add option to AtmosphereProfile.line_of_sight_integral to include curvature of Earth

[maxnoe]

Please describe the use case that requires this feature.

This is important for high-zenith-angle observations

Describe the solution you'd like

Include a formula / numeric method for computing the line of sight integral assuming spherical earth.

Describe alternatives you've considered

There is not really an alternative, for high zenith angles, the current solution will produce nonsensical results.

Additional context

This problem will have a solution somewhere, didn't check yet

[kosack]

yes, I only implemented the simplest case which is fine for lowish zenith angles. Certainly it's not sufficient at >60° or so. We should probably at least implement the Corsika model, which I think assumes a Earth is a perfect sphere, and the atmosphere is assumed to be spherically symmetric around the Earth's center.

EDIT: this is wrong I think, see next comment

[kosack]

Actually, no I'm wrong, looking in the corsika paper (Heck et al 1998), they say (paraphrasing): only a flat atmosphere is used... When the zenith angle is ≥75°, the difference between flat and spherical is no longer negligible and an approximation is made. They approximate nearly horizontal showers by setting the zenith angle to 0 and using a flat atmosphere profile that is basically the value at the observation level.

This is also backed up by Konrad's presentation here http://hess.in2p3.fr/aaa/02__shower.pdf

[kosack]

Since CTA's requirements only are to observe to 60°, it may not be necessary to support spherical atmosphere integration. However, for cases like MAGIC observing near or below the horizon (due to being on a mountain), it would be necessary, but perhaps that is too special a case to require implementing something fancier.

from Berlöhr's presentation at Atelier ASTROPARTICULES et ATMOSPHÈRE ASTROPARTICLES and ATMOSPHERE (AA) workshop Paris, 26-28 May 2003

[kosack]

So perhaps we can close this