First and foremost, what forms of radiation are encountered in the space environment? There are two sources of ionizing radiation of concern to us here: high energy electromagnetic radiation and fast moving particles, called cosmic rays. Most commonly these particles are ionized hydrogen nuclei, protons.
In the solar environment the primary source of cosmic rays is from solar particle events. These particles are primarily lower energy protons, on the order of 3 keV - 10 MeV. More energetic galactic cosmic rays are slowed or diverted by the solar wind and the sun's own magnetic field.
Close to the sun our chief concern will be low energy but more common solar cosmic rays. However between star systems it is higher energy cosmic rays that will present the largest danger to a vehicle.
Cosmic ray particles travel with energy anywhere from 10 - 10^14 MeV. Fortunately their average number density is inversely proportional to their energy. Particles with energies higher than those produced by the sun up to about 10^9 MeV are thought to be galactic in origin, likely from supernova remnants, and are bound to the galaxy. Those with higher energies have enough energy to escape galactic pull and are thought to be extragalactic in origin, produced in high energy active galaxies. Because these later are extremely rare we'll focus more on protection from galactic cosmic rays. These are most common at around 10^15 eV, we'll use a range from 1 GeV to 10 EeV (10^9 - 10^19).
There are a number of models of the cosmic ray spectrum in use now. Differential spectrum gives the number of particles of a given energy passing through a differential area from a differential solid angle per time. Here a modified form of the Badhwar-O'Neil model is used for spectrum in terms of kinetic energy per nucleon:
Where the beta term is the particle velocity in terms of the speed of light, E0 is the particle rest mass per nucleon (938 MeV/n), and gamma and delta are constants for the atom under consideration. Spectrum is very low, resulting in an extremely low average exposure. As any lotto winner or shark attack victim can tell you, however, averages mean nothing to individuals. That low average exposure is the result of a very few particle collisions, each of which imparts a huge amount of energy. Worse the effect that a single very high energy particle has on living tissue is poorly understood, as most particle radiation damage information is for a larger number of particles striking with significantly lower energy.
We'll integrate intensity in a range of delta-E about a set of energies E (integrate from E - delta-E to E + delta-E), to find the number of particles passing through a differential surface area element per time. Since cosmic rays can only enter the vehicle, not exit, we'll assume the solid angle is limited to the exterior of the vehicle and convert differential solid angle to differential surface area/radius.
Consider for the moment a particle collector on the exterior of a large starship. We'll assume for the moment that the starship is large enough that it blocks any cosmic rays coming through it and that the starship is large enough compared to the collector that the hull appears flat. So the only rays that intersect the collector must come from a hemisphere over it which has an equator at the starship hull. Integrating over the solid angle in this case gives us simply 2-pi. Now let's assume that the collector in question is, in fact, a person. Or rather, a rectangular prism into which a human being would fit. Such a prism would be about 2x0.6x0.6m (the approximate dimensions of a casket). We'll further approximate the human body as water. So in fact our human stand-in / particle collector more closely resembles a Houdini water torture tank. We'll use one face of that tank as our collector surface for the moment:
We know from the above the number of particles in a given energy range passing through our "crew member" per time. Now standard guidelines state that the increase in mortality risk from radiation exposure shall not exceed 3% over the lifetime of a worker. If we assume that every interaction with a high energy heavy particle is fatal this means that each crew member must have less than a 3% chance of such an interaction during the course of his life. This means that any particle which has a less than 3% chance of interacting with our particle collector (which you recall is mounted on the exterior wall of the starship) is one that we will ignore. Anything more common than that we will need to shield from.
In the first chapter of the nuclear spacecraft textbook, which I posted previously, I outline the concept of nuclear interaction cross-section. That analysis assumes a neutral interacting particle, specifically a neutron. Here the interacting particle is a more complicated positively charged ion, which loses energy as it passes through a material due to interactions with electrons in the material according to the Bethe-Bloch equation:
This equation can give us two very useful pieces of information. Fist it tells us how much energy an incoming particle imparts to our particle collector. Second it tells us how much material we would need to reduce the incoming particle energy to an acceptable level. Now we combine the two recalling that beta is related to total energy by:
Together these two tell us how much of our chosen material we would need to stop a high energy particle and the number of such particles we are likely to encounter. If we imagine that the collector is now situated on the interior wall we can find the energies of all the particles that pass through that thickness and the amount of energy they impart into it. Divide that by the density and we have a approximate guess as to the radiation dose our crewman will suffer standing just inside that wall his entire life. In addition to preventing any very high energy particles from interacting with him, the shield must also keep this value below certain thresholds.
A brief look at our model verifies the assumptions we've already made: only hydrogen nuclei toward the lower GCR range (10 - 10,000 MeV) have a more than 3% likelihood of impacting our example crew member in his lifetime. And any such interaction will impart more energy than a human being could tolerate. We'll need some way of slowing these particles to a tolerable velocity, which is what the next post will deal with.
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