A method for predicting a satellite orbit

Abstract

This application relates to a satellite orbit prediction method. The method includes: constructing an expression for the gravitational force acting on the satellite based on its mass and distance from Earth; constructing a differential equation for the satellite's acceleration based on Newton's second law and the expression for the gravitational force; solving the differential equation based on the conservative force field characteristics of mechanics, calculating the gravitational potential energy function and its derivative with respect to time; transforming the derivative expression based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, to obtain the derivative expression for acceleration; converting the time relationship in the derivative expression of acceleration to a radial position relationship based on the principle of conservation of angular momentum; integrating the radial position relationship to obtain the satellite's orbit equation. This method enables satellite orbit prediction.

Description

Technical Field

[0001] This application relates to the field of satellite orbit prediction technology, and in particular to a satellite orbit prediction method. Background Technology

[0002] In recent years, with the advancement of the aerospace industry, an increasing number of satellites have been launched into space and are operating in predetermined orbits to perform various missions. During satellite operation, various factors influence their trajectory, making accurate prediction extremely difficult. Furthermore, the engines used to control satellite motion cannot operate with absolute precision. Therefore, how to predict satellite orbits has become a pressing problem to be solved. Summary of the Invention

[0003] Therefore, it is necessary to provide a satellite orbit prediction method that can achieve satellite orbit prediction in response to the above-mentioned technical problems.

[0004] A satellite orbit prediction method, the method comprising:

[0005] Obtain the satellite's mass and launch angle; construct the expression for the gravitational force acting on the satellite based on its mass and distance from Earth; construct the differential equation for the satellite's acceleration based on Newton's second law and the expression for the gravitational force acting on the satellite.

[0006] The differential equation is solved based on the conservative force field characteristics of mechanics, and the gravitational potential energy function and its derivative with respect to time are calculated. Based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, the derivative expression is transformed to obtain the derivative expression of acceleration.

[0007] Based on the principle of conservation of angular momentum, the time relationship in the derivative expression of acceleration is converted into a radial position relationship. By integrating the radial position relationship, the orbital equation of the satellite is obtained.

[0008] In one embodiment, the expression for the gravitational force acting on the satellite is calculated based on the satellite's mass and its distance from Earth, including:

[0009] The expression for the gravitational force acting on a satellite, calculated based on its mass and distance from Earth, is as follows:

[0010]

[0011] Where G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, and r represents the distance between the Earth and the satellite.

[0012] In one embodiment, the differential equation for the satellite's acceleration is constructed based on Newton's second law and the expression for the gravitational force acting on the satellite, including:

[0013] Based on Newton's second law and the expression for gravity acting on the satellite, the differential equation for the satellite's acceleration can be constructed as follows:

[0014]

[0015] Where F represents the expression for the gravitational force acting on the satellite, G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, and r represents the distance between the Earth and the satellite.

[0016] In one embodiment, calculating the gravitational potential energy function and its derivative with respect to time includes:

[0017] Calculate the gravitational potential energy function and its derivative with respect to time.

[0018]

[0019] Where F represents the expression for the gravitational force acting on the satellite, G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, r represents the distance between the Earth and the satellite, and t represents the launch time.

[0020] In one embodiment, the process expression is based on the fact that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero. Where F represents the expression for the gravitational force acting on the satellite, m represents the satellite's mass, a represents the satellite's acceleration, r represents the distance between the Earth and the satellite, and t represents the launch time.

[0021] In one embodiment, the derivative expression is transformed based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, to obtain the derivative expression of acceleration, including:

[0022] because Where F = ma, that is The derivative expression for acceleration is:

[0023]

[0024] Where C represents the integration constant and G represents the gravitational constant.

[0025] In one embodiment, the time relationship in the derivative expression of acceleration is converted into a radial position relationship based on the principle of conservation of angular momentum, including:

[0026] The derivative expression for acceleration is: in It is radial velocity. It is angular velocity; according to the law of conservation of angular momentum, It is a constant, where L is the angular momentum, therefore Therefore That is, the radial positional relationship is: C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, t represents the launch time, and M represents the Earth mass.

[0027] In one embodiment, integrating the radial positional relationship yields the satellite's orbital equations, including:

[0028] Integrating the radial positional relationship, we obtain the satellite's orbital equation as follows:

[0029]

[0030] or

[0031] Where C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, θ is the launch angle, M represents the Earth mass, L is the angular momentum, and r0 represents the Earth radius.

[0032] The aforementioned satellite orbit prediction method first constructs an expression for the gravitational force acting on the satellite based on its mass and distance from Earth. Then, it constructs a differential equation for the satellite's acceleration based on Newton's second law and the gravitational force expression. The differential equation is solved using the conservative force field characteristics of mechanics, calculating the gravitational potential energy function and its derivative with respect to time. Based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, the derivative expression is transformed to obtain the derivative expression for acceleration. Finally, based on the principle of conservation of angular momentum, the time relationship in the derivative expression of acceleration is converted to a radial position relationship. Integrating this radial position relationship yields the satellite's orbital equation. This application achieves satellite orbit prediction by utilizing a conservative approach, Newton's second law, and the law of conservation of momentum, and by transforming related equations based on these laws. The calculation process is simple, fast, and accurate, making it more suitable for satellite orbit prediction in scheme design or engineering missions. Attached Figure Description

[0033] Figure 1 This is a flowchart illustrating a satellite orbit prediction method in one embodiment;

[0034] Figure 2 This is an internal structural diagram of a computer device in one embodiment. Detailed Implementation

[0035] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0036] In one embodiment, such as Figure 1 As shown, a satellite orbit prediction method is provided, including the following steps:

[0037] Step 102: Obtain the satellite's mass and launch angle; construct the expression for the gravitational force acting on the satellite based on the satellite's mass and distance from Earth; construct the differential equation for the satellite's acceleration based on Newton's second law and the expression for the gravitational force acting on the satellite.

[0038] First, calculate the gravitational force acting on the satellite at any time t. In the polar coordinate system, we can construct an expression for the gravitational force acting on the satellite based on its mass and distance from Earth. Then, according to Newton's second law, the radial acceleration of the satellite can be expressed as:

[0039]

[0040] Step 104: Solve the differential equation based on the conservative force field characteristics of mechanics, calculate the gravitational potential energy function and its derivative with respect to time, and transform the derivative expression based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, to obtain the derivative expression of acceleration.

[0041] The differential equation relates to the distance between the satellite and the Earth, i.e., its position *r*. This application solves the equation using the properties of a conservative force field, in which the potential energy function satisfies the following relationship:

[0042]

[0043] Where U(r) is the potential energy function, which can be found from the gravitational potential energy function. The gravitational potential energy function is: Therefore, the total mechanical energy of a satellite can be expressed as the sum of its kinetic and potential energy:

[0044]

[0045] The derivatives of kinetic and potential energy with respect to time are:

[0046]

[0047] According to the law of conservation of mechanical energy because Where F = ma, that is

[0048] The derivative expression for acceleration is:

[0049]

[0050] Where C represents the integration constant and G represents the gravitational constant.

[0051] Step 106: Based on the principle of conservation of angular momentum, the time relationship in the derivative expression of acceleration is converted into a radial position relationship. The radial position relationship is then integrated to obtain the orbital equation of the satellite.

[0052] The derivative expression for acceleration is: in It is radial velocity. It is angular velocity; according to the law of conservation of angular momentum, It is a constant, where L is the angular momentum, therefore Therefore That is, the radial positional relationship is: C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, t represents the launch time, and M represents the Earth mass.

[0053] Integrating the radial positional relationship yields the satellite's orbital equations, including:

[0054] Integrating the radial positional relationship, we obtain the satellite's orbital equation as follows:

[0055]

[0056] or Where C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, θ is the launch angle, M represents the Earth mass, L is the angular momentum, and r0 represents the Earth radius.

[0057] In one embodiment, the expression for the gravitational force acting on the satellite is calculated based on the satellite's mass and its distance from Earth, including:

[0058] The expression for the gravitational force acting on a satellite, calculated based on its mass and distance from Earth, is as follows:

[0059]

[0060] Where G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, and r represents the distance between the Earth and the satellite.

[0061] In one embodiment, the differential equation for the satellite's acceleration is constructed based on Newton's second law and the expression for the gravitational force acting on the satellite, including:

[0062] Based on Newton's second law and the expression for gravity acting on the satellite, the differential equation for the satellite's acceleration can be constructed as follows:

[0063]

[0064] Where F represents the expression for the gravitational force acting on the satellite, G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, and r represents the distance between the Earth and the satellite.

[0065] In one embodiment, calculating the gravitational potential energy function and its derivative with respect to time includes:

[0066] Calculate the gravitational potential energy function and its derivative with respect to time.

[0067]

[0068]

[0069] Where F represents the expression for the gravitational force acting on the satellite, G represents the gravitational constant, M represents the mass of the Earth, m represents the mass of the satellite, r represents the distance between the Earth and the satellite, and t represents the launch time.

[0070] In one embodiment, the process expression is based on the fact that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero. Where F represents the expression for the gravitational force acting on the satellite, m represents the satellite's mass, a represents the satellite's acceleration, r represents the distance between the Earth and the satellite, and t represents the launch time.

[0071] In one embodiment, the derivative expression is transformed based on the mechanical conservation law that the sum of the derivatives of kinetic energy and gravitational potential energy with respect to time is zero, to obtain the derivative expression of acceleration, including:

[0072] because Where F = ma, that is The derivative expression for acceleration is:

[0073]

[0074] Where C represents the integration constant and G represents the gravitational constant.

[0075] In one embodiment, the time relationship in the derivative expression of acceleration is converted into a radial position relationship based on the principle of conservation of angular momentum, including:

[0076] The derivative expression for acceleration is: in It is radial velocity. It is angular velocity; according to the law of conservation of angular momentum, It is a constant, where L is the angular momentum, therefore Therefore That is, the radial positional relationship is: C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, t represents the launch time, and M represents the Earth mass.

[0077] In one embodiment, integrating the radial positional relationship yields the satellite's orbital equations, including:

[0078] Integrating the radial positional relationship, we obtain the satellite's orbital equation as follows:

[0079]

[0080] or

[0081] Where C represents the integral constant, m represents the satellite mass, G represents the gravitational constant, r represents the distance between the Earth and the satellite, θ is the launch angle, M represents the Earth mass, L is the angular momentum, and r0 represents the Earth radius.

[0082] It should be understood that, although Figure 1 The steps in the flowchart are shown sequentially as indicated by the arrows, but these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise specified herein, there is no strict order in which these steps are executed, and they can be performed in other orders. Figure 1 At least some of the steps in the process may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed in turn or alternately with other steps or at least some of the sub-steps or stages of other steps.

[0083] In one embodiment, a computer device is provided, which may be a terminal, and its internal structure diagram may be as follows: Figure 2 As shown, the computer device includes a processor, memory, network interface, display screen, and input devices connected via a system bus. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The network interface is used to communicate with external terminals via a network connection. When the computer program is executed by the processor, it implements a satellite orbit prediction method. The display screen can be an LCD screen or an e-ink screen. The input devices can be a touch layer covering the display screen, buttons, a trackball, or a touchpad mounted on the computer device casing, or an external keyboard, touchpad, or mouse.

[0084] Those skilled in the art will understand that Figure 1 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0085] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0086] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0087] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method of predicting the orbit of a satellite, characterized in that, The method comprises: acquiring the mass and launch angle of the satellite; constructing a gravitational force expression of the satellite according to the mass of the satellite and the distance between the satellite and the earth; constructing a differential equation of acceleration of the satellite according to Newton's second law and the gravitational force expression of the satellite; solving the differential equation according to the conservation field characteristics of mechanics, calculating a gravitational potential energy function and a derivative expression of the gravitational potential energy function with respect to time, converting the derivative expression based on the mechanical conservation law that the sum of kinetic energy and gravitational potential energy with respect to time is zero, and obtaining a derivative expression of acceleration; converting the time relationship in the derivative expression of acceleration into a radial position relationship according to the principle of conservation of angular momentum, and integrating the radial position relationship to obtain an orbit equation of the satellite; converting the time relationship in the derivative expression of acceleration into a radial position relationship according to the principle of conservation of angular momentum, and integrating the radial position relationship to obtain an orbit equation of the satellite; The derivative expression of acceleration is expressed as wherein is a radial velocity, is an angular velocity; according to the law of conservation of angular momentum, is a constant, wherein L is an angular momentum, thus , so , that is, the radial position relationship is: , C represents an integral constant, represents a satellite mass, represents a gravitational constant, represents a distance between the earth and the satellite, t represents a launch time, represents an earth mass; wherein the derivative expression of acceleration is: wherein C represents an integration constant, represents the gravitational constant; integrating the radial position relationship to obtain an orbit equation of the satellite, comprising: integrating the radial position relationship to obtain an orbit equation of the satellite, comprising: or wherein C represents the integral constant, represents the satellite mass, represents the gravitational constant, represents the distance between the earth and the satellite, is the launch angle, represents the earth mass, L is the angular momentum, represents the earth radius.

2. The method of claim 1, wherein, integrating the radial position relationship to obtain an orbit equation of the satellite is: calculating a gravitational force expression of the satellite according to the mass of the satellite and the distance between the satellite and the earth, comprising: wherein, G represents the gravitational constant, M represents the mass of the earth, m represents the mass of the satellite, r represents the distance between the earth and the satellite.

3. The method of claim 1, wherein, calculating a gravitational force expression of the satellite according to the mass of the satellite and the distance between the satellite and the earth is: constructing a differential equation of acceleration of the satellite according to Newton's second law and the gravitational force expression of the satellite, comprising: wherein, represents an expression of the gravity received by the satellite, represents a gravitational constant, represents the mass of the earth, represents the mass of the satellite, represents the distance between the earth and the satellite.

4. The method of claim 1, wherein, constructing a differential equation of acceleration of the satellite according to Newton's second law and the gravitational force expression of the satellite is: calculating a gravitational potential energy function and a derivative of the gravitational potential energy function with respect to time, comprising: wherein, represents an expression of gravity received by the satellite, represents a gravitational constant, represents the mass of the earth, represents the mass of the satellite, represents the distance between the earth and the satellite, t represents the launch time.

5. The method of claim 1, wherein, The process expression based on the sum of kinetic and gravitational potential energy derivatives with respect to time being zero is where, represents the gravitational expression to which the satellite is subjected, represents the satellite mass, represents the satellite acceleration, represents the distance of the Earth from the satellite, t represents the launch time.

6. The method of claim 5, wherein, calculating a gravitational potential energy function and a derivative of the gravitational potential energy function with respect to time are respectively: converting the derivative expression based on the mechanical conservation law that the sum of kinetic energy and gravitational potential energy with respect to time is zero, and obtaining a derivative expression of acceleration, comprising: Due to wherein i.e. The derivative expression for acceleration is then: wherein C represents the integral constant, represents the gravitational constant.

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