Model a simple circular satellite orbit in time

So Im essentially trying to graph an orbit which will be a circle on a polar plot it is to represent the motion of a 500kg satellite when there is no force applied to it. Im neglecting the fact it should be falling over time for now. However my code does not produce the desired result so any input would be aprriciated , thank you all in advance

clc;
clear all;
G = 6.673e-11; %Gravitational constant
M = 5.98e24; %mass of earth in (kg)
ra = 100000; %orbit distance in (m)
r = 6.37e6 + ra; %total radius of orbit in (m)
m = 500; % mass satelite (kg)
a = (G*M)/(r^2); % check for acceleration
v_orb = sqrt((G*M)/r); % orbital velocity (m/s)
T = sqrt(((4*(pi^2))*r^3)/(G*M)); % period (s)

%lets graph it for the period in steps of 150 to reduce the computation
simt = T;
for t = 1:150:simt
  v_o(t) = sqrt((G*M)/r);% array of velocity corresponding to time in steps of 150 .. should not change and be 1x34 - but ERROR
  rn(t) = 6.37e6 + ra; % array of radius corresponding to time in steps of 150 .. should not change and be 1x34 - but ERROR
  disp(t) % goes from 1 to 5101 in steps of 150
  %%% create array of the form 1x34 corresponding to t ???? %%% 
  end
  
figure
plot(t, v_o); xlabel('Simulation Time', 'FontSize', 12);
ylabel('Velocity', 'FontSize', 12);
grid;  %%% expecting a graph of straight line %%%

figure
plot(t, rn); xlabel('Simulation Time', 'FontSize', 12);
ylabel('Radius', 'FontSize', 12);
grid;  %%% expecting a graph of straight line %%%

%%% take the points of radius and time convert to polar coordinates and plot %%%
%%% expecting a circle as satellite moves around the orbit of constant radius in a sice of time of the period%%%
%[theta,rho] = cart2pol(t,rn);
theta = atan2(rn,t);
rho = sqrt((t.^2)+(rn.^2));
theta= theta*(180/pi); % to degrees

figure
polarplot(theta,rho)
title('Orbit')
  

Answer 1

[James Tursa]

Since you are setting up a circular orbit, just scale the time by the period to get theta. E.g., since one period would be an angle of 2pi,

theta = 2*pi * t / T;

Then just use that and rn for your plot.

[Michal Sleszynski]

So what you mean is instead of

theta = atan2(rn,t); -> theta = 2*pi * t / T;

still though t is not an array and v_o and rn are the wrong size instead f 1x34 , 1x5101

both v_o and rn have thier first value and then follow by 5100 zeros.

[Michal Sleszynski]

Ive changed a bit and took your advice thank you very much. My final answer is bellow.

Answer 2

[Michal Sleszynski]

This is my final code and it seems to work now:

clc;
clear all;

G = 6.673e-11; %Gravitational constant
M = 5.98e24; %mass of earth in (kg)
ra = 100000; %orbit distance in (m)
r = 6.37e6 + ra; %total radius of orbit in (m)
m = 500; % mass satelite (kg)
a = (G*M)/(r^2); % check for acceleration
v_orb = sqrt((G*M)/r); % orbital velocity (m/s)
T = sqrt(((4*(pi^2))*r^3)/(G*M)); % period (s)


%lets graph it for the period in steps of 150 to reduce the computation
steps = T/35;
simt = -steps;
for i = 1:1:36
  simt = simt+steps;
  t(i) = simt;
  v_o(i) = sqrt((G*M)/r);% array of velocity corresponding to time in steps of 150 .. should not change and be 1x34 - but ERROR
  rn(i) = 6.37e6 + ra; % array of radius corresponding to time in steps of 150 .. should not change and be 1x34 - but ERROR
  %disp(t2) % goes from 1 to 5101 in steps of 150
  
  
  end
  
figure
plot(t, v_o); xlabel('Simulation Time', 'FontSize', 12);
ylabel('Velocity', 'FontSize', 12);
grid;  %%% expecting a graph of straight line %%%

figure
plot(t, rn); xlabel('Simulation Time', 'FontSize', 12);
ylabel('Radius', 'FontSize', 12);
grid;  %%% expecting a graph of straight line %%%


%%% take the points of radius and time convert to polar coordinates and plot %%%
%%% expecting a circle as satellite moves around the orbit of constant radius in a sice of time of the period%%%
%[theta,rho] = cart2pol(t,rn);
%th = atan2(rn,t);
th = 2*pi * t / T;
rho = sqrt((t.^2)+(rn.^2));

%disp(th);

figure
polar(th,rho)
title('Orbit')

Answer 3

[Meysam Mahooti]

https://www.mathworks.com/matlabcentral/fileexchange/55777-two-body-orbit-propagation-non-spherical-earth?s_tid=srchtitle