Delayed Jet Breakouts from Binary Neutron Star Mergers

Short gamma-ray bursts (sGRBs) are a class of gamma-ray bursts (GRBs) whose duration is less than 2 s (see Nakar 2007; Berger 2014, for reviews). They are believed to be powered by relativistic jets launched from compact binary mergers (Eichler et al. 1989). This model is strongly supported by the detection of the gravitational waves (GWs) from the merging binary neutron star (NS; GW170817; Abbott et al. 2017) and by the extensive follow-ups of electromagnetic counterparts, especially very long baseline interferometry (VLBI) observations (Ghirlanda et al. 2018; Mooley et al. 2018).

Some binary–NS mergers may fail to produce sGRBs even if they launch relativistic jets. In order to produce an sGRB, the relativistic jet should break out of the matter ejected by the binary coalescence (Murguia-Berthier et al. 2014; Nagakura et al. 2014). When the ejecta are too massive or the jet opening angle is too large, the jet is choked and fails to emit prompt γ-rays. Choked-jet events are supported by a threshold timescale to produce sGRBs that appears in the duration distribution of sGRBs (Moharana & Piran 2017). Furthermore, the observed binary–NS merger rate larger than the local sGRB rate also suggests choked events.

In addition to prompt γ-rays, some (or most) sGRBs show long-lasting high-energy emissions (Kisaka et al. 2017). They are classified as an extended emission with the duration of ∼10^2–3 s (Norris & Bonnell 2006) and a plateau emission with ∼10^4–5 s (Gompertz et al. 2013, 2014). Because it is difficult to explain them using the standard afterglow theory (Ioka et al. 2005), their origins are attributed to prolonged central-engine activities that launch jets or outflows (Metzger et al. 2008; Nakamura et al. 2014; Kisaka & Ioka 2015).

Even if a prompt jet is choked, a late jet may penetrate ejecta.⁷ Late-time jets can be more powerful than prompt ones because some extended emissions have larger amounts of energy than that of prompt emissions (e.g., Perley et al. 2009). The ejecta's expansion also helps the late jet to break out by reducing the ejecta density. Hereafter, we refer to this scenario as "delayed jet breakout." In Figure 1, we show a schematic picture. We calculate the propagation of the late-time jet in the ejecta and find that the delayed jet breakout is realized with a typical late-time engine activity. In such an event, we cannot detect prompt γ-rays because the prompt jet is choked. Instead, a late-time jet breaks out of ejecta ≳10^1–2 s after the merger, and produces extended and plateau emissions. This can be observed as a soft-long GRB. We also discuss the event rate of the delayed jet breakouts and argue that they might have been observed by the Monitor of All-sky X-ray Image (MAXI).

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We calculate the jet propagation in the ejecta of a binary–NS merger by using a semi-analytical formula along Bromberg et al. (2011; see also Margalit et al. 2018). A jet launched from a central engine collides with ejecta and produces a jet head and cocoon. The cocoon surrounds the jet and collimates it (see below). The jet head velocity is given by the momentum balancing at the head as (Matzner 2003; Bromberg et al. 2011)

$$\begin{eqnarray}&&{\beta }_{{\rm{h}}}=\displaystyle \frac{{\tilde{L}}^{1/2}{\beta }_{{\rm{j}}}+{\beta }_{{\rm{a}}}}{{\tilde{L}}^{1/2}+1},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\tilde{L}\simeq \displaystyle \frac{{L}_{{\rm{j}}}}{{{\rm{\Sigma }}}_{{\rm{j}}}{\rho }_{{\rm{a}}}{{\rm{\Gamma }}}_{{\rm{a}}}^{2}{c}^{3}},\end{eqnarray} \tag{ 2 }$$

where β_j ≃ 1, β_a, L_j, Σ_j, ρ_a, ${{\rm{\Gamma }}}_{{\rm{a}}}={(1-{\beta }_{{\rm{a}}}^{2})}^{-1/2}$, and c are the velocity of the jet material, ejecta velocity, one-sided jet luminosity,⁸ jet cross section, ejecta density, ejecta's Lorentz factor, and speed of light, respectively. The ejecta's quantities are evaluated at the head.

We assume that the ejecta are homologous and have a power-law density profile (Hotokezaka et al. 2013; Nagakura et al. 2014). Even after a prompt jet is choked, although the profile is modified from the original one, these assumptions may hold. Note that if a prompt jet succeeds in breaking out, it produces a cavity and a late jet (or even a spherical outflow) easily emerges from the ejecta (see also Section 3.1). For homologous ejecta, the velocity is given by ${\beta }_{{\rm{a}}}=({R}_{{\rm{h}}}/{R}_{\mathrm{ej}}){\beta }_{\mathrm{ej}}$, where R_h, ${R}_{\mathrm{ej}}={\beta }_{\mathrm{ej}}c(t+{t}_{\mathrm{lag}})$, and β_ej are the jet head position and the radius and velocity of ejecta edge, respectively. We set the origin of time t as the jet-launching time, which is t_lag after the merger. The density is given by

$$\begin{eqnarray}&&{\rho }_{{\rm{a}}}=\displaystyle \frac{{M}_{\mathrm{ej}}}{4\pi {R}_{\mathrm{ej}}^{3}}f{\left(\displaystyle \frac{{R}_{{\rm{h}}}}{{R}_{\mathrm{ej}}}\right)}^{-k}\,\,\mathrm{for}\,\,{R}_{\mathrm{in}}\leqslant R\leqslant {R}_{\mathrm{ej}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}f=\left\{\begin{array}{ll}\tfrac{3-k}{1-{\left(\tfrac{{\beta }_{\mathrm{esc}}}{{\beta }_{\mathrm{ej}}}\right)}^{3-k}} & k\ne 3,\\ \tfrac{1}{\mathrm{ln}({\beta }_{\mathrm{ej}}/{\beta }_{\mathrm{esc}})} & k=3,\end{array}\right.\end{eqnarray} \tag{ 4 }$$

where ${M}_{\mathrm{ej}}$ is the ejecta mass. The inner boundary is set by the innermost unbound ejecta at the jet launch as

$$\begin{eqnarray}&&{\beta }_{\mathrm{esc}}={\left(\displaystyle \frac{{{GM}}_{{\rm{c}}}{\beta }_{\mathrm{ej}}}{{c}^{2}{R}_{\mathrm{ej}}}\right)}^{1/3}\simeq 0.023{\left(\displaystyle \frac{{t}_{\mathrm{lag}}}{{\rm{s}}}\right)}^{-1/3}{\left(\displaystyle \frac{{M}_{{\rm{c}}}}{2.6{\text{}}{M}_{\odot }}\right)}^{1/3},\end{eqnarray} \tag{ 5 }$$

where G and M_c are the gravitational constant and the merger-remnant mass, respectively.

The cocoon pressure determines whether the jet is collimated or conical. The jet cross section is given as

$$\begin{eqnarray}{{\rm{\Sigma }}}_{{\rm{j}}}=\left\{\begin{array}{ll}\pi {\theta }_{{\rm{j}}}^{2}{R}_{{\rm{h}}}^{2} & \mathrm{conical}\ \mathrm{jet},\\ \displaystyle \frac{{L}_{{\rm{j}}}{\theta }_{{\rm{j}}}^{2}}{4{{cP}}_{{\rm{c}}}} & \mathrm{collimated}\ \mathrm{jet}.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

The cocoon pressure is given by

$$\begin{eqnarray}&&{P}_{{\rm{c}}}=\displaystyle \frac{{E}_{{\rm{c}}}}{3{V}_{{\rm{c}}}}=\displaystyle \frac{\int {L}_{{\rm{j}}}(1-{\beta }_{{\rm{h}}}){dt}}{\pi {R}_{{\rm{c}}}^{2}{R}_{{\rm{h}}}},\end{eqnarray} \tag{ 7 }$$

where the cocoon is radiation-pressure dominated and conical with a height R_h and radius R_c. The cocoon radius is obtained by integrating the cocoon's lateral-expansion velocity of (Begelman & Cioffi 1989)

$$\begin{eqnarray}&&{\beta }_{{\rm{a}}}=\sqrt{\displaystyle \frac{{P}_{{\rm{c}}}}{{\bar{\rho }}_{{\rm{a}}}{c}^{2}}},\end{eqnarray} \tag{ 8 }$$

where ${\bar{\rho }}_{{\rm{a}}}$ is the cocoon's mean density. When a converging position of the jet's collimation shock (Komissarov & Falle 1997)

$$\begin{eqnarray}&&\hat{z}=\sqrt{\displaystyle \frac{{L}_{{\rm{j}}}}{\pi {{cP}}_{{\rm{c}}}}}\end{eqnarray} \tag{ 9 }$$

is lower than the jet head ${R}_{{\rm{h}}}\gtrsim \hat{z}$, the jet is collimated.

We integrate the above equations numerically and obtain the jet breakout time t_br for various constant jet luminosities. Because Equation (1) overestimates the jet head velocity for $\tilde{L}\lesssim 1$ compared with numerical simulations (Mizuta & Ioka 2013; Harrison et al. 2018), we correct Equation (1) in line with Harrison et al. (2018). We also modify the collimation condition to ${R}_{{\rm{h}}}\gtrsim \hat{z}/2$ to get a continuous jet cross section.

In Figure 2, we show the result. Each thick red curve shows the breakout time for each lag time. The other parameters are fixed as β_ej = 0.3, θ_j = 15° ≃ 0.26 rad, ${M}_{\mathrm{ej}}={10}^{-2}\,{\text{}}{M}_{\odot }$, and k = 2. We convert the jet luminosity to the radiation luminosity by adopting an efficiency of _γ = 0.1 as ${L}_{\gamma ,\mathrm{iso}}={\epsilon }_{\gamma }{L}_{{\rm{j}},\mathrm{iso}}$. The ejecta velocity and mass are motivated by numerical simulations (Hotokezaka et al. 2013) and the observations of the macronova in GW170817 (e.g., Coulter et al. 2017; Utsumi et al. 2017). The opening angle is based on the observations of sGRBs (Fong et al. 2015), while the observed value may be different from the jet-injection angle. The index k = 2 is relevant for wind-like ejecta and a larger indices give shorter breakout times. The thin red curve shows the result for conical jets with t_lag = 1 s, which give conservative (longer) breakout times. The emission timescale t_em and isotropic luminosity of observed sGRBs' emissions are plotted. As the observed sGRBs have successful prompt jets, we can regard the observed emission timescales as engine-working timescales, which ensure that the engine activity (jet launching) duration is long enough for a delayed breakout, t_engine ≳ t_em.

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For a large jet luminosity (e.g., ${L}_{{\rm{j}},\mathrm{iso}}\gtrsim {10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ for t_lag = 1 s), the breakout time is smaller than the lag time t_br ≲ t_lag and insensitive to the jet luminosity. This is because a large jet luminosity gives a large jet parameter $\tilde{L}\gtrsim 1$ and a jet head velocity becomes almost independent of the jet luminosity β_h ∼ 1. The breakout time is evaluated by equating the jet head radius ${\beta }_{{\rm{h}}}{ct}$ and ejecta radius ${\beta }_{\mathrm{ej}}c(t+{t}_{\mathrm{lag}})$ as (Murguia-Berthier et al. 2014),

$$\begin{eqnarray}&&{t}_{\mathrm{br}}\sim \displaystyle \frac{{\beta }_{\mathrm{ej}}{t}_{\mathrm{lag}}}{{\beta }_{{\rm{h}}}-{\beta }_{\mathrm{ej}}}.\end{eqnarray} \tag{ 10 }$$

With β_h = 1 and β_ej = 0.3, this equation reasonably reproduces our result as ${t}_{\mathrm{br}}\simeq 0.4\,{\rm{s}}\,{t}_{\mathrm{lag},0}$. Hereafter, we use the convention ${Q}_{x}=Q/{10}^{x}$ (cgs). A shorter breakout time than a lag time enables us to regard the envelope as static. In particular, the jet head velocity is constant for the index of k = 2, which we assumed to derive Equation (10).

For a small jet luminosity, the jet breakout time gets longer than Equation (10) due to a small jet head velocity. After the lag time, the expansion of ejecta affects the jet head dynamics by reducing the ejecta density and accelerating the jet head (see Equation (2)). A much longer breakout time than the lag time is inversely proportional to the jet luminosity ${t}_{\mathrm{br}}\propto {L}_{{\rm{j}},\mathrm{iso}}^{-1}$. There is a critical energy for a jet to break out of ejecta (Duffell et al. 2018). For a conical jet, this energy is simply given by the ejecta energy ${M}_{\mathrm{ej}}{(c{\beta }_{\mathrm{ej}})}^{2}/2\gtrsim {E}_{{\rm{j}},\mathrm{iso}}\sim {L}_{{\rm{j}},\mathrm{iso}}t$, which reasonably reproduces our result $t\gtrsim {10}^{2}\,{\rm{s}}\,{M}_{\mathrm{ej},-2}{\beta }_{\mathrm{ej},-0.5}^{2}{L}_{\gamma ,\mathrm{iso},48}^{-1}$. For a collimated jet with a small ejecta mass in front of the jet head, the required energy is smaller. In the Appendix, we derive an analytical scaling law (Equation (26) and the black line in Figure 2). Note that unless the ejecta expansion is taken into account precisely, the breakout time is significantly overestimated except for the parameter dependence (compare Kimura et al. 2018).

In particular, the jet breakout time for a small jet luminosity should be compared with emission timescales of extended (${t}_{\mathrm{em}}\sim {10}^{2\mbox{--}3}\,{\rm{s}}$) and plateau emissions (${t}_{\mathrm{em}}\sim {10}^{4\mbox{--}5}\,{\rm{s}}$). These emission times are longer than the required breakout time and guarantee that if these emissions are produced by jets, the jets can break out of ejecta.

We discuss the observational prospects of the delayed breakout events. In the following, we mainly consider that a late jet producing an extended emission breaks out. By combining a GW observation and follow-ups, we can check whether or not a delayed jet breakout occurs for a binary merger. First, such a combination tells us whether or not the event is on-axis (Abbott et al. 2017; Finstad et al. 2018; Mandel 2018). For an on-axis event, a detection of prompt γ-rays tells us the fate of its prompt jet. If we detect not a prompt emission but an extended (plateau) emission-like signature i.e., a flat light curve up to ∼10^2–3 s (10^4–5 s) and an abrupt shut down, it strongly supports the theory that the late-time jet does punch out a hole in the ejecta. Therefore, we should set X-ray detectors to the merger event regardless of whether or not prompt γ-rays are detected. In particular, because plateau emissions last for a very long time, they can be a good target for X-ray detectors such as the Swift X-ray Telescope (XRT) and MAXI (Nakamura et al. 2014; Kisaka et al. 2017).

3.1. A Probe of Late-time Engine Activity in Binary NS Mergers

3.2. Event Rate

We estimate the event rate of the delayed jet breakout. The binary–NS merger rate is evaluated as ${{ \mathcal R }}_{\mathrm{NSM}}\,\simeq {1550}_{-1220}^{+3220}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ by the observation of GW170817 (Abbott et al. 2017). The merger rate for on-axis events is estimated by assuming the jet opening angle to be

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{on}}\simeq \displaystyle \frac{{\theta }_{{\rm{j}}}^{2}}{2}{{ \mathcal R }}_{\mathrm{NSM}}\simeq {54}_{-42}^{+110}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}\,{\left(\displaystyle \frac{{\theta }_{{\rm{j}}}}{0.26\mathrm{rad}}\right)}^{2}.\end{eqnarray} \tag{ 11 }$$

The central value is larger than the local sGRB rate of $\simeq 4.1\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Wanderman & Piran 2015), and supports the hypothesis that many merger events produce choked jets. For the Laser Interferometer Gravitational-Wave Observatory's (LIGO)'s full sensitivity, the detectable range of on-axis binary–NS mergers is d_L ≃ 1.6 × 200 Mpc, where the factor 1.6 accounts for an enhancement of GWs (Kochanek & Piran 1993), and the comoving volume is V_com ≃ 1.1 × 10⁻¹ Gpc³. The on-axis event rate for the observation by LIGO is evaluated as

$$\begin{eqnarray}&&{N}_{\mathrm{on}}={V}_{\mathrm{com}}{{ \mathcal R }}_{\mathrm{on}}\simeq {6.0}_{-4.6}^{+12}\,{\mathrm{yr}}^{-1}\,.\end{eqnarray} \tag{ 12 }$$

The fraction of the delayed-jet-breakout events to the total on-axis events (we denote f_delay) is constrained by the current sky monitors in X-rays and γ-rays. We adopt a luminosity and duration of a late-time jet as ${L}_{{\rm{X}},\mathrm{iso}}={10}^{48}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and t_em = 300 s as fiducial values. A detector with a sensitivity f_sen can be triggered by the jet inside the luminosity distance of ${d}_{{\rm{L}}}={({L}_{{\rm{X}},\mathrm{iso}}/4\pi {f}_{\mathrm{sen}})}^{1/2}\simeq 0.94\,\mathrm{Gpc}\,{L}_{{\rm{X}},\mathrm{iso},48}^{1/2}{f}_{\mathrm{sen},-8}^{-1/2}$. We estimate the detection rate of the extended emissions in delayed jet breakouts by Swift Burst Alert Telescope (BAT) and MAXI Gas Slit Camera (GSC).

BAT has a sensitivity ${f}_{\mathrm{sen}}\sim {10}^{-8}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ and field of view (FOV) 1.4 str. The detection horizon is d_L ≃ 0.94 Gpc (${V}_{\mathrm{com}}\simeq 2.1\,{\mathrm{Gpc}}^{3}$), and the sky coverage is 1.4/4π ≃ 0.11. The detection rate is ${N}_{\mathrm{BAT}}\simeq 0.1\times 0.11\times {V}_{\mathrm{com}}{f}_{\mathrm{delay}}{{ \mathcal R }}_{\mathrm{on}}\,\simeq {1.3}_{-0.98}^{+2.6}\,{\mathrm{yr}}^{-1}\,{f}_{\mathrm{delay}}$, where we take that BAT has ever detected only ∼10% of extended emissions (Kisaka et al. 2017) into account. This implies that some soft-long GRBs detected by BAT may be the extended emissions in the delayed jet breakouts. These events do not accompany supernova signatures.

MAXI has a sensitivity ${f}_{\mathrm{sen}}\sim {10}^{-9}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ for soft bands (2–30 keV) and FOV 7.3 × 10⁻² str (Sugizaki et al. 2011). The horizon and sky coverage are evaluated as d_L ≃ 3.0 Gpc (V_com ≃ 32 Gpc³) and ≃5.8 × 10⁻³, respectively. The detection rate is ${N}_{{MAXI}}\simeq 0.0058\,\times \,{V}_{\mathrm{com}}{f}_{\mathrm{delay}}{{ \mathcal R }}_{\mathrm{on}}\,\simeq {10.0}_{-7.8}^{+21}\,{\mathrm{yr}}^{-1}\,{f}_{\mathrm{delay}}$. This value also suggests that some long GRBs detected by MAXI are extended emissions in delayed-jet events. Actually, some GRBs are detected only by MAXI and their detection rate is ∼5 yr⁻¹ (Serino et al. 2014).⁹ Therefore, MAXI constrains the fraction of delayed events as f_delay ≲ 0.5 for the central value. Interestingly, for the lower value of ${{ \mathcal R }}_{\mathrm{NSM}}$, the fraction is not constrained due to its small event rate. With the on-axis merger rate (Equation (12)), we expect that the rate of delayed jet breakouts coincident with GWs is at most $\sim 1\mbox{--}3\,\,{\mathrm{yr}}^{-1}\,{({\theta }_{{\rm{j}}}/0.26\mathrm{rad})}^{2}$.¹⁰ Future wide-field X-ray monitors such as ISS-Lobster (Camp et al. 2013) and Einstein Probe (Yuan et al. 2015) will detect delayed events or constrain f_delay more tightly. We also remark that newly discovered X-ray transients (Bauer et al. 2017) may be related to delayed events.

In GW170817, although the VLBI observations revealed a relativistic jet with E_j,iso ≳ 10⁵² erg (Mooley et al. 2018; Ghirlanda et al. 2018), this jet is not necessarily the origin of the low-luminosity prompt γ-rays (Matsumoto et al. 2018b). We can argue that a prompt jet is choked and the resulting cocoon produces the γ-rays (Kasliwal et al. 2017; Gottlieb et al. 2018; Lazzati et al. 2018), and that the relativistic jet is originated from a delayed breakout. Actually, some extended emissions show ${L}_{{\rm{X}},\mathrm{iso}}\sim {10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ with t_em ∼ 10² s (Figure 2), which suggests a large jet energy of ∼10⁵² erg. The energetic late jet penetrates the ejecta ∼10 s after the prompt jet is choked, and produces a cocoon with E_c ∼ 10⁵¹ erg. Interestingly, this cocoon's cooling emission reproduces the observed macronova in the first few days (Matsumoto et al. 2018a). Future GW observations will test this possibility.

In a delayed jet breakout, we have a chance to observe the moment that the late jet emerges from the ejecta. When the jet head reaches the ejecta edge, it may produce a shock-breakout emission (Gottlieb et al. 2018). Even if a merger event occurs outside of the FOV of γ-ray telescopes, the detection of this breakout signature may support the delayed jet breakout.

The delayed-breakout events can be a source of neutrinos. A choked prompt jet can be a powerful neutrino emitter (Kimura et al. 2018). Moreover, a delayed jet emits neutrinos efficiently if it has a low Lorentz factor (Kimura et al. 2017). Detections of these neutrinos can constrain the Lorentz factors and the baryon loading of these jets.

Lamb & Kobayashi (2016) proposed another scenario where on-axis binary–NS mergers do not produce γ-rays. They consider that a low-Lorentz-factor prompt jet breaks out of ejecta but does not emit γ-rays due to its compactness, and discuss the detectability of its afterglow. On the other hand, our scenario predicts that an extended or plateau emission accompanies the merger. In particular, a flat light curve and a sudden drop are unique signatures of a central-engine activity.

Finally, we discuss the breakout condition of prompt jets. A comparison of the breakout time (red curves in Figure 2) with the prompt-emission timescale (purple points) suggests that the lag time should be smaller than ${t}_{\mathrm{lag}}\lesssim 1\,{\rm{s}}$ to produce a sGRB (see also Murguia-Berthier et al. 2014). For longer lag times, a jet cannot catch up with the ejecta edge within the engine-working time. Note that an event with a short emission time (${t}_{\mathrm{em}}\ll {t}_{\mathrm{br}}$) does not constrain the lag time because such a burst is produced by a bare breakout ${t}_{\mathrm{engine}}={t}_{\mathrm{br}}+{t}_{\mathrm{em}}\sim {t}_{\mathrm{br}}$

If each merger event has a similar ejecta velocity and a common lag time, a characteristic breakout time is introduced (see Equation (10)). Intriguingly, Moharana & Piran (2017) found a typical jet breakout time ${t}_{\mathrm{br}}\simeq 0.2\mbox{--}0.5\,{\rm{s}}$. In the collapsar scenario, such a timescale is understood as a time for a jet to reach the progenitor's edge whose size may not change significantly among progenitors (Bromberg et al. 2012). However, binary mergers do not have a characteristic size because ejecta expand. A common lag time introduces such a special length $\sim {\beta }_{\mathrm{ej}}{{ct}}_{\mathrm{lag}}$ into the systems. Therefore, the typical breakout time may suggest that there is a favored lag time to produce sGRBs, which may be related to the jet-launch mechanism such as the formation of global magnetic fields or a BH.

We thank Motoko Serino for useful comments on the observations by MAXI. We are also grateful to Peter Mészáros for helpful comments. This Letter is supported by JSPS Overseas Challenge Program for Young Researchers, Grant-in-Aid for JSPS Research Fellow 17J09895 (T.M.), JSPS Oversea Research Fellowship, and the IGC post-doctoral fellowship program (S.S.K.).

We derive analytical formulae for jet propagations in homologously expanding media.

A.1. Conical Jet

For a conical jet, the jet parameter is given by

$$\begin{eqnarray}&&\tilde{L}=\displaystyle \frac{{L}_{{\rm{j}}}}{\pi {\theta }_{{\rm{j}}}^{2}{R}_{{\rm{h}}}^{2}{\rho }_{{\rm{a}}}{c}^{3}}=\displaystyle \frac{{L}_{{\rm{j}}}}{\pi { \mathcal A }{\theta }_{{\rm{j}}}^{2}{c}^{3}}{R}_{{\rm{h}}}^{k-2},\end{eqnarray} \tag{ 13 }$$

where we define ${\rho }_{{\rm{a}}}={ \mathcal A }{R}_{{\rm{h}}}^{-k}$. For $\tilde{L}\lesssim 1$, we can approximate Equation (1) as ${\beta }_{{\rm{h}}}\simeq {\tilde{L}}^{1/2}+{\beta }_{{\rm{a}}}$ and rewrite it as

$$\begin{eqnarray}&&t\displaystyle \frac{d({R}_{{\rm{h}}}/t)}{{dt}}=c{\tilde{L}}^{1/2},\end{eqnarray} \tag{ 14 }$$

where we used ${\beta }_{{\rm{a}}}\simeq {R}_{{\rm{h}}}/t$ for a homologously expanding media at later than $t\gt {t}_{\mathrm{lag}}$. Due to a term β_a, the left-hand side has a different form than that for static media cases. This gives a different numerical coefficient than a static one. By integrating Equation (14), we obtain a formula

$$\begin{eqnarray}&&{R}_{{\rm{h}}}={\left[\displaystyle \frac{{L}_{{\rm{j}}}}{{ \mathcal A }{\theta }_{{\rm{j}}}^{2}c}\displaystyle \frac{1}{\pi }{\left(\displaystyle \frac{4-k}{k-p-2}\right)}^{2}\right]}^{\tfrac{1}{4-k}}{t}^{\tfrac{2}{4-k}}\propto {t}^{\tfrac{2-p}{4-k}},\end{eqnarray} \tag{ 15 }$$

for $2+p\lt k\lt 4$, where the constant p is defined as ${ \mathcal A }\,=\tilde{{ \mathcal A }}{t}^{p}$. In our model, the quantities are k = 2, ${ \mathcal A }={M}_{\mathrm{ej}}f/4\pi {{R}_{\mathrm{ej}}}^{3-k}$, and $p=k-3$, and give

$$\begin{eqnarray}&&{R}_{{\rm{h}}}=7.6\times {10}^{6}\,\mathrm{cm}\,{M}_{\mathrm{ej},-2}^{-1/2}{L}_{{\rm{j}},\mathrm{iso},45}^{1/2}{\beta }_{\mathrm{ej},-0.5}^{1/2}{(t/{\rm{s}})}^{3/2}.\end{eqnarray} \tag{ 16 }$$

By equating this radius with the ejecta edge ${R}_{\mathrm{ej}}\sim {\beta }_{\mathrm{ej}}{ct}$, we obtain the critical luminosity for successful jet breakouts as

$$\begin{eqnarray}&&{L}_{{\rm{j}},\mathrm{iso}}\simeq 1.5\times {10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{M}_{\mathrm{ej},-2}{\beta }_{\mathrm{ej},-0.5}{(t/{\rm{s}})}^{-1}.\end{eqnarray} \tag{ 17 }$$

A.2. Collimated Jet

Next, we consider a collimated jet. At t > t_lag, we denote the cocoon radius as ${R}_{{\rm{c}}}=\xi {\beta }_{{\rm{c}}}{ct}$, where a numerical coefficient ξ is given later. The cocoon pressure is rewritten by using Equations (7) and (8) as

$$\begin{eqnarray}&&{P}_{{\rm{c}}}\simeq \displaystyle \frac{{L}_{{\rm{j}}}t}{3\tfrac{\pi }{3}{R}_{{\rm{c}}}^{2}{R}_{{\rm{h}}}}=\displaystyle \frac{{L}_{{\rm{j}}}}{\pi {\xi }^{2}{\beta }_{{\rm{c}}}^{2}{R}_{{\rm{h}}}{c}^{2}t}={\left(\displaystyle \frac{{L}_{{\rm{j}}}{\bar{\rho }}_{{\rm{a}}}}{\pi {\xi }^{2}{R}_{{\rm{h}}}t}\right)}^{1/2}.\end{eqnarray} \tag{ 18 }$$

The jet parameter is given as

$$\begin{eqnarray}&&\tilde{L}=\displaystyle \frac{{L}_{{\rm{j}}}}{\tfrac{{L}_{{\rm{j}}}{\theta }_{{\rm{j}}}^{2}}{4{{cP}}_{{\rm{c}}}}{\rho }_{{\rm{a}}}{c}^{3}}={\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{ \mathcal A }{\theta }_{{\rm{j}}}^{4}{c}^{4}}\displaystyle \frac{16\varrho }{\pi {\xi }^{2}}\displaystyle \frac{{R}_{{\rm{h}}}^{k-1}}{t}\right)}^{1/2},\end{eqnarray} \tag{ 19 }$$

where we use ${\bar{\rho }}_{{\rm{a}}}=\varrho {\rho }_{{\rm{a}}}$ and $\varrho =3/(3-k)$ for a conical cocoon. By substituting this for Equation (14), we obtain

$$\begin{eqnarray}&&{R}_{{\rm{h}}}={\left[\displaystyle \frac{{L}_{{\rm{j}}}}{{ \mathcal A }{\theta }_{{\rm{j}}}^{4}}\displaystyle \frac{16\varrho }{\pi {\xi }^{2}}{\left(\displaystyle \frac{5-k}{k-2-p}\right)}^{4}\right]}^{\tfrac{1}{5-k}}{t}^{\tfrac{3}{5-k}}\propto {t}^{\tfrac{3-p}{5-k}},\end{eqnarray} \tag{ 20 }$$

for p − 2 < k < 5. The cocoon velocity is given as

$$\begin{eqnarray}&&{\beta }_{{\rm{c}}}\propto {({P}_{{\rm{c}}}/{\bar{\rho }}_{{\rm{a}}})}^{1/2}\propto {({\rho }_{{\rm{a}}}{R}_{{\rm{h}}}t)}^{-1/4}\propto {({t}^{p+1}{R}_{{\rm{h}}}^{1-k})}^{-1/4}\propto {t}^{\tfrac{p+2-k}{k-5}},\end{eqnarray} \tag{ 21 }$$

which gives the coefficient as $\xi =(5-k)/(3-p)$. Finally, we get analytical expressions as

$$\begin{eqnarray}&&{R}_{{\rm{h}}}={N}_{{\rm{s}}}^{\tfrac{5}{5-k}}{\left[\displaystyle \frac{3{L}_{{\rm{j}}}}{{ \mathcal A }{\theta }_{{\rm{j}}}^{4}}\displaystyle \frac{{2}^{4}}{\pi {(3-p)}^{2}}\displaystyle \frac{{(5-k)}^{2}}{(3-k)}{\left(\displaystyle \frac{3-p}{p+2-k}\right)}^{4}\right]}^{\tfrac{1}{5-k}}{t}^{\tfrac{3}{5-k}},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\beta }_{{\rm{h}}} & = & {N}_{{\rm{s}}}^{\tfrac{5}{5-k}}\left[\displaystyle \frac{3{L}_{{\rm{j}}}}{{ \mathcal A }{\theta }_{{\rm{j}}}^{4}}\displaystyle \frac{{2}^{4}{(3-p)}^{3-k}}{\pi }\displaystyle \frac{{(5-k)}^{k-3}}{(3-k)}\right.\\ & & \times \,{\left.{\left(\displaystyle \frac{3-p}{p+2-k}\right)}^{4}\right]}^{\tfrac{1}{5-k}}\displaystyle \frac{{t}^{\tfrac{k-2}{5-k}}}{c},\end{array}\end{eqnarray} \tag{ 23 }$$

where we introduce a correction factor ${N}_{{\rm{s}}}(=0.35\,\mathrm{for}\,\tilde{L}\lt 1)$ given by Harrison et al. (2018). The time dependences are the same as Margalit et al. (2018). We compare these forms with Equations (A2) and (A3) in Harrison et al. (2018). They do not consider the time-dependent ${ \mathcal A }$, which modifies numerical coefficients. Furthermore, in expanding media, the jet head velocity is determined by not only $\tilde{L}$ but also β_a (see Equation (1)). In particular, Equation (14) gives another factor ${[(3-p)/(p+2-k)]}^{4}$.

For our case, the jet head velocity and radius are given by

$$\begin{eqnarray}\begin{array}{rcl}{\beta }_{{\rm{h}}} & = & 3.0\times {10}^{-2}\,{M}_{\mathrm{ej},-2}^{-1/3}{L}_{{\rm{j}},\mathrm{iso},45}^{1/3}{\beta }_{\mathrm{ej},-0.5}^{1/3}{\theta }_{\mathrm{ej},-0.5}^{-2/3}\\ & & \times \,{(t/{\rm{s}})}^{1/3}{\left(\displaystyle \frac{{N}_{{\rm{s}}}}{0.35}\right)}^{5/3},\end{array}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{h}}} & = & 6.8\times {10}^{8}\,\mathrm{cm}\,{M}_{\mathrm{ej},-2}^{-1/3}{L}_{{\rm{j}},\mathrm{iso},45}^{1/3}{\beta }_{\mathrm{ej},-0.5}^{1/3}{\theta }_{\mathrm{ej},-0.5}^{-2/3}\\ & & \times \,{(t/{\rm{s}})}^{4/3}{\left(\displaystyle \frac{{N}_{{\rm{s}}}}{0.35}\right)}^{5/3},\end{array}\end{eqnarray} \tag{ 25 }$$

and the critical luminosity is given by

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{j}},\mathrm{iso}} & = & 5.4\times {10}^{48}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{M}_{\mathrm{ej},-2}{\beta }_{\mathrm{ej},-0.5}^{2}{\theta }_{\mathrm{ej},-0.5}^{2}\\ & & \times \,{(t/{\rm{s}})}^{-1}{\left(\displaystyle \frac{{N}_{{\rm{s}}}}{0.35}\right)}^{-5}.\end{array}\end{eqnarray} \tag{ 26 }$$

It should be noted that the different numerical factors introduce a large difference in the critical luminosity. Actually, Kimura et al. (2018) used the equations in Harrison et al. (2018) and obtained a much larger critical luminosity than Equation (26). The discrepancy between ours and theirs are reasonably attributed to the different numerical factor that they adopted as ${[(3-p)/3]}^{2}{[(p+2-k)/(3-p)]}^{4}\simeq 7\times {10}^{-3}$.