The CNES has been a great user of Scilab, and a long-term 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 step-by-step, to demonstrate the capacities of Scilab in this field:

Download the script:

myEarthRotation.sci 1,77 kB

Download the full tutorial:

Scilab Orbite Simulation.pdf 1,35 MB

### 1. Express the physics problem

The problem is based on the universal law of gravitation:

We write down Newton’s third law of motion in an earth-centred 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:

### 2. Translate your problem into Scilab

Scilab is a matrix-based 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 = 6.67D-11; //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 Saint-Louis, G. Sallet, 2004

You can also model and simulate it with Xcos:

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