Searching for Hadronic Signatures in the Time Domain of Blazar Emission: The Case of Mrk 501

Abstract

Blazars—a subclass of active galaxies—are intrinsically time-variable broadband sources of electromagnetic radiation. In this contribution, we explored relativistic proton (hadronic) signatures in the time domain blazar emission and searched for those parameter combinations that unveil their presence during flaring epochs. We generated time series for key model parameters, like magnetic field strength and the power-law index of radiating particles, which were motivated from a simulated time series with statistical properties describing the observed GeV gamma-ray flux. We chose the TeV blazar Mrk 501 as our test case, as it had been the study ground for extensive investigations during individual flaring events. Using the code LeHaMoC, we computed the electromagnetic and neutrino emissions for a period of several years that contained several flares of interest. We show that for both of those particle distributions the power-law index variations that were tied to moderate changes in the magnetic field strength of the emitting region might naturally lead to hard X-ray flares with very-high-energy $\gamma$-ray counterparts. We found spectral differences measurable by the Cherenkov Telescope Array Observatory at sub-TeV energies, and we computed the neutrino fluence over 14.5 years. The latter predicted ∼0.2 muon and anti-muon neutrinos, consistent with the non-detection of high-energy neutrinos from Mrk 501.

1. Introduction

Blazars, a prominent subclass of active galaxies with jets aligned with our line of sight [1], are sources of broadband electromagnetic radiation that exhibit intrinsic flux variability. The flux variability is observed across the electromagnetic spectrum, from radio frequencies to very-high-energy gamma rays (for a review, see [2]). The flux may vary on both short (minutes to hours) and long (months to years) timescales [3,4].

The Spectral Energy Distribution (SED) of blazars typically features two broad components: a low-energy peak attributed to synchrotron radiation from relativistic pairs, and a high-energy peak often explained by the inverse Compton scattering of synchrotron photons [5,6] or by external photon fields [7,8] in leptonic models. In hadronic models, on the other hand, high-energy emission results from the interaction of relativistic protons with ambient photon fields, producing secondary particles, such as pions and muons, that decay into secondary pairs, gamma-rays, and neutrinos. While time-averaged SEDs provide valuable insights into the average physical conditions within the blazar’s emission region, they do not capture the dynamic aspects of blazar behavior. During variability epochs, we can obtain more precise constraints on source parameters, such as the radius of the emission region or the magnetic field strength, rather than average values. This approach allows for a more detailed understanding of the physical conditions inside the source [9,10].

The correlation between flux variations in different energy bands is a tool for distinguishing between emission models. In leptonic models, the Synchrotron Self-Compton (SSC) mechanism predicts a quadratic relationship between the X-ray and TeV gamma-ray fluxes, as both emissions originate from the same electrons. For example, ref. [11] was able to reproduce the SED of Mrk 501 during its April–May 1997 outburst with a time-dependent SSC model by varying the electron luminosity, the maximal Lorentz factor of the electrons, and the Doppler factor. In leptohadronic models, the correlations can be more complex, due to the additional processes involved, such as photohadronic interactions, synchrotron radiation from secondary particles, and photon–photon annihilation. These models may predict different time lags and flux–flux correlations compared to leptonic models [12], providing a means to distinguish between the two scenarios.

Mrk 501 is a well-known TeV blazar that has been extensively studied, due to its remarkable flaring activity and the wealth of observational data available. Previous studies, such as the detailed analysis of the 1997 flaring period [13], have revealed significant correlations between X-ray and TeV gamma-ray fluxes, consistent with SSC predictions [11]. During X-ray flares, narrow spectral features in the very high energy (VHE, $E>100$ GeV) can be explained by the emission of neutral pion decay, as shown in ref. [14]. However, the variability when combined with the multi-wavelength campaigns suggests that the observed emission from Mrk 501 could be due to a complex superposition of two different emission zones [15].

In this study, we aimed to explore the time-dependent behavior of blazar emission, focusing on Mrk 501 as our test case. Using the LeHaMoC code [16], we simulated time series for key model parameters, such as electron–proton distribution and magnetic field strength, motivated by the observed gamma-ray flux variability. Our primary objective was to identify parameter combinations that revealed hadronic signatures during flaring epochs and to search for energy windows where hadronic emissions might dominate. We analyzed electromagnetic and neutrino emissions over several years, encompassing two simulated flaring events, to investigate flux–flux correlations between different energy bands.

The structure of this paper is as follows: In Section 2, we describe briefly the physical problem and the numerical code. Then, we focus on the case study of Mrk 501, detailing the observational data and the modeling of the time-average SED that served as the baseline of our time-dependent model. In Section 2.4, we describe how we created synthetic light curves. In Section 3, we present our results on the variability of the broadband emission, discussing the identified hadronic signatures and flux correlations. Section 4 provides predictions for future observations with the Cherenkov Telescope Array (CTA), and Section 5 discusses our findings in the context of neutrino observations by IceCube. In Section 6, we discuss our findings, we present some caveats of our analysis, and we outline potential improvements. We conclude in Section 7.

2. Methods

2.1. Numerical Code

We modeled our emitting region as a spherical blob of size R within the jet of the AGN, and we assumed a homogeneous magnetic field of strength B. In blazars, the angle $\theta$ between the line of sight to the observer and the bulk plasma velocity is a few degrees [17,18]. Emissions produced in the blob’s rest frame will therefore appear to be Doppler-boosted to an external observer. For a bulk plasma motion with Lorentz factor $\Gamma$ and velocity $\beta =v/c$, we defined the Doppler factor as $\delta ={\Gamma }^{-1}{(1-\beta \cos \theta )}^{-1}$. Throughout this work, all parameters were measured in the rest frame of the blob, unless explicitly stated otherwise. We assumed the presence of an acceleration region from which particles (primary electrons and protons) escaped upon their acceleration into the blob where they radiated. The injected distribution of the accelerated particles was described by a power law with slope $p_j$, $d{N}_j/d\gamma \propto {\gamma }^{-p_j}$, between Lorentz factors ${\gamma }_{min,j}$ and ${\gamma }_{max,j}$, where $j=e,p$. Non-thermal radiation was produced by the particle distributions inside the blob, due to various processes, such as synchrotron radiation and inverse Compton scattering. Inelastic collisions between relativistic protons and photons also resulted in the creation of secondary particles, such as relativistic electron–positron pairs, neutrinos, anti-neutrinos, and ${\pi }^0$ which decay into $\gamma \gamma$ pairs. To quantify and track the time evolution of a population j in the blob, with  differential distribution $N_j\left(\gamma \right)$, we solved a system of integrodifferential equations, also known as kinetic equations. The equation of each particle species can be parameterized as

$${\displaystyle \frac{\partial N_j(\gamma ,t)}{\partial t}}+{\displaystyle \frac{N_j(\gamma ,t)}{{\tau }_{esc}}}+\sum _i{\mathcal{L}}_j^i(\gamma ,t)=\sum _i{\mathcal{Q}}_j^i(\gamma ,t),$$

(1)

assuming a physical escape from the source on a timescale ${\tau }_{esc}=R/c$. In the above, symbols ${\mathcal{Q}}_j^i$ and ${\mathcal{L}}_j^i$ denote the mathematical operators for the sources/injections and sinks/losses of particles species j due to the physical process i, respectively. In our calculations, we accounted for electron and proton synchrotron emission and absorption, inverse Compton scattering (for electrons only), $\gamma \gamma$ pair production, photopair (Bethe–Heitler pair production), and photopion production processes. For primary electrons and protons, there was an additional source term ${\mathcal{Q}}_{e,p}^{ext}$ that described the rate of particle injection into the blob. The associated energy injection rate in relativistic electrons or protons was then obtained as $L_{e,p}\left(t\right)=m_{e,p}{c}^2\int d\gamma \gamma {\mathcal{Q}}_{e,p}^{ext}(\gamma ,t)$. To solve the kinetic equations of the stable particles in the blob—namely, electrons (and positrons), protons, photons, and neutrinos (and anti-neutrinos)—we utilized the code LeHaMoC1 [16], a versatile time-dependent leptohadronic modeling code.

2.2. Mrk 501 as a Case Study

Mrk 501 is a High-Synchrotron Peaked (HSP) blazar at a luminosity distance of $d_L=149.4$ Mpc that corresponds to a redshift of $z=0.034$ [19,20]. Due to its proximity and time variability, it has been the focus of numerous observational campaigns ([21,22,23,24,25,26]). Its well-defined flares in X-rays and VHE (Very-High Energy; >100 GeV) $\gamma$-rays allow for precise dissections in the time domain for modeling its low and high activity. Furthermore, Mrk 501 features different flux variability across the multi-wavelength spectrum and exhibits spectral changes, especially in X-rays, between low-activity (quiescence) and high-activity (flaring) epochs. It is also characterized as a transient extreme HSP blazar. The 1997 flare is an excellent example of such behavior. During this outburst, at X-ray energies, the peak synchrotron energy exceeded 100 keV, whereas at VHEs, Mrk 501 exhibited drastic flux variations confirming a spectrum above 10 TeV (e.g., [27,28,29,30]). Characteristics of an extreme blazar were also reported during non-flaring epochs in Mrk 501 [31], suggesting that this was not a permanent state of the source.

For our analysis, we used the archival observations available to us (accessed through the SED builder website2), supplemented by additional X-ray spectral data. Our final data set for Mrk 501 is presented in Figure 1, where we use color to differentiate between archival observations (shown in gray) and specific datasets, which will be discussed in the next paragraph.

The VHE spectral data of the 1997 outburst observed by HEGRA [30] alongside observations from the ARGO-YBJ experiment during the flare of 2011 [32] define the VHE flaring state. These are highlighted in Figure 1 as transparent gray points. Non-flaring-(quiescent)-state VHE data were provided for Mrk 501 during 2006 by the MAGIC [33] and TACTIC [34] observatories, during 2009 by VERITAS [35], between 2008 and 2012 by the ARGO-YBJ experiment [32], and between 2008 and 2015 by the Fermi Large-Area Telescope (LAT) [36]. For the HE range (High Energies; >0.1 GeV), we used data provided by the various releases of the Fermi Gamma-ray Space Telescope. In particular, we display data from the 1-, 2-, 4-, and 14-year Fermi LAT Sources Catalogs [37,38,39,40], as well as the monthly spectral data of the 2-year release. We highlight the 14-year observations of 4FGL with green color, as these provide the most reliable long-term representation of Mrk 501 in the HE range.

In the X-ray range, the archival data consists mainly of BeppoSAX observations between 1997 and 2001 [41,42], including the extreme outburst of 1997 (highlighted in light gray in Figure 1). The rest of the X-ray archival observations are shown with a faded black color. The archival data in the X-rays are only used for display purposes. To obtain the time-average broadband X-ray spectrum that we used for modeling, we supplemented our analysis with long-term observations from the X-ray Telescope (XRT, 0.2–10 keV) and the Burst Alert Telescope (BAT, 14–195 keV) on board the Neil Gehrels Swift Observatory. In particular, we used the Swift-BAT 157-Month Hard X-ray Survey Catalogue3 and XRT data from the Neil Gehrels Swift Observatory analysis tool4. We divided the XRT Light Curve (LC) (see Appendix A) into 3 segments, $T<$ 55,500 MJD, $\mathrm{55,500}$ MJD < $T<\mathrm{58,000}$ MJD, and $T>\mathrm{58,000}$ MJD. The first region exhibited low activity while the latter 2 capture the rise and decay of a high-activity period. We also used only the Windowed Timing (WT) mode data and not the Photon Counting (PC) mode, as the latter was saturated. For each region, we built the spectrum using online tools5, and we fitted an unabsorbed power law to the data, resulting in 15 energy flux data points for each epoch. The mean and standard deviation of the epochs was included in our data set as the representative XRT observations of the average spectrum. We also fitted an unabsorbed power law to the BAT spectrum provided by the BAT database. The Swift XRT and BAT data are highlighted with pink and orange colors, respectively, in Figure 1.

Lastly, we remove the optical bulge of the host galaxy from the archival data, which dominates the total emission at eV energies. One way of removing the bulge is to use SED templates of elliptical galaxies, which are fitted to the data and then removed. We opted for a model-independent way of removing the bulge. We isolated the optical, infrared, and radio parts of the spectrum, and we fitted a third-degree polynomial to the data outside of the visually selected bounds of the bulge assuming a non-thermal broadband component (e.g., attributed to the synchrotron emission from the jet). To define flux points belonging to the non-thermal radiation in the region of the optical bulge, we calculated the mean value of the squared residuals of the archival data and compared it to the individual residual value. We only included the data point if its flux was smaller than the mean, and we highlight the excluded observations with a faded black color in Figure 1.

2.3. Computation of Time-Average SED

To derive the baseline blazar parameters for our time variability analysis with LeHaMoC we followed a two-step approach. Recent scientific findings support that the average state of blazar emission is mostly of leptonic origin without distinct hadronic signatures [14,43,44]. Therefore, we first derived the parameters of a purely leptonic model, where the emission from infrared to X-rays is attributed to synchrotron radiation of primary electrons, and the HE to VHE $\gamma$ rays are produced via SSC. The emission from relativistic protons, if present in the jet at all times, remains hidden below the leptonic spectral components. In the second step, we motivated the parameters for such a proton population.

For the leptonic average state, we did not include any flare data in our fitting procedure [21,30,32]. We also excluded low-frequency data (e.g., ≲300 GHz) for the fit, assuming that this emission originated from a more extended jet region6. This treatment was consistent with previous studies where excluded observations have been used as upper limits [44]. Additionally, observational studies have found that the radio region and the blob region are distinct [45,46], while other studies have modeled the radio region as a separate emission region [47].

We ran the Markov Chain Monte Carlo (MCMC) sampler emcee [48] for 32 parallel walkers on the prepared data. All our parameter priors used uniform distributions in logarithmic space except for the power-law index, which used a uniform distribution in linear space. We varied the step number of our fitting process throughout our testing, to allow for a thorough investigation of the parameter space. We concluded that 10,000 steps and a burn-in phase of 2000 steps were sufficient for the model to converge and have a meaningful posterior sample. In this fitting process, we chose to restrict the upper bound of the uniform radius prior to log($R_{max}$[cm]) = 16.5. This approach was conservative, in the sense that it predicted a variability timescale of $t_{var}=1$d utilizing $\delta \approx 20$, the median value of the Doppler factor posterior distribution (see Appendix B for a discussion of an unbound radius prior). We present in

Table 1. MCMC fitting results for the leptonic model describing the average SED of Mrk 501. The errors indicate the 68% range of values from the posterior distributions. The priors are uniform distributions in ${log}_{10}$ space (except for the $p_e$, where the distribution is in linear space).

Parameter Description	Parameter Symbol	Value	Uniform Priors
Blob radius	$log\left(R\right[\mathrm{cm}\left]\right)$	$16.46\pm 0.03$	$[13,16.5]$
Magnetic field strength	$log\left(B\right[\mathrm{G}\left]\right)$	$-1.42\pm 0.05$	$[-8,8]$
Doppler factor	$log\left(\delta \right)$	$1.31\pm 0.02$	$[0,5]$
Min. electron Lorentz factor	$log\left({\gamma }_e^{min}\right)$	$2.75\pm 0.05$	$[0,5]$
Max. electron Lorentz factor	$log\left({\gamma }_e^{max}\right)$	$7.4\pm 0.4$	$[6,9]$
Electron compactness 1	$log\left(l_e\right)$	$-4.75\pm 0.03$	$[-6,-1]$
Electron power law index	$p_e$	$2.50\pm 0.02$	$[1,5]$

¹ The injected isotropically distributed luminosity of the electron population can be quantified via the compactness [49], a measure comparing the energy contents of a region to its spatial dimensions, $l_e={\sigma }_T{L}_e/\left(4\pi R{m}_e{c}^3\right)$.

our parameter results and in Figure 2 a representative sample of 100 leptonic SEDs computed from the posterior distribution of parameters. A corner plot showing the posterior distributions of the model parameters can be found in Appendix B.

Describing the relativistic proton population required four additional parameters. To reduce the number of free parameters, we assumed that both populations were accelerated before injection by the same mechanism into a power law distribution with the same slope and the same extent in Lorentz factor, namely, $p_p=p_e$ and ${\gamma }_p^{min/max}={\gamma }_e^{min/max}$. In our treatment, we fixed these hadronic parameters to the median values of the posterior distributions listed in Table 1 (in Section 6, we discuss how another choice for relating the power law indices or the electron and proton energies would have affected our results). From our leptonic posterior analysis, we calculated the upper 1% limit, meaning 99% of our solutions at each frequency were below this SED (depicted in Figure 2 with a dashed line). To set an upper bound on the injection power of the relativistic proton component we expressed the latter as $L_p={10}^{\alpha }{L}_e$ and we increased $\alpha \ge 0$ until the computed SED exceeded by 10% the maximal leptonic SED (at any frequency). Our choice was conservative in the sense that it did not result in a leptohadronic SED with a distinct hadronic component. Our parameters for the underlying proton population are summarized in

Table 2. Parameters describing the relativistic proton component in the leptohadronic model describing the average SED of Mrk 501.

Parameter Description	Parameter Symbol	Value
Min. proton Lorentz factor	$log\left({\gamma }_p^{min}\right)$	$2.75$
Max. proton Lorentz factor 1	$log\left({\gamma }_p^{max}\right)$	$7.4$
Proton power law index	$p_p$	$2.50$
Injected proton luminosity	$L_p$ [erg/s]	≤${10}^{5.61}{L}_e$

¹ The Hillas criterion [50] imposes a geometrical limit on the maximum particle energy in a region of a given size R and magnetic field strength B. In particular, it requires the Larmor radius of the particle to be smaller than the confinement region, $r_L\le R$. Thus, ${\gamma }^H=qBR/\left(m{c}^2\right)$ gives the maximum allowed Lorentz factor. For Mrk 501, using the lower 68% ranges of the radius and magnetic field strength listed in Table 1 yields the result of $log\left({\gamma }_p^H\right)=8.5>log\left({\gamma }_p^{max}\right)$.

, while the SED components of the resulting average-state leptohadronic model are presented in Figure 3. As expected, the observed SED was dominated by synchrotron and IC emission of primary electrons. Synchrotron emission from secondary pairs from $\gamma \gamma$ pair production and photopion production contributed to about 3% of the HE $\gamma$-ray flux. Similarly, the synchrotron emission of secondary pairs from Bethe–Heitler pair production, which peaked at about 100 keV, was a small fraction (∼6%) of the BAT flux. We note that the synchrotron photons from primary electrons provided the main seed photons for IC scattering by the secondary pairs.

2.4. Generation of Time Series

To model the multi-wavelength variability of Mrk 501 we first derived its statistical properties in the Fermi-LAT energy range and then created synthetic Light Curves (LCs) that were mapped into model parameter time series. The software celerite2 [51,52] was used to extract the statistical properties from the observational data, namely, the logarithmic energy fluxes (without upper limits) taken from the Fermi Light Curve Repository (LCR, [53]) with a mean cadence of 7 days, spanning a period of 15.4 years. Key features of celerite2 are speed and scalability, while its employed method, Gaussian process fitting, has been used to study the behavior of a variety of blazar sources [54,55,56,57,58]. Following the available literature on general celerite usage [51], as well as on applications on blazar time series specifically [54], we modeled the observed LC with a stochastically driven damped Simple Harmonic Oscillator (SHO), which is effective in detecting periodic signatures within a stochastic process. The differential equation of an SHO is

$$\left[{\displaystyle \frac{d^2}{d{t}^2}}+{\displaystyle \frac{{\omega }_0^2}{Q}}{\displaystyle \frac{d}{dt}}+{\omega }_0^2\right]y\left(t\right)=ϵ\left(t\right),$$

(2)

where ${\omega }_0$ is the undamped oscillator (or natural) frequency and Q is a dampening factor (also named quality factor). Assuming $ϵ\left(t\right)$ to be a white noise term, the Power Spectral Density (PSD) is given by

$$S\left(\omega \right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{S_0{\omega }_0^4}{({\omega }^2-{\omega }_0^2)2+{\omega }^2{\omega }^2/Q^2}},$$

(3)

where $S_0$ is proportional to the power at ${\omega }_0$ and is given by $S\left({\omega }_0\right)=\sqrt{2/\pi }{S}_0{Q}^2$.

Although a mixture of SHOs can be used to describe and capture stochastic processes in an astrophysical setting we limited ourselves to the simplest case of one SHO. To derive its parameters, ${\omega }_0$, $S_0$, and Q, we ran the Markov Chain Monte Carlo (MCMC) sampler emcee [48] for 32 parallel walkers with an initial 2000-step burn-in phase followed by a 5000-step phase for the posterior sampling used in our analysis. We found the walker and step size sufficient for the solutions to converge. As such, for the starting guesses for the MCMC sampler we utilized an L-BFGS-B optimization routine on a random starting guess.

The resulting parameters of the SHO alongside the mean $\mu$ of the analyzed time series can be seen in

Table 3. celerite parameters for the Fermi-LAT LC of Mrk 501.

Source Name	$\mathit{\mu }$  1	$ln\left({\mathit{S}}_0\right)$	$ln\left(\mathit{Q}\right)$	$ln\left({\mathit{\omega }}_0\right)$  2
Mrk 501	$-4.{12}_{-0.03}^{+0.03}$	$0.6_{-0.4}^{+0.5}$	$-3.2_{-1.1}^{+0.8}$	$-1.4_{-0.8}^{+1.1}$

¹ Where $\mu$ is the mean of the ${log}_{10}$(Flux) in the 0.1–100 GeV range in units of MeV cm⁻² s⁻¹. ² Where ${\omega }_0$ is in units of rad/days.

, while a visual representation of the fit and PSD is showcased in Figure 4. Furthermore, a detailed exploration of the goodness of our fit is presented in Appendix C. Thus, it is possible now to create a synthetic LC that accurately reflects the behavior of Mrk 501 in the Fermi energy range.

3. Results on Variability

To investigate time-variable SED models of Mrk 501 we used the celerite parameters listed in Table 3 and created a synthetic LC with statistical properties consistent with the observed LC in the Fermi energy band of 0.1–100 GeV. We selected a number of points equal to that of the real LC and multiplied them by the mean cadence of 7 days. However, since we did not include the upper limits of the Fermi LC in our analysis, the total number of points and, consequently, the total duration of the derived LC were shorter than those observed for about 230 days. As such, our method introduces uncertainties by neglecting the information provided by upper-limit detections. Additionally, while the generated LC accurately reflects the overall behavior of Mrk 501 in the Fermi energy band, it may display features that surpass the observed maximum or minimum flux within this range. Therefore, the LC produced by celerite2 should be interpreted as a potential scenario for the blazar’s behavior rather than a precise representation of its current state. Mitigation of this uncertainty will be discussed in Section 6.

With black stars, we showcase in Figure 5 the entirety of the zero-mean synthetic LC. As we were interested in the relative changes of the logarithmic flux (indicated as y), the flux units were disregarded. The horizontal line in all three panels indicates the average state of the source. Additionally, we highlight two flares, using gray bands in the upper panel, and we show zoomed-in versions in the lower panels. The first flaring event, labeled “extreme”, had a peak flux that exceeded by about a factor of two the maximum observed flux of Mrk 501. In contrast, the second flaring event, labeled “typical”, reached a peak flux comparable to the maximum observed flux amplitude in the variability of Mrk 501. With blue markers, we indicate the interpolation of this LC at time steps of one light crossing time, $t_{cr}=0.55$ days (in the observer’s frame), creating our Time Series (TS). Thus, we accurately captured changes in the parameter space of our source7.

In our parametric study of variability, therefore, we examined long- and short-term changes with the complete TS and the two flaring events, respectively. For each case, the respective interpolated TS started close to zero with an initial 10-step “burn-in” phase where no changes occurred in any parameter value. This allowed the system to accurately converge to its average state before starting deviations from it.

3.1. Particle Energy Injection Rate and Magnetic Field Strength

To motivate variations of the injected luminosity $L_{e,p}$ of the primary electron and proton populations or the magnetic field strength B of the emission region, we connected them to the observed flux variability in HE $\gamma$-rays probed by Fermi-LAT. In particular, we followed the principles outlined in the previous section and assumed a predominantly leptonic description of Mrk 501 based on the SSC model. In this description, we correlated the peak flux of each component to our desired parameters. In the $\delta$-function approximation for the synchrotron and Inverse Compton (in the Thomson limit) emissivities, the synchrotron component at peak energy $ϵ$ and the SSC component at peak energy ${ϵ}_s$ scaled with the model parameters as [59]

$$\begin{array}{c}{f}_{ϵ}^{syn}\propto B^2{L}_e,\\ f^{Fermi}\propto f_{{ϵ}_s}^{SSC}\propto {\left(f_e^{syn}\right)}^2{B}^{-2}\Rightarrow f^{Fermi}\propto {\left(B{L}_e\right)}^2.\end{array}$$

(4)

where the symbol $f_{ϵ}$ denoted the flux in $ϵF\left(ϵ\right)$ units. We then adopted the same scaling relation between the $\gamma$-ray flux and $L_p$, assuming both populations were injected into the source via the same mechanism. Therefore, we could connect the previously derived average-state parameters of Mrk 501 listed in Table 1 and Table 2 to the logarithmic flux variations of Figure 5, as follows:

$$\begin{array}{c}log\left(L_{e,p}\left(t_i\right)\right)=log\left(\langle L_{e,p}\rangle \right)+{\displaystyle \frac{1}{2}}y\left(t_i\right),\mathrm{or}\\ log\left(B\left(t_i\right)\right)=log\left(\langle B\rangle \right)+{\displaystyle \frac{1}{2}}y\left(t_i\right),\end{array}$$

(5)

with $t_i=i·t_{cr}$ being the i-th time step of the time series. Both variations (when applied independently) failed to create flux changes that approximated the observed features of Mrk 501. We illustrate this via two arguments.

First, we present in Figure 6 the SED components for the extreme flare at the peak time of the injected TS. The average-state SED (Figure 3) from which the variations were initiated is highlighted with a black dotted line. The complete time evolution of the SEDs during this event is discussed in Appendix D, while the SED density profiles of the complete numerical runs for both parameter variations are provided in Appendix E. Furthermore, we provide histograms of the parameter variations in Appendix F. With the applied parameter variations, we could not recover the spectral hardening required to explain the most luminous X-ray flare ever observed [41]. Moreover, as demonstrated in Figure 6 for the peak of the flare (and in Appendix D for the entirety of the event), we found no distinct hadronic signatures in the SED of Mrk 501 throughout the 14.6 year-long period that we studied. All hadronic-related spectral components were hidden below the synchrotron and SSC emission of primary electrons, as illustrated in Figure 6. Moreover, the ${\pi }^0$ bump was suppressed when the attenuation by the extragalactic background light (EBL) was taken into account.

Secondly, we computed the flux histograms of the complete numerical runs, comparing our time-variation models to observations in different energy bands. We used the LC from the Fermi LCR for the time series analysis and, thus, chose the Fermi band of 0.1–100 GeV for the HE $\gamma$-ray band. Having utilized Swift XRT and BAT data in our average-state analysis, we used their respective energy ranges of 0.2–10 keV and 14–195 keV to describe the soft and hard X-ray emissions. For the VHE range, we used 0.3–3 TeV to describe the region of high flux sensitivity of the upcoming Cherenkov Telescope Array Observatory [60,61]. Lastly, we chose the R-band (138–658 nm) to represent the low-energy emission in the optical. In particular, we utilized observations from the GASP program of the Whole Earth Blazar Telescope (WEBT) and data from the Tuorla observatory using the KVA telescope. The resulting LC had a total length of 4 years, a binning time of 1 day, and it has been adopted by ref. [45]. The shorter duration of this observational LC is a caveat that needs to be taken into consideration when interpreting our results. Specifically, the R-band observations are from a historically low-activity period in X-rays and $\gamma$-rays for Mrk 501. As a result, with fewer fluctuations between low- and high-states their variability remained confined within narrower flux ranges. For this reason, they might not have been representative of the LC’s behavior during high-activity periods or while studying long-term behavior.

We present in Figure 7 and Figure 8 the flux histograms of the full (14.6 yr) leptohadronic runs for $L_{e,p}$ and B variations, respectively. In both figures, the observational histograms (whenever available) for each energy band are overplotted for comparison. We also included the histogram for the synthetic LC used for the time variability. There is agreement in the Fermi band between the model and observations, as expected (the time series were created in a way that reproduced the observed GeV variability). However, we did not recover the expected variability in the other energy bands. The observational X-ray histograms were broader, while our models predicted narrower flux distributions. Changes in the injection power of radiating particles and/or magnetic field strength alone did not result in large spectral changes in X-rays, which are needed to produce a larger spread in the derived X-ray fluxes. On the contrary, the predicted flux distribution in the R-band was broader than the observed one. This was possibly due to the shorter duration of the R-band observations used for this plot. Including data from longer time periods might have resulted in a wider range of R-band fluxes, similar to what is observable in the other energy bands. If, however, the R-band histogram had remained narrow, this would suggest a variability mechanism operating in Mrk 501 that produces less variable optical fluxes than in the other ranges in the long term.

3.2. Power-Law Slope of Particle Distribution

Changes in the power-law slope of the radiating particle distribution are typically invoked to explain spectral changes (especially during flares) that cannot be attributed to cooling effects [26].

While we did not model the acceleration zone, we note that the power-law index of the injected particles into the radiation zone is determined by the acceleration and escape processes from the former zone. To motivate variations of the power law index in our study, we noted its relation to the acceleration and escape timescales [62]:

$$p_{e,p}=1+{\displaystyle \frac{t_{acc}}{t_{esc}}}.$$

(6)

Expressions for $t_{acc}$ depended on the underlying acceleration mechanism. For example, in a Fermi acceleration scenario, we expected dependence on the magnetic field, either implicitly or explicitly through the gyroradius of a relativistic particle, $r_g\approx m_{e,p}\gamma c^2/\left(eB\right)$. In particular, for first- and second-order Fermi acceleration the dependencies of the acceleration timescales are $t_{FI}\propto r_g\propto B^{-1}$ and $t_{FII}\propto r_g/B^2\propto B^{-3}$, respectively [63]. Assuming the magnetic field strength in the acceleration and radiation zones to be similar, we were able to parameterize the power-law index of the injected particle distributions as

$$p_{e,p}\left(t\right)=1+\langle p_{e,p}\rangle {\left({\displaystyle \frac{B\left(t\right)}{\langle B\rangle }}\right)}^{-m},$$

(7)

where $\langle \dots \rangle$ denoted the parameter values of the time-average model. Therefore, the power-law index variations were tied to small amplitude changes in the magnetic field strength. Given that we did not model the acceleration physics, we explored the impact of different values of m. Therefore, utilizing Equation (5) and choosing a value of m we transformed the TS of Figure 5 into a time series of the power law index $p_{e,p}$.

To demonstrate the impact of the parameter m we present in Figure 9 two scenarios. We compare mild ($m=0.5$) to extreme ($m=4$) power-law index variability for the extreme flare introduced in Figure 5. We note that in the former scenario, the values of $p_{e,p}$ vary between 2 and 2.5, while the range of the latter is 1.05–2.5. We contrast the leptohadronic SED (left column) to the leptonic SED (right column) to exemplify the clear hadronic impact in this scenario. For the scenario with $m=0.5$ the spectral hardening is not enough to describe the hard spectrum of the extreme X-ray flare of 1997 (see lower panels). On the contrary, for the extreme slope variations with $m=4$ the electron synchrotron spectrum becomes as hard as the observed one. Due to this spectral change, the number of low-energy seed photons used for IC scattering decreases compared to the average state (dotted line), leading to suppressed IC emission (from the primary electrons) at the HE and VHE bands. However, secondary electrons produced from photomeson and Bethe–Heitler interactions, as well as $\gamma \gamma$ pair creation have a dominant contribution to the HE and VHE emissions, which exceeds by 5.6 and 6.7 times the maximum luminosity observed by Fermi in that range (0.1–100 GeV) for $m=4$ and $m=0.5$, respectively. Furthermore, when $m=4$ we notice a prominent pion bump emerging at ${ϵ}_{\gamma }=4·{10}^{16}$ eV, due to the low intrinsic $\gamma \gamma$ opacity at the peak time of the X-ray flare8. The hardening of the electron synchrotron spectrum during the flare reduces the number of low-energy photons (${ϵ}_t\approx {10}^{-5}$ eV) that are targets for $\gamma \gamma$ pair creation. However, this hadronic spectral component would not be detectable due to the EBL attenuation (not included here). Still, an accompanying high-energy neutrino component is expected. We discuss our model predictions in the context of the IceCube observations in Section 5.

Motivated by the results above, we are sought slope variations that could lead to significant spectral hardening, but without producing extreme hadronic contributions at (V)HE. Therefore, we set $m=2$ and reduced the injected proton luminosity $L_p$ by two orders of magnitude compared to the upper limit listed in Table 2. The choice of $m=2$ was consistent with second-order Fermi acceleration ($t_{FII}\propto r_g/B^{-2}$) being at work in the acceleration region Mrk 501 [25] and with an escape timescale (from the acceleration region) being proportional to the particle gyroradius. Lastly, in the long-term numerical run, we imposed an upper limit of 2.6 on the power-law index of the injected distributions whenever Equation (7) predicted very soft power laws. This ensured that our analysis was not affected by pathological cases, e.g., the distributions were effectively mono-energetic. From this point on, we will refer to these simulations as the “modified power-law index variations”. Our results at the peak of the injected TS are shown for both flares in Figure 10 and as a density plot for the complete numerical run in Appendix E. Moreover, we present the histogram of the power-law index variability in Appendix F. By reducing the injected proton luminosity the emission of secondary electrons from Bethe–Heitler, photo-meson, and $\gamma \gamma$ pair creation interactions was reduced to being comparable to or below the primary electron IC emission. The notable exception is the photo-meson contribution in the GeV range during the extreme flaring event during the peak of the injected TS (compare the left columns of Figure 9 and Figure 10). A detailed discussion on the behavior of these three SED components will be presented in Section 6.

In Figure 11, we present the result for the flux histograms for the complete numerical run. We report discrepancies in the flux ranges between model predictions and observations at all energies except the Fermi band. To highlight this result, we integrated the 1997 flare observations in the XRT and BAT energy range, and we present the respective integrated flux values with a black dashed line in the appropriate histogram panels. We thus conclude that our model over-predicted the observed flux in the X-ray range even when accounting for the major X-ray flare of 1997. However, our analysis did not attempt to fit the observational flux histograms. Rather, our time variations explored the behavior of Mrk 501 in case of a future extreme flare. As such, small deviations from the 1997 flare, as seen in Figure 11, may not be indicative of disagreement between our model and observations, as discussed in Section 6. Furthermore, the distribution shapes deviated at HE. Compared to the observations, we noticed a peak at higher flux values when the power-law index was fixed at $p=2.6$, whereas the distribution peaked at lower flux values when the IC contribution of primary electrons was reduced compared to the average-state emission. This is exemplified in Figure 12, where we highlight the LCs in different energy bands as obtained from our model for the two individual flares of interest. We report a convoluted LC picture for the flaring states. This was expected as during periods of high activity (as defined by our injected TS, shown in black) the targets for IC scattering were reduced, leading to a decrease in the flux of the SSC component. Simultaneously, we see that the expected extreme flux increased in the X-ray band, as a result of the spectral hardening, and the resulting increased synchrotron emission. These two results can also be contrasted by comparing the upper to the lower panel of Figure 12. We see how the VHE flux initially “mimics” the behavior of the X-ray flux corresponding to the initial increase of high-energy electrons (due to the hardening of the injected power law). The sudden decrease around the peak of the injected TS corresponded to the spectral hardening in the low-energy synchrotron region and the resulting decrease of available SSC targets.

In Figure 13, we present the flux–flux diagrams, comparing (a) the Fermi flux to the flux in the VHE range, (b) the harder X-ray range probed by the BAT, and (c) the softer X-ray range probed by XRT; furthermore, in (d) we compare the BAT flux to the R-band flux, comparing hard X-ray to low-energy photons. We present the variability for the complete leptohadronic run (blue circles) and the results for the extreme (yellow stars) and typical flare (purple squares). Each of these diagrams consists of a linear segment (at low fluxes) and a parabolic-like shape (at higher fluxes). In each panel, the parabola section to the right of its vertex corresponds to flaring states, as a result of reduced seed photons for SSC emission. Increasing the available $L_p$ is a method of counteracting this decrease. Furthermore, we suggest that the parabola occurs during joint magnetic field and power-law index variations while the linear segment corresponds to the magnetic field variations when $p_{e,p}=2.6$ (upper bound on slope).

4. Predictions for CTAO

The Cherenkov Telescope Array Observatory (CTAO) will be the new-generation ground-based observatory for $\gamma$-ray astronomy [61]. It will cover a wide range of energies, from 20 GeV to 300 TeV, and it will offer better sensitivity and spectral resolution than the VHE instruments currently in operation [61]. With an expected start of operations in 2025, model predictions in the spectral and time domain for CTAO are timely and informative.

For this section, we were interested in creating a simulated LC for the CTAO North array and determining the impact of the chosen exposure time. Furthermore, we have created simulated spectra to test for statistically significant differences between a leptohadronic and leptonic scenario. As our test case, we chose the modified power-law index variations and focused on the extreme flare, as it exhibited strong anti-correlation behavior during the peak of the injected TS (Figure 12). Additionally, at peak time, the SED exhibited a dominant hadronic component in the HE range (synchrotron from photomeson secondaries; Figure 10).

For this analysis, we used the standard procedures of Gammapy9 [64], a community-developed open-source Python library and core package for CTA. Thus, first, we assigned the coordinates of Mrk 501 as the source position in the sky and set an 11-degree extraction region centered on this location. To this region, we assigned our leptohadronic model (with modified power-law index variations) multiplied by $e^{-{\tau }_{EBL}(z,E)}$. Here, ${\tau }_{EBL}(z,E)$ was the optical depth to $\gamma \gamma$ pair production on EBL photons and was adopted from [65]). We chose as the energy axis the range 0.05–10 TeV, in order to study the source’s behavior in both the HE and VHE bands. Then, we simulated the $\gamma$-ray spectra for each time step of the extreme flare of our variation model, using the available instrument response functions of the CTAO northern array10. Finally, we extracted the predicted LC of this event, and we present our results in Figure 14 for exposure times of 30 min, 1 h, and 5 h. For visual clarity purposes, we shifted the 1 h and 5 h points by 0.25 and 0.5 days to the right, respectively. Highlighted with gray dashed lines are two indicative time values (t = 11, 43 days) that are used in our subsequent discussion of the spectra. We conclude that all three exposure times yielded comparable results in the 0.05–10 TeV range, while increasing it only resulted in marginally lower error values. Thus, within our model framework, low exposure times are sufficient to observe Mrk 501 during a flaring event of brightness similar to our extreme flare.

To investigate differences between leptonic and leptohadronic scenarios within our model description, we fixed the exposure time to 5 h and extracted for both scenarios the LCs in the “lower” energy range of 0.05–1 TeV and the “higher” range of 1–10 TeV. We compare our findings in Figure 15. We found no significant discrepancy between the two models in the higher energy range, as expected; the 1–10 TeV range was dominated by the SSC emission of primary electrons, as shown in Figure 10. On the contrary, the LCs of the leptonic and leptohadronic models in the 0.05–1 TeV range exhibited deviations in the decay part of the flare (see, e.g., dip at 43 days).

To further exemplify and study this hadronic impact, we generate spectra for the complete energy range of 0.05–10 TeV, for a 5 h exposure, and we fitted the simulated data with an empirical model of a Log Parabola11 (LP):

$${\varphi }_{LP}\left(E\right)={\varphi }_0{\left({\displaystyle \frac{E}{E_0}}\right)}^{-\alpha -\beta log\left({\displaystyle \frac{E}{E_0}}\right)}.$$

(8)

This model had a reference energy frozen at $E_0=1$ TeV during the fitting process and an amplitude ${\varphi }_0$ in units of (cm⁻² s⁻¹ TeV⁻¹). The fitting of our simulated spectral data to the empirical models utilized cash statistics [66] integrated within the standardized procedures of gammapy. This approach involved performing a log-likelihood minimization, returning a fit statistic (−2LogL), and calculating the residuals (residuals = (data-model)/model). An example of simulated spectra computed for both leptonic and leptohadronic models at $t=11$ and 43 days is presented in Figure 16 (left and right panels, respectively), while their respective best-fit values are summarized in

Table 4. Best-fit parameters of the log-parabolic model shown in Figure 16.

Log-Parabola Parameter	Leptohadronic (43 Days)	Leptonic (43 Days)	Leptohadronic (11 Days)	Leptonic (11 Days)
${\varphi }_0$1	$5.2\pm 0.1$	$4.8\pm 0.1$	$39.9\pm 0.3$	$39.8\pm 0.3$
$\alpha$	$2.19\pm 0.02$	$2.11\pm 0.03$	$2.353\pm 0.001$	$2.351\pm 0.01$
$\beta$	$0.10\pm 0.01$	$0.14\pm 0.02$	$0.121\pm 0.0001$	$0.123\pm 0.0001$

¹ in units of ${10}^{-12}$ cm⁻² s⁻¹ TeV⁻¹.

. No difference between the two models is observable during the onset of the plateau at $t=11$ days. In contrast, at the dip of the simulated LC at $t=43$ days, a clear hadronic contribution is present at energies below 1 TeV. This was expected from our previous LC analysis (Figure 15), and it is confirmed by visual inspection of the left panel of Figure 10. This SED highlights the dominant hadronic component in the HE region at lower fluxes, and the negligible contribution to the VHE region (which is dominated by primary electron IC emission at all times).

To quantify the statistical significance of this deviation between the two models, we created 1000 leptonic and leptohadronic synthetic CTA spectra for $t=43$ days, and $t=11$ days recording their best-fit log-parabola parameters. The resulting parameter distributions are presented as histograms and contour plots in Figure 17. Based on the contour plots, for $t=43$ days the model parameters were different at least at a 3$\sigma$ level, indicating a clear discrepancy between the leptonic and hadronic models. On the contrary, the distributions for $t=11$ days overlapped significantly and, thus, the two models must be assumed indistinguishable.

The pure leptonic model resulted in a larger range of observed flux (between day 11 and 43) and had a more evident change in spectral shape (see the range of $\alpha$ and $\beta$ parameters), whereas there was less evident spectral change in the leptohadronic model. We note, however, that from $\gamma$-ray spectral modeling alone it was challenging to distinguish between models and to secure a hadronic contribution. The reason is that the hadronic emission in our model does not have a unique spectral feature, as in the SSC+${\pi }^0$ model introduced by [14]. To that end, simultaneous X-ray and HE/VHE $\gamma$-ray observations were needed.

5. Predictions for IceCube

IceCube is a cubic-km neutrino detector located at the South Pole. In 2013, IceCube made a breakthrough with the discovery of a very high energy (TeV-PeV) flux of cosmic neutrinos [67], which led to the detection of the first likely source of high-energy neutrinos, the blazar TXS 0506+056 [68]. So far, no other blazar has been associated with high-energy neutrinos at a confidence level greater than $3\sigma$. The non-detection of neutrinos from the nearby and electromagnetically bright blazar, Mrk 501, can place strong constraints on leptohadronic models. For this section, therefore, we computed the number of events expected from Mrk 501 during the IceCube lifetime.

Using the neutrino energy flux at time t, $F_{\nu +\overline{\nu }}({\epsilon }_{\nu },t)$, where ${\epsilon }_{\nu }$ was the observed neutrino energy, computed during the modified power-law index variations (see Section 3.2), we could make predictions regarding the number of neutrino events detected by IceCube and assess the potential presence of hadronic signatures expected from this model. During these modifications in the slope of the particles $p_{\mathrm{e},\mathrm{p}}$ (see Equation (7) for $m=2$) and the magnetic field $B\left(t\right)$ the power injected into the electron and proton distributions remained constant. Therefore, the applied changes altered the particle number per energy bin.

The expected number of muon plus antimuon neutrinos from Mrk 501 could be calculated as

$${\mathcal{N}}_{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}={\displaystyle \frac{1}{3}}{\int }_{t_{\mathrm{ini}}}^{t_{\mathrm{end}}}\mathrm{d}t{\int }_{E_{\nu ,min}}^{E_{\nu ,max}}\mathrm{d}{\epsilon }_{\nu }{A}_{\mathrm{eff}}({\epsilon }_{\nu },DEC){\displaystyle \frac{F_{\nu +\overline{\nu }}({\epsilon }_{\nu },t)}{{\epsilon }_{\nu }}},$$

(9)

where we assumed vacuum neutrino mixing and used 1/3 to convert the all-flavour flux to muon neutrino flux. Moreover, $t_{\mathrm{ini}}$ and $t_{\mathrm{end}}$ defined the duration of the simulated LC, and $A_{\mathrm{eff}}({\epsilon }_{\nu },DEC)$ was the energy-dependent and declination-dependent point-source effective area of IceCube [69,70].

To account for the varying IceCube configurations over different operation seasons (see

Table 5. Point-source effective areas for different configurations used in our analysis. Data are taken from [69,70].

IceCube Configuration	Season (MJD)
IC40	54,562–54,971
IC59	54,971–55,347
IC79	55,347–55,694
IC86-I	55,694–56,062
IC86-II	>56,062

), we initialized our calculation by assuming that the last day of the Light Curve shown in Figure 5 corresponded to 1 January 2024. In the bottom panel of Figure 18, we present the expected muon neutrino number as a function of time (see blue solid line). When considering only the most recent IceCube configuration (IC86 II), the final neutrino number showed negligible variation ∼3% (see the magenta dashed line in the bottom panel of Figure 18). Our analysis indicates that the neutrino signal over this approximately 14.5-year period was less than one event (${\mathcal{N}}_{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}\sim 0.2$). In this synthetic LC, we set the proton luminosity $L_p$ lower by two orders of magnitude compared to the upper limit listed in Table 2. An increase in the proton luminosity to this value can result in about ∼20 high-energy muon neutrinos in 14.5 years (or 1.4 neutrinos per year, on average). Mrk 501 belongs to a list of predefined $\gamma$-ray sources used in IceCube point-source searches (e.g., [69,71]). By analyzing 10 years of IceCube data [69], the Collaboration obtained 10.3 best-fit signal events from Mrk 501 (see Table III in ref. [69]), which, however, did not secure a statistically significant detection. Instead, they corresponded to an upper limit of one neutrino per year. The model predictions shown in Figure 18 are consistent with the IceCube upper limit (see also [45]).

The top panel of Figure 18 shows the ratio between the flux at 100 MeV (Fermi-LAT) and the flux at 14–195 keV (Swift-BAT) as a function of time. The power-law index of the two distributions $p_{\mathrm{e},\mathrm{p}}$ is plotted against time in the middle panel. We found that during the hardening of the distributions, the ratio $L_{100\mathrm{MeV}}/L_{14-195\mathrm{keV}}$ decreased while the number of neutrinos was increasing (see, for example, the time window from 10 to 11 years). The emission at 14–195 keV primarily arose from primary electrons, and the hardening of their distribution resulted in an increased flux in this energy band. Additionally, the hardening reduced the number of low-energy target photons available for upscattering to the sub-GeV band, as described in Section 3.2. During these hardening phases, this band was predominantly influenced by the synchrotron emission of secondaries produced through photomeson interactions (see, in the left panel of Figure 3, the sub-GeV band). The increase in high-energy protons during these periods also enhanced the source’s neutrino production, a trend evident when comparing the middle and bottom panels of Figure 18.

6. Summary and Discussion

In this work, we derived a leptonic-average-state description of the HSP blazar Mrk 501, using publicly available multi-wavelength archival observations, and utilizing data from the BAT 157-Month Hard X-ray Survey, the 18.5 yr Swift-XRT LC, and the 14-year Fermi-LAT 4FGL catalog. To derive the parameters of the baseline leptonic model, we used the MCMC sampler emcee and the numerical code LeHaMoC. In our subsequent arguments for the proton population parameters, we assumed $p_p=p_e$, ${\gamma }_p^{min/max}={\gamma }_e^{min/max}$, and we searched for a conservative upper limit on the injected proton luminosity $L_p$. Using celerite2, we analyzed the 0.1–100 GeV LC of Mrk 501 and created TS to describe key parameter variations ($L_{e,p}$, B, and $p_{e,p}$). Based on these, we modeled the long-term evolution of the multi-wavelength jet emission of Mrk 501 in the context of a leptohadronic model, contrasting our findings, in the form of flux histograms, to the available observations. Lastly, we searched for detectable hadronic signatures on simulated VHE $\gamma$-ray spectra for CTAO in the investigated power-law index variations and discussed our model in light of the neutrino non-detection of Mrk 501 by IceCube.

To test the validity of our initial average-state parameters, we calculated the average jet power of Mrk 501 in the context of a leptohadronic model as $P_j\approx \pi R^2{\delta }^2\beta c(u_e+u_p+u_B+u_{rad})\approx \pi R^2{\delta }^2\beta c{u}_p$ [72], where $\Gamma \approx \delta$, $\beta =\sqrt{1-1/{\Gamma }^2}\approx 1$, $u_{e,p}$ were the energy densities of the relativistic particle populations, $u_B$ was the magnetic energy density, and $u_{rad}$ was the radiation energy density. In the above, all energy densities were measured in the jet co-moving frame, while the relativistic proton energy density, defined as $u_p=3L_p/\left(4\pi R^{2}c\right)$, was the dominant component. In particular, for the derived parameters listed in Table 1 and Table 2 we calculated $P_j\approx 3·{10}^{49}$ erg/s, or $P_j\approx 3·{10}^{47}$ erg/s for $L_p={10}^{3.61}{L}_e$ used in the modified power-law index variations of Section 3.2. The latter was comparable to the Eddington luminosity, $L_{\mathrm{Edd}}\approx 1.3·{10}^{47}$ erg/s, of a black hole with a mass of $M_{BH}\approx {10}^9{M}_{\odot }$ [73] and, as such, was favored over the preceding jet power estimate. We also noted that the inferred jet powers were comparable to those found in previous studies of blazar leptohadronic emission models [74,75].

To limit the number of free parameters in the leptohadronic model we assumed identical values for the power-law slopes and extent of the electron and proton distributions. An alternative choice for ${\gamma }_p^{min,max}$ was to relate the the minimum and maximum energies of the particle populations as $E_p^{min/max}=E_e^{min/max}$ or ${\gamma }_p^{min/max}=(m_e/m_p){\gamma }_e^{min/max}$. This choice resulted in ${\gamma }_p^{min}=1$ and ${\gamma }_p^{max}={10}^{4.12}$, where the lower value was limited by the minimum physically acceptable Lorentz factor. Repeating the method outlined in Section 2.3, we found the limit on the injected proton luminosity to be $L_p\lesssim {10}^9{L}_e$. Consequently, the upper bound on the jet power was $P_j=7.4·{10}^{52}$ erg/s, which was non-physical, based on energetic arguments for the central engine. We note, however, that combining a reduced proton luminosity $L_p={10}^{3.61}{L}_e$ (as in Section 3.2) with this alternative choice for the proton Lorentz factors resulted in a model with jet power comparable to the Eddington luminosity that can create a pion bump at VHE during the modified power-law index variations discussed in this work. This model bore many similarities to the SSC+${\pi }^0$ model presented recently by ref. [14] to explain a narrow spectral feature in the flaring VHE spectrum of Mrk 501 [19]. The implications of relaxing the assumption $\langle p_e\rangle =\langle p_p\rangle$ would strongly depend on the specifics of our choice. For example, if $\langle p_e\rangle >2$ (as found by the leptonic fit, see Table 1) and $\langle p_p\rangle <2$ (hard power law), then the limit obtained for $L_p$ would be lower than the one listed in Table 2. For a fixed target photon field (which is produced by the primary electrons), the energy injected into secondaries produced via proton–photon interactions increases with decreasing $p_p$, because more power is carried by protons of the distributions with higher energies that are relevant for Bethe–Heitler and photomeson interactions (see, e.g., Figure 12 in ref. [76] and Appendix B in ref. [77]).

To initiate parameter variations from the average-state description we used a synthetic LC of Mrk 501, motivated by its observational behavior in the Fermi energy range of 0.1–100 GeV (as this is the best observationally sampled LC over year-long timescales). Specifically, we conducted a time series analysis with celerite2 on data from the Fermi LCR with a mean cadence of 7 days. A similar methodology has been recently employed by ref. [78] to study the flux variability in single parameter variations of a one-zone SSC model of blazar sources. In this study, the authors used the Emmanoulopoulos method [79], a combination of the Timmer and Koenig procedure [80] and the Schreiber and Schmitz Fourier transform algorithm [81], which produces a TS with the same PSD and PDF as the input TS. In this approach, the input data must be an equally spaced TS. However, astrophysical observations are rarely without gaps and, thus, the input TS must be either artificially connected at the gaps or interpolated between the gap points. The first can result in stiff jumps in the synthetic LCs [82], while the latter has been shown to contaminate the power spectrum of the TS [83]. In contrast, celerite is a Gaussian process method that does not have the above restrictions. Thus, we used the provided Fermi LC without further modifications that could insert uncertainties into our problem description. Furthermore, this difference would allow for investigations similar to ours using, instead of $\gamma$-ray LCs, X-ray LCs that are known to have gaps [84,85]. Observations, such as from the Swift-XRT or Swift-BAT instruments (see Appendix A), can be modeled by celerite2 and used to create synthetic LCs motivating key parameter variations. Additionally, unlike in Fourier transform methods, it is possible to increase the model complexity of a Gaussian process, e.g., by increasing the number of SHOs in celerite2, and to sample the resulting parameter space with nested sampling methods, such as Ultranest12 [86]. This could allow for the search of secondary features in the PSD and degeneracies in the parameter space. Cross-correlations between such studies and ours could provide insights and constraints into the underlying variability mechanisms in Mrk 501. However, we note that celerite2 is restrictive in the models it can use. For example, the SHO model of Equation (3) cannot be evaluated with celerite for the choice of $Q=1/2$, corresponding to a critically damped oscillator, as the elements of the covariance matrix of the Gaussian log-likelihood become infinite (Appendix C for log-likelihood; see [51] for more details). As such, this can impose restrictions on the celerite method when exploring the parameter space, which must be considered.

Having created a synthetic LC for Mrk 501, we instigated parameter variations from their previously derived average-state values. In particular, we studied the effect of variations in electron and proton injection luminosity, the magnetic field strength, and the power-law index of both populations motivated by the magnetic field strength variations. We found that none of those scenarios could explain the complete observational behavior of Mrk 501. In particular, the first two scenarios failed to produce the broad observational X-ray flux distributions of the source and the hardness of the X-ray spectrum during bright X-ray flares, while they over-predicted the variability in the optical R-band. In the power-law index variations, we investigated the impact of the parameter m, which modeled the hardness of the power law in terms of the magnetic field strength, and we showed the hadronic impact, using the upper limit of the injected proton luminosity $L_p^{\mathrm{up}.\lim .}$. Subsequently, we modeled the predicted long-term behavior of our source, using a physically motivated choice of $m=2$, a conservative value for the injected proton luminosity of $L_p={10}^{-2}{L}_p^{\mathrm{up}.\lim .}$, and an upper limit on $p_{e,p}$ (≤2.6) to ensure we were not affected by pathological cases. We showed that this model can produce bright hard X-ray flares with a VHE counterpart, with only moderate changes in the magnetic field strength (see Figure 8). A hadronic component was needed, to avoid a large depletion of the GeV flux at the peak of the X-ray/VHE flare due to the reduction of the primary SSC emission.

When considering the predictions of the power-law index variation scenario on year-long timescales, we found that the obtained flux distributions in the XRT and BAT bands extended to higher values than the observed ones (see Figure 11). This was related to the presence of the extreme flare, and similar ones to it, in the synthetic LC we used to construct our TS (see Figure 5). The extreme flare exceeded the maximum-ever-observed flux in the Fermi-LAT energy range by a factor of 2. While this could be used as a predictive tool for future flaring activity of Mrk 501, caution must be exercised when making conclusive statements comparing the upper X-ray flux ranges between observations and the model in Figure 11. A more meaningful comparison against the observational flux histograms would require the creation of a sample of synthetic LCs and repetition of our analysis for each entry of this sample. This approach would mitigate the effect of having a single synthetic LC that might be characterized by extreme flux occurrences. The impact of the extreme flare on the R-band flux histogram of our model was its extension to lower values; the extreme flare translated to a hardening of the injected electron distribution and, hence, a hardening of the synchrotron spectrum (see, e.g., Figure 10 and Figure A7). Moreover, the observational flux distribution was clustered to values at the upper end of our distribution (see Figure 11), even though these observations corresponded to a historically low activity period of Mrk 501 [45] and represented the lowest recorded flux values of Mrk 501 in this range. As discussed in the next paragraph, this discrepancy may suggest that the optical emission originated from a different region than the one producing the X-ray/VHE flares.

One-zone models are often used to describe the broadband blazar emission, especially when there is evidence of correlated flux variability between energy bands in the low- and high-energy SED humps. Given the small number of free parameters, one-zone models can be constrained by the data (in the spectral and time domain). In several cases, emission from a single zone (or a single particle population) is not sufficient to explain the whole SED [15,31,45,47]. In particular, one-zone models for Mrk 501 fail to capture the observations below the optical band when modeling simultaneous X-ray and $\gamma$-ray data. Another leading argument for a second emission zone is the inability to explain the entirety of the GeV spectrum (in the LAT band) with a one-zone SSC description. Specifically, in ref. [47] the authors utilized hard power laws for the electron distribution, to model the simultaneous X-ray and (V)HE emissions of the extreme flare of Mrk 501 in 1997. Simultaneously, they argued in favor of a continuous jet model (as well as the contribution of the host galaxy) to account for the emission from radio frequencies up to ultraviolet. In that work, an SSC model was sufficient to explain the entirety of the $\gamma$-ray spectrum, a fact attributed, in retrospect, to the poor quality of the available instruments of the time (i.e., comparison between EGRET and Fermi-LAT). In more recent studies, such as ref. [15], the authors have proposed two distinct regions to explain the flux states of Mrk 501 during 2011. They attributed the X-ray and VHE emission to a compact “inner” blob, while the larger “outer” blob explained the radio and GeV observations. Similarly, in the work of ref. [31] the authors modeled Mrk 501 while it exhibited extreme HSP behavior in 2012. In addition to the region responsible for the X-ray and VHE data, a second slower emission zone with variability timescales of weeks was suggested, to explain the optical and GeV emission. Lastly, ref. [45] investigated the behavior of our source during a historically low period of X-ray and $\gamma$-ray activity. They reported the larger second region in the two-zone description to correspond to the baseline emission of Mrk 501. They also found the size of this region to be in agreement with millimeter-VLBI observations. We report that our work arrived at similar conclusions for the radio emission and the GeV component. In our average-state modeling, we did not include measurements below 300 GHz. Moreover, the model SED of the most probable parameters could not capture the entirety of the Fermi observations (see the high-energy flux points of the 4FGL spectrum in Figure 3). In our framework, precise modeling of the acceleration region could provide a physically motivated, second emission zone (similar to the outer/inner blob scenario) while simultaneously constraining the variations imposed on the compact zone. Furthermore, to probe whether the emission from the historically low activity period is attributable to one zone or two components of highly variable and baseline emission one could repeat our methodology and initiate variations from the parameters of a leptohadronic model describing this region. Lastly, we suggest that an alternative to the above could be simultaneous uncoupled variations of the primary electron and proton parameters in the one-zone leptohadronic model. A scenario of constant electron and variable proton parameters could lead to a description similar to the two-zone SSC models.

In the leptohadronic models we explored, hadronic contributions to the emitted electromagnetic spectrum were sub-dominant by construction, except during hard X-ray flares (see Section 3.2). Thanks to the LeHaMoC code we were able to display all hadronic-related spectral components of the SED separately, despite them being most of the times hidden below the primary leptonic emission (see, e.g., Figure 3 and Figure 6). This feature of the code allowed us to explore the temporal evolution of the various spectral components13 and to compare them against the primary leptonic emission components. For the Bethe–Heitler component, the secondary pair distribution extends from ${\gamma }_e\approx {\gamma }_p^{min}$ up to ${\gamma }_e\sim {10}^{11}$, with the shape of the distribution at injection depending on both the shape of the proton power-law index $p_p$ and the photon target spectrum, as shown in ref. [87] (see, for example, Figure 6 therein). As a result, the synchrotron spectra of primary and secondary pairs from the Bethe–Heitler process are generally different, as exemplified in Figure 6, Figure 9, and Figure 10. Additionally, we note how the secondary pair emission from photo-meson and $\gamma \gamma$ absorption interactions are of similar shape and peak at similar energies in the $\gamma$-ray range. This can be understood by examining the energies at which the secondary electrons are injected into the source. During neutral pion production and subsequent decay, the energy transfer from the parent proton of energy ${ϵ}_p$ to each high-energy photon is given by ${ϵ}_{\gamma }^{high}={\displaystyle \frac{1}{2}}{K}_{p\gamma }{ϵ}_p$ [88]. These high-energy photons interact with the low-energy emission (${ϵ}_{\gamma }^{low}$) close to the threshold of the $\gamma \gamma$ absorption process, creating electron–positron pairs with total energy approximately equal to the energy of the photon from the ${\pi }^0$ decay, namely, $2{ϵ}_e={ϵ}_{\gamma }^{high}$ [88]. Thus, secondary pairs produced through the $\gamma \gamma$ process are injected with typical energies of ${ϵ}_e={\displaystyle \frac{1}{4}}{K}_{p\gamma }{ϵ}_p$, which corresponds to the energy of secondary electrons from charged pion decays (in photo-meson interactions) [88]. Given that both channels inject secondary electrons with similar energies, their synchrotron spectra will also peak at similar energies. Lastly, additional features in the $\gamma \gamma$ pair synchrotron spectrum below its peak energy (e.g., see bump at X-rays) are attributable to $\gamma \gamma$ attenuation of SSC photons from primary electrons. This is also highlighted by their presence in the leptonic SSC models of Figure 9.

To further investigate the effect of hadronic interactions on the SEDs, we simulated results for the upcoming CTAO creating LCs and spectra for our power-law index variation model. Subsequently, we investigated the impact of exposure time in the former and the significance of the hadronic component in the latter. We found that within our model description, short exposure times (i.e., 30 min) are sufficient to observe Mrk 501 while increasing the livetime (e.g., 5 h) only marginally reduces the resulting errors. Moreover, we investigated the hadronic impact on the simulated spectra and found a statistically significant contribution in the GeV range when a decrease of the primary IC emission is present (i.e., a reduction of seed photons). To test the statistical significance of our findings, we generated 1000 leptonic and leptohadronic spectra. For each data set, we fitted the empirical model of a log-parabola and recorded the resulting best-fit parameters. Contrasting these results for the two indicative periods of $t=11$ and $t=43$ days revealed a clear hadronic impact during the latter, where the contour plots between the two scenarios did not overlap at a 3-$\sigma$ level. Simultaneously, in the former case of $t=11$ days, both histograms and contour plots featured near-identical values and, thus, the two datasets must be treated as indistinguishable.

Lastly, we computed the time-dependent neutrino energy flux $F_{\nu +\overline{\nu }}({\epsilon }_{\nu },t)$ during variations into the power-law index, to predict the number of muon neutrino events detected by IceCube and assess the potential hadronic signatures from Mrk 501. While we maintained constant luminosity for electrons and protons we altered the spectral shape of the particle distributions. During particle distribution hardening, we observed an increase in neutrino production and a decrease in the flux ratio $L_{100\mathrm{MeV}}/L_{14-195\mathrm{keV}}$ shown in the top panel of Figure 18, primarily due to the increased flux at 14–195 keV from primary electrons and reduced low-energy targets for upscattering to the sub-GeV band, which was also the band at which we could detect a hadronic component in the SED during flaring episodes. Our analysis, accounting for varying IceCube configurations, indicated that the neutrino signal over 14.5 years was less than one event (${\mathcal{N}}_{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}\sim 0.2$), aligning with the non-detection of high-energy neutrinos from Mrk 501. This result was derived assuming a proton luminosity $L_{\mathrm{p}}$ two orders of magnitude lower than the upper limit in Table 2, which was also discussed in Section 3.1. A higher proton luminosity would result in a neutrino expectation in tension with IceCube observations. This was also exemplified in ref. [85], where the neutrino expectation from blazars, including Mrk 501, was computed under the assumption of hadronic X-ray flares [89]. In this scenario, the X-ray flares were powered by proton synchrotron radiation, and neutrinos were produced through photomeson interactions between protons with their synchrotron X-ray photons. The conditions in the flaring region were such as to maximize the neutrino production rate. The 0.5–10 keV fluence of each flare in the X-ray LC of Mrk 501 was used as a proxy for the all-flavour neutrino fluence, and the rate of neutrinos was found to be $(1990\pm 80)\times {10}^{-4}{\mathrm{yr}}^{-1}$ (the prediction for 14.5 years was $2.9\pm 0.1$ muon neutrino events). These results illustrate how neutrino observations can be used to constrain the relativistic hadronic component of blazar jets.

7. Conclusions

We explored the predictions of a leptohadronic model for blazar emission, searching for hadronic signatures in the spectral and time domains, and using Mrk 501 as a test case. By utilizing a synthetic LC motivated by Fermi-LAT observations in the 0.1–100 GeV range over 14.5 years, we created time series for the particle injection luminosity, magnetic field strength, and power-law index of the injected particle distributions. None of the individual scenarios could fully capture the observed variability across the electromagnetic spectrum. However, power-law index variations that are tied to moderate changes in the magnetic field strength of the emitting region may naturally lead to hard X-ray flares with VHE counterparts, as with those observed in Mrk 501. Our analysis revealed that secondary synchrotron emission from hadronic interactions can dominate the sub-TeV energy band during hard X-ray flares. This would have a measurable difference when compared to the primary SSC emission below 1 TeV. Coordinated observations of CTAO and X-ray satellites during X-ray/VHE flares can provide valuable insights into the origin of the HE and VHE $\gamma$-ray emissions of HSP blazars.