地球物理学报  2013, Vol. 56 Issue (3): 842-847 PDF

1. 地下信息探测技术与仪器教育部重点实验室, 北京 100083;
2. 中国地质大学(北京)地球物理与信息技术学院, 北京 100083;
3. 密西根大学 安娜堡, 美国 48109

The lower limit of variation of the Earth's oblateness in geological epoch
WANG Jun-Heng1,2, LI Xin-Jun1,2, ZHANG Yu-Ying1,2, WANG Xiao-Bin1,2, WANG Jing-Yuan3
1. Key Laboratory of Geo-detection, Ministry of Education, Beijing 100083, China;
2. School of Geophysics and Information Technology, China University of Geosciences, Beijing 100083, China;
3. University of Michigan Ann Arbor 48109, USA
Abstract: A variation in the earth's shape is one of the basic problems of earth evolution. A variation in the earth's oblateness can be used to understand the variation of the earth's shape.Assuming the Earth is flexible ellipsoid, according to the theory of elastic dynamics, we derived the formula for minimum oblateness of the Earth and conclude that oblateness of the Earth is a function of the average density of the Earth ρ, the gravitational acceleration g, rotation rate ω, average radius R, elastic modulus E and Poisson's ratio v. Apply parameters such as the radius, mass, and the change in angular velocity of the Earth, which are based on the new nebular hypothesis, into the formula to calculate the lower limit of oblateness of the Earth in different geological epoch. The oblateness of the Earth has signs of reduction in its magnitude since the formation of the Earth..
Key words: Earth's oblateness lower limit      Gravitational acceleration      Rotation angular velocity      Average density      Average radius
1 引言

2 地球均匀各项同性弹性力学模型

 (1)

 (2)

Rρ分别为地球平均半径和平均密度, g=GM/R2为球表面的引力加速度, M为地球质量, G为引力常数, ω为自转角速度.(1)式满足边界条件: ur=0处有限、r=Rr方向主应力trr=0的解[4]

 (3)

 (4)

(4)式中, 当r=R时, 为地球表面上拉格郎日应变张量, 根据拉格郎日应变张量E(Eij=eij)和格林应变张量C(Cij)的下列关系

 (5)

 (6)

 (7)

 (8a)

 (8b)

 (9)

 (10)

(10)式中表示第二类完全椭圆积分[6], 为定值, 可以通过代人函数或查表获取.不同纬度的经向周长为

 (11)

2.1 当Cg≠0, Cω=0, 即只有引力的状况

R'是地球经引力形变后的半径.显然, 弹性地球变形后仍是球体, 只是半径减小, 由于R'不能为虚数, 所以有A≥0, 再根据(8a)和(9)式有

 (14)

 (15)

(15)式类似广义相对论给出的密度均匀星球质量上限.通过上述分析可知, 在地球弹性模型中, 可以认为引力是等平衡的, 引力作用没有改变地球的形状.

2.2 当Cg=0, Cω≠0, 即只有自转的状况

Cg=0, 得A=5, 则可将仅有自转时地球表面不同纬度的经向周长和极向周长分别表示为

 (16)

 (17)

Cg≠0, Cω≠0, 在既有引力又自转的情况, 这时赤道周长为

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 (24)

(24)式为计算地球扁率变化下限的最终公式, 只要确定各个地质历史时期的ACω, 就可以研究扁率变化的下限.

4 各参数值的确定

 (25)

 (26)

5 计算实例(新星云假说)

 图 1 不同泊松比对应地球扁率下限变化曲线 Fig. 1 The lower limit variation curves of the flexible Earth model'oblatenesswith different Poisson's ratio v
 图 2 不同弹性模量对应地球扁率下限变化曲线 Fig. 2 The lower limit variation curves of the flexible Earth model's oblateness with different elastic modulus E

6 结论

R的取值范围有改进的可能.

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