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I wonder how to calculate the duration of the Sun crossing the line of the horizon on equinox on Pole. For equator I used this formula:

$t = \frac{2 \times p}{\cos(\phi) \times u}$, where $u$ - is velocity of the Sun in the sky ($15'$ per minute), $p$ is Sun's visual diameter ($30'$), $\phi$ - latitude of observer

On equator everything is easy:

$\cos(0) = 1$

But on Pole:

$\cos(90) = 0$, but I can't divide by zero.

Is there any other formula?

I used this drawing

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    $\begingroup$ Sunrise/set at the poles depends on the Sun's changing declination $\endgroup$
    – Barry Carter
    Commented Jan 23, 2022 at 14:00
  • $\begingroup$ The duplicate gives an an answer (32hr 50 min) and explains why. $\endgroup$
    – James K
    Commented Jan 23, 2022 at 17:28
  • $\begingroup$ I know that declination of the Sun is zero as it is equinox $\endgroup$
    – ALiCe P.
    Commented Jan 25, 2022 at 11:44

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