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## Homework Statement

The specific power received by a radiotelescope is given by:

[itex]P_\nu = A \int {I_\nu (θ,\phi) f (θ,\phi) dθ d\phi}[/itex]

where:

θ, [itex]\phi [/itex] are the angular coordinates on the celestial sphere around the pointing direction;

[itex]I_\nu [/itex] is the angular distribution of the specific intensity of the source (in units of W/m

^{2}Hz std);

A f(θ, [itex]\phi [/itex]) is the effective area of the antenna in the direction (θ, [itex]\phi [/itex]).

The radiotelescope, operating at a frequency of 3 GHz, of area A= 1000 m

^{2}and with f(θ, [itex]\phi [/itex])= 1 for θ

_{0}>θ and f(θ, [itex]\phi [/itex])= 0 for θ>θ

_{0}, with θ

_{0}=1.9 arcminutes, is pointed in the direction of Mars (distance from the Earth d= 56 x 10

^{6}km, diameter D= 6794 km), which has an emission approximated with a black body at T=210 K.

Calculate the detected power by the antenna in a band with Δ[itex]\nu[/itex]=30 GHz around the working frequency.

[The specific intensity of a black body is: [itex]\frac{2h}{c^2}\frac{\nu^3}{e^{h\nu/KT}-1}[/itex]

K=1.381 x 10

^{-23}J/K

h = 6.6 x 10

^{-34}J s ]

## Homework Equations

The intensiy of a black body

## The Attempt at a Solution

I can not go so far.

Just I have calculated the angular diameter of Mars: δ=arctan D/d = 25"=0.417'.

Any suggestion please?

I can not how to carry out the result, but I think that this is a particular educational exercise.

Thanks in advance.