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Looking at sunrise and sunset data around 2017 vernal equinox (because of GMT timezone I took London) https://www.timeanddate.com/sun/uk/london

Date  Sunrise      Sunset        Day Length  Solar Noon
16    06:12 (92°)  18:07 (269°)  11:54:55    12:09 (37.0°)
17    06:09 (91°)  18:08 (269°)  11:58:53    12:08 (37.4°)
18    06:07 (90°)  18:10 (270°)  12:02:52    12:08 (37.8°)
19    06:05 (90°)  18:12 (271°)  12:06:50    12:08 (38.2°)
20    06:03 (89°)  18:13 (271°)  12:10:48    12:07 (38.6°)
21    06:00 (88°)  18:15 (272°)  12:14:47    12:07 (39.0°)
22    05:58 (88°)  18:17 (273°)  12:18:45    12:07 (39.3°)

We see that 12-hours day happens around 17-18 of March, however the Vernal Equinox takes place on 20 March 2017, 10:29 GMT.

I understand that due to refraction in the Earth's atmosphere and due to the fact that sunrises and sunsets are defined by the top of the Sun's disk, sunrises are "earlier" and sunsets are "later", so the 12-hours day indeed should happen before the equinox.

But why the directions of sunrise and sunset are east--west on that same day? The above factors should not influence directions, should they?

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Due to the atmospheric refraction the Sun rises earlier and sets later as you correctly wrote. Refraction however is perpendicular to the horizon. So if the apparent sun rise is earlier, than the true (actual) position of the Sun will be more to the north east (smaller azimuth) than when the true sun rise occurs. As refraction is perpendicular the apparent sun rise will also be more to the north east and hence the azimuth will have a lower value (88-89° instead of 90°).

NB. This is of course only true from the northern hemisphere.

NB2. The Sun is not to scale in the figure below. enter image description here

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