Line of sight on surface of Earth

LINE OF SIGHT ON SURFACE OF EARTH

INTRODUCTION

Earth is the third planet in our solar system which has a mean radius of 6371 km. Although the equatorial radius is not equal to its polar radius, we can approximate the earth as a sphere. Any object on the surface of sphere has a finite view due to the curvature of the sphere. Thus any one can view only up to a finite distance before the horizon. The horizon is itself defined on the height of the object, the greater the height the more the view. In this article, we intend to determine the line of sight for an average human being on the surface of Earth.

ASSUMPTIONS

1.      The surface of Earth is smooth

2.      Earth is a homogeneous sphere

3.      The sky is clear and vision is not obscured

4.      Light does not undergo diffraction and refraction

5.      Space time around Earth is not curved but flat

6.      The observer is at sea level

CALCULATION

Figure.1 Kansas flat plain with maximum line of sight

Figure.2 Line of sight on Earth

From figure 2,

R – Radius of Earth [m]                                                                                                                            

R = 6371 Km = 6371000 m (Eq. 1)

h – Height of the observer [m]

h = 5 feet

   = 1.5 m (Eq. 2)

                                                                                                                               {⸪ 1 feet = 0.3 m}

d – Observable distance by observer [m]                                                                                         

We can apply Pythagorean Theorem,

d² = (R+h)² – R² (Eq. 3)

d² = 2Rh + h²

d = √ (2Rh+h²) (Eq. 4)

Now substitute equations (1), (2) in equation (4)

d = √ (2*6371000*1.5+1.5²)

d = 4371.8419 m

d = 4.3718 Km [2.7148 miles]

This is the distance that can be viewed by an observer on the surface of Earth provided the weather is clear.

CONCLUSION

We thus determined the line of sight or field of view for an observer on the surface of Earth.