In an (alpha) Chapman layer for which photochemical equilibrium has been established, the following equation represents the electron density distribution as a function of reduced height z:

where a is the recombination coefficient, % is the solar zenith angle, and qo is the maximum production rate in the layer.

Recall that a is the recombination coefficient (see Section 3.3). The quantity (qo/a) is dependent upon sunspot number and is specific to the region involved, in this case the E-region. The maximum rate of electron production q0 occurs only for the overhead sun. However, it may be shown that actual maxima for other zenith angles are simply related by this expression:

qmax^qocosx (3.12)

It may be shown that the ordinary ray critical frequency for the E-region, which is directly related to the E-region maximum electron density through Equation 3.10, may be found from Equation 3.11, and is given by:

where at is a constant of proportionality, which is dependent upon the sunspot number. The exponent n tends to a value 0.25 in terms of the long-term seasonal behavior, and in compliance with Chapman theory, but some workers have found that a value for n = 0.3 better represents the diurnal dependence. The constant of proportionality k ranges between about 3 and 4 MHz, bearing in mind that equation 3 represents a climatological median value.

The solar activity dependence of the ratio of peak production to the effective loss (recombination) coefficient has been studied by a number of workers, and the results enable values of foE to be deduced. There have also been direct measurements of foE using vertical incidence sounders. While there is some variability to be considered, it is possible to develop a relationship between the median value of foE, and the solar zenith angle, and the 12-month running-mean sunspot number. A generally accepted candidate for the daytime E-region critical frequency is:

where foE is given in MHz and Rj2 is the running 12-month sunspot number that may range between roughly 10 and 150.

Equation 3.14 provides an excellent agreement with observation during the daytime, but alternative expressions are found to be more appropriate during the nighttime hours [Davies, 1990]. Moreover, it has been found that Equation 3.14 is inaccurate in the very high latitudes where other means of electron production become important, invalidating the Chapman hypothesis. Internationally adopted relations for monthly median foE are due to Muggleton [1975]; and an alternative relation, specific to the European region, has been published [Bradley, 1999].

Figure 3-8a contains an E region critical frequency map for summer solstice conditions in 1958, a period of high solar activity (i.e., R12 large). The contours are representative of median conditions as a function of geographic latitude and local time. It is seen that the E-region critical frequencies (and consequently the electron densities) are vanishingly small in those regions devoid of solar illumination. This summer solstice behavior is consistent with Equation 14, and other seasons have also been shown to behave in conformance with (cos %) °'25 as well.

Figure 3-8b shows the monthly variation offoE for a specified station (i.e., Ft. Belvoir, Virginia) for the year 1958. The solar control is obvious in the median data being plotted.

Figure 3-8: (a) Depiction of the local time (LST) and the latitude dependence oi' foE for solar maximum conditions in summer, (b) Contours offoE at Fort Bel voir, Virginia in 1958 (solar maximum), showing seasonal variations. The contours are in MHz. From Davies [1965].

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