NavList:

You asked:
"My question is how can I calculate the time for specific Hc, position and day for the Sun? Let’s say at land there is a building and the altitude of the building is five degrees as seen from my position. When the sun will rise over the building?"

If you just want a specific case, I would recommend exactly the procedure Gary described: find a calculator (and I do recommend the USNO page) and just try a few times until you get the altitude that you want. For a practical case, that is almost certainly good enough. Note that there are endless tools for this sort of thing, and you could just as easily do this with a tool like Stellarium on a desktop computer or any number of apps on a smartphone or tablet.

For the general theoretical case, it's worth recognizing that the standardized, "normal" time of sunrise is asking exactly the same question: when does the Sun reach an arbitrary altitude relative to the true horizon? For standardized sunrise, that arbitrary altitude places the center of the Sun something like 50 minutes of arc below the true horizon (I would have to check the exact value, but it's 50' if I remember correctly), and this REGARDLESS of where you are on the Earth. This value puts the Sun's refracted upper limb on the horizon for an observer at sea level with zero height of eye. Even if you're in Denver, a mile high in the Rocky Mountains, where there's nothing remotely resembling a sea horizon and refraction is significantly different, this is how standardized sunrise time is calculated. Mathematically, the algorithm for the general calculation is simple enough: we draw the usual spherical triangle for altitude and hour angle. But instead of using watch time to determine the local hour angles and then calculate altitude, we set the altitude to the required arbitrary value, and then solve for hour angle. Note that this gives local hour angle and in the case of the Sun this corresponds to Local Apparent Time ("sundial time") so you have to add/subtract the Equation of Time to convert to mean time and then adjust for longitude and zone time just as you would for a standard sunrise calculation. The lesson here is that you can take any standard piece of software that calculates standard sunrise and replace the idealized 50' below the horizon altitude with any other special altitude that you want including, for example, the 5° number that you specified.

Needless to say, in real cases of nearby buildings, a small change in the obsrver's location could make a big difference in the actual time. Also, the angular altitude would vary depending on azimuth for a typical building (or a line of hills), so there are other geometric calculations that really ought to be included to do this right if you're trying to do some general situation.

-FER

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