PASA 2024

# Deepest far ultraviolet view of a central field in the Coma cluster by
AstroSat UVIT

Smriti Mahajan1 Corresponding author; Email<EMAIL_ADDRESS>Kulinder Pal Singh1 Joseph E. Postma2 Kala G. Pradeep1 Koshy George3 Patrick
Côté4 1Department of Physical Sciences, Indian Institute for Science Education
and Research Mohali- IISERM, Knowledge City, Manauli, 140306, Punjab, India  2
Department of Physics and Astronomy, University of Calgary, 2500 University Dr
NW, Calgary, Alberta T2N 1N4, Canada  3 Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679, Munich, Germany  4 Herzberg Astronomy and Astrophysics
Research Centre, National Research Council of Canada, 5071 W. Saanich Road,
Victoria, BC V9E 2E7, Canada

###### Abstract

We present analysis of the far ultraviolet (FUV) emission of sources in the
central region of the Coma cluster ($z=0.023$) using the data taken by the
UVIT aboard the multi-wavelength satellite mission AstroSat. We find a good
correlation between the UVIT FUV flux and the fluxes in both wavebands of the
Galex mission, for the common sources. We detect stars and galaxies, amongst
which the brightest ($r\lesssim 17$ mag) galaxies in the field of view are
mostly members of the Coma cluster. We also detect three quasars
($z=0.38,0.51,2.31$), one of which is likely the farthest object observed by
the UVIT so far. In almost all the optical and UV colour-colour and colour-
magnitude planes explored in this work, the Coma galaxies, other galaxies and
bright stars could be separately identified, but the fainter stars and quasars
often coincide with the faint galaxies. We have also investigated galaxies
with unusual FUV morphology which are likely to be galaxies experiencing ram-
pressure stripping in the cluster. Amongst others, two confirmed cluster
members which were not investigated in the literature earlier, have been found
to show unusual FUV emission. All the distorted sources are likely to have
fallen into the cluster recently, and hence have not virialised yet. A subset
of our data have optical spectroscopic information available from the
archives. For these sources ($\sim 10\%$ of the sample), we find that 17
galaxies identify as star-forming, 18 as composite and 13 as host galaxies for
active galactic nuclei, respectively on the emission-line diagnostic diagram.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

galaxies: evolution; galaxies: fundamental parameters; galaxies: star
formation

##  1 Introduction

The broadband ultraviolet (UV) light (1000–3000 Å) from galaxies offers a
unique window into their evolution. The far UV light (FUV; 1000–2000 Å) in
normal star-forming galaxies is produced by intermediate-mass (2-5
$M_{\odot}$) short-lived stars ($\leq 1$ Gyr), and hence is a good tracer of
the star formation rate (SFR; Kennicutt, 1998; Mahajan et al., 2019). On the
other hand, low-mass helium burning stars which evolve through the horizontal
branch produce FUV emission in the more evolved elliptical galaxies ($>8-10$
Gyr old), resulting in a UV upturn in the spectra of such galaxies at
wavelengths shorter than 2000 Å. In either case, the FUV emission can
therefore be used as an age-dating tool to assess the evolution history for a
variety of galaxies.

The Indian multi-wavelength satellite AstroSat launched in 2015 (Singh et al.,
2014) greatly surpassed its predecessors in observing the UV sky. The
Ultraviolet Imaging Telescope (UVIT; Tandon et al., 2017, 2020) on board the
AstroSat offers the highest spatial resolution of $\lesssim
1.8^{\prime\prime}$ relative to similar missions such as the Swift–UVOT, XMM-
OM and Galaxy evolution explorer mission Galex. With a $28^{\prime}$ diameter
field of view, UVIT is second only to Galex ($1.2\degree$), although the
latter had a spatial resolution of $\sim 5^{\prime\prime}$. The large field of
view together with the unprecedented resolution therefore makes UVIT a great
tool for exploring the recent star formation history of galaxies in galaxy
clusters.

The Coma cluster ($\sim 100h^{-1}$ Mpc, $z=0.023$) is one of the richest and
the most well studied cluster of galaxies in the nearby Universe. Hence it is
not surprising that this cluster has extensive panchromatic coverage from
X-rays to the radio continuum. Prior to the launch of the AstroSat mission,
the deepest UV observations ($\sim 26$ ks) of an off-centre field near the
core of the Coma cluster were obtained by the Galex mission (Hammer et al.,
2010). This region lies $\sim 1\degree$ away from the core, and is linked with
the secondary peak in the X-ray emission from the Coma cluster (Finoguenov et
al., 2003; Neumann et al., 2003), associated with the infalling galaxy group
NGC 4839 and several post-starburst galaxies (Poggianti et al., 2004; Mahajan
et al., 2010).

Amongst others, the balloon-borne imaging FOCA instrument (2000 Å) was used to
conduct a UV survey of the core of the Coma cluster with a spatial resolution
of $\sim 20^{\prime\prime}$ which resulted in the first ever UV luminosity
function for a cluster of galaxies (Donas et al., 1991). Follow up
observations of several other nearby clusters together with the Coma cluster
showed that the UV luminosity functions are also well fitted by a Schechter
function, and are similar to the ones obtained for the general ‘field’ (e.g.
Cortese et al., 2003). These observations indicate that the mechanisms
responsible for quenching star formation of galaxies falling into clusters
influence the giant and dwarf galaxies in a similar fashion, therefore
preserving the shape of the luminosity function, albeit with different
normalization.

In this work we utilise the merits of the UVIT to explore a region $\sim
5.5^{\prime}$ away ($\sim 0.7$ Mpc at $z=0.023$) from the core of the Coma
cluster. The deviation from the core is necessitated by the technical
requirements of the UVIT instruments. In our knowledge, this is the first
statistical study of galaxies based on UVIT data, and hence a benchmark for
the ones to follow from the arsenal of good quality data which is now becoming
available.

In the next section we describe the UV and optical data used in this work,
followed by the properties of the detected UV sources in Sec. 3. In Sec. 4 we
discuss some individual sources with distorted UV morphology, followed by a
discussion of our findings in Sec. 5. We finally conclude with the summary of
our work in Sec. 6. Throughout this work we use concordance $\Lambda$ cold
dark matter cosmological model with $H_{0}=70$ km s-1 Mpc-1,
$\Omega_{\Lambda}=0.7$ and $\Omega_{m}=0.3$ to calculate distances and
magnitudes.

Figure 1: The UVIT field of view with the red circular region centred at
$\alpha=13:00:10.240$ and $\delta=28:01:01.498$, and having a radius of
$14.5^{\prime}$ analysed in this paper. The blue cross denotes the x-ray
centre of the Coma cluster ($\alpha=12:59:31.900,\delta=27:54:10.000$; Rines
et al., 2003). The image is oriented such that North is up and east is on the
left. The giant elliptical galaxy NGC 4889 is also shown for reference.

Figure 2: This sky plot shows the distribution of 1308 sources (blue points)
detected in the UVIT’s FUV BaF2 filter. The 134 sources with redshifts from
the SDSS or NED are shown as red squares, and those also detected by GALEX are
represented by (black crosses), respectively. This figure clearly shows that
some UVIT detected sources are observed for the first time, while many UV
sources lack redshift information.

##  2 data

###  2.1 UV observations and data processing

In this paper we utilise the data from the UVIT on-board the AstroSat (Kumar
et al., 2012). The UVIT comprises two 35-cm Ritchey-Chrétien telescopes with
one of them dedicated to the far ultraviolet (FUV; 1250–1830 Å) and the other
split between the near ultraviolet (NUV; 1900–3040 Å) and the visible (VIS;
3040–5500 Å) channels, respectively. The raw data from the mission are
processed by the Indian Space Science Data Centre (ISSDC) and the Level 1 as
well as Level 2 (pre-processed events and images) from each instrument are
provided to the users.

The data used for the Coma cluster used in this work were taken as part of the
guaranteed time observations in cycle 6 taken over 17 orbits of the AstroSat
(Observation ID: G06-077T01-9000001090; PI: K P Singh). Each orbit included
simultaneous observations by VIS, FUV and X-ray instruments. The NUV channel
could not be used due to the presence of a bright star in the field. The FUV
observations were done in the broad-band BaF2 filter centred at 1541Å (Tandon
et al., 2017, 2020) with a total exposure time of 22669.5 seconds. The final
science image required manual source-tracking techniques to determine the
drift, as detailed below, due to the field having only a single source bright
enough for tracking in the VIS channel, as well as fainter sources which were
interfered with by electronic artefacts in the VIS image data. The VIS data
are not recommended for scientific analysis and are primarily used for
tracking the spacecraft. Fig. 1 shows the position of the UVIT field relative
to the x-ray centre ($\alpha=13:00:10.240$ and $\delta=28:01:01.498$; Rines et
al., 2003), along with the giant elliptical galaxy, NGC 4889.

AstroSat UVIT is a unique telescope because it oscillates its pointing on the
orthogonal UVIT image axis at a rate of a few arcseconds every second. These
oscillations having an amplitude of a few arcminutes protect the detector
components from bright objects. The original image field for the UVIT is then
recovered from the effects of these “drifts" by de-shifting and co-adding the
count centroids of the detected sources, as a function of the pointing
oscillations. For this work we processed the Level 1 FUV data with the CCDLAB
software (Postma & Leahy, 2017), following the procedure described in Postma &
Leahy (2021). The CCDLAB pipeline takes the L1 zip file as input and then
performs various corrections on individual images from each orbit after
extraction. The CCDLAB pipeline automatically corrects for the translational
drift using the VIS data along with other procedures such as subtracting the
flat field, detector corrections, and exposure array corrections, amongst
others.

The corrected images are then co-aligned with manual intervention and stacked
to create a deep field image. Following this, the automatic procedure for
optimising the point source function (PSF) of the sources was initiated in
order to remove the residual drift effects from the image, yielding a final
full width half maximum of $1.2^{\prime\prime}$ for this image. Finally, the
world coordinate system (WCS) solution was obtained by mapping the pixel
coordinates of sources in the image to the GAIA catalogue using the Astroquery
utility (Ginsburg et al., 2019) through CCDLAB, and its implementation of the
automated WCS-solver as detailed in Postma & Leahy (2020). The image was then
de-rotated to match the sky coordinates, thus eliminating the effect of the
telescope’s field rotation, and used for science.

The FUV source catalogue was created using the sextractor software (Bertin &
Arnouts, 1996). Sources were detected on a convolved version of the image
(Gaussian kernel of FWHM $=4.0$ pixels) in order to prevent shredding of
larger galaxies. Other de-blending and detection parameters used by sextractor
were chosen after extensively testing a wide range of values and visually
testing the processed image. The background was detected by sextractor in the
“AUTO" mode. The full set of sextractor parameters used for this image are
provided in Table 1 in order to aid fellow UVIT users in processing similar
imaging data. We detect 1308 sources in the UVIT BaF2 filter (Fig. 1), which
are analysed further in this paper.

Table 1: Sextractor (version v1.2b14) parameters used for the analysis of the UVIT FUV image.  Parameter | Value
---|---
THRESH_TYPE | RELATIVE
DETECT_MINAREA | 20
DETECT_THRESH | 2.0
ANALYSIS_THRESH | 5.0
FILTER | Y
FILTER_NAME | gauss_4.0_7x7.conv
DEBLEND_NTHRESH | 30
DEBLEND_MINCONT | 0.18
CLEAN | Y
CLEAN_PARAM | 1.0
MASK_TYPE | CORRECT
PHOT_APERTURES | 4.0
PHOT_AUTOPARAMS | 0.3, 2.5
PHOT_FLUXFRAC | 0.9
PHOT_PETROPARAMS | 0.5, 2.5
SATUR_LEVEL | 55000
GAIN | 1.0
PIXEL_SCALE | 0
SEEING_FWHM | 1.5
STARNNW_NAME | default.nnw
BACK_TYPE | AUTO
BACK_SIZE | 8
BACK_FILTERSIZE | 5
BACKPHOTO_TYPE | GLOBAL
BACKPHOTO_THICK | 24

The integrated counts for all the 1308 sources identified in this image were
normalized by the total integration time, and then converted to fluxes using
the conversion factors provided by Tandon et al. (2020). The measured flux was
corrected for galactic extinction assuming a Milky way-like extinction curve
(Calzetti et al., 2000). Following Bianchi (2011) we assume
$A_{FUV}=8.06E(B-V)$. The $E(B-V)$ values are obtained for positions of all
the identified sources from https://irsa.ipac.caltech.edu/applications/DUST/.
This resource provided the Schlegel et al. (1998) and Schlafly & Finkbeiner
(2011) estimates for reddening excess, of which we employed the latter
estimate of $E(B-V)$ in this work. We however note that both estimates
correlate very well for the sample studied in this work. The reddening
excesses for these Coma cluster data are in the range $0.0067\leq E(B-V)\leq
0.0112$ with a median at $0.0090$ mag. The distribution of extinction-
corrected fluxes and the corresponding FUV magnitude for all the UVIT sources
are shown in Fig. 3. This figure shows that our data spans almost three orders
of magnitude in FUV flux, and a range of more than five magnitudes for the
sources detected by the UVIT.

Figure 3: The distribution of UVIT FUV flux corrected for the Milkyway
extinction for all the 1308 sources analysed in this work. The top axis shows
the AB magnitudes for the same.

###  2.2 Optical data

Figure 4: The distribution of 50% Petrosian radius as a function of extinction
corrected $r$-band magnitude for the optical counterparts of our UVIT sources.
The sources are separated into stars (grey asteriks), galaxies (red points)
and quasars (black star) by using the SDSS classification. The Coma cluster
galaxies are highlighted as blue triangles. The dot-dashed line represents the
completeness limit for the SDSS photometric data.

The SDSS photometric catalogue is complete to $r=22.7$
mag111https://www.sdss.org/dr14/imaging/other_info/. The optical counterparts
for 968/1308 UVIT sources were found in the Sloan Digital Sky Survey (SDSS,
data release 14) database, of which only 923 ($\sim 70$ per cent) fall within
the SDSS completeness limit. We considered a maximum radius of
$5^{\prime\prime}$ to find an optical counterpart for each UVIT source in our
catalogue. Of these, 96.3 per cent of the optical counterparts were found
within $3^{\prime\prime}$ of the UVIT source position.

Amongst others, the SDSS photometric pipeline provides object type for most of
the matched sources brighter than the magnitude completeness limit. Combining
information from the SDSS and the NASA Extragalactic Database (NED)
222https://ned.ipac.caltech.edu/, we could identify 969/1308 objects: 114
stars, 852 galaxies and 3 quasi-stellar objects (QSO), respectively. However,
only 957 of these sources meet the magnitude completeness limit for SDSS. We
also note that the SDSS object type for some sources was modified based on
visual inspection, spectroscopic information or additional data obtained from
NED. Furthermore, spectroscopic redshifts for 95 sources were found in the
SDSS database, while an additional 39 were obtained from NED. In Table 2 we
show a sample of few rows from the full catalogue of 1308 UVIT sources
detected in this image, the complete version of which is available online.

Table 2: The object ID, sky coordinates, UVIT flux and the uncertainty in the flux, object type (1: QSO; 3: galaxy; 6: star), $r$-band magnitude, plate, MJD and Fiber ID from SDSS, redshift and the source of redshift (0: Not available; 1: SDSS; 2: NED) for all the 1308 sources detected in our image. (A complete version of this table is available online) ID | $\alpha$ | $\delta$ | Flux ($10^{-15}$ |  $\Delta$(Flux) ($10^{-15}$ | Object | $m_{r}$ | Plate | MJD | Fiber ID | $z$ | Source of
---|---|---|---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) | ergs/s/cm2/Å) | ergs/s/cm2/Å) | type | mag |  |  |  |  | redshift ($z$)
1065 | 195.063 | 27.893 | 0.1150 | 0.0046 | 3 | 19.00 | - | - | - | 0.281 | 2
238 | 194.819 | 28.132 | 0.1976 | 0.0062 | 3 | 18.75 | - | - | - | - | 0
790 | 195.159 | 27.989 | 0.1197 | 0.0047 | 3 | 18.36 | - | - | - | 0.168 | 2
72 | 195.015 | 28.219 | 0.1502 | 0.0054 | 3 | 19.11 | - | - | - | - | 0
150 | 195.019 | 28.184 | 0.1720 | 0.0058 | 3 | 18.92 | - | - | - | - | 0
884 | 194.899 | 27.959 | 0.1820 | 0.0059 | 3 | 11.98 | 2240 | 53823 | 585 | 0.024 | 1
307 | 194.844 | 28.129 | 0.1907 | 0.0061 | 3 | 17.90 | 2240 | 53823 | 576 | 0.194 | 1
1195 | 195.091 | 27.845 | 0.1249 | 0.0048 | 3 | 18.91 | - | - | - | - | 0
49 | 195.106 | 28.229 | 0.2224 | 0.0064 | 3 | 18.23 | - | - | - | - | 0
1 | 195.026 | 28.253 | 1.6038 | 0.0179 | 3 | 17.18 | 2241 | 54169 | 407 | 0.021 | 1

In Fig. 4, the Petrosian radius containing 50 per cent of the total flux in
the $r$-band ($R_{50}$) is shown as a function of the extinction corrected
$r$-band magnitude for all the optical counterparts, sub-classified into the
Coma cluster galaxies, other galaxies, quasars and stars, respectively. The
Coma cluster members have been identified assuming $z_{Coma}=0.023$ and
velocity dispersion ($\sigma_{v}$) of 1000 km s-1. These criteria yield 79
member galaxies within $\pm 3\sigma_{v}$ as members of the Coma cluster. All
the other extended sources, with or without redshift, are addressed as ‘other
galaxies’ henceforth, although it is possible that some of the fainter
galaxies in this category might be members of the Coma cluster.

Fig. 4 shows that most of the bright galaxies found in these UVIT data are
members of the Coma cluster. These also happen to be the largest galaxies in
our sample. Most of the stars brighter than $r\sim 20$ mag can be separated
out from the galaxies in this plane. While one of the quasars coincides with
stars, the second follows the galaxies, and the third one lies in the region
occupied by the faint stars in this $r$-band magnitude-radius plane.

It is also worth mentioning here that the three QSOs have $z=0.383,0.513$ and
$2.315$, respectively. While the last one has spectroscopic redshift available
in the SDSS database, redshift for the other two were obtained from NED. The
quasar B1257$+$280 ($z=2.315$) shown in Fig. 5 deserves a special mention,
since this might be the highest redshift object observed by the UVIT to date.
The multi-band optical image shows a blue star-like object, while the faint
FUV emission at the edges show elongation in flux contours in the north-south
direction. This QSO is known to be a Lyman $\alpha$ absorber (Garnett et al.,
2017).

Figure 5: The UVIT FUV (left) and FUV contours overlaid on the optical
$r$-band image (right) of the quasar B1257$+$280 ($z=2.315$).

##  3 Properties of the detected FUV sources

Figure 6: The distribution of magnitude of GALEX NUV sources and (b) GALEX FUV
sources plotted as a function of the FUV magnitude of the corresponding UVIT
source, respectively. All magnitudes are corrected for extinction due to
Milkyway. The red points denote galaxies, grey asterisks represent stars and
QSOs are shown as stars, respectively. The dot-dashed line in both panels
represents a perfect correlation, while the dashed lines represent the least
square fit to the data along with $1\sigma$ deviation in it. Typical
uncertainty along each axis is shown in the bottom right corner of the panels.
This figure therefore shows that the FUV fluxes of the matched sources are
consistent between the UVIT BaF2 band and both the wavebands of the GALEX
mission. A marginal discrepancy of increasing magnitude in GALEX FUV, can be
noticed towards the faint end, which is however well within $\sim 3\sigma$ for
most of the sources.

The Coma field studied here has previously been observed with the Galex
mission. So as a sanity check we compared our catalogue to the one created
from Galex images. In order to do so, we made use of the processed images
downloaded from the galexview333https://galex.stsci.edu/GalexView/ page. The
comparison images have an exposure time of 25656.55 seconds in $NUV$ and
18620.45 seconds in the $FUV$ band, respectively (Object ID:
GI5_025001_COMA_0002). sextractor was used to identify the sources in the
Galex images. We then searched for a Galex sources within $3^{\prime\prime}$
of each of the UVIT sources independently in the near and far UV catalogues.
This resulted in 464 FUV and 599 NUV sources matched between the UVIT and
Galex catalogues.

The distributions of magnitudes of all the matched sources in the two GALEX
wavebands are shown in Fig. 6. Both stars and galaxies as identified in the
previous section have well correlated FUV fluxes in the UVIT and the GALEX
bands, albeit with the expected scatter which is a consequence of mismatched
wavelength range covered by the three wavebands analysed here. We note that in
the $FUV_{UVIT}$–$NUV_{Galex}$ plane, bright ‘stars’ are well separated from
the galaxies. The best-fit least square relation between the UVIT and GALEX
bands for these data as shown in Fig. 6 are:

$\begin{split}Galex_{NUV}=0.3808+0.9514UVIT_{FUV}~{}~{}\pm 0.8995,\\\
Galex_{FUV}=-1.3338+1.0672UVIT_{FUV}~{}~{}\pm 0.3851\end{split}$ (1)

In Fig. 6 we show that despite good general agreement between the UVIT $FUV$
and GALEX bands, there are subtle discrepancies, particularly at the faint
end. We note that the scatter is higher at the faint end in both the
distributions. In fact, the standard deviation in the
$FUV_{UVIT}$–$NUV_{Galex}$ correlation seems to be dictated by the faint end
scatter. The correlation between $FUV_{UVIT}$ and $FUV_{Galex}$ is relatively
better, but we observe a slight upturn in the distribution for galaxies
fainter than $FUV_{UVIT}\sim 22.6$ mag. The trend implies that these faint
galaxies detected by both UVIT and Galex, are fainter in GALEX relative to
UVIT. This might be a consequence of the poor resolution of GALEX relative to
UVIT, but further investigation may be required if similar trends are observed
in other UVIT fields as well. It should be noted, however, that the typical
uncertainty in the GALEX  bands is $\sim 0.44$ and $0.57$ magnitude in the
$FUV$ and $NUV$ bands, respectively, which is an order of magnitude higher
than the uncertainty of $\sim 0.03$ magnitude estimated for the UVIT BaF2
band.

Figure 7: The colour-colour and colour–magnitude distributions for all UVIT
sources along with their optical and GALEX counterparts, respectively in the
(a) $NUV-r$ vs $r$-band magnitude, (b) $NUV-r$ vs $FUV-r$, (c) $FUV-r$ vs
$g-r$, and (d) $g-r$ vs $r$-band magnitude plane. The symbols are same as in
Fig. 4. The $FUV$ magnitudes are from UVIT, while the NUV magnitudes are from
the GALEX mission. Given the heterogeneous nature of the data plotted here,
the number of data points plotted in each quadrant vary as mentioned in Sec.
2. The Coma cluster galaxies are distributed coherently in the brighter,
redder region of the respective planes, while other galaxies are relatively
fainter and bluer in comparison, with more scatter. It is notable that the
other galaxies and stars are not easily distinguishable in all of these
planes, particularly at the faint end with one exception. A handful of stars
redder than $NUV-r\sim 2$ mag and $FUV-r\sim 5$ mag, are somewhat separated
from galaxies (panels (a), (b) and (c)).

In Fig. 7 we show the distribution of the UVIT sources in various optical, UV
and composite colour-magnitude and colour-colour planes. Note that information
in all wavebands is not available for all the sources, hence different number
of sources are plotted in each of the four panels. The Coma cluster galaxies
form a well defined sequence in the optical g-r vs $r$-band colour-magnitude
plane (Fig. 7 (d)), but the other fainter galaxies show a larger scatter. On
the other hand, all galaxies follow a continuous sequence in the plane mapped
by the NUV-r colour and $r$-band magnitude (Fig. 7 (a)). It is interesting to
note that most of the stars brighter than $r\sim 19$ mag span similar range of
r-band magnitudes as the bright Coma cluster galaxies, yet have relatively
bluer g-r, NUV-r and FUV-r colours.

The Coma cluster galaxies also follow a well defined sequence in the g-r vs
FUV-r colour-colour plane (Fig. 7 (c)), and the NUV-r–FUV-r plane (Fig. 7
(b)), respectively. All other galaxies, although fainter than their Coma
cluster counterparts, seem to follow the same correlation in the NUV-r vs
FUV-r plane. Just like in panels (a) and (d), in the plane mapped by the NUV-r
and FUV-r colours, a small subset of bright stars seems to follow a
correlation just like the galaxies but have FUV-r colours redder by $\sim 2$
magnitude at fixed NUV-r colour, relative to the galaxies. As mentioned
earlier, many of the objects at the faint end classified as stars by the SDSS
photometric pipeline may be high-z quasars. Also, many of the ‘other’ galaxies
falling in the same region of the colour-colour and colour-magnitude planes as
the Coma cluster galaxies may also be member galaxies of the Coma cluster.

Figure 8: The distribution of all the UVIT sources having optical and GALEX
counterparts in the (a) FUV-NUV vs $r$-band and (b) FUV-NUV vs g-r colour-
colour plane, respectively. The symbols are same as in Fig. 4. Most of the
Coma cluster galaxies fall in well defined regions of the space dictated by
their optical properties, while stars and quasars co-exist with the other,
relatively fainter galaxies.

In Fig. 8 we show the distribution of various sources in the plane mapped by
the UV colour and the $r$-band magnitude and the g-r colour, respectively. The
difference in the $r$-band luminosity of the Coma cluster galaxies and other
galaxies leads to a clear separation between the two sub-samples of galaxies
in the $r$ vs $FUV-NUV$ colour-magnitude plane. On the other hand, the Coma
cluster galaxies form a tight sequence in the $g-r$ vs $FUV-NUV$ colour-colour
plane (Fig. 8 (b)), an expected signature from the optical colour-magnitude
diagram of clusters. We note however, that stars and quasars remain
indistinguishable from galaxies in these planes.

To summarise, this analysis shows that (i) the FUV flux from the UVIT BaF2
filter data are well correlated with the GALEX bands, (ii) Coma cluster
galaxies are much brighter, and hence well separated from the other galaxies
in the image in almost all the colour-colour and colour-magnitude planes
explored here, and (iii) stars and quasars have identical photometric
properties as the other, relatively fainter galaxies.

##  4 FUV sources with unusual morphology

We found many galaxies with unconventional morphology in the UVIT $FUV$ image
by visualizing the UVIT image simultaneously with the optical $r$-band image.
Objects showing tails, rings or other features suggesting the impact of
environment-related mechanism, and galaxies which showed morphology very
different from their optical counterparts, were shortlisted and analysed
further. The galaxies for which subsequent investigation showed that the
‘features’ were background objects, were then dropped, finally leaving a set
of sixteen bright and seven faint galaxies with distorted morphology. Some of
these galaxies have been subjects of other optical and ultraviolet studies in
the literature (e.g. Yagi et al., 2010; Smith et al., 2010), with most of them
showing an ‘unusual’ morphology in the UV bands only. We describe each of them
briefly below, incorporating relevant information from the literature where
available. All the objects are listed in Table 3, and their UVIT FUV and
optical $r$-band images are shown in Figs. 9-15. The GMP identification in the
first column of Table 3 is from Godwin et al. (1983). Redshift information is
available only for the first sixteen bright galaxies with unusual FUV
morphology in the Table. Where available, Table 3 also includes the stellar
masses of galaxies from Mendel et al. (2014) or the GSWLC, and HI-mass
fraction estimated by applying artificial neural network on SDSS optical data
(Teimoorinia et al., 2017).

Table 3: The common name, sky coordinates and redshift for the objects with distorted morphology found in the FUV image. Where available, the last two columns show the stellar mass and the HI-to-stellar mass ratio of the objects. The last seven objects are optically-faint galaxies without any redshift information. Common | Right Ascension | Declination | Redshift | log $M*$ | log $M_{HI}/M*$
---|---|---|---|---|---
name | (J2000) | (J2000) |  | ($M_{\odot}$) |
GMP 3016 | 195.005 | 28.082 | 0.0259 | - | -
PGC 044679 | 194.979 | 28.128 | 0.0252 | 10.253 | 0.0878
GMP 2787 | 195.076 | 28.059 | 0.0332 | 9.390 | -0.2226
GMP 3523 | 194.843 | 28.129 | 0.1943 | - | -
GMP 2910 | 195.038 | 27.866 | 0.0177 | 9.568 | 0.1665
NGC 4867 | 194.813 | 27.971 | 0.0161 | 10.17 | -0.6905
NGC 4895 | 195.075 | 28.202 | 0.0283 | - | -
GMP 2584 | 195.148 | 28.146 | 0.0182 | 9.972 | -0.3045
GMP 2559 | 195.158 | 28.058 | 0.0255 | - | -
GMP 2347 | 195.247 | 27.899 | 0.0230 | - | -
NGC 4907 | 195.204 | 28.159 | 0.0194 | 10.766 | -
GMP 2943 | 195.026 | 28.253 | 0.0211 | 8.530 | -
GMP 2989 | 195.012 | 28.241 | 0.0257 | 9.900 | -0.0648
GMP 3317 | 194.901 | 28.042 | 0.2747 | - | -
GMP 3170 | 194.945 | 27.974 | 0.0314 | 10.674 | -0.9705
GMP 2956 | 195.023 | 27.807 | 0.0219 | 10.255 | -0.3113
GMP 2717 | 195.095 | 27.821 | - | - | -
Subaru-UDG | 195.064 | 27.836 | - | - | -
- | 194.953 | 27.797 | - | - | -
GMP 2476 | 195.190 | 27.918 | - | - | -
GMP 2454 | 195.199 | 27.994 | - | - | -
GMP 2320 | 195.259 | 28.102 | - | - | -
- | 195.205 | 28.064 | - | - | -

###  4.1 GMP 3016

This is an extremely blue, faint irregular galaxy as observed in the optical
wavebands. Yagi et al. (2010) have reported ‘fireball’ features i.e., extended
H$\alpha$ emission following knots of blue stars (Yoshida et al., 2008) from
GMP 3016 (see the right panel of Fig. 9(a)). On the other hand, Smith et al.
(2010) found this to be a ‘less convincing case’ of a stripping galaxy in the
combined NUV$+$FUV image obtained with Galex. Our UVIT image (Fig.9(a))
however, shows a linear structure in the north-eastern part of the FUV image
clearly extending towards GMP 3016. In FUV, this trail is detached from the
galaxy and extends upto $\sim 26$ kpc in the north-east direction444All
galaxies shown in Figs. 9, 10 and 11 are oriented such that north is up and
east is left of the image., with the cluster centre $<200h^{-1}$ kpc away
parallel to the direction of the alignment of knots. In our opinion, the
morphology of the narrow structure extending in the north-east suggests that
it is likely a part of GMP 3016, but without any optical counterpart.

###  4.2 PGC 044679

This is a barred S0 galaxy member of the Coma cluster (Lansbury et al., 2014).
The FUV image (Fig. 9(b)) shows two discrete components, one centred at the
galaxy’s nucleus and the other in the north-east. The nuclear component (C1 in
Fig. 9(b)) has an interesting morphology pointing in orthogonal directions,
one of which points in the direction of the cluster centre which is at a
distance of $\sim 256h^{-1}$ kpc from the galaxy. The second component C2,
however, remains to be further investigated to confirm whether it also
corresponds to PGC 044679, is a satellite galaxy or a random background
object. Unlike GMP 3016, this is therefore one of our less convincing cases of
multiple component UV emission. Table 3 shows that this galaxy is detected in
HI (Table 3).

###  4.3 GMP 2787

This is an irregular galaxy ($z=0.033;9974$ km s-1), having radial velocity
slightly above the maximum threshold considered here for selecting the members
of the Coma cluster. It has a stellar mass of $9.39M_{\odot}$ (Teimoorinia et
al., 2017). This galaxy appears to be very faint in the Galex images. The UVIT
image (Fig. 9(c)) of this galaxy shows an irregular morphology, very different
from the optical extent of the galaxy, and pointing in a direction away from
the cluster centre which is $\sim 236h^{-1}$ kpc away.

###  4.4 GMP 3523

This is a star-forming galaxy at $z=0.194$ with tail like features observed on
the top and south-western edge of the galaxy (Fig. 10(a)). All the tail like
features are invisible in the optical image, and may be tidal tail(s) of this
galaxy, although this needs to be confirmed in further investigation. The
south-western extension was also seen in the Galex image (e.g. Donas et al.,
1995), but appeared as a fainter feature relative to the UVIT image.

###  4.5 GMP 2910

This is the most well studied of our sources with unusual FUV morphology.
Several authors have reported extended emission associated with this post-
starburst galaxy commonly known as D100 ($z=0.0177$; Yagi et al., 2007; Smith
et al., 2010; Yagi et al., 2010; Cramer et al., 2019). This source was also
reported as KUG $1257+281$ in the Kiso Survey for UV-excess galaxies (Takase &
Miyauchi-Isobe, 1993). The FUV emission observed in the UVIT image (Fig.
10(b)) coincides well with the extended H$\alpha$ emission (Yagi et al.,
2007).

GMP 2910 is rich in molecular (Jáchym et al., 2017) and atomic gas
(Teimoorinia et al., 2017), with a considerable HI-to-stellar mass fraction
($>0.16$; Table 3). The extended FUV emission points in the northeast
direction almost perpendicular to the vector pointing in the direction of the
cluster centre. The spectacular narrow tail observed with the Subaru
telescope’s Suprime-Cam in the H$\alpha$ narrow band filter (see Fig. 2 of
Cramer et al., 2019) is well aligned with the FUV feature observed with the
UVIT.

This galaxy is only $\sim 240h^{-1}$ kpc away from the cluster centre, and
hence likely a classic case of ram-pressure stripping. However, the narrow,
linear structure emerging from the galaxy is difficult to explain via ram
pressure stripping alone. The other possibility is that the structure is
formed by a dwarf galaxy or gas cloud which was disrupted due to ram-pressure
stripping by the ICM, or tidal forces of GMP 2910 (Yagi et al., 2007). In a
recent study, Peluso et al. (2022) explored the connection between active
galactic nucleus (AGN) and ram pressure stripping, in a sample of 115 ram-
pressure stripped galaxies (also see, George et al., 2019a). Peluso et al.
(2022) find that the incidence of AGN increases among ram-pressure stripped
galaxies, relative to a stellar mass matched sample of cluster galaxies,
thereby suggesting that GMP 2910, showing a Seyfert type 2 emission based on
the emission-line diagnostic diagram (Baldwin et al., 1981), may also harbour
an AGN at its centre (see table 4 of Peluso et al., 2022, also, Mahajan et
al., 2010).

###  4.6 NGC 4867

This is a typical S0 galaxy in the Coma cluster ($z=0.016$), around 1.6 kpc
($5{{}^{\prime\prime}}$) from the most dominant galaxy at the centre. NGC 4867
(seen as component C1 in Fig. 10) shows FUV emission extending towards west
and southeast, but invisible in the optical image. A weak feature also appears
to extend towards another FUV source C2 towards the left, which has no optical
counterpart. The western tail from C1 seem to be extending towards NGC 4864
($z=0.022$; component C3), which is also a Coma cluster member, separated in
redshift space by 1800 km s-1 from NGC 4867. Hence, it is possible that the
two galaxies are gravitationally influencing each other. It is notable that
the cluster centre lies at a distance of $\sim 200h^{-1}$ kpc eastward of this
galaxy. The third galaxy (component C4) in the image panel is a background
object at $z=0.283$, but shows an interesting UV morphology.

###  4.7 NGC 4895

NGC 4895 ($z=0.0283$) comprises multiple components in the FUV (Fig. 11(a)).
While the dominant component (C1) in centred at the galaxy’s nucleus, the
northwestern component C3 is a blue foreground star observed in the multi-band
SDSS image. The relatively smaller, northeastern component C2 however, does
not have an optical counterpart. The two extraplanar concentrations in
southwest and southeast (C4 and C5, respectively) are likely background
objects, with no optical counterparts available in the SDSS database. Hence,
it will be interesting to explore the latter three in follow-up observations.
We also note that the major axis of this galaxy is orthogonal to the direction
of the cluster centre. Nisbet & Best (2016) found this massive galaxy
(log$M*/M_{\odot}=11.21$) to be hosting a low-ionization nuclear emission-line
region (LINER) type AGN.

###  4.8 GMP 2584

This edge-on S0 galaxy in the Coma cluster, also known as PGC 044784, shows a
central FUV emission reported here for the first time (Fig. 11(b)). The FUV
emission is tilted in the direction of the cluster centre towards northwest
relative to the optical disk of the galaxy.

###  4.9 GMP 2559

Also known as PGC 044789 and IC4040, this is a well studied blue, Sd type
spiral galaxy in the Coma cluster. It is known to be an HI-deficient galaxy,
having asymmetric HI distribution, such that most of the HI gas lies in the
south-eastern part of the galaxy, while the north-western part appears
depleted (Bravo-Alfaro et al., 2000). The major axis of this galaxy is almost
orthogonal to the cluster centre. GMP 2559 exhibits an HI tail extending
towards south-east (Miller et al., 2009; Chen et al., 2020), almost coincident
with the extra-planar FUV emission as shown in Fig. 11(c). It is also
noteworthy that the three “knots" of star-forming region seen extending
towards south-west in the UVIT image are also observed in the H$\alpha$
emission (see fig. 2 of Chen et al., 2020), but not in the radio continuum.

###  4.10 GMP 2347

This early-type spiral galaxy in the Coma cluster shows a central FUV
component (C1), along with an off-centre component C2. The latter might be a
dwarf galaxy merging with GMP 2347. This galaxy is $\sim 0.45$ Mpc away from
the cluster centre. Another faint galaxy GMP 2333 (component C3) can also be
seen in the north-east direction relative to the galaxy in Fig. 12(a). In the
absence of redshift information, however, it is not possible to test if this
faint blue galaxy is interacting with GMP 2347 or is a member of the Coma
cluster.

###  4.11 NGC 4907

This face-on barred spiral galaxy in the Coma cluster, also known as GMP 2441,
shows distinct FUV emission, unlike the symmetric spiral arms observed in the
optical waveband. This is a LINER galaxy (Mahajan et al., 2010; Toba et al.,
2014). Fig. 12(b) shows that unlike the optical, the UVIT FUV image shows
clear asymmetry between the northeast and southwestern regions of the galaxy.
It is possible that gas is being pushed to the northeastern face as galaxy
moves towards the cluster centre $\sim 0.44$ Mpc away. Another interesting
feature is the C-shaped hole seen in the FUV emission surrounding the bar at
the centre. This cavity may be a consequence of the bar suppressing star
formation in the centre just like in Messier 95 (George et al., 2019b).

###  4.12 GMP 2943 and 2989

GMP 2943 (C1 in Fig. 12(c)) is a distorted spiral galaxy in the Coma cluster,
whose optically-red nuclear region is observed below the blue spiral arms
which take on a spider-like morphology in the optical image as shown in Fig.
14. However, in the FUV image this galaxy appears to be a face-on spiral, much
unlike it’s optical facade.

GMP 2989, which is also a Coma cluster member, is also seen alongside GMP 2943
in the image (Fig. 12(c)) as C2. This spiral galaxy has it’s major axis
pointed along the vector pointing in the direction of the cluster centre. GMP
2943 and 2989 are $\sim 1380$ km s-1 away in redshift, hence unlikely to be
interacting with each other, but within $\sim 0.45$ Mpc from the cluster
centre. GMP 2989 is also the only bonafide AGN host galaxy (see Fig. 19)
amongst the galaxies with unusual FUV morphology discussed in this section.

###  4.13 GMP 3317

GMP 3317 appears as a small, red, faint spheroid in the optical image, very
close to a bright star. Not much is known about this distant galaxy
($z=0.2748$). However, the FUV image inevitably shows that this is likely a
much larger galaxy, only whose nucleus is observed in the optical images.
Interestingly though, there appears to be a hole in the FUV emission
surrounding the nuclear region as seen in Fig. 13(a).

###  4.14 GMP 3170

This Coma cluster galaxy is only $\sim 0.04$ Mpc from the cluster centre, and
hence very likely experiencing the hostile environmental mechanisms. Besides
the central component C1 in Fig. 13(b), one can clearly see a tail of FUV
blobs (components C2, C3 and C4) in the direction perpendicular to the vector
pointing towards the cluster centre. Although follow-up spectroscopic
observations are required to confirm our speculation that the blobs are
connected to GMP 3170.

###  4.15 GMP 2956

This S0 Coma cluster galaxy lies at a distance of $\sim 0.31$ Mpc from the
cluster centre. The FUV emission appears to be divided into at least two
components: C1 is centered at the core of the galaxy, while the component C2
does not show any obvious optical counterpart. C2 also appears to be tilting
in the north-east direction away from the direction of the cluster centre.

Figure 9: This figure shows (left): the UVIT FUV image, and (right): the
$r$-band optical image with FUV contours overlaid for (a) GMP 3016, (b) PGC
044679, and (c) GMP 2787, respectively from top to bottom. The contours are
smoothed over 4 pixels and created such that the outermost contour is
2-3$\sigma$ above the typical background of the UVIT image. The arrow points
towards the cluster centre. Images are oriented such that north is up and east
is on the left of the image. These images evidently show the difference in
morphology of the young, UV emitting stars and the generic distribution of
other stellar populations in these galaxies. The impact of the cluster
environment can also be noticed for galaxies which show UV emission away from
the plane of the galaxy, often with no optical counterpart.

Figure 10: Same as Fig. 9, but for (a) GMP 3523, (b) GMP 2910, and (c) NGC
4867.

Figure 11: Same as Fig. 9, but for (a) NGC 4895 , (b) GMP 2584, and (c) GMP
2559.

Figure 12: Same as Fig. 9, but for (a) GMP 2347 , (b) NGC 4907, and (c) GMP
2943 and GMP 2989.

Figure 13: Same as Fig. 9, but for (a) GMP 3317, (b) GMP 3170, and (c) GMP
2956.

Figure 14: This figure shows the SDSS gri image of the galaxy GMP 2943 (left),
and the same with UVIT FUV contours overlaid (right). Also, see Fig. 12(c).

###  4.16 Optically-faint candidates

Besides the above mentioned galaxies, we also detected a few optically faint
galaxies for which redshift information is not available, but their FUV
morphology is markedly different from the optical image. These galaxies are
listed at the bottom of Table 3, shown in Fig. 15, and are briefly described
below.

  * •

GMP 2717: This faint irregular galaxy shows an extended emission pointing
opposite to the cluster centre on the southwestern side of the image.

  * •

Subaru-UDG: This ultra-diffuse galaxy (UDG) was first reported by Yagi et al.
(2016) in their Subaru imaging survey of UDGs in the Coma cluster region. The
FUV extent of the galaxy is evidently larger than it’s optical size.

  * •

Galaxy 1: This is also an irregular low surface brightness galaxy, which is
much larger in FUV relative to the optical extent.

  * •

GMP 2476: This is component C1 in Fig. 15 (d), and like other faint dwarf
galaxies, appears to be much smaller in the optical image. Also seen in the
field GMP 2457 which shows nuclear FUV emission (component C2) and an off-
centre component C3. The entire system is aligned normal to the vector
pointing in the direction of the cluster centre.

  * •

GMP 2454: This faint dwarf galaxy appears as a small, red spheroid in the
multi-band SDSS data, but shows a distinct double-object morphology in the FUV
image.

  * •

GMP 2320: Only the nuclear region of this dwarf galaxy seems to be observable
in the optical image, while the larger extent is apparent in the FUV data. The
point source in the southeastern corner may be a star unrelated to the galaxy.

  * •

Galaxy 2: This dwarf galaxy shows a very interesting morphology in the FUV
image, but interestingly appears as two separate faint spheroidal objects in
the optical data. A distinct tail of FUV emitting blobs can also be observed
extending towards southwest of the field.

In summary, at least 13 of the 23 galaxies discussed in this section are Coma
cluster members likely to be experiencing ram-pressure stripping. Although
gravitational influence of the neighbouring galaxies may also have contributed
to their unusual FUV morphology, it is worth noting that all of them are
$\lesssim 0.5h^{-1}$ Mpc of the centre of the Coma cluster. The seven low
surface brightness dwarf galaxies will be explored further in future work.

Figure 15: Same as Fig. 9, but for (from top to bottom:) (a) GMP 2717, (b)
Subaru ultra-diffuse galaxy, (c) Galaxy1, (d) GMP 2476, (e) GMP 2454, (f) GMP
2320 and (g) Galaxy2.

##  5 Discussion

The aim of this paper is to analyse the sources detected in our deep FUV UVIT
data for the central region of the Coma cluster. In order to do so, we have
employed the NUV and FUV data from the GALEX mission, and optical photometric
and spectroscopic data from the SDSS as well. We have presented various
statistical properties of the sources detected in our data, classifying them
into Coma cluster members, other galaxies, stars and quasars, respectively,
where possible. In the following, we compare our data and findings with the
existing literature, and further discuss the properties of the galaxies having
unusual morphology in the FUV band.

###  5.1 Phase space distribution in Coma cluster

Figure 16: This figure shows the distribution of all galaxies (grey asterisks)
and those with distorted UV morphology (blue points) in the phase space around
the Coma cluster centre. It is evident that most of the UV galaxies lie away
from the cluster core in spatial as well as velocity space.

The phase space diagram mapped by the distance of a galaxy to the cluster
centre normalized by $r_{200}$555$r_{200}$ is the radius at which the mean
interior over-density in a sphere of radius $r$ is 200 times the critical
density of the Universe. and the line-of-sight velocity of the galaxy scaled
by the velocity dispersion666Although there are multiple values of
$\sigma_{v}$ present in the literature, for simplicity we assume
$\sigma_{v}=1000$ km s-1 here. is shown in Fig. 16. This figure shows that
most of the Coma cluster galaxies having distorted UV morphologies are far
away from the cluster centre, in spatial as well as velocity space. This
suggests that these galaxies have recently entered the cluster, and are
experiencing the impact of the intra-cluster medium (ICM) for the first time.

###  5.2 Spectroscopic properties of UVIT sources

Figure 17: This figure shows the distribution of (a) log SFR, (b) sSFR, (c)
log $M^{*}$ and (d) $A_{FUV}$ for the galaxies matched to our UVIT sources in
the GSWLC database. The solid black lines are for all the sources, while the
dotted blue line represents the Coma cluster members, respectively.

We found 88 galaxies from our UVIT sample in the GALEX-SDSS-WISE Legacy
catalogue (GSWLC henceforth; Salim et al., 2016), 79 of which are members of
the Coma cluster. The GSWLC provides estimates of physical properties for
$\sim 700,000$ galaxies in the Galex footprint, also observed by the SDSS. The
physical properties of galaxies are obtained by fitting the UV and optical
spectral energy distribution following a Bayesian methodology. In Fig.17 we
show the distributions of star formation rate (SFR), stellar mass ($M^{*}$),
SFR/$M^{*}$ and extinction in the FUV band ($A_{FUV}$), for all the matched
galaxies and the Coma cluster members, respectively.

Figure 18: This figure shows the distribution of all UVIT detected galaxies
found in the GSWLC catalogue (black points), and Coma cluster galaxies with
unusual morphology (blue asterisks) in the stellar mass–SFR space. The dot-
dashed line represents the main sequence of star-forming field galaxies
(Roberts & Parker, 2020). While three distorted galaxies lie $\lesssim 0.5$
dex away from the main sequence, in the region where ram-pressure stripping
candidates are expected to lie (see fig. 7 of Roberts & Parker, 2020),
majority of the galaxies with unusual morphology populate the region of this
space expected to be inhabited by the red sequence and normal star-forming
galaxies of the Coma cluster.

In Fig. 18 we show the SFR of the matched GSWLC galaxies as a function of
their stellar mass. It is well known that the SFR of galaxies correlate well
with their stellar mass, such that more massive galaxies have higher SFR
relative to their lower mass counterparts. In a recent work, Roberts & Parker
(2020) have used a bayesian-based regression approach on the SDSS data release
12 data to find the main-sequence relationship for star-forming field galaxies
as $\rm{log~{}SFR}=0.55\times{\rm log}~{}M^{*}-5.7$. This relation777The
stellar masses and SFR for their work are in turn taken from the GSWLC-2 SED
fitting catalogue (Salim et al., 2016, 2018). is shown for reference as a
dotted line in Fig. 18, along with the UVIT galaxies, and galaxies with
unusual FUV morphology found in the GSWLC catalogue. Most of our UVIT detected
galaxies are found in the region occupied by the passive galaxies (log
sSFR$<-11$ yr-1; where, sSFR $\equiv$ SFR/$M^{*}$) in the SFR-$M^{*}$ space,
evidently showing that star formation activity is suppressed in the cluster
environment, even for UV-bright galaxies. We also note that nine of the twelve
galaxies with unusual FUV morphology are found below the field main sequence
relation, in the region occupied by passive galaxies in the sample of Roberts
& Parker (2020). These authors however, found their ram-pressure stripping
candidates to lie above the main sequence, where only three of our distorted
galaxies make an appearance (Roberts & Parker, 2020, see their fig. 7). This
analysis suggests that beside ram-pressure stripping galaxies, unusual FUV
morphology may be found among transitional galaxy populations. We also note
that the low-surface brightness dwarf galaxies do not make an appearance on
this plot due to lack of spectroscopic data.

Recently, Chen et al. (2020) presented high spatial resolution HI and deep
1.4GHz continuum data for 20 galaxies which are being ram-pressure stripped
(RPS) in the Coma cluster. Three of these galaxies (GMP 2910, GMP 2559 and GMP
3016) are also part of our UVIT data. While GMP 2910 and GMP 2559 exhibit RPS
tails matching their H$\alpha$ observations (Smith et al., 2010; Yagi et al.,
2010), GMP 3016 does not show RPS tails in the radio continuum.

Figure 19: This figure shows the distribution of 48 UVIT detected galaxies
found in the GSWLC catalogue (grey asterisks), and those with distorted
morphology (blue points) in the BPT plane. The solid and dot-dashed lines mark
the limiting criteria to distinguish between star-forming and AGN galaxies,
respectively. The galaxies inhabiting the region between these two lines are
known as composites, i.e. part of their emission comes from an AGN-like
component. We note that 18 and 17 galaxies respectively classify as star-
forming and composites, while the remaining 13 as AGN. While amongst the
galaxies with distorted UV morphology, five are star-forming, two are
composite and one is an AGN.

There are 48 galaxies matched with the GSWLC, and having good signal-to-noise
in all four emission lines (H$\alpha$, [NII], [OIII] and H$\beta$) in the SDSS
databse, which allows us to distinguish the active galactic nuclei (AGN) from
the star-forming galaxies using the Baldwin et al. (1981) criteria. The
Baldwin, Phillip, Terlevich, popularly known as the BPT diagram is shown in
Fig. 19, with the galaxies with distorted UV morphology highlighted where
possible. This space can be further divided into the regions occupied by star-
forming, AGN and composites (i.e. galaxies with a weak AGN-like emission
component) using the well used criteria put forward by Kauffmann et al. (2003)
and Kewley et al. (2001). As shown in Fig. 19, we find that 18 UVIT sources
with optical spectra are star-forming galaxies, 17 are composites and the
remaining 13 are AGN host galaxies. Amongst the galaxies with unusual UV
morphology, five are star-forming galaxies, two are found in the region
occupied by the composite galaxies, and one is an AGN. Since star-forming
galaxies are in general comparable in number to the composite galaxies, it
would be interesting to explore the FUV emission in AGN host galaxies for a
similar, larger sample in future.

We conclude this discussion by summarising some of the limitations of the data
analysed in this paper. Our UVIT data are limited to the FUV band only. The
NUV band was not operated due to the presence of bright point sources in the
field. Instead, we have used the NUV data from GALEX. The large difference in
the resolution and the sensitivity of the two instruments, however, made a
comparison of the FUV and NUV emission from the galaxies difficult. We wish to
point out that since this is the first time that this field has been observed
and analysed to such a depth in the FUV waveband, many interesting sources
detected in the UVIT field lack redshift information, and require follow-up
optical observations to throw more light on their origin and properties.

##  6 Summary

In this paper we have examined a deep FUV image of the central region of the
Coma cluster, observed by the UVIT onboard the Indian multi-wavelength
satellite mission AstroSat. For the sources detected by both missions, we find
a good correlation between the flux derived from the UVIT BaF2 filter and the
two GALEX wavebands. The UVIT fluxes of the detected sources (galaxies, stars
and quasars) span two orders of magnitude in flux. We find that most of the
galaxies brighter than $r\sim 17$ mag are members of the Coma cluster, and
occupy the bright end of the distributions in all colour-magnitude and, red
end in the colour-colour space, respectively. Amongst others, we have also
detected three quasars, one of which at $z=2.315$ is likely the farthest
object observed by the UVIT so far.

Furthermore, we investigated sources with unusual FUV morphology, twenty three
of which are discussed in detail here. Amongst others, we have identified
several new candidate galaxies having unusual $FUV$ morphology, many of which
could be members of the Coma cluster. New, spectroscopic data are however
needed to confirm this hypothesis. Our analysis indicates that distorted FUV
sources may have recently entered the Coma cluster, and hence undergoing
stripping events under the influence of the cluster-related environmental
mechanisms. While five of these galaxies could be classified as star-forming
on the basis of their emission line properties, two are found to be composites
and one is an AGN.

To the best of our knowledge, this is the first study of a galaxy cluster
field being carried out with the UVIT data. Hence, despite several limitations
mentioned in the previous section, this work provides an idea of the finesse
with which UVIT has captured the features of nearby extra-galactic objects,
and a flavour of the problems which, together with the multi-band archival
data may be addressed using the UVIT. In order to facilitate this, we have
also provided detailed information for data handling and reduction for fellow
UVIT users who intend to work with similar data.

##  7 Acknowledgements

S. Mahajan was funded by the INSPIRE Faculty award
(DST/INSPIRE/04/2015/002311) and the SERB Research Scientist (SRS) award (SB-
SRS/2020-21/56/PS), Department of Science and Technology (DST), Government of
India. K. P. Singh thanks the Indian National Science Academy for support
under the INSA Senior Scientist Programme. Authors are very grateful to Prof.
S. Tandon for his valuable comments on earlier versions of this manuscript. We
are very grateful to the anonymous reviewer for their constructive critique
which helped in improving this manuscript.

This publication uses data from the AstroSat mission of the Indian Space
Research Organisation (ISRO), archived at the Indian Space Science Data Centre
(ISSDC). UVIT project is a result of collaboration between IIA (Bengaluru),
IUCAA (Pune), TIFR (Mumbai), several centres of ISRO, and the Canadian Space
Agency (CSA). This research has made use of the NASA/IPAC Infrared Science
Archive, which is funded by the National Aeronautics and Space Administration
and operated by the California Institute of Technology. The topcat software
(Taylor, 2005) was used for some of the analysis presented in this paper.

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# On Geometry, Arithmetics and Chaos

Lars Andersen

###### Abstract.

Our main result is that chaos in dimension $n+1$ is a one-dimensional
geometrical object embedded in a geometrical object of dimension $n$ which
corresponds to a $n$ dimensional object which is either singular or non-
singular. Our main result is then that this chaos occurs in the first case as
either on an isolated or non-isolated singularity. In the first case this
chaos is either boundary chaos or spherical chaos which is what happens also
in the non-singular case. In the case of an isolated singular geometry one has
chaos which can either be boundary, spherical or tubular chaos. We furthermore
prove that the prime number display quantum behaviour.

###### Contents

  1. 1 Introduction
  2. 2 Recollections
    1. 2.1 Topological stability of functions
    2. 2.2 Topological stability of dynamical systems
  3. 3 Arithmetics
  4. 4 Geometry
  5. 5 Chaos

Classification: 14-XX, 94-XX

##  1\. Introduction

Since Newtons unification [10] of Geometry, Analysis and Physics as was then
understood by these subjects dynamical systems have been studied such as those
concerning the motions of the geometry of the spheres of the heavens. The idea
of solving a dynamical system is already apparent in the many propositions
concerning on how to draw a trajectory with given properties defined
geometrically111One could make it clear that Leibniz in [8] only defined $dx$
and $\int x$ as differences respectively sums of $x$ with an ’etc etc’ added
to it. This is not mathematics, has nothing to do with Euclid and that is
probably why Newton never bothered to respond. and furthermore the idea that
light is a reverberation in matter as expounded by Newton [11] is again a hint
that the bifurcations of dynamical systems were something to study. Yet the
author would argue that Newton invented bifurcation theory since he moves
trajectories by changing the given geometrical properties. In modern parlance
one speaks of the study of _chaos_ as the study of how solutions to dynamical
systems change under such changes (see e.g [12] and the subject is classically
ascribed to Poincaré with important contributions by a enormity of
mathematicians since his masterpiece ’Analysis Situs’ [12].

We now give a brief outline of what will be meant by chaos. This varies
slightly from author to author but we follow in essense [6]. By chaos is meant
the sudden change in the topology of solutions of a dynamical system
$dx/dt=V(x(t),\alpha)$ under a change called a bifurcation of parameter values
$\alpha\in T$.
By a singularity means the sudden change in topology of the fibers
$f_{\alpha},\alpha\in T$ of a family of smooth functions $f:M\to T$ under a
bifurcation of parameter values. Here $M$ and $T$ are smooth manifolds. A
singularity is usually represented by a map germ
$f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{m},0)$.

Our objectives with these notes is to connect these two notions and to apply
the results to the ”singularities” of $\mathbb{N}$ namely the primes.

##  2\. Recollections

###  2.1. Topological stability of functions

A function $f$ is _topologically equivalent_ to $g$ if $f=u\circ g\circ v$ for
homeomorphisms $u$ and $v$. If every function sufficiently close to $f$ in the
$C^{\infty}$ topology is topologically equivalent to $f$ then one says that
$f$ is topologically stable.

If $M$ is not compact then the set of topologically stable functions is not
dense. If $M$ is dense however then they are, however.

###  2.2. Topological stability of dynamical systems

Two dynamical systems are said to be topologically equivalent if there is a
homeomorphism mapping orbits into orbits and preserves the time direction.
Introduce a metric on the set of dynamical systems by saying that the distance
between $dx/dt=V$ and $dx/dt=W$ is the $C^{1}$ distance $d(V,W)$. The
dynamical system $dx/dt=V$ is structurally stable if every dynamical system
sufficiently close to it is topologically equivalent to it.

##  3\. Arithmetics

Consider the dynamical system

$dx/dt=1,$ $dy/dt=1/z\qquad(\ast)$ $dz/dt=1$

on
$\mathbb{R}^{+}\times\mathbb{R}^{+}\setminus\\{0\\}\times\mathbb{R}^{+}\setminus\\{0\\}.$
and let $z=m$ be fixed, $m\in\mathbb{N}$. For a choice of initial values it
describes unit speed motion along the number line because

$dx/dt=1,\qquad dy/dt=\frac{1}{m}dx/dt$

gives $x=my+c$ where $c\in\mathbb{N}$ only depends on the initial conditions.
When $z$ is no longer fixed the multiple moves along the number line with the
same speed as $x$. A solution that is an orbit therefore consists of a triple
$(x,y,z)=(my+c,y,m)$.

######  Lemma 0.1.

The primes are the orbits which are unstable.

###### Proof.

An orbit which is stable is independent of small perturbations of the initial
conditions so must be representable by a solution which has first coordinate a
multiple $m_{0}y$ for some $m_{0}\in\mathbb{N}$. ∎

Let $k$ be a precision parameter to be specified later. The main theorem of
this section is

######  Theorem 1 (Quantum Phenomena of Primes).

The primes can be approximated as $A_{k}$\- catastrophes with precision $k+1$
for $k=1,2$.

###### Proof.

Rewrite $(\ast)$ as

$dx/dt=1,$ $dy/dt=dlog(z)/dt,$ $dz/dt=1$

and write

$log(z)=log(z-1)+\sum_{k=1}^{\infty}(-1)^{k}\frac{(z-1)^{-l}}{k}$
$=\sum_{k=1}^{\infty}(-1)^{k}\frac{(z-N)^{k}}{k}+(z^{-1}+\dots)$

which we for $z$ large will approximate to the second order as

$=-\frac{x^{2}}{2}+(N+1)x-\frac{N^{2}}{2}-N$

hence as a $A_{1}$-catastrophe. The order $k=3$ approximations are of the form

$-\frac{x^{2}}{2}+(N+1)x-\frac{N^{2}}{2}-N+\frac{x^{3}-3Nx^{2}+3Nx-N^{3}}{3}$

with derivative

$(x-N-1/2)^{2}+N^{2}+N-(N+1/2)^{2}$

hence a $A_{2}$ fold catastrophe. ∎

######  Remark 1.

For $k$ large we might have more difficult catastrophes.

The meaning of the theorem is that primes exhibit quantum behaviour. Each time
one looks closer one has has a different behaviour. And the dynamics of the
primes changes chaotically. So one can say that the chaos of the primes is
itself chaotic.

######  Theorem 2.

One-way functions exist.

###### Proof.

Now stability is generic since transversality is generic by Thom’s
Transversality Theorem [6]. Therefore the multiplication map $(m,n)\mapsto nm$
of integers is hard to invert probabilistically. It follows that it is a one-
way function. In particular the $N\neq NP$ problem is resolved. ∎

##  4\. Geometry

Geometry can be divided into global and local geometry. The main objects of
interest in local geometry are the Milnor fibers [9]. There is and has been an
enormous amount of research into their topology. We refer to the authors
thesis work ([4], [5] and [2]) for further information and references. In the
subsequent section we return to local geometry and discusses its connection to
analysis and we continue by discussing global geometry. Given an embedded real
analytic manifold $M\subset\mathbb{R}^{n}$ there exist a covering
$U=\sum_{i\in I}U_{i}$ and functions of the form
$f_{i}:\mathbb{R}^{n}\to\mathbb{R}$ such that $M\cap U_{i}$ is the set

$y_{1}=f_{1}(x_{1},\dots,x_{n})$ $\dots$ $y_{k}=f_{k}(x_{1},\dots,x_{n}).$

We will assume that $f_{i}(0)=0$ for $i=1,\dots,k$. Either $f_{i}$ has non-
degenerate critical points or it has not.

  1. (1)

If $f_{i}$ has a degenerate critical point then the hypersurface
$\\{y_{i}=f_{i}\\}$ has zero Gaussian curvature by definition.

  2. (2)

If $f_{i}$ has a non-degenerate critical point then there are coordinates such
that $f_{i}\circ\phi_{i}^{-1}(x_{1},\dots,x_{n})=\sum\pm x_{i}^{2}$. In
particular in new coordinates
$(y_{1},\dots,y_{k},\phi^{-1}(x_{1}),\dots,\phi^{-1}(x_{k}))$ we have
intersected with a quadric

$y_{i}=\sum_{i=1}^{n}\pm x_{i}^{2}$

In the second case we furthermore know that the critical point is isolated. We
therefore conclude since the curvature tensor $R_{\mu\lambda}^{\nu}$ is
algebraic the following Let $M\subset\mathbb{R}^{n}$ be a differentiable
manifold. There is a covering $V=\bigcup V_{i}$ of $M$ such that $M\cap V_{i}$
is described by intersecting flat hypersurfaces and at most $codim(M)$
quadrics. In particular the curvature tensor can take at most $\lceil
n/2\rceil+1$ types described by the curvature tensor of a quadric
$y=\sum_{i=1}^{n}\pm x_{i}^{2}$ and by flat space

######  Theorem 3 (Geometry of Manifolds).

There are $\lceil n/2\rceil+1$ geometries of differentiable manifolds
$M\subset\mathbb{R}^{n}$.

##  5\. Chaos

Given a dynamical system

$dx/dt=V(x(t),\alpha)$

depending on $\alpha\in T$ for $T$ a real analytic manifold and $x(t)\in M$ a
smooth manifold we consider

$\pi:M\times T\to T,\qquad\pi(x,\alpha)=\alpha.$

and it’s restriction to the real analytic set $\mathcal{F}_{V_{\alpha}}$ given
by

$V(x,\alpha)=\vec{\eta},\quad
dx/dt=\vec{\eta},\quad\lVert\vec{\eta}\rVert=\eta,\quad
x\in\mathbb{B}_{\delta}\qquad(\ast)$

and we can always find a Whitney stratification such that
$\pi_{|\mathcal{F}_{V}}$ is locally trivial above each stratum. In particular
the topological type of $\mathcal{F}_{V_{\alpha}}$ and
$\mathcal{F}_{V_{\alpha^{\prime}}}$ are the same whenever $\alpha$ and
$\alpha^{\prime}$ belong to the same stratum. Indeed by Sard’s Theorem [7] the
set of critical values is of measure zero and one can apply the Isotopy Lemma
of Thom-Mather [7] to its complement. In particular if $x(t)$ is a solution
over $\alpha$ then there exist a stratum preserving homeomorphism $h:M\times
T\to M\times T$ such that $\mathcal{F}_{V_{\alpha^{\prime}}}\cong
h(\mathcal{F}_{V_{\alpha}})$ and $h(x(t))$ is then a solution over
$\alpha^{\prime}$ which is topologically equivalent to $x(t)$.

######  Definition 1.

One says that $dx/dt=V(x(t),\alpha)$ undergoes major chaos whenever a solution
is mapped to a non-topologically equivalent solution when $\alpha$ changes
from one stratum to another.

######  Definition 2.

One says that $dx/dt=V$ undergoes minor chaos whenever a solution is mapped to
a non-topologically equivalent solution when $\alpha$ changes within a
stratum.

######  Definition 3.

One says that $dx/dt=V(x(t),\alpha)$ undergoes chaos if it undergoes minor or
major chaos under perturbation of the parameter.

######  Definition 4.

The real analytic set $\mathcal{F}_{V_{\alpha}}$ is called the Milnor fiber of
the dynamical system $dx/dt=V(x,\alpha)$

It is the main object of study in the understanding of chaos. The following
theorem says that all chaos in dimension $n$ lives in a manifold of dimension
$n$.

######  Theorem 4.

A system $dx/dt=V(x,\alpha)$ undergoes chaos if and only the outward unit
normal $\nu_{x}(t,\alpha)$ to $\mathcal{F}_{V_{\alpha}}$ traces out a curve
which is non-topologically equivalent to the curve traced out by the outward
unit normal $\nu_{x}(t,\alpha)$ to $\mathcal{F}_{V_{\alpha}}$.

###### Proof.

We can always assume that the trajectories to $dx/dt=V(x,\alpha)$ are
parametrised by unit speed, since $M$ is a smooth manifold. That $x(t)$ are
paths on the Milnor fiber then gives the claim. That one can work instead with
the unit normal is because the normal space is orthogonal and of the same
dimension. ∎

An interesting form of chaos is the case of a perturbed radial vector field
$V_{\alpha}=\sum_{k=1}^{n}\alpha_{k}x_{k}^{2}$ in local coordinates on $M$.

######  Definition 5.

One says that $dx/dt=\sum_{k=1}^{n}\alpha_{k}x_{k}^{2}$ undergoes spherical
chaos if it undergoes chaos.

The main result in this section is that in the simplest case where the
dimension is one chaos is either spherical or lives in dimension zero.

######  Theorem 5.

Suppose that $\dim V^{-1}(0)>0$, that $V^{-1}(0)$ has only isolated singular
points and that the hypothesis of [1, Theorem 5]) hold. There is a stratum
preserving diffeomorphism

$\bar{\mathcal{F}}_{V,\alpha}\cong\partial\bar{\mathcal{F}}_{V,\alpha}\cup\bigcup_{i=1}^{l}\mathbb{D}^{\lambda(p_{i})}\times\mathbb{D}^{n-\lambda(p_{i})}$

In particular chaos either is spherical or appear on the boundary of the
Milnor fiber.

###### Proof.

The proof is identical to the proof of [1, Theorem 5] except that in the last
step one uses that

$V_{\alpha}^{-1}((x,x+\epsilon])\cap\mathbb{B}_{\gamma}(x)\cong
V_{\alpha}^{-1}(x+\epsilon)\cap\mathbb{B}_{\gamma}\times(x,x+\epsilon]$

by a stratum-preserving diffeomorphism, which follows from Ehresmann’s
Fibration Theorem [7]. In the same way

$\partial
V_{\alpha}^{-1}((x,x+\epsilon])\cap\mathbb{B}_{\gamma}(x)\cong\partial
V_{\alpha}^{-1}(x+\epsilon)\cap\mathbb{B}_{\gamma}\times(x,x+\epsilon]$

by a stratum-preserving diffeomorphism. Then one identifies the strata one
wants and get the wanted stratum-preserving diffeomorphism. For the last step
one simply remarks that the Milnor fiber of $(V_{1},\dots,V_{n})$ is identical
to the Milnor fiber of $\sum_{i=1}^{n}V_{i}^{2}$. ∎

######  Definition 6.

A stationary point of a dynamical system is called a differential singular
point.

######  Definition 7.

A critical point of a function is called a topological singular point.

It would be interesting to ask whether this result generalises to when
$V^{-1}(0)$ has non isolated critical points. It turns out that apart from
boundary chaos and spherical chaos one also gets what one might call tubular
chaos. As before the Milnor fiber in the above sense is that of the function
$\sum_{i=1}^{n}V_{i}^{2}$.

######  Definition 8.

One says that $dx/dt=V(x,\alpha)$ undergoes tubular chaos if
$dx/dt=V(x,\alpha)-\alpha^{\prime}$ undergoes chaos.

######  Lemma 5.1.

If $V=\sum_{i=1}^{n}V_{i}^{2}$ as a isolated critical value in the origin and
satisfies the transversality condition of D.B Massey and if $\dim V^{-1}(0)>0$
then chaos of $dx/dt=V(x,\alpha)$ consists of boundary chaos of $V$, tubular
and boundary chaos of $V-c(\sum_{i=1}^{n}x_{i}^{2})^{k}$ and spherical chaos.

###### Proof.

Apply [3, Theorem 2] and Lemma 5 to obtain that the chaos consists of chaos
along the link of $V$, chaos along the link of

$\sum_{i=1}^{n}V_{i}^{2}-c(\sum_{i=1}^{n}x_{i}^{2})^{k}$

, and chaos along $(\sum_{i=1}^{n}V_{i}^{2})^{-1}([0,\epsilon])$. The first
two are thus boundary chaos and the latter is clearly tubular chaos and
boundary chaos again by [3, Theorem 2] and Lemma 5 by the definition (for
$\alpha^{\prime}$ varying in $(0,\epsilon)$. Then there is only singular chaos
corresponding to chaos at $V^{-1}(0)$. But $dx/dt=0$ is non-chaotic. ∎

When the unit normal $\nu_{x}(t,\alpha)\in\mathcal{F}_{V_{\alpha}}$ undergoes
change of parameter $\alpha$ the Milnor fiber ”breathes” intuitively speaking
and one can then ask what kind of figures are possible? For instance for
spherical chaos $\mathcal{F}_{V_{\alpha}}$ is a sphere and when $\alpha$
changes one can in the case of $n=3$ get a ellipsoid
$\alpha_{1}x^{2}+\alpha_{2}y^{2}+\alpha_{3}z^{2}=\epsilon$ or a one-sheeted
hyperboloid $\alpha_{1}x^{2}+\alpha_{2}y^{2}-\alpha_{3}z^{2}=\epsilon$ and a
two-sheeted hyperboloid
$\alpha_{1}x^{2}+\alpha_{2}y^{2}-\alpha_{3}z^{2}=\epsilon$ where for
simplicity $\alpha_{i}\in\mathbb{R}^{+}.$ Then if $t$ is fixed at
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ varies then there is only linear
motion possible. So here we have the most trivial Milnor fibers and only
linear motion. On the other hand for $\alpha_{1}x^{3}+y^{2}+z^{2}=\epsilon$
which corresponds to a cusp singularity (see [4, 5] for the homology groups of
$ADE$-singularities in general) we can guess that a small change in parameter
gives a larger effect.

Therefore, to measure the amount of chaos persistent we consider a small ball
$x(t,\alpha)\in\mathbb{B}_{\delta}^{loc}$ intersecting the Milnor fibers
transversally for $|t|\leq\tau,\tau\in\mathbb{R}^{+}$ and for
$|\alpha|\leq\alpha_{0}$.

######  Definition 9.

The first energy measure of chaos of a solution $x(t,\alpha)$ is

$l_{loc}^{I}=\lVert\nu_{x}(t,\alpha)\rVert+\det\nabla_{x}\nu_{x}(t,\alpha)$

######  Definition 10.

The second energy measure of chaos of a solution $x(t,\alpha$ is

$l_{loc}^{I}I=\lVert\nu_{x}(t,\alpha)\rVert+\lVert\nabla_{x}\nu_{x}(t,\alpha)\rVert$

Here the determinant appears as the natural measure of size of a matrix in the
first definition and in the second definition $\lVert\sum_{i,j\in
I}a_{ij}\rVert=\sum_{i,j\in I}|a_{ij}|$ where $I\subset\mathbb{N}.$ Let
$K_{loc}$ and $\chi_{loc}$ denote the local Gaussian curvature respectively
Euler characteristics.

######  Lemma 5.2.

$\int_{x\in\\{\lVert
V\rVert=\eta\\}\cap\mathbb{B}_{\delta}^{loc}}l_{loc}^{I}dx=2\pi\chi_{loc}+Vol(\mathbb{B}_{\delta}^{loc})$

and

$\int_{x\in\\{\lVert
V\rVert=\eta\\}\cap\mathbb{B}_{\delta}^{loc}}l_{loc}^{II}dx=2Vol(\mathbb{B}_{\delta}^{loc})$

###### Proof.

For the first claim this is just the Gauss-Bonnet theorem stated locally. For
the second claim one uses the exponential map to transport the question to the
tangent space of the Milnor fiber in question, then one integrates on the
tangent spaces giving the volume in the first summand and then in the second
summand one again gets the volume since one is just integrating the ball in a
normal direction. ∎

We now come to the main result which is the classification result mentioned in
the abstract.

######  Theorem 6 (Classification Theorem of Chaos).

Chaos in dimension $n$ is one dimensional and occurs on the $n-1$-dimensional
Milnor fiber of a function on $\mathbb{R}^{n}$ which is either singular
isolated, singular non-isolated or non-singular. In the first case this chaos
is either boundary chaos or spherical chaos. In the second case this chaos is
either boundary chaos, spherical chaos or tubular chaos. In the third chaos
this chaos is tubular chaos.

## References

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