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…