This project is funded by the NSF through grant PHY and the Universities of JINA. The Joint Institute for Nuclear Astrophysics Electron Capture Rates for Neutron Star Crusts. Ana Becerril-Reyes, Sanjib S. Gupta and Hendrik Schatz. Comparison between the two compilations of electron capture rates: We start by comparing the EC rates at the lowest values of temperature and density (T9=0.01 and Log 10 (  Ye) = 1.0). Then we compare rates for higher values of density, while keeping T9 constant. Below are shown the absolute values of for T9=0.01 and Log 10 (  Ye) = 1.0, 8.0, 9.0 and The color code for these plots is as follows: R= R  1.0 Thick black borders denote stable nuclei. Comparison between the two compilations of electron capture rates: We start by comparing the EC rates at the lowest values of temperature and density (T9=0.01 and Log 10 (  Ye) = 1.0). Then we compare rates for higher values of density, while keeping T9 constant. Below are shown the absolute values of for T9=0.01 and Log 10 (  Ye) = 1.0, 8.0, 9.0 and The color code for these plots is as follows: R= R  1.0 Thick black borders denote stable nuclei. EC in crusts of accreting neutron stars (models of crust evolution)  Even at low accretion rates of M solar yr -1 a neutron star can accrete enough material from the secondary to replace its entire crust with ashes of H/He burning not in NSE. Rising electron chemical potential with density as the ashes are pushed deeper in the crust will switch on energetically unfavorable EC transitions. This lowers nuclear charge and generates heat.  Threshold effects at low temperature cannot be captured accurately in tables. Our analytic implementation is fast enough to be used in real-time inside a reaction network.  This implementation is not susceptible to low-temperature inaccuracies due to Fermi-Dirac distribution shape (as Gauss-Laguerre quadrature schemes are in the grid region T 9 = when compared to trapezoidal rule schemes). Inaccuracies are only introduced because we evaluate distortion of electron wave function at only one “effective” electron energy. Most approximations ignore the Coulomb Correction altogether by setting it to unity over the integration range. By retaining an effective correction we retain the effects of a varying nuclear radius on the phase space (important for a rate compilation sufficiently global over the nuclear chart). EC in crusts of accreting neutron stars (models of crust evolution)  Even at low accretion rates of M solar yr -1 a neutron star can accrete enough material from the secondary to replace its entire crust with ashes of H/He burning not in NSE. Rising electron chemical potential with density as the ashes are pushed deeper in the crust will switch on energetically unfavorable EC transitions. This lowers nuclear charge and generates heat.  Threshold effects at low temperature cannot be captured accurately in tables. Our analytic implementation is fast enough to be used in real-time inside a reaction network.  This implementation is not susceptible to low-temperature inaccuracies due to Fermi-Dirac distribution shape (as Gauss-Laguerre quadrature schemes are in the grid region T 9 = when compared to trapezoidal rule schemes). Inaccuracies are only introduced because we evaluate distortion of electron wave function at only one “effective” electron energy. Most approximations ignore the Coulomb Correction altogether by setting it to unity over the integration range. By retaining an effective correction we retain the effects of a varying nuclear radius on the phase space (important for a rate compilation sufficiently global over the nuclear chart). Where we are now:  EC rates successfully implemented in Neutron Star Crust simulation. (Gupta, Brown, Schatz, Möller, Kratz. TBP). Where we are headed:  Calculating rates with excited states in parent nuclei.  Using these rates in high temperature (T9), high density (Log 10 (  Ye)) conditions in core – collapse supernova simulations. Where we are now:  EC rates successfully implemented in Neutron Star Crust simulation. (Gupta, Brown, Schatz, Möller, Kratz. TBP). Where we are headed:  Calculating rates with excited states in parent nuclei.  Using these rates in high temperature (T9), high density (Log 10 (  Ye)) conditions in core – collapse supernova simulations. Analytic formulation of rate from state “I” in parent to state “J” in daughter: R=(ln2*f)/(ft_value for transition from I to J) Phase space factor: f(T 9,u,w,q) = G eff * F u,w,q=electron chemical potential, capture threshold, capture q-value in m e c 2 units (electron rest mass) Rising “u” in NS crust allows electrons to overcome unfavorable capture thresholds. For pre-threshold captures important in NS crusts: F(u < w,w = 1) = (w 4 +2qw 3 +q 2 w 2 )(T 9 /5.93)f 1 (z)+(4w 3 +6qw 2 +2q 2 w) (T 9 /5.93) 2 f 2 (z) + (12w 2 +12qw+2q 2 )(T 9 /5.93) 3 f 3 (z)+(24w+12q) (T 9 /5.93) 4 f 4 (z)+24(T 9 /5.93) 5 f 5 (z) For pre-threshold when q < -1 : F(u < w,w = - q) = 2w 2 (T 9 /5.93) 3 f 3 (z)+12w (T 9 /5.93) 4 f 4 (z)+ 24(T 9 /5.93) 5 f 5 (z) Analytic formulation of rate from state “I” in parent to state “J” in daughter: R=(ln2*f)/(ft_value for transition from I to J) Phase space factor: f(T 9,u,w,q) = G eff * F u,w,q=electron chemical potential, capture threshold, capture q-value in m e c 2 units (electron rest mass) Rising “u” in NS crust allows electrons to overcome unfavorable capture thresholds. For pre-threshold captures important in NS crusts: F(u < w,w = 1) = (w 4 +2qw 3 +q 2 w 2 )(T 9 /5.93)f 1 (z)+(4w 3 +6qw 2 +2q 2 w) (T 9 /5.93) 2 f 2 (z) + (12w 2 +12qw+2q 2 )(T 9 /5.93) 3 f 3 (z)+(24w+12q) (T 9 /5.93) 4 f 4 (z)+24(T 9 /5.93) 5 f 5 (z) For pre-threshold when q < -1 : F(u < w,w = - q) = 2w 2 (T 9 /5.93) 3 f 3 (z)+12w (T 9 /5.93) 4 f 4 (z)+ 24(T 9 /5.93) 5 f 5 (z) G eff = Coulomb Correction = (p/w)F C (Z,A,w) evaluated at effective electron energy w=w eff extracted from w eff 2 (w eff +q) 2 =5.93*F/(T 9 f 1 (z)). P = electron momentum, Z = nuclear charge of captor F C (Z,A,w)= Distortion of electron wave function (for a given electron energy w) due to nuclear charge and finite size of nucleus. F= analytic expression above G eff = Coulomb Correction = (p/w)F C (Z,A,w) evaluated at effective electron energy w=w eff extracted from w eff 2 (w eff +q) 2 =5.93*F/(T 9 f 1 (z)). P = electron momentum, Z = nuclear charge of captor F C (Z,A,w)= Distortion of electron wave function (for a given electron energy w) due to nuclear charge and finite size of nucleus. F= analytic expression above Where f n (z) is a generalization of the Logarithm function: f 1 (z)=ln(1+z) f n (z)=  [ {(-1) k-1 z k }/k n ] (k=1,…N<300 for convergence when n<6) and z=exp{-(5.93|w-u|)/ T 9 } < 1 Where f n (z) is a generalization of the Logarithm function: f 1 (z)=ln(1+z) f n (z)=  [ {(-1) k-1 z k }/k n ] (k=1,…N<300 for convergence when n<6) and z=exp{-(5.93|w-u|)/ T 9 } < 1 Comparison of EC rates to those calculated by FFN FULLER, FOWLER & NEWMAN (FFN) GUPTA & MÖLLER (G&M) Spherical Independent Particle Model. Experimental input (gs->gs or gs-> low-lying forbidden transitions) included. Excited states of parent included (required for Core-Collapse Supernovae, not for Neutron Star Crusts) Nilsson Model. Allowed (GT) only. Ground state of parent only (thermal population effects in Neutron Star Crusts very mild). No quenching of strength.Residual interactions using QRPA (Peter Möller). Neutron star EXO (blue sphere) is part of a binary star system, and its neighboring star (yellow-red sphere) supplies the fuel for the thermonuclear bursts. (Image Credit: NASA) Results of comparisons At Log(rhoYe)=1.0:  EC Rates for neutron rich nuclei agree fairly well between the two compilations.  For proton rich nuclei differences may arise due to low lying structure (experimental data vs. QRPA). At higher values of Log 10 (  Ye) (e.g. 8.0, 9.0, 10.0) a larger fraction of strength in daughter nuclei is accessed. Thus, the observed changes in the calculated rates may be due to:  In FFN deformation for neutron rich nuclei is not taken into account, but it is in (G & M). Therefore, calculated structure is very different.  For rates in which the initial nucleus is even-even and the final is an odd-odd, g.s.  g.s. may not be allowed via (GT) transition.  Gupta & Möller calculations do not include experimental input. Results of comparisons At Log(rhoYe)=1.0:  EC Rates for neutron rich nuclei agree fairly well between the two compilations.  For proton rich nuclei differences may arise due to low lying structure (experimental data vs. QRPA). At higher values of Log 10 (  Ye) (e.g. 8.0, 9.0, 10.0) a larger fraction of strength in daughter nuclei is accessed. Thus, the observed changes in the calculated rates may be due to:  In FFN deformation for neutron rich nuclei is not taken into account, but it is in (G & M). Therefore, calculated structure is very different.  For rates in which the initial nucleus is even-even and the final is an odd-odd, g.s.  g.s. may not be allowed via (GT) transition.  Gupta & Möller calculations do not include experimental input. Z Log 10 (  Ye) = 8.0 N Log 10 (  Ye) = 1.0 N Z Log 10 (  Ye) = 10.0 Z N Log 10 (  Ye) = 9.0 N Z National Superconducting Cyclotron Laboratory. Michigan State University.