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From Gaisser's Cosmic Rays and Particle Physics (ed. 2016).

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I know $\theta$ and I have the vertical depths $X_{v}$. I want to solve for $X$ via a simple trigonometric relation. However, theta evolves as $\theta^*$ as you go up the atmosphere.

As Gaisser points out, the curvature of the Earth is the cause for the evolution of $\theta$. How would I go about correcting for this as a function of vertical depth?

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