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Satellites: Orbital velocity, Time period and Energy

Satellites: Orbital velocity, Time period and Energy
A body moving in an orbit around a planet is called satellite. The moon is the natural satellite of the Earth. It moves around the Earth once in 27.3 days in an approximate circular orbit of radius 3.85 105 km. The first artificial satellite Sputnik was launched in 1956. India launched its first satellite Aryabhatta on April 19, 1975.

Satellites

 

A body moving in an orbit around a planet is called satellite. The moon is the natural satellite of the Earth. It moves around the Earth once in 27.3 days in an approximate circular orbit of radius 3.85 105 km. The first artificial satellite Sputnik was launched in 1956. India launched its first satellite Aryabhatta on April 19, 1975.

Orbital velocity

 

Artificial satellites are made to revolve in an orbit at a height of few hundred kilometres. At this altitude, the friction due to air is negligible. The satellite is carried by a rocket to the desired height and released horizontally with a high velocity, so that it remains moving in a nearly circular orbit.

 

The horizontal velocity that has to be imparted to a satellite at the determined height so that it makes a circular orbit around the planet is called orbital velocity.

 

Let us assume that a satellite of mass m moves around the Earth in a circular orbit of radius r with uniform speed vo. Let the satellite be at a height h from the surface of the Earth. Hence, r = R+h, where R is the radius of the Earth.

 

The centripetal force required to keep the satellite in circular orbit is F = mv02/r = mv02 / R+h

The gravitational force between the Earth and the satellite is

F = GMm / r2 = GMm/(R+h)2

For the stable orbital motion,

v0 = root(gR2/R+h)

If the satellite is at a height of few hundred kilometers (say 200 km), (R+h) could be replaced by R.

orbital velocity, vo = rt(gR)

If the horizontal velocity (injection velocity) is not equal to the calculated value, then the orbit of the satellite will not be circular. If the injection velocity is greater than the calculated value but not greater than the escape speed (ve = 2 vo), the satellite will move along an elliptical orbit. If the injection velocity exceeds the escape speed, the satellite will not revolve around the Earth and will escape into the space. If the injection velocity is less than the calculated value, the satellite will fall back to the Earth.

 

Time period of a satellite

 

Time taken by the satellite to complete one revolution round the Earth is called time period.

Time period, T = circumference of the orbit / orbital velocity

T = 2πr / v0  = 2π(R +h) / v0

where r is the radius of the orbit which is equal to (R+h).

v0 = root[GM/R+h] so

T = 2π (R+h) root[(G+h)/GM]

As GM = gR2, T = 2π root[(R+h)3/gR2]

If the satellite orbits very close to the Earth, then h << R

T = 2π root[R/g]

Energy of an orbiting satellite

A satellite revolving in a circular orbit round the Earth possesses both potential energy and kinetic energy. If h is the height of the satellite above the Earth's surface and R is the radius of the Earth, then the radius of the orbit of satellite is r = R+h.

If m is the mass of the satellite, its potential energy is,

EP = - GMm/r

where M is the mass of the Earth. The satellite moves with an orbital velocity of vo = root[GM/(R+h)]

Hence, its kinetic energy is,

EK = 1/2 mvo2 

 EK = GMm / 2(R+h)

The total energy of the satellite is, E = EP + EK

E = − GMm / 2(R+h)

The negative value of the total energy indicates that the satellite is bound to the Earth.

Geo-stationary satellites

 

A geo-stationary satellite is a particular type used in television and telephone communications. A number of communication satellites which appear to remain in fixed positions at a specified height above the equator are called synchronous satellites or geo-stationary satellites. Some television programmes or events occuring in other countries are often transmitted 'live' with the help of these satellites.

 

For a satellite to appear fixed at a position above a certain place on the Earth, its orbital period around the Earth must be exactly equal to the rotational period of the Earth about its axis.

 

Consider a satellite of mass m moving in a circular orbit around the Earth at a distance r from the centre of the Earth. For synchronisation, its period of revolution around the Earth must be equal to the period of rotation of the Earth (ie) 1 day = 24 hr = 86400 seconds.

 

The speed of the satellite in its orbit is

v = Circumference of orbit / Time period

v = 2π r / T

The centripetal force is F = mv2/r

F = 4mπ2r /T2

The gravitational force on the satellite due to the Earth is

F = GMm/r2

For the stable orbital motion 4mπ2r / T2  = GMm / r2

We know that, g = GM/R2

r3 = gR2T2 / 4 π2

The orbital radius of the geo- stationary satellite is, r = [gR2T2 / 4 π2]1/3

Substituting T = 86400 s, R = 6400 km and g = 9.8 m/s2, the radius of the orbit of geo-stationary satellite is calculated as 42400 km.

 

The height of the geo-stationary satellite above the surface of the Earth is h = r - R = 36000 km.

 

If a satellite is parked at this height, it appears to be stationary. Three satellites spaced at 120o intervals each above Atlantic, Pacific and Indian oceans provide a worldwide communication network.

 

Polar satellites

 

The polar satellites revolve around the Earth in a northsouth orbit passing over the poles as the Earth spins about its north - south axis.

 

The polar satellites positioned nearly 500 to 800 km above the Earth travels pole to pole in 102 minutes. The polar orbit remains fixed in space as the Earth rotates inside the orbit. As a result, most of the earth's surface crosses the satellite in a polar orbit. Excellent coverage of the Earth is possible with this polar orbit. The polar satellites are used for mapping and surveying.

 

 

Uses of satellites

 

(i) Satellite communication

 

Communication satellites are used to send radio, television and telephone signals over long distances. These satellites are fitted with devices which can receive signals from an Earth - station and transmit them in different directions.

 

(ii) Weather monitoring

 

Weather satellites are used to photograph clouds from space and measure the amount of heat reradiated from the Earth. With this information scientists can make better forecasts about weather. You  might have seen the aerial picture of our country taken by the satellites, which is shown daily in the news bulletin on the television and in the news papers.

 

(iii) Remote sensing

 

Collecting of information about an object without physical contact with the object is known as remote sensing. Data collected by the remote sensing satellities can be used in agriculture, forestry, drought assessment, estimation of crop yields, detection of potential fishing zones, mapping and surveying.

 

(iv) Navigation satellites

 

These satellites help navigators to guide their ships or planes in all kinds of weather.


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