Electrostatic Screening of Nuclear Reactions in Stellar Plasmas

1. Electrostatic Screening of Nuclear Reactions in Stellar Plasmas Erice

2. What is electrostatic screening of nuclear reaction and what is the dynamic effect Do we understand the simple picture (and theory)? The difference between a test particle and a particle in statistical equilibrium What is the right definition for ‘screening of nuclear reactions’. What did we calculate so far? Can we get the screening from a simple picture which we can understand? The Molecular Dynamic calculations - results Conclusion - so far Plan of talk I :

3. The connection between the screening and relaxation processes in the gas The Fokker-Planck equation and screening The Langevin equation How to calculate the relaxation in plasma streams like jets Plan of talk II:

4. What is the problem? Where it arises? Why bother?

5. In core of stars - where the nuclear reactions take place- the matter is in a form of plasma: a sea of bare protons, bare He nuclei as well as (almost) fully stipped heavier ions with free electrons. An ion in the plasma feels a potential created by the sum of the interaction of all particles. The sum of all interactions changes the simple Coulomb force (and potential).

6. The motion of a charge particle in plasma We search the electrostatic potential of a proton moving in the plasma Let: be the single particle distribution function The electric potential of the proton with speed vp relative to the laboratory The undisturbed number density of particles

7. Vlassov equation Poisson equation  is the plasma dialectric function The governing equations: Solution: first linearize, then Fourrier transform to get:

8. For vanishing velocity one finds Here RD is the Debye radius and it expresses the rate at which the simple Coulomb potential decays in plasma This is the Debye Huckel potential. It is the effective potential in the plasma. Note: the potential does not depend on velocity (or kinetic energy) of the particle in this limit.

9. The classical picture of the screening: Shatzmann 1948 Salpeter 1954.

10. The enhancement factor f=exp(-Epot/kT) > 1 But this is all static: the potential does not depend on the relative velocity between the particles

11. The Debye potential is caused by e and p. Does the massive p gain energy from the e’s. Should we take the complete Debye potential (e+p) or only p’s? What about the potential well of the incoming particle? Is the problem symmetric target-projectile? What about p-p reaction Particles enter the potential well and gain energy. What happens when the particles separate? Do they give all the energy gained back? Is the potential well rigid? The plasma loses energy to the pair and never regains? Simple questions:

12. Here is the binary distribution function. The BBGKY hierarchy for the N particle distribution function: Define: Define now: Getting the effect on the nuclear reaction

13. Substitute now: Then the rate becomes: is the effective potential. If the effective potential does not depend on the velocities we can take the exponent out of the integral This is the Salpeter result.

14. For the Sun Typical numbers: Enhancement factor f=exp(G) Weak screening limit G << 1 fBe7p = 1.4-1.5 In WD’s (SNIa) the screening correction is about 1018!

15. Limitations: The BBGKY is an expansion in 1/ND (ND - the number of particles in a Debye sphere) In the solar core ND ~ 3-4 The expansion is not valid and three body and higher correlations may be important

16. Assume the Hamiltonian of the system is: Then the canonical ensemble is: and The sum is replaced with an effective potential Finding the screening potential

17. What is the effective potential? What is the effective interaction between two particles in the plasma? Does it depend on X and V? Can the effective potential depend on V when the basic interaction depends on X only? Various authors used different approximations to get the effective potential. The controversy is about the dependence of the potential on the velocity.

18. Nuclear cross-section MB distribution The Gamow peak for Be7+p We need the effect at the Gamow peak What is the Gamow peak

19. So, let us find the effective potential

20. The ensemble average Take the canonical ensemble: is separable: the potential does not depend on the velocity Define the operator

21. The ensemble average of the potential is: The kinetic energy factors out The average potential energy of a particle does not depend on its kinetic energy Ensemble average: the average energy of every particle is on a long time scale the same. This is so because every particle must go through all energy states of the system. Hence, the long time means long enough for it to go through all states. But we want only particles in the Gamow peak, we do not care what happens in other energies….

22. Time averages The time average of the potential energy of a particle is: In the limit of very long time The limit does not depend on the particular particle. All particles have the same limit. For thermodynamics we need Time average Canonical average

23. The ergodic assumption:The two averages are equal If we assume the ergodic assumption we have thermodynamics and canonical averaging. But what we need is: Or alternatively: The sum is only over the relevant periods, when the kinetic energy is in the Gamow peak.

25. The basic question: Mean potential energy of particles in the Gamow peak Mean potential energy of a particle If there is dynamic screening a la Koonin et al. there should be no equality!

26. If The Molecular Dynamic calculation Classical MD: 512 - 1024 particles + periodic boundary conditions (Ewald sums) The returned particle forgets the conditions around it. Actual calculations with 2*363 = 93312 to 2*403 = 128,000 particles You must have a large number of particles to probe the high energy tail of the Maxwell Boltzmann distribution The interaction with up to ~ 1200 particles is included

27. Results for a high density low T case: The main reason for these conditions: to enhance the effects. In the Sun We have results also for the sun

28. A snapshot at the distribution of potential energies in the system. n=1029#/cc, T=1.5 x 107K

29. The potential energy per particle as a function of kinetic energy for protons. Also shown, the standard deviation, the error and the number of particles. n=1029#/cc, T=1.5 x 107K

30. The force The potential The distribution of the potential energy, <f2>1/2 and |<f>| as a function of the kinetic energy of the proton. n=1029#/cc, T=1.5 x 107K

31. In complete agreement with statistical mechanics A comparison of the potential energy per particle distribution for two assumed masses of the proton. n=1029#/cc, T=1.5 x 107K

32. Comparison between the kinetic energy distribution and the MB distribution. n=1029#/cc, T=1.5 x 107K

33. The restricted ergodic theorem is fully satisfied Conclusions: The ergodic assumption is fully satisfied The conditional average is equal to the ensemble one. There is no dynamic screening

34. Did we really calculate the screening? Does a proton get the entire <U> energy from the plasma around the proton it scatters with? Is there time for energy exchange even for protons in the Gamow peak? If the number of particles in the Debye sphere is 3-5, how the energy gained by the incoming particle is returned to the plasma? What does the particle do when it moves away? Is the mean field ok for ND~3-5?

35. Is it possible to calculate the screening effect (the plasma interaction with a scattering pair) from first principles? ...and in a clear and obvious way?

36. How to calculate directly the screening Consider the energy of the entire plasma: Consider the total energy of just a pair of mutually scattering protons (Z=1).

37. and is replaced by the Debye potential or by another mean field approximation The classical approach:

38. close Far away Define: screening energy energy transfer between the pair and the plasma during a collision.

39. The description of the path of two particles.

40. The potential energy of particle 1, The relative kinetic energy and the distance from the paired particles as a function of time.

41. The long time average potential (The Debye potential) This is the mean field….

42. The change of the energy of the pair due to plasma interaction during the collision for collisions with maximum distance of closest approach rmax<0.25.

43. The dependence of the assumed max(rmin) on the energy exchange. The number in brackets give the maximum distance of closest approach

44. The effect of the proton mass on the screening energy as a function of the relative kinetic energy when far away. The red is mp=0.001amu and the blue mp=1amu.

45. The change in the energy of the center of mass during The scattering of two protons

46. The Gamow peak Results for the pp in the core of the Sun

47. The Gamow peak The Be7+p reaction

48. The Gamow peak The results for the He3+He3

49. The mean energy exchange between the scattering particles in the plasma vanishes! ~0! As it should in a plasma in equilibrium, namely zero!

50. Power Plasma frequency w tun The power spectrum of the fluctuations