# Energetic requirements of the gamma-ray emission from pulsars: A
nonparametric analysis of the data in the Fermi-LAT 12-Year Catalog

Houshang Ardavan
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3
0HA, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The prevalent view that the radio-loud gamma-ray pulsars have gamma-ray
luminosities that exceed their radio luminosities by several orders of
magnitude is based on the assumption that the decay with distance of their
gamma-ray fluxes obeys the inverse-square law as does that of their radio
fluxes. The results presented here, of testing the hypothesis of independence
of luminosities and distances of gamma-ray pulsars by means of the
Efron–Petrosian statistic, do not uphold this assumption however: they imply
that the observational data in the Fermi-LAT 12-Year Catalog are consistent
with the dependence $S\propto D^{-3/2}$ of the flux densities $S$ of the
gamma-ray pulsars on their distances $D$ at substantially higher levels of
significance than they are with the dependence $S\propto D^{-2}$. These
results, which were theoretically predicted in Ardavan (2021, MNRAS, 507,
4530), are not incompatible with the requirements of the conservation of
energy because the radiation process by which the superluminally moving
current sheet in the magnetosphere of a neutron star has been shown to
generate the slowly decaying gamma-ray pulses is intrinsically transient: the
difference in the fluxes of power across any two spheres centred on the star
is balanced by the change with time of the energy contained inside the shell
bounded by those spheres. Once the over-estimation of their values is
rectified, the luminosities of gamma-ray pulsars turn out to have the same
range of values as do the luminosities of radio pulsars. This conclusion
agrees with that reached earlier on the basis of the smaller data set in the
Second Fermi-LAT Catalog of Gamma-ray Pulsars.

###### keywords:

gamma-rays: stars – pulsars: general – stars: neutron – stars: statistics –
methods: data analysis – radiation mechanisms: non-thermal

††pubyear: 2022††pagerange: Energetic requirements of the gamma-ray emission
from pulsars: A nonparametric analysis of the data in the Fermi-LAT 12-Year
Catalog–Energetic requirements of the gamma-ray emission from pulsars: A
nonparametric analysis of the data in the Fermi-LAT 12-Year Catalog

##  1 Introduction

Numerical computations based on the force-free and particle-in-cell formalisms
have now firmly established that the magnetosphere of a non-aligned neutron
star entails a current sheet outside its light cylinder whose rotating
distribution pattern moves with linear speeds exceeding the speed of light in
vacuum (see the review article Philippov & Kramer, 2022, and the references
therein). A study of the characteristics of the radiation that is generated by
this superluminally moving current sheet has in turn provided an all-
encompassing explanation for the salient features of the radiation received
from pulsars: its brightness temperature, polarization, spectrum and profile
with microstructure and with a phase lag between the radio and gamma-ray peaks
(Ardavan, 2021, 2022a). A fit to the exceptionally broad gamma-ray spectrum of
the Crab pulsar, for example, is provided by the spectral distribution
function of this radiation for the first time (Ardavan, 2022b).

The radiation field generated by a constituent volume element of the current
sheet in the magnetosphere of a neutron star embraces a synergy between the
superluminal version of the field of synchrotron radiation and the vacuum
version of the field of Čerenkov radiation. Once superposed to yield the
emission from the entire volume of the source, the contributions from the
volume elements of the current sheet that approach the observation point with
the speed of light and zero acceleration at the retarded time interfere
constructively and form caustics in certain latitudinal directions relative to
the spin axis of the neutron star. The waves that embody these caustics are
more focused the further they are from their source: as their distance from
their source increases, two nearby stationary points of their phases draw
closer and eventually coalesce at infinity. As a result, flux densities of the
pulses that are generated by this current sheet diminish with the distance $D$
from the star as $D^{-3/2}$ (rather than $D^{-2}$) in the latitudinal
directions along which the pulses are most tightly focused (Ardavan, 2021,
Section 5.5). By virtue of their extremely narrow peaks in the time domain,
such pulses in addition have broad spectra that encompass X-ray and gamma-ray
frequencies (Ardavan, 2021, Table 1 and Section 5.4).

It is expected, therefore, that the class of progenitors of the non-
spherically decaying radiation in question would include, inter alia, the
gamma-ray pulsars.

To see whether this expectation is supported by the observational data on
fluxes and distances of gamma-ray pulsars, we analyse the latest version of
these data (Abdollahi et al., 2022) here on the basis of the fact that, in a
statistical context, luminosities and distances of the sources of a given type
of radiation represent two independent random variables. A generalization of
the nonparametric rank methods for testing the independence of two random
variables (Hájek, 1969) to cases of truncated data, such as the flux-limited
data on gamma-ray pulsars, is the method developed by Efron & Petrosian (1992,
1994): a method that has been widely used in astrophysical contexts (see e.g.
Bryant et al., 2021, and the references therein).

From the raw data on fluxes $S$ of gamma-ray pulsars in the Fermi-LAT 12-Year
Catalog111https://fermi.gsfc.nasa.gov/ssc/data/access/lat/12yr_catalog/, the
distances $D$ of those of them that are listed in the ATNF Pulsar
Catalogue222http://www.atnf.csiro.au/research/pulsar/psrcat, and the candidate
decay rates of flux density with distance ($S\propto D^{-\alpha}$ for various
values of $\alpha$) we compile a collection of data sets on the prospective
luminosities of these pulsars (Section 2). We then test the hypothesis of
independence of luminosity and distance by evaluating the Efron–Petrosian
statistic (Section 3.1) for the data sets on prospective luminosities and
pulsar distances choosing a wide range of values for the flux threshold (i.e.
the truncation boundary below which the data set on $S$ may be incomplete).
The resulting values of the Efron–Petrosian statistic for differing values of
$\alpha$ in each case determine the significance levels at which the
hypothesis of independence of luminosity and distance (and hence a given value
of $\alpha$) can be rejected (Sections 3.2 and 3.3).

We analyse the two homogeneous data sets on the young and the millisecond
pulsars in the Fermi-LAT 12-Year Catalog separately, each with both the NE2001
(Cordes & Lazio, 2002) and the YMW16 (Yao et al., 2017) distances. We also
assess the effects of random and (if any) systematic errors in the estimates
of flux on the test results by means of a Monte Carlo simulation (Section
3.4). Whether the sizes of the currently available data sets are sufficient
for the purposes of the present rank analysis is ascertained in Section 3.5
where we augment these data sets by the inclusion of their permissible
permutations.

Figure 1: (a) Histogram of the $120$ millisecond gamma-ray pulsars listed in
the Fermi-LAT 12-Year Catalog. The bins in darker brown contain the $114$ of
these pulsars whose distances are listed in the ATNF Pulsar Catalogue. (b)
Distribution of logarithm of flux versus logarithm of distance for the
millisecond pulsars in the darker brown bins based on the YMW16 distances in
the ATNF Catalogue. The dashed lines $a$ and $b$ respectively designate the
flux thresholds $\log S_{\rm th}=-9.112$ and $\log S_{\rm th}=-9.07$, i.e. the
thresholds marked in Fig. 4a. (c) Distribution of logarithm of flux versus
logarithm of distance for the same set of pulsars based on the NE2001
distances in the ATNF Catalogue. The dashed lines $c$ and $d$ respectively
designate the flux thresholds $\log S_{\rm th}=-9.092$ and $\log S_{\rm
th}=-9.005$, i.e. the thresholds marked in Fig. 4b.

Figure 2: (a) Histogram of the $135$ young gamma-ray pulsars listed in the
Fermi-LAT 12-Year Catalog. The bins in darker brown contain the $93$ of these
pulsars whose distances are listed in the ATNF Pulsar Catalogue. (b)
Distribution of logarithm of flux versus logarithm of distance for the young
pulsars in the darker brown bins based on the YMW16 distances in the ATNF
Catalogue. The dashed lines $a$, $b$ and $c$ respectively designate the flux
thresholds $\log S_{\rm th}=-9.365$, $\log S_{\rm th}=-9.08$ and $\log S_{\rm
th}=-8.95$, i.e. the thresholds marked in Fig. 6a. (c) Distribution of
logarithm of flux versus logarithm of distance for the same set of pulsars
based on the NE2001 distances in the ATNF Catalogue. The dashed lines $d$, $e$
and $f$ respectively designate the flux thresholds $\log S_{\rm th}=-9.066$,
$\log S_{\rm th}=-8.911$ and $\log S_{\rm th}=-8.8$, i.e. the thresholds
marked in Fig. 6b.

##  2 Observational data

The Fermi-LAT 12-Year Catalog lists the observational data on $120$
millisecond and $135$ young gamma-ray pulsars. Of these, there are $6$
millisecond and $42$ young pulsars with no distance estimates in the ATNF
Pulsar Catalogue. Histograms of the fluxes of these two sets of pulsars are
shown in Figs. 1a and 2a. The histogram bins in darker brown contain the
pulsars for which both fluxes and distances are known. Figures 1 and 2 also
show the distributions of logarithm of flux (in units of erg cm-2 s-1) versus
logarithm of distance (in units of pc) for these two sets of pulsars. Parts
(b) and (c) of these figures are based on the distances in the ATNF Catalogue
that are listed under YMW16 and NE2001, respectively. The dashed lines in them
each designate the value of a flux threshold $S_{\rm th}$ below which the
plotted data set may be incomplete.

The isotropic gamma-ray luminosity of each pulsar is given, in terms of its
gamma-ray flux density $S$ and its distance $D$, by

$L=4\pi\ell^{2}(D/\ell)^{\alpha}S,$ (1)

where $\alpha=2$ if $S$ diminishes with distance as predicted by the inverse-
square law and $\ell$ is a constant with the dimension of length whose value
only affects the scale of $L$. The corresponding distribution of the logarithm
of $L$ (in units of erg s-1) versus logarithm of $D$ (in units of pc) for the
data set shown in Fig. 1b is plotted in Fig. 3 for $\alpha=2$. The solid line
in Fig. 3 shows an example of a flux threshold: that corresponding to the
threshold $S=2.82\times 10^{-10}$ erg cm-2 s-1 which excludes three elements
of this data set.

Figure 3: The luminosity-distance data set that follows from the flux-distance
data set of Fig. 1b and equation (1) for $\alpha=2$. The solid line (in blue)
is the image $L=L_{\rm th}(D)$ of the flux threshold $\log S_{\rm th}=-9.55$.
Those elements of this data set that lie within (and on the boundary of) the
rectangular area bounded by the vertical axis and the vertical and horizontal
dashed lines (in green) comprise the set comparable to the element
$(2.84,35.46)$ on the vertical dashed line.

##  3 Testing the hypothesis that the data sets on luminosity and distance are
independent

###  3.1 The Efron–Petrosian rank statistic

In this section we only outline the procedure by which the Efron–Petrosian
statistic is calculated for two given data sets; lucid expositions of the
theoretical basis of this procedure can be found in Efron & Petrosian (1992),
Maloney & Petrosian (1999) and Petrosian (2002).

If we let $S_{\rm th}$ stand for the threshold value of flux density, then the
corresponding truncation boundary for the values of luminosity (e.g. that
shown as a solid line in Fig. 3) is given, according to equation (1), by
$L=L_{\rm th}(D)$ with

$\log L_{\rm th}=\log[4\pi(3.085\times 10^{18})^{2}\ell^{2-\alpha}S_{\rm
th}]+\alpha\log D,$ (2)

in which the numerical factor $3.085\times 10^{18}$ converts the units of $D$
and $\ell$ from pc to cm. The data set on luminosity is thus regarded as
complete only in the sector $\log L\geq\log L_{\rm th}$ of the $(\log D,\log
L)$ plane.

The set comparable to any given element $(\log D_{i},\log L_{i})$ of the
bivariate distance-luminosity data set (such as the data set plotted in Fig.
3) is defined to comprise all those elements for which

$\log D\leq\log D_{i},\qquad i=1,\cdots n,$ (3)

and

$\log L\geq\log[4\pi(3.085\times 10^{18})^{2}\ell^{2-\alpha}S_{\rm
th}]+\alpha\log D_{i},$ (4)

where $n$ is the number of elements in the part of the data set that is not
excluded by the chosen flux threshold. For instance, the set comparable to the
data point $(2.84,35.46)$ on the green dashed line in Fig. 3 consists of the
elements of the data set that lie within (and on the boundary of) the
rectangular region delineated by the vertical axis and the dashed lines
coloured green in this figure. We denote the number of elements in the set
comparable to $(\log D_{i},\log L_{i})$ by $N_{i}$.

To determine the rank ($1\leq R_{i}\leq N_{i}$) of the element $(\log
D_{i},\log L_{i})$, we now order the $N_{i}$ elements of its comparable set by
the ascending values of their coordinates $\log L_{i}$ and equate $R_{i}$ to
the position at which $(\log D_{i},\log L_{i})$ appears in the resulting
ordered list. Coordinates of the elements of the bivariate data set $(\log
D,\log L)$ are in the present case all distinct. So, a rank $R_{i}$ can be
assigned to every element of this set unambiguously.

The Efron–Petrosian statistic is given by

$\tau=\frac{\sum_{i=1}^{n}(R_{i}-E_{i})}{\sqrt{\sum_{i=1}^{n}V_{i}}},$ (5)

in which

$E_{i}=\textstyle{\frac{1}{2}}(N_{i}+1),\qquad
V_{i}=\textstyle\frac{1}{12}(N_{i}^{2}-1).$ (6)

Note that the value of $\tau$ is independent of that of the scale factor
$\ell$ that appears in equation (1): the value of the Efron–Petrosian
statistic does not change if we make monotonically increasing transformations
on the values of distance and/or flux (Efron & Petrosian, 1992).

The hypothesis of independence is rejected when the value of $\tau$ falls in
one of the tails of a Gaussian distribution whose mean is $0$ and whose
variance is $1$. More precisely, the hypothesis of independence is rejected
when the value of

$p=(2/\pi)^{1/2}\int_{|\tau|}^{\infty}\exp(-x^{2}/2){\rm d}x={\rm
erfc}(|\tau|/\sqrt{2})$ (7)

is smaller than an adopted significance level between $0$ and $1$, where ${\rm
erfc}$ denotes the complementary error function. If $\tau$ equals zero, for
instance, then $p$ would assume the value one and the hypothesis in question
cannot be rejected at any significance level.

###  3.2 Test results for the millisecond pulsars

We have used the data on millisecond pulsars (Fig. 1) to evaluate the
expression in equation (5) for the Efron–Petrosian statistic $\tau$ as a
function of the flux threshold $S_{\rm th}$, once letting $\alpha=2$ and
another time letting $\alpha=3/2$. The results that follow from the two
alternative data sets depicted in Figs 1b and 1c (i.e. from YMW16 and NE2001
distances) are plotted in parts (a) and (b) of Fig. 4, respectively. The flux
thresholds designated by the letters $a$ to $d$ in these figures are the ones
for which the value of $\tau$ for $\alpha=3/2$ is $1$ (thresholds $a$ and
$c$), $0$ (threshold $b$) or minimum (threshold $d$). The corresponding values
of $\tau$ for $\alpha=2$ at the same flux thresholds are listed, together with
their associated $p$ values, in Table 1. Dependence of the Efron–Petrosian
statistic $\tau$ on the exponent $\alpha$ at fixed values of the flux
threshold $S_{\rm th}$ is shown in Fig. 5.

It can be seen from Figs 4 and 5 that the values of $|\tau|$ for $\alpha=2$
lie above those for $\alpha=3/2$ throughout the displayed thresholds. In the
case of threshold $a$ in Fig. 4a, which excludes $30$ of the $114$ elements in
the data set on millisecond pulsars, for example, the $p$-values (defined in
equation 7) for $\alpha=3/2$ and $\alpha=2$ are respectively given by $0.317$
and $0.147$: while the hypothesis of independence of luminosity and distance
can be rejected at any significance level exceeding $14.7\%$ when $\alpha=2$,
the rejection of this hypothesis requires a significance level exceeding
$31.7\%$ when $\alpha=3/2$. For threshold $b$ in Fig. 4a, which excludes $31$
of the $114$ elements in the data set, the $p$-values corresponding to
$\alpha=3/2$ and $\alpha=2$ are respectively given by $1$ and $0.617$: while
it can be rejected at any significance level exceeding $61.7\%$ when
$\alpha=2$, the present hypothesis cannot be rejected at even a $100\%$
significance level when $\alpha=3/2$. Similar results are obtained for the
flux thresholds $c$ and $d$ in Fig. 4b (see Table 1).

There is an upper limit to how high the chosen value of the flux threshold
$S_{\rm th}$ can be. The Kolmogorov–Smirnov statistic $p_{KS}$ yields the
probability that the truncated data set with the elements $S\geq S_{\rm th}$
and the uncut $114$-element data set depicted in Fig. 1 are drawn from the
same distribution: from the unknown distribution that is complete over all
values of the flux density. Evaluation of $p_{KS}$ for the millisecond pulsars
shows that the value of the Kolmogorov–Smirnov statistic for any truncated
data set with a flux threshold larger than $\log S_{\rm th}=-8.9$ is lower
than $3.54\times 10^{-9}$. In other words, any truncated data set for which
the value of $S_{\rm th}$ lies outside the range of flux thresholds plotted in
Fig. 4 is extremely unlikely to have the same origin as the uncut
$114$-element data set provided by the observation of millisecond pulsars.

Table 1: The Efron–Petrosian statistic $\tau$ and its associated $p$ and $p_{\rm perm}$ values for $\alpha=1.5$ and $2$ versus the flux threshold $S_{\rm th}$ for the millisecond pulsars. The rows YMW16 and NE2001 refer to the alternative distances listed in the ATNF Pulsar Catalogue and the letters $a$–$d$ refer to the flux thresholds shown in Fig. 4. | $\log[S_{\rm th}$ | $\tau|_{\alpha=1.5}$ | $p|_{\alpha=1.5}$ | $p_{\rm perm}|_{\alpha=1.5}$ | $\tau|_{\alpha=2}$ | $p|_{\alpha=2}$ | $p_{\rm perm}|_{\alpha=2}$
---|---|---|---|---|---|---|---
| (erg cm-2 s${}^{-1})]$ |  |  |  |  |  |
YMW16 | -9.112 ($a$) | 1 | 0.317 | 0.311 | 1.45 | 0.147 | 0.150
| -9.070 ($b$) | 0 | 1 | 1 | 0.5 | 0.617 | 0.605
NE2001 | -9.092 ($c$) | 1 | 0.317 | 0.258 | 1.55 | 0.121 | 0.116
| -9.005 ($d$) | 0.114 | 0.909 | 0.865 | 0.822 | 0.411 | 0.392

Figure 4: The Efron–Petrosian statistic $\tau$ versus the logarithm of the
flux threshold $S_{\rm th}$ for the millisecond pulsars. The two curves are
plotted for the values $2$ (coloured red) and $3/2$ (coloured blue) of
$\alpha$. (a) For the distances listed under YMW16 in the ANTF Catalogue. The
dashed vertical lines $a$ and $b$ (in green) designate the same thresholds as
those in Fig. 1b, i.e. $\log S_{\rm th}=-9.112$ and $\log S_{\rm th}=-9.07$,
respectively. The values of $\tau$ corresponding to threshold $a$ are $1$ for
$\alpha=3/2$ and $1.45$ for $\alpha=2$. The values of $\tau$ corresponding to
threshold $b$ are $0$ for $\alpha=3/2$ and $0.5$ for $\alpha=2$. (b) For the
distances listed under NE2001 in the ANTF Catalogue. The dashed vertical lines
$c$ and $d$ (in green) designate the same thresholds as those in Fig. 1c, i.e.
$\log S_{\rm th}=-9.092$ and $\log S_{\rm th}=-9.005$, respectively. The
values of $\tau$ corresponding to threshold $c$ are $1$ for $\alpha=3/2$ and
$1.55$ for $\alpha=2$. The values of $\tau$ corresponding to threshold $d$ are
$0.114$ for $\alpha=3/2$ and $0.822$ for $\alpha=2$.

Figure 5: Contours of the function $\log S_{\rm th}(\alpha,\tau)$ over
$-9.3\leq\log S_{\rm th}\leq-9.0$ at intervals of $0.05$ for the millisecond
pulsars. (a) Based on the distances listed under YMW16 in the ANTF Catalogue
and for the following values of $\log S_{\rm th}$: -9.3 ($a$, green), -9.25
($b$, magenta), -9.2 ($c$, brown) , -9.15 ($d$, red), -9.1 ($e$, purple),
-9.05 ($f$, blue), -9 ($g$, black). (b) The same as in part (a) but based on
the distances listed under NE2001 in the ATNF Catalogue.

Figure 6: The Efron–Petrosian statistic $\tau$ versus the logarithm of the
flux threshold $S_{\rm th}$ for the young pulsars. The two curves are plotted
for the values $2$ (red) and $3/2$ (blue) of $\alpha$. (a) For the distances
listed under YMW16 in the ANTF Catalogue. The dashed vertical lines $a$, $b$
and $c$ (in green) designate the thresholds $\log S_{\rm th}=-9.365$, $\log
S_{\rm th}=-9.08$ and $\log S_{\rm th}=-8.95$ which result in
$\tau|_{\alpha=1.5}=1$, $\tau|_{\alpha=1.5}=0$ and
$|\tau|_{\alpha=1.5}=|\tau|_{\alpha=2}$, respectively. (b) For the distances
listed under NE2001 in the ANTF Catalogue. The dashed vertical lines $d$, $e$
and $f$ (in green) designate the thresholds $\log S_{\rm th}=-9.066$, $\log
S_{\rm th}=-8.911$ and $\log S_{\rm th}=-8.8$ which result in
$\tau|_{\alpha=1.5}=1$, $\tau|_{\alpha=1.5}=0$ and
$|\tau|_{\alpha=1.5}=|\tau|_{\alpha=2}$, respectively.

Figure 7: Contours of the function $\log S_{\rm th}(\alpha,\tau)$ over
$-9.4\leq\log S_{\rm th}\leq-8.8$ at intervals of $0.1$ for the young pulsars.
(a) Based on the distances listed under YMW16 in the ANTF Catalogue and for
the following values of $\log S_{\rm th}$: -9.4 ($a$, cyan), -9.3 ($b$,
green), -9.2 ($c$, magenta) , -9.1 ($d$, brown), -9 ($e$, red), -8.9 ($f$,
purple), -8.8 ($g$, blue). (b) The same as in part (a) but based on the
distances listed under NE2001 in the ATNF Catalogue.

###  3.3 Test results for the young pulsars

The counterparts of Figs 4 and 5 for the data on young pulsars (Fig. 2) are
plotted in Figs 6 and 7. In Fig. 6, whose parts (a) and (b) are based on the
YMW16 and NE2001 distances respectively, the flux thresholds designated by the
letters $a$ to $f$ are chosen such that thresholds $a$ and $d$ each result in
$\tau|_{\rm\alpha=1.5}=1$, thresholds $b$ and $e$ each result in
$\tau|_{\rm\alpha=1.5}=0$ and thresholds $c$ and $f$ each result in
$|\tau|_{\rm\alpha=1.5}=|\tau|_{\rm\alpha=2}$. It can be seen from Figs 6 and
7 and Table 2 that in the case of young pulsars, too, there is a wide range of
significance levels at which the hypothesis of independence of luminosity and
distance can be rejected for $\alpha=2$ but not for $\alpha=3/2$.

In this case, there do exist higher flux thresholds (higher than those
designated $c$ in Fig. 6a and $f$ in Fig. 6b) which result in a greater value
of $|\tau|$ for $\alpha=3/2$ than for $\alpha=2$. However, the thresholds $c$
($\log S_{\rm th}=-8.95$) and $f$ ($\log S_{\rm th}=-8.8$) at which
$|\tau|_{\alpha=1.5}=|\tau|_{\alpha=2}$ result in truncated data sets
consisting, respectively, of only $82$ and $79$ elements. The probability that
these truncated data sets have the same origins as the $93$-element data sets
depicted in Figs 2b and 2c is comparatively low. According to the
Kolmogorov–Smirnov test, the probabilities $p_{KS}$ that the resulting $82$
and $79$-element data sets and the uncut $93$-element data set are drawn from
the same distribution are $0.425$ and $0.18$, respectively (see Fig. 8). In
contrast, the probabilities that the truncated data sets resulting from the
thresholds $a$ and $d$ in Fig. 6 and the uncut $93$-element data set are drawn
from the same distribution are given by $p_{KS}=0.975$ and $p_{KS}=0.53$,
respectively (see Fig. 8). It is essential that the observationally obtained
data set and the part of it that lies above the chosen flux threshold could be
regarded as drawn from the same distribution: from the unknown distribution
that is complete over all values of the flux density.

Figure 8: The Kolmogorov–Smirnov statistic $p_{KS}$ versus the logarithm of the flux threshold $S_{\rm th}$ for the young pulsars depicted in Fig. 2. The values of $p_{KS}$ corresponding to the flux thresholds $a$ ($\log S_{\rm th}=-9.365$), $b$ ($\log S_{\rm th}=-9.08$), $c$ ($\log S_{\rm th}=-8.95$), $d$ ($\log S_{\rm th}=-9.066$), $e$ ($\log S_{\rm th}=-8.911$) and $f$ ($\log S_{\rm th}=-8.8$), i.e. the thresholds designated by the same letters in Fig. 6, are respectively $0.975$, $0.55$, $0.425$, $0.53$, $0.38$ and $0.18$. Table 2: The Efron–Petrosian statistic $\tau$ and its associated $p$ and $p_{\rm perm}$ values for $\alpha=1.5$ and $2$ versus the flux threshold $S_{\rm th}$ for the young pulsars. The rows YMW16 and NE2001 refer to the alternative distances listed in the ATNF Pulsar Catalogue and the letters $a$–$f$ refer to the flux thresholds shown in Fig. 6. The last column shows the Kolmogorov–Smirnov statistic $p_{KS}$ associated with each flux threshold. | $\log[S_{\rm th}$ | $\tau|_{\alpha=1.5}$ | $p|_{\alpha=1.5}$ | $p_{\rm perm}|_{\alpha=1.5}$ | $\tau|_{\alpha=2}$ | $p|_{\alpha=2}$ | $p_{\rm perm}|_{\alpha=2}$ | $p_{KS}$
---|---|---|---|---|---|---|---|---
| (erg cm-2 s${}^{-1})]$ |  |  |  |  |  |  |
YMW16 | -9.365 ($a$) | 1 | 0.317 | 0.315 | 2.706 | 0.007 | 0.010 | 0.975
| -9.080 ($b$) | 0 | 1 | 1 | 1.541 | 0.124 | 0.125 | 0.550
| -8.950 ($c$) | -0.795 | 0.427 | 0.406 | 0.795 | 0.427 | 0.428 | 0.425
NE2001 | -9.066 ($d$) | 1 | 0.317 | 0.309 | 2.75 | 0.006 | 0.001 | 0.53
| -8.911 ($e$) | 0 | 1 | 1 | 1.49 | 0.136 | 0.134 | 0.38
| -8.800 ($f$) | -0.78 | 0.435 | 0.416 | 0.78 | 0.435 | 0.455 | 0.18

###  3.4 The effect of observational errors on the test results

Even if present, a purely systematic error in the estimates of distance and/or
flux would not alter the results reported in Sections 3.2 and 3.3: the value
of the Efron–Petrosian statistic does not change if we make monotonically
increasing transformations on the values of distance and/or flux (Efron &
Petrosian, 1992). For instance, the dependence of $\tau$ on $\alpha$ (depicted
by the curves in Fig. 5) remains exactly the same if the distances in the data
sets shown in Figs 1 and 2 are all multiplied by a positive factor. In this
section we perform a Monte Carlo simulation with $10^{3}$ random samplings to
assess the effect on the test results of (the known) random errors in the
estimates of flux.

Fluxes are listed in the Fermi-LAT 12-year Catalog each with an uncertainty
$\pm\sigma$ (Abdollahi et al., 2022). The distribution of the error in the
listed value of flux, $\mu$, can accordingly be modelled by a Gaussian
probability density:

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi}}\sigma^{-1}\left[1+{\rm
erf}\left(\frac{\mu}{\sqrt{2}\sigma}\right)\sigma\right]^{-1}\exp\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right],$
(8) $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad 0<x<\infty,$

where erf and H denote the error function and the Heaviside step function,
respectively.

The sampling domain of the Monte Carlo method we use consists of the
collection of the intervals $\mu-\sigma\leq x\leq\mu+\sigma$ which enclose
each flux coordinate (whose listed value we have denoted by $\mu$) of the
elements of the data sets shown in Figs 1 and 2. In a given sampling, the flux
coordinate of each element of these data sets is replaced by a randomly chosen
value of $x$ from the probability distribution $f(x)$ over the interval
$\mu-\sigma\leq x\leq\mu+\sigma$, where $\mu$ and $\sigma$ are the parameters
of the flux coordinate of the data point in question as they appear in the
Fermi-LAT 12-Year Catalog. The modified data set thus obtained in a given
sampling is then used, in conjunction with a choice of the flux threshold
$S_{\rm th}$, to calculate the Efron–Petrosian statistic $\tau$ for both
$\alpha=2$ and $\alpha=3/2$. Repeating this procedure $10^{3}$ times and
aggregating the two resulting sets of values of $\tau$ for differing values of
$\alpha$, we arrive at the distributions shown in Fig. 9 in the case of YMW16
distances. These, as well as the corresponding distributions for NE2001
distances and other flux thresholds, show that the uncertainties in the
estimates of pulsar fluxes, though bringing about a spreading of the values of
$\tau$, do not alter the conclusions reached in Sections 3.2 and 3.3.

In the case of the millisecond pulsars, the mean and standard deviation of the
$\tau$-distribution for $\alpha=2$ in part ${\bf b}$ of Fig. 9 have the values
$1.42$ and $0.17$, respectively, so that the uncertainties in the estimates of
flux result in replacing the value of the Efron–Petrosian statistic found in
Section 3.2, i.e. $1.45$, by the interval $1.25\leq\tau\leq 1.59$. According
to equation (7), the $p$-values corresponding to this range of values of
$\tau$ occupy the interval $0.11\leq p\leq 0.21$. In contrast, the mean and
standard deviation of the $\tau$-distribution for $\alpha=3/2$ in part ${\bf
a}$ of Fig. 9 have the values $0.86$ and $0.17$, respectively, so that the
uncertainties in the estimates of flux result in replacing the value of the
Efron–Petrosian statistic found in Section 3.2, i.e. $1$, by the interval
$0.69\leq\tau\leq 1.03$. The $p$-values corresponding to this range of values
of $\tau$ occupy the interval $0.30\leq p\leq 0.49$.

In the case of the young pulsars, the mean and standard deviation of the
$\tau$-distribution for $\alpha=2$ in part ${\bf d}$ of Fig. 9 have the values
$1.60$ and $0.14$, respectively, so that the uncertainties in the estimates of
flux result in replacing the value of the Efron–Petrosian statistic found in
Section 3.3, i.e. $1.54$, by the interval $1.46\leq\tau\leq 1.74$. The
$p$-values corresponding to this range of values of $\tau$ occupy the interval
$0.08\leq p\leq 0.14$. In contrast, the mean and the standard deviation of the
$\tau$-distribution for $\alpha=3/2$ in part ${\bf c}$ of Fig. 9 have the
values $0.07$ and $0.14$, respectively, so that the uncertainties in the
estimates of flux result in replacing the value of the Efron–Petrosian
statistic found in Section 3.3, i.e. $0$, by the interval $-0.07\leq\tau\leq
0.21$. The $p$-values corresponding to this range of values of $\tau$, which
includes $\tau=0$, occupy the interval $0.83\leq p\leq 1$.

Figure 9: The histograms (obtained by the Monte Carlo simulation described in
Section 3.4) of the distributions of the values of the Efron–Petrosian
statistic $\tau$ that result from the inclusion of observational errors. The
distributions for the millisecond pulsars are plotted in parts ${\bf a}$ and
${\bf b}$ and those for the young pulsars in parts ${\bf c}$ and ${\bf d}$ of
this figure. The computations are based on the YMW16 distances and on the flux
threshold $\log S_{\rm th}=-9.112$ (designated $a$ in Fig. 4a) for the
millisecond pulsars and on the flux threshold $\log S_{\rm th}=-9.08$
(designated $b$ in Fig. 6a) for the young pulsars. The mean and standard
deviation of each distribution are denoted by $\bar{\tau}$ and $\bar{\sigma}$,
respectively.

Figure 10: The normalized probability distribution functions of the
Efron–Petrosian statistic $\tau$ for young pulsars (the data set shown in Fig.
2a) and the flux threshold $\log S_{\rm th}=-9.08$ (designated $b$ in Fig.
6a). (a) The permutation distribution of $\tau$ for $\alpha=3/2$ based on an
augmented data set consisting of $2749$ elements. The shaded area under this
distribution function yields $p_{\rm perm}=1$ for $\tau_{\rm obs}=0$, i.e. for
the value of $\tau$ that is selected by the flux threshold $b$ and the blue
curve in Fig. 6a. (b) The permutation distribution of $\tau$ for $\alpha=2$
based on an augmented data set consisting of $2434$ elements. The shaded area
under this distribution function yields $p_{\rm perm}=0.125$ for $\tau_{\rm
obs}=1.541$, i.e. for the value of $\tau$ that is selected by the flux
threshold $b$ and the red curve in Fig. 6a.

###  3.5 The effect of the finiteness of the size of the data set on the test
results

The only step in the analysis described in Section 3.1 at which the size of
the data set is assumed to be large is where the resulting value of the
Efron–Petrosian statistic $\tau$ is appraised by comparison with a normalized
Gaussian distribution, i.e. the very last step at which the $p$ value
associated with $\tau$ is calculated by means of equation (7). By relaxing
that assumption, one can in fact apply the Efron–Petrosian method to data sets
of any size: the normalized Gaussian distribution should be replaced, in the
general case, by the normalized permutation distribution of the data set (see
Efron & Petrosian, 1992, Section 2.2).

A permissible permutation of a truncated luminosity-distance data set, such as
that shown in Fig. 3, is obtained by replacing an element $(\log D_{i},\log
L_{i})$ of that data set by any one of the $N_{i}$ elements that constitute
the set comparable to $(\log D_{i},\log L_{i})$. The number of permuted data
sets thus obtained from an original $n$-element data set is therefore given by
$N=\sum_{1}^{n}N_{i}$. In the case of the young pulsars shown in Fig. 2b, for
example, the value of $N$ for the $83$-element data set corresponding to the
threshold $\log S_{\rm th}=-9.08$ (designated $b$ in Fig. 6a) is $2749$ when
$\alpha=3/2$ and $2434$ when $\alpha=2$.

If we calculate the Efron–Petrosian statistic $\tau$ for each one of the $N$
permuted data sets (corresponding to a given threshold and a given value of
$\alpha$) and normalize the distribution of the resulting values of $\tau$ (by
replacing each $\tau$ with $(\tau-{\bar{\tau}})/\sigma$, where ${\bar{\tau}}$
and $\sigma$ are the mean and standard deviation of the distribution), we
arrive at a normalized probability distribution function representing the
permutation distribution of $\tau$. In the case of a data set with a limited
number of elements, the observed value $\tau_{\rm obs}$ of $\tau$ (i.e. the
value of $\tau$ determined by the original data set) should be appraised by
comparison with this permutation distribution of $\tau$ instead of the
normalized Gaussian distribution used at the end of Section 3.1. The size of
the area that lies under a normalized permutation distribution in
$-\infty<\tau\leq-|\tau_{\rm obs}|$ and $|\tau_{\rm obs}|\leq\tau<\infty$,
here denoted by $p_{\rm perm}$, plays the role of the $p$-value defined in
equation (7) (see Efron & Petrosian, 1992, Section 2.2).

As an example, the normalized probability distribution functions of the
Efron–Petrosian statistic for the $83$-element data set on young pulsars (Fig.
2a) that corresponds to the flux threshold $\log S_{\rm th}=-9.08$ (designated
$b$ in Fig. 6a) and to the values $3/2$ and $2$ of $\alpha$ are shown in Fig.
10. The mean and standard deviation of each of the shown distributions are
given by $0$ and $1$, respectively. The parameters $({\bar{\tau}},\sigma)$,
i.e. the mean and standard deviation, of the un-normalized versions of these
distributions have the values $(0,0.150)$ for $\alpha=3/2$ and $(1.547,0.141)$
for $\alpha=2$. The shaded regions in this figure, whose areas determine
$p_{\rm perm}|_{\alpha=3/2}$ and $p_{\rm perm}|_{\alpha=2}$, include all
values of $\tau$ in $\tau\leq-|\tau_{\rm obs}|$ and $\tau\geq|\tau_{\rm obs}|$
with $\tau_{\rm obs}=0$ for $\alpha=3/2$ and $\tau_{\rm obs}=1.541$ for
$\alpha=2$ (see Fig. 6a). In this case, $p|_{\alpha=3/2}=1$ and
$p|_{\alpha=2}=0.124$ found in Section 3.3 on the basis of a normalized
Gaussian distribution are respectively replaced by $p_{\rm
perm}|_{\alpha=3/2}=1$ and $p_{\rm perm}|_{\alpha=2}=0.125$ (see Table 2).

Figure 10 and its counterparts for other flux thresholds and alternative
distances thus show that taking account of the finiteness of the sizes of the
present data sets does not alter the outcomes of the test results obtained
before: the values of $p_{\rm perm}$ only slightly differ from those of $p$ in
most cases (see Tables 1 and 2).

##  4 Discussion

The conclusion to be drawn from the above results is that the observational
data in the Fermi-LAT 12-Year Catalog on gamma-ray pulsars are consistent with
the dependence $S\propto D^{-3/2}$ of the flux densities $S$ of these pulsars
on their distances $D$ at substantially higher levels of significance than
they are with the dependence $S\propto D^{-2}$: a conclusion that agrees with
those arrived at earlier (Ardavan, 2022c, d) on the basis of the smaller data
sets in the Second Fermi-LAT Catalog of Gamma-ray Pulsars (Abdo et al., 2013)
and the McGill Magnetar Catalog (Olausen & Kaspi, 2014). To the list of
observed features of the pulsar emission (brightness temperature,
polarization, spectrum and profile with microstructure and with a phase lag
between the radio and gamma-ray peaks) that the analysis of the radiation by
the current sheet in the magnetosphere of a neutron star has decoded (Ardavan,
2021, 2022b), we can therefore add another feature of this emission: the non-
spherical decay of its high-frequency flux density.

The violation of the inverse-square law encountered here is not incompatible
with the requirements of the conservation of energy because the radiation
process by which the superluminally moving current sheet in the magnetosphere
of a neutron star generates the observed gamma-ray pulses is intrinsically
transient. Temporal rate of change of the energy density of the radiation
generated by this process has a time-averaged value that is negative (instead
of being zero as in a steady-state radiation) at points where the envelopes of
the wave fronts emanating from the constituent volume elements of the current
sheet are cusped (Ardavan, 2021, Fig. 1 and Section 3.2). The difference in
the fluxes of power across any two spheres centred on the star is thus
balanced by the change with time of the energy contained inside the shell
bounded by those spheres (see Ardavan, 2019, Appendix C, where this is
demonstrated for each high-frequency Fourier component of a superluminally
rotating source distribution).

Luminosities of gamma-ray pulsars are over-estimated when the decay of their
flux density $S$ is assumed to obey the inverse-square law $S\propto D^{-2}$
instead of $S\propto D^{-3/2}$ by the factor $(D/\ell)^{1/2}$ (see equation
1). The value of the scale factor $\ell$ is of the same order of magnitude as
the values of the light-cylinder radii of these pulsars (Ardavan, 2021,
Section 5.5). Hence, the factor by which the luminosity of a $100$ ms gamma-
ray pulsar at a distance of $2.5$ kpc is over-estimated is approximately
$4\times 10^{6}$. Once this is multiplied by the ratio $\sim 1/13$ of the
latitudinal beam-widths of gamma-ray and radio pulsars (implied by the
fraction of known pulsars that are detected in gamma-rays), we obtain a value
of the order of $10^{5}$ for the over-estimation factor in question: a result
that implies that the range of values of the correctly-estimated luminosities
of gamma-ray pulsars is no different from that of the luminosities of radio
pulsars.

For any given value of the angle between the magnetic and spin axes of the
neutron star, there are four critical colatitudes (denoted by $\theta_{P1S}$,
$\theta_{P2S}$, $\pi-\theta_{P1S}$ and $\pi-\theta_{P2S}$ in Ardavan, 2021)
with respect to the spin axis of the star along which the flux density of the
radiation decays non-spherically. The gradual change in the rate of decay of
flux density with distance, from $D^{-3/2}$ to $D^{-2}$, away from a critical
colatitude takes place over a latitudinal interval of the order of a radian.
But the latitudinal width of the tightly focused part of the radiation beam
whose flux density decreases as $D^{-3/2}$ with distance $D$ has a much lower
value: it is of the order of $(D/R_{lc})^{-1}$, where $R_{lc}$ is the radius
of the star’s light cylinder (Ardavan, 2021, Section 5.5). Hence, the solid
angle centered on the neutron star within which the flux density of the
present radiation decays at a significantly lower rate than that predicted by
the inverse-square law is only a small fraction of $4\pi$.

It follows that our lines of sight to only a small fraction of the total
number of known neutron stars are expected to coincide with the privileged
directions along which the flux density of the radiation we observe would
decay as $D^{-3/2}$. This is borne out by the fact that the known X-ray and
gamma-ray emitting neutron stars comprise a small fraction of all observed
pulsars.

## Data availability

The data used in this paper can be found at
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/12yr_catalog/ and
http://www.atnf.csiro.au/research/pulsar/psrcat.

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