Transcript Document

The Joint Institute for Nuclear Astrophysics
Electron Capture Rates for Neutron Star Crusts.
Ana D. Becerril Reyes1,2, Sanjib S. Gupta1 and Hendrik Schatz1,2.
1 National
Superconducting Cyclotron Laboratory, Michigan State University, 2 Dept. of Physics and Astronomy, Michigan State University
EC in crusts of accreting neutron stars
(models of crust evolution)
Artist’s conception of
neutron star EXO 0748676 (blue sphere). It is
part of a binary star
system, and its
neighboring star
(yellow-red sphere)
supplies the fuel for the
thermonuclear bursts.
(Image Credit: NASA)
Even at the low accretion rates of ~10-10 Msolar 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. The rising electron chemical potential
with density as the ashes are pushed deeper in the crust will switch on
energetically unfavorable EC transitions. This lowers the nuclear charge
and generates heat.
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) = (w4+2qw3+q2w2)(T9 /5.93)f1(z)+(4w3+6qw2+2q2w)
(T9 /5.93)2f2(z) + (12w2+12qw+2q2)(T9 /5.93)3f3(z)+(24w+12q)
(T9 /5.93)4f4(z)+24(T9 /5.93)5f5(z)
FULLER, FOWLER & NEWMAN
(FFN) (Ap.J. Suppl.42,447 (1980))
GUPTA & MÖLLER
(GM)
Spherical Independent Particle Model.
Log10(Ye) = 10.0
UF(T9=0.01)=11.119 MeV
Nilsson Model.
Allowed (GT) only.
Experimental input (gs->gs or gs-> lowlying forbidden transitions) included.
Z
Excited states of parent included
(required for Core-Collapse Supernovae,
not for Neutron Star Crusts)
Ground state of parent only (thermal
population effects in Neutron Star Crusts
very mild).
No quenching of strength.
Residual interactions using QRPA (Peter
Möller).
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Results of comparisons
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 Log10(Ye) = 1.0). Then we compare rates for higher
values of density, while keeping T9 constant.
Below are shown the ratios
 ECR(GM ) 

R  Log10 
 ECR( FFN) 
Log10(Ye) = 1.0, 8.0, 9.0 and 10.0.
for T9=0.01 and
factor of 2
The color code for these plots is as follows:
R  -1.0
-0.8
-0.6 -0.4 -0.2
0.0
0.2
0.4
0.6
factor of 5
Thick black borders denote stable nuclei.
0.8
R1.0
At the lowest density some differences arise due to low lying structure
(experimental data vs. QRPA).
At higher values of Log10(Ye) (e.g. 8.0, 9.0, 10.0) a larger fraction of
strength in daughter nuclei is accessed, and more n-rich nuclei can EC.
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 GM. Therefore, calculated structure is very different.
Significantly deformed neutron orbits can increase the number of states
accessible by the electron chemical potential.
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.
For pre-threshold when q < -1 :
F(u < w,w = - q) = 2w2 (T9 /5.93)3f3(z)+12w (T9 /5.93)4f4(z)+
24(T9 /5.93)5f5(z)
Where fn(z) is a generalization of the Logarithm function:
f1 (z)=ln(1+z)
fn(z)=[ {(-1)k-1zk}/k n ] (k=1,…N<300 for convergence when n<6)
and z=exp{-(5.93|w-u|)/ T9 } < 1
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Comparison of EC rates to those calculated by FFN
This implementation is not susceptible to low-temperature inaccuracies
due to the Fermi-Dirac distribution shape (as Gauss-Laguerre quadrature
schemes are in the grid region T9 = 0.01-0.1 when compared to
trapezoidal rule schemes). Inaccuracies are only introduced because we
evaluate the distortion of the 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).
R=(ln2*f)/(ft_value for transition from I to J)
Phase space factor:
f(T9 ,u,w,q) = Geff * F
Where: u, w, q are electron chemical potential, capture threshold and
capture q-value respectively, in mec2 units (electron rest mass).
UF(T9=0.01)=5.182 MeV
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Threshold effects at low temperature cannot be captured accurately in
tables. Our analytic implementation of the electron capture phase space is
fast enough to be used in real-time inside a reaction network.
Analytic formulation of rate from state I in parent to
state J in daughter:
Log10(Ye) = 9.0
Log10(Ye) = 1.0
UF(T9=0.01)=0.508 MeV
Gupta & Möller calculations do not include experimental input. We
intend to compare the FFN and GM with available experimental level
information.
Where we are now:
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EC rates successfully implemented in Neutron Star Crust simulation.
(Gupta, Brown, Schatz, Möller, Kratz. TBP).
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Geff = Coulomb Correction = (p/w)FC (Z,A,w)
evaluated at effective electron energy w=weff extracted from
Where we are headed:
weff 2 (weff +q)2 =5.93*F/(T9 f1(z)).
Calculating rates with excited states in parent nuclei.
Using these rates in high temperature (T9), high density (Log10(Ye))
conditions in core – collapse supernova simulations.
P = electron momentum, Z = nuclear charge of captor
FC (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
Log10(Ye) = 8.0
UF(T9=0.01)=2.447 MeV
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This project is funded by the NSF through grants PHY0216783
(JINA), the NSCL, and by Michigan State University.
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