Signatures of r-process Enrichment in Supernovae from Collapsars

Very late-time observations have been suggested (SBM19) as a key strategy for testing the r-process collapsar hypothesis, since by the nebular phase the ejecta is transparent and lines of sight extend into the inner regions where, for core-envelope models of rCCSNe, the r-process material resides. However, even in the nebular phase, the strength of the signal depends on the degree to which the nebular spectrum of the r-process-rich layers diverges from that of the r-process-free envelope, as well as on the brightness of each component.

In the case of a completely optically thin ejecta heated by uniformly distributed radioactive ⁵⁶Ni/Co, the ratio of luminosities from the r-process-rich and r-process-free layers is simply ${L}_{\mathrm{neb}}^{rp}/{L}_{\mathrm{neb}}^{\mathrm{sn}}={\psi }_{\mathrm{mix}}/(1-{\psi }_{\mathrm{mix}})$. (Since we are ignoring heating from r-process decay, this estimate is a conservative lower limit.)

We assign to the r-process-free envelope a spectral energy distribution (SED) derived from late-time photometry of the SN Ic SN 2007gr (Hunter et al. 2009; Bianco et al. 2014), one of the few SNe Ic observed up to ∼ 150 days after peak in both optical and near-infrared (NIR) bands. We assume that the r-process-enriched layers shine like a blackbody regardless of the r-process mass fraction in the core. (Thus, the models of Section 2.1 depend only on ψ_mix, and not (directly) on M_rp.) We leave the effective temperature T_rp as a free parameter, deferring a more detailed discussion of our treatment of emission from optically thin material (whether r-process-enriched or not) to Section 3.

With these simplifications, we can determine how the spectrum of a totally optically thin rCCSN differs from the r-process-free case, for a given ψ_mix and T_rp. For the range of T_rp we consider, which are broadly consistent with (admittedly limited) constraints from both theory (Hotokezaka et al. 2021) and observation (Kasliwal et al. 2022), the impacts of the r-process are most visible in the NIR. We therefore characterize the signal strength in terms of Δ(R − X), the change in R − X color relative to the r-process-free SN case (M_rp = 0), where X ∈ {J, H, K}. All magnitudes are calculated using the AB system and generic Bessel filters.

Figure 2 shows how each color changes for 400 K ≤ T_rp ≤ 3000 K and 0 ≤ ψ_mix ≤ 0.95. The signal strength increases with the mixing coordinate ψ_mix and for a given ψ_mix is maximal for T_rp with blackbody functions peaking at wavelengths within the NIR band under consideration. Regardless of T_rp and ψ_mix, the signal becomes easier to observe at longer wavelengths. However, for cases of low to moderate mixing (ψ_mix ≲ 0.3) the color difference even in R − K is ≲1.5 mag for all T_rp.

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While this difference may still seem substantial, it is important to bear in mind that models of nebular-phase emission are not well constrained in either the standard or r-process-enriched case. Relying exclusively on nebular observations may make it difficult to evaluate collapsars as r-process sites, particularly if the r-process matter is centrally concentrated (low ψ_mix). We therefore turn our attention to the pre-nebular (photospheric) phase.

Observations in the photospheric phase (before the ejecta is fully transparent) can sidestep some of the complications inherent in the acquisition and analysis of nebular-phase data. First, r-process emission in the photospheric phase is better understood, thanks both to extensive theoretical studies of r-process elements' atomic structures, which enable descriptions of their behavior in local thermodynamic equilibrium (e.g., Kasen et al. 2013; Fontes et al. 2020; Tanaka et al. 2020), and to observations of kn170817 (Arcavi et al. 2017; Coulter et al. 2017; Drout et al. 2017; Evans et al. 2017; Kilpatrick et al. 2017; McCully et al. 2017; Nicholl et al. 2017; Shappee et al. 2017; Smartt et al. 2017; Soares-Santos et al. 2017; Kasliwal et al. 2022).

Second, the SN is brighter during the photospheric phase, and it is therefore easier to obtain high signal-to-noise ratio photometry across a range of wavelengths. Finally, early observations do not preclude late-time follow-up. To the contrary, they may be useful for filtering out the events most worthy of further attention during nebular epochs.

Despite these advantages, the photospheric phase presents its own set of challenges. While the ejecta remains optically thick, the r-process signal may be more difficult to discern than it would be during the nebular phase, due to overlapping, highly broadened spectral absorption features (e.g., Chornock et al. 2017; Kasen et al. 2017), and/or contamination from the r-process-free envelope. This is of particular concern early on, when the envelope is opaque and obscures emission from the enriched core underneath it. The position of the photosphere (the surface that divides optically thick from optically thin material) is therefore a good indicator of how observable an r-process signature is at a given time.

To model the photospheric phase, we build on the simple ejecta structure introduced in Section 2.1. In addition to M_ej, we now describe the ejecta in terms of its kinetic energy E_k and explicitly consider the r-process mass fraction in the enriched layers, ζ = M_rp/M_mix. This model directly corresponds to Figure 1 and makes use of the full set of parameters defined there and in Table 1.

As in Section 2.1, our analysis here rests on the distinct optical properties of r-process elements. We assume that the high opacity of the enriched regions causes them to shine in the NIR, significantly redder than both photospheric and nebular-phase emission from the r-process-free layers. (The assumption of an NIR-dominated SED is a good one for cores composed primarily of r-process elements; however, with increased mixing, both dilution and enhanced ionization due to energy from ⁵⁶Ni/Co decay may reduce the opacity, resulting in somewhat bluer emission; e.g., Barnes et al. 2021.)

While the effect of the high-opacity core will be subtle during the light curve's early stages, it will become more apparent as the photosphere, which forms at ever-lower mass coordinates, sinks into the r-process layers. At this point, the enriched core becomes "visible," and its higher opacity exerts a greater influence on the overall SED of the rCCSN.

This argument suggests that r-process enrichment will be easier to detect in explosions with larger ψ_mix and M_rp. Simple one-zone light-curve models allow us to map out this dependence. We focus here on times after the outer layers have become transparent. During this period, radiation from the photosphere originates entirely from r-process-enriched material, while optically thin emission comes predominantly from the r-process-free envelope. Both enriched and unenriched regions contain radioactive material, and both continue to radiate after becoming optically thin. However, the recession of the photosphere slows dramatically after it reaches the inner high-opacity r-process layers, ensuring that nebular emission from the envelope dominates nebular emission from the core (${L}_{\mathrm{neb}}^{\mathrm{sn}}\gg {L}_{\mathrm{neb}}^{r{\rm{p}}}$) out to fairly late times. We therefore ignore for the time being the contribution of ${L}_{\mathrm{neb}}^{r{\rm{p}}}$ and assume ${L}_{\mathrm{neb}}={L}_{\mathrm{neb}}^{\mathrm{sn}}$.

We define the ratio of the photospheric (r-process) and nebular (r-process-free) luminosities to be

$\begin{eqnarray}&&{{ \mathcal R }}_{{\rm{L}}}\equiv {L}_{\mathrm{ph}}^{r{\rm{p}}}/{L}_{\mathrm{neb}}^{\mathrm{sn}},\end{eqnarray} \tag{ 1 }$

and we adopt it as a rough indicator of the detectability of an r-process-enrichment signature.

We estimate (see Section 3 for more detail) that optically thin r-process-free material emits ∼ 55% of its energy at NIR wavelengths (1 μm ≤ λ ≤ 2.5 μm). The strength of the NIR excess from the photosphere can then be approximated as $\approx {L}_{\mathrm{ph}}^{r{\rm{p}}}/0.55{L}_{\mathrm{neb}}^{\mathrm{sn}}$. To produce an NIR signal ∼ 50% stronger than expected from r-process-free nebular emission alone requires ${{ \mathcal R }}_{{\rm{L}}}\gtrsim 0.3$.

The r-process-detectability metric ${{ \mathcal R }}_{{\rm{L}}}$ depends on a few fundamental timescales. For a constant-density ejecta of mass M_ej and kinetic energy E_k, the r-process-free envelope becomes transparent at

$\begin{eqnarray}&&\begin{array}{l}{\tau }_{\mathrm{tr}}=53\ \mathrm{days}\times \frac{{M}_{\mathrm{ej}}/{M}_{\odot }}{{\left({E}_{k}/1\ \mathrm{foe}\right)}^{1/2}}\sqrt{1-{\psi }_{\mathrm{mix}}^{1/3}},\\ \,\,\,\,\mathrm{or}\ {\left(\frac{{\tau }_{\mathrm{tr}}}{{\tau }_{\mathrm{pk}}}\right)}^{2}=1.2\frac{(1-{\psi }_{\mathrm{mix}}^{1/3})}{{\beta }_{\mathrm{ej}}},\end{array}\end{eqnarray} \tag{ 2 }$

where 1 foe = 10⁵¹ erg and β_ej is the characteristic velocity in units of c (${E}_{{\rm{k}}}={M}_{\mathrm{ej}}{\beta }_{\mathrm{ej}}^{2}{c}^{2}/2$). In the second line, we have normalized to the standard light-curve peak time, ${\tau }_{\mathrm{pk}}\,=9\ \mathrm{days}\,{\left({M}_{\mathrm{ej}}/{M}_{\odot }\right)}^{3/4}{\left({E}_{{\rm{k}}}/1\ \mathrm{foe}\right)}^{-1/4}$, which, like the transparency time ${\tau }_{\mathrm{tr}}$, was calculated assuming an ejecta opacity of ${{\rm{\kappa }}}_{\mathrm{sn}}=0.05$ cm² g⁻¹, much lower than the opacity of r-process compositions. This choice presumes that the early light-curve evolution is driven by the unenriched layers, an assumption that becomes less reliable with increased outward mixing (higher ψ_mix).

Once the unenriched layers are optically thin, their luminosity reflects the rate at which the material in those layers produces energy via radioactivity. Thus, for $t\gt {\tau }_{\mathrm{tr}}$, ${L}_{\mathrm{neb}}^{\mathrm{sn}}$ declines as ⁵⁶Ni/Co decay away. In contrast, emission from the r-process layers evolves on a distinct timescale set by their opacity and mass. It may be rising, maximal, or in decline at ${\tau }_{\mathrm{tr}}$. A second key timescale is therefore τ_rp, the time over which ${L}_{\mathrm{ph}}^{r{\rm{p}}}$ rises to a maximum. This can be estimated as the time to peak for a transient consisting solely of the inner enriched layers,

$\begin{eqnarray}&&\begin{array}{l}{\tau }_{r{\rm{p}}}=9\ \mathrm{days}\ \times \frac{{\left({M}_{\mathrm{ej}}/{M}_{\odot }\right)}^{3/4}{\psi }_{\mathrm{mix}}^{1/3}}{{\left({E}_{k}/1\ \mathrm{foe}\right)}^{1/4}}{\left[\zeta ({ \mathcal K }-1)+1\right]}^{1/2},\\ \,\,\,\,\ \mathrm{or}\ {\left(\frac{{\tau }_{r{\rm{p}}}}{{\tau }_{\mathrm{tr}}}\right)}^{2}\,=\,0.9\frac{{\beta }_{\mathrm{ej}}{\psi }_{\mathrm{mix}}^{2/3}}{(1-{\psi }_{\mathrm{mix}}^{1/3})}\left[\zeta ({ \mathcal K }-1)+1\right],\end{array}\end{eqnarray} \tag{ 3 }$

where ζ = M_rp/M_mix is the r-process mass fraction of the enriched layers and ${ \mathcal K }={{\rm{\kappa }}}_{r{\rm{p}}}/{{\rm{\kappa }}}_{\mathrm{sn}}$ is the ratio of the r-process opacity to the opacity of the (unenriched) SN ejecta. In normalizing to ${\tau }_{\mathrm{tr}}$, we have again assumed ${{\rm{\kappa }}}_{\mathrm{sn}}=0.05$ cm² g⁻¹. Since heavy r-process compositions have κ_rp ≈ 10 cm² g⁻¹ (Barnes & Kasen 2013; Tanaka & Hotokezaka 2013; Grossman et al. 2014), ${ \mathcal K }\approx 200$.

If ${\tau }_{\mathrm{tr}}$ represents the first chance to observe emission from the r-process-enriched layers, τ_rp is a proxy for the best chance—the point at which that emission component glows brightest. Thus, systems for which ${\tau }_{r{\rm{p}}}\approx {\tau }_{\mathrm{tr}}$ are close to ideal from an observability standpoint; the r-process layers are shining most strongly around the time they first come into view, when the SN overall is still fairly bright. If instead ${\tau }_{r{\rm{p}}}\ll {\tau }_{\mathrm{tr}}$, the luminosity from the inner r-process core begins its decline before it is even visible. (At the other extreme, for ${\tau }_{r{\rm{p}}}\gg {\tau }_{\mathrm{tr}}$, there is a danger that ${L}_{\mathrm{ph}}^{r{\rm{p}}}$ will climb to its peak only after the rCCSN overall has grown faint.) Thus, as we will see, the observability parameter ${{ \mathcal R }}_{{\rm{L}}}$ is sensitive to τ_rp/${\tau }_{\mathrm{tr}}$, and a successful observation is more likely when this ratio ≳ 1.

The value of ${\tau }_{r{\rm{p}}}/{\tau }_{\mathrm{tr}}$ depends on the interplay between ψ_mix, ζ, and β_ej, as expressed in Equation (3). Not surprisingly, for constant β_ej, τ_rp/${\tau }_{\mathrm{tr}}$ increases with both the mixing coordinate (ψ_mix) and the r-process mass fraction in the core (ζ). However, if β_ej is sufficiently high, τ_rp/${\tau }_{\mathrm{tr}}$ can approach unity even for low ψ_mix and ζ. This can be seen in Figure 3, which plots the contours at which ${\tau }_{r{\rm{p}}}/{\tau }_{\mathrm{tr}}=1$ for different β_ej. As Figure 3 illustrates, higher-velocity outflows will offer opportunities to observe r-process enrichment for a greater variety of enrichment parameters (ψ_mix and ζ).

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However, even if the timescales are favorable (${\tau }_{r{\rm{p}}}\approx {\tau }_{\mathrm{tr}}$), the r-process-visibility parameter ${{ \mathcal R }}_{{\rm{L}}}$ is limited by ψ_mix; only energy deposited behind the photosphere can be emitted from behind the photosphere. Our models have uniformly distributed ⁵⁶Ni, which establishes ψ_mix/(ψ_mix − 1) as a fundamental scale for ${{ \mathcal R }}_{{\rm{L}}}$. According to Arnett's law (Arnett 1980, 1982), ${{ \mathcal R }}_{{\rm{L}}}$ reaches a maximum of ψ_mix/(1 − ψ_mix) at t = τ_rp.

We estimate ${L}_{\mathrm{ph}}^{r{\rm{p}}}$ at t ∼ τ_rp by Taylor-expanding the analytic light-curve solution to a one-zone model of a radioactively powered transient (Chatzopoulos et al. 2012), ⁴

$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{ph}}^{r{\rm{p}}}(t\sim {\tau }_{r{\rm{p}}}) & \approx & {\psi }_{\mathrm{mix}}{M}_{56}\left[{\dot{\epsilon }}_{56}({\tau }_{r{\rm{p}}})-\displaystyle \frac{1}{2}\displaystyle \frac{{\left(t-{\tau }_{r{\rm{p}}}\right)}^{2}}{{\tau }_{r{\rm{p}}}{\tau }_{\mathrm{Ni}}}\right.\\ & & \times \left.\left({\dot{\epsilon }}_{56}({\tau }_{r{\rm{p}}})-{\dot{\epsilon }}_{\mathrm{Co},\mathrm{eq}}({\tau }_{r{\rm{p}}})\right)\Space{0ex}{3.55ex}{0ex}\right],\end{array}\end{eqnarray} \tag{ 4 }$

where

$\begin{eqnarray*}&&{\dot{\epsilon }}_{\mathrm{Co},\mathrm{eq}}(t)=\displaystyle \frac{{q}_{\mathrm{Co}}}{{\tau }_{\mathrm{Co}}}\exp \left[\displaystyle \frac{-t}{{\tau }_{\mathrm{Co}}}\right].\end{eqnarray*}$

In Equation (4), M₅₆ is the ⁵⁶Ni mass, ${\dot{\epsilon }}_{56}$ is the rate of specific energy production by ⁵⁶Ni and ⁵⁶Co decay, and τ_Ni (τ_Co) is the ⁵⁶Ni (⁵⁶Co) lifetime. The quantity q_Co is the decay energy of ⁵⁶Co divided by its mass. Since we approximate the nebular luminosity from the r-process free layers as

$\begin{eqnarray*}&&{L}_{\mathrm{neb}}^{\mathrm{sn}}=(1-{\psi }_{\mathrm{mix}}){M}_{56}{\dot{\epsilon }}_{56},\end{eqnarray*}$

Equation (4) enables a straightforward determination of ${{ \mathcal R }}_{{\rm{L}}}$, once all model parameters have been specified.

This framework allows us to estimate, given M_ej and E_k, the minimum ψ_mix for which some r-process mass M_rp can produce an "observable" signal—i.e., one that produces a strong NIR excess on a timescale not too delayed relative to the light-curve peak. This estimate is shown in Figure 4 for M_rp = 0.05 M_⊙, an r-process mass comparable to what was produced in GW170817 (Abbott et al. 2017a; Kasen et al. 2017; Kasliwal et al. 2017; Tanaka et al. 2017). In calculating ${\psi }_{\min }$, we defined an observable signal as one for which (a) ${\tau }_{\mathrm{tr}}\leqslant 3{\tau }_{\mathrm{pk}}$ and (b) ${{ \mathcal R }}_{{\rm{L}}}$ reaches a maximum ≥ 0.3 at some point in the interval ${\tau }_{\mathrm{tr}}\leqslant t\leqslant 3{\tau }_{\mathrm{pk}}$. The first criterion addresses whether the r-process signal will appear before the light curve has dimmed significantly, while the second requires the signal to be strong enough to appreciably alter the SED from the optically thin r-process-free ejecta.

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Applying these conditions, we find (for M_rp = 0.05 M_⊙) that SNe with low mixing coordinates (ψ_mix ≲ 0.4) generate a visible r-process signal only for β_ej ≳ 0.04c. This is somewhat larger than typical SN Ib/c velocities, but not uncommon for higher-energy GRB-SNe, whose velocities cluster in the range β_ej = 0.057 ± 0.013 (Modjaz et al. 2016). To better illustrate this point, we have plotted in Figure 4 the inferred E_k and M_ej for several observed GRB-SNe, which we take from Cano et al. (2017). Fortuitously, our analysis suggests that the high-velocity SNe Ic-BL connected to long GRBs— i.e., the SNe most likely to be associated with collapsars, and hence significant r-production—are the same events for which r-process enrichment is detectable even for low ψ_mix.

Despite the simplifications invoked in the preceding analysis, these trends suggest that rCCSN search strategies need not be limited to the nebular phase. Indeed, they motivate a more rigorous consideration of the full evolution of rCCSN light curves.

We begin with ejecta composed of concentric, homologously expanding spherical shells. The density and composition of each shell are parameters of the model, and the frequency-independent (gray) opacity is a function of the local temperature and composition.

The internal energy of a shell i evolves as

$\begin{eqnarray}&&\frac{{{dE}}_{\mathrm{int},i}}{dt}={\dot{Q}}_{\mathrm{rad},i}-\frac{{E}_{\mathrm{int},i}}{t}-{L}_{i},\end{eqnarray} \tag{ 5 }$

where ${\dot{Q}}_{\mathrm{rad},i}$ is the power injected by radioactive decays (in this case, of ⁵⁶Ni, ⁵⁶Co, and r-process nuclei), and E_int,i /t accounts for adiabatic losses. The radiated luminosity L _i depends on the local diffusion (t_diff) and light-crossing (t_cross) timescales,

$\begin{eqnarray*}&&{L}_{i}=\frac{{E}_{\mathrm{int},i}}{{t}_{\mathrm{diff},i}+{t}_{\mathrm{cross},i}},\end{eqnarray*}$

with

$\begin{eqnarray*}&&{t}_{\mathrm{diff},i}=\frac{2}{c}\displaystyle \sum _{j\geqslant i}{\rho }_{j}{\kappa }_{j}{r}_{j}{\rm{\Delta }}{r}_{j}\end{eqnarray*}$

and

$\begin{eqnarray*}&&{t}_{\mathrm{cross},i}=\frac{{r}_{i}}{c},\end{eqnarray*}$

where ρ _i is the mass density of shell i, κ_i its opacity, r_i its current radius, and Δr _i its radial width. The emerging bolometric luminosity at any time t is a sum over L _i ,

$\begin{eqnarray*}&&{L}_{\mathrm{bol}}(t)=\displaystyle \sum _{i}{L}_{i}.\end{eqnarray*}$

In an optically thick system, radiation tends toward a blackbody distribution. For a gray-opacity medium like ours, the photosphere is well defined and the relationship between the luminosity and the SED is straightforward,

$\begin{eqnarray*}&&L(t)=4\pi {r}_{\mathrm{ph}}^{2}{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^{4}\end{eqnarray*}$

and

$\begin{eqnarray}&&{L}_{\nu }=4{\pi }^{2}{r}_{\mathrm{ph}}^{2}{B}_{\nu }({T}_{\mathrm{eff}}),\end{eqnarray} \tag{ 6 }$

where σ_SB is the Stefan–Boltzmann constant, r_ph is the radius of the photosphere, T_eff is the effective temperature, and B_ν is the Planck function.

As the ejecta becomes increasingly transparent, the blackbody approximation becomes less and less reliable. We are interested in modeling the emission of the SN as it transitions from optically thick to optically thin, which motivates a modification of the blackbody prescription of Equation (6). As in Section 2, we categorize the emerging luminosity as photospheric if it originates interior to the radius r_ph defined by $\tau ={\int }_{{r}_{\mathrm{ph}}}^{\infty }\rho (r^{\prime} )\kappa (r^{\prime} )dr^{\prime} =2/3$, and nebular otherwise.

The photospheric component, L_ph, is translated to an SED via Equation (6). The rigorous numerical modeling required to predict the emission of the nebular component is beyond the scope of this work. Instead, we assume that the SED of radiation from optically thin regions depends only on the composition of the zone where it originates. Building on the discussions in Sections 2.1 and 2.2, we associate one characteristic nebular SED with r-process-free SN ejecta and a second one with the r-process. The net SED from an optically thin zone is a scaled sum of the two, as explained in Section 3.2.

We apply this blueprint to regular and rCCSN models characterized by the same parameters as in Section 2: ejecta mass (M_ej) and velocity (β_ej), ⁵⁶Ni mass (M₅₆), the r-process mass M_rp (equal to zero for unenriched SNe), and, for M_rp > 0, the mixing coordinate ψ_mix. All models have a uniform distribution of ⁵⁶Ni and a broken power-law mass density profile, $\rho \propto {\left(v/{v}_{\mathrm{tr}}\right)}^{-\alpha }$, where α = 1 (10) for $v\lt \ (\geqslant ){v}_{\mathrm{tr}}$, and the transition velocity ${v}_{\mathrm{tr}}$ is chosen to ensure that ρ(v) integrates to the desired M_ej and E_k ($={M}_{\mathrm{ej}}{\beta }_{\mathrm{ej}}^{2}{c}^{2}/2$).

The decays of ⁵⁶Ni/Co and, if present, r-process nuclei supply the energy ultimately radiated by the SN. We assume that γ-rays from ⁵⁶Ni and ⁵⁶Co, which constitute most of the energy from that decay chain, are deposited in the zone that produced them with an efficiency f_dep,γ that depends on τ_γ , the ejecta's global optical depth to γ-rays. We adopt the functional form of Colgate et al. (1980) for f_dep,γ (τ_γ ) and calculate τ_γ using their suggested γ-ray opacity κ_γ = 1.0/35.5 cm² g⁻¹. The fast positrons from the β⁺-decay of ⁵⁶Co are assumed to thermalize locally and instantaneously.

For the M_rp and M₅₆ we consider, the energy from r-process decay is subdominant to that from ⁵⁶Ni on the timescales of interest (SBM19). Thus, while we include energy from r-process radioactivity in ${\dot{Q}}_{\mathrm{rad},\ i}$ (Equation (5)), we forgo complex treatments of the decay phase (e.g., Barnes et al. 2021) in favor of a power-law model (Metzger et al. 2010; Korobkin et al. 2012),

$\begin{eqnarray*}&&{\dot{\epsilon }}_{\mathrm{rp}}=3\times {10}^{10}\,{\left(\displaystyle \frac{t}{1\ \mathrm{day}}\right)}^{-1.3}\ \ \mathrm{erg}\ {{\rm{s}}}^{-1}\,{{\rm{g}}}^{-1}.\end{eqnarray*}$

We assume that 40% of this energy is in γ-rays, which thermalize in the same way as γ-rays from ⁵⁶Ni and ⁵⁶Co. We divide the remaining energy between β-particles (35%), which thermalize with perfect efficiency, and neutrinos, which do not thermalize at all. This simplified treatment is justified by the subdominance of ${\dot{\epsilon }}_{r{\rm{p}}}$, and by the high densities in the enriched regions compared to the densities expected in kilonovae (Bauswein et al. 2013; Hotokezaka et al. 2013; Kyutoku et al. 2015; Bovard et al. 2017; Radice et al. 2018), which supports efficient thermalization (Barnes et al. 2016).

We model only emission derived from radioactivity. In the case of a GRB-SN, the GRB afterglow could contribute to, or even dominate, optical and NIR emission at some epochs. However, as we will show, the timescales of interest for r-process detection are generally much longer than the timescales on which the afterglow fades away, and contamination is not a major concern.

Opacity in our model is wavelength independent but varies with temperature and composition. Ejecta free of both ⁵⁶Ni and r-process elements is assigned a baseline opacity κ_sn $=\,0.05$ cm² g⁻¹, consistent with Section 2.2. The effects on opacity of ⁵⁶Ni and its daughter products are accounted for with a simplified scheme, in which

$\begin{eqnarray*}{\kappa }_{56}(T)=\left\{\begin{array}{ll}{\kappa }_{\mathrm{56,0}} & \ \mathrm{for}\ T\lt {T}_{\kappa }\\ {\kappa }_{\mathrm{56,0}}{\left(\displaystyle \frac{T}{{T}_{\kappa }}\right)}^{4/3} & \ \mathrm{for}\ {T}_{\kappa }\leqslant T\leqslant 2.5\times {10}^{4}\ {\rm{K}}\\ 0.1\ {\mathrm{cm}}^{2}{{\rm{g}}}^{-1} & \ \mathrm{for}\ T\gt 2\times {10}^{4}\ {\rm{K}},\end{array}\right.\end{eqnarray*}$

where T_κ = 3500 K and the lower limit κ_56,0 = 0.01 cm² g⁻¹ reflects the dominance of electron-scattering opacity at low temperatures with a limited number of bound-bound transitions. This approximation is based on Planck mean opacities calculated for a mixture of ⁵⁶Ni, ⁵⁶Co, and ⁵⁶Fe (e.g., Kasen et al. 2013).

The total opacity in a zone is then given by

$\begin{eqnarray}&&{\kappa }_{i}={\kappa }_{\mathrm{sn}}(1-{X}_{r{\rm{p}},i}-{X}_{56})+{\kappa }_{r{\rm{p}}}{X}_{r{\rm{p}},i}+{\kappa }_{56}({T}_{i}){X}_{56},\end{eqnarray} \tag{ 7 }$

where κ_rp = 10 cm² g⁻¹ is the opacity of a pure r-process composition (Kasen et al. 2013; Tanaka & Hotokezaka 2013; Grossman et al. 2014), the r-process mass fraction X_rp,i is ζ within the enriched core and zero elsewhere, and the ⁵⁶Ni mass fraction X₅₆ equals M₅₆/M_ej in all zones. The zone's temperature T _i is a function of its internal energy density.

Equation (7) allows the determination of the photosphere and the demarcation of the optically thin region. As alluded to in Section 3.1, emission from optically thin zones is modeled as the linear combination of two distinct SEDs associated with r-process and r-process-free material (see Figure 1). The SEDs are empirically derived and independent of time and therefore elide the complex physics of nebular-phase spectral formation and evolution (e.g., Jerkstrand 2017). Nevertheless, r-process modeling (Hotokezaka et al. 2021) and SN observations (Gómez & López 2002; Tomita et al. 2006; Taubenberger et al. 2009) suggest that emission in the nebular phase may be fairly uniform in time and across different events, at least at the level of photometry. Furthermore, we find that this approach reproduces the photometry of SNe Ic with reasonable fidelity.

As mentioned in Section 2, we construct the r-process-free SED from the late-time B- through K-band photometry of SN 2007gr (Hunter et al. 2009; Bianco et al. 2014), accessed via the Open Supernova Catalog ⁵ (OSC; Guillochon et al. 2017). We consider data from t ≈ 120 days after B-band maximum and perform a spline fit to convert photometry-derived monochromatic luminosities to a continuous SED, ${{ \mathcal F }}_{\nu }^{07\mathrm{gr}}$, as shown in Figure 5. To improve the agreement between our model and SN Ic/Ic-BL observations, we assume that 30% of the energy in this SED falls blueward of U or redward of K.

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The SED associated with optically thin r-process compositions is highly uncertain. Spitzer observations (Villar et al. 2018; Kasliwal et al. 2022) of kn170817 (the only definitive r-process transient detected to date) at 43 days post-explosion yielded one mid-infrared (MIR) photometric point and one upper limit. As Kasliwal et al. (2022) pointed out, if the kilonova's spectrum is a blackbody, these measurements constrain its temperature to ≲440 K.

On the other hand, nebular-phase spectra are not expected to conform to a Planck function. Indeed, numerical modeling by Hotokezaka et al. (2021) of the emission from a nebula with properties similar to a collapsar disk wind and composed purely of neodymium (a high-opacity element synthesized by the r-process) predicted a spectrum with far more complexity than a blackbody. While their results showed MIR emission consistent with the observations of Kasliwal et al. (2022), they also found significant flux at lower wavelengths. However, while nebular-phase spectra are unlikely to resemble a Plank function, we found that a blackbody at T_rp = 2500 K nonetheless reproduces the photometric colors predicted by the optical and NIR regions of the spectrum of Hotokezaka et al. (2021). (The MIR component accounts for ∼ 60% of the total energy but is too red to impact the photometry; it can be incorporated into the model SED simply by scaling the Plank function.) Figure 5 shows both the observations and the numerical model, as well as blackbody spectra for select temperatures, which we present for comparison. In the bottom panel, we show broadband magnitudes calculated for the Hotokezaka et al. (2021) spectrum and for our blackbody approximation to it, which agree well despite our simplifications.

The calculation of Hotokezaka et al. (2021) relies on a simplified model of pure r-process ejecta; neither the assumed composition nor the heating rate (due exclusively to r-process decay) maps directly onto the collapsar context. However, the argument for a bimodal spectrum is supported by simple arguments about r-process elements' atomic structures, which may be robust against increasing model complexity due to the distribution in energy space of permitted and dipole-forbidden atomic transitions (Hotokezaka et al. 2021). In the absence of additional data, we therefore approximate the r-process-associated SED as a blackbody at 2500 K, which we scale to account for the out-of-band emission. Though some of our zones are r-process-free, none are purely r-process; at a minimum, each zone contains a mixture of r-process elements and ⁵⁶Ni. The appropriate way to model nebular emission from zones with complex compositions is an additional uncertainty. Nebular spectra are dominated by the species that cool most efficiently, which may be distinct from the most abundant elements.

Here, we move away from the simpler approach of Section 2.1 and allocate the luminosity of an optically thin zone according to the fraction of the total optical depth a given component provides across that zone. For r-process material, that fraction is

$\begin{eqnarray*}&&{f}_{r{\rm{p}},i}=\frac{{\kappa }_{r{\rm{p}}}{X}_{r{\rm{p}},i}}{{\kappa }_{i}},\end{eqnarray*}$

with κ_i defined by Equation (7). Thus, the luminosity L_neb,i from a zone outside the photosphere is converted to an SED following

$\begin{eqnarray*}&&{L}_{\mathrm{neb},i,\nu }={L}_{\mathrm{neb},i,\nu }^{r{\rm{p}}}+{L}_{\mathrm{neb},i,\nu }^{\mathrm{sn}},\end{eqnarray*}$

where

$\begin{eqnarray*}\begin{array}{rcl}{L}_{\mathrm{neb},i,\nu }^{r{\rm{p}}} & = & 0.41\times {f}_{r{\rm{p}},i}\,\frac{\pi {L}_{\mathrm{neb},i}}{{\sigma }_{\mathrm{SB}}{T}_{r{\rm{p}}}^{4}}\,{B}_{\nu }({T}_{r{\rm{p}}}),\\ {T}_{r{\rm{p}}} & = & 2500\ {\rm{K}},\end{array}\end{eqnarray*}$

and

$\begin{eqnarray*}&&{L}_{\mathrm{neb},i,\nu }^{\mathrm{sn}}=0.7\times (1-{f}_{r{\rm{p}},i})\,{L}_{\mathrm{neb},i}\,{{ \mathcal F }}_{\nu }^{07\mathrm{gr}}.\end{eqnarray*}$

We validated our semianalytic model against a handful of SNe Ic/Ic-BL with late-time multiband photometry. In Figure 6, we show our predictions alongside observations for one ordinary and one broad-lined SN Ic. The model parameters we used, as well as inferred ejecta properties reported in the literature, are recorded in Table 2. Despite its simplifications, our model reproduces the basic features of the SNe, suggesting that we are accounting for the most important physical processes driving the light curve's evolution.

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Table 2. Model Parameters Adopted in Figure 6

SN	Mej/M⊙	Ek/1051 erg	M56/M⊙
2007gr a	2.0 (2.0–3.5)	3.2 (1.0–4.0)	0.08 (${0.076}_{-0.02}^{+0.02}$)
2002apb	3.2 (2.5–5.0)	4.4 (4.0–10.0)	0.11 (0.07)

Notes.

^a Inferred ejecta properties for SN 2007gr (in parentheses) from Hunter et al. (2009). ^b Inferred ejecta properties for SN 2002ap (in parentheses) from Deng et al. (2003).

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Around maximum light, our model is most accurate in ultraviolet and optical bands. Though a lack of data makes it harder to gauge model performance in the NIR, the data that are available suggest that we may overpredict J, H, and K magnitudes in the light curve's early phases. However, we focus on optical−NIR color difference as an r-process signature (e.g., Section 2.1, Section 4). Thus, slightly overestimating the NIR emission of r-process-free SNe near peak will merely make our calculation more conservative. Enriched models that are redder (in R − J, R − H, and/or R − K) than our unenriched models would be much redder than true SNe, meaning that the color differences we predict are likely to be underestimates, and the actual r-process signal may be stronger than we forecast.

We also found that our model is slightly less successful at reproducing the photometry of SNe Ic-BL. This is perhaps not surprising considering that the extreme kinetic energies (∼ 10⁵² erg) of SNe Ic-BL, as well as the GRBs sometimes observed in conjunction with them, point to a nonstandard (e.g., "engine-driven") explosion mechanism that may induce ejecta asymmetries or unusual density profiles (e.g., Maeda et al. 2003; Tanaka et al. 2008; Barnes et al. 2018). They are also the SNe most closely linked, theoretically, to the collapsar explosion model (e.g., Woosley & Bloom 2006), and therefore the most likely to produce emission-altering r-process elements. Having validated our approach, we next use it to model the emission from enriched and unenriched SNe across a broad parameter space.

We construct multiband light curves extending to t = 200 days after explosion for r-process-enriched and unenriched SNe with a range of explosion and r-process-enrichment properties. The parameter values used in the model suite are presented in Table 3. We require that radioactive material not dominate the total ejecta mass (i.e., we enforce M₅₆ ≤ 0.5M_ej and M_rp ≤ 0.5(M_ej − M₅₆)), and we do not consider models with kinetic energies E_k > 5 × 10⁵² erg. Beyond these constraints, all parameter combinations are explored, even those that produce SNe with kinetic energies somewhat lower than would be expected for a collapsar origin. The r-process-free models reproduce the range of luminosities and timescales of observed SNe Ic and SNe Ic-BL (Drout et al. 2011; Perley et al. 2020), which validates both our modeling framework and the range of parameter values we adopt.

We do not consider here the case of complete mixing (ψ_mix = 1). For fully mixed models, the high opacity of the r-process elements would affect the SN emission from the explosion onward, resulting in a transient very distinct from ordinary SNe at all phases of its evolution. The question of detecting fully mixed rCCSNe (or of recognizing one should it turn up in a blind search) is therefore better deferred to a separate work (though it has been at least partially addressed by Siegel et al. 2021, who discuss electromagnetic counterparts from very massive, uniformly mixed, r-process-enhanced "superkilonovae"). We focus instead on rCCSNe with at least a small outer shell of r-process-free material. This conceals the effects of the r-process initially, allowing such rCCSNe to masquerade (if only for a limited time) as unenriched SNe.

Enriching SN ejecta with r-process elements extends the photospheric phase and alters the SED from the optically thin layers. Each of these effects shifts the emitted spectrum from the optical toward the NIR; however, the strength of this shift and the timescale at which it occurs depend on the mass of r-process material (M_rp) and how extensively it is mixed in the ejecta (ψ_mix).

The response of bolometric and broadband light curves to ψ_mix is illustrated in Figure 7. When the r-process is concentrated in the ejecta's center, its influence is minimal, since only a negligible fraction of the radiation originates in the enriched layers. At higher ψ_mix, the effects are more visible. The extended high-opacity core limits diffusion from the interior, producing lower L_bol and L_ph near peak. After the outer layers reach transparency, the opaque core slows—or even reverses—the recession of the photosphere, sustaining a higher L_ph at the expense of ${L}_{\mathrm{neb}}^{\mathrm{sn}}$. As the enriched layers (slowly) become transparent, their nebular emission begins to contribute to ${L}_{\mathrm{neb}}^{r{\rm{p}}}$, and for high enough ψ_mix or late enough epochs, ${L}_{\mathrm{neb}}^{r{\rm{p}}}$ can overpower ${L}_{\mathrm{neb}}^{\mathrm{sn}}$.

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The long-lived photosphere and the r-process nebular component each provide a luminous source of low-temperature emission that impacts the evolution of the SED and, therefore, the broadband light curves. The net effect is a transfer of energy from optical to NIR bands, as can be seen in the bottom panels of Figure 7. The greater the mixing, the more dramatic the redistribution.

As seen in Figure 7, mixing also affects light-curve shapes, though not always in a straightforward way. In most cases, the opacity of the core is high enough that emission from the r-process-rich and r-process-free layers effectively becomes decoupled, rising to distinct peaks on distinct timescales (e.g., Section 2.2). Unless ψ_mix is very high, diffusion from the core is suppressed to the degree that the peak of L_ph is driven mainly by the r-process-free ejecta. Increasing ψ_mix reduces the mass and increases the average velocity of this ejecta component, producing a narrower peak in L_ph and sharper light curves in optical bands, despite the increasing spatial extent of the high-opacity region.

To better understand how r-process-enriched SNe may be distinguished from their r-process-free counterparts, we expand the parameter space of Figure 7, enriching a single explosion model (M_ej, β_ej, M₅₆) = (4.0 M_⊙, 0.04, 0.25 M_⊙) with a range of M_rp at various ψ_mix. We calculate the broadband evolution for each combination (M_rp, ψ_mix) and compare the colors to those of an r-process-free SN with the same M_ej, β_ej, and M₅₆.

Because the effects of enrichment are seen primarily in the NIR, we use R − X color as a proxy for the r-process signal strength, with X ∈ {J, H, K}. As in Section 2.1, we focus on color difference: we determine the earliest time, t_Δ, at which the colors of each rCCSN model differ from those of the unenriched model by at least one magnitude. Since this divergence occurs at a range of R − X, the focus on color difference rather than absolute color is useful for making comparisons across a diverse set of models. (However, we provide more concrete predictions of color itself in Sections 4.1.1 and 4.2.2.) To demonstrate the importance of the NIR bands, we perform the same calculation for the optical color V − R and find no models for which Δ(V − R) is ever greater than 1 mag.

The top three panels of Figure 8 plot, as contours, t_Δ for R − J, R − H, and R − K. Models with high ψ_mix and high M_rp diverge from the r-process-free model earlier than models with less extreme enrichment parameters. This is consistent with earlier analytic arguments (e.g., Figure 3). The divergence also occurs sooner for R − K than for R − J or R − H. (Crucially, however, any choice of R − X is a more reliable r-process indicator than optical colors, as seen in the bottom panel, which shows the maximum difference in V − R for each model.)

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Figure 8 also hints at when signs of more modest enrichment may appear; for ψ_mix ≤ 0.3, the color difference for all R − X is <1 mag until ∼2–3 months after explosion. In fact, for many (M_rp, ψ_mix), Δ(R − X) does not exceed our threshold within the time frame of the simulation, or exceeds it for only one of the three colors considered.

Still, Figure 8 reinforces the potential of pre-nebular-phase observations for r-process detection. The dashed black curves in the top three panels show which rCCSNe, once their ejecta were fully transparent, would have Δ(R − X) = 1 mag. (Here, in contrast to Figure 2, we have determined the nebular SED following the prescription of Section 3.2.) The swaths of parameter space that lie between the light-gray regions and the dashed lines contain models whose enrichment is more visible late in the photospheric phase, when at least the enriched core remains opaque, than during the nebular phase, after the emission has achieved its asymptotic colors.

4.1.1. Supernova Case Studies

We will argue later that, for questions of r-process detectability, classifying models based on M_ej, β_ej, and M₅₆ is of limited utility. Regardless, before proceeding to a more observationally motivated schema, we present detailed color evolution predictions for a handful of models based on analyses of stripped-envelope SN demographics by Barbarino et al. (2021) and Taddia et al. (2019). (We note that many groups (Drout et al. 2011; Prentice et al. 2019; Perley et al. 2020) have contributed to efforts to uncover the distributions of SN Ic/Ic-BL properties, and that the characteristics of an "average" SN in a given category remain uncertain.)

In addition to cases representing typical SNe Ic/Ic-BL, we consider models based on individual SNe Ic-BL with inferred ejecta masses much higher and much lower than average, in order to explore how the signal may vary within the SN Ic-BL population. The explosion properties of our four models, along with the event or analysis on which each is based, can be found in Table 4. We enrich each of these models with 0.03 M_⊙ of r-process material, spread out to varying mixing coordinates ψ_mix.

Table 4. Explosion Properties of the Models of Figure 9

Type	Mej (M⊙)	βej	M56	Reference
Typical Ic-BL	3.97	0.044	0.33	T2019 a (average values)
High-mass Ic-BL	10.45	0.029	0.85	T2019 (PTF10ysd)
Low-mass Ic-BL	1.51	0.050	0.21	T2019 (PTF10tqv)
Typical Ic	3.97	0.020	0.21	B2021 b (average values)

Notes.

^a T2019: Taddia et al. (2019). ^b B2021: Barbarino et al. (2021).

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Figure 9 shows the R − X color evolution for each set of explosion parameters as a function of ψ_mix. We display for comparison the colors of an SN with the same explosion properties but no r-process enrichment. To better summarize the data, and to situate the r-process-induced changes in color more directly in the context of observational SN properties, we also plot in the bottom row of Figure 9 the maximum color for each model and the time at which that maximum occurs, formulated as a multiple of the R-band rise time, t_R,pk. We adopt this normalization because a given SN's time to peak defines the timescale on which it evolves; expressing times in terms of t_R,pk makes it easier to compare SNe with a range of characteristic durations.

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Regardless of ψ_mix, the colors of the rCCSN models track those of their unenriched counterparts at early times, becoming noticeably redder only later on (around t ≳ 25 days for this set of transients). The extent of the reddening depends on ψ_mix and on M_ej, β_ej, and M₅₆; nevertheless, certain general trends are apparent.

While the effects of the r-process are present for all colors considered, they increase in prominence with the wavelengths of the NIR band. This theme is also apparent in the bottom panels of Figure 9. For all colors, models with at least moderate ψ_mix occupy a distinct region of parameter space compared to poorly mixed or unenriched models. The separation is strongest for R − K.

More than color, the signal strength depends on the explosion parameters, particularly velocity. R-process-induced changes in color are far more noticeable for the SN Ic-BL models with typical and low total ejecta masses (which have β_ej = 0.044 and 0.050, respectively) than for the typical SN Ic model or the SN Ic-BL with a higher ejecta mass (which have β_ej < 0.03). This can be seen in both the full color evolution and the position of each model in parameter space of the bottom panels.

However, consistent with Figure 8, the color change is most sensitive to the mixing coordinate ψ_mix. For nearly all models with ψ_mix ≥ 0.1, r-process enrichment produces color extrema redder than the asymptotic colors of the corresponding unenriched SN. For some models, this is a global maximum occurring at a delay relative to the unenriched SN. This is the case for ψ_mix ≳ 0.2–0.3 for the typical and low-mass Ic-BL models and for ψ_mix ≳ 0.4 for the high-mass Ic-BL and the typical Ic model. (Even in the case of strong mixing, though, the difference between the enriched and unenriched SN colors is much smaller for the latter two cases than for the former two.)

At lower levels of mixing, the colors of the rCCSNe track those of their unenriched SN counterparts for a longer period of time and rise only to a local maximum after diverging. In these cases, as the bottom panels indicate, the maximum values of R − X would not be sufficient to distinguish enriched from unenriched SNe; longer-term precision photometry would be required.

Nevertheless, it is reassuring that only the slowest and/or most poorly mixed models considered here fail entirely to form a distinct R − X peak prior to the nebular phase. For typical SN Ic-BL explosion parameters, rCCSNe with ψ_mix ≥ 0.1 exhibit either a global maximum R − K color > 0 or a secondary maximum < 0 at 50 days ≲ t ≲ 100 days. In contrast, without enrichment we predict a peak value of R − K = 0.2 at t = 40 days. This again points to the value of pre-nebular-phase observations for evaluating collapsars as sites of r-production, particularly for ψ_mix ≥ 0.2.

While the discussion of Sections 4.1 and 4.1.1 is useful for illustrating trends, it may overstate the differences between r-process-rich and r-process-free SNe. Emission from models with large M_rp and ψ_mix—i.e., the models Figures 8 and 9 suggest should be easiest to identify—is likely to be so impacted by enrichment that it bears little resemblance to the emission from an r-process-free counterpart of the same M_ej, β_ej, and M₅₆. Since observers have no way of knowing a priori the physical properties of an SN explosion, a more appropriate reference case for an rCCSN—particularly if it is highly enriched or very well mixed—is an r-process-free SN with similar observed properties.

For the following analysis, we therefore categorize our models in terms of observable, rather than physical, parameters. Specifically, we classify them according to their R-band rise time, t_R,pk; peak R-band magnitude, M _R ; and velocity, β_ej. (While β_ej is not an observed property in the same sense as t_R,pk and M _R , measurements of absorption features in SN spectra can provide estimates of average ejecta velocities.) SNe (whether r-process-enriched or not) with comparable t_R,pk, M _R , and β_ej will not evolve in perfect synchronicity; still, this procedure allows us to at least compare models with similar behavior near peak light, when most observations are obtained.

Figure 10 demonstrates the advantage of this approach. Its histogram shows the diversity of M _R and t_R,pk that characterize models with fixed explosion parameters—in this case, (M_ej, β_ej, M₅₆) = (4.0 M_⊙, 0.04, 0.25 M_⊙)—but variable M_rp and ψ_mix. If r-process material is present, the enrichment parameters (M_rp and ψ_mix) influence the properties of the light curves. More than 30% of the models in Figure 10 have a t_R,pk (M _R ) that differs from the r-process-free case by more than 1 day (0.5 mag). This highlights the risk of assuming that M_ej, β_ej, and M₅₆ can be extracted from light-curve data independent of the amount (or existence) of r-process enrichment.

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However, a challenge of this framework is that there is no way to identify a single r-process-free model to which an rCCSN model should be compared. (Unlike M_ej, β_ej, and M₅₆, we cannot force ordinary SNe and rCCSNe to have the same t_R,pk and M _R .) As in earlier sections, we focus on color as a diagnostic and continue to use Δ(R − X) ≥ 1 mag as the criterion for detectability, with X ∈ {J, H, K}. Now, however, instead of making one-to-one comparisons, we sort our models into bins of size Δt_R,pk = 2 days, ΔM _R = 0.25 mag, and Δβ_ej/β_ej = 0.37 and then contrast the colors of individual rCCSNe with an average color evolution constructed from the r-process-free SNe in the same bin.

This introduces some uncertainty into the comparison, as unenriched models in a given bin do not produce identical SNe. Indeed, the time-dependent standard deviation of the R − X colors for r-process-free SNe, σ_R−X (t), can reach ∼0.7 mag at certain times for certain bins. However, σ_R−X depends on velocity. This can be seen in Figure 11, which presents the cumulative distribution function of the maximum σ_R−X for bins with a particular β_ej. For bins corresponding to typical SN Ic/Ic-BL velocities (0.03 ≲ β_ej ≲ 0.1), σ_R−X < 0.5 mag at all times for all X considered. Moreover, in this same velocity range, the average bin has at all times σ_R−X ≲ 0.25 mag, much less than our detection threshold Δ(R − X) ≥ 1. This increases our confidence that models flagged as detectable truly do differ in significant ways from the unenriched reference cases.

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An additional complication is the fact that enriched and ordinary SNe do not populate the exact same regions in the t_R,pk−M _R −β_ej parameter space. In the few instances where rCCSNe occupy a bin containing no unenriched SNe, we prioritize comparing events of similar t_R,pk and M _R , since these are directly measured, while β_ej must be inferred from observations. In these cases, we define the reference color evolution for the calculation of Δ(R − X) by selecting, from all the r-process-free SNe within the empty bin's range of t_R,pk and M _R , those with velocities closest to the empty bin's central velocity and averaging over that subset. If there are no r-process-free SNe (of any velocity) in the desired bin in (t_R,pk, M _R ), we do not calculate Δ(R − X).

4.2.1. Minimum Observable Mixing Coordinate

Having established a new method for comparing enriched and unenriched SN models, we return to the rCCSNe and determine, as a function of t_R,pk, M _R , and β_ej, the minimum mixing coordinate that produces a detectable signal. We define this minimum, ${\psi }_{\min }$, as the value of ψ_mix for which ≥50% of models in a given bin satisfy Δ(R − X) ≥ 1 mag before some threshold time, with the threshold time a parameter of the calculation.

We find that ${\psi }_{\min }$ depends primarily on the ejecta velocity β_ej of the rCCSNe, the time frame over which observations are carried out, and the color considered (i.e., the choice of X). The effect of each of these is illustrated in Figure 12, which shows ${\psi }_{\min }$ for each bin, or, for bins in which no ψ_mix qualified as detectable, the maximum ψ_mix represented in the bin. For simplicity, we present in Figure 12 slices from the parameter space; each panel corresponds to one observation time frame, one choice of X, and one (bin-centered) β_ej. We have also restricted the models of Figure 12 to those with M_rp = 0.03 M_⊙. However, the same trends apply to other choices of M_rp.

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As expected from the discussion of Section 2.2 (e.g.,Figure 4), the top row of Figure 12 shows that r-process signatures are more easily detected for rCCSNe with higher β_ej. Higher β_ej values characterize SNe associated with the GRBs that serve as indirect evidence of the accretion disks predicted to support r-process production. This means that the most straightforward version of the r-process collapsar hypothesis, in which an accretion disk wind injects r-process elements into a high-velocity SN, should be the easiest to test. On the other hand, alternate modes of r-production in SNe, such as lower-energy "failed-jet" SNe (Grichener & Soker 2019) or neutrino-driven winds from newly born magnetars (Vlasov et al. 2017; Thompson & ud-Doula 2018), will be more difficult to evaluate.

While incomplete mixing and slower expansion velocities hinder r-process detection, observing strategy can at least partially compensate. The middle row of Figure 12 demonstrates that extending observations to later times increases the likelihood of a detection, even for models with lower ψ_mix. For rCCSNe with β_ej ≈ 0.06, we find that r-process signals are not reliably detected in R − H at times ≤ 2t_R,pk, even if mixing is extensive. (Nearly all bins in the middle row contain models with ψ_mix = 0.9.) In contrast, for t ≤ 4t_R,pk (6t_R,pk), ${\psi }_{\min }\leqslant 0.3$ for 31% (46%) of the bins. That said, according to our definition, a successful detection implies only that a color difference Δ(R − X) ≥ 1 mag is obtained at some point during the observation; it reveals nothing about the duration of that signal. Thus, cadence is also important.

Finally, as can be seen in the bottom row of Figure 12, the passbands chosen for the color comparison also matter. We find that r-process detection for minimally mixed ejecta is easier for R − K than R − H and R − J. However, particularly if observations are limited to photometry, multiple NIR bands may be required to rule out spurious emission features unrelated to r-process enrichment (e.g., overtones of carbon monoxide; Gerardy et al. 2002). In that sense, detection prospects may be limited by the performance of the weakest color considered.

Not included in Figure 12 is the effect of r-process mass, which we find has a comparatively minor impact on the minimum detectable mixing coordinate. In Figure 13, we show ${\psi }_{\min }$ for models with bin center β_ej = 0.06 (the fiducial velocity of Figure 12), t_obs ≤ 4t_R,pk, and two r-process masses, M_rp = 0.03 and 0.15 M_⊙. We calculate ${\psi }_{\min }$ based on the R − K color, for which the effects of r-process enrichment are strongest.

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Despite the fivefold increase in M_rp, values of ${\psi }_{\min }$ are fairly stable. In part because of the velocity dependence described above, the greatest gains in detectability are for fast-evolving transients (t_R,pk ≈ 5–10 days), for which ${\psi }_{\min }$ falls from ≃0.2 to ≃0.1. For events with longer rise times, even a large M_rp is invisible except in cases of strong mixing.

The insensitivity of ${\psi }_{\min }$ to M_rp is due to the high opacity of r-process material relative to the unenriched ejecta and to the dominance of ⁵⁶Ni as an energy source in our model. Because ${{\rm{\kappa }}}_{r{\rm{p}}}\gg {{\rm{\kappa }}}_{\mathrm{sn}}$, core ejecta polluted with even trace amounts of r-process elements acquires an opacity much higher than that of the surrounding envelope. Although increasing M_rp in the enriched core further enhances the opacity, the opacity difference between enriched and unenriched ejecta is larger than that between minimally and highly enriched ejecta. In this way, the effects of κ_rp diminish with increasing M_rp. Moreover, even for high M_rp, ⁵⁶Ni supplies most of the radioactive energy to the core, and raising M_rp does not meaningfully alter the energy budget of the enriched layers. Were the reverse true, increasing M_rp would cause the r-process-rich layers to shine more brightly (at NIR wavelengths) and, presumably, would produce a tighter correlation between M_rp and ${\psi }_{\min }$.

4.2.2. Colors and Timescales

In order to make our analysis more concrete, we next explore the color evolution of a subset of our model suite. As in Section 4.2.1, we categorize the models according to their R-band rise times and maxima, t_R,pk and M _R . We focus on rCCSNe of velocity β_ej = 0.06 (typical for the SNe Ic-BL most likely to produce r-process elements in disk outflows) and a moderate level of mixing, ψ_mix = 0.3. We use the same binning procedure as in Section 4.2.1 .

For each model, we find the earliest time, t_RX2, at whichΔ(R − X) ≥ 1 mag for at least two X ∈ {J, H, K}. (As before, the color difference is calculated with respect to an average of unenriched SNe in the same bin.) We adopt the more stringent, two-color standard to protect against false positives due to, e.g., emission features that affect the flux in only one band. However, we acknowledge that this stricter criterion will in general be met at later times, when finite (and band-dependent) detector sensitivities pose more of a challenge for observations. In addition to t_RX2, we also record R − K at t = t_RX2, which we denote (R − K)_RX2. This establishes for every model a characteristic timescale and R − K color associated with r-process enrichment signals.

In Figure 14, we present for different M_rp the latest t_RX2 and the lowest value of (R − K)_RX2 in each bin. (We note that the model associated with the latest time is not necessarily also associated with the lowest color.) In other words, the grids of Figure 14 can be read as saying that an rCCSN with the specified parameters should produce a signal with R − K at least as red as the indicated color by no later than the indicated time.

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The r-process signal appears first for rCCSNe that are fast evolving and/or bright. In extreme cases (t_R,pk ≲ 10 days), the signal appears within ∼20 days, or ∼2t_R,pk, and is associated with a moderate R − K ≈ 0.5 mag. For low M_rp, these fast/bright rCCSNe are the only ones that meet our two-band criterion. For higher M_rp, detections happen over a broader swath of the parameter space. Detections in events with higher t_R,pk and lower M _R occur much later (75–100 days after explosion) but produce stronger R − K colors (≈ 1mag).

While the details of Figure 14 depend on the choice of ψ_mix and β_ej, the results nonetheless suggest that, absent a high degree of mixing, minimal r-process enrichment (M_rp ≲ 0.03 M_⊙) will be difficult to detect except in the fastest-evolving transients. In contrast, if r-production occurs at a higher level (as SBM19 argued based on the presumed more massive accretion disks formed in collapsars vis-à-vis NSMs), a signal should be visible for a much wider range of observational parameters.

Though the quantity of r-process material determines the breadth of the parameter space over which a detection is feasible, Figure 14 suggests that it may be difficult, once a detection is made, to precisely constrain the r-process mass M_rp. More specifically, it appears that a detection can more easily be used to derive a lower limit for M_rp than an upper limit.

However, longer-term monitoring can provide additional information, allowing a better estimate of the r-process mass produced in a given rCCSN. In Figure 15, we show, for the same models represented in Figure 14 (β_ej = 0.06, ψ_mix = 0.3), the time at which the rCCSN reaches a maximum in R − K and the value of that maximum. While we do not apply any criterion for Δ(R − X) in calculating these quantities, when determining ${(R-X)}_{\max }$, we consider only times after the photosphere has reached the r-process layers, ensuring that our color maxima are associated with the enriched ejecta. As a result, in some cases the maxima identified in Figure 15 are local rather than global extrema.

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For bins in which detections are possible for multiple values of M_rp (in each panel, light-colored hatching indicates the regions where r-process signatures are not detectable according to the two-color standard established for Figure 14), increasing M_rp results in a higher maximum R − K, occurring at a later time. Thus, while the r-process mass, at least in this mixing regime, has a small effect on the signal in its earliest stages, diligent photometric follow-up may still successfully constrain M_rp. In fact, the relative insensitivity of the early characteristics of the signal to M_rp may be an advantage, as it decouples the determination of ideal observing strategies from assumptions about the level of r-production in collapsars.

As we have seen, even for SNe with optimal explosion properties, the odds of confidently detecting r-process signatures depend on the observing strategy employed. The observing window and the bands selected are particularly important. If we require for detection a color difference of at least 1 mag for at least two colors R − X, we find that 48% of our rCCSN models with SN Ic-BL-like velocities (0.05 ≤β_ej ≤ 0.1) are detectable before t = 50 days if the chosen colors are R − J and R − K. This percentage encompasses all mixing coordinates and smooths over considerable variation with ψ_mix. If we consider only rCCSNe with ψ_mix ≥ 0.6, 74% are detectable, while for ψ_mix < 0.3 that value drops to 0.5%.

Observing in redder bands improves the situation, as does extending the observing period. If R − H and R − K are instead used for the color comparison, 58% of the high-velocity models are detectable by t = 50 days, and 65% by t = 80 days. If we again differentiate by ψ_mix, we find that 14% of the poorly mixed models and 79% of the well-mixed models are detectable by t = 50, and 17% and 85%, respectively, by t = 80.

Lower-velocity rCCSNe pose a greater challenge. Of models with β_ej < 0.05, r-process signatures are visible for only 10% before day 50, even when Δ(R − X) is calculated with respect to R − H and R − K. The percentage is a dismal 1% for models with ψ_mix < 0.3 and rises to only 15% for ψ_mix ≥ 0.6.

To clarify these trends, we show in Figure 16 the fraction of rCCSN models of different ψ_mix that satisfy a two-color detection criterion before a given time. Consistent with the discussion above, we divide our models into low-velocity (0.01 ≤ β_ej < 0.05) and high-velocity (0.05 ≤ β_ej ≤ 0.1) subsets and consider the effect of the colors used to calculate Δ(R − X).

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While Figure 16 shows that only a limited fraction of enriched models are detectable under this framework, it also identifies the period most likely to yield a successful detection, if one is to be forthcoming. For example, considering rCCSNe with higher β_ej (i.e., the top two panels), nearly all models that become detectable within our simulation time frame are detectable by t = 70 days post-explosion. In contrast, the detectable fraction of lower-velocity models continues to rise steadily well past t = 100 days. (We caution that while these quoted percentages reveal important relationships between observing strategies and the ability to discern r-process-enrichment signatures, their exact values depend on the distribution of explosion and enrichment parameters in our model suite, which does not necessarily reflect the distributions within the cosmological population of SNe Ic/Ic-BL. Rather than the percentage of rCCSNe that can be identified by a certain observation, they more accurately measure the fraction of the parameter space an observation can probe.)

Observing r-process signatures in poorly mixed explosions will be challenging regardless of other factors; still, Figure 16 suggests that the odds of success will be maximized if high-velocity targets are followed up in multiple bands—ideally including multiple NIR bands covering the reddest wavelengths possible—for at least 2 months after explosion. By t ≈ 2.5 months, the odds that a signal will appear for the first time decline steeply. If no signal has been detected by this point, observing resources would be better spent on new targets.

What Figure 16 does not elucidate is the cadence of observations needed to catch r-process signals, which are often transient. To provide some sense of the signals' longevity, we calculate the fraction of models that are instantaneously detectable as a function of time, as well as the distribution of signal lifetimes, Δt_sig. Here, as elsewhere, "detectability" refers only to the color difference relative to an r-process-free SN baseline; we do not consider telescope sensitivity, or other technicalities that would in practice constrain the acquisition of data. We show the results in Figure 17 for rCCSNe with 0.05 ≤ β_ej ≤ 0.1, for a signal constituted by Δ(R − X) ≥ 1 mag with X = H and K. (In other words, we selected the SNe and the R − X colors that favor detection.) The top panel shows the fraction of models, as a function of ψ_mix, detectable at a given time, while the bottom panel shows the fraction of models, f_>, with a signal lifetime exceeding a given duration Δt_sig.

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A large fraction of well-mixed models are detectable starting at t ∼ 30 days and remain detectable thereafter; of models with ψ_mix ≳ 0.5, ≲60% are detectable over a period of ≥100 days. For lower mixing coordinates, the epoch at which the largest fraction of models is detectable is similar (∼50 days after explosion), but that fraction is lower, and Δt_sig decreases in concert with ψ_mix. For example, while 47% of models with ψ_mix = 0.3 have signals lasting Δt_sig ≥ 30 days, only 13% of those with ψ_mix = 0.2 do. Figure 17 suggests that long-lived signals enduring out into the nebular phase can be expected in a majority of cases only if ψ_mix ≳ 0.5.

Taken together, Figures 16 and 17 indicate that observations targeting high-velocity SNe in the first few months after explosion and at reasonably high cadence will maximize the chances of identifying rCCSNe. The most poorly mixed rCCSNe that offer any hope of detection (those with ψ_mix = 0.1) have signal durations sharply concentrated at Δt_sig ≲ 25 days. Ideally, observing campaigns would return to such a target multiple times during this window to maximize confidence in a detection, suggesting a delay of no greater than ∼1 week (and optimally ∼3–4 days) between consecutive visits.