负荷勒夫数渐近表达式的直接证明

DIRECT PROOF OF THE ASYMPTOTIC EXPRESSION OF THE LOADING LOVE NUMBERS

负荷勒夫数ｈ＇_ｎ、ｌ＇_ｎ、和ｋ＇_ｎ用以表征地球在表面点负荷作用下的变形．当ｎ→∞时，ｈ＇_ｎ、ｎｌ＇_ｎ和ｎｋ＇_ｎ的渐近值均为常数，这是Ｆａｒｒｅｌｌ通过比较Ｂｏｕｓｓｉｎｅｓｑ问题的解与地球在表面负荷作用下变形的球函数级数展开式得出的．本文直接由负荷勒夫数满足的微分方程推导出了它们的渐近表达式，方法更为直观和简明，而且更易于理解。

The loading love numbers h'_n,l'_n and k'_n are used to describe Earth's deformation under surface load of punctual mass. The asymptotic values of h'_n, nl'_n and nk'_n for n→∞ are all constant. Farred (1972) obtained these results by comparing the solution of the Boussinesq problem and the spherical harmonic series development of Earth's deformation under surface load of punctual mass. For obtaining these results, farrel used Hankel transformation and the asymptotic relation between Legendre functions and Bessel functions near the pole. In this paper, the same results are derived directly from the differential equations satisfied by the loading Love numbers. Only the knowledge of differential equations and the asymptotic expression of Legendre functions are used. The method in this paper is relatively straightforward, it is also easier to be understood.