Transient orbital element and average orbital element conversion method
By converting the number of instantaneous track roots into the number of orbit roots at the vernal equinox point and calculating the amplitude of the short-period term, the problem of high-precision flat-instant conversion in the existing technology is solved, and a high-precision conversion effect suitable for different types of tracks and track perturbations is achieved.
Patent Information
- Application Number
- CN202510198045.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-21
- Publication Date
- 2025-06-10
AI Technical Summary
The prior art is difficult to achieve high-precision conversion of instantaneous orbit roots and average orbit roots, especially when facing the perturbation effects of different types of orbits and various orbits.
By obtaining the instantaneous orbital root number of the spatial target and converting it into the orbital root number of the vernal equinox point, then, based on the perturbation function and the Lagrangian perturbation equation, the differential equation is obtained and the amplitude of the short-period term is calculated, and the accurate average orbital root number is finally obtained.
High-precision flat-instant conversion under the influence of perturbation of different types of tracks and various tracks is realized, avoiding the singular problems of classical methods under zero inclination or zero eccentricity tracks, and improving the conversion accuracy and applicability.
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Figure CN120124285A_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the technical field of orbital element conversion, and particularly to a method for converting instantaneous orbital elements to mean orbital elements. Background Art
[0002] With the development of space technology and the increasing commercial demands, human space launch activities have become increasingly frequent, and the number of on-orbit spacecraft has been continuously increasing. In spacecraft mission planning and control tasks such as orbit correction, maintenance, and maneuvering, mean orbital elements, as variables describing the long-term motion law of spacecraft, are key parameters for mission planning. Therefore, accurately converting instantaneous orbital elements of a spacecraft to mean orbital elements is an important basis for various orbital missions, and it is very necessary to carry out research on the conversion method between mean orbital elements and instantaneous orbital elements.
[0003] For the conversion between instantaneous orbital elements and mean orbital elements, its essence is to eliminate the amplitudes of short-period terms in the instantaneous orbital elements, so as to obtain the mean orbital elements. However, in practical applications, how to perform high-precision mean-instantaneous conversion for various orbital perturbation effects and different types of orbits is a research problem in the field of orbital dynamics and control.
[0004] Currently, the conversion methods between instantaneous orbital elements and mean orbital elements are mainly divided into three categories: batch processing methods, recursive filtering methods, and analytical methods.
[0005] However, batch processing methods and recursive filtering methods cannot achieve mean-instantaneous conversion for a single point; analytical methods cannot be applied to orbits with arbitrary inclinations and eccentricities, have weak applicability to different types of orbits, and have low conversion accuracy. Summary of the Invention
[0006] Based on this, in view of the above technical problems, it is necessary to provide a method for converting instantaneous orbital elements to mean orbital elements, which can eliminate the short-period amplitudes of the instantaneous orbital elements and obtain accurate mean orbital elements.
[0007] A method for converting instantaneous orbital elements to mean orbital elements includes:
[0008] Obtain the instantaneous orbital elements / mean orbital elements of a space target and convert them into corresponding equinox orbital elements;
[0009] Obtain the perturbation functions of the space target under different perturbation forces, substitute the perturbation functions into the Lagrangian perturbation motion equations of the equinox orbital elements, and obtain the differential equations of the equinox orbital elements according to the substituted Lagrangian perturbation motion equations;
[0010] Obtain the corresponding amplitudes of short-period terms according to the differential equations of the equinox orbital elements;
[0011] Based on the orbital elements of the vernal equinox point and the corresponding amplitudes of the short-period terms, the mean orbital elements / instantaneous orbital elements of the space target are obtained.
[0012] In one embodiment, both the instantaneous orbital elements and the mean orbital elements include: semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee, and mean anomaly.
[0013] In one embodiment, the orbital elements of the vernal equinox point include: semi-major axis, mean longitude, eccentricity vector, and inclination vector.
[0014] In one embodiment, different perturbing forces include: the non-spherical gravitational force of the Earth, the three-body gravitational force of the sun and the moon, and the solar radiation pressure.
[0015] In one embodiment, according to the differential equations of the orbital elements of the vernal equinox point, the corresponding amplitudes of the short-period terms are obtained, including:
[0016] The short-period terms in the differential equations of the semi-major axis, mean longitude, eccentricity vector, and inclination vector are respectively integrated indefinitely, and the non-periodic terms after integration are removed, and the amplitudes of the short-period terms corresponding to the semi-major axis, mean longitude, eccentricity vector, and inclination vector are respectively obtained.
[0017] In one embodiment, based on the orbital elements of the vernal equinox point and the corresponding amplitudes of the short-period terms, the mean orbital elements / instantaneous orbital elements of the space target are obtained, including:
[0018] Based on the instantaneous orbital elements of the vernal equinox point / mean orbital elements of the vernal equinox point and the corresponding amplitudes of the short-period terms, the mean orbital elements of the vernal equinox point / instantaneous orbital elements of the vernal equinox point are obtained; based on the mean orbital elements of the vernal equinox point / instantaneous orbital elements of the vernal equinox point, the mean classical orbital elements / instantaneous classical orbital elements of the space target are obtained.
[0019] In one embodiment, based on the instantaneous orbital elements of the vernal equinox point / mean orbital elements of the vernal equinox point and the corresponding amplitudes of the short-period terms, the mean orbital elements of the vernal equinox point / instantaneous orbital elements of the vernal equinox point are obtained; based on the mean orbital elements of the vernal equinox point / instantaneous orbital elements of the vernal equinox point, the mean classical orbital elements / instantaneous classical orbital elements of the space target are obtained, including:
[0020] The instantaneous orbital elements of the vernal equinox point minus the corresponding amplitudes of the short-period terms to obtain the mean orbital elements of the vernal equinox point, and perform conversion to obtain the mean classical orbital elements of the space target;
[0021] Or, the mean orbital elements of the vernal equinox point plus the corresponding amplitudes of the short-period terms to obtain the instantaneous orbital elements of the vernal equinox point, and perform conversion to obtain the instantaneous classical orbital elements of the space target.
[0022] The above conversion method between the instantaneous orbital elements and the mean orbital elements takes into account the influences of three perturbations, namely the non-spherical gravitational force of the Earth, the three-body gravitational force of the sun and the moon, and the solar radiation pressure, on the instantaneous orbital elements. A perturbation differential equation of the equinox orbital elements under the influence of the three perturbation forces is established. The indefinite integral is performed on the short-period terms therein, and the non-periodic terms included in the expression after integration are removed, obtaining an analytical expression for the amplitude of the short-period terms of each equinox orbital element. Subtracting the amplitude of the short-period terms from the instantaneous equinox orbital elements gives the mean equinox orbital elements. The mean-instantaneous conversion method of this application has a complete and accurate model. In the conversion process, the form of the equinox orbital elements is used, avoiding the singularity problem of the classical orbital elements in the orbit with zero inclination (i = 0) or zero eccentricity (e = 0). It is applicable to the analytical conversion requirements of the mean orbital elements and the instantaneous orbital elements of space targets with arbitrary eccentricity and inclination orbits. When obtaining the amplitude of the short-period terms of the equinox orbital elements in this application, the indefinite integral method is adopted and the non-periodic terms included in the expression after integration are removed. This treatment removes the non-periodic part remaining in the short-period amplitude obtained by directly integrating the short-period part of the differential equation, making the obtained short-period amplitude closer to the true value, improving the conversion accuracy between the instantaneous orbital elements and the mean orbital elements, and having strong applicability and high calculation efficiency, and can be widely applied to the field of orbital dynamics and control. BRIEF DESCRIPTION OF THE DRAWINGS
[0023] Figure 1 FIG. is a schematic flow chart of a conversion method between the instantaneous orbital elements and the mean orbital elements in an embodiment;
[0024] Figure 2 FIG. is one of the comparison curves of the mean orbital elements and the instantaneous orbital elements obtained by the method of this application in a specific embodiment;
[0025] Figure 3 FIG. is another comparison curve of the mean orbital elements and the instantaneous orbital elements obtained by the method of this application in a specific embodiment;
[0026] Figure 4 FIG. is yet another comparison curve of the mean orbital elements and the instantaneous orbital elements obtained by the method of this application in a specific embodiment;
[0027] Figure 5 FIG. is the fourth comparison curve of the mean orbital elements and the instantaneous orbital elements obtained by the method of this application in a specific embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS
[0028] To make the objectives, technical solutions and advantages of this application more clear and understandable, the following further elaborates on this application in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely used to explain this application and are not intended to limit this application. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in this application without creative efforts fall within the scope of protection of this application.
[0029] In addition, in this application, descriptions such as "first" and "second" are only for descriptive purposes and should not be construed as indicating or implying their relative importance or implicitly specifying the quantity of the indicated technical features. Thus, features defined with "first" and "second" may explicitly or implicitly include at least one such feature. In the description of this application, the meaning of "multiple groups" is at least two groups, such as two groups, three groups, etc., unless otherwise specifically and clearly defined.
[0030] In this application, unless otherwise clearly specified and limited, terms such as "connection" and "fixation" should be understood in a broad sense. For example, "fixation" can be a fixed connection, a detachable connection, or integrated; it can be a mechanical connection, an electrical connection, a physical connection, or a wireless communication connection; it can be directly connected or indirectly connected through an intermediate medium, and can be the communication inside two components or the interaction relationship between two components, unless otherwise clearly limited. For those of ordinary skill in the art, the specific meanings of the above terms in this application can be understood according to specific circumstances.
[0031] In addition, the technical solutions between various embodiments of this application can be combined with each other, but it must be based on the ability of those of ordinary skill in the art to implement. When the combination of technical solutions results in contradictions or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection required by this application.
[0032] This application provides a method for converting instantaneous orbital elements and mean orbital elements, as Figure 1 shown in the schematic flowchart. In one embodiment, it includes:
[0033] Step 101, obtain the instantaneous orbital elements / mean orbital elements of a space target and convert them into corresponding equinox orbital elements.
[0034] Specifically:
[0035] Obtain the instantaneous classical orbital elements / mean classical orbital elements of a space target (such as a satellite) and convert them into instantaneous equinox orbital elements / mean equinox orbital elements;
[0036] The conversion process is as follows:
[0037]
[0038] In the formula, a is the semi-major axis, e is the eccentricity, i is the orbital inclination, Ω is the right ascension of the ascending node, ω is the argument of perigee, M is the mean anomaly, λ is the mean longitude, e x is the component of the eccentricity vector in the x direction, e y is the component of the eccentricity vector in the y direction, i x is the component of the inclination vector in the x direction, i y is the component of the inclination vector in the y direction.
[0039] In this step, the instantaneous orbital elements include: semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of perigee, and mean anomaly. The equinox orbital elements include: semi-major axis, mean longitude, eccentricity vector, and inclination vector.
[0040] Step 102: Obtain the perturbation function of the space target under different perturbing forces, substitute the perturbation function into the Lagrangian perturbation motion equation of the equinox orbital elements, and obtain the differential equation of the equinox orbital elements according to the substituted Lagrangian perturbation motion equation.
[0041] Specifically:
[0042] Obtain the perturbation function of the space target under the three perturbing forces of the non-spherical gravity of the Earth, the three-body gravity of the sun and the moon, and the solar radiation pressure. Substitute the perturbation function into the Lagrangian perturbation motion equation of the equinox orbital elements, and obtain the differential equation of the equinox orbital elements under the influence of the three perturbing forces.
[0043] More specifically:
[0044] Obtain the perturbation function of the space target under the perturbation of the non-spherical gravity of the Earth:
[0045]
[0046] In the formula, R E is the perturbation function of the space target under the perturbation of the non-spherical gravity of the Earth, μ is the Earth's gravitational constant, r is the distance from the Earth's center to the satellite in the J2000 coordinate system, r e is the Earth's equatorial radius, is the geocentric latitude, J 2 、J 22 、J 3 、J 31 、J 32 、J 33 、J 4 are different spherical harmonic coefficients, λ c 、λ c,22 、λ c,31 、λ c,32 、λc,33 are different gravitational model parameters;
[0047] Obtain the perturbation function of a space target under the gravitational perturbation of the sun, moon, and the target itself:
[0048]
[0049] Among them, performing Legendre polynomial decomposition on the 1 / ||r k - r‖ term gives:
[0050]
[0051] In the formula, R ls is the perturbation function of the space target under the gravitational perturbation of the sun, moon, and the target itself; μ k is the gravitational constant of the perturbing body. When the subscript k = s, μ s is the solar gravitational constant. When the subscript k = m, μ m is the lunar gravitational constant; r k = r k (x k , y k , z k ) T is the position vector of the perturbing body in the J2000 coordinate system, obtained through the JPL ephemeris; r = r(x, y, z) T is the position vector of the satellite in the J2000 coordinate system, calculated from the orbital elements; r k is the distance from the center of the earth to the perturbing body in the J2000 coordinate system, cosθ k = (r·r k ) / (r·r k ) is the cosine of the angle between the two position vectors r k and r;
[0052] Obtain the perturbation function of the space target under the solar radiation pressure perturbation:
[0053]
[0054] In the formula, R sp is the perturbation function of the space target under the solar radiation pressure perturbation, C R is the solar radiation pressure coefficient, A is the surface area of the satellite facing the sun, m is the mass of the satellite, P 0 is the solar radiation pressure near the earth, x is the x - component of the position vector of the satellite in the J2000 coordinate system, y is the y - component of the position vector of the satellite in the J2000 coordinate system, z is the z - component of the position vector of the satellite in the J2000 coordinate system, x s is the x - component of the position vector of the sun in the J2000 coordinate system, y sis the y - component of the position vector of the sun in the J2000 coordinate system, z s is the z - component of the position vector of the sun in the J2000 coordinate system;
[0055] Substitute the perturbation function of the space target under the non - spherical gravitational perturbation of the earth, the perturbation function of the space target under the three - body gravitational perturbation of the sun and moon, and the perturbation function of the space target under the solar radiation pressure perturbation into the Lagrangian perturbation motion equation of the equinox orbital elements, and obtain the Lagrangian perturbation motion equation after substitution:
[0056]
[0057] In the formula, t is time, and n is the mean motion, that is, the average angular velocity required for the satellite to complete a full orbital period;
[0058] According to the Lagrangian perturbation motion equation after substitution, the differential equation of the equinox orbital elements is obtained; among them, the differential equation of the equinox orbital elements includes: the differential equation of the semi - major axis, the differential equation of the mean longitude, the differential equation of the eccentricity vector, and the differential equation of the inclination vector;
[0059] The differential equation of the semi - major axis is:
[0060]
[0061] Among them,
[0062]
[0063] In the formula, a is the semi - major axis, the subscript E indicates that this term is affected by the non - spherical gravitational force of the earth, the subscript ls indicates that this term is affected by the three - body gravitational force of the sun and moon, and the subscript sp indicates that this term is affected by the solar radiation pressure;
[0064] The differential equation of the mean longitude is:
[0065]
[0066] Among them,
[0067]
[0068]
[0069] In the formula, λ is the mean longitude, f is the true anomaly; the differential equation of the eccentricity vector is:
[0070]
[0071]
[0072] In the formula, e xis the x - component of the eccentricity vector, e y is the y - component of the eccentricity vector;
[0073] The differential equation of the inclination vector is:
[0074]
[0075] where, i x is the x - component of the inclination vector, i y is the y - component of the inclination vector.
[0076] In this step, the Lagrangian - type perturbation motion equations of the equinox orbital elements are prior art and will not be elaborated here.
[0077] Step 103: Obtain the corresponding short - period term amplitudes according to the differential equations of the equinox orbital elements.
[0078] Specifically:
[0079] Perform indefinite integration on the short - period terms in the differential equations of the equinox orbital elements, and remove the non - periodic terms after integration to obtain the corresponding short - period term amplitudes respectively;
[0080] Among them, the instantaneous orbital elements are:
[0081]
[0082] The short - period term amplitudes are:
[0083]
[0084] where, E is the orbital element, is the change of the mean orbital element, (dE / dt) short is the change of the short - period term corresponding to the orbital element, ΔE short is the short - period term amplitude, and C is the non - periodic term in the expression after integrating the short - period term.
[0085] More specifically:
[0086] Perform indefinite integration on the short - period terms in the differential equation of the semi - major axis, and remove the non - periodic terms after integration to obtain the short - period term amplitude corresponding to the semi - major axis:
[0087]
[0088] Among them,
[0089]
[0090]
[0091] where, Δashort is the amplitude of the short-period term corresponding to the semi-major axis, C 1 is the corresponding non-periodic term, is the short-period term part of is the short-period term part of is the short-period term part of is the short-period term part of is the short-period term part of (1 / r) 5 short is (1 / r) 5 the short-period term part of the corresponding non-periodic term, (cos 4 θ k ) short is (cos 4 θ k ) the short-period term part of (cos 2 θ k ) short is (cos 2 θ k ) the short-period term part of C 3 is the corresponding non-periodic term;
[0092] Integrate the short-period terms in the differential equation of the mean longitude indefinitely and remove the non-periodic terms after integration to obtain the amplitude of the short-period term corresponding to the mean longitude:
[0093]
[0094] where,
[0095]
[0096]
[0097]
[0098]
[0099] In the formula, Δλ short is the amplitude of the short-period term corresponding to the mean longitude, C 4 is the non-periodic term corresponding to ∫(n) short dt, C 5 is the corresponding non-periodic term, C 6 is the corresponding non-periodic term, C7 is the corresponding aperiodic term, C 8 is the corresponding aperiodic term, C 9 is the corresponding aperiodic term, C 10 is the corresponding aperiodic term, C 11 is the corresponding aperiodic term, C 12 is the corresponding aperiodic term, C 13 is the corresponding aperiodic term, C 14 is the corresponding aperiodic term, C 15 is the corresponding aperiodic term, C 16 is the corresponding aperiodic term, C 17 is the corresponding aperiodic term, C 18 is the corresponding aperiodic term, is the short-period term part of, is the short-period term part of, is the short-period term part of;
[0100] Indefinite integration is performed on the short-period terms in the differential equation of the eccentricity vector, and the aperiodic terms after integration are removed to obtain the short-period term amplitude of the corresponding eccentricity vector:
[0101]
[0102] In the formula, Δe x,short is the short-period term amplitude of the component of the corresponding eccentricity vector in the x direction, Δe y,short is the short-period term amplitude of the component of the corresponding eccentricity vector in the y direction;
[0103] Indefinite integration is performed on the short-period terms in the differential equation of the inclination vector, and the aperiodic terms after integration are removed to obtain the short-period term amplitude of the corresponding inclination vector:
[0104]
[0105] In the formula, Δi x,short is the short-period term amplitude of the component of the corresponding inclination vector in the x direction, Δi y,short is the short-period term amplitude of the component of the corresponding inclination vector in the y direction.
[0106] In this step, the amplitudes of the short-period terms corresponding to the orbital elements of the vernal equinox are obtained respectively.
[0107] Step 104: Obtain the mean orbital elements / instantaneous orbital elements of the space target according to the orbital elements of the vernal equinox and the corresponding amplitudes of the short-period terms.
[0108] Specifically:
[0109] Obtain the mean orbital elements of the vernal equinox / instantaneous orbital elements of the vernal equinox according to the instantaneous orbital elements of the vernal equinox / mean orbital elements of the vernal equinox and the corresponding amplitudes of the short-period terms; obtain the mean classical orbital elements / instantaneous classical orbital elements of the space target according to the mean orbital elements of the vernal equinox / instantaneous orbital elements of the vernal equinox.
[0110] More specifically:
[0111] Subtract the corresponding amplitude of the short-period term from the instantaneous orbital elements of the vernal equinox to obtain the mean orbital elements of the vernal equinox:
[0112]
[0113] In the formula, is the mean value of the semi-major axis, is the mean value of the mean longitude, is the mean value of the x-component of the eccentricity vector, is the mean value of the y-component of the eccentricity vector, is the mean value of the x-component of the inclination vector, is the mean value of the y-component of the inclination vector;
[0114] And perform a transformation on the mean orbital elements of the vernal equinox to obtain the mean classical orbital elements of the space target;
[0115] Or, add the corresponding amplitude of the short-period term to the mean orbital elements of the vernal equinox to obtain the instantaneous orbital elements of the vernal equinox, and perform a transformation on the instantaneous orbital elements of the vernal equinox to obtain the instantaneous classical orbital elements of the space target.
[0116] In this step, the mean orbital elements include: the mean value of the semi-major axis the mean value of the eccentricity the mean value of the orbital inclination the mean value of the right ascension of the ascending node the mean value of the argument of perigee and the mean value of the mean anomaly
[0117] The specific inverse transformation and transformation process belong to the prior art and will not be elaborated here.
[0118] The above conversion method between instantaneous orbital elements and mean orbital elements considers the influences of three perturbations, namely the non-spherical gravitational force of the Earth, the gravitational force of the Sun-Moon-Earth three-body system, and the solar radiation pressure, on the instantaneous orbital elements. A perturbation differential equation of the equinox orbital elements under the influence of the three perturbation forces is established. The indefinite integral is performed on the short-period terms therein, and the non-periodic terms included in the expression after integration are removed, obtaining an analytical expression for the amplitudes of the short-period terms of each equinox orbital element. Subtracting the amplitude of the short-period term from the instantaneous equinox orbital elements gives the mean equinox orbital elements. The mean-instantaneous conversion method of this application has a complete and accurate model. In the conversion process, the form of the equinox orbital elements is used, avoiding the singularity problem of the classical orbital elements in the orbit with zero inclination (i = 0) or zero eccentricity (e = 0). It is applicable to the analytical conversion requirements of the mean orbital elements and instantaneous orbital elements of space targets with arbitrary eccentricity and inclination orbits. When obtaining the amplitudes of the short-period terms of the equinox orbital elements in this application, the indefinite integral method is adopted and the non-periodic terms included in the expression after integration are removed. This treatment removes the non-periodic part remaining in the short-period amplitude obtained by directly integrating the short-period part of the differential equation, making the obtained short-period amplitude closer to the true value, improving the conversion accuracy between the instantaneous orbital elements and the mean orbital elements, and having strong applicability and high calculation efficiency, and can be widely applied to the field of orbital dynamics and control.
[0119] It should be understood that although Figure 1 the steps in the flowchart of Figure 1 are shown in sequence according to the indication of the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless there is a clear indication in this article, the execution of these steps has no strict order limit, and these steps can be executed in other orders. Moreover,
[0120] In a specific embodiment, the current epoch of the space target is set as t 0 which is 12:00:00 (UTC) on March 12, 2018. The instantaneous orbital elements at this epoch are set as follows: the semi-major axis a is 42164.8 km, the eccentricity e is 0.0003, the orbital inclination i is 0.009°, the right ascension of the ascending node Ω is 54°, the argument of perigee ω is 162°, and the mean anomaly M is 280°. The instantaneous classical orbital elements are converted into the corresponding equinox orbital elements: the semi-major axis a is 42164.8 km, the mean longitude λ is 145.912 deg, the eccentricity e x is -0.000242705, e yis -0.000176336, the inclination angle i x is 0.000461646, i y is 0.000635400.
[0121] Regarding the conversion problem between the mean orbital elements and the instantaneous orbital elements of a space target, the influences of three perturbation forces are considered: the non-spherical gravitational force of the Earth, the three-body gravitational force of the sun and the moon, and the solar radiation pressure. Dynamic modeling is carried out for the three perturbation effects.
[0122] The perturbation term of the non-spherical gravitational force of the Earth is considered up to the 4th order, and the perturbation function is as follows:
[0123]
[0124] Among them, r e is the equatorial radius of the Earth, which is 6378.137 km, μ is the gravitational constant of the Earth, which is 3.986×10^14 m 3 / s 2 , r is the distance from the center of the Earth to the satellite in the J2000 coordinate system, which is 42164.8 km, J 2 , J 22 , J 3 , J 31 , J 32 , J 33 and J 4 are spherical harmonic coefficients, and λ c,22 , λ c,31 , λ c,32 , λ c,33 The gravity model parameters all adopt the parameters of the JGM3 Earth gravitational field model, and the specific parameter values are prior art.
[0125] The perturbation function of the three-body gravitational force of the sun and the moon is as follows:
[0126]
[0127] Among them, r k = r k (x k , y k , z k ) T and r = r(x, y, z) T respectively define the position vectors of the satellite and the perturbing body in the J2000 coordinate system;
[0128] At the epoch of the current scenario:
[0129] r m = [167013432.428251, -342210000.283064, -134765571.245830] km,
[0130] r s = [147065439602.912, -19902966061.0253, -8628482516.08462] km,
[0131] cosθ m = -0.885932, cosθ s = -0.804644.
[0132] For 1 / ||r k - r|| term, perform Legendre polynomial decomposition, and the solar-lunar three-body gravitational perturbation function is converted to:
[0133]
[0134] The solar radiation pressure perturbation function is as follows:
[0135]
[0136] where C R represents that the solar radiation pressure coefficient is 1.5, A / m is 0.02 m 2 / kg, P 0 is the solar radiation pressure near the Earth, which is 4.56e-6 N / m 2 .
[0137] Substitute the established perturbation function into the Lagrangian perturbation motion equation of the equinox orbital elements, as shown in the following formula:
[0138]
[0139] Perform indefinite integration on the perturbation differential equations of each equinox orbital element and remove the non-periodic terms to obtain the analytical expression of the corresponding short-period term amplitude.
[0140] Obtain the analytical expression of the short-period term amplitude of the semi-major axis a:
[0141]
[0142] Obtain the analytical expression of the short-period term amplitude of the mean longitude λ:
[0143]
[0144] Obtain the eccentricities e x and e y of the short-period term amplitude analytical expressions:
[0145]
[0146] Obtain the inclinations i x and iy Analytical expression for the amplitude of the short-period term:
[0147]
[0148]
[0149] The orbital elements of the instantaneous vernal equinox minus the corresponding amplitude of the short-period term: Δa short is 740.734 m, Δλ short is -1.96720e-05 rad, Δe x,short is -2.79397e-05, Δe y,short is 3.55871e-05, Δi x,short is -5.61658e-06, Δi y,short is 7.72498e-06, obtaining the orbital elements of the mean vernal equinox: is 42164.059 km, is 145.913 deg, is -0.000214765, is -0.000211923, is 0.000467262, is 0.000627676.
[0150] Regarding the mean orbital element values calculated by the batch processing method as reference values, the conversion errors of the mean orbital elements in this embodiment are calculated as follows: a is 36.10 m, λ is 4.219e-4 deg, e x is 2.444e-5, e y is 2.083e-5, i x is 4.474e-6, i y is 6.545e-6.
[0151] Meanwhile, in order to verify the superiority of this application in conversion accuracy, it is compared with three traditional mean-instantaneous conversion methods: Brouwer-Lyddane Long method, Brouwer-Lyddane Short method, and Kozai-Izsak method.
[0152] As shown in Table 1, the comparison of the mean orbital element conversion accuracy between the method of this application and the three classical methods is given. It can be seen that the conversion accuracy of the conversion method of this application is much higher than that of the three classical methods.
[0153] Table 1: Comparison of the mean orbital element conversion accuracy between the method of this application and the three classical methods
[0154]
[0155] As Figures 2 to 5 shown, the comparison curve between the mean equinox orbital elements and the instantaneous orbital elements obtained by converting the method of the present application is given. It can be seen that the method of the present application can eliminate the short-period amplitude of the instantaneous orbital elements and convert to obtain accurate mean orbital elements.
[0156] The content not described in detail in this specification belongs to the prior art well known to those skilled in the art.
[0157] The technical features of the above embodiments can be combined arbitrarily. For the sake of concise description, 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, it should be considered as the scope described in this specification.
[0158] The above-described embodiments merely represent several implementation manners of the present application. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the present application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several deformations and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.
Claims
1. A method for converting instantaneous orbital elements to average orbital elements, characterized in that: include: Obtain the instantaneous orbital element number / average orbital element number of the space target and convert it into the corresponding vernal equinox orbital element number; Obtain the perturbation function of the space target under different perturbation forces, substitute the perturbation function into the Lagrangian perturbation motion equation of the vernal equinox orbital element, and obtain the differential equation of the vernal equinox orbital element based on the Lagrangian perturbation motion equation after substitution; According to the differential equation of the orbital roots of the vernal equinox, the corresponding amplitude of the short-period term is obtained; According to the orbital elements of the vernal equinox and the corresponding short-period term amplitude, the average orbital elements / instantaneous orbital elements of the space target are obtained.
2. The method for converting instantaneous orbital elements to average orbital elements according to claim 1, characterized in that the instantaneous The orbital elements and mean orbital elements include: semi-major axis, eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly.
3. The method for converting instantaneous orbital elements to average orbital elements according to claim 2, characterized in that: The orbital elements of the vernal equinox include: semi-major axis, mean longitude, eccentricity vector, and inclination vector.
4. The method for converting instantaneous orbital elements to average orbital elements according to claim 3, characterized in that: Different perturbations include: the Earth's non-spherical gravity, the gravity of the Sun and the Moon, and solar pressure.
5. The method for converting instantaneous orbital elements to average orbital elements according to any one of claims 1 to 4, characterized in that: According to the differential equation of the orbital roots of the vernal equinox, the corresponding short-period term amplitude is obtained, including: The short-period terms in the differential equations of the semi-major axis, mean longitude, eccentricity vector and inclination vector are respectively integrated indefinitely, and the non-periodic terms after integration are removed to obtain the amplitudes of the short-period terms corresponding to the semi-major axis, mean longitude, eccentricity vector and inclination vector.
6. A method for converting instantaneous orbital elements to average orbital elements according to any one of claims 1 to 4, characterized in that: According to the orbital elements of the vernal equinox and the corresponding short-period term amplitude, the average orbital elements / instantaneous orbital elements of the space target are obtained, including: According to the instantaneous equinox orbital elements / average equinox orbital elements and the corresponding short-period term amplitude, the average equinox orbital elements / instantaneous equinox orbital elements are obtained; according to the average equinox orbital elements / instantaneous equinox orbital elements, the average classical orbital elements / instantaneous classical orbital elements of the space target are obtained.
7. The method for converting instantaneous orbital elements to average orbital elements according to claim 6, characterized in that: According to the instantaneous equinox orbital elements / average equinox orbital elements and the corresponding short-period term amplitude, the average equinox orbital elements / instantaneous equinox orbital elements are obtained; According to the mean vernal equinox orbital elements / instantaneous vernal equinox orbital elements, the mean classical orbital elements / instantaneous classical orbital elements of the space target are obtained, including: The instantaneous equinox orbital element minus the corresponding short-period term amplitude is obtained to obtain the average equinox orbital element, and then converted to obtain the average classical orbital element of the space target; Or, the mean equinox orbital element is added to the corresponding short-period term amplitude to obtain the instantaneous equinox orbital element, and then converted to obtain the instantaneous classical orbital element of the space target.