## 1 Introduction  
Solar flares (SFs) are giant eruptions on the surface of the Sun [1; 2] that release huge amounts of electromagnetic energy over the whole electromagnetic spectrum [3; 4; 5; 6; 7]. Levels of photoionization processes in the ionosphere depend on plasma composition along with radiation spectral ranges at specific altitudes [8]. Information on certain solar spectral lines could be of importance in research of solar flares [9; 10]. The enhanced Extreme Ultraviolet (EUV) radiation is absorbed at higher terrestrial altitudes additionally ionizing E and F regions of the ionosphere [11]. In addition, Lyman-alpha and X-rays penetrate more deeply into the ionosphere, reaching the D-region and causing enhanced ionization and absorption of electromagnetic (EM) waves there [12; 13; 14]. Solar flares can be classified into different classes based on their peak emission in the X-ray 0.1-0.8 nm spectral range as B (\(\geq\)10\({}^{-7}\) Wm\({}^{-2}\)), C (\(\geq\)10\({}^{-6}\) Wm\({}^{-2}\)), M (\(\geq\)10\({}^{-5}\) Wm\({}^{-2}\)), and X (\(\geq\)10\({}^{-4}\) Wm\({}^{-2}\)) classes [15; 16; 17].  
The abrupt increase in X-radiation and EUV emission following solar flares causes additional ionization and increased absorption of EM waves in the sunlit part of the Earth's ionosphere. At the time of SFs and consequently during sudden ionospheric disturbances (SIDs), the gain of the atmosphere electron density at all heights is noticeable [18; 19]. As a consequence of radiation influence, SFs create SIDs and induce disturbance in the monitored amplitude and phase of Very Low-Frequency (VLF in narrow band 3-30 kHz) radio signals, primarily in the D layer, which is located between the Earth's lower atmosphere, which has dense air, and its strongly conducting ionosphere [20; 21]. These events of X-ray SFs have been monitored by Geostationary Operational Environmental Satellite (GOES) [15].  
In this paper we will focus on the VLF technique--on amplitude and phase signals of worldwide transmitters of signals monitored by BEL VLF system (Belgrade, Serbia) [22]. VLF signals from the emitters located all over the world are continuously recorded by the BEL system. Events of X-ray SFs monitored by GOES satellites are further identified using a radio station's system of receivers. For these events, VLF wave enhancements are measured and analysed for the daytime atmosphere. In continuation of our previous research, the aim of this contribution is to present our study of solar X-ray flare effects and to obtain the daytime atmospheric parameters and ionization rate induced by this extreme radiation, and to provide a simple equation for altitude-dependent electron density of D-region plasma which depends on X-ray spectral intensity and sluggishness of the medium. Finally, we discuss how the approximately obtained altitude-dependent electron density relies on the shape of X-ray flux. The python scripts for calculating ionosphere parameters can be found on GitHub: [[https://github.com/sambolino/flarED](https://github.com/sambolino/flarED)]([https://github.com/sambolino/flarED](https://github.com/sambolino/flarED)) (accessed on 17 October 2021).  
The text is organized as follows: Section 2 briefly presents methods for observing the D-region and introduces a methodology. Section 3 provides the details of numerical results, analyses, and introduces a simple expression of electron density of the D-region. In Section 4 the results are discussed together with further directions of research.