RADIATION, HIGH-ENERGY INTERACTION WITH MATTER DEMOSTHENESS KAZANAS Our knowledge and understanding of astrophysical phenomena is attained by observation and deepened by modeling and interpretation of the observational data. Led by observation, astronomers have come to the realization that certain astrophysical phenomena and objects involve the interaction among particles and photons at high energies. These high-energy particles have either been measured directly (e.g., in cosmic rays and solar flares) or their presence has been inferred from their observed photon emission (as in quasars, extragalactic jets, accreting black holes, and neutron stars in our galaxy). Therefore, the detailed knowledge of the high-energy interactions between photons and matter is an indispensable tool in order to successfully model these sources. The qualification of an interaction as "high energy" requires a qualifier (a scale) with respect to which the interactions will be considered as "high energy." This scale is determined by the fundamentals of each interaction. Thus for the interactions between electrons and photons (the electromagnetic interactions), this scale is the energy corresponding to the electron rest mass, m***, where c is the velocity of light. For the interactions between protons or protons and photons (hadronic interactions), this scale is the rest energy corresponding to the mass of the pion (m***), the particle responsible for transmitting the nuclear force. Interactions are therefore termed "high energy" when the energy of at least one of the participating particles is larger than its characteristic energy scale. In practice, however, interactions involving particles (or photons) of energies greater than m*** are termed "high-energy" ones. The great depth and breadth of the subject do not allow a detailed, comprehensive study of all these processes in the limited space of this entry. However, their general characteristics (cross sections, range of validity, etc.) can be obtained in a qualitative fashion by general considerations of their kinematics, classical electrodynamics, and quantum mechanics. Similar general considerations can also be applied to obtain the energy loss rates for particles associated with each such process; these loss rates are of great importance in modeling and are given in a separate section. The division of the interactions into electromagnetic and hadronic (strong) is a natural one and the two sets are discussed separately. Also, because electromagnetic interactions are better understood and more prevalent, they are discussed in greater detail than hadronic ones. ELECTROMAGNETIC INTERACTIONS The electromagnetic interactions are by far the best understood interactions in physics to date. Given the fact that there exists a quantum theory of the electromagnetic field in the form of a perturbation theory which is convergent and finite (renormalizable), one can in fact calculate all the relevant quantities of the interaction (i.e., cross section, energy loss, etc.) from first principles; the agreement of the theory with the experimental findings has so far been excellent. We will discuss individually the electron-photon, electron-magnetic field, photon-photon, and bremsstrahlung interactions. The case of photon-electron scattering is used to introduce the general notions involved and is therefore presented in greater detail. ELECTRON-PHOTON INTERACTIONS One of the most common processes in high-energy astrophysics is the collision of a relativistic electron of Lorentz factor * (i.e., of energy *****) with a photon of energy *, more commonly known as inverse Compton (IC) scattering. In the center of momentum (CM) frame (this is the electron rest frame if ******), which moves with Lorentz factor * relative to the lab, the photon energy ***** (where h is Planck's constant and *** is the frequency) is * times larger than seen in the lab (i.e., **); in addition, the collision in this frame is almost elastic (the electron preserves its energy, **, during the collision). Transformed to the lab frame, the photon energy after the collision appears larger by yet another factor *; that is, the scattered photon energy in the lab is ***. This energy gain is hence the result of the two successive Lorentz transformations and the elastic scattering in the center of momentum. This energy is much smaller than the total energy of the electron *****, So that an electron has to suffer a large number collisions before it loses a, sizable fraction of its energy. The cross section for this interaction can be calculated classically by considering the electron as a free point-like charge that responds to an external electric field according to the laws of classical electrodynamics. This cross section is roughly the square of the classical electron radius r**=(**/****)* (the only length scale in the classical theory; * is the charge on the electron); it is the so-called Thomson cross section, and its precise value is **=(8**/3)r**=6.65x10** cm**. As the energy of the photon or the electron increases, a break-down of the preceding arguments is expected. Kinematic modifications are needed when ***=****, or when the photon energy in the center of momentum frame ***=** approaches ****. The CM frame is no longer the electron rest frame and the collision is not elastic in this frame. The electron recoil is significant and the photons lose energy in these collisions. As seen in the lab frame, the electrons lose a large fraction of their energy in a single collision. In addition, quantum mechanical effects become important when the wavelength of the photon in the CM frame becomes comparable to the Compton wavelength of the electron **=*******. The electron can no longer behave as a single, structureless, point particle in interactions with radiation; according to quantum theory there exists a "cloud" of virtual particles around the electron of size =**, whose constituents are "felt" by the photon when its wavelength becomes of order **. The response of the electron to the electric field of the incident radiation is no longer coherent; the different parts of the electron interfere destructively with each other, causing a decrease of the cross section. The corresponding photon energy is ************************; that is, the quantum mechanical effects become important at the same energy as the electron recoil effects. This regime in the energetics of the collisions, in which both quantum and electron recoil effects become important, is the so-called Klein-Nishina regime and the cross section for interaction drops substantially from that of the Thomson value. ELECTRON-MAGNETIC FIELD INTERACTIONS Bearing in mind that a magnetic field of strength B can be considered as a collection of virtual photons of energy, ****(******,**** is the Larmor frequency of the electron), the interactions of relativistic electrons with the magnetic field (synchrotron radiation) are qualitatively very similar to their interactions with real photons as described above (IC scattering). All the arguments of the collision kinematics and the onset of the Klein-Nishina modifications in the cross section are qualitatively applicable to this process, though the specific details are quite different. There is, however, an important distinction: Synchrotron radiation converts "virtual" photons into "real" ones and hence manifests itself as a photon-producing process, whereas IC involves the scattering of "real" photons and preserves their number. PHOTON-PHOTON INTERACTIONS This is a purely quantum mechanical process in which a collision between two photons of energies ** and ** yields an electron-positron pair. This process does not exist within the framework of classical electrodynamics, because this theory, being linear, does not allow for interactions between photons. However, quantum mechanics (and the uncertainty principle) allow for a "cloud" of virtual ***** (electron-positron) pairs around each photon; it thus becomes possible for the other photon to "knock off" such a virtual pair, thus allowing the interaction between photons. In the simplest case both photons disappear producing an electron-positron pair. Because of energy conservation, the energies of the two photons in the CM frame ****(**/**)*****=(**/**)*****=(****)** must be at least equal to **** for the process to occur (threshold condition). At threshold, the cross section is roughly that of electron-photon scattering, that is, the Thomson cross section, whereas as ***(****)******** the process moves into the Klein-Nishina regime and the cross section drops accordingly. This process, which allows for the absorption of a=*** MeV gamma ray by a 30-50 keV x-ray, is the main source of opacity for gamma rays in accreting compact objects, as discussed below. BREMSSTRAHLUNG This process consists of the emission of a photon in the Coulomb scattering of two charges (in most astrophysical applications this occurs between an electron of charge * and a nucleus of charge **). In analogy with synchrotron radiation, bremsstrahlung can be considered as the scattering of a high-energy electron by the virtual photons that make up the electrostatic field of the nucleus. As with synchrotron radiation, because this scattering converts a virtual photon into a real one, bremsstrahlung is also a photon-producing process. The relevant cross section is therefore the Thomson one, **, multiplied by the probability ** of emitting a photon of energy * within an interval ** (or the virtual photon spectrum). This probability is dN=(****/****/*), where *=**/**=1/137 is the fine-structure constant and *=*/2**. The effective cross section is therefore * times smaller than **, and despite the apparent 1/* divergence in the number of photons the total emission is finite, because the energies of the emitted photons correspondingly decrease. In a large number of astrophysical applications, one usually considers bremsstrahlung emission from a thermal electron distribution. The resulting differential photon spectrum diverges at low energies, as argued previously and cuts off exponentially for energies much larger than the gas temperature, reflecting the cutoff in the electron distribution. STRONG (HADRONIC) INTERACTIONS Our lack of a quantum theory of the strong interactions as complete and successful as that of quantum electrodynamics does not allow the a priori calculation of interactions between strongly interacting particles. However, the fact that the nuclear force is mediated by the exchange of pions does provide a scale to the theory, namely the Compton wavelength of the pion, ************* cm. An estimate for the cross section of order ***** is in good agreement with the experimental value of 3x10** cm*. For strong interactions, high-energy collisions are hence those with CM energies larger than the rest energy corresponding to the mass of the pion, ****. The result of such collisions is the copious production of pions (*********), which, however, are very short lived and upon their decay produce relativistic electrons and neutrinos (from the decay of ******) or photons (from the decay of **). Because the resulting electrons and photons interact electromagnetically, it is very difficult to trace their origin to hadrons. In the absence of detectors capable of observing these neutrinos from pion decay, the presence of strong interactions in astrophysics can be unequivocally inferred only in rare occasions (e.g., interactions of high-energy cosmic rays in the atmosphere). Interactions of photons (of energy *) with high-energy protons (of Lorentz factor **), resulting in pion production, are also possible when allowed kinematically, that is, ***** ****=140 MeV. However, these require protons of extremely high energies [**********)*****=10**-10** eV for **keV], whose presence has been indicated in only a small number of sources. ENERGY LOSS RATES-ASTROPHYSICAL APPLICATIONS The apparently nonthermal character of observed radiation from most high-energy sources precludes the use of thermal particle distributions in modeling their photon emission. The determination of the emitting particle distributions requires, in general, the solution of the kinetic equation. In its simplest form this is the continuity equation in energy space **********************************, where * is the unknown distribution, Q(*,*) is the high-energy particle injection rate which is determined by the dynamics and the acceleration mechanism, and * is the total particle energy loss rate. In most cases one is interested in the particle distribution * under steady-state conditions (*******), for a given injection Q(*). Under these conditions, the energy loss rate * is the single most important quantity in determining * and hence the emitted photon spectrum. The loss rate can be estimated by multiplying the energy emitted per collision with the number of collisions a particle suffers per unit time, ***** (*** is the number density of scatterers, * is the cross section for the particular interaction, and * is the speed of light). For inverse Compton scattering *** is the number density of photons n(*)** in an energy interval **, * is the Thomson cross section, and the energy loss per scattering is ***. The total energy loss rate can be estimated by integrating over the photon distribution (assuming that Klein-Nishina effects do not become important), that is, *******************************. The integral ****** ** is equal to the total energy density in radiation **** which for a source of luminosity L and size R is equal to ******. Owing to the analogy of synchrotron radiation to inverse Compton scattering, the energy loss for synchrotron radiation has precisely the same form with the energy density of the radiation **** replaced by the energy density in the magnetic field ********. For bremsstrahlung the number density of scatterers is equal to the number density of the ambient particles *, and the cross section is roughly **, whereas the energy per scattering is equal to the energy of the emerging photon *, integrated over virtual photon spectrum **** that is, ******************************. The energy loss rate is proportional to the energy of the electron, because the energy of the emitted photons * extends up to the energy of the electron ****. The need to use the kinetic equation for calculating the function *, rather than simply using the injection Q(*), can be assessed by comparing the electron loss time scales (**********) with the fastest possible dynamical time, namely the light-crossing time ****** across the source. If ******* the source cannot be considered to be in a quasistatic state and solution of the kinetic equation is in order. (In cases where more than one loss process is involved, one should consider the dominant process, i.e., the one with the shortest loss time scale.) The dominant loss process depends on the conditions encountered in the particular astrophysical source. As is apparent from the previous expressions, bremsstrahlung dominates in sources of high particle densities, whereas inverse Compton and synchrotron dominate in sources of high photon densities and magnetic fields. However, even in the same source, due to their different energy dependence, alternate processes may dominate in different energy regimes. Inverse Compton and synchrotron losses dominate in sources such as active galaxies, quasars, and extragalactic jets, whereas bremsstrahlung is important in high-energy events such as solar flares, where the particle densities are high relative to those of photons or magnetic fields. The physical parameters of the sources cannot in general be deduced from observation, and thus it is not possible to determine the importance of the various radiation loss mechanisms. This is possible, however, for IC losses because the **** can be estimated from the observed luminosity L and the size of the source R, which can be inferred from variability measurements. The requirement that the light-crossing time be longer than the IC loss time scale (*******) provides a relation (************************ erg s** cm**) which can determine the dynamical importance of IC losses. So when the ratio L/R (called the compactness of the source) is greater than =1.15x10** erg s ** cm**, IC loss effects are important even for subrelativistic (*=1) electrons. Another important process that can be estimated easily in terms of measurable source parameters is the photon opacity to pair production, ***. If the source luminosity in gamma rays is L, the number of photons of energy ********* is ********* and the *-* opacity is ************************. This opacity, like the ratio ***** of light crossing to IC loss times, depends only on the ratio L/R, that is, the compactness of the source. It is roughly unity when L/R=1.15x10** erg s** cm** (i.e., the value at which ****** for subrelativistic electrons) and in most sources it increases with the energy of the gamma ray (more precisely it depends on the spectrum of photons at ********). Therefore, a source with L/R********** erg s** cm** will convert all its high-energy radiation into **** pairs which will also have to be incorporated into the kinetic equation when modeling these sources. The compactness parameter is of paramount importance for modeling high-energy sources in which IC scattering is the dominant loss mechanism, such as accreting compact sources (black holes and neutron stars). For these sources, in fact, L/R turns out to be roughly independent of the mass of the accreting object, because both their luminosity and size scale linearly with its mass M. The luminosity is believed to be a fraction, *********, of the Eddington luminosity [***************(M/M*) erg s**], whereas the radius is a multiple, x=10, of the Schwarzschild radius [***********(M/M*) cm], so that L/R=4x10** (F/x)=14x10**-4x10** erg s** cm**, independent of the mass. Moreover, these values for the compactness are larger than the critical one for which *******. It is therefore expected that these sources will be pair dominated and will have suppressed gamma ray emission. It is somehow ironic that in the most luminous high-energy sources, the high-energy (****) radiation remains hidden, precisely because of their high luminosity! Additional Reading Blumenthal, G.R. and Gould, R.J.(1970). Bremsstrahlung, synchrotron radiation, and Compton scattering of high-energy electrons traversing dilute gases. Rev. Mod. Phys. 42 (No. 2) 237. Gaisser, T.K.(1991). Cosmic Rays and Particle Physics. Cambridge University Press, Cambridge, U.K. Jackson, J.D.(1962). Classical Electrodynamics. Wiley, New York. Jauch, J.M. and Rohlich, F.(1980). The Theory of Photons and Electrons. Springer-Verlag, New York.