# Help plotting a circular orbit

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Adam Lacey on 4 Jun 2022
Edited: Adam Lacey on 6 Jun 2022
Hi so basically i would like to be able to produce a circular graph of sattelite position around earth. The code attached below starts with acceleration and use Eulers method to integrate once for velocity and again for displacement. The code works and produces results i am just unsure how to convert the outputs into a circular graph that updates displacement around earth as a result of the calculated velocity.
Code:
clear all
clc
G = 6.6743*10^-11; %Gravitational Constant, Units: m^3 kg^-1 s^-2
Mc = 5.972*10^24; %Mass of central body (earth), Units: kg
rE = 6371; %Radius of earth, Units: Km
height = 4000 %Altitude of sattelites orbit, Units: km
r = height + rE; %Radius of circular orbits, Units: km
mu = 3.986004418*10^5; %Earths Gravitational parameter, Units: km^3s^-2
x_initial = 0;
v_initial = 12000; %km/h
dt = 0.1;
t_vector = 0:dt:100000;
x(1) = x_initial;
v(1) = v_initial;
%%Math working out
%Fx = -G*x*(Mc/r^3);
%a = Fx/Mc
%a = (-G*x)/r^3
%a = -5.9833e-23*x m/s^2
%v(t) = 12000 + a*t
%y(t) = r + 12000t + 0.5a*t^2
x_initial = 0;
v_initial = 12000;
x(1)=x_initial;
v(1)=v_initial;
for i=1:length(t_vector)-1
a = -G/r^3;
v(i+1) = v(i) + a*dt;
xslope = v(i);
x(i+1) = x(i)+xslope*dt;
end
figure(1)
plot(t_vector,x)
figure(2)
plot(t_vector,v)
##### 1 CommentShowHide None
Walter Roberson on 4 Jun 2022
You need to track x and y separately.

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### Accepted Answer

Lateef Adewale Kareem on 4 Jun 2022
G = 6.6743*10^-11; %Gravitational Constant, Units: m^3 kg^-1 s^-2
Mc = 5.972*10^24; %Mass of central body (earth), Units: kg
rE = 6371; %Radius of earth, Units: Km
height = 4000; %Altitude of sattelites orbit, Units: km
r = height + rE; %Radius of circular orbits, Units: km
x_initial = 0;
y_initial = r*1e3;
dt = 1;
T = sqrt(4*pi^2*(r*1e3)^3/(G*Mc)); % period in sec
t_vector = 0:dt:T;
dsdt = @(s)[s(3), s(4), -G*Mc*s(1)/norm(s([1,2]))^3, -G*Mc*s(2)/norm(s([1,2]))^3]; %satelite orbitaal dynamics
x = x_initial; y = y_initial; v = sqrt(G*Mc/(r*1e3));
xv = v*y_initial/(r*1e3); yv = -v*x_initial/(r*1e3); % initial velocities
state = [x_initial, y_initial, xv, yv];
for i=2:length(t_vector)
state = state + rk4(dsdt, state, dt);
x = [x, state(1)];
y = [y, state(2)];
xv = [xv, state(3)];
yv = [yv, state(4)];
end
figure
subplot(2,2,1)
plot(t_vector,x)
title('x position of satellite')
subplot(2,2,2)
plot(t_vector,y)
title('y position of satellite')
subplot(2,2,3)
plot(t_vector,xv)
title('x velocity of satellite')
subplot(2,2,4)
plot(t_vector,yv)
title('y velocity of satellite')
figure
plot(x, y)
title('orbit of satellite')
function dy = rk4(dydt, y, dt) %runge kutta integrator
k1 = dydt(y); k2 = dydt(y + dt*k1/2);
k3 = dydt(y + dt*k2/2); k4 = dydt(y + dt*k3);
dy = dt*(k1+2*k2+2*k3+k4)/6;
end
##### 1 CommentShowHide None
Adam Lacey on 4 Jun 2022
This is amazing thank you so much

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