# The Galactic neutron star population I - an extragalactic view of the Milky
Way and the implications for fast radio bursts

A. A. Chrimes,1 A. J. Levan1,2, P. J. Groot1,3,4, J. D. Lyman2 and G.
Nelemans1,5,6
1Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010,
6500 GL Nijmegen, The Netherlands
2Department of Physics, University of Warwick, Coventry CV4 7AL, UK
3Inter-University Institute for Data Intensive Astronomy, Department of
Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South
Africa
4South African Astronomical Observatory, P.O. Box 9, 7935 Observatory, South
Africa
5Institute for Astronomy, KU Leuven, Leuven, Belgium
6SRON, Netherlands Institute for Space Research, Sorbonnelaan 2, NL-3584 CA
Utrecht, The Netherlands
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A key tool astronomers have to investigate the nature of extragalactic
transients is their position on their host galaxies. Galactocentric offsets,
enclosed fluxes and the fraction of light statistic are widely used at
different wavelengths to help infer the nature of transient progenitors.
Motivated by the proposed link between magnetars and fast radio bursts (FRBs),
we create a face-on image of the Milky Way using best estimates of its size,
structure and colour. We place Galactic magnetars, pulsars, low mass and high
mass X-ray binaries on this image, using the available distance information.
Galactocentric offsets, enclosed fluxes and fraction of light distributions
for these systems are compared to extragalactic transient samples. We find
that FRBs follow the distributions for Galactic neutron stars closest, with 24
(75 per cent) of the Anderson-Darling tests we perform having a p-value
greater than 0.05. This suggests that FRBs are located on their hosts in a
manner consistent with Galactic neutron stars on the Milky Way’s light,
although we cannot determine which specific neutron star population is the
best match. The Galactic distributions are consistent with other extragalactic
transients much less often across the range of comparisons made, with type Ia
SNe in second place, at only 33 per cent of tests exceeding 0.05. Overall, our
results provide further support for FRB models invoking isolated young neutron
stars, or binaries containing a neutron star.

###### keywords:

stars: neutron – stars: magnetars – transients: fast radio bursts –
supernovae: general – Galaxy: structure

††pubyear: 2021††pagerange: The Galactic neutron star population I - an
extragalactic view of the Milky Way and the implications for fast radio
bursts–B

##  1 Introduction

Several classes of astrophysical transients are associated with neutron stars,
ranging from their formation in core-collapse supernovae to their mergers in
short ${\gamma}$-ray bursts (GRBs). Magnetars are a subset of neutron stars
(Duncan & Thompson, 1992), distinguished by their extreme magnetic fields
(1014-1015 G), high radio/X-ray/${\gamma}$-ray luminosities despite spin-down
luminosities comparable to pulsars (Thompson & Duncan, 1996; Kouveliotou et
al., 1998), and young characteristic ages of ${\sim}$103-105 yrs (Olausen &
Kaspi, 2014). Magnetars are observed in the Milky Way and Magellenic clouds as
soft ${\gamma}$ repeaters (SGRs, Mazets et al., 1979a; Mazets et al., 1979b)
and anomalous X-ray pulsars (Fahlman & Gregory, 1981), detectable primarily
through their high energy flares, and in some cases persistent emission
(Olausen & Kaspi, 2014). Giant magnetar flares, detectable at extragalactic
distances, have been identified as an alternative origin for some short GRBs
(Hurley et al., 2005; Ofek et al., 2006, 2008; Svinkin et al., 2021; Burns et
al., 2021). Magnetars have also been invoked as central engines in
superluminous supernovae (Kasen & Bildsten, 2010; Woosley, 2010) and GRBs
(Metzger et al., 2008, 2011). More recently, they have become a promising
explanation for the fast radio burst (FRB) phenomenon.

This paper is partly motivated by the FRB-magnetar connection, following the
detection of FRB-like bursts from the Galactic magnetar SGR 1935$+$2154
(hereafter SGR 1935, Bochenek et al., 2020b; CHIME/FRB Collaboration et al.,
2020). Although the SGR 1935 bursts are fainter than extragalactic FRBs, they
are orders of magnitude brighter than other radio transients with comparable
frequencies and millisecond durations (e.g. pulsar pulses, Keane, 2018). Even
before this detection, magnetar flaring was a leading theory among the dozens
offered to explain FRBs (Platts et al.,
2019)111frbtheorycat.org/index.php/Main_Page. See also Chatterjee (2020),
Baring et al. (2020), Xiao et al. (2021) and Zhang (2020) for reviews. Despite
this association, there are still open questions around the relationship
between the X-ray/${\gamma}$-ray and radio emission (Katz, 2016; Lin et al.,
2020; De et al., 2020a; Tavani et al., 2021; Ridnaia et al., 2021; Li et al.,
2021; Bailes et al., 2021; Verrecchia et al., 2021; Younes et al., 2021). It
is unclear if all FRBs repeat, or if there are two distinct populations of
single and repeating events. Results from the first CHIME/FRB catalogue (The
CHIME/FRB Collaboration et al., 2021; Pleunis et al., 2021) suggests that
bursts from repeaters have larger temporal widths and narrow bandwidths,
although the origin of this difference is currently uncertain. The nature of
the weak periodicity/clustering seen in repeating FRB bursts also remains to
be understood (Beniamini et al., 2020; Pastor-Marazuela et al., 2020; Li &
Zanazzi, 2021), although claims of similar periodicity have now also been made
for SGR 1935 (Grossan, 2021).

If we assume that FRBs are caused by magnetar flares, and that SGR 1935
produces FRBs, then we must conclude that the Milky Way is an FRB host galaxy.
If we also assume that all magnetars are capable of producing FRBs, then it
follows that the distribution of FRBs on their host galaxies should match the
distribution of magnetars on the Milky Way, at least for morphologically
similar spiral galaxies. FRB sample size increases (now $\sim$600 single FRBs
and 20 repeaters, The CHIME/FRB Collaboration et al., 2021) have lead to more
host identifications (Eftekhari & Berger, 2017; Heintz et al.,
2020)222frbhosts.org/, and consequently, population studies of these hosts
have now begun.

One way to study transient hosts is through their spectral energy
distributions (SED), from which SED fitting can be used to infer the
constituent stellar populations. Such work has found that FRB hosts span a
range of masses and star formation rates, and correspondingly, morphological
types. The mean stellar mass is around ${\sim}$1010M⊙, and star formation
rates are typically moderate ${\sim}0.1-1$M⊙yr-1 (Li & Zhang, 2020; Bhandari
et al., 2020; Heintz et al., 2020), comparable to the Milky Way. These
properties disfavour very massive stellar progenitors, such as those of
superluminous supernovae (SLSNe) and long GRBs, but are consistent with
regular core-collapse supernova (CCSN) hosts. To complicate matters further,
Fong et al. (2021) find that the host galaxy of FRB 20201124A assembled $>90$%
of its stellar mass $>1$ Gyr ago, perhaps favouring a long-delay time
progenitor. Even more striking was the identification of a repeating FRB in an
M81 globular cluster (Kirsten et al., 2021), for which an old underlying
population is overwhelmingly likely. Such scenarios are not straightforward to
ascribe to magnetars given their young ages, and presumed origin in massive
stars, although alternative models invoking accretion induced collapse have
been put forward (e.g. Usov, 1992; Levan et al., 2006).

The projected offset of transients from their host centre is another commonly
used measure. At its most basic level, galaxies tend to have star formation
rate gradients such that more star formation occurs towards the centre
(neglecting metallicity effects and morphological differences). Furthermore,
large offsets can provide insight into natal kicks, most prominently seen in
short GRBs (Fong et al., 2010; Church et al., 2011; Tunnicliffe et al., 2014).
Heintz et al. (2020) and Mannings et al. (2020) measure the physical offsets
${\delta}r$ of FRBs from their hosts, and also the normalised offsets
${\delta}r/r_{e}$, where $r_{e}$ is the half-light radius. Using the
normalised offset accounts for different radial intensity profiles (for
example, the underlying stellar mass at a given offset can be different in two
otherwise identical galaxies, if their physical size and compactness differ).
Heintz et al. (2020) and Mannings et al. (2020) find that CCSNe, SGRBs and
type Ia SNe are the most consistent with FRB offsets.

We can also measure the enclosed fraction of total galaxy flux within the
galactocentric radius of the transient (James & Anderson, 2006; Anderson &
James, 2009; Anderson et al., 2015; Audcent-Ross et al., 2020). A cumulative
distribution of the fractional enclosed fluxes will produce a 1:1 relation if
the transient traces the light. Mannings et al. (2020) calculate this
distribution for 8 FRBs on their hosts in the IR, again finding consistency
with a stellar mass tracing progenitor.

Another metric is the fraction of light (Flight, Fruchter et al., 2006). This
technique ranks host-associated pixels and normalises their cumulative
distribution, such that the brightest pixel on a galaxy is assigned the value
1, representing the cumulative fraction of total host flux in regions of
surface brightness ‘below’ the pixel that contains the transient. Unbiased
tracers of light sample Flight values uniformly, whereas biased datasets
over/under sample from brighter/fainter pixels. Different wavelengths are used
to probe different stellar populations: shorter, UV/blue bands trace young
stars and thus star formation, IR/red bands better trace older stars and
stellar mass. Mannings et al. (2020) compute Flight for 8 FRB hosts using UVIS
and IR Hubble Space Telescope/WFC3 data. They compare to other transients,
finding that type Ia SNe and SGRBs are the best match, but noting that the
uncertainties are large. Nevertheless, this result suggests a stellar-mass
tracing progenitor.

As magnetars have been suggested as both the source of FRBs, and as the
central engines/remnants of SLSNe and GRBs, there have been several targeted
radio searches at the locations of recently (i.e. in the last few decades)
observed transients. Searches of this nature have yet to detect an FRB
(Metzger et al., 2017; Mondal et al., 2020; Hilmarsson et al., 2020;
Palliyaguru et al., 2021), but deep optical searches have been able to rule
out an association between FRBs and SLSNe/bright SNe Ia, GRB afterglows and
tidal disruption events, on timescales of 1 day to 1 year post-burst (Nuñez et
al., 2021).

Table 1: The Galactic neutron star samples used in this work. Listed are the source reference, initial sample size, size following a luminosity cut (if applicable), and the final size, after restricting the sample to $y<8.3$ kpc and removing objects in the Magellenic clouds (if previously included).  Population | Source | Init. size | Subset with | Luminosity cut | Size after | Final size after
---|---|---|---|---|---|---
|  |  |  $d$ estimate |  |  $L$ cut |  $y<8.3$k̇pc cut
Magnetars | McGill - Olausen & Kaspi (2014) | 31 | 26 | - | 26 | 20
Pulsars | ANTF - Manchester et al. (2005) | 2872 | 2822 |  $>65$ mJy kpc2 | 229 | 127
LMXBs | INTEGRAL - Sazonov et al. (2020) | 166 | 119 | - | 119 | 84
HMXBs | INTEGRAL - Kretschmar et al. (2019) | 64 | 57 | - | 57 | 36

A final method to explore the nature of FRB progenitors is to compare their
redshift distribution to the cosmic star formation rate (SFR) history. FRB
population synthesis has again found conflicting results, with some work
finding that their rate is consistent with the SFR (and therefore magnetar
production, Gardenier & van Leeuwen, 2020; James et al., 2021), and others
claiming that they better trace stellar mass (Luo et al., 2020)

As can be seen from this overview of host populations, offsets (host level and
resolved), Flight, targeted searches for new FRB sources, and redshift
distribution studies, the sole origin of FRBs in giant magnetar flares is
still far from certain. Indeed, these results have led to conflicting
interpretations over whether magnetars can explain all FRBs (Bochenek et al.,
2020a; Safarzadeh et al., 2020), and if so, what the progenitor channels
should be (Margalit et al., 2019; Margalit et al., 2020). Our comparison of
magnetar (and other neutron star) positions in the Milky Way, versus FRBs
locations within their hosts, is therefore a valuable alternative diagnostic.

The aims of this paper require the ability to map both the Milky Way itself,
and the neutron star populations within in. Galactic magnetars have been
studied in detail, in terms of their distances, activity, magnetic fields,
ages and more (Olausen & Kaspi, 2014; Beniamini et al., 2019); pulsar
(Manchester et al., 2005; Yao et al., 2017) and X-ray binary (XRB) catalogues
are also maintained (Kretschmar et al., 2019; Sazonov et al., 2020). Much of
the previous literature has focused on modelling the Galaxy and neutron star
populations in a statistical sense (e.g. Sartore et al., 2010; Sharma et al.,
2011), or if measured remnant positions are used, in terms of longitude,
latitude or scale height distributions (van Paradijs & White, 1995; White &
van Paradijs, 1996; Jonker & Nelemans, 2004; Manchester et al., 2005; Olausen
& Kaspi, 2014; Repetto et al., 2017; Verberne & Vink, 2021). However, there
have been some attempts to measure correlations between neutron stars and
Galactic structures in the plane of the Galaxy (e.g. clustering between OB
associations and high-mass X-ray binaries, Bodaghee et al., 2012; Coleiro &
Chaty, 2013). Crucially, it is now possible to map spiral arms and star
forming complexes throughout the Milky Way using masers (Reid et al., 2014;
Reid et al., 2016; Reid et al., 2019). Doing this at other wavelengths, such
as the optical, is difficult due to dust extinction, particularly further away
from the Solar neighbourhood (e.g. Russeil, 2003; Hou & Han, 2014; Gaia
Collaboration et al., 2016). There have also been studies looking at Milky Way
global properties in the context of Milky Way analogues (e.g. Licquia et al.,
2015). With precise localisations and distance estimates, we can now see how
neutron stars would appear distributed on the Milky Way, if viewed externally.

This paper is motivated by the magnetar-FRB connection, and as such there is a
focus on magnetars and FRBs. However, studying how neutron stars are
distributed throughout the Milky Way has a wider ranging relevance, for many
classes of core-collapse and merger transients. This will be the focus of a
follow-up paper in this series.

This paper is structured as follows. In Section 2 the Galactic magnetar,
pulsar and X-ray binary samples are described. Section 3 describes how the
basic components of the Milky Way are assembled in order to produce a face-on
‘image’ of the Galaxy. The neutron star projected offsets, host normalised
offsets and enclosed fluxes on the face-on disc of the Milky Way are discussed
in Section 4. Fraction of light measurements follow in Section 5. The arm
offsets, host offsets, enclosed fluxes and fraction of light distributions are
compared to samples of extragalactic transients in these sections. Section 6
presents a summary of the results and their implications, followed by
concluding remarks in Section 7.

##  2 Mapping Neutron Stars in the Milky Way

Neutron stars can be identified observationally in several ways. Here, we use
publicly available catalogues for four categories of neutron star/systems
containing neutron stars. These are magnetars (including SGRs and AXPs,
Olausen & Kaspi, 2014), pulsars (including binaries and millisecond pulsars,
Manchester et al., 2005), and X-ray binaries (high and low mass, Kretschmar et
al., 2019; Sazonov et al., 2020). These initial catalogues are reduced by
selecting those sources with a distance estimate, and a luminosity cut is also
applied to the pulsars, for increased uniformity of coverage across the
Galaxy.

The samples are then restricted to the half of the Galaxy this side of
Galactic centre (GC). This is because the neutron star samples have biases
against detection at large distances, particularly towards and beyond Galactic
centre. Distance uncertainties also tend to be larger further away, for
example, Gaia parallax-inferred distances (for XRBs) can only be estimated in
the local 4-5 kpc. Galactic centre is approximately 8 kpc away (e.g. Gravity
Collaboration et al., 2019), but throughout this paper we adopt a
$y$-coordinate cut of $<8.3$ kpc (we define the $y$-axis as parallel to the
Sun-Galactic centre line in a face-on Galactic coordinate system, with the Sun
at 0 and GC at 8.3 kpc). This cut approximately halves the galaxy at Galactic
centre, but ensures that the centrally located magnetar SGR 1745$-$2900 is not
excluded from any distributions. This would be unrepresentative given that we
know of at least one magnetar in the central ${\sim}100$ pc of the Galaxy.
Because we expect that Galactic populations on our side, versus the other
side, of Galactic centre should be similar, these half-Galaxy samples should
be broadly representative of the Galaxy as a whole. Any objects in the
Magellenic clouds are also removed in this final cut.

The input catalogues, cuts and final samples used are summarised in Table 1.
We now describe each neutron star data set in turn.

###  2.1 Magnetars

We use the McGill magnetar catalogue of Olausen & Kaspi (2014). The distances
for each source are the most recent values in the literature, and the
reasoning for each distance is listed in Table 7 of Olausen & Kaspi (2014).
These include estimating the distance to an associated SNR, using the tip of
the red giant branch in associated clusters, and X-ray inferred column
densities. Heliocentric distances uncertainties and are given by the upper and
lower values listed in the McGill catalogue (and distance references therein).
Where an uncertainty has not been quantified, a 15 per cent error on the
distance is assumed, the mean of the uncertainties that are available.

The distance for SGR 1935 is highly uncertain, with estimates from dispersion
measures, dust extinction, a potentially associated SNR, and nearby molecular
clouds (see Bailes et al., 2021, for an overview). We adopt the Bailes et al.
(2021) best distance estimate of 6.5 kpc, with a lower limit from their
dispersion measure/dust analysis of 1.2 kpc, and an upper limit of 9.5 kpc the
value assumed by Bochenek et al. (2020b).

Although the lack of a strong local bias in the magnetar sample suggests that
selection effects are not significant (Olausen & Kaspi, 2014), there is a
dearth of magnetars on the far side of the Galactic centre. This is despite
the fact that ${\gamma}$-ray observatories have sufficient sensitivity to
detect giant magnetar flares at extragalactic distances (Burns et al., 2021).
A similar effect has been noted before in the X-ray binary population (Jonker
& Nelemans, 2004), although in that case, detections are in softer X-rays
which are more liable to HI absorption and source confusion effects that
preferentially occur along sightlines towards the Galactic centre.

There are three possibilities: that many of the magnetars have mis-assigned
distances, that the sample really is incomplete, or that the effect is real
and there fewer magnetars on the far side of the Galaxy (unlikely, given that
spiral galaxies tend not to have large scale asymmetries of this nature).
Beniamini et al. (2019) discuss the Milky Way magnetar sample, explaining that
calculating completeness is non trivial when SGRs are detected in different
modes (quiescence versus flaring) with different observatories. They conclude
that the missing fraction is around 0.3 for currently active magnetars.
Similarly, Gullón et al. (2015) find that the Galactic X-ray pulsar population
is complete above an absorbed flux of 3$\times$10-12 erg s-1 cm-2. The sample
used is therefore unlikely to be seriously incomplete for currently active
sources this side of Galactic centre. The magnetars in the final sample are
listed in Table 7 of appendix A.

Figure 1: ATNF catalogue pulsar 1.4 GHz luminosities versus the best distance
estimates. A cut is applied at 65 mJy kpc2, which removes much of the strong
local disc bias in the sample and gives (approximately, with the exception of
the Galactic centre) equal coverage across the Galaxy. Two alternative cuts
($\pm$0.5 dex) are shown by dashed lines, these are later used to demonstrate
the impact of different choices. Selecting the most luminous pulsars slightly
shifts the sample towards younger sources - in log10(age/yrs), the full sample
has a median and standard deviation of 6.8$\pm$1.4, compared to 6.5$\pm$0.9
for the bright pulsars.

###  2.2 Pulsars

For pulsars, we use the ATNF catalogue (version 1.64, Manchester et al., 2005,
2016). Pulsars are far more numerous and have substantially weaker magnetic
fields than active magnetars, typically in the range 1011-1013 G. The
catalogue provides best estimate distances, inferred from parallaxes, HI
absorption, globular cluster associations, nebular lines and stellar
companions. Otherwise, the dispersion measure is used to infer a distance, in
this case using the Yao et al. (2017) electron density model. We note that the
majority of the sample are reliant on this model for a distance estimate, and
will therefore be concentrated into the high electron-density spiral arms.
This is less of an issue in the local disc, where other distance measurements
are more prevalent.

Incompleteness is likely higher in this sample than the magnetars, as shown in
Fig. 1. However, pulsars are sufficiently numerous that a cut can be placed at
high luminosity, such that a large region is covered in a less biased way,
with the exception of the known dearth of pulsars in the Galactic centre. This
detection bias along central sightlines arises due to the high dispersion
measure here (Rajwade et al., 2017). We will later demonstrate the effect this
has by comparing pulsar fraction of light distributions that include and
exclude the Galactic bulge region.

The chosen cut is 65 mJy kpc2. This is the lowest cut that yields
approximately constant numbers across the whole Galaxy (specifically, it
yields numbers either side of the GC heliocentric distance that are consistent
within Poisson uncertainties). To make the numbers exactly equal across both
halves of the Galaxy requires a harsh cut, reducing the total to only a
handful of sources. Our chosen cutoff is a compromise between sample size and
completeness. We later vary this threshold to determine the impact on our
fraction of light results. The $y<8.3$ kpc half-Galaxy cut is applied after
the luminosity threshold is applied to the whole catalogue.

###  2.3 Low-mass X-ray binaries

XRBs are systems containing an accreting black hole or neutron star; low mass
XRBs (LMXBs) have a donor star masses that are typically ${\lesssim}2.5$M⊙.
Although still subject to luminosity-distance completeness issues, XRBs
benefit from having a maximum theoretical luminosity (the Eddington
luminosity) and therefore distances can be approximated from X-ray
observations alone, in the absence of other indicators (Jonker & Nelemans,
2004). We use the INTEGRAL (Winkler et al., 2003) sample of Sazonov et al.
(2020), consisting of LMXBs detected by the hard X-ray sensitive telescope
IBIS. Some distances in the catalogue are estimated from Gaia parallaxes of
the optical counterpart, which may be introducing a local bias in the sample
with measured distances.

Krivonos et al. (2017) show the distance contours at which X-ray sources of
given luminosities can be detected in the INTEGRAL survey. The majority of the
Galaxy is covered with high completeness down to hard X-ray luminosities of
2${\times}$1035 ergs-1, which is the lower end of the LMXB luminosity
distribution. The sensitivity map provided by Krivonos et al. (2017) shows
that coverage is not uniform, as total survey time varies with Galactic
longitude. However, the contours can be approximated as circular from a
location that is ${\sim}3-5$ kpc closer to the Galactic centre than the Sun
(see figure 2 of Krivonos et al., 2017). Since the 2${\times}$1035 ergs-1
limit is ${\sim}$20 kpc away from this point in all directions, only the
outskirts of the Galaxy are affected by survey biases, which should be of
minimal impact. As the accretor type (i.e. neutron star or black hole) is
often ambiguous, we do not attempt to separate them. The XRB populations in
this paper therefore include a small contribution from black hole systems.

###  2.4 High-mass X-ray binaries

A review and catalogue of INTEGRAL-detected high-mass X-ray binaries (HMXBs)
is presented by Kretschmar et al. (2019). HMXBs are typically neutron stars
(although black hole systems also exist), accreting material from a
${\gtrsim}5$M⊙ companion. They are among the brightest X-ray sources in the
Galaxy, as such the sample should have high completeness across the region
being considered. As for the LMXBs, where available, distances are estimated
from Gaia DR2 parallaxes. The projected locations therefore reflect both
actual HMXB locations and survey/catalogue biases (the impact of which will be
quantified in Section 5). When a distance is uncertain and a range is given,
we use the mean of the upper and lower estimates. As for the LMXBs, we do not
separate black hole and neutron star systems.

Figure 2: Normalised distributions in Galactic latitude, longitude and radial
offset for the four Galactic neutron star samples described in Section 2, with
cuts applied. These are compared to star forming complexes (SFCs, Russeil,
2003), supernova remnants (SNRs Verberne & Vink, 2021), hot luminous stars
(Zari et al., 2021) and the pulsar distribution of Yusifov & Küçük (2004). The
distributions of the neutron star samples are generally in good agreement with
the previous works shown here, and others as discussed in the text. Perhaps
the most prominent difference is the excess - particularly of magnetars - at
9-10 kpc, above the smooth Yusifov & Küçük (2004) and Verberne & Vink (2021)
profiles, possibly associated with the Perseus arm.

###  2.5 Comparison to other Galactic population studies

We now discuss the broad spatial distributions of our final samples in the
context of previous mapping efforts. Distributions in Galactic latitude $b$,
longitude $l$ and radial offset $r$ from Galactic centre are shown in Fig. 2.
Also shown on the $b$ and $l$ panels are star forming complexes from Russeil
(2003), and the hot luminous star catalogue of Zari et al. (2021), which is
restricted to the local 3-4 kpc. Faucher-Giguère & Kaspi (2006) and Yusifov &
Küçük (2004) studied the Galactic pulsars population and found similar $b$ and
$l$ distributions to both the pulsar sample shown here and young stars/SFCs,
as expected.

The LMXB study of Jonker & Nelemans (2004) produces a similar range in $b$
(with the majority in the central 15 degrees) and $l$ (concentrated in the 30
degrees towards GC). Arnason et al. (2021) show that LMXBs do not correlate
with spiral arm structure, this can be seen in Fig. 2 in their smooth $l$
distribution compared to magnetars, pulsars and HMXBs. A number of studies
have noted HMXB correlations with spiral arms and SFCs (Bodaghee et al., 2007;
Bodaghee et al., 2012; Coleiro & Chaty, 2013; Arnason et al., 2021), which can
be seen here in $l$ distributions of the HMXB (and magnetar and pulsar)
samples, where over-densities correspond to tangents to the spiral arms. These
samples also have slightly increased numbers towards central longitudes
compared to SFCs and hot stars, which may be an artefact of X-ray and radio
surveys spending more time in this region.

On the right hand panel of Fig. 2, we show the Galactocentric radial offset
distribution for the four neutron star samples, plus the pulsar distribution
of Yusifov & Küçük (2004) and the supernova remnant distribution inferred by
Verberne & Vink (2021). Our Manchester et al. (2005) derived pulsar sample
appears shifted to higher offsets by ${\sim}$1 kpc compared to Yusifov & Küçük
(2004), similar to the simulated distribution of Faucher-Giguère & Kaspi
(2006). The SNR distribution, which should in principle trace the locations of
young neutron stars, is similarly more centrally concentrated. However, for
the magnetars ($N=20$) and HMXBs ($N=36$) in particular, the samples are small
enough that Poisson uncertainties may be producing noticeable differences
between these distributions. The largest discrepancy is the excess of
magnetars at 9-10 kpc, over the smooth Yusifov & Küçük (2004) and Verberne &
Vink (2021) curves. While this may partly be due to low number statistics, an
over-density here could also be due to the Perseus arm. Nevertheless, the
final half-Galaxy samples summarised in Table 1 broadly follow the expected
trends when compared to previous studies of various Galactic populations.

##  3 Building a Face-on Milky Way Image

In this section we construct an image of the Milky Way in two photometric
bands, on which the four neutron star populations can be placed, and measured
in the same way as extragalactic transients on their hosts. As in the previous
section, we restrict the mapping to $y<8.3$ kpc. This is to match the spatial
extent of the neutron star samples, and because the extrapolation of the local
spiral arm structure to the far side of the disc is highly uncertain. We now
describe the construction of the half-Galaxy image from three core components
- the spiral arms, a smooth underlying disc, and the bulge/bar.

###  3.1 Spiral arms

In order to map out where ongoing star formation is occurring in the Milky
Way, we follow the approach of Reid et al. (2019), who use water and methanol
masers to trace out the spiral arm structure in 3D. Masers are uniquely suited
to this, as they are numerous, bright, and their detection is relatively
unaffected by foreground absorption. In brief, Reid et al. (2016) and Reid et
al. (2019) use parallax measurements of various maser sources (star forming
complexes, young stellar objects, AGB stars and others, Valdettaro et al.,
2001; Pestalozzi et al., 2005; Green et al., 2017; Anderson et al., 2012;
Urquhart et al., 2014) to infer the spiral arm positions and parameters
locally, including pitch angle and tightness. The arms are then extrapolated
around the entire Galaxy. Because radio observations of masers yield radial
velocities, which can be converted into a local standard of rest velocity
$V_{\mathrm{LSR}}$, the velocity structure of the Galaxy can be mapped out to
distances beyond where parallaxes can be measured. They develop a Bayesian
code, with the estimated arm positions as a prior, that can infer the
likeliest distance given only a Galactic latitude, longitude and LSR velocity.
The warp of the disc is accounted for in the model (see Reid et al., 2016).
The code includes a prior (0 to 1) for whether the source is near or far, as
the VLSR measurements contain a distance degeneracy for sightlines along a
circular path.

We follow the same methodology as Reid et al. (2019) but restrict ourselves to
the Red MSX catalogue of Urquhart et al. (2014, see also ()). Where a near or
far distance is favoured in the Urquhart et al. (2014) catalogue, we use
P${}_{\mathrm{far}}=$0 or 1 as appropriate, otherwise
P${}_{\mathrm{far}}=$0.5. Although less Galactic structure is mapped out by
using a single input catalogue, the luminosity distribution is well
understood, as is the incompleteness as a function of distance. Limiting
ourselves to a single, well understood catalogue also avoids double counting
star forming complexes or single sources that appear in multiple maser
catalogues. We further limit the sample to HII regions and young stellar
objects (YSOs), in order to trace out star formation specifically. The
resultant star forming region catalogue comprises 1644 sources.

Taking the 1644 HII region and YSO masers of Urquhart et al. (2014), we run
the Bayesian distance fortran code of Reid et al. (2019) and obtain a Milky
Way spiral arm map. Beyond 3 kpc, the completeness of the input catalogues
drops off as a function of distance from the Solar System (inverse square law)
and Galactocentric radius (mainly due to source confusion towards the Galactic
centre). We want to correct for this, so that the the following three
requirements are satisfied:

  * •

The mean distance from each HII region/YSO maser to the next nearest
neighbouring maser as a function of Galactocentric radius does not depend on
Galactocentric azimuthal angle,

  * •

the mean number of masers within a fixed distance of each maser, again as a
function of Galactocentric radius, does not depend on Galactocentric azimuthal
angle,

  * •

the luminosity function of the masers is the same across the map.

To compensate for the high level of incompleteness at larger distances, we aim
to match the density of sources in the local (${\sim}$3 kpc) neighbourhood
according to the above metrics, across the (half) Galaxy.

We again follow the method of Reid et al. (2019). Artificial masers
(sprinkles) are added at random $x$ and $y$ offsets from each of the 1644 real
maser positions. The number of sprinkles added per real HII region/YSO maser
varies as a function of distance from Earth and the Galactocentric radius.
More random sprinkles are added at larger distances, and the width of the
Gaussian used to determine the random $x$ and $y$ offsetss from the existing
maser positions increases with Galactocentric radius (to reflect the widening
spiral arm width). This process is described by the following. The number of
sprinkles added for each of the 1644 masers is given by,

$N_{\mathrm{sprinkle}}=\Big{(}\frac{D}{3\mathrm{kpc}}\Big{)}^{2}-N_{\mathrm{corr}}$
(1)

at distances greater than 3 kpc. Below this, no sprinkles are added. The
maximum number added is 10, and the minimum 1. $N_{\mathrm{corr}}$ is a
correction to the form given by Reid et al. (2019), which is necessary given
our smaller input catalogue size. The $N_{\mathrm{sprinkle}}$ sources are
added following Gaussians in $x$ and $y$. The standard deviation in each is
given by,

$\sigma_{xy}=\frac{336+36(R_{g}-8.15)}{W_{\mathrm{corr}}}$ (2)

where $W_{\mathrm{corr}}$ is a correction we have added, again needed in order
to tune the random sprinkling to match the local 3 kpc density of points. In
Reid et al. (2019), with their large input catalogue, $N_{\mathrm{corr}}=0$
and $W_{\mathrm{corr}}=1$.

A final issue is that there is a dearth of sources in a cone of around
${\sim}$30 degrees toward the Galactic centre. On our side of the Galactic
centre, this is corrected for by fitting a parabola to join the gap in the
arms and sprinkling points along these curves so that their spatial density is
comparable to the arms either side of the gap. No correction is made to the
far side of the disc, as we only consider the half-Galaxy at $y<8.3$ kpc going
forwards.

Figure 3: Upper panel: the mean distance to the nearest neighbouring maser,
following the method of Reid et al. (2019) as described in the text. The 3
heliocentric distance ranges have approximately equal maser density as a
function of Galactocentric distance. Lower panel: The mean number of masers
within 1 kpc of each maser, again as a function of radial distance from
Galactic centre. The arm width and Nsprinkle functions used to add artificial
masers are given in equations 1, and 2. Figure 4: The three Galaxy components
used, and their combination (with appropriate scaling) to produce the $I$ and
$B$-band images. The spiral arm locations are derived from YSO and HII region
masers in the Urquhart et al. (2014) catalogue, assigned a distance using the
Reid et al. (2019) distance calculator. The disc is an exponential with scale
length 2.6 kpc, and the bulge is a Sérsic profile convolved with the bar
profile of Grady et al. (2020), at an angle of 27 degrees to the Sun-Galactic
centre line.

In order to tune the incompleteness corrections described by equations 1 and
2, so that the first two bullet-point criteria above are met, we vary
$N_{\mathrm{corr}}$ and $W_{\mathrm{corr}}$. The Galaxy is split into three
regions: sources within 3 kpc, those in the helicoentric distance range $3-8$
kpc, and those beyond 8 kpc. In Fig. 3, the target radial profiles are shown
in black, representing the maser distribution in the local 3 kpc. We aim to
find $N_{\mathrm{corr}}$ and $W_{\mathrm{corr}}$ values such that, when the
artificial maser sprinkles are added, the $3-8$ kpc and $>8$ kpc profiles
match those at $<3$ kpc. This will correct for incompleteness, so that it is
at least uniform across the half-Galaxy region in consideration. We find that
the default $N_{\mathrm{corr}}=0$ and $W_{\mathrm{corr}}=1$ values from Reid
et al. (2019) provide a poor agreement between local sources and the rest of
the Galaxy, unsurprising given our smaller input catalogue. Sampling values in
the range 0-8, $N_{\mathrm{corr}}=6$ and $W_{\mathrm{corr}}=5$ provide the
best match in terms of ${\chi}^{2}$ between the local and more distant
regions. The result is shown in Fig. 3.

Each sprinkle is also assigned a bolometric luminosity. Inside 3 kpc, the
Urquhart et al. (2014) catalogue is complete down to bolometric luminosities
of 1000L⊙. Therefore, outside 3 kpc, the randomly sprinkled points are
assigned a luminosity drawn randomly from the distribution below the
completeness limit at that distance. In this way, the luminosity function is
approximately constant across the half-Galaxy map, complete down to 1000L⊙
everywhere.

The resultant maser map is shown in the left-hand panel of Fig. 4 (and can
also been seen in the final image, see Fig. 5). To compare the arm positions
to other Milky Way spiral arm maps, references include Vallée (2002), Russeil
(2003), Faucher-Giguère & Kaspi (2006), Vallée (2008), Churchwell et al.
(2009), in addition to more local mapping efforts using Gaia (e.g., Castro-
Ginard et al., 2021; Zari et al., 2021) and electron density models (e.g.
Cordes & Lazio, 2002).

###  3.2 The disc, bulge and bar

In addition to the arms, the Milky Way has a disc (in which the arms are
embedded) and a central bulge/bar. Our basic model follows Widrow et al.
(2008) and Bland-Hawthorn & Gerhard (2016). We model the disc with an
exponential profile, adopting a scale length of R${}_{d}=2.6$ kpc. This value
is based on the range in the literature, and is around the mean of thin disc
(which dominates the disc stellar content) values. The bulge is modelled as a
Sérsic profile with index $n=1.32$ and half-light radius R${}_{d}=0.64$ kpc
(Widrow et al., 2008). To account for the bar, this is convolved with the bar
profile as mapped by Mira variables in Gaia (Grady et al., 2020). The angle
between the Sun-Galactic centre line and bar semi-major axis is taken to be 27
degrees (e.g. Binney et al., 1997; Bland-Hawthorn & Gerhard, 2016).

###  3.3 Weighting the components

The three components - a smooth disc, embedded arms and bulge/bar - now need
weighting to represent their respective contributions to the total luminosity
of the Milky Way. This will also be wavelength dependent.

As a starting point, we take the bulge and disc luminosities of Flynn et al.
(2006), where the disc includes the arms. The $I$-band bulge and disc
luminosities are 1010L⊙ and 3$\times$1010 L⊙ respectively.

The procedure described in Section 3.1 means that we have a value for the
total bolometric luminosity of the masers above the 1000L⊙ limit,
1.78$\times$108L⊙. This is twice the Urquhart et al. (2014) estimate for the
total contribution of high mass embedded star formation to the Galactic
luminosity (due to differences in incompleteness corrections). However, for
our purposes we are only interested in the spiral arm locations, and the
relative flux contributions of the galaxy components.

To determine what fraction of the disc light should arise from the spiral arms
versus the smooth underlying exponential, we adopt an arm strength of 0.15 in
the $I$-band, typical for Milky Way mass galaxies (e.g. Díaz-García et al.,
2019; Yu & Ho, 2020). Arm strength is defined as the ratio of disc surface
density arising from the $m=2$ spatial Fourier component, versus the $m=0$
component (Seigar & James, 1998). It therefore quantifies the proportion of
disc light arising from short spatial scales (i.e., the spiral arm structure).
Arm strength varies in different bands, as the spiral arms are usually
slightly bluer than the inter-arm regions - for $I$-band strengths of 0.15,
typical $B$-band strengths are ${\sim}$0.2, values which we use going forward
(Yu et al., 2018).

We now assume that $I$-band luminosity of the maser-associated HII regions is
proportional to their maser-inferred bolometric luminosity. If the $I$-band
arm strength is 0.15, the $I$-band exponential disc must have a luminosity
$\sim 5.5$ times that of the arms. The $I$-band bulge/bar luminosity is then
one third of the disc$+$arm total (as the Flynn et al. (2006) disc value
includes spiral arms). The constants in the exponential disc and Sérsic
profiles are scaled to reflect these respective contributions to the $I$-band
light.

We now have an $I$-band image with the components appropriately scaled.
However, the $I$-band is a better tracer of stellar mass than star formation.
To obtain a bluer version of the image, we use the $B$–$I$ colours of Licquia
et al. (2015), who estimate a total integrated Galaxy colour - including dust
- of 1.77. Bland-Hawthorn & Gerhard (2016) obtain an alternative value of
1.85$\pm$0.1, but this is consistent with Licquia et al. (2015). Note that all
$B$ and $I$ magnitudes and colours quoted in this paper are in the Vega
system, but they are converted to AB magnitudes to perform any calculations.

We now need the $B$–$I$ colour of either the disc or bulge in order to
determine the $B$-band luminosities of all three components. Flynn et al.
(2006) provide a local disc $B$–$I$ colour of 1.48, but this has been
corrected for dust and is therefore artificially blue. We instead use the ‘as
observed’ (not dust-corrected) Milky Way colour estimates of Licquia et al.
(2015), which are based extragalactic analogues chosen for their mass and SFR
similarity with the Milky Way. Within these analogues they identify a bulge-
dominated sub-sample, which we use to estimate the Milky Way bulge colour.
This has a mean $B$-band mass-to-light ratio (MLR) of 4.1. We also use their
global MLR estimates in $I$ and $B$ of 1.29 and 1.89, and a bulge-to-total
mass ratio of 0.3 (Bland-Hawthorn & Gerhard, 2016). Taking the $I$-band
component luminosities from Flynn et al. (2006), this implies a bulge/bar
$B$–$I$ of 2.41.

We now have a bulge/bar $B$–$I$ of 2.41 a global $B$–$I$ of 1.77, and $I$-band
luminosities for the total Galaxy and the components. We can therefore infer
the colour of the final component (disc including arms) - 1.62 - and hence we
have the luminosities of all three components in each band. This disc colour
is bluer than the total integrated colour as expected, and 0.14 redder than
the Flynn et al. (2006) dust-corrected value for the local disc.

###  3.4 The final Milky Way image

The construction of the final Galaxy image is shown in Fig. 4, in both bands.
These images have a pixel scale of 250 pc/pix, and we have applied a Gaussian
blur with a FWHM of 2 pixels to simulate Hubble Space Telescope, (HST) quality
imaging (${\sim}$0.05 arcsec pix-1). The masers are treated as point sources,
and their luminosity is assigned to whichever pixel they fall under. In
reality they are not point sources, but this unlikely to be a significant
issue, as the majority of star forming regions have radii $<250$ pc, and
almost all $<500$ pc (Wisnioski et al., 2012). This effect is further
diminished as they tend to have approximately Gaussian density profiles, so
that the light contributed to neighbouring pixels will not usually be a
significant fraction of the total star forming region luminosity. The final
$B$ and $I$-band images are similar, as the three input components have the
same profiles, but the different scaling of these components with respect to
each other means that there are differences (e.g. the light is more centrally
concentrated in the $I$-band image). The final half-Galaxy $B$ and $I$-band
images are available in fits format at https://github.com/achrimes2/MW-NS-
Flight.

###  3.5 Placing Galactic neutron stars on the image

The final neutron star samples compiled in Section 2 are now placed on the
half-Galaxy image constructed in Fig. 4. Their positions have been computed
from the Galactic latitude, longitude, and given distance, correcting for a
Solar System height above the Galactic plane of 15 pc (Olausen & Kaspi, 2014,
although this negligible for our purposes). Their positions on the $B$-band
image are shown in Fig. 5. The selected magnetars, bright pulsars, INTEGRAL
LMXBs and HMXBs are shown as blue circles, magenta circles, cyan squares and
orange triangles respectively.

Figure 5: The Milky Way map overlaid with magnetars positions (top left, blue
dots with distance uncertainties indicated, Olausen & Kaspi, 2014), pulsars
(top right, magenta dots, Manchester et al., 2005), LMXBs (Sazonov et al.,
2020, bottom left, cyan squares,) and HMXBs (bottom right, orange triangles,
Kretschmar et al., 2019). The Solar system is located at (0,0), and Galactic
centre at (0,8.3). The FRB source SGR 1935+2154 is indicated in the magnetar
panel by a triangle. Only the half-Galaxy region shown is used for
measurements throughout, to limit the impact of heliocentric distance
uncertainties and detection biases.

##  4 Offsets and Enclosed fluxes

###  4.1 Offsets

Having selected our neutron star samples in Section 2, and constructed Milky
Way model in Section 3, we now move on to measuring how neutron stars are
distributed with respect to Galactic structures and light. We start with their
offsets from Galactic centre.

Figure 6: Projected (left) and host normalised (right) offsets for Milky Way
magnetars, compared to other Galactic neutron star systems (top row) and the
offsets of extragalactic transients on their hosts (bottom row, shown
separately for clarity). FRB 20200120E has been added to the FRB offset
dataset, assuming association with M81 (Bhardwaj et al., 2021). The light grey
shaded region is bounded by the extreme upper and lower distributions for the
magnetars, assuming the maximum and minimum offsets possible along that line
of sight (and that the magnetar lies ‘in’ the Galaxy, at ${\delta}r<12$ kpc).
The narrower dark grey shading indicates the maximum and minimum allowed
within the quoted magnetar distance uncertainties. A 15 per cent error is used
where one is not available. The $I$-band host-normalised offset is shown (with
half-light radius r${}_{h}=2.22$ kpc), similar results are obtained using the
$B$-band rh of 2.37 kpc (see Table 7 of appendix A).

We now look at the offsets of the neutron stars from the Galactic centre. Our
offsets measurements are 2D projections of the 3D distance from Galactic
centre. Both offsets ${\delta}r$ and host-normalised offsets ${\delta}r/r_{e}$
are shown in the upper panels of Fig. 6, where $r_{e}$ is the half-light
radius. For our Milky Way image, $r_{e}=2.37$ kpc in the $B$-band and $2.22$
kpc in the $I$-band. In principle, normalising by the half-light radius should
make comparisons between galaxies of difference physical sizes fairer.

The four Galactic samples are compared to extragalactic transient offsets
distributions, shown in the lower two panels of Fig. 6. The comparison data
are from Mannings et al. (2020, FRBs), Blanchard et al. (2016, LGRBs); Lyman
et al. (2017, LGRBs), Fong et al. (2013, SGRBs), Uddin et al. (2020, type Ia
SNe), Kelly & Kirshner (2012, CCSNe); Schulze et al. (2020, CCSNe), Lunnan et
al. (2015, SLSNe); Schulze et al. (2020, SLSNe), and Lunnan et al. (2017, Ca-
rich SNe); De et al. (2020b, Ca-rich SNe). We have also added the M81 repeater
to the FRB distribution Bhardwaj et al. (2021). The results of AD-tests
between the magnetars/pulsars/XRBs offsets and the comparison samples are
listed in Table 2. Results which do not round up to 0.01 are simply listed as
0.00. The maximum value is 0.25, as outputs are capped at this value by the
scipy Anderson function.

Across the four Galactic samples, FRBs most frequently have a p-value $>0.05$.
Only the $I$-band normalised results are shown in Fig. 6, using the $B$-band
half-light radius results in slightly less offset distributions (we refer the
reader to appendix A, where the full magnetar results are listed in Table 7,
and access to the data for the other samples is described).

To quantify the uncertainty on the magnetar distribution, we take the the
maximum and minimum offsets possible along the line of sight, assuming that
the magnetars lie ‘inside’ the Galaxy (i.e. $<$12 kpc from Galactic centre).
The region bounded by the minimum and maximum possible distributions is shaded
light grey in Fig. 6. We also take the listed uncertainties, shading between
the quoted upper and lower bounds produces the narrower dark grey band. Even
if we consider the lowest offsets that the magnetars could possibly have along
their sightlines, they cannot match the CCSN offset distribution.

Table 2: Offset and host-normalised offset AD test results, for the comparisons made in Fig. 6. Top: offsets, middle: $B$-band host normalised offset (the comparison data is also normalised in a UV/blue band where available), bottom: $I$-band host normalised offset (comparison data normalised in an $r$ or $I$-band). p-values of 0.05 or greater, indicating statistical consistency, are highlighted in bold and have a grey background. None of the Galactic populations are consistent with LGRBs, SLSNe or CCSNe according to this measure. Note that host-normalised data for SNe Ia are not available. | LGRB | SLSN | CCSNe | FRB | SNe Ia | SGRB
---|---|---|---|---|---|---
Offset |  |  |  |  |  |
Magnetars | 0.00 | 0.00 | 0.00 | 0.07 | 0.17 | 0.00
HMXRBs | 0.00 | 0.00 | 0.00 | 0.02 | 0.02 | 0.00
Pulsars | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
LMXRBs | 0.00 | 0.00 | 0.04 | 0.25 | 0.01 | 0.00
$B$-band normed |  |  |  |  |  |
Magnetars | 0.00 | 0.00 | 0.00 | 0.04 | n/a | 0.08
HMXRBs | 0.00 | 0.00 | 0.00 | 0.00 | n/a | 0.01
Pulsars | 0.00 | 0.00 | 0.00 | 0.00 | n/a | 0.00
LMXRBs | 0.00 | 0.00 | 0.00 | 0.25 | n/a | 0.02
$I$-band normed |  |  |  |  |  |
Magnetars | 0.00 | 0.00 | 0.00 | 0.02 | n/a | 0.06
HMXRBs | 0.00 | 0.00 | 0.00 | 0.00 | n/a | 0.01
Pulsars | 0.00 | 0.00 | 0.00 | 0.00 | n/a | 0.00
LMXRBs | 0.00 | 0.00 | 0.00 | 0.25 | n/a | 0.03

Looking at Fig. 6, it is evident that the specific morphology of the Milky Way
dictates the shape of the Galactic distributions. The choice of disc scale
length has an impact of the normalised offsets, but literature estimates only
vary by at most a factor of ${\sim}$1.5-2 (Bland-Hawthorn & Gerhard, 2016).
Much of the light is concentrated centrally in the bulge (in both bands), a
predominantly older population where fewer young neutron stars reside.
Consequently, the LMXBs (known to be a older population) appear distributed on
the Milky Way’s $I$-band light in a similar way to supernovae on their hosts,
although this does not imply a connection. Likewise, the younger pulsars,
HMXBs and magnetars are more offset than the short-delay time extragalatic
transients. This arises because we are dealing with a single Galaxy: in the
extragalactic samples, the range of host morphologies, sizes and viewing
angles removes any such trends.

There may also be issues related to faint, halo light that is not detected in
extragalactic samples Perets & Beniamini (2021), and other Galaxy-specific
morphological effects that influence the offsets and flux profile, such as the
specific arm locations. For example, the Galactic ring of star formation at
3-5 kpc has a high density of magnetars, relative to other arm structures.
This is reflected as a dearth of objects at $<3$ kpc and a rapid climb in the
cumulative distribution at $3-5$ kpc. Interestingly, magnetar locations appear
to favour this inner, nuclear ring, over spiral arms further out. This does
not appear to be a distance-sensitivity effect as similarly bright arms at
comparable heliocentric distances still have fewer magnetars. This may be
suggestive of other factors playing a role in their production, beyond a high
star formation rate (such as metallicity or IMF variations) and warrants
further investigation.

###  4.2 Enclosed fluxes

Another way to measure how the neutron stars relate to light is to measure the
fraction of the total Galactic flux enclosed at their radial distance. We
calculate this for the magnetar, pulsar and XRB samples, comparing to
supernova (Anderson & James, 2009; Anderson et al., 2015) and FRB
distributions (Mannings et al., 2020) in Fig. 7. The grey shaded regions are
the magnetar uncertainties, calculated in the same way as for the offsets. We
again restrict the samples and pixels to $y<8.3$ kpc, and measure the fraction
of flux enclosed in the semicircle produced.

Table 3: As for Table 2, but for enclosed fluxes. The distributions are shown in Fig. 7. Only the $I$-band is used for a fair comparison to the extragalactic samples. | Type Ia SNe | FRB | SNe II | SNe Ibc
---|---|---|---|---
Magnetars | 0.04 | 0.25 | 0.02 | 0.00
HMXRBs | 0.00 | 0.11 | 0.00 | 0.00
Pulsars | 0.00 | 0.07 | 0.00 | 0.00
LMXRBs | 0.25 | 0.25 | 0.25 | 0.01

AD-test results are given in Table 3. FRBs are the transient most often
consistent with the various neutron star populations. The enclosed flux
measurements, like the offsets, are biased by the specific morphology of the
Milky Way, specifically because the neutron star distributions are being drawn
from just one Galaxy. To disentangle these effects, we now look at the
fraction of light statistic, which operates on a pixel-by-pixel basis and can
be restricted to sub-regions within the Galaxy.

Figure 7: Top: enclosed fluxes for the four Galactic samples. Bottom: the
fraction of Milky Way flux (at $y<8.3$ kpc) enclosed within the Galactocentric
radius of the magnetars, the $I$-band version is used for a fairer comparison
to the extragalactic samples. The maximum and minimum distributions possible
for the magnetars are shaded in light grey, the same is shown in dark grey
using their given distance uncertainties (or a 15 per cent error where one is
not available).

Figure 8: Upper panels: half-Galaxy Flight colour maps with magnetars
positions and uncertainties indicated ($B$-band left, $I$-band right). The 250
pc/pixel scale (with a 2 pixel FWHM) is broadly representative of HST spatial
resolution for cosmological transients. Pixels within a 12 kpc Galactocentric
radius are selected for calculating the Flight distributions. Lower panels:
the corresponding fraction of light distributions for the magnetars (solid
black lines). The light grey shaded regions represent the minimum and maximum
distributions that arise from taking the minimum and maximum Flight values
along the magnetar sightlines. This places a strong upper limit on how
concentrated magnetars can be on the Galactic light. The darker grey bands use
linear sampling between the McGill catalogue upper and lower distance
estimates, taking the mean uncertainty of 15 per cent on a distance where a
range is not provided. The distributions for various extragalactic transients
on their host galaxies are also shown.

##  5 Fraction of light

We now turn to the fraction of light (Flight, Fruchter et al., 2006)
statistic. The Flight distribution is calculated by ranking host-associated
pixels by brightness, and assigning each their value in the cumulative sum,
normalised by the total (half) galaxy cumulative flux. The brightest pixel
therefore has the value 1. Transients that arise from the light in an unbiased
way produce an Flight cumulative distribution that follows a 1:1 line, those
that are concentrated on brighter regions have a distribution below the 1:1
line, and those are that offset or avoid bright regions lie above this line.
UV/blue bands are assumed to trace young stars and thus star formation,
whereas IR/red bands better trace stellar mass.

The fraction of light method has the benefit of being less dependent on host
morphology. For example, if a transient occurs on a bright star forming
region, but at a large offset from the host centre, both the offset and
enclosed flux measurements will indicate a low likelihood of association with
star formation/stellar mass (depending on the wavelength). The Flight value,
however, will assign the proportionate ranking in the distribution of host
pixels, ranking it highly, and correctly identifying a close association. It
is also possible to restrict the region considered, for example, if we want to
look solely at the disc (Lyman et al., 2017). This enables use to probe how
transients (and in our case neutron stars) trace light in different stellar
populations.

Because Flight relies on pixel values, the choice of spatial resolution is
important. The adopted resolution of 250 pc per pixel is broadly typical of
HST resolution for targets at cosmological ($z\sim 1$) distances.

###  5.1 Half-Galaxy results

We select all pixels at $y<8.3$ kpc and within 12 kpc of the Galactic centre
as being associated with the galaxy. Pixels outside this semicircular region
are assigned F${}_{\mathrm{light}}=0$. The pixels in the half-Galaxy are
ranked by their cumulative value and normalised. The pixel selection and
Flight colour maps are shown in the upper panels of Fig. 8. In the lower
panels, magnetar Flight distributions are compared with extragalactic
transients. The comparison data are from Mannings et al. (2020, FRBs),
Blanchard et al. (2016, LGRBs); Lyman et al. (2017, LGRBs), Fong et al. (2013,
SGRBs), Wang et al. (2013, type Ia SNe), Svensson et al. (2010, CCSNe) and
Lunnan et al. (2015, SLSNe).

The light and dark grey uncertainty bands on the magnetar distributions are
again derived from the maximum and minimum pixel values possible along the
line of sight, and within the quoted distance ranges (or with a 15 per cent
distance uncertainty if unavailable). Where an errorbar takes us past $y=8.3$
kpc, or when calculating the full range of values along a sightline, we have
to quantify the expected pixel values on the far side of the disc. To do this,
we simply mirror the galaxy across $y=8.3$ kpc, sampling the mirrored values
out to the appropriate distance (either the upper distance estimate, or the
’edge’ of the Galaxy). This should give a reasonable estimate of the range of
pixel values expected for sightlines beyond $y=8.3$ kpc, assuming the Galaxy
is approximately symmetric on large scales.

The dark grey uncertainty band is somewhat larger here than for the offsets or
enclosed fluxes. This is because we are now dealing with pixel values, which
can have far more variation along the same line of sight than offsets (no
dependence on pixel values) or enclosed fluxes (summing pixel values within a
radius). We note that, because the quoted uncertainties are typically not
formally quantified, we have simply shaded between the upper and lower
heliocentric distances (i.e., assuming constant probability density in this
range). In reality, the probability density closer to the assumed magnetar
distances is likely greater than at the edges of these ranges.

The uncertainties are slightly larger in the $B$-band (this can be seen in
Fig. 8), because the arms contribute a higher fraction of the total flux.
Random variations in location due to distance uncertainties therefore sample a
wider range of pixel values along a given line of sight, providing a wider
range of possible distributions. Based solely the range possible along a line-
of-sight, it is possible that many magnetars actually lie on highly-ranked
pixels and are not inconsistent with CCSNe or even SLSNe. However, the LGRB
distribution cannot be reached, independent of the heliocentric distances
assumed.

AD tests between the magnetars/pulsars/XRBs and extragalactic transients are
listed in Table 4. The equivalent figures for the pulsars and XRBs are
available in Figs 10, 11 and 12 of Appendix B. Across the four Galactic
samples, as we found for the offset and enclosed flux measurements, FRBs fail
to reject the null hypothesis that they are drawn from the same distribution
as Galactic neutron stars the most often. SGRBs are the least consistent with
Galactic neutron star populations (SGRBs are much less concentrated on the
light).

The choice of pixel selection radius (12 kpc) has only a small impact on the
resultant Flight values. For the interested reader, we have made python
scripts and notebooks available which generate a similar Flight figure, given
user defined input parameters333https://github.com/achrimes2/MW-NS-Flight.

Figure 9: Exploring the effect of restricting the Flight analysis to the local disc ($<4$ kpc). This region has more accurate distances estimates for both the Galactic neutron stars and spiral-arm tracing masers. It also allows us to quantify the effect of the bulge on the distributions shown in Fig. 8, sampling a more uniform stellar population. Top: the pixel selection for Flight when a 4 kpc heliocentric distance restriction is applied. Bottom: magnetar Flight distributions for the half Galaxy as shown in Fig. 8, and the local region results in each band. Excluding the bulge pushes the distribution towards the 1:1 light-tracer line. Table 4: As for Tables 2 and 3, but for half-Galaxy $B$-band (top four rows) and $I$-band (bottom four rows) Flight comparisons, as shown in Figs. 8, 10, 11 and 12.  | LGRB | SLSN | FRB | SN Ia | SGRB
---|---|---|---|---|---
Magnetars | 0.02 | 0.01 | 0.25 | 0.01 | 0.01
HMXRBs | 0.01 | 0.00 | 0.11 | 0.00 | 0.00
Pulsars | 0.00 | 0.00 | 0.21 | 0.00 | 0.00
LMXRBs | 0.25 | 0.20 | 0.25 | 0.00 | 0.00
| LGRB | CCSNe | FRB | SN Ia | SGRB
Magnetars | 0.00 | 0.00 | 0.25 | 0.03 | 0.00
HMXRBs | 0.00 | 0.00 | 0.25 | 0.00 | 0.00
Pulsars | 0.00 | 0.00 | 0.25 | 0.00 | 0.00
LMXRBs | 0.00 | 0.10 | 0.25 | 0.25 | 0.00
Table 5: Flight AD-test results for comparisons within the local 4 kpc only, as shown in Figs. 9, 10, 11 and 12. Top four rows: $B$-band, bottom four: $I$-band. | LGRB | SLSN | FRB | SN Ia | SGRB
---|---|---|---|---|---
Magnetars | 0.25 | 0.25 | 0.22 | 0.19 | 0.00
HMXRBs | 0.25 | 0.24 | 0.25 | 0.04 | 0.00
Pulsars | 0.07 | 0.04 | 0.25 | 0.01 | 0.00
LMXRBs | 0.22 | 0.11 | 0.15 | 0.02 | 0.00
| LGRB | CCSNe | FRB | SN Ia | SGRB
Magnetars | 0.09 | 0.23 | 0.25 | 0.25 | 0.00
HMXRBs | 0.06 | 0.25 | 0.25 | 0.25 | 0.00
Pulsars | 0.00 | 0.14 | 0.25 | 0.25 | 0.00
LMXRBs | 0.13 | 0.25 | 0.19 | 0.25 | 0.00

###  5.2 Local region results

In Sections 2 and 3, it was noted that sample completeness is higher, and
distance estimates more reliable, in the local disc (within 3-5 kpc of our
location). This is because trigonometric parallax measurements are possible in
this region (both for the neutron stars and the masers), and survey
sensitivities are less of a issue. There are also other biases, such as an
absence of pulsars around the Galactic centre due to the high dispersion
measures here (Rajwade et al., 2017, and see Fig. 5). Furthermore, using the
half-Galaxy map mixes the older population of the bulge and the younger,
higher star formation rate disc. In summary, our half-Galaxy comparisons to
extragalactic samples may be suffering from biases which have not been fully
corrected.

In order to quantify these effects on our results, we present Flight AD-test
results for the local 4 kpc region versus extragalactic transient samples. In
this region, completeness should be higher and distance uncertainties lower,
as a higher proportion of the sample (particularly XRBs) have parallax-based
distance estimates. We select pixels and neutron stars within 4 kpc of the Sun
and $<12$ kpc from Galactic centre, and re-calculate the Flight distribution
for this area only. The magnetar results are shown in Fig. 9, alongside the
half-Galaxy results to demonstrate the difference. The corresponding pulsar
and XRBs distributions are again shown in Figs. 10, 11 and 12 of Appendix B.

Restricting the Flight analysis to the local disc has a larger impact in the
$I$-band than the $B$-band, but in both cases has shifted the
magnetars/pulsars/XRBs closer towards being light-tracers. This makes sense in
terms of the Galactic morphology: younger magnetars and HMXB systems avoid the
older (but bright) bulge, so the inclusion of the bulge in Flight calculations
pushes them away from being unbiased tracers. Similarly, LMXBs - which are
concentrated in the bulge - are shifted the least by its exclusion from
Flight. This also explains why the shift between half-Galaxy and local results
is greater in $I$, as the bulge contributes a higher proportion of the total
light in that band.

Table 5 lists the local region AD-test results. The results are less clear
compared to the half-Galaxy comparisons. FRBs are again a good match to
Galactic neutron stars, and the significance of this is even more apparent
when the p-values are considered (reaching the 0.25 cap 6/8 times). However,
unlike the half-Galaxy case, LGRBs, SLSNe, CCSNe and thermonuclear SNe are
also consistent with several Galactic populations. Plausibly, the smaller
Galactic sample sizes that result from restricting ourselves to $<4$ kpc
reduces the power of the AD-test, making consistency with other samples harder
to rule out.

###  5.3 Fraction of light uncertainties

Aside from the impact of the multi-component nature of the Galaxy on these
results, uncertainties in both neutron star distances and spiral arm structure
also play a role in shaping the distributions. For example, it is possible
that the neutron stars appear artificially less concentrated with respect to
light due to distance uncertainties. This should preferentially scatter them
away from their natal star forming regions, assuming they were born in star
forming regions, given that inter-arm regions take up more volume than the
arms. The local region results should address this somewhat - in addition to
removing the effect of the Galactic bulge, parallax measurements are possible
here (both in the optical for XRBs and radio for pulsars).

A reason to suspect that arm position inaccuracies are not a major issue is
that the spiral arm model is derived directly from trigonometric parallax
measurements of CO/HI emission in the local disc (Reid et al., 2016). CO/HI
tracers have previously been demonstrated to better trace arm mid-points than
masers (Vallée, 2014) \- we are simply populating the CO/HI derived spiral arm
model with the masers. Furthermore, the images we construct have a spatial
resolution of 250 pc/pixel and a PSF to replicate HST-quality imaging, which
is insufficient to resolve the offset between arm mid-points and typical CO/HI
offsets of 100-200 pc (Vallée, 2014). In the local few kiloparsecs the spiral
arm tracer distances are directly measured, beyond this the arms are
extrapolated, but throughout we have neglected the far side of the disc where
this extrapolation would be particularly problematic.

While the local 4 kpc comparisons are inherently less affected by
uncertainties, the full half-Galaxy results are still valuable, as many
transients do occur in galaxies with mixed populations and multiple
components. This is particularly notable for FRBs, whose host galaxies are
varied and include several Milky Way-like spirals and barred spirals.

The other assumptions we have made about the Galactic structure and the colour
of the different components also contribute uncertainty, such as the adopted
arm widths, which may have been underestimated. Zari et al. (2021) show that
young stars are only loosely concentrated in the spiral arms (Arnason et al.,
2021, find a similar result for HMXBs). Widening the arms by changing Equation
2 in Section 3 would tend to increase the association of Galactic neutron
stars with light. This is because any distance uncertainty induced offsets
from nearby bright pixels will be reduced. However, the pixel values would
also decrease (there would be fewer masers per pixel).

Overall, specific Galaxy mapping choices appear to be less important than
Galactic neutron star sample incompleteness and distance uncertainties. We
refer the reader to the interactive tools at https://github.com/achrimes2/MW-
NS-Flight, which can be used to vary the relative light contributions of the
bulge/bar, disc and arms, to visualise the impact on Flight. The spatial
resolution of the image can also be degraded below the 250 kpc per pixel level
used in this paper.

##  6 Discussion

###  6.1 Summary of results

Table 6 gives an overview of our results, in terms of the percentage of times
that the different extragalactic transients agree at the 2${\sigma}$ level
($p>0.05$) with Galactic neutron stars, as measured on the Milky Way face-on
image. The comparisons considered are the Galactocentric offsets and host-
normalised offsets in both bands, the $I$-band enclosed flux, the $B$ and
$I$-band half-Galaxy Flight and the local 4 kpc $B$ and $I$-band Flight. The
total number of AD-tests made against each transient is counted across Tables
2, 3, 4 and 5.

It is clear from Table 6 that the distribution of Galactic neutron stars on
the Milky Way best matches, of all of the transients, FRBs on their hosts.
However, we are unable to clearly differentiate which class of neutron star is
the best fit - within the FRB AD-tests, magnetars return $p>0.05$ on 6
occasions, HMXBs/pulsars 5 times each, and LMXBs 8 times.

Of the other extragalactic transients, SGRBs (known to be mergers involving at
least one neutron star, Barthelmy et al., 2005; Fong et al., 2010; Church et
al., 2011; Tunnicliffe et al., 2014; Abbott et al., 2017), are the worst
match. SRGBs have extended offset distributions, due to the combination of
natal kicks and long gravitation wave inspiral times. The implication is that,
in the Milky Way, old, kicked systems are missing from our catalogues,
possibly because they are distant and not bright emitters like magnetars,
pulsars or XRBs. Of the other samples, none stand out as being more or less
consistent with the Galactic systems.

Table 6: For each extragalactic transient, the number of comparisons made to magnetar, pulsar and XRB systems on the Milky Way is listed, along with the number and fraction of those tests which return at p-value$>0.05$. The results used to populate this table are listed in Tables 2 (offsets and host normalised offsets in both bands), 3 (enclosed fluxes, $I$-band only), 4 (half-Galaxy Flight) and 5 (Flight in the local 4 kpc). Overall, FRBs are clearly distributed on their hosts in a similar manner to neutron stars on the Milky Way, and are a better match than the other transients tested. Transient | N${}_{\mathrm{AD-test}}>0.05$ | NAD-test | Fraction $>0.05$
---|---|---|---
LGRB | 8 | 28 | 0.29
SLSN | 4 | 20 | 0.20
CCSNe | 6 | 28 | 0.21
FRB | 24 | 32 | 0.75
SNe Ia | 8 | 24 | 0.33
SGRB | 2 | 28 | 0.07

###  6.2 Implications for the neutron star-FRB connection

Initial FRB host population studies (Heintz et al., 2020; Mannings et al.,
2020) led to differing interpretations over whether magnetar flares can
explain all FRBs (Bochenek et al., 2020a; Safarzadeh et al., 2020; Margalit et
al., 2020). It is also unclear whether the identification of FRB-like flares
from SGR 1935 (Bochenek et al., 2020b; CHIME/FRB Collaboration et al., 2020)
definitively establishes a connection between magnetars and extragalactic FRBs
(Tavani et al., 2021; Ridnaia et al., 2021; Bailes et al., 2021).

Our results, looking at magnetars, pulsars and XRBs on the Milky Way’s light,
show that Galactic neutron stars are consistent with FRB locations on their
hosts. This result holds across various types of comparison (distance to the
star forming region, host offset, enclosed flux and fraction of light),
although the significance varies substantially between these. Other
extragalactic transients are also consistent with the Milky Way neutron star
distribution, depending on the comparison made, but FRBs are the transient
most frequently in agreement. Our results cannot differentiate between most
FRB progenitor models which invoke neutron stars (e.g. magnetars from core-
collapse and accretion induced collapse, ultra luminous X-ray binaries,
combing models etc, Margalit et al., 2019; Ioka & Zhang, 2020; Lu et al.,
2020; Sridhar et al., 2021; Zhao et al., 2021; Deng et al., 2021). If FRBs do
arise from neutron star systems, we disfavour the scenario where they
originate from nascent magnetars born in LGRBs or SLSNe (see also Heintz et
al., 2020; Mannings et al., 2020). A caveat is that the Milky Way would be an
atypical LGRB or SLSN host galaxy, so the current Galactic magnetar population
may not have arisen through this pathway. Furthermore, the stellar masses and
star formation rates of FRB hosts are also, on average, similar to the Milky
Way (Li & Zhang, 2020; Bhandari et al., 2020; Heintz et al., 2020).

It is interesting to note that the Galactic neutron star population is less
consistent with CCSNe than FRBs, perhaps surprising given that CCSNe are
expected to be the dominant production channel for neutron stars. This could
be reflecting the offset between where supernovae occur and where neutron
stars are observed. Given typical natal kick velocities (e.g. Hobbs et al.,
2005; Bray & Eldridge, 2016), for young magnetars (ages ${\sim}$103-105 yrs)
this is expected to be a small distance, but it could be hundreds of parsec
for XRBs. Indeed, this has been used to argue for a XRB-like origin for FRB
180916B, which lies ${\sim}$250 pc from a nearby star forming region in its
host (Tendulkar et al., 2021).

Alternative pathways have also been put forward, particularly for magnetar
production, including merger and accretion induced collapse of white dwarfs
(Margalit et al., 2019). The recent discovery of repeating FRBs in a host
galaxy dominated by a ${\sim}$ Gyr old stellar population (FRB 20201124A, Fong
et al., 2021), and in a globular cluster (FRB 20200120E, Kirsten et al.,
2021), suggests that at least some FRB progenitors/magnetars have long delay
times.

Finally, we acknowledge that the FRB host sample is currently small and
therefore statistical consistency with other data by means of an AD-test is
more likely (or rather, it is harder to rule out). This does not diminish the
results summarised in Table 6, but reflects the limitations of the current
data sets. Looking solely at Table 6, neutron star systems appear to be a more
plausible origin for FRBs than for other transients (if we were to ignore all
other knowledge that we have about the nature of the other transients). To
increase the significance of this study’s results, we require better accuracy
and precision in Galactic neutron star distance measurements, a deeper
understanding of the biases affecting these samples, and a larger sample of
FRB hosts. In future, it may be possible to use these methods to determine not
only whether extragalactic FRBs arise from neutron stars, but which specific
systems are the progenitors. The apparent difference between single and
repeating bursts (Pleunis et al., 2021) could also be investigated in this
way, both in terms of their global host properties, and the environments
sampled within them.

##  7 Conclusions

Motivated by the possibility that magnetar activity is the origin of fast
radio bursts, we have created an image of the Milky Way, simulating its face-
on appearance from an extragalactic distance. We then placed magnetar, pulsar
and XRB populations on the image, according to their best distance estimates,
and measured how these systems are distributed with respect to Galactic light
in terms of Galactocentric offsets, host normalised offsets, enclosed fluxes
and the fraction of light statistic. Distributions of these measurements for
extragalactic transients, including fast radio bursts, are compared to the
Galactic neutron star results with AD-tests. There are $\sim$20-30 AD-tests
for each transient, across the range of Galactic populations and literature
measurements available for each. We find that Galactic neutron stars are
distributed on the Milky Way in a similar manner to FRBs on their hosts, with
75 per cent of AD-tests returning a p-value $>0.05$. FRBs also stand out as
being in better agreement with the Galactic neutron star population than other
extragalactic transients. These results appear robust against incompleteness,
uncertainties in Galaxy modelling and distance uncertainties. We cannot
distinguish whether isolated magnetars or accreting/interacting binaries
containing a neutron star are the best match, but nevertheless, these results
provide further support for FRB source models which invoke neutron star
systems.

To make further progress with this method for understanding FRBs, an improved
understanding of distance uncertainties and incompleteness in Galactic neutron
star populations is required. Larger FRB host samples will also be key, but
this population will surely grow over the coming years.

## Acknowledgements

AAC is supported by the Radboud Excellence Initiative. AJL has received
funding from the European Research Council (ERC) under the European Union’s
Seventh Framework Programme (FP7-2007-2013) (Grant agreement No. 725246). JDL
acknowledges support from a UK Research and Innovation Fellowship
(MR/T020784/1).

This work has made use of ipython (Perez & Granger, 2007), numpy (Harris et
al., 2020), scipy (Virtanen et al., 2020); matplotlib (Hunter, 2007) and
astropy,444https://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013; Price-Whelan et al., 2018).
We have also made use of Ned Wright’s cosmology calculator Wright (2006).

Finally, we thank the referee for their constructive feedback, which has
substantially improved the clarity of this manuscript.

## Data Availability

The input masers used in this paper were selected from the RMS survey
(Urquhart et al., 2014). Their distances were calculated using version 2 of
the parallax-based BeSSel (Brunthaler et al., 2011) distance calculator (Reid
et al., 2016; Reid et al., 2019), available at https://bessel.vlbi-
astrometry.org/node/378. Neutron star data were obtained from the McGill
catalogue (for magnetars, Olausen & Kaspi, 2014), available at
https://www.physics.mcgill.ca/~pulsar/magnetar/main; the ANTF database (for
pulsars, Manchester et al., 2005), available at
https://www.atnf.csiro.au/research/pulsar/psrcat; Sazonov et al. (2020, for
LMXBs) and Kretschmar et al. (2019, for HMXBs).

We have made interactive python scripts and notebooks available online, which
generate the Flight results presented in this paper, but also allow the user
to change various parameters. Along with images of the Galaxy in fits format,
these can be accessed at https://github.com/achrimes2/MW-NS-Flight. Also
available in this repository are the Flight, offset and enclosed flux values
for the magnetar, pulsar and XRB samples used in this paper. These data can
also be found on the journal website as supplementary materials.

## References

  * Abbott et al. (2017) Abbott B. P., et al., 2017, ApJ, 848, L12
  * Anderson & James (2009) Anderson J. P., James P. A., 2009, MNRAS, 399, 559
  * Anderson et al. (2012) Anderson L. D., Bania T. M., Balser D. S., Rood R. T., 2012, ApJ, 754, 62
  * Anderson et al. (2015) Anderson J. P., James P. A., Habergham S. M., Galbany L., Kuncarayakti H., 2015, Publ. Astron. Soc. Australia, 32, e019
  * Arnason et al. (2021) Arnason R. M., Papei H., Barmby P., Bahramian A., Gorski M. D., 2021, arXiv e-prints, p. arXiv:2102.02615
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Audcent-Ross et al. (2020) Audcent-Ross F. M., Meurer G. R., Audcent J. R., Ryder S. D., Wong O. I., Phan J., Williamson A., Kim J. H., 2020, MNRAS, 492, 848
  * Bailes et al. (2021) Bailes M., et al., 2021, MNRAS,
  * Baring et al. (2020) Baring M. G., Wadiasingh Z., Gonthier P. L., Harding A. K., Hu K., 2020, arXiv e-prints, p. arXiv:2012.10815
  * Barthelmy et al. (2005) Barthelmy S. D., et al., 2005, Nature, 438, 994
  * Beniamini et al. (2019) Beniamini P., Hotokezaka K., van der Horst A., Kouveliotou C., 2019, MNRAS, 487, 1426
  * Beniamini et al. (2020) Beniamini P., Wadiasingh Z., Metzger B. D., 2020, MNRAS, 496, 3390
  * Bhandari et al. (2020) Bhandari S., et al., 2020, ApJ, 895, L37
  * Bhardwaj et al. (2021) Bhardwaj M., et al., 2021, arXiv e-prints, p. arXiv:2103.01295
  * Binney et al. (1997) Binney J., Gerhard O., Spergel D., 1997, MNRAS, 288, 365
  * Blanchard et al. (2016) Blanchard P. K., Berger E., Fong W.-f., 2016, ApJ, 817, 144
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529
  * Bochenek et al. (2020a) Bochenek C. D., Ravi V., Dong D., 2020a, arXiv e-prints, p. arXiv:2009.13030
  * Bochenek et al. (2020b) Bochenek C. D., Ravi V., Belov K. V., Hallinan G., Kocz J., Kulkarni S. R., McKenna D. L., 2020b, Nature, 587, 59
  * Bodaghee et al. (2007) Bodaghee A., et al., 2007, A&A, 467, 585
  * Bodaghee et al. (2012) Bodaghee A., Tomsick J. A., Rodriguez J., James J. B., 2012, ApJ, 744, 108
  * Bray & Eldridge (2016) Bray J. C., Eldridge J. J., 2016, MNRAS, 461, 3747
  * Brunthaler et al. (2011) Brunthaler A., et al., 2011, Astronomische Nachrichten, 332, 461
  * Burns et al. (2021) Burns E., et al., 2021, arXiv e-prints, p. arXiv:2101.05144
  * CHIME/FRB Collaboration et al. (2020) CHIME/FRB Collaboration et al., 2020, Nature, 587, 54
  * Castro-Ginard et al. (2021) Castro-Ginard A., et al., 2021, arXiv e-prints, p. arXiv:2105.04590
  * Chatterjee (2020) Chatterjee S., 2020, arXiv e-prints, p. arXiv:2012.10377
  * Church et al. (2011) Church R. P., Levan A. J., Davies M. B., Tanvir N., 2011, MNRAS, 413, 2004
  * Churchwell et al. (2009) Churchwell E., et al., 2009, PASP, 121, 213
  * Coleiro & Chaty (2013) Coleiro A., Chaty S., 2013, ApJ, 764, 185
  * Cordes & Lazio (2002) Cordes J. M., Lazio T. J. W., 2002, arXiv e-prints, pp astro–ph/0207156
  * De et al. (2020a) De K., et al., 2020a, ApJ, 901, L7
  * De et al. (2020b) De K., et al., 2020b, ApJ, 905, 58
  * Deng et al. (2021) Deng C.-M., Zhong S.-Q., Dai Z.-G., 2021, arXiv e-prints, p. arXiv:2102.06796
  * Díaz-García et al. (2019) Díaz-García S., Salo H., Knapen J. H., Herrera-Endoqui M., 2019, A&A, 631, A94
  * Duncan & Thompson (1992) Duncan R. C., Thompson C., 1992, ApJ, 392, L9
  * Eftekhari & Berger (2017) Eftekhari T., Berger E., 2017, ApJ, 849, 162
  * Fahlman & Gregory (1981) Fahlman G. G., Gregory P. C., 1981, Nature, 293, 202
  * Faucher-Giguère & Kaspi (2006) Faucher-Giguère C.-A., Kaspi V. M., 2006, ApJ, 643, 332
  * Flynn et al. (2006) Flynn C., Holmberg J., Portinari L., Fuchs B., Jahreiß H., 2006, MNRAS, 372, 1149
  * Fong et al. (2010) Fong W., Berger E., Fox D. B., 2010, ApJ, 708, 9
  * Fong et al. (2013) Fong W., et al., 2013, ApJ, 769, 56
  * Fong et al. (2021) Fong W.-f., et al., 2021, arXiv e-prints, p. arXiv:2106.11993
  * Fruchter et al. (2006) Fruchter A. S., et al., 2006, Nature, 441, 463
  * Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A1
  * Gardenier & van Leeuwen (2020) Gardenier D. W., van Leeuwen J., 2020, arXiv e-prints, p. arXiv:2012.06396
  * Grady et al. (2020) Grady J., Belokurov V., Evans N. W., 2020, MNRAS, 492, 3128
  * Gravity Collaboration et al. (2019) Gravity Collaboration et al., 2019, A&A, 625, L10
  * Green et al. (2017) Green J. A., et al., 2017, MNRAS, 469, 1383
  * Grossan (2021) Grossan B., 2021, PASP, 133, 074202
  * Gullón et al. (2015) Gullón M., Pons J. A., Miralles J. A., Viganò D., Rea N., Perna R., 2015, MNRAS, 454, 615
  * Harris et al. (2020) Harris C. R., et al., 2020, arXiv e-prints, p. arXiv:2006.10256
  * Heintz et al. (2020) Heintz K. E., et al., 2020, ApJ, 903, 152
  * Hilmarsson et al. (2020) Hilmarsson G. H., et al., 2020, MNRAS, 493, 5170
  * Hobbs et al. (2005) Hobbs G., Lorimer D. R., Lyne A. G., Kramer M., 2005, MNRAS, 360, 974
  * Hou & Han (2014) Hou L. G., Han J. L., 2014, A&A, 569, A125
  * Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
  * Hurley et al. (2005) Hurley K., et al., 2005, Nature, 434, 1098
  * Ioka & Zhang (2020) Ioka K., Zhang B., 2020, ApJ, 893, L26
  * James & Anderson (2006) James P. A., Anderson J. P., 2006, A&A, 453, 57
  * James et al. (2021) James C. W., Prochaska J. X., Macquart J. P., North-Hickey F., Bannister K. W., Dunning A., 2021, arXiv e-prints, p. arXiv:2101.08005
  * Jonker & Nelemans (2004) Jonker P. G., Nelemans G., 2004, MNRAS, 354, 355
  * Kasen & Bildsten (2010) Kasen D., Bildsten L., 2010, ApJ, 717, 245
  * Katz (2016) Katz J. I., 2016, ApJ, 826, 226
  * Keane (2018) Keane E. F., 2018, Nature Astronomy, 2, 865
  * Kelly & Kirshner (2012) Kelly P. L., Kirshner R. P., 2012, ApJ, 759, 107
  * Kirsten et al. (2021) Kirsten F., et al., 2021, arXiv e-prints, p. arXiv:2105.11445
  * Kouveliotou et al. (1998) Kouveliotou C., et al., 1998, Nature, 393, 235
  * Kretschmar et al. (2019) Kretschmar P., et al., 2019, New Astron. Rev., 86, 101546
  * Krivonos et al. (2017) Krivonos R. A., Tsygankov S. S., Mereminskiy I. A., Lutovinov A. A., Sazonov S. Y., Sunyaev R. A., 2017, MNRAS, 470, 512
  * Levan et al. (2006) Levan A. J., Wynn G. A., Chapman R., Davies M. B., King A. R., Priddey R. S., Tanvir N. R., 2006, MNRAS, 368, L1
  * Li & Zanazzi (2021) Li D., Zanazzi J. J., 2021, arXiv e-prints, p. arXiv:2101.05836
  * Li & Zhang (2020) Li Y., Zhang B., 2020, ApJ, 899, L6
  * Li et al. (2021) Li C. K., et al., 2021, Nature Astronomy,
  * Licquia et al. (2015) Licquia T. C., Newman J. A., Brinchmann J., 2015, ApJ, 809, 96
  * Lin et al. (2020) Lin L., et al., 2020, Nature, 587, 63
  * Lu et al. (2020) Lu W., Kumar P., Zhang B., 2020, MNRAS, 498, 1397
  * Lumsden et al. (2013) Lumsden S. L., Hoare M. G., Urquhart J. S., Oudmaijer R. D., Davies B., Mottram J. C., Cooper H. D. B., Moore T. J. T., 2013, ApJS, 208, 11
  * Lunnan et al. (2015) Lunnan R., et al., 2015, ApJ, 804, 90
  * Lunnan et al. (2017) Lunnan R., et al., 2017, ApJ, 836, 60
  * Luo et al. (2020) Luo R., Men Y., Lee K., Wang W., Lorimer D. R., Zhang B., 2020, MNRAS, 494, 665
  * Lyman et al. (2017) Lyman J. D., et al., 2017, MNRAS, 467, 1795
  * Manchester et al. (2005) Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, AJ, 129, 1993
  * Manchester et al. (2016) Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2016, VizieR Online Data Catalog, p. B/psr
  * Mannings et al. (2020) Mannings A. G., et al., 2020, arXiv e-prints, p. arXiv:2012.11617
  * Margalit et al. (2019) Margalit B., Berger E., Metzger B. D., 2019, ApJ, 886, 110
  * Margalit et al. (2020) Margalit B., Beniamini P., Sridhar N., Metzger B. D., 2020, ApJ, 899, L27
  * Mazets et al. (1979a) Mazets E. P., Golenetskij S. V., Guryan Y. A., 1979a, Soviet Astronomy Letters, 5, 343
  * Mazets et al. (1979b) Mazets E. P., Golentskii S. V., Ilinskii V. N., Aptekar R. L., Guryan I. A., 1979b, Nature, 282, 587
  * Metzger et al. (2008) Metzger B. D., Quataert E., Thompson T. A., 2008, MNRAS, 385, 1455
  * Metzger et al. (2011) Metzger B. D., Giannios D., Thompson T. A., Bucciantini N., Quataert E., 2011, MNRAS, 413, 2031
  * Metzger et al. (2017) Metzger B. D., Berger E., Margalit B., 2017, ApJ, 841, 14
  * Mondal et al. (2020) Mondal S., Bera A., Chandra P., Das B., 2020, MNRAS, 498, 3863
  * Nuñez et al. (2021) Nuñez C., et al., 2021, arXiv e-prints, p. arXiv:2104.09727
  * Ofek et al. (2006) Ofek E. O., et al., 2006, ApJ, 652, 507
  * Ofek et al. (2008) Ofek E. O., et al., 2008, ApJ, 681, 1464
  * Olausen & Kaspi (2014) Olausen S. A., Kaspi V. M., 2014, ApJS, 212, 6
  * Palliyaguru et al. (2021) Palliyaguru N. T., Agarwal D., Golpayegani G., Lynch R., Lorimer D. R., Nguyen B., Corsi A., Burke-Spolaor S., 2021, MNRAS, 501, 541
  * Pastor-Marazuela et al. (2020) Pastor-Marazuela I., et al., 2020, arXiv e-prints, p. arXiv:2012.08348
  * Perets & Beniamini (2021) Perets H. B., Beniamini P., 2021, arXiv e-prints, p. arXiv:2101.11622
  * Perez & Granger (2007) Perez F., Granger B. E., 2007, Computing in Science and Engineering, 9, 21
  * Pestalozzi et al. (2005) Pestalozzi M. R., Minier V., Booth R. S., 2005, A&A, 432, 737
  * Platts et al. (2019) Platts E., Weltman A., Walters A., Tendulkar S. P., Gordin J. E. B., Kandhai S., 2019, Phys. Rep., 821, 1
  * Pleunis et al. (2021) Pleunis Z., et al., 2021, arXiv e-prints, p. arXiv:2106.04356
  * Price-Whelan et al. (2018) Price-Whelan A. M., et al., 2018, AJ, 156, 123
  * Rajwade et al. (2017) Rajwade K. M., Lorimer D. R., Anderson L. D., 2017, MNRAS, 471, 730
  * Reid et al. (2014) Reid M. J., et al., 2014, ApJ, 783, 130
  * Reid et al. (2016) Reid M. J., Dame T. M., Menten K. M., Brunthaler A., 2016, ApJ, 823, 77
  * Reid et al. (2019) Reid M. J., et al., 2019, ApJ, 885, 131
  * Repetto et al. (2017) Repetto S., Igoshev A. P., Nelemans G., 2017, MNRAS, 467, 298
  * Ridnaia et al. (2021) Ridnaia A., et al., 2021, Nature Astronomy,
  * Russeil (2003) Russeil D., 2003, A&A, 397, 133
  * Safarzadeh et al. (2020) Safarzadeh M., Prochaska J. X., Heintz K. E., Fong W.-f., 2020, ApJ, 905, L30
  * Sartore et al. (2010) Sartore N., Ripamonti E., Treves A., Turolla R., 2010, A&A, 510, A23
  * Sazonov et al. (2020) Sazonov S., et al., 2020, New Astron. Rev., 88, 101536
  * Schulze et al. (2020) Schulze S., et al., 2020, arXiv e-prints, p. arXiv:2008.05988
  * Seigar & James (1998) Seigar M. S., James P. A., 1998, MNRAS, 299, 685
  * Sharma et al. (2011) Sharma S., Bland-Hawthorn J., Johnston K. V., Binney J., 2011, ApJ, 730, 3
  * Sridhar et al. (2021) Sridhar N., Metzger B. D., Beniamini P., Margalit B., Renzo M., Sironi L., 2021, arXiv e-prints, p. arXiv:2102.06138
  * Svensson et al. (2010) Svensson K. M., Levan A. J., Tanvir N. R., Fruchter A. S., Strolger L. G., 2010, MNRAS, 405, 57
  * Svinkin et al. (2021) Svinkin D., et al., 2021, arXiv e-prints, p. arXiv:2101.05104
  * Tavani et al. (2021) Tavani M., et al., 2021, Nature Astronomy,
  * Tendulkar et al. (2021) Tendulkar S. P., et al., 2021, ApJ, 908, L12
  * The CHIME/FRB Collaboration et al. (2021) The CHIME/FRB Collaboration et al., 2021, arXiv e-prints, p. arXiv:2106.04352
  * Thompson & Duncan (1996) Thompson C., Duncan R. C., 1996, ApJ, 473, 322
  * Tunnicliffe et al. (2014) Tunnicliffe R. L., et al., 2014, MNRAS, 437, 1495
  * Uddin et al. (2020) Uddin S. A., et al., 2020, ApJ, 901, 143
  * Urquhart et al. (2014) Urquhart J. S., Figura C. C., Moore T. J. T., Hoare M. G., Lumsden S. L., Mottram J. C., Thompson M. A., Oudmaijer R. D., 2014, MNRAS, 437, 1791
  * Usov (1992) Usov V. V., 1992, Nature, 357, 472
  * Valdettaro et al. (2001) Valdettaro R., et al., 2001, A&A, 368, 845
  * Vallée (2002) Vallée J. P., 2002, ApJ, 566, 261
  * Vallée (2008) Vallée J. P., 2008, AJ, 135, 1301
  * Vallée (2014) Vallée J. P., 2014, AJ, 148, 5
  * Verberne & Vink (2021) Verberne S., Vink J., 2021, arXiv e-prints, p. arXiv:2103.16973
  * Verrecchia et al. (2021) Verrecchia F., et al., 2021, arXiv e-prints, p. arXiv:2105.00685
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wang et al. (2013) Wang X., Wang L., Filippenko A. V., Zhang T., Zhao X., 2013, Science, 340, 170
  * White & van Paradijs (1996) White N. E., van Paradijs J., 1996, ApJ, 473, L25
  * Widrow et al. (2008) Widrow L. M., Pym B., Dubinski J., 2008, ApJ, 679, 1239
  * Winkler et al. (2003) Winkler C., et al., 2003, A&A, 411, L1
  * Wisnioski et al. (2012) Wisnioski E., Glazebrook K., Blake C., Poole G. B., Green A. W., Wyder T., Martin C., 2012, MNRAS, 422, 3339
  * Woosley (2010) Woosley S. E., 2010, ApJ, 719, L204
  * Wright (2006) Wright E. L., 2006, PASP, 118, 1711
  * Xiao et al. (2021) Xiao D., Wang F., Dai Z., 2021, arXiv e-prints, p. arXiv:2101.04907
  * Yao et al. (2017) Yao J. M., Manchester R. N., Wang N., 2017, ApJ, 835, 29
  * Younes et al. (2021) Younes G., et al., 2021, Nature Astronomy, 5, 408
  * Yu & Ho (2020) Yu S.-Y., Ho L. C., 2020, ApJ, 900, 150
  * Yu et al. (2018) Yu S.-Y., Ho L. C., Barth A. J., Li Z.-Y., 2018, ApJ, 862, 13
  * Yusifov & Küçük (2004) Yusifov I., Küçük I., 2004, A&A, 422, 545
  * Zari et al. (2021) Zari E., Rix H. W., Frankel N., Xiang M., Poggio E., Drimmel R., Tkachenko A., 2021, A&A, 650, A112
  * Zhang (2020) Zhang B., 2020, Nature, 587, 45
  * Zhao et al. (2021) Zhao Z. Y., Zhang G. Q., Wang Y. Y., Tu Z.-L., Wang F. Y., 2021, ApJ, 907, 111
  * van Paradijs & White (1995) van Paradijs J., White N., 1995, ApJ, 447, L33

##  Appendix A Milky Way magnetar measurements

In Table 7 of this appendix, we provide a table of measurements for the 20
Milky Way magnetars which lie this side of Galactic centre ($y<8.3$ kpc on our
map). The measurements are typical of those used in extragalactic transient
studies, including host offset, host normalised offset, Flight and the
enclosed flux fraction at their radius. The heliocentric distances, their
uncertainties, and the type of measurement, are also provided (partially
reproduced from the McGill magnetar catalogue, Olausen & Kaspi, 2014).

Table 8 contains the same offset, enclosed flux and fraction of light data for
the pulsar and XRB samples. Only a subset of pulsar data is shown here, the
full results for all samples are available at https://github.com/achrimes2/MW-
NS-Flight or as supplementary materials on the journal website. These results
can be used in future comparative studies.

Table 7: The 20 magnetars that satisfy the criteria of having (i) a distance
estimate, (ii) lying at $y<8.3$ kpc. Magnetar distance information is from the
McGill catalogue (Olausen & Kaspi, 2014, and reference therein). The Flight
values are calculated using a 12 kpc pixel selection radius and pixels from
the whole Galaxy.

Magnetar | Dist./Assoc. | d⊙ [kpc] |  ${\delta}r$ [kpc] |  ${\delta}r$/$r_{e}$ (B) |  ${\delta}r$/$r_{e}$ (I) | Flight (B) | Flight (I) | Fenc (B) | Fenc (I)
---|---|---|---|---|---|---|---|---|---
4U0142+61 | RC | 3.6${}^{+0.4}_{+0.4}$ | 10.85 | 4.57 | 4.88 | 0.02 | 0.01 | 0.98 | 0.98
SGR0418+5729 | PA |  ${\sim}$2 | 9.95 | 4.20 | 4.47 | 0.07 | 0.05 | 0.95 | 0.96
SGR0501+4516 | PA/SNR |  ${\sim}$2 | 10.12 | 4.27 | 4.55 | 0.03 | 0.03 | 0.96 | 0.96
1E1048.1-5937 | RC | 9${}^{+1.7}_{-1.7}$ | 10.10 | 4.26 | 4.54 | 0.03 | 0.03 | 0.96 | 0.96
1E1547.0-5408 | SNR | 4.5${}^{+0.5}_{-0.5}$ | 5.04 | 2.13 | 2.27 | 0.35 | 0.31 | 0.57 | 0.64
PSRJ1622-4950 | DM/SNR | 9${}^{+1.4}_{-1.4}$ | 3.97 | 1.68 | 1.78 | 0.44 | 0.39 | 0.44 | 0.53
CXOUJ164710.2-455216 | CA | 3.9${}^{+0.7}_{-0.7}$ | 4.75 | 2.00 | 2.14 | 0.37 | 0.32 | 0.54 | 0.61
1RXSJ170849.0-400910 | RC | 3.8${}^{+0.5}_{-0.5}$ | 4.59 | 1.94 | 2.06 | 0.42 | 0.36 | 0.52 | 0.59
SGRJ1745-2900 | HI | 8.3${}^{+0.3}_{-0.3}$ | 0.10 | 0.04 | 0.04 | 0.99 | 0.99 | 0.00 | 0.00
XTEJ1810-197 | HI | 3.5${}^{+0.5}_{-0.4}$ | 4.81 | 2.03 | 2.16 | 0.34 | 0.31 | 0.54 | 0.61
SwiftJ1818.0-1607 | DM | 4.8${}^{+1.65}_{-1.65}$ | 3.77 | 1.59 | 1.69 | 0.49 | 0.47 | 0.42 | 0.50
SwiftJ1822.3-1606 | HI/HII | 1.6${}^{+0.3}_{-0.3}$ | 6.67 | 2.81 | 3.00 | 0.39 | 0.32 | 0.74 | 0.78
SwiftJ1834.9-0846 | SNR | 4.2${}^{+0.3}_{-0.3}$ | 4.65 | 1.96 | 2.09 | 0.38 | 0.34 | 0.52 | 0.59
1E1841-045 | SNR | 8.5${}^{+1.3}_{-1}$ | 3.96 | 1.67 | 1.78 | 0.60 | 0.49 | 0.42 | 0.50
3XMMJ185246.6+003317 | HI/SNR |  ${\sim}$7.1 | 4.54 | 1.92 | 2.04 | 0.41 | 0.36 | 0.49 | 0.57
SGR1935+2154 | F/SNR | 6.5${}^{+3.0}_{-5.3}$ | 7.20 | 3.04 | 3.24 | 0.13 | 0.13 | 0.77 | 0.81
1E2259+586 | SNR/PA | 3.2${}^{+0.2}_{-0.2}$ | 9.73 | 4.11 | 4.37 | 0.17 | 0.13 | 0.94 | 0.95
AXJ1845.0-0258 | HI/SNR |  ${\sim}$8.5 | 4.27 | 1.80 | 1.92 | 0.59 | 0.48 | 0.45 | 0.54
SGR2013+34 | HII |  ${\sim}$8.8 | 10.04 | 4.24 | 4.51 | 0.03 | 0.03 | 0.95 | 0.96
PSRJ1846-0258 | SNR | 6${}^{+1.5}_{-0.9}$ | 4.22 | 1.78 | 1.90 | 0.46 | 0.42 | 0.46 | 0.54

RC - distance from red clump stars, PA - Perseus arm association, SNR -
supernova remnant association, DM - dispersion measure distance, CA - cluster
association, HI - distance from HI column density, HII - HII region
association, F - distance estimate from burst flux

Table 8: An extract of a table containing offset, enclosed flux and Flight results for the pulsar sample. The full data for the pulsars, XRBs and magnetars are available as .txt files on the journal website, and at https://github.com/achrimes2/MW-NS-Flight, with the same columns and in the same format as below. The Flight values quoted are for half-Galaxy measurements. ${\delta}r$ [kpc] |  ${\delta}r$/$r_{e}$ (B) |  ${\delta}r$/$r_{e}$ (I) | Flight (B) | Flight (I) | Fenc (B) | Fenc (I)
---|---|---|---|---|---|---
9.64 | 4.06 | 4.33 | 0.04 | 0.04 | 0.94 | 0.95
15.65 | 6.59 | 7.03 | 0.00 | 0.00 | 1.00 | 1.00
8.76 | 3.69 | 3.94 | 0.06 | 0.06 | 0.88 | 0.90
9.26 | 3.90 | 4.16 | 0.04 | 0.04 | 0.91 | 0.93
10.99 | 4.63 | 4.94 | 0.02 | 0.02 | 0.98 | 0.99

##  Appendix B Fraction of light distributions for pulsars and XRBs

The fraction of light method described for the magnetars in Section 5 is
repeated here for the ATNF luminous pulsars and the INTEGRAL XRBs. Their
positions on the half-map, the local sub-region, and the resultant Flight
distributions, are shown in Figs. 10, 11 and 12.

In Fig. 10, the result of varying the pulsar luminosity cut shown in Fig. 1 is
also demonstrated. Moving the cut within the region denoted by the dashed
lines in Fig. 1, the maximum and minimum distributions that occur in this
range are shaded between, demonstrating that the precise choice of cut has a
minimal impact.

As for the magnetars, restricting the analysis to the local 4 kpc pushes the
pulsar and HMXB distributions closer towards the 1:1 line, with the effect
greater in the $I$-band than the $B$-band. Again, this is because these
samples avoid the Galactic bulge (with an additional effect due to
survey/detection biases in the pulsar case). The effect occurs to a much
lesser extent for LMXBs, which being an older population, already favour the
Galactic bulge.

Figure 10: The same information as in Figs. 8 and 9, but for the bright ANTF
pulsars. Top: Flight $B$-band and $I$-band colour maps with the half-Galaxy
and 4 kpc heliocentric distance restriction. The pulsar locations are
indicated. Bottom: the corresponding Flight distributions. The grey shading
bounds the minimum and maximum distributions that arise from varying the
luminosity cut by $\pm$0.5 dex (see Fig. 1).

Figure 11: As in Fig. 10, but for the LMXBs. Only in the half-Galaxy case do
we consider objects outside the selected pixels, assigning them
F${}_{\mathrm{light}}=0$.

Figure 12: As in Figs. 10 and 11, but for HMXBs.