{"title":"带有地球扰动引力势新展开的 G 修正亥姆霍兹方程、其函数和等重力面研究","authors":"G. Manoussakis","doi":"10.3390/appliedmath4020032","DOIUrl":null,"url":null,"abstract":"The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.","PeriodicalId":503400,"journal":{"name":"AppliedMath","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces\",\"authors\":\"G. Manoussakis\",\"doi\":\"10.3390/appliedmath4020032\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.\",\"PeriodicalId\":503400,\"journal\":{\"name\":\"AppliedMath\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"AppliedMath\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/appliedmath4020032\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"AppliedMath","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/appliedmath4020032","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
G 修正亥姆霍兹方程是一个偏微分方程,它使我们能够将重力强度 g 表示为一系列径向距离为 r 的无理幂球面谐波。三维空间中的拉普拉斯方程(在直角坐标中,未知量的二阶偏导数之和等于零)用于表示地球重力势能(扰动势能和法向势能),以便将其他有用的量--也称为扰动势能的函数--如重力扰动、重力异常和大地水准面起伏表示为一系列球面谐波。我们证明,通过使用 G 修正亥姆霍兹方程,不仅重力强度,而且扰动势及其函数都可以用一系列球面谐波来表示。将重力强度表示为一系列球面谐波使我们能够创建新的全球重力模型。此外,我们还对地球等重力面进行了更详细的研究。最后,我们以表格形式列出了我们的研究结果,这些结果清楚地表明,重力强度 g 的新全球重力模型将对许多地球物理和大地测量应用非常有用。
A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces
The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.