with a 90% C.L. Other limits on the mixing angle φ reported in the literature are given in Refs. [26, 28, 38, 39]. 

In Figure 1 we show the emissivity as a function of the degeneration parameter β for several values representative of the mixing angle φ = − 0 . 0016 , 0 , 0 . 0011. We observe that emissivity decreases when β increases, which is due to the reduction in the number of positrons available necessary to cause the collision. Also we see that the emissivity is affected by the φ parameter. There are other effects which may change the emissivity, for example, the radiative corrections at one-loop level. 

To analyze the effects of the φ parameter of the left-right symmetric model on the emissiv- ity Q LRSM ν ¯ ν ( φ, β ) of the neutrinos, in Fig. 2 we show the ratio Q LRSM ν ¯ ν ( φ,β ) Q SM ν ¯ ν ( β ) (ratio of emissivity calculated in the LRSM to the emissivity calculated in the SM [34]), as a function of the φ parameter for several values representative of degeneration parameter β = 0 , 5 , 12. 

According to collapse theories, the full energy loss in a stellar collapse is L$_{ν}$$_{¯}$ ν = Q$_{ν}$$_{¯}$ ν · V$_{star}$ · t$_{col}$ ≈ 10 53 erg , where V$_{star}$ is the volume of a star and t$_{col}$ the collapse time. Therefore, if we assume that t$_{col}$ ∼ 1 − 10 s , the stellar collapse temperature T$_{col}$ ∼ (1 . 1 − 1 . 4) × 10 11 $^{o}$K will be obtained. On the contrary, if we know the exact stellar collapse temperature, the collapse time can be obtained. For instance, the value T$_{col}$ ∼ × 10 12 $^{o}$K corresponds to t$_{col}$ ∼ 10 − 7 s . 

In summary, we have analyzed the effects of the mixing angle φ of a left-right symmetric model on the emissivity of the neutrinos via the process e $^{+}$e − → ν ¯ ν . We find that the emissivity is dependent of the mixing angle φ of the model in the allowed range for this parameter. As expected, in the limit of vanishing φ we recover the expression for the emissivity Q SM ν ¯ ν ( β ) for the SM previously obtained in the literature [29–33]. In addition, the analytical and numerical results for the emissivity have never been reported in the literature before and could be of relevance for the scientific community. 

## Acknowledgments 

This work was supported by CONACyT, SNI and PROMEP (M´ exico). 

