Inclination Dependence of Kilonova Light Curves from Globally Aspherical Geometries

We study the emission from a homologously expanding ejecta using the time-dependent Monte Carlo radiative transfer code sedona (Kasen et al. 2006). We study an ejecta with mass M and kinetic energy E_k = Mv_ch²/2, where v_ch = β_ch c is the characteristic velocity and c is the speed of light. We use a one-component, constant gray opacity κ to parameterize the strength of the interaction between the thermal photons and matter.

The light-curve evolution is determined by two timescales. The first is the effective diffusion timescale t_d = (Mκ/v_ch c)^1/2, which characterizes the time at which photons escape the ejecta faster than the ejecta can expand. We take this as our definition of t_d but note that other definitions exist that contain additional numerical factors (Arnett 1982). We use this to define the dimensionless time τ = t/t_d, where t is the physical time. The second is the thermalization timescale t_e, which characterizes the time at which the absorption of radioactive decay products begins to become inefficient. This can be written in terms of the dimensionless parameter τ_e = t_e/t_d. For neutron-rich ejecta, the radioactive power from the r-process alone can be approximated by a power law (Metzger et al. 2010; Lippuner & Roberts 2015; Hotokezaka et al. 2017) and thus does not itself have an intrinsic timescale. We note that ejecta with Y_e ≳ 0.35 can exhibit a different time dependence (Wanajo et al. 2014).

The light curves are powered by the radioactive decay of neutron-rich elements synthesized by the r-process (Metzger et al. 2010). The fraction of the radioactive power that will thermalize and contribute to the electromagnetic emission is determined by its distribution in the different decay channels (beta, alpha, and fission) and their thermalization efficiencies (Metzger et al. 2010; Barnes et al. 2016), though the total contribution can be reduced to a simple approximate prescription (Kasen & Barnes 2019). We adopt the latter, yielding the specific heating rate

$$\begin{eqnarray}&&q(\tau ;{\tau }_{{\rm{e}}})={q}_{0}{t}_{{\rm{d}},\mathrm{day}}^{{\sigma }_{1}}{\tau }^{{\sigma }_{1}}{\left(1+\displaystyle \frac{\tau }{{\tau }_{{\rm{e}}}}\right)}^{{\sigma }_{2}},\end{eqnarray} \tag{ 1 }$$

where τ_e is the thermalization timescale, t_d,day is the diffusion time in days, and we take q₀ = 10¹⁰ erg s⁻¹ g⁻¹, σ₁ =−1.3, and σ₂ =−1.2. The term τ^σ₁ gives the time dependence of the radioactive power from an ensemble of r-process nuclei (Metzger et al. 2010; Lippuner & Roberts 2015; Hotokezaka et al. 2017), and the term in parentheses is the thermalization efficiency (Kasen & Barnes 2019). We take τ_e = 10 for our canonical example but also examine the effects of different values. The total heating rate is then Q = Mq.

Given the above specific heating rate, we define the dimensionless luminosity Λ = L/L_n, where L is the physical luminosity and ${L}_{{\rm{n}}}={{Mq}}_{0}{t}_{{\rm{d}},\mathrm{day}}^{{\sigma }_{1}}$ is the scale factor. If all parts of the outflow are in the diffusive regime, then ejecta profiles with the same τ_e would be degenerate and exhibit the same dimensionless light curves Λ(τ); i.e., the light-curve shape and anisotropy will not depend on the individual values of M, κ, β_ch, and q₀ but rather only on the combinations t_d = (Mκ/β_ch c²)^1/2 and ${L}_{{\rm{n}}}={{Mq}}_{0}{t}_{{\rm{d}},\mathrm{day}}^{{\sigma }_{1}}$ (as well as the variables τ_e, σ₁, etc.). In practice, this scaling breaks down for outflows with a sufficiently high β_ch or low M, for which the outer ejecta layers are of low density and become optically thin. However, for a given τ_e, we find that outflows with β_ch ≲ 0.1 generally remain in the diffusive regime.

With the above framework and qualifications, we run our simulations with the fiducial ejecta parameters M = 10⁻² M_⊙, β_ch = 0.1, and κ = 1 cm² g⁻¹. We start our simulations at the initial time τ₀ = 0.01 and nondimensionalize our results.

We study several idealized axisymmetric geometries that can represent the ejecta from a kilonova as described in Section 1. We summarize these geometries in Figure 1. We primarily study an ellipsoid and a ring torus and extend our analysis to a conical section embedded in a sphere.

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These geometries are robust and versatile and can replicate the shape of the different ejecta components described in Section 1. An ellipsoid is one of the simplest geometries that has dipolar asymmetry and thus serves as a generic model for deviations from sphericity. A modestly prolate ellipsoid with an axial ratio R (defined in Section 2.2.1) near unity can serve as a model for the disk wind, which magnetohydrodynamics (MHD) simulations have shown can have a global distortion (Fernández et al. 2019), with the amount of mass ejection varying by ∼2 from pole to equator. An oblate ellipsoid with R ≳ 2 can serve as a model for the tidal tail, and a prolate ellipsoid with R ≲ 1/2 can serve as a model for the collision interface ejecta. A torus with a radius ratio K (defined in Section 2.2.2) near unity can serve as a model for the wind in the aftermath of jet puncture, and one with K ≳ 2 can also serve as a model for the tidal tail. A conical section embedded in a sphere can serve as a model for the low-opacity polar ejecta from the collision enshrouded by both the high-opacity equatorial ejecta from the tidal tail and the postmerger wind.

Since the geometries we study are axisymmetric, we parameterize the viewing angle with $\mu =\cos \theta$, where θ is the polar angle measured from the z-axis. We record the emission in 20 uniformly spaced bins in μ ∈ [−1, 1]. In Appendix A, we calculate the parallel projected areas as a function of μ for each of these geometries, which are useful for understanding the light curves.

2.2.1. Ellipsoid

We study an ellipsoid with velocity coordinates (v_x, v_y, v_z) and velocity space semimajor axes (a_x, a_y, a_z), where a_x = a_y (axisymmetric, spheroid). We define the axial ratio R = a_x/a_z and examine several values in the range R ∈ [1/4, 6]. We define the dimensionless velocity coordinate

$$\begin{eqnarray}&&s={\left[{\left(\displaystyle \frac{{v}_{x}}{{a}_{x}}\right)}^{2}+{\left(\displaystyle \frac{{v}_{y}}{{a}_{y}}\right)}^{2}+{\left(\displaystyle \frac{{v}_{z}}{{a}_{z}}\right)}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 2 }$$

where 0 ≤ s ≤ 1. A surface of constant s is the surface of an ellipsoid with its center at the origin and semimajor axes $({{sa}}_{x},{{sa}}_{x},{{sa}}_{z});$ s = 1 corresponds to the outer surface of the ejecta.

We adopt a density profile of the form ρ = ρ(s, τ), in which surfaces of constant density correspond to surfaces of constant s. We study two types: (1) a constant density profile with a sharp cutoff at the surface and (2) a broken power-law density profile. The constant profile is given by ${\rho }_{c}(s,\tau )\,={{\rho }_{c0}({\tau }_{0})({\tau }_{0}/\tau )}^{3}$. The broken power-law profile has been successfully used to model kilonovae and more general transients (Chevalier & Soker 1989; Barnes & Kasen 2013; Kasen et al. 2017) and is given by

$$\begin{eqnarray}{\rho }_{b}(s,\tau )={\rho }_{b0}({\tau }_{0}){\left(\displaystyle \frac{{\tau }_{0}}{\tau }\right)}^{3}\left\{\begin{array}{ll}{\left(\tfrac{s}{{s}_{b}}\right)}^{{\delta }_{1}}, & s\lt {s}_{b}\\ {\left(\tfrac{s}{{s}_{b}}\right)}^{{\delta }_{2}}, & {s}_{b}\leqslant s\leqslant 1.\end{array}\right.\end{eqnarray} \tag{ 3 }$$

Here s_b is the location of the break in the power law, and δ₁ and δ₂ are the exponents in the broken power law that satisfy $0\gt {\delta }_{1}\gt -3$ and δ₁ > δ₂ to have a declining profile that decreases more precipitously after s_b and has finite M and E_k. We take s_b = 0.5, δ₁ =−1, and δ₂ =−10. We examine these two types of profiles in order to identify the trends in our results that are robust to changes in the form or parameters of the density profile. We use the values of M, β_ch, and τ₀ given in Section 2.1 and choose a value for R, and this sets the parameters a_x, a_z, and either ρ_c0(τ₀) or ρ_b0(τ₀).

2.2.2. Torus

We study a torus with velocity coordinates (v_x, v_y, v_z), velocity space spine radius a_c, and tube radius a_t, where a_c ≥ a_t (ring torus). We define the radius ratio K = a_c/a_t ≥ 1 and examine several values in the range K ∈ [1, 5]. We define the dimensionless velocity coordinate

$$\begin{eqnarray}&&s=\displaystyle \frac{{\left[{\left({\left({v}_{x}^{2}+{v}_{y}^{2}\right)}^{1/2}-{a}_{c}\right)}^{2}+{v}_{z}^{2}\right]}^{1/2}}{{a}_{t}},\end{eqnarray} \tag{ 4 }$$

where 0 ≤ s ≤ 1. A surface of constant s is the surface of a torus with spine radius a_c and tube radius sa_t; s = 1 corresponds to the outer surface of the ejecta.

We adopt the same types of density profile as in Section 2.2.1, though with two modifications: the variable s has the present definition, and the exponents satisfy 0 > δ₁ > −2 and δ₁ > δ₂. We take the same values for s_b, δ₁, and δ₂. We use the values of M, β_ch, and τ₀ given in Section 2.1 and choose a value for K, and this sets the parameters a_c, a_t, and either ρ_c0(τ₀) or ρ_b0(τ₀).

Figure 2 shows the isotropic-equivalent light curves L(τ, μ; R) for the ellipsoidal kilonova with a broken power-law density profile and an axial ratio R = 4. The viewing angle is defined by $\mu =\cos \theta$ (where θ is the polar angle measured from the z-axis), and each value of μ has equal probability of being observed. The light curves are invariant under the transformation $\mu \to -\mu$ due to reflection symmetry about z = 0, so the figure shows only the viewing angles μ > 0.

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The light curves of the ellipsoidal model are brightest along the pole (θ = 0, or μ = 1) and dimmest along the equator (θ = π/2, or μ = 0). The total variation in the luminosity at peak is a factor of ∼9. The time to peak τ_p(μ; R) ∼ 0.1, though, remains largely unchanged with viewing angle. The viewing angle dependence is most pronounced at early times but decreases over time until the light curves become mostly isotropic by τ ∼ 5 and converge to the total heating rate Q(τ; τ_e). This is because at late times, the ejecta becomes fully optically thin, so that all parts of the ejecta radiate equally in all directions (apart from small relativistic corrections).

Figure 2 also shows the angle-averaged light curve L_av(τ; R) of the ellipsoidal model and the isotropic-equivalent light curves normalized to this. The light curve of an intermediate viewing angle μ_ref ≈ 0.55 (θ_ref ≈ 57°) roughly equals the angle-averaged light curve, $L(\tau ,{\mu }_{\mathrm{ref}};R)\simeq {L}_{\mathrm{av}}(\tau ;R)$, with an error  < 0.1. The polar light curves are brighter than these, and the equatorial ones are dimmer. The angle-averaged light curve is comparable to the light curve L_s(τ) of the equivalent spherical model (R = 1 with the same τ_e), L_av(τ; R) ≃ L_s(τ), with an error  < 0.5 (Figure B3).

The basic viewing angle dependence of the light curves can be understood by considering the parallel projected surface area of the ejecta at different inclinations. If we consider a simple approximation in which an ellipsoidal photospheric shell emits blackbody radiation at a fixed temperature T_bb, then the luminosity would be proportional to the projected area, which is found to be (Appendix A.1)

$$\begin{eqnarray}&&{A}_{\mathrm{proj}}(\mu ;R)=\pi {{Ra}}_{z}^{2}{\left[\left({R}^{2}-1\right){\mu }^{2}+1\right]}^{1/2}.\end{eqnarray} \tag{ 5 }$$

For an oblate ellipsoid, the projected area is a maximum along the pole and decreases monotonically to a minimum at the equator. The pole-to-equator projected area ratio is

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\mathrm{proj}}(\mu =1;R)}{{A}_{\mathrm{proj}}(\mu =0;R)}=R,\end{eqnarray} \tag{ 6 }$$

and this gives a rough scale for the luminosity variation of the light curve from pole to equator.

In reality, the kilonova ejecta is not described by a constant temperature ellipsoidal photosphere. Rather, the photosphere recedes with time, and there will be emission from a surface of nonconstant temperature plus emission from the optically thin volume outside the photosphere. The different viewing angles will have different effective diffusion times and effective photospheric parameters. Nevertheless, we show below that the projected area appears to capture the primary geometric effects on the light curve.

Figure 3 shows the spectra as a function of viewing angle at two different times for R = 4. The spectra for different μ are well described by an effective blackbody,

$$\begin{eqnarray}&&L(\nu ,\tau ,\mu ;R)=4\pi {R}_{\mathrm{surf}}^{2}(\tau )B(\nu ,{T}_{\mathrm{eff}}(\tau )),\end{eqnarray} \tag{ 7 }$$

where ν is the photon frequency and B is the blackbody spectrum,

$$\begin{eqnarray}&&B(\nu ,{T}_{\mathrm{eff}})=\displaystyle \frac{2h{\nu }^{3}/{c}^{2}}{{e}^{h\nu /{{kT}}_{\mathrm{eff}}-1}},\end{eqnarray} \tag{ 8 }$$

with effective temperature T_eff and effective areal radius R_surf. The blackbody form arises because we used a constant gray opacity (Section 2.1). The effective temperatures and radii increase with projected surface area: they are larger for μ toward the poles and smaller for μ toward the equator. The temperatures decrease and the radii increase with time until τ ∼ 2; the temperatures then converge to one constant value, and the radii decrease exponentially with time while retaining their dependence on μ. The viewing angle variation of the spectra thus decreases with time, as expected from the light curves.

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Figure 4 shows the variation of the luminosity ratio $L(\tau ,\mu ;R)/L(\tau ,{\mu }_{\mathrm{ref}};R)$ with viewing angle at different times for R = 4. At the time τ ≃ 0.65 (not shown in the figure), the angular dependence is remarkably well fit by the ratio of projected surface areas ${A}_{\mathrm{proj}}(\mu ;R)/{A}_{\mathrm{proj}}({\mu }_{\mathrm{ref}};R)$ (Equation (5)). At other times, the viewing angle dependence has the same shape (Figure A1) but with a different scale. This suggests that we can roughly describe the basic behavior of the light curve with a simple time-dependent function of the projected area (Equation (5)), such as

$$\begin{eqnarray}&&\displaystyle \frac{L(\tau ,\mu ;R)}{L(\tau ,{\mu }_{\mathrm{ref}};R)}\simeq 1+k(\tau ;R)\left(\displaystyle \frac{{A}_{\mathrm{proj}}(\mu ;R)}{{A}_{\mathrm{proj}}({\mu }_{\mathrm{ref}};R)}-1\right),\end{eqnarray} \tag{ 9 }$$

where k(τ; R) is a dimensionless fitting parameter that describes the scale of the viewing angle variation as a function of time. The dashed lines in Figure 4 show the fits to $L(\tau ,\mu ;R)/L(\tau ,{\mu }_{\mathrm{ref}};R)$ using this function.

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The fitting parameter k(τ; R) rises with time to a peak around τ ∼ 0.3 and then decays to zero roughly exponentially (Figure B1). We can see its effect in Figure 2, which shows the ratio L(τ, μ; R)/L_av(τ; R) as a function of time for R = 4. The curves show that the orientation effects are largest at early times, when the ejecta is optically thick, and converge at late times, when the ejecta becomes optically thin and the emission becomes isotropic (apart from Doppler shift effects, which are small for the values of β_ch of interest). The effects fall off roughly exponentially with time.

A more detailed inspection suggests that we can approximate k(τ; R) with the analytic expression

$$\begin{eqnarray}&&k(\tau ;R)\simeq {k}_{0}(R)\displaystyle \frac{2+\tau /{\tau }_{b}(R)}{1+{e}^{\tau /{\tau }_{b}(R)}}\end{eqnarray} \tag{ 10 }$$

with the fitting parameters k₀(R) and τ_b(R), where k₀(R) sets the value at τ = 0 and τ_b(R) is an estimate of the timescale for the anisotropy to decay. At early times τ ≪ τ_b(R), the expression becomes k(τ; R) ∼ k₀(R) with ∂_τ k = 0. At late times τ ≫ τ_b(R), it becomes $k(\tau ;R)\sim \tau {e}^{-\tau /{\tau }_{b}(R)}$, which is close to the observed exponential falloff. Using this expression for R = 4, we find the values k₀ = 1.4 and τ_b = 0.59 and the fit residual $| \delta | \lt 0.1$ (Figure B1).

We can thus obtain a rough projection factor from L_s(τ) to L(τ, μ; R) by using Equation (9), with the replacement $L(\tau ,{\mu }_{\mathrm{ref}};R)\to {L}_{s}(\tau )$, and Equation (10), with the values given for k₀ and τ_b. Figure 2 shows the semianalytic light curves for R = 4 obtained by applying this projection factor, along with their errors. The projected light curves are not impeccable; they typically have lower peaks and converge more quickly to the heating rate at late times. However, they are accurate to within an error of  ≲ 0.4 before peak and  ≲ 0.3 after peak and thus roughly capture the temporal evolution of the emission and the correct scale of the variation.

The projection factor obtained from this approach provides a workable estimate of the viewing angle dependence, though it has limitations and should not be used cavalierly. First, the fitting parameter k(τ; R) rises to a peak and decays, and is bounded above because Equation (9) must be positive; in contrast, the analytic approximation in Equation (10) starts at a peak with zero slope and decreases monotonically with increasing τ. Second, the angle-averaged light curves L_av(τ; R) are not quite equal to the equivalent spherical light curves L_s(τ), and the divergence between the two is more pronounced for larger axial ratios. Appendix B quantifies the scale of these effects and provides more accurate but more involved parameterizations.

The features presented above for R = 4 generalize over our range of R ∈ [0.25, 6]. Ejecta with higher R have a larger spread in brightness since the projected area changes more from pole to equator (Equation (6)). The time to peak τ_p(μ; R) is largely insensitive to R in addition to μ and falls within 0.05 ≤ τ_p(μ; R) ≤ 0.12. The various fits and approximations are more accurate for R closer to unity and grow less accurate with increasing asphericity. The relation $L(\tau ,{\mu }_{\mathrm{ref}};{\text{}}R)\simeq {L}_{\mathrm{av}}(\tau ;R)$ for the reference value μ_ref = 0.55 has errors  ≲ 0.1. The relation L_av(τ; R) ≃ L_s(τ) has an error  < 0.1 for 0.25 ≤ R ≤ 2, rising to  < 0.9 for R = 6 (Figure B3). For different R, the fitting parameter k(τ; R) has a similar time evolution, and the analytic approximation for k(τ; R) remains robust (Figure B1). The semianalytic light curves have errors  < 0.2 for 0.25 ≤ R ≤ 2, rising to  < 0.6 for R = 6, which are comparable to those from uncertainties in the opacity (Barnes & Kasen 2013) and heating rate (Metzger et al. 2010; Korobkin et al. 2012; Lippuner & Roberts 2015). More simply, we find that the constant values k₀(R) ≃ 1.2 and τ_b(R) ≃ 0.7 produce semianalytic light curves within these error bounds.

We summarize the inclination dependence for different R with the pole-to-equator luminosity ratio, shown in Figure 5. The curves inherit the behavior described in Figures 2 and 4 and the related text. In particular, at early times, the luminosity ratio is larger than the corresponding projected area ratio (∼R; Equation (6)), and at late times, it approaches unity as the emission becomes isotropic. The dashed colored curves show fits using the analytic approximations in Equations (9) and (10) with the fitting parameters k₀(R) and τ_b(R) in Figure B1. The fits are accurate within errors  < 0.3 during the rise phase τ ≲ τ_p(μ; R) ∼ 0.1 and  < 0.2 after this.

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The constant density case is similar to the broken power-law case, with some systematic differences. Figure 6 shows the isotropic-equivalent light curves for a constant density profile and an axial ratio R = 4 as an example. The light curves begin at a peak and decline monotonically, since the constant density ejecta has more mass at larger distances. The pole-to-equator projected area ratio again gives the rough scale of the luminosity variation at intermediate times. The relation $L(\tau ,{\mu }_{\mathrm{ref}};R)\simeq {L}_{\mathrm{av}}(\tau ;R)$ has errors  ≲ 0.1 for the reference value μ_ref = 0.55. This suggests that μ_ref is a function primarily of the geometry. The relation L_av(τ; R) ≃ L_s(τ) has an error  < 0.1 for 0.25 ≤ R ≤ 2, rising to  < 0.4 for R = 6 (Figure B3). The fitting parameters k(τ; R) begin at lower values at early time but exhibit similar peak values and decline rates as the broken power-law case, and the analytic approximation for k(τ; R) has similar parameters (Figure B1). The projection factor then yields semianalytic light curves with errors of  ≲ 0.3 for 0.25 ≤ R ≤ 2, rising to  ≲ 0.8 for R = 6.

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The results here strictly apply only to an ellipsoidal geometry. However, since the light curves arise as an integration of the emission over the entire ejecta, they depend primarily on the global morphology and only weakly on small-scale distortions. The orientation effects of general, compact, contiguous geometries can thus be roughly estimated by considering a closely fitting ellipsoid with an effective axial ratio R_eff. In what follows, though, we analyze other geometries using the geometry-specific parallel projected area method, since this approach turns out to be fairly robust.

The light curves of a torus ejecta are analogous to those of an ellipsoid ejecta. In particular, we can explain the inclination variation of the light curves using the same projected surface area method, with two main geometry-specific differences. The first is that the torus has a more complicated projected area (Appendix A.2). In particular, it has a characteristic break at $\pm {\mu }_{\mathrm{break}}=\pm 1/K$, which corresponds to the viewing angle at which the torus begins to fully block the central hole. For $| \mu | \geqslant {K}^{-1/2}$, the torus does not block the hole; for ${K}^{-1}\lt | \mu | \lt {K}^{-1/2}$, it partially blocks the hole; and for $| \mu | \leqslant {K}^{-1}$, it fully blocks the hole. The projected area thus transitions from a shallower to a steeper dependence on viewing angle at μ = K⁻¹ (Figure A1). The scale of the variation is again given by the pole-to-equator projected area ratio,

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\mathrm{proj}}(\mu =1;K)}{{A}_{\mathrm{proj}}(\mu =0;K)}=\displaystyle \frac{4\pi K}{4K+\pi }.\end{eqnarray} \tag{ 11 }$$

The second is that the torus has a different reference direction, μ_ref = 0.45. The relation $L(\tau ,{\mu }_{\mathrm{ref}};K)\,\simeq \,{\text{}}{L}_{\mathrm{av}}(\tau ;K)$ has errors  ≲ 0.1 with this μ_ref for both the broken power-law and constant density profiles. As before, this suggests that μ_ref is a function primarily of the geometry.

To illustrate these differences, we present the signatures for a torus ejecta with a constant density profile and a radius ratio K = 3. Figure 7 shows the isotropic-equivalent light curves. Figure 8 shows the variation of the luminosity ratio L(τ, μ; K)/L(τ, μ_ref; K) with viewing angle at different times.

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The features shown in this specific case generalize to both density profiles (broken power-law and constant) and over our range of K ∈ [1, 5] in a way analogous to the ellipsoid ejecta. The light curves in the broken power-law case have a rise-peak-fall time evolution. The relation L_av(τ; K) ≃ L_s(τ) for the constant density case has errors  ≲ 0.2 for 1 ≤ K ≤ 2 rising to  ≲ 2.2 for K = 5, and for the broken power-law case, it has errors  ≲ 0.3 for 1 ≤ K ≤ 2 rising to  ≲ 3.7 for K = 5. The fitting parameters k(τ; K) have similar shapes but different values compared to the ellipsoidal case, and thus the analytic approximation for k(τ; K) has different values for k₀(K) and τ_b(K) (Figure B2). It is interesting to compare the values for K ∈ [1, 5] to those for R ∈ [2, 6], since they have similar degrees of asphericity; in general, the parameters k₀ and τ_b appear to depend on both the geometry and density profile. The semianalytic light curves for the broken power-law case have errors  < 0.4 for 1 ≤ K ≤ 2 rising to  < 0.8 for K = 5, and those for the constant density case have errors  < 0.55 for 1 ≤ K ≤ 2 rising to  < 0.75 for K = 5.

We study an ellipsoid with semimajor axes (a_x, a_y, a_z), where a_x = a_y (axisymmetric, spheroid), and we define the axial ratio R = a_x/a_z (Figure 1). A straightforward analysis of the geometry shows that the parallel projected area of the ellipsoid in the direction μ is an ellipse with semimajor axes $({a}_{x},{a}_{z}{[({R}^{2}-1){\mu }^{2}+1]}^{1/2})$. The parallel projected area in the direction μ is thus

$$\begin{eqnarray}&&{A}_{\mathrm{proj}}(\mu ;R)=\pi {{Ra}}_{z}^{2}{\left[\left({R}^{2}-1\right){\mu }^{2}+1\right]}^{1/2}.\end{eqnarray} \tag{ A1 }$$

Figure A1 shows a plot of A_proj(μ; R). The pole-to-equator ratio is

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\mathrm{proj}}(\mu =1;R)}{{A}_{\mathrm{proj}}(\mu =0;R)}=R.\end{eqnarray} \tag{ A2 }$$

We study a torus with spine radius a_c and tube radius a_t, where ${a}_{c}\gt {a}_{t}$ (ring torus), and we define the radius ratio K = a_c/a_t ≥ 1 (Figure 1). A straightforward analysis of the geometry shows that the parallel projected area of the torus in the direction μ is the area bounded by the parallel curves of an ellipse with semimajor axes (a_c, a_c μ). The outer (+) and inner (−) parallel curves can be expressed with the parametric equations

$$\begin{eqnarray}&&\tilde{x}(t)={a}_{t}\left(K\pm \displaystyle \frac{\mu }{{\left(1-(1-{\mu }^{2}){\cos }^{2}t\right)}^{1/2}}\right)\cos t,\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}&&\tilde{y}(t)={a}_{t}\left(K\cos \theta \pm \displaystyle \frac{1}{{\left(1-(1-{\mu }^{2}){\cos }^{2}t\right)}^{1/2}}\right)\sin t.\end{eqnarray} \tag{ A4 }$$

The parallel projected area in the direction μ can be written as

$$\begin{eqnarray}&&{A}_{\mathrm{proj}}(\mu ;K)={A}_{\mathrm{outer}}(\mu ;K)-{A}_{\mathrm{inner}}(\mu ;K),\end{eqnarray} \tag{ A5 }$$

where A_outer(μ; K) and A_inner(μ; K) are the areas contained inside the outer and inner parallel curves, respectively. The following integrals will be useful for evaluating these areas:

$$\begin{eqnarray}&&{I}_{1}({t}_{1},{t}_{2})={\int }_{{t}_{1}}^{{t}_{2}}{\sin }^{2}{tdt},\end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray}&&{I}_{2}(\mu ;{t}_{1},{t}_{2})={\int }_{{t}_{1}}^{{t}_{2}}\displaystyle \frac{{\sin }^{2}t}{{\left(1-\left(1-{\mu }^{2}\right){\cos }^{2}t\right)}^{3/2}}{dt},\end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray}&&{I}_{3}(\mu ;{t}_{1},{t}_{2})={\int }_{{t}_{1}}^{{t}_{2}}\displaystyle \frac{{\sin }^{2}t}{{\left(1-\left(1-{\mu }^{2}\right){\cos }^{2}t\right)}^{1/2}}{dt},\end{eqnarray} \tag{ A8 }$$

$$\begin{eqnarray}&&{I}_{4}(\mu ;{t}_{1},{t}_{2})={\int }_{{t}_{1}}^{{t}_{2}}\displaystyle \frac{{\sin }^{2}t}{{\left(1-\left(1-{\mu }^{2}\right){\cos }^{2}t\right)}^{2}}{dt}.\end{eqnarray} \tag{ A9 }$$

We note that ${I}_{1}(0,\pi )=\pi /2$ and ${I}_{4}(\mu ;0,\pi )=\pi /2| \mu |$. The outer projected area can be written as

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{outer}}(\mu ;K) & = & 2{a}_{t}^{2}\left[{K}^{2}| \mu | {I}_{1}(0,\pi )+K{\mu }^{2}{I}_{2}(\mu ;0,\pi )\right.\\ & & \left.+\,{{KI}}_{3}(\mu ;0,\pi )+| \mu | {I}_{4}(\mu ;0,\pi )\right].\end{array}\end{eqnarray} \tag{ A10 }$$

The form of the inner projected area depends on the range of μ: for different μ, the torus blocks the central hole to a different degree, which leads to a different degree of overlap of the inner parallel curve with itself. The inner projected area can be divided into three domains as follows.

• 1.  
  $| \mu | \geqslant {K}^{-1/2}$: no blocking/overlap,

  $$\begin{eqnarray}&&\begin{array}{l}{A}_{\mathrm{inner}}(\mu ;K)=2{a}_{t}^{2}\left[{K}^{2}| \mu | {I}_{1}(0,\pi )-K{\mu }^{2}{I}_{2}(\mu ;0,\pi )\right.\\ \,\left.-{{KI}}_{3}(\mu ;0,\pi )+| \mu | {I}_{4}(\mu ;0,\pi )\right].\end{array}\end{eqnarray} \tag{ A11 }$$
• 2.  
  ${K}^{-1/2}\gt | \mu | \gt {K}^{-1}$: partial blocking/overlap,

  $$\begin{eqnarray}&&\begin{array}{l}{A}_{\mathrm{inner}}(\mu ;K)=2{a}_{t}^{2}\left[{K}^{2}| \mu | {I}_{1}({t}_{r},\pi -{t}_{r})\right.\\ \quad -\,K{\mu }^{2}{I}_{2}(\mu ;{t}_{r},\pi -{t}_{r})-{{KI}}_{3}(\mu ;{t}_{r},\pi -{t}_{r})\\ \quad \left.+\,| \mu | {I}_{4}(\mu ;{t}_{r},\pi -{t}_{r})\right],\end{array}\end{eqnarray} \tag{ A12 }$$

  where ${t}_{r}(\mu ;K)={\cos }^{-1}\left({\left(1-{\mu }^{2}\right)}^{-1/2}{\left[1-{K}^{-2}{\mu }^{-2}\right]}^{1/2}\right).$
• 3.  
  ${K}^{-1}\geqslant | \mu |$: full blocking/overlap,

  $$\begin{eqnarray}&&{A}_{\mathrm{inner}}(\mu ;K)=0.\end{eqnarray} \tag{ A13 }$$

All of the integrals above can be evaluated either explicitly or in terms of elliptic integrals. Figure A1 shows a plot of A_proj(μ; K). The curves show a characteristic break at ±μ_break = ± 1/K, which corresponds to the viewing angle at which the torus begins to fully block the central hole. The pole-to-equator ratio is

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\mathrm{proj}}(\mu =1;K)}{{A}_{\mathrm{proj}}(\mu =0;K)}=\displaystyle \frac{4\pi K}{4K+\pi }.\end{eqnarray} \tag{ A14 }$$

We study a cone with a half-opening angle θ_c ∈ [0, π/2] (${\mu }_{c}=\cos {\theta }_{c}\in [0,1]$) embedded inside a sphere of radius a (Figure 1). The parallel projected area has a different form depending on the viewing angle and can be divided into four domains. In the interval $-{(1-{\mu }_{c}^{2})}^{1/2}\lt \mu \lt {(1-{\mu }_{c}^{2})}^{1/2}$, the following variables will be useful:

$$\begin{eqnarray}&&{\theta }_{{\rm{p}}}(\mu ;{\mu }_{c})={\cos }^{-1}\left(\displaystyle \frac{{\mu }_{c}}{{\left(1-{\mu }^{2}\right)}^{1/2}}\right)\in [0,{\theta }_{c}],\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&{\theta }_{{\rm{d}}}(\mu ;{\mu }_{c})={\tan }^{-1}\left(\displaystyle \frac{\sin {\theta }_{{\rm{p}}}}{{\mu }_{c}}\displaystyle \frac{{\left(1-{\mu }^{2}\right)}^{1/2}}{| \mu | }\right)\in \left[0,\displaystyle \frac{\pi }{2}\right].\end{eqnarray} \tag{ A16 }$$

The parallel projected area of the top and bottom caps together, when the rest of the cone is blocked by the sphere, can be written as

$$\begin{eqnarray}&&{A}_{\mathrm{proj}}(\mu ;{\mu }_{c})={A}_{\mathrm{proj}}^{\mathrm{top}}(\mu ;{\mu }_{c})+{A}_{\mathrm{proj}}^{\mathrm{top}}(-\mu ;{\mu }_{c}),\end{eqnarray} \tag{ A17 }$$

where A_proj^top(μ; μ_c) is the parallel projected area of the top cap alone, when the rest of the cone is blocked by the sphere, which takes the following form in the four domains.

• 1.  
  $1\geqslant \mu \geqslant {\left(1-{\mu }_{c}^{2}\right)}^{1/2}$:

  $$\begin{eqnarray}&&{A}_{\mathrm{proj}}^{\mathrm{top}}(\mu ;{\mu }_{c})=\pi {a}^{2}(1-{\mu }_{c}^{2})\mu .\end{eqnarray} \tag{ A18 }$$
• 2.  
  (1−μ_c²)^1/2 > μ ≥ 0:

  $$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{proj}}^{\mathrm{top}}(\mu ;{\mu }_{c}) & = & {a}^{2}({\theta }_{{\rm{p}}}-\sin {\theta }_{{\rm{p}}}\cos {\theta }_{{\rm{p}}})\\ & & -\,{a}^{2}{\left(1-{\mu }_{c}^{2}\right)}^{1/2}\mu ({\theta }_{{\rm{d}}}-\sin {\theta }_{{\rm{d}}}\cos {\theta }_{{\rm{d}}}-\pi ).\end{array}\end{eqnarray} \tag{ A19 }$$
• 3.  
  $0\gt \mu \gt -{\left(1-{\mu }_{c}^{2}\right)}^{1/2}$:

  $$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{proj}}^{\mathrm{top}}(\mu ;{\mu }_{c}) & = & {a}^{2}({\theta }_{{\rm{p}}}-\sin {\theta }_{{\rm{p}}}\cos {\theta }_{{\rm{p}}})\\ & & +\,{a}^{2}{\left(1-{\mu }_{c}^{2}\right)}^{1/2}\mu ({\theta }_{{\rm{d}}}-\sin {\theta }_{{\rm{d}}}\cos {\theta }_{{\rm{d}}}).\end{array}\end{eqnarray} \tag{ A20 }$$
• 4.  
  $-{\left(1-{\mu }_{c}^{2}\right)}^{1/2}\geqslant \mu \geqslant -1$:

  $$\begin{eqnarray}&&{A}_{\mathrm{proj}}^{\mathrm{top}}(\mu ;{\mu }_{c})=0.\end{eqnarray} \tag{ A21 }$$

Figure A1 shows a plot of A_proj(μ; μ_c). The pole-to-equator ratio is

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\mathrm{proj}}(\mu =1;{\mu }_{c})}{{A}_{\mathrm{proj}}(\mu =0;{\mu }_{c})}=\displaystyle \frac{\pi {\sin }^{2}{\theta }_{c}}{2({\theta }_{c}-\sin {\theta }_{c}\cos {\theta }_{c})}.\end{eqnarray} \tag{ A22 }$$

Figure B1 shows the fitting parameter k(τ; R) from Equation (9) for an ellipsoid outflow over our range $R\in [0.25,6]$. The curves have a similar behavior for different R. They rise to peak values >1 at times τ ≲ 1 and then decay to zero roughly exponentially. The ejecta with higher R take longer to converge. Figure B1 also shows the fitting parameters k₀(R) and τ_b(R) from the analytic approximation to k(τ; R) given in Equation (10). We find that the constant values k₀(R) ≃ 1.2 and τ_b(R) ≃ 0.7 yield semianalytic light curves with errors  < 0.2 for 0.25 ≤ R ≤ 2 rising to  < 0.6 for R = 6.

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The light curves L(τ, μ; R) > 0, and thus k(τ; R), must lie in the range $0\leqslant k(\tau ;R)\lt {k}_{\mathrm{upper}}(\tau ;R)$, where ${k}_{\mathrm{upper}}(\tau ;R)\,={(1-{\min }_{\mu }{A}_{\mathrm{proj}}(\mu ;R)/{A}_{\mathrm{proj}}({\mu }_{\mathrm{ref}};R))}^{-1}$. The lower bound ensures that the semianalytic approximation for L(τ, μ; R) in Equation (9) satisfies L(τ, μ₁; R) > L(τ, μ₂; R) when ${A}_{\mathrm{proj}}({\mu }_{1};R)\gt {A}_{\mathrm{proj}}({\mu }_{2};R)$, and the upper bound ensures that it satisfies $L(\tau ,\mu ;R)\gt 0$ when ${A}_{\mathrm{proj}}(\mu ;R)/{A}_{\mathrm{proj}}({\mu }_{\mathrm{ref}};R)\lt 1$.

An analytic approximation for k(τ; R) should remain within the above bounds as well. However, the expression given in Equation (10) increases monotonically with decreasing τ and can potentially surpass ${k}_{\mathrm{upper}}(\tau ;R)$ at early times if the fitting parameters k₀(R) and τ_b(R) are chosen carelessly and thereby produce negative values for L(τ, μ; R). This additional constraint can potentially restrict the range of R over which the analytic approximation is accurate. The values for k₀(R) and τ_b(R) given above preserve a positive value for L(τ, μ; R).

The torus outflow has analogous results, shown in Figure B2.

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Figure B3 shows the curves ${L}_{\mathrm{av}}(\tau ;R)/{L}_{s}(\tau )$ for an ellipsoid outflow over our range R ∈ [0.25, 6], which we can examine to quantify the deviation of L_av(τ; R) from L_s(τ). There is a time τ_cr(R) at which ${L}_{\mathrm{av}}(\tau ;R)/{L}_{s}(\tau )=1$, i.e., where ${L}_{\mathrm{av}}(\tau ;R)$ and L_s(τ) cross, and it falls in the range 0.3 ≤ τ_cr(R) ≤ 0.37. For τ > τ_cr(R), L_av(τ; R) and L_s(τ) differ by an error  < 0.2, and we can roughly take L(τ; R)/L_s(τ) ≃ 1. For τ < τ_cr(R), L_av(τ; R) is larger than L_s(τ), and the deviation is greater for ejecta with higher R. This is because photons can escape more easily in the z-direction for the ellipsoidal ejecta, leading to a lower effective diffusion time. In this region, we can roughly take ${L}_{\mathrm{av}}(\tau ;R)/{L}_{s}(\tau )\simeq 1$ for R ≲ 3, since we only incur an error  < 0.2, but we should use a more accurate expression for R ≳ 3. It is important to be accurate in this region, since the light curves experience their peak here.

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Motivated by this behavior, we adopt the following analytic approximation:

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{av}}(\tau ;R)}{{L}_{s}(\tau )}\simeq 1+\alpha (R){e}^{-{\tau }^{2}/{\tau }_{g}^{2}(R)}.\end{eqnarray} \tag{ B1 }$$

Here α(R) is the amplitude and τ_g(R) is the decay time, which falls in the range τ_g(R) ≲ τ_cr(R). The function thus provides a fit in the region τ ≲ τ_cr(R) and is ${L}_{\mathrm{av}}(\tau ;R)/{L}_{s}(\tau )\simeq 1$ in the region $\tau \gtrsim {\tau }_{\mathrm{cr}}(R)$. Figure B3 shows the fitting parameters α(R) and τ_g(R) obtained from fitting the curves ${L}_{\mathrm{av}}(\tau ;R)/{L}_{s}(\tau )$ with g(τ; R). For R ≲ 3, we can approximate L_av(τ;R)/L_s(τ) ≃ 1, incurring an error of  ≲ 0.2 for all τ. For R ≳ 3, one should use the full expression for L_av(τ; R)/L_s(τ) if an error ≲0.2 is needed.

We can thus obtain a more accurate projection factor from L_s(τ) to L(τ, μ; R) by combining Equations (9) (with the replacement $L(\tau ,{\mu }_{\mathrm{ref}};R)\to {L}_{\mathrm{av}}(\tau ;R)$), (B1), and (10). This parameterization is more involved. The numerical models are also available if accurate light curves are needed.

The torus outflow has analogous results, shown in Figure B4.

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