Laws of artificial satellites motion (Lecture 1)

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 :

Laws of artificial satellites motion (Lecture 1)

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

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

Laws of artificial satellites motion (Lecture 1)

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