基于径向泰勒级数展开的月球表面磁场快速球谐解算方法

Fast Spherical Harmonic Synthesis of Magnetic Field at Lunar Surface Based on Radial Taylor’s Series Expansion

在利用球谐模型解算月球起伏表面的磁场时,传统算法计算效率较低且存在不稳定性,因此提出一种基于球谐分析的径向泰勒级数展开法。首先采用450阶次的月球内源磁场球谐模型在月表地形起伏剧烈的背面区域进行不同泰勒级数展开次数范围以及不同平均半径的解算精度实验,验证了所提方法的有效性,并且通过与传统算法的对比证明了所提方法具有更高的计算效率和计算稳定性;然后将计算得到的月球全球起伏表面磁场分布与月球参考球面上的磁场分布进行了对比分析,结果表明,月表地形起伏面与参考球面上的磁场差异较大,磁场总强度在实际地形起伏面与参考球面上的差异和地形起伏呈现反相关性,磁场径向分量与地形起伏不存在相关性,这些说明月球内源磁场磁性载体埋深较浅且磁化方向并非径向。

Objectives:For spherical harmonic synthesis of magnetic field at lunar irregular surface,traditional calculating algorithm has a low computing efficiency and the possible instability.Therefore,the radial Taylor’s series expansion method is proposed.Methods:Firstly,the formulae of the traditional and newly presented methods are given.Then,we adopt a spherical harmonic model of the lunar magnetic field with maximum degree and order of 450,the numerical tests are performed in the lunar highland region with a drastic topographic relief.Finally,the global magnetic fields on lunar relief surface and on the reference sphere are calculated,respectively.Results:The results of numerical tests and practical application show that the calculating accuracy of our method mainly depends on the maximum order of the Taylor’ s series expansion,and if the observing surface changes more dramatically and the spherical harmonic model of the magnetic field has higher maximum degree and order,a higher maximum order of the Taylor’ s series expansion is required.Moreover,the average radius of the algorithm is suggested to be the area-weighted mean of the radius values of all calculating points.The computing time varies linearly with the maximum order of the Taylor’ s series expansion.If more calculating points and higher maximum degree and order of spherical harmonic model of the magnetic field,the computational efficiency will be improved more significantly by our proposed method.The differences between magnetic fields on lunar relief surface and on the reference sphere are very large,which suggests that the source depth of the internal magnetic field of the Moon is relatively shallow.Especially,the differences of the intensity of the magnetic field have a negative correlation with the topography variations,but there is no correlation between the radial component of the magnetic field and the topography variations.These indicate that the sources’ magnetized directions of the internal magnetic field of the Moon are not radial.Conclusions:In brief,this study significantly certifies that our proposed method has a high computing efficiency and a very weakly instability.