A radio-astronomy receiver observes a pulsar

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Plasma Physics · Waves in plasmas

Modes, Packets and Media, Ch Transmission lines and electromagnetic waves in matter, plasmas and guides

Problem

A radio-astronomy receiver observes a pulsar. Its dipole feed presents a resistive impedance

Z_{\rm L} = 300\,\Omega

and is matched to the

Z_0 = 50\,\Omega

coaxial line of the receiver by a quarter-wave transformer designed for

f_0 = 800\,\mathrm{MHz}

, built from a coaxial line with velocity factor

0.70

, giving a wave speed of

0.70c

in the transformer. ((a)) Find the transformer's characteristic impedance

Z_{\rm m}

and its physical length

\ell

. ((b)) The receiver specification requires the voltage reflection coefficient at the

50\,\Omega

input,

\Gamma = (Z_{\rm in}-Z_0)/(Z_{\rm in}+Z_0)

, to satisfy

|\Gamma| \le 0.10

down to

750\,\mathrm{MHz}

. Using the general (not the quarter-wave-only) input-impedance formula for a lossless line of length

\ell

and characteristic impedance

Z_{\rm m}

terminated in

Z_{\rm L}

, find

|\Gamma|

750\,\mathrm{MHz}

and state whether the specification is met. ((c)) The pulsar's radiation crosses about

1.5\,\mathrm{kpc}

of interstellar plasma whose mean electron density gives a plasma frequency of order

1.3\,\mathrm{kHz}

, far below the

400

800\,\mathrm{MHz}

band used here. Using the free electron's equation of motion, explain (i) why this plasma does not measurably absorb the signal at these frequencies, and (ii) why, if a wave's frequency were instead below the plasma frequency, the same plasma would turn it back completely rather than absorb it. ((d)) A two-channel spectrogram of one pulse shows the

800\,\mathrm{MHz}

channel leading the

400\,\mathrm{MHz}

channel by

584\,\mathrm{ms}

. Taking the source distance as

D = 1.5\,\mathrm{kpc}

, find the dispersion measure and the mean electron density along the path, and state briefly why the delay scales as

1/f^2

rather than

1/f

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