Satellite orbit around the earth
The CNES has been a great user of Scilab, and a longterm partner of us. Upon their advice, we presented one of their use case at the the 6th International Conference on Astrodynamics Tools and Techniques (ICATT).
The question is a classical problem of space mechanics solved stepbystep, to demonstrate the capacities of Scilab in this field:
Download the script (myEarthRotation  PDF)
Download the full tutorial (Scilab Orbite Simulation  PDF)
1. Express the physics problem
The problem is based on the universal law of gravitation:
We write down Newton’s second law of motion in an earthcentred referential:
(1)
(2)
Position of the satellite is at a distance r [x; y]
Earth mass centre is at O [0; 0]
Gravitational constant:
Mass of the earth:
Radius of the Earth:

Translate your problem into Scilab
Scilab is a matrixbased language. Instead of expressing the system as set of 4 independent equations (along the x and y axis, for position and speed), we describe it as a single matrix equation, of dimension 4×4:
This method is a classical trick to switch from a second order scalar differential equation to a first order matrix differential equation.
with
To simplify the equation, we define the variable:
Open scinotes with edit myEarthRotation.sci
Define the skeleton of the function:
function udot=f(t, u) G = 6.67D11; //Gravitational constant M = 5.98D24; //Mass of the Earth c = G * M; r_earth = 6.378E6; //radius of the Earth r = sqrt(u(1)^2 + u(2)^2); // Write the relationhsip between udot and u if r < r_earth then udot = [0 0 0 0]'; else A = [[0 0 1 0]; [0 0 0 1]; [c/r^3 0 0 0]; [0 c/r^3 0 0]]; udot = A*u; end endfunction
The condition defined by the distance r of the satellite with the centre of earth stops the simulation if it’s colliding with earth’s surface.
Try out the final script with the following initial conditions in speed and altitude:
> geo_alt = 35784; // in kms > geo_speed = 1074; // in m/s > simulation_time = 24; // in hours > U = earthrotation(geo_alt, geo_speed, simulation_time);
3. Compute the results and create a visual animation
With this function, we go to the core of the problem:
function U=earthrotation(altitude, v_init, hours) // altitude given in km // v_init is a vector [vx; vy] given in m/s // hours is the number of hours for the simulation r_earth = 6.378E6; altitude = altitude * 1000; U0 = [r_earth + altitude; 0; 0; v_init]; t = 0:10:(3600*hours); // simulation time, one point every 10 seconds U = ode(U0, 0, t, f); // Draw the earth in blue angle = 0:0.01:2*%pi; x_earth = 6378 * cos(angle); y_earth = 6378 * sin(angle); fig = scf(); a = gca(); a.isoview = "on"; plot(x_earth, y_earth, 'b'); plot(0, 0, 'b+'); // Draw the trajectory computed comet(U(1,:)/1000, U(2,:)/1000, "colors", 3); endfunction
The resolution of the ordinary differential equation (ODE) is computed with the Scilab function ode.
ode solves Ordinary Different Equations defined by:
where y is a real vector or matrix
The simplest call of ode is: y = ode(y0,t0,t,f) where y0 is the vector of initial conditions, t0 is the initial time, t is the vector of times at which the solution y is computed and y is matrix of solution vectors y=[y(t(1)),y(t(2)),…].
Go further
To go further in numerical analysis, find out more about the solvers:
Ordinary Differential Equations with Scilab, WATS Lectures, Université de SaintLouis, G. Sallet, 2004