t a magnetic field and negligible loss of energy is given by equation 7.88, where N is the number of electrons per cubic centimeter and f is the frequency in kilohertz. 12-1-81N (7.88) If the Earth's magnetic field and electron collisions are included, the complex refractive index, u, is given by the Appleton-Hartree formula, as in equation 7.89, where Y is the gyrofrequency divided by the wave frequency, O is the angle between magnetic field and ray, and 2Z is the collision divided by wave frequencies. (7.89) The square root term takes the positive sign for the ordinary wave and the negative sign for the extraordinary wave. The Critical Frequency The maximum electron density of an ionospheric layer determines the critical frequency, fc kHz. This is the highest frequency for which a vertically incident ray can be returned to Earth and corresponds to a refractive index of zero. The Earth's magnetic field and collisions can usually be neglected, resulting in equation 7.90. fc=9VN (7.90) lonospheric Cross Modulation The oscillatory energy given to an electron by a wave is normally small compared with the ambient thermal energy, but for a very strong wave this might not be SO. This increases the probability of electrons colliding with molecules SO that all other waves entering the iono- sphere will have increased attenuation. If the strong wave is amplitude modulated, the ionospheric attenuation will vary with the instantaneous amplitude of this wave and thus Chapter 7 Electromagnetic Waves Escapes lonosphere (ref. index no) Skip Distance Multiple Surface Wave Fig. 7.26 Sky wave and Transmitter surface wave transmission showing the skip distance. the modulation. All other waves are therefore modulated in amplitude by the strong wave. This is called ionospheric cross-modulation. Skip Distance and Maximum Usable Frequency When a wave has a frequency greater than the critical frequency of the layer, it can still be returned to Earth if the angle of incidence, a, is such that inequality 7.91, where no is the refractive i