalized by a
mixture of O and OH groups (top) and the corresponding pseudohydrogenated
structure (bottom). (b-e) Effective band structures for all materials (left
panel) and band structures of pseudohydrogenated surfaces overlaid on the EBS
(right panel). Fermi-level is set to zero.

More realistic structural models for the distribution of functional groups
were developed by us in Refs. Ibragimova et al., 2019, 2021. As an example,
the structure of Ti3C2 with O0.5OH0.5 surface composition is shown in Fig.
1(a). In the left panels of Fig. 1(b-e) we show the effective band structures
for the four Ti-based MXenes with O0.5OH0.5 surface composition. From these
results, and from comparison to the band structure of pure surfaces (see Figs.
2–3 below), it is clear that the band structures undergo a gradual evolution
and there are no extraneous bands arising from the surface groups. The band(s)
at the Fermi-level arise from the non-bonding t2g states of Ti-d character Hu
et al. (2017). Consequently, to a very good approximation, the surface groups
only control the electron concentration and consequently the Fermi-level
position in the metal d-bands, as has been previously assumed Khazaei et al.
(2013, 2017). This also suggests that other charge acceptor or donor species
could be used just as well for controlling the Fermi-level position.

Given that effective band structures are unwieldy for detailed studies of
these materials’ fermiology, we propose to use pseudohydrogenated surfaces to
control the Fermi-level position. In practice, we use the primitive cell of a
fully OH-terminated structures, but with the H atom replaced by a
pseudohydrogen atom with atomic number $Z=0.25$, $0.50$, or $0.75$ (i.e.,
proton charge is $Ze$ and total electron charge $-Ze$). The pseudohydrogen
band structures for $Z=0.5$ are shown on the right panels of Fig. 1(b-e),
overlaid on top of the effective band structures. In all cases, the band
structures of pseudohydrogenated MXenes agree extremely well with the EBS,
thereby further corroborating the physical picture described above.

It is worth noting that we also calculated the band structures for systems
with one side fully terminated by O and the other by OH, but the resulting
band structures did not agree favorably with the EBS due to broken symmetry.
Moreover, such approach would obviously be limited to a O0.5OH0.5 composition.

Figure 2:  Evolution of the band structures of Ti3C2 and Ti2C with surface
group composition. The Fermi-surfaces are plotted under each panel, with the
color denoting Fermi-velocity.  Figure 3:  Evolution of the band structures of
Ti4N3 and Ti2N with surface group composition. The Fermi-surfaces are plotted
under each panel, with the color denoting Fermi-velocity.

The band structure evolution with O/OH composition, for all materials and five
different O/OH compositions, are collected in Fig. 2 and 3. From these plots,
the gradual change in the band energies is easily resolved.

Based on the analysis by Hu et al. Hu et al. (2017), the states below Fermi-
level (red colored band and below in Figs. 2 and 3) arise from bonding state
between Ti-d and C-p/N-p ($e_{\rm g}$), and similar to bulk carbides and
nitrides Häglund et al. (1993). The states at and above Fermi-level (orange
colored band and above) arise from non-bonding $t_{\rm 2g}$ states of Ti-d. In
most cases there is a gap between these two band manifolds, but the Fermi-
level resides in the Ti-d manifold in all cases except for the fully
O-terminated Ti2C. Due to the larger electronegativity of N, as compared to C,
the Ti-N bonding states are clearly lower in energy and lead to a larger gap
below the nonbonding states. The Ti-C bands cross the Fermi-level around
$\Gamma$-point only for O-rich Ti3C2, but thereby prevent it from becoming
semiconductor as happens with Ti2CO2.

Due to the additional electron(s) in N, as compared to C, there is one
additional electron per X atom to fill the M-d bands and consequently Fermi-
level resides higher within this band. For instance, the lowest energy band
(orange) is exactly half-filled by the one additional electron in the case of
Ti${}_{2}N$, whereas the band is empty in the case of Ti2C. Similar half-
filling of the band can be obtained in the case of O0.5OH0.5 surface
composition of Ti3C2 or Ti2C.

The band fillings can also be readily obtained via electron counting. In the
case of Ti3C2O2 and Ti2CO2, Ti, C, and O are in the oxidation states +4, -4,
and -2, respectively, as in TiC and TiO2. Upon addition of H atoms (oxidation
state +1) and the subsequent occupation of the non-bonding Ti-d band, the
charge balance is achieved by decreasing the oxidation state of Ti. The
nitrogen atoms (oxidation state -3) have a similar effect.

O/OH composition controls the occupation of the t2g band, but since it is
nonbonding, it should have relatively minor effect on the energetics. Among
the purely terminated surfaces, O-terminated surfaces have been predicted to
be the most stable, but the mixed surfaces arise in practice due to the strong
attractive interactions between the opposite type surface groups Ibragimova et
al. (2019, 2021). Consequently, since the surface composition is dominated by
the interactions between surface groups, it is rather independent on the M and
X species, as we observed in Ref. Ibragimova et al., 2021 (we also note, that
the energy gain upon mixing is not captured in the pseudohydrogen models).
Even if the O/OH mixing cannot be easily avoided, we expect that the Fermi-
level position can still be tuned by introducing alloying in the M and X
sublattices.

###  III.2 Fermi-surface properties

The Fermi-surfaces are plotted below the band structures in Figs. 2 and 3, and
colored by the Fermi-velocity. When Fermi-level crosses the lowest energy Ti-d
band, it leads to hexagonally symmetric ellipsoids around the M-points. These
have fairly large Fermi-velocities, reaching values above $0.5\cdot 10^{6}$
m/s, which is only about factor of two lower than in the highly conducting
elemental metals. At large OH concentrations (or any concentration in the case
of Ti4N3) when the Fermi-level rises to cross the set of states around
$\Gamma$-point, the Fermi-surfaces develop more complex features.

Figure 4:  Fermi-surface properties: (a) sheet conductivity, (b) conductivity,
(c) Fermi-surface length, (d) Fermi-velocity, and (e) density of states.

With access to Fermi-surfaces of (effectively) mixed surface MXenes, we can
next estimate their properties. Since MXenes are known to exhibit excellent
electrical conductivity, we evaluated the sheet conductivity within the
constant relaxation time approximation (in x-direction) as

$\displaystyle\sigma_{\rm 2D}$
$\displaystyle=\frac{e^{2}}{2\pi^{2}\hbar}\tau_{0}\int_{E=E_{F}}\frac{v_{x}^{2}(k)}{|v(k)|}dk_{F}$
(1a) $\displaystyle=\frac{e^{2}}{2\pi^{2}\hbar}\tau_{0}\frac{\langle
v_{F}\rangle}{2}\cdot l_{F}$ (1b)

see Supplemental Material for more detailed description.

The sheet conductivities are shown in Fig. 4(a). The solid lines show the
results from the full integral (Eq. 1a) and the dashed lines from average
Fermi-velocity approximation (Eq. 1b). All four considered MXenes show similar
$\sigma_{\rm 2D}/\tau_{0}$, mostly falling within 2–4 $1/\Omega\square$s. The
approximation (Eq. 1b) works very well and thus by inspecting the $\langle
v_{F}\rangle$ and $l_{F}$ shown in Fig. 4(c,d), we can study their roles in
governing the conductivity. Fermi-surface lengths and average Fermi-velocity
are similar in Ti3C2, Ti2C, and Ti2N. Ti4N3 shows larger total length due to
large number of states crossing the Fermi-level, but the average Fermi-
velocities in these states are low. Overall, our results indicate that by
simply tuning the surface composition the sheet conductivity cannot be changed
by more than about a factor of two.

Trends in electrical conductivity are often evaluated from the density of
states (DOS) at Fermi-level, since $l_{F}\sim D_{\rm 2D}(E_{F})\langle
v_{F}\rangle$ (cf. Supplemental Material). The DOS at Fermi-level are
presented in 4(e) and shows that the trends in the conductivity changes with
composition or between different materials cannot be extracted solely from the
DOS due to missing the effect of $\langle v_{F}\rangle$.

Experimentally measuring the sheet conductivity of MXene monolayers is
challenging, but the conductivity of MXene films has been reported in several
publications. In the case of Ti3C2 films, conductivities above $10^{5}$ S/m
have been reported often Halim et al. (2014); Ling et al. (2014); Shahzad et
al. (2016); Liu et al. (2020). Recently a value as high as $1.1\cdot 10^{6}$
S/m was achieved for a high crystalline quality Ti3C2 monolayer Lipatov et al.
(2021). For other considered materials, the reports are more scarce, but
nevertheless fairly similar values can be found: $5\cdot 10^{5}$ S/m for Ti2N
Soundiraraju and George (2017), and $2.3\cdot 10^{5}$ S/m for Ti2C Halim et
al. (2019). It is worth noting, that the conductivity may be governed by
interflake contacts and thus also sensitive to the presence of any
intercalated species Hart et al. (2019).

In order to estimate the film conductivity, we calculate $\sigma=(\sigma_{\rm
2D}/\tau_{0})\cdot\tau_{0}/t$, where $\tau_{0}$ is the relaxation time and $t$
is the interlayer separation taken from experiments: 10.3, 7.5,Naguib et al.
(2012) 14,Urbankowski et al. (2016) and 8.7 ÅSoundiraraju and George (2017)
for Ti3C2, Ti2C, Ti4N3, and Ti2N, respectively. Note, that the interlayer
separation depends on the preparation method, as, e.g., intercalation of Li
can increase the spacing by a few Å. Here we do not attempt to evaluate
$\tau_{0}$ from first-principles calculations, but instead from the comparison
to experimentally measured conductivities. In Fig. 4(b) we show the results
for $\tau_{0}=1$ fs, which yields $\sigma$ values comparable to majority of
the experimental reports. The result of Ref. Lipatov et al., 2021 can be
reproduced by taking slightly larger $\tau\approx 2-3$ fs, depending on the
surface composition. The lower $\tau_{0}$ in earlier reports may then be
ascribed to larger number of defects caused by the strong etching, although
the interflake contacts may also contribute.

###  III.3 Angle-resolved photoemission spectrum (ARPES)

Figure 5:  (a) Calculated effective band structure of Ti3C2(O0.5OH0.5)2
overlaid with the band structure from pseudohydrogenated system. (b) Effective
band structure of Ti3C2(O0.25OH0.25F0.5)2. (c) The same as panel (b), but
showing only the $\Gamma$-K line, broadened, and colormap chosen to match that
in the experiments. (d) Experimentally measured ARPES spectrum of Ti3C2Tx,
adapted from Ref. Schultz et al., 2019.

To verify that our calculations properly capture the main features in the
electronic structures of MXenes, we compare them to the only available
experimental piece of information: the ARPES results for Ti3C2 reported in
Ref. Schultz et al., 2019. The experimental data is reproduced in Fig.
LABEL:arpes(d) and shows: (i) flat band at the Fermi-level, (ii) band with
negative dispersion starting at -1.5 eV, (iii) band with negative dispersion
starting at -2 eV, (iv) flat band at -3 eV, (v) flat band segment at -4 eV,
(vi) broad flat band between -5 and -6 eV, and (vii) broad flat band between
-8 and -9 eV. The composition was determined to be Ti3C2F0.8O0.8.

In Fig. 5(a) we replot the effective band structure of Ti3C2(O0.5OH0.5)2
overlaid with the pseudohydrogenated band structure from Fig. 1(a), but in a
wider energy range. As discussed, the non-bonding Ti bands at and above 0 eV
and the Ti-C bonding bands between -4 and 0 eV are well reproduced in the
pseudohydrogenated systems. The Ti-O bands between -7 and -4 eV are not
reproduced very well, and the O-H states at -9 eV are understandably poorly
described by pseudohydrogens. Also some parts of the effective band structure
are more strongly smeared by the surface group disorder than others, whereas
the pseudohydrogen band structures contain no disorder. These factors limit
the use of the pseudohydrogenated structures for reproducing ARPES results of
deeper states.

Since the features close to Fermi-level [features (i)–(iii) above] are
properly described by pseudohydrogen models, we can start by comparing them to
the band structures in Fig. LABEL:fig:bs. Feature (i) indicates that the non-
bonding band at $\Gamma$-point must be occupied, and therefore O content must
be less than about $0.25$. The gap of about 1.5 eV between features (i) and
(ii) would match to the pure OH-terminated surface. We then calculated
effective band structures for few different mixtures with small O content and
found overall best agreement with Ti3C2(O0.25OH0.25F0.5)2, as shown in Fig.
5(b) and processed such that it is more comparable to the experimental
spectrum in Fig. 5(c). In this case, features (i)–(iv) are all reproduced very
well. The EBS also shows flat band away from the $\Gamma$-point at about -4
eV, matching well with feature (v) in the experimental spectrum. We also
reproduce fairly well the broad band between -5 and -6 eV, feature (vi). The
lowest energy feature (vii) is not reproduced in our calculations. The band at
-7 eV in EBS originates from Ti-F bonds and the band at below -9 eV from O-H
bonds. Since semilocal functionals, such as PBEsol one used here, are known to
underestimate band widths and experimentally measured sample should contain
significant amount of F, we ascribe feature (vii) to Ti-F bonds, in agreement
with the discussion in Ref. Schultz et al., 2019.

###  III.4 Conclusions

In conclusion, we presented a study of the fermiology of titanium carbide and
nitride MXenes via density-functional theory calculations. We first calculated
the effective band structures for surfaces with mixtures of O and OH groups.
The electronic structure near the Fermi-level was found to evolve gradually
with the composition, and we further showed they are very well reproduced by
surfaces fully terminated with pseudohydrogenated OH groups. With the
pseudohydrogenated surfaces we could readily access Fermi-surface properties,
such as Fermi-surface area (or length in the case of 2D materials) and Fermi-
velocity. We calculated the electrical conductivity within the constant
relaxation time approximation and by comparing to the experimentally
determined conductivities we obtained relaxation time of about 1 fs. For the
same relaxation time, the conductivities for all studied MXenes were rather
similar and also fairly independent of the surface composition, changing at
most by a factor of two. Finally, we compared our effective band structures to
those obtained from recent ARPES experiments Schultz et al. (2019) for Ti3C2,
where the best agreement was obtained using Ti3C2(O0.25OH0.25F0.5)2 model.

The approaches demonstrated here can be used also for other MXenes, including
double MXenes Anasori et al. (2015) and even solid solutions in the M or X
sublattice if their positions and those of the surface functional groups are
uncorrelated. With the pseudohydrogenation approach it should be possible to
calculate many other material properties of MXenes with an effective mixed
surface composition, such as optical properties and electron-electron and
electron-phonon scattering rates. It will also allow investigating how
instabilities in electronic structure, leading to e.g. magnetism or charge
density waves, depend on the surface composition.

## Acknowledgments

We thank Prof. Norbert Koch and Dr. Thorsten Schultz for providing us the
original ARPES data, previously published in Ref. Schultz et al., 2019. We are
grateful to the Academy of Finland for support under Academy Research Fellow
funding No. 311058. We also thank CSC–IT Center for Science Ltd. for generous
grants of computer time.

## References

  * Naguib et al. (2012) M. Naguib, O. Mashtalir, J. Carle, V. Presser, J. Lu, L. Hultman, Y. Gogotsi, and M. W. Barsoum, ACS Nano 6, 1322 (2012), eprint http://dx.doi.org/10.1021/nn204153h, URL http://dx.doi.org/10.1021/nn204153h.
  * Gogotsi and Anasori (2019) Y. Gogotsi and B. Anasori, ACS Nano 13, 8491 (2019), pMID: 31454866, eprint https://doi.org/10.1021/acsnano.9b06394, URL https://doi.org/10.1021/acsnano.9b06394.
  * Naguib et al. (2011) M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi, and M. W. Barsoum, Advanced Materials 23, 4248 (2011), ISSN 1521-4095, URL http://dx.doi.org/10.1002/adma.201102306.
  * Anasori et al. (2017) B. Anasori, M. R. Lukatskaya, and Y. Gogotsi, Nature Reviews Materials 2, 16098 (2017), URL https://www.nature.com/articles/natrevmats201698.
  * Fu et al. (2019) Z. Fu, N. Wang, D. Legut, C. Si, Q. Zhang, S. Du, T. C. Germann, J. S. Francisco, and R. Zhang, Chemical Reviews 119, 11980 (2019), pMID: 31710485, eprint https://doi.org/10.1021/acs.chemrev.9b00348, URL https://doi.org/10.1021/acs.chemrev.9b00348.
  * Lee and Kim (2020) E. Lee and D.-J. Kim, Journal of The Electrochemical Society 167, 037515 (2020), URL https://doi.org/10.1149%2F2.0152003jes.
  * Matsuda and Hasegawa (2007) I. Matsuda and S. Hasegawa, Journal of Physics: Condensed Matter 19, 355007 (2007), URL https://doi.org/10.1088/0953-8984/19/35/355007.
  * Lasek et al. (2021) K. Lasek, J. Li, S. Kolekar, P. M. Coelho, L. Guo, M. Zhang, Z. Wang, and M. Batzill, Surface Science Reports p. 100523 (2021), ISSN 0167-5729, URL https://www.sciencedirect.com/science/article/pii/S016757292100008X.
  * Schultz et al. (2019) T. Schultz, N. C. Frey, K. Hantanasirisakul, S. Park, S. J. May, V. B. Shenoy, Y. Gogotsi, and N. Koch, Chemistry of Materials 31, 6590 (2019), eprint https://doi.org/10.1021/acs.chemmater.9b00414, URL https://doi.org/10.1021/acs.chemmater.9b00414.
  * Chen et al. (2017) L. Chen, P. Cheng, and K. Wu, Journal of Physics: Condensed Matter 29, 103001 (2017), URL https://doi.org/10.1088/1361-648x/aa54da.
  * Ouisse and Barsoum (2017) T. Ouisse and M. W. Barsoum, Materials Research Letters 5, 365 (2017), eprint https://doi.org/10.1080/21663831.2017.1333537, URL https://doi.org/10.1080/21663831.2017.1333537.
  * Xie and Kent (2013) Y. Xie and P. R. C. Kent, Phys. Rev. B 87, 235441 (2013), URL http://link.aps.org/doi/10.1103/PhysRevB.87.235441.
  * Khazaei et al. (2013) M. Khazaei, M. Arai, T. Sasaki, C.-Y. Chung, N. S. Venkataramanan, M. Estili, Y. Sakka, and Y. Kawazoe, Advanced Functional Materials 23, 2185 (2013), ISSN 1616-3028, URL http://dx.doi.org/10.1002/adfm.201202502.
  * Khazaei et al. (2017) M. Khazaei, A. Ranjbar, M. Arai, T. Sasaki, and S. Yunoki, J. Mater. Chem. C 5, 2488 (2017), URL http://dx.doi.org/10.1039/C7TC00140A.
  * Hu et al. (2015) T. Hu, H. Zhang, J. Wang, Z. Li, M. Hu, J. Tan, P. Hou, F. Li, and X. Wang, Scientific Reports 5, 16329 (2015).
  * Ibragimova et al. (2019) R. Ibragimova, M. J. Puska, and H.-P. Komsa, ACS Nano 13, 9171 (2019), eprint https://doi.org/10.1021/acsnano.9b03511, URL https://doi.org/10.1021/acsnano.9b03511.
  * Ibragimova et al. (2021) R. Ibragimova, P. Erhart, P. Rinke, and H.-P. Komsa, The Journal of Physical Chemistry Letters 12, 2377 (2021), eprint https://doi.org/10.1021/acs.jpclett.0c03710, URL https://doi.org/10.1021/acs.jpclett.0c03710.
  * Ku et al. (2010) W. Ku, T. Berlijn, and C.-C. Lee, Phys. Rev. Lett. 104, 216401 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.104.216401.
  * Popescu and Zunger (2010) V. Popescu and A. Zunger, Phys. Rev. Lett. 104, 236403 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.104.236403.
  * Kresse and Furthmüller (1996a) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996a).
  * Kresse and Furthmüller (1996b) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996b).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994), URL https://link.aps.org/doi/10.1103/PhysRevB.50.17953.
  * Perdew et al. (2008) J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008), URL https://link.aps.org/doi/10.1103/PhysRevLett.100.136406.
  * Medeiros et al. (2014) P. V. C. Medeiros, S. Stafström, and J. Björk, Phys. Rev. B 89, 041407 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.041407.
  * Madsen et al. (2018) G. K. Madsen, J. Carrete, and M. J. Verstraete, Computer Physics Communications 231, 140 (2018), ISSN 0010-4655, URL https://www.sciencedirect.com/science/article/pii/S0010465518301632.
  * Ganose et al. (2021) A. M. Ganose, A. Searle, A. Jain, and S. M. Griffin, Journal of Open Source Software 6, 3089 (2021), URL https://doi.org/10.21105/joss.03089.
  * Hu et al. (2017) T. Hu, Z. Li, M. Hu, J. Wang, Q. Hu, Q. Li, and X. Wang, The Journal of Physical Chemistry C 121, 19254 (2017), eprint https://doi.org/10.1021/acs.jpcc.7b05675, URL https://doi.org/10.1021/acs.jpcc.7b05675.
  * Häglund et al. (1993) J. Häglund, A. Fernández Guillermet, G. Grimvall, and M. Körling, Phys. Rev. B 48, 11685 (1993), URL https://link.aps.org/doi/10.1103/PhysRevB.48.11685.
  * Halim et al. (2014) J. Halim, M. R. Lukatskaya, K. M. Cook, J. Lu, C. R. Smith, L.-Å. Näslund, S. J. May, L. Hultman, Y. Gogotsi, P. Eklund, et al., Chemistry of Materials 26, 2374 (2014), pMID: 24741204, eprint https://doi.org/10.1021/cm500641a, URL https://doi.org/10.1021/cm500641a.
  * Ling et al. (2014) Z. Ling, C. E. Ren, M.-Q. Zhao, J. Yang, J. M. Giammarco, J. Qiu, M. W. Barsoum, and Y. Gogotsi, Proceedings of the National Academy of Sciences 111, 16676 (2014), ISSN 0027-8424, eprint https://www.pnas.org/content/111/47/16676.full.pdf, URL https://www.pnas.org/content/111/47/16676.
  * Shahzad et al. (2016) F. Shahzad, M. Alhabeb, C. B. Hatter, B. Anasori, S. Man Hong, C. M. Koo, and Y. Gogotsi, Science 353, 1137 (2016), ISSN 0036-8075, eprint https://science.sciencemag.org/content/353/6304/1137.full.pdf, URL https://science.sciencemag.org/content/353/6304/1137.
  * Liu et al. (2020) J. Liu, Z. Liu, H.-B. Zhang, W. Chen, Z. Zhao, Q.-W. Wang, and Z.-Z. Yu, Advanced Electronic Materials 6, 1901094 (2020), eprint https://onlinelibrary.wiley.com/doi/pdf/10.1002/aelm.201901094, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/aelm.201901094.
  * Lipatov et al. (2021) A. Lipatov, A. Goad, M. J. Loes, N. S. Vorobeva, J. Abourahma, Y. Gogotsi, and A. Sinitskii, Matter 4, 1413 (2021), ISSN 2590-2385, URL https://www.sciencedirect.com/science/article/pii/S2590238521000540.
  * Soundiraraju and George (2017) B. Soundiraraju and B. K. George, ACS Nano 11, 8892 (2017), pMID: 28846394, eprint http://dx.doi.org/10.1021/acsnano.7b03129, URL http://dx.doi.org/10.1021/acsnano.7b03129.
  * Halim et al. (2019) J. Halim, I. Persson, E. J. Moon, P. Kühne, V. Darakchieva, P. O. Å. Persson, P. Eklund, J. Rosen, and M. W. Barsoum, Journal of Physics: Condensed Matter 31, 165301 (2019), URL https://doi.org/10.1088%2F1361-648x%2Fab00a2.
  * Hart et al. (2019) J. L. Hart, K. Hantanasirisakul, A. C. Lang, B. Anasori, D. Pinto, Y. Pivak, J. T. van Omme, S. J. May, Y. Gogotsi, and M. L. Taheri, Nature Communications 10, 522 (2019).
  * Urbankowski et al. (2016) P. Urbankowski, B. Anasori, T. Makaryan, D. Er, S. Kota, P. L. Walsh, M. Zhao, V. B. Shenoy, M. W. Barsoum, and Y. Gogotsi, Nanoscale 8, 11385 (2016), URL http://dx.doi.org/10.1039/C6NR02253G.
  * Anasori et al. (2015) B. Anasori, Y. Xie, M. Beidaghi, J. Lu, B. C. Hosler, L. Hultman, P. R. C. Kent, Y. Gogotsi, and M. W. Barsoum, ACS Nano 9, 9507 (2015), pMID: 26208121, eprint https://doi.org/10.1021/acsnano.5b03591, URL https://doi.org/10.1021/acsnano.5b03591.


# The impact of environment on the lives of disk galaxies as revealed by SDSS-
IV MaNGA

Shuang Zhou1, Michael Merrifield1, Alfonso Aragón-Salamanca1, Joel R.
Brownstein2, Niv Drory3, Renbin Yan4, Richard R. Lane5
1School of Physics & Astronomy, University of Nottingham, University Park,
Nottingham, NG7 2RD, UK
2Department of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt
Lake City, UT 84112, USA
3McDonald Observatory, The University of Texas at Austin, 1 University
Station, Austin, TX 78712, USA
4Department of Physics, The Chinese University of Hong Kong, Shatin, N.T.,
Hong Kong, China
5Centro de Investigación en Astronomía, Universidad Bernardo O’Higgins,
Avenida Viel 1497, Santiago, Chile
Contact e-mail<EMAIL_ADDRESS>
(Last updated ???; in original form ???)

###### Abstract

Environment has long been known to have significant impact on the evolution of
galaxies, but here we seek to quantify the subtler differences that might be
found in disk galaxies, depending on whether they are isolated, the most
massive galaxy in a group (centrals), or a lesser member (satellites). The
MaNGA survey allows us to define a large mass-matched sample of 574 galaxies
with high-quality integrated spectra in each category. Initial examination of
their spectral indices indicates significant differences, particularly in low-
mass galaxies. Semi-analytic spectral fitting of a full chemical evolution
model to these spectra confirms these differences, with low-mass satellites
having a shorter period of star formation and chemical enrichment typical of a
closed box, while central galaxies have more extended histories, with evidence
of on-going gas accretion over their lifetimes. The derived parameters for gas
infall timescale and wind strength suggest that low-mass satellite galaxies
have their hot halos of gas effectively removed, while central galaxies retain
a larger fraction of gas than isolated galaxies due to the deeper group
potential well in which they sit. S0 galaxies form a distinct subset within
the sample, particularly at higher masses, but do not bias the inferred lower-
mass environmental impact significantly. The consistent picture that emerges
underlines the wealth of archaeological information that can be extracted from
high-quality spectral data using techniques like semi-analytic spectral
fitting.

###### keywords:

galaxies: fundamental parameters – galaxies: stellar content –galaxies:
formation – galaxies: evolution

††pubyear: 2022††pagerange: The impact of environment on the lives of disk
galaxies as revealed by SDSS-IV MaNGA–The impact of environment on the lives
of disk galaxies as revealed by SDSS-IV MaNGA

##  1 Introduction

Galaxies are complex systems, and their formation and evolution are affected
by a variety of processes. In the current paradigm, galaxies form at the
centre of dark matter halos whose formation and evolution are described by the
Lambda cold dark matter cosmology (White & Rees, 1978). Baryonic matter
accretes into these dark matter halos, fueling star formation. During this
initial gas accretion and subsequent evolution, galaxies can experience a wide
range of interactions with their environment, on scales ranging from mergers
with individual galaxies to hydrodynamic interactions with galaxy clusters and
the cosmic web.

It is therefore perhaps unsurprising that galaxy properties depend on the
environment that they inhabit. Most famously, high-density regions such as
groups and clusters contain a significantly higher fraction of red galaxies,
often associated with earlier morphological types (e.g. Oemler, 1974;
Dressler, 1980; Postman & Geller, 1984). With the advent of large scale
surveys such as the Sloan Digital Sky Survey (SDSS,York et al. 2000), the
environmental dependence of the galaxies’ star formation histories has been
extensively explored. Such surveys have revealed that satellite galaxies in
groups and clusters (e.g. Pasquali et al., 2010; Peng et al., 2012; Wetzel et
al., 2012; Wetzel et al., 2013) and galaxies that live in high-density regions
(e.g. Kauffmann et al., 2003; Baldry et al., 2006) are more likely to have
their star formation quenched than isolated galaxies. Other properties related
to the galaxies’ star-formation histories show related environmental
dependence: galaxies are redder (Blanton et al., 2005b; Li et al., 2006) and
have a higher D4000 index (Kauffmann et al., 2004; Li et al., 2006) in denser
environments. Further observations have shown that this environmental
dependence may persist to high redshifts (e.g. Peng et al., 2010; Darvish et
al., 2016; Kawinwanichakij et al., 2017). More recently, spatially-resolved
observations allow the investigation of such environmental effects within the
galaxies themselves, although several studies indicate that the environment
seems to work mainly globally, with very little effect on the radial profiles
of stellar population properties (e.g. Zheng et al., 2017; Spindler et al.,
2018).

There is some evidence that, in addition to affecting the star-formation
activity of galaxies, the environment may also impact on their chemical
evolution, although the effects seem more subtle, complex, and uncertain.
Early-type galaxies in low-density environments are more metal rich than their
counterparts in high-density environments (Sánchez-Blázquez et al., 2006).
Further, Pasquali et al. (2010) found that satellite galaxies seem to be more
enriched in metals than central ones, especially in low-mass systems. However,
the opposite trend was found by Zheng et al. (2017): in large-scale sheets and
voids, satellite galaxies are relatively more metal poor than central ones.
Zheng et al. (2019) also found that the environment appears to have a modest
effect on the $\alpha$-element abundances of galaxies, especially at low and
average galaxy masses. Finally, as with stellar ages, although the global
metallicity of galaxies correlates with their environment, metallicity
gradients show weak or no environmental dependence (e.g. Zheng et al., 2017;
Goddard et al., 2017).

In trying to understand the physical drivers of these differences, it is
apparent that varying processes are at work depending on whether a galaxy is
central within a halo or a satellite system. Indeed, analysing SDSS data, Peng
et al. (2012) concluded that the quenching of satellite galaxies in galaxy
groups provides the starting point of environmental effects. Kovač et al.
(2014) further confirmed the dominant role of satellite quenching over time by
analysing samples from zCOSMOS-bright data; they found little evidence that
the red fraction of central galaxies depends on over-density, while satellites
are commonly redder at all over-densities, at least back to $z\sim 0.7$.

Currently, it is still unclear what distinction in physical mechanism is
responsible for these differences in what is termed "environmental quenching".
A range of processes have been suggested that can contribute, which all
involve depriving a galaxy of the cold gas that it needs to keep forming
stars. For example, a galaxy may experience ram-pressure stripping when it
moves through a hot gaseous medium, which can rapidly remove the fuel for star
formation and quench the galaxy (e.g. Gunn & Gott, 1972; Abadi et al., 1999).
Alternatively, if only the gas in the outer part of the galaxy is removed, or
is prevented from falling to the galaxy through heating, the galaxy would
experience ‘strangulation’ and gradually quench when the existing fuel is
exhausted (e.g. Larson et al., 1980; Balogh et al., 2000). Tidal interactions
between galaxies in a group or between galaxies and the dark matter halo of
the group or cluster can also cause stripping of the gas (e.g. Read et al.,
2006). In each case, the net effect is the same, so the observable
consequences of quenching will be similar, making it difficult to determine
which processes are important in any particular situation. However, given the
physical differences in these processes, we might expect to be able to
distinguish between them on the basis of subtler indicators such as the
timescale over which star formation shuts down, and the way that gas loss can
alter a galaxy’s chemical evolution.

Fortunately, we are now in a position to investigate these subtler factors to
distinguish between such effects. The recently-completed Mapping Nearby
Galaxies at Apache Point Observatory (MaNGA) survey (Bundy et al., 2015) has
provided us with high-quality spectra from a well-defined sample of over
10,000 nearby galaxies, which allows us to select a large set of galaxies in
different environments, to search for quite subtle variations. The size of the
survey also means that we can focus on a homogeneous subset of galaxies,
matched in mass between environments, and chosen to have a shared underlying
morphology – in this case, all disk galaxies – to seek to minimize the
variations that might be expected for galaxies of very different types in
different environments. Moreover, we have recently shown that, by fitting
relatively simple stellar and chemical evolution models directly to the high-
quality spectra that MaNGA provides, it is possible to determine physical
parameters related to the timescales of star formation and rates of gas
accretion and loss that should allow us to distinguish between quenching
mechanisms (Zhou et al. 2022, hereafter Paper I).

To this end, we have set out the analysis in this paper as follows. The data
and sample selection, and a preliminary look at the evidence it presents for
environmental influence, are described in §2. The "semi-analytic spectral
fitting" process, by which we fit a simple physically-motivated evolutionary
model directly to the spectra, is summarized in §3. The inferred evolutionary
parameters for central, satellite and isolated galaxies, and their
implications for the drivers of environmental quenching, are described in §4.
Finally, we summarise the key findings in §Data availability.

##  2 Data

###  2.1 The MaNGA Survey

As part of the fourth generation of SDSS (SDSS-IV, Blanton et al. 2017), MaNGA
has completed its mission to collect spatially-resolved high-quality spectra
for a well-defined sample of over 10,000 nearby galaxies (redshift
$0.01<z<0.15$, Yan et al. 2016b; Wake et al. 2017). MaNGA targets were
selected to span the stellar mass range $5\times 10^{8}h^{-2}{M}_{\odot}\leq
M_{*}\leq 3\times 10^{11}h^{-2}{M}_{\odot}$ (Wake et al., 2017), using stellar
masses from the NASA Sloan Atlas catalogue 111http://www.nsatlas.org/ (NSA,
Blanton et al. 2005a). Using two dual-channel BOSS spectrographs (Smee et al.,
2013) mounted on the 2.5 m telescope (Gunn et al., 2006), intermediate
resolution ($R\sim 2000$, Drory et al. 2015) spectra that cover $3600-10300$Å
in wavelength were obtained out to at least 1.5 effective radii for all the
target galaxies (Law et al., 2015). The raw spectra were reduced and
calibrated (Yan et al., 2016a) via the Data Reduction Pipeline (DRP; Law et
al. 2016) to produce science-ready spectra with flux calibrations batter than
$5\%$ across most of the wavelength range. The Data Analysis Pipeline
(DAP;Westfall et al. 2019; Belfiore et al. 2019) processed these reduced data
and provided the measurements of stellar kinematics, emission-line properties
and spectral indices employed in this work.

###  2.2 Sample selection and data reduction

Figure 1: The star formation rate as a function of stellar mass for our sample
galaxies. Isolated, central and satellite galaxies are shown in green, red and
blue, respectively. The black dash line shows the star forming main sequence
obtained by Renzini & Peng (2015) as reference.

The intent of this paper is to investigate how the stellar population
properties in galaxies depend on the environment that they inhabit,
specifically whether they are central or satellite galaxies in groups, or are
isolated objects. To identify such systems, we make use of the 11th MaNGA
Product Launch (MPL11) data, which comprises the full 10,010 unique galaxies
observed, and was released with the SDSS-IV Data Release 17 (DR17, Abdurro’uf
et al. 2022). We are also looking to select a sample in which mergers are
unlikely to have confused the evolutionary picture significantly, so focus on
disk-like systems (Mo et al., 1998). To this end, we make use of the value-
added catalogue (VAC), the MaNGA Morphology Deep Learning DR17 Catalogue
(Domínguez Sánchez et al., 2022), which provides classifications for all the
MaNGA MPL11 galaxies. In this catalogue, several indicators are provided to
describe the morphology of the galaxy. $\rm P_{LTG}$ is the probability of
being late-type galaxies from the deep-learning approach. T-Type is another
indicator with T-type$\leq$0 for early-type galaxies, T-type$>$0 for late-type
galaxies, and T-type=0 for S0s. Visual class (VC) comes from visual inspection
of the image of the galaxy, in which late-type galaxies are marked with VC=3
and S0 with VC=2. In addition, a visual classification flag (VF) is used to
indicate the robustness of the visual classification, in which certain visual
classification results have VF=0. There are multiple ways in which these
indicators can be combined to create a single measure of morphology; here, we
follow the criteria suggested in Domínguez Sánchez et al. (2022) to select
disk galaxies as those in the VAC labeled with

  * •

spiral: ($\rm P_{LTG}>$0.5) and (T-Type$>$0) and (VC=3) and (VF=0),

  * •

S0: ($\rm P_{LTG}<$0.5) and (T-Type$<$0) and (PS0$>$0.5) and (VC=2) and
(VF=0),

which yields 5125 spirals and 891 S0s. To ensure the robustness of
morphological classifications and to minimize dust attenuation effects, we
also require the sample galaxies to be reasonably face-on; after applying an
axis-ratio cut of $b/a>0.5$, we obtain a sample of 4416 reasonably face-on
disk galaxies.

To obtain the requisite environment information, we make use of another VAC,
the Galaxy Environment for MaNGA Value Added Catalog (GEMA-VAC). In this VAC,
the environment information of MaNGA galaxies is derived using the methods
described in Argudo-Fernández et al. (2015), Etherington & Thomas (2015) and
Wang et al. (2016). The final catalogue will be described in more detail in
Argudo-Fernádez et al. (in prep.), but the VAC itself has been released with
SDSS DR17. We use the group environment of MaNGA galaxies provided in the VAC,
in which galaxies are allocated to groups using the catalogue of Yang et al.
(2007). Using this information, we divide the sample into three classes:

  * •

isolated: number of galaxies NG=1 in the group

  * •

central: the most massive galaxy in a group with NG$>1$

  * •

satellite: not the most massive galaxy in a group with NG$>1$.

One further issue is that galaxies selected in different environments could
have systematically different properties. In particular, their mass
distributions are likely different, with more massive systems found in higher-
density regions. Since mass is likely the biggest single intrinsic driver of
evolutionary differences, this effect would mask the extrinsic environmental
effects that we are looking for. To mitigate this issue, we have mass-matched
the sub-samples of galaxies in each environmental category in the stellar mass
range $9.5<\log(M_{*}/{\rm M}_{\odot})<11.5$. This range is divided into bins
of width $\log(M_{*}/{\rm M}_{\odot})=0.1$, which is approximately equal to
the uncertainty in the mass measurement, and within each bin we select all $N$
galaxies from the least-popular category, and a random sub-sample of $N$
galaxies from the other two categories. In this way, we construct a mass-
matched sample in the three categories, which ends up with 574 galaxies in
each environmental class.

In this sample, most of the central galaxies come from relatively small
groups, with a median number of group members NG=3, and 90% of the groups have
NG<10\. Satellite galaxies come from bigger groups with a median NG=16. In
addition, the GEMA-VAC provides halo mass estimations for MaNGA galaxies
obtained from Yang et al. (2007). We use these halo masses to characterise in
more detail the dark matter halo properties of the sample galaxies. Isolated
galaxies have halo masses in the range $10^{11.5}<M_{\rm h}/{\rm
M}_{\odot}<10^{13}$ with a median of $10^{12.4}{\rm M}_{\odot}$. Central
galaxies generally reside in relatively larger dark matter halos with halo
masses in the range $10^{11.5}<M_{\rm h}/{\rm M}_{\odot}<10^{14}$ and a median
of $10^{12.6}{\rm M}_{\odot}$. As might be expected for satellite galaxies
that do not define the mass of the halo in which they reside, the range for
these systems is much larger, spanning from $10^{12}{\rm M}_{\odot}$ to
$10^{14.5}{\rm M}_{\odot}$ with a median of $10^{13.7}{\rm M}_{\odot}$. In
what follows we will explore how this range of environments has affected the
evolution of the galaxies they host.

###  2.3 Environmental dependence of the galaxies’ empirical properties

Before getting into the detailed modelling that will tell us about the
physical processes involved in environmental quenching, it is helpful to look
for clues in the simpler empirical data available directly from the MaNGA DAP.
In Figure 1 we present the star formation activity of the sample galaxies as a
function of their masses. For this analysis, we use H$\alpha$ flux
measurements provided by MaNGA DAP (where detected) to estimate the star
formation rate. Using the Calzetti extinction law (Calzetti et al., 2000) and
assuming a intrinsic H${\alpha}$/H${\beta}$ ratio of 2.87 (Osterbrock &
Ferland, 2006), we correct the H$\alpha$ flux for dust attenuation. A
conversion factor from Murphy et al. (2011) calibrated with a Chabrier
(Chabrier, 2003) IMF,

$\rm\frac{SFR}{M_{\odot}yr^{-1}}=5.37\times 10^{-42}\frac{L(H\alpha)}{erg\
s^{-1}},$ (1)

is then applied to calculate the SFR, which is shown as a function of the
stellar mass from the NSA. From the plot, we see that most of the sample
galaxies lie close to the star-forming main sequence. However, significant
deviations from the main sequence towards lower SFRs are also seen in many
galaxies, particularly the satellites systems. Clearly, this offers evidence
that the present-day star-formation rates in disk galaxies are significantly
affected by environment.

To probe further back into the star-formation histories of galaxies of
different types, we can look for empirical evidence in various absorption-line
indices that have been measured for these galaxies. In particular, the Dn4000
index is often used as an indicator of the stellar population, and
specifically a value of Dn4000$<1.6$ will be an indicator of star formation
within the past 1–2 Gyr (e.g. Kauffmann et al., 2003). In addition, the $\rm
Mgb/\langle Fe\rangle\equiv Mgb/(0.5*Fe5270+0.5*Fe5335)$ index is a proxy for
the relative abundance of alpha elements, for which higher values are found in
galaxies whose stars formed on a shorter timescale (e.g. Worthey, 1994; Thomas
et al., 2005; Zheng et al., 2019). Figure 2 shows the average values for these
indices, integrated out to the effective radius of all the sample galaxies, as
a function of their stellar masses. Although there is a good deal of scatter,
the average lines on these plots show clear systematic differences between
galaxies in different environments. The satellite galaxies show stronger
values of both indices, indicating relatively little recent star formation,
consistent with what we have learned from H$\alpha$, and a shorter timescale
for their formation. Interestingly, though, we also see consistent, if subtle,
differences between central and isolated galaxies, even in these relatively
crude measures of galaxy evolution, which motivates a more through
investigation of its causes.

Figure 2: Dn4000 (top) and $\rm Mgb/\langle Fe\rangle$ (bottom) as functions
of stellar mass for our sample galaxies. Isolated, central and satellite
galaxies are shown in green, red and blue, respectively. Triangles linked by
lines are mean results in different stellar mass bins, with error bar showing
the error estimated from the jackknife resampling method.

##  3 Analysis

###  3.1 The chemical evolution model

In order to investigate the physical processes that underlie the spectral
differences between galaxies in different environment, we fit a chemical
evolution model directly to the spectra. The model is designed to be
sufficiently general to capture the main physical processes in a wide variety
of formation scenarios, but simple enough to produce robust fits to the
spectra. This semi-analytic spectral fitting approach is described in detail
in Paper I, but we summarize the main steps in the approach here.

To construct a sufficiently-general picture of the star-formation and chemical
evolution of a galaxy, we have to take into account gas inflow and outflow
processes as well as the star formation triggered by the resulting gas
reservoir. We therefore consider a model in which the gas mass evolution of
the galaxy can be described by

$\dot{M}_{\rm g}(t)=\dot{M}_{\rm in}(t)-\psi(t)+\dot{M}_{\rm
re}(t)-\dot{M}_{\rm out}(t).$ (2)

The first term characterises the gas inflow, which is assumed to have an
exponentially decaying form

$\dot{M}_{\rm in}(t)=Ae^{-(t-t_{0})/\tau},\ \ \ t>t_{0}.$ (3)

The second term represents the gas lost to star formation, with the third term
accounting for the mass ejection from dying stars, and the final term
describing mass lost from the system. We assume a linear Schmidt law (Schmidt,
1959) to model the star formation activity, so

$\psi(t)=S\times M_{\rm g}(t),$ (4)

where $S$ is the star-formation efficiency. We assume this efficiency to be
constant throughout the evolution and estimate it with the extended Schmidt
law proposed by Shi et al. (2011),

$S(yr^{-1})=10^{-10.28\pm
0.08}\left(\frac{\Sigma_{*}}{M_{\odot}pc^{-2}}\right)^{0.48},$ (5)

where $\Sigma_{*}$ is the stellar mass surface density of the galaxy. We
obtain an approximate average value for $\Sigma_{*}$ from the current stellar
mass and effective radius in the NSA, $\Sigma_{*}=0.5\times M_{*}/(\pi R_{\rm
e}^{2})$.

In this model, a constant mass return fraction of $R=0.3$ is assumed, so that
the stars formed in each generation will return 30% of their stellar masses to
the ISM. For the mass-loss term, we follow the canonical assumption that the
outflow strength is proportional to the star formation activity:

$\dot{M}_{\rm out}(t)=\lambda\psi(t),$ (6)

where the dimensionless quantity $\lambda$ is the relative outflow strength,
often called the ‘wind parameter.’

In Paper I, we assumed the outflow to be constant, but with a sharp cut-off at
some time (to represent when a galaxy’s potential well became too deep for the
winds to escape). This assumption allows a secondary star-formation episode
during the evolution of the galaxy, but the sudden cutoff leads to
discontinuity in the chemical and star-formation evolution. Here, we consider
a somewhat gentler model for the changes in the wind parameter, which is
assumed to vary linearly between an initial value $\lambda_{\rm b}$ at the
beginning of galaxy formation (14Gyr ago,) and a value of $\lambda_{\rm e}$ at
the present day:

$\lambda(t)=\lambda_{\rm b}+(\lambda_{\rm e}-\lambda_{\rm b})t/(14\,{\rm
Gyr}).$ (7)

We found that this simple refinement avoids the likely-unphysical
discontinuity in properties, while leaving the statistical results largely
unchanged. Using all those ingredients, the equation that characterise the gas
mass evolution can be written as

$\dot{M}_{\rm g}(t)=Ae^{-(t-t_{0})/\tau}-S(1-R+\lambda(t))M_{\rm g}(t).$ (8)

For the chemical evolution, we adopt the usual instantaneous mixing
approximation, in which the gas in a galaxy is always well mixed during its
evolution. The equation that characterize the chemical evolution can then be
written as

$\displaystyle\dot{M}_{Z}(t)=$ $\displaystyle Z_{\rm
in}Ae^{-(t-t_{0})/\tau}-Z_{\rm g}(t)(1-R)SM_{\rm g}(t)+$ (9) $\displaystyle
y_{Z}(1-R)SM_{\rm g}(t)-Z_{\rm g}(t)\lambda(t)SM_{\rm g}(t),$

where $Z_{\rm g}(t)$ is the gas phase metallicity and $M_{Z}(t)\equiv M_{\rm
g}\times Z_{\rm g}$. The first term is the inflow, which brings in gas with
metallicity $Z_{\rm in}$, which is simply assumed to be $Z_{\rm in}=0$
throughout this work. The second term characterizes the metal locked up in
long-lived stars, while the third term represents the metal-enriched gas
returned by dying stars. The last term characterise how the outflow blows away
metal-enriched gas, with a time-dependent outflow strength specified by
Equation 7.

###  3.2 Fitting the model to observed data

Now that we have a very general model for the star-formation history (SFH) and
chemical evolution history (ChEH) of a galaxy, we are in a position to
constrain its parameters using the spectral data from each galaxy. We do so by
adopting a Bayesian framework, using an updated version of the Bayesian
Inference of Galaxy Spectra (BIGS), as presented in Paper I. In short, a set
of model parameters are generated from an appropriate prior distribution
(listed in Table 1), which are used to calculate the SFH and ChEH following
Equation 8 and Equation 9. The present-day gas phase metallicity $Z_{\rm g}$
is obtained from the ChEH. Using the Bruzual & Charlot (2003) stellar
population models, we calculate a model spectrum corresponding to the SFH and
ChEH following the standard stellar population synthesis approach (see review
by Conroy 2013). We use Bruzual & Charlot (2003) models constructed with the
‘Padova1994’ isochrones, the Chabrier (Chabrier, 2003) IMF and the STILIB (Le
Borgne et al., 2003) stellar templates, which covers metallicities from
$Z=0.0001$ to $Z=0.05$, and ages from $0.0001\,{\rm Gyr}$ to $20\,{\rm Gyr}$.
The library models have a FWHM resolution of 3Å in the wavelength range $3200$
– $9500$Å, which makes them well-suited to fitting MaNGA spectra observed at
$3600$ – $10300$Å with low redshifts ($z\sim 0.03$, Wake et al. 2017), but we
must also broaden the stellar templates to match the broadening due to stellar
velocity dispersion and instrumental effects, which we infer from an initial
pPXF fit to the spectra (Cappellari, 2017). We can then compare the inferred
model spectrum to the MaNGA data and vary the parameters to find the best-fit
physically-motivated model.

This method was extensively tested in Paper 1, in which we showed that, with
spectral data at a sufficient signal-to-noise ratio (SNR), it can reliably
reproduce the overall SFH and ChEH of galaxies even when the simplified model
does not exactly match the detailed history of a galaxy. However, it does
require quite a high SNR to fit reliably. Fortunately, here we are interested
in the global properties of galaxies, so, as described in Paper 1, can combine
all the data within 1$R_{\rm e}$ of each galaxy to obtain a spectrum with a
SNR of $\sim 70$ per Å, which is sufficient to robustly reproduce the system’s
history.

In Paper I, we used emission-line diagnostics to further constrain the
present-day star-formation rate and gas-phase metallicity. In this work,
however, as we no longer focus only on spiral galaxies, some objects do not
offer these constraints. However, we can use a population-wide relation
instead, at least for the metallicity constraint. Specifically, Andrews &
Martini (2013) demonstrated that there is quite a tight relationship between
mass and gas-phase metallicity, so we can assign an expected gas-phase oxygen
abundance using

$12+{\rm log(O/H)}=8.798-{\rm
log}\left(1+\left(\frac{10^{8.901}}{M_{\star}/{\rm
M}_{\odot}}\right)^{0.64}\right).$ (10)

This oxygen abundance is then converted to total metallicity using a solar
metallicity of 0.02 and oxygen abundance of $12+\log({\rm O/H})=8.83$ (Anders
& Grevesse, 1989). We note that this relation is chosen to be consistent with
the calibration of the Bruzual & Charlot (2003) SSP models; alternative
calibrations can also be applied but appropriate caution is urged when
comparing with our results. The likelihood function used in deriving a best
fit can then be written as:

$\ln{L(\theta)}\propto-\sum_{i}^{N}\frac{\left(f_{\theta,i}-f_{\rm
D,i}\right)^{2}}{2f_{\rm err,i}^{2}}-\frac{(Z_{\rm g,\theta}-Z_{\rm
g,D})^{2}}{2\sigma_{Z}^{2}},$ (11)

where $f_{\theta,i}$ is the flux predicted from the model with parameter set
$\theta$, while $f_{\rm D,i}$ is the observed flux from the stacked spectrum.
$f_{\rm err,i}$ represents the corresponding error spectrum, and the sum is
made over all $N$ wavelength points. $Z_{\rm g,\theta}$ is the current gas
phase metallicity from the model, while $Z_{\rm g,D}$ is the gas phase
metallicity estimated from the empirical relation of Equation 10. $\sigma_{Z}$
is the uncertainty in the gas phase metallicity estimates, which was set to
$\sigma_{Z}=0.1Z_{g,D}$ (Zhou et al., 2022).

Table 1: Priors of model parameters used to fit galaxy spectra Parameter | Description | Prior range
---|---|---
$y_{Z}$ | Yield parameter | $[0.0,0.08]$
$\tau$ | Gas infall timescale |  $[0.0,14.0]$Gyr
$t_{0}$ | Start time of gas infall |  $[0.0,14.0]$Gyr
$\lambda_{b}$ | The wind parameter $14\,$Gyr ago | $[0.0,10.0]$
$\lambda_{e}$ | The current wind parameter | $[0.0,10.0]$
$E(B-V)$ | Dust attenuation parameter | $[0.0,0.5]$
Figure 3: Environment dependence of galaxy properties derived from best-fit
models for our sample galaxies. The first row shows the cumulative SFH while
the second row shows the cumulative metallicity distribution, averaged over
galaxies in each bin and category respectively. To interpret the derived mass
and chemical evolution, the third and fourth panel show the cumulative
distributions of the gas infall time-scale $\tau$ and mass-weighted averaged
wind parameter $\overline{\lambda}$, as indicators of gas infall and outflow
properties respectively in our sample galaxies. In each row, from left to
right, different panels show results from different stellar mass bins as
indicated. In each panel, green, red and blue lines are results obtained from
isolated, central and satellite galaxies respectively.

##  4 Results and discussion

Having performed the semi-analytic spectral fitting on the full mass-matched
sample of central, satellite and isolated galaxies, we are now in a position
to quantify and interpret, at least in broad terms, the empirical differences
in their properties found in Section 2.3 in terms of the basic physical
parameters that this analysis provides. In Figure 3, we present the average
derived properties and underlying physical parameters for the mass-matched
samples of isolated, central and satellite galaxies, divided into four mass
bins. We present these properties in cumulative form, in part because in some
cases it has a simpler physical interpretation (e.g. Greener et al., 2021),
but also because it makes clearer the systematic differences between the sub-
samples.

In terms of star-formation history (top row of Figure 3), it is clear that in
the lowest stellar mass bin ($10^{9.5}<M_{*}/{\rm M}_{\odot}<10^{10.0}$),
central galaxies accumulate their stellar mass significantly later than
satellite galaxies: on average less than 30% of the stellar mass in central
galaxies formed earlier than 5 Gyrs ago, while in the same period satellite
galaxies had accumulated more than 60% of their stellar mass. The evolutionary
history of isolated galaxies lie between these two extremes. These SFHs are in
line with indications from the lower panel of Figure 2, in which the higher
$\alpha$ element abundance in low-mass satellites flags their shorter star
formation timescales. The difference between central and satellite galaxies
becomes weaker in higher mass galaxies, and in galaxies with $M_{*}/{\rm
M}_{\odot}>10^{10.5}$ the difference is almost negligible. The different
formation history of central and satellite galaxies, as well as its variation
with stellar mass, is consistent with previous investigations (e.g. Pasquali
et al., 2010; Wetzel et al., 2012; Peng et al., 2012), but the evolutionary
model fitted here means that we will be able to go on to attribute a physical
interpretation to these differences.

The second row of Figure 3 shows the associated average cumulative metallicity
distribution (CMDF) in the different mass bins. Once again, it is the low mass
galaxies that display the biggest differences, with satellite galaxies having
$\sim 80\%$ of stars with $Z<0.005$, while in central galaxies of similar
stellar mass this fraction decreases to $\sim 20\%$. The low-metal-rich CMDF
of low-mass satellite galaxies is characteristic of the so-call ‘closed-box’
or ‘leaky-box’ model, in which there is no significant gas infall to the
system (Tinsley, 1974); in contrast, the CMDFs in central galaxies are closer
to the predictions of ‘accreting box’ models (Tinsley, 1974, 1980), in which a
replenishing supply of new gas is provided, which suppresses the relative
fraction of low-metallicity stars. This effect is still present in higher mass
bins, but is much reduced, indicating that environment is a much less
important factor in the chemical evolution of these galaxies.

We can now quantify plausible drivers of these differences using the physical
parameters inferred in the semi-analytic spectral fitting process. The third
row of Figure 3 shows one of the primary drivers of differences between
galaxies, the gas infall timescale, as a function of both environment and
mass. It is apparent that, at low and intermediate masses, central galaxies
tend to have longer gas infall timescales, while the gas infall occurs over a
systematically shorter period in satellite galaxies. In the lowest mass bin
($10^{9.5}<M_{*}/{\rm M}_{\odot}<10^{10.0}$), almost all of the satellite
galaxies have gas infall timescale less than $\sim$5 Gyr, while around $40\%$
of central galaxies have timescales greater than $\sim$5 Gyr, with isolated
galaxies somewhere in-between. The median gas infall timescales for satellite,
isolated and central galaxies are 2.6, 2.9 and 3.9 Gyr, respectively. In more
massive galaxies ($M_{*}/M_{\odot}>10^{10.5}$) these differences become less
systematic: the median gas infall timescales for all the three categories
becomes shorter, and all converge to around 2.5 Gyr in the highest mass bin
($10^{11}<M_{*}/{\rm M}_{\odot}$). Such variations with stellar mass are in
line with the down-sizing formation scenario, with timescales compressed for
more massive objects. In addition, The environment effects found in low-mass
galaxies fit with suggested physical mechanisms for quenching galaxy
formation, in which satellites falling into a larger dark matter halo can
experience the removal of their hot gaseous halo through tidal “strangulation”
processes (Balogh et al., 2000; Pasquali et al., 2010) or even the direct ram-
pressure stripping of their cold star-forming gas (Gunn & Gott, 1972; Abadi et
al., 1999), resulting in an early end to their gas supply. The fitted
timescales for gas depletion in satellites in Figure 3 are relatively long
($\sim 3$ Gyr), which argues for the gentler process of strangulation,
consistent with the conclusions from entirely independent lines of evidence
reached by van den Bosch et al. (2008), Pasquali et al. (2010) and Peng et al.
(2015).

One other interesting phenomenon apparent in the distribution of $\tau$ in
Figure 3 is the step in its cumulative distribution that is particularly
apparent in more massive galaxies. This feature represents a separate
population of galaxies in which the gas infall timescale is less than $\sim
1$Ġyr. Further investigation of these galaxies reveals that they are almost
all classified as S0 or lenticular galaxies. These systems accumulated their
stellar masses very early, so are barely resolved in the current analysis. It
is nonetheless interesting to note that they make up a higher fraction of
satellite than central galaxies, again implying that the process responsible
for their creation is dependent on environment.

The other major component in the semi-analytic model is outflow, as described
by the wind parameter $\lambda$. Since this parameter is allowed to vary over
the lifetime of each galaxy, we quantify it in each system using an average
value that reflects when it will have most affected the star-formation history
of the galaxy,

$\overline{\lambda}\equiv\frac{\int_{0}^{t_{\rm
u}}\psi(t)\lambda(t)dt}{\int_{0}^{t_{\rm u}}\psi(t)dt},$ (12)

where $\psi(t)$ is the star formation rate at time $t$ and ${t_{\rm u}}=14.0$
Gyr is the age of the Universe. The bottom row of Figure 3 shows how this
average wind parameter varies with galaxy mass and environment. Once again, it
is only in the lowest-mass galaxies where we see a significant effect, with
satellite galaxies showing much stronger wind parameters than central ones,
with isolated galaxies somewhere in between. The median average wind-parameter
for satellite, isolated and central galaxies in the lowest mass bin
($10^{9.5}<M_{*}/{\rm M}_{\odot}<10^{10.0}$) are 4.5, 3.5 and 2.6
respectively, while in relatively massive galaxies ($10^{10}<M_{*}/{\rm
M}_{\odot}$), the median values all converge to around 4. Although this
parameter was formulated in terms of a star-formation-driven wind, it really
just measures the effectiveness of the loss of processed gas from the galaxy.
The relatively high rate of loss from satellites therefore fits with a picture
in which a system falling into a more massive halo might be expected to lose
its own extended halo of hot processed gas through interaction with other
group members (Gunn & Gott, 1972; Abadi et al., 1999). Similarly, the greater
ability of a central galaxy to hold on to this processed gas when compared to
an isolated system presumably reflects the deeper potential well in which it
sits. The disappearance of difference between the wind-parameters in galaxies
with stellar masses higher than ${10^{10.0}\rm M}_{\odot}$ implies that
massive satellite galaxies do not lose more gas than centrals. This
convergence in properties can likely be attributed to the greater ability of
massive galaxies to hold on to processed gas against external influences, so
that their evolution is mostly determined by internal processes rather than
their environment.

In summary, after controlling for mass, we find strong evidence for
environmental influence in the chemical and star-formation history of disk
galaxies, particularly in low-mass systems. We can also quantify these
differences in terms of the underlying physical processes of pristine gas
accretion and processed gas loss, in the sense that low-mass satellites have
their gas accretion shut off at an earlier stage, and are more likely to lose
their processed gas rather than recycling it into further generations of
stars. We also find a clear difference between central and isolated galaxies,
in that central galaxies continue to accrete gas over longer timescales, and
are better at holding on to that gas once processed through a generation of
stars. We note that these physically-plausible differences were not at all
imposed by the semi-analytic spectral fitting process, which had no way of
knowing whether a particular spectrum came from a central, satellite, or
isolated galaxy. The differences therefore seem to be astrophysical in origin,
and offer further confirmation of the reliability of this archaeological
approach to studying galaxy evolution.

##  5 Conclusions

In this work we have applied the semi-analytic spectral fitting approach to a
sample of disk galaxies selected from the SDSS-IV/MaNGA survey to investigate
the dependence of such galaxies’ evolution on their environment. The sample
comprised three mass-matched sets of isolated, central, and satellite
galaxies, the latter two categories defined by whether or not the galaxy is
the most massive member of a group. The data for each galaxy within one
effective radius were combined to create a high signal-to-noise ratio spectrum
representing the system’s global properties. The semi-analytic spectral
fitting process not only reproduces the properties of the spectrum by fitting
a star-formation and chemical evolution history, but also determines physical
parameters of gas accretion and outflow that generate the model. From these
fits, we have determined that:

  * •

The cumulative SFHs derived from the sample galaxies reveal an earlier
cessation of star formation activities in low-mass ($10^{9.5}<M_{*}/{\rm
M}_{\odot}<10^{10.0}$) satellite galaxies than their central equivalents: less
than 40% of the stellar masses in satellite galaxies formed within the most
recent 5 Gyrs, while central galaxies of similar mass accumulated more than
70% of their stellar masses over the same period. Isolated galaxies have
intermediate properties. This environmental dependence becomes weaker in more
massive galaxies, and essentially disappears for systems with $M_{*}/{\rm
M}_{\odot}>10^{10.5}$. These differences also fit with a more qualitative
analysis looking at the abundances of $\alpha$ elements in different galaxies,
with low-mass satellites showing the sort of $\alpha$ enhancement associated
with a briefer period of star formation.

  * •

The environmental dependence of disk galaxy evolution, as well as its mass
dependence, is also seen in the chemical evolution of the galaxies. Low-mass
($10^{9.5}<M_{*}/{\rm M}_{\odot}<10^{10.0}$) satellite galaxies contain a high
fraction of low metallicity stars compared to central galaxies, indicating
that they are more likely to be ‘closed-box’ or ‘leaky-box’ systems, while
central galaxies of similar masses contain lower fractions of low metallicity
stars, suggesting that on-going accretion played a more important role in
their formation. Once again, isolated galaxies lie between central and
satellite galaxies, and environmental dependence becomes negligible in
galaxies more massive than $\sim 10^{10.5}{\rm M}_{\odot}$.

  * •

The gas accretion history inferred from the semi-analytic fitting process
drives the variations in timescale: in low-mass central galaxies, the average
gas infall timescale is found to be systematically longer than comparable
satellite galaxies, leading to their more extended star formation histories.
Again, this variation disappears in more massive galaxies, where we also start
to see a different sub-population of S0 galaxies, whose gas was accreted over
a very short timescale. However, the timescales are not that short for the
low-mass satellite systems, suggesting the slow ‘strangulation’ of their star
formation caused by the removal of their extended hot halo of gas rather than
more abrupt ram-pressure stripping.

  * •

This scenario is further supported by the inferred mass-loss parameters in
low-mass galaxies, where satellite systems are found to have stronger wind
parameters, indicating that their processed gas has been effectively removed.
By contrast, central galaxies have weaker wind parameters than their isolated
equivalents, as would be expected if the deeper potential well of a group were
better at retaining such processed material.

The consistent story obtained from the semi-analytic spectral fitting process
offers real confidence that even the simple chemical evolution model adopted,
with its parameterisation of gas accretion and mass loss, provides a credible
physical description of the histories of these galaxies. This new direct model
fitting approach, combined with the large sample size and high quality data in
the MaNGA survey, have revealed the significant impact of environment on lower
mass galaxies, and allowed us to quantify it in a way that has not been
possible previously.

## Acknowledgements

SZ, MRM and AAS acknowledge financial support from the UK Science and
Technology Facilities Council (STFC; grant ref: ST/T000171/1).

For the purpose of open access, the authors have applied a creative commons
attribution (CC BY) to any journal-accepted manuscript.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-
Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatário
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University, and Yale University.

## Data availability

The data underlying this article were accessed from: SDSS DR17
https://www.sdss.org/dr17/manga/. The derived data generated in this research
will be shared on request to the corresponding author.

## References

  * Abadi et al. (1999) Abadi M. G., Moore B., Bower R. G., 1999, MNRAS, 308, 947
  * Abdurro’uf et al. (2022) Abdurro’uf et al., 2022, ApJS, 259, 35
  * Anders & Grevesse (1989) Anders E., Grevesse N., 1989, Geochimica Cosmochimica Acta, 53, 197
  * Andrews & Martini (2013) Andrews B. H., Martini P., 2013, ApJ, 765, 140
  * Argudo-Fernández et al. (2015) Argudo-Fernández M., et al., 2015, A&A, 578, A110
  * Baldry et al. (2006) Baldry I. K., Balogh M. L., Bower R. G., Glazebrook K., Nichol R. C., Bamford S. P., Budavari T., 2006, MNRAS, 373, 469
  * Balogh et al. (2000) Balogh M. L., Navarro J. F., Morris S. L., 2000, ApJ, 540, 113
  * Belfiore et al. (2019) Belfiore F., et al., 2019, AJ, 158, 160
  * Blanton et al. (2005a) Blanton M. R., et al., 2005a, AJ, 129, 2562
  * Blanton et al. (2005b) Blanton M. R., Eisenstein D., Hogg D. W., Schlegel D. J., Brinkmann J., 2005b, ApJ, 629, 143
  * Blanton et al. (2017) Blanton M. R., et al., 2017, AJ, 154, 28
  * Bruzual & Charlot (2003) Bruzual G., Charlot S., 2003, MNRAS, 344, 1000
  * Bundy et al. (2015) Bundy K., et al., 2015, ApJ, 798, 7
  * Calzetti et al. (2000) Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, ApJ, 533, 682
  * Cappellari (2017) Cappellari M., 2017, MNRAS, 466, 798
  * Chabrier (2003) Chabrier G., 2003, PASP, 115, 763
  * Conroy (2013) Conroy C., 2013, ARA&A, 51, 393
  * Darvish et al. (2016) Darvish B., Mobasher B., Sobral D., Rettura A., Scoville N., Faisst A., Capak P., 2016, ApJ, 825, 113
  * Domínguez Sánchez et al. (2022) Domínguez Sánchez H., Margalef B., Bernardi M., Huertas-Company M., 2022, MNRAS, 509, 4024
  * Dressler (1980) Dressler A., 1980, ApJ, 236, 351
  * Drory et al. (2015) Drory N., et al., 2015, AJ, 149, 77
  * Etherington & Thomas (2015) Etherington J., Thomas D., 2015, MNRAS, 451, 660
  * Goddard et al. (2017) Goddard D., et al., 2017, MNRAS, 465, 688
  * Greener et al. (2021) Greener M. J., Merrifield M., Aragón-Salamanca A., Peterken T., Andrews B., Lane R. R., 2021, MNRAS, 502, L95
  * Gunn & Gott (1972) Gunn J. E., Gott J. Richard I., 1972, ApJ, 176, 1
  * Gunn et al. (2006) Gunn J. E., et al., 2006, AJ, 131, 2332
  * Kauffmann et al. (2003) Kauffmann G., et al., 2003, MNRAS, 341, 33
  * Kauffmann et al. (2004) Kauffmann G., White S. D. M., Heckman T. M., Ménard B., Brinchmann J., Charlot S., Tremonti C., Brinkmann J., 2004, MNRAS, 353, 713
  * Kawinwanichakij et al. (2017) Kawinwanichakij L., et al., 2017, ApJ, 847, 134
  * Kovač et al. (2014) Kovač K., et al., 2014, MNRAS, 438, 717
  * Larson et al. (1980) Larson R. B., Tinsley B. M., Caldwell C. N., 1980, ApJ, 237, 692
  * Law et al. (2015) Law D. R., et al., 2015, AJ, 150, 19
  * Law et al. (2016) Law D. R., et al., 2016, AJ, 152, 83
  * Le Borgne et al. (2003) Le Borgne J.-F., et al., 2003, A&A, 402, 433
  * Li et al. (2006) Li C., Kauffmann G., Jing Y. P., White S. D. M., Börner G., Cheng F. Z., 2006, MNRAS, 368, 21
  * Mo et al. (1998) Mo H. J., Mao S., White S. D. M., 1998, MNRAS, 295, 319
  * Murphy et al. (2011) Murphy E. J., et al., 2011, ApJ, 737, 67
  * Oemler (1974) Oemler Augustus J., 1974, ApJ, 194, 1
  * Osterbrock & Ferland (2006) Osterbrock D. E., Ferland G. J., 2006, Astrophysics of gaseous nebulae and active galactic nuclei
  * Pasquali et al. (2010) Pasquali A., Gallazzi A., Fontanot F., van den Bosch F. C., De Lucia G., Mo H. J., Yang X., 2010, MNRAS, 407, 937
  * Peng et al. (2010) Peng Y.-j., et al., 2010, ApJ, 721, 193
  * Peng et al. (2012) Peng Y.-j., Lilly S. J., Renzini A., Carollo M., 2012, ApJ, 757, 4
  * Peng et al. (2015) Peng Y., Maiolino R., Cochrane R., 2015, Nature, 521, 192
  * Postman & Geller (1984) Postman M., Geller M. J., 1984, ApJ, 281, 95
  * Read et al. (2006) Read J. I., Wilkinson M. I., Evans N. W., Gilmore G., Kleyna J. T., 2006, MNRAS, 366, 429
  * Renzini & Peng (2015) Renzini A., Peng Y.-j., 2015, ApJ, 801, L29
  * Sánchez-Blázquez et al. (2006) Sánchez-Blázquez P., Gorgas J., Cardiel N., González J. J., 2006, A&A, 457, 809
  * Schmidt (1959) Schmidt M., 1959, ApJ, 129, 243
  * Shi et al. (2011) Shi Y., Helou G., Yan L., Armus L., Wu Y., Papovich C., Stierwalt S., 2011, ApJ, 733, 87
  * Smee et al. (2013) Smee S. A., et al., 2013, AJ, 146, 32
  * Spindler et al. (2018) Spindler A., et al., 2018, MNRAS, 476, 580
  * Thomas et al. (2005) Thomas D., Maraston C., Bender R., Mendes de Oliveira C., 2005, ApJ, 621, 673
  * Tinsley (1974) Tinsley B. M., 1974, ApJ, 192, 629
  * Tinsley (1980) Tinsley B. M., 1980, Fundamentals Cosmic Phys., 5, 287
  * Wake et al. (2017) Wake D. A., et al., 2017, AJ, 154, 86
  * Wang et al. (2016) Wang H., et al., 2016, ApJ, 831, 164
  * Westfall et al. (2019) Westfall K. B., et al., 2019, AJ, 158, 231
  * Wetzel et al. (2012) Wetzel A. R., Tinker J. L., Conroy C., 2012, MNRAS, 424, 232
  * Wetzel et al. (2013) Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2013, MNRAS, 432, 336
  * White & Rees (1978) White S. D. M., Rees M. J., 1978, MNRAS, 183, 341
  * Worthey (1994) Worthey G., 1994, ApJS, 95, 107
  * Yan et al. (2016a) Yan R., et al., 2016a, AJ, 151, 8
  * Yan et al. (2016b) Yan R., et al., 2016b, AJ, 152, 197
  * Yang et al. (2007) Yang X., Mo H. J., van den Bosch F. C., Pasquali A., Li C., Barden M., 2007, ApJ, 671, 153
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Zheng et al. (2017) Zheng Z., et al., 2017, MNRAS, 465, 4572
  * Zheng et al. (2019) Zheng Z., et al., 2019, ApJ, 873, 63
  * Zhou et al. (2022) Zhou S., Merrifield M., Aragón-Salamanca A., 2022, MNRAS, 513, 5446
  * van den Bosch et al. (2008) van den Bosch F. C., Aquino D., Yang X., Mo H. J., Pasquali A., McIntosh D. H., Weinmann S. M., Kang X., 2008, MNRAS, 387, 79