#### Satellite:

• Any object that revolves around a given planet in circular orbit under the influence of planet’s gravitational force is called as a satellite.
• The moon is the natural satellite of earth. The Earth, the Venus, and Jupiter are the natural satellites of the sun. INSAT-B, INSAT-IC are artificial satellites of the earth. Sputnik was the first artificial satellite put in the orbit by U.S.S.R. (Russia).

#### Projection of Satellite:

• To launch a satellite in an orbit around the earth minimum two-stage rocket is used.  The launching involves two steps. In the first step the satellite is taken to the desired height and then in the second step, it is projected horizontally with calculated speed in a definite direction.
•  The satellite is kept at the tip of the two-stage rocket. Initially, the first stage of the rocket is ignited on the ground so that rocket is raised to the desired height, the first stage is detached and falls back on the earth. Then the rocket is rotated by remote control to point it in the horizontal direction. Then the second stage is ignited so rocket gets push in the horizontal direction and acquires certain horizontal velocity (Vh).  When the fuel is completely burnt second stage also gets detached. Satellite starts orbiting around the earth.

#### Possibilities of Orbits of Satellites:

• Depending upon Magnitude of Horizontal Velocity following four cases can arise for Motion of Satellite
• Case – 1 (vh < vc): If the horizontal velocity imparted to the satellite is less than critical velocity Vc, then the satellite moves in long elliptical orbit with the centre of the Earth as the further focus. If the point of projection is apogee and in the orbit, the satellite comes closer to the earth with its perigee point lying at 180o. If enters earth’s atmosphere while coming towards perigee, it will lose energy and spiral down on the earth. Thus it will not complete the orbit. If it does not enter the atmosphere, it will continue to move in elliptical orbit.
• Case – 2 (vh = vc): If the horizontal velocity imparted to the satellite is exactly equal to the critical velocity Vc then satellite moves in stable circular orbit with earth as centre as shown in the diagram.
• Case – 3 (Ve > Vh > Vc): If the horizontal velocity Vh is greater than critical velocity Vc but less than escape velocity Ve then satellite orbits in
the elliptical path around the earth with the centre of the earth as one of the foci (nearer focus) of the elliptical orbit as shown.
• Case – 4 (vh = vc): If the horizontal velocity imparted to the satellite is exactly equal to escape velocity of satellite Ve then satellite moves in a parabolic trajectory.
• Case – 5 (Vh > Ve): If the horizontal velocity is greater than or equal to escape velocity Ve then satellite overcomes the gravitational
attraction and escapes into infinite space along a hyperbolic trajectory.

#### Geostationary or Geosynchronous or Communication Satellite:

• A communication satellite is an artificial satellite which revolves around the earth in a circular orbit in the equatorial plane such that,
• its direction of motion is the same as the direction of rotation of the ‘earth about its axis.
• its period is the same as the period of rotation of the earth, i.e. 24 hours.
• When observed from the earth’s surface, this satellite appears stationary. Therefore, it is called a geostationary satellite.
• The height of communication satellite above the surface of the earth is about 36,000 km.
• e.g. INSAT series satellites, APPLE.

#### Uses of Communication Satellites /Satellites:

• Satellites are used for sending microwave and TV signals from one place to another.
• They are used for weather forecasting and to give warnings of cyclones to coastal villages.
• They are used for detecting water resource locations and areas rich in ores.
• They are used for spying In enemy countries i.e. It can be used for military purposes
• They are used for global positioning system (GPS).
• They are used for astronomical observations and to study of solar and cosmic radiations.

#### Weightlessness in Satellite:

• By the definition, the weight of a body is equal to the gravitational force with which the body is attracted towards the centre of the earth. When the astronaut is on the surface of the earth, his weight acts vertically downwards.  At the same time, the earth’s surface exerts an equal and opposite force of reaction on the astronaut.  Due to this force of reaction, the astronaut feels his weight.
• When the astronaut is in an orbiting satellite, a gravitational force still acts upon him.  However, in this case, both the astronaut as well as the satellite are now attracted towards the earth and have the same centripetal acceleration due to gravity at that place. It is the condition of free fall of the body.
• As both astronaut and the surface of the satellite are attracted towards the earth centre with the same acceleration, and hence the astronaut can’t produce any action on the floor of the satellite.  So the floor does not give any reaction on the astronaut.  Hence the astronaut has a feeling of weightlessness.

#### Critical Velocity of Satellite:

• The minimum horizontal velocity of projection that must be given to a satellite at a certain height, so that it can revolve in a circular orbit round the earth is called critical velocity or orbital velocity.

#### Expression for Critical Velocity of Satellite:

Let m = mass of the satellite

M = Mass of the earth

R = Radius of the earth

h = Height of satellite above the surface of the earth

r = R + h = Radius of orbit of the satellite

G = Universal gravitational constant

• The necessary centripetal force for the circular motion of satellite is provided by the gravitational attraction between the satellite and the earth.

Now, Centripetal force = Gravitational force

• This is the expression for the critical velocity of a satellite orbiting around the earth at height h from the surface of the earth.
• If a satellite is orbiting very close to the earth’s surface i.e. h < < R then h may be neglected in comparison of R and Critical velocity may be given as

#### Expression for the Critical Velocity of Satellite in Terms of Acceleration Due to Gravity:

• We have

Where gh = acceleration due to gravity at height h from the surface of the earth.

• This is an expression for the critical velocity of a satellite orbiting at height h in terms of acceleration due to gravity at that height.
• If the satellite is orbiting very close to the earth’s surface then h < < R  and thus h can be neglected. Similarly near the surface of the earth gh =  g i.e. the acceleration due to gravity on the surface of the earth.

• This is an expression for the critical velocity of a satellite orbiting very close to the Earth’s surface in terms of acceleration due to gravity.

Period of Satellite:

#### Period of a Satellite

• The time taken by the satellite to complete one revolution around the planet is known as the period of revolution of the satellite.
• It is denoted by ‘T’.  It’s S.I. unit is second.

#### Expression for Period of a Satellite:

Let m = mass of the satellite

M = Mass of the earth

R = Radius of the earth

h = Height of satellite above the surface of the earth

r = R + h = Radius of orbit of the satellite

G = Universal gravitational constant

• The necessary centripetal force for the circular motion of satellite is provided by the gravitational attraction between the satellite and the earth.

Now, Centripetal force = Gravitational force

Now the period of satellite ‘T’ is given by

This is the expression for the time period of a satellite orbiting around the earth at height h from the surface of the earth.

Squaring both sides of above equation, we get

For the given planet the quantity in the bracket is constant hence we can conclude that

T2  ∝  r3

• Thus the square of the period of a satellite is directly proportional to the cube of the radius of Its orbit. (Kepler’s third law of planetary motion)

#### Expression for the Periodic Time in Terms of Acceleration Due to Gravity:

• Let gh be the acceleration due to gravity at a point on the orbit i.e. at a height h above the earth’s surface.

• This is an expression for the period of a satellite orbiting at height h in terms of acceleration due to gravity at that height.
• If satellite orbiting very close to the earth (i.e. h < < R) then h can be neglected. Then r = R and    gh = g

This is an expression for the time period of a satellite orbiting very close to the earth’s surface.