Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
Module 1
Satellite Orbits and Trajectories
DEFINITION OF AN ORBIT AND A TRAJECTORY
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.
The motion of different planets of the solar system around the sun and the motion of artificial satellites
around Earth (Figure 2.1) are examples of orbital motion.
The term ‘trajectory’, on the other hand, is associated with a path that is not periodically revisited.
The path followed by a rocket on its way to the right position for a satellite launch (Figure 2.2) or the
path followed by orbiting satellites when they move from an intermediate orbit to their final destined
orbit (Figure 2.3) are examples of trajectories.
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 1
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
BASIC PRINCIPLES
Orbiting Satellites
The motion of natural and artificial satellites around Earth is governed by two forces. One of them is the
centripetal force directed towards the centre of the Earth due to the gravitational force of attraction of
Earth and the other is the centrifugal force that acts outwards from the centre of the Earth (Figure 2.4).
The centrifugal force is the force exerted during circular motion, by the moving object upon the other
object around which it is moving.
In the case of a satellite orbiting Earth, the satellite exerts a centrifugal force. However, the force that is
causing the circular motion is the centripetal force. In the absence of this centripetal force, the satellite
would have continued to move in a straight line at a constant speed after injection.
The centripetal force is directed at right angles to the satellite’s velocity towards the center of the Earth
that transforms the straight line motion to the circular or elliptical one, depending upon the satellite
velocity.
Centripetal force further leads to a corresponding acceleration called centripetal acceleration as it causes
a change in the direction of the satellite’s velocity vector.
The centrifugal force is simply the reaction force exerted by the satellite in a direction opposite to that of
the centripetal force.
This is in accordance with Newton’s third law of motion, which states that for every action there is an
equal and opposite reaction.
This implies that there is a centrifugal acceleration acting outwards from the centre of the Earth due to
the centripetal acceleration acting towards the centre of the Earth.
The only radial force acting on the satellite orbiting Earth is the centripetal force. The centrifugal force
is not acting on the satellite; it is only a reaction force exerted by the satellite.
The two forces can be explained from Newton’s law of gravitation and Newton’s second law of motion.
Newton’s Law of Gravitation
According to Newton’s law of gravitation, every particle irrespective of its mass attracts every other
particle with a gravitational force whose magnitude is directly proportional to the product of the masses
of the two particles and inversely proportional to the square of the distance between them and written as
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 2
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
m1, m2 = masses of the two particles
r = distance between the two particles
G = gravitational constant = 6.67 × 10−11 m3/kg s2
The force with which the particle with mass m1 attracts the particle with mass m2 equals the force with
which particle with mass m2 attracts the particle with mass m1.
The forces are equal in magnitude but opposite in direction (Figure 2.5). The acceleration, which is
force per unit mass, experienced by the two particles, however, would depend upon their masses.
A larger mass experiences lesser acceleration.
Newton also explained that although the law strictly applied to particles, it is applicable to real objects
as long as their sizes are small compared to the distance between them. He also explained that a uniform
spherical shell of matter would behave as if the entire mass of it were concentrated at its centre.
Newton’s Second Law of Motion
According to Newton’s second law of motion, the force equals the product of mass and acceleration.
In the case of a satellite orbiting Earth, if the orbiting velocity is v, then the acceleration, called
centripetal acceleration, experienced by the satellite at a distance ‘r’ from the center of the Earth would
be v2/r.
If the mass of satellite is ‘m’, it would experience a reaction force of mv2/r.
This is the centrifugal force directed outwards from the center of the Earth and for a satellite is equal in
magnitude to the gravitational force.
If the satellite orbited Earth with a uniform velocity v, which would be the case when the satellite orbit
is a circular one, then equating the two forces mentioned above would lead to an expression for the
orbital velocity ‘v’ as follows:
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 3
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
In the case of an elliptical orbit, the forces governing the motion of the satellite are the same.
The velocity at any point on an elliptical orbit at a distance ‘d’ from the center of the Earth is given by
the formula
Kepler’s Laws
Johannes Kepler, based on his lifetime study, gave a set of three empirical expressions that explained
planetary motion. These laws were later vindicated when Newton gave the law of gravitation. Though
given for planetary motion, these laws are equally valid for the motion of natural and artificial satellites
around Earth or for any body revolving around another body.
Here, these laws are with reference to the motion of artificial satellites around Earth.
Kepler’s First Law
The orbit of a satellite around Earth is elliptical with the center of the Earth lying at one of the foci of
the ellipse (Figure 2.6).
The elliptical orbit is characterized by its semi-major axis ‘a’ and eccentricity ‘e’.
Eccentricity is the ratio of the distance between the center of the ellipse and either of its foci (= ae) to
the semi-major axis of the ellipse ‘a’.
A circular orbit is a special case of an elliptical orbit where the foci merge together to give a single
central point and the eccentricity becomes zero.
Other important parameters of an elliptical satellite orbit include its apogee (farthest point of the orbit
from the Earth’s center) and perigee (nearest point of the orbit from the Earth’s center) distances.
For any elliptical motion, the law of conservation of energy is valid at all points on the orbit.
The law of conservation of energy states that energy can neither be created nor destroyed; it can only be
transformed from one form to another.
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 4
Module 1
Satellite Orbits and Trajectories
DEFINITION OF AN ORBIT AND A TRAJECTORY
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.
The motion of different planets of the solar system around the sun and the motion of artificial satellites
around Earth (Figure 2.1) are examples of orbital motion.
The term ‘trajectory’, on the other hand, is associated with a path that is not periodically revisited.
The path followed by a rocket on its way to the right position for a satellite launch (Figure 2.2) or the
path followed by orbiting satellites when they move from an intermediate orbit to their final destined
orbit (Figure 2.3) are examples of trajectories.
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 1
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
BASIC PRINCIPLES
Orbiting Satellites
The motion of natural and artificial satellites around Earth is governed by two forces. One of them is the
centripetal force directed towards the centre of the Earth due to the gravitational force of attraction of
Earth and the other is the centrifugal force that acts outwards from the centre of the Earth (Figure 2.4).
The centrifugal force is the force exerted during circular motion, by the moving object upon the other
object around which it is moving.
In the case of a satellite orbiting Earth, the satellite exerts a centrifugal force. However, the force that is
causing the circular motion is the centripetal force. In the absence of this centripetal force, the satellite
would have continued to move in a straight line at a constant speed after injection.
The centripetal force is directed at right angles to the satellite’s velocity towards the center of the Earth
that transforms the straight line motion to the circular or elliptical one, depending upon the satellite
velocity.
Centripetal force further leads to a corresponding acceleration called centripetal acceleration as it causes
a change in the direction of the satellite’s velocity vector.
The centrifugal force is simply the reaction force exerted by the satellite in a direction opposite to that of
the centripetal force.
This is in accordance with Newton’s third law of motion, which states that for every action there is an
equal and opposite reaction.
This implies that there is a centrifugal acceleration acting outwards from the centre of the Earth due to
the centripetal acceleration acting towards the centre of the Earth.
The only radial force acting on the satellite orbiting Earth is the centripetal force. The centrifugal force
is not acting on the satellite; it is only a reaction force exerted by the satellite.
The two forces can be explained from Newton’s law of gravitation and Newton’s second law of motion.
Newton’s Law of Gravitation
According to Newton’s law of gravitation, every particle irrespective of its mass attracts every other
particle with a gravitational force whose magnitude is directly proportional to the product of the masses
of the two particles and inversely proportional to the square of the distance between them and written as
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 2
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
m1, m2 = masses of the two particles
r = distance between the two particles
G = gravitational constant = 6.67 × 10−11 m3/kg s2
The force with which the particle with mass m1 attracts the particle with mass m2 equals the force with
which particle with mass m2 attracts the particle with mass m1.
The forces are equal in magnitude but opposite in direction (Figure 2.5). The acceleration, which is
force per unit mass, experienced by the two particles, however, would depend upon their masses.
A larger mass experiences lesser acceleration.
Newton also explained that although the law strictly applied to particles, it is applicable to real objects
as long as their sizes are small compared to the distance between them. He also explained that a uniform
spherical shell of matter would behave as if the entire mass of it were concentrated at its centre.
Newton’s Second Law of Motion
According to Newton’s second law of motion, the force equals the product of mass and acceleration.
In the case of a satellite orbiting Earth, if the orbiting velocity is v, then the acceleration, called
centripetal acceleration, experienced by the satellite at a distance ‘r’ from the center of the Earth would
be v2/r.
If the mass of satellite is ‘m’, it would experience a reaction force of mv2/r.
This is the centrifugal force directed outwards from the center of the Earth and for a satellite is equal in
magnitude to the gravitational force.
If the satellite orbited Earth with a uniform velocity v, which would be the case when the satellite orbit
is a circular one, then equating the two forces mentioned above would lead to an expression for the
orbital velocity ‘v’ as follows:
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 3
, Regulation – 2022 Scheme SATELLITE AND OPTICAL COMMUNICATION – BEC515D
In the case of an elliptical orbit, the forces governing the motion of the satellite are the same.
The velocity at any point on an elliptical orbit at a distance ‘d’ from the center of the Earth is given by
the formula
Kepler’s Laws
Johannes Kepler, based on his lifetime study, gave a set of three empirical expressions that explained
planetary motion. These laws were later vindicated when Newton gave the law of gravitation. Though
given for planetary motion, these laws are equally valid for the motion of natural and artificial satellites
around Earth or for any body revolving around another body.
Here, these laws are with reference to the motion of artificial satellites around Earth.
Kepler’s First Law
The orbit of a satellite around Earth is elliptical with the center of the Earth lying at one of the foci of
the ellipse (Figure 2.6).
The elliptical orbit is characterized by its semi-major axis ‘a’ and eccentricity ‘e’.
Eccentricity is the ratio of the distance between the center of the ellipse and either of its foci (= ae) to
the semi-major axis of the ellipse ‘a’.
A circular orbit is a special case of an elliptical orbit where the foci merge together to give a single
central point and the eccentricity becomes zero.
Other important parameters of an elliptical satellite orbit include its apogee (farthest point of the orbit
from the Earth’s center) and perigee (nearest point of the orbit from the Earth’s center) distances.
For any elliptical motion, the law of conservation of energy is valid at all points on the orbit.
The law of conservation of energy states that energy can neither be created nor destroyed; it can only be
transformed from one form to another.
Prepared by: Prof. Martin Joel Rathnam & Prof. Mohanthi K Department of ECE, SSCE, Anekal, Bengaluru. Page | 4
