Satellite Orbit

Make sure to use the correct initial conditions and units for your calculations. In summary, the correct equations of motion for a satellite traveling from Earth to Jupiter should include the gravitational forces of both Earth and Jupiter. Use the Runge-Kutta algorithm to solve the equations numerically, and make sure to use the correct initial conditions and units for your calculations.
  • #1
SolStis
12
0
Hi, i am trying to solve the following differential equations for a satellite traveling from Earth to Jupiter using a runge kutta algorithm.

The equations are:

dr/dt=+-(h/r)[((r/r0)-1)(1-(r/r1))]^(0.5)

and

dr/d(theta)=+-r[((r/r0)-1)(1-(r/r1))]^(0.5).

where r0 and r1 are the radii of Earth's and Jupiters orbits respectively.

The problem is when trying to solve with the runge kutta when the initial conditions are set (ie r = r0) the function returns 0 and hence there is nochange in further steps. The equations in polar co ords. and my RK is as follows:

double dr(double r)
{
double h,v0=38534.7, r0=0.1495*(pow(10.0,12)), r1=0.7778*(pow(10.0,12));
h=v0*r0;
return h/pow(r,2);
}

int main()
{
double t=0.0, s=10000,r1,th1,kr1,kr2,kr3,kr4; //Declaration and initiation of vars for use in 4th order runge kutta
int n=0;
while (t<=8000000)
{
t=n*s; //time
cout << t << "\t" << r0 << endl; //output data to terminal and file
outs1 << t << "\t" << r0 << endl;

//Runge Kutta Method
kr1=s*dr(r0);
kr2=s*dr(r0+kr1/2);
kr3=s*dr(r0+kr2/2);
kr4=s*dr(r0+kr3);
r1=r0+(kr1+2*kr2+2*kr3+kr4)/6.0;

r0=r1;
n++;
}
}

I am unsure if i am using the correct equations of motion to solve this and any help would be greatly appreciated

Cheers

Sol
 
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  • #2
:The equations of motion you have provided are not correct. The equations of motion for a satellite traveling from Earth to Jupiter should include the gravitational forces of both Earth and Jupiter acting on the satellite. The equations of motion can be written as:dr/dt = vx dvx/dt = -(GM_earth/r^2) + (GM_jupiter/r^2) where G is the gravitational constant, M_earth and M_jupiter are the masses of Earth and Jupiter respectively, and r is the distance between the satellite and the barycenter of the Earth-Jupiter system. Using the Runge-Kutta algorithm, you can then solve the equations of motion numerically.
 

What is a satellite orbit?

A satellite orbit is the path that a satellite takes around a celestial body, such as the Earth or another planet.

How are satellite orbits determined?

Satellite orbits are determined by the laws of physics, specifically Kepler's laws of planetary motion. These laws take into account the mass and gravitational pull of the celestial body, as well as the velocity and trajectory of the satellite.

What factors affect the shape and size of a satellite orbit?

The shape and size of a satellite orbit are primarily affected by the altitude of the satellite above the celestial body, as well as the velocity and direction of the satellite's motion. Other factors such as atmospheric drag, solar radiation, and gravitational forces from other bodies can also play a role.

How does a satellite maintain its orbit?

A satellite maintains its orbit through a balance of its velocity and the gravitational pull of the celestial body it is orbiting. If the satellite's velocity is too low, it will fall towards the celestial body. If the velocity is too high, it will fly away from the celestial body. This balance is known as orbital speed.

What are the different types of satellite orbits?

There are several types of satellite orbits, including low Earth orbit, medium Earth orbit, geostationary orbit, and polar orbit. Each type has its own specific altitude, velocity, and purpose, such as weather observation or communication.

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