[Physics] Lunar twilight and sixth magnitude stars

Question

Summary :

When the Moon is $x$ degrees below the horizon, it interferes
with stargazing the same as astronomical twilight would. What is $x$ (as a
function of the Moon's phase)?

We define civil, nautical, and astronomical twilight as when the sun
is $0-6$ , $6-12$ , and $12-18$ degrees below the horizon respectively. This
corresponds roughly to what most people call twilight, the ability to distinguish a horizon at sea, and the ability to see 6th magnitude stars at the zenith.

However, even when below the horizon, the Moon shines brightly enough to interfere with stargazing. What are the equivalent twilight angles for the Moon? I realize that even the full Moon overhead isn't bright enough for civil twilight, so my real interest is in astronomical twilight. Of course, this will vary greatly with the Moon's phase, and slightly with Moon's distance.

Answer 1

It was really difficult to find any mentioning of the "lunar twilight" in astronomical literature. But I managed to find one in an old book by G.V.Rosenberg called "General picture of twilight phenomena". The book is in russian, but there is not too much about the moon. So I'll just translate all the relevant stuff.

First of all there is this plot for the solar twilight (I've translated it):

Where E is the illuminance of a horizontal surface in luxes. And lg -- is a logarithm base 10.

And then there is a small chapter:

Lunar twilight
Lunar twilight is similar to solar twilight, but it is much more bleak. The illumination from the moon varies from $10^{-9}$ (new moon) to $2\cdot 10^{-8}$ (full moon) from the illumination from the sun at the same point in the sky. Therefore the illumination from the full moon high in the sky corresponds approximately to the middle of nautical twilight. And lunar twilight ends virtually when the moon gets under the horizon.