  It is known that the dipole moments of the neutrino lead to important
astrophysical and cosmological effects. In this regard, within the context of a
$U(1)_{B-L}$ model, we develop and present novel analytical formulas to assess
the effects of the anomalous magnetic moment and electric dipole moment of the
neutrino on the stellar energy loss rates through some common physical process
of pair-annihilation $e^+e^-\to(\gamma, Z, Z^{\prime})\to\nu\bar\nu$. Our
results show that the stellar energy loss rates strongly depends on the
effective magnetic moment of the neutrino, but also on the parameters which
characterize the adopted $U(1)_{B-L}$ model.
