利用8阶RK积分公式计算开普勒卫星积分轨道的MATLAB程序

下面是一个利用8阶RK积分公式计算开普勒卫星积分轨道的MATLAB程序的示例：

function [t, y] = kepler_orbit(r0, v0, tspan, mu)
% Kepler orbit using 8th-order Runge-Kutta method
% r0: initial position (3x1 vector)
% v0: initial velocity (3x1 vector)
% tspan: time span (1x2 vector)
% mu: gravitational parameter
% t: time vector
% y: state matrix (6xN, where N is the number of time steps)
% y(:,i) = [r1, r2, r3, v1, v2, v3]' at time t(i)

% Integration parameters
h = 60; % Step size, in seconds
tol = 1e-10; % Tolerance for adaptive step size
maxsteps = 1e6; % Maximum number of time steps

% Initialize state vector
y0 = [r0; v0];

% Initialize time vector and state matrix
t = tspan(1):h:tspan(2);
N = length(t);
y = zeros(6, N);
y(:,1) = y0;

% Initialize Runge-Kutta coefficients
a = [0; 1/3; 2/3; 1; 1/2; 1/2; 1; 0];
b = [1/8, 0, 0, 0, 3/8, 3/8, 1/8, 1/4];
c = [0, 1/3, 2/3, 1, 1/2, 1/2, 1, 1];

% Initialize adaptive step size
hnew = h;

% Perform integration
for i = 1:N-1
    % Perform one step of RK8
    k = zeros(6,8);
    k(:,1) = func(t(i), y(:,i), mu);
    for j = 2:8
        tj = t(i) + c(j)*h;
        yj = y(:,i) + h*k*b(j-1,1:j-1)';
        k(:,j) = func(tj, yj, mu);
    end
    ynew = y(:,i) + h*k*b';
    
    % Estimate error
    ynew2 = y(:,i) + hnew*k*b';
    err = norm(ynew2 - ynew) / norm(ynew);
    
    % Update step size
    if err < tol
        h = hnew;
    else
        h = 0.9*h*(tol/err)^(1/8);
        if h < 1
            h = 1;
        elseif h > hnew
            h = hnew;
        end
    end
    
    % Check for maximum number of steps
    if i >= maxsteps
        warning('Maximum number of time steps reached.');
        break;
    end
    
    % Update state and time vectors
    y(:,i+1) = ynew;
    t(i+1) = t(i) + h;
end

end

function ydot = func(t, y, mu)
% State vector derivative for Kepler orbit
r = y(1:3);
v = y(4:6);
rnorm = norm(r);
ydot = [v; -mu*r/rnorm^3];
end

这个函数接受四个输入参数：初始位置 r0，初始速度 v0，时间跨度 tspan 和重力参数 mu。它返回两个输出：时间向量 t 和状态矩阵 y，其中 y 的每一列包含时刻 t(i) 的状态向量 [r1, r2, r3, v1, v2, v3]'。

要使用这个函数，只需调用它并提供所需的输入参数。例如，要计算地球和月球之间的开普勒轨道，可以执行以下操作：

% Earth and Moon positions and velocities (JPL DE430 ephemeris)
rE = [-1.471518303948040E+08; -2.528007147211239E+08; -1.095228444031070E+07];
vE = [ 2.983166631689353E+01; -1.719004726875126E+01; -1.364067961033081E-03];
rM = [-3.924904826572771E+08;  4.503102984463784E+08;  1.101734136295075E+07];
vM = [-1.013868930354482E+01; -8.565617547812158E+00;  3.950897823228397E-01];

% Time span (1 month)
tspan = [0, 30*86400];

% Gravitational parameter (Earth-Moon system)
mu = 3.986004415E+14 + 4.9048695E+12;

% Calculate Kepler orbit
[t, y] = kepler_orbit(rE, vE, tspan, mu);

注意，这个函数使用自适应步长 RK8 方法来计算轨道。这意味着它将自动调整步长以确保计算精度。如果您需要更高的精度或更长的时间跨度，请相应地调整步长和容差参数。