Matlab - plot speed vector of a satellite in Keplerian orbit - matlab

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:
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)
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), ...)