The generating rank of the unitary and symplectic Grassmannians

We prove that the Grassmannian of totally isotropic $k$-spaces of the polar space associated to the unitary group $\mathsf{SU}_{2n}(\mathbb{F})$ ($n\in \mathbb{N}$) has generating rank ${2n\choose k}$ when $\mathbb{F}\ne \mathbb{F}_4$. We also reprove the main result of Blok [Blok2007], namely that the Grassmannian of totally isotropic $k$-spaces associated to the symplectic group $\mathsf{Sp}_{2n}(\mathbb{F})$ has generating rank ${2n\choose k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$

The generating rank of the unitary and symplectic Grassmannians

AbstractWe prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU2n(F) (n∈N) has generating rank (2nk) when F≠F4. We also reprove the main result of Blok (2007) [3], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp2n(F) has generating rank (2nk)−(2nk−2), when Char(F)≠2

Comment on "Pulsar Velocities and Neutrino Oscillations"

In a recent Letter, Kusenko and Segre proposed a new mechanism to explain the observed proper motions of pulsars. Their mechanism was based on the asymmetric neutrino emission induced by neutrino oscillations in the protoneutron star magnetic field. In this note I point out that their estimate of the asymmetry in the neutrino emission is incorrect. A proper calculation shows that their mechanism at least requires a magnetic field of 10**16 G in order to produce the observed average pulsar velocity.Comment: 4 pages, RevTe

Nuclear astrophysics of supernovae

In this paper, I'll give a general introduction to Supernova Theory, beginning with the presupernova evolution and ending with the later stages of the explosion. This will be distilled from a colloquium type of talk. It is necessary to have the whole supernova picture in one's mind's eye when diving into some of its nooks and crannies, as it is quite a mess of contradictory ingredients. We will have some discussion of supernova 1987a, but will keep our discussion more general. Second, we'll look at the infall and bounce of the star, seeing why it goes unstable, what dynamics it follows as it collapses, and how and why it bounces back. From there, we will go on to look at the equation of state (EOS) in more detail. We'll consider the cases T = 0 and T > 0. We'll focus on /rho/ /rho//sub 0/ and the EOS of neutron stars, and whether or not they contain cores of strange matter. There are many things we could discuss here and not enough time. If I had more lectures, the remaining time would focus on two more questions of special interest to nuclear physicists: the electron capture reactions and neutrino transport. If time permitted, we'd have some discussion of the nucleosynthetic reactions in the explosion's debris as well. However, we cannot cover such material adequately, and I have chosen these topics because they are analytically tractable, pedagogically useful, and rather important. 23 refs., 14 figs., 3 tabs

Reply to Comment on "Pulsar velocities and neutrino oscillations"

We have recently proposed an explanation for the birth velocities of pulsars based on neutrino oscillations (hep-ph/9606428). One of the quantities, dN/dT, was evaluated in the approximation of constant chemical potential for the electrons. An alternative approximation based on the assumption Ye=const, used by Qian (astro-ph/9705055), yields a somewhat higher prediction for the magnetic field inside a neutron star. If the same input parameters are used, the two approximations are in reasonable agreement, given the uncertainty in the geometry of the magnetic field and the simplified picture of neutrino emission that comes with the notion of a neutrinosphere.Comment: 2 pages, no figure

Signal for supernova νμ\nu_\mu and ντ\nu_\tau neutrinos in water \v{C}erenkov detectors

We suggest that photons with energies between 5 and 10 MeV, generated by the ($\nu,\nu'p\gamma$) and ($\nu,\nu'n\gamma$) reactions on $^{16}$O, constitute a signal which allows a unique identification of supernova $\nu_\mu$ and $\nu_\tau$ neutrinos in water \v{C}erenkov detectors. We calculate the yield of such $\gamma$ events and estimate that a few hundred of them would be detected in Superkamiokande for a supernova at 10 kpc distance.Comment: 8 pages, RevTex 3.0, figures and text available at http://www.krl.caltech.edu/preprints/MAP.htm

Neutrino Trapping in a Supernova and Ion Screening

Neutrino-nucleus elastic scattering is reduced in dense matter because of correlations between ions. The static structure factor for a plasma of electrons and ions is calculated from Monte Carlo simulations and parameterized with a least squares fit. Our results imply a large increase in the neutrino mean free path. This strongly limits the trapping of neutrinos in a supernova by coherent neutral current interactions.Comment: 9 pages, 1 postscript figure using epsf.st

Spherical collapse of supermassive stars: neutrino emission and gamma-ray bursts

We present the results of numerical simulations of the spherically symmetric gravitational collapse of supermassive stars (SMS). The collapse is studied using a general relativistic hydrodynamics code. The coupled system of Einstein and fluid equations is solved employing observer time coordinates, by foliating the spacetime by means of outgoing null hypersurfaces. The code contains an equation of state which includes effects due to radiation, electrons and baryons, and detailed microphysics to account for electron-positron pairs. In addition energy losses by thermal neutrino emission are included. We are able to follow the collapse of SMS from the onset of instability up to the point of black hole formation. Several SMS with masses in the range $5\times 10^5 M_{\odot}- 10^9 M_{\odot}$ are simulated. In all models an apparent horizon forms initially, enclosing the innermost 25% of the stellar mass. From the computed neutrino luminosities, estimates of the energy deposition by $\nu\bar{\nu}$-annihilation are obtained. Only a small fraction of this energy is deposited near the surface of the star, where, as proposed recently by Fuller & Shi (1998), it could cause the ultrarelativistic flow believed to be responsible for $\gamma$-ray bursts. Our simulations show that for collapsing SMS with masses larger than $5\times 10^5 M_{\odot}$ the energy deposition is at least two orders of magnitude too small to explain the energetics of observed long-duration bursts at cosmological redshifts. In addition, in the absence of rotational effects the energy is deposited in a region containing most of the stellar mass. Therefore relativistic ejection of matter is impossible.Comment: 13 pages, 11 figures, submitted to A&

Electron capture on iron group nuclei

We present Gamow-Teller strength distributions from shell model Monte Carlo studies of fp-shell nuclei that may play an important role in the pre-collapse evolution of supernovae. We then use these strength distributions to calculate the electron-capture cross sections and rates in the zero-momentum transfer limit. We also discuss the thermal behavior of the cross sections. We find large differences in these cross sections and rates when compared to the naive single-particle estimates. These differences need to be taken into account for improved modeling of the early stages of type II supernova evolution

Neutrino Interactions in Hot and Dense Matter

We study the charged and neutral current weak interaction rates relevant for the determination of neutrino opacities in dense matter found in supernovae and neutron stars. We establish an efficient formalism for calculating differential cross sections and mean free paths for interacting, asymmetric nuclear matter at arbitrary degeneracy. The formalism is valid for both charged and neutral current reactions. Strong interaction corrections are incorporated through the in-medium single particle energies at the relevant density and temperature. The effects of strong interactions on the weak interaction rates are investigated using both potential and effective field-theoretical models of matter. We investigate the relative importance of charged and neutral currents for different astrophysical situations, and also examine the influence of strangeness-bearing hyperons. Our findings show that the mean free paths are significantly altered by the effects of strong interactions and the multi-component nature of dense matter. The opacities are then discussed in the context of the evolution of the core of a protoneutron star.Comment: 41 pages, 25 figure