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2.7.7 Cosmic rays
There is in nearby interstellar space a flux of particles—mostly protons and atomic nuclei—travelling at almost the speed of light, having kinetic energies from below 108 eV to above 1020 eV. The flux reaching the solar system is virtually isotropic (to within 0.1% below 1014 eV) and unchanging, but the flux observed at the Earth varies somewhat because: (a) at energies below a few GeV, particles are affected by interplanetary magnetic fields, which cause intensity variations with an irregular 11-year period; and (b) the geomagnetic field deflects low-energy particles away from low-latitude regions. The particles incident at the top of the atmosphere are mainly protons and bare atomic nuclei: encounters with nuclei in the air generate secondary particles, and the less energetic particles are attenuated on traversing the atmosphere, so that near sea level the dominant particles are the penetrating secondary muons and the rapidly generated secondary electrons, positrons and photons.
Occasional bursts of particles originating in solar flares can reach a few GeV (usually having little effect at most sea level locations); there is also a variable anomalous component of nuclei below 0.1 GeV per nucleon originating in the solar system. These are not tabulated.
Cosmic rays at the top of the atmosphere
The flux of particles per unit time, area and solid angle has a variation with energy of the general form
J(E) dE E −γ dE,
where E is the particle's kinetic energy (usually quoted in GeV). γ varies between 2.5 and 3.1 in different energy ranges (see below). The integral flux, I(E), gives the flux of particles of kinetic energy > E : J = −dI/dE. The table below gives estimates of the particle flux at a time of minimum sunspot activity (when the flux is highest, as in 1965, 1977, 1987) and also (in italics) for a period of near maximum solar activity (low flux—though there is no well-defined absolute minimum). J and I are quoted for protons and I for the aggregate of all nuclear particles (including protons). Some compromises between inconsistent data mean that J and I are not always exactly consistent: errors of ~15% may be present in the fluxes for nuclei and protons: the uncertainties for electrons are much larger. To interpolate between quoted fluxes, a power law as quoted above is satisfactory: see also formulae given below. (The coverage is extended by empirical formulae given later.)
Fluxes observed (above atmosphere) where particles are not excluded by geomagnetic field
Differential flux J in m−2 s −1 sr−1 GeV−1, I in m−2 s−1 sr−1.
The ‘electron’ flux includes positrons—about 30% of the total below 0.3 GeV, but only ~7% above 4 GeV. The photon flux above 0.1 GeV is 0.6 m−2 s−1 sr−1 averaged over the sky, but 50% is from galactic latitudes <10°; about 20% is probably of extragalactic origin.
The more common nuclei heavier than protons have very similar spectra when expressed in terms of magnetic rigidity R = c· momentum/charge. (When v ~ c, R ≈ (E + Mc2)/eZ, Z being the nuclear charge number: thus a 100 GeV He nucleus has a rigidity of 52 GV.) Somewhat more convenient is to express the fluxes of cosmic ray nuclei in terms of kinetic energy-per-nucleon, U = E/A, where A is the atomic mass number. Over the range ~2 to 104 GeV/nucleon, the flux of some types of particles may be expressed to within ~15% by
j(U) = C(U + V)−γ particles m−2 s−1 sr−1 (GeV/nucleon)−1
where the following table gives values for C, V and γ for various nuclei—and it also gives as F the accurately measured relative flux j(U) at U = 5 GeV/n. (Multiplying F by 0.001 65 m−2 s−1sr−1 (GeV/nucleon)−1 will give the absolute value of j(U) at U = 5 GeV/n.)
This fit over a very wide energy range disguises slightly different forms over a smaller range: for great detail over 1–20 GeV/n see Engelmann et al. (1990), Astronomy & Astrophysics, 223, 96–111. For elements given in italics, the fits do not extend to such high energy and the exponents γ may be inaccurate. (These figures refer to conditions not far from maximum flux—minimum solar activity: near maximum of the sunspot cycle, the constant V would be increased, by perhaps 1 GeV/n.) The relative abundances F are closely related to the abundance of the elements in normal matter, though modified by the effect of fragmentation in nuclear collisions: these provide a supply of what would otherwise be uncommon nuclei (e.g. Li, Be, B), though less abundantly at the highest energies. If the fluxes are compared at the same actual kinetic energy E = AU, protons are seen to be much less dominant—about 42% of all particles at 1012 eV and 27% at 1014 eV.
The geomagnetic field prevents the arrival in a vertical direction of particles of rigidity less than 15 cos4 λ GV, where λ is the geomagnetic latitude: particles with rigidity slightly higher than this are admitted at full intensity.
Above about 105 GeV, the nuclear composition of cosmic rays is uncertain, but the total flux of particles of all types has been determined. From 10 to 106 GeV, the flux J(E) found by adding the above components may be expressed as 3.1 × 104(E + 1.35)−2.68 m−2 s−1 sr−1 GeV−1. Above this, the flux has been obtained from studying cascades of particles generated in the atmosphere (‘extensive air showers’): from 106 to 5 × 106 GeV, J = 9.33 × 103E −2.60; from 5× 106 GeV (the ‘knee of the spectrum’) to 109 GeV, J = 6.08 ×106 E −302, then more approximately, from 109 to 5 × 109 GeV, J = 8.91 ×107 E −3.15, and above this, very roughly, J= 3.09 × 104 E −2.8. (Generally ~ 15% accuracy.) The spectrum extends at least as far as 3 × 1020 eV.
Cosmic rays near sea level
The flux of particles (in m−2 s−1 sr−1) arriving nearly vertically above various threshold values of kinetic energy is tabulated below for the more common types of particle (at geomagnetic latitudes > ~40°). R gives also the median range in lead of muons of the specified energy, allowing for track scattering. Other charged particles are less penetrating.
* 40–50% are
positrons, above 0.1 GeV (but only ~ 5% at 1
Above 100 GeV, pions will have fluxes comparable to nucleons: they are less important at lower energies.
Fluxes are averaged over the 11-year cycle. The total muon flux will vary about 3% either way: above 1 GeV the variation is slight. The muon fluxes are the best determined—a few percent at low E (20% say at 100 GeV); the proton flux is uncertain to tens of percent above a few GeV.
Away from the vertical direction, the muon flux per unit solid angle varies with zenith angle θ, approximately as I α cosn θ, with n = 2.15, out to 80° (these refer to the total flux including all energies: muons of tens of GeV vary little with zenith angle). For the protons and neutrons, however, at least above tens of MeV, n ~ 8. About 20% of the electrons striking the ground will be distributed like nucleons, the rest like muons. With a zenith-angle variation of this form, the flux passing through unit horizontal surface, integrated over a hemisphere of angles, is 2π/(n + 2) times the vertical flux per unit solid angle, as tabulated above. Thus a thin, horizontal plane detector will record a flux per m2 per second of about 150 muons and, if its wall has 1 MeV electron stopping power, 70 electrons, and 1 proton.
Nuclear interactions of cosmic rays in the atmosphere generate about 6 × 104 neutrons s−1 per m2 of the Earth at 45° latitude, of which 4 × 104 m−2s−1 are absorbed by nitrogen to generate 14C when solar activity is low: the long-term all-Earth average would be about 60% of this. 0.05–0.13 (or 0.2) neutrons s−1 are generated per kg of air (or Pb) near sea level, at latitudes >40° at times of low solar activity.
Showers. Primary cosmic rays above ~ 1012eV generate extensive showers of secondary particles in the atmosphere and above ~ 1014eV these penetrate to sea level. At this level, such a shower contains about 1 charged particle per 10 GeV of primary energy (at 1014eV, or 1 per 3 GeV at 1017eV, 1 per 1.6 GeV at 1020eV): 5% of the particles are within 3 m of the centre, 50% are within 40 m and 90% within ~ 250 m. Most of the particles are electrons and positrons; a few percent are penetrating muons. A burst of >100 particles m−2 over a few m2 due to a shower in the air would be seen about once per hour near sea level under a very thin cover (mass can add local showers), and >650m−2 once per day, the rate of showers doubling per 100 mb reduction in atmospheric overlay.
Variation with latitude. The number of vertical muons above 0.2 GeV is typically about 13% lower at the equator than at high geomagnetic latitudes; above 50° the flux does not change much. The component generating neutrons by nuclear interactions (largely the neutrons below 1 GeV)—long used to monitor cosmic ray variations—falls by about 24% in going from latitude 55° to the equator.
Variations with time. During the course of the 11-year sunspot cycle, the flux of neutron-generating particles near sea level varies by about 20%, being highest in years of low solar activity (e.g. 1954, 1965, less well-defined in 1977, 1987), though the variation is far from smooth. Flux minima occurred in 1947, 1958, 1969, 1982. The muon flux varies much less.
Cosmic rays as background radiation. Near sea level the dose equivalent rate of cosmic rays, describing their medical (whole body) effect, is about 0.31 mSv per year (somewhat less at latitudes < 30°), but this might be only a third or less of the total natural dose. The dose increases by about 3% per 100 m up in the lower atmosphere. (Above the atmosphere the dose due to low-energy nuclei is very much larger.)
Cosmic radiation is more penetrating than radioactive emissions. The particles penetrating more than 5 cm of lead are mostly muons (only above 0.7 GeV does an electron produce more than 0.5 particle under such a shield), so the particle flux penetrating a thickness x of absorber normally may be judged from the table above—shielding materials of ligher elements are more effective than lead (for a given mass per unit area) by about a factor 1.4, but lead is much more effective in cutting out electrons and photons.
Cosmic rays at other levels
Some muons have extraordinary penetrating power. The following table gives their flux in the vertical direction (in m−2 s−1 sr−1 ) under specified masses h of ‘standard rock’ (having effective Z2/A = 5.5). The mass is given in tonnes m−2, equivalent in mass to metres of water, and, roughly, to feet of rock. The actual depth of water which is estimated to have the same stopping power is also given, though measurements at great depths of water are not yet available.
At great depths, the chemical composition of the rock (Z2/A) can have a large effect. The flux at angle θ to the vertical, under rock, may be estimated quite well from the formula
H being the mass overlay measured along the slant direction from the top of the atmosphere: H = hrock, slant + (10 sec θ). For standard rock, k = 7.1 × 10−4, but if the rock has Z2/A = 6.37, as at the Kolar Gold Fields, where many of the observations were made, k = 8 × 10−4. The formula fits the observations to about 15% for small angles and for depths of more than a few metres, and probably for θ up to 45°.
Above ground level, the absorption length, in which the intensity increases by a factor e, is roughly as follows, for the lower half of the atmosphere:
Nucleons above a few GeV, 110 g cm−2; total muon flux, 550 g cm−2;
total electronic flux, 180 g cm−2; nuclear disintegrations 165 g cm−2.
The rate of nuclear disintegration reaches a peak near the 100 millibar level, several hundred times the sea-level rate (depending on latitude). At this level, though, large increases due to solar flares occasionally occur, lasting for a few hours.
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