Questions and Probems

Matter inside protoneutron stars Hydrostatic equilibrium in the protoneutron star: Rough estimate of the central pressure is: Note that for neutron stars special and general relativistic corrections are important. This modifies the structure equation. The correct equation for structure in this case is (TOV eqn.), To obtain the structure of a protoneutron star we need to specify the equation of state: Estimate the speed of sound at the center of a neutron star whose density profile is given by

Introducing new degrees of freedom: Hyperons Extend the model to include hyperons by coupling them to the mesons. Typically on defines the following ratios. Naïve quark counting rules suggest they are about 2/3. If the binding energy of  in a large symmetric nucleus is -28 MeV, what is the resulting constraint on the hyperon-meson coupling constants ? Binding energy of  hypernuclei provides a constraint. For eg. energy to introduce a  particle is When this is less than  n hyperons appear in the ground state. Three independent chemical potentials are: Baryon number chemical potential:  n Electric Charge chemical potential:  e Lepton number chemical potential:  e

Temperature and specific heat Number of degrees of freedom and interactions affect the specific heat and influence the ambient temperature inside the protoneutron star. For a degenerate multi-component Fermi-gas show that Or

Competition between surface and Coulomb Energy Why do stable nuclei have a preferred size ? Coulomb energy Surface energy If the surface tension between nuclear matter and vacuum is 2 MeV/fm 2 what is the preferred size of a symmetric nucleus ?

Reason nuclei exist Nuclear matter pressure has a Taylor expansion in the charge chemical potential Estimate the critical surface tension above which all nuclei become unstable to fusion. When one tries to make the object bigger than the Debye screening length, electric charges rearrange to minimize the Coulomb energy. This effectively drives to become charge neutral. Surface Energy LossBulk Energy Gain Charge densityCharge susceptibility

Time-scales We shall find later that the electron neutrino mean free paths has the form:  0 ~ 100 cm The nuclear symmetry energy affects the change in lepton number when the neutrino fraction changes. Use the fact that the differential equation has a solution to show that Y   t) = Y (0) Exp(-t//  D ) where Attempt a similar analysis for the cooling time scale assuming that At an intermediate step you should find that the energy transport equation can be written as

Metastability of Protoneutron Stars Neutrinos delay phase transitions that can soften the EoS. This leads to the intriguing possibility that the protoneutron star is hydrostatically stable until its core deleptonizes. Explain the above trend using the equation Prakash et al (1996) Ellis, Prakash and Lattimer (1996)

Pauli-blocking of cross sections The cross section can be written as (neglecting   ) as : Where the function Derive this result for the case when  2 =  4.

Degeneracy matters Cross-section suppressed roughly as (T/E F ) 2 compared to the non-degenerate case. Derive the absorption cross- section for: (i)non-degenerate mater (ii) in the elastic limit.

Neutrino Mean Free Path in a Heterogeneous Phase ’ Neutrino’s with wavelength large compared to the bubbles scatter coherently. Coherent scattering will enhances cross sections. The bubble size is limited by screening length. Typically D ~ 5 fm and we finds Q W ~200 Reddy, Bertsch & Prakash, (2000) Scattering from quark droplets in a first-order quark-hadron transition. What is S q in the formula for the differential cross-section?

CFL matter is opaque. Reddy, Sadszikowski & Tachibana (2003) The radiation of H mesons dominates the opacity. The mean free path for this process is given by The integrations can be done to obtain Derive the above result.