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Earth Satellites

Ptolemy, a Greek astronomer who lived in the second century AD, was one of the earliest to make a detailed study of the motion of planets and their relative positions in space.
He advanced a theory called the Geocentric theory which considered the earth as the centre of the universe, around which all the planets, the moon and the stars revolved in various orbits. The orbits proposed by the theory were very complex. Despite its complexities, the theory was universally accepted for more than fourteen centuries.
In the year 1543, a Polish astronomer Copernicus proposed a new theory called the Heliocentric theory, according to which the sun was at the centre with planets revolving around it in various orbits.
Copernicus proposed a new theory called the heliocentric theory, according to which the sun was at the centre with planets revolving around it in various orbits. Copernicus also believed that the earth rotated on its axis once each day.
The great Indian mathematician and astronomer Aryabhat who lived in the fifth century AD had made astronomical observations and said that the earth moved about an axis.
The two scientists whose contributions made the most profound-impact on the understanding of the universe where a Danish astronomer Tycho Brabe (1546-1601) and a German theorist Johannes Kepler (1571 - 1630).
Tycho Brabe made very accurate observations of the motion of plants and Kepler analysed these observations carefully and systematically and modified the heliocentric theory.
Kepler was able to unify the observations into three empirical laws relating to planetary motion. These laws formed the basis of the famous Newton's law of universal gravitation.

Kepler's Laws of planetary motion

Kepler (1571-163) made a careful study of Tycho Brahe's records of observations on planetary motion and proposed the following three laws, now known after his name, as kepler's laws.
  1. Each planet revolved round the sun in an elliptical orbit with the sun at one focus of the ellipse (the law of orbits).
  2. The line joining the sun and the planet sweeps out equal areas in equal times (the law of areas).
  3. The squares of the periods of revolution of the planet are proportional to the cubes of their mean distances from the sun (the law of periods).

Kepler's laws of planetary motion

The speed of the planet in its orbit varies but the variation is such that the areas swept out (shown shaded) in equal times are equal. The gravitational force between the sun and the planet provides the necessary inward force to keep it in orbit.
Kepler's law was empirical. Kepler could not give a theoretical proof of his laws. He obtained them by inductive reasoning. it is indeed remarkable that the predictions of the positions of planets based on Kepler's laws agree very well with observation.

Derivation of Kepler's Law of periods from the Law of Gravitation

Let us assume that the planet of mass m moves around the sun of mass M in a circular orbit of radius r. hence, from Newton's law of gravitation, the force of attraction of the sun on the planet is given by
If this force is the centripetal force keeping the planet in orbit, then,
Where u is the speed of the planet in the circular orbit. Therefore, equaling the two, we have
If T is the period of revolution of the planet around the circle,
Since GM is constant for any planet, it follows that is constant, which is Kepler's third law.
If the earth were a perfect sphere of radius 6.37 x 106 m, rotating about its axis with a period of 1 day (-8.64 x 104 s), how much would the acceleration due to gravity (g) differ from the poles to the equator?
Let us take an object of mass m and measure its weight at the poles and at the equator with a spring balance. At the north or South Pole, the object has no rotation about the polar axis. The restoring force Fp in the spring measures the weight mgp of the object, where gp is the acceleration due to gravity at the poles.
When the same object is taken to the equator, it rotates with the earth at an angular velocity ωγιϖενβψ
w = 
where T is the rotational period of the earth. The object, therefore, has a centripetal acceleration Rω2 which is directed to the centre of the earth. Here R is the radius of the earth (Fig (b)). The restoring force Fe on the spring acting upwards is, therefore given by


But Fe = mge,
where ge is the acceleration due to gravity at the equator.
Thus we have
M ge = m gp - m R w 2
gp-ge = Rω2
=0.0337 ms-2

Natural and artificial satellites

A is a heavenly body revolving around a planet in a stable orbit is called a natural satellite.
In solar system, the nine planets namely mercury, venus, earth, mars, Jupiter, Saturn, Uranus, Neptune and Pluto are the main planets, which revolve around the sun and each may be called a satellite of the sun. 
These planets too have got satellites revolving around them. Whereas mercury, venus and Pluto do not have any satellite, the earth has one, mars and neptune - two each, uranus - five, saturn - ten and jupiter has twelve.
These heavenly bodies going around a planet are called natural satellites. They are named so, because they were created by nature. Thus, moon is natural satellite of the earth. 
It goes round the earth in about 27.3 days in an almost cicular orbit of radius 3.84 105 km.
A satellite put to in its orbit around a planet by the man is called the artificial satellite.
Since 1957, many artificial satellites of the earth have been launched in their orbits. On October 4, 1957, man entered the space age, when Russian scientists put Sputnik-I weighing 83.6 kg into an orbit around the earth. Its period of revolution was 96.2 minutes and it continued revolving round the earth for 58 days. 
About one month after, on November 3,1957, Sputnik-II was launched with a dog 'Laika' on board. On January 31, 1958, USA launched its first satellite Explorer-I. It made the important discovery of Van Allen Radiation Belt (a belt of electrically charged particles trapped in earth's magnetic field). 
On April 12, 1961, Russia launched the first manned satellite. Major Yuri Gagarin got the unique honour of piloting the flight. After this, there started an undeclared competition between the Russians and Americans. 
For some time, it appeared that Russian technology is far ahead of the American technology, but in the due course of time, the Americans not only came at par with Russians but surpassed them and succeeded in making man land on the moon and return safe and sound with what are known as Appolo flights. 
On July 16, 1969, Appolo - 11 was launched by U.S.A. with astronauts Neil A. Armstrong, Edwin E. Aldrin and Michael Collins on board. On the historic day of July 20, 1969, Neil Armstrong planted first human foot on the surface of moon.
France launched its first artificial satellite of earth with a pay load of 42 kg on November 1,1965. Later on, Japan launched a 24 kg satellite in February, 1970. China started its space programme by launching a satellite weighing 173 kg. During 1970-1980 period, the space programmes remained a closed circuit of these five countries.
India also joined space club on April 19, 1975 by putting in orbit Aryabhatta, a 350 kg satellite with Russian rockets and from Russian soil. On June 7, 1979, Bhaskara-I was put into orbit around the earth by India as its second satellite. It weighed 444 kg and made a complete round of the earth in 95.2 minutes.
India launched the SLV-3 (Satellite Launch Vehicle) on August 10, 1979, which failed. On July 18, 1980, second SLV-3 was launched from Sri Harikota, island 1000 km off the Madras coast. This four stages rocket with a pay load of 35 kg was named Rohini–I. It was followed by Rohini-II on May 31, 1981.
On June 19,1981, India put its first experimental communication satellite- APPLE (Ariane Passenger Pay Load Experiment). On November 20, 1981, Bhaskara-II was launched.
India launched the first multipurpose satellite INSAT-1 A (Indian National Satellite) on September 4, 1982. 
It is a stationary satellite revolving at a height of 36,000 km above the earth surface and weighed 150 kg. The second satellite of this programme INSAT-1B was launched on August 30, 1983. 
India has been continuing its space programme with the launching of INSAT-1C (on July 12, 1988);INSAT-1D (on June 12, 1990); INSAT-2A (on July 10, 1992), INSAT-2B (on July 23, 1993); INSAT-1E (on September 20, 1993) and INSAT-2C (on December 7, 1995). The satellite IRS-1C was launched by India on December 28, 1995.

Orbital velocity and period of revolution of earth satellites

Moon is a natural satellite of earth moving around earth in an approximate circular orbit of 3.84 105 km radius. Other artificial satellites are made to revolve in an orbit of few hundred kilometers from the surface of earth. These satellites are called earth satellites. 
They are carried by rockets to the desired heights and then given a horizontal velocity so that they remain moving in the orbits. The orbital velocity is the horizontal velocity that has to be given to the satellite at a particular height so that it makes a circular orbit around the planet.

Orbital Velocity (vo)

Let us consider a planet of mass 'm' revolving the earth at a height 'h'. The radius of the earth is 'R'. The mass moving with an orbital velocity 'vo' and the centripetal force
keeping the planet in orbit, is given by
The force of attraction between two masses is given by =
By equating the above two equations, we get vo2 = , since 'h' is very small it is neglected. vo = =

Time period of revolution

Time period T=
We know that orbital velocity is given by vo =
Since, vo =
Time period of revolution is given by
By substituting g = 9.8 ms-2 and R = 6.4 106 m in the equation, it comes out that the time period of a satellite revolving around the earth just close to its surface is about 85 minutes.

Height of satellite above earth's surface

squaring both sides of the equation, we have

T2 = or (R + x)3 = or (R + x) =

x =

Thus, knowing the time period of the satellite T, radius of the earth R, we can calculate the height x of the satellite above the surface of the earth.

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