Matlab - plot speed vector of a satellite in Keplerian orbit

Question

I have to plot the speed vector of an object orbiting around a central body. This is a Keplerian context. The trajectory of object is deduced from the classical formula ( r = p/(1+e*cos(theta)) with e=eccentricity.

I manage into plotting the elliptical orbit but now, I would like to plot for each point of this orbit the velocity speed of object.

To compute the velocity vector, I start from classical formulas (into polar coordinates), below the 2 components :

v_r = dr/dt and v_theta = r d(theta)/dt

To take a time step dt, I extract the mean anomaly which is proportional to time.

And Finally, I compute the normalization of this speed vector.

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)]);

Unfortunately, once plot performed, it seems that I get a bad vector for speed. Here for example the 30th element of vrNorm and vthetaNorm arrays :

As you can see, the vector has the wrong direction (If I assume to take 0 for theta from the right axis and positive variation like into trigonometrics).

If someone could see where is my error, this would nice.

UPDATE 1: Has this vector representing the speed on elliptical orbit to be tangent permanently to the elliptical curve ?

I would like to represent it by taking the right focus as origin.

UPDATE 2:

With the solution of @MadPhysicist, I have modified :

% 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)]);

I get the following result :

On some parts of the orbit, I get wrong directions and I don't understand why ...

Answer 1

There are two issues with your code:

1. The normalization is done incorrectly. norm computes the generalized p-norm for a vector, which defaults to the Euclidean norm. It expects Cartesian inputs. Setting p to 1 means that it will just return the largest element of your vector. In your case, the normalization is meaningless. Just set vrNorm as

   vrNorm = vr ./ max(vr)
   

2. It appears that you are passing in the polar coordinates vrNorm and vthetaNorm to quiver, which expects Cartesian coordinates. It's easy to make the conversion in a vectorized manner:

   vxNorm = vrNorm * cos(vtheta);
   vyNorm = vrNorm * sin(vtheta);
   

This assumes that I understand where your angle is coming from correctly and that vtheta is in radians.

Note

The entire loop

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

can be rewritten in a fully vectorized manner:

vr = (2 * pi) .* diff(r) ./ (P .* diff(M))
vtheta = (2 * pi) .* r .* diff(theta) ./ (P .* diff(M))
vrNorm = vr ./ max(vr)
vxNorm = vrNorm * cos(vtheta);
vyNorm = vrNorm * sin(vtheta);

Note 2

You can call quiver in a vectorized manner, on the entire dataset, or on a subset:

quiver(x(20:199:20), y(20:199:20), vxNorm(20:199:20), vyNorm(20:199:20), ...)