# Re-examination of constraints on the Maxwell-Boltzmann distribution by Helioseismology

• Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution, the well-known Maxwell-Boltzmann Distribution (MBD). Strong constraints on such deviations were made previously with the detailed helioseismic information of the solar structure. With a small deviation parameterized with a factor exp[-δ (E/kT)2], it was shown δ restricted between -0.005 and +0.002. These constraints have been carefully re-examined in the present work. We find that a normalization factor was missed in the previous modified MBD. In this work, the normalization factor c is calculated as a function of δ. It shows the factor c is almost unity within the range 0< δ ≤0.002, which supports the previous conclusion. However, it demonstrates that δ cannot take a negative value from the normalization point of view. As a result, a stronger constraint on δ is defined as 0≤ δ ≤0.002. The astrophysical implication on the solar neutrino fluxes is simply discussed based on a positive δ value of 0.003. The reduction of the 7Be and 8B neutrino fluxes expected from the modified MBD can possibly shed alternative light on the solar neutrino problem. In addition, the resonant reaction rates for the 14N(p, γ)15O reaction are calculated with a standard MBD and a modified MBD, respectively. It shows that the rates are quite sensitive even to a very small δ. This work demonstrates the importance and necessity of experimental verification or testing of the well-known MBD at high temperatures.
PCAS:
•  [1] Degl'Innocenti S et al. Phys. Lett. B, 1998, 441: 291-298[2] Landau L D, Lifshitz E M. Statistical Physics. Oxford: Pergamon Press, 1980[3] Huang K. Statistical Mechanics (2nd Edition). New York: John Wiley Sons, Inc., 1987[4] Rolfs C E, Rodney W S. Cauldrons in the Cosmos. Chicago: Chicago University Press, 1988[5] Fowler W A, Caughlan G R, Zimmerman B A. Ann. Rev. Astron. Astrophys., 1967 5: 525-570[6] Clayton D D. Principles of Stellar Evolution and Nucleosynthesis. Chicago: Chicago University Press, 1983[7] Ciacio F, Degl'Innocenti S, Ricci B. Astr. Astrophy. Suppl. Ser., 1997, 123: 449-454[8] Bahcall J N, Pinsonneault M. Rev. Mod. Phys., 1992, 64: 885-926[9] Angulo C et al. Nucl. Phys. A, 1999, 565: 3-183

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HE Jian-Jun, ZHANG Li-Yong, HOU Su-Qing and XU Shi-Wei. Re-examination of constraints on the Maxwell-Boltzmann distribution by Helioseismology[J]. Chinese Physics C, 2013, 37(10): 104001. doi: 10.1088/1674-1137/37/10/104001
HE Jian-Jun, ZHANG Li-Yong, HOU Su-Qing and XU Shi-Wei. Re-examination of constraints on the Maxwell-Boltzmann distribution by Helioseismology[J]. Chinese Physics C, 2013, 37(10): 104001.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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## Re-examination of constraints on the Maxwell-Boltzmann distribution by Helioseismology

###### Corresponding author: XU Shi-Wei,

Abstract: Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution, the well-known Maxwell-Boltzmann Distribution (MBD). Strong constraints on such deviations were made previously with the detailed helioseismic information of the solar structure. With a small deviation parameterized with a factor exp[-δ (E/kT)2], it was shown δ restricted between -0.005 and +0.002. These constraints have been carefully re-examined in the present work. We find that a normalization factor was missed in the previous modified MBD. In this work, the normalization factor c is calculated as a function of δ. It shows the factor c is almost unity within the range 0< δ ≤0.002, which supports the previous conclusion. However, it demonstrates that δ cannot take a negative value from the normalization point of view. As a result, a stronger constraint on δ is defined as 0≤ δ ≤0.002. The astrophysical implication on the solar neutrino fluxes is simply discussed based on a positive δ value of 0.003. The reduction of the 7Be and 8B neutrino fluxes expected from the modified MBD can possibly shed alternative light on the solar neutrino problem. In addition, the resonant reaction rates for the 14N(p, γ)15O reaction are calculated with a standard MBD and a modified MBD, respectively. It shows that the rates are quite sensitive even to a very small δ. This work demonstrates the importance and necessity of experimental verification or testing of the well-known MBD at high temperatures.

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