The forecasting uses astronomical ephemerides that calculate the relative positions of the sun and the earth, and the orientation of the earth. The process is complex, but it’s based on predicting motion based on the inverse square law of gravitational attraction. Some simpler maths can then work out the position of the sun in the sky for any location on earth.

For forecasting sunrise and sunset, and also the start and end of twilight, the important measure is the sun’s “zenith distance.” This is just the angle between the centre of the sun and the “zenith”, which is the point vertically overhead in the sky.

You’d think that a zenith distance of 90° would be sunset but actually the definition used is 90.83°. The extra distance is for two reasons: the sun appears as a disc with a radius of about 0.27° so you need the centre of the sun to be that far below the horizon for the top edge of the disc to be level with the horizon; and the atmosphere bends light a little, allowing us to see objects about 0.56° below the horizon.

The calculations are, in principle, trying to work out the time the top of the sun is just on the horizon as seen at sea level with a sea horizon. Even just climbing up a few metres will allow you to see farther over the sea horizon and alter the times. And, of course, a land horizon with hills will have a different observed time of sunset. The weather can have an impact too because the amount the atmosphere bends light depends on air pressure. There’s even a tiny impact due to the time of year, because in January the sun is closest and so its disc is largest and in July its disc is smallest; the adopted forecast method ignores this effect and just uses the average size.

The times of the beginning and end of twilight is very similar; the zenith angle used is simply increased. The definitions are arbitrary so there are no corrections applied. “Civil” twilight uses 102°, “nautical” twilight 105°, and “astronomical” twilight 108°.

Sunrise is the time when the center of the sun crosses the horizon, and sunset is the time when the center of the sun drops below the horizon.

The sun rising and setting takes a lot longer but the time quoted is when the center line crosses the horizon at your location(assuming you’re at sea level with no obstructions)

The time before sunrise and the time after sunset when its still somewhat light are called twilight. There are three stages of twilight(Civil, Nautical, Astronomical) which are all defined by the angle made with respect to the horizon and the center line of the sun.

Just “Its kinda bright now” isn’t a good enough definition of sunrise when you want to list it for everywhere in the world

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