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For a given date, longitude, and latitude:

Popular apps and websites give the time when the sun's center or edge is on the horizon (i.e., the time of sunrise and sunset) or its transit (i.e., noontime), but I want to ask how to calculate the time needed for the solar disk to pass the horizon's line and the transit's line (preferably in c# or java)?

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The time it takes the sun to cross the horizon is given at:

Re the time it takes to cross the transit line, this doesn't really answer your question, but, to a good approximation, the time varies between 128 seconds at the equinoxes and 140 seconds at the solstices, regardless of latitude or longitude.

More specifically, the formula is $\frac{128}{\cos (\text{dec})}$ seconds, where $\text{dec}$ is the Sun's declination. You can calculate the Sun's declination itself using the formulas at https://en.wikipedia.org/wiki/Position_of_the_Sun

The calculation here is relatively simple:

  • The Sun travels $360 \cos (\text{dec})$ degrees in a 24-hour day, where $\text{dec}$ is the Sun's declination

  • When the Sun is transiting, the motion is perpendicular to the transit line (the Sun's motion is entirely in azimuth, not in altitude)

  • Therefore, all of the Sun's angular motion translates to motion across the transit line; in contrast, the Sun rises and sets a (non-perpendicular) angle (except at tropical latitudes on the two days where the Sun passes directly overhead), so sunsets and sunrises take longer than $\frac{128}{\cos (\text{dec})}$ seconds

  • Since the Sun has an angular diameter of 32 minutes or $\frac{8}{15}$ degrees, it takes $\frac{\frac{8}{15}}{360 \cos (\text{dec})}$ of a day for the Sun to cross the transit line

  • Since a day is 86400 seconds, this works out to $\frac{128}{\cos (\text{dec})}$ seconds.

Caveats and nitpicks:

  • I assume there are 86400 seconds between successive noons. This is incorrect for two reasons:

    • The time between noons is not exactly one day. The cumulative difference forms the Equation of Time but the day to day difference is quite small

    • The Earth's day is slightly longer than 86400 seconds, which is why we need leap seconds

  • The Sun's angular diameter actually varies based on Earth's distance from the Sun, but 32 minutes is a good approximation

  • I assume the Sun's altitude doesn't change while it's transiting. While this is a good approximation, the altitude does change slightly

  • I assume the Sun's declination doesn't change while it's transiting. The change in declination is very small, so this is a reasonable assumption

  • There are probably other assumptions I made implicitly that I am not noting here.

  • Refraction is not an issue, since the motion we are discussing is azimuthal and not in altitude.

  • I used the "one over cosine" form above to make things easier for non-mathematicians. The more compact form would use "secant".

  • The calculations I did for this problem are disorganized, but available at: https://github.com/barrycarter/bcapps/blob/master/STACK/bc-solar-transit.m

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