Visibility Function--Cosine Power -- from Eric Weisstein's World of Physics

V v

Visibility Function--Cosine Power

V v

Visibility Function--Cosine Power

Let the intensity emitted from a point on a spherical planet be

(1)

where is the angle between the normal to the surface and the line of sight. From the diagram,

(2)

(3)

(4)

(5)

Then

(6)

But

(7)

(8)

so

(9)

Expanding in powers of ,

(10)

since . Therefore since

(11)

is a radial function, the Fourier transform Eric Weisstein's World of Math becomes a Hankel transform Eric Weisstein's World of Math

(12)

Now use

(13)

(14)

to obtain

(15)

Use Gradshteyn and Ryzhik (6.683.6, p. 740)

(16)

with

(17)

(18)

(19)

gives

(20)

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 4th ed. San Diego, CA: Academic Press, 1979.