Stars as Laboratories for Fundamental Physics - MPP Theory Group

408 Chapter 11
scattering cross section on nonrelativistic nucleons is given by
σ ≈ G 2 FE 2 ν/π = 1.7×10 −42 cm 2 (E ν /10 MeV) 2 . (11.3)
Nuclear density (ρ 0 ≈ 3×10 14 g cm −3 ) corresponds to a nucleon density
of about 1.8×10 38 cm −3 so that λ ≈ 300 cm for 30 MeV neutrinos. This
yields a diffusion time scale t diff = O(1 s).
In summary, one expects an energy of about 0.5×10 53 erg to be
emitted in each (anti)neutrino degree of freedom over a time scale of
order 1 sec with typical energies of order several 10 MeV.
11.2.2 Energies and Spectra
These global properties of the expected neutrino signal are broadly
confirmed by detailed numerical calculations of neutrino transport. 65
However, there are a number of important “fine points” to keep in
mind. First, the nonelectron neutrino degrees of freedom ν µ,τ and ν µ,τ
have smaller opacities; their energies are too low for charged-current
reactions of the sort ν µ + p → n + µ + because of the large masses of the
µ and τ leptons. These flavors decouple at higher densities and temperatures
than ν e and ν e and so they are emitted with higher average
energies. Equally important, ν e ’s have lower energies than ν e ’s because
the opacities are dominated by ν e + n → p + e − and ν e + p → n + e + ,
respectively, and because there are fewer protons than neutrons. Typically
one finds (Janka 1993)
⎧
⎪⎨ 10−12 MeV for ν e ,
⟨E ν ⟩ = 14−17 MeV for ν e ,
(11.4)
⎪⎩
24−27 MeV for ν µ,τ and ν µ,τ ,
i.e. typically ⟨E νe ⟩ ≈ 2⟨E 3 ν e
⟩ and ⟨E ν ⟩ ≈ 5⟨E 3 ν e
⟩ for the other flavors.
The number fluxes of the nonelectron flavors are smaller than those
of ν e because the energy is found to be approximately equipartitioned
between the flavors: the total E νe +ν e
lies between 1 and 1 of E 3 2 b. Similarly,
the number flux of ν e is larger than that of ν e (the lepton number
is carried away in ν e ’s!) so that, again, the energy is approximately
equipartitioned between ν e and ν e . The total E νe is found to lie between
1
and 1 of E 6 4 b. The SN 1987A observations were almost exclusively sensitive
to the ν e flux. The total E νe inferred from these measurements
65 See, e.g., Burrows and Lattimer (1986); Bruenn (1987); Mayle, Wilson, and
Schramm (1987); Burrows (1988); Janka and Hillebrandt (1989a,b); Myra and
Bludman (1989); Myra and Burrows (1990). For reviews see Cooperstein (1988)
and Burrows (1990a,b).