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Cosmic Rays Are Cosmic Particles
First things first: cosmic rays are particles, consisting primarily of protons, electrons and alpha particles (helium nuclei), and not rays in the comic book science sense of the word. The particles enter the atmosphere individually, and not in any kind of coherent group.
From there, we can divide cosmic rays into two groups: primary and secondary. Primary cosmic rays are derived from some cosmic source, while secondary cosmic rays are created when primary cosmic rays strikes against interstellar matter, breaking it up into more cosmic rays, a process known as cosmic ray spallation.
Another subgroup of cosmic rays are the ultra-high-energy cosmic rays, which have energies either at the theoretic limit—or even higher. This particular puzzle is currently being teased out by physicists.
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Sources Of Cosmic Rays
Cosmic rays come from a wide variety of sources, from the innermost workings of the Sun to active galactic cores, hypernovae to gamma ray bursts. Anyplace where there are vigorous nuclear reactions going on is a potential source of cosmic rays. However, there's still some puzzling over the exacts of cosmic ray formation, so don't expect any concrete answers quite yet.
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Protection From Cosmic Rays
What affects the amount of cosmic rays that enter our atmosphere? Well, the Sun's solar wind and the Earth's magnetic field help shield our little planet by deflecting some particles and decelerating others. Indeed, scientists have been able to note differences in the number of cosmic rays that different parts of the globe receive based on these sources of protection: the poles get more, as they are less well protected by the magnetic field, while the equator gets less. With the current lull in solar activity, the Sun's magnetic field is not as strong as it is when our star is more active and the solar wind is diminished, consequently we are measuring a marked increase in the intensity of cosmic rays entering our solar system from outer space.
Cosmic rays react strongly with matter, so physical shields will stop many of them. For instance, there is a literal shadow of fewer cosmic rays between the Earth and the Moon! Our atmosphere provides last-ditch protection against cosmic rays, providing increasingly more matter for cosmic rays to interact with and lose something of their energy, through Compton Scattering. As a result, most of the cosmic rays that actually reach the surface are secondary cosmic rays.
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Impact! Cosmic Rays & Planet Earth
Cosmic rays influence Earthlings in a number of innocuous ways. For instance, cosmic rays are a cause of memory corruption and CPU malfunction as the highly energetic particle interacts with the delicate electronics. They've even been cited as a possible reason for an unexplained malfunction that almost led to a plane crash! They've also been suggested as a possible trigger for lightning.
While cosmic rays are not an obvious health risk for us terrestrials, there is some worry that they may cause serious damage to any interplanetary astronomers who will be spending long periods of time unprotected in space. Little is known about the effect of cosmic rays on the body, however, due to the rather low sample size of astronauts who have spent considerable amount of time in space. There is ongoing research to develop shielding systems for spacecraft, NASA's predictably named Space Radiation Shielding Program, as well as into possible drugs to repair radiation damage.
There's been some argument that the amount of cosmic rays that enter the atmosphere is correlated with climate change. This is somewhat difficult to prove, however, and is the subject of considerable debate. This goes along with other theories that attempt to account for climate change with solar variation.
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Cosmic Ray Observation
Due to their mysterious nature, there are many research initiatives and specialized observatories dedicated to cosmic rays, including a wide array of ground, satellite and balloon experiments such as the Piere Auger Cosmic Ray Observatory, CHICOS, PAMELA, TRACER, MARIACHI, and more. Plenty of research is underway to learn more!