The length of summer extreme period

Let us assume that the summer extreme period starts when one of the astronomical phenomena necessary for prayer times calculation does not exist. For example, when the sun never goes below 18° below the horizon (other definition of extreme is described here). Consequently, the white and/or red twilight is always in the sky, such that the sky never dark throughout the night. It means no Isha. It also means that the end of Maghrib and the beginning of Subh is not clear. Nevertheless, the best estimate of it leads to the middle of the night. Still, Isha cannot be determined. Therefore, when this situation occurs, we may defined it as extreme period.

Length of days when Isha Fajr merge.

Fig.1. The length of summer extreme period as function of latitude.

There are two questions. First, which area in the world experience summer extreme period. Second, how long the extreme period for the respective area. The answers of both questions are given in Fig.1, which presents the length of summer extreme period as function of latitude. As long as you know the latitude of a particular area, then by utilizing Fig.1. you can determine how long the summer extreme period for the area.

The absica of the figure is latitude. Its ordinate is the length of summer extreme period (in days). The figure was developed by the following procedure:

The result was plotted in Fig.1 as blue dots, which form a sort of curve. Note that L=0 for 0°-48°. For 48.4°-60° the blue dots form a non linear curve and then for 60°-90° the curve is linear. The curve may be approximated by a function (or a set of functions) as presented in Fig.1 as well. They are single variable polynomial of degree three (for 48.4°-60°) and degree one (for 60°-90°). Both functions meet at (ψ=60°, L=121 days). Small error is expected, especially for latitude ψ close to 90°.

L=\begin{cases} \left | \left | 0.075\psi^{*3} + 0.747\psi^{*2} + 8.337\psi^* + 121 \right | \right | & \text{ if } 48.4^o \leq \psi^* \leq 60^o\\ \left | \left | 5.346\psi^* + 121 \right | \right | & \text{ if } 60^o \leq \psi^* \leq 90^o \end{cases}
\text{ where }\psi^* = \psi - 60 \text{ and } ||x|| \text{ means the nearest integer to } x

Example of several cities located at different latitudes and their corresponding length of summer extreme period is listed in the following table.

Table 1. The length of shadow at noon relative to object’s length for selected dates.
Cities Latitude Days Months
Makkah 21°25′N 0 0
Stuttgart 48°47′N 15 0.5
Paris 48°51′N 16 0.5
Vancouver 49°15′N 24 0.8
Luxembourg 49°37′N 30 1.0
Prague 50°05′N 38 1.3
Kiev 50°27′N 44 1.5
Brussels 50°51′N 49 1.6
Dresden 51°02′N 52 1.7
Calgary 51°03′N 52 1.7
Astana 51°10′N 53 1.8
London 51°30′N 58 1.9
Rotterdam 51°56′N 63 2.1
Berlin 52°31′N 69 2.3
Kristiansand 58°10′N 108 3.6
Oslo 59°57′N 121 4.0
Trondheim 63°26′N 140 4.7
Mo i Rana 66°18′N 155 5.2
Tromsø 69°41′N 173 5.8
Longyearbyen 78°13′N 218 7.3