XRF 100316D/SN 2010bh AND THE NATURE OF GAMMA-RAY BURST SUPERNOVAE^*

2.1.1. Faulkes Telescope South and Gemini-South

2.1.2. Hubble Space Telescope

Using ISIS (Alard 2000), we performed image subtraction on the FTS and Gemini-S images, using the last epoch of images taken in each filter and on each telescope as the respective templates. Subtracting the template from the early FTS images gave a clear detection of the OT and an accurate position, and the same was also observed for the Gemini-S images.

Aperture photometry was then performed on the subtracted images using a small aperture and an aperture correction was computed from the pre-subtracted images and applied. The instrumental magnitudes obtained from the FTS subtracted images are calibrated to BVR_ci. The Gemini-S subtracted images taken in filters griz are calibrated to BR_ciz, respectively. Images in filters gr are calibrated using a zero point and a color term, while images taken in i are calibrated using only a zero point. Magnitudes of the secondary standards in z are computed using transformation equations from Jester et al. (2005), and the Gemini-S data are calibrated to these using a zero point. All of the image-subtracted, foreground-corrected magnitudes obtained from the FTS and Gemini-S data are displayed in Table 2, where the quoted errors (added in quadrature) are determined from the photometry and calibration.

As already noted by Starling et al. (2011), the host galaxy system of XRF 100316D/SN 2010bh has a highly disturbed morphology, with a bright central region, a possible spiral arm and a number of bright knots (see Figure 8 in Starling et al. 2011). SN 2010bh is located on top of a bright knot close to the bright central region, which is perhaps the galactic nucleus. In our final epoch of Gemini-S images, the host complex appears as a blended, elliptical object due to the lower resolution of Gemini-S relative to HST.

We have performed photometry on the final Gemini-S images using SExtractor (Bertin & Arnouts 1996). We obtained (1) corrected isophotal magnitudes (MAG_ISOCOR) and (2) Kron-like elliptical aperture magnitudes (MAG_AUTO), with the best of these values being used (MAG_BEST). The host magnitudes in filters griz have been calibrated to secondary standards in the field of XRF 100316D, where the magnitudes of the secondary standards in filters grz are computed using transformation equations from Jester et al. (2005), and the host photometry are calibrated to these using a zero point. The Gemini-S photometry, which have been corrected for foreground extinction, are listed in Table 3, where the quoted errors of the Gemini-S magnitudes are added in quadrature and are derived from the photometry and calibration.

The image-subtracted LCs of XRF 100316D/SN2010bh, which have been corrected for foreground and host extinction (see Section 4), are displayed in Figure 1. Plotted for comparison are the multi-wavelength LCs of two known GRB-SNe: SN 1998bw and SN 2006aj, as well as SN 1994I which is a local Type Ic SN that was not associated with a GRB. All of the comparison LCs have been shifted in magnitude to match SN 2010bh around maximum light. No other changes have been made to the LCs apart from the LCs of SN 1998bw, which have been stretched in each filter by the values listed in Table 5 (see Section 3.5), and the early-time data (t − t₀ < 3 days) for SN 2006aj, for which the light was dominated by flux coming from the very bright shock breakout, was not included to avoid confusion between different filters. The stretch factors have been applied to SN 1998bw for clarity.

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The peak times and magnitudes in each filter are estimated from low-order polynomial fits. We have determined that SN 2010bh peaked at t − t₀ = 8.02 ± 0.12 days in the B filter, and at subsequently later times in the redder filters. The quoted errors are statistical and are determined from the distribution of peak times determined from the different-order polynomials. The behavior of the LC peaking at later times in redder filters is typical behavior for core-collapse SNe, and SN 2010bh appears no different in this respect. The main photometric parameters of XRF 100316D/SN 2010bh in filters BVR_ci are presented in Table 4.

Comparing the optical LCs of SN 2010bh with those of SN 2006aj, it appears they evolve at roughly the same rate in the B filter. However, it appears that the U-band LC of SN 2006aj evolves faster than SN 2010bh, while in the redder V, R_c, and i filters it evolves more slowly. It is also worth noting that the U-band measurement of SN 2010bh at t − t₀ = 24.8 days is ≈0.5 mag brighter than the shifted LC of SN 2006aj, and perhaps even more at t − t₀ = 47.3 days. As the fluxes are measured in a fixed, and albeit small, aperture, the flux we are detecting is from both the SN and the underlying host galaxy. When performing photometry on the third and fourth epoch U-band images, it appears that the background/host flux is becoming dominant and affecting the overall shape of the LC (i.e., making it appear that it is fading slower than is expected).

The color curves of SN 2010bh are plotted in Figure 2, and have been corrected for foreground and host extinction (see Section 4). Plotted for comparison are the color curves of GRB-SNe 1998bw and 2006aj, as well as local SN 1994I, all of which have been corrected for foreground and host extinction using the values quoted in the literature: SN 1994I: E(B − V)_total = 0.45 mag (Richmond et al. 1996; Iwamoto et al. 1994; though also see Sauer et al. 2006 who find E(B − V) = 0.30 mag from spectral modeling); SN 1998bw: E(B − V)_total = 0.07 mag (Patat et al. 2001); SN 2006aj: E(B − V)_total = 0.14 mag (Sollerman et al. 2006; though also see Campana et al. 2006 who find E(B − V)_host = 0.20 mag from their Swift/UVOT observations). No other changes have been made to the color curves apart from those of SN 1998bw, which have been stretched by s = 0.63 in B−V and V–R_c and by s = 0.61 in R_c–i (see Table 5). The stretch factors, which are the average values of s determined in the individual filters (i.e., s = 0.62, 0.64 for respective filters B and V; thus, the average value is s = 0.63), have been applied so as to present a consistent analysis of the properties between SN 2010bh and SN 1998bw (i.e., as the stretch factors were also applied to the LCs of SN 1998bw in Figure 1).

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Starting before maximum light, all of the colors of SN 2010bh become redder over time, with similar behavior seen in the comparison SNe. Additionally, SN 2010bh appears to be redder than SN 1998bw and SN 1994I in the pre-maximum phases, while in the post-maximum phase SN 2010bh appears to have colors similar to those seen in SN 1994I.

We have determined the Δm₁₅ parameter (i.e., the amount that the LCs fade by from peak to 15 days later) of the BVR_ci LCs of XRF 100316D/SN 2010bh. We have used low-order polynomials to determine the peak times of each LC, and in turn the Δm₁₅ parameter. It is seen that the LCs decay more slowly in the redder filters, with the Δm₁₅ parameter in filters BVR_ci, respectively, being 1.80 ± 0.14, 0.95 ± 0.10, 0.90 ± 0.05, and 0.77 ± 0.20 (which are also listed in Table 5). The uncertainties in the Δm₁₅ parameters are derived from the uncertainties in determining the peak times from the photometry, which, for example, in the i band are less certain due to the sparseness of photometry around and just after the peak, as well as from the distribution of the peak times determined from fitting different-order polynomials.

The V- and R-band values of Δm₁₅ measured for SN 2010bh are somewhat larger (i.e., fade faster) than the average values measured for a sample of local Ibc SNe observed in the V band and R band by Drout et al. (2011, their Table 4). It is also seen that SN 2010bh fades faster, on average, than Ia SNe in the B band (e.g., Phillips 1993; Riess et al. 1998).

The breakout of the blast/shock wave and/or emission coming from the shock-heated stellar envelope has been detected in both Type Ibc and Type II SNe, including Type IIp SN 1987A (early peaks in optical LCs: Hamuy et al. 1988, and PCyg-like feature around 1500 Å in UV light echo off of a dust cloud ∼300 pc from the SN: Gilmozzi & Panagia 1999); Type IIb SN 1993J (early peak in optical LCs: Wheeler et al. 1993; Schmidt et al. 1993; Richmond et al. 1994; Lewis et al. 1994); Type Ibc SN 1999ex (early optical peaks: Stritzinger et al. 2002); Type IIP SNe SNLS-04D2dc and SNLS-06D1jd (UV-flash: Gezari et al. 2008); Type IIP GALEX supernova SNLS-04D2dc (UV-flash: Schawinski et al. 2008); Type IIn PTF 09UJ (UV-flash: Ofek et al. 2010); and Type IIP SN 2010aq (early UV and optical peaks: Gezari et al. 2010).

One SN, in particular, has been the subject of much interest and debate when interpreting the high-energy properties of the event. SN 2008D (Modjaz et al. 2009; Soderberg et al. 2008; Mazzali et al. 2008; Malesani et al. 2009) occurred in nearby galaxy NGC 2770 while Swift was observing SN 2007uy within the same galaxy. The resultant high-energy emission detected by Swift was interpreted differently in the literature, with the hot, blackbody X-ray spectrum and early peak in optical LC being interpreted by Soderberg et al. (2008) as being due to the shock breakout, while Mazzali et al. (2008) suggest that the emission is due to a "choked" jet. Interestingly, a recent paper by Van der Horst et al. (2011) has perhaps put the debate to rest. Using observations of the radio emission of SN 2008D during the first year after the explosion, the authors showed that there was no evidence of a relativistic jet contributing to the observed radio flux, thus strongly suggesting that the high-energy emission was due to the shock breakout and not to a GRB- or XRF-like event.

Emission coming from the shock-heat, expanding stellar envelope has also been detected in GRB/XRF-SNe, though the topic is still contentiously debated in the literature. For XRF 060218, the hot, blackbody X-ray spectrum and early peak in UV and optical LCs were interpreted by Campana et al. 2006 (hereafter C06), and later Waxman et al. (2007), as being due to the shock breakout. However, the emission was interpreted by Ghisellini et al. (2007) as being due to a relativistic jet.

Figure 3 shows the B-band LC of SN 2010bh, which has been corrected for foreground and host extinction. Plotted for comparison are the LCs of two GRB-SNe, 1998bw and 2006aj, which have been shifted to match SN 2010bh in peak brightness, and the LC of SN 1998bw has also been stretched by a factor s = 0.62 (see Table 5).

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Upon inspection, similarities between SN 2010bh and SN 2006aj are seen in the early-time LC, where a bright component is detected at t − t₀ = 0.598 days. The magnitude at this epoch is similar to that at t − t₀ = 3.489 days, indicating that SN 2010bh did not increase linearly in brightness from detection, as was seen for SN 1998bw. In the case of SN 2006aj, early peaks in the UV and optical LCs (t − t₀ ⩽ 10⁴ s) were explained by C06 as light coming from the low-energy tail of the thermal X-ray emission (see below) produced by the radiation shock driven into the stellar wind. At later times, the optical and UV emission is attributed to the expanding envelope of the progenitor star that is heated as the shock wave passes through it. Initially the envelope is opaque due to the dense stellar wind, but as the star and wind expand, the photosphere propagates inward, exposing the shocked stellar material. For SN 2010bh, though we are not able to ascertain the exact shape of the early B-band LC due to the paucity of our early-time observations, it is not unreasonable to envision a scenario similar to that observed for SN 2006aj in which the brightness of the flux at this early epoch is due to flux coming from the shock-heated, expanding stellar envelope.

In addition to the bright B-band detection, inspection of the spectral energy distribution (SED) at this epoch, which has been corrected for foreground and host extinction, gives additional clues to the origin of the flux. At the first epoch, the SED is very blue and does not resemble the shape of the SEDs at later times (see Figure 4).

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To determine whether the flux at this time is synchrotron in origin, the optical SED was fit with a power law (i.e., F_ν∝ν^β), for which we find β = +0.94 ± 0.05 (χ²/degrees of freedom (dof) = 2.17/2). The slope of the spectrum is harder than that expected for light emitted as synchrotron radiation, where, when neglecting effects due to self-absorption (see below), a maximum value of β = +1/3 is allowed (e.g., Sari et al. 1998). Thus, the shape of the SED at this early epoch is consistent with emission coming from the shock-heated stellar envelope.

Additionally, we have constructed SEDs for SN 2006aj at t − t₀ = 0.50 days and 0.75 days and fit them with a single power law. Quite interestingly, at t − t₀ = 0.50 days we find β = +0.97 ± 0.05 (χ²/dof = 2.13/2), while at t − t₀ = 0.75 days we find β = +0.94 ± 0.03 (χ²/dof = 2.03/2). These values for the spectral index are akin to that found for SN 2010bh at a similar epoch, and are both harder than that expected for synchrotron radiation.

The assumption that the optical flux is above the self-absorption frequency (ν_a) cannot be proved with our current set of observations; however, it has been seen in other GRB events that the synchrotron self-absorption frequency at typical observing epochs of about one day after the GRB is expected to be of order ∼10⁹–10¹⁰ Hz (i.e., at radio frequencies). In the literature there are several events where there has been enough multi-wavelength observations to determine the self-absorption frequency, including GRB 970508: ν_a ∼ 3 × 10⁹ Hz (Granot et al. 1999); ν_a ∼ 2.5 × 10⁹ Hz (Galama et al. 1998b); GRB 980329: ν_a ∼ 13 × 10⁹ Hz (Taylor et al. 1998); GRB 991208: ν_a ∼ 4–11 × 10⁹ Hz (Galama et al. 2000). As the frequency range of the optical observations lies between ∼3–10 × 10¹⁴ Hz (i.e., ∼10⁴–10⁵ times higher frequency), it is unlikely that the optical observations suffer from self-absorption effects.

Further evidence that the light at t − t₀ = 0.598 days is coming from the shock-heated stellar envelope is the analysis performed by Starling et al. (2011) on the X-ray spectrum at 144 s ⩽t − t₀ ⩽ 737 s. The authors found evidence for a soft, hot blackbody component that contributes ≈3% to the total 0.3–10.0 keV flux (though see also Fan et al. 2011 for a different interpretation). With this in mind, the presence of the blackbody component, with kT = 0.14 keV (T ≈ 1.6 × 10⁶ K), is similar in temperature to the soft, hot blackbody component seen in the X-ray spectrum of XRF 060218/SN 2006aj (kT ≈ 0.17 keV; T ≈ 2.0 × 10⁶ K; C06).

According to C06, the thermal component is key to understanding XRF 060218. It is known that the signature of shock breakout is a hot blackbody X-ray spectrum immediately after the explosion, and C06 argued that the hot temperature of the thermal X-ray component is indicative of radiation emitted by a shock-heated plasma. Furthermore, the characteristic radius of the emitting region was ∼5 × 10¹² cm (Waxman et al. 2007 also find the emitting region to be at a radius of 7.8 × 10¹² cm), which, though quite large, (i.e., of order of the radius of a blue supergiant), was explained by C06 and Waxman et al. (2007) by the presence of a massive, dense stellar wind surrounding the progenitor, which is common for Wolf–Rayet stars (the supposed progenitors of Type Ibc SNe and possibly long-duration GRBs). Both authors argue that the thermal radiation is observed once the shock, which is driven into the wind, reaches a radius where the wind becomes optically thin.

For XRF 100316D/SN 2010bh, the temperature of the blackbody component in the X-ray spectrum implies an emitting radius of ∼8 × 10¹¹ cm, which is almost an order of magnitude larger than the radius of a Wolf–Rayet star (∼10¹¹ cm). As for XRF 060218, the large radius of the X-ray emitting region could be the result of the shock being driven into the dense stellar wind, and the thermal radiation is observed once the wind density decreases and becomes optically thin.

Thus, the brightness of the B-band detection at t − t₀ = 0.598 days, when taken in tandem with the hard value of the power-law index (β = +0.94 ± 0.05) of the SED at this epoch, as well as the presence of the extremely hot thermal component (T ≈ 1.6 × 10⁶ K) in the X-ray spectrum at 144 s ⩽t − t₀ ⩽ 737 s, suggests that we have detected optical emission coming from the cooling, expanding envelope that was heated by the passage of the shock wave through it.

We have determined the stretch factor (s) and luminosity factor (k) of SN 2010bh in relation to the archetype GRB-SN, SN 1998bw. To do this we have created synthetic flux SN 1998bw LCs in filters BVR_ci as they would appear if they occurred at z = 0.0591. At each epoch, an SED of the observed SN 1998bw LCs is built and interpolated using polynomials. This allows us to cover the temporal-frequency space for SN 1998bw and build synthetic LCs for other specified observed frequencies. When constructing the synthetic LCs we have accounted for the inverse-squared distance (i.e., luminosity distance) suppression of flux, as well as rest-frame extinction and time dilation effects. We then fit the flux SN 1998bw LCs with an empirical relation of the form

$$\begin{equation} U(t) = A + st\left(\frac{e^{({-t^{\alpha _{1}}}/{F})}}{1 + e^{({p-t}/{R}) }}\right) + t^{\alpha _{2}}\log (t^{-\alpha _{3}}), \end{equation} \tag{ 1 }$$

where A is the intercept of the line and s and F are related to the respective amplitude and width of the function. The exponential cutoff function for the rise has a characteristic time R and a phase zero-point p, while α₁, α₂, and α₃ are free parameters. All of the parameters are allowed to vary during the fit.

Once the flux LC of SN 1998bw was fit by Equation (1), the stretch and luminosity factors of SN 2010bh were determined by fitting the equation

$$\begin{equation} W(t)=k \times U(t/s) \end{equation} \tag{ 2 }$$

to the flux LC of SN 2010bh in each filter.

The stretch factors of SN 2010bh relative to SN 1998bw were determined using data taken several days after the initial burst (t − t₀ ⩾ 7.0 days) as the early LCs, whose shapes are altered by light coming from an additional shock breakout component, are very different to that of SN 1998bw. The results of the fit are listed in Table 5, where the quoted errors are statistical only.

For GRB and XRF events where an optically bright SN has been detected (i.e., not including events such as SN-less GRBs 060505 and 060614; e.g., Fynbo et al. 2006), it is seen that SN 2010bh is the faintest SN to date that has been linked spectroscopically with a GRB or XRF (see Table 8), and is of similar peak brightness as the SN associated with GRB 970228 (k = 0.40 ± 0.29, though the host extinction is unknown), but not as faint as the (possible) SN that was photometrically linked to GRB 101225A (k = 0.08 ± 0.03; Thöne et al. 2011). The slower evolution of SN 2006aj relative to SN 2010bh seen upon inspection of the optical LCs is echoed in the computed stretch factors for the two SNe. In the B filters they are approximately the same value (SN 2010bh: s = 0.62 ± 0.01; SN 2006aj: s = 0.60 ± 0.01); however, in the redder filters SN 2010bh has smaller stretch factors relative to SN 2006aj.

It should be mentioned that SN 1998bw is not the ideal template for SN 2010bh. At early times light from the shock breakout changes the shape of the LC, which affects how well SN 1998bw, for which no shock breakout was observed, can be used as a template. Even at late times it is seen in Figure 1 that SN 2010bh decays more slowly than the stretched LC of SN 1998bw, thus showing that the two SNe evolve quite differently. The limitations of using SN 1998bw as a template was also noted by Ferrero et al. (2006), where they used an additional power-law component when fitting their SN 1998bw template to their multi-filter observations of SN 2006aj.

It was shown in the preceding section that, despite ejecting only a moderate amount of material, SN 2010bh was an energetic explosion, mainly due to the large photospheric velocities. To put the results of our bolometric LC modeling into context, we first compare SN 2010bh with SN 2006aj and then with the other GRB-SNe.

The ultraviolet, optical, and IR (UVOIR) bolometric LC (in the 1900–16660 Å wavelength regime) of XRF 060218/SN 2006aj, which has been constructed from the data published in the literature (Campana et al. 2006; Sollerman et al. 2006; Cobb et al. 2006; Ferrero et al. 2006), is displayed in Figure 6. It can be seen that during the first few days the fractional contribution of UV flux (i.e., light detected in the 1900–3000 Å wavelength regime) to the total bolometric LC (Figure 7) is considerable due to the shock breakout of the progenitor star (Campana et al. 2006), which decays at a linear rate after ≈4 days. It is important to note that the contribution of flux at UV wavelengths is not negligible, and at roughly 10 days still contributes 15% of the total bolometric flux.

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By making the assumption that the fraction of UV flux of the total bolometric flux at a given epoch is the same in SN 2010bh as it is in SN 2006aj, we have used SN 2006aj as a template/proxy for creating a "UV-corrected" bolometric LC for SN 2010bh, which is also displayed in Figure 6. The bolometric LC of SN 2010bh is shown twice: (1) first, by considering flux only from the optical (UBVR_ci) and IR (JH) filters and (2) by including a contribution of flux at UV wavelengths.

Our assumption that the fraction of UV flux is similar in each event is perhaps not unreasonable when we also consider the percentage of the bolometric flux detected in the optical and IR regimes for SN 2010bh and SN 2009bb. Plotted in Figure 8 is a plot of the amount of light detected in the optical and IR as a fraction of the total bolometric flux in each epoch for SN 2009bb and SN 2010bh. The data included in this plot do not include all of the data obtained for these two events, but only those where there is contemporaneous optical and IR data. The first IR detection of SN 2010bh was during the first HST epoch at t − t₀ = 8.6 days, and the first IR detection of SN 2009bb was at t − t₀ = 7.7 days (Pignata et al. 2011). Inspection of the optical and IR filter pairs reveals similar behavior in both events, though it is seen that more flux is emitted at IR wavelengths in SN 2010bh relative to SN 2009bb.

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The fractional contribution of the optical and IR flux to the bolometric LCs for these two events, as well as the large UV contribution seen in SN 2006aj, is similar to that seen in XRF 080109/SN 2008D. Modjaz et al. (2009) also compared the relative contribution of the UV, optical, and IR flux in the bolometric LC of SN 2008D for the first ≈30 days (Figure 9 of Modjaz et al. 2009), and noted a large contribution of flux from 2008D at UV wavelengths (≈40% at t − t₀ ≈ 0.8 days). This is not quite as high as for SN 2006aj, which is ≈70% at t − t₀ ≈ 0.8 days, but their UV contribution does not include far-UV flux, whereas SN 2006aj does. Indeed at t − t₀ = 1.8 days, when they do include the far-UV flux, the total UV flux is seen to increase. Modjaz et al. (2009) also showed that the amount of IR flux increased over time, while the optical flux increased linearly for the first 10 days or so, then decreased. While there is scant data plotted in Figure 8 before t − t₀ = 10 days, our plot does show that the optical contribution decreases during 8.6 days ⩽t − t₀ ⩽ 24.8 days for SN 2010bh, whereas for SN 2009bb it decreases until ≈25–30 days, then appears to contribute a nearly constant, if slightly increasing, amount to the total bolometric flux. Indeed, when comparing Figure 8 with Figure 9 of Modjaz et al. (2009), the general behavior of SNe 2008D, 2009bb, and 2010bh appears to follow similar trends in their relative bolometric fluxes during the first ≈30 days.

Next, we applied our analytical model to our constructed bolometric LC of XRF 060218/SN 2006aj. Taking the peak photospheric velocity to be v_ph ≈ 20, 000 km s⁻¹ (Pian et al. 2006), we find M_Ni = 0.16 ± 0.01 M_☉, M_ej = 1.63 ± 0.05 M_☉, and E_k = 6.47 ± 0.20 × 10⁵¹ erg. In comparison, Mazzali et al. (2006) found when modeling the spectra and photometry of SN 2006aj parameters of M_Ni ≈ 0.2 M_☉, M_ej ≈ 2 M_☉, and E_k ≈ 2 × 10⁵¹ erg. It appears that our simple model is consistent with the values of the physical parameters of the SN 2006aj explosion determined via detailed modeling, but overestimates the explosion energy by a factor of ∼3. As mentioned in Section 5, our estimate of the energy assumes that all of the mass moves at the photospheric velocity. Here is a clear example where this assumption overestimates the explosion energy, implying that a bulk of the ejecta are moving at slower velocities.

Alternatively, if we exclude the contribution of flux at UV wavelengths, similar to the UBVR_cI_cJH bolometric LCs presented by Pian et al. (2006), we find for SN 2006aj M_Ni = 0.14 ± 0.01 M_☉, M_ej = 1.92 ± 0.12 M_☉, and E_k = 7.64 ± 0.49 × 10⁵¹ erg, which is again consistent with those found by Mazzali et al. (2006), but the explosion energy is again overestimated, this time by a factor of ∼4. In both cases the errors are estimated as per Section 5. We note that by including the contribution of flux at UV wavelengths, which is considerable at early times (i.e., ⩽5 days), this increases the total luminosity of the SN, but makes the bolometric LC peak earlier, thus decreasing the overall width of the LC (i.e., less mass is ejected). The results of our modeling are displayed in Table 6.

When we apply our model to the bolometric LC of 2010bh that includes a UV correction, we find M_Ni = 0.12 ± 0.01 M_☉, M_ej = 1.93 ± 0.07 M_☉, and E_k = 1.20 ± 0.05 × 10⁵² erg. The inclusion of the UV flux makes the overall bolometric LC brighter, implying more ⁵⁶Ni was created during the explosion. However, as for SN 2006aj, the inclusion of the UV flux, which is dominant at early times, causes the LC to peak earlier, thus making the overall shape of the bolometric LC narrower, and implying that less mass was ejected during the explosion (and thus a less energetic explosion).

Nevertheless, whether an estimated contribution of the UV flux is included or not, the results of bolometric LC modeling imply that SN 2010bh ejected a mass of M_ej = 1.9–2.2 M_☉, has a nickel mass of M_Ni = 0.10–0.12 M_☉, and has a kinetic energy of roughly E_k = 1.2–1.4 × 10⁵² erg.

It is observed that, despite the similarities of the high-energy properties of XRF 100316D and XRF 060218 (Starling et al. 2011), the accompanying SNe are somewhat different, with SN 2010bh being much more energetic. In this section we compare the bolometric properties of SN 2010bh with other GRB-SNe, as well as non-GRB-SNe: Ic SN 1994I and broad-lined Ic SN 2009bb. We investigate whether clear differences exist in the bolometric properties of stripped-envelope, core-collapse SNe that produce a GRB and those that do not.

To date, numerous authors have determined physical parameters of Ibc SNe by modeling photometry and spectroscopy. Many authors employ analytical models that are derived from the original prescription of Arnett (1982) to describe bolometric LCs; however, very often these models only supplement the derivation of physical parameters that are determined by comparing synthetic spectra created by synthesis codes to observations. In this work we are attempting to obtain physical parameters of previously studied events by modeling only the bolometric LCs, and compare our results with those obtained via more detailed analyses.

Figure 9 shows the optical and IR bolometric LCs of XRF 100316D/SN 2010bh in relation to three spectroscopically confirmed GRB-SNe: 1998bw, 2003dh, and 2006aj, as well as broad-lined Ic SN 2009bb and Ic SN 1994I, neither of which were accompanied by a GRB trigger. All of the bolometric LCs have been determined by integrating the flux in the optical and IR UBVR_cI_cJH filters, and have been corrected for foreground and host extinction. We have constructed the bolometric LCs of SN 1998bw (Galama et al. 1998a; McKenzie & Schaefer 1999; Sollerman et al. 2002; Patat et al. 2001), SN 2006aj (Campana et al. 2006; Sollerman et al. 2006; Cobb et al. 2006; Ferrero et al. 2006), and SN 2009bb (Pignata et al. 2011) from the individual LCs presented in the literature. We have also incorporated the bolometric data presented in Deng et al. (2005) for SN 2003dh and the bolometric data presented in Richmond et al. (1996) for SN 1994I. When constructing the bolometric LC of SN 1998bw, for which very little IR data are available in the literature, we estimated the amount of flux emitted at IR wavelengths using the behavior of SNe 2006aj, 2009bb, and 2010bh as a proxy. Note also that the bolometric LCs of SN 1994I and SN 2003dh are in filters BVR_cI_c and UBVR_cI_c, respectively. Finally, using the same method that we applied to SN 2010bh in Section 6.1, we have estimated the amount of flux emitted at UV wavelengths for SN 1998bw by using SN 2006aj as a template/proxy.

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As for SN 2006aj, we fit our analytical model to the bolometric LCs of the comparison SNe. We have used values of the photospheric velocities given in the literature for each event. The results of our fits are presented in Table 6, where the quoted errors are statistical only.

When we compare the values of the nickel mass (M_Ni), ejected mass (M_ej), and explosion energy (E_k) determined from our model with those in the literature, it can be seen that our model is able to recover the physical parameters of these events that are determined by more detailed spectral and photometric modeling. We note that on average, we tend to underestimate the amount of ejected mass, which in turn underestimates the explosion energy in these events, apart for SN 2006aj, where we overestimate the explosion energy. There are several assumptions that go into our model, such as no asphericity and the central location of the nickel mass during the explosion (that does not mix with the expanding ejecta), as well as a constant value for the optical opacity. Even if the opacity is approximately constant during the photospheric phase, it may be a slightly different value to that which we are using here.

Finally, the nickel mass determined from our fit of SN 2010bh (Section 5) reiterates the faint nature of SN 2010bh in relation to the other GRB-SNe. While the nickel mass produced by SN 2010bh is slightly greater than that of SN 1994I, it is less than that seen in other GRB-SNe. Indeed, for all of the GRB-SNe that have been investigated through spectral and photometric modeling, SN 2010bh appears to be the faintest (i.e., has created the least amount of nickel during the explosion).

In conclusion, the energetics of GRB-SNe and Ic-BL SNe such as SN 2009bb and SN 2002ap are comparable, and are considerably higher than that of other Ibc SNe such as Ic SN 1994I and Ib SN 1999ex. Additionally, the ejected masses of the GRB-SNe and Ic-BL SNe are commensurate and somewhat higher than some Ibc SNe (e.g., SN 1994I and SN 1999ex), though some Ibc SNe do eject comparable masses (e.g., SN 2009jf and SN 1999dn). Finally, the nickel masses of all these events span a range of 0.07–0.52 M_☉, with no clear distinction between GRB-SNe, Ic-BL SNe, and Ibc SNe.

In an attempt to put the luminosity and stretch factors of SN 2010bh (Table 5) into context, we assembled from the literature all of the photometric data available for most of the previously detected GRB-SNe. Note that we have not pursued every event but only included those where several measurements were made of the SN "bump" that would allow us to constrain the properties of the associated SN.

For every GRB-SN event we assume that light is coming from three sources: (1) the afterglow, (2) the supernova, and (3) the host galaxy. First we remove the constant source of flux from the host galaxy, either by using magnitudes determined by image subtraction by the authors (e.g., XRF 020903; Bersier et al. 2006), or by mathematically subtracting the flux from each measurement using observations made of the host galaxy long after the OT had faded below the brightness of the galaxy (e.g., GRB 090618; Cano et al. 2011).

Then, we use the early-time data for each event, and model the behavior of the afterglow using single or broken power laws

$$\begin{equation} {\rm flux}\ (t) = f_{0} \times \left(\left(\frac{t}{T_{{\rm break}}}\right)^{\alpha _{1}} + \left(\frac{t}{T_{{\rm break}}}\right)^{\alpha _{2}}\right)^{-1}, \end{equation} \tag{ 7 }$$

where f₀ is the flux zero point, unique to each filter, t is the time since the burst, and T_break is the time when the power law changes from temporal index α₁ to α₂. The afterglow parameters f₀, T_break, α₁, and α₂ were allowed to vary in the fit for each event, and the results of our modeling are listed in Table 7.

In all cases, the values we derived are similar to those seen in the literature. For example, for GRB 990712 we find that a single power law fits both the V- and R-band data well with a decay constant of α = −0.95 ± 0.02. In comparison, Björnsson et al. (2001) find that the V-band data can be fit with either a single or broken power law, where the value in the former is α_V − 0.82 ± 0.03. They find that the R-band photometry can be fit with a single power law with a slightly steeper decay index of α_R = −0.91 ± 0.06. Both of these values are fully consistent with our own fit of the optical data.

Additionally, for GRB 020405, Price et al. (2003) found from their R-band photometry a decay constant of α = −1.41 when they limited the range of the fit up to 10 days after the burst, while Masetti et al. (2003) find α = −1.54 ± 0.06 in all optical filters for the same time period. However, when Price et al. (2003) re-fit the data with a broken power law, they find a better fit with parameters α₁ = −0.94 ± 0.25, α₂ = −1.93 ± 0.25, and the time the LC breaks from α₁ to α₂ of T_break = 1.67 ± 0.52 days (χ²/dof = 35.9/28). Similarly, Masetti et al. (2003) find that at times >20 days, α = −1.85 ± 0.15. When we restrict our fit to 2–10 days, we find that a single power law provides a suitable fit to the data with α = 1.72 ± 0.08, χ²/dof = 6.16/4. This value for the temporal index is consistent with the value of α₂ found by Price et al. (2003) and the value of α found by Masetti et al. (2003) at late times.

Once the afterglow parameters were determined, we then subtract Equation (7) from the OT (e.g., afterglow and supernova) flux LC to obtain host- and afterglow-subtracted "supernova" LCs (e.g., Cano et al. 2011). The resultant LCs are displayed in Figure 10. Note that in Figure 10 for XRF 060218/SN 2006aj, we have included an additional power-law component of α = −0.2, which was fixed in the fit, and is similar to that determined by Ferrero et al. (2006) of α = −0.20 ± 0.10.

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For each of the host- and afterglow-subtracted "supernova" LCs, we determined the stretch and luminosity factor for each GRB-SN in relation to SN 1998bw in various filters, repeating the same process we followed for SN 2010bh (Section 3.5). The results of our fit are listed in Table 8, with the R-band stretch and luminosity factors displayed in Figure 11, and the host- and afterglow-subtracted "supernova" LCs displayed in Figure 10.

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A previous study by Zeh et al. (2004), which was extended by Ferrero et al. (2006, hereafter F06), also determined the stretch and luminosity factors of GRB-SN in relation to SN 1998bw²³ by modeling the SN "bumps" seen in the GRB LCs. For many events where we have determined the stretch and luminosity factors from the afterglow- and host-subtracted "supernova" LCs, and F06 from the supernova "bumps," we find similar results, e.g., GRB 050525A/SN 2005nc—this paper: k = 0.69 ± 0.03, s = 0.83 ± 0.03; F06²⁴: k = 0.66^+0.10_−0.08, s = 0.77 ± 0.04; and XRF 060218/SN 2006aj (V band): k = 0.72 ± 0.02, s = 0.65 ± 0.02; F06: k = 0.724 ± 0.007, s = 0.682 ± 0.005; with similar results for the other filters.

However, for some events we find different properties of the GRB-SN: for GRB 011121/SN 2001ke, we find similar stretch factor to F06 (this paper: s = 0.84 ± 0.01; F06: s = 0.80 ± 0.02); however, we find SN 2001ke to be much brighter once correcting for host extinction (this paper: k = 2.11 ± 0.03; F06: k = 0.88^+0.08_−0.07). Additionally, for GRB 020405 we find a similar luminosity factor to F06 (this paper: k = 0.82 ± 0.14; F06: k = 0.90^+0.15_−0.11); however, we find that the SN evolves more quickly (this paper: s = 0.62 ± 0.03; F06: s = 0.97 ± 0.07). However, GRB 020405 aside, for every other GRB-SN event we find the same value for the stretch factor as F06 within our respective error bars.

We note that when using this method to determine the luminosity factor relative to SN 1998bw, the SN brightness depends on how accurately we have modeled the afterglow. For instance, if we overestimate the relative contribution of the afterglow, then we will correspondingly underestimate the SN brightness.

We find that the stretch factor of each GRB-SN event is approximately the same in each filter (e.g., GRB 990712, GRB 031203, XRF 060218, GRB 060729, and GRB 090618), though this trend applies less in the bluer filters. For example, for XRF 060218/SN 2006aj, we find stretch factors in the R and I filters of s = 0.68 ± 0.02, while in V and B we find slightly smaller values of s = 0.65 ± 0.02 and s = 0.60 ± 0.02, respectively.

We have investigated whether there might be a correlation between the k and s factors of the GRB-SNe, perhaps something akin to that established for Type Ia SNe (i.e., the possibility of using GRB-SNe as "standard candles"). We have used all of the R-band data, of which there are 18 GRB-SNe. For GRB/XRFs 021211, 040924, 080319B, and 091127, for which no R band data exist, we have assumed that the stretch and luminosity factors in the V band (GRB 021211) and the I band (GRBs 040924, 080319B, and 091127) are approximately the same as in the R band. As already noted for events such as XRF 060218 and GRB 090618, the stretch and luminosity factors in filters VR_cI_c are approximately the same and differ by only a few percent. The inclusion of these events brings the total sample size up to N = 22, and for the entire sample we find a correlation/Pearson coefficient of c = −0.05 and a 95% confidence interval of (−0.46, 0.38) for a significance of p > 0.05. As the critical value for the Pearson correlation coefficient for N = 22, dof = 20, and p > 0.05 is 0.492, we can conclude that a statistically significant correlation does not exist.

Stanek et al. (2005) noted that a relation may exist between LC shape and luminosity, which the results of F06 could neither confirm nor refute (total sample of N = 13). F06 noted that there might be a trend of rising k with rising s; however, the addition of nine more GRB-SNe to the total sample shows that it is highly unlikely that such a correlation exists. When this result is taken in tandem with the results of Drout et al. (2011), who found no statistically significant correlation between peak luminosity and LC decay shapes for a sample of Ibc SNe, it indicates that Ibc SNe (including GRB-SNe and Ic-BL SNe) cannot be used as standardizable candles.