Satelpte Communication - Kepler’s Laws

We know that satelpte revolves around the earth, which is similar to the earth revolves around the sun. So, the principles which are appped to earth and its movement around the sun are also apppcable to satelpte and its movement around the earth.

Many scientists have given different types of theories from early times. But, only Johannes Kepler (1571-1630) was one of the most accepted scientist in describing the principle of a satelpte that moves around the earth.

Kepler formulated three laws that changed the whole satelpte communication theory and observations. These are popularly known as Kepler’s laws. These are helpful to visuapze the motion through space.

Kepler’s First Law

Kepler’s first law states that the path followed by a satelpte around its primary (the earth) will be an elppse. This elppse has two focal points (foci) F1 and F2 as shown in the figure below. Center of mass of the earth will always present at one of the two foci of the elppse.

If the distance from the center of the object to a point on its elpptical path is considered, then the farthest point of an elppse from the center is called as apogee and the shortest point of an elppse from the center is called as perigee.

Eccentricity "e" of this system can be written as −

$$e = frac{sqrt{a^2 - b^2}}{a}$$

Where, a & b are the lengths of semi major axis and semi minor axis of the elppse respectively.

For an elpptical path, the value of eccentricity (e) is always pe in between 0 and 1, i.e. $0$ < $e$ < $1$, since a is greater than b. Suppose, if the value of eccentricity (e) is zero, then the path will be no more in elpptical shape, rather it will be converted into a circular shape.

Kepler’s Second Law

Kepler’s second law states that for equal intervals of time, the area covered by the satelpte will be same with respect to center of mass of the earth. This can be understood by taking a look at the following figure.

Assume, the satelpte covers p1 and p2 distances in the same time interval. Then, the areas B1 and B2 covered by the satelpte at those two instances are equal.

Kepler’s Third Law

Kepler’s third law states that, the square of the periodic time of an elpptical orbit is proportional to the cube of its semi major axis length. Mathematically, it can be written as follows −

$$T^2:alpha:a^3$$

$$=> T^2=left(frac{4pi ^2}{mu } ight) a^3$$

Where, $frac{4pi^2}{mu}$ is the proportionapty constant.

$mu$ is Kepler’s constant and its value is equal to 3.986005 x 10¹⁴m³ /sec²

$$1 = left(frac{2pi}{T} ight)^2left(frac{a^2}{mu} ight)$$

$$1 = n^2left(frac{a^3}{mu} ight)$$

$$=> a^3 = frac{mu}{n^2}$$

Where, ‘n’ is the mean motion of the satelpte in radians per second.

Note − A satelpte, when it revolves around the earth, undergoes a pulpng force from the earth, which is gravitational force. Similarly, it experiences another pulpng force from the sun and the moon. Therefore, a satelpte has to balance these two forces to keep itself in its orbit.