**3. Models for LGRB and SLSNe**

Several observations and theoretical predictions lead to similar origins for both LGRB and SLSNe, given the similarities in their environments, spectra, and energetics [11]. For LGRB, one needs a central engine that can drive a collimated relativistic jet that produces beamed emission [49–51] of energetic *<sup>γ</sup>*-rays. The jet typically has a power of <sup>∼</sup>10<sup>50</sup> erg s−<sup>1</sup> within a narrow opening angle of 0.1 radian. Therefore, models should provide &10<sup>52</sup> ergs of energy in a wider angle of ∼1 radian for LGRB to be accompanied with SN like SN 1998bw and SN 2003dh [52] along with the jet emission. This energy is &10 times the "typical" SN energy. The jet head generally travels at subrelativistic speed with power <sup>∼</sup><sup>3</sup> <sup>×</sup> <sup>10</sup><sup>48</sup> erg s−<sup>1</sup> inside the star, and it takes 8–25 s to reach the surface with power <sup>∼</sup><sup>3</sup> <sup>×</sup> <sup>10</sup><sup>50</sup> erg s−<sup>1</sup> [53]. If the jet is interrupted (there are several interruption scenarios, for example, [54]) or its direction changes in that timespan, then the flow will remain subrelativistic, and therefore will not make a LGRB. Hence, theoretical models suitable for LGRB production need to provide &10<sup>50</sup> erg s−<sup>1</sup> of relativistic, beamed power for &10 s. Considering these constraints, the most acceptable model for LGRB production is the "collapsar" model [13] 2 , where a rapidly rotating stellar core collapses to a BH. The suitable progenitor for LGRB is the metal-poor, rapidly rotating either single massive star [6,57] that undergoes quasi chemically homogeneous evolution due to rotational mixing, or, a massive star in a closely interacting binary [58,59].

SLSNe progenitors are also not yet understood completely, there are several leading theories: (i) continuous energy injection to the SN ejecta by the spin-down of the newly-formed central millisecond magnetar [60–62], (ii) accretion of surrounding ejecta onto the central compact object [63], (iii) SN ejecta-circumstellar medium interaction [64], (iv) radioactive decay of large amount of <sup>56</sup>Ni (20–30 M) produced by pair-instability explosion in very massive stars [12]. Amongst them, the widely accepted theory for SLSNe is the "magnetar" model (for details, see [11] and references therein). In the spirit of the magnetar model, a large number of SLSNe light curves have been analyzed to obtain the distribution of ejecta mass, magnetar spin period and the strength of the magnetic field to reproduce the observables [62,65–67]. The ejecta masses of SLSNe are estimated to be 3.6–40 M that is significantly different from Type Ib/Ic ejecta masses, i.e., strictly >10 M [11]. Estimated magnetar spin period and magnetic fields are 1 to 8 ms and 0.3 to <sup>10</sup> <sup>×</sup> <sup>10</sup><sup>14</sup> G, respectively [11]. Ref. [11] found that even the SLSNe progenitors are rapidly rotating, metal-poor massive stars.

Therefore, it is difficult to theoretically predict which stars will explode as SLSNe with NS remnant and which ones will collapse as BH. There are five commonly used parameters to determine a star's explodability criteria. Stellar core compactness is one such parameter, as mentioned in Section 1, is defined as,

$$\zeta\_{\rm M} = \frac{\rm{M/M}\_{\odot}}{R(M\_{\rm baryon} = \rm{M})/1000 \,\rm{km}} \,\rm{}\tag{1}$$

where *R*(Mbaryon = M) is the radius where the progenitor's core baryon mass Mbaryon = M. *ξ*<sup>M</sup> is the collapse indicator in non-rotating star. [15] found that *ξ*<sup>M</sup> is well determined at the Lagrangian mass coordinate of 2.5 M at the core collapse, where the infall velocity in the core reaches 1000 km/s. Thus *ξ*<sup>M</sup> is typically denoted as *ξ*2.5. Stellar cores with *ξ*2.5 . 3.0–4.5 explode as SLSNe (neutrino winds being the cause of the explosion) leaving behind NS and higher values *o f ξ*2.5 produce BHs [11,16,68]. The other four parameters [11,68] are:

M<sup>4</sup> = *m*(*s* = 4)/M , (2)

*m* is the Lagrangian mass at specific entropy (in the units of *k*B) *s* = 4;

*µ*<sup>4</sup> = *dm*/M *dr*/1000 km *s*=4 , (3)

where *dm* is calculated at M4, in practice, it is set as *dm* = 0.3 M, and *dr* is the change in radius between M<sup>4</sup> and M<sup>4</sup> + *dm*; the dynamo-generated magnetic field strength averaged within the innermost 1.5 M,

$$
\langle B\_{\Phi} \rangle = \frac{\int\_0^{1.5 \mathcal{M}\_{\odot}} B\_{\Phi}(m) dm}{\int\_0^{1.5 \mathcal{M}\_{\odot}} dm} \ ; \tag{4}
$$

the mass averaged specific angular momentum within the innermost mass M,

$$\bar{j}\_{\mathbf{M}} = \frac{\int\_0^{\mathbf{M}} j\_{\mathbf{M}} dm}{\int\_0^{\mathbf{M}} dm} \,. \tag{5}$$

Typical values of <sup>h</sup>*Bφ*<sup>i</sup> are <sup>∼</sup>1014–10<sup>15</sup> G for NS and an order of magnitude higher for collapsars [11]. The average specific angular momentum are ¯*j*1.5 M <sup>∼</sup>10<sup>15</sup> (within the innermost 1.5 M) and ¯*j*5 M <sup>∼</sup>10<sup>16</sup> cm<sup>2</sup> s −1 (within the innermost 5 M) for NS and BH progenitors, respectively [11].
