44B |  Create another list called ratings and for each movie that is entered (that is not equal to stop), ask the user to enter a rating from 0 to 10 (as an integer) and add the number to the ratings list. At the end, display the number of movies and the number of ratings Note: they should have the same number of elements and stop should not be included in the movies
45A |  Create an empty list called numbers and then use a for loop that repeats for a random number of times between 50 to 75 to update the list by adding a random number between 0 to 100. Then use another loop to find the largest number in the list. Finally, display the largest number after the second loop has finished
45B |  Create another variable called smallest and use the second for loop to find the smallest number in the list in addition to the largest. At the end, display both the largest and the smallest numbers


# Detection of Double Nuclei Galaxies in SDSS

Anwesh Bhattacharya
Department of Physics
Department of CSIS
Birla Institute of Technology and Science
Pilani, Rajasthan, India
<EMAIL_ADDRESS>
&Snehanshu Saha
Department of CSIS
APPCAIR
Birla Institute of Technology and Science
Goa, India
<EMAIL_ADDRESS>
&Mousumi Das
Indian Institute of Astrophysics
Bangalore, India
<EMAIL_ADDRESS>

###### Abstract

It is now well established that galaxy interactions and mergers play a crucial
role in the hierarchical growth of structure in our universe. Galaxy mergers
can lead to the formation of elliptical galaxies and larger disk galaxies, as
well as drive galaxy evolution through star formation and nuclear activity.
During mergers the nuclei of the individual galaxies come closer and finally
form a double nuclei galaxy. Although mergers are common, the detection of
double-nuclei galaxies (DNGs) is rare and fairly serendipitous. Their
detection is very important as their properties can help us understand the the
formation of supermassive black hole (SMBH) binaries, dual active galactic
nuclei (DAGN) and the associated feedback effects. There is thus a need for an
automatic/systematic survey of data for the discovery of double nuclei
galaxies. Using the Sloan digital sky survey (SDSS) as the target catalog, we
have introduced a novel algorithm $\proc{gothic}$ — Graph-bOosTed iterated
HIll Climbing[13] — that detects whether a given image of a galaxy has
characteristic features of a DNG (ASCL entry 2707) . We have tested the
algorithm on a random sample of 100,000 galaxies from the Stripe 82[2] region
in SDSS and obtained a maximum detection rate of $4.2\%$ with a careful choice
of the input catalog.

##  1 Introduction

A galaxy is a system of stars, interstellar gas, dust and dark matter bounded
by gravitational forces. Based on their morphology galaxies are categorized as
spiral, irregular or elliptical. Galaxies are not usually isolated systems but
are instead clustered into galaxy groups and large galaxy clusters [18].
Hence, galaxies often interact tidally at a distance with each other or
undergo mergers with one another. During these interactions the gravitational
field of one galaxy affects the other one. If the two galaxies do not have
enough momentum to continue to travel after interacting, they will finally
collide and merge, leading to the formation of a galaxy merger remnant
[5].Galaxy mergers are common in our universe and are one of the main drivers
of galaxy evolution [7, 1].

The outcome of galaxy mergers depends on the mass ratio of the merging
galaxies. In the cases where one galaxy is much more massive than the other
one, the more massive galaxy remains more or less intact whereas the smaller
galaxy is tidally stripped of a large fraction of its mass and finally becomes
part of the larger galaxy. Such mergers are called galaxy accretion events or
minor mergers. When the colliding galaxies are similar in mass and size, it is
called a major merger event. Major mergers usually lead to the formation of
elliptical galaxies. Another classification system for galaxy mergers depends
on the presence of cold interstellar gas. When one or both the galaxies are
gas rich, the merger is said to be a wet merger and when both galaxies are gas
free, as in ellipticals, the merger is said to be a dry merger. However, wet
or dry, major or minor, all mergers can lead to the formation of a single
envelope containing two nuclei. Such dual nuclei systems represent the final
stage of galaxy mergers [15].

As the galaxies merge, the supermassive black holes (SMBHs) in their nuclei
come closer and may start accreting mass. Such accreting SMBHs are called
active galactic nuclei (AGN) and give out large amounts of energy over the
whole electromagnetic spectrum and can be detected using optical, radio or
X-ray telescopes [31]. Alternatively, the large gas infall into the nuclear
regions can trigger starburst activity. Thus double nuclei galaxies can host
AGN pairs, AGN-starburst pairs or starburst nuclei in their centers [33]. When
two AGN are found in a merger remnant, they can form a dual AGN (DAGN) if the
nuclei are separated by $<$10 Kpc. AGN pairs at separations of a few parsecs
or $<<$1 Kpc are called binary AGN (BAGN). The number of DAGN identified till
now is less than 50 [32] and the number of binary AGN identified is less than
5 [23].

There are several reasons why we need to find more DAGN or BAGN. The first
reason is that they represent SMBHs coming closer together in galaxies. At
distances of several Mpc and beyond the local universe, the only way to detect
SMBHs is through the radiation emitted from AGN activity. The mass of the SMBH
pairs affects the stellar and gas content of the nuclear regions of the
galaxies, triggering disk instabilities and star formation. Secondly, although
DAGN are rare in the universe, their study is important for understanding the
growth of SMBHs as it is suggested that merger-triggered AGN may dominate SMBH
growth [22]. Third, SMBH binaries represent black holes that will eventually
merge and produce gravitational radiation [21]. Although the gravitational
radiation will be emitted only when the SMBHs are at very close separations, a
larger sample of DAGN will help us understand the evolutionary path for such
systems. Fourth, AGN give out enormous amounts of radiation that can push gas
out from the nuclear region and even beyond the galaxy. This is called AGN
feedback and it can result in the growth the circumgalactic medium around
galaxies [16]. The gas can also fall back onto the disks resulting in star
formation. In the past decade it has become clear that AGN feedback is
important for galaxy evolution, and if there is an AGN pair then the effect
will be even more important not only because of two AGN in the galaxy center,
but also because the AGN winds can overlap and produce additional feedback
effects. Hence, large samples of DAGN are essential for fully understanding
the role of mergers in galaxy evolution.

The earlier DAGNs have been serendipitously discovered [9, 24, 35, 37], mainly
during galaxy surveys of emission line galaxies and merger remnants. Later
studies used samples of double peaked AGN emission line galaxies (DPAGN) to
search for DAGN [44]. However, DPAGN can also be due to AGN outflows or
rotating disks. So one of the best ways in which to detect DAGN is by
identifying individual nuclei in a dual nuclei galaxy [34]. The existing
surveys of galaxy pairs are either too small [20] or the galaxy separations
are too large to include merger remnants [30]. Thus a deeper and larger
catalog is necessary in order to do statistical studies of dual nuclei
galaxies and identify DAGN. The detection of previously undiscovered samples
is essential for understanding the final stages of the galaxy merging process.
Hence, there is a need of a pipeline to sift through a large catalog of
galaxies and steadily increase the pool of candidate DAGNs. The aim of this
paper is to develop an algorithm that extracts images from the Sloan Digital
Sky Survey (SDSS) survey and identifies a pool of dual nuclei systems that we
will use later for identifying a large sample of DAGN.

##  2 Previous Studies, Test Sample and Input Catalog

As mentioned in the previous section, galaxies with two nuclei separated by
kiloparsec scales are known as DAGN, whereas, the ones in which the separation
is of the order of parsecs are called binary AGN. The heirarchical model of
structure formation predicts the formation of such gravitationally bound
binary SMBHs. The accretion of mass onto the nuclei transforms the nuclei into
AGN and is an important part of SMBH growth and galaxy evolution. Moreover,
these galaxy merger events could ultimately give rise to black-hole merger
events which are crucial for gravitational wave detection [3]. There have been
several studies of double nuclei galaxies at different wavelengths. In this
section we discuss these previous studies and describe the test sample which
we finally use for testing our algorithm.

The main catalog that we used for testing the algorithm developed in this
paper is from Gimeno et. al. [20]. The relevance of galaxy interactions and
minor mergers in the formation of double-nuclei galaxies, the correlation
between geometric and photometric parameters of the nuclei and their host
galaxies, have been investigated in this study. It also discusses the
differences or similarities between the component nuclei and their location in
the host system. The galaxies selected had systematic velocities $cz$, and
redshifts $z$ in the range $cz$ <15000 km/s, $z$ <0.05, $m_{B}$ <18\. Mezcua
et. al. [28] did further studies of this sample using observations. In their
study, the luminosity of the nuclei and their relative separations were
derived from the multi-component photometric fits to the galaxies in the
r-band SDSS optical images. A majority of the sources have projected
separations $\leq$ 4kpc. The ratio of nuclear luminosities indicated that most
of the systems were in the final stages merging. This was supported by the
existence of a single galaxy disk in 65% of the systems studied and the
siginificant correlation between nuclear luminosity and host luminosity for
the single-disk systems. The sources fitted with a single disk are in a more
evolved stage of the merging process and present an enhancement of the nuclear
luminosity compared to the double-disk systems.

Koss et. al. [26] have mentioned the difficulty of resolving each AGN in
optical and X-ray observations. The Swist Burst Alert Telescope survey was
used to create a catalog of $167$ AGN out of which $81$ have a close companion
within a $100$ kpc radius. MRK 739 (also listed as NGC 3758 in Gimeno et. al
[20]) has been well known since 1986 due to the paper by Netzer et. al. [29].
In another work by Koss et. al [27], SDSS optical imaging and observations
from the Chandra X-ray Observatory were utilized to rule out the possibility
of MRK 739W (the nuclei on the western side) being a starbust region, which
was suggested by its optical spectra. MRK 739E was already known to be an AGN
due to Netzer [29]. It is interesting to note that the optical spectra of MRK
739E and MRK 739W both have a peak around $6750$ Angstrom that falls under the
R-band of the SDSS filters.

The detection of double-nuclei galaxies is important from the point of view of
Cosmology and Structure Formation. Davis [17] and Springel [39] have suggested
that galaxy interactions play an important role in the growth of their central
SMBH. It has been suggested theoretically that DAGN must be found in relative
abundance in our universe [25]. However, such objects are rarely found as most
of them are radio-quiet and are difficult to resolve in the optical.
Previously, double-peaked narrow emission lines were used as the criteria for
a galaxy to be a candidate DAGN. However, such a criteria has received
criticism as similar observations can be drawn from other phenomenon in single
active nuclei galaxies itself (Xu & Komossa [42]). This suggests that
alternative methods must be devised to procure candidates for DAGN galaxies.
For example radio observations have used the Very Large Baseline Array (VLBA)
to identify DAGN candidates [19]. There have been efforts towards studying
individual DAGNs such as MRK 739 by Koss et al. [27], MRK 463 by Bianchi [6]
and ARP 299 by Ballo et al. [4]. Rubinur et al. [33] have conducted radio-
studies on a sample of DPAGN. Hence, it is an important task to increase the
pool of DAGN candidates for robust statistical studies. In this paper, we have
focused on detecting the presence of dual nuclei given a $$$\times$$$ r-band
image of a galaxy from the SDSS images of galaxies. This sample will form the
basic sample for a followup study aimed at deriving a large sample DAGN in
SDSS.

###  2.1 Selected Samples from Pre-Existing Catalogs

Mezcua et. al. [28] have reported that out of the 107 galaxies in the Gimeno
catalog, only 60 have imaging data in SDSS DR8. Two galaxies, NGC-3758 and
MRK-212 are shown as representatives in figure (1). The former has gained some
attention outside the astronomy community due to its symmetric morphology [40,
43], and the latter has been extensively studied in multiple wavelengths [32].
The algorithm is domain-specific and hence a pre-existing sample was necessary
for its development and testing. We have chosen the Gimeno catalog [20] as our
test sample, and Gothic has achieved 100% accuracy on this dataset. The images
have been selected from SDSS.

(a) NGC-3758

(b) MRK-212

Figure 1: Double-Nuclei Galaxies Sample

###  2.2 Input Catalog

Our input dataset is located in this public folder. It contains the entire
SDSS and Gaia catalog in the Stripe-82 region, which was used for
crossmatching (section 5.2), in the files SDSS_Stripe82_All.csv and
Gaia_Stripe82_All.csv, respectively.

##  3 The Gothic Algorithm

The FITS data of the given source, in all the SDSS bands (ugriz) were
downloaded and subsequently a cutout of $None$ centred around the source was
performed. The algorithm works individually for each band. The sequence of
steps can be described as follows -

  1. 1.

Scaling/Smoothing of the image for ease of visualisation

  2. 2.

Edge detection with Canny to approximate galaxy envelope

  3. 3.

Sersic profile fitting to infer noise/signal levels

  4. 4.

Peak detection by Iterated Hill Climbing

  5. 5.

Identification of the galaxy as single/double-nuclei or neither, based on
graph searching

Given below is the formal pseudocode for the algorithm, the details of which
have been explained in Appendix A. The result of running Gothic is shown in
figure (2)

Gothic(cood, frame) cood - Celestial coordinates of an object frame - FITS
frame containing the object img $\leftarrow$ Cutout (frame, cood, $None$)
Returns a $None$ cutout image img $\leftarrow$ Log-Normalize (img) Step 1 img
$\leftarrow$ Gaussian-Kernel (img, 5, 5) Step 1 edges $\leftarrow$ Canny (img)
Step 2 hull, reg $\leftarrow$ Convex-Hull (edges) Step 2 noise, signal
$\leftarrow$ Sersic-Fit (img, reg) Step 3 sreg $\leftarrow$ Search-Region
(img, reg, noise) Step 4 peak_list $\leftarrow$ Iterated-Hill-Climb (sreg,
img, 500) Step 4 verdict $\leftarrow$ Classify (peak_list, sreg, img, signal)
Step 5 verdict

(a) NGC-3758

(b) MRK-212

Figure 2: Detected peaks from Gothic

##  4 Mathematical Basis for Classification and Related Procedures

In this section, we detail lines 10 and 11 of Gothic. The procedure Search-
Region extracts the region inside the blue border, as shown in figure (2).
Drawing of the border, in which the galaxy of interest lies in the cutout
image, is required by the classification procedure. Moreover, it is also
dependent on the Sersic [36] light-profile parameters of the galaxy. Parameter
fitting has been performed explicitly using scipy, instead of depending on
derived parameters of the SDSS catalog. These steps have been detailed in
Appendix A.

###  4.1 Iterated Hill Climbing

To identify the intensity peaks in the image, it is tempting to sort the
pixels in descending order of pixel value and extract the top few pixels.
This, however, does not give sufficient information to classify the object as
a single-nuclei or double-nuclei galaxy. The top two pixels extracted are
likely to be pixel neighbors near the brightest peak. Due to the high chance
of foreground stars, which are brighter than the central galaxy, appearing in
the field of view, it might lead to erroneous detections. The robust
definition of a peak is a pixel whose value is greater than all of its
neighbors. Finding a peak is achieved by a discrete hill-climbing algorithm,
operating on a 2-D image. We have allowed for an angular resolution of $None$,
which translates to a separation of 3 pixels in the cutout image. Hence, the
neighbor of a pixel is defined to be the $7\times 7$ square grid centred on
it.

Hill-Climb(pt, reg, img) pt - starting point reg - search region img -
underlying image newpt $\leftarrow$ NULL pt $\neq$ newpt neighs $\leftarrow$
Neighs-in-reg(pt, reg) Returns the $7\times 7$ grid neighbors centred at pt
newpt $\leftarrow$ pt n in neighs img[n] > img[newpt] newpt $\leftarrow$ n pt
$\leftarrow$ newpt newpt

The input point pt is initialised randomly within the search region reg. The
procedure is guaranteed to terminate as the pixel value of pt strictly
increases in each iteration until no greater neighbor, centred about it, can
be found. Iterated-Hill-Climb iterates over Hill-Climb, by seeding random
initial points in the search region, and returns a list of peaks thus
obtained. It is also to be noted that a peak is accepted only if its signal to
noise ratio $>3$. Figure (3) shows the output for example double-nuclei
galaxies.

Iterated-Hill-Climb(reg, img, iter) reg - search region img - underlying image
iter - number of iteratons to perform peak_list $\leftarrow$ [ ] Empty list i
$\leftarrow$ 0 pt $\leftarrow$ Choose-Random(reg) Returns a random point in
the region reg peak $\leftarrow$ Hill-Climb(pt, reg, img) peak not in
peak_list and snr(peak) $\geq$ 3 Append(peak, peak_list) Adds peak to the list
peak_list i $\leftarrow$ i + 1 i = iter peak_list

(a) Peak on Stray Object

(b) Multiple peaks on one object

Figure 3: Optima from Iterated-Hill-Climbing

###  4.2 Graph-Searching and Classification

The peaks in peak_list comprise local maxima, the global maxima, and the
peak(s) corresponding to the central object in the image. It is the latter
which is ascertained by graph-searching. This is exhibited by the sample cases
in figure (3(a)), where Iterated-Hill-Climbing detects a peak on an offset
stray object, and figure (3(b)), where multiple peaks are detected in the
object of interest. The procedure Classify processes the list of peaks and
returns one of the three verdicts - NO_PEAK, SINGLE or DOUBLE

Classify(peak_list, reg, img, signal) reg - search region peak_list - peak
list img - underlying image comps $\leftarrow$ Connected-Components (reg) Runs
depth-first-search Sort-by-Distance (comps) group $\leftarrow$ First-Group
(comps) lcmp $\leftarrow$ Largest-Component (group) pks $\leftarrow$ Find-
Peaks-In (lcmp, peak_list) pks $\leftarrow$ Filter-by-Intensity (pks, img,
signal) len $\leftarrow$ Length (pks) len $\geq$ 2 DOUBLE len = 1 SINGLE
NO_PEAK

The cutout image is treated as a graph $G(V,E)$ with pixel coordinates $(x,y)$
as vertices. $img(x,y)$ is the function that maps pixel coordinates to
intensity values. The edge relation is also defined as $E$ -

$\displaystyle img(x,y)>\texttt{noise}\implies(x,y)\in\id{V}$ $\displaystyle
x_{2}=x_{1}\pm s_{x},y_{2}=y_{1}\pm
s_{y}\implies((x_{1},y_{1}),(x_{2},y_{2}))\in E$ $\displaystyle
s_{x},s_{y}\in\\{-1,0,1\\}$ $\displaystyle s_{x}\neq s_{y}=0$

####  4.2.1 Connected-Components and Sort-by-Distance

The image graph is not necessarily fully connected. By running depth-first-
search, the connected components are enlisted into comps. Physically, each
component represents an indivisible envelope of light intensity. comps is
sorted by Sort-by-Distance according to the metric, $m$, which takes a vertex
subset, $v$, as its input ($v\in\id{comps}$)-

$\displaystyle m(v)=\sqrt{(\bar{x}-x_{img})^{2}+(\bar{y}-y_{img})^{2}}$
$\displaystyle\bar{x}=mean(x),\bar{y}=mean(y)\hskip 2.0pt\forall\hskip
2.0pt(x,y)\in v$

The metric is essentially the radial pixel distance of the centre of a
component, from the centre of the cutout image. This is necessary SDSS
catalogs celestial coordinates of objids with respect to an object’s centre,
and any detected peaks should lie in the intensity envelope corresponding to
the given objid

####  4.2.2 First-Group and Largest-Component

$\displaystyle S_{i}=\\{c_{j}\\},\hskip 5.0pti\in\mathbb{W}$ $\displaystyle
i=\left\lfloor\frac{m(c_{j})}{10}\right\rfloor\hskip 2.0pt\forall\hskip
2.0ptc_{j}\in S_{i},\hskip 5.0ptc_{j}\in\id{comps}$

The list $\id{comps}=[c_{1},c_{2},..,c_{n}]$, sorted by metric $m(v)$, is
aggregated into disjoint subsets by the above relation. This contrasts two
opposing factors - closeness of an envelope of intensity from the centre of
the image, and the relative size of such an envelope. From manual observation
of the image samples, it is noted that the underlying image graph is usually
disconnected (true of noisy images) and it would be remiss to simply choose
the closest component to the centre. First-Group returns the set $S_{m}$ where
$m$ is the smallest number such that $S_{m}\neq\phi$ and Largest-Component
returns the largest component in $S_{m}$ which is $max(|c_{i}|),\hskip
2.0ptc_{i}\in S_{m}$, which is then stored in lcmp

####  4.2.3 Find-Peaks-In and Sort-by-Intensity

The procedure Find-Peaks-In returns $\id{peak\\_list}\cap\id{lcmp}$ as they
are both sets of the form $\\{(x,y)\\}$. This is stored in pks and any
$img(x,y)<\texttt{signal}$ for $(x,y)\in\id{pks}$ is discarded by Filter-By-
Intensity. Based on the length of the remaning list, the verdict of SINGLE,
DOUBLE or NO_PEAK is returned.

##  5 Preliminary Detections

(a) 1237657190909149335

(b) 1237663204919345642

(c) 1237663277929988198

(d) 1237663782599393457

(e) 1237663783133708437

(f) 1237663783140065522

Figure 4: Detections from Blind-Search in Stripe-82

With CasJobs, 100,000 galaxies were chosen at random from the Stripe-82 [2]
region and classified by Gothic in 4 bands (ugri). The input file is
SDSS_Stripe82_100000.csv (section 2.2). We present some positive detections
that also have spectroscopic data available in SDSS. From their cutout images,
it can be seen that their morphology is similar to that of NGC-3758 or
MRK-212, from the Gimeno sampling. These are displayed in figure (4). Samples
such as 4(b), 4(c), 4(e) have their double peaks in the same envelope of
light. In all probability, these are true double-nuclei galaxies. The
remaining 4(a), 4(d), 4(f) are two separate, interacting galaxies.

###  5.1 Blind-Searching in Stripe-82

(a) u-band

(b) g-band

(c) r-band

(d) i-band

Figure 5: Photometric Magnitudes Distribution in the Stripe-82 Blind-Search

(a) Redshift Distribution

(b) Color Diagram

Figure 6: Distributions from Blind-Search in Stripe-82

  1. 1.

2,298 galaxies were classified as double-nuclei. These were checked manually
for the presence of large foreground stars as SDSS catalouging is not error
free

  2. 2.

1,246 images were determined to not be dominated by a large foreground star.
It was noted that the images from the z-band were extremely noisy/erratic

  3. 3.

854 images had valid entries of photometric redshift at both the detected
peaks. Of these 589 galaxies had their double peaks cataloged as unique objid
in SDSS, and the remaining 206 had non-identical objid among the their two
peaks

  4. 4.

104 of the 854 had spectral information available

The results from the blind-search are available in SDSS_Stripe82_Results.csv
(section 2.2). The distributions of the photometric redshifts and the color
diagram of the detected double-nuclei galaxies is given in figure (6). The
peak is at $z=0.356$ with a standard deviation of $\sigma_{z}=0.197$, and the
Kolmogorov-Smirnov goodness-of-fit test gives a statistic of $\approx 0.05$
and p-value of $\approx 0.012$. The distribution of the photometric magnitudes
in each of the bands is also given in figure (5).

####  5.1.1 Band Statistics

The maximum simultaneous detection of a double-peaked galaxy was in 4 bands.
The maximum detections occured in the i-band with 519 detections. Detection
counts and correlations are summarised in table (2) and (2), respectively.

####  5.1.2 Detection Accuracy

The false-positive rate is $\frac{2298-854}{2298}\times 100\approx 62.8\%$.
The false negative rate is expected to be minimal as Gothic was tested to have
$100\%$ accuracy on the Gimeno [20] sample. The high false-positive rate is
due to the obstruction of the field of view by large foreground stars in SDSS
images, and also due to the fact that SDSS labels of Star/Galaxy are faulty.

Bands | u | g | r | i
---|---|---|---|---
Total Detections | 90 | 94 | 284 | 519
Exclusive Detections | 76 | 37 | 140 | 390
Maximally Correlating Band | i | g | i | r
Detections with above | 10 | 53 | 123 | 123

Table 1: Detection information from Blind-Search

Bands | u | g | r | i
---|---|---|---|---
u |  | 8 | 8 | 10
g | 8 |  | 53 | 35
r | 8 | 53 |  | 132
i | 10 | 35 | 132 |

Table 2: Band Correlation in Blind-Search

###  5.2 Cross-Matching SDSS with Gaia

In an effort to reduce the false-positive rate, it was decided to crossmatch
the SDSS catalog with Gaia in the Stripe-82 regions. By this method, we have
sampled another 100,415 galaxies that are devoid of any stars upto $None$. To
illustrate how interspersed Gaia stars are among the SDSS galaxies, each
galaxy in from SDSS is crossmatched to its nearest star in Gaia. Subsequently,
the galaxies are binned, according to the angular distance of the nearest
crossmatch, in intervals of 10 arcseconds. The resulting plot for the
Stripe-82 region is as shown in figure (7). It was done by the KD-Tree
crossmatching utility in astropy. The input file is
SDSSxGaia_Stripe82_100434.csv (section 2.2)

Figure 7: Contamination of SDSS galaxies with Gaia stars in Stripe-82 (X-axis
: arcsecond)

  1. 1.

4,215 galaxies were classified as double-nuclei by Gothic

  2. 2.

3,573 images were determined to not be dominated by a large foreground star.

  3. 3.

3,431 images had valid entries of photometric redshift at both the detected
peaks. 3,185 had unique objid in SDSS, and 246 were non-identical.

  4. 4.

230 had spectral information available

####  5.2.1 Plots and Distributions

The results from the crossmatched search are available in
SDSSxGaia_Stripe82_Results.csv (section 2.2). The photoZ distribution has
nearly the same features as that of the random-search example - a normal
distribution with mean at $z=0.38$ and $\sigma_{z}=0.184$. Due to a higher
number of galaxies with spectral information available, we have shown the
distribution of the schlegel[8] redshift and the color-diagram for the 230
samples in figure (8). The redshift distribution does not have a clear peak,
as in the previous case, and the data points in the color-diagram are not as
diverse. Double-Nuclei galaxies with spectroscopic data would be necessary to
conduct astrophysical studies, and this has been discussed in section (6).

(a) Schlegel-Z Distribution

(b) Color Diagram

Figure 8: Distributions from Crossmatched-Search in Stripe-82

####  5.2.2 Band Statistics

The maximum simultaneous detection of a double-peaked galaxy was in 3 bands.
The maximum detections occured in the r-band with 1535 detections. Detection
counts and correlations are summarised in table (4) and (4), respectively.

Bands | u | g | r | i
---|---|---|---|---
Total Detections | 188 | 441 | 1535 | 1472
Exclusive Detections | 181 | 374 | 1363 | 1327
Maximally Correlating Band | u | r | i | r
Detections with above | 3 | 54 | 134 | 134

Table 3: Detection information from Crossmatched-Search

Bands | u | g | r | i
---|---|---|---|---
u |  | 3 | 3 | 2
g | 3 |  | 54 | 28
r | 3 | 54 |  | 134
i | 2 | 28 | 134 |

Table 4: Band Correlation in Crossmatched-Search

####  5.2.3 Detection Accuracy

The results from random-search and crossmatching with Gaia is summarised in
table (5). NDFS is the number of detections that are not dominated by a large
foreground star, which is significantly higher in the case of crossmatching.
The false positive rate is reduced by 82% and the filtered detection rate
(double peaks that correspond to a valid objid in SDSS) is increased by 320%

| Random-Search | Gaia-Crossmatch
---|---|---
Input Size | 100,000 | 100,415
Gothic Detections | 2298 | 4214
NDFS | 854 | 3753
False-Positive Rate | 63% | 11%
Filtered Detection Rate | $7.95\times 10^{-3}$ | $3.34\times 10^{-2}$

Table 5: Comparison of Blind and Crossmatched Searches

(NDFS - Not Dominated by Foreground Star)

##  6 Future Scope of Work

The photoZ attribute in SDSS is itself constructed from predictive
techniques[14, 12] and does not provide a strong basis for the astrophysical
study of any sample of double-nuclei galaxies obtained via the pipeline.
Galaxies with available spectroscopic data would be crucial to further studies
and new techniques need to be developed to obtain such sample quickly, as only
a tiny fraction of galaxies in SDSS have spectral data available. It would
also be useful to develop a technique to differentiate between truly double-
nuclei galaxies and interacting galaxies, as demonstrated in figure (4)

##  7 Conclusion and Summary

A novel detection pipeline Gothic, has been proposed, that given an image of a
galaxy from SDSS, determines whether it is possibly a double-nuclei galaxy. It
enjoys 100% accuracy in the Gimeno[20] sample of double-nuclei galaxies.
Focusing on the Stripe-82 region in SDSS, we have sampled 100,000 galaxies at
random and run it through the pipeline. Due to a high false-positive rate by
the obstruction of foreground stars in the field of view, an alternative
cross-matching technique has been used to intelligently sample the input
sources, and its superiority has been demonstrated by its higher detection and
lower false-positive rate.

##  8 Acknowledgements

The first author gratefully acknowledges the support received from the Indian
Institute of Astrophysics, Bangalore, India during an internship. The second
author would like to thank the Science and Engineering research Board (SERB),
Department of Science and Technology, Government of India, for supporting our
research by providing us with resources to conduct our experiments. The
project reference number is: EMR/2016/005687.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/. The SDSS is managed by the Astrophysical Research
Consortium for the Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck- Institute
for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory, and the University of Washington.

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##  Appendix A Details of Gothic

###  A.1 Scaling/Smoothening

FITS files deliver pixel intensity in units of nanomaggies. Due to shot noise,
and possible numerical errors in the SDSS pipeline, some of the pixels have
negative values for intensity. This is unphysical and such pixels need to be
modified before the smoothening process. If this is not done, the log-
normalisation produces a heavily skewed color-scaling, which is unfit for
further processing. A $None$ cutout centred around the object is taken. Then
the maximum negative pixel value is found out in the cutout, which is
subsequently added to all the pixels. To perform the log-normalisation, a
lower and upper limit needs to be supplied, within which the normalisation
would be conducted, subsequently followed by convolution with a $5\times 5$
gaussian kernel -

$\displaystyle\id{lower}\leftarrow\id{max}(0.1,\
\frac{\id{median}(\id{cutout})}{2})$
$\displaystyle\id{upper}\leftarrow\id{max}(\id{cutout})$

(a) Before smoothening

(b) Gaussian kernel convolved

Figure 9: Smoothening Cutout Image

###  A.2 Edge Detection

Canny [10] has been used as the standard edge-detector. The edges detected do
not necessarily correspond to the physical boundary of the object, since Canny
utilizes the Sobel [41] operator which computes a first-order spatial
derivative of the image. Using two kernels, the Sobel operator approximates
$\displaystyle{\partialderivative{x}}$ and
$\displaystyle{\partialderivative{y}}$ at every pixel which is subsequently
replaced by
$\sqrt{{\left(\partialderivative{x}\right)}^{2}+{\left(\partialderivative{y}\right)}^{2}}$,
that constitutes the magnitude of the edge.

Apart from the edges detected at the border of the galaxy envelope, it is
possible that some edges of high magnitude are detected within it. Moreover,
the edges at the envelope border don’t necessarily form a closed figure.
Hence, the convex hull [38] of the edges is used to bound the light envelope
of the gal1axy in a closed polygon. The statement of the convex hull problem
is as follows - Given a set of points $S=\\{(x_{i},y_{i})\\}$, find the
smallest convex subset $H$ of $S$ such that every member of $S$ lies within
the polygon defined by $H$. The definition of a convex subset $H$ is as
follows -

$\displaystyle(x,y)\in interior(H)$ $\displaystyle\forall\hskip 2.0pt(x,y)\in
L$ $\displaystyle\forall\hskip 2.0pt(x_{1},y_{1}),(x_{2},y_{2})\in
interior(H)$

where $L$ is the line-segment between $(x_{1},y_{1})$ and $(x_{2},y_{2})$, and
$interior(H)$ being the region contained by H. In other words, all points in
any line segment within the interior of the hull, lie within the hull. Figure
(A.10) shows this operation. The procedure accurately bounds the central
galaxy within the cutout, however, it’s possible it also includes stray
foreground objects within the field of view. This is handled by the techniques
discussed in (4.2)

(a) Canny Edges

(b) Convex Hull on Canny

Figure 10: Edge Detection

###  A.3 Intensity Distribution Fitting

The convex hull method, as demonstrated above, usually encloses more than the
galaxy envelope. Frequently, stray objects are enclosed within it. For bands
with weak signals, the hull might enclose a large region without any
appreciable signal. Thus, it is important to identify the pixel value which
separates noise and signal. The Sersic [36] profile is widely used to fit the
intensity distribution of galaxies. Intensity of light of a galaxy is taken to
be a function of the radial distance from its centre. It is typically
expressed in terms of the log of the intensity function. This form of is
convenient as the original intensity data has been log-normalised. Hence,
except by a constant factor, the sersic profile is applicable for intensity
fitting -

$\displaystyle{\log I(R)=\log I_{0}-kR^{\frac{1}{n}}}$

The parameter $n$ is known as the sersic index, which characterises the shape
of the plot. It has been stated in [11] that $1\leq n\leq 15$ can fit most
galaxies. However, we have found that $0.25\leq n\leq 15$ gives a better fit
for our purposes as we are observing the galaxies in 4 different bands.

####  A.3.1 Translating Frequency to Radius

Figure 11: Solutions on Cartesian Grid

It is a simple matter to compute the histogram of the pixel intensity values.
However, the sersic profile requires intensity values to be fit against radius
from the centre of the galaxy, and not its respective frequency count. Thus,
it’s necessary to translate the frequency of pixel values to their distance
from the centre of a galaxy. This is achieved by finding the number of
solutions for (given $R$) -

$\displaystyle x+y=R$ $\displaystyle x,y\in\mathbb{Z}$

Let the number of solutions be $S_{R}$. It can be seen that the enclosing
rectangle for $S_{4}$ can be shifted one step outward, except the four
vertices, to partially cover the edges of $S_{5}$, after which 8 points would
be remaining. Thus, the recurrence is -

$\displaystyle{S_{R+1}=\left(S_{R}-4\right)+8=S_{R}+4}$

The base case is $S_{1}=4$, which gives $S_{R}=4R$. In another words, if the
number of solutions is taken to be the frequency of a particular pixel value,
then $f=4R$

####  A.3.2 Setting the fit function

The maximum pixel value enclosed in the hull region, $P_{max}$, translates to
$\log I_{0}$ as the cutout has been log-normalised. With $P(f)$ as the inverse
of the pixel value-frequency histogram $f(P)$, the function to be fit for the
parameters $k,n$ is -

$\displaystyle{P(f)=P_{max}-k\left({\frac{f}{4}}\right)^{\frac{1}{n}}}$

Figure 12: Fitting the Sersic Profile

####  A.3.3 Inferring the noise and signal levels

(a) Hull Region

(b) Region above noise

Figure 13:

From the fit, the noise and signal levels are inferred which aid in searching
the intensity peaks in the image. Peak searching is only performed in those
regions that lie above the noise level (noise). This step is necessary as the
residual coarseness of the image results in erroneous peaks. We infer the
noise level from the radius at which the Sersic fit gives 95% of the
integrated light. The signal level is also determined by the same process, but
at 50% of the intergrated light, and referred to as (signal). The regions of
the image above the noise level are marked in turquoise in figure (13(b)).