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From the Current Literature
COSMIC RAYS AND RADIO EMISSION OF THE GALAXY
- The question of the origin of galactic radio emission is of great astrophysical interest and at the same time remains unclear. Since the discovery of galactic radio emission and until recently it was believed that this emission is the thermal radiation of interstellar electrons having a temperature \(T \cong 10^4^\circ\).
The specific intensity of thermal radio emission is equal to*):
\[ I_\nu=\frac{2kT_{\mathrm{eff}}}{\lambda^2} =\frac{2.76\cdot 10^{-16}T_{\mathrm{eff}}}{\lambda^2\;(\text{in cm})}\, \frac{\mathrm{erg}}{\mathrm{cm}^2\cdot\mathrm{sec}\cdot\mathrm{steradian}\cdot\mathrm{hertz}}, \tag{1} \]
where \(\lambda\) is the wavelength, \(k=1.38\cdot 10^{-16}\), and \(T_{\mathrm{eff}}\) is the effective temperature, equal to \(T_{\mathrm{eff}}=(1-e^{-\tau_\nu})T\), where \(\tau_\nu\) is the optical thickness of the medium (in our case, of the interstellar gas) for the given frequency \(\nu\) in the given direction (for details see, for example, \({}^1\)).
The effective temperature \(T_{\mathrm{eff}}\), obviously, must be less than \(T\) and approach \(T\) only when \(\tau_\nu \gg 1\).
Measurements carried out at wavelengths \(\lambda < 5\) m, in general, did not contradict the assumption of a thermal origin of galactic radio emission, and this hypothesis, as stated, seemed very probable. However, measurements at longer wavelengths \(\lambda \gtrsim 10\) m led to values \(T_{\mathrm{eff}}\sim 10^5{}^\circ\) (see \({}^{1,2}\)). In addition, it turned out that the distribution of the “radio brightness” of the Milky Way is practically independent of frequency. It follows from this that, for all the wavelengths studied, the case \(\tau_\nu \ll 1\) obtains. Indeed, if, as was previously assumed (see \({}^1\)), already for \(\lambda=5\) m in the direction of the Galactic center \(\tau>1\), then for longer waves the optical thickness \(\tau\) would be considerable also in directions strongly deviating from the Galactic center. As a result, the galactic emission reaching the Earth should have been considerably more isotropic at long wavelengths (\(\lambda \ge 10\) m) than at short wavelengths. The absence of such an effect thus indicates that the Galaxy should be considered optically “thin” even for
*) By definition, \(I_\nu\,d\nu\) is the energy flux in the frequency interval \(d\nu\), flowing in the given direction through an area of \(1\ \mathrm{cm}^2\) in 1 sec and referred to unit solid angle.
... waves \(\lambda \gg 10\) m and already long ago for waves with \(\lambda \lesssim 5\) m\(^*\)). But it also follows from this that, if the galactic radio emission is thermal, then \(T \gg 10^9\). Meanwhile, independent astrophysical measurements indicate that, for interstellar electrons, \(T \lesssim 10^4\) (see 3). It follows, therefore, that galactic radiation must be due, at least mainly, to some mechanism different from the thermal radiation of interstellar electrons. The existence of a whole series of discrete sources of galactic radiation testifies to the same thing (see 2 and abstract 3). The total intensity of these sources reaches the value
\[ J_\nu = I_\nu \Delta\Omega \sim 10^{-19}\, \frac{\text{erg}}{\text{cm}^2 \cdot \text{sec} \cdot \text{sterad}}, \tag{2} \]
where \(\Delta\Omega\) is the solid angle under which the source is seen. From experience it is known only that the angular size of the source is less than \(6' \div 8'\), i.e. \(\Delta\Omega < 3 \cdot 10^{-6}\) steradian. If one assumes that the size of a discrete source is \(< 6'\) (this could be, for example, if the role of the source were played by some nebula), then the effective temperature of the source corresponding to the value (2) reaches \(\sim 5 \cdot 10^6\). If, however, the source of the radia-
[[unclear: several damaged lines in the left margin only; the main text is missing]]
Below we shall examine the question of the connection of galactic radio emission with cosmic rays, dwelling on note 7.
- As is known, a particle with charge \(e\) and mass \(m\), moving in a magnetic field \(H\) perpendicular to its velocity, describes a circle with angular velocity (frequency)
\[ \omega_0 = 2\pi\nu_0 = \frac{eH}{mc} \cdot \frac{mc^2}{E}, \]
where \(E\) is the energy of the particle and \(c\) is the speed of light. If \(E \ll mc^2\), then the particle, by virtue of the imparted—
\(^*\) The value \(\tau = 1\) for \(\lambda = 3\) m over a path of \(6 \cdot 10^{22}\) cm (the radius of the Galaxy) is obtained\(^1\) for \(T = 10^9\) and an interstellar-electron concentration \(N = 1\). Since \(\tau\) is proportional to
\[ \frac{N^2}{T^{3/2}}, \]
it is clear that already a slight decrease in the mean concentration \(N\) is sufficient for the inequality \(\tau \ll 1\) to be observed at \(\lambda = 3\) m.
... by the acceleration field radiates practically only electromagnetic waves with frequency \(\omega_0\). If, however, \(E \gg mc^2\), then it radiates an entire spectrum of frequencies that are overtones of the frequency \(\omega_0\). This spectrum was investigated in a number of works in different forms by various authors, and, in the form most convenient for our purposes, in \({}^{10}\). The energy radiated in 1 sec. in a unit spectral interval is equal to:
\[ \begin{gathered} P(\nu)=2\pi P(\omega)=16\,\frac{e^3H}{mc^2}\,p\!\left(\frac{\omega}{\omega_1}\right). \\[6pt] \text{For }\frac{\omega}{\omega_1}\ll 1 \\[4pt] p\!\left(\frac{\omega}{\omega_1}\right)=0.256\left(\frac{\omega}{\omega_1}\right)^{1/3}; \\[6pt] \text{for }\frac{\omega}{\omega_1}\gg 1 \\[4pt] p\!\left(\frac{\omega}{\omega_1}\right)=\frac{1}{16}\left(\pi\frac{\omega}{\omega_1}\right)^{1/2} e^{-\frac{2\omega}{3\omega_1}}, \end{gathered} \tag{3} \]
where
\[ \omega_1=\frac{eH}{mc}\left(\frac{E}{mc^2}\right)^2,\qquad \nu=\frac{\omega}{2\pi} \]
is the radiation frequency, and \(H\) is the projection of the magnetic field perpendicular to the velocity of the particle. In the interval
\[ 0.18<\frac{\omega}{\omega_1}<16 \]
the function \(p\) has been tabulated in \({}^{10}\), and it is maximal at \(\frac{\omega}{\omega_1}=0.5\), with \(p(0.5)=0.10\), \(p(0.18)=0.09\), \(p(2)=0.055\), \(p(4.05)=0.018\), \(p(6.5)=0.004\), \(p(10.4)=4\cdot10^{-4}\), and \(p(16)=10^{-5}\), where all figures are rounded, since an accuracy exceeding several percent is not needed by us. Outside the indicated interval, the limiting formulas given in (3) may be used with the same accuracy. The graph of the function
\[ p\!\left(\frac{\omega}{\omega_1}\right) \]
is presented in the figure.
At the maximum,
\[ \left\{ \begin{aligned} P(\nu_{\max})&=1.6\,\frac{e^3H}{mc^2} =2.15\cdot10^{-22}H\ \frac{\mathrm{erg}}{\mathrm{sec}\cdot\mathrm{cps}},\\[4pt] \nu_{\max}&=0.5\,\frac{\omega_1}{2\pi} =1.4\cdot10^6 H\left(\frac{E}{mc^2}\right)^2\ \mathrm{cps}, \end{aligned} \right. \tag{4} \]
where, in passing to numerical values, it is assumed that the radiating particle is an electron. From what follows it is easy to see that the assumption that protons radiate appears unrealistic.
- Let us first consider the general galactic radio emission, i.e., the radiation arising in interstellar space.
The specific intensity of this radiation in a given direction is equal to \(I_\nu=\dfrac{1}{4\pi}\int P(\nu)N\,dx\), where \(x\) is the distance from the point of observation (i.e., from the Earth) and \(N\) is the concentration of radiating particles (more precisely, \(P(\nu)N=\int P(\nu,E)N_1(E)\,dE\), where \(N_1\,dE\) is the concentration of particles in the energy interval \(dE\)). In addition, it is assumed that the radio emission of a volume element is on the average isotropic, owing to the isotropy of cosmic radiation and the chaotic character in the direction of the field \(H\) along the line of sight. Finally, by not introducing into the expression for \(I_\nu\) a factor allowing for absorption, we have already taken into account that the Galaxy is optically “thin” (see above).
The quantity \(I_\nu\) can be written in the form
\[ I_\nu=\frac{P(\nu)}{4\pi}\cdot N\cdot R, \tag{5} \]
where \(P(\nu)\) and \(N\) are averaged over the path and over the energies of the cosmic particles, and \(R\) is the size of the system in the given direction. The value of \(I_\nu\) will evidently be maximal if the given frequency \(\nu\) is the frequency \(\nu_{\max}\) in (4). In this case
\[ \left. \begin{aligned} I_{\nu,\max} &= 1.7\cdot 10^{-23}\,H\cdot NR\, \frac{\mathrm{erg}}{\mathrm{sec}\cdot\mathrm{cm}^{2}\cdot\mathrm{sterad}\cdot\mathrm{cps}},\\ \nu &= 1.4\cdot 10^{6}\,H\left(\frac{E}{mc^{2}}\right)^{2}. \end{aligned} \right\} \tag{6} \]
It has been established experimentally that for \(\lambda=\dfrac{c}{\nu}\ll 15\ \mathrm{m}\) the intensity \(I_\nu\) decreases with increasing frequency. Assuming, for example, that the radiation is maximal at a wavelength of \(20\ \mathrm{m}\), from (6) we find that \(H\left(\dfrac{E}{mc^{2}}\right)^{2}\simeq 10\), i.e., that for \(H\sim 10^{-6}\ \mathrm{oersted}\), \(E\sim 10^{9}\ \mathrm{eV}\). Comparing (6) and (1), we see that for \(T_{\mathrm{eff}}=10^{5}\), \(\lambda=10^{3}\ \mathrm{cm}\), and \(R=10^{23}\ \mathrm{cm}\) \(\left(\dfrac{1}{10}\right.\) of the diameter of the Galaxy\(\left.\right)\), in order that the intensities calculated from formulas (1) and (6) be equal, it is necessary that
\[ HN\simeq 10^{-16}. \tag{7} \]
Hence, for \(H\sim 10^{-6}\ \mathrm{oersted}\), \(N\sim 10^{-10}\ \mathrm{cm}^{-3}\). As is clear from the character of the estimate given, it gives the minimum admissible value of \(HN\).
At the boundary of the Earth’s atmosphere, in the vertical direction, the intensity of cosmic rays (mainly protons) is \(I_k\simeq 0.12\,\dfrac{\text{particles}}{\mathrm{cm}^{2}\cdot\mathrm{sec}\cdot\mathrm{sterad}}\), whence, for isotropy, \(N=\dfrac{4\pi}{c}I_k\sim 5\cdot 10^{-11}\ \mathrm{cm}^{-3}\). The number of electrons incident on the Earth is at least 100 times smaller, but in interstellar space the value \(N\sim 10^{-10}\ \mathrm{cm}^{-3}\) for electrons with \(E\sim 10^{9}\ \mathrm{eV}\) is not inadmissible.
Thus, the observed intensity of the general galactic radiation may be explained by the radiation of cosmic electrons
if only some independent considerations do not compel one to doubt the values \(H\sim 10^{-6}\) oersted along the path \(R\sim 10^{22}\ \mathrm{cm}\) at \(N\sim 10^{-3}\ \mathrm{cm}^{-3}\).
As stated above, consideration of a number of other sources of radio emission shows that they cannot explain the observed picture of the sky. Therefore at the present time, apart from the mechanism under discussion for general galactic radio emission, no other sources of it are apparent. The other possibility amounts, in essence, to denying the very existence of sources of radiation in interstellar space and to reducing the general galactic radio emission to the radio emission of discrete sources associated with stars.
4. The origin of the radio emission of discrete sources is also still unclear and in \({}^{5}\) is connected with the mechanism, considered here, of radiation by relativistic electrons in the magnetic field surrounding a star. If the radiation of the source is bremsstrahlung radiation of fast electrons in the circumstellar magnetic field, then under the most favorable conditions, i.e. using (6), we have:
\[ J_{\nu}\approx \frac{P(\nu)}{4\pi}\,N\Delta\Omega D =1.7\cdot 10^{-23}\,HN\Delta\Omega D, \tag{8} \]
where \(D\) is the diameter of the source.
Taking into account (2), putting \(\Delta\Omega=3\cdot 10^{-6}\) steradian and the distance to the source equal to \(R=5\cdot 10^{22}\ \mathrm{cm}\) (whence \(D=R\cdot 6' \sim 10^{20}\)), we obtain:
\[ (HN)_{\min}\sim 10^{-11}, \tag{9} \]
whence even for \(H\sim 10^{-4}\) oersted \(N_{\min}\sim 10^{-7}\ \mathrm{cm}^{-3}\). If, however, as in \({}^{5}\), one takes \(D\simeq 0.1\) light-year \(\simeq 10^{17}\ \mathrm{cm}\), then for \(H\sim 10^{-4}\) oersted \(N_{\min}\sim 10^{-4}\ \mathrm{cm}^{-3}\), i.e. a concentration of electrons is required approximately \(10^{6}\) times greater than the concentration of all cosmic particles near the Earth. Hence it is clear that the discrete radio-emission sources considered in \({}^{5}\) cannot in any way be similar to our “solar cosmic-ray system” with \(D\sim 10^{17}\), \(H\sim 10^{-5}\) oersted, \(N_{\text{electrons}}<5\cdot 10^{-13}\ \mathrm{cm}^{-3}\), \(N_{\text{protons}}\sim 5\cdot 10^{-11}\ \mathrm{cm}^{-3}\); see \({}^{9}\).
From the point of view of checking the correctness of the mechanism under discussion for the origin of cosmic rays, it is especially important to determine the spectrum of the general galactic radiation, primarily at \(\lambda>10\ \mathrm{m}\), and also the spectrum of radio emission and the sizes of discrete sources (the determination of the sizes is especially difficult, but known possibilities in this direction exist \({}^{12}\)).
In connection with the material discussed, the question naturally arises of the radio emission of cosmic rays in the Earth’s magnetic field \(H\sim 0.5\) oersted. It is easy to see that the radiation of an individual particle in this case is too weak for it to be observable (the total radiated energy is, in order of magnitude, no more than \(P(\nu_{\max})\dfrac{\rho}{c}\simeq\)
\[ \simeq 10^{-24}\,\frac{eH^{2}}{2ev}, \]
where \(\rho\) is the radius of the Earth). The radio emission arising in the braking and birth of cosmic particles is likewise too weak (see \({}^{7}\)). However, both these mechanisms of radio emission may in principle prove significant for explaining the radio emission of the sky and the radio emission associated with auroras (for more detail see \({}^{7}\)).
Н. И. Г.
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