Method for solving refracted starlight path in atmosphere independent of atmospheric density model

Abstract

The invention relates to a method for solving a refracted starlight path in atmosphere, and belongs to the field of celestial navigation. The method comprises the following steps: firstly, carrying out Taylor expansion on a Snell theorem by utilizing an infinitesimal thought to obtain a function of a curvature of a starlight path relative to a refractive index; then constructing a second-order differential equation of a refracted starlight path by using a Gladstone-Dale theorem and a curvature equation under a Cartesian coordinate system; and finally, carrying out numerical integration on the differential equation, and solving to obtain an expression of a refracted starlight path in the atmospheric profile. According to the method, the existing atmospheric density observation data is directly utilized, the method does not depend on an atmospheric density mathematical model, and the propagation path of the refracted starlight in the complex atmospheric environment is calculated by utilizing a numerical integration method, so that the problem of overlarge deviation from the actual working condition caused by ideal hypothesis and a simple atmospheric mathematical model is avoided. The method is especially suitable for refraction light path resolving of media with irregular refractive indexes, continuous and guidable distribution, and can improve the precision of quantity measurement and the precision of navigation and positioning in an indirect sensitive horizon navigation and positioning method.

Description

(I) Technical field

[0001] The invention belongs to the field of celestial navigation and is a method for solving the path of starlight refracted in the atmosphere which is independent of the atmospheric density model. (II) Background technology

[0002] Using starlight refraction to achieve indirect sensitive horizon is an important way of spacecraft positioning technology. How to use atmospheric observation information to solve the starlight path is the key link of the starlight refraction positioning algorithm. For the problem of light transmission in non-uniform media, it is difficult to obtain an analytical solution to the refracted starlight path equation, so it is necessary to use computer numerical methods to obtain a numerical solution to the refracted starlight path.

[0003] At present, the ray tracing research on media with known refractive index distribution has been mature, but the ray tracing research on media with irregular refractive index distribution is relatively small. In existing studies, some require pre-division of the grid, and this method requires a high degree of grid fineness; some need to specifically deal with the problem of light rotation. The general assumption of the traditional method of calculating the refraction of starlight in the atmosphere is that the atmospheric density decreases strictly exponentially with altitude and the temperature of the atmosphere is consistent at different altitudes. This assumption is not strictly valid, and the starlight refraction angle obtained under this assumption is only an approximation. Based on Snell's theorem and Gladstone-Dale's theorem, the present invention constructs a differential equation of atmospheric density with respect to the starlight path through calculation of the curvature of the light path. It does not rely on the mathematical model of atmospheric density, and can use numerical integration to obtain the propagation path of starlight in the atmosphere. (III) Summary of the invention

[0004] The object of the present invention is to provide a method for solving the path of starlight refracted in the atmosphere which is independent of the atmospheric density model.

[0005] The purpose of the present invention is achieved through the following technologies:

[0006] Firstly, the Snell theorem is Taylor expanded using the idea of ​​differential element to obtain the function of the curvature of the starlight path relative to the refractive index. Then, the Gladstone-Dale theorem and the curvature equation in the Cartesian coordinate system are used to construct the second-order differential equation of the refracted starlight path. Finally, the differential equation is numerically integrated to obtain the expression of the refracted starlight path in the atmospheric profile.

[0007] The beneficial effects of the present invention are described as follows:

[0008] In the traditional navigation and positioning method of indirect sensitive horizon of starlight refraction, the mathematical model of atmospheric density with respect to altitude is simply modeled as an exponential function, and the mathematical model of refraction angle with respect to refraction altitude is also mostly an exponential function. Compared with the traditional method, the present invention directly uses the existing atmospheric density observation data, does not rely on the atmospheric density mathematical model, and uses the numerical integration method to calculate the propagation path of refracted starlight in a complex atmospheric environment, thereby avoiding the problem of excessive deviation from actual working conditions caused by ideal assumptions and simple atmospheric mathematical models. It is particularly suitable for solving the refraction optical path of a medium with irregular continuous and derivable distribution of refractive index, and can improve the accuracy of quantitative measurement and navigation and positioning in the indirect sensitive horizon navigation and positioning method. (IV) Description of the drawings

[0009] In order to further illustrate the technical solution of the present invention, some drawings required for describing some embodiments or prior art solutions are briefly introduced. Among them:

[0010] Figure 1 The steps for implementing the method for solving the path of starlight refracted in the atmosphere that does not rely on the atmospheric density model;

[0011] Figure 2 A schematic diagram of the relevant physical quantities involved in the derivation of the curvature of the refracted starlight path;

[0012] Figure 3 is the established starlight refraction coordinate system. (V) Specific implementation methods

[0013] The present invention is described in more detail below:

[0014] Assume that the starlight path corresponding to the lens entrance pupil passes through two points M and N that are close enough in space, and the arc length between the two points is △s, and the angle of refraction is △τ. According to the definition of the radius of curvature, the radius of curvature of the light path is R L It can be expressed as

[0015]

[0016] Because the distance between points M and N is close enough, the distance from the center of the earth, the refractive index, and the starlight elevation angle at M and N can be made to differ by a small amount, denoted as r,n,i and r+△r,n+△n,i+△i. From the geometric relationship, we know that

[0017]

[0018]

[0019] Assuming that the atmosphere is spherical and the atmospheric refractive index is equal at the same altitude, Snell's theorem can be applied to points M and N to obtain

[0020] rncosi=(r+△r)(n+△n)cos(i+△i) (5)

[0021] Perform Taylor expansion and retain the first-order small quantity, and we have

[0022]

[0023] Combining equations (3), (4), and (6), we get

[0024]

[0025] For the stratospheric atmosphere, we can assume that n≈1, and substitute it into equation (1) and equation (2) to obtain

[0026]

[0027] For ease of description, create the starlight refraction coordinate system Ox r y r, , the coordinate origin O coincides with the center of mass of the earth, x r The y axis is consistent with the incident direction of starlight, r Axis perpendicular to x r The axis is in the plane where the starlight is refracted and bent toward the center of the earth. For any point on the light path in the starlight refraction coordinate system, according to the calculation formula (8) for the radius of curvature, we have

[0028]

[0029] From the geometric relationship, we can know that the cosine value of the starlight elevation angle can be expressed as

[0030]

[0031] Since the atmospheric density in the stratosphere decreases with increasing altitude, the ray will bend toward the center of the earth, so y″<0. Combining equations (8), (9), and (10), we get

[0032]

[0033] To solve equation (11), we need to first obtain the derivative of the atmospheric refractive index with respect to the distance from the center of the earth: According to the Gladstone-Dale theorem, we have

[0034]

[0035] Where: k is the Gladstone-Dale constant, which is a constant related to the wavelength λ; ρ is the atmospheric density. Using the tabulated data of air density at different altitudes in the US standard atmospheric model as input, the atmospheric density derivative can be obtained by solving the four-point central differential formula.

[0036]

[0037] Where: h is the height from the earth's surface. For the spherical atmosphere model, the earth's radius R e can be regarded as a constant, so the derivative of the atmospheric density with respect to the distance from the center of the earth is numerically equal to the derivative of the atmospheric density with respect to the height of the earth's surface. Substituting into equation (12) and in parallel with equation (13), we get

[0038]

[0039] Formula (14) shows that is a function of x and y, denoted by Substituting into equation (11), we can obtain the second-order differential equation of any point y on the optical path with respect to x:

[0040]

[0041] For a parallel light beam emitted by a navigation star, the corresponding starlight refraction coordinate system is established by arbitrarily selecting an atmospheric profile and numerically integrating the second-order differential equation described by equation (11), then the expression of the complete refraction light path can be obtained.

Claims

1. A method for solving the path of starlight refracted in the atmosphere, characterized in that: The method comprises the following steps: Step 1: Use the idea of ​​microelement to obtain the function of the curvature of the starlight path relative to the refractive index; Step 2: construct the second-order differential equation of the refracted starlight path; Step three, numerically integrate the differential equation to obtain an expression for the path of refracted starlight in the atmospheric profile.

2. The method according to claim 1, characterized in that This method uses the idea of ​​infinitesimal elements to perform Taylor expansion on Snell's theorem and obtains the function of the curvature of the starlight path relative to the refractive index.

3. The method according to claim 1, characterized in that This method uses the Gladstone-Dale theorem and the curvature equation in the Cartesian coordinate system to construct the second-order differential equation of the refracted starlight path.

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