Above the atmosphere, ionizing radiation from galactic rays and the sun become more of a threat. However, the worst of the radiation near Earth is trapped in a set of large belts surrounding the planet at very high altitudes.
These belts, known as the Van Allen Radiation Belts, contain huge amounts of trapped ionizing radiation that can damage spacecraft passing through them. Satellites heading to a geostationary orbit are often forced to fly through these rings a single time in order to reach their target.
First, let's talk about why these belts exist in the first place.
Earth's Magnetic Field and Radiation
As anyone who has used a compass knows, Earth itself produces a global magnetic field. It's not very strong (typically tens of microteslas), but it's constantly there, and covers all of the planet and the space near it. Earth's magnetic field extends far out into space, surrounding the planet and interacting with the charged solar wind. Overall, the Earth's magnetic field is somewhat close to a dipole - like a permanent bar magnet.
Artist rendition of Earth's magnetic field.
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Charged particles like electrons and protons (hydrogen atoms stripped of their electron) can interact with this magnetic field. When an object with an electric charge moves through a magnetic field, a magnetic force pushes on the object. This force pushes perpendicular to both the object's velocity and the magnetic field. The result is that the trajectories of charged particles curve in a magnetic field.
Most ionizing radiation is charged. Gamma/X-ray photons and neutrons aren't charged, but most everything else is: Electron/beta, Protons, Helium/Alpha, Positrons, Muons, and heavy ions are all forms of charged ionizing radiation. This radiation can reach Earth from various sources. The biggest ones are solar wind from the sun (waves of mostly keV-scale electrons and ions, all charged) and interplanetary/galactic cosmic rays (extremely high energy protons, electrons, and ions originating outside the solar system).
When these charged particles approach Earth, they encounter the large planetary magnetic field. This field forces the particles to curve, bending their trajectories around the planet. Some are deflected away from Earth, while some are deflected directly into it and crash into the atmosphere. Some collide with the atmosphere in concentrated jets near the poles, producing the bright polar aurora.
Southern aurora from space
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But some particles don't take either path. Given proper conditions, a charged particle can move in a circular path in a magnetic field, with a radius given by the mass and charge of the particle. Calculating this radius is quite simple (look up Larmor radius or cyclotron frequency if you're interested), as it depends only on a few values. Other than slow energy loss by cyclotron electromagnetic radiation (produced when charged accelerate along a curved path), these particles don't lose any kinetic energy, and will remain in trapped regions above the Earth for long periods of time. Such a particle might pass over the Earth many, many times the speed of satellites, because it is held in place by the electromagnetic force rather than gravitational force.
It is these trapped particles that form the Van Allen Belts.
Earth's Radiation Belts and Contents
This interaction between charged particles and Earth's magnetic field results in the formation of semi-permanent rings of strong ionizing radiation. Two main belts exist, lying primarily between low-Earth orbit (<1000 kilometers) and Geostationary orbit (~35000 kilometers above the ground). The belts are filled with high energy charged particles, mostly electrons and protons.
Scale rendition of the Van Allen Belts
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These particles are moving very quickly. While the compositions of the two belts vary, they can contain electrons ranging from keV to MeV energies (higher energy electrons tend to populate the outer of the two rings). A 5 MeV electron moves with a velocity over 99% of the speed of light! Protons and rare heavier ions within the belts move much slower but can attain much higher kinetic energies because of how much heavier they are than electrons. Protons in the belts can reach kinetic energies of 100 MeV, around ten times the energy of the fastest electrons in the outer belt. These kinetic energies are so high that they arrive at the same energy scales as the masses of the particles themselves: A 1 MeV electron has a kinetic energy just under twice its mass (as you probably know, rest mass and rest mass energy are related by the famous equation E = m*c^2).
The pink region represents the radiation belts
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Antimatter has even been detected in the belts. Antimatter particles are most simply the opposite of the everyday particles that make up the world around us. Instead of a negatively-charged electron, we have a positively charged positron. Instead of a positively charged heavy proton, we have a negatively charged antiproton of the same mass. When antimatter reacts with normal matter, all of the energy stored in the rest mass of both particles involved in violently converted into other particles. Antimatter reactions produce magnitudes more energy than even the most violent nuclear decay, fission, or fusion reactions. For example, when a positron reacts with an electron, typically two each energy gamma rays are produced. While antiprotons and positrons are technically stable, they won't last long anywhere in the universe unless they are moving very fast, as they will quickly annihilate themselves with nearby regular protons and electrons.
The Antiproton. If you held a chunk of antimatter the mass of a penny, made up of antiprotons and antineutrons, it would immediately destroy itself in the surrounding air, producing an explosion larger than the two atomic bombs dropped during WW2 combined.
Very high energy antiprotons (with energies up to 750 MeV!!) were detected in Earth's radiation belts by the PAMELA detector onboard the Resurs-DK1 spacecraft. Positrons are the most common type of antimatter as they are relatively low mass and are produced in many naturally occurring nuclear decay reactions, but antiprotons are another story entirely. Their high mass (far too high energy to be produced in nuclear decay) and lack of occurrence in nature makes them extremely rare, only produced in particle accelerations and cosmic ray collisions. It is these cosmic ray collisions that deposited antiprotons into the magnetic radiation belts of planet Earth. Collisions between super-high-energy cosmic rays and gas atoms in the atmosphere can produce antineutrons (neutron antiparticles that still have no charge), which can then fly off away from the Earth. But antineutrons, like neutrons, are unstable, and can decay via positron emission into an antiproton. Some of these antineutrons decay near the Earth, turning into an antiproton that is then subject to the whims of the magnetic field. If the particle is low enough energy, the antiproton can become trapped in the belt, creating a tiny but somewhat constant antimatter storage belt high above our heads as these particles build up.
Resurs-DK1, the satellite that carried the PAMELA detector that discovered the antiproton components of the belts
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Many of the regular high-energy protons (regular matter) were also produced in a similar way: From decaying neutrons produced in cosmic ray collisions. The lower energy protons are likely sourced from the solar wind, waves of charged particles radiating off of the surface of the sun.
Threats of the Belts
The Van Allen Belts provide an interesting problem for spacecraft. Pass through it once and you can expect electronics upsets. Pass through it many times over and over again and you'll need to protect your spacecraft, or it might fall victim to an early, radiation-induced death. Send people through this belt and you either need to do it quickly or make a well-shielded crew chamber. Ionizing radiation doesn't just affect the health of people and animals - it also damages electronics. As total ionizing dose builds up in a piece of electronics, so does the chance that the device fails. Ionizing radiation like the kind in the belts destroys chemical bonds with ease, ionizes atoms, triggers transistors, damages PN-junctions, and even degrades materials. Staying in this region too long is death sentence for any device, living or not.
Van Allen belt densities
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If you want to send something to the moon, Mars, Venus, or beyond, you probably have to take a trajectory going straight through the radiation belts. The radiation dosage acquired can be reduced by passing through the belt quickly, and only doing it once.
Other planets have Van Allen Belt -like radiation belts as well. Jupiter in particular has an extremely strong radiation belt, which happens to lie right over its innermost large moon Io (you can read more about this fascinating moon here in a great post from 
Interestingly, there have been proposals to destroy the Van Allen belts. These range from using low frequency radio electromagnetic waves to deorbit high-energy particles to dipping highly charged tethers into the belts, deflecting radiation particles and rapidly draining the belts of their stored radiation. None of these have been implemented obviously, but the technology to actually totally eliminate (at least temporarily) these radiation belts really already exists. Whether or not this happens remains to be seen - it doesn't seem like there is any momentum to do so as of now. If the belts could be drained and destroyed, it would open up tons of new orbits for spacecraft which currently must avoid much of the area between geostationary and low-Earth orbits, and would reduce risk to astronauts on lunar or interplanetary missions.
I hope you were able to learn something new about these deadly belts of radiation above our heads. Thankfully, there is nothing to worry about down here as they don't reach us (except in one spot above the Atlantic where they collide with the upper atmosphere). As for spacecraft and astronauts passing through them, precautions can be taken to minimize risk if the time in the belts is small.
Please let me know if you have any questions, comments, or corrections. Also, I just noticed that I passed 200 followers - thanks to everyone who has supported me in the short few months I've been on Steemit.
Thanks for reading!
Sources Utilized:
Antimatter Belt found circling Earth
IEEE: Hacking the Van Allen Belts
Magnetosphere of Jupiter Wikipedia Entry
Van Allen Belt Wikipedia Entry
HiVOLT Van Allen Belt Killer Spacecraft
ESA SPENVIS Tool
EDIT: I made a mistake in my original post. The particles do not travel in full circles about Earth, rather, they oscillate along the path of the belts. This has been corrected in the post.