Mary Beard University of Notre Dame

Reaction Rates Calculations in Dense Stellar Matter

mbeard@nd.eduFrontiers 2005

Aim:

•To establish a general reaction rate expression for all stellar burning regimes

•To establish uncertainties in existing reaction rate expressions

Dense Stellar EnvironmentsDense Stellar Environments

Accreting neutron starsAccreting neutron stars: The ashes of the rp : The ashes of the rp process are compressed and undergo electron process are compressed and undergo electron captures producing extremely neutron rich, light captures producing extremely neutron rich, light nucleinuclei

(WKB approximation)

(Hill-Wheeler formula)

Barrier Penetration Model (fusion cross sections)

- 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.

2 24( ) ( ) v cLE FoldV R V R e

Phys. Rev. Lett. 78 (1997) 3270Phys. Rev. Lett. 79 (1997) 5218Phys. Rev. C58 (1998) 576Phys. Rev C66 (2002) 014610

São Paulo potential

The Nuclear Potential

Density dependence

BPMextrapolation

21

21 1574.0

E

ZZ

2exp)()( EEES

Pycnonuclear Reaction Rates

Pycnonuclear reactions between neutron-rich isotopes can provide a new heat source in accreting neutron star crust.

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;

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;

d

Pycnonuclear FormalismPycnonuclear FormalismThere 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, withdimensionless parameters representing model differences

)exp(10)( 2/1exp

3464 CCESAZXR CLPYCPKiPYC

Where length parameter λ is defined by:

3/1

3112 /103574.1

11

cmgx

X

AAZi

CPYC, Cexp and CL are dimensionless model parameters

Burning Regimes

Boltzmann Gas

Coupled classical Coulomb system

Coupled quantum Coulomb system

Strongly coupled quantum system

Single analytical approximation in all Single analytical approximation in all regimesregimes

Thermonuclear reaction rate is defined by:Thermonuclear reaction rate is defined by:

)exp()(

3

24

2

2

T

ESEnR PKPKith ththPK

ith FP

emZES

nR

22

2

)(2

Where Pth and Fth are given by

3/2

3/1

3/1

23

8

T

EaPth

)exp( thF

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.

Nuclear Physics:

We are using the São Paulo potential to describe the fusion process.

Nuclear Astrophysics:

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.

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.

The next step is extend the treatment presented for one-component-plasma case towards a general formalism for the fusion rate between different isotopes.

Summary