2.5cm2.5cm2.5cm2.5cm

Necessary and sufficient conditions for a family of continuous functions to
form a Karhunen-Loève basis

Ricardo CARRIZO VERGARA
Université de Paris II Panthéon-Assas

<EMAIL_ADDRESS>
August 2022

The following fact is widely known and used in both practice and theory of
stochastic processes: if $D\subset\mathbb{R}^{d}$ is a compact set, and if
$K:D\times D\to\mathbb{R}$ is a continuous positive-definite Kernel (also
called a covariance Kernel), then $K$ has a decomposition in Fourier series of
$L^{2}(D\times D)$ of the form

$K=\sum_{n\in\mathbb{N}}\lambda_{n}f_{n}\otimes f_{n},$ (1)

where $(\lambda_{n})_{n\in\mathbb{N}}\in\ell^{1}(\mathbb{N})$ is a sequence of
non-negative numbers, and $(f_{n})_{n\in\mathbb{N}}$ is an orthonormal basis
of $L^{2}(D)$. The pair $(f_{n})_{n},(\lambda_{n})_{n}$ consists in the
eigenfunctions and eigenvalues of the associated covariance operator (Hilbert-
Schmidt Theorem [4, Theorem VI.16]), having

$\lambda_{n}f_{n}(x)=\int_{D}K(x,y)f_{n}(y)dy.$ (2)

From equation (2) it is easy to verify that for $\lambda_{n}>0$, the function
$f_{n}$ is continuous. The cases in which $\lambda_{n}=0$ do not really
intervene in the decomposition (1), hence the decomposition consists
essentially of continuous functions. For simplicity, we will suppose
$\lambda_{n}>0$ for every $n\in\mathbb{N}$ and that the family $(f_{n})_{n}$
is thus constituted of continuous functions (the family is hence not
necessarily a complete orthonormal system of $L^{2}(D)$). In addition, it is
possible to prove that the convergence (1) is not only in the sense of
$L^{2}(D\times D)$ but also uniform over $D\times D$ (Mercer’s Theorem, which
uses Dini’s Theorem to prove the result [3]). Those facts are the basis of the
well-known and used Karhunen-Loève expansion of mean-square continuous
stochastic processes over compact sets of the Euclidean space [2, 1].

The question we answer here is the one concerning the converse of such result:
given an orthonormal system of $L^{2}(D)$ consisting of continuous functions
$(f_{n})_{n}$, and given a sequence of strictly positive numbers
$(\lambda_{n})_{n}\in\ell^{1}(\mathbb{N})$, does the limit

$\lim_{n\to\infty}\sum_{j\leq n}\lambda_{j}f_{j}\otimes f_{j}$ (3)

exists uniformly over $D\times D$, defining thus a continuous covariance
Kernel? Note that the limit (3) always exists in the $L^{2}(D\times D)$ sense,
so the real question is if the resulting limit is continuous. By Dini’s
Theorem, using similar arguments as those used in the proof of Mercer’s
Theorem, one can conclude that if such limit is continuous, the convergence
must be uniform.

The next result shows that the functions $(f_{n})_{n}$ and the coefficients
$(\lambda_{n})_{n}$ must satisfy an extra criterion. Hence, not any pair
basis-coefficients can be used to construct a Karhunen-Loève expansion.

######  Theorem 0.1.

Let $D\subset\mathbb{R}^{d}$ be a compact set. Let $(f_{n})_{n\in\mathbb{N}}$
be an orthonormal system of $L^{2}(D)$ consisting of continuous functions, and
let $(\lambda_{n})_{n\in\mathbb{N}}\in\ell^{1}(\mathbb{N})$ be a sequence of
strictly positive numbers. Then, the limit (3) converges (uniformly) over
$D\times D$ to a continuous covariance Kernel if and only if the sequence of
functions

$v_{n}:=\sum_{j\leq n}\lambda_{j}f_{j}^{2},\quad n\in\mathbb{N},$ (4)

is equicontinuous over $D$.

Proof: For the necessity, suppose $K$ is a continuous covariance kernel having
the expansion (1). Using Cauchy-Schwartz inequality, one concludes for every
$x,y\in D$,

$\displaystyle|v_{n}(x)-v_{n}(y)|$ $\displaystyle=\left|\sum_{j\leq
n}\lambda_{j}(f^{2}_{j}(x)-f^{2}_{j}(y))\right|$ (5)
$\displaystyle\leq\sqrt{\sum_{j\leq
n}\lambda_{j}(f_{j}(x)-f_{j}(y))^{2}}\sqrt{\sum_{j\leq
n}\lambda_{j}(f_{j}(x)+f_{j}(y))^{2}}$
$\displaystyle\leq\sqrt{\sum_{j\in\mathbb{N}}\lambda_{j}(f_{j}(x)-f_{j}(y))^{2}}\sqrt{2\sum_{j\in\mathbb{N}}\lambda_{j}(f_{j}^{2}(x)+f_{j}^{2}(y))}$
$\displaystyle=\sqrt{K(x,x)-2K(x,y)+K(y,y)}\sqrt{2(K(x,x)+K(y,y))}$
$\displaystyle\leq 2\sqrt{\|K\|_{\infty}}\sqrt{K(x,x)-2K(x,y)+K(y,y)}.$

Since $K$ is uniformly continuous over the compact $D\times D$, for any given
$\epsilon>0$ there exists $\delta>0$ such that
$|K(u,v)-K(x,y)|<\frac{\epsilon^{2}}{8\|K\|_{\infty}}$ if
$|u-x|+|v-y|<\delta$. Hence, if $|x-y|<\delta$,

$|K(x,x)-2K(x,y)+K(y,y)|\leq|K(x,x)-K(x,y)|+|K(y,y)-K(x,y)|<\frac{\epsilon^{2}}{4\|K\|_{\infty}}.$
(6)

Using this in inequality (5), we obtain that if $|x-y|<\delta$ then

$|v_{n}(x)-v_{n}(y)|\leq\epsilon,\quad\forall n\in\mathbb{N},$ (7)

hence the family $(v_{n})_{n}$ is indeed equicontinuous.

For the sufficiency, suppose the sequence $(v_{n})_{n}$ is equicontinuous over
$D$. For $n,m\in\mathbb{N}$, using the orthonormality of the family
$(f_{n})_{n}$ and the fact that $\sum_{n\in\mathbb{N}}\lambda_{n}<\infty$, we
have

$\int_{D}|v_{n}-v_{m}|dx=\int_{D}\sum_{j=n\wedge m+1}^{n\vee
m}\lambda_{j}f_{j}^{2}dx=\sum_{j=n\wedge m+1}^{n\vee
m}\lambda_{j}\xrightarrow[n,m\to\infty]{}0.$ (8)

Hence $(v_{n})_{n}$ is a Cauchy sequence of $L^{1}(D)$, hence it converges by
completeness to a function $v\in L^{1}(D)$. Now, the converges in $L^{1}(D)$
implies the existence of a sub-sequence $(v_{n_{k}})_{k}$ which converges
almost-everywhere in $D$ to a function which must coincide with $v$ almost-
everywhere. Let $\tilde{D}\subset D$ be the full-measure set where the sub-
sequence $(v_{n_{k}})_{k}$ converges point-wise to $v$. Hence, for every
$x\in\tilde{D}$, the sequence $(v_{n}(x))_{n}$ has a sub-sequence which
converges to $v(x)$. But since the sequence $v_{n}(x)$ consists in the partial
sums of a series of positive numbers, then the convergence of a sub-sequence
implies the convergence of the entire sequence $(v_{n}(x))_{n}$. We conclude
hence that $v_{n}$ converges to $v$ point-wise over $\tilde{D}$. Since
$\tilde{D}$ has full-measure, it must be dense in $D$. Since the family
$(v_{n})_{n}$ is equicontinuous and it converges point-wise to a function $v$
in a dense subset of $D$, then the convergence is everywhere in $D$ and the
function $v$ must be continuous [4, Theorem I.26]. In addition, since the
sequence $(v_{n})_{n}$ is monotonically increasing, by Dini’s theorem the
convergence must be uniform.

The uniform convergence of $(v_{n})_{n}$ and the Cauchy-Schwarz inequality
allow us to conclude

$\begin{aligned} \sup_{(x,y)\in D\times D}\left|\sum_{j=n\wedge m+1}^{n\vee
m}\lambda_{j}f_{j}(x)f_{j}(y)\right|&\leq\sup_{(x,y)\in D\times
D}\sqrt{\sum_{j=n\wedge m+1}^{n\vee
m}\lambda_{j}f_{j}^{2}(x)}\sqrt{\sum_{j=n\wedge m+1}^{n\vee
m}\lambda_{j}f_{j}^{2}(y)}\\\
&=\|v_{n}-v_{m}\|_{\infty}\xrightarrow[n,m\to\infty]{}0\end{aligned}.$ (9)

From the Cauchy-sequence convergence criterion, the series
$\sum_{j\in\mathbb{N}}\lambda_{j}f_{j}\otimes f_{j}$ converges uniformly over
$D\times D$ to some function $K$ which must hence be continuous, and by
construction, it is also a positive-definite Kernel. $\blacksquare$

## References

  * [1] G. Giambartolomei.  The Karhunen-Loève Theorem.  Master’s thesis, Università di Bologna, Italy, 2015.
  * [2] M. Loève.  Probability theory, Vol. II.  Graduate texts in mathematics, 46:0–387, 1978.
  * [3] J. Mercer.  Functions of positive and negative type and their connection with the theory of integral equations.  Philos. Trinsdictions Rogyal Soc, 209:4–415, 1909.
  * [4] M. Reed and B. Simon.  Methods of modern mathematical analysis: Functional Analysis.  Academic Press, Singapore, 1980.


# FAST Discovery of Eight Isolated Millisecond Pulsars in NGC 6517

Dejiang Yin College of Physics, Guizhou University, Guiyang 550025, China  Li-
yun Zhang College of Physics, Guizhou University, Guiyang 550025, China
International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216,
China  Lei Qian Guizhou Radio Astronomical Observatory, Guizhou University
Guiyang 550025, China  National Astronomical Observatories, Chinese Academy of
Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, China  CAS Key
Laboratory of FAST, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  College of Astronomy and Space Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Ralph P.
Eatough National Astronomical Observatories, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beijing 100101, China  Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, D-53121, Bonn, Germany  Baoda Li  College
of Physics, Guizhou University, Guiyang 550025, China  Duncan R. Lorimer
Department of Physics and Astronomy, West Virginia University, Morgantown, WV
26506-6315, USA  Center for Gravitational Waves and Cosmology, West Virginia
University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA
Yinfeng Dai College of Physics, Guizhou University, Guiyang 550025, China
Yaowei Li  College of Physics, Guizhou University, Guiyang 550025, China
Xingnan Zhang  State Key Laboratory of Public Big Data, Guizhou University,
Guiyang 550025, China  Minghui Li  State Key Laboratory of Public Big Data,
Guizhou University, Guiyang 550025, China  Tianhao Su  College of Physics,
Guizhou University, Guiyang 550025, China  Yuxiao Wu  Chongqing University of
Posts and Telecommunications, Chongqing, 40000, China  Yu Pan  Chongqing
University of Posts and Telecommunications, Chongqing, 40000, China  Yujie
Lian Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China  Department of Astronomy, Beijing Normal
University, Beijing 100875, China  Tong Liu  National Astronomical
Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District,
Beijing 100101, China  Zhen Yan Shanghai Astronomical Observatory, Chinese
Academy of Sciences, Shanghai 200030, China  Zhichen Pan Guizhou Radio
Astronomical Observatory, Guizhou University Guiyang 550025, China  National
Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road,
Chaoyang District, Beijing 100101, China  CAS Key Laboratory of FAST, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
College of Astronomy and Space Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the discovery of 8 isolated millisecond pulsars in Globular Cluster
(GC) NGC 6517 using the Five-Hundred-meter Aperture Spherical radio Telescope
(FAST). The spin periods of those pulsars (namely PSR J1801–0857K to R, or,
NGC 6517K to R) are all shorter than 10 ms. With these discoveries, NGC 6517
is currently the GC with the most known pulsars in the FAST sky. The largest
difference in dispersion measure of the pulsars in NGC 6517 is 11.2 cm-3 pc,
the second among all GCs. The fraction of isolated pulsars in this GC (16 of
17, 94$\%$) is consistent with previous studies indicating an overabundance of
isolated pulsars in the densest GCs, especially in those undergoing cluster
core collapse. Considering the FAST GC pulsar discoveries, we modeled the GC
pulsar population using the empirical Bayesian method described by Turk and
Lorimer with the recent counts. Using this approach, we find that the expected
number of potential pulsars in GCs seems to be correlated with the central
escape velocity, hence, the GCs Liller 1, NGC 6441, M54 (NGC 6715), and
$\omega$-Cen (NGC 5139) are expected to host the largest numbers of pulsars.

Globular star clusters (656); Millisecond pulsars (1062); Radio telescopes
(1360)

##  1 Introduction

The proportion of millisecond pulsars in Globular Clusters (GCs) with spin
periods shorter than 30 ms is $\sim 97\%$ (e.g., Yin et al. 2023), much higher
than that in the field population ($\sim 12\%$111Manchester et al. 2005, 1.70
version, https://www.atnf.csiro.au/people/pulsar/psrcat/), making them rich
targets for fast spinning pulsars and exotic binary systems, such as Terzan
5ad (with the highest known spin frequency of 716 Hz, Hessels et al. 2007);
M4A (the triple system with a planet, Sigurdsson et al. 2003); M71E (a binary
in M71 with an orbit of 53 minutes, Pan et al. 2023); M15C (a double neutron
star binary, Anderson et al. 1990); and NGC 1851E (a pulsar–black hole system
candidate, Barr et al. 2024). GCs have been the subject of pulsar searches for
more than 30 years. Since the first GC pulsar was discovered (Lyne et al.,
1987), 322 radio pulsars have been detected in 41 GCs as of May
2023222https://www3.mpifr-bonn.mpg.de/staff/pfreire/GCpsr.html. Previous
surveys for GC pulsars have been carried out with telescopes across the world,
including Lovell (e.g., Lyne et al. 1987), Parkes (e.g., Robinson et al.
1995), Arecibo (e.g., Hessels et al. 2007), Green Bank Telescope (GBT, e.g.,
Ransom et al. 2005), and Giant Metrewave Radio Telescope (GMRT, e.g., Gautam
et al. 2022).

In the past 5 years alone, about 120 GC pulsars have been discovered with
newly built telescopes. MeerKAT (Jonas & MeerKAT Team 2016) found $\sim$ 80 GC
pulsars (e.g., Ridolfi et al. 2021, TRAPUM
project333http://trapum.org/discoveries/) and the telescope used in this work,
Five-hundred-meter Aperture Spherical radio Telescope (FAST, Nan et al. 2011)
discovered another $\sim$ 40 (e.g., Pan et al. 2020, FAST GC Pulsar
Discoveries444https://fast.bao.ac.cn/cms/article/65/). With a sensitivity
limit of 4 $\mu$Jy, 100 to 300 GC pulsars are expected to be discovered by
SKA1-MID when it comes online (Hessels et al., 2015).

Among 170 Galactic GCs, there are 45 in the FAST sky. One of these 45 GCs, NGC
6517 is located at $\alpha_{\rm J2000}=18^{\rm h}01^{\rm m}50.52^{\rm s}$,
$\delta_{\rm
J2000}=-08^{\rm\circ}57^{\rm{}^{\prime}}31.6^{\rm{}^{\prime\prime}}$. It is
the densest GC ($\rho_{0}=10^{5.29}L_{\odot}{\rm pc}^{-3}$) in the FAST sky.
Its core and half-light radius are $r_{\rm c}=0.06{\rm{}^{\prime}}$ and
$r_{\rm h}=0.50{\rm{}^{\prime}}$, respectively (Harris 1996, 2010 edition),
well within the FAST half-power beamwidth at L-band ($\sim$2.9′, Qian et al.
2020). The four pulsars, PSR J1801$-$0857A to D (NGC 6517A to D), were
detected by the GBT (Lynch et al., 2011). The dispersion measures (DMs) of
those pulsars range from 174.71 to 182.56 cm-3 pc. PSR J1801$-$0857E, F, and G
(NGC 6517E, F, and G) were discovered by FAST with a candidate selection
method based on DM–signal-to-noise ratio curves (Pan et al., 2021a). PSR
J1801$-$0857H and I (NGC 6517H and I) were also discovered at FAST in long
tracking observations (Pan et al., 2021b). Currently, NGC 6517B is the only
binary pulsar ($\sim$60 d orbital period) detected in this GC. Among all 41
GCs currently known to host pulsars, NGC 6517 ranks third highest in terms of
the fraction of isolated pulsars, at 94%. Only NGC 6522, where 6 out of the 6
known pulsars are isolated, and Terzan 1 (8 out of 8) have a higher fraction.
This is in spite of the fact that, as described below, our analyses of NGC
6517 include acceleration searches which aim to retain sensitivity to binary
systems.

Located at 10.4 and 10.6 kpc away from the Sun (Harris 1996, 2010 edition),
NGC 6517 and M15 (NGC 7078) respectively, are the only two core-collapsed GCs
(e.g., Verbunt & Freire 2014) in the FAST sky. The isolated pulsars in NGC
6517 and M15 dominate the pulsar population, similar to other core-collapsed
GCs outside the FAST sky, such as Terzan 1 (8 isolated, 0 binaries) and NGC
6624 (10 isolated, 2 binaries). In Verbunt & Freire (2014) a dynamical
parameter, the encounter rate per binary in a GC, $\gamma$
($\gamma\propto\rho_{0}^{0.5}r_{c}^{-1}$, $\rho_{0}$ is the central mass
density and $r_{c}$ is the core radius), was used as a factor to characterize
the differences between the pulsar populations of different GCs. Original
millisecond pulsar binaries are thought to be disrupted by an exchange
encounter, enabled by the higher $\gamma$ in core-collapsed clusters. The high
fraction of isolated pulsars in core-collapsed GCs is therefore a natural
result of the increased number of these encounter and disruption events. In
addition, several alternative formation scenarios are proposed to explain the
seeming overabundance of isolated millisecond pulsars in core-collapsed
clusters, such as the formation of single millisecond pulsars via accretion
following mergers of neutron stars and main-sequence stars (e.g., Davies et
al. 1992; Camilo & Rasio 2005; Kremer et al. 2022), the formation of single
pulsars via mergers of massive white dwarf binaries (e.g., King et al. 2001;
Schwab 2021; Kremer et al. 2023) , and the number of single millisecond
pulsars in GCs can significantly be boosted by both main-sequence star tidal
disruption events and merger-induced collapses discussed in the recent N-body
simulations (Ye et al., 2024).

Ranking third among the GCs with the most pulsars in the FAST sky, NGC 6517 is
one of pulsar search targets with the highest chance for further discoveries
(see, e.g., Turk & Lorimer, 2013). Motivated by this potentially significant
pulsar population, in this work we present the discovery of an additional
eight pulsars in NGC 6517, bringing the total number to 17 in this cluster.
The new discoveries were made during the reprocessing of FAST archival data
and increase the total number of GC pulsars discovered by FAST to 49. Pulsar
population simulations can help to predict the expected number of GC pulsars.
Due to sensitivity limits and the selection effects of binary pulsar search
algorithms, the number of known pulsars in a given GC can be much lower than
predictions from simulations. The stellar encounter rate $\Gamma$, which is
estimated with $\Gamma\propto\rho_{0}^{1.5}r_{c}^{2}$ ($\rho_{0}$ is the
central mass density and $r_{c}$ is the core radius, Pooley et al. 2003), has
also been used to estimate the GC pulsar numbers (e.g., Pooley et al. 2003;
Abdo et al. 2009). The number of potential pulsars in a GC is scaled by its
$\Gamma$ in the empirical Bayesian model of Turk & Lorimer (2013). This method
was used to estimate the total population of pulsars for all Galactic GCs and
predicted a number of 2280${}_{-1490}^{+2720}$ (Turk & Lorimer, 2013). In
light of the number of GC pulsars discovered by FAST, it is timely to revisit
the population analysis carried out a decade ago by Turk & Lorimer (2013).

The observations and data reduction are presented in Section 2. The
discoveries of pulsars and their timing solutions are given in Section 3.
Section 4 provides an updated analysis of the pulsar population in GCs using
the procedures described in Turk & Lorimer (2013). A summary of the findings
is given in Section 5.

##  2 Observations and Data Reduction

###  2.1 Observations

As a part of the project “Globular Cluster FAST: A Neutron-star Survey” (GC
FANS, Pan et al. 2021b; Wu et al. 2023), the first observation of NGC 6517 was
performed on June 24, 2019. By December 31, 2022, a total of 19 observations
with integration lengths between 1400 s and 9000 s were carried out. The data
were taken with the central beam of the FAST 19-beam receiver covering a
frequency range of 1.0-1.5 GHz. The data were channelized into 4096 channels
(channel width $0.122\,{\rm MHz}$). All observations were performed with the
tracking mode with a sampling time of $49.152\,\mu{\rm s}$ and packaged into
the standard search-mode PSRFITS data format (Hotan et al., 2004). During each
observation, the central beam was pointed at the center of NGC 6517.

###  2.2 Data Reduction

PRESTO555https://github.com/scottransom/presto (Ransom, 2001; Ransom et al.,
2002) and TEMPO666http://nanograv.github.io/tempo/ (Nice et al., 2015) were
used for pulsar search and timing respectively. In the following, we first
describe the pulsar search procedure, followed by the timing analysis.

The routine rfifind from PRESTO was used to identify and mask the detrimental
effects of terrestrial radio frequency interference (RFI) in both the time and
frequency domains. The prepdata or prepsubband routines were used to
dedisperse the PSRFITS data over a range of trial DM values. Since the DM of
previously known pulsars in NGC 6517 ranges from $174.7\,{\rm cm}^{-3}\,{\rm
pc}$ (NGC 6517D) to 185.6 cm-3 pc (NGC 6517G), with an average of
$181.3~{}{\rm cm}^{-3}\,{\rm pc}$, dedispersion was performed across a DM
range of 170 to 190 cm-3 pc, with a step of 0.05 cm-3 pc. This DM coverage was
determined to maintain sensitivity to any undiscovered pulsars in NGC 6517.

To identify periodic signals, the dedispersed time series were transformed
into the fluctuation frequency domain using the routine realfft. The routine
accelsearch was then used to perform Fourier domain acceleration searches,
with harmonic summing to account for pulses of narrow duty cycle. The
accelsearch parameter zmax represents the largest drift of the pulse
frequency, in Fourier bins, under the assumption of a constant acceleration
(Ransom et al. 2002). We performed a search for isolated pulsars with a zmax
of 20 and a segmented acceleration search (with time series of length down to
$1/5$ of the original integration time which ranged from 500 s to 1800 s) for
binaries and with a zmax of 1200.

So-called candidate “sifting” routines were then used to amalgamate multiple
detections of each pulsar candidate from the accelsearch results. Both the
routine ACCEL_sift.py of PRESTO and
JinglePulsar777https://github.com/jinglepulsar/jinglesifting were used for
this task. We also identified any candidates that appeared on different days,
but with similar spin period and DM values. This method will be described in
another publication (Yu et al., in preparation).

In the final stage of the pulsar search, the prepfold routine was used to fold
all sifted pulsar candidates using either dedispersed time series, or raw
PSRFITs data, to generate standard pulsar candidate diagnostic plots for
visual inspection.

The timing analysis of detected pulsars was performed as follows:

  1. 1.

The observational data were dedispersed to time series according to the
candidate DM value. During this process, the RFI “mask” generated earlier from
the routine rfifind was used.

  2. 2.

The dedispersed and RFI cleaned time series were folded at the pulsar spin
period in the topocentric reference frame by the routine prepfold.

  3. 3.

The PRESTO routines pygaussfit.py and get_TOAs.py were utilized to form a
standard reference template profile and extract the times of arrival (TOAs),
respectively.

  4. 4.

The initial pulsar ephemeris was then iterated using TEMPO over progressively
longer time spans until a phase-connected timing solution was obtained.

A phase-connected timing solution means that every rotation of the pulsar can
be accurately predicted by an ephemeris. The pyplotres.py routine from PRESTO
was used to realize interactive inspection of timing residuals. The
ephemerides of pulsars were gradually fitted and updated by removing the
arbitrary phase offsets (with so-called “JUMPs”) between epochs. In the
archival observations of NGC 6517, the gap between observations could be up to
more than one year. The script Dracula888https://github.com/pfreire163/Dracula
was also used to determine the exact rotation count of pulsars (Freire &
Ridolfi, 2018).

##  3 Results

Eight isolated millisecond pulsars in NGC 6517, namely PSR J1801–0857K to R
(NGC 6517K to R)999The previously announced pulsar NGC 6517J, was subsequently
identified as the 17th spin frequency harmonic of NGC 6517B., were discovered
during the reprocessing of FAST archival data. The same data set was used to
time these newly discovered pulsars and the other known pulsars in NGC 6517.
Our discoveries make NGC 6517 the GC with the most known pulsars in the FAST
sky, hosting a total of 17 pulsars. The spin periods of the new pulsars are
all shorter than 10 ms. Their pulse profiles and timing residuals are in
Figure 1. The timing solution of NGC 6517B, the only known binary in the GC,
was obtained with the “BT” model (Blandford & Teukolsky, 1976), which includes
the projected pulsar semi-major axis ($x_{\rm p}$), eccentricity ($e$), epoch
of periastron passage ($T_{\rm 0}$), orbital period ($P_{\rm b}$) and
longitude of periastron passage ($\omega$).

Table 1: The timing solutions of known pulsars NGC 6517A to I and new pulsars NGC 6517K to O. In the timing analysis, the DE438 Solar System Ephemeris and the TBD time units are used and the times were not referenced to one of the atomic time standards (UNCORR). The timing data spans from MJD 58659 to 59944 for NGC 6517A to M, MJD 58688 to 59944 for NGC 6517N and NGC 6517O.  Pulsar | J1801–0857A | J1801–0857B | J1801–0857C
---|---|---|---
Right ascension, $\alpha$ (J2000) | 18:01:50.61097(7) | 18:01:50.5654(1) | 18:01:50.73891(5)
Declination, $\delta$ (J2000) | –08:57:31.905(3) | –08:57:32.867(7) | –08:57:32.761(3)
Spin frequency, $f$ (s-1) | 139.36088855893(2) | 34.52849284047(2) | 267.47267670678(3)
1st spin frequency derivative, $\dot{f}$ (Hz s-2) | 9.9034(4)$\times 10^{-15}$ |  $-$2.6270(2)$\times 10^{-15}$ | 4.5058(8)$\times 10^{-15}$
Reference epoch (MJD) | 58871.058570 | 58871.058570 | 58871.058570
Dispersion measure, DM (pc cm-3) | 182.655(3) | 182.402(3) | 182.356(2)
Number of TOAs | 149 | 152 | 150
Post-fit residual RMS ($\mu$s) | 17.78 | 25.10 | 15.15
Binary Parameters
Binary model | – | BT | –
Projected semi-major axis, $x_{\rm p}$ (lt-s) | – | 33.87542(1) | –
Orbital eccentricity, $e$ | – | 3.82259(6)$\times 10^{-2}$ | –
Longitude of periastron, $\omega$ (deg) | – |  $-$57.8927(7) | –
Epoch of periastron passage, $T_{0}$ (MJD) | – | 54757.7229(1) | –
Orbital period, $P_{b}$ (days) | – | 59.8364531(8) | –
Derived Parameters
Spin period, $P$ (ms) | 7.175614409039(1) | 28.96158846609(1) | 3.7386996395757(5)
1st Spin period derivative, $\dot{P}$ (s s-1) |  $-$5.0992(2)$\times 10^{-19}$ | 2.2034(2)$\times 10^{-18}$ |  $-$6.298(1)$\times 10^{-20}$
|  |  |
Pulsar | J1801–0857D | J1801–0857E | J1801–0857F
Right ascension, $\alpha$ (J2000) | 18:01:55.36430(7) | 18:01:50.6240(1) | 18:01:50.7417(1)
Declination, $\delta$ (J2000) | –08:57:24.316(3) | –08:57:31.331(8) | –08:57:31.292(8)
Spin frequency, $f$ (s-1) | 236.60059797388(3) | 131.55003246018(5) | 40.17357389120(1)
1st Spin frequency derivative, $\dot{f}$ (Hz s-2) |  $-$4.211(8)$\times 10^{-16}$ | 1.7928(1)$\times 10^{-14}$ | 4.3105(3)$\times 10^{-15}$
Reference Epoch (MJD) | 58871.058570 | 58871.058570 | 58871.058570
Dispersion measure, DM (pc cm-3) | 174.537(3) | 183.184(5) | 183.785(5)
Number of TOAs | 151 | 146 | 152
Post-fit residual RMS ($\mu$s) | 17.36 | 40.32 | 34.35
Derived Parameters
Spin period, $P$ (ms) | 4.2265320061042(6) | 7.601670492196(3) | 24.891985032456(8)
1st Spin period derivative, $\dot{P}$ (s s-1) | 7.52(1)$\times 10^{-21}$ |  $-$1.03598(6)$\times 10^{-18}$ |  $-$2.6708(2)$\times 10^{-18}$
|  |  |
Pulsar | J1801–0857G | J1801–0857H | J1801–0857I
Right ascension, $\alpha$ (J2000) | 18:01:50.0979(7) | 18:01:52.5989(2) | 18:01:53.5073(2)
Declination, $\delta$ (J2000) | –08:57:27.53(3) | –08:57:45.04(1) | –08:57:41.84(1)
Spin frequency, $f$ (s-1) | 19.38308799410(3) | 177.21938880576(8) | 307.2974449774(2)
1st Spin frequency derivative, $\dot{f}$ (Hz s-2) |  $-$4.01(7)$\times 10^{-17}$ |  $-$1.078(2)$\times 10^{-15}$ | 1.550(4)$\times 10^{-15}$
Reference epoch (MJD) | 58871.058570 | 58871.058570 | 58871.058570
Dispersion measure, DM (pc cm-3) | 185.06(2) | 179.634(9) | 177.877(9)
Number of TOAs | 145 | 141 | 70
Post-fit residual RMS ($\mu$s) | 136.77 | 49.96 | 34.58
Derived Parameters
Spin period, $P$ (ms) | 51.59136667513(8) | 5.642723444307(3) | 3.254176096627(2)
1st Spin period derivative, $\dot{P}$ (s s-1) | 1.07(2)$\times 10^{-19}$ | 3.431(6)$\times 10^{-20}$ |  $-$1.642(4)$\times 10^{-20}$
|  |  |
Pulsar | J1801–0857K | J1801–0857L | J1801–0857M
Right ascension, $\alpha$ (J2000) | 18:01:50.7212(9) | 18:01:49.5847(7) | 18:01:50.6297(4)
Declination, $\delta$ (J2000) | –08:57:32.56(4) | –08:57:08.06(4) | –08:57:31.45(2)
Spin frequency, $f$ (s-1) | 104.2699714144(2) | 165.0902463911(2) | 186.6773548929(1)
1st Spin frequency derivative, $\dot{f}$ (Hz s-2) |  $-$8.893(4)$\times 10^{-15}$ | 9.21(6)$\times 10^{-16}$ |  $-$1.6932(3)$\times 10^{-14}$
Reference epoch (MJD) | 58868.079260 | 58867.093100 | 58871.058500
Dispersion measure, DM (pc cm-3) | 182.38(2) | 185.74(2) | 183.18(1)
Number of TOAs | 54 | 60 | 59
Post-fit residual RMS ($\mu$s) | 108.18 | 93.82 | 55.52
Derived Parameters
Spin period, $P$ (ms) | 9.59048886688(2) | 6.057293037354(9) | 5.356836133518(4)
1st Spin period derivative, $\dot{P}$ (s s-1) | 8.180(3)$\times 10^{-19}$ |  $-$3.38(2)$\times 10^{-20}$ | 4.8586(9)$\times 10^{-19}$
|  |  |
Pulsar | J1801–0857N | J1801–0857O |
Right ascension, $\alpha$ (J2000) | 18:01:50.6318(5) | 18:01:50.7720(3) |
Declination, $\delta$ (J2000) | –08:57:34.50(2) | –08:57:33.37(2) |
Spin frequency, $f$ (s-1) | 200.2170232280(2) | 233.2571878284(2) |
1st Spin frequency derivative, $\dot{f}$ (Hz s-2) | 2.1306(5)$\times 10^{-14}$ |  $-$7.149(4)$\times 10^{-15}$ |
Reference epoch (MJD) | 58868.079200 | 59941.134200 |
Dispersion measure, DM (pc cm-3) | 182.642(9) | 182.49(1) |
Number of TOAs | 57 | 54 |
Post-fit residual RMS ($\mu$s) | 68.11 | 46.74 |
Derived Parameters
Spin period, $P$ (ms) | 4.994580300304(5) | 4.287113333183(4) |
1st Spin period derivative, $\dot{P}$ (s s-1) |  $-$5.315(1)$\times 10^{-19}$ | 1.3139(8)$\times 10^{-19}$ |
Table 2: The information of NGC 6517P, Q, R from preliminary timing measurements made to date. The timing data spans from MJD 58689, 58867 and 58856 to 59944 for NGC 6517P, Q and R, respectively. The optical center of NGC 6517 is used as the pulsar’s position. The $\dot{f}$ of each pulsar is fixed to 0.  Pulsar | J1801–0857P | J1801–0857Q | J1801–0857R
---|---|---|---
Spin frequency, $f$ (s-1) | 180.632266(2) | 137.776526(2) | 151.8545784(9)
Reference epoch (MJD) | 59944.125910 | 58871.058570 | 59944.125910
Dispersion measure, DM (pc cm-3) | 183.05 | 182.45 | 182.50
Number of TOAs | 33 | 24 | 29
Post-fit residual RMS ($\mu$s) | 119.84 | 91.53 | 56.03
Derived Parameters
Spin period, $P$ (ms) | 5.53610949(6) | 7.25813048(9) | 6.58524762(4)
Figure 1: The average pulse profiles and timing residuals of all pulsars in
NGC 6517. The left panel of each subplot is the integrated pulse profile
obtained by summing all detections over 64 pulse phase bins. The timing
residuals from the best-fit timing model are shown in the right panel. The
phase-connected timing solutions of NGC 6517A to O were obtained, while for
NGC 6517P, Q, and R, solutions with only the spin frequencies ($f$) were
fitted, with commonly used arbitrary phase offsets (so-called “JUMPs”) between
observations.

Whilst obtaining or updating the timing solutions for all known pulsars in
(NGC 6517A to I) we find solutions that are consistent with previous studies
(Lynch et al., 2011; Pan et al., 2021a), albeit with measurements of higher
precision (e.g., $\alpha$ and $\delta$ were improved in their accuracy by at
least one decimal place). Of the newly discovered pulsars, we can only obtain
the phase-connected timing solutions for NGC 6517K, L, M, N, and O. The phase-
connected timing solutions of NGC 6517A to M were obtained by manually
removing the JUMPs between different observations. For the pulsars NGC 6517N
and O, the Dracula algorithm returned single solutions with a reduced
$\chi^{2}<2$. Presently, only NGC 6517B is confirmed as a binary pulsar and
the other pulsars appear to be isolated. Updated timing solutions for all
previously known pulsars in NGC 6517, in addition to solutions for the newly
discovered pulsars described in this work, are presented in Table LABEL:tab:1.
For the other three pulsars, NGC 6517P, Q, and R, their low flux density
(S${}_{\rm mean}\sim 1~{}\mu$Jy) led to intermittent detection through
standard search methods. However, once the observations were folded with a
basic ephemeris containing the spin period, and adopting the central location
of NGC 6517 as the pulsar position, detections could be made in all
observations longer than $\sim 2\,{\rm h}$. Continuous monitoring to increase
the number of detections of PSRs NGC 6517P, Q, and R is needed to remove JUMPs
(see Table 2 and Figure 1). Since the submission of this letter, three weaker
millisecond pulsars, NGC 6517S to U, were detected in this cluster. Their spin
periods are 3.77 ms, 3.68 ms, and 6.02 ms, respectively, and more details will
be discussed in another works (Dai et al. in preparation).

Our fits for DM, including the newly discovered pulsars, show that the
difference between the maximum and minimum DM of pulsars in NGC 6517 is
$\sim$11.2 cm-3 pc, making it the GC with the second largest spread in pulsar
DMs. Only GLIMPSE-C01, with pulsars that show a DM range of $\sim$62 cm-3 pc,
is larger. The DM difference may be caused by the intrinsic nature of the
interstellar medium (ISM) within GCs, or the turbulent nature of the Galactic
ISM on small angular scales (see e.g., Freire et al. 2005). NGC 6517, located
close to the Galactic plane, is therefore a useful yardstick for understanding
these effects. Three X-ray sources were detected in 0.3-8 keV with
luminosities ranging from $2.7\times 10^{31}$ erg s-1 to $2.4\times 10^{32}$
erg s-1 (Zhao & Heinke, 2022). Because nine of the 17 pulsars lie within the
core-radius, their possible association with X-ray counterparts will be
discussed in another paper (Yin et al. in preparation).

##  4 An update on the estimated number of GC pulsars

After making the first detections of pulsars in NGC 6517, Lynch et al. (2011)
made complete population predictions for this GC using three observed pulsar
luminosity functions, and with limits defined by the peak luminosity of the
pulsars detected and the weakest known GC pulsar at that time. They found that
12 to 17 pulsars would exist in this GC. We note the upper limit from Lynch et
al. (2011) now matches the known pulsar population in NGC 6517, however, as
discussed below, the new pulsar discoveries described in this work change
population predictions. Since then other methods have been employed that not
only predict the number of pulsars in GCs such as NGC 6517, but try to
establish possible correlation between the intrinsic cluster properties and
their pulsar count (Turk & Lorimer, 2013). The empirical Bayesian method
described by Turk & Lorimer (2013) was designed to model pulsar populations
that are heavily biased by selection effects resulting in populations with
either low to zero pulsar detections. For NGC 6517 this method results in a
prediction of $\sim 15$ pulsars.

Using the publicly available software
101010http://astro.phys.wvu.edu/gcpsrs/empbayes described in Turk & Lorimer
(2013) it was found that by incorporating recent pulsar discoveries from FAST,
the predicted number of pulsars using this method in GCs M53 and M71 differed
from earlier estimations by 3 to 14 and 1 to 4 pulsars respectively. Because
all GCs in the FAST sky have now been searched for undiscovered pulsars,
resulting in a total of 50 new
discoveries111111https://fast.bao.ac.cn/cms/article/65, in this work we used
the updated detected pulsar count, including FAST detections, to re-evaluate
the optimal models found using the method described by Turk & Lorimer (2013).
As this method can be used to investigate the largely unknown relationship
between GC properties and their pulsar population, incorporating the increased
pulsar sample is expected to elucidate this further.

We applied largely the same methods as outlined in Turk & Lorimer (2013), and
refer the reader to this work for detailed descriptions of the statistical
methods used. In the following, only the necessary definitions have been
explained. The GC parameters to be investigated included the optical $V$-band
luminosity ($L_{\rm V}$); escape velocity ($V_{\rm esc}$); metallicities
($M_{\rm L}$) from Harris 1996 (2010 edition); stellar encounter rates
($\Gamma$) from Bahramian et al. (2013); and minimum detectable flux densities
($S_{\rm min}$) from Boyles et al. (2011). In Turk & Lorimer (2013) a total of
94 GCs (with 144 known pulsars in 28 of the GCs) with all five of these
parameters measured were used. An a-priori candidate set of 240 models,
$\hat{N}_{\rm psr,i}$, were modelled with different covariate effects,
$X_{ij}$, which are combinations of the physical properties of the GCs
mentioned above and their regression coefficients, $\beta_{i}$. Here $X_{ij}$
is the $j^{\rm th}$ covariate in $i^{\rm th}$ GC ($j=1,2,...,r$) and the
regression coefficient $\beta_{j}$ characterises the effect of the $j^{\rm
th}$ covariate. These were then evaluated by the Akaike’s Information
Criterion (AIC, Burnham & Anderson, 2004) to select optimal models; where
models with smaller AIC values (by at least two AIC units, Burnham & Anderson
2004) are considered to provide a better fit to the sample data. In Turk &
Lorimer (2013) the estimated detection probability, $\hat{p}_{i}$, for the
$i^{\rm th}$ GC was considered in three prior distributions on GC pulsar
population ($\hat{N}_{\rm psr,i}$): Poisson (P), negative binomial (NB), and
zero-inflated Poisson (ZIP). Through their analysis Turk & Lorimer (2013)
found that the stellar encounter rate is the most critical parameter in
estimating the number of pulsars in a GC.

In our updated analysis we divided the GC samples into two groups. Firstly we
fitted models using all 94 GCs, but with updated pulsar counts. Our second
sample consisted of only the GCs visible from FAST. The reasoning for this is
described below. For the first sample of all 94 GCs, the results from the AIC
ranking showed that models which included escape velocity were favored over
models in which this was not a factor. The results from the most simplified
models (i.e., those with the fewest parameters) within 2 AIC units are shown
in the upper section of Table 3. The second sample of FAST only GCs was chosen
for the following reasons. To date, almost all GCs within the FAST sky have
been deeply searched for pulsars (limiting flux of $\sim 1\,\mu{\rm Jy}$, Pan
et al. 2021b) by FAST. The searches for GC pulsars outside the FAST sky have
been carried out by different telescopes with different sensitivity and
observing systems. Therefore we assume that a sample with uniform sensitivity
coverage could reduce the influence of selection bias and more accurately
reveal the underlying distribution of the pulsar population in GCs.

There are 45 GCs in the FAST sky and amongst these, 31 were used because they
have all five of the necessary GC property parameters. Results from the three
models most highly ranked by their AIC values are presented in the lower
section of Table 3. For our GC sample covering the FAST sky, models which
included the GC escape velocity $V_{\rm esc}$ are also favored over other
models in which this is not a factor. The AIC values of the three models in
the lower section of Table 3 are almost equal. Here, in order to further
briefly compare the difference between the results of the two samples in the
estimated counts of pulsars for all GCs in the Milky Way, we used the optimal
model with only one intercept term for further analysis since the estimated
detection probabilities of pulsars in some GCs were not available. We
therefore adopted the second model (Model no. 2 in the lower section of Table
3), with an AIC value of $103.88$, and in which $\beta_{0}=0.369$ and
$\beta_{1}=0.087$, for further study. The association from the selected model
is significant with a $P$-value121212The probability, $P$-value, is a measure
of evidence against the null hypothesis in favor of the alternative
hypothesis, i.e., a smaller $P$-value indicates more significant the
correlation between the parameter $V_{\rm esc}$ and $\hat{N}_{\rm psr}$ (see
Turk & Lorimer 2013 and reference therein for details). of $5.94\times
10^{-6}<0.05$. This provides further support for the potential correlation
between the population of pulsars in GCs and their escape velocities. For the
sake of brevity (i.e., only one intercept term), the third model from the
sample of all GCs was used for further comparisons with the selected model
from the FAST sky. The association of the selected model from all GCs is also
significant with a $P$-value of $1.78\times 10^{-7}$ ($\beta_{0}=-0.169$,
$\beta_{1}=0.086$, and AIC=314.60). Assuming that the correlation from the GCs
within the FAST is applicable to Galactic GCs outside of the FAST sky, the
pulsar numbers of these GCs were estimated accordingly (see Figure 2 and Table
LABEL:table:longtable_example_1). As seen in Figure 2, the estimated counts
between the FAST sky model and all sky model are slightly different, which may
be due to the higher pulsar detection rate in GCs from the FAST (e.g., Yin et
al. 2023).

Table 3: The top models within two AIC units to the smallest AIC. $\Delta$AIC is the change in AIC relative to the model with smallest AIC value and Type refers to the model prior used for abundance, in this case Negative Binomial (NB), see Turk & Lorimer (2013) for details. The models in the upper section are from all GCs, and the results in the lower section are from GCs within the FAST sky. Note the AIC value can only be used to rank models within the same sample, not between different samples.  | The sample of all GCs |  |  |
---|---|---|---|---
Model no. | Model Structure | AIC |  $\Delta$AIC | Type
1. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}+\lg(\widehat{p}_{i})$ | 313.65 | 0.37 | NB
2. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}+\widehat{p}_{i}$ | 314.16 | 0.88 | NB
3. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}$ | 314.60 | 1.32 | NB
| The sample of only GCs within FAST sky. |  |  |
1. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}+\widehat{p}_{i}$ | 103.84 | 0.000 | NB
2. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}$ | 103.88 | 0.035 | NB
3. |  $\ln\hat{N}_{\rm psr,i}$ = $\beta_{0}+\beta_{1}V_{\rm esc}+\lg(\widehat{p}_{i})$ | 103.91 | 0.063 | NB

The fact that GCs with larger total mass also have, on average, larger escape
velocities is a natural correlation. The $V_{\rm esc}$ was derived from the
best-fitting N-body models of each GC in Baumgardt’s list, which can be
estimated by $V_{\rm esc}=\sqrt{2GM/r_{\rm h,l}}$ or $V_{\rm
esc}=\sqrt{1.5GM/r_{\rm h,m}}$, where M, $r_{\rm h,l}$, and $r_{\rm h,m}$ are
the GC mass, projected half-light radius, and half-mass radius (3D),
respectively. Because neutron stars receive a high natal kick velocity (e.g.,
Hobbs et al. 2005) it was generally predicted that they would not be retained
within GCs, and therefore those within GCs have been proposed to form through
a low-velocity channel (e.g., electron-capture supernovae, Ivanova et al.
2008; Freire 2013). Assuming that the evolutionary nature of Galactic GCs are
alike, simply the total mass of the GCs may influence the number of neutron
stars originally retained, as pointed out by Ransom (2008). Therefore, more
initial neutron stars can be formed due to a larger cluster mass, while a
sufficiently high escape velocity of the cluster considerably ensures
retention of the neutron stars in this stage (e.g., Ivanova et al. 2008).
After retention, the initial neutron stars may be recycled to millisecond spin
periods by accretion. The escape velocity of cluster is related to the total
mass and size of the cluster (i.e. related to density). Higher central density
of GC leads to an increase in the formation of more low-mass X-ray binaries
and their offspring of millisecond pulsars through close stellar encounters
(e.g., Alpar et al. 1982; Pooley et al. 2003). The dependence here between the
number of pulsars and the escape velocity is consistent with previous N-body
simulation works exploring the formation of millisecond pulsars in GCs (e.g.,
Ivanova et al. 2008; Ye et al. 2019).

The range of $V_{\rm esc}$ of the 31 GCs in the FAST sky is $1.8\,{\rm
km\,s}^{-1}$ (Pal 13) to $48.9\,{\rm km\,s}^{-1}$ (M15). In Table
LABEL:table:longtable_example the results of our pulsar number predictions,
and those from Turk & Lorimer (2013) are given in order of descending $V_{\rm
esc}$. Thus, in the FAST sky M15 (13 pulsars, M15A to M) was expected to host
102 pulsars, being the GC with the largest number of predicted pulsars.
Predicted to host 64 pulsars and with at least 6 pulsars discovered by FAST
(e.g., Pan et al. 2021b), M2 is suggested to be the GC with the second largest
number of pulsars. With only one pulsar, an eclipsing redback namely M92A,
discovered in 2017 by FAST (Pan et al., 2020), the modeled pulsar number for
M92 is 34. Thus, we suggest that more pulsars should be discovered in M92. On
the other hand, the possible reasons of missing pulsars can be either the
relatively low fluxes and or compact orbits. The only exception is M71. There
are 5 pulsars in it, while the modeled pulsar number is 4.

Figure 2: The correlation between the number of pulsars and the central escape
velocity of Galactic GCs. The lines with 95$\%$ credible interval (CI) in grey
colour shows the relation between the escape velocity and the number of
estimated pulsars, while the stem plot shows the escape velocity of 39 GCs
with known pulsars and the number of known pulsars in them. The blue dotted
line and the black solid line present the model derived from all GCs and only
GCs within FAST sky, respectively. GLIMPSE-C01 (with 6 known pulsars) was not
included in the figure as its escape velocity was not available. In the
sample, Pal 13 and M15 are the minimum and maximum escape velocities in the
FAST sky, respectively. Liller 1 has the largest escape velocity in the
Galactic GCs.

With the $V_{\rm esc}$ of 87.9 km s-1, Liller 1 is predicted to host thousands
of pulsars. The prediction is supported by the previous estimation that it
could host 410${}_{-210}^{+480}$ pulsars (Abdo et al. 2010; Tam et al. 2011)
due to its highest $\gamma$-ray luminosity among all GCs. The distance of this
GC from the Sun is 8.2 kpc, close enough for finding new pulsars. Located in
the bulge of the Milky Way, the heavy dispersive smearing and scatter
broadening caused by possible high DM values (e.g., $\sim$ 697 cm-3 pc from
NE2001, Cordes & Lazio 2002; $\sim$ 1104 cm-3 pc from YMW16, Yao et al. 2017)
can be the reason for missing pulsars. The pulsars in this GC might require
higher frequency bands at large radio telescopes.

M54, NGC 6388, NGC 2808, and M75 are the other four GCs with similar or higher
$V_{\rm esc}$ than M15, but with few pulsars detected. M54 and M75 are located
more than 20 kpc away, further than M53 which is the furthest GC with known
pulsars. The distance to either NGC 6388 or NGC 2808 is around 10 kpc, similar
to Liller1. So, we suggest that the priority of pulsar surveys towards NGC
6388 and NGC 2808 should be higher than those to M54 and M75.

We caution against directly applying the above estimated numbers, as selection
bias may still exist and a further analysis is required to refine this
simulation. The flux density limit used here (Boyles et al., 2011), as
discussed in Turk & Lorimer (2013), is strictly applicable only to long-period
pulsars, and thus we focus here on the potential correlation between the
pulsar population of GCs and their physical parameters, without analyzing the
pulsar detection probability. The updating of some of those numbers may refine
the analysis. On the premise that almost all GCs within the FAST sky have been
deeply searched for pulsars at the same sensitivity limit, we assume that the
population of pulsars within these GCs is more akin to the native population,
but a selection bias may still exist by using only these GCs as a sample to
estimate the whole Galactic population distribution. For example, most GCs are
located around the Galactic center or in the Southern sky, where some of them
host the largest number of pulsars, while the GCs within the FAST sky are
mostly out of the Galactic plane. Also, most GCs in the FAST sky have smaller
amounts of DM and scattering, enabling the initial pulsars within them to be
more easily detected. Similarly, the slight difference between the estimated
counts derived from the FAST sky and all sky may also arise from the different
ranges of cluster physics parameters, such as escape velocities.

In the FAST sky, around 50 GC pulsars were discovered by either FAST (e.g.,
Pan et al. 2021b) or others (e.g., Glimpse C01 A, McCarver et al. 2023). With
known pulsars, the computational time cost for dedisperion can be largely
saved. Thus, those GCs with known pulsars should be prioritized for future
observing campaigns. After searching for almost all the GCs in FAST sky, 20
pulsars were discovered in the GCs with no previously known pulsars, including
M92, M14, M2, NGC 6712, M10, and M12. The probability of pulsar discoveries in
GCs with or without known pulsars may not differ significantly. On the other
hand, short observations should be carried out before longer ones to reduce
unnecessary processing time and to improve chances of finding compact
binaries. Based on predictions made here, there is little doubt that the
ongoing and future GC pulsar surveys with GMRT (e.g., Gautam et al. 2022),
Parkes (e.g., Zhang et al. 2022), MeerKAT (e.g., Ridolfi et al. 2021), and SKA
(Hessels et al., 2015) in the near future will bring more discoveries.

##  5 Conclusions

In this study of FAST observations of NGC 6517, we have reached the following
conclusions:

  1. 1.

Eight new isolated millisecond pulsars, namely PSR J1801$-$0857K to R, or, NGC
6517K to R, were discovered in NGC 6517. Up to the end of 2023, there are 17
pulsars detected in NGC 6517.

  2. 2.

NGC 6517J is proven to be a harmonic of NGC 6517B. The phase-connected timing
solutions of NGC 6517K to O were obtained. The timing solutions of previously
known pulsars were updated and are consistent with previous works.

  3. 3.

In the framework of the empirical Bayesian model used in this paper, the
recently observed sample of GC pulsars favored a correlation between the
number of potential pulsars and the central escape velocities unlike the
original prediction of Turk & Lorimer (2013) using the same method which
predicted the stellar encounter rate as the main factor.

  4. 4.

In the FAST sky, M2 and M92 may have more pulsars. Outside of the FAST sky,
Liller 1, NGC 6388, NGC 2808, M54, and M75 have the highest possibilities for
finding new pulsars.

This work is supported by National SKA Program of China (No. 2020SKA0120100),
the Basic Science Center Project of the National Nature Science Foundation of
China (NSFC) under Grant Nos. 11703047, 11773041, U1931128, U2031119, 12003047
and 12173053. Lei Qian is supported by the Youth Innovation Promotion
Association of CAS (id. 2018075, Y2022027), and the CAS “Light of West China”
Program. RPE is supported by the Chinese Academy of Sciences President’s
International Fellowship Initiative, Grant No. 2021FSM0004. M.L. is supported
by Guizhou Provincial Basic Research Program (Natural Science) (ZK[2023] 039),
Key Technology R&D Program([2023] 352) and National Natural Science Foundation
of China under Grand No. 12363010. This work made use of the data from FAST
(Five-hundred-meter Aperture Spherical radio Telescope). FAST is a Chinese
national mega-science facility, operated by National Astronomical
Observatories, Chinese Academy of Sciences. This work is also supported by the
International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001
and 202302AN36000104). Finally, we thank the anonymous referees for their
helpful suggestions to bring clarity to the text.

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Table 4: The 45 GCs in FAST sky. The latest central escape velocities ($V_{\rm esc}$) of GCs are from the Baumgardt’s list (The $4^{\rm th}$ Mar. 2023) version141414https://people.smp.uq.edu.au/HolgerBaumgardt/globular/parameter.html), while other physical parameters of GCs are the same as in Turk & Lorimer (2013). The $N_{\rm obs}$ is the number of observed known pulsars by the end of 2023. The expected number of potential pulsars of $\hat{N}_{\rm psr,1}$ and $\hat{N}_{\rm psr,2}$ are from the Equation 13 ($\ln\hat{\lambda}=-1.1+1.5\lg\Gamma$) of Turk & Lorimer (2013) and model no. 2 from the FAST sky only sample in this work ($\ln\hat{N}_{\rm psr,i}=0.369+0.087\ V_{\rm esc,i}$ ), respectively. The $D_{\rm Sun}$ is the distance from the Sun (Harris 1996, 2010 edition). The $L_{\rm min}$ ($L_{\rm min,i}$=$S_{\rm min,i}$$D^{2}_{\rm Sun,i}$, Boyles et al. 2011) is from the list of flux density detection limits of GCs. Note: Table 4 and Table 5 are also can be presented in an online available form.  ID | Name | $D_{\rm Sun}$ | $L_{\rm min}$ | $\Gamma$ | $V_{\rm esc}$ | $N_{\rm obs}$ | $\hat{N}_{\rm psr,1}$ | $R_{1}$ | $\hat{N}_{\rm psr,2}$ | $R_{2}$
---|---|---|---|---|---|---|---|---|---|---
|  | (kpc) | (mJy kpc2) |  | (km s-1) |  |  | ($N_{\rm obs}$/$\hat{N}_{\rm psr,1}$) |  | ($N_{\rm obs}$/$\hat{N}_{\rm psr,2}$)
NGC 7078 | M 15 | 10.4 | 2.2714 | 4510 | 48.9 | 12 | 80 | 0.15 | 102 | 0.12
NGC 7089 | M 2 | 11.5 | 0.7935 | 518 | 43.6 | 6 | 20 | 0.3 | 64 | 0.09
NGC 6517 |  | 10.6 | 0.3371 | 338 | 37.3 | 17 | 15 | 1.13 | 37 | 0.46
NGC 6341 | M 92 | 8.3 | 0.3513 | 270 | 36.4 | 1 | 13 | 0.08 | 34 | 0.03
NGC 6402 | M 14 | 9.3 | 4.7397 | 124 | 35.6 | 5 | 8 | 0.63 | 32 | 0.16
NGC 6205 | M 13 | 7.1 | 1.3611 | 68.9 | 32.4 | 6 | 5 | 1.2 | 24 | 0.25
NGC 5272 | M 3 | 10.2 | 2.1848 | 194 | 32 | 6 | 10 | 0.6 | 23 | 0.26
NGC 5904 | M 5 | 7.5 | 1.4063 | 164 | 30.4 | 7 | 9 | 0.78 | 20 | 0.35
NGC 5024 | M 53 | 17.9 | 6.0878 | 35.4 | 25.9 | 5 | 3 | 1.67 | 14 | 0.36
NGC 6760 |  | 7.4 | 2.0261 | 56.9 | 25.6 | 2 | 5 | 0.4 | 13 | 0.15
NGC 6229 |  | 30.5 |  | 47.6 | 25.3 |  | 4 | 0 | 13 | 0
NGC 5634 |  | 25.2 |  | 20.2 | 24.2 |  | 2 | 0 | 12 | 0
NGC 6254 | M 10 | 4.4 | 1.2894 | 31.4 | 23.9 | 2 | 3 | 0.67 | 12 | 0.17
NGC 6539 |  | 7.8 | 2.8595 | 42.1 | 22.5 | 1 | 4 | 0.25 | 10 | 0.1
NGC 6779 | M 56 | 9.4 | 2.209 | 27.7 | 21.9 |  | 3 | 0 | 10 | 0
NGC 6934 |  | 15.6 | 6.8141 | 29.9 | 21.4 |  | 3 | 0 | 9 | 0
NGC 6749 |  | 7.9 | 1.9971 | 38.5 | 20.5 | 2 | 4 | 0.5 | 9 | 0.22
NGC 6712 |  | 6.9 | 0.319 | 30.8 | 19.2 | 1 | 3 | 0.33 | 8 | 0.13
Pal 2 |  | 27.2 | 25.1546 | 929 | 19.2 |  | 29 | 0 | 8 | 0
NGC 2419 |  | 82.6 |  | 2.8 | 18.2 |  | 1 | 0 | 7 | 0
NGC 6218 | M 12 | 4.8 | 1.2626 | 13 | 17.6 | 2 | 2 | 1 | 7 | 0.29
NGC 6171 | M 107 | 6.4 | 2.2446 | 6.77 | 15.7 |  | 1 | 0 | 6 | 0
NGC 7006 |  | 41.2 | 32.2514 | 9.4 | 15.7 |  | 1 | 0 | 6 | 0
Pal 10 |  | 5.9 | 0.8006 | 59 | 14.6 |  | 5 | 0 | 5 | 0
IC 1276 | Pal 7 | 5.4 |  | 7.97 | 14.3 |  | 1 | 0 | 5 | 0
NGC 4147 |  | 19.3 | 7.0773 | 16.6 | 14.2 |  | 2 | 0 | 5 | 0
NGC 6981 | M 72 | 17 | 1.734 | 4.69 | 12.4 |  | 1 | 0 | 4 | 0
NGC 6426 |  | 20.6 | 10.609 | 1.58 | 10.3 |  | 0 | - | 4 | 0
NGC 6535 |  | 6.8 | 1.8958 | 0.388 | 10.2 |  | 0 | - | 4 | 0
NGC 6838 | M 71 | 4 | 0.304 | 2.05 | 10.2 | 5 | 1 | 5 | 4 | 1.25
NGC 6366 |  | 3.5 |  | 5.14 | 9.1 |  | 1 | 0 | 3 | 0
IC 1257 |  | 25 |  |  | 6.7 |  | - | - | 3 | 0
NGC 5466 |  | 16 | 5.632 | 0.239 | 6.5 |  | 0 | - | 3 | 0
NGC 5053 |  | 17.4 | 6.0552 | 0.105 | 6 |  | 0 | - | 2 | 0
Pal 15 |  | 45.1 | 77.2924 | 0.0222 | 4.3 |  | 0 | - | 2 | 0
Pal 11 |  | 13.4 |  | 20.8 | 4 |  | 2 | 0 | 2 | 0
Pal 4 |  | 108.7 |  | 0.0189 | 2.6 |  | 0 | - | 2 | 0
Pal 3 |  | 92.5 |  | 0.0409 | 2.4 |  | 0 | - | 2 | 0
Pal 14 | AvdB | 76.5 |  | 0.00186 | 2.3 |  | 0 | - | 2 | 0
Pal 5 |  | 23.2 | 17.2237 | 0.00212 | 2.1 |  | 0 | - | 2 | 0
Pal 13 |  | 26 | 14.196 | 0.00109 | 1.8 |  | 0 | - | 2 | 0
Whiting 1 |  | 30.1 |  |  | 1.1 |  | - | - | 2 | 0
GLIMPSE01 |  | 4.2 |  |  |  | 2 | - | - | - | -
Ko 1 |  | 48.3 |  |  |  |  | - | - | - | -
Ko 2 |  | 34.7 |  |  |  |  | - | - | - | -
Table 5: The 112 GCs outside of FAST aky ID | Name | $D_{\rm Sun}$ | $L_{\rm min}$ | $\Gamma$ | $V_{\rm esc}$ | $N_{\rm obs}$ | $\hat{N}_{\rm psr,1}$ | $R_{1}$ | $\hat{N}_{\rm psr,2}$ | $R_{2}$
---|---|---|---|---|---|---|---|---|---|---
|  | (kpc) | (mJy kpc2) |  | (km s-1) |  |  | ($N_{\rm obs}$/$\hat{N}_{\rm psr,1}$) |  | ($N_{\rm obs}$/$\hat{N}_{\rm psr,2}$)
Liller 1 |  | 8.2 | 0.6657 | 391 | 87.9 |  | 16 | 0 | 3030 | 0
NGC 6441 |  | 11.6 | 1.6685 | 2300 | 69.4 | 9 | 52 | 0.17 | 606 | 0.01
NGC 6715 | M 54 | 26.5 |  | 2520 | 68.9 |  | 55 | 0 | 580 | 0
NGC 5139 |  $\omega$-Cen | 5.2 | 1.306 | 90.4 | 62.2 | 18 | 6 | 3 | 324 | 0.06
NGC 6388 |  | 9.9 | 5.3709 | 899 | 60 |  | 28 | 0 | 267 | 0
NGC 6266 | M 62 | 6.8 | 1.0219 | 1670 | 59.3 | 10 | 42 | 0.24 | 252 | 0.04
Terzan 5 | Terzan 11 | 6.9 | 0.5523 | 6800 | 57.5 | 49 | 104 | 0.47 | 215 | 0.23
NGC 6440 |  | 8.5 | 0.7081 | 1400 | 55.6 | 8 | 37 | 0.22 | 182 | 0.04
NGC 2808 |  | 9.6 | 5.0504 | 923 | 53.1 |  | 28 | 0 | 147 | 0
NGC 6864 | M 75 | 20.9 |  | 307 | 48.4 |  | 14 | 0 | 97 | 0
NGC 104 | 47 Tuc | 4.5 | 3.3939 | 1000 | 47.4 | 35 | 30 | 1.17 | 89 | 0.39
NGC 6139 |  | 10.1 | 7.9058 | 307 | 46.4 |  | 14 | 0 | 82 | 0
NGC 5824 |  | 32.1 |  | 984 | 44.8 |  | 30 | 0 | 71 | 0
NGC 6273 | M 19 | 8.8 | 4.2437 | 200 | 44.8 |  | 11 | 0 | 71 | 0
NGC 6093 | M 80 | 10 | 0.57 | 532 | 44.6 | 1 | 20 | 0.05 | 70 | 0.01
NGC 5286 |  | 11.7 | 7.5016 | 458 | 41.8 |  | 18 | 0 | 55 | 0
NGC 1851 |  | 12.1 | 4.4069 | 1530 | 41.1 | 15 | 40 | 0.38 | 52 | 0.29
NGC 6626 | M 28 | 5.5 | 0.124 | 648 | 41.1 | 14 | 23 | 0.61 | 52 | 0.27
Terzan 1 | HP 2 | 6.7 | 3.479 | 0.292 | 37.9 | 7 | 0 | - | 39 | 0.18
NGC 6541 |  | 7.5 | 0.54 | 386 | 37.3 |  | 16 | 0 | 37 | 0
NGC 6656 | M 22 | 3.2 | 0.0584 | 77.5 | 35.5 | 4 | 6 | 0.67 | 32 | 0.13
Terzan 10 |  | 5.8 |  | 4430 | 35.2 |  | 79 | 0 | 31 | 0
NGC 362 |  | 8.6 |  | 735 | 34 | 12 | 25 | 0.48 | 28 | 0.43
NGC 5694 |  | 35 |  | 191 | 33.6 |  | 10 | 0 | 27 | 0
NGC 6553 |  | 6 |  | 69 | 32.8 |  | 5 | 0 | 25 | 0
NGC 6522 |  | 7.7 | 3.2491 | 363 | 32.4 | 6 | 15 | 0.4 | 24 | 0.25
NGC 6333 | M 9 | 7.9 | 3.4201 | 131 | 32.4 |  | 8 | 0 | 24 | 0
NGC 6380 | Ton 1 | 10.9 |  | 116 | 31.8 |  | 7 | 0 | 23 | 0
NGC 6752 |  | 4 | 0.504 | 401 | 31.3 | 9 | 17 | 0.53 | 22 | 0.41
NGC 6356 |  | 15.1 |  | 88.1 | 31.1 |  | 6 | 0 | 22 | 0
NGC 5986 |  | 10.4 | 0.4002 | 61.9 | 30.6 | 1 | 5 | 0.2 | 21 | 0.05
NGC 6316 |  | 10.4 |  | 77 | 30.3 | 1 | 6 | 0.17 | 20 | 0.05
Terzan 9 |  | 7.1 |  | 1.71 | 29.1 |  | 0 | - | 18 | 0
NGC 6569 |  | 10.9 |  | 53.6 | 28.4 |  | 4 | 0 | 17 | 0
NGC 6624 |  | 7.9 | 0.9986 | 1150 | 28 | 12 | 33 | 0.36 | 17 | 0.71
NGC 6453 |  | 11.6 | 10.8186 | 371 | 27.9 |  | 16 | 0 | 16 | 0
NGC 6284 |  | 15.3 | 12.8281 | 666 | 27.6 |  | 23 | 0 | 16 | 0
NGC 1904 | M 79 | 12.9 |  | 116 | 27.6 |  | 7 | 0 | 16 | 0
NGC 6293 |  | 9.5 | 4.9457 | 847 | 27.4 |  | 27 | 0 | 16 | 0
NGC 6287 |  | 9.4 | 6.8479 | 36.3 | 26.8 |  | 3 | 0 | 15 | 0
NGC 6638 |  | 9.4 |  | 137 | 26.7 |  | 8 | 0 | 15 | 0
Terzan 6 | HP 5 | 6.8 | 0.3376 | 2470 | 26.4 |  | 54 | 0 | 14 | 0
NGC 6681 | M 70 | 9 | 6.2775 | 1040 | 25.3 |  | 31 | 0 | 13 | 0
NGC 5927 |  | 7.7 | 3.2491 | 68.2 | 25.2 |  | 5 | 0 | 13 | 0
NGC 6544 |  | 3 | 0.3492 | 111 | 25.2 | 3 | 7 | 0.43 | 13 | 0.23
NGC 5946 |  | 10.6 | 6.1573 | 134 | 24.3 |  | 8 | 0 | 12 | 0
NGC 6637 | M 69 | 8.8 |  | 89.9 | 24.3 |  | 6 | 0 | 12 | 0
NGC 6401 |  | 10.6 | 8.7079 | 44 | 23.6 |  | 4 | 0 | 11 | 0
Terzan 4 | HP 4 | 7.2 |  |  | 23.1 |  | - | - | 11 | 0
NGC 6304 |  | 5.9 | 1.9076 | 123 | 22.8 |  | 8 | 0 | 11 | 0
NGC 4833 |  | 6.6 | 3.3759 | 25 | 22.2 |  | 3 | 0 | 10 | 0
FSR 1735 |  | 9.8 |  |  | 22.2 |  | - | - | 10 | 0
NGC 6355 |  | 9.2 | 6.5596 | 99.2 | 21.6 |  | 7 | 0 | 9 | 0
NGC 6397 |  | 2.3 | 0.1666 | 84.1 | 21.5 | 2 | 6 | 0.33 | 9 | 0.22
NGC 6723 |  | 8.7 | 4.1478 | 11.4 | 21.5 |  | 2 | 0 | 9 | 0
NGC 1261 |  | 16.3 | 10.3088 | 15.4 | 21.4 |  | 2 | 0 | 9 | 0
NGC 7099 | M 30 | 8.1 | 0.7086 | 324 | 21 | 2 | 14 | 0.14 | 9 | 0.22
NGC 6325 |  | 7.8 | 3.334 | 118 | 20.6 |  | 7 | 0 | 9 | 0
HP 1 | BH 229 | 8.2 |  | 0.662 | 20.3 |  | 0 | - | 8 | 0
NGC 6642 |  | 8.1 | 3.5954 | 97.8 | 20 |  | 7 | 0 | 8 | 0
NGC 6256 |  | 10.3 |  | 169 | 19.8 |  | 9 | 0 | 8 | 0
Pal 6 |  | 5.8 | 0.2254 | 15.5 | 19.7 |  | 2 | 0 | 8 | 0
NGC 6652 |  | 10 | 7.75 | 700 | 19.1 | 2 | 24 | 0.08 | 8 | 0.25
NGC 6528 |  | 7.9 | 3.4201 | 278 | 19 |  | 13 | 0 | 8 | 0
NGC 6121 | M 4 | 2.2 | 0.636 | 26.9 | 18.8 | 1 | 3 | 0.33 | 7 | 0.14
NGC 6342 |  | 8.5 | 19.5075 | 44.8 | 18.4 | 2 | 4 | 0.5 | 7 | 0.29
NGC 6809 | M 55 | 5.4 | 2.2599 | 3.23 | 18.1 |  | 1 | 0 | 7 | 0
Terzan 2 | HP 3 | 7.5 |  | 22.1 | 17.4 |  | 3 | 0 | 7 | 0
NGC 3201 |  | 4.9 | 1.3157 | 7.17 | 17.3 |  | 1 | 0 | 7 | 0
NGC 6584 |  | 13.5 | 9.9873 | 11.8 | 16.7 |  | 2 | 0 | 6 | 0
NGC 6558 |  | 7.4 | 3.4663 | 105 | 16.7 |  | 7 | 0 | 6 | 0
UKS 1 |  | 7.8 | 3.334 | 100 | 15.7 |  | 7 | 0 | 6 | 0
NGC 4372 |  | 5.8 | 1.8435 | 0.233 | 15.5 |  | 0 | - | 6 | 0
NGC 6235 |  | 11.5 | 7.2473 | 5.75 | 15.5 |  | 1 | 0 | 6 | 0
Djorg 2 | ESO456-SC38 | 6.3 |  | 46.4 | 15.4 |  | 4 | 0 | 6 | 0
NGC 2298 |  | 10.8 | 0.4782 | 4.31 | 15.3 |  | 1 | 0 | 5 | 0
NGC 6540 | Djorg 3 | 5.3 | 2.177 | 263 | 15.1 |  | 13 | 0 | 5 | 0
NGC 6144 |  | 8.9 | 4.3407 | 3.14 | 15 |  | 1 | 0 | 5 | 0
Djorg 1 |  | 13.7 |  | 9.04 | 14.8 |  | 1 | 0 | 5 | 0
NGC 4590 | M 68 | 10.3 | 5.8137 | 5.82 | 14.8 |  | 1 | 0 | 5 | 0
NGC 6352 |  | 5.6 |  | 6.74 | 13.2 |  | 1 | 0 | 5 | 0
NGC 6362 |  | 7.6 |  | 4.56 | 13.1 |  | 1 | 0 | 5 | 0
Terzan 12 |  | 4.8 |  | 35.8 | 13.1 |  | 3 | 0 | 5 | 0
Lynga 7 | BH184 | 8 |  |  | 12.8 |  | - | - | 4 | 0
NGC 5897 |  | 12.5 | 0.8594 | 0.851 | 12.5 |  | 0 | - | 4 | 0
Pal 8 |  | 12.8 |  | 4.22 | 11.7 |  | 1 | 0 | 4 | 0
Ton 2 | Pismis 26 | 8.2 |  | 4.29 | 11.6 |  | 1 | 0 | 4 | 0
NGC 6717 | Pal 9 | 7.1 | 2.7625 | 39.8 | 11.6 |  | 4 | 0 | 4 | 0
NGC 6101 |  | 15.4 |  | 0.974 | 11.5 |  | 0 | - | 4 | 0
2MS-GC01 | 2MASS-GC01 | 3.6 |  |  | 11.2 |  | - | - | 4 | 0
NGC 288 |  | 8.9 | 0.5149 | 0.766 | 10.9 |  | 0 | - | 4 | 0
NGC 6496 |  | 11.3 | 6.9974 | 0.657 | 10.8 |  | 0 | - | 4 | 0
IC 4499 |  | 18.8 |  | 0.797 | 10.5 |  | 0 | - | 4 | 0
BH 261 | AL 3 | 6.5 |  |  | 9.1 |  | - | - | 3 | 0
2MS-GC02 | 2MASS-GC02 | 4.9 |  |  | 8.4 |  | - | - | 3 | 0
Terzan 3 |  | 8.2 |  | 0.89 | 7.6 |  | 0 | - | 3 | 0
ESO-SC06 | ESO280-SC06 | 21.4 |  |  | 6.8 |  | - | - | 3 | 0
Terzan 8 |  | 26.3 |  | 0.0793 | 6 |  | 0 | - | 2 | 0
Rup 106 |  | 21.2 |  | 0.359 | 5.6 |  | 0 | - | 2 | 0
1636-283 | ESO452-SC11 | 8.3 |  | 1.72 | 5.5 |  | 0 | - | 2 | 0
Terzan 7 |  | 22.8 |  | 1.59 | 4.6 |  | 0 | - | 2 | 0
Arp 2 |  | 28.6 |  | 0.00518 | 4.5 |  | 0 | - | 2 | 0
NGC 7492 |  | 26.3 |  | 0.192 | 4.4 |  | 0 | - | 2 | 0
Pyxis |  | 39.4 |  |  | 3.6 |  | - | - | 2 | 0
AM 1 | E 1 | 123.3 |  | 0.00419 | 3.3 |  | 0 | - | 2 | 0
Pal 12 |  | 19 | 2.2382 | 0.397 | 2.5 |  | 0 | - | 2 | 0
E 3 |  | 8.1 |  | 0.0759 | 2.4 |  | 0 | - | 2 | 0
Eridanus |  | 90.1 |  | 0.0339 | 2.2 |  | 0 | - | 2 | 0
Pal 1 |  | 11.1 |  | 0.895 | 1.9 |  | 0 | - | 2 | 0
AM 4 |  | 32.2 |  | 0.0026 | 0.9 |  | 0 | - | 2 | 0
BH 176 |  | 18.9 |  | 0.146 |  |  | 0 | - | - | -
GLIMPSE02 |  | 5.5 |  |  |  |  | - | - | - | -