# Low Ionosphere under Influence of Strong Solar Radiation: Diagnostics and Modeling
## 3 Results
### Approximative Expressions  
Figure 10 shows variation of X-ray flux, as measured by GOES-15 satellite, and the corresponding electron density evaluated at reference height 74 km versus universal time UT during five successive flares on 10 June 2014. The red line presents results obtained using simple and accurate approximative Equation (3) based on a least-squares method. Circles presented as _Ne_ are obtained by the method mentioned above. For these explosive X-class solar flares, electron density grows by almost two orders of magnitude. Figure 10: Variation of X-ray flux, as measured by GOES-15 satellite, and the corresponding electron density evaluated at reference height 74 km versus universal time UT during five successive flares on 10 June 2014. The red line presents results obtained using simple approximative Equation (3). With circles are presented _Ne_ obtained by the mentioned above method. Figure 9: The height profile of electron density at peak time for five successive flares on 10 June 2014. To enable the better and more adequate use of data, we give electron density results obtained using simple approximative formula, which is logarithmic and represented by a second-degree polynomial with height-dependent coefficients \(\text{a}_{1}(h)\), \(\text{a}_{2}(h)\), \(\text{a}_{3}(h)\):  
\[\log Ne(h,Ix)=\text{a}_{1}(h)+\text{a}_{2}(h)\cdot\log Ix+\text{a}_{3}(h)\cdot \left(\log Ix\right)^{2} \tag{3}\]  
Here, \(Ix\) is solar X-ray flux (Wm\({}^{-2}\)), and \(h\) is height (km). Table 2 presents data of the selected fits. The whole table with smaller step in altitudes is available in the Supplementary material (see Table S2.) To obtain a formula that applies to all conditions and input parameters, we analyzed all investigated SF cases from this study and obtained coefficients. We have selected the most appropriate ones (Table 2 and Table S2) that cover all the examples and can honestly satisfy the accuracy. Moreover, in order to check the validity of the formula, we presented the values of the approximative formula together with the calculations from this work and also of other authors in Figure S4. One can see that it is in good agreement with all relevant data. Surely a more complicated formula could have been chosen, but there was no need for such accuracy because it is known that the values of the D-region parameters can vary literally by an order of magnitude. The idea was to make the formula easy to use as well as to enable rapid calculations and analysis. During this study, two hundred events of solar X-ray flares were examined. The results were obtained on the basis of VLF and satellite data and static analysis of numerous events. One of the main results is that we can obtain values of electron density at any height, against maximum intensity of X-ray flux by expression (3). Figure 11 shows observed amplitude and phase perturbations on GQD/22.10 kHz radio signals, due to solar X-ray flares, measured at the Belgrade station (during 2003-2017), as a function of X-ray flux. Values of electron density at reference height \(h\) = 74 km, against maximum intensity of X-ray flux obtained by Equation (3), are presented. The color of the point is proportional to the electron density values. This makes it easier to analyze the D-region during different solar disturbances. In modern times, one of the largest solar flare measured with instruments occurred on 28 October 2003 (X17.2-class, \(Ix\) = 1.72 \(\times\) 10\({}^{-3}\) Wm\({}^{-2}\)). Coronal mass ejection (CME) was associated with the 28 October 2003 X17.2 solar flare. In Figure S3 in Supplementary Material, VLF and satellite data are presented. The approximative Equation (3) allows evaluation of electron density even for this extremely large SF. Lower panels in Figure S3 present perturbations of amplitude and perturbation of phase on the same day. At the peak time, amplitude changes \(\Delta A\) = 7.2 dB and phase changes \(\Delta P\) = 76 deg (it is the farthest point on the right in Figure 11).