That's called "twilight". In the section "Atmospheric refraction and the solar limb" there's a modified sunrise equation with a variable "a" and text suggesting a correction of 50 minutes of arc there.
If you replace 50 minutes with:
- 6 degrees, you get the times of "civil twilight" (roughly speaking, when you don't need light outside). At 60 degrees north at midsummer the minimum altitude of the sun will be about -6.5 degrees so almost all of the nominal night is civil twilight.
- 12 degrees is "nautical twilight" (horizon clearly visible)
- 18 degrees is "astronomical twilight" (sky is dark enough for all astronomical observations).
(It's possible that those are defined as 6 degrees + 50 minutes, etc.)
If you replace 50 minutes with: - 6 degrees, you get the times of "civil twilight" (roughly speaking, when you don't need light outside). At 60 degrees north at midsummer the minimum altitude of the sun will be about -6.5 degrees so almost all of the nominal night is civil twilight. - 12 degrees is "nautical twilight" (horizon clearly visible) - 18 degrees is "astronomical twilight" (sky is dark enough for all astronomical observations).
(It's possible that those are defined as 6 degrees + 50 minutes, etc.)