mary beard university of notre dame
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/1.jpg)
Mary Beard University of Notre Dame
Reaction Rates Calculations in Dense Stellar Matter
[email protected] 2005
Aim:
•To establish a general reaction rate expression for all stellar burning regimes
•To establish uncertainties in existing reaction rate expressions
![Page 2: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/2.jpg)
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
![Page 3: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/3.jpg)
(WKB approximation)
(Hill-Wheeler formula)
Barrier Penetration Model (fusion cross sections)
![Page 4: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/4.jpg)
- 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
![Page 5: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/5.jpg)
BPMextrapolation
21
21 1574.0
E
ZZ
2exp)()( EEES
![Page 6: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/6.jpg)
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
![Page 7: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/7.jpg)
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
![Page 8: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/8.jpg)
Burning Regimes
Boltzmann Gas
Coupled classical Coulomb system
Coupled quantum Coulomb system
Strongly coupled quantum system
![Page 9: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/9.jpg)
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
![Page 10: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/10.jpg)
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
![Page 11: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/11.jpg)
Solid lines refer to models
![Page 12: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/12.jpg)
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.
![Page 13: Mary Beard University of Notre Dame](https://reader036.vdocuments.net/reader036/viewer/2022062323/568157e7550346895dc560f2/html5/thumbnails/13.jpg)
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