地球形成和演化过程中的分异能计算方法研究

地球形成初期,构成地球的物质在组成上是大致均一的.目前地球的地核-地幔-地壳圈层结构,是由分异作用形成的.分异过程释放的能量称为分异能.Sorokhtin和Chilingarian等人从行星吸积的定义出发,导出了基于地球内部密度分布的势能计算公式,计算出的分异能大小为1.698×1031J.本文采用计算球体势能的思路,导出分异能计算的解析公式和数值计算公式,通过求取原始地球模型与均匀分层模型、PREM模型的势能差计算分异能.两种方法的计算结果分别为1.535×1031J和1.698×1031J.前者与Sorokhtin等的结果相近,后者与之相同.本文初步分析了方法间的异同以及造成结果偏差的主要原因.

Research on calculation methods of differentiation energy during the formation and evolution of the earth

According to Dai Wensai's nebular hypothesis, the formation of the Earth was closely related to the formation of the solar system, which can be described as "primordial nebular-protoplanetary disc-konisphere-planetesimal-planet". Accretion was the last stage during the formation of the Earth. Generally considered, homogeneous accretion has a greater possibility, which means that the primordial Earth was a nearly homogeneous body without significant stratification. However, different from the primordial Earth, the present-day Earth is divided into crust, upper mantle, lower mantle, outer core and inner core. This layering structure was formed by the differentiation process. During the differentiation and adjustment of the layers, the generation, migration, conversion and consumption of the Earth's internal energy was the decisive factor that restricted the whole process. Therefore, the calculation of differentiation energy is a pivotal issue.Starting from the definition of planet accretion, Sorokhtin et al. derived a potential calculation formula which is based on the density distribution within the Earth, and the calculated differentiation energy is 1.698×10³¹J. Flasar and Birch calculated the work done by gravity in the process of the Earth's accretion in the light of primordial Earth and present-day Earth. The difference between these two quantity, 1.66×10³¹J is the potential energy loss in the process of the Earth's differentiation. Estimates given by other authors suggest that the gravitational potential energy released during the process of the Earth's differentiation is between 1.46×10³¹J and 2×10³¹J.The idea of calculating the potential energy of a sphere was adopted in this paper, and differentiation energy was calculated by evaluating the potential energy difference between primordial Earth and present-day Earth. Firstly, the analytic formula of the Earth's potential energy was derived based on a uniformly layered Earth model. The calculated differentiation energy is 1.535×10³¹J which is close to the result given by Sorokhtin et al. Further, using a more sophisticated model, the preliminary reference Earth model (PREM), and by applying the numerical formula of the Earth's potential energy, the differentiation energy was calculated to be 1.698×10³¹J, which is the same as the result of Sorokhtin et al. within the given precision.Different from the "accretion work method" in previous studies, the"uniform layered analytic method" gives the analytic formula for the potential energy of primordial Earth and present-day Earth, from which the tedious steps of numerical summation were avoided. In the actual case, the density of core, mantle and crust decreases with radius increasing. This will make the potential energy of present-day Earth under uniform layered Earth model larger than that in the actual case, which can further make the calculated differentiation energy small. Since an Earth model with more layers can bring inconvenience to the derivation and calculation of the "uniform layered analytic method", only the derivation and calculation on the "core-mantle two-layer structure" was given in this paper.Considering that the difference of pressure is smaller than the difference of density between different Earth models, the "PREM numerical summation method" uses pressure instead of density to describe the Earth's potential energy, which can reduce the error brought by the differences between models. Using the Earth models adopted in this paper, the "PREM numerical summation method" gives the same result as the method of Sorokhtin et al. Moreover, when density distributions given by different Earth models vary significantly, this method can lead to more reliable results than the method of Sorokhtin et al.At present, the Earth's differentiation has not yet stopped, but it is no longer comprehensive and large-scale activity which forms the core-mantle-crust structure. In this process, a portion of the differentiation energy was consumed by the Earth's elastic compression, while most of it was converted into the Earth's internal heat. Subsequent research should focus on the heat sources provided by other physical processes during the evolution of the Earth and other relevant issues, such as the decay of radioactive elements, the total temperature the Earth raised by absorbing this heat, and the releasing rate and releasing amount of differentiation energy with time.