An Introduction to Modern Astrophysics · Problem 6.11

Binary Systems and Stellar Parameters — Carroll & Ostlie

Physics — Astrophysics · radio-telescope-receiver-bandwidth

Problem

Suppose that a radio telescope receiver has a bandwidth of 50 MHz centered at 1.430 GHz (1 GHz =1000= 1000 MHz). Assume that, rather than being a perfect detector over the entire bandwidth, the receiver's frequency dependence is triangular, meaning that the sensitivity of the detector is 0\% at the edges of the band and 100\% at its center. This filter function can be expressed as fν={ννm−νℓ−νℓνm−νℓif νℓ≤ν≤νm−ννu−νm+νuνu−νmif νm≤ν≤νu0elsewhere.f_\nu = \begin{cases} \dfrac{\nu}{\nu_m - \nu_\ell} - \dfrac{\nu_\ell}{\nu_m - \nu_\ell} & \text{if } \nu_\ell \leq \nu \leq \nu_m \\[10pt] -\dfrac{\nu}{\nu_u - \nu_m} + \dfrac{\nu_u}{\nu_u - \nu_m} & \text{if } \nu_m \leq \nu \leq \nu_u \\[10pt] 0 & \text{elsewhere.} \end{cases} (a) Find the values of νℓ\nu_\ell, νm\nu_m, and νu\nu_u. (b) Assume that the radio dish is a 100\% efficient reflector over the receiver's bandwidth and has a diameter of 100 m. Assume also that the source NGC 2558 (a spiral galaxy with an apparent visual magnitude of 13.8) has a constant spectral flux density of S=2.5S = 2.5 mJy over the detector bandwidth. Calculate the total power measured at the receiver. (c) Estimate the power emitted at the source in this frequency range if d=100d = 100 Mpc. Assume that the source emits the signal isotropically.

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