How to calculate moon overhead and underfoot time? For hunting purposes, some hunters follow the moon phases to hunt accordingly. To give some context, here's some links:


Lunar cycle's effects on hunting: https://www.acurite.com/blog/weather-stations-with-moon-phase-for-hunting-fishing
Solunar theory: https://en.wikipedia.org/wiki/Solunar_theory


I would like to know how to calculate the time when the moon is directly overhead in the sky (moon overhead) and when the moon is directly underfoot in the opposite side of the earth's sky (moon underfoot). Those are the information that I have:


Moon rise time
Moon set time
Moon phase (first, full, last, new)


Those information come from AccuWeather API: https://apidev.accuweather.com/developers/astronomyAPIGuide#MoonPhasesRange. There are some other APIs that give the moon phase in degree format or between 0 and 1 but we chose AccuWeather. 

I tried to calculate this with only the moon rise and set time. I thought that simply calculating halfway between rise and set time will suffice. However, I didn't always get the same results as the moon overhead and underfoot in this website: http://www.solunarforecast.com/solunarcalendar.aspx

Is there a formula to get the time of moon overhead and underfoot with the information that I have?

UPDATE

I used the definition


  when the moon is directly overhead in the sky


because that is what I found from my research. However, it's seems the correct definition is when the moon is at highest point in the sky.

Also, since it's for hunting purposes, I don't need the time to be accurate to the minute (but that could be nice).
At the accuracy you need, the calculation can be quite simple:  It takes 12 hours and 26 minutes for the moon to go from "overhead" to "underfoot", and 12 hours 26 minutes to go from "underfoot to overhead".

So if you notice that the moon is overhead at 20:00 one evening, it will be overhead at 20:52 the next evening,  then 21:44, 22:36 and so on.  (Or, even easier,  approximate to "50 min" and use "add an hour, minus 10 min"

The actual motion of the moon is rather complicated, as the moon's orbit is elliptical and perturbed by the sun. If you need more accurate values you need to use a complex method of calculation, which is what the websites are using.

If you only have rise and setting time, the culmination will be roughly in the middle

The Solunar calculator page contains some obvious errors (such as on 5th May)