# Radio spectra of pulsars fitted with the spectral distribution function of
the emission from their current sheet

Houshang Ardavan
Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, United Kingdom Email address:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In their catalogue of pulsars’ radio spectra, Swainston et al. (2022, PASA,
39, e056) distinguish between five different forms of these spectra: those
that can be fitted with (i) a simple power law, (ii) a broken power law, (iii)
a low-frequency turn-over, (iv) a high-frequency turn-over or (v) a double
turn-over spectrum. Here, we choose two examples from each of these categories
and fit them with the spectral distribution function of the caustics that are
generated by the superluminally moving current sheet in the magnetosphere of a
non-aligned neutron star. In contrast to the prevailing view that the curved
features of pulsars’ radio spectra arise from the absorption of the observed
radiation in high-density environments, our results imply that these features
are intrinsic to the emission mechanism. We find that all observed features of
pulsar spectra (including those that are normally fitted with simple or broken
power laws) can be described by a single spectral distribution function and
regarded as manifestations of a single emission mechanism. From the results of
an earlier analysis of the emission from a pulsar’s current sheet and the
values of the fit parameters for each spectrum, we also determine the physical
characteristics of the central neutron star of each considered example and its
magnetosphere.

###### keywords:

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

††pubyear: 2023††pagerange: Radio spectra of pulsars fitted with the spectral
distribution function of the emission from their current sheet–Radio spectra
of pulsars fitted with the spectral distribution function of the emission from
their current sheet

##  1 Introduction

Attempts at explaining the radiation from pulsars has so far been focused
mainly on mechanisms of acceleration of charged particles (see, e.g., the
references in Melrose et al., 2021): an approach spurred by the fact that,
once the relevant version of this mechanism is identified, one can calculate
the electric current density associated with the accelerating charged
particles involved and thereby evaluate the classical expression for the
retarded potential that describes the looked-for radiation. In the present
paper, however, we evaluate the retarded potential, and hence the generated
radiation field, using the macroscopic distribution of electric charge-current
density that is already provided by the numerical computations of the
structure of a non-aligned pulsar magnetosphere (Ardavan, 2021, Section 2).
Both the radiation field thus calculated and the electric and magnetic fields
that pervade the pulsar magnetosphere are solutions of Maxwell’s equations for
the same charge-current distribution. These two solutions are completely
different, nevertheless, because they satisfy different boundary conditions:
the far-field boundary conditions with which the structure of the pulsar
magnetosphere is computed are radically different from the corresponding
boundary conditions with which the retarded solution of these equations (i.e.
the solution describing the radiation from the charges and currents in the
pulsar magnetosphere) is derived (see Section 3 and the last paragraph in
Section 6 of Ardavan 2021).

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 Spitkovsky 2006; Kalapotharakos et al. 2012; Tchekhovskoy et al.
2016; and the references in Philippov & Kramer 2022). However, the role played
by the superluminal motion of this current sheet in generating the multi-
wavelength, focused pulses of radiation that we receive from neutron stars is
not generally acknowledged. Given that the superluminally moving distribution
pattern of this current sheet is created by the coordinated motion of
aggregates of subluminally moving charged particles (see Ginzburg, 1972;
Bolotovskii & Bykov, 1990), the motion of any of its constituent particles is
too complicated to be taken into account individually. Only the densities of
charges and currents enter the Maxwell’s equations, on the other hand, so that
the macroscopic charge-current distribution associated with the magnetospheric
current sheet takes full account of the contributions toward the radiation
that arise from the complicated motions of the charged particles comprising
it.

The radiation field generated by a uniformly rotating volume element of the
distribution pattern of the current sheet in the magnetosphere of a non-
aligned 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 this
distribution pattern 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
farther they are from their source: as their distance from their source
increases, two nearby stationary points of their phases draw closer to each
other and eventually coalesce at infinity. By virtue of their narrow peaks in
the time domain, the resulting focused pulses thus procure frequency spectra
whose distributions extend from radio waves to gamma-rays (Ardavan, 2021,
Table 1 and Section 5.4).

This paper is concerned with the radio spectra of pulsars. Its task is to
ascertain whether the spectrum of the caustics generated by a pulsar’s current
sheet (Section 2) can account for all five categories of spectral shapes
catalogued111https://all-pulsar-spectra.readthedocs.io/en/latest/ by Swainston
et al. (2022). To this end, it presents fits to two examples (with the largest
number of known data points) of the catalogued spectra in each category
(Section 3) and relates the values of their fit parameters to the physical
characteristics of the central neutron star of the corresponding pulsar and
its magnetosphere (Section 4).

##  2 Radio spectrum of the caustics generated by the superluminally moving
current sheet

The frequency spectrum of the radiation that is generated as a result of the
superluminal motion of the current sheet in the magnetosphere of a non-aligned
neutron star was presented, in its general form, in equation (177) of Ardavan
(2021, Section 5.3).

In a case where the magnitudes of the vectors denoted by $\boldsymbol{\cal
P}_{l}$ and $\boldsymbol{\cal Q}_{l}$ in equation (177) of Ardavan (2021) are
appreciably larger than those of their counterparts, ${\bar{\boldsymbol{\cal
P}}}_{l}$ and ${\bar{\boldsymbol{\cal Q}}}_{l}$, and the dominant contribution
towards the Poynting flux $S_{\nu}$ of the radiation is made by only one of
the two terms corresponding to $l=1$ and $l=2$, e.g. $l=2$, that equation can
be written as

$S_{\nu}=\kappa_{0}\,k^{-2/3}\left|\boldsymbol{\cal P}_{2}\,{\rm
Ai}(-k^{2/3}\sigma_{21}^{2})-{\rm i}k^{-1/3}\boldsymbol{\cal Q}_{2}\,{\rm
Ai}^{\prime}(-k^{2/3}\sigma_{21}^{2})\right|^{2},$ (1)

where Ai and ${\rm Ai}^{\prime}$ are the Airy function and the derivative of
the Airy function with respect to its argument, respectively,
$k=2\pi\nu/\omega$ is the frequency $\nu$ of the radiation in units of the
rotation frequency $\omega/2\pi$ of the central neutron star, and $\kappa_{0}$
and $\sigma_{21}$ are two positive scalars. The coefficients of the Airy
functions in the above expression stand for $\boldsymbol{\cal
P}_{2}=k^{-1/2}\boldsymbol{\cal P}_{2}^{(2)}$ and $\boldsymbol{\cal
Q}_{2}=k^{-1/2}\boldsymbol{\cal Q}_{2}^{(2)}$ when $k\geq k_{2}$ and for
$\boldsymbol{\cal P}_{2}=\boldsymbol{\cal P}_{2}^{(0)}$ and $\boldsymbol{\cal
Q}_{2}=\boldsymbol{\cal Q}_{2}^{(0)}$ when $k<k_{2}$, in which the complex
vectors $\boldsymbol{\cal P}_{2}^{(0)}$, $\boldsymbol{\cal Q}_{2}^{(0)}$,
$\boldsymbol{\cal P}_{2}^{(2)}$ and $\boldsymbol{\cal Q}_{2}^{(2)}$ are
defined by equations (138)-(146) of Ardavan (2021) and $k_{2}$ designates a
threshold frequency.

The variable $\sigma_{21}$ determines the separation between two nearby
stationary points of the phases of the received waves: the smaller the value
of $\sigma_{21}$, the more focused is the observed radiation and the higher is
its frequency content (Ardavan, 2021, Section 4.5). The radio component of the
present radiation is mostly generated by values of $\sigma_{21}$ that range
from $10^{-4}$ to $10^{-2}$. In this paper, we replace $\boldsymbol{\cal
P}_{2}^{(0)}$, $\boldsymbol{\cal Q}_{2}^{(0)}$, $\boldsymbol{\cal
P}_{2}^{(2)}$ and $\boldsymbol{\cal Q}_{2}^{(2)}$, which only have weak
dependences on $\sigma_{21}$, by their values for $\sigma_{21}=10^{-3}$ and
treat them as constant parameters.

Evaluation of the right-hand side of equation (2) results in

$\displaystyle S_{\nu}$ $\displaystyle=$
$\displaystyle\kappa_{1}\,k^{-2/3-j/2}\Big{[}{\rm
Ai}^{2}(-k^{2/3}\sigma_{21}^{2})+\zeta_{1}^{2}k^{-2/3}{{\rm
Ai}^{\prime}}^{2}(-k^{2/3}\sigma_{21}^{2})$ (2)
$\displaystyle+2\zeta_{1}\cos\beta\,k^{-1/3}{\rm
Ai}(-k^{2/3}\sigma_{21}^{2}){\rm
Ai}^{\prime}(-k^{2/3}\sigma_{21}^{2})\Big{]},$

where $j=0$ when $k<k_{2}$ and $j=2$ when $k\geq k_{2}$,

$\kappa_{1}=\kappa_{0}\left|\boldsymbol{\cal
P}^{(j)}_{2}\right|^{2},\,\,\,\zeta_{1}=\frac{\left|{\boldsymbol{\cal
Q}^{(j)}_{2}}\right|}{\left|{\boldsymbol{\cal
P}^{(j)}_{2}}\right|},\,\,\,\cos\beta=\frac{\Im\left(\boldsymbol{\cal
Q}^{(j)}_{2}\cdot\boldsymbol{\cal P}^{(j)*}_{2}\right)}{\left|\boldsymbol{\cal
Q}^{(j)}_{2}\right|\left|\boldsymbol{\cal P}^{(j)}_{2}\right|},$ (3)

and $\Im{}$ and $*$ denote an imaginary part and the complex conjugate,
respectively. The above spectrum is emblematic of any radiation that entails
caustics (see Stamnes, 1986).

To take account of the fact that the parameter $\sigma_{21}$ assumes a non-
zero range of values across the (non-zero) latitudinal width of the detected
radiation beam (Ardavan, 2021, Section 4.5), we must integrate $S_{\nu}$ with
respect to $\sigma_{21}$ over a finite interval
$\rho\sigma_{0}\leq\sigma_{21}\leq\sigma_{0}$ with $\sigma_{0}\ll 1$ and
$0\leq\rho<1$. Performing the integration of the Airy functions in equation
(2) with respect to $\sigma_{21}$ by means of Mathematica, we thus obtain

$\displaystyle{\cal F}_{\nu}$ $\displaystyle=$
$\displaystyle\int_{\rho\sigma_{0}}^{\sigma_{0}}S_{\nu}\,{\rm d}\sigma_{21}$
$\displaystyle=$
$\displaystyle\kappa\,\chi^{-j/2}\left[f_{1}(\chi,\rho)+\frac{\zeta^{2}}{4\sqrt{3}}f_{2}(\chi,\rho)-\frac{\zeta\cos\beta}{2\sqrt{3}}f_{3}(\chi,\rho)\right],$

where

$\kappa=\frac{2^{j/2}\sigma_{0}^{3(1+j/2)}}{3^{(j+3)/2}\pi^{3/2}}\kappa_{1},\qquad\zeta=\sigma_{0}\zeta_{1},\qquad\chi=\frac{2}{3}{\sigma_{0}}^{3}k,$
(5) $\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\bigg{[}3\Gamma\left(\frac{7}{6}\right)\eta\chi^{-2/3}{}_{2}F_{3}\left(\begin{matrix}1/6&1/6&{}\\\
1/3&2/3&7/6\end{matrix};-\eta^{6}\chi^{2}\right)$
$\displaystyle+\pi^{1/2}\eta^{3}{}_{2}F_{3}\left(\begin{matrix}1/2&1/2&{}\\\
2/3&4/3&3/2\end{matrix}\,;-\eta^{6}\chi^{2}\right)$
$\displaystyle+\frac{9}{20}\Gamma\left(\frac{5}{6}\right)\eta^{5}\chi^{2/3}{}_{2}F_{3}\left(\begin{matrix}5/6&5/6&{}\\\
4/3&5/3&11/6\end{matrix}\,;-\eta^{6}\chi^{2}\right)\bigg{]}_{\eta=\rho}^{\eta=1},$
(6)
$f_{2}=\eta\chi^{-4/3}\,{}_{24}G^{31}\left(-\eta^{2}\chi^{2/3},\frac{1}{3}\left|\,\begin{matrix}5/6&7/6&{}&{}\\\
0&2/3&4/3&-1/6\end{matrix}\right)\right.\bigg{|}_{\eta=\rho}^{\eta=1},$ (7)
$f_{3}=\eta\chi^{-1}\,{}_{24}G^{31}\left(-\eta^{2}\chi^{2/3},\frac{1}{3}\left|\,\begin{matrix}5/6&1/2&{}&{}\\\
0&1/3&2/3&-1/6\end{matrix}\right)\right.\bigg{|}_{\eta=\rho}^{\eta=1},$ (8)

and ${}_{2}F_{3}$ and ${}_{24}G^{31}$ are respectively the generalised
hypergeometric function (see Olver et al., 2010) and the generalised Meijer
G-Function 222https://mathworld.wolfram.com/MeijerG-Function.html. The
variable $\chi$ that appears in the above expressions is related to the
frequency $\nu$ of the radiation via $\chi=4\pi{\sigma_{0}}^{3}\nu/(3\omega)$.

The scale and shape of the spectrum described by equation (LABEL:E4) depend on
whether $j$ equals $0$ or $2$ (i.e. on whether the dimensionless frequency $k$
lies below or above the threshold frequency $k_{2}$) and on the five
parameters $\kappa$, $\zeta$, $\beta$, $\sigma_{0}$ and $\rho$: parameters
whose values are dictated by the characteristics of the magnetospheric current
sheet (see Section 4). The parameters $\zeta$, $\beta$ and $\rho$ determine
the shape of the spectral distribution while the parameters $\kappa$ and
$\sigma_{0}$ respectively determine the position of this distribution along
the flux-density (${\cal F}_{\nu}$) and the frequency ($\nu$) axes.

##  3 Fits to the data on examples of various forms of radio spectra

In this section, we choose from each of the five galleries of pulsar spectral
shapes catalogued by Swainston et al. (2022) two examples with the largest
number of known data points and fit them with the spectral distribution
function described by equation (LABEL:E4). In the case of each example, we use
Mathematica’s ‘NonlinearModelFit’ procedure333
https://reference.wolfram.com/language/ref/NonlinearModelFit.html and the
statistical information that it provides to determine the values of the fit
parameters in equation (LABEL:E4) and their standard errors. Where, owing to
the complexity of the expression in equation (LABEL:E4), this procedure fails
to work and so the fits to the data are obtained by elementary iteration, only
the values of these parameters are specified.

The results are presented in Figs. (1)–(10) and their captions. The horizontal
and vertical axes in these logarithmic plots are marked with the values of
${\cal F}_{\nu}$ and $\nu_{\rm MHz}$, respectively, where $\nu_{\rm MHz}$
stands for the frequency of the radiation in units of MHz.

It can be seen from Figs. (1)–(10) that the fit residuals in the case of each
pulsar are smaller than the corresponding observational errors for the
majority of the data points. Since there are no two values of any of the fit
parameters for which the spectrum described by equation (LABEL:E4) has the
same shape and position, the specified values of the fit parameters are
moreover unique.

Figure 1: Spectrum of J1915+1009: an example from the simple-power-law gallery
of Swainston et al. (2022). The curve is a plot of the flux density ${\cal
F}_{\nu}$ given by equation (LABEL:E4) for $j=2$, $\kappa=8.39\times 10^{3}$
mJy, $\sigma_{0}=8.22\times 10^{-3}$, $\zeta=\beta=\rho=0$ and
$\chi=0.150\,\nu_{\rm MHz}$.

Figure 2: Spectrum of J0206-4028: a second example from the simple-power-law
gallery of Swainston et al. (2022). The curve is a plot of the flux density
${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=2$, $\kappa=2.58\times
10^{3}$ mJy, $\sigma_{0}=7.75\times 10^{-3}$, $\zeta=0.668$, $\beta=\rho=0$
and $\chi=0.112\,\nu_{\rm MHz}$.

Figure 3: Spectrum of J1024-0719: an example from the broken-power-law gallery
of Swainston et al. (2022). The curve is a plot of the flux density ${\cal
F}_{\nu}$ given by equation (LABEL:E4) for $j=2$, $\kappa=2.81\times 10^{-2}$
mJy, $\sigma_{0}=2.81\times 10^{-3}$, $\zeta=~{}0.507$, $\beta=0.5$, $\rho=0$
and $\chi=7.67\times 10^{-5}\,\nu_{\rm MHz}$.

Figure 4: Spectrum of J0452-1759: a second example from the broken-power-law
gallery of Swainston et al. (2022). The curve is a plot of the flux density
${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=2$, $\kappa=30.6\pm 10.6$
mJy, $\sigma_{0}=(1.88\pm 0.18)\times 10^{-3}$, $\zeta=1.75\pm 0.59$,
$\beta=0.64\pm 0.18$, $\rho=0.58\pm 0.34$ and $\chi=(2.42\pm 0.62)\times
10^{-3}\nu_{\rm MHz}$.

Figure 5: Spectrum of J0332+5434: an example from the low-frequency-turn-over
gallery of Swainston et al. (2022). The curve is a plot of the flux density
${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=2$, $\kappa=(1.55\pm
1.42)\times 10^{3}$ mJy, $\sigma_{0}=(1.49\pm 0.06)\times 10^{-3}$,
$\zeta=0.592\pm 0.033$, $\beta=(3.99\pm 2.00)\times 10^{-2}$, $\rho=0.927\pm
0.077$ and $\chi=(1.58\pm 0.20)\times 10^{-3}\,\nu_{\rm MHz}$.

Figure 6: Spectrum of J1845-0743: a second example from the low-frequency-
turn-over gallery of Swainston et al. (2022). The curve is a plot of the flux
density ${\cal F}_{\nu}$ given by equation (LABEL:E4) for $\kappa=2.22\pm
0.12$ mJy, $\sigma_{0}=(2.94\pm 0.08)\times 10^{-3}$, $\zeta=2.08\pm 0.29$,
$\beta=0$, $\rho=0.418\pm 0.056$ and $\chi=(1.77\pm 0.14)\times
10^{-3}\,\nu_{\rm MHz}$.

Figure 7: Spectrum of J1829-1751; an example from the high-frequency-cut-off
gallery of Swainston et al. (2022). The curve is a plot of the flux density
${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=0$, $\kappa=3.16\times
10^{-4}$ mJy, $\sigma_{0}=5.93\times 10^{-4}$, $\zeta=10^{3}$, $\beta=0$,
$\rho=0.999$ and $\chi=4.27\times 10^{-5}\,\nu_{\rm MHz}$.

Figure 8: Spectrum of J1835-0643: a second example from the high-frequency-
cut-off gallery of Swainston et al. (2022). The curve is a plot of the flux
density ${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=0$,
$\kappa=1.58\times 10^{-4}$ mJy, $\sigma_{0}=7.88\times 10^{-4}$,
$\zeta=10^{3}$, $\beta=0$, $\rho=0.999$ and $\chi=10^{-4}\,\nu_{\rm MHz}$.

Figure 9: Spectrum of J1932+1059: an example from the double-turn-over-
spectrum gallery of Swainston et al. (2022). The curve is a plot of the flux
density ${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=2$,
$\kappa=8.38\pm 7.35$ mJy, $\sigma_{0}=(1.49\pm 0.36)\times 10^{-3}$,
$\zeta=0.354\pm 0.069$, $\beta=\rho=0$ and $\chi=(5.02\pm 2.84)\times
10^{-4}\,\nu_{\rm MHz}$.

Figure 10: Spectrum of J0141+6009: a second example from the double-turn-over-
spectrum gallery of Swainston et al. (2022). The curve is a plot of the flux
density ${\cal F}_{\nu}$ given by equation (LABEL:E4) for $j=2$,
$\kappa=5.01\times~{}10^{2}$ mJy, $\sigma_{0}=7.88\times 10^{-4}$,
$\zeta=0.225$, $\beta=0$, $\rho=0.999$ and
$\chi=3.98\times~{}10^{-4}\,\nu_{\rm MHz}$.

##  4 The connection between the parameters of a fitted spectrum and the
physical characteristics of the source of the observed radiation

###  4.1 Derivation of the connecting relations

From equations (5), (3) and (1), it follows that the parameters $\kappa$,
$\zeta$, $\rho$ and $\sigma_{0}$ in equation (LABEL:E4) are related to the
characteristics of the source of the observed radiation via the quantities
$\omega$, $\kappa_{0}$, $\boldsymbol{\cal P}_{2}^{(j)}$ and $\boldsymbol{\cal
Q}_{2}^{(j)}$ that appear in the expression for the flux density ${\cal
F}_{\nu}$. In this section we use the results of the analysis presented in
Ardavan (2021) to express these quantities in terms of the inclination angle
of the central neutron star, $\alpha$, the magnitude of the star’s magnetic
field at its magnetic pole, $B_{0}=10^{12}{\hat{B}}_{0}$ Gauss, the radius of
the star, $r_{s0}=10^{6}d$ cm, the rotation frequency of the star,
$\omega=10^{2}{\hat{P}}^{-1}$ rad/s, and the spherical polar coordinates,
$R_{P}=D\,\,{\rm kpc}=3.085\times 10^{21}D$ cm, $\varphi_{P}$ and
$\theta_{P}$, of the observation point $P$ in a frame whose centre and
$z$-axis coincide with the centre and spin axis of the star.

From equations (177), (138)–(146) and (136) of Ardavan (2021) it follows that

$\kappa_{0}=4.15\times
10^{18}w_{1}^{2}w_{3}^{2}{\hat{B}}_{0}^{2}d^{4}D^{-2}{\hat{P}}^{-1}\quad{\rm
Jy},$ (9)

$\left|\boldsymbol{\cal P}_{2}^{(0)}\right|=\frac{b}{6}\left|{\tilde{\bf
P}}_{2}\right|\left(\sqrt{\frac{2\sigma_{21}}{\partial^{2}f_{2C}/\partial\tau^{2}|_{\tau=\tau_{0{\rm
min}}}}}+\sqrt{\frac{-2\sigma_{21}}{\partial^{2}f_{2C}/\partial\tau^{2}|_{\tau=\tau_{0{\rm
max}}}}}\right),$ (10)

and

$\left|\boldsymbol{\cal P}_{2}^{(2)}\right|=\left(\frac{2}{\pi
a^{3}}\right)^{1/2}\left|\boldsymbol{\cal P}_{2}^{(0)}\right|,$ (11)

where

$w_{1}=\left|1-2\alpha/\pi\right|,\qquad w_{3}=1+0.2\sin^{2}\alpha,$ (12)
$\displaystyle f_{2C}$ $\displaystyle=$
$\displaystyle({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}-{\hat{R}}_{P}-\arccos\left({\hat{r}}_{P}^{-1}{\hat{r}}_{sC}^{-1}\csc\theta\right)$
(13)
$\displaystyle-\arccos(\cot\alpha\cot\theta)+{\hat{r}}_{sC}+\varphi_{P}-{\hat{r}}_{s0}-2\pi,$
$\displaystyle a$ $\displaystyle=$
$\displaystyle({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)\left[\frac{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}+{\hat{r}}_{sC}}{{\hat{r}}_{sC}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}\right.$
(14)
$\displaystyle\left.-\frac{1}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}\right],$
$b=\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1}{{\hat{r}}_{sC}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}},$
(15)
${\hat{r}}_{sC}=\frac{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}-{\hat{z}}_{P}\cos\theta}{{\hat{r}}_{P}^{2}\sin^{2}\theta-1},$
(16) $\sigma_{21}=[{\textstyle\frac{3}{4}}(f_{2C}|_{\tau=\tau_{2{\rm
max}}}-f_{2C}|_{\tau=\tau_{2{\rm min}}})]^{1/3},$ (17)
$\theta=\arccos(\sin\alpha\cos\tau),$ (18) $\displaystyle\left|{\tilde{\bf
P}}_{2}\right|$ $\displaystyle=$
$\displaystyle\cos\alpha[{\hat{r}}_{sC}^{2}(1+\cos^{2}\theta_{P}-2\cos\theta\cos\theta_{P})+\tan^{2}\alpha$
$\displaystyle-\cot^{2}\theta+2{\hat{r}}_{sC}\sin\theta\cos\theta_{P}(\tan^{2}\alpha-\cot^{2}\theta)^{1/2}]^{1/2},\quad\,\,$

and the variables $\tau_{2{\rm min}}$ and $\tau_{2{\rm max}}$ stand for the
minimum and maximum of the function $f_{2C}$ (see also equations 7, 9, 174,
175, 93–95, 97 and 88 of Ardavan 2021). The caret on $R_{P}$ and $r_{s0}$ (and
$r_{P}=R_{P}\sin\theta_{P}$, $z_{P}=R_{P}\cos\theta_{P}$) is used here to
designate a variable that is rendered dimensionless by being measured in units
of the light-cylinder radius $c/\omega$. (Note the following two corrections:
the vector ${\bf P}_{2}$ in equation 145 and the numerical coefficient
$1.54\times 10^{19}$ in equation 177 of Ardavan 2021 have been corrected to
read ${\tilde{\bf P}}_{2}$ and $4.15\times 10^{18}$, respectively.)

The expression for $|{\tilde{\bf P}}_{2}|$ in equation (LABEL:E19) is derived
from that for ${\bf P}_{2}$ given by equations (98), (80), (78) and (62) of
Ardavan (2021). In this derivation, we have set the observation point on the
cusp locus of the bifurcation surface where ${\hat{r}}_{s}={\hat{r}}_{sC}$,
have approximated $(p_{1},p_{2},p_{3})$ by its far-field value
$2^{1/3}{\hat{R}}_{P}^{-1}({\hat{R}}_{P}^{-1},-1,1)$ and have let ${\bf
P}_{2}={\hat{R}}_{P}^{-1}{\tilde{\bf P}}_{2}$. The factor ${\hat{R}}_{P}^{-2}$
that would have otherwise appeared in the resulting expression for
$|\boldsymbol{\cal P}^{(2)}_{2}|^{2}$ is thus incorporated in the coefficient
$\kappa_{0}$ in equation (9).

For certain values of $\theta_{P}$, denoted by $\theta_{P2S}$, the function
$f_{2C}(\tau)$ has an inflection point (see Ardavan 2021, Section 4.4, and
Ardavan 2023c, Section 2). For any given inclination angle $\alpha$, the
position $\tau_{2S}$ of this inflection point and the colatitude
$\theta_{P2S}$ of the observation points for which $f_{2C}(\tau)$ has an
inflection point follow from the solutions to the simultaneous equations
$\partial f_{2C}/\partial\tau=0$ and $\partial^{2}f_{2C}/\partial\tau^{2}=0$.
(Explicit expressions for the derivatives that appear in these equations can
be found in Appendix A of Ardavan 2021.) For values of $\theta_{P}$
sufficiently close to $\theta_{P2S}$, the separation between the maximum
$\tau=\tau_{2{\rm max}}$ and minimum $\tau=\tau_{2{\rm min}}$ of $f_{2C}$ and
hence the value of $\sigma_{21}$ are small. In other words, the focused
radiation beam is centred on the colatitude $\theta_{P2S}$ plotted in Fig. 11
(Ardavan, 2021, Section 5.5). Since the fit parameter $\sigma_{0}$, which
denotes the maximum value assumed by $\sigma_{21}$, lies between $10^{-2}$ and
$10^{-4}$ for most of the examples considered in Section 3, we have chosen the
separation between $\tau=\tau_{2{\rm max}}$ and $\tau=\tau_{2{\rm min}}$ in
each example such that the value of $\sigma_{21}$ in equation (17) is of the
order of $10^{-3}$.

Equations (5), (3), and (9)–(11) jointly yield

$\kappa=1.59\times
10^{16}\sigma_{0}^{3}{\hat{B}}_{0}^{2}d^{4}D^{-2}{\hat{P}}^{-1}{\tilde{\kappa}}_{\rm
th}\qquad{\rm Jy},$ (20)

with

$\displaystyle{\tilde{\kappa}}_{\rm th}$ $\displaystyle=$ $\displaystyle
w_{1}^{2}w_{3}^{2}b^{2}\left|{\tilde{\bf
P}}_{2}\right|^{2}\sigma_{21}\left(\frac{1}{\partial^{2}f_{2C}/\partial\tau^{2}|_{\tau=\tau_{2{\rm
min}}}}\right.$ (21)
$\displaystyle\left.-\frac{1}{\partial^{2}f_{2C}/\partial\tau^{2}|_{\tau=\tau_{2{\rm
max}}}}\right),$

when $j=0$, and

$\kappa=6.76\times
10^{15}\sigma_{0}^{6}{\hat{B}}_{0}^{2}d^{4}D^{-2}{\hat{P}}^{-1}{\hat{\kappa}}_{\rm
th}\qquad{\rm Jy},$ (22)

with

${\hat{\kappa}}_{\rm th}=a^{-3}{\tilde{\kappa}}_{\rm th},$ (23)

when $j=2$. Equations (20) and (22) can be written as

${\tilde{\kappa}}_{\rm th}={\tilde{\kappa}}_{\rm obs}\quad{\rm
and}\quad{\hat{\kappa}}_{\rm th}={\hat{\kappa}}_{\rm obs},$ (24)

respectively, where

${\tilde{\kappa}}_{\rm obs}=6.27\times
10^{-17}\sigma_{0}^{-3}({\hat{B}}_{0}d^{2})^{-2}D^{2}{\hat{P}}\kappa$ (25)

and

${\hat{\kappa}}_{\rm obs}=1.48\times
10^{-16}\sigma_{0}^{-6}({\hat{B}}_{0}d^{2})^{-2}D^{2}{\hat{P}}\kappa.$ (26)

While ${\tilde{\kappa}}_{\rm obs}$ and ${\hat{\kappa}}_{\rm obs}$ only contain
the observed parameters of the pulsar and its emission, the values of
${\tilde{\kappa}}_{\rm th}$ and ${\hat{\kappa}}_{\rm th}$ are determined by
the physical characteristics of the magnetospheric current sheet that acts as
the source of the observed emission.

For a given value of $\sigma_{21}$, the right-hand sides of equations (21) and
(23) are functions of the inclination angle $\alpha$ and the observer’s
distance ${\hat{R}}_{P}$ only (see Fig. 12). The values of the fit parameters
together with the relations in equation (24) and the plots of
${\tilde{\kappa}}_{\rm th}$ and ${\hat{\kappa}}_{\rm th}$ in Fig. 12 thus
enable us to connect the parameters of the fitted spectra to the physical
characteristics of their sources.

Figure 11: The relationship between the colatitude $\theta_{P2S}$ on which the
focused radiation beam is centred and the star’s inclination angle $\alpha$.

Figure 12: Plots of the dependences of the functions ${\hat{\kappa}}_{\rm th}$
and ${\tilde{\kappa}}_{\rm th}$, defined in equations (21) and (23), on the
inclination angle $\alpha$ for $\sigma_{21}\simeq 10^{-3}$ and an observation
point in the far zone. The red dots and the blue dots depict the values
assumed by ${\hat{\kappa}}_{\rm th}$ and ${\tilde{\kappa}}_{\rm th}$,
respectively.

###  4.2 Application of the connecting relations to the fitted spectra

In this section, we use the relations derived in Section 4.1, the values of
the fit parameters given in the captions to Figs. 1–10 and the data listed in
the ATNF Pulsar Catalogue (Manchester et al., 2005) to determine (or set
limits on) certain attributes of the central neutron stars of the pulsars
considered in Section 3 and their magnetospheres.

Once the values of the fit parameters $\sigma_{0}$ and $\kappa$ given in the
caption to Fig. 1, the period ($0.405$ s) and the distance ($7$ kpc) of the
pulsar J1915+1009 are inserted in equation (26), the resulting value of
${\hat{\kappa}}_{\rm obs}$ and the second member of equation (24) yield

${\hat{B}}_{0}d^{2}=1.13\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J1915+1009.$ (27)

If the central neutron star of this pulsar has the radius $10^{6}$ cm (i.e.
$d=1$) and the magnetic field $2.51\times 10^{12}$ Gauss at its magnetic pole
(i.e. ${\hat{B}}_{0}=2.51$) as predicted by the formula for magnetic dipole
radiation, then equation (27) and the curve delineated by the red dots in Fig.
12 imply that the value of the angle $\alpha$ between the rotation and
magnetic axes of J1915+1009 is either $3.8^{\circ}$ or $42.3^{\circ}$. (Here,
and in other similar cases, a choice between the two possible values of
$\alpha$ can be made by comparing the pulse profile of the pulsar in question
with the theoretically predicted ones presented in Section 5.1 of Ardavan
2021.) Depending on whether $\alpha=3.8^{\circ}$ or $42.3^{\circ}$, the
direction along which the radiation is observed forms the angles
$\theta_{P2S}=3.8^{\circ}$ or $35.1^{\circ}$ with the spin axis of this pulsar
(see Fig. 11).

Within the framework of the present emission mechanism, the value of
${\hat{B}}_{0}$ can be significantly different from that given by the formula
for magnetic dipole radiation, in which case equation (27) and Fig. 12 merely
determine the required value of ${\hat{B}}_{0}d^{2}$ as a function of
$\alpha$.

In the same way, the values of $\sigma_{0}$ and $\kappa$ in the caption to
Fig. 2 together with the period $0.361$ s and the distance $1.26$ kpc yield

${\hat{B}}_{0}d^{2}=0.127\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J0206-4028.$ (28)

If $d$ is set equal to $1$ and the value of ${\hat{B}}_{0}$ is assumed to be
that given by the formula for magnetic dipole radiation, i.e. $0.879$, then
equation (28) and the red curve in Fig. 12 imply that $\alpha$ equals either
$0.29^{\circ}$ or $62.8^{\circ}$. According to Fig. 11, the values of
$\theta_{P2S}$ corresponding to $\alpha=0.29^{\circ}$ and $62.8^{\circ}$ are
$0.29^{\circ}$ and $97.2^{\circ}$, respectively.

In the case of the broken-power-law example in Fig. 3, the fit parameters
$\sigma_{0}=2.81\times 10^{-3}$ and $\kappa=2.81\times 10^{-2}$ mJy together
with the period $5.16\times 10^{-3}$ s and the distance $1.22$ kpc yield

${\hat{B}}_{0}d^{2}=1.02\times 10^{-3}\,{\hat{\kappa}}_{\rm
th}^{-1/2}\quad{\rm for}\quad J1024-0719.$ (29)

Since ${\hat{\kappa}}_{\rm th}^{-1/2}\geq 1.42$ (see Fig. 12), equation (29)
implies that either $d\geq 2.15$ if ${\hat{B}}_{0}$ is given by its magnetic-
dipole-radiation value $3.13\times 10^{-4}$ or ${\hat{B}}_{0}\geq 1.45\times
10^{-3}$ if $d=1$.

In contrast, the fit parameters for the broken-power-law example in Fig. 4
together with the period $0.549$ s and the distance $0.4$ kpc yield

${\hat{B}}_{0}d^{2}=0.38\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J0452-1759,$ (30)

an equation that is satisfied by the magnetic-dipole-radiation value of
${\hat{B}}_{0}$ (i.e. $1.8$) and $d=1$ if $\alpha=57.7^{\circ}$ and so
$\theta_{P2S}=84.6^{\circ}$.

The values of the fit parameters $\sigma_{0}$ and $\kappa$ used for plotting
Fig. 5 together with the period $0.715$ s and the distance $1.67$ kpc yield

${\hat{B}}_{0}d^{2}=25.8\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J0332+5434.$ (31)

The constraint ${\hat{\kappa}}_{\rm th}^{-1/2}\geq 1.42$ (see Fig. 12)
therefore implies that either $d\geq 5.58$, if ${\hat{B}}_{0}$ has its
magnetic-dipole-radiation value $1.22$, or ${\hat{B}}_{0}\geq 36.6$, if $d=1$.

Next, the values of $\sigma_{0}$ and $\kappa$ in the caption to Fig. 6
together with the period $0.105$ s and the distance $7.11$ kpc yield

${\hat{B}}_{0}d^{2}=0.207\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J1845-0743.$ (32)

In this case, too, the constraint ${\hat{\kappa}}_{\rm th}^{-1/2}\geq 1.42$
(see Fig. 12) implies that either $d\geq 1.22$, if ${\hat{B}}_{0}$ has its
magnetic-dipole-radiation value $0.198$, or ${\hat{B}}_{0}\geq 0.294$, if
$d=1$.

Unlike the examples in Figs. 1–6 for which the value of $j$ in equation
(LABEL:E4) equals $2$, the example in Fig. 7 is fitted with a flux density for
which $j=0$. The first member of equation (24), the values of the fit
parameters $\sigma_{0}$ and $\kappa$ in the caption to Fig. 7 and the period
$0.307$ s and the distance $5.92$ kpc jointly yield

${\hat{B}}_{0}d^{2}=4.03\times 10^{-6}\,{\tilde{\kappa}}_{\rm
th}^{-1/2}\quad{\rm for}\quad J1829-1751.$ (33)

The resulting value of ${\tilde{\kappa}}_{\rm th}^{-1/2}$ for
${\hat{B}}_{0}=1.32$ and $d=1$, i.e. $3.28\times 10^{5}$, implies that
$\alpha\simeq 90^{\circ}$ in this case (see the curve delineated by the blue
dots in Fig. 12).

In the case of the example shown in Fig. 8, too, $j$ equals zero so that the
corresponding values of the fit parameters $\sigma_{0}$ and $\kappa$ together
with the period $0.306$ s and the distance $5.03$ kpc yield

${\hat{B}}_{0}d^{2}=1.58\times 10^{-6}\,{\tilde{\kappa}}_{\rm
th}^{-1/2}\quad{\rm for}\quad J1835-0643.$ (34)

If ${\hat{B}}_{0}$ has the value $3.56$ that is obtained from the formula for
magnetic dipole radiation, then ${\tilde{\kappa}}_{\rm th}^{-1/2}=2.25\times
10^{6}$ for $d=1$ and so $\alpha\simeq 90^{\circ}$ according to Fig. 12.

For the double-turn-over spectrum shown in Fig. 9, $j$ equals $2$ and the
second member of equation (24) in conjunction with the values of the fit
parameters $\sigma_{0}$ and $\kappa$, the period $0.277$ s and the distance
$0.31$ kpc yield

${\hat{B}}_{0}d^{2}=0.198\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm for}\quad
J1932+1059.$ (35)

If $d$ is set equal to $1$ and the value of ${\hat{B}}_{0}$ is assumed to be
that given by the formula for magnetic dipole radiation, i.e. $0.518$, then
equation (35) and the red curve in Fig. 12 imply that $\alpha$ equals
$46.5^{\circ}$. Moreover, the colatitude along which this pulsar is observed
has the value $\theta_{P2S}=60.8^{\circ}$ according to Fig. 11.

Finally, the values of the fit parameters in the caption to Fig. 10, the
period $1.22$ s and the distance $2.3$ kpc yield

${\hat{B}}_{0}d^{2}=1.78\times 10^{2}\,{\hat{\kappa}}_{\rm th}^{-1/2}\quad{\rm
for}\quad J0141+6009.$ (36)

Given the constraint ${\hat{\kappa}}_{\rm th}^{-1/2}\geq 1.42$ (see Fig. 12),
this implies that either $d\geq 19.0$, if ${\hat{B}}_{0}$ has its magnetic-
dipole-radiation value $0.70$, or ${\hat{B}}_{0}\geq 2.53\times 10^{2}$, if
$d=1$.

##  5 Concluding remarks

No emission mechanism is as yet identified in the published literature on
pulsars whose spectral distribution function can fit the data on all five
categories of spectral shapes depicted in Figs. 1–10 (see Jankowski et al.,
2018; Swainston et al., 2022, and the references therein). Curved or
gigahertz-peaked spectra are generally thought to reflect the free-free
absorption of the pulsar radiation in ionised high-density environments rather
than being intrinsic to the emission mechanism (see Rajwade et al., 2016;
Kijak et al., 2021, and the references therein). As we have seen, however, the
spectral distribution function of the caustics that are generated by the
superlluminally moving current sheet in the magnetosphere of a non-aligned
neutron star single-handedly accounts for all observed features of pulsar
spectra (including those that are normally fitted with simple or broken power
laws).

A study of the characteristics of the radiation that is generated by this
superluminally moving current sheet has already provided an all-encompassing
explanation for the salient features of the radiation received from pulsars:
its brightness temperature, polarization, spectrum, profile with
microstructure and with a phase lag between the radio and gamma-ray peaks
(Ardavan, 2021, 2022) and the discrepancy between the energetic requirements
of its radio and gamma-ray components (Ardavan, 2023b). Fits to the
exceptionally broad gamma-ray spectra of the Crab, Vela and Geminga pulsars,
for example, are provided by the spectral energy distribution of this
radiation over the entire range of photon energies so far detected from them
(Ardavan, 2023c, a).

Detailed analyses of the structure of the magnetospheric current sheet and the
coherent emission mechanism by which this sheet creates the caustics
underlying the present spectral distribution function can be found in Ardavan
(2021). A heuristic account of the mathematical results of those analyses in
more transparent physical terms is presented in Ardavan (2022) and Ardavan
(2023c, Section 2).

Finally, the following cautionary remark concerning a common misconception is
in order: it is often presumed that the plasma equations used in the numerical
simulations of the magnetospheric structure of an oblique rotator should, at
the same time, predict any radiation that the resulting structure would be
capable of emitting (see, e.g. Spitkovsky, 2006; Kalapotharakos et al., 2012).
This presumption stems from disregarding the role of boundary conditions in
the solution of Maxwell’s equations. As we have already pointed out, the far-
field boundary conditions with which the structure of the pulsar magnetosphere
is computed are radically different from the corresponding boundary conditions
with which the retarded solution of these equations (i.e. the solution
describing the radiation from the charges and currents in the pulsar
magnetosphere) is derived (see Section 3 and the last paragraph in Section 6
of Ardavan 2021).

## Acknowledgements

I thank N. A. Swainston for helpful correspondence.

## Data availability

The data used in this paper are available in the public domain.

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