## 3 Results and Discussion
### Approximative Expressions and Simulations
From Figure 11b (see rug plot), we certainly have the largest grouping between 1-4 min, which is also the case with the results of [PERSON] et al. [38] i.e., the results are in accordance. Certainly, there are other results in the literature such as from the paper of [PERSON] [39], but our results and also the recent findings of [PERSON] et al. [38] differ from findings of [PERSON] & [PERSON], which surely does not affect the main goal of the present work. Figure 11: **(a)** The time delay \(\Delta t\) between the VLF amplitude max and the peak of SF X-ray flux as a function of peak flux (black circles are data from present investigation, red stars data from [27]; **(b)** Distribution of \(\Delta t\) data points (a rug plot). The conclusion is that the maximum of the signal, and consequently the electron density, is shifted by an average value of three minutes in relation to the maximum of the flux during the flare event due to ionosphere sluggishness. In addition, detailed statistics could have been performed but the main goal was to obtain a useful and easy to use expression for \(N_{e}\). Variation in X-ray flux (upper panel), measured by GOES-15, and \(N_{e}\) (h = 74 km) as a function of UT during two successive flares on 12 June 2014 (lower panel) are presented in Figure 12. The red line in the lower panel shows results acquired by approximative Equation (5). The circles present values of electron density \(N_{e}\) obtained in this research by the method mentioned above using Equation (3). One can see that the results are in good agreement except the post-flare wings (\(N_{e}\) shape is more stretched). It can be noted that due to the sluggishness of the ionosphere, the shape of electron density over time does not totally follow the solar flux shape and it returns more slowly (post-flare wings) to a calm state due to the D-region relaxation time. In [29], the database contains parameters \(\beta\) and \(H^{\prime}\) for different values of \(I_{x}\), i.e., different classes of solar flares (during the period of ascending phase and maximum of the solar cycle 24 and 25). Solar flares are monitored and analysed by VLF technique and parameters \(\beta\) and \(H^{\prime}\) in database are obtained by method described in [20]. In [[https://github.com/sambolino/flarED](https://github.com/sambolino/flarED)]([https://github.com/sambolino/flarED](https://github.com/sambolino/flarED)) (accessed on 19 September 2021) [29] for the input values of solar X-ray flux, [PERSON]'s parameters \(\beta\) and \(H^{\prime}\) can be evaluated and altitude values of electron density for the low terrestrial ionosphere can be calculated. Figure 13 presents a scatter plot of a pair of D-region parameters, \(H^{\prime}\) and \(\beta\), for SID cases analysed in this study together with data obtained using methods from [29]. Calculated parameters \(H^{\prime}\) and \(\beta\) are in the range of 60-74 km and 0.30-0.60 km\({}^{-1}\), respectively. Black circles represent data from present work and red circles are related to the data obtained using the flared method (see Section 2.2.2). In the bottom right corner (for M- and X-classes of SF) data are more scattered. Figure 12: X-ray flux from GOES-15, and \(N_{e}\) (\(h\) = 74 km) as a function of UT during two M-class SFs on 12 June 2014. The dashed red line shows results acquired by approximative Equation (5). \(N_{e}\) obtained by the above-mentioned method is presented by circles.