Expert: Philip A. Stahl - 2/20/2008

Question
I am hoping this is an astrophysics question, you can help me solve, though it is to do with radio astronomy. The question is: A wave of 10^-10 jansky and normalized Stokes parameters: 1, 0, 1/3, 0 is incident on an antenna of 10^3 m^2 effective aperture and normalized Stokes parameters: 1, 0, 1, 0. Find the received spectral power.

Answer
Hello,

The normalized Stokes parameters are given as:

S[s_i] =  S x

(s0)
(s1)
(s2)
(s3)


where ‘S’ is the total Poynting vector in watts/m^2 and s0….s3 are the normalized Stokes parameters as a column matrix, in this case:

s0 = 1  

s1 = 0

s2 = 1

s3 = 0

It should be emphasized that these parameters – call them [s_i], apply to the *receiving* antenna (REM: the question asks for "power received").

Meanwhile, [a_i]  =

(a0)  
(a1)
(a2)
(a3)

where, a0 = 1, a1 = 0, a2 = 1/3, and a3 = 0 all apply to the transmitting antenna.and the transmitted wave of 10^-10 jansky

The (received) spectral power would then be given by:

W = ½ [S x A_e] [a^T][s_i}

where S is the Poynting vector, A_e the effective aperture of the antenna. and [a^T] is the transpose of the column matrix for [a_i]

Thus, we will have:

[a^T] = [a0…  a1…  a2…  a3] = [1…0…   1/3…   0]

Of course, [s_i] remains a column matrix which will now have to be multiplied by [a^T]

From here you ought to be able to solve the problem, finish it off. As a helpful hint just bear in mind that the unit jansky = watts m^-2 s^-1.

If your solution is correct, you should come out with the received spectral power in units of watts s^-1.

Good luck!