Download Toward a Global Description of the Nucleus

Reaction Rates Calculations in Dense
Stellar Matter
Mary Beard
University of Notre Dame
[email protected]
Frontiers 2005
Aim:
•To establish a general reaction rate expression for
all stellar burning regimes
•To establish uncertainties in existing reaction rate
expressions
Dense Stellar Environments
Accreting neutron stars: The ashes of the rp process
are compressed and undergo electron captures
producing extremely neutron rich, light nuclei
Barrier Penetration Model
(fusion cross sections)
(Hill-Wheeler formula)
(WKB approximation)





The Nuclear Potential
VLE (R )  VFold (R ) e
 4v 2 c 2
Phys. Rev. Lett. 78 (1997) 3270
Phys. Rev. Lett. 79 (1997) 5218
Phys. Rev. C58 (1998) 576
Phys. Rev C66 (2002) 014610
São Paulo potential
Density dependence
- Densities obtained through theoretical calculations (RMF, for example).
M. Stoitsov et al., Phys. Rev. C58 (1998) 2086.
A. V. Afanasjev et al., Phys. Rev. C60 (1999) 051303.
extrapolation
BPM
S ( E)   ( E) E exp 2

12
  0.1574 Z1Z 2 

E
Pycnonuclear Reaction Rates
In a neutron star crust, ions form a Coulomb lattice structure surrounded by a
degenerate electron gas.
Electron screening effects become so strong that rates of nuclear reactions
increase considerably even at low energies;
Pycnonuclear reactions take place under very high density conditions and are
more sensitive to density than to temperature – from the Greek, pyknos means
compact, dense;
d
Pycnonuclear reactions between neutron-rich isotopes can provide a new heat
source in accreting neutron star crust.
Pycnonuclear Formalism
There are a couple of models available for pycnonuclear calculations,
(eg Salpeter and Van Horn Astrophys. J. 155, 183 1969)
All can be written in one general (user-friendly) way, with
dimensionless parameters representing model differences
RPYC  X i AZ 4 S ( EPK )CPYC 10 46 3CL exp( Cexp 1 / 2 )
Where length parameter λ is defined by:

1
AZ 2
1
X i

11
3
A
1
.
3574
x
10
g
/
cm




1/ 3
CPYC, Cexp and CL are dimensionless model parameters
Burning Regimes
Boltzmann Gas
Coupled quantum
Coulomb system
Coupled classical
Coulomb system
Strongly coupled
quantum system
Single analytical approximation in all regimes
Thermonuclear reaction rate is defined by:
ni2
2 E PK S ( E PK )
Rth 
4
exp(  )
2
3
T
ni2

Rth 
S ( E PK )
P F
2 2 th th
2
mZ e
Where Pth and Fth are given by
8 1 / 3  Ea 
Pth 


1/ 3
32  T 
2/3
Fth  exp(  )
By Analogy, the thermally enhanced pycnonuclear rate can be
written as:
Where P and F are given by:
Reaction rate approximation is then given by:
This reduces to appropriate expression in all burning
regimes; when T>>Tp ΔR  Rth >> Rpyc retrieve Rth
T  0 ΔR  0 retrieve Rpyc
Solid lines refer to models
The rates involving isotopes with identical charge number show only minor
differences which are entirely due to the difference in S-factor;
For higher Z-values the rates decrease steeply at density values less than
1012 g/cm3 because of the strong Z-dependence in the pycno equation.
Summary
Nuclear Physics:
We are using the São Paulo potential to describe the fusion process.
Nuclear Astrophysics:
We are proposing a single analytical expression for the fusion rate, which
is valid in all regimes. The parameters reflect theoretical uncertanties of
the reaction rates.
An exact calculation should take into account many effects as lattice
impurities and imperfections, classical motion of plasma ions, related
structure of Coulomb plasma fields, etc.
The next step is extend the treatment presented for one-componentplasma case towards a general formalism for the fusion rate between
different isotopes.