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Given the local sidereal time (LST) of a star when it rises, how can I find out the time when it will rise (according to my watch) on a specific date?

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You'll need to convert local sidereal time to local time on the specific date, using the longitude of your location. This can be done with the formula described here:

$$ UT = \frac{LST - \frac{L}{15} - 6.697374558 - 2400.051336*T - 0.000025862*T^2)}{1.0027379093} $$

where $UT$ is the time in Universal Coordinated Time, $LST$ is the Local Sidereal Time, $L$ is the longitude, and $T$ is:

$$ T = \frac{JD-2451545.0}{36525.0} $$ where JD is the Julian Date at noon UTC. Once I've calculated the time in UT, I then add or subtract 24 until the result is within 0 - 24 hours. Then I can convert from UT to local time.

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