matlab编程求卫星轨道长度,matlab-绘制开普勒轨道卫星的速度矢量
我必须画出一个物体围绕中心物体旋转的速度矢量。这是开普勒的背景。物体的轨道由经典公式(r=p/(1+e*cos(theta))推导出,其中e=偏心率。
我设法画出了椭圆轨道,但现在,我想画出这个轨道上每一点物体的速度。
为了计算速度矢量,我从经典公式(转换为极坐标)开始,下面是两个分量:
v_r=d r/dt和v_theta=r d(theta)/dt
为了计算时间步长dt,我提取了与时间成正比的平均异常。
最后,我计算了速度向量的归一化。
clear % clear variables
e = 0.8; % eccentricity
a = 5; % semi-major axis
b = a*sqrt(1-e^2); % semi-minor axis
P = 10 % Orbital period
N = 200; % number of points defining orbit
nTerms = 10; % number of terms to keep in infinite series defining
% eccentric anomaly
M = linspace(0,2*pi,N); % mean anomaly parameterizes time
% M varies from 0 to 2*pi over one orbit
alpha = zeros(1,N); % preallocate space for eccentric anomaly array
%%%%%%%%%%
%%%%%%%%%% Calculations & Plotting
%%%%%%%%%%
% Calculate eccentric anomaly at each point in orbit
for j = 1:N
% initialize eccentric anomaly to mean anomaly
alpha(j) = M(j);
% include first nTerms in infinite series
for n = 1:nTerms
alpha(j) = alpha(j) + 2 / n * besselj(n,n*e) .* sin(n*M(j));
end
end
% calcualte polar coordiantes (theta, r) from eccentric anomaly
theta = 2 * atan(sqrt((1+e)/(1-e)) * tan(alpha/2));
r = a * (1-e^2) ./ (1 + e*cos(theta));
% Compute cartesian coordinates with x shifted since focus
x = a*e + r.*cos(theta);
y = r.*sin(theta);
figure(1);
plot(x,y,'b-','LineWidth',2)
xlim([-1.2*a,1.2*a]);
ylim([-1.2*a,1.2*a]);
hold on;
% Plot 2 focus = foci
plot(a*e,0,'ro','MarkerSize',10,'MarkerFaceColor','r');
hold on;
plot(-a*e,0,'ro','MarkerSize',10,'MarkerFaceColor','r');
% compute velocity vectors
for i = 1:N-1
vr(i) = (r(i+1)-r(i))/(P*(M(i+1)-M(i))/(2*pi));
vtheta(i) = r(i)*(theta(i+1)-theta(i))/(P*(M(i+1)-M(i))/(2*pi));
vrNorm(i) = vr(i)/norm([vr(i),vtheta(i)],1);
vthetaNorm(i) = vtheta(i)/norm([vr(i),vtheta(i)],1);
end
% Plot velocity vector
quiver(x(30),y(30),vrNorm(30),vthetaNorm(30),'LineWidth',2,'MaxHeadSize',1);
% Label plot with eccentricity
title(['Elliptical Orbit with e = ' sprintf('%.2f',e)]);
不幸的是,一旦执行了绘图,我似乎得到了一个坏的速度向量。例如,这里
30th
元素
vrNorm
和
vthetaNorm
数组:
如你所见,向量的方向是错误的(如果我假设从右轴取θ为0,正变分为三角函数)。
如果有人能看到我的错误在哪里,那就太好了。
更新1:
这个表示椭圆轨道上速度的向量是否与椭圆曲线永久相切?
我想用正确的焦点作为原点来表示它。
更新2:
有了@mad物理学家的解决方案,我修改了:
% compute velocity vectors
vr(1:N-1) = (2*pi).*diff(r)./(P.*diff(M));
vtheta(1:N-1) = (2*pi).*r(1:N-1).*diff(theta)./(P.*diff(M));
% Plot velocity vector
for l = 1:9 quiver(x(20*l),y(20*l),vr(20*l)*cos(vtheta(20*l)),vr(20*l)*sin(vtheta(20*l)),'LineWidth',2,'MaxHeadSize',1);
end
% Label plot with eccentricity
title(['Elliptical Orbit with e = ' sprintf('%.2f',e)]);
我得到以下结果:
在轨道的某些部分,我得到了错误的方向,我不明白为什么…