Angular aberration from observer motion

Motivated by this success, we try the method of characteristics also on the angular aberration corrections. In this case, the characteristic is described by . We write the prefactor of the angular aberration correction in Eq. (21) as a function of the quantities and defined in Eq. () to obtain with ,

$$O_{\mu }=\left( A+B/\zeta \right) \frac{\partial }{\partial \mu }\left[ \zeta \mu F\right] .$$

As before, we try to eliminate the partial derivative with respect to the angle cosine with a time derivative along the characteristic in the angle part of the momentum phase space Liebendoerfer_00. For the quantity , we find the relation

$$\frac{\partial \ln R_{a}}{\alpha \partial t}=A+B/\zeta .$$

With this, we transform from the ``Eulerian'' variable to the ``Lagrangian'' variable . After a multiplication with , the angular aberration correction becomes:

$$0 = \zeta \mu \left[ \left( \frac{\partial F}{\partial t}\right) _{\m... ...a}}{\partial t}\frac{\partial }{\partial \mu }\left[ \zeta \mu F\right] \right]$$

$$= \left( \frac{\partial }{\partial t}\left[ \zeta \mu F\right] \rig... ...times \zeta ^{3/2}R_{a}\frac{\partial }{\partial \mu }\left[ \zeta \mu F\right]$$

$$= \left( \frac{\partial }{\partial t}\left[ \zeta \mu F\right] \rig... ...tial }{\partial t}\left[ \zeta \mu F\right] \right) _{\zeta ^{-1/2}\mu /R_{a}}.$$

The particles initially residing in the angular interval are shifted by angular aberration along characteristics with constant :

$$\left( \frac{\partial }{\partial t}\left[ \left( 1-3\mu ^{2}\right) F\Delta \mu \right] \right) _{\zeta ^{-1/2}\mu /R_{a}}=0.$$ (79)

We have to keep in mind that any correction to the particle propagation direction also affects the energy conservation in the frame of a distant observer. Therefore, it is desirable to construct a numerical implementation of angular aberration that prescribes the changes in the specific luminosity to machine precision. Hence, we evaluate the change of the specific luminosity, , along the characteristic in analogy to Eq. (),

$$\left( \frac{\partial }{\partial t}\left[ \left( 1-3\mu ^{2}... ...1/2}/R_{a}}=\zeta \frac{\partial \ln R_{a}}{\partial t}d\ell .$$ (80)

We identify the bin size with our Gaussian quadrature weights. In the finite difference representation, Eq. () leads to the condition for number conservation

$$F_{i',j',k'}w_{j'}-\left[ \left( F_{i',j',k'}w_{j'}-n^{-}_{i',j',k'}\right) +n^{+}_{i',j'+dj,k'}\right] =0$$

and Eq. () to a prescription for the numerical evolution of the specific luminosity

$$\begin{eqnarray*} F_{i',j',k'}\mu _{j'}w_{j'} & - & \left[ \left( F_{i',j',k'}\m... ...}+B_{i',j',k'}\right) F_{i',j',k'}\mu _{j'}w_{j'}\alpha _{i'}dt. \end{eqnarray*}$$

The direction of the differencing can be chosen by . The change in the neutrino distribution function from angular aberration is then:

$$F_{i',j',k'}=\bar{F}_{i',j',k'}+\left( n_{i',j',k'}^{+}-n_{i',j',k'}^{-}\right) /w_{j'}$$

with

$$\begin{eqnarray*} n^{-}_{i',j',k'} & = & \left( A_{i',k'}+B_{i',j',k'}/\zeta _{j... ...k'}\alpha _{i'}dt\\ n^{+}_{i',j',k'} & = & n^{-}_{i',j'-dj,k'}. \end{eqnarray*}$$

Combined into one update, this leads to the following finite difference representation of the angular aberration term in the Boltzmann Eq. (15):

$$O_{\mu } = \frac{1}{w_{j'}}\left[ \left( A_{i',k'}+B_{i',j'-dj,k'}/\zeta _{j... ...j'-dj}}{\mu _{j'}-\mu _{j'-dj}}\zeta _{j'-dj}\mu _{j'-dj}F_{i',j'-dj,k'}\right.$$

$$- \left. \left( A_{i',k'}+B_{i',j',k'}/\zeta _{j'}\right) \frac{w_{j'}}{\mu _{j'+dj}-\mu _{j'}}\zeta _{j'}\mu _{j'}F_{i',j',k'}\right] .$$ (81)

We apply the aberration corrections with for and for . This is not upwind differencing and, therefore, runs the risk of producing negative particle distribution functions. However, there are two good 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) The chosen direction of guarantees that no particles are shifted off the grid. This is a prerequisite for number and energy conservation. If we calculate the contribution of to the energy evolution, , along the lines of Eq. (), this choice of causes the perfect vanishing of boundary terms in the discrete ``integration by parts'' with respect to the angle cosine. We obtain

$$O^{1-4}_{\mu }:\qquad \sum _{j,k}u_{i+1}\mu _{j'}\left( -A_{... ...',k'}-B_{i',j',k'}\right) F_{i',j',k'}w_{j'}E_{k'}^{3}dE_{k'}.$$ (82)

The second and third terms in the parenthesis cancel exactly with the yet unmatched terms from in Eq. (). Thus, also the cancellations and are guaranteed in Eq. (). However, in failed supernova simulations, the shock recedes at late time to very small radii and becomes very strong because of high infall velocities. We have encountered numerical fluctuations in the neutrino distribution function at the shock position at this late time that are due to this fixed choice of differencing in the angular advection in the aberration terms. They disturb the progress of the simulation with large time steps. In future long term simulations, we will use strict upwind differencing also in the angular aberration terms and allow a deviation from perfect energy conservation in these terms (we still conserve particle number). In section , we demonstrate that the expected energy violations from angular aberration are unlikely to dominate the violations of energy conservation by the adaptive grid or a mismatch in .