SEEING THE COLLISION OF A SUPERNOVA WITH ITS COMPANION STAR

Observations of supernova light curves and spectra have allowed us to characterize the outcome of the explosion—the burned and ejected stellar debris—in remarkable detail. But we still know very little about the starting point. In the most widely considered scenario, Type Ia supernovae (SNe Ia) result from carbon/oxygen white dwarfs that reach a critical mass by accreting material from a non-degenerate companion star. Observational confirmation of the binary nature of the progenitor system is lacking, however, and almost nothing is known about the properties and diversity of the companion stars. The proposed progenitors should be rather dim, so it is not surprising that we have so far failed to find them in pre-explosion images of the host galaxies (e.g., Maoz & Mannucci 2008). One therefore seeks other means of inferring the presence of a stellar companion.

A few minutes to hours after the supernova eruption, the debris ejected in the explosion is expected to overrun the companion. The star is shocked by the impact, and its envelope partially stripped and ablated, but it survives the ordeal (Wheeler et al. 1975; Fryxell & Arnett 1981; Chugai 1986; Livne et al. 1992; Marietta et al. 2000; Pakmor et al. 2008). Observations of the remnant of Tycho's 1572 supernova turned up a high-velocity G star, claimed to be the runaway companion (Ruiz-Lapuente et al. 2004). This identification is still debated (Kerzendorf et al. 2009). Meanwhile, several attempts to look for evidence of stripped hydrogen in the spectra of SNe Ia have detected nothing (Mattila et al. 2005; Leonard 2007). The supernova ejecta is distorted by the collision, which should lead to polarization of the supernova light (Kasen et al. 2004). But although polarization has been detected in several SNe Ia (Wang & Wheeler 2008), there is nothing to unambiguously indicate that this asymmetry relates to companion interaction.

While the previous investigations have focused on the long term consequences, one might wonder: could we see the collision itself? Here, we can draw an interesting parallel with core collapse supernova explosions, in which a shock wave propagates through the envelope of a massive star. When that shock front nears the stellar surface at radius R ≈ 10¹¹–10¹³ cm, the post shock energy vents in an X-ray breakout burst lasting minutes to hours (Klein & Chevalier 1978; Matzner & McKee 1999). In the days that follow, optical/UV radiation continues to diffuse from the deeper layers of shock-heated ejecta. Eventually, the energy deposition from radioactive ⁵⁶Ni decay takes over. The early shock luminosity phase, however, has been observed in several events, e.g., SN 1987A (Arnett et al. 1989) and SN 1993J (Wheeler et al. 1993), while the shock breakout itself was caught for SN 2008D (Soderberg et al. 2008; Modjaz et al. 2009).

The same physics applies to Type Ia supernovae, but because the radius of the progenitor white dwarf is so small (R_wd ≈ 2 × 10⁸ cm) the breakout emission should be extremely brief and the early luminosity remarkably dim. The problem is that when energy input occurs at small radii, adiabatic losses in the rapidly expanding (v ≈ 10⁹ cm s⁻¹) ejecta are overwhelming, and the thermal shock deposited energy is converted to kinetic energy on the expansion timescale (R_wd/v ∼ 0.1s), which is much shorter than the diffusion timescale.

As it turns out, the separation distance between the white dwarf and its companion star is presumed to be a ∼ 10¹¹–10¹³ cm, comparable to the radii of core collapse progenitors. When the supernova ejecta collides with its companion, the impacting layers are re-shocked and the kinetic energy partially dissipated. If the geometry is favorable, some of this energy might escape straightaway in a prompt burst, which will be followed by a longer-lasting tail of diffusive emission—the analogs of shock breakout and its aftermath in core collapse events. In this case, the emission provides a measure not of the stellar radius R_wd, but of the separation distance a.

Here, we develop an analytic description of the collision dynamics and subsequent radiation transport which suggests that early time observations of supernovae at X-ray through optical wavelengths should offer a powerful means of confirming the presence of a companion star and constraining its parameters.

In the single degenerate scenario of SNe Ia, the companion stars are thought to be either slightly evolved main-sequence (MS) stars or red giants (RG; Branch et al. 1995; Hachisu et al. 1996). For the most promptly exploding systems, the companions may be 5–6 M_☉ MS subgiants, with radii R_⋆ ∼ 5 × 10¹¹ cm. More commonly, the MS companions may be 1–3 M_☉ subgiants with radii ∼1–3 ×  10¹¹ cm. In the RG case, the stars are evolved ∼1–2 M_☉ stars with R_⋆ ∼ 10¹³ cm. In most scenarios, the companion is believed to be in Roche lobe overflow. The separation distance, a, is then comparable to the companion radius; for typical mass ratios, a/R_⋆ = 2–3.

After the supernova explodes, the ejected debris expands freely for some time before hitting the companion. The flow becomes homologous, and the radius of a fluid element r = vt, where v is the velocity and t is the time since explosion. We will describe the ejecta density profile by a broken power law (Chevalier & Soker 1989) with a shallow inner region ρ_i ∝ r^−δ and a steep outer region ρ₀ ∝ r⁻ⁿ. The profiles join at the transition velocity

$$\begin{equation} v_{\rm t}= 6 \times 10^8 \zeta _{v}(E_{51}/M_{\rm c})^{1/2}\;{\rm cm\;s}^{-1}, \end{equation} \tag{ 1 }$$

where E₅₁ = E/10⁵¹ ergs is the explosion energy, M_c = M/M_ch is the ejecta mass in units of the Chandrasekhar mass, and ζ_v is a numerical constant. The density in the outer layers (v>v_t) is

$$\begin{equation} \rho _0(r,t) = \zeta _{\rho }\frac{M}{v_{\rm t}^3 t^3} \biggl (\frac{r}{ v_{\rm t}t } \biggr)^{-n}, \end{equation} \tag{ 2 }$$

with a similar expression for the inner layers. The numerical constants follow from the requirement that the density integrate to the specified mass and kinetic energy

$$\begin{eqnarray} \zeta _{v}&= \biggl [ \frac{2(5-\delta)(n-5)}{(3-\delta)(n-3)} \biggr ]^{1/2}, \nonumber\\ \zeta _{\rho }&= \frac{1}{4 \pi } \frac{(n-3)(3-\delta)}{n-\delta }. \end{eqnarray} \tag{ 3 }$$

The broken power-law profile was originally derived for core collapse supernovae, but it fits multi-dimensional delayed-detonation models of SNe Ia remarkably well. For the models of Kasen et al. (2009), we find typical values of δ = 1 and n = 10, in which case the constants are ζ_v = 1.69 and ζ_ρ = 0.12.

The characteristic timescale for the supernova ejecta to interact with the companion is

$$\begin{equation} t_{\rm i}= a/v_{\rm t}= 10^4 a_{\rm 13}v_9^{-1}\,\mbox{s}, \end{equation} \tag{ 4 }$$

where a₁₃ = a/10¹³ cm and v₉ = v_t/10⁹ cm s⁻¹. For RG companions, the interaction timescale t_i ≈ 5 hr, while for MS subgiants, t_i ≈ 5–30 minutes.

The ejecta is highly supersonic when it collides with the companion, with mach number $\mathcal {M} \approx (a/R_{\rm wd})^{1/2} \gg 1$. Figure 1 illustrates the hydrodynamics of the interaction in a two-dimensional numerical calculation using the FLASH code (Fryxell et al. 2000) and assuming a polytropic γ = 4/3 equation of state, appropriate for a radiation dominated gas. As the flow sweeps over the companion star, a bow shock forms. Ejecta passing through the shock is heated and compressed into a thin shell, and its velocity vector is redirected.

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We will rely here on an approximate analytic description of the collision dynamics. Material moving at velocity v interacts at a time t_v ≈ a/v. The ejecta properties immediately after being shocked are given by the Rankine–Hugoniot jump conditions in the hypersonic limit. The density of the shocked gas is

$$\begin{equation} \rho _{\rm s}(v) = \frac{\gamma + 1}{\gamma -1} \rho _0(a,t_v) = 7 \zeta _{\rho }\frac{M}{a^3} \biggl (\frac{v}{v_{\rm t}} \biggr)^{-n+3}, \end{equation} \tag{ 5 }$$

taking γ = 4/3. The pressure of the shocked gas is of the order of the incoming ram pressure

$$\begin{equation} p_{\rm s}(v) = \frac{2}{1 + \gamma } \rho _0 v^2 \sin ^2 \chi = \frac{6}{7} \zeta _{\rho }\sin ^2\chi \frac{M v_{\rm t}^2}{a^3} \biggl (\frac{v}{v_{\rm t}} \biggr)^{-n+5}, \end{equation} \tag{ 6 }$$

where χ is the angle of the oblique shock front relative to the flow direction. The actual value of χ varies along the bow shock, but for simplicity we take a constant, characteristic value near the maximal turning angle for hypersonic flows, χ ≈ 45°.

For a radiation dominated gas, p_s = a_RT⁴_s/3, where a_R is the radiation constant, which gives the equilibrium temperature of the shocked debris

$$\begin{equation} T_{\rm s}(v) = 2.8 \times 10^6 M_{\rm c}^{1/4} v_9 ^{1/2} a_{\rm 13}^{-3/4} \bigg (\frac{v}{v_{\rm t}} \biggr)^{-(n-5)/4} \;{\rm K}. \end{equation} \tag{ 7 }$$

For an RG companion at a = 2.5 × 10¹³ cm, the outer layers of ejecta (v ∼ 3v_t) have T_s ≈ 3 × 10⁵ K, a value confirmed by the numerical calculation (Figure 1(a)). MS companions will have higher shock temperatures with T_s ≈ 10⁶–10⁷ K. To check the assumption of radiation domination in the shocked region, we note that Equations (5) and (6) imply a ratio of radiation energy to electron/ion energy

$$\begin{equation} \frac{ a T_{\rm s}^4}{\rho _{\rm s}k_B T_{\rm s}/m_p} = 707 a_{\rm 13}^{3/4} M_{\rm c}^{-1/4} v_9^{3/2}, \end{equation} \tag{ 8 }$$

which is ≫1 for the scenarios under consideration. The details of thermalization, however, deserve further investigation. Initially, the electrons/ions are shocked to very high temperatures (∼10⁹–10¹⁰ K), then radiatively cool toward the equilibrium value of Equation (7) by processes similar to those in ordinary supernova shocks—i.e., bremsstrahlung followed by Compton up-scattering, and in some cases pair production (Weaver 1976). If the timescale for the gas to radiate lags the dynamical timescale, non-equilibrium temperatures significantly greater than T_s can be realized in the relaxation region. This is likely to occur in the highest velocity, lowest density outermost layers (Katz et al. 2009), and may allow for harder radiation.

The interaction with the companion diverts the incoming flow, carving out a conical hole in the supernova ejecta. The half opening angle of the hole is roughly θ_h = arctan(r_b/a), where r_b is the extant of the bow shock. Simulations find r_b ∼ 2R_⋆ and θ_h = 30°–40° (see Figure 1 and Marietta et al. 2000). The solid angle, Ω_h, of the hole is

$$\begin{equation} \frac{\Omega _{\rm h}}{4\pi } = \frac{1}{2} (1 - \cos \theta _{\rm h}) = \frac{1}{2} \biggl [ 1 - \frac{1}{1 + (r_{\rm b}/a)^2} \biggr ] \approx \frac{1}{10}. \end{equation} \tag{ 9 }$$

This hole will provide a channel for radiation to quickly escape from the otherwise optically thick ejecta.

The ejecta displaced from the hole piles up into a compressed shell along the cone surface. Assuming this shell layer is thin, its thickness l_sh can be estimated by mass conservation. The volume of a region of radial extent dr within the conical cavity is V_i = Ω_ha²dr. The gas swept out of this region occupies a volume V_f = 2πal_shdr. The condition ρ₀V_i = ρ_sV_f gives

$$\begin{equation} \frac{l_{\rm sh}}{a} = \frac{\Omega _{\rm h}}{4 \pi } \frac{2 \rho _0}{\rho _{\rm s}} \approx \frac{1}{35}. \end{equation} \tag{ 10 }$$

A dense shell of roughly this thickness is seen in Figure 1(a). The actual dynamics can become quite complex, with the shell broken in pieces by shear instabilities and the companion envelope shredded.

After passing by the companion star, the shocked gas can expand laterally to try to refill the evacuated hole. The situation resembles the isentropic expansion of a gas cloud into vacuum (Zel'Dovich & Raizer 1967); the front of the rarefaction wave moves outward at the maximum escape velocity v_l = 2/(γ − 1)c_s, where c_s = (γp_s/ρ_s)^1/2 = (8/49)^1/2sin χv is the sound speed of the shocked material. The net velocity in the direction perpendicular to the symmetry axis is then v_x = vsin θ_h − v_lcos θ_h ≈ −0.2v. The ejecta moving at velocity v thus re-closes at a time R_⋆/|v_x| ≈ 5R_⋆/v after passing by the companion, or at a time t_h = (a + 6R_⋆)/v after the explosion. In the adiabatic calculation (Figure 1(b)), the rarefaction softens the density gradient in the polar direction, but fails to refill the shadowcone region uniformly before freezing out. If radiative cooling is significant during these phases (see Section 3), the sound speed will be reduced, which would further delay or halt the closing of the hole.

The energy density of the shocked gas is _s = 3p_s and so the total energy dissipated in the collision shock is found to be

$$\begin{equation} E_{\rm th} = \frac{18}{49} \sin ^2\chi \frac{\Omega _{\rm h}}{4 \pi } E \approx 1.5 \times 10^{49} E_{51} \;{\rm ergs}. \end{equation} \tag{ 11 }$$

Much of this thermal energy will be lost again to adiabatic expansion, but if even a fraction is radiated the collision luminosity should be quite bright.

As shocked supernova ejecta flows past the companion star, the hot surface layers of the dense shell become exposed (Figure 1(a)). At this time, some radiation may be able to escape straightaway through the evacuated shadowcone hole, giving rise to a sudden burst of emission. In general, only a fraction of the energy in the shell can be radiated promptly—i.e., before suffering significant loses due to adiabatic expansion. This prompt emission arises from the surface layer of the shell with a thickness, l_d, determined by requiring that the diffusion time through that layer,

$$\begin{equation} t_d = \tau \frac{l_{\rm d}}{3 c} = \frac{l_{\rm d}^2 \kappa \rho _{\rm s}}{ 3c}, \end{equation} \tag{ 12 }$$

be less than the timescale for expansion, given by the shell sound crossing time l_sh/c_s (which is typically shorter than the dynamical timescale a/v). Using Equations (5) and (10) gives

$$\begin{equation} \frac{l_{\rm d}}{l_{\rm sh}} = \frac{a}{v_{\rm t}t_{\rm sn}} \biggl (\frac{4 \pi }{\Omega _{\rm h}} \biggr)^{1/2} \biggl (\frac{\sqrt{8}}{14 \zeta _{\rho }\sin \chi } \biggr)^{1/2} \biggl [\frac{v}{v_{\rm t}} \biggr ]^{(n-4)/2}, \end{equation} \tag{ 13 }$$

where the quantity

$$\begin{equation} t_{\rm sn}= \biggl (\frac{\kappa M}{3 c v_{\rm t}} \biggr)^{1/2} = 29 M_{\rm c}^{1/2} v_9^{-1/2} \kappa _{e}^{1/2} \;{\rm days} \end{equation} \tag{ 14 }$$

is the familiar "effective" diffusion time (Arnett 1982) that sets the duration of the ordinary ⁵⁶Ni-powered SN Ia light curve. We have assumed a constant opacity κ_e = 0.2 cm² g⁻¹, appropriate for electron scattering in fully ionized A/Z = 2 elements. For RG companions at a ≈ 10¹³ cm, we find l_d ≃ l_sh for the layers v ≳ 2v_t. In this case, most of the energy dissipated in the outer layers can be radiated in the prompt burst. For MS companions with a ≃ 10¹¹–10¹² cm, the ratio l_d/l_sh ≃ 0.01–0.1 and only a fraction of the photons escape promptly.

While the bulk of the SN ejecta remains extremely optically thick at this phase, photons can initially escape through the hole carved out in the interaction. This channel will close, however, once the outer layers of ejecta have re-expanded to fill the shadowcone, which happens at a time t_h ≈ (a + 6R_⋆)/v_max, where v_max is the maximum ejecta velocity (see Section 2). A given layer of ejecta can contribute to the burst only if it passes the companion at a time less than t_h, which holds for material moving faster than v_min ≃ v_max(a + R_⋆)/(a + 6R_⋆). For a/R_⋆ = 3, we find v_min ≈ v_max/2.

By integrating the energy density, _s = 3p_s, of the shocked gas within the diffusion length l_d, we can estimate the total energy escaping in the burst

$$\begin{eqnarray} E_{\rm x}=& 2\pi \int _{v_{\rm min}}^{v_{\rm max}} 3 p_{\rm s}(v) l_{\rm d}(v) a \biggl (\frac{a}{v} \biggr) dv \nonumber\\ =& 7.72 \times 10^{47} \biggr (\frac{\Omega _{\rm h}}{4 \pi } \zeta _{\rho }\sin ^3 \chi \biggl)^{1/2} \biggr (\frac{62}{n-6} \biggl)\nonumber\\ &\times a_{13} M_{\rm c}^{1/2} v_9^{3/2} \kappa _{e}^{-1/2} \biggl (\frac{v_{\rm t}}{v_{\rm min}} \biggr)^{(n-6)/2}\;{\rm ergs}, \end{eqnarray} \tag{ 15 }$$

where we approximated the upper limit as v_max → ∞. The duration of the burst is the time it takes the emitting ejecta to flow past the companion, or Δt_x = (a + R_⋆)/v_min − (a + R_⋆)/v_max. For typical values (v_min ≈ 2v_t; v_max ≈ 4v_t; R_⋆ ≈ 3a), this duration is Δt_x ≈ t_i/3. Assuming the radiation is emitted into a solid angle Ω_h, the isotropic equivalent luminosity is L_x,iso = (4π/Ω_h)(E_x/Δt_x). Taking characteristic values (n, δ, χ, θ_h, v_min) = (10, 1, 45°, 40°, 2v_t), we find

$$\begin{equation} L_{\rm x,iso}= 5.8 \times 10^{44} M_{\rm c}^{1/2} v_9^{5/2} \kappa _{e}^{-1/2}\;{\rm ergs\;s}^{-1}. \end{equation} \tag{ 16 }$$

This luminosity is independent of a, and so roughly comparable for all types of companions. The value is similar to that of shock breakout in core collapse SNe, which is not surprising given that the shock temperature and emitting surface area are comparable in the two phenomena. The collision burst will only be visible for viewing angles peering down the hole, θ ≲ θ_h. Such an orientation occurs Ω_h/4π ≈ 10% of the time.

The spectrum of the prompt burst may be approximated as a blackbody at the equilibrium shock temperature (Equation (7)), implying emission peaking in the soft X-ray with typical energies T_x ∼ 0.05–0.1 keV for RG and T_x ∼ 0.2–2 keV for MS companions. Non-equilibrium effects (see Section 2) could lead to some emission at significantly higher energies (10–100 keV), while non-thermal particle acceleration may also contribute a power-law continuum of hard radiation. X-rays emitted in the direction of the companion star will ionize its surface layers and be reprocessed, likely giving rise to substantial line recombination/fluorescence emission (e.g., Ballantyne & Ramirez-Ruiz 2001).

According to Equation (7), the luminosity of the burst initially rises sharply as L_x,iso ∝ tⁿ⁻⁵ while the temperature of the spectrum evolves as

$$\begin{equation} T_x(t) = 0.1 a_{\rm 13}^{-3/4} M_{\rm c}^{1/4} v_9 ^{1/2} \bigg (\frac{t}{t_{\rm i}/2} \biggr)^{(n-5)/4} \;{\rm keV}, \end{equation} \tag{ 17 }$$

which predicts that, at least initially, the spectrum becomes harder with time, as interior layers of ejecta have higher densities and shock temperatures. On the other hand, deviations from equilibrium are likely to be greatest in the highest velocity layers, which may counteract this trend. Eventually, as the supernova ejecta gradually refills the shadowcone, the diffusion time through the hole region becomes significant. At this point, the effective photosphere moves to a larger radius and the emission must decline and soften (see Section 4).

The actual structure of the collision region will be more complex and inhomogeneous than that imagined here, due either to the inherent clumpiness of the supernova ejecta, or to secondary shocks and hydrodynamic instabilities developing in the interaction (e.g., Cid-Fernandes et al. 1996). This may lead to fluctuations in the burst light curve on a timescale δR/v, where δR is the typical clump size. Multi-dimensional radiation-hydrodynamics calculations will be needed to characterize the light curve and spectra in detail.

Thermal energy not radiated in the prompt burst can diffuse out in the hours and days that follow, but will suffer loses from adiabatic expansion. At t ∼ 1 day, this emission will be primarily at UV/optical wavelengths.

We consider here times ≫t_i such that homology has been re-established in the ejecta. The final ejecta structure will be asymmetric, but for now we neglect angular dependencies. The density profile is taken to be ρ_f(r, t) = 7f₀ρ₀(r, t), where the constant f₀ < 1 accounts for the lateral expansion and radial readjustment that occur during the transition to homology.

Assuming the evolution is adiabatic, the pressure profile of a shocked fluid element at these times is p_f(t) = p_s[ρ_f(t)/ρ_s]^γ. For the region affected by the collision (θ < θ_h),

$$\begin{equation} p_{\rm f}(r,t) = \frac{6}{7} f_0^{4/3} \zeta _{\rho }\sin ^2 \chi \frac{M a}{v_{\rm t}^2 t^4} \biggl (\frac{r}{v_{\rm t}t} \biggr)^{-n+1}. \end{equation} \tag{ 18 }$$

The pressure is negligible outside θ_h. As time progresses, the adiabatic profile, Equation (18), will continue to describe the opaque inner layers of ejecta, but the outer layers will be modified by radiative diffusion. The evolution can be calculated using a self-similar diffusion wave analysis (Chevalier 1992). At a time t, diffusion will have affected the ejecta above a radius r_d determined by setting the diffusion time t_d = r²_dκρ_f/3c equal to the elapsed time t

$$\begin{equation} r_{\rm d}(t) = \biggl [ \frac{\zeta _{\rho }}{3} t_{\rm sn}^2 \biggr ]^{1/(n-2)} v_{\rm t}t^{(n-4)/(n-2)}. \end{equation} \tag{ 19 }$$

Over time, the diffusion wave recedes into the ejecta in a Lagrangian sense. However, for times t ≲ 5 days, r_d remains in the steep outer layers of ejecta (v>v_t).

Considering only the radial transport, the isotropic equivalent luminosity in the comoving frame is given by the diffusion approximation

$$\begin{equation} L_{\rm c,iso}(r,t) = -4 \pi r^2 \frac{c}{\kappa \rho _{\rm f}} \frac{\partial p_{\rm f}}{\partial r}. \end{equation} \tag{ 20 }$$

We continue to assume a constant opacity, even though at UV wavelengths the line expansion opacity may exceed electron scattering and will, to some extant, be temperature dependent.

In the outer layers of ejecta (r>r_d), the comoving luminosity L_c,iso is constant with radius. Its value is therefore set by processes near the diffusion wave radius. A reasonable estimate of L_c,iso is derived by evaluating Equation (20) at r_d, using the pressure profile Equation (18),

$$\begin{eqnarray} L_{\rm c,iso}=& \frac{16 \pi }{49} (n-1) \eta \sin ^2\chi f_0^{1/3} \biggl [ \frac{\zeta _{\rho }}{3} \biggr ]^{2/(n-2)}\nonumber\\ &\times \biggl (\frac{t_{\rm i}}{t_{\rm sn}^2} \biggr) \frac{1}{2} M v_{\rm t}^2 \biggl (\frac{t}{t_{\rm sn}} \biggr)^{-4/(n-2)}, \end{eqnarray} \tag{ 21 }$$

where η is a constant of order unity that must be determined by solving the full diffusion equation (Chevalier 1992). Written in this form, the resemblance to core collapse supernova light curves is clear: apart from constants, the luminosity is of order E/t_sn times a factor t_i/t_sn that accounts for adiabatic loses.

Taking (n, δ, χ, f₀, η) = (10, 1, 45°, 0.5, 0.5), we find that from appropriate viewing angles (θ ≲ θ_h)

$$\begin{equation} L_{\rm c,iso}= 10^{43} a_{\rm 13}M_{\rm c}^{1/4} v_9^{7/4} \kappa _{e}^{-3/4} t_{\rm day}^{-1/2}\;{\rm ergs\;s}^{-1}, \end{equation} \tag{ 22 }$$

where t_day is the time since explosion measured in days. The derivation applies only at times significantly greater than t_i, but Equation (22) may provide a workable estimate at earlier times. The observer frame luminosity differs from Equation (22) by an additional term accounting for the advected luminosity, of order v/c ≲ 10% compared to L_c,iso.

The collision luminosity will only be readily discernible when it exceeds the luminosity, L_ni, of the ordinary ⁵⁶Ni powered light curve. At early times, t ≪ t_sn, the approximate analytic light curves of SNe Ia give L_ni = _niM_ni(t/t_sn)², where _ni = 4.8 × 10¹⁰ ergs s⁻¹g⁻¹ and M_ni is the mass of ⁵⁶Ni (Arnett 1982). We then find that L_c,iso>L_ni for times

$$\begin{equation} t_{\rm c} < 7.3 a_{\rm 13}^{2/5} M_{\rm c}^{1/2} v_9^{3/10} \kappa _{e}^{1/10} \biggl (\frac{\kappa _{\rm ni}}{\kappa _{e}} \biggr)^{2/5} M_{{\rm ni},0.6}^{-2/5}\;{\rm days}, \end{equation} \tag{ 23 }$$

where M_ni,0.6 = M_ni/0.6 M_☉. Note that the opacity in the ⁵⁶Ni region, κ_ni, is heavily affected by iron group line blanketing, and so may be greater than the opacity κ_e in the outer layers. This would help delay the ⁵⁶Ni luminosity takeover.

The wavelength of the early emission depends on the photospheric radius, r_p, defined as the location where the optical depth $\tau = \int _{r_p}^\infty \rho _{\rm f}\kappa dr = 1$, or

$$\begin{equation} r_p = t^{(n- 3)/(n-1)} \biggl (\frac{\zeta _{\rho }\kappa M v_{\rm t}^{(n-3)} }{n-1} \biggr)^{1/(n-1)}. \end{equation} \tag{ 24 }$$

The effective temperature of the emission T_eff = (L_c,iso/4πr²_pσ)^1/4 is then, for n = 10

$$\begin{equation} T_{\rm eff} = 2.5 \times 10^4 a_{\rm 13}^{1/4} \kappa _{e}^{-35/36} t_{\rm day}^{-37/72} \;{\rm K}. \end{equation} \tag{ 25 }$$

At t = 1 day, the emission peaks at wavelengths λ ≈ 1000 Å; however, the Rayleigh–Jeans tail of the blackbody extends into the near UV and optical.

While the analytic solution captures the essential physical effects, it neglects the ejecta asymmetry and non-radial transport. We have, therefore, calculated numerical light curves using a three-dimensional non-gray radiation transfer code, which also accounts for the effects of line opacity and ⁵⁶Ni heating (Kasen et al. 2006). For simplicity, we use an artificial ejecta model with density profile ρ_f(r, θ, t) = ρ₀(r, t)f₀(θ), where the function f₀(θ) now describes the angular structure of the conical hole region

$$\begin{equation} f_0(\theta) = f_h + (1 - f_h) \frac{x^m}{1 + x^m} \biggl (1 + A \exp \biggl [- \frac{(x-1)^2}{(\theta _p/\theta _{\rm h})^2} \biggr ] \biggr), \end{equation} \tag{ 26 }$$

where x = θ/θ_h. This formula approximates the results of simulation in the homologous phase choosing m = 8, f_h = 0.1, θ_h = 30°, θ_p = 15°, and A = 1.8. The pressure profile in the shocked region was taken from Equation (18).

The light-curve calculation (Figure 2) shows that the early collision luminosity is dramatic when the companion is a RG at a = 2 × 10¹³ cm. The numerical results agree with the analytic estimate (Equation (22)), and also illustrate the anisotropy of the radiation. The collision luminosity is brightest for viewing angles looking down upon the shocked region (θ < θ_h), but a significant amount of radiation diffuses out at angles θ ≈ 90°, and a few percent is even back-scattered along θ ≈ 180°.

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While the companion interaction produces a conspicuous signature (a kink) in the early bolometric light curve, most of the emission is in the UV; in the optical bands, the effect is less dramatic (Figure 3). For a RG companion, the B-band light curve shows a distinct bump at t < 5 days, which should be relatively easy to probe observationally. At redder wavelengths, or for smaller separation distance a, the collision simply modifies the shape of the light-curve rise. However, because SN Ia light curves are quite standard, a statistical analysis of (good quality) early time photometry should be able to pull out these subtle differences.

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The results derived here suggest a new means for constraining supernova companions using early photometric observations. Table 1 summarizes the analytic estimates of the collision emission for various SNe Ia progenitors. The basic predicted signatures appear to be quite robust, as they rely only on established physics familiar from the core collapse SNe context. However, further numerical studies using multi-dimensional radiation-hydrodynamics calculations (and including non-equilibrium effects) will be needed to refine the detailed light-curve and spectrum predictions.

For all companion types, signatures of the collision will be prominent only for viewing angles looking down upon the shocked region, or ∼10% of the time. Detection will, therefore, require high cadence observations of many supernovae at the earliest phases (≲5 days) and at the bluest wavelengths possible. Ironically, these observations may sometimes be easier for distant SNe. At redshifts z ≳ 0.5, the UV flux would be redshifted into the U-band, while cosmological time dilation would prolong the light curve by a factor (1 + z).

Detecting the collision signatures becomes significantly easier for larger separation distances. Current optical and UV data sets likely already constrain RG companions (a ≃ 10¹³ cm). Ongoing or upcoming surveys could be tuned to probe the larger (M ≳ 3 M_☉) MS companions (a ≃ 10¹² cm). Optical detection of the smallest ∼1 M_☉ MS companions (a ≃ 10¹¹ cm) will be challenging, requiring measurement of subtle differences in the light curves at t ≲ 2 day. However, in all cases the prompt X-ray burst should be bright. Proposed X-ray surveys (e.g., EXIST; Grindlay 2005) may then detect a large number of collision bursts every year, at least in the case of MS companions that produce harder radiation. If non-equilibrium or non-thermal shock effects are operative, some hard radiation may accompany all bursts.

The most compelling reason for studying the collision emission is that it offers a straightforward measure of the separation distance between the stars. This value can be determined from the duration of the X-ray burst (Δt_x ≃ a/3v_t), or its temperature (Equation (7)), or from the luminosity of the early optical/UV emission (Equation (22)). If we, in turn, assume that the companion is in Roche lobe overflow, its radius can be inferred R_⋆ ∼ a/3 − a/2. In principle, the ratio a/R_⋆ itself could be constrained using a statistical sample of SNe Ia, as the anisotropy of the emission depends on the opening angle, θ_h, of the shocked region of ejecta.

While our discussion has focused on Type Ia supernovae, similar signatures of companion interaction should apply to Type Ib/Ic and some Type II SNe that may arise from close massive binaries (Podsiadlowski et al. 1992). In these cases, the shock breakout from the exploding star will contribute to the luminosity on comparable timescales, likely producing two peaks in the X-ray emission. Some gamma-ray bursts might also come from binary systems, and the interaction of the relativistic material with a stellar companion may produce another type of X-ray flare (MacFadyen et al. 2005).

It is also possible that early emission from SNe stems from a collision not with the companion star, but with a surrounding circumstellar medium (CSM). To substantially decelerate the ejecta, the CSM would need to have a mass ∼0.01–0.1 M_☉ located at radii ∼10¹¹–10¹³ cm. A slow (10 km s⁻¹) stellar wind moves beyond these distances in less than a year, so it may be difficult to realize these conditions in a single degenerate scenario of SNe Ia. In the double degenerate merger scenario, the total mass of the system can exceed M_ch, and a few 0.1 M_☉ of excess carbon/oxygen may linger in the vicinity. If this material remains at the tidal radius ∼10⁹ cm, the resulting emission will be extremely brief (∼1 s); however, if some mass is puffed out to larger radii in the super-Eddington accretion phase of the merger, the emission may be similar to that discussed here. Interaction with a spherical CSM is distinguishable from companion interaction by its luminosity function; in the former case, the emission should be nearly the same from all viewing angles.

In either case, the early time emission of supernovae provides much needed insight into the nature of the progenitor system. Observational surveys could be designed, either from space or the ground, to acquire the collision signatures in a systematic way. If one collects a significant number of events, it will be possible to correlate the measured separation distances with the properties of the ordinary ⁵⁶Ni powered light curve and spectra. Such observations would provide direct, empirical insight into how the parameters of the progenitor system influence the outcome of supernova explosions.

The author is grateful for comments from E. Ramirez-Ruiz, T. Plewa, S. E. Woosley, and the referee C. Matzner. Support was provided by NASA through Hubble fellowship grant No. HST-HF-01208.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. This research has been supported by the DOE SciDAC Program (DE-FC02-06ER41438). Computing time was provided by ORNL through an INCITE award and by NERSC.