# Composite Bulges – II. Classical Bulges and Nuclear Discs in Barred
Galaxies: The Contrasting Cases of NGC 4608 and NGC 4643

Peter Erwin,1,2 Anil Seth,3 Victor P. Debattista,4 Marja Seidel,5 Kianusch
Mehrgan,1,2 Jens Thomas,1,2 Roberto Saglia,1,2 Adriana de Lorenzo-Cáceres,6,7
Witold Maciejewski,8 Maximilian Fabricius,1,2 Jairo Méndez-Abreu,6,7 Ulrich
Hopp,1,2 Matthias Kluge,1,2 John E. Beckman,6,7,9 Ralf Bender,1,2 Niv Drory,10
and Deanne Fisher11
1Max-Planck-Insitut für extraterrestrische Physik, Giessenbachstrasse, 85748
Garching, Germany
2Universitäts-Sternwarte München, Scheinerstrasse 1, D-81679 München, Germany
3Department of Physics and Astronomy, University of Utah, 115 South 1400 East,
Salt Lake City, Utah 84112, USA
4Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1
2HE, UK
5IPAC, California Institute of Technology, 1200 East California Boulevard,
Pasadena, CA 91125, USA
6Instituto de Astrofísica de Canarias, C/ Via Láctea s/n, 38200 La Laguna,
Tenerife, Spain
7Departamento de Astrofísica, Universidad de La Laguna, Avda. Astrofísico Fco.
Sánchez s/n, 38200, La Laguna, Tenerife, Spain
8 Astrophysics Research Institute, Liverpool John Moores University, Twelve
Quays House, Egerton Wharf, Birkenhead CH41 1LD, UK
9Consejo Superior de Investigaciones Científicas, Spain
10McDonald Observatory, The University of Texas at Austin, 1 University
Station, Austin, Texas 78712, USA
11Center for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Australia E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present detailed morphological, photometric, and stellar-kinematic analyses
of the central regions of two massive, early-type barred galaxies with nearly
identical large-scale morphologies. Both have large, strong bars with
prominent inner photometric excesses that we associate with boxy/peanut-shaped
(B/P) bulges; the latter constitute $\sim 30$% of the galaxy light. Inside its
B/P bulge, NGC 4608 has a compact, almost circular structure (half-light
radius $R_{e}\approx 310$ pc, Sérsic $n=2.2$) we identify as a classical
bulge, amounting to 12.1% of the total light, along with a nuclear star
cluster ($R_{e}\sim 4$ pc). NGC 4643, in contrast, has a nuclear disc with an
unusual broken-exponential surface-brightness profile (13.2% of the light),
and a very small spheroidal component ($R_{e}\approx 35$ pc, $n=1.6$; 0.5% of
the light). IFU stellar kinematics support this picture, with NGC 4608’s
classical bulge slowly rotating and dominated by high velocity dispersion,
while NGC 4643’s nuclear disc shows a drop to lower dispersion, rapid
rotation, $V$-$h_{3}$ anticorrelation, and elevated $h_{4}$. Both galaxies
show at least some evidence for $V$-$h_{3}$ correlation in the bar (outside
the respective classical bulge and nuclear disc), in agreement with model
predictions. Standard 2-component (bulge/disc) decompositions yield $B/T\sim
0.5$–0.7 (and bulge $n>2$) for both galaxies. This overestimates the true
“spheroid” components by factors of four (NGC 4608) and over 100 (NGC 4643),
illustrating the perils of naive bulge-disc decompositions applied to massive
barred galaxies.

###### keywords:

galaxies: structure – galaxies: bulges – galaxies: spiral

††pubyear: 2021††pagerange: Composite Bulges – II. Classical Bulges and
Nuclear Discs in Barred Galaxies: The Contrasting Cases of NGC 4608 and NGC
4643–D

##  1 Introduction

Disc galaxies have traditionally been said to have two main stellar
components: a flattened, rotationally supported disc and (optionally) a
rounder, kinematically hot bulge. The latter – often characterized as akin to
a small elliptical galaxy embedded within the disc – is thought to be
especially prominent in the earliest spirals and in lenticular/S0 galaxies.
Older studies have suggested that the bulge constitutes $\sim 40$–60% of the
total stellar light in such systems (e.g., Simien & de Vaucouleurs, 1986), a
result shared by some recent decompositions of large, SDSS-based samples
(e.g., Oohama et al., 2009; Kim et al., 2016).

Bulges are important for understanding galaxy evolution for a number of
reasons. First, their formation is supposed to involve early mergers – with
the possibility of further growth via minor mergers – so their presence and
size provide potential clues about the early stages of galaxy formation (e.g.,
Brooks & Christensen, 2016, and references therein). In addition, a number of
studies have shown that star formation anticorrelates with the presence of
bulges, which suggests that the presence or growth of bulges is a possible
prerequisite for the cessation of star formation, or even a possible cause of
it, a phenomenon sometimes termed “morphological quenching” (e.g., Martig et
al., 2009; Martig et al., 2013; Bluck et al., 2014; Lang et al., 2014; Eales
et al., 2020). Finally, there are the well-known correlations between bulge
characteristics – particularly central velocity dispersions or bulge
luminosity/mass – and supermassive black hole (SMBH) masses (e.g., Kormendy &
Ho, 2013; Saglia et al., 2016, and references therein). In this context, the
idea that bulges are in effect smaller, lower-mass elliptical galaxies
embedded within larger discs seems to match up quite well with the idea of
SMBH-galaxy correlations that extend from massive elliptical galaxies down to
the small bulges of disc galaxies.

However, our understanding of how bulges form, grow, and correlate with other
galaxy properties has been complicated in the last couple of decades by the
realization that bulges are not all alike, and that not all bulges are
necessarily similar to small elliptical galaxies. This began with the
discovery that some bulges were unusually flattened or dominated by rotation,
or marked by the presence of clearly disclike phenomena such as spiral arms or
nuclear bars – almost as if they were more like small discs embedded within
the main disc (see Kormendy & Kennicutt, 2004, and references therein).

A second development was the demonstration that some bulges in edge-on
galaxies had characteristics – boxy or peanut shapes, cylindrical stellar
rotation, associated in-plane gas and stellar kinematics indicative of bars –
indicating that even though they were clearly not disclike structures, they
were the vertically thickened inner parts of bars (e.g., Kuijken & Merrifield,
1995; Merrifield & Kuijken, 1999; Bureau & Freeman, 1999; Lütticke et al.,
2000; Chung & Bureau, 2004; Bureau et al., 2006). Such structures are usually
referred to as “boxy/peanut-shaped” (B/P) bulges; although they are thicker
than discs, in origin, structure, and kinematics they are emphatically
different from the traditional idea of a bulge. More recent imaging studies
have shown that these structures can be identified in galaxies with
intermediate or low inclinations as well, appearing in the form of “box +
spurs” or “barlens” morphologies inside bars (Erwin & Debattista, 2013;
Laurikainen et al., 2014; Athanassoula et al., 2015; Herrera-Endoqui et al.,
2017; Li et al., 2017; Laurikainen & Salo, 2017; Erwin & Debattista, 2017;
Kruk et al., 2019); their frequency is a strong function of galaxy stellar
mass, becoming almost ubiquitous in the most massive barred galaxies (Erwin &
Debattista, 2017; Li et al., 2017).

All such “non-classical” bulges have tended to be lumped together under the
term “pseudobulges”, with the complementary term “classical bulges” denoting
the more traditional kinematically hot, elliptical-like spheroids. Although
there continues to be a tendency to interpret these developments as meaning
that the central regions of disc galaxies are dichotomous – that they can have
either a classical bulge or some (single) kind of pseudobulge – there has also
been a growing awareness that “bulges” may sometimes be made up of multiple,
coexisting stellar components. This was first clearly articulated from a
theoretical perspective by Athanassoula (2005), who suggested that the centers
of discs could harbor potentially any combination of classical bulges, “disc-
like” bulges, and the B/P bulges of bars. Recent observational work has
clearly shown individual examples of galaxies with bulges that are made up of
at least two distinct components, either from a morphological perspective
based on profiles or image decompositions (e.g., Laurikainen et al., 2014;
Athanassoula et al., 2015; Läsker et al., 2016), or from combinations of
decompositions and analyses of the stellar kinematics (e.g., Méndez-Abreu et
al., 2014; Erwin et al., 2015).

Evidence for multiple types of bulges has been found for the Milky Way itself:
recent work has established that the dominant bulge in the Milky Way is the
B/P bulge of its bar (e.g., Shen et al., 2010; Bland-Hawthorn & Gerhard, 2016;
Debattista et al., 2017b), with good evidence for a low-mass nuclear disc
(e.g., Launhardt et al., 2002; Schönrich et al., 2015; Nogueras-Lara et al.,
2019; Gallego-Cano et al., 2020; Sormani et al., 2020) – but also little
evidence for any significant classical bulge (e.g., Bland-Hawthorn & Gerhard,
2016). In addition, careful analysis of M31 shows that its bulge consists of
two components: a dominant B/P bulge belonging to its bar and a sub-dominant
classical bulge, with the former having about twice the stellar mass of the
latter (Athanassoula & Beaton, 2006; Opitsch et al., 2018; Blaña Díaz et al.,
2017; Blaña Díaz et al., 2018).

Figure 1: SDSS $gri$ colour-composite images (David W. Hogg, Michael R.
Blanton, and the Sloan Digital Sky Survey Collaboration) for the two galaxies
studied in this paper. The light grey boxes outline the large-scale regions
shown (using Spitzer IRAC1 isophotes) in the leftmost panels of Figures 2 and
3.

The question of what “bulges” in disc galaxies actually are is now a critical
one. In particular, we would like to know how often disc galaxies have
spheroidal, kinematically hot classical bulges, how often they have nuclear
discs or discy pseudobulges, and how often they have B/P bulges. We would also
like to know how often, and in which ways, these different structures can
coexist in the same galaxy, and what fraction of a galaxy’s stars can be found
in the different inner components.

Identifying, enumerating, and understanding these distinct forms of bulges is
important, because they have very different formation mechanisms, and
potentially different effects on their host galaxies. As noted above,
classical bulges are supposed to be the result of violent, gas-rich mergers
very early in galaxy’s history. Discy pseudobulges (nuclear discs) are thought
to form from bar-driven gas inflow and subsequent star formation, and are thus
the result of internal (“secular”) processes that are unrelated to mergers,
and which require the prior formation of a bar (e.g., Kormendy & Kennicutt,
2004; Wozniak & Michel-Dansac, 2009; Cole et al., 2014; Baba & Kawata, 2020).
B/P bulges also require the formation of a bar, since they form out of (and
remain part of) the bar; they are also the result of internal processes,
albeit ones rather different from gas inflow (e.g., Combes & Sanders, 1981;
Combes et al., 1990; Raha et al., 1991; Quillen, 2002; Martinez-Valpuesta &
Shlosman, 2004; Debattista et al., 2005, 2006; Quillen et al., 2014; Sellwood
& Gerhard, 2020). All of this is clearly relevant to more general bulge–host-
galaxy correlations and processes. For example, it is easy to see how
classical bulges might correlate with SMBHs, while a connection with nuclear-
disc formation is less obvious, and the formation of B/P bulges would at first
glance seem quite unrelated to SMBH growth.111Fragkoudi et al. (2016) do
suggest that bars with B/P bulges could be less efficient (than bars without
B/P bulges) at promoting gas inflow to the central kpc, leading to lower
nuclear fueling rates.

To address these issues, we have undertaken a multi-wavelength imaging and
spectroscopic observing campaign – the Composite Bulges Survey (Erwin et al.,
2020, hereafter “Paper I”) – aimed at identifying and characterizing all the
different components that may reside in the centers of massive disc galaxies,
from bars – and their boxy/peanut-shaped inner components – down to nuclear
discs (and nuclear rings and nuclear bars) and even to nuclear star clusters
(NSCs; Neumayer et al., 2020). This study is based around a mass- and volume-
limited sample of approximately fifty nearby, early- and intermediate-type
(S0–Sbc) disc galaxies, each observed in the optical and near-IR with the
Hubble Space Telescope (HST), and with planned observations of all galaxies
using the MUSE integral field spectrograph.

In this paper we present a detailed study of inner structures of two galaxies
from this sample. This paper both serves as a model for analysis of the rest
of our sample (using both high-resolution near-infrared imaging and 2D stellar
kinematics), and enables a comparison of how our detailed approach contrasts
with the simpler approaches frequently used in galaxy decompositions,
especially for large samples. The galaxies we study here are early-type disc
galaxies with similar masses and morphologies, seen at very similar
orientations: NGC 4608 (SB0, $\log\,(M_{\star}/M_{\sun})=10.38$) and NGC 4643
(SB0/a, $\log\,(M_{\star}/M_{\sun})=10.79$); see Figure 1 for large-scale
optical views of both.222See Section 3 for the source of their classifications
and stellar masses. Both galaxies have inclinations $i\sim 35$–40° and are
strongly barred, with the bar oriented close to the minor axis of its parent
galaxy. In a general morphological sense, they are close to being twins, and
there is evidence that both galaxies’ bars contain B/P bulges, making them
even more similar.333Evidence for B/P bulges can be found in Laurikainen et
al. (2014), Athanassoula et al. (2015), Laurikainen & Salo (2017), and Section
4.4. And yet, as we will show, their inner regions are quite different: one
galaxy (NGC 4608) hosts a classical bulge (and probably a nuclear star
cluster) inside its B/P bulge, while the other (NGC 4643) has a massive
nuclear disc444Previously suggested by Erwin & Sparke (2003) and Erwin (2004).
with an unusual broken-exponential surface-brightness profile. Inside this
nuclear disc is a smaller spheroidal structure that is probably too large to
be a nuclear star cluster, but is about an order of magnitude smaller and less
massive than the first galaxy’s classical bulge.

Section 2 of this paper discusses the parent sample and the data sources for
the two galaxies, while Section 3 summarizes their general characteristics.
Section 4 is devoted to our detailed, 2D decompositions of near-IR images of
both galaxies, while Section 5 uses published and archival 2D stellar-
kinematic data to test and validate the morphological results of the preceding
decompositions. Section 6 includes discussion of the nuclear disc in NGC 4643
and the implications of our analysis for simplistic, two-component
decompositions of large galaxy samples; Section 7 summarizes our findings.

##  2 Parent Sample and Data Sources

A fuller description of our sample and observing strategy is presented in
Paper I; here, we provide a brief overview.

The Composite Bulges Survey is designed to probe the morphology and stellar
kinematics and populations of the bulge regions (e.g., the inner 1–2 kpc) of
massive disc galaxies. It is based on a volume- and mass-limited sample with
an upper limit on distances of 20 Mpc, so that with HST imaging we can resolve
down to $\sim 15$ pc or better in the near-IR (assuming a PSF FWHM $\approx
0.15\arcsec$ in the F160W band).

The sample is defined so as to include all S0–Sbc galaxies in de Vaucouleurs
et al. (1993, RC3) with Galactic latitude $|b|>20\degr$, distances $\leq 20$
Mpc, stellar masses $\geq 10^{10}M_{\sun}$, and inclinations between $35\degr$
and $60\degr$. Since we plan to augment our dataset with archival and future
spectroscopy from the Very Large Telescope (VLT), we limited the positions on
the sky to $\delta\leq 20\degr$. The result was a total of 54 galaxies
spanning a range of environments from the local field to the Virgo Cluster.
One galaxy (NGC 5363) was subsequently identified as a probable elliptical
rather than an S0, so we are left with a total of 53 disc galaxies.

###  2.1 HST Data from the Composite Bulges Survey

The primary dataset is a consistent set of optical and near-IR images obtained
with HST, using the Wide Field Camera 3 (WFC3) in both its UVIS and IR modes
(Cycle 25, Proposal ID 15133). Since we are first and foremost interested in
the underlying stellar structure, with as little confusion due to dust as
possible, the main emphasis of the proposal was on full-field WFC3-IR imaging
using the F160W filter, with a total integration time of 600s divided into
four dithered exposures. For the sake of efficiency in readout time and data
storage, the optical imaging was restricted to the C1K1C aperture, which is a
$1024\times 1024$-pixel subset of the full WFC3-UVIS array; we obtained four
dithered exposures in each of the F475W and F814W filters, for total
integration times of 700s in F475W and 500s in F814W.555See Paper I for notes
on slight variations in this scheme for six of the galaxies, which did not
apply to the galaxies studied in this paper. This enabled us to obtain all
exposures for a given galaxy within a single HST orbit.

Individual exposures in each band were combined using the Python-based
Drizzlepac code. After some experimentation, we adopted output image scales of
0.03″ pixel-1 for the optical images and 0.06″ pixel-1 for the F160W images,
with pixfrac = 0.7. Since the galaxies are larger than the WFC3-IR field of
view, it is impossible to accurately estimate the sky background from the HST
images; accordingly, we turned sky subtraction off during the processing. (We
account for the sky background as part of our modeling process; see Section
4.)

For purposes of photometric calibration, we started with the standard
calibration to the Vega-magnitude (VEGAMAG) F160W system in the F160W filter.
We converted these to 2MASS $H$-band equivalents following the suggestion of
Riess (2011), which uses the 2MASS $J-H$ colours. For red galaxies like NGC
4608 and NGC 4643, the correction works out to $m_{H}\approx m_{\rm
160w,vega}+0.025$.

###  2.2 Other Imaging Data

For large-scale decompositions, we make use of Spitzer IRAC1 (3.6µm) images of
NGC 4608 and NGC 4643. The image for NGC 4643 comes from the Spitzer Survey of
Stellar Structure in Galaxies (S4G; Sheth et al., 2010), with the latter’s
final mosaic pixel scale of 0.75 ″/pixel. NGC 4608 was not part of S4G so we
use the archive-generated mosaic image (Program ID 10043, PI Kartik Sheth),
with the default archive mosaic pixel scale of 0.6 ″/pixel.666The observation
of NGC 4608 is part of an extension to S4G meant to fill in the missing
elliptical and S0 galaxies (Sheth et al., 2013; Knapen et al., 2014). For both
galaxies, we estimated the residual sky background as the mean of the median
values of $\sim 50$–100 $20\times 20$-pixel boxes located well away from the
galaxies and from bright stars.

For the purpose of estimating how both galaxies would fare under bulge-disc
decompositions at redshifts typical of large, SDSS-based studies (Section
6.3), we also used $r$-band images from Data Release 7 of SDSS (York et al.,
2000; Abazajian et al., 2009).

###  2.3 Integral Field Unit Spectroscopic Data: SAURON and MUSE

Two-dimensional stellar-kinematic information, obtained with the SAURON
instrument, is available for the central regions of both galaxies from ATLAS3D
(Cappellari et al., 2011; Krajnović et al., 2011). SAURON (Bacon et al., 2001)
is an integral field spectrograph composed of 0.94″ lenslets with a
$33\arcsec\times 41\arcsec$ field of view, which spans the inner $3\times 3$
kpc of each galaxy. We used the stellar-kinematics tables (containing the
Gauss-Hermite $V$, $\sigma$, $h_{3}$ and $h_{4}$ parameters) from the main
ATLAS3D web site777http://www-astro.physics.ox.ac.uk/atlas3d/tables . The
instrumental resolution $\sigma_{\rm inst}$ is $\approx 105$ ${\rm
km\;s}^{-1}$, which is comfortably below the central velocity dispersions of
both galaxies (130 ${\rm km\;s}^{-1}$ for NGC 4608 and 147 ${\rm km\;s}^{-1}$
for NGC 4643, according to HyperLEDA).

We additionally made use of SAURON observations from Seidel et al. (2015) for
NGC 4643. These consist of seven separate pointings which together cover the
whole of the bar region ($\sim 100\times 80\arcsec$) rather than just the
single-pointing ATLAS3D observation, which only covers the innermost region.

NGC 4643 has been observed with the VLT’s MUSE instrument as part of the TIMER
project (Gadotti et al., 2019). MUSE (Bacon et al., 2010) is an integral field
spectrograph using image slicers to sample a $1\arcmin\times 1\arcmin$ field
of view (in the Wide Field Mode) at 0.2″ per spaxel (spatial pixel). The
spectrograph covers a wavelength range of 4800–9300 Å, with a resolution that
ranges from $R=\lambda/\Delta\lambda=1770$ to 3590 over that same range and a
spectral sampling of 1.25 Å/pixel.

Preliminary MUSE stellar and gas kinematics, along with a stellar-population
analysis, have been presented for NGC 4643 in Gadotti et al. (2019, 2020) and
Bittner et al. (2020), which we refer to later. We also performed our own
stellar-kinematic analysis of the publicly available datacube (ESO Phase 3
reduced datacube, part of the “DEEP MUSE” datastream). We inspected the
datacube for possible sky-subtraction errors, but did not find any; in any
case, our interest is in the bright central region of NGC 4643, dominated by
the galaxy light. To do this, we used the penalized, pixel-fitting pPXF code
(Cappellari & Emsellem, 2004; Cappellari, 2017) to derive stellar kinematics
in the usual form of a parameterized line-of-sight velocity distribution
(LOSVD) – mean stellar velocity $V$, velocity dispersion $\sigma$, and the
Gauss-Hermite $h_{3}$ and $h_{4}$ parameters – using the MILES stellar-
template library (Sánchez-Blázquez et al., 2006; Falcón-Barroso et al., 2011).
We first performed a fit to the combined spectrum of the full datacube in
order to identify a subset of template spectra with non-zero weights. Using
the latter as a restricted template library, we then fit spectra (using the
wavelength range 4800–7400 Å) from all the individual spaxels, masking out
narrow regions around emission lines. This differed somewhat from the analysis
of Gadotti et al. (2019, 2020), who used Voronoi-binned spectra, fitted
wavelength ranges of 4750–5500 Å (Gadotti et al., 2019) or 4800–8950 Å
(Gadotti et al., 2020), and the E-MILES model library of SSP template spectra
(Vazdekis et al., 2015), but our results are almost identical to theirs.

We also performed a kinematic analysis using our own non-parametric pixel-
fitting code (Mehrgan et al. 2019, J. Thomas et al. 2020, in preparation), in
part to verify that the results were not dependent on the particulars of which
code was used; further details of this analysis will be presented elsewhere.

Figure 2: Imaging overview of NGC 4608. Top row: from left to right,
logarithmically spaced Spitzer IRAC1 isophotes (left, median-smoothed with
width = 5 pixels, contour spacing of 0.1 des) and two close-ups showing HST
WFC3-IR F160W isophotes (middle, median-smoothed with same width; right,
unsmoothed). Bottom row: HST WFC3-UVIS colour maps (F475W $-$ F814W, converted
to $g-i$) on the same spatial scales and smoothing as the corresponding F160W
isophote plots. (Unless otherwise explicitly noted, all figures in this paper
have standard astronomical orientation: N = up, E = left) Figure 3: Same as
Figure 2, but now showing NGC 4643.

##  3 An Overview of NGC 4608 and NGC 4643

NGC 4608 and NGC 4643 are both massive, strongly barred, early-type disc
galaxies. The former is located in the Virgo Cluster at a distance of $17.3\pm
0.8$ Mpc (surface-brightness-fluctuation distance; Cantiello et al., 2018);
the latter is in the nearby field (approximately 2.4 Mpc away from NGC 4608),
with a redshift-based distance estimate of $19.3\pm 2.9$ Mpc.888Assuming
$H_{0}=72$ ${\rm km\;s}^{-1}$ kpc-1 and a Virgocentric-infall-corrected
redshift of 1392 ${\rm km\;s}^{-1}$ from HyperLEDA, along with an assumed
uncertainty of 15%. These distances give scales of 83.9 pc/″ for NGC 4608 and
93.6 pc/″ for NGC 4643.

NGC 4608 is classified in RC3 as SB0 and NGC 4643 as SB0/a; very faint, wispy
spiral arms account for the latter classification (e.g., Erwin & Sparke,
2003). They are very similar in overall orientation as well, with inclinations
of 36° and 38°, respectively; in both cases their bars are positioned almost
perfectly along each galaxy’s minor axis. The position angles of their discs
are 100° and 53° for NGC 4608 and NGC 4643, respectively. (Disc orientations
are based on the isophote shapes of their outer discs and are taken from Erwin
2005 and Erwin et al. 2008.)

Their stellar masses are also similar. For both galaxies, we estimated
$H$-band $M/L$ ratios from their global $g-i$ colours, using the relation of
Roediger & Courteau (2015). The colours are based on the SDSS Model $g$ and
$i$ magnitudes from Data Release 6, while the $H$-band magnitudes are from the
2MASS Extended Source Catalog (all values retrieved from NED, corrected for
Galactic extinction using the Schlafly & Finkbeiner 2011 values). This gives
us $\log\,(M_{\star}/M_{\sun})=10.381$ for NGC 4608 and 10.789 for NGC 4643,
so the latter galaxy is about 2.5 times more massive than the former.

They thus present themselves (aside from the factor of two difference in
stellar mass) as very much alike, as noted by, e.g., the Carnegie Atlas
(Sandage & Bedke, 1994): “NGC 4643 is nearly identical to NGC 4608 … The
description there applies to NGC 4643 as well.”

The bars and inner (“bulge”) regions appear at first glance to be very similar
as well. Both bars show strong, narrow isophotes which transition from being
sharply pointed (“discy”) to having rectangular (“boxy”) ends as one moves out
in radius. The inner isophotes are significantly rounder, which encourages the
idea (mostly erroneous, as we will argue) that we are seeing dominant
classical bulges in the centers of both galaxies.

Nonetheless, there are hints of differences when the inner isophotes are
examined more carefully. In particular, the isophotes of NGC 4643 switch from
the bar orientation (PA $\approx 135\degr$) to a mildly elliptical shape
nearly perpendicular to the bar – and closely aligned with that of the outer
disc: PA $\approx 52\degr$ for $r\la 5\arcsec$. While this is in principle
consistent with an oblate classical bulge, previous analysis, including
unsharp masking, of ground-based images has found evidence for a nuclear disc
with a possible stellar nuclear ring in this region (Erwin & Sparke, 2003;
Erwin, 2004). Gadotti et al. (2019) have used MUSE data (see Section 5.2) to
show that the stellar kinematics in this region is dominated by rotation,
arguing for the presence of a nuclear disc.

Figures 2 and 3 show large-scale (IRAC1) and close-up views (F160W) of the
near-IR isophotes for both galaxies, along with HST colourmaps using the F475W
and F814W WFC3-UVIS images. Figure 4 shows the position angle and ellipticity
of ellipses fitted to the IRAC1 and F160W isophotes using the iraf ellipse
package. The colourmaps show that NGC 4608 has no evidence for dust or recent
star formation, just a slight inward reddening trend. NGC 4643, on the other
hand, shows spiral dust lanes in the inner $r\la 10\arcsec$. The latter are
not strong enough to cause significant distortions in the F160W image, though
they may be responsible for the small variations in ellipticity and position
angle at $a\la 2\arcsec$ visible in the ellipse fits for this galaxy.

Figure 4: Ellipse fits to Spitzer IRAC1 images (black) and HST F160W images
(red) of both galaxies (top: NGC 4608; bottom: NGC 4643), along with ellipse
fits to the best-fitting model images (blue = model fit to IRAC1 image, green
= model fit to F160W image).

##  4 Morphological Decompositions

###  4.1 General Outline

Our general goal is to dissect each galaxy into astrophysically distinct
stellar components, which might have different dynamics and/or formation
mechanisms. We do this by modeling each image as the sum of multiple 2D image
functions, with the functions chosen to represent plausible galaxy components
(discs, bars, rings, nuclear star clusters, etc.). We put particular emphasis
on the accurate modeling of the inner regions (roughly, the inner 1–2 kpc), in
order to identify distinct “bulge” components and sub-components.

We combine modeling of low-resolution, whole-galaxy images from Spitzer with
the high-resolution, small-field-of-view WFC3-IR F160W images from our HST
observations. This emphasis on near-IR images stems from our interest in the
stellar structure, undistorted (as much as possible) by dust extinction.
Although NGC 4608 appears to be dust-free (e.g., Figure 2), NGC 4643 has some
dust lanes in the inner $\sim 1$ kpc. These are not strong enough to
noticeably distort the F160W image, but would make fitting the optical images
problematic (Figure 3). We note that we are implicitly assuming no strong
colour gradients involving the F160W and IRAC1 bands, so that we can treat
them as describing the same stellar structure.

Our fits for each galaxy are done in a two-stage process. First, we focus on
fitting the large-scale, sky-subtracted IRAC1 image using disc- and bar-
related components; the innermost components (e.g., bulge, nuclear disc,
nuclear star cluster) are treated approximately, since the IRAC1 images lack
the spatial resolution to properly sample these parts of the galaxy. When an
acceptable fit has been achieved, we switch to fitting the F160W image, where
we can refine and better constrain the innermost components of the model. The
F160W modeling begins with the best-fitting model and parameter values from
the IRAC1 fits, translating size and orientation parameters appropriately.
Since the galaxies are much larger than the F160W field of view, we fix the
outermost component (the main disc) to the best-fit values from the IRAC1
fits, with the exception of intensity parameters (i.e., $I_{0}$ for the
BrokenExponential component we use for the main disc in both galaxies), which
are left as free parameters of the F160W fit. We also included a constant-
background (“FlatSky”) term to account for the sky background in the F160W
image (since the galaxies are larger than the WFC3-IR field of view, the
background cannot be determined directly).

For each galaxy, we start with very simple models, such as bulge + disc (i.e.,
an elliptical Sérsic component + an elliptical Exponential component). We
inspect the residuals to find where they are particularly bad (data much
brighter than model or vice-versa) and to look for possible shapes suggestive
of missing components. We then gradually add additional components to the
model, using basic morphological analysis and prior studies as a guide. For
example, since both NGC 4608 and NGC 4643 are strongly barred, including bar
components in the models is a logical step. Since NGC 4643 has previously been
identified as having a possible nuclear stellar ring (e.g., Erwin, 2004)
and/or a nuclear disc (e.g., Gadotti et al., 2019), we also test both of these
as possible additional components. The specific sequence of models for NGC
4608 is discussed in Section 4.5; the process for NGC 4643 is described in
Section 4.6.1.

The “goodness” of these successive fits is evaluated in a relative fashion: by
comparing how well a new model reduces systematic, patterned residuals
produced by previous models; by judging how well a new model reproduces
certain features in the isophotes; and by looking for significant improvement
in the Akaike Information Criterion (AIC, Akaike 1974). The AIC is useful
because, unlike the reduced $\chi^{2}$, it can be used to compare different
models fit to the same data. Most discussions of model comparison argue that
$\Delta\mathrm{AIC}<-10$ indicate a clearly better fit for the model with the
lower AIC value. However, from our experience in fitting models to galaxy
images, we find that most additions to a model, even relatively trivial ones,
can usually produce large negative values of $\Delta\mathrm{AIC}$, and we only
consider improvements of $\Delta\mathrm{AIC}<-1000$ to be truly useful.

Fits were done with the Imfit package (Erwin,
2015)999https://www.mpe.mpg.de/~erwin/code/imfit, using standard $\chi^{2}$
minimization and the (default) Levenberg-Marquardt minimizer; we used the data
values to estimate the per-pixel uncertainties under the usual Gaussian
approximation of Poisson statistics. For the IRAC1 images, pixel values were
converted back to ADUs, with an assumed A/D gain of 3.7 used in the fit to
convert ADUs to detected photons. For the F160W images, our reduction
generated final images with detected photons (“electrons”) as the final pixel
value, so no further processing was necessary. Because the images include
foreground stars, background galaxies, and occasional image defects, we
prepared masks based on running SExtractor (Bertin & Arnouts, 1996) to
identify individual sources, with detected sources transformed into circles
(for stars) or ellipses (for background galaxies) with radii or semi-major
axes equal to small multiples of the SExtractor A_IMAGE parameter.

All fits include convolution with the appropriate PSF image. For the IRAC1
images, we used the official in-flight Pixel-Response-Function
images101010https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/psfprf/,
downsampled to the appropriate pixel from the PRF scale of 0.24″/pixel. We
used the PRF image at column,row = 129,129 as an approximation to the central
location of the galaxy in the image; since our final modeling of the central
regions of each galaxy is based on the HST images, more accuracy than this for
the IRAC1 PSF is not needed. For the F160W images, we generated appropriate
PSF images using the grizli software,111111https://github.com/gbrammer/grizli
which inserts the “empirical PSF” images of Anderson
(2016)121212http://www.stsci.edu/~jayander/STDPSFs/WFC3IR/ into the four
individual F160W exposures and then runs them through the same drizzling
process we use to prepare our final F160W images, extracting the final PSF
image from the combined, drizzled image (Mowla et al., 2019).

###  4.2 Determining Best-Fit Models: The Example of NGC 4608

As an example of our successive modeling approach, consider Figure 5, which
shows residuals from fitting successively more complex models to the IRAC1 and
F160W images of NGC 4608.

The top row of the figure shows the results of our fits to the IRAC1 image.
Panel b shows a simplistic reference (“bulge+disc”) decomposition, where we
use only an Exponential (disc) and a Sérsic (bulge) component. The residuals
indicate this is a terrible fit, both due to the lack of a bar component and
because the disc is not a simple exponential. Panel c shows the next stage,
where we add our new, two-component bar model (Appendix B) and change the disc
component from a simple exponential to a broken-exponential. This is a
dramatic improvement, with much-reduced residuals and a much lower AIC
($\Delta\mathrm{AIC}\sim-3.5\times 10^{5}$). In the next stage (panel c to
panel d), we use the strong, ringlike residual surrounding the bar in panel c,
along with the clear evidence from the isophotes (and previous classifications
of this galaxy) for an inner ring, to motivate adding a GaussianRing component
to the model. As panel d shows, this clearly reduces the ringlike residual and
the deficits immediately inside the ring, though the residuals in the center
and the disc outside the ring are unchanged; the improvement in AIC
($\Delta\mathrm{AIC}\approx-13500$) is significant, but less than the previous
stage. Finally, in panel e we show what happens when we replace the uniform
GaussianRing component with one that has an azimuthally varying surface
brightness (the GaussianRingAz component, described in Appendix C). This is
formally a meaningful improvement ($\Delta\mathrm{AIC}\approx-1700$), though
the residuals appear almost unchanged (except for the partial spiral residuals
immediately north and south of the east and west ends of the bar,
respectively). Comparison of the isophotes of the data and model images (e.g.,
upper and middle panels of Figure 6) shows that this change does make the
model match the data better in the inner-ring region (e.g., the fact that the
inner ring visibly decays as the angle with respect to the bar major axis
increases). Nonetheless, at this stage we seem to be reaching the limits of
significant improvements for modeling the IRAC1 image.

We then shift to modeling the F160W image, starting with the final IRAC1
model. Since most of the main disc is outside the field of view of the F160W
image, we re-use the BrokenExponential component by holding its parameters
fixed to their best-fit values from the IRAC1 fit, with the exception of the
central surface brightness, which was left as a free parameter. We also
include a uniform-background component (Imfit’s FlatSky component) to account
for the (unknown) sky background in the F160W image. The resulting fit is
generally excellent, except for clear nuclear excess in data (panel g; also
suggested by nuclear excess in higher-resolution WFC3-UVIS optical images).
The addition of a compact Gaussian to represent a possible nuclear star
cluster clearly improves the fit and removes the nuclear excess from the
residuals (panel h). Since the central isophotes are essentially circular, and
since this component is only partially resolved, we fix its shape to be
circular.

###  4.3 Estimating Parameter Uncertainties

Determining the uncertainty of fitted parameters for fits such as these is
problematic. Although Imfit provides nominal error estimates when run with the
(default) Levenberg-Marquardt minimization algorithm, we elected to use the
bootstrap-resampling option, which produced slightly more generous (and
probably accurate) results, including asymmetric uncertainties; these are
reported in our best-fit parameter tables (Tables 1 and 2). Bootstrap
resampling also allows for plotting uncertainty distributions between
parameters, which allows the identification of correlated uncertainties.
Samples of such correlations are presented in Appendix D.

As inspection of the tables will show, these error estimates are in general
extremely small – typically $<1$%. This is probably because the data we work
with is very high $S/N$, and the error estimates assume that the model is a
correct representation of the data, with the uncertainty coming only from
random, per-pixel errors (e.g, from Poisson processes and Gaussian readout
noise). As an example of how misleading this can be, consider the simple
bulge/disc (Sérsic + exponential) decomposition of the IRAC1 image of NGC 4608
(panel b of Figure 5). The best-fit position-angle and ellipticity of the disc
component are $111.4\degr\pm 0.2\degr$ and $0.209\pm 0.003$, respectively –
while the best-fit values for the (BrokenExponential) disc in our final IRAC1
model are $106.6\degr\pm 0.2\degr$ and $0.127\pm 0.001$. Clearly, the formal
errors of the fit do not adequately represent uncertainties in what we might
consider general, model-independent properties of the galaxy or its main
subcomponents. (A similar point was made by Gao & Ho 2017 regarding the
dependence of “bulge” properties on the inclusion or exclusion of other
components in their 2D models of galaxies.)

We investigated the possible influence of uncertainties in the sky background
subtraction by fitting versions of the IRAC1 images that had been perturbed
with the $\pm 1$-$\sigma$ sky uncertainties (the latter determined by 1000
rounds of bootstrap resampling applied to the sky-level determination method).
These produced variations in the fitted parameters roughly equivalent to, or
smaller than, the bootstrap-based uncertainties.

###  4.4 Some Particulars: Broken-Exponential Discs and a Two-Component Bar
Model

Preliminary fitting experiments with the IRAC1 images indicated that the best
results were obtained when we treated the main disc as having a broken-
exponential radial surface-brightness profile (e.g., Erwin et al., 2008;
Erwin, 2015), where an inner, shallow exponential zone breaks to a steeper
outer exponential profile. This is handled with the BrokenExponential
component in Imfit, which combines elliptical isophotes with a broken-
exponential radial profile. We note that in Erwin et al. (2008) neither galaxy
shows a clear “Type II” profile (though NGC 4608 does in fact have a weak Type
II.i profile). However, those profile classifications were based on the
azimuthally averaged profile outside the bar region, while our interest here
is in 2D modeling of the entire galaxy, including the bar and its interior.
The necessity for a broken-exponential (i.e., Type II) profile for 2D fits to
images of NGC 4608 was previously pointed out by Gadotti (2008).

Laurikainen et al. (2014) suggested, based on 2D fits to a ground-based
$K$-band image, that NGC 4643 possessed a “barlens”, which is their term for
the B/P bulge seen at low inclinations. They fit the bar with a “thin bar”
component (a 2D projection of a Ferrers ellipsoid) and an exponential-like
component for the B/P bulge, along with an extra sub-exponential (Sérsic
$n=0.7$) component for the “bulge”. For NGC 4608, Laurikainen et al. (2005)
noted “a large oval inside the bar, and apparently a small spherical bulge” in
their $K$-band image. Athanassoula et al. (2015) modeled the same image using
a separate “barlens” component for the B/P bulge and a projected Ferrers
ellipsoid for the outer, thin part of the bar. Both galaxies are listed as
“good examples” of a thin bar plus a barlens in Laurikainen & Salo (2017). We
note that both galaxies are also massive enough for the presence of B/P bulges
to be statistically likely: using the logistic-regression analysis of Erwin &
Debattista (2017), which models the probability that a barred galaxy has a B/P
bulge as a function of the galaxy’s stellar mass, the probabilities are 0.52
for NGC 4608 and 0.85 for NGC 4643.

Given this evidence, we work from the assumption that the bars of these two
galaxies do have B/P bulges, and that the latter are best considered as
components with rounder isophotes and steeper surface-brightness profiles,
added to the more elongated, shallow-surface-brightness-profile components of
the outer (vertically thin) part of the bar.

Our general logic is thus similar to that of Laurikainen et al. (2014) and
Athanassoula et al. (2015), with the bar represented as the sum of two
components. (A very similar approach was followed by Neumann et al. 2019 for
some of their barred galaxies.) We differ from them in using a new component
for the outer part of the bar, designed to better represent (than is possible
with Sérsic or Ferrers functions) the broken-exponential major-axis profile
and the transition from discy to boxy isophotes near the end of the bar. This
new component (“FlatBar”), is described in Appendix B.

Unlike Athanassoula et al. (2015), we put more emphasis on modeling the main
disc of the galaxy as well, in order to better constrain the modeling of the
bar region; this includes (as noted above) treating the main discs of both
galaxies as having broken-exponential surface-brightness profiles; it also
means using a separate component to model the prominent inner ring surrounding
the bar in NGC 4608. This allows us to fit most or all of the parameters
freely, in contrast to having to fix such parameters as the bar length and the
B/P bulge ellipticity by hand, as Athanassoula et al. did.

###  4.5 Fitting NGC 4608: Summary

Figure 5: Progressively more complex and accurate 2D image modeling of NGC
4608. Upper row: logarithmically spaced, median-smoothed isophotes of the
fitted region of the Spitzer IRAC1 image (a) and “residual-ratio” images (data
/ model) from models fit to this image (b–e). These are, from left to right:
simple bulge/disc (Sérsic + exponential) model (b); addition of our two-
component bar model (Appendix B) and replacement of exponential (disc) with
BrokenExponential component (c); addition of GaussianRing component to model
the inner ring surrounding the bar (d); and replacement of the GaussianRing
with a GaussianRingAz (Appendix C) component (e). Numbers below each image
show the relative improvement of the fit (change in Akaike Information
Criterion [AIC] relative to the previous fit). Note that image orientation in
this row is as observed. Lower row: Isophotes from the inner region of the HST
WFC3-IR F160W image (f) and residual-ratio images from fits to the (full)
F160W image. From left to right, these are: the “final model” from the IRAC1
fits (g) and addition of a central Gaussian component to represent a nuclear
star cluster (h). Note that in this row, image orientation is standard (N =
up, E = left).  Figure 6: Best-fitting model for NGC 4608, compared with
Spitzer IRAC1 image. Upper and middle panels show logarithmically spaced
isophotes from IRAC1 image (top) and best-fitting model (middle); bottom
panels show residuals (data values divided by model values). The large-scale
images (left) were smoothed using a $5\times 5$-pixel-wide median filter.
Figure 7: Best-fitting model for NGC 4608, compared with HST WFC3-IR F160W
image. Upper and middle panels show logarithmically spaced isophotes from
F160W image (top) and best-fitting model (middle); bottom panels show
residuals (data values divided by model values). The large-scale images (left)
were smoothed using a $9\times 9$-pixel-wide median filter.

(See Section 4.2 and Figure 5 for how we proceeded in constructing and
successively refining our best-fit model for this galaxy.)

Our final model for NGC 4608 uses the combination of Imfit’s BrokenExponential
component and the new GaussianRingAz component (Appendix C) for the disc and
the inner ring surrounding the bar; the new FlatBar component for the outer
part of the bar and a mildly elliptical Sersic_GenEllipse component for the
inner, B/P part of the bar; a central round Sérsic component; and a compact,
circular Gaussian. Table 1 summarizes the best-fitting parameter values, while
Table 3 shows the luminosities and relative fluxes of the different
components, and Figures 6 and 7 compare this model with the IRAC1 and F160W
data. In addition, the upper half of Figure 4 compares the ellipticity and
position-angle profiles of the best-fit model with those of the data. This
shows generally excellent agreement, except possibly for the inner
$r<2\arcsec$, where the model isophotes are slightly too round and differ in
PA by $\sim 20\degr$.

The main/outer disc (the combination of the BrokenExponential and
GaussianRingAz components) accounts for 45% of the total light; most of this
is in the BrokenExponential, with the ring component being only 3.3%. The
inner part of the BrokenExponential is effectively constant (the fits converge
to the upper limit we set on the parameter value); this changes to an
exponential scale length of $\approx 27\arcsec$ (2.16 kpc) beyond the break
radius ($R_{\rm brk}=70\arcsec$, 5.58 kpc). This outer scale length is very
similar to the value of 29″ reported by Erwin et al. (2008) for the disc
outside the bar, based on an azimuthally averaged $r$-band profile from an
SDSS image.

The bar is represented by the combination of an inner Sérsic component (with
very weakly boxy isophote shapes) for the B/P bulge and a FlatBar component
for the outer part of the bar; the latter has major-axis inner and outer scale
lengths of 50.8″ and 3.7″, respectively, and a break radius of $\sim
41\arcsec$, which is very similar to the maximum-ellipticity length of
$44\arcsec$ reported for the bar by Erwin (2005). The B/P-bulge part of the
bar has an essentially exponential surface-brightness profile ($n=0.96$); it
is slightly misaligned with respect to the FlatBar component ($\Delta$PA
$\approx 7\degr$, with the B/P bulge oriented slightly closer to the galaxy
major axis); this is consistent with the effects of projection operating on a
bar close to, but not perfectly aligned with, the galaxy minor axis (see Erwin
& Debattista 2013 for a discussion of how projection effects produce
misalignments between the projected B/P bulge and the “spurs” of the outer
part of the bar). Together, the two components are 43% of the total galaxy
light, only slightly less than the main disc; the B/P bulge by itself is 29%
of the light.

The inner, round (ellipticity $\approx 0.04$) Sérsic component has $n\approx
2.2$ and $R_{e}\approx 3.7\arcsec=310$ pc. Given that it is significantly
rounder than the disc (ellipticity $\approx 0.18$), we view this component as
a candidate classical bulge; it amounts to $\approx 12.1$% of the total galaxy
light. The Gaussian component inside this is a plausible nuclear star cluster,
especially considering its compact size ($R_{e}\approx 4$ pc); its luminosity
is only $\sim 0.056$% of the total galaxy light. This combination of size and
stellar mass ($\sim 1.3\times 10^{7}M_{\sun}$) makes it reasonably typical for
a nuclear star cluster (e.g., Fig. 7 in Neumayer et al., 2020).

We note that including this last (NSC) component has a moderate effect on the
derived parameters of the round, inner Sérsic component (which we identify as
a potential classical bulge). Without the NSC component, the latter has
$n=2.55$ and $R_{e}=4.14\arcsec$ (347 pc), both of which are about $\sim
13$–15% larger than their values when the NSC component is included. There are
minor changes to the B/P bulge component, at the level of $\sim 2$% for $n$
and $R_{e}$.

Table 1: NGC 4608: 2D Decomposition

Component | Parameter | Value | Units
---|---|---|---
Gaussian | PA | — | °
(NSC) | $\epsilon$ | $0^{1}$ |
| $\mu_{0}$ | $12.27\;[+0.14,-0.23]$ | SB
| $\sigma$ | $0.058\;[+0.004,-0.006]$ | ″
|  | $4.8\;[+0.4,-0.5]$ | pc
Sersic | PA | $82.9\pm 0.5$ | °
(bulge) | $\epsilon$ | $0.0359\pm 0.0005$ |
| $n$ | $2.214\;[+0.007,-0.008]$ |
| $\mu_{e}$ | $16.514\;[+0.006,-0.007]$ | SB
| $r_{e}$ | $3.67\pm 0.02$ | ″
|  | $308.0\pm 1.5$ | pc
Sersic_GenEll | PA | $33.00\;[+0.07,-0.06]$ | °
(bar: BP bulge) | $\epsilon$ | $0.1129\pm 0.0004$ |
| $c_{0}$ | $0.099\pm 0.002$ |
| $n$ | $0.960\pm 0.002$ |
| $\mu_{e}$ | $17.348\pm 0.003$ | SB
| $r_{e}$ | $10.342\pm 0.008$ | ″
|  | $867.4\pm 0.7$ | pc
FlatBar | PA | $26.520\pm 0.007$ | °
(bar: outer) | $\epsilon$ | $0.8708\;[+0.0003,-0.0004]$ |
|  $\Delta$PAm | $40.0\pm 0.3$ |
| $\mu_{0}$ | $17.831\;[+0.002,-0.003]$ | SB
| $h_{1}$ | $46.3\pm 0.2$ | ″
|  | $3886\;[+13,-14]$ | pc
| $h_{2}$ | $3.75\pm 0.02$ | ″
|  | $315\pm 1$ | pc
| $R_{\rm brk}$ | $40.7\pm 0.02$ | ″
|  | $3412\pm 2$ | pc
| $\alpha$ | $0.498\pm 0.002$ | ″-1
GaussianRingAz | PA | $106.2\pm 0.2$ | °
(inner ring) | $\epsilon$ | $0.0850\pm 0.0005$ |
| $A_{\rm maj}$ | $21.667\;[+0.006,-0.005]$ | SB
| $A_{\rm min,rel}$ | $2.26\pm 0.02$ |
| $R_{\rm ring}$ | $47.99\pm 0.02$ | ″
|  | $4025\pm 1$ | pc
| $\sigma$ | $6.37\pm 0.03$ | ″
|  | $534\pm 2$ | pc
BrokenExp | PA | $106.6\pm 0.2$ | °
(main disc) | $\epsilon$ | $0.127\pm 0.001$ |
| $\mu_{0}$ | $20.366\pm 0.004$ | SB
| $h_{1}$ | $6000^{1}$ | ″
|  | $50300$ | pc
| $h_{2}$ | $27.0\;[+0.1,-0.2]$ | ″
|  | $2262\;[+12,-19]$ | pc
| $R_{\rm brk}$ | $69.7\;[+0.2,-0.3]$ | ″
|  | $5847\;[+19,-24]$ | pc
| $\alpha$ | $0.32\;[+0.02,-0.03]$ | ″-1

Summary of the final 2D decomposition of NGC 4608. Column 1: Imfit component
names. Column 2: Parameter names. Column 3: Best-fit parameter value; note
that for size parameters, we include sizes in arcsec and also in pc. Errors
are nominal 68% confidence intervals from bootstrap resampling, and should be
considered underestimates; errors for linear sizes do not include
uncertainties in the distance. Column 4: Units of the parameter (“SB” =
$H$-band mag arcsec-2). For all parameters except the final BrokenExponential
(“main disc”), values come from fitting the HST WFC3-IR F160W image; for the
latter, values come from fitting the Spitzer IRAC1 image. Surface-brightness
values are for $\mu_{H}$. Notes: 1 = at limit of parameter-value boundaries;
uncertainty undefined.

###  4.6 Fitting NGC 4643

####  4.6.1 Determining the Best-Fit Model

In Figure 8 we show progressively more complex fits to the images of NGC 4643.
Panel b shows a basic bulge+disc decomposition, which – as we saw for NGC 4608
– is completely inadequate. Panel c shows a more complicated fit with our two-
component bar model (using, for simplicity, an Exponential for the B/P bulge
rather than a Sérsic function) and a first attempt at a nuclear disc model,
consisting of an elliptical exponential and a GaussianRing component (the
latter for the putative nuclear ring). This is a significantly better fit
(e.g, $\Delta\mathrm{AIC}\approx-9.8\times 10^{6}$), but the residuals are
still strong and systematic.

Panel d shows what is almost the same model, except that we have replaced the
outer Exponential with a BrokenExponential component. This brings a dramatic
improvement in the residuals (and a further
$\Delta\mathrm{AIC}\approx-6.3\times 10^{6}$), which illustrates the
importance of getting the main disc component right. (Note that the fit in the
bar region has improved as well.) Finally, panel e shows what happens when we
replace the B/P-bulge exponential component with one having a Sérsic profile.
This is an improved fit, with the best-fitting Sérsic index for the B/P-bulge
component $n\approx 0.52$.

In panel g we show this “best” model from the IRAC1 image fits (panel e) fit
to the F160W image. The residuals are generally small, but there are some
troublesome systematics in the inner $r\la 10\arcsec$, with alternating radial
deficits and excesses. In addition, we found that the GaussianRing component
in this fit converged to a very small radius ($\sim 0.06\arcsec$) and a very
broad width ($\sigma\sim 2.5\arcsec$). This disagrees with the apparent size
of the ring ($r\sim 3.0\arcsec$) and indicates that the Exponential +
GaussianRing model for the center of this galaxy is not a good match to the
data. Careful inspection of the F160W image (panel f) and profile cuts through
the center hinted that the morphology might be better represented by the
combination of a broken-exponential structure for the nuclear disc and a
compact, round component at $r\la 0.5\arcsec$. Replacing the inner Exponential
+ GaussianRing with a BrokenExponential component and a central, round Sérsic
component produced our final model (panel h), with clearly better residuals in
the central region and $\Delta\mathrm{AIC}\sim-1.0\times 10^{6}$. There remain
some very faint systematic patterns in the residuals at this stage, perhaps
indicative of very subtle ring or spiral morphology within the nuclear disc.

Figure 8: As for Figure 5, but now showing progressive modeling of NGC 4643.
Upper row: logarithmically spaced, median-smoothed isophotes of the fitted
region of the Spitzer IRAC1 image (a) and “residual-ratio” images (data /
model) from models fit to the image (b–e). These are (from left to right):
simple bulge/disc (Sérsic + exponential) model (b); addition of bar + nuclear
disc + nuclear ring (c); replacement of outer exponential with
BrokenExponential (d); replacement of exponential B/P-bulge component with
Sérsic component (e). Lower row: Isophotes from the HST WFC3-IR F160W image
(f) and residual-ratio images from fits to the (full) image. From left to
right, these are: the “final model” from the IRAC1 fits (g) and replacement of
inner exponential + GaussianRing with BrokenExponential + Sersic components to
better model nuclear disc and compact classical bulge (or NSC) (h). Figure 9:
Best-fitting model for NGC 4643, compared with Spitzer IRAC1 image. Upper and
middle panels show logarithmically spaced isophotes from IRAC1 image (top) and
best-fitting model (middle); bottom panels show residuals (data values divided
by model values). The large-scale images (left) were smoothed using a $5\times
5$-pixel-wide median filter. Figure 10: Best-fitting model for NGC 4643,
compared with HST WFC3-IR F160W image. Upper and middle panels show
logarithmically spaced isophotes from F160W image (top) and best-fitting model
(middle); bottom panels show residuals (data values divided by model values).
The large-scale images (left) were smoothed using a $9\times 9$-pixel-wide
median filter.

####  4.6.2 Summary

Our final model for NGC 4643 uses Imfit’s BrokenExponential component for the
main disc; the new FlatBar component for the outer part of the bar and a
mildly elliptical Sersic_GenEllipse component for the inner, B/P part of the
bar; another BrokenExponential component for the nuclear disc; and a central
compact, nearly circular Sérsic component. Table 2 summarizes the best-fitting
parameter values, with the absolute and fractional luminosities in Table 3,
and Figures 9 and 10 compare this model with the IRAC1 and F160W data. In the
lower half of Figure 4 the ellipticity and position-angle profiles of the
best-fit model are compared with those of the data. This shows excellent
agreement, even better than that between NGC 4608 and its model, especially in
terms of how the PA of the model inside the bar agrees with the data.

The bar is, as for NGC 4608, the combination of a FlatBar component (17% of
the total galaxy light, with a break radius of $\approx 44\arcsec$) and a
Sersic_GenEllipse component (with slightly discy isophotes) for the B/P bulge
(31% of the total light). In contrast to NGC 4608, the B/P bulge surface-
brightness profile is closer to a Gaussian than to an exponential (Sérsic
$n=0.61$). As in the case of NGC 4608, there is a slight offset between the
position angles of the FlatBar and B/P-bulge components, with the latter about
10° closer to the galaxy major axis. Again, this agrees with the expected
projection effects acting on a bar with a vertically thin outer component and
an inner B/P component (Erwin & Debattista, 2013).

Finally, the center of NGC 4643 is modeled by the combination of two elements.
The first, which we argue represents a nuclear disc, is an elliptical
BrokenExponential component with a position angle of 52.6°; this is
beautifully consistent with the global PA of 53°. (The main-disc
BrokenExponential component of our model has a slightly different PA of $\sim
59\degr$, which may represent the influence of weak spirals in the region just
outside the bar.) The ellipticity of this component is 0.13, which is somewhat
rounder than the outer disc ($\epsilon\approx 0.2$); this may indicate that
the nuclear disc is thicker than the outer disc. The profile, as noted, is a
broken exponential, with inner and outer scale lengths of $\approx 300$ and
160 pc, respectively, and a break radius of $\approx 265$ pc.

The innermost component models the steep central rise in the surface
brightness interior to $r\sim 0.5\arcsec$, using a circular Sérsic component
with $n=1.63$ and $R_{e}\approx 35$ pc.

A question remains: can our model really explain the strikingly ringlike
appearance that unsharp masks of NGC 4643 images display? Figure 11 compares
unsharp masks of the inner region of NGC 4643 and the best-fitting model
image, showing how the appearance of a ring can indeed be created by the sharp
break in the broken-exponential profile of the nuclear disc. (See also
Appendix A for evidence that the broken-exponential nature of the nuclear
disc’s profile is robust against changes to how we model this part of the
galaxy.)

Table 2: NGC 4643: 2D Decomposition

Component | Parameter | Value | Units
---|---|---|---
Sersic | PA | $40\pm 6$ | °
(Bulge) | $\epsilon$ | $0.020\;[+0.005,-0.004]$ |
| $n$ | $1.6\pm 0.1$ |
| $\mu_{e}$ | $14.3\;[+0.1,-0.09]$ | SB
| $r_{e}$ | $0.37\pm 0.02$ | ″
|  | $35\pm 2$ | pc
BrokenExp | PA | $52.97\;[+0.05,-0.04]$ | °
(Nuclear Disc) | $\epsilon$ | $0.1269\;[+0.0002,-0.0003]$ |
| $\mu_{0}$ | $13.62\;[+0.02,-0.01]$ | SB
| $h_{1}$ | $3.20\;[+0.06,-0.05]$ | ″
|  | $300\;[+6,-5]$ | pc
| $h_{2}$ | $1.687\;[+0.002,-0.001]$ | ″
|  | $157.9\;[+0.1,-0.09]$ | pc
| $R_{\rm brk}$ | $2.841\pm 0.009$ | ″
|  | $265.9\pm 0.8$ | pc
| $\alpha$ | $12\;[+2,-1]$ | $\arcsec^{-1}$
Sersic_GenEllipse | PA | $123.12\pm 0.02$ | °
(bar: BP bulge) | $\epsilon$ | $0.1534\pm 0.0002$ |
| $c_{0}$ | $-0.1396\;[+0.0006,-0.0008]$ |
| $n$ | $0.6147\;[+0.0003,-0.0005]$ |
| $\mu_{e}$ | $16.9134\;[+0.0006,-0.0005]$ | SB
| $r_{e}$ | $13.349\;[+0.006,-0.004]$ | ″
|  | $1249.1\;[+0.5,-0.4]$ | pc
FlatBar | PA | $133.276\pm 0.004$ | °
(bar: outer) | $\epsilon$ | $0.91756\pm 0.00003$ |
|  $\Delta$PAm | $32.84\;[+0.03,-0.02]$ |
| $\mu_{0}$ | $17.5342\;[+0.0009,-0.0007]$ | SB
| $h_{1}$ | $90.0\pm 0$ | ″
|  | $8421.2\pm 0$ | pc
| $h_{2}$ | $5.01\pm 0.01$ | ″
|  | $469\pm 1$ | pc
| $R_{\rm brk}$ | $44.55\pm 0.01$ | ″
|  | $4168\pm 1$ | pc
| $\alpha$ | $0.401\pm 0.002$ | $\arcsec^{-1}$
BrokenExp | PA | $58.7\pm 0.1$ | °
(main disc) | $\epsilon$ | $0.178\pm 0.008$ |
| $\mu_{0}$ | $19.1226\;[+0.0008,-0.0007]$ | SB
| $h_{1}$ | $81.3\;[+0.7,-0.6]$ | $\prime\prime$
|  | $7600\pm 60$ | pc
| $h_{2}$ | $10.70\;[+0.05,-0.04]$ | $\prime\prime$
|  | $1001\;[+5,-4]$ | pc
| $R_{\rm brk}$ | $105.7\pm 0.2$ | $\prime\prime$
|  | $9890\pm 20$ | pc
| $\alpha$ | $0.13^{1}$ | $\arcsec^{-1}$

Summary of the final 2D decomposition of NGC 4643. Column 1: Imfit component
names. Column 2: Parameter names. Column 3: Best-fit parameter value; note
that for size parameters, we include sizes in arcsec and also in pc. Errors
are nominal 68% confidence intervals from bootstrap resampling, and should be
considered underestimates; errors for linear sizes do not include
uncertainties in the distance. Column 4: Units of the parameter (“SB” =
$H$-band mag arcsec-2). For all parameters except the final BrokenExponential
(“main disc”), values come from fitting the HST WFC3-IR F160W image; for the
latter, values come from fitting the Spitzer IRAC1 image. Surface-brightness
values are for $\mu_{H}$. Notes: 1 = at limit of parameter-value boundaries;
uncertainty undefined.

Table 3: Model Component Fluxes and Fractions

Component | $M_{H}$ | Fraction
---|---|---
NGC 4608
Gaussian | $-14.74$ | 0.00056
(NSC) |  |
Sersic | $-20.57$ | 0.121
(bulge) |  |
Sersic_GenEll | $-21.50$ | 0.286
(bar: B/P bulge) |  |
FlatBar | $-20.74$ | 0.142
(bar: outer) |  |
GaussianRingAz | $-19.17$ | 0.0333
(inner ring) |  |
BrokenExp | $-21.91$ | 0.417
(main disc) |  |
NGC 4643
Sersic | $-17.94$ | 0.0050
(bulge/NSC) |  |
BrokenExp | $-21.51$ | 0.134
(nuclear disc) |  |
Sersic_GenEll | $-22.42$ | 0.310
(bar: B/P bulge) |  |
FlatBar | $-21.74$ | 0.166
(bar: outer) |  |
BrokenExp | $-22.66$ | 0.386
(main disc) |  |

Absolute and relative luminosities of components in our best-fitting 2D
decompositions for the two galaxies studied in this paper. Column 1: Imfit
component names (corresponding galaxy component in second row; NSC = nuclear
star cluster). Column 2: $H$-band absolute magnitude of component (assuming
$D=17.3$ Mpc for NGC 4608 and 19.3 Mpc for NGC 4643). Column 3: Fraction of
total galaxy luminosity.

Figure 11: Comparison of the nuclear-disc region of NGC 4643 with our best-
fitting model. Left two panels: Logarithmically spaced isophotes for the F160W
image of NGC 4643 (left) and our model (right). Right two panels: Unsharp
masks of the same (made using a Gaussian with $\sigma=15$ pixels). Note the
clearly ringlike feature in the unsharp masks, produced (in the model, at
least) by the abrupt change in slope of the nuclear disc rather than by a
separate nuclear ring.

###  4.7 Contrasting NGC 4608 and NGC 4643: Isolating the Central Bulge and
Nuclear-Disc Components

Our modeling suggests that NGC 4608 and NGC 4643 have very similar structures
– except inside their B/P bulges. The former appears to have something rather
like a modest-sized, nearly spherical classical bulge (ellipticity $\approx
0.04$, Sérsic $n=2.2$, $R_{e}\sim 310$ pc, with a nuclear star cluster in its
center), while the latter has (in projection) an elliptical structure with a
broken-exponential profile and a compact, spherical structure (Sérsic $n=1.6$,
$R_{e}\sim 35$ pc) in the very center.

These parameters are taken from our best-fitting models; to see what the
centers might look like if we could remove the rest of the galaxy surrounding
them, Figure 12 shows isophotes and major-axis profiles from the F160W image
after subtracting the disc and bar components of our best-fitting models.

Figure 12: Isolation of the central stellar components of NGC 4608 and NGC
4643\. Upper left: The classical bulge (+ NSC) in NGC 4608. Contours show
logarithmically spaced isophotes from the HST WFC3-IR F160W image after
subtracting the best-fitting model components corresponding to the disc and
bar. The red circle marks the effective radius ($R_{e}=3.67\arcsec=310$ pc) of
the Sérsic component from the full fit; the dashed grey line indicates the
galaxy major axis. Upper right: Same, except now showing the nuclear disc and
compact classical bulge/NSC in NGC 4643. The blue ellipse marks the break
radius of the broken-exponential component from the full fit ($a_{\rm
brk}=2.86\arcsec\approx 270$ pc), while the small red circle marks the
effective radius ($R_{e}=0.38\arcsec\approx 35$ pc) of the inner Sérsic
component from the same fit. Bottom panels: Profiles along the major axis
through both isolated components.

##  5 The Stellar Kinematics of the Central Structures

The preceding decompositions indicate that both NGC 4608 and NGC 4643 have
central regions ($r\la 1.5$ kpc) dominated by the B/P bulges of their bars. In
the inner few hundred parsecs, however, the galaxies differ significantly. Our
preceding, photometry-based claim – that NGC 4608 hosts a modest classical
bulge while NGC 4643 has instead a nuclear disc – implies rather different
stellar kinematics: we would expect a “classical bulge” to be slowly rotating
and dominated by velocity dispersion, while a “nuclear disc” ought to be
rapidly rotating, with lower velocity dispersion. In addition, our argument
that the mildly elongated isophotes at $r\sim 1$ kpc in both galaxies are due
to the B/P bulges of bars – rather than to massive, large-scale classical
bulges – suggests that the kinematics there should be barlike rather than
spheroidal. In this section, we use archival and published IFU data to probe
the stellar kinematics in the inner regions of both galaxies to test these
predictions.

###  5.1 SAURON Stellar Kinematics

We begin with published stellar kinematics from observations with the SAURON
integral field unit. Although VLT-MUSE data with superior coverage and spatial
and spectral resolution is available for NGC 4643, the SAURON data is all we
have for NGC 4608, so it makes sense to compare the kinematics from the SAURON
observations of the two galaxies first. We can then see how well these compare
with the MUSE data for NGC 4643.

####  5.1.1 NGC 4608

Figure 13 shows the SAURON stellar kinematics for NGC 4608: stellar velocity,
velocity dispersion, $h_{3}$, and $h_{4}$, along with isophotes from our F160W
image to show the underlying stellar structure. We can see a clear pattern of
rotation, along with velocity dispersion that increases toward the center of
the galaxy; the dispersion appears elongated along the major axis of the bar,
though it becomes rounder in the very center. The $h_{3}$ map shows relatively
little structure, though there is weak evidence for a slight correlation
between velocity and $h_{3}$, with positive velocities and $h_{3}$ values on
the NW side of the galaxy center and negative values of both on the SE side.
This is potentially significant, since $V$-$h_{3}$ correlation is a prediction
of bar models (e.g., Iannuzzi & Athanassoula, 2015; Li et al., 2018) due to
the presence of elliptical, bar-supporting orbits; this suggests that the
region from $r\sim 5$–20″ along the bar minor axis is dominated by bar orbits.

In Figure 14 we show a close-up of the same kinematic data. Here, instead of
the observed F160W isophotes, we plot isophotes which show the ratio between
the classical-bulge component in our best-fitting Imfit model and the rest of
the model. The solid lines show where the former component is brighter than
the rest of the model; this is where the classical-bulge component is the
dominant stellar structure in our model. This region ($r\la 2.5\arcsec$) shows
weak rotation, uniformly high velocity dispersion, and a tenuous hint of weak
$V$-$h_{3}$ anticorrelation (though the statistical significance of the latter
is dubious). All of this is consistent with a slowly rotating classical bulge.

Figure 13: NGC 4608 SAURON kinematic maps, with data from ATLAS3D (colour)
plotted on top of HST F160W isophotes. In the $h_{3}$ panel, there is evidence
for a weak asymmetry: negative $h_{3}$ on the left side, positive $h_{3}$ on
the right. This matches the velocity pattern in the first panel; such a
$V$-$h_{3}$ correlation is a signature of bar orbits. Figure 14: A close-up of
Figure 13, but with contours now showing the ratio of the classical-bulge
(Sérsic) component to the rest of our best-fitting model of NGC 4608 (e.g.,
Figure 7). Solid black contours indicate where the classical-bulge component
is brighter, dashed black contours where its brightness is 50–100% of the rest
of the model, and thin grey contours where it is fainter. The region dominated
by the bulge component has almost no rotation, centrally increasing velocity
dispersion, and little or no signature in $h_{3}$ or $h_{4}$; this is
suggestive of a classical bulge.

####  5.1.2 NGC 4643

Here, we examine the SAURON velocity fields for NGC 4643. Figure 15 shows
SAURON data for this galaxy, along with isophotes from our F160W image of the
galaxy. In addition to the ATLAS3D kinematics (lower panel), we also show (top
panel) kinematics from the study of Seidel et al. (2015), which used multiple
pointings to sample a larger field of view.

Outside $r\sim 6\arcsec$, the stellar-velocity pattern is similar to that in
NGC 4608: clear stellar rotation, an elongated region of higher velocity
dispersion aligned with the bar and increasing toward the center, and a
tenuous $V$-$h_{3}$ correlation (extending to $r\sim 25\arcsec$ along the
major axis of the galaxy, which is approximately the minor axis of the bar).
As with NGC 4608, the evidence for $V$-$h_{3}$ correlation in the bar is very
weak; however, for this galaxy the VLT-MUSE kinematics provide clear
confirmation of this pattern (e.g., Gadotti et al. 2019 and Section 5.2).

But for $r\la 6\arcsec$ ($r\la 500$ pc), the stellar kinematics is very
different from that of NGC 4608. There is a region of rapid rotation
accompanied by a clear anticorrelation with $h_{3}$. There is also a clearly
elliptical region of distinctly lower velocity dispersion, and a similar
region of elevated $h_{4}$.

From the bottom panel of Figure 15, it is clear that this inner region of
rapid, disclike rotation is associated with the elliptical inner isophotes we
previously identified as due to a nuclear disc (e.g., Section 4.6). Figure 16
demonstrates this more clearly by comparing the ATLAS3D kinematics with the
isophotes from an image which is the ratio of the nuclear-disc component of
our model to the rest of the model (similar in spirit to Figure 14). Solid
contour lines show where the nuclear-disc component is the dominant component,
and this is precisely where we see the rapid rotation, $V$-$h_{3}$
anticorrelation, lower dispersion, and elevated $h_{4}$.

Our conclusion is that the inner $r\la 500$ pc of NGC 4643 is dominated by the
rapid rotation, $V$-$h_{3}$ anticorrelation, and lower dispersion typical of a
disc rather than a classical bulge. The only really puzzling part of this is
the rather high $h_{4}$ in the nuclear-disc region, since a pure disc should
probably have $h_{4}$ close to zero (see, e.g., Debattista et al., 2005). As
we will argue in Section 5.2, this is probably due to the superposition of two
components along our line of sight: the dominant, low-$\sigma$ nuclear-disc
kinematics and the innermost part of the bar’s B/P bulge, with higher
$\sigma$.

Figure 15: NGC 4643 SAURON kinematic maps, plotted on top of HST F160W
isophotes. Top panels show maps from Seidel et al. (2015), bottom panels show
maps from ATLAS3D. Note the rapid rotation in the inner $r\la 5\arcsec$,
accompanied by lower velocity dispersion, $V$-$h_{3}$ anticorrelation, and
positive $h_{4}$. (A very weak indication of $V$-$h_{3}$ correlation can be
seen at $r\sim\pm 20\arcsec$ along the minor axis of the bar, suggesting the
presence of bar orbits.) Figure 16: A close-up of the lower panels (ATLAS3D
stellar kinematics) of Figure 15, but with contours now showing the ratio of
the nuclear-disc component to the rest of our best-fitting model of NGC 4643
(e.g., Figure 10). Solid black contours indicate where the nuclear disc is
brighter, dashed black contours where its brightness is 50–100% of the rest of
the model, and thin grey contours where it is fainter. Note that the region
dominated by the nuclear-disc component matches almost perfectly with the fast
rotation, lower velocity dispersion, strong $V$-$h_{3}$ anticorrelation, and
enhanced $h_{4}$.

###  5.2 NGC 4643: Additional Evidence from MUSE

NGC 4643 is one of the barred galaxies studied by the TIMER project (Gadotti
et al., 2019). Fig. 3 of that paper shows the VLT-MUSE-based stellar
kinematics for this galaxy, and evinces the same stellar-kinematic patterns we
noted in the SAURON data, albeit with higher spatial resolution and
considerably better $S/N$. In particular, the nuclear-disc region is singled
out by a pattern of high rotation, clear $V$-$h_{3}$ anticorrelation, lower
velocity dispersion, and positive $h_{4}$. Outside the nuclear disc, the
velocity dispersion is clearly higher along the major axis of the bar. In
addition, the maps show that near the minor axis in the outer part of the bar
(e.g., between the fourth or fifth and the seventh or eighth contour lines in
their figure) $h_{3}$ is correlated with the velocity, as hinted at in the
SAURON data. The combination of the latter two features is, as Gadotti et al.
(2019) noted, strong kinematic evidence for a B/P bulge within a bar.

In Figure 17, we show unbinned (single-spaxel) stellar kinematics derived from
the same MUSE data, showing both the full MUSE field of view and the region
containing the nuclear disc. Here, we can see that the nuclear-disc region has
two extra kinematic features. The first is the fact that at smaller radii
(e.g., $r\la 2\arcsec$), the velocity dispersion increases to $\sigma\sim
125$–130 ${\rm km\;s}^{-1}$, versus $\sigma\sim 110$–115 kms in the region
$r\sim 1.5$ to 3.5″. This corresponds with a nearly circular region of
slightly lower $h_{4}$.

The second interesting feature is that the dispersion drops again in the very
center ($r<0.5\arcsec\approx 45$ pc), to a value of $\sim 110$–115 ${\rm
km\;s}^{-1}$. This corresponds almost exactly with the round inner Sérsic
component in our decomposition. We note that there is some evidence for
nuclear stellar-velocity-dispersion depressions on similar scales in other
early-type spirals. In NGC 3368, which has a similar-sized ($R_{e}\approx 25$
pc) compact “classical-bulge” component (Nowak et al., 2010; Erwin et al.,
2015), the velocity dispersion observed with VLT-SINFONI in AO mode shows a
drop in the dispersion in the inner $0.5\arcsec\approx 25$ pc (Nowak et al.
2010, their Fig. 14). Erwin et al. (2015) argued for a compact central stellar
component with $R_{e}\approx 35$ pc in NGC 1068, where the SINFONI AO stellar
kinematics of Davies et al. (2007) showed a central decrease (albeit to an
implausible value of 0 ${\rm km\;s}^{-1}$ at the center, presumably due to
problems subtracting the contribution of the Seyfert nucleus). Finally,
although we do not (currently) have any evidence for a similar round, compact
structure in NGC 1097, the SINFONI AO data of Davies et al. (2007) shows a
clear drop from from 150 ${\rm km\;s}^{-1}$ to 100 ${\rm km\;s}^{-1}$ for
$r<0.5\arcsec\approx 35$ pc.

So there is some evidence for central drops in the stellar velocity dispersion
in the inner 25–35 pc of massive, early-type spirals. The underlying dynamical
origin of the phenomenon needs further investigation.

Figure 18 shows the inner MUSE stellar kinematics again, this time with the
nuclear-disc/model ratio isophotes, as in Figure 16. From this, we can again
see that the elliptical region of high velocities, low dispersion and
anticorrelated $h_{3}$ matches almost perfectly with where the nuclear-disc
component dominates in our model. Fitting the velocity field in this region
using the kinemetry code of Krajnović et al. (2006) gives a kinematic position
angle of $52.0\degr\pm 0.9\degr$, identical within the uncertainties with the
position angle of the BrokenExponential component we use to model the nuclear
disc (52.6°, Table 2).

This is also where the elevated $h_{4}$ is found. By plotting the $h_{4}$
values against the ratio of the nuclear-disc component to the bar model in
Figure 19, we can see a clear trend: the kurtosis values increase as the ratio
increases, peaking where the ratio is $\sim 1.3$, and then decline as one
moves to even higher ratio values. We interpret this as the result of changing
relative contributions from the B/P bulge (high velocity dispersion) and from
the nuclear disc (low velocity dispersion) along the line of sight. Outside
the nuclear disc (ND ratio $<1$), the LOSVD is dominated by the high-
dispersion B/P bulge LOSVD, and so the kurtosis is low. In the region where
both the B/P bulge and the nuclear disc contribute almost equally, the
kurtosis reaches a maximum. At smaller radii, where the nuclear disc
dominates, the kurtosis declines.

We will explore this phenomenon further in a follow-on paper.

Figure 17: MUSE stellar kinematics for NGC 4643. Top panels: Isophotes from
IRAC1 and F160W images of NGC 4643. Middle panels: Median-smoothed stellar
kinematics, from single-spaxel analysis of VLT-MUSE data (full MUSE field of
view). Bottom panels: Close-up of unsmoothed stellar kinematics in the
nuclear-disc region. Figure 18: MUSE stellar kinematics for the nuclear-disc
region of NGC 4643. Leftmost panel: Isophotes from F160W image of NGC 4643.
Middle and right panels: Stellar kinematics for NGC 4643, from single-spaxel
analysis of VLT-MUSE data. Contours in these panels are as in Figure 16: solid
contour lines indicate where the nuclear-disc component of our best-fitting
model is brighter, dashed black contours where its brightness is 50–100% of
the rest of the model, and thin grey contours where it is fainter (note that
the inner four solid contour lines indicate decreasing levels of relative
brightness toward the center, due to the increasing importance of the compact
classical-bulge/NSC component). The nuclear-disc ratio image has been
convolved to match the FWHM $\approx 0.75\arcsec$ seeing of the MUSE
kinematics. Figure 19: MUSE stellar-kinematic $h_{4}$ values for $12\times
12\arcsec$ central region of NGC 4643, plotted against the local ratio of our
nuclear-disc model to the rest of the bar model. The $h_{4}$ values peak where
the ratio is $\sim 1$, suggesting that the high kurtosis is the result of
nearly equal contributions from nuclear-disc and bar LOSVDs with significantly
different dispersions.

###  5.3 Summary of Stellar Kinematics Results

We can describe the results of our analysis of the stellar kinematics data
from published SAURON observations (both galaxies) and from MUSE observations
(NGC 4643 only) as follows:

  * •

The outer isophotes of the bars in both galaxies show rotation with possible
(NGC 4608) or clear (NGC 4643) $V$-$h_{3}$ correlation, as expected for bars.

  * •

The B/P-bulge regions (the vertically thickened inner parts of the bars) of
both galaxies show higher velocity dispersion, aligned with the projected B/P
bulges and increasing towards the center.

  * •

In NGC 4608, the region we photometrically identified as a possible classical
bulge ($r\la 2.5\arcsec\approx 200$ pc) shows slow rotation, high velocity
dispersion ($\sim 140$ ${\rm km\;s}^{-1}$), and no clear $V$-$h_{3}$
correlation or anticorrelation, consistent with a slowly rotating classical
spheroid.

  * •

By contract, in NGC 4643 the inner region that we photometrically identified
as a nuclear disc ($r\la 5\arcsec\approx 500$ pc) shows up as an elliptical
zone of significantly lower dispersion, with rapid rotation and strong
$V$-$h_{3}$ anticorrelation, consistent with a rotation-dominated disc. The
dispersion has a minimum value of $\sim 110$ ${\rm km\;s}^{-1}$, increasing
mildly toward the center.

  * •

The nuclear-disc region of NGC 4643 also shows very high $h_{4}$ values,
declining toward the center; we tentatively interpret this as a signature of
overlapping LOSVDs from the B/P bulge (higher dispersion) and the nuclear disc
(lower dispersion), with the decline in $h_{4}$ toward the center happening as
the nuclear-disc component becomes more dominant (and possibly also as the
dispersion of the nuclear-disc component becomes higher).

  * •

Finally, there is a drop to lower dispersion in the very innermost region of
NGC 4643 ($r<0.5\arcsec\approx 45$ pc, from $\sim 130$ ${\rm km\;s}^{-1}$ at
the edge of this region to $\sim 110$ ${\rm km\;s}^{-1}$ in the center),
coincident with the photometrically identified compact classical bulge (or
large NSC). Although there are some similar near-nuclear dispersion drops seen
in a few other massive spirals, the cause of this remains unclear.

##  6 Discussion

Outside of the central few hundred parsecs, the two galaxies studied in this
paper are remarkably similar. In both cases, their bars amount to $\sim 45$%
of the stellar light, with dominant B/P bulges comprising $\sim 30$% of the
light. They both have distinct stellar structures inside the B/P bulges, each
of which is $\sim 12$–13% of the total light. But while this structure in NGC
4608 is nearly circular, centrally concentrated (Sérsic index $\sim 2.2$),
slowly rotating, and dominated by velocity dispersion – matching a traditional
classical bulge – the inner structure of NGC 4643 is a nuclear disc:
elliptical and aligned with the outer disc, with rapid rotation and lower
velocity dispersion.

The photometric and kinematic evidence for a nuclear disc inside the B/P bulge
of NGC 4643 is a clear demonstration of a pattern Chung & Bureau (2004) and
Bureau et al. (2006) suggested based on their studies of edge-on disc
galaxies: that the B/P bulges of barred galaxies are frequently accompanied by
nuclear discs. The case of NGC 4608, on the other hand, demonstrates that this
is not a universal pattern: some B/P bulges contain structures much more like
classical bulges, rather than nuclear discs.

###  6.1 The Nuclear Disc in NGC 4643

The prominent nuclear disc in NGC 4643 makes up 13% of the galaxy’s near-IR
light. Assuming no strong gradients in $M/L$ and a total galaxy stellar mass
of $6.2\times 10^{10}M_{\sun}$ (Section 3), the nuclear disc has a stellar
mass of $\sim 7\times 10^{9}M_{\sun}$. This is about seven times the mass
estimated for Milky Way’s nuclear disc ($\sim 1\times 10^{9}M_{\sun}$;
Nogueras-Lara et al., 2019; Sormani et al., 2020). The stellar-population
analysis of the MUSE data in Bittner et al. (2020) indicates that the nuclear
disc is old ($\sim 10.5$ Gyr) – but slightly less so than the stars in the B/P
bulge ($\sim 11.5$ Gyr) in which it is embedded. It is also slightly more
metal-rich than the B/P bulge (Seidel et al., 2015; Gadotti et al., 2019;
Bittner et al., 2020). This suggests that it was formed not long after the bar
itself, from gas that was slightly enriched compared to that from which the
stars in the bar were formed.

The most peculiar thing about NGC 4643’s nuclear disc is its surface-
brightness profile: instead of the standard single-exponential form one
(perhaps naively) expects of a disc, it has a downbending, broken-exponential
profile (a Freeman Type II profile, in the scheme used for large-scale discs)
– with a break which is sufficiently sharp to appear ringlike in unsharp masks
(e.g., Figure 11); see Appendix A. We are not aware of any similar structures
in other galaxies, though it is quite possible they have escaped notice
because few nuclear discs have been analyzed with sufficient resolution or in
sufficient detail. Broken-exponential profiles are, of course, quite common in
large-scale discs, but it is unclear whether the proposed mechanisms for those
discs – e.g., radial truncations in star-formation in concert with spiral-
driven radial migration (e.g., Debattista et al., 2017a, and references
therein) – would apply here. One speculative possibility is that this is an
end-state for the combination of a nuclear disc and a nuclear ring, with an
initially narrow ring having broadened over time, blending with the nuclear
disc to produce the broken-exponential structure.

Most theoretical discussions of nuclear rings and discs inside bars suggest
that the stars on them form from gas which has settled onto the x2 orbits of a
bar potential, which implies that the stars should themselves be trapped
around the same family of orbits. Since x2 orbits are elongated perpendicular
to the x1 orbits that support the bar, we might expect nuclear discs to
themselves be elongated perpendicular to their host bars. For the case of NGC
4643, where the bar has a PA of 133°, the line of nodes is at 53°, and the
inclination is 38°, the minor axis of the bar should have an observed PA of
47°; an elliptical structure oriented along this axis would have an observed
PA in between that and line of nodes. However, the ellipse fits to the
observed isophotes in the nuclear-disc region (Figure 4 and the nuclear-disc
component in our best-fitting model of the F160W image (Table 2) both agree on
a PA of 52–53°, essentially identical to that of the main disc – and to the
kinematic PA of the nuclear-disc region (52°; Section 5.2). The implication is
that the nuclear disc is intrinsically close to circular in the plane of the
galaxy, rather than significantly elongated perpendicular to the bar.

The observed ellipticity of the nuclear disc is $\approx 0.13$, which is
rounder than that of the main disc (0.20). This in turn suggests that the
nuclear disc, if circular, is intrinsically somewhat thicker than the main
disc – or perhaps that it is slightly elongated parallel to the bar. The
latter possibility might mean that the stars are trapped around round inner x1
orbits of the bar, and even that the nuclear disc could predate the bar. As
noted above, the stellar ages estimated from the analysis of the MUSE data by
Gadotti et al. 2019 show that the stars in the nuclear disc are approximately
the same age as the stars in the bar surrounding it (mass-weighted ages $\sim
10$ Gyr); given the inherent uncertainties in age estimates for older stellar
populations, we probably cannot tell whether the nuclear disc is truly younger
or older than the bar.

###  6.2 Speculations About Formation Histories

We have emphasized that both NGC 4608 and NGC 4643 are quite similar in their
large-scale morphology: massive, early-type discs with very strong bars that
contain prominent B/P bulges. But their inner structure is rather different:
NGC 4608 has what appears to be a modest, kinematically hot classical bulge
(and a very compact NSC), while NGC 4643 has a kinematically cool nuclear disc
and what is either a very compact classical bulge or a large NSC. This
suggests two questions. First, why does NGC 4608 have a significant classical
bulge, while NGC 4643 does not? Second, why is there a nuclear disc present in
NGC 4643 but not in NGC 4608?

The formation of a classical bulge in NGC 4608 could potentially be ascribed
to local variations in the very early formation stages, leading to more merger
activity in the central region of one galaxy than the other. The question then
becomes why NGC 4608 was not able to form a nuclear disc, despite having a
strong bar very much like NGC 4643’s.

One possibility is that this dichotomy reflects the different environments of
the two galaxies: NGC 4643 is a field galaxy, while NGC 4608 is a member of
the Virgo Cluster. In this scenario, we could argue that NGC 4608 could have
undergone ram-pressure stripping within the cluster environment early enough
in its history that its bar was unable to drive significant gas inflow and
form a nuclear disc. NGC 4643, by contrast, must have retained enough gas
after its bar formed for bar-driven inflow and star formation to have formed
its nuclear disc. NGC 4643 in fact still retains some gas, with H i detections
(e.g., van Driel et al., 2000) and H$\alpha$ emission within the bar which is
consistent with the optical dust lanes (Figure 3) and with kinematics
consistent with that of the stars (Gadotti et al., 2019).

We should note, however, that another barred S0 in the Virgo Cluster – NGC
4371 – contains evidence for both a modest classical bulge (Erwin et al.,
2015) and a significant nuclear disc/ring with a slightly younger and more
metal-rich population in its MUSE spectra (Gadotti et al., 2015; Bittner et
al., 2020). So it is clear that the cluster environment doesn’t exclude
nuclear-disc formation.

###  6.3 Classical Bulges and B/T Ratios

As mentioned in the Introduction, large samples of galaxies, imaged at
moderate to low resolution, are typically analyzed in terms of simple
bulge/disc decompositions, treating the galaxies as consisting of just an
(exponential) disc and an optional (Sérsic-profile) bulge; the latter is often
assumed to be a classical bulge, or else treated in a dichotomous fashion as
either a classical bulge or a pseudobulge (e.g., depending on whether the
Sérsic index $n$ is greater or less than 2). Given the complexity in the
central regions we find for these two galaxies, it is instructive to consider
how they would be analyzed as part of a large, low-spatial-resolution sample.

To do this, we performed simple bulge/disc decompositions with Imfit on two
sets of images. The first was the same Spitzer IRAC1 images (Section 2.2) used
as part of our multi-component decompositions in Section 4. For NGC 4643, this
can be compared with the two-component fit done to the same image by Salo et
al. (2015). The second sets of images consisted of SDSS $r$-band images,
artificially redshifted to $z=0.04$ (chosen as a redshift typical of large,
SDSS-based samples); this corresponds to an angular-diameter distance almost
exactly ten times further away than the galaxies’ actual distances. We
redshifted each image by first convolving it with a Gaussian so that it had
same effective resolution it would have under the original observing
conditions, but with the galaxy at $z=0.04$; we then binned the pixels 10:1.
We did not attempt to include any $k$-correction or surface-brightness
dimming, since these are minor effects for the target redshift.

For NGC 4608, the two-component fit of the IRAC1 image gives $B/T=0.49$, with
Sérsic $n=2.4$ and $R_{e}\approx 1.1$ kpc for the bulge component.
Decomposition of the redshifted $r$-band image also gives $B/T=0.49$; the
Sérsic component now has a higher value of $n=3.8$ but essentially the same
$R_{e}$.

For NGC 4643, we find even more “bulge-dominated” fits, with $B/T=0.65$
(Sérsic $n=2.4$, $R_{e}\approx 1.3$ kpc) for the IRAC1 image; this is similar
to the two-component fit to the same image by Salo et al. (2015): $B/T=0.72$,
Sérsic $n=3.0$ and $R_{e}\approx 2.3$ kpc.131313Retrieved from webpage
https://www.oulu.fi/astronomy/S4G_PIPELINE4/P4STORE/ For the redshifted
$r$-band image, we find $B/T=0.61$, with Sérsic $n=2.9$ and $R_{e}\approx 1.6$
kpc.

Clearly, the simplistic, two-component B/D decomposition that is still
standard for large surveys fails dramatically for galaxies like these. It
significantly (even catastrophically) overestimates the fraction of galaxy
light that is part of the “bulge”, especially when that component is assumed
to be a classical spheroid. Even the somewhat ad-hoc use of Sérsic indices to
discriminate between classical bulges and pseudobulges would fail, since for
both galaxies the Sérsic indices are $>2$. (These galaxies would satisfy at
least two of the criteria for classical bulges in Kormendy 2016, since they
have $n>2$ and $B/T\ga 0.5$.) This is misleading because both galaxies have
significant pseudobulges: the B/P bulge in NGC 4608 and the B/P bulge and the
nuclear disc in NGC 4643.

We have argued that NGC 4608 has a classical bulge that is only $\sim 12$% of
the galaxy light, while NGC 4643 has (at best) a compact classical bulge that
is $\sim 0.5$% of the total light; this latter component could also be
interpreted as a large NSC. This means that two-component fits would
overestimate the spheroid fraction of NGC 4608 by roughly a factor of four,
and by roughly a factor of at least one hundred for NGC 4643.

It is now (somewhat) common to include optional bars in decompositions of
moderately large samples (e.g., Gadotti, 2009; Salo et al., 2015; Méndez-Abreu
et al., 2017; Kruk et al., 2018), though this is still not true of the largest
surveys (e.g., Simard et al., 2011; Mendel et al., 2014; Kim et al., 2016;
Lange et al., 2016; Bottrell et al., 2019). This does tend to reduce the
fraction of light assigned to the “classical bulge”, but can still
significantly overestimate things.

In the case of NGC 4608, the “spheroid/T” value drops from the two-component
value of 0.49 to 0.33 when a separate (single-Sérsic) bar is included
(Gadotti, 2008). For NGC 4643, the value drops from $\sim 0.65$ (various two-
component fits) to 0.25 in the three-component fit of Salo et al. (2015). So
three-component decompositions, with a simple Sérsic or Ferrers component for
the bar, still result in overestimating the spheroid fraction by factors of
three (NGC 4608) or fifty (NGC 4643).

##  7 Summary

We have presented a detailed morphological and stellar-kinematic analysis of
two similar massive, early-type, barred galaxies, NGC 4608 and NGC 4643\. We
find that images of both galaxies can be fit using a new two-component bar
model which assigns $\sim 14$–17% of the total galaxy light to the narrow,
vertically thin outer part of the bar and $\sim 30$% of the total light to the
bar’s B/P bulge (the vertically thickened inner part of the bar). A further
$\sim 39$–41% of the light can be assigned to the main disc of each galaxy,
leaving $\sim 12$–13% of the light for the central stellar components inside
the bars.

In NGC 4608, this central component is apparently a classical bulge, with
Sérsic $n=2.2$ and half-light radius $R_{e}\approx 310$ pc; there is also
evidence for a possible nuclear star cluster, though this is only $\sim 0.06$%
of the galaxy light. Published SAURON stellar kinematics show slow rotation
and high velocity dispersion in this region, consistent with the idea that
this is a classical bulge.

By contrast, the inner light in NGC 4643 is dominated by a stellar nuclear
disc, with a somewhat unusual broken-exponential surface-brightness profile;
the break in this profile produces a ringlike feature in unsharp masks, though
there seems to be little in the way of an actual ring in addition to the
nuclear disc. There is also evidence for a compact, round structure at the
very center, with Sérsic $n\approx 1.6$ and $R_{e}\approx 35$ pc, amounting to
$\sim 0.5$% of the light; this is probably either a compact classical bulge
(such as some of those identified in Erwin et al., 2015) or a very large
nuclear star cluster.

The stellar kinematics for NGC 4643, including both published SAURON and MUSE
data (and our reanalysis of the latter), show strong rotation, lower velocity
dispersion, strong $V$-$h_{3}$ anticorrelation, and elevated $h_{4}$ in the
inner region, consistent with a rapidly rotating, kinematically cool nuclear
disc. The increased $h_{4}$ is, we suggest, due to the overlapping
contributions from the (more slowly rotating, kinematically hot) B/P bulge and
the (faster-rotating, kinematically cool) nuclear disc. The inner $r<50$ pc
shows a distinct drop in the velocity dispersion, in approximately the same
region as where the central compact classical bulge (or large NSC) dominates.
Although there are a handful of other examples of similar nuclear dispersion
drops in some other early-type, barred galaxies, the explanation for this
remains unclear.

The dramatically different central structures in these two otherwise very
similar galaxies indicates that seemingly identical galaxies can have
significantly different centers and thus different formation histories.

We also note that both galaxies have quite large $B/T$ ratios when they are
fit using the same two-component (bulge + disc) models as are used for large
galaxy surveys. If these galaxies were included in such surveys, they would
most likely be assigned $B/T$ values of $\sim 0.5$–0.7, even though their
actual spheroid fractions are only 0.12 for NGC 4608 and 0.005 for NGC 4643;
the high Sérsic indices for the “bulges” in such 2-component fits ($n\sim
2.5$–3.5) would (erroneously) suggest the existence of dominant classical
bulges in these galaxies. Even three-component fits (disc + Sérsic/Ferrers bar
+ bulge) produce $B/T$ values of 0.25–0.3, which significantly overestimates
the spheroid fraction.

This paper is a preview of the analysis we plan to apply to the other $\sim
50$ galaxies in our volume- and mass-limited sample of nearby S0 and
early/intermediate-type spirals, as part of a project to determine the true
nature and characteristics of the central stellar structures of disc galaxies.

## Acknowledgments

We thank the referee for their comments on this paper, which have led to
several improvements. We are also happy to thank Dimitri Gadotti, Matias Blaña
Diaz and Mattia Sormani for comments on earlier drafts of this paper. VPD was
supported by Science and Technology Facilities Council Consolidated grant
ST/R000786/1. AdLC acknowledges support from grant AYA2016-77237-C3-1-P and
JMA acknowledges support from grant AYA2017-83204-P, both from the Spanish
Ministry of Economy and Competitiveness (MINECO).

This research is based on observations made with the NASA/ESA Hubble Space
Telescope, obtained at the Space Telescope Science Institute. Support for
Program number GO-15133 was provided by NASA through a grant from the Space
Telescope Science Institute. The Space Telescope Science Institute is operated
by the Association of Universities for Research in Astronomy, Incorporated,
under NASA contract NAS5-26555.

This work is based in part on observations made with the Spitzer Space
Telescope, obtained from the NASA/IPAC Infrared Science Archive, both of which
are operated by the Jet Propulsion Laboratory, California Institute of
Technology under a contract with the National Aeronautics and Space
Administration. This paper also makes use of data obtained from the Isaac
Newton Group Archive which is maintained as part of the CASU Astronomical Data
Centre at the Institute of Astronomy, Cambridge.

Based on observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere under ESO programme 097.B-0640(A).

Funding for the creation and distribution of the SDSS Archive has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Aeronautics and Space Administration, the National Science
Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and
the Max Planck Society. The SDSS Web site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium (ARC) for the
Participating Institutions. The Participating Institutions are The University
of Chicago, Fermilab, the Institute for Advanced Study, the Japan
Participation Group, The Johns Hopkins University, the Korean Scientist Group,
Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University,
University of Pittsburgh, University of Portsmouth, Princeton University, the
United States Naval Observatory, and the University of Washington.

This research also made use of both the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration, and the Lyon-Meudon Extragalactic Database (LEDA;
http://leda.univ-lyon1.fr).

## Data Availability

The raw HST, Spitzer, and MUSE data for both galaxies are publicly available
from the respective archives. Reduced, mosaic Spitzer images for individual
observations of both galaxies are also available from the Spitzer archive, and
the reduced S4G mosaic image for NGC 4643 is available at the NASA/IPAC
Infrared Science Archive (https://irsa.ipac.caltech.edu/data/SPITZER/S4G/).
The reduced, combined MUSE datacube for NGC 4643 is available at
http://archive.eso.org/wdb/wdb/adp/phase3_spectral/form?collection_name=MUSE_DEEP;
our kinematic analyses of this datacube are available on request. Finally, the
SAURON kinematic data for both galaxies is available at the ATLAS3D site
(http://www-astro.physics.ox.ac.uk/atlas3d/.

Reduced images of our WFC3-IR F160W observations of both galaxies, along with
sky-subtracted versions of the IRAC1 images, mask and PSF images, ellipse-fit
files, Imfit config files, and Python code for generating the figures in this
paper can all be found in a Github repository at
https://github.com/perwin/n4608-n4643 and also at
https://doi.org/10.5281/zenodo.4235501.

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##  Appendix A The Broken-Exponential Nuclear Disc in NGC 4643

For NGC 4643, we argue that the inner region is best understood as a nuclear
disc with a somewhat unusual broken-exponential radial surface-brightness
profile. The evidence for a distinct nuclear disc of some kind is relatively
unambiguous. First, there is set of isophotes (semi-major axes $\sim 1$–4.5″)
with similar ellipticities and, most significantly, a common position angle
($\sim 52\degr$) which is clearly distinct from that of the bar ($\sim
133\degr$); see, e.g., Figures 3 and 4. Second, unsharp masking shows a clear,
narrow “ring-like” zone at $r\sim 3\arcsec$ where the surface-brightness
changes abruptly (e.g., Figure 11). And, of course, there is very clear
stellar-kinematic evidence for a kinematically cool and thus disclike
structure in this region (Section 5).

However, the exact profile of this structure is perhaps not as clear. For
example, might the ringlike feature in the unsharp mask be the result of a
narrow ring added to a smoother profile (e.g., that of an exponential)? Figure
20 plots major-axis profiles (folded about $r=0$) of residual images obtained
by subtracting the bar+main-disc components of three different best-fit
models. The goal is to see how much variation in the residual profile is due
to variations in the total model. All three models use our standard composite
Sérsic + FlatBar model for the bar and a small Sérsic component for the
central classical bulge/NSC; they differ in how we model the nuclear-disc
region itself. The top panel shows the profile when we use a Sérsic function
for the nuclear disc. The middle panel shows the same when the model uses the
sum of an exponential and an asymmetric Gaussian ring (Imfit’s
GaussianRing2Side component) – a plausible model for the sum of a star-forming
nuclear ring and an underlying nuclear disc. Finally, the bottom panel shows
the result for our preferred best-fit model (nuclear disc modeled as
BrokenExponential), and is the same as what is shown in the right-hand panels
of Figure 12.

In all three cases, we see the same basic structure in the residual profile:
the compact central excess (which we associate with the CB/NSC) and a broken-
exponential profile outside, which dominates the residual profile for $r\ga
1\arcsec$. This indicates that our identification of a broken-exponential
structure for the nuclear disc in NGC 4643 is robust: it persists regardless
of the specific components used in the modeling of this region, and is not an
artifact of simply assuming the profile must be modeled with a
BrokenExponential component.

It is also worth noting that in the Exponential + GaussianRing2Side model, the
best-fit “ring” radius is only 1.22″, which is clearly much smaller than the
visible break radius ($r\sim 3\arcsec$); in addition, the inner part of this
component is essentially flat, so it is not really a ring. Although this more
complicated model provides a formally better fit with lower AIC than the
BrokenExponential model, its parameters are less useful as descriptions of the
structure. Since the best-fit BrokenExponential break radius of 2.9″ is a much
closer match to the visible break, we prefer it as an approximation of the
nuclear disc.

Figure 20: Profiles from major-axis cuts through the F160W image of NGC 4643
(thick grey lines) and through residual images (black lines), formed by
subtracting the bar components of the best fitting model. All models are
identical except for how the nuclear-disc region is represented. In all three
cases, the residuals in the nuclear-disc region show similar broken-
exponential profiles with a break at $r\sim 3\arcsec$, indicating that this is
a robust feature of the galaxy. Upper panel: Best-fitting model uses Sérsic
component for nuclear disc. Middle panel: Best-fitting model uses Exponential
\+ GaussianRing2Side components for nuclear disc. Lower panel: Best-fitting
model uses BrokenExponential component for nuclear disc (this is the model
discussed in the main text of the paper).

##  Appendix B A Two-Dimensional Model of a Bar with a Boxy/Peanut-shaped
Bulge

Although it is standard to model bars in 2D decompositions as simple
elliptical141414With the option of describing the isophotes as having fixed
deviations from pure ellipticity in the discy/boxy sense. structures with
either Sérsic profiles or variations on the Ferrers (1877) ellipsoid, recent
analysis of bars in simulations and real galaxies (e.g., Laurikainen et al.,
2014; Athanassoula et al., 2015) has suggested that many bars are better
described as two-component systems. In this scheme, one combines an outer,
narrow component with a shallow (sometimes constant) radial surface-brightness
profile and an inner, rounder component with a steeper profile. These can be
readily associated with the dual three-dimensional structure of bars with B/P
bulges: the outer, vertically thin part of the bar has (when seen face-on)
narrow isophotes and a shallow surface-density profile, while the vertically
thick B/P bulge has rounder isophotes and a steeper surface-density profile.

Laurikainen et al. (2014) and Athanassoula et al. (2015) modeled this
composite structure using a slightly modified version of the Ferrers ellipsoid
for the outer part of the bar and a mildly elliptical Sérsic component,
typically with Sérsic index $n$ close to 1, for the B/P bulge (the “barlens”
in their terminology). While this is a clear improvement on past models for
bars, it is only a weak approximation of the outer-bar structure. Our model
attempts to capture three characteristics of the outer parts of bars seen at
low inclinations:

  1. 1.

The narrow isophote shape of the outer part of the bar;

  2. 2.

The fact that the surface-brightness profile along the bar’s major axis has a
broken-exponential nature, with a shallow (or even constant) profile breaking
sharply to a steeper, quasi-exponential falloff near the end of the bar;

  3. 3.

The transition between discy isophotes for most of the outer part of the bar
and rectangular isophotes near and at the end of the bar.

We achieve this by creating a model for the outer part of the bar (“FlatBar”)
with intrinsically elliptical isophotes and a surface-brightness profile which
has a broken-exponential form (Erwin et al., 2008; Erwin, 2015). This is then
modified by interpolating the profile as a function of angle relative to the
bar major axis. Along the major axis, we have the default broken-exponential
profile, with inner scale length $h_{1}$ and outer scale length $h_{2}<h_{1}$,
and with the transition region specified by the break radius $R_{\rm brk}$ and
the sharpness parameter $\alpha$. As we move away from the major axis – i.e.,
considering radial profiles with angles relative to the major axis
$\Delta\mathrm{PA}>0$ – we gradually increase the value of $h_{2}$ using
parabolic interpolation. For
$\Delta\mathrm{PA}\geq\Delta\mathrm{PA}_{\mathrm{max}}$, $h_{2}=h_{1}$ and the
profile becomes a simple exponential. In addition, we fix $R_{\rm brk}$ so
that it is constant regardless of $\Delta\mathrm{PA}$, as opposed to the
normal pattern of reducing $R_{\rm brk}$ to match the ellipticity of the image
function away from the major axis (as is done in Imfit’s BrokenExponential
function.)

As Figure 21 shows, this result in a structure where the inner isophotes are
clearly discy but transition to boxy near the break radius. At very large
radii – i.e., well outside the bar proper – the isophotes can become
implausibly polygonal, but this will usually be in regions dominated by the
main disc and so the effects in a typical combined model will be minimal.

We combine the FlatBar component with a mildly elliptical Sérsic component
(for the B/P bulge) to represent the whole bar (Figure 21). Preliminary tests
for a small number of barred galaxies indicates that best-fit Sérsic indices
for the B/P bulge component range between 0.5 and 1.1. For the two galaxies
studied in this paper, we use a Sersic_GenEllipse component (with generalized
ellipses for the isophotes) to account for the possibility of slightly boxy
(or discy) isophotes in the B/P bulge component.

We emphasize that this is an ad-hoc solution which appears to work well when
the galaxy is close to face-on, and to a certain degree for mildly inclined
systems when the bar is close to either the major or minor axis – the latter
being the case for the two galaxies studied here. For intermediate bar
position angles and higher inclinations, the projected appearance of bars with
B/P bulges becomes strongly asymmetric (see, e.g. Erwin & Debattista, 2013,
2017) and this model will be less suitable.

This function has been added to the most recent release (v1.8) of Imfit; the
code for it can be found on Imfit’s GitHub site
(https://github.com/perwin/imfit).

Figure 21: The Imfit components of our bar model. Left: the new “FlatBar”
component, representing a face-on or low-inclination view of the vertically
thin part of the bar. Middle: a mildly elliptical Sersic component ($n=1$),
representing the face-on/low-inclination view of the B/P bulge. Right: the sum
of the two. (All isophotes are logarithmically spaced.)

##  Appendix C A Two-Dimensional Model of a Ring with Azimuthally Varying
Intensity

NGC 4608 has a prominent inner ring surrounding its bar (e.g., Figures 6 and
7). Imfit includes an image function (GaussianRing) describing an elliptical
ring with a Gaussian radial profile, which we initially used in our modeling
of the images of this galaxy. Although this produced reasonable results, it
was clear from both the images and residuals of the fit that the galaxy’s ring
did not have a constant surface brightness, but rather decayed in brightness
away from the major axis of the bar.

To deal with this, we created a modification of the GaussianRing component in
which the peak intensity $A$ of of the (Gaussian) radial profile varies as a
function of angle $\theta$, with $A(\theta)=A_{\rm maj}$ along the ring’s
major axis ($\theta=0\degr$), and $A(\theta)=A_{\rm min}$ at $\theta=90\degr$
(along the minor axis), interpolating smoothly between the two values as a
function of $\cos(2\theta)$ for intermediate angles. The surface brightness at
scaled radius $r$ and angle $\theta$ is thus given by

$I(r,\theta)\;=\;A(\theta)\exp(-(r-R_{\rm ring})^{2}/\sigma^{2})$ (1)

where the ring has semi-major axis $R_{\rm ring}$ and radial width $\sigma$,
and

$A(\theta)=(A_{\rm maj}\,+\,A_{\rm min})/2+(A_{\rm maj}\,-\,A_{\rm
min})/2\,\cos(2\theta).$ (2)

(In practice, $A_{\rm min}=A_{\rm min\\_rel}A_{\rm maj}$, where $A_{\rm
min\\_rel}$ is one of the fitted parameters.)

This function – GaussianRingAz – has been added to the most recent release
(v1.8) of Imfit; the code for it can be found on Imfit’s GitHub site
(https://github.com/perwin/imfit).

##  Appendix D Parameter Correlations from Bootstrap Resampling

The parameter uncertainties for our best-fit models for both galaxies (Tables
1 and 2), based on bootstrap resampling, present a slightly misleading
picture, because they suggest the uncertainties for individual parameters are
independent of each other. In reality, the uncertainties may include
correlations between different parameters.

In Figure 22 we plot parameter distributions from 500 rounds of bootstrap
resampling for our fit to the F160W image of NGC 4643. Since the model has
over 30 free parameters, a full corner plot would need hundreds of individual
panels. We show instead a subset of parameters from the Sérsic and
BrokenExponential components which fit the innermost structures of NGC 4643:
the compact bulge (or NSC) and the nuclear disc surrounding it. We can see
that some parameters are quite strongly correlated: for example, the Sérsic
index $n$ correlates with the Sérsic effective radius and with the inner slope
($h_{1}$) of the BrokenExponential, and anti-correlates with the break radius
of the latter. Although this has no effect on our conclusions in this paper,
we note that future analysis of larger samples of galaxies with similar
structures should be careful to ensure that any apparent correlations between
parameters of different components are not simply side-effects of parameter
degeneracies in the modeling.

Figure 22: Corner plot showing bootstrap resampling analysis for our model of
the F160W image of NGC 4643 (e.g., Figure 10, Table 2), based on 500 rounds of
bootstrap resampling. Panels show best-fit parameter values from fits to the
resampled data from each iteration (grey points) or histograms of parameter
values; vertical lines in the histogram panels show the best-fit parameter
values for the fit to the original data. The full model contains 33 free
parameters for five components; we show a subsample of parameters
corresponding to the Sérsic component for the classical bulge/nuclear star
cluster (“CB/NSC”) and the BrokenExponential component for the nuclear disc
(“ND”).