Cosmic Rays and Astroparticle Physics

Derivation. Derives Phenomenon 115.2. Let the ionising agent be attenuated exponentially in the mass of absorber traversed, \(I=I_{0}\exp\left(-X/\lambda\right)\), with \(X\) the column density of Equation (115.2) and \(\lambda\) an attenuation length; this is the behaviour of any penetrating radiation with a fixed interaction cross section per unit mass. Two sources are in play and they see opposite overburdens.

For a source in the ground the absorber between source and instrument is the air below the instrument, whose column density increases with height; write it \(\ell(h)\) with \(\ell(0)=0\) and \(\dd\ell/\dd h=\rho(h)>0\). For a source above the atmosphere the absorber is the depth \(X(h)\) of Equation (115.2), which decreases with height, \(\dd X/\dd h=-\rho(h)<0\). The measured rate is the sum,

and differentiating,

The bracket is negative at \(h=0\) whenever the terrestrial term dominates there, and it grows monotonically with \(h\) because the first exponential decreases and the second increases. It therefore has exactly one zero, at the altitude where

and that zero is a minimum. So the observed shape — fall, minimum, rise — is not merely consistent with an external source; it is what an external source superposed on a terrestrial one must produce, and a purely terrestrial source cannot produce it at all, since for \(q_{c}=0\) the bracket is negative everywhere and \(q\) falls monotonically. The two attenuation lengths are then read off the two limbs of the curve, and the upper limb gives a \(\lambda_{c}\) far larger than any then-known gamma radiation possessed, which is the sense in which the radiation is “penetrating”.

Derivation. Derives Phenomenon 115.4. A photon is undeflected by a magnetic field, so a photon primary produces no latitude effect of any size; the observation of one excludes photons as the dominant component. For a charged primary, the Earth's field acts as a momentum filter, and the scale of the filter follows from a single comparison of lengths.

Model the geomagnetic field as a dipole of moment \(m\), so that in the equatorial plane at distance \(r\) from the centre

the value of \(M\) being fixed by the measured surface field at the magnetic equator, \(B\left(R_{\oplus}\right)\approx3.1\times 10^{-5}\,\mathrm{T}\) with \(R_{\oplus}=6.371\times 10^{6}\,\mathrm{m}\). A particle of momentum \(p\) and charge \(Ze\) moving perpendicular to a field \(B\) turns on a circle of radius

with \(R\) the rigidity of Equation (115.1); the second form shows that \(r_{L}\) depends on the particle only through \(R\), which is the whole reason rigidity is used. A particle can reach the Earth from outside only if its trajectory is not turned round before it arrives, which requires the turning radius in the field it crosses to be at least of order the distance over which that field acts. Setting \(r_{L}=r\) with \(B=M/r^{3}\) gives the rigidity scale

that is \(60\,\mathrm{GV}\). The exact treatment is Störmer's: the motion of a charge in a dipole field has a first integral, and it partitions directions at each point into an allowed cone, from which particles of a given rigidity can arrive, and a forbidden one. The resulting cutoff rigidity for vertical incidence at geomagnetic latitude \(\lambda_{m}\) is [Lemaitre:1933]

which is Equation (115.9) with the numerical coefficient \(\tfrac{1}{4}\) that the exact integration supplies. No particle of rigidity below \(R_{c}\) arrives vertically at that latitude.

The latitude effect follows at once. At \(\lambda_{m}=50\,^\circ\) the cutoff is \(14.9\,\mathrm{GV}\times\cos^{4}50\,^\circ =2.5\,\mathrm{GV}\), while at the geomagnetic equator it is the full \(14.9\,\mathrm{GV}\): a detector at high latitude sees the primary spectrum down to a rigidity six times lower than one at the equator, and therefore sees more particles. The size of the effect is then a measurement of the spectrum. If the integral intensity above rigidity \(R\) behaves as \(R^{-\Gamma}\), the ratio of equatorial to polar rates is \(\left(14.9/2.5\right)^{-\Gamma}\), so a \(10\,\mathrm{\%}\) deficit requires \(\Gamma\approx0.06\) for the ionisation-chamber response — that is, the ionisation is dominated by secondaries from primaries well above both cutoffs, which is why the effect is a ten-percent effect and not a factor of six. The qualitative conclusion is independent of all of this: only a charged primary can produce a latitude effect at all.

Derivation. Derives Phenomenon 115.5. The sign can be got without Equation (115.11), from the Lorentz force alone. Set up axes at the geomagnetic equator: \(\hat{\vect{z}}\) upward, \(\hat{\vect{x}}\) east, \(\hat{\vect{y}}\) north. There the geomagnetic field is horizontal and points north, \(\vect{B}=B\hat{\vect{y}}\) with \(B\) of order \(3\times 10^{-5}\,\mathrm{T}\). A primary of charge \(q\) arriving vertically has velocity \(\vect{v}=-v\hat{\vect{z}}\), and feels

since \(\hat{\vect{z}}\times\hat{\vect{y}}=-\hat{\vect{x}}\). For \(q>0\) the force is eastward: as the particle descends, its velocity is deflected towards the east, so that it reaches the ground travelling east of vertical — that is, arriving from the west. Positive primaries therefore produce a western excess and negative primaries an eastern one, and the sign of the observed excess fixes the sign of the charge. A neutral primary, which was still the alternative hypothesis in 1933, produces no azimuthal asymmetry whatever. The measurement thus decides between the two hypotheses without any assumption about the energy spectrum or the composition, which is exactly what the absorption curves of Section 115.1.2 could not do.

The size of the asymmetry follows from the cutoffs. Writing the integral intensity above rigidity \(R\) as \(J(>R)\propto R^{-\Gamma}\), the west-to-east ratio at the horizon is

so that even a modest \(\Gamma\) gives a large effect: the observed excess at Mexico City was of order \(10\,\mathrm{\%}\) at moderate zenith angle [Johnson:1933], growing towards the horizon, and it was found independently and with the same sign by Alvarez and Compton [Alvarez:1933]. The primaries are positive. They were later identified as protons and heavier nuclei, the composition measurement being the subject of Section 115.3.4.

Derivation of the force-field intensity. Liouville's theorem, applied along the characteristics of the transport equation, states that the phase-space density is unchanged, so \(f_{\oplus}(p)=f_{\mathrm{LIS}}(p_{\mathrm{LIS}})\) where the subscript denotes the local interstellar value and the two momenta are connected by Equation (115.12). The differential intensity per unit kinetic energy \(T\) is related to \(f\) by the phase-space volume element: the number of particles per unit volume with momentum in \(\dd^{3}p\) is \(f\,\dd^{3}p=4\pi p^{2}f\,\dd p\), and \(J(T):=p^{2}f\) up to constants, since \(\dd T=v\,\dd p\). Hence

the ratio being \(p^{2}/p_{\mathrm{LIS}}^{2}\) written out with the relativistic relation \(p^{2}c^{2}=T\left(T+2mc^{2}\right)\) of Section 40.2. For a proton of \(T=100\,\mathrm{MeV}\) and \(\Phi=500\,\mathrm{MV}\) the factor is severe — the observed intensity is a small fraction of the interstellar one — while for \(T\gg Ze\Phi\) it tends to unity. This is why the spectrum below a few \(\mathrm{GeV}\) per nucleon measured at Earth is not a source spectrum and must not be read as one, and why the acceleration arguments of Section 115.5 are tested above rather than below that energy.

Derivation. Derives Phenomenon 115.8. Take the segment with \(B\rho=0.21\,\mathrm{T}\,\mathrm{m}\). By Equation (115.14) with \(Z=1\),

so that

Now test the two candidate species against the observed grain density, which was that of a minimum-ionising particle, i.e. one with \(\beta\approx1\) in Equation (115.15).

If the particle is a proton, \(mc^{2}=938.272\,\mathrm{MeV}\) [Navas:2024], then

and Equation (115.15) then demands an ionisation \(1/\beta^{2}=223\) times the minimum. A track ionising two hundred times more heavily than the observed one is not a subtle discrepancy; it is a different photograph. If instead the particle has the electron mass, \(mc^{2}=0.511\,\mathrm{MeV}\), then \(\beta=1-3.3\times 10^{-5}\) and the track is minimum ionising, which is what was seen. Anderson's stated bound from the ionisation was a mass below twenty electron masses; the modern value is exactly one, since the positron and electron masses are equal by the \(CPT\) theorem of Discrete Symmetries and CPT.

A second, independent, check comes from the range. The track was followed for several centimetres of chamber gas. A proton of \(pc=63\,\mathrm{MeV}\) has kinetic energy \(T=E-mc^{2}=2.1\,\mathrm{MeV}\) and, by Equation (115.15), stops within a few centimetres of air; a \(63\,\mathrm{MeV}\) positron crosses the whole chamber and the lead plate as well.

Derivation: the penetrating particles are not electrons. An electron of energy \(E\) crossing matter loses energy chiefly by bremsstrahlung, at a rate proportional to \(E\) itself,

where \(X\) is the traversed depth of Equation (115.2) and \(X_{0}\) is the radiation length of the material, an SI areal mass density: \(X_{0}=365\,\mathrm{kg}/\mathrm{m}^{2} =36.5\,\mathrm{g}/\mathrm{cm}^{2}\) for air and \(63.7\,\mathrm{kg}/\mathrm{m}^{2}=6.37\,\mathrm{g}/\mathrm{cm}^{2}\) for lead. The exponential is the content of the Bethe–Heitler cross-section [Bethe:1934], whose derivation belongs to the bremsstrahlung section of Radiation and Scattering of Electromagnetic Waves, where the heading is reserved and the calculation is not yet written; what matters here is that the loss is proportional to \(E\) and is therefore catastrophic at high energy.

A metre of lead has depth \(X=11.35\times 10^{3}\,\mathrm{kg}/\mathrm{m}^{3}\times1\,\mathrm{m} =1.135\times 10^{4}\,\mathrm{kg}/\mathrm{m}^{2}\), i.e. \(X/X_{0}=178\) radiation lengths. An electron crossing it would emerge with \(\ee^{-178}\) of its energy: nothing survives, at any incident energy. The observed penetrating particles cross such absorbers losing energy at the far slower, nearly energy-independent rate characteristic of ionisation alone, \(-\dd E/\dd X\approx0.2\,\mathrm{MeV}\,\mathrm{m}^{2}/\mathrm{kg}\), i.e. about \(2\,\mathrm{MeV}\) per \(\mathrm{g}/\mathrm{cm}^{2}\). Since the radiative rate carries a factor \(1/m^{2}\) from the recoil of the radiating particle against the nucleus, a particle heavier than the electron radiates less, and much less: the penetrating component is a particle of mass well above \(m_{e}\) and below \(m_{p}\), because a proton of the observed momenta would ionise far too heavily by Equation (115.15).

Derivation: the mesotron is not strongly interacting. A negative particle stopped in matter cascades into a Bohr orbit about a nucleus. The mesic Bohr radius scales as the inverse mass and the inverse nuclear charge,

for carbon, \(Z=6\). The nuclear radius is \(R=r_{0}A^{1/3}=1.2\times 10^{-15}\,\mathrm{m}\times12^{1/3}=2.7\times 10^{-15}\,\mathrm{m}\), so the probability of finding the particle inside the nucleus is of order

If the particle interacts strongly, the rate at which it is absorbed once inside is of order the only rate the strong interaction has, \(c/R=1.1\times 10^{23}\,/\mathrm{s}\). The capture rate is then

a capture time of order \(3\times 10^{-20}\,\mathrm{s}\), against the observed decay rate \(1/\tau_{\mu}=4.6\times 10^{5}\,/\mathrm{s}\). The two differ by fourteen orders of magnitude: a strongly interacting particle stopped in carbon would never be seen to decay. It was seen to decay. Hence the mesotron does not interact strongly and is not Yukawa's meson — the same conclusion reached, with the nuclear-physics apparatus, in Phenomenon 107.18.

Derivation. Derives Phenomenon 115.9. Let a particle of mass \(m_{\pi}\) decay at rest into a charged particle of mass \(m_{\mu}\) and \(n\) unseen particles. Conservation of four-momentum, Equation (40.11), gives for the charged secondary

with \(X\) the unseen system. Squaring the four-momentum of \(X\),

so that

If \(n=1\) then \(m_{X}\) is a fixed mass and \(E_{\mu}\) is a single number: the secondary is monoenergetic and its range is constant. If \(n\geq2\) then \(m_{X}\) ranges continuously from \(\sum m_{i}\) upwards, \(E_{\mu}\) is spread over a continuum, and no fixed range can occur. A constant range therefore establishes a two-body decay — which is the same logic that identifies the beta spectrum as three-body in Section 101.1.1, read in the opposite direction.

Putting \(m_{X}=0\) and the modern masses \(m_{\pi^{+}}c^{2}=139.57039\,\mathrm{MeV}\), \(m_{\mu}c^{2}=105.6583755\,\mathrm{MeV}\) [Navas:2024] into Equation (115.17),

in SI \(T_{\mu}=6.60\times 10^{-13}\,\mathrm{J}\), and the range of a \(4.12\,\mathrm{MeV}\) muon in emulsion is indeed about \(600\,\mu\mathrm{m}\). The unseen partner is the muon neutrino, whose own story is Flavour Physics and Neutrinos.

Derivation. Derives Phenomenon 115.10. The only time scale the strong interaction possesses is the light transit time of a hadron. With a hadronic radius \(r_{h}\approx10^{-15}\,\mathrm{m}\),

and any decay proceeding through the strong interaction into kinematically allowed hadronic channels must take a time of that order. The observed lifetime is \(10^{-10}\,\mathrm{s}\), larger by \(3\times 10^{13}\). Since the production is strong — the cross sections are millibarns, not microbarns — the initial and final states are connected by the strong interaction in one direction and not in the other, which is impossible for a single interaction. The resolution is an additive quantum number \(S\), conserved by the strong interaction and violated by the weak one: strange particles are then produced only in pairs of opposite \(S\), so that production conserves \(S\), and each must then wait for a weak decay, whose rate is smaller by the square of the Fermi constant. The classification of the resulting spectrum by \(\SU(3)\) is Section 102.1.1, and the identification of \(S\) as a quark flavour is Section 102.1.3; the weak decay itself is quark mixing, Weak Interactions.

Derivation of the integral rates. Derives Phenomenon 115.11. For a power law \(J(E)=J_{0}\left(E/E_{0}\right)^{-\gamma}\) with \(\gamma>1\) the integral intensity above \(E\) converges and is

Inserting Equation (115.19) with \(E_{0}=1\,\mathrm{GeV}\), \(J_{0}=1.8\times 10^{4}\,/\mathrm{m}^{2}/\mathrm{s}/\mathrm{sr}/\mathrm{GeV}\) and \(\gamma=2.7\),

which is the first anchor, and

which is the second. For the third the index must be changed at the knee of Section 115.3.2. Matching a second power law \(\gamma_{2}=3.1\) continuously at \(E_{k}=3\times 10^{15}\,\mathrm{eV}\) gives, for \(E>E_{k}\),

and at \(E=10^{19}\,\mathrm{eV}=10^{10}\,\mathrm{GeV}\) this evaluates to \(3.3\times 10^{-15}\,/\mathrm{m}^{2}/\mathrm{s}/\mathrm{sr}\). Multiplying by \(10^{6}\,\mathrm{m}^{2}/\mathrm{km}^{2}\) and by \(3.156\times 10^{7}\,\mathrm{s}/\mathrm{yr}\),

so that an array accepting some \(\pi\) steradians records of order one event per square kilometre per year — the third anchor, and the reason the observatories of Section 115.4.3 are the size of a small country. The total dynamic range follows in the same way: \(J\left(10^{20}\,\mathrm{eV}\right)/J\left(1\,\mathrm{GeV}\right) =3.1\times 10^{-32}\), thirty-one and a half orders of magnitude.

Derivation: any magnetic origin of the knee is ordered by rigidity. Derives Phenomenon 115.13. Suppose the break is produced by a magnetic process, either at the accelerator (a maximum energy the source can reach) or in transport (a maximum energy the Galaxy can retain). Both are governed by the equation of motion of a charge in a magnetic field, and that equation, written for the momentum,

depends on the species only through \(Ze\). Dividing by \(Ze\) and using Equation (115.1),

so two particles of the same rigidity and the same velocity follow identical trajectories in any magnetic field whatever, regardless of their charge and mass. Any limit imposed by magnetic fields is therefore a limit on rigidity, \(R\leq R_{\max}\), and for an ultrarelativistic nucleus of charge \(Z\) this is a limit on energy

The prediction is sharp: the knee of the proton component sits at \(E_{k}\), that of helium at \(2E_{k}\), and that of iron at \(26E_{k}\), so that the all-particle spectrum steepens gradually as one species after another drops out, and the composition becomes progressively heavier across the knee region. With \(E_{k}=3\times 10^{15}\,\mathrm{eV}\) the iron knee is predicted at \(7.8\times 10^{16}\,\mathrm{eV}\). This is what KASCADE found by unfolding the measured shower-size and muon-number distributions into elemental groups: the light component breaks first, the heavy components later, in the rigidity-ordered sequence Equation (115.22) demands [Antoni:2005].

Derivation. Derives Phenomenon 115.15. The Galactic magnetic field measured through Faraday rotation and synchrotron polarisation has a regular component of order \(B=3\times 10^{-10}\,\mathrm{T}\), that is \(0.3\,\mathrm{nT}\) or three microgauss in the older literature. An ultrarelativistic proton of energy \(E\) has \(p=E/c\), so Equation (115.8) gives

For \(E=10^{19}\,\mathrm{eV}=1.602\,\mathrm{J}\) and \(Z=1\),

using \(1\,\mathrm{pc}=3.0857\times 10^{16}\,\mathrm{m}\). The half-thickness of the Galactic disc is about \(0.15\,\mathrm{kpc}\) and the distance to the Galactic centre about \(8\,\mathrm{kpc}\), so the orbit is twenty-four times the disc thickness and about half the size of the whole Galaxy. At \(10^{20}\,\mathrm{eV}\) the same formula gives \(r_{L}=36\,\mathrm{kpc}\), larger than the Galaxy itself: such a particle crosses the disc essentially in a straight line. A population confined by the Galactic field therefore cannot extend to these energies, and conversely a Galactic source of them would produce a strong anisotropy towards the Galactic plane, which Section 115.6.3 shows is not observed. The particles are extragalactic.

Derivation of the traversed grammage. Derives Phenomenon 115.16. Let \(N_{C}\) be the number of carbon nuclei in a beam and \(N_{B}\) the number of boron nuclei produced from them by spallation, and let \(X\) be the traversed grammage of Equation (115.2), now measured through interstellar gas rather than air. Over a path element the number of fragmentations is the number of target nuclei per unit area times the cross section,

with \(\bar{m}\) the mean target mass, essentially the proton mass \(1.673\times 10^{-27}\,\mathrm{kg}\) for interstellar hydrogen. As long as the secondary fraction stays small this integrates to

The measured boron-to-carbon ratio at a few \(\mathrm{GeV}\) per nucleon is \(N_{B}/N_{C}\approx0.3\), and the fragmentation cross section is \(\sigma_{C\to B}\approx6\times 10^{-30}\,\mathrm{m}^{2}\), which is \(60\) millibarn in the customary non-SI unit (one barn is \(10^{-28}\,\mathrm{m}^{2}\) exactly), whence

That is the whole of the matter a cosmic ray meets between its source and the Earth. It is a small quantity — less than one percent of the atmosphere of Equation (115.3) — and it is nevertheless enough to make the secondary nuclei, because their solar-system abundance is essentially zero. The corresponding path length in an interstellar medium of number density \(n\approx3\times 10^{5}\,/\mathrm{m}^{3}\) is \(L=X/\left(n\bar{m}\right)=1.7\times 10^{23}\,\mathrm{m}\), some \(5\,\mathrm{Mpc}\) of path — folded, of course, into a volume a few kiloparsecs across, since the trajectory is a random walk and not a straight line. Since \(X\) is by Equations (115.2) and (115.24) the matter traversed along the path, \(L/c=5.6\times 10^{14}\,\mathrm{s}\), that is \(1.8\times 10^{7}\,\mathrm{yr}\), is already the residence time; it is recomputed directly as Equation (115.53) and confirmed there by the independent radioactive clock of Section 115.5.5.

Derivation. Derives Phenomenon 115.18. Two counters with singles rates \(r_{1}\) and \(r_{2}\) and a coincidence resolving time \(\tau\) register accidental coincidences at the rate

since counter \(2\) must fire within \(\pm\tau\) of counter \(1\). For counters of area \(100\,\mathrm{cm}^{2}\) at sea level, where the vertical muon intensity is about \(70\,/\mathrm{m}^{2}/\mathrm{s}/\mathrm{sr}\), the singles rate is \(r\approx1.7\,/\mathrm{s}\); with the \(\tau\approx10\,\mu\mathrm{s}\) of the period, Equation (115.26) gives \(r_{\mathrm{ch}}=5.8\times 10^{-5}\,/\mathrm{s}\), about five accidental coincidences a day.

The decisive point is not the size of this number but its independence of the separation: Equation (115.26) contains no distance at all. The measured coincidence rate, by contrast, falls steeply and smoothly as the counters are drawn apart, and remains above the accidental rate out to \(300\,\mathrm{m}\). A distance-dependent excess over a distance-independent background can only be a correlation, and a correlation over \(300\,\mathrm{m}\) at the speed of light means a single common ancestor.

The energy follows from the particle number. If the front contains \(N\) particles at the ground and each was produced by dividing the primary energy down to the point where production stops — the critical energy \(E_{c}\approx85\,\mathrm{MeV}\) of Section 115.4.2 — then \(E_{0}\gtrsim NE_{c}\). Auger's estimate of \(N\approx10^{6}\) particles gives \(E_{0}\gtrsim8.5\times 10^{13}\,\mathrm{eV}\), and correcting for the fact that the ground lies well past the shower maximum, where most of the particles have already been absorbed, raises this to of order \(10^{15}\,\mathrm{eV}=1.6\times 10^{-4}\,\mathrm{J}\).

Derives Theorem 115.19. After \(n\) splittings the depth is \(X=nd\) and the number of particles is \(2^{n}\), which is Equation (115.28) with \(n=X/d\). Energy is conserved and shared equally, so \(E(X)N(X)=E_{0}\). Multiplication continues while \(E(X)>E_{c}\) and stops when equality is reached; at that depth \(N=E_{0}/E_{c}\), which is the first of Equation (115.29), and \(2^{X_{\max}/d}=E_{0}/E_{c}\) gives

which is the second. Differentiating with respect to \(\log_{10}E_{0}=\ln E_{0}/\ln10\) gives Equation (115.30).

Derivation of the pion critical energy. A charged pion of Lorentz factor \(\gamma\) travels a mean distance \(\gamma c\tau_{\pi}\) before decaying, with \(\tau_{\pi}=2.6033\times 10^{-8}\,\mathrm{s}\) and \(c\tau_{\pi}=7.80\,\mathrm{m}\). It travels \(\lambda_{I}/\rho\) before interacting. At an altitude of \(10\,\mathrm{km}\), where most of the shower develops, the air density is \(\rho\approx0.41\,\mathrm{kg}/\mathrm{m}^{3}\) and the interaction length is

Equating the two lengths gives the Lorentz factor at which the two fates are equally likely,

in SI \(3.8\times 10^{-9}\,\mathrm{J}\). Pions above this energy interact and keep the hadronic cascade going; pions below it decay and become muons. The value depends on altitude through \(\rho\) and on energy through \(\lambda_{I}\), and is conventionally taken as \(E_{c}^{\pi}\approx20\,\mathrm{GeV}\) for the shower as a whole [Matthews:2005]; the derivation above shows where that number comes from and how much it can be trusted. Note how sensitive it is to the cross section: the older tabulated interaction length for air, \(90\,\mathrm{g}/\mathrm{cm}^{2}\), which is a few-\(\mathrm{GeV}\) value, would give \(39\,\mathrm{GeV}\) instead and turn a twenty-percent agreement into a factor-of-two puzzle.

Derives Theorem 115.21. Of the \(\tfrac{3}{2}N_{\mathrm{ch}}\) pions made in one interaction, \(N_{\mathrm{ch}}\) are charged and carry \(2/3\) of the energy onward; after \(n\) generations the hadronic energy is \(E_{0}\left(2/3\right)^{n}\) spread over \(N_{\mathrm{ch}}^{n}\) pions, so the energy per pion is \(E_{0}\left(\tfrac{3}{2}N_{\mathrm{ch}}\right)^{-n}\). Setting this equal to \(E_{c}^{\pi}\) determines the number of generations,

and \(N_{\mu}=N_{\mathrm{ch}}^{n}=\exp\left(n\ln N_{\mathrm{ch}}\right)\), which is Equation (115.32). With \(N_{\mathrm{ch}}=10\), \(\beta=\ln10/\ln15=0.850\); the exponent is insensitive to \(N_{\mathrm{ch}}\), moving only from \(0.80\) to \(0.88\) as \(N_{\mathrm{ch}}\) runs from \(5\) to \(20\), which is why it is quoted as a number rather than as a function.

Derives Corollary 115.22. Each of the \(A\) nucleons initiates a sub-shower of energy \(E_{0}/A\); the muon numbers add, giving the first of Equation (115.33) directly from Equation (115.32), and \(1-\beta=0.15\). The depths of maximum do not add: the shower maximum of the superposition is that of a single sub-shower, so \(\avg{X_{\max}}\) is evaluated at \(E_{0}/A\) instead of at \(E_{0}\), and by the definition Equation (115.30) of the elongation rate this shifts it by \(-D_{10}\log_{10}A\).

Derivation of the angular resolution of an array. Treat the shower front as a plane advancing at speed \(c\) along the unit vector \(\hat{\vect{n}}\). Two detectors separated by the baseline \(\vect{d}\) are struck at times differing by

for a baseline in the plane of incidence, so that measuring \(\Delta t\) measures \(\theta\). Differentiating, \(\sigma_{\theta}=c\,\sigma_{t}/\left(d\cos\theta\right)\). With a timing precision \(\sigma_{t}=10\,\mathrm{ns}\), achievable with photomultipliers and satellite time transfer, and a baseline \(d=1500\,\mathrm{m}\),

at the zenith. The realised resolution of a large array is about \(1\,^\circ\), an order of magnitude worse, because the front is not a plane but a curved and thick shell whose sampled thickness is tens of nanoseconds; Equation (115.34) sets the floor, not the answer.

Derivation of the fluctuation scaling. Derives Phenomenon 115.25. Two independent sources of fluctuation enter. The first is the depth of the first interaction, which is exponentially distributed with mean \(\lambda_{I}\), so it contributes a standard deviation \(\lambda_{I}\) to \(X_{\max}\). The relevant \(\lambda_{I}\) is the one at the energy of the measurement, and by Equation (115.31) it shrinks as the cross section grows: near \(10^{19}\,\mathrm{eV}\) the inelastic proton–air cross section is about \(5.0\times 10^{-29}\,\mathrm{m}^{2}\), that is \(500\) millibarn, so

The second source is the fluctuation in the subsequent cascade, of comparable size; the two add in quadrature, and \(\left(48^{2}+36^{2}\right)^{1/2}=60\) reproduces the measured proton width of some \(60\,\mathrm{g}/\mathrm{cm}^{2}\).

For a nucleus of mass \(A\), superposition (Corollary 115.22) replaces one shower by \(A\) sub-showers of energy \(E_{0}/A\), each with its own first-interaction depth. The shower maximum of the sum is essentially the average of the \(A\) sub-maxima, and the variance of an average of \(A\) independent variables is \(1/A\) of the individual variance:

For iron \(\sqrt{56}=7.5\), so the rule predicts a collapse from the proton value of some \(60\,\mathrm{g}/\mathrm{cm}^{2}\) to \(8\,\mathrm{g}/\mathrm{cm}^{2}\). The measured iron width is about \(20\,\mathrm{g}/\mathrm{cm}^{2}\): the superposition rule gets the direction and the order of the effect right and underpredicts the width by a factor of about two and a half, which is the same crudeness Remark 115.20 concedes for the model it rests on. The qualitative conclusion — heavy primaries fluctuate much less than light ones — is robust; the number is not. Mean and width are thus two independent composition observables, and they must agree: a mean lying between the proton and iron expectations can be produced either by an intermediate species, which would have a narrow distribution, or by a mixture of protons and iron, which would have a very wide one. Auger sees the mean move heavier above the ankle and the width narrow, which favours a genuinely intermediate composition over a mixture [Aab:2014].

Derives Proposition 115.26. For (i), \(J(>E)=AE^{-\left(\gamma-1\right)}/\left(\gamma-1\right)\) by Equation (115.21). A scale bias means the flux reported at \(E\) is the true flux at \(E/\left(1+\epsilon\right)\), so

For (ii), change variable to \(x=\ln E\). The number of events per unit \(x\) is \(J(E)E=Ae^{-\left(\gamma-1\right)x}\), and smearing convolves this with a Gaussian of width \(\sigma\):

by completing the square — this is the moment generating function of a Gaussian evaluated at \(\gamma-1\). Dividing by \(E_{r}\) to return to a differential intensity gives Equation (115.41). The result is exact, not perturbative, and holds for any \(\sigma\).

Derivation. Derives Phenomenon 115.29. Take the confinement volume to be a disc of radius \(R=15\,\mathrm{kpc}=4.63\times 10^{20}\,\mathrm{m}\) and half-thickness \(h=300\,\mathrm{pc}=9.26\times 10^{18}\,\mathrm{m}\):

so the stored energy is

In a steady state the injected power equals the stored energy divided by the residence time \(\tau_{\mathrm{esc}}\approx10^{7}\,\mathrm{yr} =3.16\times 10^{14}\,\mathrm{s}\) of Section 115.5.5:

The available power is that of the supernovae. A core-collapse supernova delivers of order \(10^{44}\,\mathrm{J}\) to the kinetic energy of its ejecta — three tenths of one percent of the \(3\times 10^{46}\,\mathrm{J}\) of gravitational binding energy released, the remaining ninety-nine and more percent leaving as the neutrino burst of Section 115.7.3 — and the Galactic rate is about two per century:

The required efficiency is therefore

ten percent of the mechanical energy of the blast wave. That is demanding but not absurd, and it is exactly what the shock acceleration of Section 115.5.3 predicts when the accelerated particles are dynamically important. Every alternative Galactic reservoir — stellar winds, pulsar rotational energy, the interstellar magnetic field — falls short of \(P_{\mathrm{CR}}\) by an order of magnitude or more, which is why the supernova hypothesis has survived ninety years without a rival.

Derivation of the pion-decay signature. Derives Phenomenon 115.30. A neutral pion at rest decays to two photons each of energy \(m_{\pi^{0}}c^{2}/2=67.5\,\mathrm{MeV}\). Boost it to Lorentz factor \(\gamma_{\pi}\): by the transformation of a null four-momentum, the photon energy in the laboratory is

with \(\theta^{*}\) the emission angle in the pion frame. Because the decay is isotropic in that frame, \(\cos\theta^{*}\) is uniform on \([-1,1]\), so \(E_{\gamma}\) is uniformly distributed between \(\gamma_{\pi}\left(1-\beta_{\pi}\right)m_{\pi}c^{2}/2\) and \(\gamma_{\pi}\left(1+\beta_{\pi}\right)m_{\pi}c^{2}/2\). The distribution is flat, and in \(\log E_{\gamma}\) it is symmetric about \(\log\left(m_{\pi}c^{2}/2\right)\), since the two limits multiply to \(\left(m_{\pi}c^{2}/2\right)^{2}\) whatever \(\gamma_{\pi}\) is. Summing over any pion spectrum preserves that symmetry point. The resulting photon spectrum therefore rises to a maximum at \(E_{\gamma}=67.5\,\mathrm{MeV}\) and falls below it, whatever the parent proton spectrum — the “pion bump”.

No leptonic process produces such a feature. Bremsstrahlung from electrons follows the electron spectrum with no scale in it; inverse-Compton scattering likewise. The observed turnover below \(200\,\mathrm{MeV}\) in IC 443 and W44 is thus a direct measurement of a hadronic population, and it is the observation that converts Phenomenon 115.29 from a budget into evidence [Ackermann:2013].

Derives Theorem 115.31. Write \(\beta_{V}=V/c\) and \(\gamma_{V}=\left(1-\beta_{V}^{2} \right)^{-1/2}\), and let \(\theta\) be the angle in the laboratory between the particle's direction and the cloud's velocity. Transforming the particle's energy into the cloud frame (a boost, as in Section 40.1),

for an ultrarelativistic particle, \(pc=E\). The scattering is elastic in the cloud frame — the field does no work there — so \(E''=E'\) with a new direction \(\theta''\), and transforming back,

Isotropisation in the cloud frame gives \(\avg{\cos\theta''}=0\), so

The laboratory angles are not uniformly sampled: the rate of encounters with a cloud approaching at angle \(\theta\) is proportional to the relative closing speed, \(\propto\left(1-\beta_{V}\cos\theta\right)\), so

Head-on encounters, which gain energy, are commoner than overtaking ones, which lose it. Hence

which is Equation (115.44).

Derives Theorem 115.33. The gain. Transform to the frame of the upstream fluid. The downstream fluid approaches it at speed \(u_{1}-u_{2}\), so a particle that crosses into the downstream region and is isotropised there undergoes exactly the calculation of Theorem 115.31 with \(\beta_{V}=\left(u_{1}-u_{2}\right)/c\), but with one difference: the crossing is always head-on, so the average over \(\cos\theta\) is taken over the flux of particles crossing a plane, whose distribution is \(\propto\mu\,\dd\mu\) on \([0,1]\), giving \(\avg{\mu}=\tfrac{2}{3}\). Each of the two crossings that make a cycle contributes \(\beta_{V}\avg{\mu}=\tfrac{2}{3}\beta_{V}\), so \(\xi=\tfrac{4}{3}\beta_{V}\), which is Equation (115.45). The result is first order because the approach is systematic: there are no overtaking encounters to cancel it.

The escape. Downstream the particles are isotropic in the fluid frame and their flux onto the shock from behind is \(n c/4\), the standard one-sided flux of an isotropic gas. They are simultaneously carried away from the shock by the downstream flow at the rate \(nu_{2}\). The fraction lost per cycle is the ratio,

which is Equation (115.46). Note that neither \(\xi\) nor \(P_{\mathrm{esc}}\) depends on the particle's energy. That is the whole reason the spectrum is a power law.

The spectrum. After \(k\) cycles a particle has energy \(E_{k}=E_{0}\left(1+\xi\right)^{k}\) and the number surviving is \(N_{k}=N_{0}\left(1-P_{\mathrm{esc}}\right)^{k}\). Eliminating \(k\),

so \(N(>E)\propto E^{-P_{\mathrm{esc}}/\xi}\) with

which is Equation (115.47); the differential index is one greater.

The compression ratio. Work in the shock frame, where the flow is steady, and impose conservation of mass, momentum and energy across the discontinuity — the Rankine–Hugoniot conditions, which belong to the fluid mechanics of Section 31.10.2:

with the specific enthalpy of an ideal gas \(h=\gamma_{\mathrm{ad}}P/\left[\left(\gamma_{\mathrm{ad}} -1\right)\rho\right]\). A strong shock is one whose upstream pressure is negligible, \(P_{1}\to0\) and hence \(h_{1}\to0\); the momentum condition then gives \(P_{2}=j\left(u_{1}-u_{2}\right)\), and substituting this and \(\rho_{2}=j/u_{2}\) into the energy condition,

Cancelling the common factor \(u_{1}-u_{2}\), which is non-zero for a genuine shock, leaves \(u_{1}+u_{2}=2\gamma_{\mathrm{ad}}u_{2}/ \left(\gamma_{\mathrm{ad}}-1\right)\) and therefore

for a monatomic ionised gas, whence \(\gamma_{s}=2\).

Derives Proposition 115.34. In a steady state the number density at energy \(E\) satisfies injection equals escape: \(Q(E)=n(E)/\tau_{\mathrm{esc}}(E)\), so \(n(E)=Q(E)\tau_{\mathrm{esc}}(E)\), and the intensity is \(J=nc/4\pi\) for an isotropic distribution. Since the particles are ultrarelativistic, \(R=E/\left(Ze\right)\) and the rigidity dependence is an energy dependence at fixed \(Z\).

Derivation of the maximum energy. Derives Equation (115.51). The mechanism has a limit, and it is the reason the knee exists in the first reading of Remark 115.14. The time to complete one cycle is set by how far the particle diffuses from the shock before returning. Consider the upstream side. The flow sweeps the particles onto the shock at speed \(u_{1}\) while diffusion carries them away from it, and the balance \(u_{1}n=D_{1}\,\dd n/\dd x\) gives a precursor of scale length \(D_{1}/u_{1}\), hence a column \(nD_{1}/u_{1}\) of particles per unit area waiting to recross. They recross at the one-sided isotropic flux \(nc/4\) already used in Equation (115.46), so the mean time spent upstream per crossing is \(\left(nD_{1}/u_{1}\right)/\left(nc/4\right)=4D_{1}/\left(u_{1}c\right)\), and the same argument downstream gives \(4D_{2}/\left(u_{2}c\right)\). One complete cycle therefore takes

and since each cycle multiplies the energy by \(1+\xi\) with the \(\xi\) of Equation (115.45), the time to raise the energy by one \(e\)-folding is \(t_{\mathrm{acc}}=t_{\mathrm{cyc}}/\xi\), that is

Taking the same diffusion coefficient \(D\) on both sides and \(r=4\), so that \(u_{2}=u_{1}/4\) and \(u_{1}-u_{2}=3u_{1}/4\), the bracket is \(5D/u_{1}\) and

the second expression being the smallest diffusion coefficient a magnetic field permits, the Bohm value, obtained when the scattering mean free path equals the gyroradius of Equation (115.8). Setting \(t_{\mathrm{acc}}\) equal to the lifetime \(T\) of the shock,

With \(Z=1\), \(u_{1}=10^{7}\,\mathrm{m}/\mathrm{s}\), \(T=1000\,\mathrm{yr}=3.16\times 10^{10}\,\mathrm{s}\) and the ambient interstellar field \(B=3\times 10^{-10}\,\mathrm{T}\),

a factor of twenty below the knee at \(3\times 10^{15}\,\mathrm{eV}\). The mechanism as stated therefore fails to reach the knee. It succeeds if the field at the shock is amplified above the ambient value by a factor of order \(30\): with \(B=10^{-8}\,\mathrm{T}\), Equation (115.51) gives \(E_{\max}=4.7\times 10^{15}\,\mathrm{eV}\), which does reach it. Such amplification is observed — the X-ray synchrotron filaments at young remnant shocks are thin, and their thinness measures a downstream field of order \(10^{-8}\,\mathrm{T}\) — and the physical mechanism proposed for it is a streaming instability driven by the accelerated particles themselves.

Derives Theorem 115.36. By Equation (115.8) a particle of rigidity \(R\) has a gyroradius \(r_{L}=R/\left(cB\right)\). It remains in the accelerating region only while \(r_{L}\leq L\), i.e. while \(R\leq BLc\); beyond that it leaves in less than one gyration and cannot be turned back. The factor \(\beta\) enters because the energy gain per crossing is first order in the speed of the scattering centres (Equation (115.45)), so a slow shock requires many crossings and loses a corresponding factor. Converting rigidity to energy with \(E=ZeR\) gives Equation (115.52). Dimensionally, \(\left[Ze\right]\left[B\right]\left[L\right]\left[c\right] =\mathrm{C}\cdot\mathrm{T}\cdot\mathrm{m} \cdot\mathrm{m}/\mathrm{s}=\mathrm{J}\), as required.

Derivation of the threshold. Derives Phenomenon 115.40. Let a proton of energy \(E\) and momentum \(p=\sqrt{E^{2}/c^{2} -m_{p}^{2}c^{2}}\) meet a photon of energy \(E_{\gamma}\) head-on. The invariant mass of the pair is given by Equation (40.12):

the limit being the ultrarelativistic one, \(pc\to E\); for a general collision angle \(\theta\) the last term is \(2EE_{\gamma}\left(1-\cos\theta\right)\), maximal head-on. Photopion production \(p+\gamma\to N+\pi\) is kinematically possible once \(\sqrt{s}\geq\left(m_{N}+m_{\pi}\right)c^{2}\), and the lowest such threshold is the \(p\pi^{0}\) channel. Solving Equation (115.55),

The photon bath is measured, not assumed. Its temperature is \(T_{0}=2.72548(57)\,\mathrm{K}\) [Fixsen:2009], so

and the mean energy of a Planck photon is

while their number density is

With \(m_{p}c^{2}=938.272\,\mathrm{MeV}\) and \(m_{\pi^{0}}c^{2}=134.977\,\mathrm{MeV}\) [Navas:2024], Equation (115.56) gives

and the same calculation with the \(\Delta(1232)\) mass in place of \(\left(m_{p}+m_{\pi}\right)\) puts the resonance peak at \(2.5\times 10^{20}\,\mathrm{eV}\).

Derivation of the onset against the Planck tail. A threshold computed with the mean photon energy is not the energy at which the effect begins, because the Planck distribution has a tail. At proton energy \(E\) the reaction proceeds on any photon with \(E_{\gamma}\geq\varepsilon_{\mathrm{th}}(E)\), where by Equation (115.56) \(\varepsilon_{\mathrm{th}}=m_{\pi}c^{2}\left(2m_{p}c^{2} +m_{\pi}c^{2}\right)/(4E)\). Writing \(x=\varepsilon_{\mathrm{th}}/\left(k_{B}T_{0}\right)\), the fraction of the bath above that energy is

the approximation being the leading term of the expansion of \(\left(\ee^{u}-1\right)^{-1}\), excellent for \(x\gtrsim3\). At \(E=5\times 10^{19}\,\mathrm{eV}\), \(\varepsilon_{\mathrm{th}}=1.36\times 10^{-3}\,\mathrm{eV}\) and \(x=5.78\), so Equation (115.59) gives \(6.0\,\mathrm{\%}\): six percent of the bath, that is \(2.5\times 10^{7}\,/\mathrm{m}^{3}\) photons, are already above threshold. At \(E=10^{20}\,\mathrm{eV}\), \(x=2.89\) and the fraction is \(37\,\mathrm{\%}\). The onset of the suppression is therefore expected in the range \(4\times 10^{19}\,\mathrm{eV}\) to \(10^{20}\,\mathrm{eV}\), and not at the nominal \(1.07\times 10^{20}\,\mathrm{eV}\) — which is where the observed break in Phenomenon 115.42 lies.

Order-of-magnitude derivation of the horizon. The mean free path is \(\ell=1/\left(n_{\mathrm{eff}}\sigma\right)\) with \(n_{\mathrm{eff}}\) the density of above-threshold photons and \(\sigma\) the photopion cross section, which near the \(\Delta\) peak is about \(5\times 10^{-32}\,\mathrm{m}^{2}\), that is \(500\) microbarn. At \(E=10^{20}\,\mathrm{eV}\) the fraction above threshold is \(37\,\mathrm{\%}\), so \(n_{\mathrm{eff}}=1.5\times 10^{8}\,/\mathrm{m}^{3}\) and

Each interaction removes only the fraction \(\kappa\approx0.2\) of the proton's energy — the inelasticity, fixed by the kinematics of Equation (115.17) applied to the \(\Delta\) decay — so the energy-loss length is \(\ell/\kappa\approx20\,\mathrm{Mpc}\). This estimate is crude in two ways that both push it upward: it uses the resonance-peak cross section for every above-threshold photon, and it treats every collision as head-on rather than averaging over angle. The proper calculation, integrating the measured cross section over the Planck spectrum and over angle, gives an energy-loss length of order \(100\,\mathrm{Mpc}\) at \(10^{20}\,\mathrm{eV}\), falling to a few tens of megaparsecs above it [Stecker:1968].

Either number is small in the only comparison that matters. The Hubble radius is

with \(H_{0}\) from [Aghanim:2020]. A particle above the threshold therefore reaches us from at most one or two percent of the radius of the observable universe: the sky at these energies is a local sky, and if a source is identified it will be a nearby one.

Derivation and its limits. Derives Phenomenon 115.42. The predicted position is Equation (115.56) softened by the Planck tail, Equation (115.59): an onset between \(4\times 10^{19}\,\mathrm{eV}\) and \(10^{20}\,\mathrm{eV}\). The two measurements bracket that range. The HiRes fluorescence detectors found the spectrum to break at \(5.6\times 10^{19}\,\mathrm{eV}=9.0\,\mathrm{J}\), the break being significant at about five standard deviations against a continued power law [Abbasi:2008]; the Pierre Auger Observatory, with a larger exposure and the hybrid calibration of Section 115.4.3, found the flux above \(4\times 10^{19}\,\mathrm{eV}=6.4\,\mathrm{J}\) to be suppressed relative to the extrapolation at more than six standard deviations [Abraham:2008]. The two techniques share no systematic, which is why the joint statement is stronger than either.

That events exist above the break is not a contradiction; it is required. Attenuation is exponential in distance, not a wall, so a source within \(20\,\mathrm{Mpc}\) can deliver a particle of any energy its accelerator reaches. The Fly's Eye event of \(3.2(9)\times 10^{20}\,\mathrm{eV}=51(14)\,\mathrm{J}\) [Bird:1995] is such a particle, and its existence constrains its own source to lie within the horizon of Phenomenon 115.40 — which is a strong statement, since no candidate accelerator is known within that distance.

The inference is nevertheless not closed, and the reason is Remark 115.41 together with Phenomenon 115.25. If the primaries at these energies are nuclei rather than protons, the same break is produced, equally well, by the maximum rigidity of the sources through Equation (115.22): an accelerator limited to \(R_{\max}\approx5\times 10^{18}\,\mathrm{V}\) delivers protons to \(5\times 10^{18}\,\mathrm{eV}\) and iron to \(1.3\times 10^{20}\,\mathrm{eV}\), and the superposition of the elemental cutoffs mimics a propagation suppression. The measurement therefore establishes that the spectrum is suppressed; it does not by itself establish that the mechanism is the one Greisen, Zatsepin and Kuz'min predicted. Distinguishing them requires the composition above the break, which is where the current data are weakest, and the appearance of the cosmogenic neutrinos that photopion production must also produce, which have not been detected.

Derivation: why a few percent is a detection. Derives Phenomenon 115.44. Let \(N\) events arrive with right ascensions \(\varphi_{i}\), and form the first-harmonic amplitudes

Under isotropy each \(\varphi_{i}\) is uniform, so \(\sum\cos\varphi_{i}\) and \(\sum\sin\varphi_{i}\) are sums of \(N\) independent zero-mean terms of variance \(\tfrac{1}{2}\) each; by the central limit theorem (Theorem 11.46) they are Gaussian of variance \(N/2\), and independent. Then \(Na^{2}/2\) and \(Nb^{2}/2\) are independent \(\chi^{2}\) variables of one degree of freedom (Definition 11.80), so \(Nr^{2}/2\) is \(\chi^{2}\) with two, i.e. exponential, and

This is the Rayleigh probability. With the \(N\approx3.2\times 10^{4}\) events recorded above \(8\times 10^{18}\,\mathrm{eV}\) and \(r=0.065\), Equation (115.60) gives \(\exp(-33.8)\), a number so small that it is not the quoted result: the published chance probability is far more conservative, because a three-dimensional dipole must be reconstructed from an exposure that does not cover the whole sky, and the declination component is measured much less well than the right-ascension component. Equation (115.60) shows why a percent-level anisotropy is measurable at all with tens of thousands of events — the sensitivity goes as \(N^{-1/2}\), so \(3\times 10^{4}\) events probe amplitudes near \(1\,\mathrm{\%}\) — and the published \(10^{-6}\) is the honest number after the exposure is accounted for [Aab:2017].

Derivation of the parent energy, and of the ambiguity. Derives Phenomenon 115.46. Suppose first the emission is hadronic. A proton of energy \(E_{p}\) striking interstellar gas produces pions carrying, on average, some \(20\,\mathrm{\%}\) of its energy each, and a \(\pi^{0}\) decays to two photons sharing its energy, so

and a \(1.4\,\mathrm{PeV}\) photon requires a proton of about \(1.4\times 10^{16}\,\mathrm{eV}\) — above the knee. Suppose instead the emission is leptonic, by inverse-Compton scattering of electrons on the microwave background. In the Thomson regime the scattered photon energy is \(E_{\gamma}\approx\tfrac{4}{3}\gamma_{e}^{2}\avg{E_{\gamma}^{\mathrm{ CMB}}}\), which for \(E_{\gamma}=1.4\,\mathrm{PeV}\) and Equation (115.57) would need \(\gamma_{e}=1.3\times 10^{9}\), i.e. \(E_{e}=0.66\,\mathrm{PeV}\). But at that Lorentz factor the scattering is deep in the Klein–Nishina regime, since \(\gamma_{e}\avg{E_{\gamma}^{\mathrm{CMB}}}=0.8\,\mathrm{MeV}\) exceeds \(m_{e}c^{2}=0.511\,\mathrm{MeV}\); there the electron transfers nearly all its energy in one scattering and the cross section is suppressed, so the required electron energy rises to \(E_{e}\gtrsim1.4\,\mathrm{PeV}\) and the emission becomes inefficient.

Either way the source accelerates particles to at least a petaelectronvolt. Which species it accelerates the gamma-ray spectrum alone cannot say, because both mechanisms produce power-law photon spectra of similar index over a limited band. This is the standing hadronic–leptonic ambiguity, and it is why the detection of neutrinos from a source would be decisive: Section 115.7.4 shows that neutrinos at these energies have no leptonic production channel at all.

Derivation of the oscillation phase in SI units. Derives Phenomenon 115.47. Take two neutrino mass eigenstates of masses \(m_{1}\), \(m_{2}\) and write the flavour states as \(\ket{\nu_{\alpha}}=\cos\theta\ket{\nu_{1}}+\sin\theta\ket{\nu_{2}}\), \(\ket{\nu_{\beta}}=-\sin\theta\ket{\nu_{1}}+\cos\theta\ket{\nu_{2}}\); the unitarity of the mixing matrix is licensed by Linear Algebra and Representation Theory and the physics of the mixing itself belongs to Flavour Physics and Neutrinos. A state created with definite flavour and definite momentum \(p\) evolves with the phase \(\exp\left(-\ii E_{i}t/\hbar\right)\), where

the expansion being valid because \(m_{i}c^{2}\ll pc\) for every neutrino ever detected. After a flight time \(t=L/c\) over a baseline \(L\), the accumulated phase difference is

writing \(E\approx pc\) for the neutrino energy and \(\Delta m^{2}:=m_{2}^{2}-m_{1}^{2}\). Every symbol is SI: \(\hbar c\) has dimension \(\mathrm{J}\,\mathrm{m}\), \(\Delta m^{2}c^{4}\) has dimension \(\mathrm{J}^{2}\), and \(L/E\) has dimension \(\mathrm{m}/\mathrm{J}\), so \(\Delta\Phi\) is a pure number as it must be.

The transition probability follows from the amplitude \(A_{\alpha\alpha}=\cos^{2}\theta\,\ee^{-\ii\Phi_{1}} +\sin^{2}\theta\,\ee^{-\ii\Phi_{2}}\):

so that, defining

the survival and appearance probabilities are

Equation (115.63) is the formula the literature quotes as \(1.267\,\Delta m^{2}L/E\), and the number is derived and not conventional. Put \(\Delta m^{2}c^{4}\) in \(\mathrm{eV}^{2}\), \(L\) in kilometres and \(E\) in \(\mathrm{GeV}\), and use \(\hbar c=1.9732698\times 10^{-7}\,\mathrm{eV}\,\mathrm{m}\):

Derivation of the zenith-angle deficit. Atmospheric neutrinos are produced at an altitude of about \(15\,\mathrm{km}\) over the whole Earth. One arriving from directly overhead has travelled \(L\approx15\,\mathrm{km}\); one arriving from directly below has crossed a diameter, \(L\approx2R_{\oplus}=1.27\times 10^{4}\,\mathrm{km}\). The path length varies by three orders of magnitude with zenith angle, and the neutrino energy does not.

Take the atmospheric mass splitting \(\Delta m^{2}c^{4}\approx2.5\times 10^{-3}\,\mathrm{eV}^{2}\) and \(E=1\,\mathrm{GeV}\). For the downward-going neutrino,

so no oscillation is observable. For the upward-going one,

which is many complete oscillations; averaged over the spread in \(L\) and \(E\) within any experimental bin, \(\avg{\sin^{2}\Phi}=\tfrac{1}{2}\) and

which for maximal mixing is one half. The prediction is therefore specific and asymmetric: the upward-going muon-neutrino rate is halved while the downward-going rate is untouched, and the electron-flavour rate is unaffected at either zenith angle because the corresponding splitting is thirty times smaller and the phase is correspondingly negligible over these baselines. That is exactly the pattern Super-Kamiokande measured [Fukuda:1998], and it is the first compelling evidence that neutrinos have mass.

Derivation: the energy is the binding energy of a neutron star. Derives Phenomenon 115.48. If a stellar core of mass \(M\) collapses to a uniform sphere of radius \(R\), the gravitational energy released is

the standard self-energy integral. With \(M=1.4M_{\odot} =2.78\times 10^{30}\,\mathrm{kg}\) and \(R=10\,\mathrm{km}\),

which is \(12\,\mathrm{\%}\) of the rest energy \(Mc^{2}=2.50\times 10^{47}\,\mathrm{J}\) of the collapsing core. Essentially none of it emerges as light: the total optical output of the supernova is smaller by three orders of magnitude, and the kinetic energy of the ejecta by two — the \(10^{44}\,\mathrm{J}\) used in Phenomenon 115.29. It emerges as neutrinos, because at the densities of the collapsing core nothing else escapes. The total energy inferred from the nineteen detected events, given the detector masses, the cross section for \(\bar{\nu}_{e}p\to ne^{+}\) and the assumption of equipartition among the six neutrino species, is of order \(3\times 10^{46}\,\mathrm{J}\). The agreement is the confirmation, and it is the only direct observation ever made of the mechanism of a core-collapse supernova.

Derivation: the bound on the neutrino speed. The neutrinos preceded the first optical light by about \(\Delta t=3\,\mathrm{h}=1.1\times 10^{4}\,\mathrm{s}\), and that lead is astrophysical — the shock takes hours to reach the stellar surface after the core collapses, whereas neutrinos leave immediately. Taking the lead as an upper bound on any difference in travel time over the flight time

gives

The bound is weak as a fractional velocity but it is obtained over a baseline of a hundred and sixty thousand light years, which no terrestrial measurement can approach, and it is the reason a claim of superluminal neutrino propagation is checked against SN 1987A first.

Derivation: the bound on the neutrino mass. A massive neutrino is dispersive. From \(E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}\) the group velocity is

so a neutrino of energy \(E\) arrives later than a massless one by

Two neutrinos of energies \(E_{1}<E_{2}\) emitted simultaneously therefore arrive separated by

The observed burst spanned \(\Delta t\approx13\,\mathrm{s}\) with energies from about \(7.5\,\mathrm{MeV}\) to \(35\,\mathrm{MeV}\). Requiring the dispersion not to exceed the observed spread,

whence

The bound is honest only under an assumption that cannot be checked: that the burst was emitted in a time short compared with \(13\,\mathrm{s}\). Since the emission is expected to last of order that long by itself, the true limit is weaker, and careful analyses that model the emission profile obtain a few electronvolts rather than seventeen. It is quoted here because it is the only kinematic neutrino mass bound ever obtained over an astronomical baseline, and because the method — dispersion over a known distance — is the one used again for photons in Section 115.8.2.

Derivation: why the background falls away above \(100\) teraelectronvolts. Derives Phenomenon 115.50. The atmospheric neutrino flux of Section 115.7.2 is not simply a copy of the primary spectrum. A pion of energy above the critical energy \(E_{c}^{\pi}\approx24\,\mathrm{GeV}\) derived in Section 115.4.2, conventionally rounded to \(20\,\mathrm{GeV}\), interacts before it decays, and the probability that it decays instead is the ratio of the two lengths,

falling as \(1/E\). The neutrino flux is therefore the parent spectrum \(E^{-2.7}\) multiplied by \(E^{-1}\):

one power steeper than the cosmic-ray spectrum that makes it. An astrophysical flux produced by shock acceleration, by contrast, carries the source index of Equation (115.47), near \(E^{-2}\) and certainly harder than \(E^{-3}\), and suffers no such suppression because its pions decay in vacuum. Two power laws differing by more than one unit of index must cross, and they cross near \(100\,\mathrm{TeV}\): below it the sky is atmospheric, above it the atmospheric flux has fallen away and an astrophysical component, if it exists, dominates. That is why the IceCube analysis selects events above \(30\,\mathrm{TeV}\) and why the excess is at the high-energy end of the selection.

A second discriminant is available and is used. An atmospheric neutrino is made in an air shower, and the same shower makes muons that arrive at the detector at the same time. Selecting only events whose interaction vertex is inside the instrumented volume, with no entering track, therefore vetoes atmospheric neutrinos arriving from above against their own accompanying muons — the self-veto — while an astrophysical neutrino, which has no accompanying shower, passes. The veto is what reduces the expected background of the Phenomenon 115.50 sample to about ten events.

Derivation: a neutrino detection is evidence of hadronic acceleration. Neutrinos at these energies have exactly one production channel: the weak decay of pions and kaons, which are made only in hadronic collisions. Electrons produce photons — by synchrotron radiation, by bremsstrahlung, by inverse-Compton scattering — and produce no neutrinos at all, because none of those processes has a weak vertex. Detecting a teraelectronvolt neutrino from a direction therefore establishes that protons or nuclei are accelerated there, and resolves the hadronic–leptonic ambiguity of Phenomenon 115.46 in one measurement.

The same chain fixes the expected ratio of neutrinos to gamma rays. In a \(pp\) collision the three pion charge states are produced in roughly equal numbers. The \(\pi^{0}\) gives two photons carrying all its energy; each \(\pi^{\pm}\) gives, through \(\pi^{+}\to\mu^{+}\nu_{\mu}\) and \(\mu^{+}\to e^{+}\nu_{e}\bar{\nu}_{\mu}\), three neutrinos carrying about three quarters of its energy. Hence

a source's all-flavour neutrino luminosity is comparable to its hadronic gamma-ray luminosity, which is what makes a joint detection feasible at all. The flavour composition at the source is \(\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:2:0\) from the same counting, and oscillation over astronomical baselines — where Equation (115.63) gives an enormous \(\Phi\) and every \(\sin^{2}\Phi\) averages to one half — converts this to approximately \(1:1:1\) at Earth. The measured flavour ratio is consistent with that, which is a non-trivial check that the sources are pion sources.

Derives Proposition 115.52. By Equation (40.12) for a projectile of energy \(E\) on a target of mass \(m_{p}\) at rest,

so

in SI \(6.9\times 10^{-5}\,\mathrm{J}\). The square root is the whole story: a fixed-target collision converts energy to centre-of-mass energy only as \(\sqrt{E}\), which is why colliders were built (Example 40.13), and why even a beam seven orders of magnitude above the LHC's buys only a factor of thirty in \(\sqrt{s}\).

Derivation of the bound from the photopion threshold. Redo the threshold calculation of Equation (115.56) with species-dependent limiting speeds. Writing \(\delta:=\delta_{\pi}-\delta_{p}\), the invariant that must reach \(\left(m_{p}+m_{\pi}\right)^{2}c^{4}\) acquires an extra term of order \(\delta E^{2}\), so the threshold condition becomes

For \(\delta>0\) the reaction is forbidden above a maximum energy rather than allowed above a minimum one, and the photopion suppression would not occur at all. Requiring the second term not to dominate the first at the energy where the suppression is observed, \(E\approx5\times 10^{19}\,\mathrm{eV}\), gives

a bound on a fractional velocity difference at the level of \(10^{-23}\) [Coleman:1999]. For comparison, the best terrestrial Michelson–Morley-type experiments — rotating optical cavities, the modern descendants of [Brillet:1979] — bound anisotropies of the speed of light at the level of \(10^{-17}\) [Herrmann:2009]; the cosmic-ray bound is better by six orders of magnitude, and it is obtained not by building anything but by observing that a predicted suppression occurs.

Derivation. Derives Phenomenon 115.55. The PDG quotes the muon width as \(\Gamma_{\mu}=2.9959836(30)\times 10^{-19}\,\mathrm{GeV}\) [Navas:2024], which is a lifetime

and the distance a muon covers in one lifetime at a speed arbitrarily close to \(c\) is \(c\tau=658.6\,\mathrm{m}\). Were the laboratory lifetime the same as the proper one, the surviving fraction over \(15\,\mathrm{km}\) of atmosphere would be

and no muons would reach the ground in any detectable number. What the flight consumes is instead the muon's proper time, shorter than the laboratory time by the factor \(\gamma\) (Experiment: Time Dilation and Relativistic Kinematics), so the laboratory decay length is \(\gamma\beta c\tau\); for a typical sea-level muon of a few gigaelectronvolts, \(\gamma\approx20\) — that is \(E=20\,m_{\mu}c^{2}=2.1\,\mathrm{GeV}\), which is also about the energy a muon needs to survive the \(2\,\mathrm{MeV}\) per \(\mathrm{g}/\mathrm{cm}^{2}\) ionisation loss across the atmosphere of Equation (115.3) — and

The observed flux is of this order. The nine and a half orders of magnitude between the two estimates are what make the sea-level muon flux a continuously running test of relativistic kinematics rather than a curiosity, and the momentum dependence of the surviving fraction is the form in which the test was first made quantitative [Rossi:1941].