Proc. Neutrino Astrophysics - MPP Theory Group

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scopic asymmetry can build up because the anisotropy of the neutrino flux increases with the
average number of scatterings per neutrino, i.e. with the optical depth of the medium. The
polarization term in the cross section represents a slightly different chance for a neutrino to be
scattered into the direction of the external magnetic field than into the opposite direction, or
means a tiny enhancement of the scattering probability of neutrinos moving into (or opposite
to) the field direction. Formally, the neutrino flux is not only described by a diffusive propagation
mode (proportional to the gradient of the neutrino energy density), but an additional
“advective” vector component in the magnetic field direction (see [17, 18]). The ratio between
advective and diffusive component grows approximately linearly with the scattering optical
depth τ. Physically, this can be understood [15] by recalling that a diffusing particle has
travelled a mean distance d = √ Nλ after N steps with (constant) scattering mean free path
λ. If the particle has a tiny chance P to be scattered into the field direction, the distance
it has propagated along the field after N scatterings is l = PNλ. Therefore the relative
anisotropy increases as l/d ∝ P √ N ∝ τ (for P √ N ≪ 1). The second proportionality holds
because the optical depth is defined as τ = d/λ, thus τ = √ N.
The discussed process would be highly interesting if the magnetic fields required to create
a few per cent anisotropy of the neutrino emission of a neutron star were much smaller
than the very strong fields (> ∼ several 10 15 G) where the neutrino opacity is affected by the
change of the electron phase space distribution. Horowitz and Li [15] have only considered
very simple situations of scattering media to show the fundamental characteristics of the
cumulative process and have disregarded essential complications of neutrino transport in
neutron stars, e.g., neutrino absorption, transport of different neutrino types with opposite
signs of the polarization terms, and the feedback of the transport on the energy distribution
in the neutron star. The question must be asked whether the proposed cumulative process
leads to a global anisotropy of the neutrino emission of the nascent neutron star and how
large this anisotropy can be for a given magnetic field.
The neutron star situation was more realistically investigated by Lai and Qian [18] who estimated
a maximum recoil velocity of about 200km/s for average magnetic fields of ∼ 10 14 G,
provided the magnetic field in a nascent neutron star possesses a strong dipole component.
Their analysis, however, is based on the assumption that neutrinos and antineutrinos, in particular
νe and ¯νe, have the same opacity for interactions with the stellar medium. Therefore
their kicks are only produced by the deleptonization neutrinos which carry away only a minor
fraction (∼ 10%) of the gravitational binding energy of the neutron star which is released by
neutrino emission. This assumption, however, leads to a large underestimation of the recoil
acceleration [19].
Monte Carlo simulations [19] which include neutrino production and absorption, different
opacities and polarizations for different neutrino and antineutrino flavors (νe, ¯νe, and heavy
lepton neutrinos νx and ¯νx), and the evolution of the stellar background in a simplified but reasonable,
approximative description 1 , suggest that a polarization term in the neutrino-nucleon
scattering opacity of κs,pol/κs ∼ 10 −5 can lead to a global asymmetry q of the integrated
neutrino luminosity of about 16% for the one-dimensional (plane) case (Fig. 1). Taking
into account projection effects by multiplying this result with 1/2, the corresponding threedimensional
asymmetry q3D is estimated to about 8%, i.e. q3D ∼ 0.08· � (κs,pol/κs)/10 −5� . This
1 Monte Carlo simulations become extremely expensive and CPU-time consuming for high optical depths.
Instead of considering a neutron star with neutrino optical depth of 10000–100000, a toy model with optical
depth τ = 100 and correspondingly rescaled polarization terms of the neutrino interactions was investigated.
The scaling of the results was tested by comparative computations for τ = 30 and for τ = 300.