1. Mary Beard University of Notre Dame Reaction Rates Calculations in Dense Stellar Matter mbeard@nd.edu Frontiers 2005 Aim: To establish a general reaction rate expression for all stellar burning regimes To establish uncertainties in existing reaction rate expressions

2. Dense Stellar Environments Accreting neutron stars: The ashes of the rp process are compressed and undergo electron captures producing extremely neutron rich, light nuclei

3. (WKB approximation) (Hill-Wheeler formula) Barrier Penetration Model (fusion cross sections)

4. - 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. 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 The Nuclear Potential Density dependence

5. BPMextrapolation

6. 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

7. 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 Where length parameter λ is defined by: C PYC, C exp and CL are dimensionless model parameters

8. Burning Regimes Boltzmann Gas Coupled classical Coulomb system Coupled quantum Coulomb system Strongly coupled quantum system

9. Single analytical approximation in all regimes Thermonuclear reaction rate is defined by: Where P th and F th are given by

10. 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>>T p ΔR  R th >> R pyc retrieve R th T  0 ΔR  0 retrieve R pyc

11. Solid lines refer to models

12. 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 10 12 g/cm 3 because of the strong Z-dependence in the pycno equation.

13. 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