Satellite orbital instability generated by the perturbing factors.

Chiru, Anghel ; Pirna, Ion ; Candea, Ioan 等

1. INTRODUCTION

Due to the increased dependence of humankind on space technologies, the impact of malfunctioning satellites would be great. In this situation, research on the perturbing factors and the calculation of their influence upon the orbit should be a prime step for the orbital correction problems.

Until not long ago the shape of any orbit (as demonstrated by Johannes Kepler in 1609 in "Astronomia nova") was thought to be an ellipse with the central body in one of the foci. Analyzing the real orbital trajectory using data collected during actual space missions it became clear that there is a discrepancy between the predicted and the real orbit.

The new problem was to find what exterior forces are acting upon the spacecraft and then to try to calculate them in order to make better predicaments for the trajectory.

A series of perturbing factors were presented as influencing the movement on the orbit: gravity of other massive celestial bodies, friction with the upper atmosphere, solar wind and solar pressure. Some other insignificant factors as the flattening of the Earth or the variable magnetic field are not taken into account in this paper. The induced perturbations are very small, but taken a long period of time, the impact upon the satellite orbit is significant

Each of these perturbing factors is at the moment well known and can be calculated using specific formulas.

The novelty presented in this paper is the development of a method that takes into account the added effect of all this factors. Based on this model a recursive step-by-step computer program was created and used to accurately predict the orbits.

2. GRAVITY OF CELESTIAL BODIES

The Sun, Moon and other planets can modify the trajectory of a satellite by their gravity. The most significant bodies are the Sun (huge mass) and the Moon (due to proximity). Equation 1 determines the gravity forces that are acting on the satellite in a system with n bodies (Schaub & Junkins, 2002).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 1 OMITTED]

Where G = 6.67e-11 [m.sup.3]/[s.sup.2]kg is the universal constant of attraction, [a.sub.g] is the gravity induced acceleration, [m.sub.sat] is the mass of the satellite, [m.sub.j] is the mass of the n'th body and [r.sub.sat j] is the distance between the satellite and the n'th body.

It can be noticed form the figure 1 that the gravitational acceleration is decreasing rapidly. This acceleration is directly proportional with the attracting body mass, therefore the largest celestial bodies will induce higher accelerations. But the acceleration is inverse proportional with the distance, therefore the closest bodies will induce larger accelerations and more distant planets will induce only small accelerations.

Perturbations coming from these forces are relevant for satellites. The acceleration induced by the Sun is almost constant (5.9e-3 m/[s.sup.2]), the acceleration induced by the Moon varies with the satellite's position around the Earth (3e-5 m/[s.sup.2] .. 4e-5 m/[s.sup.2]). The accelerations induced by the other planets are very small (1e-7 m/[s.sup.2]).

3. ATMOSPHERIC DRAG

The atmospheric drag is acting as a breaking force. The orbital height of the satellite will decrease slightly.

The drag force is dependent on the air density (which decreases with altitude), therefore it is inversely proportional with the altitude. The atmospheric drag can be one of the main perturbations if working on a satellite in LEO (Low Earth Orbit).

The acceleration from the atmospheric drag force can be expressed as presented in equation 2 (Deliu, 2003):

[a.sub.d] = 1/2 [rho] x [C.sub.d] x A x [v.sup.2]/[m.sub.sat] (2)

where: p is the density of the air, [C.sub.d] is the drag coefficient, A is the reference transversal area and v is the velocity of the satellite relative to the air.

As a reference, the Space Shuttle (2008) with a mass of 92000 kg and a sectional area of 362 [m.sup.2] at an altitude of 350 km has an induced drag acceleration of 5e-3 m/[s.sup.2].

[FIGURE 2 OMITTED]

The variation of the induced drag acceleration upon the Space Shuttle can be also deduced from figure 2.

4. SOLAR RADIATION PRESSURE

The Sun's radiation causes a small force on the spacecraft that is exposed to it. This is because the Sun emits photons that are either absorbed or reflected by the satellite. Therefore, the force experienced by the satellite depends upon the surface area of the satellite. The acceleration can be expressed as presented in equation 3 (Campbell & McCandless, 1996).

[a.sub.sp] = s x (1 - f) x k x P x A/c x 1/[m.sub.sat] (3)

where: k is the reflectivity factor and can be set between 1 and 2 (1 means that the radiation is totally absorbed, 2 means that it is totally reflected; P is the solar power (=1400W/m2 at Earth's location), A is the reference transversal area and c is the speed of light.

The s factor is used to take into account the solar wind influence (1< s < 1.2). It is dependent of solar activities. For maximum solar activity, the coefficient can be taken as 1.2 and for low solar activities the coefficient can be set to 1.

A normal satellite has a solar pressure induced acceleration of 1e-8m/[s.sup.2].

The f coefficient represents the time when the Earth is between the Sun and the satellite and therefore the solar pressure and the solar wind are no longer active on the craft. In figure 3 is presented the light period for a family of orbits, from one can extract the f coefficient.

In figure 3 can be observed the variation of the light function of the orbital radius. Curve 1 represents an orbit with an inclination of 0 deg. (orbit in the ecliptic plane). Curve 2, 3 and 4 are inclined with 10, 30 and 50 deg. with respect at the ecliptic plane.

[FIGURE 3 OMITTED]

5. ONBOARD THRUSTERS SYSTEM

The satellite's own thrusters can change the orbit. During a thrust orbital maneuver, the mass of the satellite will change as propellant is consumed.

A simple, constant thrust model is however often sufficient to describe the motion of a spacecraft during thrust arcs.

When a propulsion system ejects a mass m per time interval dt at a velocity [v.sub.e], the spacecraft of mass [m.sub.sat] experiences a thrust F which results in the acceleration presented in equation 4 (Frank, 1998).

[a.sub.t] = F/[m.sub.sat] = [absolute value of [??]]/[m.sub.sat] [v.sub.e] (4)

6. IMPACT ON THE MOTION EQUATION

Starting from the gravitational force equation expression, the acceleration of the satellite is expressed as in equation 5.

a = G x M/[r.sup.2] + [a.sub.g] + [a.sub.d] + [a.sub.sp] + [a.sub.t] (5)

where M is the mass of the Earth.

Based on this equation, a computer program was written using recursive functions and dividing time into small steps. Starting with the known positions and speeds of all bodies, and giving values to different parameters, the program computes the new position and speeds over a small time step. Now this new positions and speeds are used as the input for the next step.

Results were validated using as input data available online information for the Hubble Space Telescope (***, 2008). Actual calculated perturbations, for a time range of 1 year, are much accurate, in the range of 1% difference from the real orbit compared with the theoretical orbit. The previous calculations based on older model give differences of 5% or more.

Future research could take into consideration some other smaller perturbing factors and try to reduce even more the difference between the predicted orbit and the real orbit. Also, the model has to be validated using real data.

7. CONCLUSIONS

Using the equation presented above, one can calculate with an iterative computer program the positions and acceleration for a satellite. The iteration steps should be small in order to keep the errors as low as possible. Then, the perturbation can be calculated knowing both the real position and the theoretical position. With these calculations, the correction of the orbit can be computed.

8. REFERENCES

Campbell, B. A.; McCandless W. (1996). Introduction to Space Sciences and Spacecraft Applications, Gulf Publishing, ISBN 0-88415-411-4, Houston

Deliu, G. (2003). Aircraft Mechanics, Ed. Albastra, ISBN 973-650-029-2, Cluj-Napoca, Romania

Frank, W. B. (1998). Propulsion Flight Research at NASA, Dryden Flight Research Center, ISBN 0-8027-1427-7, California

Schaub, H. & Junkins, J. L. (2002). Analytical Mechanics of Aerospace Systems, Dover Publications, ISBN 1563475634, New York

*** (2008) http://hubble.nasa.gov/- National Aeronautics and Space Administration, Hubble Space Telescope, Accessed on: 2008-04-11