WHAT CAN WE LEARN FROM THE RISING LIGHT CURVES OF RADIOACTIVELY POWERED SUPERNOVAE?

Just prior to emission, a shock is traveling through the envelope. This heats and accelerates the material, unbinding it from the star. A radiation-dominated shock accelerates in the decreasing density of the outer edge of the star with a shock velocity that scales with the density as v_s∝ρ^−β where β ≈ 0.19 (Sakurai 1960). The shock continues to shallower regions until the optical depth falls to τ ≈ c/v_s, where c is the light speed. At this point, the photons are no longer trapped; they stream away and the shock dies. This is what is typically referred to as the "shock breakout" UV/X-ray flash, and it has been frequently studied because of its strong dependence on the radius (which determines both the energy budget of the shock and the timescale of the emission), allowing the progenitor star to be studied from its detection (Colgate 1974; Falk 1978; Klein & Chevalier 1978).

Recently, it was realized that the radiation behind the shock falls out of thermal equilibrium for small radii hydrogen-stripped massive stars with high shock velocities, and that the breakout will be in X-rays (Katz et al. 2010; Nakar & Sari 2010). In SNe Ia, the shock achieves relativistic velocity, and its breakout emission is in the MeV range (Nakar & Sari 2012). In any case, although shock breakout is the first indication of the explosion, its impact for SNe I in the optical/UV bands is negligible. For this reason, it is not discussed further in this paper and not plotted in Figure 1.

Immediately after shock breakout, the observed radiation is out of thermal equilibrium until roughly the time when the gas doubles its radius at t ≈ R_*/v_f (Nakar & Sari 2010), where v_f ≈ 2v_s is the final velocity of the material (Matzner & McKee 1999), which is achieved within minutes or less. The observed photons then gain thermal equilibrium and the expansion enters its spherical homologous phase (Chevalier 1992; Piro et al. 2010; Nakar & Sari 2010; Rabinak & Waxman 2011). As the material that has been heated by the shock expands, a thermal diffusion wave begins backing its way through the ejecta. Above the depth of the diffusion wave, material cools via photon diffusion. Below this depth, material evolves adiabatically. For each fluid element, there is then a competition between adiabatic cooling and diffusive cooling that controls the energy density of that material at the moment its photons begin streaming out of the star. This determines the observational signature of shock-heated cooling, leading to a bolometric luminosity (Nakar & Sari 2010)⁴

$$\begin{equation} L \approx 2 \times 10^{41} \frac{E_{51}^{0.91}R_1}{\kappa _{0.2}^{0.82}M_1^{0.73}} \left(\frac{t}{1\ {\rm hr}}\right)^{-0.35} {\rm erg\ s^{-1}}, \end{equation} \tag{ 1 }$$

and an observed (color) temperature of

$$\begin{equation} T_c \approx 4 \frac{E_{51}^{0.11}R_1^{0.38}}{\kappa _{0.2}^{0.23}M_1^{0.11}} \left(\frac{t}{1\ {\rm hr}}\right)^{-0.61} {\rm eV}, \end{equation} \tag{ 2 }$$

where E = E₅₁10⁵¹ erg is the explosion energy, M = M₁ M_☉ is the ejecta mass, R₁ = R_*/R_☉, and κ is the opacity with κ_0.2 = κ/0.2 cm² g⁻¹ being the canonical value during the cooling phase for fully ionized hydrogen-free gas. For all the scalings in this paper, we assume a polytropic index of n = 3, as is relevant for compact, radiative stars forming SNe Ib/c, or relativistic, degenerate WDs exploding as SNe Ia. The key result is that the luminosity during the shock-heated cooling is directly proportional to the progenitor radius (as was used to derive radius constraints for SN 2011fe by Bloom et al. 2012).

In the optical/UV, the luminosity is rising as long it is in the Rayleigh–Jeans tail, resulting in (see Appendix A)

$$\begin{equation} L_{\rm opt/UV} \propto t^{1.5}. \end{equation} \tag{ 3 }$$

When optical/UV luminosity is observed to rise more steeply than this (as in PTF 12gzk; Ben-Ami et al. 2012), it indicates that the rise from shock-heated cooling is not being seen, and the time of explosion is not confidently constrained. This phase continues for several hours in a core-collapse SN until T_c crosses the UV (at which point the UV flux starts to drop) and reaches the optical band. This is also the point that recombination starts in cases that radioactive decay does not play an important role (as further discussed the next paragraph). In SNe Ia, this phase is terminated after about an hour when the diffusion wave reaches material where the shock was not radiation-dominated and the cooling envelope emission drops significantly (Rabinak et al. 2012).

Once T_c drops sufficiently in a core-collapse SN, a recombination wave begins backing its way in from the surface. The temperature drop becomes more gradual, settling at ≈5000–8000 K and the luminosity drop stops or even gently rises (as discussed in Dessart et al. 2011; also see T. Goldfriend et al., in preparation). As far as optical photometry is considered, both the temperature and the luminosity are roughly constant and the SN enters a "plateau phase." This plateau is similar to the one observed in SN II-P (Popov 1993; Kasen & Woosley 2009), except that for SNe I it is dimmer and it starts earlier due to the smaller progenitor radius. Setting T_c ≈ 0.6 eV as found for the plateau in the SN Ib/c models of Dessart et al. (2011), Equation (2) implies that the optical plateau starts at

$$\begin{equation} t_p \approx 20 \frac{E_{51}^{0.18}R_1^{0.62}}{\kappa _{0.2}^{0.38}M_1^{0.18}}\, {\rm hr}, \end{equation} \tag{ 4 }$$

and together with Equation (1) we find that

$$\begin{equation} L_p \approx 7\times 10^{40} \frac{E_{51}^{0.85}R_1^{0.78}}{\kappa _{0.2}^{0.69}M_1^{0.67}}\, {\rm erg\,{\rm s}^{-1}}, \end{equation} \tag{ 5 }$$

is roughly the plateau luminosity.⁵

To conclude, the detection of the shock-heated cooling phase in core-collapse SNe I is very challenging given its low luminosity and short duration, but the rewards are high. Photometry alone of the rising phase provides tight constraints on the explosion time, and both the optical rising and subsequent plateau phase provide constraints on the progenitor radius (if not overcast by radioactive heating as we discuss next). A convenient method to identify that a rising optical emission is due to shock-heated cooling emission (and not radioactively powered emission) is that the optical emission rises (as predicted by Equation (3)) while the bolometric luminosity drops. This can be seen by having simultaneous UV and optical coverage in the time that T_c sits between these two frequency windows.

If there was no radioactive energy input, this would be the end of the story for the electromagnetic signal from SNe, and most SNe I would be too dim to have ever been detected. But eventually continuous energy release by the radioactive decay of ⁵⁶Ni starts to dominate the observed luminosity, and this is where their light curves begin rising in earnest. When the thermal diffusion wave reaches the shallowest deposits of ⁵⁶Ni, the energy generation from ⁵⁶Ni roughly goes directly into the observed bolometric luminosity (as shown with the red curves in Figure 1), so that

$$\begin{eqnarray} &&L_{56}\approx M_{56}\epsilon , \end{eqnarray} \tag{ 6 }$$

where the specific heating rate from ⁵⁶Ni is

$$\begin{eqnarray} &&\epsilon (t) = \epsilon _{\rm Ni} e^{-t/t_{\rm Ni}} + \epsilon _{\rm Co}(e^{-t/t_{\rm Co}}- e^{-t/t_{\rm Ni}}), \end{eqnarray} \tag{ 7 }$$

where _Ni = 3.9 × 10¹⁰ erg g⁻¹ s⁻¹, t_Ni = 8.76 days, _Co = 7.0 × 10⁹ erg g⁻¹ s⁻¹, and t_Co = 111.5 days and M₅₆ is the mass of ⁵⁶Ni that is exposed by the diffusion wave.

The depth of the diffusion wave tells us which part of the exploding star is being probed by the observations, and it is related to the time after explosion by (see Appendix C)

$$\begin{eqnarray} &&\Delta M_{\rm diff} \approx 8\times 10^{-2} \frac{E_{51}^{0.44}}{\kappa _{0.1}^{0.88}M_1^{0.32}} \left(\frac{t}{1\ {\rm day}} \right)^{1.76}\,M_\odot , \end{eqnarray} \tag{ 8 }$$

where κ_0.1 = κ/0.1 cm² g⁻¹ is the canonical value we use for the radioactively powered phase. This value for the opacity is motivated by the more detailed analysis of Pinto & Eastman (2000). We discuss how this diffusion depth is adjusted by differences in the opacity, especially when the shallow layers become partially ionized, in more detail in Appendix C. For the estimates in this paper, we use quantities that roughly correspond to the SNe Ib/c, but the same general arguments apply to SNe Ia (as well as the rare SNe II that are radioactively powered) with different prefactors. To see how much these can vary for other progenitors, see Appendix C. Also note that Equation (8) becomes less accurate closer to the peak of the light curve, when ΔM_diff grows more slowly with time.

When powered by radioactive heating, the observed luminosity is no longer sensitive to the progenitor radius as can be seen by the lack of an explicit dependence on R_* in Equation (8). Instead, it provides a direct measurement of the amount of ⁵⁶Ni (via Equation (6)) at the location of the diffusion wave. If the explosion time is known, Equation (8) provides ΔM_diff into which this ⁵⁶Ni is mixed. From this, the mass fraction X₅₆ ≈ M₅₆/ΔM_diff can be inferred as a function of the depth ΔM_diff. Thus, in principle, detection of the rise of the ⁵⁶Ni light curve should provide an estimate of the distribution of ⁵⁶Ni (as attempted for the SN Ia 2011fe by Piro 2012), which is helpful for understanding the nature of the explosive burning and the outer structure of the progenitor.

In practice, the exercise described above is not as straightforward as one might think. The reason is that if there is a steep increase in the ⁵⁶Ni abundance, then the assumption that the observed emission is generated only by the composition at the location of the diffusion wave may not be valid. Instead a "diffusive tail" of the energy released in ⁵⁶Ni-rich layers deeper than the diffusion depth (shown as purple curves in Figure 1) may actually dominate over the energy released in ⁵⁶Ni-poor layers shallower than the diffusion depth (shown as red curves in Figure 1). Next we estimate the contribution of this diffusive tail. To do this, we neglect the effect of expansion and adiabatic losses of the trapped radiation. This is appropriate since at any time the observed luminosity is dominated by energy that was deposited over the last dynamical timescale, during which expansion and adiabatic losses introduce only small corrections.

Consider first a deposit of ⁵⁶Ni at some depth d in the star at t = 0. The ⁵⁶Ni decays to produce gamma rays, which are absorbed and create thermal photons. The photons then spread due to diffusion, creating roughly a Gaussian distribution around the ⁵⁶Ni. At each time t, the Gaussian has a width $\sqrt{Kt}$, where K is the diffusion coefficient, which is related to the diffusion time by t_diff = d²/K. For such a distribution, there is a small, but non-zero, fraction of photons that escape from the star because they have diffused by a distance >d from the ⁵⁶Ni depth. The fraction of photons released at t = 0 that have reached the surface by time t is roughly the integral of a Gaussian from d to infinity, which results in

$$\begin{eqnarray} &&{\rm Escaping\ fraction} \approx {\rm erfc}\sqrt{t_{\rm diff}/2t}, \end{eqnarray} \tag{ 9 }$$

where erfc is the complementary error function.

Photons are continuously emitted from the depth d, and thus the total number of photons that diffuse a distance >d is the integral of ${\rm erfc}\sqrt{t_{\rm diff}/2(t-t^{\prime \prime })}$ with time from t'' = 0 to t'' = t. The luminosity scales as the derivative of this integral total number of photons. Taking the derivative of an integral, we find that the luminosity scales proportional ${\rm erfc}\sqrt{t_{\rm diff}/2t}$. Putting these arguments together, if at some time t' the diffusion wave reaches a layer of ⁵⁶Ni that would be producing a luminosity L₅₆(t'), this implies for previous times t < t' that the diffusive tail from this layer also produces a luminosity,

$$\begin{eqnarray} &&L_{\rm tail}(t<t^{\prime }) \approx L_{56}(t^{\prime }) \frac{\epsilon (t)}{\epsilon (t^{\prime })} \frac{{\rm erfc}(t^{\prime }/\sqrt{2}t)}{ {\rm erfc}(1/\sqrt{2})}, \end{eqnarray} \tag{ 10 }$$

where we have used t_diff∝ρr²∝1/t, so that t_diff = t'²/t. This is the appropriate normalization because, as we have described, the diffusion wave reaches the layer at time t' and therefore t_diff = t' when t = t'.

From this discussion, one can see what we meant at the beginning of this section when we mentioned "a steep increase in the abundance of ⁵⁶Ni." Consider the two ⁵⁶Ni distributions shown in Figure 2. In the top panel, the ⁵⁶Ni has a rather shallow distribution which produces the L₅₆(t) (shown by the red curve). Therefore, if we consider the luminosity L₅₆ at some time t', and then trace back the diffusive tail implied from that depth using Equation (10) (shown as a purple curve), the diffusive tail always falls below the L₅₆ light curve. This is a case where the shallow ⁵⁶Ni prevents the diffusive tail from having a noticeable impact, and M₅₆ can be approximated from the observations. Furthermore, X₅₆ ≈ M₅₆/ΔM_diff can be inferred as a function of time if the explosion time is well constrained. In the bottom panel of Figure 2, the ⁵⁶Ni has a steeper distribution. Now, when the diffusive tail is drawn back from a point at time t', it exceeds the L₅₆ light curve. In this latter case, the diffusive tail will dominate the observed rise, and this must be accounted for before attempting to infer M₅₆.

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These conclusions are only approximate since we are focusing on the diffusive tail from a single depth. This is likely not too bad of a simplification, since the diffusive tails from ⁵⁶Ni at even larger depths are exponentially suppressed (by the scaling of the complementary error function). In detail, though, the luminosity of the diffusive tail should depend on the sum of contributions from all depths. We calculate and discuss the impact of this in Appendix B.

Our discussion thus far makes it clear that it is very important to know the time of explosion. It sets the time at which the thermal diffusion wave begins backing its way into the expanding ejecta, and from this at all later times it is roughly known what depths of the exploding star are being probed by the observations via Equation (8).

When the time of explosion is not known from direct detection of shock breakout or shock heating of the surface layers, our discussion of the early light curve should also provide some reason for caution. To illustrate the problem, in the top and middle panels of Figure 1 we compare the light curves for different ⁵⁶Ni depositions. In the top panel, the ⁵⁶Ni is deposited into rather shallow layers, therefore the timescale between the beginning of the explosion and the rise of the light curve is fairly short. In this case, the time of explosion could be reasonably well approximated by extrapolating the ⁵⁶Ni light curve back in time. In the middle panel, the ⁵⁶Ni is deposited in deeper layers, and correspondingly the delay between the shock heating and rising light curve is longer. In this case, extrapolating the ⁵⁶Ni light curve back in time would provide a poor estimate for the time of explosion. We are led to the following important conclusion: one cannot simply estimate the time of explosion by extrapolating the rising light curve back in time because its position relative to the moment of explosion depends on the depth of radioactive heating. This means that for events like SN 2011fe, the constraints on R_* cannot be as tight as previously reported (Bloom et al. 2012) when the explosion time is not known.

The earliest detection of the rising light curve from ⁵⁶Ni does not probe the shallowest layers of the star, but merely the radioactive heating of the diffusive tail from the shallowest deposits of ⁵⁶Ni. For example, if the diffusion wave only reaches the heating after traveling ∼0.1 M_☉ below the exploding star's surface, then it will take ∼2 days to detect this depth (using Equation (8)). Similar delays between the moment of explosion and the rising light curve are seen in the numerical work of Dessart et al. (2011).

When the shock breakout and shock heating are not detected (as is often the case), additional information from color or, even better, spectroscopic observations can be used to break the degeneracy between the depth of ⁵⁶Ni and the time of explosion. We summarize some of the properties of SNe that are most useful for doing this in this section and the next.

In the bottom panel of Figure 1, we schematically show the time-dependent radius of the photosphere (orange curve) during an SN. For a polytropic index of n = 3, the photospheric radius is (Rabinak & Waxman 2011)

$$\begin{eqnarray} &&r_{\rm ph}(t) \approx 3\times 10^{14}\frac{\kappa _{0.1}^{0.11}E_{51}^{0.39}}{M_1^{0.28}} \left(\frac{t}{1\ {\rm day}}\right)^{0.78}\ {\rm cm}, \end{eqnarray} \tag{ 11 }$$

where we have suppressed the dependence on the density structure factor f_ρ and set it to 0.01 (see Appendix D). Even though r_ph is moving into the star as the ejecta expands, it is always moving out in an Eulerian frame. The observed color temperature is

$$\begin{eqnarray} &&T_c \approx \left(\frac{L\tau _c}{4\pi r_{\rm c}^2\sigma _{\rm SB}}\right)^{1/4}\gtrsim \left(\frac{L}{4\pi r_{\rm ph}^2\sigma _{\rm SB}}\right)^{1/4}, \end{eqnarray} \tag{ 12 }$$

where r_c is the color radius and τ_c is the optical depth at the color (thermalization) depth. The inequality comes from the fact that τ_c ⩾ 1 and r_c ⩽ r_ph. For typical parameters τ_c is not much larger than unity (e.g., Nakar & Sari 2010) and so is r_ph/r_c, so this inequality is also a fair approximation of T_c. Although r_ph roughly always scales as a power law with time (Equation (11)), L(t) can vary greatly depending on the ⁵⁶Ni deposition. Thus, the color evolution during the rising phase provides the first clue on the ⁵⁶Ni depth. For deep ⁵⁶Ni, the initial rising phase of L is exponential, so from Equation (12) we see that T_c should be rising as well. On the other hand, if ⁵⁶Ni is shallow then L rises more gradually and T_c is roughly constant or even slowly decreasing during the lightcurve rise.

In the bottom panel of Figure 1, we also show the photospheric velocity (green curve). Taking v_ph = r_ph/t, we find

$$\begin{eqnarray} &&v_{\rm ph}(t)\approx 35{,}000\frac{\kappa _{0.1}^{0.11}E_{51}^{0.39}}{M_1^{0.28}} \left(\frac{t}{1\ {\rm day}}\right)^{-0.22}\ {\rm km\ s^{-1}}. \end{eqnarray} \tag{ 13 }$$

Unlike the photospheric radius, the photospheric velocity is decreasing with time. This is because the photosphere is moving into the star in a Lagrangian sense, and reaching material that is moving at successively lower velocities. Furthermore, the time derivative of the photospheric velocity is dv_ph/dt∝t^−1.22. Therefore, we expect the change in the photospheric velocity to become more gradual with time.

Using the above arguments, we conclude that there are four main ways in which the depth of ⁵⁶Ni heating may be qualitatively inferred. All other things being equal, a larger ⁵⁶Ni depth implies the following.

• 1.  
  The photospheric radius is larger when the ⁵⁶Ni heating gets out, and therefore the temperature will be lower.
• 2.  
  The luminosity at early times is increasing faster than the radius expands, causing temperature to evolve bluer.
• 3.  
  The photospheric velocities are lower.
• 4.  
  The photospheric velocities evolve more slowly with time.

In additional to the qualitative correlations listed above, a single measurement of the photosphere velocity and color temperature during the rise can provide a strict, model-independent, upper limit to the time of explosion before that measurement. This is seen by first rewriting Equation (12) as

$$\begin{eqnarray} &&L \approx 4\pi r_c^2 \sigma _{\rm SB} T_c^4/\tau _c. \end{eqnarray} \tag{ 14 }$$

If we define v_c as the velocity of material at the color depth, then r_c ≈ v_ct_exp where t_exp is the time since the explosion began. Putting these together, we estimate

$$\begin{eqnarray} &&t_{\rm exp} \approx \left(\frac{L\tau _c}{4\pi v_c^2\sigma _{\rm SB}T_c^4} \right)^{1/2}. \end{eqnarray} \tag{ 15 }$$

In general, T_c can be measured directly from the observations, but any velocity that can be measured will be from an absorption feature shallower than the photosphere and generally v_ph ≳ v_c. Furthermore, τ_c will be greater than unity but also difficult to infer just from the observation. It is therefore useful to have a quantity that can simply be estimated directly in terms of observable quantities,

$$\begin{eqnarray} &&t_{\rm min} \equiv \left(\frac{L}{4\pi v_{\rm ph}^2\sigma _{\rm SB}T_c^4} \right)^{1/2} = \frac{t_{\rm exp}}{\tau _c^{1/2}}\left(\frac{v_c}{v_{\rm ph}} \right)\lesssim t_{\rm exp}. \end{eqnarray} \tag{ 16 }$$

Substituting typical values we find

$$\begin{eqnarray} &&t_{\rm min} = 4.3 \left(\frac{L}{10^{42}\rm \ {erg\ s^{-1} }}\right)^{1/2} \left(\frac{T_c}{10^4\ \rm {K}} \right)^{-2} \nonumber \\ &&\times \left(\frac{v_{\rm ph}}{10^4\ {\rm km\ s^{-1}}}\right)^{-1}{\rm days}. \end{eqnarray} \tag{ 17 }$$

Therefore, t_min is an observable quantity that provides a model-independent lower limit to the time of explosion. Another reason that t_min is especially useful is that it only requires that the velocity and temperature of the SN be obtained at a single time. Thus, when resources are limited, using Equation (17) is a helpful technique for learning a lot about the SN with minimal additional investment.

This concludes our discussion of the qualitative features of rising SN I light curves. In general, specific events may have details that we have not addressed, like non-spherical explosions or complicated velocity profiles. But by clearly spelling out the chain of reasoning behind our conclusions, we provide rules of thumb that can be used to build intuition, even in cases that are somewhat more complicated. To illustrate how our arguments can be used in practice, in the next section we apply them to a recently discovered, well-studied SNe Ic.

Even though shock-heated cooling was not observed, PTF 10vgv did have an early detection of the rising light curve and upper limits in the time before this. Using this information, interesting constraints on the progenitor's radius are still possible (Corsi et al. 2012). In Figure 3, we plot the observed R-band absolute magnitude of PTF 10vgv (circles), along with upper limits (triangle). These all assume a distance modulus of 34.05 and galactic extinction of A_R = 0.445 mag (Schlegel et al. 1998). We then calculate the shock-heated cooling according to Equation (1), which transitions to the plateau phase given by Equation (5). We infer the corresponding R-band absolute magnitude using Equation (2), or a plateau temperature of ≈0.6 eV, along with P48-calibrated bolometric corrections provided by E. O. Ofek (2012, private communication). The resulting light curves are plotted alongside the data from PTF 10vgv in Figure 3 for a range of explosion times and progenitor radii. The conclusion is that without knowing the time of explosion, the radius can only be constrained to be R_* ≲ 1–20 R_☉. In general, the earlier the explosion time is the more stringent the constraints on the radius.

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Next we consider what can be learned about the mass and distribution of ⁵⁶Ni from the observed light curve. Before this can be done, there are two considerations that must be made. First, we need to convert the observed R-band magnitudes to a bolometric luminosity. We solve for this using

$$\begin{eqnarray} &&L \approx 4\pi r_c^2 \sigma _{\rm SB}T_c^4/\tau _c\approx 10^{(M_{R,\odot }-M_R-{\rm BC})/2.5}\, L_\odot. \end{eqnarray} \tag{ 18 }$$

Since BC depends on T_c, we can self-consistently find T_c and thus L at any given time. For our estimates here we simply take τ_c ∼ 1 and r_c ∼ r_ph as given by Equation (11) to produce the thick, solid curves in Figure 4. Although not physically accurate, these estimates are adequate since the bolometric luminosity is found to be fairly robust. Once again, we must take into account that the time of explosion is not known, so we consider three example explosion times.

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The second consideration we must make before attempting to associate the bolometric light curve with the ⁵⁶Ni distribution is the impact of a diffusive tail. Assuming that the bolometric light curves in Figure 4 are representative of the ⁵⁶Ni distribution, we can then ask, for a given luminosity at a given time, what is the implied diffusive tail at earlier times? The dotted lines in Figure 4 show the diffusive tails originating from various times using Equation (10). If the dotted line sits below the solid curve, this means that the diffusive tail is insufficient to explain the bolometric light curve at this depth, and thus the bolometric light curve represents direct heating from ⁵⁶Ni (as shown in the top diagram in Figure 2). On the other hand, if the dotted line sits above the solid curve, then the diffusive tail from that point overestimates the earlier bolometric light curve and luminosity at that time cannot be from direct heating of ⁵⁶Ni at the diffusion depth (as shown in the bottom diagram in Figure 2).

The conclusion from this comparison is that it is difficult to determine which of these three explosion times are more accurate from just this information. If the SN occurred recently (top panel of Figure 4), then only the earliest heating is explained by a diffusive tail and we would infer that the majority of the rising light curve directly probes the ⁵⁶Ni distribution. On the other hand, if the explosion occurred further in the past (the middle and bottom panels of Figure 4), then the majority of the rising light curve is explained as a diffusive tail due to deep deposits of ⁵⁶Ni. This is consistent with our discussion in Section 2.6 that there is a degeneracy between these two limits unless more information is available.

Nevertheless, we can still try to constrain the mass and distribution of ⁵⁶Ni as a function of the explosion time, the results of which are shown in Figure 5. In the top panel, we plot the mass of ⁵⁶Ni inferred from Equation (6). This is only plotted for depths at which the bolometric luminosity cannot be explained by a diffusive tail using Figure 4. In the middle panel, we plot the mass fraction X₅₆, and in bottom panel the thermal diffusion depth using Equation (8).

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We next consider what additional information can be learned about PTF 10vgv from temperature and velocity measurements. Unfortunately, the available data during the rise are rather limited. Corsi et al. (2012) mention a 16, 000 km s⁻¹ Si ii absorption feature (which is typically associated with the photosphere; Tanaka et al. 2008) observed on 2010 September 16, which corresponds to ≈2 days after discovery. From the spectrum taken on that same day, by eye one can see that it roughly peaks at ∼4300 Å, which, assuming a blackbody spectrum, corresponds to a temperature of ∼6700 K. But we must be cautious. In the study of SN 1994I by Sauer et al. (2006), a spectrum with a similar peak is observed at eight days post-explosion, and their detailed models infer a temperature of ∼10, 000 K. As we describe next, if the correct temperature were known, then tight constraints could be placed on the time of explosion and thus the other properties of PTF 10vgv. But, given that the temperature cannot be extracted from the data without a detailed spectral modeling, which is beyond the scope of this paper, we separately consider both the cases of 6700 K and 10, 000 K and discuss the implications.

Using Equation (17), we estimate the minimum time of the explosion. For T_c = 6700 K, v_ph = 16, 000 km s⁻¹, and L ≈ 1.5 × 10⁴² erg s⁻¹ (taken from Figure 4), we find t_min ∼ 7 days. Therefore, if the cooler temperature is the correct one then the explosion must have occurred ∼5 days or more before the first detection. On the other hand, for T_c = 10, 000 K we estimate t_min ∼ 3 days and the explosion must have occurred ∼1 day or more before the first detection.

To test these conclusions, we perform a detailed fit of the temperature, velocity, and bolometric luminosity constraints simultaneously. This is done by varying E, M, and the time of explosion over a wide range of values and identifying which best fit the constraints. The result for either 6700 K or 10, 000 K is that a fit can only be obtained when E∝M^0.72 because this fixes the normalization of the velocity (see Equation (13)). But, for a given temperature, only a very narrow range of explosion times are consistent with the data. In Figures 6 and 7, we plot the results of our fitting. These show that for a temperature of 6700 K or 10, 000 K, the explosion much have occurred ∼5 days or ∼2 days prior to first detection, respectively. These match the values of t_min estimated before.

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Besides showing how the explosion time can be constrained, Figures 6 and 7 highlight many of the general trends that we discussed in Section 2 when considering how different explosion times impact various observables of the SN. For a more recent explosion time, the gradient in v_ph is greater and the temperature is higher and decreasing. In contrast, for an explosion time more distant in the past, T_c is lower and increasing for a longer time. If merely a couple of data points were available, these trends could be identified in the observations, and even tighter constraints could be placed on PTF 10vgv. Nevertheless, it is powerful that even a single velocity and temperature measurement provides such stringent constraints. SNe Ia tend to have more early data available than SNe Ib/c, and we consider some of these events in forthcoming work.

Our conclusions from the previous sections show just how greatly our inferences about the progenitor can vary depending on the explosion time. These are summarized as follows.

• 1.  
  If the temperature at 2 days past first detection is ∼6700 K, then the explosion occurred ∼5 days or more before first detection. The radius is constrained to be ≲ 1 R_☉ and the ⁵⁶Ni must be located much deeper in the ejecta. Also the ejecta is likely of higher mass since it takes longer to reach the SN peak.
• 2.  
  If the temperature at 2 days past first detection is ∼10, 000 K, then the explosion occurred merely ∼2 days before first detection. The radius is constrained to be ≲ 6 R_☉ and the ⁵⁶Ni must be located fairly shallowly. Also the ejecta is likely of lower mass.

Thus we are left with two seemingly opposite conclusions about almost all the aspects of the ejecta just from opposite assumptions about the time of explosion. We hope that a proper modeling of the available spectrum, which we do not carry out here, can provide the correct temperature and decide between these two options. We next discuss the implications for the progenitor star that produced PTF 10vgv in each case.

Since the progenitors of such SN Ic are hydrogen-stripped stars, the models of Yoon et al. (2010) are a helpful starting place. The most striking feature of these progenitors is the inverse relationship between mass and radius that is shown in their Figure 12. If PTF 10vgv had a radius ≲ 1 R_☉, it is strongly inconsistent with any of the binary models below M_f ≲ 7 M_☉, where M_f is the final mass at the time of explosion, which all have radii larger than ∼2 R_☉. Their helium star models (i.e., Wolf–Rayet progenitors) have smaller radii and may be consistent with M_f ≳ 6–8 M_☉, depending of the mass-loss prescription employed.⁶ Other progenitors that are consistent with such small radii are the gamma-ray burst models of Yoon et al. (2006) with M_f ≳ 12 M_☉, but these probably are not applicable to PTF 10vgv.

Combining the generally large mass inferred for a small-radius progenitor with an ejecta mass of ∼3–5 M_☉ (estimated from ΔM_diff at peak), the remnant mass must be ≳ 2–3 M_☉. This is near the range of the maximum mass of normal neutron stars (Lattimer & Prakash 2001). Would this imply that this event led to the formation of a black hole? Given the number of estimates that have been made throughout our analysis, it is difficult to make this conclusion with such certainty. But this does demonstrate how our analysis can make interesting connections between observations on one hand and the detailed simulations of progenitors on the other.

For the case where R_* ≲ 6 R_☉, Yoon et al. (2010) show that such radii are naturally expected for progenitors in a mass range of M_f ∼ 4–5 M_☉, and it is easier to reconcile the ejecta mass estimate with formation of a neutron star remnant. A less massive progenitor with a large radius has been predicted to have mixing of ⁵⁶Ni into the outer layers by Rayleigh–Taylor instability during the SN explosion (Hachisu et al. 1991; Joggerst et al. 2009), similar to what we infer for a recent explosion (Figure 5). So even in this case there are interesting correlations and implications for the explosion mechanism that can be investigated.

Finally, it is worth discussing the inferences that can be made from the fact that PTF 10vgv was classified as an SN Ic. Dessart et al. (2012) studied the influence of ⁵⁶Ni mixing on the ejecta of SNe Ib/c, and found that it strongly impacts the observed spectral features. Chief among these are the He i lines, which require non-thermal electrons that are excited by γ-ray lines from ⁵⁶Ni and ⁵⁶Co when the helium mass fraction is ≲ 0.5 (Dessart et al. 2011). This means that the same ejecta with a significant amount of helium can produce either an SN Ib if the ⁵⁶Ni mixing is strong or an SN Ic if the ⁵⁶Ni mixing is weak. If PTF 10vgv were constrained to have a recent explosion with strong ⁵⁶Ni mixing, we would be forced to conclude that its outer layers are very helium-poor. In the future, it would be informative to repeat our analysis on an SN Ib to see if shallow ⁵⁶Ni can be inferred to make the He i visible.