Determine at what latitudes approximately the 23rd February the astronomical night does not occur

Determine at what latitudes approximately the 23rd February the astronomical night does not occur (the astronomical night begins after the sun plunges 18 degrees below the horizon).

Astronomical night will not come in those latitudes where the Sun, even at its lowest climax, does not drop below 18º below the horizon. When solving the problem, it is convenient to use a schematic representation of the celestial sphere.
Consider the northern hemisphere of the Earth. The height of the luminary with declination δ at latitude ϕ in the lower culmination is h = 90-ϕ-δ. Let’s define δ of the Sun on February 23. December 22, on the day of the winter solstice, the declination of the Sun is negative and is numerically equal in magnitude to the angle of inclination of the ecliptic to the celestial equator, i.e. –23, º5. On March 22, the vernal equinox, the declination is 0º. We will assume that the rate of declination change is constant (this is not the case, but the error will be small). Then the declination of the Sun on February 22 will be approximately equal to 23.5 * 60 / 90-23.5≈-8º (according to the astronomical calendar –10º). Those. the latitude at which the Sun plunges 18 degrees below the horizon will be ϕ = 90-h-δ = 90-18 – (- 8) = 80º. To the north, the depth of the sun’s immersion will be less than 18º.
Similarly, we will draw the moment of the lower climax of the Sun in the southern hemisphere. We get that in the southern hemisphere the astronomical night will come at latitudes north of 64º S. sh. The final answer is that the astronomical night of February 23 does not occur in the polar regions of the Earth (approximately at latitudes –90º <ϕ <- 64º and 80º <ϕ <90º).

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