model the earth's orbit around the sun using keplers laws and equations of motion in matlab in matlab

To model the Earth's orbit around the Sun using Kepler's laws and equations of motion in MATLAB, you can follow these steps:

1. Define the necessary parameters:

   • Gravitational constant G
   • Mass of the Sun M
   • Initial position and velocity vectors of the Earth r0 and v0
   • Time step dt
   • Total simulation time t

2. Calculate the gravitational force acting on the Earth at each time step using Newton's law of universal gravitation:

   F = -(G * M * m) / r^2

   where m is the mass of the Earth and r is the distance between the Earth and the Sun.

3. Update the position and velocity vectors of the Earth using the equations of motion:

   a = F / m v = v0 + a * dt r = r0 + v * dt

4. Repeat steps 2 and 3 until your desired simulation time.

Here's an example implementation in MATLAB:

main.m
% Define parameters
G = 6.674e-11;  % Gravitational constant
M = 1.989e30;  % Mass of the Sun
m = 5.972e24;  % Mass of the Earth
r0 = [147e9; 0];  % Initial position vector of the Earth (x, y)
v0 = [0; 30e3];  % Initial velocity vector of the Earth (vx, vy)
dt = 3600;  % Time step (seconds)
t = 365 * 24 * 3600;  % Total simulation time (seconds)

% Initialize arrays to store positions and velocities
num_steps = t / dt;
r = zeros(2, num_steps);
v = zeros(2, num_steps);

% Perform simulation
for i = 1:num_steps
    % Calculate distance between the Earth and the Sun
    distance = norm(r0);
    
    % Calculate gravitational force
    F = -(G * M * m) / distance^2;
    
    % Calculate acceleration
    a = F / m;
    
    % Update velocity and position vectors
    v0 = v0 + a * dt;
    r0 = r0 + v0 * dt;
    
    % Store values in arrays
    v(:, i) = v0;
    r(:, i) = r0;
end

% Plot the Earth's orbit
figure;
plot(r(1,:), r(2,:));
axis equal;
xlabel('x (m)');
ylabel('y (m)');
title('Earth''s Orbit around the Sun');
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This code will simulate the Earth's orbit around the Sun for a year (365 days) using a time step of one hour. The resulting orbit will be plotted in a figure. Please note that the scale of the plot may not be realistic; you may need to adjust the scaling factor for visual clarity.

Make sure to set the parameters and adjust the time step and simulation time according to your needs.