Consider a right-hand circularly polarized wave, of angular frequency , whose electric field takes the form

(See Section 7.7.) Let us search for solutions of Equations (9.54) and (9.55) of the form(9.58) | ||

(9.59) |

(9.62) |

(9.69) | ||

(9.70) |

(9.71) |

(9.73) |

Consider a left-hand circularly polarized wave whose electric field takes the form

(9.74) | ||

(9.75) |

(9.76) |

(9.78) |

According to the previous analysis, in the presence of a longitudinal equilibrium magnetic field, the refractive indices of right-hand and left-hand circularly polarized electromagnetic waves propagating through a plasma are slightly different. Consider what happens when a linearly polarized electromagnetic wave, whose electric field is initially of the form

(9.79) | ||

(9.80) |

(9.81) | ||

(9.82) |

(9.83) | ||

(9.84) |

(9.85) |

(9.86) | ||

(9.87) |

(9.88) |

(9.89) |

*Pulsars* are rapidly rotating neutron stars that emit regular blips of
highly polarized radio waves (Longair 2011). Hundreds of such objects have been found
in our galaxy since the first was discovered in 1967. By measuring
the variation of the angle of polarization, , of radio emission from a pulsar with frequency, , astronomers can effectively determine
the line integral of along the straight line joining the
pulsar to the Earth using formula (9.90) (ibid.). Here, is the number
density of free electrons in the interstellar medium, whereas
is the parallel (to the line joining the pulsar to the Earth) component of the galactic magnetic field. In order
to perform this calculation, astronomers must make the reasonable assumption that the
radiation was emitted by the pulsar with a common angle of polarization, ,
over a wide range of different frequencies. By fitting Equation (9.90)
to the data, and then extrapolating to large , it is possible to
determine , and, hence, the amount,
,
through which the polarization angle of the radiation has rotated, at a given frequency, during its passage to Earth.

Astronomers can also determine the line integral of by looking at the variation of the arrival time of the various components of a pulsar radio blip with frequency (Longair 2011). This calculation depends on the reasonable assumption that the components were emitted simultaneously, and then traveled through interstellar space at the frequency dependent group velocity . [See Equation (9.34).] It follows that the arrival time can be written

By fitting Equation (9.91) to the data, and then extrapolating to large , it is possible to determine , and, hence, at a given frequency. Finally, once the line integrals of and have been independently determined, estimates can be made of the mean electron number density, and the mean galactic magnetic field, along the straight line joining the pulsar to the Earth.