@Christoph Graf

The best introduction to the subject of "measuring the shadows" is at the website dedicated entirely to the problems of atmospheric optics:

http://www.atoptics.co.uk

(Please go to the section: Rays/shadows and on the left hand side you will find:

Mountain shadows, formation, more shadows)

The images presented in these sections are very helpful in explaining why we do not have a closed analytical formula to answer the original question.

A good approximation could be derived, however, from the eikonal equation

https://en.m.wikipedia.org/wiki/Eikonal_equation

The main difficulty is the definition of the average atmospheric conditions for this problem. The best strategy is to solve the eikonal equation for several typical stratifications and zenith angles. The resulting formulae could be used later on in order to estimate the shadow of the mountains at different times of the day.

In a more general perspective I would like also to recommend a book on Kepler's optic of invisibility, "Measuring Shadows"

http://www.psupress.org/books/titles/978-0-271-07098-8.html

The text is very relevant to your original question and it shows also how crucial was the concept of the "shadow" for formulation of the foundations of physics.

Sincerely, Janusz

thank you so much!