5. Asymptotic Estimates of the Attenuation

[24]  Since the conditions of the radio wave propagation through LREI are determined mainly by the attenuation, the main amount of calculations falls on the calculation of the latter, the simulation of the attenuation with the relative error of the order of a few percent takes a lot of time. It should be noted that such a high accuracy of calculations is needed to provide a control of the errors of the numerical simulation of the electron concentration in LREI on the basis of the physical measurements. At the same time, to perform mass calculations of the attenuation with the errors not exceeding the accuracy of the N simulation in LREI [Kozlov, 1967, 1971; Kozlov and Kudimov, 1969; Kozlov and Raizer, 1966], one can obtain simple asymptotic formulae making possible to evaluate the attenuation value with the accuracy of the order of a few tens of percent.

[25]  To do that, neglecting the terms of the order of h²/R_E², we integrate equation (23) with respect to h and (taking into account equation (2)) rewrite the integrand in the full form

(25)

where N(h,l,t) and n _{eff, en} (h) are determined by equations (11) and (24), respectively, and

(26)

Evaluating integral (25) by the mountain pass method, we obtain for the radio waves with w 10⁹ s ^-1 the value of the attenuation in LREI expressed in decibels

(27)

Here

and l₀ with the given coordinates (M, F) of the observed object, geographic coordinates of the explosion (j₁, l₁ ) and the radio engineering device (j₂, l₂ ) are calculated for h = h₀ using formulae (12) and (13), where

[26]  The parameters entering equation (27) and other formulae and used to calculate attenuation are fitted to provide the best approximation of the LREI calculation results obtained on the basis of the Kozlov method [Kozlov, 1967, 1971; Kozlov and Kudimov, 1969; Kozlov and Raizer, 1966]. These values for a wide range of equivalents q=300 10,000 kT and explosion altitudes H =150 500 km are shown in Table 2. Moreover, the interpolation formulae are obtained for the values J_Mb, s cm^-1; J_MN, s cm^-1; and h_m, km:

(28)

(29)

(30)

where q = 47 + 0.05H, g = 625 + 0.5 H, x = 7.1- 0.00133H, and z = 8.5- 0.01H. The values of t, H, and q are measured in sec, km, and kT, respectively.

[27]  As an example, the asymptotic estimates of the attenuation in LREI are calculated using equation (27) and formulae for the values included into equation (27). The calculations were performed for the working frequency w =10⁹ s^-1 for the explosion with q=5000 kT conducted at H = 150 km and distanced from the radio engineering device by b = 335 km and are presented in Figure 8 (dashed curves).