利用克來勞方程式估計LLSVP造成之CMB非自轉扁率

Abstract

克來勞定理(Clairaut's theory)是描述受牛頓古典重力與相對小的自轉離心力之天體達靜力平衡時，自轉離心力與天體內部扁率兩者關係之理論，為大地測量學(geodesy)中歷久不衰之經典定理。本文中，我們以Chao & Shih (2024)與Poisson & Will (2014)為基礎(i) 擴張經典的克來勞定理，使其能夠計算天體於不連續面以上擁有微小的側向密度變化且達靜力平衡時，天體內部之扁率或形變係數隨半徑的變化，並嘗試使用此理論計算地球內部LLSVP (large low shear velocity province) 對於CMB (core mantle boundary) 扁率、地表扁率、地球 J_2參數之影響；(ii) 計算克來勞定理於球諧函數(spherical harmonics)任何一次之形狀變化，並討論潮汐鎖定即自轉與公轉同步之天體，受重力與相對微小之引潮力的二次項後達靜力平衡時的形變大小；(iii) 證明經典的克來勞定理於等密度層中有解析解，並嘗試將PREM (preliminary reference earth model) 換算成n層密度模型(n-layer density model)，避開數值積分，以疊代的方式直接計算地球內部扁率之強近似值。透過結合克來勞定理與地殼均衡學說(isostasy)即能夠以相對精簡的方式計算天體內部之不連續面，包括天體表面之上有側向密度變化時天體內部的扁率或形變係數隨半徑的變化，並且此方法能夠同時計算側向密度變化之荷重造成的形變與重力位變化造成的形變，因此相較於傳統的地殼均衡學說(isostasy)，此方法能夠完整計算天體達靜力平衡後之各深度的形狀，尤其是地殼均衡學說從未考慮之側向密度變化對遠處之扁率或形變係數的影響。

Clairaut's theory describes the relationship between the internal flattening of a celestial body and the rotational effects under hydrostatic equilibrium governed by Newtonian gravity and small rotational centrifugal force. As a foundational theory in geodesy, Clairaut's theorem has remained a cornerstone for centuries. In this study, we build upon the classical formulation following Chao & Shih (2024) and Poisson & Will (2014) to extend its applicability in three directions: (i) We generalize the classical Clairaut's theory to accommodate celestial bodies with small lateral density variations above discontinuities, still under hydrostatic equilibrium. This generalized formulation allows us to compute the radial profiles of internal flattening or deformation coefficients. We apply this theory to estimate the influence of Large Low Shear Velocity Provinces (LLSVPs) in Earth’s deep mantle on the flattening of the core-mantle boundary (CMB), surface flattening, and the Earth's J₂ gravitational coefficient; (ii) We compute the shape deformation of celestial bodies expressed in spherical harmonics of arbitrary degree, based on the generalized Clairaut equation. We further analyze tidally locked bodies—those in synchronous rotation with their orbital motion—under the influence of gravity and second-order tidal force, to assess the magnitude of resulting equilibrium deformations; (iii) We demonstrate that the classical Clairaut equation admits analytical solutions in constant-density layers. Based on this, we attempt to approximate the Preliminary Reference Earth Model (PREM) with an n-layer density model, thereby avoiding direct numerical integration and instead employing an iterative approach to obtain accurate approximations of Earth’s internal flattening profile. By integrating Clairaut’s theory with the principle of isostasy, we offer a relatively simplified yet robust method for computing radial variations in internal flattening or deformation coefficients within celestial bodies, including scenarios with lateral density heterogeneities at the surface. This method simultaneously accounts for deformation due to lateral mass loading and the resulting gravitational potential perturbations. Compared to traditional isostatic models, which do not consider the gravitational effects of lateral density variations on remote deformation, our approach provides a more complete description of the hydrostatic shape of a celestial body across all depths.