Reexamination of constrains 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 (textsf{MBD}). In a previous paper published in Physics Letters 441B(1998)291, DeglInnocenti {it et al}. made strong constrains on such deviations with the detailed helioseismic information of the solar structure. With a small deviation parameterized with a factor exp$[{-delta (E/kT)^2}]$, it was shown $delta$ restricted between -0.005 and +0.002. These constrains have been carefully reexamined in the present work. We find that a normalization factor was missed in the previous modified textsf{MBD}. In this work, the normalization factor $c$ is calculated as a function of $delta$. It shows the factor $c$ is almost unity within the range 0$< delta leq$0.002, which supports the previous conclusion. However, it demonstrates that $delta$ cannot take a negative value from the normalization point of view. As a result, a stronger constraint on $delta$ is defined as 0$leq delta leq$0.002. The astrophysical implication on the solar neutrino fluxes is simply discussed based on a positive $delta$ value of 0.003. The reduction of the $^7$Be and $^8$B neutrino fluxes expected from the modified textsf{MBD} can possibly shed alternative light on the solar neutrino problem. In addition, the resonant reaction rates for the $^{14}$N($p$,$gamma$)$^{15}$O reaction are calculated with a standard textsf{MBD} and a modified textsf{MBD}, respectively. It shows that the rates are quite sensitive even to a very small $delta$. This work demonstrates the importance and necessity of experimental verification or test of the well-known textsf{MBD} at high temperatures.