Question

If the radius of the earth is  $ R $ and the height of a satellite above earth’s surface is  $ R $ then find the minimum co-latitude (in degree) which can directly receive a signal from the satellite.

Answer 1

Hint: In order to solve this question, we are going to first define what a colatitude of an angle is, after that we need to find the latitude of the satellite by considering an arbitrary angle and finding its sine from the triangle so formed and finally calculating the angle. Then, we can calculate the co-latitude. The colatitude is  $ {90^ \circ } $ minus the latitude angle of the sphere.

Complete Step By Step Answer:

In the spherical system, the colatitude is defined as the complimentary angle of the given latitude. Mathematically it is  $ {90^ \circ } $ minus the latitude angle of the sphere.

Now, if  $ \theta  $ is the latitude of the given satellite corresponding to the earth, then sine of this angle can be calculated as:

$   \sin \theta  = \dfrac{R}{{R + R}} = \dfrac{R}{{2R}} $

$   \Rightarrow \theta  = {\sin ^{ - 1}}\dfrac{1}{2} = {30^ \circ } $ 

Now, as we know that the colatitude of an angle is its complementary angle, thus, finding the colatitude, we get

 $ {90^ \circ } - {30^ \circ } = {60^ \circ } $ 

Hence, the minimum colatitude which can directly receive the signal from the satellite is  $ {60^ \circ } $.

Note:
The significance of the colatitude is that it gives the Zenith distance for the celestial bodies, which refers to the distance of the zenith i.e. the overhead point from the center of that celestial body. The right angled triangle formed between the earth and the point of the satellite at the distance  $ R $ above is the only key to find the latitude of the earth and hence colatitude as well.