On June 22, the Sun rose to a maximum height of 60 ° above the horizon. How high will it rise at this point on December 21?

As you know, the height of the star at the upper culmination h is expressed through its declination δ and the latitude of the observation site ϕ by the ratio
h = 90 ° – ϕ + δ,
where δ> 0 north of the celestial equator and δ <0 south. However, here it is necessary to make a reservation that this formula is valid for the luminaries culminating south of the zenith, and if the culmination occurs in the north, then this formula gives the value of the height subtracted from 180 °.
The statement of the problem does not say in which side of the horizon the Sun culminated – in the north or in the south.
If the culmination occurred in the south, then h = 60 °, and it happened in the northern hemisphere, at latitude
ϕ = 90 ° + ε – h = 53.5 °.
Here ε = 23.5 ° is the angle of inclination of the equator to the ecliptic, equal to the declination of the Sun on the summer solstice on June 22. In this case, on December 21, when the declination of the Sun is equal to –ε, the Sun will rise to its maximum height
h = 90 ° – ϕ – ε = 13 °.
If the culmination took place in the north, then h = 120 ° must be substituted in the formula, the latitude is
ϕ = 90 ° + ε – h = –6.5 °,
which means that this point is in the southern hemisphere of the Earth. On December 21, the maximum height of the Sun will be
h = 90 ° – ϕ – ε = 73 °,
with the Sun culminating in the south.

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