A long-term study of Mrk 50 : Appearance and disappearance of soft excess

Swift (Gehrels et al., 2004) is a multi-wavelength satellite consisting of three onboard telescopes, operating in the optical/UV (Ultra-violet Optical Telescope; UVOT Roming et al. 2005), X-ray (X-ray Telescope; XRT Burrows et al. 2005) and hard X-ray (Burst Alert Telescope; BAT Barthelmy et al. 2005) wavebands. Mrk 50 was observed with Swift  at several epochs from 2007 to 2022, with the most recent observation on 27 June 2022 in simultaneous with NuSTAR  (see Table 1). We extracted the light curves and spectra from the observed data by using the web tool ‘XRT product builder’¹¹1http://swift.ac.uk/user_objects/(Evans et al., 2009). Additional finer steps, such as pile-up correction on the given observation ID/IDs, are also applied while using the web tool. The tool processes and calibrates the data and produces final spectra and light curves of Mrk 50 in two modes, e.g., window timing (WT) and photon counting (PC) modes. In 2007, Swift  observed Mrk 50 in three consecutive days from 14 November 2007 to 16 November 2007 with XRT for exposure times of $\sim 7$ ks, $\sim 4.6$ ks, and $\sim 8.7$ ks, respectively. Due to the short exposures, the signal-to-noise ratios (SNR) during these observations are poor. The hardness ratios (ratio between count rates in the 1.5–10 keV and the 0.3–1.5 keV band) were also found to be unchanged during these observations. We, therefore, considered these three observations together and produced a combined observation (S1) for an exposure time of $20.4$ ks.

The Swift/UVOT provides data in three optical filters (V, B, and U) and three UV filters( UVW1, UVM2, and UVW2). We started our UVOT data reduction from the level II image files, performing photometry using the tool UVOTSOURCE. To obtain the source counts, we assumed a circular region of 5 arcsec radius centered at the source position, whereas a circular region with 20 arcsec radius, away from the source position, was considered for background counts. We used the UVOT2PHA tool to create XSPEC-readable source and background spectra and used response files provided by the Swift  team. In this work, we used only data from the UV filters (UVW1, UVM2, and UVW2) and excluded the optical (V, B, and U) data due to the contribution from the host galaxy and starburst in this band.

For Swift/BAT, we downloaded the hard X-ray spectrum of Mrk 50 from the 105-Month Swift/BAT  All-sky Hard X-Ray Survey²²2https://swift.gsfc.nasa.gov/results/bs105mon/. In addition, a response matrix appropriate for the BAT spectra was used in our work.

XMM-Newton  (Jansen et al., 2001) observed Mrk 50 at two epochs in July 2009 and December 2010. The details of the observations are mentioned in Table 1. Data from the European Photon Imaging Camera (EPIC), Reflection Grating Spectrometer (RGS) and Optical Monitor (OM) instruments are used in the present work. The EPIC includes three X-ray CCD cameras covering the 0.3 – 10 keV energy bandpass: two Metal Oxide Semiconductors (MOS1 and MOS2; Turner et al. (2001)) and a p-n CCD (PN; Strüder et al. (2001)). The two RGS detectors (RGS1 and RGS2) provide high-resolution spectroscopy over the 0.35–2 keV energy range  (den Herder et al., 2001). The OM (Mason et al., 2001) provides photometric data from optical to UV bands (U, B, V, UVW2, UVM2, and UVW1).

We analyzed the Observation Data File (ODF) using XMM-Newton Science Analysis System ($\tt SAS~{}v18.0.0$) and updated calibration files as of 23 October 2019. We processed all the EPIC data using the tasks $\tt EMPROC$ and $\tt EPPROC$ to obtain the calibrated and integrated event lists for the MOS and PN detectors, respectively. We filtered the data using the standard filtering criterion. Data during high background flaring events were removed before extracting the spectral products by creating and choosing appropriate good time intervals ($\tt GTI$) using the task $\tt TABGTIGEN$. In this work, we considered only the unflagged events with PATTERN $\leq 4$ and PATTERN $\leq 12$ for the PN and MOS detectors, respectively. The data were checked for photon pile-up using the task $\tt EPATPLOT$ and corrected accordingly. The source photons were extracted by considering an annular region with outer and inner radii of 30 and 5 arcsecs, respectively, centered at the source coordinates. We used a circular region of 60 arcsec radius, away from the source position, for the background products. The response files for PN and MOS detectors were generated using $SAS$ tasks $\tt ARFGEN$ and $\tt RMFGEN$ for arf and rmf files, respectively.

We reduced the RGS data using the standard $SAS$ task $\tt RGSPROC$ and the updated calibration files to produce the source and background spectra. The cleaned event lists were generated by applying the standard filtering criteria. The response matrices were generated using the $\tt RGSRMFGEN$ task. The individual RGS1 and RGS2 spectra were combined into a single merged spectrum using the task $\tt RGSCOMBINE$ to achieve better signal-to-noise for the purpose of spectral fitting.

The Optical Monitor (OM) (Mason et al., 2001) was also used simultaneously to observe Mrk 50 in 2009 in imaging mode with UVW1 and UVW2 filters. For the 2010 observation (0650590401), however, the optical/UV data from the OM are not available. We processed the OM data using the task $\tt OMCHAIN$. The $\tt om2pha$ command was used to generate $\tt XSPEC$-readable spectral files for all the available filters. For the OM data, we used the canned response files available in the ESA XMM-Newton website³³3https://sasdev-xmm.esac.esa.int/pub/ccf/constituents/extras/responses/OM/. We obtained background corrected count rates from the source list for each OM filter and converted the count rates into respective flux densities⁴⁴4https://www.cosmos.esa.int/web/xmm-newton/sas-watchout-uvflux.

NuSTAR  is a hard X-ray focusing telescope consisting of two identical focal plane modules, FPMA and FPMB, and operates in the 3–79 keV energy range (Harrison et al., 2013). Mrk 50 was observed with NuSTAR  simultaneously with Swift  in June 2022. The observation details are presented in Table 1. We use standard NuSTAR Data Analysis Software (NuSTARDAS v2.1.2⁵⁵5https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/) package to extract data. The standard NUPIPELINE task with the latest calibration files CALDB ⁶⁶6http://heasarc.gsfc.nasa.gov/FTP/caldb/data/nustar/fpm/ is used to generate the cleaned event files. The NUPRODUCTS task is utilized to extract the source spectra and light curves. We consider circular regions of radii 60 and 120 arcsecs for the source and background products, respectively. The circular region for the source is selected with the center at the source coordinates, and the region for the background is chosen far away from the source to avoid contamination.

We conducted the timing analysis of Mrk 50 using light curves from the Swift/XRT, XMM-Newton, and NuSTAR observations (see Table 1). The time resolution of the light curves used in our analysis is 200 s. The light curves in the 0.3-10 keV range, generated from the XMM-Newton  and Swift/XRT  observations, and in the 3-60 keV range from NuSTAR  are shown in Figure A1. Further, we extract light curves in the soft (0.3–3 keV range) and hard (3-10 keV range) X-ray bands for the variability and correlation studies. The light curve obtained from the NuSTAR observation is used to explore the variability of the source in the high-energy domain (3–60 keV). The entire energy range (3–60 keV) is further subdivided into two energy bands: band 1 (3–10 keV) and band 2 (10–60 keV), for variability studies.

The spectral analysis is carried out using Swift (XRT & UVOT) and XMM-Newton (EPIC-pn, MOS & OM) observations in 0.001-10 keV range and simultaneous Swift (UVOT, XRT & BAT) and NuSTAR  observations in 0.001–120 keV range (see Table 1). The NuSTAR data beyond 60 keV are not considered in the present analysis as it is dominated by background. We use XSPEC v12.12.1 (Arnaud, 1996) software package for spectral fitting. We binned the spectral data at a minimum of 25 counts/bin for both XMM-Newton  and NuSTAR  observations and 20 counts in each bin for the Swift/XRT  observations. The GRPPHA task is used for binning the spectral data. The model likelihoods are determined using the $\chi^{2}$ statistic. However, it is to be noted that the Swift/BAT data are Gaussian⁹⁹9https://swift.gsfc.nasa.gov/analysis/threads/batspectrumthread.html, and we continue to present the $\chi^{2}$-statistic here for simplicity. We also ignored the bad channels in our analysis.

The best-fit parameters are reported in the rest frame of the source. The uncertainties corresponding to the model parameters are quoted at the 90% confidence level using the Monte Carlo Markov Chain (MCMC) method embedded in $\tt XSPEC$. The MCMC technique simultaneously determines the errors in the model parameters and renders better parameter space sampling than other methods. We used the Goodman & Weare sampler (Goodman & Weare, 2010) to deal with degeneration in the model parameters. For sampling, we choose the number of walkers to be more than twice the number of free model parameters. In all cases, we consider the chain length of 200000 for the chains to converge in the same parameter values. The first 10000 steps are discarded for the burn-in period to remove the bias introduced by the choice of the starting location. To ensure that the walkers have enough sampling in the parameter space, we checked whether the steps were rejected less than 75% of the time or not.

The unabsorbed X-ray luminosity from each spectrum is estimated using clumin task on the powerlaw model. While estimating the luminosity, we use the redshift, $z$=0.023. We also calculate the X-ray flux by using cflux command in XSPEC. The observed UV flux is corrected for reddening and Galactic extinction using the reddening coefficient $E(B-V)=0.0147$ obtained from the Infrared Science Archive ¹⁰¹⁰10http://irsa.ipac.caltech.edu/applications/DUST/ and $R_{V}=A_{V}/E(B-V)=3.1$ following Schlafly & Finkbeiner (2011). The Galactic extinction ($\rm A_{\lambda}$) value used is $0.140$. The X-ray and UV monochromatic flux are quoted in Table 7. Throughout this work, we use the Cosmological parameters as follows: ${H}_{0}=70\,\text{kms}^{-1}\,\text{Mpc}^{-1},\Lambda_{0}=0.73,\text{and}\sigma_{M}=0.27$ (Bennett et al., 2003).

As Mrk 50 has not been explored in the high-energy domain to date, our first motivation is to characterize the spectrum of this source. For this purpose, we begin our spectral fitting with a set of phenomenological models to characterize the source spectra at different epochs and determine the spectral features quantitatively. Further, these phenomenological models are replaced with more sophisticated physical models to better understand the physical properties of the source.

Initially, we consider the 3–10 keV X-ray continuum spectra of the source for the spectral fitting. According to current understanding, the X-ray continuum photons are produced through the inverse Compton scattering of thermal photons from the accretion disk in a hot electron cloud (Shakura & Sunyaev, 1973), though the geometry and location of this hot electron cloud are poorly understood. This non-thermal process leads to a power-law-type spectrum. Therefore, we applied a simple Powerlaw model to fit the spectrum of each observation. Along with this, we use the Galactic line of sight hydrogen column density ($N_{\rm H,Gal}$) as the multiplicative model TBabs (Wilms et al., 2000) in XSPEC. The value of $N_{\rm H,Gal}$¹¹¹¹11https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl used in our work is $1.71\times 10^{20}\rm cm^{-2}$. We note that we did not observe any prominent (positive) residual in the 6–7 keV energy range during the continuum fitting of any of the five sets of spectra (see left panel of Figure 2). This indicates the absence of the Fe-line in the X-ray continuum of Mrk 50. So, for the continuum, the baseline model is ${\tt Const\times TBabs\times Powerlaw}$ for all observations. The Constant component is used as a cross-normalization factor while using data from different instruments in simultaneous spectral fitting (see Table A1).

After successfully parameterizing the primary continuum in the 3-10 keV range, we extend the spectrum into the low-energy domain (below 3 keV), where we find distinct results across different epochs of observations (see right panel of Figure 2). In the case of S1, we notice deviation in the low-energy data points from the primary continuum, which is attributed to the presence of soft excess at $\rm E<2keV$. To address the presence of soft excess in this observation, we consider another power-law component below 3 keV (Walter & Fink, 1993; Nandi et al., 2021). To ensure the significance of this additional $\tt Powerlaw$ component, we conducted an F-test¹²¹²12https://heasarc.gsfc.nasa.gov/xanadu/XSpec/manual/node82.html for the additive component, and find the $F_{value}$ to be 9.86 with a probability of chance improvement $1.40\times 10^{-5}$. Using the F-test, we verified that an additional component is required. We also find evidence of soft excess in X1 & X2 observation.

However, for observations S2 and S3+NU, it is surprising that the low energy spectra (below 3 keV) do not show the presence of any excess emission over the continuum. The primary continuum fits the extended low energy range of these observations. The exposure times for S2 and S3 observations are shorter compared to the other observations, which may explain the non-detection of the soft excess component below 3 keV. However, for further clarification, we performed an F-test on these spectra. The F-test confirms that adding the extra power-law component is insignificant and did not improve the fit.

To investigate the presence of any intrinsic absorption along the line of sight of Mrk 50, we initially used the neutral absorption model component, zTbabs, with our baseline model and found that the estimated value of $N_{H}$ is $<0.001\times 10^{22}cm^{-2}$ for $z=0.024$. To investigate further, we replace zTbabs with a partially covering absorption model Pcfabs to check for the presence of any partial absorbers present in the line of sight. However, this model is also found to be insensitive during spectral fitting and consistently yields the lowest value of the absorption parameter. We considered UV data for these observations in our spectral fitting to ensure our findings. However, the inclusion of UV data in the fitting did not detect the presence of any extragalactic absorption component in Mrk 50. As a result, we drop this zTbabs component from our composite model. The corresponding model in XSPEC read as ${\tt Const\times TBabs\times(powerlaw+powerlaw})$ and is used to fit all the spectra in the 0.3 – 10 keV range. Later, we extend the low energy part into the UV domain whenever the UV observations are available.

In the high energy domain ($>$10 keV), we combine spectra from the Swift/XRT, NuSTAR, and Swift/BAT to obtain a broad-band X-ray spectrum up to 100 keV. While extrapolating the primary continuum model to the high-energy range, we did not find any deviation in the data points from the primary model (see Figure 2). This suggests that the reflection component in the X-ray spectra above 10 keV is either absent or insignificant during this observation.

After characterizing each spectrum of all the observations with simple models, such as power-law, we use the phenomenological model Diskbb to address the soft X-ray excess (see Section 3.3.3) in this source. To better understand the physical nature of the soft X-ray excess emission in Mrk 50, we use Optxagnf and Relxillcp as physical models. The results are discussed in the following sections.

In this section, our motivation is to understand the physical origin of each spectral component and its evolution over time. The origin of the soft excess component is yet to be understood. To explore this, we considered two possible physical scenarios, warm Comptonization (Done et al., 2012) and relativistic blurred reflection (García et al., 2018), on the observed spectra to investigate the physical origin of this component.

In our spectral analysis of Mrk 50 (Section 3.2), we found that the emission in the low energy domain, starting from 1 eV to $\sim$3 keV, is unaffected by the intrinsic hydrogen column densities along the line of sight. We used various absorption models to detect the contribution of neutral or ionized hydrogen along the line of sight to the observed spectra. However, each model failed to detect any trace of extragalactic hydrogen along the line of sight. Therefore, all observations of Mrk 50 from 2007 to 2022 are free from neutral and/or ionized absorptions in the UV/X-ray domain. Similar results are reported for other AGNs with a ‘bare’ nucleus at the center (Vaughan et al., 2004; Walton et al., 2010; Nandi et al., 2023, 2024). The lack of intrinsic absorption has also been previously reported for Mrk 50 by Vasudevan et al. (2013). Our preliminary examination of these observations confirms that Mrk 50 is indeed a ‘bare’ Seyfert 1 nucleus during the observational period from 2007 to 2022. For further clarification, one would need to examine optical observations, which is beyond the scope of this work.

In the previous sections, we presented spectral fit results using several phenomenological and physical models applied to the data of Mrk 50. We observed that in S1 observation, the total X-ray flux (0.3–10 keV) was $2.04^{+0.08}_{-0.05}\times 10^{-11}$  erg cm^-2 s^-1, which increased by approximately 1.4 times ($\sim 2.85\pm 0.03\times 10^{-11}$  erg cm^-2 s^-1) in the X1 observation. However, within a year, the flux dropped by a factor of approximately 1.5 in the X2 observation. The calculated flux in X2 observation was $1.76^{+0.02}_{-0.03}\times 10^{-11}$  erg cm^-2 s^-1. After that, the total X-ray flux decreased to $1.12\pm 0.03\times 10^{-11}$  erg cm^-2 s^-1  in the most recent observation S3+NU, which is approximately 2.5 times less than the flux in the X1 observation. From Table 7, it is evident that both the hard X-ray and UV monochromatic flux varied significantly between the observations. The UV monochromatic flux (UVW1) was $7.23\pm 0.14\times 10^{-15}$  erg cm^-2 s^-1 Å^-1in S1 and then declines to $3.80\pm 0.11\times 10^{-15}$  erg cm^-2 s^-1 Å^-1in S3+NU observation.

The variation in the X-ray flux over time may arise due to changes in the strength of Comptonization inside the Compton cloud. It is also believed that these variations may be attributed to changes in the UV photon flux. Simultaneous UV and X-ray observations allow us to probe these types of spectral variability. For this purpose, a hypothetical power-law between 2500 Å and 2 keV, known as $\alpha_{OX}$ (Tananbaum et al., 1979), is estimated from the simultaneous UV and X-ray observations. The parameter $\alpha_{OX}$ is defined as $\alpha_{OX}$=$-0.384\rm log[L_{2500}/L_{2keV}]$. We used the UVW1 band to compute $\alpha_{OX}$ because it has the closest effective wavelength to 2500 Å . The observed $\alpha_{OX}$ values remain nearly constant at $\sim 1.24$ across all observations, suggesting that the X-ray and UV photons vary in a similar manner. The steeper value of this parameter indicates that the X-ray emission is dimmed more than the UV emission. This phenomenon is commonly observed in the case of most AGNs (Strateva et al., 2005). According to Gallo (2006), steep $\alpha_{OX}$ is often accompanied by X-ray spectral complexity such as enhanced blurred reflection. Due to relativistic effects, most of the emission from the primary power-law component bends towards the black hole, preventing it from reaching the observer (Miniutti & Fabian, 2004; Pal et al., 2018). In comparison, the UV emission (produced by the colder accretion disc) remains relatively unchanged. A relatively weaker X-ray and constant UV emission makes $\alpha_{OX}$ steep.

During the spectral fitting, the softest spectrum is observed in the S1 observation with the primary continuum power-law slope of $1.88^{+0.27}_{-0.28}$ (see Table 3). The normalized accretion rate for this observation is estimated to be $-0.87\pm 0.03$ (see Table 5), the highest among all observations. As the accretion rate is high, the emission from the disk is significant, indicating an increased supply of seed photons. As a result, a soft spectrum is observed for this observation. With the increase in the supply of seed photons, more photons interact within the Compton cloud, increasing the luminosity of the primary continuum. For this observation, the luminosity of the primary continuum is found to be $\rm logL_{PC}=43.58^{+0.02}_{-0.04}$, the highest among all the observations used in the present work. Due to the high accretion rate, the UV flux also increased. We observed the highest UV fluxes of $10.94\pm 0.20\times 10^{-15}$, $8.77\pm 0.19\times 10^{-15}$ and $7.23\pm 0.14\times 10^{-15}$  erg cm^-2 s^-1 Å^-1(see Table 7) for the wavelength 1928 Å(UVW2), 2310 Å(UVM2)  and 2930 Å(UVW1), respectively, for this observation. As both UV and X-ray fluxes increased, we found the UV to X-ray slope, $\alpha_{OX}=1.20$ (see Table 7), which is comparable with other observations.

From the spectral analysis, a marginal presence of soft excess was observed in the S1 observation (see the first panel of Figure 2). Initially, this excess component was characterized by a power-law with a photon index ($\Gamma_{SE}$) of $1.91^{+0.03}_{-0.04}$ (see Table 3). Later, it was described by a multicolour disk blackbody ($\tt Diskbb$) with inner disk temperature ($kT_{e}$) at $0.16^{+0.02}_{-0.01}$ keV (see Table 4). If the soft excess is attributed to the Comptonization of seed photons from a warm corona, the optical depth of the corona is estimated to be $29^{+4}_{-3}$, with a dimension of $62^{+19}_{-9}~{}R_{g}$ (see Table 5). On the other hand, if the soft excess results from reflection processes, the reflection coefficient is determined to be $R_{f}=2.83^{+0.66}_{-0.21}$ with a photon index $\Gamma_{Ref}=1.91^{+0.12}_{-0.04}$ (see Table 6).

After a two-year gap from the S1 observation, the next X-ray observation was conducted with XMM-Newton  in 2009 (X1). The X-ray continuum became hard during this observation with a photon index of $\Gamma_{PC}=1.72\pm 0.05$. The luminosity of the primary continuum marginally decreased from $\rm logL_{PC}=43.58^{+0.02}_{-0.04}$ to $\rm logL_{PC}=43.36^{+0.02}_{-0.01}$ (see Table 3) within two years. A decrease in luminosity suggests a decrease in the accretion rate, estimated to be $-1.62\pm 0.01$, lower than the previous observation (see Table 5).

With decreasing accretion rate, the number of seed photons decreases. The decreasing number of seed photons is unable to cool the warm corona efficiently due to less scattering and is most likely responsible for the expansion of the warm corona. In this work, we found an increase in the value of $r_{cor}$ and a decrease in the accretion rate as found in other AGNs, e.g., Mrk 1018 (Noda & Done, 2018) and NGC 1566 (Tripathi & Dewangan, 2022). This decrease in accretion rate also results in a reduction in UV flux, measured at $7.26\pm 0.07\times 10^{-15}$ and $6.23\pm 0.10\times 10^{-15}$  erg cm^-2 s^-1 Å^-1for wavelengths 2310Å  and 2930Å, respectively (see Table 7). Although both UV and X-ray fluxes decrease, the slope remains constant ($\alpha_{OX}=1.20$, as noted in Table 7).

From spectral analysis, a strong soft excess is observed in this observation (second panel of Figure 2). This soft excess is characterized by a power-law with photon index and luminosity of $\Gamma_{SE}=2.63\pm 0.04$ and $\rm logL_{SE}=43.64\pm 0.01$, respectively. This represents the highest soft excess luminosity among all observations from 2007 (S1) to 2022 (S3+NU). The optical depth and coronal radius are also found to maximum ($\tau=36^{+3}_{-2}$ and $r_{cor}=98^{+1}_{-3}~{}R_{g}$; see Table 5) during this observation. This higher optical depth and larger corona generate more soft X-ray photons, resulting in the strong presence of soft excess. The reflection model suggests a stronger reflection during this observation, with the inner disk deviating from Newtonian geometry (inner emissivity index $q_{1}=3.52^{+1.00}_{-0.57}$), producing a steeper power-law with photon index $\Gamma_{Ref}=2.78\pm 0.04$.

During the X2 observation, the observed soft excess is relatively low, irrespective of the similar nature of the primary power-law continuum. The slope of the primary continuum and corresponding luminosity during this observation are $\Gamma_{PC}=1.70\pm 0.05$ and $\rm logL_{PC}=43.38^{+0.03}_{-0.02}$, respectively, consistent with the X1 observation. The soft excess luminosity declined from $\rm logL_{SE}=43.64^{+0.02}_{-0.01}$ during X1 observation, to $\rm logL_{SE}=43.15^{+0.02}_{-0.01}$ during X2 observation. This decrease in the soft excess can be explained by the warm Comptonization model. Using this model, we found that the optical depth decreased from $\tau=36^{+3}_{-2}$ to $\tau=18\pm 2$, while the size of the warm corona contracted from $98^{+1}_{-3}~{}R_{g}$ to $50^{+31}_{-20}~{}R_{g}$. A smaller size of warm corona with a relatively small optical depth produces less number of X-ray photons in soft X-ray band. Therefore, the observed decrease in the soft excess luminosity during observation X2, compared to that during X1, is due to the decrease in the size and optical depth of the warm corona. We also attempted to describe the soft excess using the reflection model. In this model, we also noticed that the value of the reflection coefficient decreased from $6.53^{+1.43}_{-1.21}$ to $5.73^{+2.86}_{-1.38}$. As a result, fewer photons were effectively reflected to produce excess emission below 2 keV.

During the S2 observation, we encountered the hardest spectrum of Mrk 50 with an index and corresponding luminosity of the primary continuum as $\Gamma_{PC}=1.54^{+0.42}_{-0.40}$, and $\rm logL_{PC}=43.46^{+0.21}_{-0.36}$, respectively. From the physical model fitting, the accretion rate is estimated to be ($-1.77^{+0.05}_{-0.03}$), the lowest among all the observations. We also noticed that the amount of soft excess is remarkably reduced and falls below the detection limit. This is possibly due to the low exposure time of this observation. In this case, the slope of the power-law representing the soft excess is comparable with the slope of the primary continuum. Although the effective size of the warm corona was not reduced at the time of observation, the optical depth decreased from $\tau=18\pm 3$ (X2 observation) to $\tau<5$ (S1 observation). Consequently, very few excess photons are produced in the soft X-ray range. Therefore, most of the photons observed in the soft X-ray regime are from the primary continuum through the process of Comptonization in the hot corona. From the reflection model perspective, we found that the reflection coefficient is $0.52^{+0.72}_{-0.49}$, much lower than other observations. As a result, fewer reflected photons are produced to contribute towards the excess emission in the soft X-ray domain. Additionally, we found that the dimensions of the Compton cloud or corona are of the same order as calculated from different models for this observation.

The latest observation in the X-ray domain was conducted with the Swift  and NuSTAR  observatories in 2022 (S3+NU). Using these observations, we obtained a broadband spectrum of Mrk 50 starting from UV to hard X-ray range. We noticed that while extending the primary continuum up to 100 keV, no deviation was seen in the spectra, indicating the absence of a reflection hump above 10 keV. During this observation, the primary continuum luminosity $\rm logL_{PC}=43.14^{+0.02}_{-0.06}$, lowest among all the observations, with $\Gamma_{PC}=1.77\pm 0.07$. The normalized accretion rate increased from $-1.77^{+0.05}_{-0.03}$ (S2 observation) to $-1.20^{+0.03}_{-0.05}$ during this observation. Since the soft excess component is below the detection limit, determining parameters associated with the warm corona is challenging. However, spectral analysis indicates an optical depth of $\tau<3$ and a transitional radius of $r_{cor}>10~{}R_{g}$ for this observation. Therefore, the observed broadband spectrum (from UV to hard X-ray range) is dominated by the primary continuum with power-law fraction $f_{pl}>0.95$. From the reflection model perspective, the whole disk followed the Newtonian geometry with a negligible amount of reflection ($R_{f}=0.48^{+0.36}_{-0.24}$). Thus, photon contribution in the soft X-ray band is nearly negligible. Consequently, a single power-law model is sufficient to fit the broadband spectrum from 1 eV to 100 keV.

For the overall picture, we observed that the slope of the primary continuum varied from $1.54^{+0.42}_{-0.40}$ to $1.88^{+0.27}_{-0.28}$. However, the primary continuum luminosity remained nearly constant, with an average value of $\rm logL^{avg}_{PC}=43.38^{+0.19}_{-0.18}$. In contrast, for the soft excess component (below 3 keV), we found that both the power-law index($\Gamma_{SE}$) and luminosity ($\rm logL_{SE}$) varied within the ranges of $1.77$ to $2.63^{\pm}0.04$ and $42.91$ to $43.64^{+0.02}_{-0.01}$, respectively. We explored the possible physical reasons for this spectral variability and found that the variation in the primary continuum can be explained in terms of accretion dynamics. Depending on the accretion rates and the efficiency of the Compton cloud, the slope of different spectral components and the corresponding luminosity change. The long-term variations of a few spectral parameters are presented in Figure 7.

In the previous section, we explained the observed spectral variabilities in Mrk 50 using various spectral parameters derived from different model fittings (see Section 3.2). Although all of the model parameters contribute to the overall spectral variability, they are not entirely independent of each other. In this subsection, we examine the correlations between different spectral parameters. We calculated the Correlation Coefficient¹³¹³13https://www.statskingdom.com/correlation-calculator.html (CC) to calculate the degree of correlation between the parameters. The correlations between some selected parameters are shown in Figure 10. This limited number of observations (five) in a period of 15 years makes it challenging to draw definitive physical conclusions about the correlations between different parameters. However, we discuss the observed trends in the correlation between different parameters on this 15-year timescale. Based on these correlation studies, we infer the nature of the source and understand the overall spectral properties of Mrk 50.

We began our spectral analysis by parameterizing the observed spectra using a power-law model, calculating the spectral slopes and luminosities and/or fluxes of different spectral components, including the soft excess and the primary continuum. We observed hints of correlations between luminosities and fluxes in the hard and soft X-ray regions, with correlation coefficients (CC) of 0.66 ($p-value=0.22$) and 0.78 ($p-value=0.11$), respectively, as shown in Figure 10 panels (a) and (b). Similar correlations have been observed in other AGNs, such as IC 4329A  (Tripathi et al., 2021), PKS 0558-504  (Gliozzi et al., 2013), NGC 7469  (Nandra & Papadakis, 2001) and have been interpreted in terms of thermal Componization of soft photons in the hot corona. As the number of soft photons increases, the number of scattering in the hot corona also increases, leading to an enhancement in the Comptonized X-ray flux. This common feature is also observed in ‘bare’ AGNs (Nandi et al., 2021, 2023).

Additionally, we found that the UV flux correlates with both soft and hard X-ray fluxes, with CC=0.61 ($p-value=0.39$) for soft X-ray vs. UV, and CC=0.79 ($p-value=0.20$) for hard X-ray vs. UV. These correlations indicate a relationship between the disk and the hot corona, and a link between the disk and the medium producing the soft excess. Furthermore, we observed a correlation trend between the spectral slope of the primary continuum ($\Gamma_{PC}$) and the normalized accretion rate $(\lambda_{Edd})$, with a CC of 0.89 ($p-value=0.09$). The $\Gamma_{PC}$-$\lambda_{Edd}$ correlation indicates the connection between the accretion disk and the X-ray corona. The physics behind this correlation is not clear. One possible explanation is that a higher accretion rate increases the supply of soft photons; an increase in the supply of soft photons can produce more hard photons by interacting with the Compton cloud. Consequently, the power-law index steepens, leading to the observed correlation between $\Gamma_{PC}$ and $\lambda_{Edd}$ (Fabian et al., 2015; Ricci et al., 2018; Vasudevan & Fabian, 2007; Layek et al., 2024).

Next, we applied physical models to the observed spectra to explain the nature and origin of the different spectral components. We found a hint of a correlation between the reflection coefficient ($R_{f}$) and the slope of the soft excess component ($\Gamma_{SE}$). Since the $relxillcp$ model fits the X-ray spectra well, this correlation may suggest a link between $R_{f}$ and $\Gamma_{SE}$ in the X-ray band, implying that the soft excess component may be associated with the reflection phenomena. We also explored the correlation between the UV to X-ray index, $\alpha_{OX}$, and various spectral parameters, finding that $\alpha_{OX}$ is anti-correlated with $\lambda_{Edd}$, with $CC=-0.87$ ($p$-value=0.11). Since $\alpha_{OX}$ remains nearly constant at 1.24 across our observations, and the values from different observations fall within the range of uncertainties, it is challenging to comment on this anti-correlation conclusively. The dependence of $\alpha_{OX}$ with $\lambda_{Edd}$ suggests that the disc/corona relative intensity also depends on the accretion rate (Fanali et al., 2013).

In this study, we explored correlations between various parameters and found hints of correlations between the spectral parameters. These correlations can be explained from a physical perspective. However, it is important to note that the $p$-values associated with these correlations are greater than 0.05, indicating that the correlations may not be statistically significant at the 5% level. This suggests that the observed relationships could potentially be due to random variability. Further investigation or a larger sample size may be needed to draw more accurate conclusions.

An enhancement of flux in the soft X-ray range (below $\sim 2$ keV) over the primary power-law continuum is commonly observed in most of the Seyfert 1 AGNs. This excess in the low energy range is known as soft excess. Our spectral analysis revealed the presence of a prominent soft excess below 2 keV in Mrk 50. However, this soft excess component is found to change over time. From the flux calculations, we observed significant variability in soft excess flux between observations of Mrk 50 (see Table 7). This appearance and disappearance of the soft excess component intrigued us to explore its origin. Although the origin of this excess emission in the soft X-ray domain is a long-standing and unsolved puzzle in AGN studies, we attempted to unravel this mystery using various models discussed in Section 3.3.