Neutrino Transport in Core Collapse Supernovae 55
This leads to the following finite difference representation of the angular
aberration term in the Boltzmann equation (2):
1
w
j
(A
i
,k
+ B
i
,j
−dj, k
/ζ
j
−dj
)
w
j
−dj
µ
j
− µ
j
−dj
ζ
j
−dj
µ
j
−dj
F
i
,j
−dj, k
− (A
i
,k
+ B
i
,j
,k
/ζ
j
)
w
j
µ
j
+dj
− µ
j
ζ
j
µ
j
F
i
,j
,k
. (38)
We apply the aberration corrections with dj =+1forµ ≤ 0anddj = −1for
µ>0. This is not upwind differencing and, therefore, runs the risk of produc-
ing negative neutrino distribution functions. However, there are three reasons
to accept this shortcoming: (i) The angular aberration correction is generally
small, with the exception of aberration in the vicinity of strong shocks with
large velocity gradients. (ii) As long as the angle-integrated neutrino density
is positive, transiently negative contributions from backward directions on the
Gaussian angular grid (µ<0) allow the computer representation of strongly
forward peaked neutrino fluxes that would not be representable with positive
neutrino distribution functions on a grid with limited angles µ
max
< 1. (iii)
The chosen direction of dj guarantees that no neutrinos are shifted off the
grid. This is a prerequisite for number and energy conservation.
4 The General Case: The Multidimensional Neutrino
Transport Equations
As the previous sections illustrate, one of the greatest challenges encoun-
tered when developing the numerical methods to obtain a physically reliable
solution of the neutrino transport equations in core collapse supernovae is to
achieve simultaneous lepton number and energy conservation or balance,the
former for massless neutrinos and Newtonian gravity, where lepton number is
conserved and where the concept of a conserved energy can be defined in mul-
tiple spatial dimensions, the latter for the general relativistic case in two and
three spatial dimensions. The first step in meeting this challenge is to derive
the conservative lepton number and energy conservation/balance equations,
which at a more fundamental level means finding the corresponding evolved
variables that naturally lend themselevs to a conservative decription of the
neutrino radiation field. As we will see, both the conservation/balance equa-
tions and the fundamental variables can be extracted from the underlying
kinetic theory at a point in the theory where the fundamental physical laws
are manifestly conservative [20].
The distribution function f represents the density of neutrinos in phase
space. The phase space for neutrinos of definite mass m, M
m
, is filled with
trajectories
x
µ
(λ),p
ˆ
i
(λ)
, or “states”. As a collection of neutrinos evolves,
the number of neutrinos in each state changes due to collisions. If one consid-
ers a 6-dimensional hypersurface Σ in M
m
, the ensemble-averaged number