How to Calculate Detected Power by a Radiotelescope?

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SUMMARY

The calculation of detected power by a radiotelescope involves integrating the specific intensity of a source over angular coordinates. For a radiotelescope operating at 3 GHz with an effective area of 1000 m², pointed at Mars, the detected power can be calculated using the formula Pν = A ∫ Iν(θ, φ) f(θ, φ) dθ dφ. Given Mars' temperature of 210 K and a frequency bandwidth of Δν = 30 GHz, the specific intensity of Mars, modeled as a black body, is essential for this calculation. The angular diameter of Mars is determined to be 25" or 0.417'.

PREREQUISITES
  • Understanding of radiotelescope operation and parameters
  • Familiarity with black body radiation and Planck's law
  • Knowledge of angular coordinates and integration in spherical coordinates
  • Basic proficiency in physics, particularly thermodynamics and electromagnetic theory
NEXT STEPS
  • Study the derivation of Planck's law for black body radiation
  • Learn about the integration of functions over spherical coordinates
  • Explore the concept of effective area in antenna theory
  • Investigate the impact of frequency and bandwidth on radiotelescope sensitivity
USEFUL FOR

Astronomy students, astrophysicists, and engineers involved in radio astronomy or radiotelescope design will benefit from this discussion.

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Homework Statement



The specific power received by a radiotelescope is given by:

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

where:
θ, \phi are the angular coordinates on the celestial sphere around the pointing direction;
I_\nu is the angular distribution of the specific intensity of the source (in units of W/m2 Hz std);
A f(θ, \phi) is the effective area of the antenna in the direction (θ, \phi).

The radiotelescope, operating at a frequency of 3 GHz, of area A= 1000 m2 and with f(θ, \phi)= 1 for θ0>θ and f(θ, \phi)= 0 for θ>θ0, with θ0=1.9 arcminutes, is pointed in the direction of Mars (distance from the Earth d= 56 x 106 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\nu=30 GHz around the working frequency.

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