Satellites

Satellites are bodies that orbit around a celestial body. They can be classified as:

• Natural : celestial bodies that orbit around a planet, for example, the Moon;
• Artificial : These are man-made objects placed in orbit around a celestial body. There are several artificial satellites around our planet and with different functions, such as communication, meteorological, military and astronomical satellites.

The movement of satellites around a planet obeys Kepler’s Laws and Universal Gravitation. 

Considering the planet in the figure with mass M and a satellite of mass m in a circular orbit, of radius r, around this planet, we can obtain the satellite’s velocity with the Law of Universal Gravitation and an expression for the satellite’s period through the third Kepler’s Law.

The gravitational force of attraction between the satellite and the planet is centripetal, so we can obtain two equations:

F = G. M .m and F = mv ²
 r ²                 r

Since the two forces are equal, we can equate the two equations, obtaining the expression:

G. M.m = mv ²
 r ²           r

Simplifying the equations, we find an expression for the orbital velocity of the satellite:

v ² = G . Mr
       _

To calculate the period of the satellite, which is the time it takes to go around the planet, we can use the expression found for the velocity:

v ² = G . Mr
      _

and relate it to the equation of velocity in circular motion:

v = ω . r

getting the expression:

ω ² . r ² = G . Mr
_

being ω = 2 π , substituting in the above equation, we have:
                    T

4π ^2. r ² = G . M
 T ²                 r   

We can also find the period of the satellite’s orbit:

T ² = 4π ^2. . r ³
GM

If we define K as 4π ² , we get Kepler’s Third Law or Law of periods:
                                      GM

T ² = k
r ³       

Moon: Earth’s natural satellite

The moon is Earth’s natural satellite. The most accepted hypothesis for its formation is that it was the result of a collision between a body the size of Mars and Earth about 4.4 billion years ago.

The Moon is located at a distance of 384,400 km from Earth and has an orbital period of 27 days. Its mass is equal to 7.349. 10 ²² kg.

The Moon has always beautified our sky, but there are studies that indicate that, each year, it moves about 4 cm away from our planet, which could cause changes in the climate and seasons of the year in the future, generating serious changes in style. life of the earth’s population. But we don’t need to worry yet, as these changes can take up to millions of years.