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redpajama_set_name: RedPajamaCommonCrawl


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\section{Introduction} \label{sec:intro}

Ultraluminous X-ray sources (ULXs) are some of the brightest known  X-ray sources ($\rm L_x > 10^{39} \ensuremath{\mathrm{erg}\,\mathrm{s}^{-1}}$). Their luminosities exceed the classical Eddington limit of a $10$ \ensuremath{M_{\odot}} black hole (see \citealt{Kaaret2017} for a recent review). In addition, many  ULXs display a unique spectral curvature at energies $\simeq 10$~keV as shown  by broadband X-ray data (e.g., \citealt{Bachetti2013, Walton2013}). This spectral feature is one of the distinctive characteristics of ULXs compared to the hard state of Galactic X-ray binaries (XRBs) and active galactic nuclei (AGNs). The discovery of a neutron star ULX \citep{Bachetti2014} changed the perception of these sources, and a foremost conjecture is that a large fraction of the ULX population is neutron stars (e.g., \citealt{King2016, King2017, King2020}). Many theoretical models have been studied to explain emission mechanisms that generate such high luminosity from neutron stars (e.g., \citealt{Mushtukov2017, Mushtukov2018, Mushtukov2019}). The origin of spectral cutoff, however, has no compelling theoretical model. Recent observational studies provide phenomenological models which invoke physical scenarios like compton scattering in the coronal region in low magnetic sources like black holes or compton scattering in the accretion column in highly magnetized neutron stars (see \citealt{Walton2018apr, Walton2020} and references therein). In this paper, we propose an alternative model based on synchrotron radiation from different latitudes to explain the observed spectral cutoff. We explore the possible physical scenarios of this phenomenon in the context of ULXs and estimate physical parameters related to both the luminosity and the spectral cutoff in ULXs.

Synchrotron radiation is one of the most prevalent radiative processes in astrophysics \citep{Ribicky}. While the non-relativistic synchrotron radiation, the cyclotron radiation, provides a discrete spectrum, the relativistic radiation regime yields near continuum emission owing to the higher harmonics contributing more predominantly to the observed spectrum (see \citealt{Landau} for a review). The astrophysical implications of synchrotron radiation are well studied in multiple wavelengths, including soft to hard X-rays (e.g., \citealt{Longair, Heinz2004, Maccarone2005, Markoff2005, Kisaka, Kisaka2017, Reigler}).

In the next section, we briefly review the physics of synchrotron radiation relevant to our work. We also provide
approximate analytical expressions that allow one to study the emission from semi-relativistic to highly relativistic electrons for a range of latitudes. In \S~3, we provide details of the data we use and its pre-processing. The main results are presented in \S~4. In \S~5, we discuss how our proposed method can be distinguished from other models.

\section{Synchrotron radiation and approximations}\label{sec:Theory}
We assume a geometric construct in which the incoherent synchrotron radiation originates close to the surface of a neutron star. As the length scale of magnetic fields is much larger than the curvature of the gravitating body, we can assume that the magnetic field lines are straight on scales from which the observed synchrotron emission occurs. Without loss of generality, we assume that the magnetic field is in the -z direction of cartesian geometry and the charged particles move in a circular motion around the uniform magnetic field lines in the xy plane.

The angular distribution of the radiated power in $n$th harmonic (or an angular frequency of observation, $\omega$) for a single electron   ($\rm erg \, sec^{-1}$)  per unit solid angle ($d\Omega$) can be expressed as \citep{Landau}:
\begin{equation}
  dI_n = \frac{e^2\omega^2}{2\pi c} \left[\tan^2\theta J^2_n(n\beta\cos\theta)  +\beta^2 J'^2_n(n\beta\cos\theta) \right] {d\Omega}
  \label{eq:synrad_ang}
\end{equation}
Here $\beta = v/c$,   $B$ is the magnetic field strength, and $\theta$ is the angle between radiated emission and the particle's orbital plane. $J_n(x)$ is the Bessel function  and $J_n'(x)$  its derivative. The integer  $n$ denotes the discrete energy levels of electron's energy with  $\omega = n \omega_B$, where $\omega_B = eB/\gamma m c$, $\gamma$ being the relativistic boost $1/\sqrt{1-\beta^2}$.

Eq.~(\ref{eq:synrad_ang}) gives the synchrotron  spectrum for a single electron of energy $E = m_e c^2 \gamma$. We consider a range of electron energies and model the electron energy distribution using an exponential cutoff power law $f(\gamma) = N\gamma^{-p} e^{-\frac{\gamma}{\gamma_{\rm max}}}$ in the range $\gamma_{\rm min}$ and $\gamma_{\rm max}$ (e.g., \citealt{Reynolds1999}). $N$ gives the overall normalization. For our work, we treat $\gamma_{\rm min}$ as a free parameter and $\gamma_{\rm max} = 1000 \gamma_{\rm min}$. We use the energy spectral index $p =2.2$, which is consistent with the shock acceleration mechanism (e.g., \citealt{Allen2001}). For this case, if $\gamma_{\rm max}$ is larger than $\gamma_{\rm min}$ by more than a few factors of 10, its impact on our results is found to be negligible.   The factors needed for conversion to flux units for comparison with the data are absorbed in the definition of $N$: $N = \rho_N V /D^2$, where $\rho_N$ is the number density
of relativistic electrons, $V$ is the volume of the emitting region, and $D$ is the luminosity distance to the source.

Eq.~(\ref{eq:synrad_ang}) allows one to analyze the transition from cyclotron to synchrotron radiation. For large $\gamma$, the emission is dominated by large $n$ and is restricted to an angle $\theta \simeq 1/\gamma$ centered on the plane of the orbit. For intermediate
$\gamma$ or semi-relativistic electrons ($\gamma \lesssim 10$), it is possible to have substantial emissions from higher
latitudes.
In this paper, we explore the possibility that the observed radiation could emanate from high latitudes with respect to the plane of the orbit. In this case,  $\beta' = \beta \cos\theta$ acts as the effective velocity parameter in  Eq.~(\ref{eq:synrad_ang}) and determines the cut-off frequency, the frequency at which the synchrotron emission begins to fall exponentially; this behavior is seen in Figure ~\ref{fig:diff_angle_spectrum}.

While analytic expressions can approximate the angle-averaged emission for $\beta \simeq 1$ (e.g., \citealt{Schwinger}), such approximations are not valid here as the relevant parameter for us is $\beta \cos\theta$, which can deviate significantly from unity for large angles. However, it is possible to approximate the Bessel function and its derivative in Eq.~(\ref{eq:synrad_ang}) using the stationary phase approximation (e.g.,  \citealt{Schwinger} employs in the angle-averaged case). This allows us to obtain the following approximate expressions for the Bessel function and its derivative:

\begin{align}
  J_n(n\beta') & \simeq  0.447 n^{-1/3} \beta'^{-1/3} \> \> {\rm for}\> \> n \le n_c \nonumber \\
  J_n(n\beta') & \simeq 0.335 n^{-1/2} (1-\beta')^{-1/4}\beta'^{-1/4} \nonumber \\
  &\times \exp\left(-2\frac{\sqrt{2}}{3} n \beta'^{-1/2}(1-\beta')^{3/2} \right ) \> \> {\rm for}\> \> n \ge n_c \nonumber \\
   J'_n(n\beta') & \simeq  0.411 n^{-2/3} \beta'^{-2/3} \> \> {\rm for}\> \> n \le n_c \nonumber \\
   J'_n(n\beta') & \simeq 0.474 n^{-1/2} (1-\beta')^{1/4}\beta'^{-3/4}  \nonumber \\
   &\times \exp\left (-2 \frac{\sqrt{2}}{3} n \beta'^{-1/2}(1-\beta')^{3/2} \right ) \> \> {\rm for}\> \> n \ge n_c
   \label{eq:bess_appro}
\end{align}

Here $n_c \simeq \frac{\beta'^{1/2}}{(1-\beta')^{3/2}}$.  $n_c \ge 1$, hence, $0.3 \lesssim \beta \lesssim 1$ and $0.3 \lesssim \cos\theta \lesssim 1$ \footnote{\url{https://www.wolfram.com/mathematica/}}. For fitting to X-ray continuum spectral data, we require $n_c \gg 1$, which constrains the latitude $\theta \lesssim 70^{\circ}$. The exponential terms in Eq.~(\ref{eq:bess_appro}) adequately capture the cut-off frequency in the entire parameter range of interest, which is key to modeling the ULX cut-off frequency. Also, we note that for
$\beta' \simeq 1$ ($\beta \simeq 1$ and $\theta \simeq 0$), $1/(1-\beta') \simeq 2\gamma^2$, which yields the expression derived in
Schwinger for the extreme relativistic case.  While we compute Bessel functions numerically for data analysis, these analytic expressions help us interpret our results.

In Figure~\ref{fig:diff_angle_spectrum}, we display synchrotron spectra for emission from different latitudes. Eq.~(\ref{eq:bess_appro}) allows us to understand the spectral shapes seen in the figure. The spectral cut-off occurs
at an angular frequency $\omega \simeq n_c \omega_B$. For emission close to the plane of the rotation ($\theta \simeq 0$), $n_c \simeq \gamma^3$. However, for larger angles $n_c < \gamma^3$, as Eq.~(\ref{eq:bess_appro}) shows, and the spectral cut-off shifts to smaller frequencies. As we discuss later, the spectral cut-off in the data we analyze occurs at $E \simeq 10~\, \rm keV$, which is possible for a range of $\gamma$, $B$, and $\theta$ as will be discussed below in more detail. Even though we only assume electron motion
in the plane perpendicular to the magnetic field, our results do not qualitatively change if the electron has a $z$-component of velocity. This case can be incorporated into our analysis by altering $B$ to $B_\perp \equiv B \cos\chi$, where $\chi$ is the angle between the velocity vector and the magnetic field (e.g., \citealt{Landau}).

\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{angular_distribution_of_power.pdf}
    \caption{For $B=5 \times 10^{11}$ Gauss and $\gamma=10$, the synchrotron spectra are shown as a function of multipoles  (Eq.~(\ref{eq:synrad_ang}))  for different latitudes. The figure shows the role of high-latitude emission  in  introducing spectral curvature.}
    \label{fig:diff_angle_spectrum}
\end{figure}

\section{Data} \label{sec:data}

 We utilize the \textit{NuSTAR}\ observations of two bright pulsar ULXs, NGC 5907 ULX1 (RA:15 15 58.62, DEC: +56 18 10.3; \citealt{Israel2017feb,Walton2015,Furst2017}) and NGC 7793 P13 (RA: 23 57 50.9, DEC: -32 37 26.6; \citealt{Furst2016,Israel2017mar,Walton2018feb}) to compare  our  theoretical model against  data. The distances to the host galaxies are $\simeq 17.1$ Mpc (e.g.,  \citealt{Furst2017}) and $\simeq 3.5$~Mpc (e.g., \citealt{Walton2018feb}),  respectively. These two sources were observed by \textit{NuSTAR}\ several times in the past decade, which provide us an opportunity to verify the consistency of the theory over long-term spectral evolution of the sources. In figure \ref{fig:data_spectra}, we have plotted the spectra from the \textit{NuSTAR}\ observations for both the sources. In particular, for NGC 7793 P13, we detect two distinct flux states. The choice of the instrument is motivated by its energy coverage, which allows us to model the spectral curvature of the source. The broadband spectra of ULXs are generally fitted with multiple components: neutral absorption, accretion disk (geometrically thin or slim), and a phenomenological model of either magnetic or non-magnetic Comptonization processes (see e.g., \citealt{Kaaret2017}). The thermal disk component and neutral absorption mostly play a role in the soft energy regime ($E \lesssim 5 \, \rm keV$). Our aim in this paper is to explain the spectral cutoff in ULXs, which occurs in a higher energy range ($E \simeq \rm 10 keV$). To minimize contamination from soft components and to adequately model the spectral break, we study  the energy  range $\simeq 5\hbox{--}25$~keV in this paper.  In our study, we consider all the available \textit{NuSTAR}\ data sets for both the sources. However, for NGC 5907 ULX1, there are a few observations for which  the  signal-to-noise ratio (S/N) is poor owing to the faintness of the source. We do not utilize these data for our analysis.

\subsection{Data reduction process}\label{subsec:data_reduction}
The \textit{NuSTAR}\ data are extracted using the \texttt{HEASOFT} version 6.29 \footnote{\url{https://heasarc.gsfc.nasa.gov/docs/software/heasoft/}}. We use \texttt{nupipeline} tool to extract cleaned products and \texttt{nuproducts} tool to extract the source and background spectra and the response files from both FPMA and FPMB modules. In general we follow the method outline in previous works (e.g., \citealt{Israel2017feb,Walton2015,Furst2017,Furst2016,Israel2017mar,Walton2018feb,Lin2022}) for data reduction of these two sources. We choose the source photon extraction region as $50$ arcsec radius circle for both sources. The background regions are selected as $100$ arcsec radius circle in all the cases. The number of counts per energy bin for grouping the spectra are a minimum of $30$ counts per energy bin for all NGC 5907 ULX1 spectra and $50$ counts per bin for NGC 7793 P13 spectra where the source is in high flux state, and $20$ counts per energy bin for low flux state of the source (see figure \ref{fig:data_spectra}).

After we obtain the spectra, we use \texttt{XSPEC} \citep{XSPEC} spectral analysis package to convert the spectra into flux units for further analysis. \textit{NuSTAR}\ spectra beyond $\sim 25$~keV are dominated by the background for both the sources, and therefore spectral data above this energy are not utilized in our analysis. In some low flux state observations, the background starts dominating well below $\sim 20$ keV; however, to provide similar treatment to all observations, we take spectra up to $\sim 25$ keV for all cases. We fit the \textit{NuSTAR}\ spectra with a cutoff power-law model (in \texttt{XSPEC} the syntax is \texttt{constant*cutoffpl}). The \texttt{constant} model represents the instrumental cross-calibration differences, and \texttt{cutoffpl} is the continuum representing an exponentially cutoff power-law spectrum. This model for $5.0\hbox{--}25.0$~keV spectra give statistically good fit for both the sources. For the cutoffpl model, we fix the index to $0.59$, a typical value for ULX pulsars (see e.g., \citealt{Walton2020}). We do not consider neutral absorption, since it plays a role  only in softer regime of the spectra. We then convert the spectral counts into flux $\rm \nu F_\nu$ (ergs/cm$^2$/sec) by the \texttt{eeufspec} command and take the data points in the energy range $5.0\hbox{--}25.0$~keV ($\simeq 1.2\hbox{--}6.0 \times 10^{18}$~Hz) to perform further analysis described in the \S~\ref{sec:Results}. We have further verified the robustness of this data extraction procedure with another model such as a simple \texttt{powerlaw} of photon index $0$, instead of the \texttt{cutoffpl} model, and extract the spectra in flux unit using \texttt{eeufspec}. We find that our results are insensitive to the  choice of \texttt{XSPEC} models used to generate the flux data points.

\begin{figure*}
    \centering
    \hspace{-2cm}
    \includegraphics[width=0.55\linewidth]{NGC5907_NuSTAR_observations_spectra.pdf}
    \hspace{-1cm}
    \includegraphics[width=0.55\linewidth]{NGC7793_NuSTAR_observations_spectra.pdf}
    \hspace{-2cm}
    \caption{The spectra of NGC 5907 ULX1 (left) and NGC 7793 P13 (right) are displayed for  multiple \textit{NuSTAR}\ observations.}
    \label{fig:data_spectra}
\end{figure*}

\section{Analysis and Results} \label{sec:Results}

From Eq.~\ref{eq:synrad_ang}, one can verify that $\gamma$ and $B$ are degenerate with each other if $\beta \simeq 1$. As we wish to explore a range of electron speeds from semi-relativistic to ultra-relativistic, $\beta \simeq 1$ is a reasonable approximation in the equation. {\footnote{$\beta \simeq 1$ approximation is appropriate for ultra-relativistic case or semi-relativistic case in higher latitudes.}} Thus we choose $\gamma_{\rm min}/B$ as one of the parameters in the analysis. We use three parameters---$\gamma_{\rm min}/B$, $\theta$ and $N$---in our analysis and later fix one more parameter to deal with residual degeneracies. We also convert the model to $\rm \nu F_\nu$ unit by multiplying Eq.~\ref{eq:synrad_ang} by $n = \omega/\omega_B$, where $\omega = 2\pi\nu$.

To explore the congruence of the data and the model, we choose two different statistical methods-- frequentist approach and Bayesian analysis. Given that it is hard to determine the best-fit and the errors on the three parameters simultaneously, we fix the angle $\theta$ and keep the other two parameters $\gamma_{\rm min}/B=x_{\rm min}$ and $N$ free to vary. For the frequentist approach, we first carry out a minimum $\chi^2$ analysis using the \texttt{scipy} \citep{2020SciPy-NMeth} ``curve\_fit" tool \footnote{\url{https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html}}. The best fit parameters, $\chi^2$ values, and 1$\sigma$ errors (computed using covariance matrix) are given in Table~\ref{tab:parametersNGC5907} and ~\ref{tab:parametersNGC7793}. For each data set, we consider three values of $\theta = 1^{\circ}, 15^{\circ}, 30^{\circ}$ for our analysis.  As discussed in the foregoing, this choice is based on the acceptable range of latitudes to ensure $n_c \gg 1$. On the Bayesian approach method, we utilize the Markov chain Monte Carlo (MCMC) analysis with \texttt{python} \texttt{emcee} package \footnote{\url{https://emcee.readthedocs.io/en/stable/}} \citep{emcee} by maximum likelihood technique. Here, we present the result of MCMC analysis for a sub-set of data. The MCMC analysis is done with $32$ random walkers and $500$ burn-in and $5000$ post burn-in iterations to reach convergence. We use combination of two moves, namely, \texttt{moves.DEMove} and \texttt{moves.DESnookerMove} with $80\%$ and $20\%$ probability, respectively, at each step. The marginalized posterior probabilities of estimated parameters for one observation of each source are shown in Figure~\ref{fig:posteriors}.

We next discuss the physical implications of the parameter ranges we find in our analysis. The estimated range of  $x_{\rm min}$ encompasses a large range of $\gamma_{\rm min}$, e.g. highly relativistic particle,  $\gamma_{\rm min}\simeq 10^5$, for emission close the plane or semi-relativistic electrons $\gamma_{\rm min} \simeq 10$ for higher latitude emission, with typical NS magnetic field, $B \simeq 10^{12}\, \rm G$ (e.g., \citealt{Caballero2012, Petri2016}). The overall normalization $N = \rho_N V/D^2$ is highly uncertain as both the relativistic electron density $\rho_N$ and the volume of emission region $V$ are very poorly determined even theoretically. Typically, in neutron star magnetosphere, the minimum limit of plasma density is given by the Goldreich-Julian limit \citep{Goldreich} which depends on pulsar spin period, magnetic field strength, and alignment of spinning axis with magnetic field line. Depending on the volume of emission region, we find that the estimated number density is comparable to the Goldreich-Julian limit \citep{Goldreich} for a $1$ sec spinning pulsar with $B \simeq 10^{12}$~G. This means our results are consistent with this theoretical expectation. We also determine  that, for a range of acceptable parameters, the emitting region is  optically thin to synchrotron self-absorption and compton scattering.

If we restrict the maximum limit of magnetic field on the NS surface to the Schwinger limit of $4.4 \times 10^{13}$~G, then the maximum value of $\gamma_{\rm min}$ can be estimated. For lower $x_{\rm min}$ (i.e., lower $\gamma_{\rm min}$ or higher $B$), we get emission at higher latitudes which requires lower value of $N$ to explain the observed ULX flux. On the other hand, when $x_{\rm min}$ is higher, we get emission closer to the plane of orbit and higher $N$ is required to generate such high flux in these sources. It would be possible in the future to constrain all the parameters adequately if we can have at least one parameter determined  from other analyses. Our results point to the possibility that the spectral curvature in ULXs might have a common origin and all the ULXs are possibly highly magnetized neutron stars. This theoretical model can also be employed to explain  the high energy cutoff in  X-ray binary pulsar sources.

\begin{deluxetable*}{CCCCC}
\tablenum{1}
\tablecaption{The best fit parameters and $\chi^2$ from Python curve fit \label{tab:parametersNGC5907} for 7 \textit{NuSTAR}\ observations of NGC 5907 ULX1 are listed. The errors are calculated for  1-$\sigma$ confidence from the covariance matrix using the parameter \texttt{absoulte\_sigma=True}.}
\tablewidth{0pt}
\tablehead{
\colhead{Observations}  & \colhead{$\theta^\circ$} & \colhead{$x_{\rm min} (G^{-1})$} &
 \colhead{$N (cm^{-2})$} & \colhead{$\chi^2$/d.o.f} \\
}
\startdata
30002039005 & 1 & (9.54 \pm 0.56) \times 10^{-7} & (1.10 \pm 0.18) \times 10^{-17} & 57/49\\
 & 15 & (2.81 \pm 0.16) \times 10^{-10} & (6.99 \pm 1.16) \times 10^{-19} & 57/49\\
 & 30 & (3.45 \pm 0.20) \times 10^{-11} & (2.97 \pm 0.49) \times 10^{-19} & 57/49\\
 \hline
 30302004006 & 1 & (6.38 \pm 0.36) \times 10^{-7} & (9.99 \pm 1.25) \times 10^{-18} & 66/52\\
  & 15 & (1.88 \pm 0.11) \times 10^{-10} & (6.35 \pm 0.79) \times 10^{-19} & 66/52\\
  & 30 & (2.31 \pm 0.13) \times 10^{-11} & (2.72 \pm 0.34) \times 10^{-19} & 66/52\\
 \hline
 30302004008 & 1 & (5.94 \pm 0.35) \times 10^{-7} & (8.10 \pm 1.01) \times 10^{-18} & 62/49\\
  & 15 & (1.75 \pm 0.10)\times 10^{-10} & (5.15 \pm 0.64) \times 10^{-19} & 62/49\\
  & 30 & (2.15 \pm 0.12)\times 10^{-11} & (2.21 \pm 0.27) \times 10^{-19} & 62/49\\
 \hline
 80001042002 & 1 & (7.69 \pm 0.27) \times 10^{-7} & (2.52 \pm 0.23) \times 10^{-17} & 80/78\\
  & 15 & (2.27 \pm 0.08) \times 10^{-10} & (1.60 \pm 0.14) \times 10^{-18} & 80/78\\
  & 30 & (2.78 \pm 0.10) \times 10^{-11} & (6.84 \pm 0.61) \times 10^{-19} & 80/78\\
 \hline
 80001042004 & 1 & (6.57 \pm 0.26) \times 10^{-7} & (1.69 \pm 0.15)\times 10^{-17} & 82/77\\
  & 15 & (1.94 \pm 0.08)\times 10^{-10} & (1.07 \pm 0.10) \times 10^{-18} & 82/77\\
  & 30 & (2.38 \pm 0.09)\times 10^{-11} & (4.58 \pm 0.41) \times 10^{-19} & 83/77\\
 \hline
 90501331002 & 1 & (5.97 \pm 0.46) \times 10^{-7} & (4.70 \pm 0.77) \times 10^{-18} & 34/39\\
  & 15 & (1.76 \pm 0.14) \times 10^{-10} & (2.99 \pm 0.49) \times 10^{-19} & 34/39\\
  & 30 & (2.16 \pm 0.16) \times 10^{-11} & (1.28 \pm 0.21) \times 10^{-19} & 34/39\\
 \hline
 90601323002 & 1 & (7.23 \pm 0.29)\times 10^{-7} & (1.66 \pm 0.16) \times 10^{-17} & 89/88\\
  & 15 & (2.13 \pm 0.09) \times 10^{-10} & (1.05 \pm 0.10) \times 10^{-18} & 90/88\\
  & 30 & (2.62 \pm 0.10) \times 10^{-11} & (4.50 \pm 0.43) \times 10^{-19} & 90/88\\
\enddata
\end{deluxetable*}

\begin{deluxetable*}{CCCCC}
\tablenum{2}
\tablecaption{The best fit parameters and $\chi^2$ from Python curve fit \label{tab:parametersNGC7793} for 10 \textit{NuSTAR}\ observations of NGC 7793 P13 are listed. The errors are calculated for  1-$\sigma$ confidence from the covariance matrix using the parameter \texttt{absoulte\_sigma=True}.}
\tablewidth{0pt}
\tablehead{
\colhead{Observations}  & \colhead{$\theta^\circ$} & \colhead{$x_{\rm min} (G^{-1})$} &
 \colhead{$N (cm^{-2})$} & \colhead{$\chi^2$/d.o.f} \\
}
\startdata
30302005002 & 1 & (7.66 \pm 0.20) \times 10^{-7} &  (3.07 \pm 0.20) \times 10^{-17} & 77/82\\
 & 15 & (2.26 \pm 0.06)\times 10^{-10} & (1.95 \pm 0.13) \times 10^{-18} & 77/82\\
 & 30 & (2.77 \pm 0.07)\times 10^{-11} & (8.30 \pm 0.55) \times 10^{-19} & 77/82\\
 \hline
 30302005004 & 1 & (7.46 \pm 0.15) \times 10^{-7} & (4.58 \pm 0.23) \times 10^{-17} & 111/117\\
  & 15 & (2.20 \pm 0.04) \times 10^{-10} & (2.90 \pm 0.15) \times 10^{-18} & 111/117\\
  & 30 & (2.70 \pm 0.05) \times 10^{-11} & (1.24 \pm 0.06) \times 10^{-18} & 111/117\\
 \hline
 30302015002 & 1 & (7.53 \pm 0.14) \times 10^{-7} & (5.95 \pm 0.27) \times 10^{-17} & 147/133\\
  & 15 & (2.22 \pm 0.04) \times 10^{-10} & (3.77 \pm 0.17) \times 10^{-18} & 147/133\\
  & 30 & (2.73 \pm 0.05) \times 10^{-11} & (1.61 \pm 0.07) \times 10^{-18} & 148/133\\
 \hline
 30302015004 & 1 & (7.53 \pm 0.17) \times 10^{-7} & (4.75 \pm 0.26) \times 10^{-17} & 130/123\\
  & 15 & (2.22 \pm 0.05) \times 10^{-10} & (3.02 \pm 0.17) \times 10^{-18} & 130/123\\
  & 30 & (2.72 \pm 0.06) \times 10^{-11} & (1.29 \pm 0.07) \times 10^{-18} & 130/123\\
 \hline
 30502019002 & 1 & (7.41 \pm 0.53) \times 10^{-7} & (6.52 \pm 1.14) \times 10^{-18} & 42/61\\
  & 15 & (2.18 \pm 0.16) \times 10^{-10} & (4.14 \pm 0.72) \times 10^{-19} & 42/61\\
  & 30 & (2.68 \pm 0.19) \times 10^{-11} & (1.77 \pm 0.30) \times 10^{-19} & 42/61\\
 \hline
 30502019004 & 1 & (6.93 \pm 0.62) \times 10^{-7} & (5.71 \pm 1.19) \times 10^{-18} & 32/42\\
  & 15 & (2.04 \pm 0.18) \times 10^{-10} & (3.62 \pm 0.75) \times 10^{-19} & 32/42\\
  & 30 & (2.51 \pm 0.22) \times 10^{-11} & (1.55 \pm 0.32) \times 10^{-19} & 32/42\\
 \hline
 50401003002 & 1 & (7.37 \pm 0.79) \times 10^{-7} & (5.91 \pm 1.53) \times 10^{-18} & 25/31\\
  & 15 & (2.17 \pm 0.23) \times 10^{-10} & (3.75 \pm 0.97) \times 10^{-19} & 25/31\\
  & 30 & (2.67 \pm 0.28) \times 10^{-11} & (1.60 \pm 0.41) \times 10^{-19} & 25/31\\
 \hline
 80201010002 & 1 & (7.26 \pm 0.10) \times 10^{-7} & (6.20 \pm 0.22) \times 10^{-17} & 156/204\\
  & 15 & (2.14 \pm 0.03) \times 10^{-10} & (3.94 \pm 0.14) \times 10^{-18} & 156/204\\
  & 30 & (2.63 \pm 0.04) \times 10^{-11} & (1.68 \pm 0.06) \times 10^{-18} & 156/204\\
 \hline
 90301326002 & 1 & (7.20 \pm 0.14) \times 10^{-7} & (7.11 \pm 0.33) \times 10^{-17} & 104/123\\
  & 15 & (2.12 \pm 0.04)\times 10^{-10} & (4.51 \pm 0.21) \times 10^{-18} & 104/123\\
  & 30 & (2.61 \pm 0.05)\times 10^{-11} & (1.93 \pm 0.09) \times 10^{-18} & 105/123\\
 \hline
 90601327002 & 1 & (9.65 \pm 1.01) \times 10^{-7} & (10.00 \pm 2.98) \times 10^{-18} & 35/34\\
  & 15 & (2.84 \pm 0.30) \times 10^{-10} & (6.33 \pm 1.89) \times 10^{-19} & 35/34\\
  & 30 & (3.49 \pm 0.36) \times 10^{-11} & (2.70 \pm 0.80) \times 10^{-19} & 35/34\\
\enddata
\end{deluxetable*}

\begin{figure*}
    \centering
    \hspace{-1.cm}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC5907_angle01.pdf}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC5907_angle15.pdf}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC5907_angle30.pdf}\\
    \hspace{-1.cm}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC7793_angle01.pdf}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC7793_angle15.pdf}
    \includegraphics[width=0.33\textwidth]{posteriors_two_params_NGC7793_angle30.pdf}
    \caption{Based on the MCMC analysis described in the text, we display corner plots  for two  parameters $N$ and $x_{\rm min}$ for emission angles:
    $1^{\circ}$,$15^{\circ}$, $30^{\circ}$ (from left to right). The top and bottom panels correspond to NGC 5907 ULX1 (Observation - 80001042002) and NGC 7793 P13 (Observation - 90301326002), respectively. The 0.16, 0.5 and 0.84 quantiles are shown as vertical lines in the figures. $50$ bins are used in the histograms which are smoothed with $3 \sigma$ Gaussian kernel.}
    \label{fig:posteriors}
\end{figure*}

\section{Discussion}\label{sec:discussion}
In this paper, we seek to explain the X-ray spectra of two known pulsar ULXs. In particular, we focus on the  cut-off in such spectra at $E \simeq 10$~keV, which is a generic feature of  many pulsar ULXs (see e.g., Figure~\ref{fig:data_spectra}).
We propose synchrotron radiation (at a range of latitudes) as the possible physical process to explain this observed spectral shape. Our main results are summarized in Table~\ref{tab:parametersNGC5907} and~\ref{tab:parametersNGC7793} and Figure~\ref{fig:posteriors}. We are able to explain the data
for an acceptable range of theoretical parameters.

Other  models that have been explored  to explain the dominant emission in hard  X-ray range in ULXs, invoke the comptonization from coronal region of a non-magnetic sources and comptonization from magnetized column in neutron stars (see e.g., \citealt{Walton2020} for a detailed discussion on these two scenarios). In principle, there could be two possible ways to distinguish our proposed scenario from these models.

Table~\ref{tab:parametersNGC5907} and ~\ref{tab:parametersNGC7793} show that the best-fit value $\gamma_{\rm min}/B \simeq 10^{-11}$ for $\theta \simeq 30^\circ$. This could correspond, for instance, to a semi-relativistic electron ($\gamma_{\rm min} \simeq 10$) along with a magnetic field $B\simeq 10^{12} \, \rm G$. In such cases, the fundamental mode of emission, $\nu_B = eB/(2\pi mc\gamma_{\rm min}) \simeq  1\, \rm keV$. As this is larger than the spectral resolution of \textit{NuSTAR}\ in energy range of interest, the observed spectrum could be a set of discrete cyclotron lines. In practice, such an interpretation could be difficult owing to mixing with larger $\gamma$ values and the width of spectral lines, which are  difficult to ascertain. This would also require either re-analysis of the data or new data which is beyond the scope of the paper. Our analysis raises the intriguing possibility that the discreteness of the spectrum could probe the latitude of the emission. As we have already discussed in \S~\ref{sec:Theory}, an  upper limit on the  latitude  of emission can be obtained by  requiring $n_c \gg 1$; this yields a stringent upper bound $\theta \simeq 70{^\circ}$. For fitting  continuum X-ray spectral data  in the energy range of interest,  this requirement    motivates the upper limit of   $\theta \simeq 30^{\circ}$ we use in this paper.

Another possible probe of our model could be the polarization of received photons. The photons emerging from higher latitudes would be elliptically polarized while those from closer to the plane of the orbit would be linearly polarized. While the non-magnetic comptonization will not show polarized emission, the magnetic comptonization and the synchrotron radiation could display different degrees of polarization. Modern X-ray polarimeters such as IXPE \citep{Ixpe} and upcoming mission POLIX \citep{Polix} might be able to address these questions.

\begin{acknowledgments}
We would like to thank Keith Arnaud from HEASARC helpdesk for his valuable suggestions regarding eeufspec tool in XSPEC.
\end{acknowledgments}

\vspace{5mm}
\facilities{\textit{NuSTAR} ; \citealt{NuSTAR} }

\software{HEASOFT (\url{https://heasarc.gsfc.nasa.gov/docs/software/heasoft/}; \citet{Heasoft2014}), Mathematica (\url{https://www.wolfram.com/mathematica/}; \citet{Mathematica})
          }

redpajama_set_name: RedPajamaArXiv


text:
Q: How to group and handle priority extraction in pandas Now I would like to handle dataframe
df
A B
1 A0
1 A1
1 B0
2 B1
2 B2
3 B3
3 A2
3 A3

First, I would like to group by df.A
sub1
A B
1 A0
1 A1
1 B0

Second, I would like to extract first rows which contains letter A
A B
1 A0

If there is no A
sub2
A B
2 B1
2 B2

I would like to extract the first rows
A B
2 B1

So, I would like to get the result below
A B
1 A0
2 B1
3 A2

I would like to handle priority extraction,I tried grouping but Couldnt figure out. How to handle this?

A: You can groupby column A and for each group use idxmax() on str.contains("A"), then if there is A in column B, it will get the first index which contains letter A, otherwise it falls back to the first row as all values are False:
df.groupby("A", as_index=False).apply(lambda g: g.loc[g.B.str.contains("A").idxmax()])

#   A   B
#0  1   A0
#1  2   B1
#2  3   A2

In cases where you may have duplicated index, you can use numpy.ndarray.argmax() with iloc which accepts integer as position indexing:
df.groupby("A", as_index=False).apply(lambda g: g.iloc[g.B.str.contains("A").values.argmax()])

#   A   B
#0  1   A0
#1  2   B1
#2  3   A2

redpajama_set_name: RedPajamaStackExchange


Maintains regular book of accounts of their receipts and expenses.
If there is business related income, there should be separate accounts maintained for it. The donations received by the NGO should not be used for such businesses.
12A and 80G Registration procedure typically need to be done by an NGO immediately after its incorporation.
80G Certificate is issued to a non-profit organization or non-governmental organizations (NGO), a charitable trust or a Section 8 Company by the Income Tax Department. The objective behind the 80G certificate is to encourage more and more donors to donate funds to such organizations. The benefit that the donor avails by donating to such an NGO is that he gets tax exemption on 50% of his donation as the donor is allowed to deduct their donations from their Gross Total Income. To avail the tax exemption, the donor needs to attach the stamped receipt against the donation that is issued by the NGO – the receipt needs to include the name, date, and PAN of the organization.
The application for 80G certificate needs to be submitted to the Commissioner of Income Tax (Exemption) in the jurisdictional area of the institution.
Once the form and the required documents for 80G registration are submitted, an on-the-premise inspection is done by the Income Tax department.
The officials might ask for additional documents and evidence that are required to be made available by the institution.
On satisfactory verification and scrutinizing of documents and the office of the NGO, the 80G certificate is awarded to the institution by the Commissioner.
Validity: 80G Certificate valid for a lifetime.
Registration certificate and MoA in case of Section 8 companies and Societies; Trust Deed in case of a Trust.
No objection certificate from the owner of the property where the registered office of the Institution is located.
PAN card copy of the NGO.
Copy of Utility Bills – Electricity bills, water bills or House Tax Receipt.
Donor list along with their complete address and PAN.
Book of Accounts & Income Tax Return documents, of last three years.
List of welfare activities being carried out & the progress report for last three years.
Detailed list of the board of trustees.
Deed for Verification – Original RC, MoA or Trust Deed.
Now that we have covered 80G Registration we will discuss 12A registration and 12A registration process.
By virtue of 12A registration, Trusts and NGOs and other Section 8 companies enjoy exemption from paying income tax. NGOs are basically organizations that are meant for charitable and non-profit activities. However, they do have income and would be required to pay tax as per normal rates if not registered under section 12A of the Income Tax Act. The Section 12A of the IT Act, 1962 does not differentiate between charitable and religious Trusts. Hence 12A registration is applicable to both the kinds of organizations.
The application needs to be done using Form 10A, as per Rule 17A of the Income Tax Act, 1962. The application needs to be filed with the Jurisdictional Commissioner of Income Tax (Exemptions).
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12A Registration can be canceled at any point in time subject to proof that the activities of the organization being carried out is against the object of the institution or are not genuine or is benefitting a particular religion or caste or the funds are being invested in prohibited modes or income of the institution is being used for the benefit of specific persons.
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PAN card of the organization.