A Sensitive Search for Supernova Emission Associated with the Extremely Energetic and Nearby GRB 221009A

We used the GIT located at the Indian Astronomical Observatory (IAO), Hanle-Ladakh, to acquire observations of the GRB 221009A optical afterglow (Kumar et al. 2022b). The source was observed in Sloan g', r', i', and z' bands. Data were downloaded and processed in real time by the GIT data reduction pipeline. We used individual exposures for photometry in the early stages when the afterglow was bright. Later, we stacked images with SWarp (Bertin 2010) to increase the signal-to-noise ratio (S/N) of the detections.

The data were reduced in a standard manner using the GIT pipeline (Kumar et al. 2022c). All images were preprocessed by subtracting bias, flat fielding, and cosmic-ray removal via the Astro-SCRAPPY (McCully & Tewes 2019) package. Astrometry was performed on the resulting images using the offline solve-field astrometry engine. Sources were detected using SExtractor (Bertin & Arnouts 1996) and were crossed-matched against the Pan-STARRS1 DR1 catalog (PS1; Flewelling et al. 2020) through VizieR to obtain the zero-point. Finally, the pipeline performed point-spread function (PSF) fit photometry to obtain the magnitudes of the GRB 221009A afterglow (Table 1).

We also observed GRB 221009A in ${r}^{{\prime} }$, ${i}^{{\prime} }$, and ${z}^{{\prime} }$ with the 4.3 m Large Monolithic Imager (LMI) on the LDT through an approved Target-of-Opportunity (ToO) program. We reduced the images using a custom Python-based image analysis pipeline (Toy et al. 2016), that can perform data reduction, astrometry, registration, source extraction, and PSF photometry using SExtractor, which was calibrated using point sources from the PS1 catalog (Table 1). These observations were also reported in O'Connor et al. (2022a, 2022b, 2023). Figure 1 shows both the wider field of view of GRB 221009A's position on the sky, as well as the individual evolution of its flux over time in both filters.

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Two additional publicly available i'-band observations obtained with GMOS-S mounted on the 8.1 m Gemini-South Telescope (PIs: Rastinejad, O'Connor; Rastinejad & Fong 2022) were also analyzed. The data were reduced using DRAGONS ²⁷ (Labrie et al. 2019) to align and stack individual frames. PSF photometry was calibrated using nearby point sources in the PS1 catalog (Table 1).

We draw upon three results from previous studies of the broadband afterglow of GRB 221009A in order to inform our approach to characterizing the associated SN emission. First, fits to the broadband spectral energy distribution (SED) revealed that the frequency range from optical to hard X-rays is not well fit by a single power law (i.e., f_ν ∝ ν^−β ). Instead, a change in the spectral slope, physically attributed to the synchrotron cooling frequency (ν_c ), is inferred around the X-ray bandpass (O'Connor et al. 2023; Williams et al. 2023). As a result, in standard synchrotron afterglow theory, we do not expect the optical afterglow to decay with the same power-law index as the X-rays (e.g., Sari et al. 1998). Thus, to remove the afterglow contribution we must model the optical decay separately from the X-rays (see Fulton et al. 2023 and the first approach in Shrestha et al. 2023).

Second, in addition to significant extinction due to dust in the Milky Way galaxy, broadband SED fits indicate the possible existence of absorption beyond the nominal value of E(B − V) = 1.30 mag reported in Schlafly & Finkbeiner (2011). Given the low redshift, it is difficult to disentangle a larger than expected Galactic extinction (e.g., due to small-scale variations in Galactic dust), or extinction in the GRB host galaxy. Existing works differ on the significance of this extinction component, with inferred values ranging from E(B − V) = 1.30 mag (i.e., no additional extinction; O'Connor et al. 2023) to E(B − V) = 1.80 mag (Fulton et al. 2023; Williams et al. 2023). Given these uncertainties, we consider the implications of differing host extinction levels throughout this work.

Finally we correct for underlying host galaxy light in our analysis, derived by Levan et al. (2023) through GALFITM modeling of late-time Hubble Space Telescope (HST) observations. We use their measurements of F625W = 24.88 ± 0.08 mag and F775W = 23.80 ± 0.14 mag, which approximately correspond to the r' and i' bands. Previous analyses (Fulton et al. 2023; Kann et al. 2023; Shrestha et al. 2023) did not incorporate the host contribution explicitly in their analysis (though Pan-STARRS and DECam photometry in Fulton et al. 2023, and DECam photometry in Shrestha et al. 2023 are subjected to template subtraction, which negates the host contribution to a degree).

Assuming the optical afterglow is powered by synchrotron emission from the forward shock (Mészáros & Rees 1997), we fit the early-time light curve with power-law models (f_ν ∝ t^−α ) to attempt to isolate the contribution from the SN. Prior to T₀ + 1 day, the optical data display a shallow initial slope with α = 0.88 ± 0.05 (Kann et al. 2023; O'Connor et al. 2023). Beyond this time there is a clear steepening in the decay, and a single power law does not provide a good fit to all optical data (D'Avanzo et al. 2022).

Since a SN will contribute negligibly (compared to the bright afterglow here) in the first days postexplosion, we perform a power-law fit of all g', r', i', and z' data between 1 and 6 days postexplosion. We find a best-fit decay index of α = 1.434 ± 0.004. As found by other authors, this index is appreciably shallower than the X-ray decay found at this time (Laskar et al. 2023; O'Connor et al. 2023; Williams et al. 2023). Figure 2 shows the best-fit power law derived, along with the addition of the host galaxy emission to that power law.

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The late-time behavior shows possible deviations from this power law, even after the host galaxy emission is considered. Though it is possible that central-engine activity can cause rebrightening of the optical afterglow in excess of what is expected from a power-law decay, this usually occurs directly after the prompt emission in the early-time evolution of the GRB (Kann et al. 2007; Oates et al. 2009). Because the brightening occurs weeks after the initial prompt emission (see Figure 1), we determine that there is the possibility that an associated SN is contributing excess flux to the optical emission, and further investigate this in Section 3.3.

In order to evaluate the statistical significance of the excess late-time emission, we perform a Bayesian model selection using the PyMultiNest package (Feroz et al. 2009; Buchner et al. 2014). We consider two empirical models—one where the optical emission is explained as the sum of the afterglow (i.e., synchrotron radiation) and a (constant) host galaxy emission, and one where an additional SN component is added. For each model, we calculate the Bayesian evidence, and use it to calculate the Bayes factor to see which is statistically preferred (Trotta 2008). We limit the analysis to the r' and i' bands, as the large Galactic extinction makes g' band uninformative, and z' band SN templates are less available due to a lack of spectral coverage in that wavelength range.

For the afterglow+host model, we assume a single power-law decay in both bands at times after T₀ + 1 day. While the index is fixed to be identical in both the r' and i' bands, we do not require its value be equal to that derived from 1 to 6 days postexplosion (Section 3.2); rather, we allow the index to vary to provide the best fit to the entire data set. The host galaxy emission is incorporated as a free parameter in the fit, using a Gaussian prior with the mean and standard deviation measured by HST (Levan et al. 2023).

For the SN model, an additional component is added to the afterglow+host model to mimic SN behavior. We take the light curve of SN 1998bw (Clocchiatti et al. 2011), and (de)redden and K-correct it to match the relevant properties of GRB 221009A, using SNCosmo (Barbary et al. 2016). Specifically we adopt E(B − V)_MW = 1.30 mag (Schlafly & Finkbeiner2011), E(B − V)_host = 0.3 mag, which is the most conservative value found in Williams et al. (2023), and apply the Milky Way extinction law from Cardelli et al. (1989) with R_V = 3.1. The resulting r' and i'-band light curves for a SN 1998bw-like source at z = 0.151 [and behind an extinction of E(B − V)_MW = 1.30 mag and E(B − V)_host = 0.3 mag] are shown in Figure 2.

The final SN model used in the model selection has two free parameters: a flux-stretching factor ${k}_{\mathrm{SN}\ \,\ 1998\mathrm{bw}}$ and a time-stretching factor ${s}_{\mathrm{SN}\ \,\ 1998\mathrm{bw}}$ (e.g., Klose et al. 2019). Both of these parameters have priors drawn from a bivariate normal distribution fit to values derived for previous GRB-SNe (Cano et al. 2017). We also allow the afterglow power-law decay index (α) to be free with uniform priors, the flux constants of proportionality (a_AG) to be free with uniform priors in log space, and add the host galaxy emission in both bands utilizing the same Gaussian priors. Therefore, the full SN model is:

$$\begin{eqnarray}&&\begin{array}{l}{f}_{\nu }({t}_{\mathrm{obs}})={k}_{\mathrm{SN}}({f}_{\nu }^{\mathrm{SN}\ 1998\mathrm{bw}}({t}_{\mathrm{obs}}/{s}_{\mathrm{SN}\,\ 1998\mathrm{bw}}))\\ +{a}_{\mathrm{AG}}{\left({t}_{\mathrm{obs}}\right)}^{-\alpha }+{f}_{\nu }^{\mathrm{host}},\end{array}\end{eqnarray} \tag{ 1 }$$

where ${f}_{\nu }^{\mathrm{SN}\,1998\mathrm{bw}}({t}_{\mathrm{obs}})$ is the flux seen of the SN at a time in the observer frame, t_obs is the time in the observer frame, and ${f}_{\nu }^{\mathrm{host}}$ is the flux contribution from the host galaxy.

Initially, we fit the two models concurrently to the r' and i' bands, while assuming the errors are the nominal values reported in Table 1 and Laskar et al. (2023) for the LT photometry. We calculate the Bayes factor (K_Bayes) between the two models to determine the likelihood that an SN is contributing excess flux to the optical afterglow in addition to the host galaxy emission, and find K_Bayes = 10^4.0, which indicates that the SN model is strongly favored. However, when we calculate the χ² statistic for each of the models, we find that ${\chi }_{\mathrm{PL}+\mathrm{host}}^{2}=120.83$ for 62° of freedom and ${\chi }_{\mathrm{PL}+\mathrm{host}+\mathrm{SN}}^{2}=112.62$ for 60° of freedom. Though the Δχ² = 8.21 for two additional degrees of freedom is also indicative of a preference for the SN model, the χ² statistics themselves indicate that neither model adequately fits the data. This is likely because in the initial fitting, we did not account for systematic uncertainties that arise from combining data from multiple telescopes in the full photometric data set, or S-corrections (Stritzinger et al. 2002).

Therefore, we modify the fitting procedure to optimize the likelihood function numerically, where we assume that the reported errors underestimate the true uncertainty. To account for this, we include an error-stretching factor (β) in the fitting procedure to represent the S-correction effect, and recalculate K_Bayes. The log-likelihood function we minimize, with the addition of the error-stretching factor, is:

$$\begin{eqnarray}&&\mathrm{ln}\,p(y\,| \,m,{s}_{n})=-\displaystyle \frac{1}{2}\displaystyle \sum _{n}\displaystyle \frac{{\left({y}_{n}-{m}_{n}\right)}^{2}}{{s}_{n}^{2}}+\mathrm{ln}(2\pi {s}_{n}^{2}),\end{eqnarray} \tag{ 2 }$$

where y_n are the observed data, m_n are the modeled data, and s_n ² are:

$$\begin{eqnarray}&&{s}_{n}^{2}={\sigma }_{n}^{2}+{\beta }^{2}({m}_{n}^{2}),\end{eqnarray} \tag{ 3 }$$

where σ_n are the nominal errors to the observed data.

The new Bayes factor we find is K_Bayes = 10^1.2, which indicates the SN model is moderately, but not conclusively, preferred. We report the median parameters with their 1σ errors, along with the best-fit parameters which minimize the log-likelihood function in Table 2. The best-fit models are shown in Figure 3, along with their associated χ values. According to the model selection analysis, the optimal error-stretching factor increases the error bars by ∼3% with respect to the model value at the observed time, for both the afterglow+host and SN model. We also calculated the Bayesian evidence for the two models, while incorporating independent error-stretching factors (β_i ) for each telescope rather than a single factor across all data sets. We did so to investigate if the way we account for systematic uncertainties plays a role in biasing the model preferences. We still find similar results, as the SN model is favored by a Bayes factor of K_Bayes = 10^0.7.

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Table 2. Bayesian Model Selection Fitting Parameters for the Afterglow+host and the Afterglow+Host+SN Models

	${a}_{{\rm{r}}^{\prime} }$	${a}_{{\rm{i}}^{\prime} }$	α	${f}_{\nu ,{\rm{r}}^{\prime} }^{\mathrm{host}}$	${f}_{\nu ,{\rm{i}}^{\prime} }^{\mathrm{host}}$	${k}_{\mathrm{SN}\,1998\mathrm{bw}}$	${s}_{\mathrm{SN}\,1998\mathrm{bw}}$	$\mathrm{ln}$ (β)
	(μJy)	${a}_{{\rm{i}}^{\prime} ,\ \mathrm{AG}}$		(μJy)	(μJy)
Afterglow+Host (Best fit)	146	388	1.43	0.44	1.64	⋯	⋯	−3.37
Afterglow+Host (Median ±1σ)	144 ± 2	${384}_{-6}^{+5}$	1.42 ± 0.01	0.41 ± 0.03	1.33 ± 0.11	⋯	⋯	−3.30 ± 0.21
Afterglow+Host+SN (Best fit)	147	394	1.46	0.35	1.17	0.39	0.69	−3.51
Afterglow+Host+SN (Median ±1σ)	146 ± 2	${395}_{-6}^{+5}$	1.47 ± 0.01	0.39 ± 0.03	${1.12}_{-0.12}^{+0.11}$	0.43 ± 0.10	0.63 ± 0.09	$-{3.50}_{-0.24}^{+0.23}$

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Next, we analyze the LDT and LT photometry separately using their nominal error bars, to identify if the combination of photometry from different telescopes plays a role in biasing the model preferences. We do not fit the Gemini-South and GIT photometry separately, as there are only two Gemini-South photometry points, and the GIT photometry are all at early times where any SN excess contributing to the afterglow would be negligible. We find that the afterglow+host model is favored with a Bayes factor of K_Bayes = 10^0.9 for the LDT photometry, while the two models are indistinguishable for the LT photometry. Figure 2 shows that the majority of the LDT photometry is at late times, where excess flux from a SN would be identifiable. This, in addition to only five and six photometry points being available respectively in the r' and i' bands leads to the model selection converging toward a shallow power-law decay slope in its fitting procedure, in order to fit for the excess flux. The best-fit power-law decay index is α = 1.36 (median ± 1σ = 1.36 ± 0.02), which is shallower than what is derived from the early-time optical photometry fitting in Section 3.2. Therefore, although the afterglow+host model is preferred for the LDT data, the lack of early-time data biases the fit toward a power-law decay index that is shallower than expected in order to fit for the excess flux.

On the other hand, the majority of LT data are at early times and at a significantly higher cadence than the LDT photometry. The best-fit power-law decay index for the afterglow+host model is α = 1.46 (median ± 1σ = 1.45 ± 0.01), which is consistent with the power-law decay index derived in Section 3.2. The two models are indistinguishable due to a few photometry points that do show excess emission at later times, for which the afterglow model cannot account. Therefore, it is necessary to incorporate a combination of data sets in the modeling, in order to have optimal temporal coverage such that a sufficient amount of photometry points at both early and late times is accounted for in the fitting. After accounting for S-corrections modeled through our error-stretching factor, it is clear that the most accurate physical description of the light curve comes from incorporating the entire data set, which we find favors the SN model moderately.

As a final sanity check, we perform a similar analysis using the observed light curve of SN 2013dx associated with GRB 130702A (D'Elia et al. 2015; Toy et al. 2016) with the extinction values described above, again optimizing the likelihood function and allowing for an error-stretching factor. Due to the similar redshift (z = 0.145) to GRB 221009A, this avoids the requirement of calculating K-corrections (Fulton et al. 2023; Shrestha et al. 2023). In this case we find a comparable Bayes factor to the analysis using SN 1998bw: K_Bayes = 10^0.7, preferring the SN model. Thus we infer that the SN model preference is relatively insensitive to the details of the template SN used. We also emphasize that the detailed value of the host extinction adopted has little impact on the model selection. Since the SN flux is scaled by the free parameter ${k}_{\mathrm{SN}}$, increasing or decreasing the host extinction is largely offset in the modeling by a corresponding change in ${k}_{\mathrm{SN}}$.

Our preference for the SN model agrees with Fulton et al. (2023), as they find significant evidence of excess emission in the optical afterglow that is well modeled by an additional SN component. However, they did not account for host galaxy emission explicitly in their analysis, and Figure 2 shows that the host galaxy makes nonnegligible contributions to the optical afterglow at late times. This is likely why they were able to find significant evidence of excess emission, while our preference for the SN model is moderate. Shrestha et al. (2023) and Levan et al. (2023) report no evidence for bright SN emission, while Kann et al. (2023) does not find any strong evidence for or against SN emission. None of them rule out the possibility of a faint associated SN, and one of the conclusions of Levan et al. (2023) is that an associated SN to GRB 221009A must be either substantially (∼10%–40%) fainter or bluer than SN 1998bw. Our findings point toward the former being true, as the best-fit flux-scaling factor for SN 2022xiw with respect to SN 1998bw is ${k}_{\mathrm{SN}\,\ 1998\mathrm{bw}}=0.39$, with an upper limit at the 99% confidence interval of ${k}_{\mathrm{SN}\,\ 1998\mathrm{bw}}\lt 0.67$.

After demonstrating a preference for models including a SN component, we derive flux measurements for SN 2022xiw by subtracting the the best-fit optical afterglow model (see Section 3.2) from the observed r'- and i'-band photometry. We only use photometry starting from T₀ + 7 days, and convert negative flux values after the subtractions to 3σ upper limits. The resulting SN light curve is shown in Figure 4. The SN photometry shown in the Figure is not host subtracted, because we allow the host galaxy emission to vary as a free parameter within the Gaussian priors corresponding to the values from Levan et al. (2023) when extracting physical parameters from the light curve.

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The decay of ⁵⁶Ni to ⁵⁶Co and to ⁵⁶Fe releases the energy that powers the optical light curve of Type I SNe, so the ⁵⁶Ni mass is a key physical parameter that can provide insight into the properties of the explosion and progenitor (Arnett 1982; hereafter A82). Therefore, we fit semianalytic light-curve models from A82 to the observed r'- and i'-band photometry after day 7 to constrain the ⁵⁶Ni mass. Equation (36) in A82 provides an analytic expression for the bolometric luminosity of Type I SNe, assuming full γ-ray trapping of the ejecta along with further radioactive inputs (Valenti et al. 2008). We use the infrastructure from the Hybrid Analytic Flux FittEr for Transients (HAFFET; Yang & Sollerman 2023) to perform the fits, where the two free parameters are the nickel mass (M_Ni) and the photon diffusion timescale (τ_m).

Given a bolometric light curve from the model, it is necessary to extract the associated r'- and i'-band light curves to compare to the observed photometry. It is possible to derive light curves in individual bands from a bolometric magnitude light curve using bolometric correction (BC) coefficients:

$$\begin{eqnarray}&&{\mathrm{BC}}_{x}={M}_{\mathrm{bol}}-{M}_{x},\end{eqnarray} \tag{ 4 }$$

where x is the relevant filter.

For stripped-envelope SNe, Lyman et al. (2014) derive a g-band BC coefficient of:

$$\begin{eqnarray}&&{\mathrm{BC}}_{g}=0.054-0.195\times (g-r)-0.719\times {\left(g-r\right)}^{2}.\end{eqnarray} \tag{ 5 }$$

Here we assume that the color evolution of SN 2022xiw is identical to that of SN 1998bw. We use the BC coefficient, along with the color evolution of SN 1998bw, to generate r'- and i'-band light curves to fit to the SN photometry. First, we convert the BVRI photometry of SN 1998bw from Clocchiatti et al. (2011) to Sloan Digital Sky Survey (SDSS) filters using conversions from Jester et al. (2005). At the time of each observation, we compute BC_g , along with the g – r and g – i colors. Given a bolometric absolute magnitude light curve from A82, we derive a g-band absolute magnitude light curve from BC_g . We then compute the associated r- and i-band light curves from the g – r and g – i colors derived above. Finally we apply the distance modulus, host galaxy emission, and extinction corrections described below, along with conversions from r and i band to r' and i' band, ²⁸ to produce observed r'- and i'-band light curves for comparison with the data. Best-fit models with associated uncertainties are generated using Markov Chain Monte Carlo (MCMC) techniques.

Given the large uncertainty in the host extinction (Section 3.1), we perform the fitting under three different assumptions: (1) E(B − V)_host allowed to vary freely between 0.3 and 0.5 mag; (2) E(B − V)_host fixed to a value of 0.3 mag; and, (3) E(B − V)_host fixed to a value of 0. For all three scenarios we allow M_Ni, τ_m, and E(B − V)_MW to vary as additional free parameters. We allow E(B − V)_MW to vary due to the high uncertainty in the Galactic extinction at the location of GRB 221009A, and adopt uniform priors corresponding to the minimum and maximum values from Schlafly & Finkbeiner (2011). Additionally, we allow the host galaxy emission in the r' and i' bands to vary as free parameters, with Gaussian priors corresponding to the values presented in Levan et al. (2023). We adopt uniform priors for M_Ni and τ_m based on values from previous Type Ic-BL SN studies (Corsi et al. 2016, 2022; Taddia et al. 2019).

We note that there are varying interpretations of the intrinsic extinction and hydrogen column density of the host that are model dependent on the spectral shape of the afterglow (e.g., E(B − V)_host < 0.1 mag, O'Connor et al. 2023; N_H ≈ 4 × 10²¹ cm⁻², Tiengo et al. 2023). However, we believe that the most likely physically plausible situation is the first where E(B − V)_host is allowed to vary freely between 0.3 and 0.5 mag, due to the possibility of a significant amount of intrinsic hydrogen column density in the host that may be up to N_H ≈ 1.29 × 10²² cm⁻² (Williams et al. 2023). We present the best-fit model from this scenario and 100 random samples from the posterior distribution in Figure 4, along with the associated corner plots for each parameter.

The best-fit values derived in all three scenarios are displayed in Table 3, and we find M_Ni = 0.05–0.25 M_⊙, depending on the scenario. Because the SN flux is only marginally detectable with respect to the afterglow and host galaxy emission, these best-fit values should be taken with a grain of salt. However, we can more robustly determine an upper limit, and find that at the 99% confidence level, M_Ni < 0.36 M_⊙. The M_Ni we find is systematically lower than that of SN 1998bw (M_Ni = 0.3–0.9 M_⊙; Sollerman et al. 2000), which agrees with the results from Section 3.3, that SN 2022xiw must be substantially fainter than SN 1998bw.

Table 3. Best-fit Physical Parameters and Their Statistical 1σ Errors Corresponding to the Three Different Host Extinction Scenarios Presented in Section 4.1

	MNi	τm	E(B − V)MW	E(B − V)host	Mej	EKE	${m}_{\mathrm{host}},\ r^{\prime}$	${m}_{\mathrm{host}},\ i^{\prime}$
	(M⊙)	(days)	(mag)	(mag)	(M⊙)	(erg)	(mag)	(mag)
E(B − V)host = 0.3–0.5 mag	${0.18}_{-0.06}^{+0.07}$	${15.98}_{-5.36}^{+2.77}$	${1.31}_{-0.07}^{+0.06}$	${0.41}_{-0.07}^{+0.06}$	${7.93}_{-4.43}^{+2.99}$	${3.71}_{-2.07}^{+1.40}\times {10}^{52}$	${24.88}_{-0.07}^{+0.08}$	${23.83}_{-0.12}^{+0.14}$
E(B − V)host = 0.3 mag	${0.14}_{-0.04}^{+0.05}$	${16.56}_{-3.24}^{+2.35}$	${1.31}_{-0.07}^{+0.06}$	0.3	${8.53}_{-3.01}^{+2.59}$	${3.99}_{-1.41}^{+1.21}\times {10}^{52}$	${24.88}_{-0.07}^{+0.08}$	${23.84}_{-0.12}^{+0.13}$
E(B − V)host = 0 mag	0.07 ± 0.02	${16.26}_{-3.34}^{+2.52}$	${1.31}_{-0.07}^{+0.06}$	0	${8.22}_{-3.03}^{+2.74}$	${3.84}_{-1.42}^{+1.28}\times {10}^{52}$	${24.89}_{-0.07}^{+0.08}$	${23.82}_{-0.11}^{+0.13}$

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Finally, we note that if we repeat the analysis under the same assumptions as Fulton et al. (2023), who do not explicitly account for host galaxy emission, and assume an additional 0.8 mag of extinction in the optical when compared to the nominal galactic extinction from Schlafly & Finkbeiner (2011), we find M_Ni = 0.59 ± 0.04 M_⊙. This M_Ni is lower, but marginally consistent with their value of ${M}_{\mathrm{Ni}}={1.0}_{-0.4}^{+0.6}\,{M}_{\odot }$. The lower M_Ni in comparison to their work is expected, due to their use of a steeper optical afterglow power-law decay slope (f_ν ∝ t^{−1.556±0.002}, see Section 3.1), which in turn leads to more luminous SN emission, and a higher M_Ni.

Given the photon diffusion timescale τ_m, it is possible to derive the total ejecta mass (M_ej) of a SN, through Equation A1 of Valenti et al. (2008):

$$\begin{eqnarray}&&{\tau }_{{\rm{m}}}^{2}=\displaystyle \frac{2{\kappa }_{\mathrm{opt}}{M}_{\mathrm{ej}}}{\beta {{cv}}_{\mathrm{sc}}},\end{eqnarray} \tag{ 6 }$$

where κ_opt = 0.07 cm⁻² g⁻¹ is a constant, average opacity that is able to produce consistent results with hydrodynamical light-curve modeling of stripped-envelope SNe (Taddia et al. 2018), β = 13.8 is a constant, c is the speed of light, and v_sc is a scale velocity, which is set observationally to the photospheric velocity v_ph, which is roughly related to the line velocity at the peak epoch.

Given that we assumed the color and spectral evolution of SN 2022xiw were identical to that of SN 1998bw, we also assume it has a comparable photospheric velocity: v_ph = 28,000 km s⁻¹ (Iwamoto et al. 1998). We note that SN 1998bw's photospheric velocity is high with respect to the population of GRB-SNe in the literature, which possess an average of v_ph = 20,200 km s⁻¹, with a dispersion σ = 8500 km s⁻¹ (Cano et al. 2017). However, a spectrum taken of the optical afterglow at T₀ + 8 days reported the possible existence of broad features with velocities slightly larger than SN 1998bw (de Ugarte Postigo et al. 2022b). Therefore, our assumption of SN 2022xiw's peak photospheric velocity is valid, and may possibly underrepresent the true photospheric velocity. We report the derived values in Table 3, and find M_ej = 3.5–11.1 M_⊙.

Given M_ej and v_sc, it is possible to derive the kinetic energy E_KE of the explosion, assuming that it is a constant density sphere undergoing homologous expansion (Lyman et al. 2016):

$$\begin{eqnarray}&&{v}_{\mathrm{sc}}^{2}\equiv {v}_{\mathrm{ph}}^{2}=\displaystyle \frac{5}{3}\displaystyle \frac{2{E}_{{\rm{K}}}}{{M}_{\mathrm{ej}}}.\end{eqnarray} \tag{ 7 }$$

The derived values are again reported in Table 3, and we find E_KE = (1.6–5.2) × 10⁵² erg. These values are consistent with the values Fulton et al. (2023) derive, who find ${M}_{\mathrm{ej}}={7.1}_{-1.7}^{+2.4}\,{M}_{\odot }$ and E_KE = (2.7–6.3) × 10⁵² erg. We note that both M_ej and E_K largely depend on the photon diffusion timescale, and the corner plots in Figure 4 show that τ_m is close to hitting the bounds of the priors. This is unsurprising given the nature of the data set, and the faintness of the SN with respect to the afterglow and host galaxy emission. Therefore, both the M_ej and E_KE we derive should also be considered as estimates, and we are unable to derive robust upper limits in this case due to the nature of the posterior of τ_m.

We note that throughout this analysis, systematic uncertainties likely arise from the assumptions made with the color and spectral evolution of SN 2022xiw being identical to that of SN 1998bw, as well as assuming that the explosion is undergoing homologous expansion, as the presence of a relativistic jet likely impacts the spherical symmetry of the explosion. An asymmetric explosion would likely impact κ_opt and the assumptions made in Equation (6) from Taddia et al. (2019). However, despite these caveats, the statistical uncertainties we find are large and likely dominate over any of these systematic uncertainties.