What are the sources of error in the formula I derived to estimate sunrise and sunset times, and how can I improve its accuracy? [closed]

Question

I derived the following formula to estimate sunrise and sunset times:

$$ t=\pm\cos^{-1}\frac{\sin\theta-\sin L_{loc}\sin L_{sun}}{\cos L_{loc}\cos L_{sun}} $$

• $L_{loc}$ is the local latitude
• $L_{sun}$ is the solar latitude
• $\theta$ is the desired solar altitude angle (for sunset and sunrise, I used $\theta=0$)
• $t$ is the time, expressed as an angle of a 24-hour clock, and measured from solar noon

I used the solar latitude from here, and the times and local latitude for New York from here (it was October 20, 2020; link to correct month).

I tried this with my location and was about 4 minutes off with sunrise and sunset, and I tried it for New York and was also about 4 minutes off (I got 7:18:15am for sunrise and 6:02:55pm for sunset).

I'm wondering what are the main sources of error in my formula? My goal is for it to be accurate at least within a minute, and hopefully even down to a few seconds. What factors do I need to take into account in order to reach this level of accuracy?

Thanks!

Answer 1

Your error is assuming that sunrise and sunset occur at $\theta = 0$, this is incorrect for two reasons. The first being the sun's disk has a diameter of $0.5^{\circ}$, and the second is atmospheric refraction causes distortion which change the sun's apparent location on the horizon. Altogether, at sea level this leads to an angle of correction of about $-0.83^{\circ}$, use this as your theta.