orbit - How to calculate geocentric conjunction time and moon altitude at a given time

Umm al-Qura calendar of Saudi Arabia determine the new moon if these criteria meet at 29th day:

• The geocentric conjunction occurs before sunset.


• The moon sets after the sun.

I've got the formula to calculate when the sun (seen by the observer) sets, considering observer's latitude, longitude, height (altitude from sea level), and the date (Julian Day). I would like to ask the formula to calculate:

1. When (the time) geocentric conjunction happens.


2. How much (the angle) the altitude of the moon (seen by the observer) at a given time is.

Thank you