Calculation Method for Orbital Evolution of Debris Below Centimeter Level

Abstract

The method for calculating the orbital evolution of sub - centimeter debris disclosed by the present invention includes the following steps: Step 1, calculate the debris distribution characteristics; Step 2, calculate the initial position and velocity of the debris; Step 3, superimpose the accurate acceleration model of the debris and extrapolate to calculate the orbital evolution of the debris. The method for calculating the orbital evolution of sub - centimeter debris of the present invention can effectively reduce the calculation error of the orbital evolution of existing sub - centimeter debris, which is beneficial to the long - term prediction of the orbits of sub - centimeter debris during the daily supervision of spacecraft, extend the possible collision prediction time of debris to spacecraft, and increase the effective time of collision warning.

Description

Technical Field

[0001] The present invention belongs to the field of space technology, and particularly relates to a method for calculating the orbital evolution of debris below centimeter level. Background Art

[0002] The waste debris left in space during space activities and orbiting due to the earth's gravity is called space debris. The debris includes the payloads of out-of-control spacecraft, the wreckage and debris scattered during the operation of spacecraft, the debris generated by the explosion and impact of space targets, as well as the aluminum flakes and paint flakes ejected by the engine, and the detachment of the spacecraft coating film, etc. Debris below centimeter level has a much larger surface area, and the accelerations caused by atmospheric drag, radiation pressure, Lorentz force, etc. are much larger than those of large-sized debris and spacecraft in the same orbit. This results in much larger errors in the calculation of the orbital evolution of debris below centimeter level compared to that of large-sized debris and spacecraft. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for calculating the orbital evolution of debris below centimeter level, which solves the problem of relatively large errors in the calculation of the orbital evolution of existing debris below centimeter level.

[0004] The technical solution adopted by the present invention is: a method for calculating the orbital evolution of debris below centimeter level, including the following steps:

[0005] Step 1: Calculate the debris distribution characteristics;

[0006] Step 2: Calculate the initial position and velocity of the debris;

[0007] Step 3: Superimpose the precise acceleration model of the debris and extrapolate to calculate the orbital evolution of the debris.

[0008] The present invention is also characterized in that,

[0009] In Step 1, calculating the debris distribution characteristics is to take the distribution density function of the debris velocity as:

[0010]

[0011] In formula (1), v Col is the target collision velocity that generates a large amount of debris, v m is the maximum relative velocity of the debris generated by the collision with respect to the parent body, β = 8.69, δ = 7.2×10 -3 s·m -1 .

[0012] In Step 2, the initial position and velocity of the debris are calculated through formula (2):

[0013]

[0014] In Equation (2), r and v are the reference position and velocity of the debris, r d and v d are the initial position and velocity of the debris, and δv is the unit direction, pointing to a random distribution.

[0015] Specifically, in Step 3, the orbital evolution of the debris is calculated by computing the gravitational force of the Earth, the gravitational forces of the sun and the moon, the atmospheric drag, the solar radiation pressure, the Lorentz force, and the Poynting-Robertson drag force acting on the debris. The extrapolation of the motion of the debris represented in the inertial space is calculated by integrating according to Equation (3):

[0016]

[0017] In Equation (3), r is the position vector from the center of the Earth to the debris relative to the inertial space, f S and f M are the gravitational accelerations produced by the Earth, the sun, and the moon on the debris, V is the gravitational potential of the Earth; f A and f SR are the accelerations from the atmospheric drag and the solar radiation pressure, respectively; f L is the acceleration produced by the Lorentz force of the Earth's magnetic field; f PR is the acceleration produced by the Poynting-Robertson drag force.

[0018] In Step 3, the gravitational potential of the Earth is calculated according to Equation (4):

[0019]

[0020] In Equation (4), G is the gravitational constant, M E is the mass of the Earth, r = |r|, R e is the equatorial radius of the Earth, λ and ψ are the geocentric longitude and latitude of the debris position, and are the normalized spherical harmonic coefficients of degree n and order m, is the normalized Legendre polynomial, and N is the maximum order used in the formula.

[0021] In Step 3, the gravitational accelerations produced by the sun and the moon on the debris are calculated according to Equations (5) and (6):

[0022]

[0023]

[0024] In Equations (5) and (6), G is the gravitational constant, M S and M M are the masses of the sun and the moon, respectively, r DS and rDM are the position vectors from the sun and the moon to the debris, r S and r M are the position vectors from the geocenter to the heliocenter and the geocenter to the barycenter of the moon, r DS and r DM is r DS and r DM are the magnitudes of r S =|r S |, r M =|r M |.

[0025] The acceleration of the solar radiation pressure in step 3 is calculated according to formula (7):

[0026]

[0027] In formula (7), the solar constant is S0 = 1.3608×10 6 ergs·cm -2 ·s -1 A s is the radiation pressure cross-sectional area of the debris, m d is the debris mass, c is the speed of light, r s0 =1AU, r sd is the distance from the sun to the debris, Q pr is the solar radiation pressure coefficient, r SR is the unit vector in the direction from the sun to the debris.

[0028] The acceleration of the atmospheric drag force on the debris in step 3 is calculated according to formula (8):

[0029]

[0030] In formula (8), the atmospheric drag coefficient is C d =2.2, A is the atmospheric drag cross-sectional area of the debris, ρ is the atmospheric density at the debris orbit position, v is the debris velocity vector in the inertial space, v = |v|.

[0031] The acceleration of the Lorentz force on the debris in step 3 is calculated according to formula (9):

[0032]

[0033] In formula (9), the permittivity is ε0 = 8.854187817×10 12 F·m -1 , the voltage of the debris surface after charging to charge balance in the space physical environment is U = +5V, s is the average radius of the debris, m d is the debris mass, ω is the angular velocity vector of the earth's spin; the magnetic field strength at the debris position is calculated according to formula (10):

[0034]

[0035] In Equation (10), the magnetic scalar potential energy is V mag Calculate according to Equation (11):

[0036]

[0037] In Equation (11), the average reference radius of the geomagnetic field is a = 6371.2 km, θ and are respectively the geocentric inertial coordinate system longitude and latitude of the position where the debris is located, t is the time, n and m are respectively the degree and order of the magnetic field strength calculation, and are respectively the Gaussian coefficients of the geomagnetic field, is the normalized Legendre function, N gm is the maximum order of the magnetic field strength calculation used.

[0038] The Poynting-Robertson drag acceleration on the debris in Step 3 is calculated according to Equation (12):

[0039]

[0040] In Equation (12), the solar constant is S0 = 1.3608×10 6 ergs·cm -2 ·s -1 , A is the atmospheric drag cross-sectional area of the debris, m d is the mass of the debris, c is the speed of light, r s0 = 1 AU, r sd is the distance from the sun to the debris, Q pr is the solar radiation pressure coefficient, r DS is the position vector from the sun to the debris.

[0041] The beneficial effects of the present invention are as follows: The method for calculating the orbital evolution of debris below centimeter level in the present invention can effectively reduce the calculation error of the orbital evolution of existing debris below centimeter level, which is beneficial to the long-term prediction of the orbits of debris below centimeter level during the daily supervision of spacecraft, extend the possible collision prediction time of debris to spacecraft, and increase the effective time of collision warning. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 is the distribution diagram of the debris orbit at a certain moment in the inertial xz plane;

[0043] Figure 2 is the distribution diagram of the debris orbit at a certain moment in the inertial yz plane. DETAILED DESCRIPTION OF THE INVENTION

[0044] The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0045] The present invention provides a method for calculating the orbital evolution of debris below centimeter level, including the following steps:

[0046] Step 1, calculate the debris distribution characteristics, and take the distribution density function of the debris velocity as:

[0047]

[0048] In formula (1), v Col is the target collision velocity that generates a large number of debris, v m is the maximum relative velocity of the debris generated by the collision with respect to the parent body, β = 8.69, δ = 7.2×10 -3 s·m -1 .

[0049] Step 2, calculate the initial position and velocity of the debris through formula (2):

[0050]

[0051] In formula (2), r and v are the reference position and velocity of the debris, r d and v d are the initial position and velocity of the debris, and δv is the unit direction, pointing to a random distribution.

[0052] Step 3, superimpose the accurate acceleration model of the debris, and extrapolate to calculate the orbital evolution of the debris; specifically, calculate the orbital evolution of the debris by calculating the gravitational force of the earth, the gravitational force of the sun and the moon, the atmospheric drag, the solar radiation pressure, the Lorentz force, and the Poynting-Robertson drag force. The extrapolation of the motion of the debris represented in the inertial space is calculated by integrating according to formula (3):

[0053]

[0054] In formula (3), r is the position vector from the center of the earth to the debris relative to the inertial space, f S and f M are the gravitational accelerations generated by the earth, the sun and the moon on the debris, V is the gravitational potential of the earth; f A and f SR are the accelerations from the atmospheric drag and the solar radiation pressure respectively; f L is the acceleration generated by the Lorentz force of the earth's magnetic field; f PR is the acceleration generated by the Poynting-Robertson drag force.

[0055] Among them, the gravitational potential of the earth is calculated according to formula (4):

[0056]

[0057] In Equation (4), G is the gravitational constant, M E is the mass of the Earth, r = |r|, R e is the equatorial radius of the Earth, and λ and ψ are the geocentric longitude and latitude of the debris position, and are the normalized spherical harmonic coefficients of degree n and order m, is the normalized Legendre polynomial, and N is the maximum order used in the formula.

[0058] Among them, the gravitational accelerations generated by the sun and the moon on the debris are calculated according to Formulas (5) and (6):

[0059]

[0060]

[0061] In Formulas (5) and (6), G is the gravitational constant, M S and M M are the masses of the sun and the moon respectively, r DS and r DM are the position vectors from the sun and the moon to the debris respectively, r S and r M are the position vectors from the geocenter to the heliocenter and the geocenter to the lunar center respectively, r DS and r DM are the moduli of r DS and r DM respectively, r S = |r S |, r M = |r M |.

[0062] Among them, the acceleration of the solar radiation pressure is calculated according to Formula (7):

[0063]

[0064] In Formula (7), the solar constant is S0 = 1.3608×10 6 ergs·cm -2 ·s -1 , A s is the light pressure cross-sectional area of the debris, m d is the mass of the debris, c is the speed of light, r s0 = 1 AU, r sd is the distance from the sun to the debris, Q pr is the solar light pressure coefficient, and r SR is the unit vector in the direction from the sun to the debris.

[0065] Among them, the atmospheric drag acceleration suffered by the debris is calculated according to formula (8):

[0066]

[0067] In formula (8), the atmospheric drag coefficient is C d = 2.2, A is the atmospheric drag cross-sectional area of the debris, ρ is the atmospheric density at the debris orbit position, v is the debris velocity vector in the inertial space, and v = |v|.

[0068] Among them, the Lorentz force acceleration suffered by the debris is calculated according to formula (9):

[0069]

[0070] In formula (9), the permittivity is ε0 = 8.854187817×10 12 F·m -1 , the voltage at which the debris surface reaches charge balance after charging in the space physical environment is U = +5V, s is the average radius of the debris, m d is the mass of the debris, ω is the angular velocity vector of the Earth's spin; the magnetic field strength at the debris position is calculated according to formula (10):

[0071]

[0072] In formula (10), the magnetic scalar potential is V mag is calculated according to formula (11):

[0073]

[0074] In formula (11), the average reference radius of the geomagnetic field is a = 6371.2km, θ and are the geocentric fixed coordinate system longitude and latitude at the debris position respectively, t is the time, n and m are the degree and order of the magnetic field strength calculation respectively, and are the Gaussian coefficients of the geomagnetic field respectively, is the normalized Legendre function, N gm is the maximum order of the magnetic field strength calculation used.

[0075] Among them, the Poynting-Robertson drag acceleration suffered by the debris is calculated according to formula (12):

[0076]

[0077] In formula (12), the solar constant is S0 = 1.3608×10 6 ergs·cm -2 ·s -1 , A is the atmospheric drag cross-sectional area of the debris, md where \(m\) is the debris mass, \(c\) is the speed of light, and \(r\) s0 \( = 1\ AU\), and \(r\) sd is the distance from the Sun to the debris, \(Q\) pr is the solar radiation pressure coefficient, and \(r\) DS is the position vector from the Sun to the debris.

[0078] In addition, the size distribution of the number of debris can also be estimated according to Equation (13):

[0079]

[0080] In Equation (13), \(N\) debris is the number of debris with a size larger than \(d\), \(M\) tot is the total weight of the debris, \(\beta = 1.32\) is the correction coefficient, \(\rho = 2.7\times10\) 3 \(kg\cdot m\) -3 , and \(d\) is the size of the debris.

[0081] Example

[0082] Let the total mass of the disintegrated debris be \(M\) tot \( = 740\ kg\), calculate the number of debris with different sizes \(N\) debris \((\geq0.2\ m)=14\), \(N\) debris \((\geq0.1\ m)=60\), \(N\) debris \((\geq0.01\ m)=6587\), \(N\) debris \((\geq0.0018\ m)=2.18\times10\) 5 , and \(N\) debris \((\geq0.001\ m)=7.22\times10\) 5 . Calculate the initial position and velocity of the debris, extrapolate to calculate the orbital evolution of the debris, and the distribution of the debris orbit at a certain moment in the inertial \(xz\) plane is as shown in Figure 1 shown, and the distribution in the inertial \(yz\) plane is as shown in Figure 2 shown.

Claims

1. A method for calculating the orbital evolution of debris below centimeter level, characterized in that, Including the following steps: Step 1: Calculate the debris distribution characteristics, and take the distribution density function of the debris velocity as: （1） In formula (1), , is the target collision velocity that generates a large number of fragments, is the maximum relative velocity of the fragments generated by the collision with respect to the parent body, ; Step 2: Calculate the initial position and velocity of the debris; Step 3: Superimpose the accurate acceleration model of the debris and extrapolate to calculate the orbital evolution of the debris; specifically, calculate the orbital evolution of the debris by calculating the gravitational force of the Earth, the gravitational forces of the sun and the moon, the atmospheric drag, the solar radiation pressure, the Lorentz force, and the Poynting-Robertson drag force. The extrapolation of the motion of the debris represented in the inertial space is calculated by integrating according to formula (3): （3） In Equation (3), is the position vector from the center of the Earth to the debris relative to the inertial space, , and are the gravitational accelerations generated by the Earth, the Sun, and the Moon on the debris, is the gravitational potential of the Earth; and are the accelerations from atmospheric drag and solar radiation pressure, respectively; is the acceleration generated by the Lorentz force of the Earth's magnetic field; is the acceleration generated by the Poynting-Robertson drag force.

2. The method for calculating the orbital evolution of sub - centimeter debris as described in claim 1, characterized in that, In step 2, the initial position and velocity of the debris are calculated by formula (2): ，（2） In formula (2), and are the fragment reference position and velocity, and are the fragment initial position and velocity, is the unit direction, pointing to a random distribution.

3. The method for calculating the orbital evolution of sub - centimeter debris as claimed in claim 1, wherein In step 3, the gravitational potential of the Earth is calculated according to formula (4): （4） In Equation (4), G is the gravitational constant, is the mass of the Earth, , is the equatorial radius of the Earth, and the geocentric longitude and latitude of the debris position, and are n the m degree normalized spherical harmonic coefficients of order is the normalized Legendre polynomial, N is the maximum order used in the formula.

4. The method for calculating the orbital evolution of fragments below centimeter level according to claim 1, wherein In step 3, the gravitational accelerations generated by the sun and the moon on the debris are calculated according to formulas (5) and (6): （5） （6） In equations (5) and (6), G is the gravitational constant, and are the masses of the sun and the moon respectively, and are the position vectors of the sun and the moon to the debris respectively, and are the position vectors from the geocenter to the heliocenter and the lunar center respectively, and are and the magnitudes of, , .

5. The method for calculating the orbital evolution of sub - centimeter debris as claimed in claim 1, wherein In step 3, the acceleration of the solar radiation pressure is calculated according to formula (7): （7） In Equation (7), the solar constant is ergs∙cm -2 ∙s -1 , is the radiation pressure cross-sectional area of the debris, is the mass of the debris, is the speed of light, , is the distance from the Sun to the debris, is the solar radiation pressure coefficient, is the unit vector in the direction from the Sun to the debris.

6. The method for calculating the orbital evolution of sub - centimeter debris as described in claim 1, characterized in that, In step 3, the atmospheric drag acceleration suffered by the debris is calculated according to formula (8): （8） In Equation (8), the atmospheric drag coefficient is , is the atmospheric drag cross-sectional area of the debris, is the atmospheric density at the debris orbit position, is the debris velocity vector in the inertial space, .

7. The method for calculating the orbital evolution of sub - centimeter debris as described in claim 1, characterized in that, In step 3, the Lorentz force acceleration suffered by the debris is calculated according to formula (9): （9） In formula (9), the dielectric constant is , the voltage at which the surface of the debris reaches charge balance after charging in the space physical environment is , is the average radius of the debris, is the mass of the debris, is the angular velocity vector of the Earth's spin; the magnetic field strength at the location of the debris is calculated according to formula (10): （10） In formula (10), the magnetic scalar potential energy is Calculated according to formula (11): （11） In Equation (11), the average reference radius of the geomagnetic field is a = 6371.2 km, and φ are the geocentric-fixed coordinate system longitude and latitude of the location where the debris is located, is the time, n and m are the degree and order of the magnetic field strength calculation, respectively, and are the Gauss coefficients of the geomagnetic field, respectively, is the normalized Legendre function, N gm is the maximum order of the magnetic field strength calculation used.

8. The method for calculating the orbital evolution of sub - centimeter debris as claimed in claim 1, wherein In step 3, the Poynting-Robertson drag force acceleration suffered by the debris is calculated according to formula (12): （12） In Equation (12), the solar constant is ergs∙cm -2 ∙s -1 , is the atmospheric drag cross-sectional area of the debris, is the mass of the debris, is the speed of light, , is the distance from the Sun to the debris, is the solar radiation pressure coefficient, is the position vector from the Sun to the debris.

Images