calculating rising time from culmination time

Question

I want to know how to calculate the rising time of a star, if culmination and setting times are available with me?

Say Culmination time is 03:00 on 28-Nov-2018 and Setting time is 05:30 on 28-Nov-2018.

Answer 1

For a star, the time from rising to culmination is the same length as the time from culmination to setting. From knowing the time of culmination and setting, you can calculate that difference and subtract it from the time of culmination to obtain the time of rising.

This method is generally accurate for the planets and Sun, but it is not always accurate for the Moon. Culmination refers to when an object is highest in the sky. For a star, this is when the object is on the meridian. For the Moon, there can be a significant difference between the time the Moon crosses the meridian (transits) and when it culminates. See Transit and Culmination (although the images on that site were not working today).

P.S. I have never seen tables that give the time of culmination for the Sun and Moon. Times are generally given for transits (when the object is on the meridian).

Answer 2

From the book "Fundamental Astronomy" by H. Karttunen et al. we have the following expression (to calculate the hour angle when the altitude is $0$):

$\cos(h) = -\tan(\delta)\tan(\phi)$ ($\phi$ is the latitude, $\delta$ the declination)

Since $\Theta = \alpha + h$ ($\Theta$ is the Mean Sidereal Time, $\alpha$ the right ascension), we can use this formula to calculate rising and setting times.

$\Theta$ matches GMT at the time of the Autumnal Equinox (roughly, and we're talking about Mean Sidereal Time), and every sidereal day is roughly 23h 56min so if you know the time of the observation, you can approximate the rising/setting time.

This of course does not take refraction, precession and other effects into account, which I assumed you do not need since you gave rather vague culmination times (stars rarely culminate around such "exact" times).