Lecture 1.

Laws of artificial satellites motion

General information about coarse

Satellite Navigation Systems:

16 lectures – (1 lecture per week)

4 laboratory works with max grade 10 (total 40)

2 modular tests with max grade 15 (total 30)

term paper (max grade 18)

Exam (max grade 12)

Physical foundations of satellites motion

An artificial satellite is a manufactured object

or vehicle intended to orbit the Earth, the Moon,

or another massive celestial body. So, satellite is

an object which has been placed into orbit by

human endeavor.

Orbital mechanics is the study of the motions of artificial satellites

moving under the influence of main forces. Orbital mechanics is a

modern offshoot of celestial mechanics which is the study of the

motions of natural celestial bodies such as the moon and planets.

The main force acting upon an orbiting satellite is the force of gravity

and due to influence of gravity satellite have to fall towards the Earth.

The Earth curves approximately 5 meters downward for every 8000

meters along its horizon. In order for a satellite to successfully orbit

the Earth, it must travel a horizontal distance of 8000 meters before

falling a vertical distance of 5 meters .

A satellite launched with speed less than 8000

m/s (1 and 2) would eventually fall to the Earth.

A satellite launched with a speed of 8000 m/s

(3) would orbit the Earth in a circular path.

Since gravitational influences decrease with the height above the Earth,

the orbital speed required for a circular orbit is less than 8000 m/s at

significantly greater heights above Earth's surface. And the exact value

of speed for each height of satellite orbit could be found.

Equating the formulas for gravitational force and centripetal force

we can solve for :

While a trajectory is a path traced by a moving body, an orbit is a

trajectory that is periodically repeated. While the path followed by the

motion of an artificial satellite around Earth is an orbit, the path followed

by a launch vehicle is a trajectory called the launch trajectory.

Difference between trajectory and orbit.

Unperturbed motion of satellite

The motion of an Earth artificial satellite is the motion of a body with

very small mass and negligible dimensions with respect to the planet. If

there are no other forces acting on such point mass (satellite) except

Earth attraction, it will repeat same track over and over. Such motion of

satellite is called unperturbed motion.

Integration of equations of unperturbed motion leads to the laws of

unperturbed motion discovered by German mathematician and

astronomer Johannes Kepler (1571 – 1630).

First Kepler’s Law

The orbit of each satellite is an ellipse with the Earth mass

center at one focus.

Kepler’s First Law states that the path of each planet around the

sun is an ellipse with the sun at one focus. This law in context of

satellite motion has follow interpretation:

Kepler’s laws (as we now know them)

allow shapes of orbit of all conic sections:

circle, ellipse, parabola and hyperbola.

Ellipse elements

F1 and F2 - foci of ellipse

a – semi-major axis

ε = – eccentricity

b – semi-minor axis 2

1

a

b

Second Kepler’s Law

A line joining a satellite and Earth’s mass center sweeps out equal areas during equal intervals of time.

Kepler’s Second Law relates a planet’s speed to its distance from the

sun. Because the planets’ orbits are elliptical, the distance from the sun

varies. The Second Law states that if a line is drawn from the sun to the

orbiting planet, then the area swept out by this line in a given time

interval is constant. Interpretation of it for satellite motion is follow:

In consequence of law in an elliptical orbit, the

satellite's velocity changes depending on where

it is in its orbital path. When the satellite is in

the part of its orbit closest to the Earth, it moves

faster because the Earth's gravitational pull is

stronger. The satellite is moving the fastest at

the low point of an elliptical orbit. The low

point of the orbit is called the perigee. The high

point of the orbit, when the satellite is moving

the slowest, is called the apogee.

Third Kepler’s Law

The square of the orbital period of a satellite is directly proportional to the cube of the semi-major axis of its orbit.

Kepler’s Third Law sometimes referred to as the law of harmonies -

compares the orbital period and radius of orbit of a planet to those of

other planets. The comparison being made is that the ratio of the squares

of the periods to the cubes of their average distances from the sun is the

same for every one of the planets. For satellites application Kepler’s

Third Law states that:

Perturbed motion of satellite

In reality satellites experience many forces other than gravity attraction

from the Earth. The accuracy of the unperturbed model of satellites’

motion tends to decrease over time as these forces perturb the orbit.

Perturbations over the satellite orbit resulted from

additional influences.

Gravitational forces:

- Non sphericity of the Earth gravitational potential

- Third body effect: direct attraction of Moon and Sun

Non-gravitational forces:

- Solar radiation pressure

- Atmospheric drag

-Satellite maneuver

Main significant perturbations are:

Conclusion:

Laws of physics (orbital mechanics) give us mathematical model

of orbital motion for any satellite. With the help of these model we can

calculate orbit and predict satellite position at any moment of time.

Additional influences perturb orbit of real satellite and decrease

over time accuracy of calculated (unperturbed) model.

If we have facilities for satellite motion monitoring we could

correct initial model and keep the accuracy of it.

Types of satellite orbits

As described earlier satellites travel around Earth along

predetermined repetitive paths called orbits. The satellite orbits can

be classified on the basis of:

1. Orientation of the orbital plane

2. Eccentricity

3. Distance from Earth

Orientation of the Orbital Plane

The orbital plane of the satellite can have various orientations with

respect to the equatorial plane of Earth. The angle between the two

planes is called the angle of inclination (i) of the satellite.

On this basis, the orbits can be classified as equatorial orbits,

polar orbits and inclined orbits.

i = 0° i = 90° 0° < i < 90°

Eccentricity of the Orbit

ε = 00 < ε < 1

On the basis of eccentricity, the orbits are classified

as elliptical and circular orbits.

Distance from Earth

Depending upon the distance

orbits are classified as:

low Earth orbits (LEOs)Height =160 to 1000 km

medium Earth orbits (MEOs) and

Height =10 000 to 20 000 km

geostationary Earth orbits (GEOs)

Height = 36 000 km

Geostationary orbit

Apogee: Point on the satellite orbit farthest from the centre of the Earth. The apogee distance is the distance of the apogee

point from the centre of the Earth

Centrifugal force: The force acting outwards from the centre of the Earth on any body orbiting it

Centripetal force: A force that is directed towards the centre of the Earth due to the gravitational force of attraction of Earth

Equatorial orbit: An orbit in which the satellite’s orbital plane coincides with the Earth’s equatorial plane

Geostationary Earth orbit (GEO): A satellite orbit with an orbit height at 35 786 km above the surface of the Earth. This

height makes the satellite’s orbital velocity equal to the speed of rotation of Earth, thus making the satellite look stationary

from a given point on the surface of the Earth

Inclination: Inclination is the angle that the orbital plane of the satellite makes with the Earth’s equatorial plane

Inclined orbit: An orbit having an angle of inclination between 0◦ and 180◦

Kepler’s laws: Laws that originally describes motion of planets around Sun

Low Earth orbit (LEO): A satellite orbit with an orbital height of around 150 km to 500 km above the surface of Earth.

These orbits have lower orbital periods, shorter propagation delays and lower propagation losses

Medium Earth orbit (MEO): A satellite orbit with an orbital height around 10 000 km to 20 000 km above the surface of

the Earth

Orbit: A trajectory that is periodically repeated

Polar orbit: An orbit having an angle of inclination equal to 90◦

Satellite: A natural or artificial body moving around a celestial body

Trajectory: A path traced by a moving body

Glossary