Explaining recently studied intermediate luminosity optical transients (ILOTs) with jet powering

The basic flow structure of the jet-powered ILOT scenario is as follows (Soker 2020a). A binary interaction leads to the ejection of a shell, spherical or not, at velocities of tens to hundreds of km s⁻¹. The shell ejection period can last from a few weeks to several years. In a delay of about days to several months (or even a few years) the binary system launches two opposite jets. The jets collide with the shell, an interaction that converts kinetic energy, mainly from the jets, to radiation.

There are two types of evolutionary channels to launch jets. (1) The more compact secondary star accretes mass from the primary star and launches the jets, as in the jet-powered ILOT scenario of the Great Eruption of Eta Carinae (e.g., Soker 2007; Kashi & Soker 2010). The binary stellar system might stay detached, might experience grazing envelope evolution, and/or enter a common envelope evolution. In this case the binary systems might experience several jet-launching episodes. (2) The primary star gravitationally destroys the secondary star to form an accretion disk around the primary star, and this accretion disk launches the jets. In this case there is one jet-launching episode, although the jets' intensity can vary with time.

Soker (2020a) obtains the following approximate relations for jets that interact with a slower spherically symmetric shell and power an ILOT. We refer to this model as the spherical shell model. Soker (2020a) considers jets-shell interaction that (1) transfers a large fraction of the kinetic energy from the outflow to radiation, and (2) radiates much more energy than what recombination of the outflowing gas can supply. Soker (2020a) considers two opposite fast jets that hit a uniform spherical shell and accelerate the entire shell. In Section 3 we build a toy model where the jets penetrate into the shell and only interact with the shell's material in the polar directions.

In the simple flow structure that Soker (2020a) considers, the relevant properties of the jets are their half opening angle α_j ≳ 10°, velocity v_j ≈ 10³ km s⁻¹ and their total mass M_2j ≈ 0.01 − 1 M_⊙. With a conversion efficiency of jet kinetic energy to radiation f_rad, the total energy in radiation is

$$\begin{eqnarray}&&{E}_{{\rm{rad}},{\rm{j}}}={10}^{48}{f}_{{\rm{rad}}}\left(\displaystyle \frac{{M}_{2{\rm{j}}}}{0.1{M}_{\odot }}\right){\left(\displaystyle \frac{{v}_{{\rm{j}}}}{1000{{\rm{km}}{\rm{s}}}^{-1}}\right)}^{2}\,{\rm{erg}}.\end{eqnarray} \tag{ 1 }$$

The relevant properties of the spherical shell are its velocity v_s ≪ v_j, mass M_s, radius r_s and width Δr_s.

The jet-shell interaction converts kinetic energy, mainly f{rom} the jets, to thermal energy. The hot bubbles that the jets inflate lose their energy adiabatically by accelerating the shell and non-adiabatically by radiation. The adiabatic cooling proceeds on a typical timescale that is the expansion time t_exp, while energy losses to radiation occur during a typical photon-diffusion time t_diff. Namely, the relative rates, Ė/E, of adiabatic and radiative energy losses are ${t}_{{\rm{\exp }}}^{-1}$ and ${t}_{{\rm{diff}}}^{-1}$, respectively. This implies that the fraction of energy that ends in radiation is

$$\begin{eqnarray}&&{f}_{{\rm{rad}}}\simeq \displaystyle \frac{{t}_{{\rm{diff}}}^{-1}}{{t}_{{\rm{diff}}}^{-1}+{t}_{{\rm{\exp }}}^{-1}}={\left(1+\displaystyle \frac{{t}_{{\rm{diff}}}}{{t}_{{\rm{\exp }}}}\right)}^{-1}.\end{eqnarray} \tag{ 2 }$$

For the simple spherically symmetric geometry he assumes, Soker (2020a) estimates the two timescales to be

$$\begin{eqnarray}\begin{array}{lll}{t}_{{\rm{\exp }}} & \approx & 73\left(\displaystyle \frac{{r}_{{\rm{s}}}}{{10}^{14}\,{\rm{cm}}}\right){\left(\displaystyle \frac{{v}_{{\rm{j}}}}{1000{{\rm{km}}{\rm{s}}}^{-1}}\right)}^{-1}\\ & & \times {\left[\displaystyle \frac{{M}_{2{\rm{j}}}}{0.1({M}_{2{\rm{j}}}+{M}_{{\rm{s}}})}\right]}^{-1/2}{\rm{d}},\end{array}\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}\begin{array}{ll}{t}_{{\rm{diff}}} & \simeq \displaystyle \frac{3\tau \Delta {r}_{{\rm{s}}}}{c}\simeq 55\left(\displaystyle \frac{{M}_{{\rm{s}}}}{1{M}_{\odot }}\right)\left(\displaystyle \frac{\kappa }{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)\\ & \times {\left(\displaystyle \frac{{r}_{{\rm{s}}}}{{10}^{14}{\rm{cm}}}\right)}^{-1}\left(\displaystyle \frac{\Delta {r}_{{\rm{s}}}}{0.3{r}_{{\rm{s}}}}\right){\rm{d}},\end{array}\end{eqnarray} \tag{ 4 }$$

where τ = ρ_s κ Δr_s is the optical depth of the shell, κ is the opacity and c is the speed of light. The relevant ratio to substitute in Equation (2) is

$$\begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{t}_{{\rm{diff}}}}{{t}_{{\rm{\exp }}}} & \approx & 0.75\left(\displaystyle \frac{{M}_{{\rm{s}}}}{1{M}_{\odot }}\right)\left(\displaystyle \frac{\kappa }{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)\left(\displaystyle \frac{{v}_{{\rm{j}}}}{1000{{\rm{km}}{\rm{s}}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{{r}_{{\rm{s}}}}{{10}^{14}{\rm{cm}}}\right)}^{-2}\left(\displaystyle \frac{\Delta {r}_{{\rm{s}}}}{0.3{r}_{{\rm{s}}}}\right){\left[\displaystyle \frac{{M}_{2{\rm{j}}}}{0.1\left({M}_{2{\rm{j}}}+{M}_{{\rm{s}}}\right)}\right]}^{1/2}.\end{array}\end{eqnarray} \tag{ 5 }$$

We emphasise that we do not assume any value for the jets' energy E_2j. We rather take the jets' velocity from observations, and rely on the timescale of the ILOT together with an assumed opacity to find the mass in the shell (Eq. (4)). We then calculate the efficiency f_rad together with the mass in the jets (or their energy) to fit the total radiated energy (Eqs. (2) and (5)).

Soker (2020a) applies this spherical shell model of the jet-powered ILOT radiation to the ILOT (LRN) V838 Mon, to the Great Eruption of Eta Carinae which is an LBV and to the ILOT V4332 Sgr. He could find a plausible set of shell and jets parameters that might explain these ILOTs. Here we apply the spherical shell model to the ILOT (ILRT) SNhunt120 and to the ILOT (LRN) AT 2014ej. We summarise the plausible physical parameters of the ILOT events in Table 1, and explain their derivation in Sections 2.2 and 2.3. We emphasise that due to the very simple model we apply here, e.g., we use a spherical shell and we keep the opacity and shell thickness constant, the properties of the jets and shells we derive are very crude, and might even not be unique. Nonetheless, they demonstrate the potential of the jet-powered ILOT scenario to account for many ILOTs. The opacity of a fully ionised gas that is appropriate for ILOTs is κ ≃ 0.3 cm² g⁻¹ (e.g., Ivanova et al. 2013; Soker & Kashi 2016). We expect that in the outer parts of the shell, hydrogen is partially neutral, and that opacity is therefore lower. Therefore, we scale with κ = 0.1 cm² g⁻¹.

Stritzinger et al. (2020b) study the ILOT (ILRT) SNhunt120 and find the following relevant properties. The velocities of different emission lines are in the range of ≃ 300 − 1800 km s⁻¹, with a typical velocity of ≈ 10³ km s⁻¹. The typical photospheric radius is R_BB ≃ 2 × 10¹⁴ cm. The time to double the luminosity at rise is about 10 d, and the decline time to half the maximum luminosity is about 20 d. The total energy in radiation is E_rad ≃ 4 × 10⁴⁷ erg. Stritzinger et al. (2020b) further find that existing electron capture supernova models over-predict the energy in radiation. We do not consider this event to be a supernova, but rather an ILOT.

Following these parameters we scale the parameters for SNhunt120 with v_j ≃ 1000 km s⁻¹ and r_s ≃ 2 × 10¹⁴ cm. To get a photon diffusion time of about the decline time of 20 d, we find from Equation (4) for κ = 0.1 cm² g⁻¹ and Δr_s = 0.3 r_s that M_s ≈ 0.7 M_⊙. For an opacity of κ = 0.3 cm² g⁻¹ and a somewhat thicker shell with Δ r_s = 0.5 R_s, the required shell mass is only M_s ≈ 0.15 M_⊙.

Equation (5) then gives t_diff/t_exp ≈ 0.1, and from Equation (2) f_rad ≃ 0.9. To account for the emitted energy, we find from Equation (1) that the mass in the two jets is M_2j ≈ 0.045 M_⊙ (v_j/1000 km s⁻¹)⁻².

In this analysis there is no reference to the shell velocity, except that it should be much lower than the jets' velocity. This implies here 100 km s⁻¹ ≲ v_s ≲ 500 km s⁻¹. To reach a distance of r_s = 2 × 10¹⁴ the binary system ejected the shell about Δt_s ≃ 0.6 (v_s /100 km s⁻¹)⁻¹ yr before detection. The kinetic energy in the shell for these parameters of M_s ≃ 0.7 M_⊙ and v_s ≃ 100 km s⁻¹ is about 15% of the jets' energy. In any case, most of the kinetic energy of the shell does not convert to radiation.

In case that the secondary star launches the jets with a mass of M_2j ≃ 0.045 M_⊙, it should accrete a mass of M_acc,2 ≃ 0.45 M_⊙ from a more evolved primary star, possibly a giant. This implies that the secondary star should be a massive star itself. We are therefore considering a massive binary system. Alternatively, it is possible that the primary star destroyed the secondary star with mass M₂ ≃ 0.3 − 1 M_⊙ to form an accretion disk that launched the jets. The primary is then a massive main sequence star, and the secondary is not yet settled on the main sequence, such that its average density is lower than that of the primary star (as in the merger model of V838 Mon; Tylenda & Soker 2006). In any case, the primary star mass can be in the range of M₁ ≈ 10 M_⊙, similar to the range that Stritzinger et al. (2020b) consider. Since there is only one jet-launching episode, the jetpowered ILOT scenario does not directly refer, in the case of SNhunt120, to the question of which of these two evolutionary routes applies here.

Stritzinger et al. (2020a) study the ILOT (LRN) AT 2014ej. They find that the light curve of models of equatorial collision (Metzger & Pejcha 2017; Sect. 1) under-predict the luminosity. We therefore consider powering by jets, i.e., polar collision.

Stritzinger et al. (2020a) find that AT 2014ej has slow component(s) moving at ≈ 100 km s⁻¹ and fast component(s) moving at ≈ 1000 km s⁻¹. The total radiated energy is E_rad ≈ 2 × 10⁴⁸ erg, with two large peaks in the light curve. From discovery to first minimum 20 d later, the luminosity decreased from L₀ = 3.2 × 10⁴¹ erg s⁻¹ to L_min,1 = 1.2 × 10⁴¹ erg s⁻¹. Over the next 35 d the luminosity increased to L_AT ≡ L_peak,2 ≃ 2.6 × 10⁴¹ erg s⁻¹, after which the luminosity decreased over a timescale of several weeks. The photosphere was hotter in the first peak than in the second one. The photosphere (blackbody surface) moderately followed the behavior of the luminosity, and first declined somewhat and then increased somewhat. Its approximate average value was R_BB ≃ 2.5 × 10¹⁴ cm.

In the jet-powered ILOT scenario such multiple-peaks can be accounted for by multiple jet-launching episodes. From Stritzinger et al. (2020a) we find that the radiated energy from detection to first minimum (0 to 20 d) is ≃ 4 × 10⁴⁷ erg. If we take a similar energy at rise, the energy in the first peak is E_rad,1p ≈ 10⁴⁸ erg. The energy in the second peak, from 20 to about 95 d, is E_rad,2p ≈ 1.4 × 10⁴⁸ erg. The outburst of V838 Mon has a similar qualitative behavior with three peaks and three declines in the photospheric radius (Tylenda 2005).

In AT 2014ej the two peaks have about the same energy (under our assumption), but the second peak is slower by a factor of about two. From Equation (4) the mass in the shell should be larger in the second peak by a factor of two, ≃ 2 M_⊙ instead of ≃ 1 M_⊙. We do not expect the system to lose much more slow mass in that short time. The difference in the timescales of the two peaks might come from different values of κ and/or Δr_s between the two peaks, rather than from different shell masses. This can also be accompanied by precessing jets, i.e., the jets' axes in the two jet-launching episodes have different directions. In the present study we apply a simple model and do not calculate the opacity, and so we simply take M_s ≃ 1.5 M_⊙ for both peaks.

From the equations in Section 2.1, we derive crude plausible values of the shell mass, jets' energy and emission efficiency for the two peaks, as we list in Table 1.

According to the jet-powered ILOT scenario, the two distinct peaks result from two jet-launching episodes. This suggests that the secondary star, possibly in an eccentric orbit, accreted mass and launched the jets. Most likely, the secondary star survived the interaction.

In the simple spherical-shell model that we apply in Section 2, the jets interact with the entire shell (Soker 2020a). We now turn to a more realistic toy model where the jets interact only with the shell segments along the polar directions. In this 'cocoon toy model' the jetshell interaction inflates a 'cocoon', i.e., a relatively hot bubble composed of the post-shock shell material and postshock jet material. We further simplify the interaction by assuming that the jets' activity time period is short, such that we can treat the jet-shell interaction that creates the cocoon as a 'mini explosion'. We base the cocoon toy model on our usage of this model to account for peaks in the light curves of core collapse supernovae (Kaplan & Soker 2020; for the geometry of a jet-ejecta interaction in core collapse supernova see the three-dimensional simulations of Akashi & Soker 2021). In the cocoon toy model we only calculate the timescale of the emission peak (eruption) and its maximum luminosity (or total energy). We do not calculate the shape of the light curve, but rather assume a simple shape for the light curve. We then calculate the total radiated energy by integrating the luminosity over time.

We assume that each mini-explosion that results from jet-shell interaction is spherically symmetric around the jet-shell interaction point (Akashi & Soker 2021), and that cooling is due to photon diffusion and adiabatic expansion. These assumptions allow us to determine the luminosity and the timescale of each mini-explosion. As we deal with ILOTs where the total radiated energy is larger than the recombination energy of the outflowing gas, we neglect the recombination energy. Like Kaplan & Soker (2020), we utilize equation (4) from Kasen & Woosley (2009) to calculate the time of maximum luminosity t_j and the maximum luminosity L_j for one jet. These expressions read

$$\begin{eqnarray}&&\begin{array}{l}{t}_{{\rm{j}}}={\left(\displaystyle \frac{3}{{2}^{5/2}{\pi }^{2}c}\right)}^{1/2}{E}_{{\rm{j}}}^{-1/4}{M}_{{\rm{js}}}^{3/4}{\kappa }_{{\rm{c}}}^{1/2},\\ {L}_{{\rm{j}}}=\displaystyle \frac{2\pi c}{3}\sin {\alpha }_{{\rm{j}}}\beta {M}_{{\rm{js}}}^{-1}{E}_{{\rm{j}}}{\kappa }_{{\rm{c}}}^{-1}{R}_{{\rm{BB}}},\end{array}\end{eqnarray} \tag{ 6 }$$

where E_j, M_js, κ_c, α_j, β and R_BB are the energy that one jet deposits into the shell, the mass in the interaction region of one jet with the shell, the opacity in the cocoon, the half opening angle of the jet, the distance of the jet-ejecta interaction relative to the shell's outer edge (the photosphere radius R_BB) and the photospheric radius of the shell, respectively. Namely, in this model the mini-explosion takes place at a radius (measured from the center of the binary system) of r_me = β R_BB. There is one mini-explosion on each of the two polar regions. The value of r_me is constant and does not change with time. What increases with time is the radius of the cocoon itself, a_c, that is measured from the place of the mini-explosion.

We build the light curve of the jet as follows. We assume that the shape of the rise of the peak to maximum luminosity is similar to the rise to maximum of the light curve of a core collapse supernova (based on photometric data of SN 2008ax, taken from The Open Supernova Catalog Guillochon et al. 2017). Since the light curve of the jet does not have a tail powered by radioactive processes and recombination, we take the decline of the mini-explosion from maximum to be symmetric to its rise. Again, we do not try to fit the light curve. We rather only derive the properties of the jets that might lead to an event that has the same timescale, luminosity and radiated energy. We assume a light curve, but our results are not sensitive to the exact shape of the light curve we assume.

We turn to estimate the jets' properties that according to the cocoon toy model might fit the eruption times and luminosities of the ILOTs SNhunt120 (Sect. 3.2) and AT 2014ej (Sect. 3.3).

First we extend the observed light curve of SNhunt120 (Stritzinger et al. 2020b; thick-red line in Fig. 1) by taking a linear fit before discovery and beyond t = 30 d after discovery, on both sides down to L = 0. This is the solidblue line in Figure 1. The observed light curve of this ILOT has a break at about 40 d post-discovery, where the decline becomes shallower. This might result from a second and weaker jet-launching episode or from matter collision in the equatorial plane. We are interested here only in the light curve around the maximum, so we continue the steep decline beyond 30 d post-discovery down to L = 0. We then find the radiated energy of SNhunt120 from our fit to the peak to be E_rad,hunt = 3.8 × 10⁴⁷ erg. As we explained in Section 3.1, we then build a toy model symmetric light curve that has the same maximum luminosity as SNhunt120, L_hunt = 1.4 × 10⁴¹ erg s⁻¹, and the same total radiated energy. This is the green line in Figure 1 (for Case 1 that we describe next).

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We calculate the energy of one jet E_j and the mass in the region of interaction of one jet with the shell (the cocoon), M_js, as follows. We build a symmetric toy model light curve (one example is the green line in Fig. 1) that is characterised by its maximum luminosity L_j and by its timescale from start to maximum t_j by Equations (6). We then calculate the total radiated energy according to this light curve (area under the green light curve). We iterate the values of E_j and M_js until we obtain the luminosity due to the two jets together of L_2j = L_hunt = 1.4 × 10⁴¹ erg s⁻¹, and the total radiated energy from the two jets is E_rad,2j = E_rad,hunt = 3.8 × 10⁴⁷ erg. We note that the cocoon toy model is not sensitive to the expansion velocities of the shell or of the jets, as long as v_j ≫ v_s.

We do not vary the values of the photosphere radius R_BB = 2 × 10¹⁴ cm that we take from Stritzinger et al. (2020b) or of β = 0.7 in Equations (6). We do vary the values of the jets' half opening angle α_j and of the opacity κ_c. We continue with the wide jets that we discussed in Section 2 (Soker 2020a) and scale by α_j = 60°, but we consider narrower jets as well. We scale the opacity by κ_c = 0.1 cm² g⁻¹ but also examine κ_c = 0.05 cm² g⁻¹ and κ_c = 0.3 cm² g⁻¹ to demonstrate the model sensitivity to opacity. The relevant scalings of Equations (6) for SNhunt120 (for one jet) read

$$\begin{eqnarray}&&\begin{array}{l}{t}_{{\rm{j}}}=22.7{\left(\displaystyle \frac{{E}_{{\rm{j}}}}{2\times {10}^{47}{\rm{erg}}}\right)}^{-1/4}\\ \times {\left(\displaystyle \frac{{M}_{{\rm{js}}}}{0.1{M}_{\odot }}\right)}^{3/4}{\left(\displaystyle \frac{{\kappa }_{{\rm{c}}}}{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)}^{1/2}{\rm{d}},\end{array}\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{L}_{{\rm{j}}}=7.3\times {10}^{40}\left(\displaystyle \frac{\sin {\alpha }_{{\rm{j}}}}{0.87}\right)\left(\displaystyle \frac{\beta }{0.7}\right)\\ \times {\left(\displaystyle \frac{{M}_{{\rm{js}}}}{0.1{M}_{\odot }}\right)}^{-3/2}{\left(\displaystyle \frac{{E}_{{\rm{j}}}}{2\times {10}^{47}{\rm{erg}}}\right)}^{3/2}\\ \times {\left(\displaystyle \frac{{\kappa }_{{\rm{c}}}}{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)}^{-1}\left(\displaystyle \frac{{R}_{{\rm{BB}}}}{2\times {10}^{14}\,{\rm{cm}}}\right){{\rm{erg}}{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 8 }$$

As with the spherical shell model, we do not assume the energy of the jets. The input variables to the fitting process are the light curve, the half opening angle of the jets, the opacity and the values of β and sin α_j. We take the radius of the continuum blackbody photosphere from observations. We then substitute the observed ILOT's (or one peak of the ILOT) duration t_j and the energy radiated from one jet-shell interaction L_j in Equations (9) and (10), and solve for the one jet's energy E_j and the mass of the shell that one jet interacts with M_js.

In Table 2 we present six sets of values in the cocoon toy model for SNhunt120. We emphasise that we do not try to fit the shape of the light curves, and only try to explain the amount of radiated energy, the timescale and the maximum luminosity of the peak. In Figure 1 we signify Case 1 by the green line.

The energy of the jets and the mass they interact with vary between the cases. The energy range is E_2j ≃ 4 × 10⁴⁷ erg − 11 × 10⁴⁷ erg. In the spherical-shell model of Section 2.2, the jets' energy is 4:5 × 10⁴⁷ erg. From the cases of Tables 1 and 2, we crudely take the jets' energy for this ILOT to be E_2j(SNhunt120) ≃ 5 × 10⁴⁷ erg.

In the cocoon toy model, the jets interact with a fraction of the shell. After the 'mini-explosion' the assumed spherical interaction zone (cocoon) expands from its initial cocoon-radius a_c,0 = sin α_j β R_BB to larger radii. The mass in the interaction zone is then M_2js > (1 − cos α_j ) M_s. Namely, the shell mass is M_s < M_2js/(1 − cos α_j ). From Table 2 we find the shell masses of the different cases to be M_s (Case 2) < 0.3 M_⊙ to M_s (Case 6) < 2.2 M_⊙. In the spherical shell model the shell mass is 0.7 M_⊙ (Table 1). We crudely take for this ILOT M_s(SNhunt120) ≃ 0.5 − 1 M_⊙, but we note that the model can accommodate somewhat lower shell masses. As we discussed in Section 2.2, the progenitor binary system of this ILOT might have a combined mass of M₁ + M₂ ≈ 10 M_⊙.

Because at discovery AT 2014ej was already in its decline from the first peak in its light curve, we only try to fit the maximum luminosity and the radiated energy of the second peak. In Figure 2 we plot the blackbody light curve of AT 2014ej by the thick-red line, as Stritzinger et al. (2020a) estimate (their fig. 4). The maximum luminosity of the second peak is L_AT = 2.6 × 10⁴¹ erg s⁻¹. In our cocoon toy model this value implies L_j = L_2j/2 = L_AT/2 = 1.3 × 10⁴¹ erg s⁻¹. We examine only the time near maximum luminosity before the break around t ≃ 70 d. We therefore extend the blackbody light curve near maximum (solid-blue line in Fig. 2) by taking a linear fit before t = 42 d and beyond t = 67 d after discovery, in both sides down to L_AT = 1.2 × 10⁴¹, which is the minimum in the light curve between the two peaks. We find that the total energy that this ILOT radiated in its second peak according to our fit (solid-blue line in Fig. 2) is E_rad,AT = 1.1 × 10⁴⁸ erg. We note that in Section 2.3 where we apply the spherical shell model we include the 'hump' at t ≃ 90 d, and therefore the radiated energy is somewhat larger. The hump can result from a weak third jet-launching episode or from mass collision in the equatorial plane.

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We recall that our cocoon toy model does not fit a light curve, but rather fits only the maximum luminosity and total radiated energy (or timescale). We rather assume a symmetric light curve (green line in Fig. 2 for Case 1). We proceed as in Section 3.2 and solve Equations (6) iteratively for several combinations of the input parameters α_j and κ_c. We can scale Equations (6) with typical values for AT 2014ej (Case 1). The scaled equations read

$$\begin{eqnarray}&&\begin{array}{l}{t}_{{\rm{j}}}=31{\left(\displaystyle \frac{{E}_{{\rm{j}}}}{1.14\times {10}^{48}{\rm{erg}}}\right)}^{-1/4}\\ \times {\left(\displaystyle \frac{{M}_{{\rm{js}}}}{0.46{M}_{\odot }}\right)}^{3/4}{\left(\displaystyle \frac{{\kappa }_{{\rm{c}}}}{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)}^{1/2}{\rm{d}},\end{array}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{L}_{{\rm{j}}}=6.9\times {10}^{40}\left(\displaystyle \frac{\sin {\alpha }_{{\rm{j}}}}{0.5}\right)\left(\displaystyle \frac{\beta }{0.7}\right)\\ \times {\left(\displaystyle \frac{{M}_{{\rm{js}}}}{0.46{M}_{\odot }}\right)}^{-3/2}{\left(\displaystyle \frac{{E}_{{\rm{j}}}}{1.14\times {10}^{48}{\rm{erg}}}\right)}^{3/2}\\ \times {\left(\displaystyle \frac{{\kappa }_{{\rm{c}}}}{0.1{{\rm{cm}}}^{2}{{\rm{g}}}^{-1}}\right)}^{-1}\left(\displaystyle \frac{{R}_{{\rm{BB}}}}{2.5\times {10}^{14}\,{\rm{cm}}}\right){{\rm{erg}}{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 10 }$$

We examine four cases with different values of α_j and κ_c that we summarise in Table 3.

We find that we can better fit the second peak in the light curve of AT 2014ej with moderately wide jets α_j ≃ 20 − 30°. Fitting with wide jets does not yield acceptable results. For the parameters we list in Table 3, the jets' energy range is E_2j ≃ 1.14 × 10⁴⁸ − 3 × 10⁴⁸ erg. In the spherical shell model for the second peak, we found this energy to be 1.6 × 10⁴⁸ erg (Table 1). We take the jets' energy for this ILOT to be E_2j(AT 2014ej) ≈ 1.5 × 10⁴⁸ erg. For jet velocity of v_j = 1000 km s⁻¹ the mass in the jets is then M_2j ≃ 0.15 M_⊙.

We proceed as in Section 3.2 to put an upper limit on the shell mass M_s < M_2js/(1 − cos α_j ). We calculate from Table 3 M_s < 2 M_⊙, 1 M_⊙, 3.7 M_⊙ and 1.2M_⊙ for Cases 1, 2, 3 and 4 respectively. In the spherical shell model we crudely estimate (Table 1) the shell mass to be M_s ≈ 1.5 M_⊙. We take the slow shell mass for this ILOT to crudely be M_s(AT 2014ej) ≈ 1 − 2 M_⊙. If this shell mass holds, then the progenitor binary system of this ILOT cannot be a low mass system, and requires the combined mass to be M₁ + M₂ > 5 M_⊙, but more likely M₁ + M₂ ≳ 10 M_⊙.