If cosmic rays reach the Earth more or less equally from all directions, why do we detect many more coming from the vertical?

As mentioned in a previous question, it has been shown that cosmic rays reach the Earth in a quite isotropic way, i.e. more or less equally from all directions. The Earth's magnetic field partially breaks this symmetry, but this does not explain why the vast majority of the particles arrive "from above" and almost none near the horizon, as can be seen in the frequency histogram of the angles of arrival on the main page of this website.

The figure shows (in yellow) the trajectory of two secondary cosmic ray particles moving towards the surface at a speed very close to that of light in a vacuum, about 300 000 km/s. One is vertical and the other horizontal: the second one has to travel a length about 30 times longer than the first one to reach the surface.

_{Fig. 1 Particles of cosmic rays pass through the atmosphere}

_{Two secondary cosmic ray particles pass through the atmosphere to our detector (the drawing is not to scale because the radius of the Earth is more than 6000 km and the secondaries are created at an average height of about 15 km). Image: Wearbeard}

In this much longer path, the "horizontal" particle is more likely to be "absorbed" by the Earth's atmosphere - that is, to disappear when it interacts with the atomic nuclei in the atmosphere. In addition, there is an even more important effect: most secondary cosmic rays are muons, unstable particles that spontaneously decay into other particles with a mean-life of about 2 millionths of a second. Without going into detail, this means that, if we start from a very large sample of muons, about half of them will have disintegrated in that interval, half of the remaining ones in the next interval and so on. Therefore, with a longer path, many more "horizontal" particles will have disintegrated.

However, we have not yet explained everything: with such a short mean-life, why don't virtually all secondary cosmic ray muons disintegrate before they reach the Earth's surface? After all, a particle that moves at almost the maximum possible speed, that of light in a vacuum, and lives on average 2x10^–6 s , will be able to travel at most an average of L ~ 300 000 km/s × 2x10^–6 s ≈ 600 m.

According to the above and taking into account that every two microseconds (or 600 m) on average about half of the muons in a sample disintegrate, after 15 km of travel, which is about 25 times more, almost no muons should reach the surface, except that they do, and are quite abundant, but that's another story…