INTERSTELLAR POLARIZATION OF LIGHT

S. B. Pikel’ner

1. INTRODUCTION

In recent years the great role that magnetic fields must play in cosmic physics has become clear. This role is due to the comparatively large value of the conductivity $\sigma$ (of order $10^{12}$ CGSE) and the very large dimensions of gas masses, which makes the interstellar medium similar to a superconducting liquid. The relaxation time of the magnetic flux in a conductor of size $R$ is, in order of magnitude,$^{1}$

\[ t \simeq \frac{\sigma R^{2}}{c^{2}} \tag{1.1} \]

and under the conditions of interstellar gas it exceeds $10^{12}$ years. Taking account of the mobility of the medium, as A. Ya. Kipper$^{2}$ has shown, considerably reduces the time of field decay; nevertheless it remains greater than the lifetime of a gas cloud.

The combination of superconductivity with the mobility of the interstellar medium creates very specific properties of it, which are studied by the recently emerged branch of electrodynamics and hydrodynamics—magnetohydrodynamics.$^{3,4,5,6}$ These properties are sometimes described visually as the “frozen-in” character of magnetic lines of force in matter, expressed in the fact that a segment of a line of force always passes through one and the same gas element. The motion of the gas as it were deforms the lines of force (the motion of a conductor in a magnetic field induces currents whose field, combining with the original one, changes it in the manner described above), and the character of the deformation depends on the ratio of the densities of magnetic $\frac{1}{8\pi}H^{2}$ and kinetic $\frac{1}{2}\rho v^{2}$ energy. If $\frac{1}{8\pi}H^{2} \ll \frac{1}{2}\rho v^{2}$, then the motion of the matter occurs just as without a magnetic field; the latter

follows the motion of the medium and thereby becomes “tangled.” If, however, the motion occurs, for example, in a uniform field with

\[ \frac{1}{8\pi}H^2 \gg \frac{1}{2}\rho v^2, \]

then even small deformations of the field create magnetic forces that hinder the further advance of the matter.

The motion of an incompressible fluid in a uniform field of arbitrary intensity \(H_0\) was considered by Alfvén\(^3\), who showed that a certain state of motion must be transported along \(H_0\) with velocity

\[ V=\frac{H_0}{\sqrt{4\pi\rho}} \]

(a magnetohydrodynamic wave). In the presence of turbulent motions the field will be tangled by chaotic motions and thereby amplified until its energy becomes comparable with the energy of small-scale motions.\(^4\) In this process the strength of the initial field from which the amplification begins is immaterial, and, since small fields can always arise through diffusion under the action of density and temperature gradients, the presence of a chaotic magnetic field in the interstellar medium is very probable. The formation of a regular field is harder to imagine.

The final answer to the question of the presence and character of the magnetic field in the interstellar medium can be given only by observations. At present there is no direct proof of the existence of a magnetic field in the Galaxy, since its strength is too small to give a measurable Zeeman effect. However, there are several indirect proofs.

The first was put forward by Richtmyer and Teller\(^7\), who proceeded from the observed isotropy and energy density of cosmic rays. If there were no magnetic field and the particles moved rectilinearly, then their isotropy at the surface of the Earth would testify to isotropy in a volume much larger than our Galaxy, and the energy density of cosmic rays in the Metagalaxy would be equal to their density near the Earth, outside its magnetic field. The latter quantity is four orders of magnitude greater than the mean energy density of radiation in the Metagalaxy; i.e., for the formation of cosmic rays, sources would be needed four orders of magnitude more powerful than the light sources of the stars. Since such sources are not known at present, one must assume that cosmic rays are confined within the Galaxy by a magnetic field. This ensures the isotropy of cosmic rays and removes the energetic difficulties, since the power of the sources necessary to compensate for their losses in collisions with the nuclei of atoms of the interstellar gas proves to be several orders of magnitude smaller than the energy emitted by the stars of the Galaxy in the form of light. The radius of curvature of the trajectory of a particle with energy \(E\) ev in a field \(H\) is numerically equal to \(r=E/300H\) cm. For the confinement of cosmic rays, \(r\) must be less than the mean size of the Galaxy; hence Alfvén obtained the estimate \(H \gg 10^{-8}\) er-

stellar medium^3. A more detailed analysis of the confinement conditions led^8 to the conclusion that there exists in the Galaxy an almost spherical subsystem of rarefied gas (in comparison with the mean density near the plane of the Galaxy) with a large velocity dispersion. This gas carries chaotic magnetic fields of strength \(H \simeq 3 \cdot 10^{-6} \div 10^{-5}\), which can confine cosmic rays within the Galaxy.

The second indirect proof of the presence of a magnetic field in the Galaxy is provided by observations of radio emission at meter wavelengths. Since there are detailed reviews^9, ^10, we shall not dwell on this question here; we note only that the radio emission, to a considerable extent, is bremsstrahlung radiation of relativistic electrons and indicates the presence of a field \(H \simeq 10^{-5}\), while it occupies in the Galaxy a region approximately coinciding with the spherical subsystem of rarefied gas.

The third proof is based on the observed phenomenon of polarization of starlight, to which the present review is devoted. As will be seen below, this phenomenon makes it possible to determine not only the field strength, but also its configuration on the average over distances of several hundred parsecs.

The fourth method for investigating the field, proposed by G. A. Shajn^11, is based on the fact that very elongated nebulae, bright and dark, exist, while the directions of elongation of nearby nebulae differ little and on the average tend to be situated in the plane of the Galaxy. The elongated nebulae have acquired their form, evidently, because of the presence of an almost homogeneous field with a strength sufficient to restrain their expansion (as a result of thermal and macroscopic motions) in directions not coinciding with the direction of the field.

This method has the advantage that it gives not averaged, but local characteristics of the field, pertaining to a comparatively small volume. The existence of an almost homogeneous field does not, generally speaking, follow from the general properties of a conducting medium and poses a rather difficult problem for its explanation. However, its existence is confirmed by the polarization of starlight.

2. OBSERVATIONAL DATA

It was first noticed by Hiltner^12 and Hall^13 in 1949 and independently by V. A. Dombrovsky^14 in 1950 that the light of some stars is polarized. The polarization is different for different stars, and its maximum value reaches 8–10%. The polarization is noticeable, mainly, in sufficiently distant stars, which must be absolutely bright for them to be studied. These are chiefly hot stars and cooler giants. The degree of polarization, within the accuracy of the measurements^1, does not depend on wavelength up to

Fig. 1.

Fig. 2.

$\lambda = 8000\,\text{\AA}$ ^15*). The plane of oscillation of the electric vector is close to the plane of the galactic equator; however, this effect depends on galactic longitude. It is most strongly expressed in directions where the line of sight crosses the spiral arm at a large angle, and least of all in directions where the line of sight skims along the arm. This is clearly seen in Figs. 1 and 2, borrowed from ^15; the axis

Fig. 3.

of abscissae is parallel to the galactic equator; the length of the segment corresponds to the degree of polarization of the given star, and the direction coincides with the preferred direction of oscillation of the electric vector. Fig. 2 refers to the constellation Cygnus, where we are looking along the arm; there is no preferred orientation here. Figs. 1 and 3, borrowed from ^16, show good agreement in orientation. However, for different stars it is somewhat different;

*) The latest data show a small (up to 20%) decrease in polarization in the red part of the spectrum compared with its blue part.

according to the estimate of Chandrasekhar and Fermi17 the dispersion is about \(10^\circ\), and according to a more accurate estimate18—about \(7^\circ\). In the same work it is noted that there exists a statistical dependence between the dispersion of directions and the magnitude of the polarization in the given region; on the average they are inversely proportional: the greater the polarization, the smaller the dispersion of directions. The presence of polarization only in the more distant stars and the connection between the orientations of the polarization vectors of different stars immediately led to the hypothesis that the polarization is caused by

Fig. 4.

the conditions under which light passes through the interstellar medium. Light, passing through this medium, undergoes absorption by dust particles having sizes of the order of \(10^{-5}\)—\(10^{-4}\) cm. More will be said below about the properties of dust. If the polarization is caused by unequal absorption of light with differently oriented electric vectors, then there must exist a dependence between the polarization and the reddening of the star associated with the amount of absorption. Such a comparison has been made by several authors. Fig. 4 shows the relation between polarization and absorption in visual rays for stars located in the direction of the center of the Galaxy19. The dependence is undoubtedly present, but the scatter is considerable. At the same time, from consideration of more extensive material it was concluded that the relation is not reversible, i.e. absorption is necessary for the presence of polarization, but not all stars with significant absorption exhibit noticeable polarization. The dependence becomes more definite if one selects stars located within a small area15 (Fig. 5). Hogg20 investigated the polarization of stars in five so-called open star clusters. It turned out that over an area with a diameter of \(10'\) the dispersion in the directions of polarization is small. Polari-

...zation and absorption for individual stars are somewhat different, but the ratio of polarization to color excess \(E\) in the international system (characterizing the reddening of the star and proportional to absorption) is almost constant for the stars of each cluster and varies depending on the galactic longitude of the cluster. This dependence may be represented in the form

\[ \frac{p}{E}=a\{1-|\cos(l-l_s)|\}, \tag{2.1} \]

where \(p\) is the degree of polarization, \(a=10.8\%\), and \(l_s=35^\circ\). The value \(l=35^\circ\) corresponds to the direction of the spiral arm in the vicinity of the Sun (the constellation Cygnus); here the polarization is smallest and its direction is random.

All the indicated facts—the dependence of polarization on absorption, its connection with galactic longitude, the connection of the orientation with the galactic

Fig. 5.

equator—make the hypothesis of the interstellar origin of polarization almost beyond doubt. Qualitatively, the observational data can be explained if it is assumed that the dust grains have an elongated shape (light with the electric vector directed along the long axis is absorbed more strongly than perpendicularly polarized light), their long axes are oriented approximately perpendicular to the spiral arm, and in individual dust clouds the orientation is often the same, while different clouds have somewhat different polarization. If light passes through several clouds with different

oriented dust grains, the polarization produced will be insignificant, whereas the absorption will be large. This explains the one-sided relation between the indicated quantities. The scatter of polarization directions in the constellation Cygnus is explained by the fact that the long axes of the grains lie almost in the picture plane, and small fluctuations strongly change the direction of the greatest inclination to this plane. In addition, the polarization in this constellation is in general small and depends only weakly on absorption.

V. A. Dombrovskii^16 believes that polarization is inherent in the stars themselves and is not of interstellar origin. He bases this on the difference in the polarization of some nearby stars in the constellation Perseus (Fig. 3), on the absence of an unambiguous dependence between polarization and absorption, and on the presence of a weak dependence between the absolute brightness of a star and polarization. However, these facts can be explained by the clumpy structure of the dust clouds, which is also visible in direct photographs of them, and by small systematic errors of observation. In any case, the rejection of an interstellar origin of polarization creates incomparably greater difficulties for explaining it.

3. PROPERTIES OF INTERSTELLAR GAS AND DUST

The first data on the presence of gas in interstellar space (apart from nebulae with bright lines in their spectra) were obtained from observations of the absorption lines Ca II and Na I, present in the spectra of almost all stars of classes O and B (temperature above \(20\,000^\circ\)). In cooler stars these lines are masked by the star’s own lines. In observations with high dispersion, the interstellar lines usually split into several components with different velocities.^21 Analysis of this fact led to the conclusion that the gas is not distributed uniformly, but forms clouds with concentrations of the order of \(10\)—\(20\) atoms · \(cm^{-3}\). The clouds participate in the rotation of the Galaxy and, in addition, move chaotically; moreover, the velocity dispersion depends on the density and size of the cloud: more massive clouds, which give strong lines, have a dispersion of the order of \(10\ km/sec\), while less massive ones have a dispersion of the order of \(35\ km/sec\). The sizes of the clouds are of the order of 10 parsecs (1 parsec equals \(3.08 \cdot 10^{18}\ cm\)); along the line of sight in the plane of the Galaxy there are 8—10 of them per 1000 parsecs. Thus, clouds occupy about 5% of the volume near the plane of the Galaxy, and the mean gas concentration there is about one atom per \(1\ cm^3\). The small velocity dispersion of most clouds leads to their being concentrated in a thin layer (half-thickness 100 parsecs) near the plane of the Galaxy. The width of the absorption lines indicates the presence within the clouds of chaotic motions with velocities of \(1\)—\(2\ km/sec\). The velocities of motions in emission nebulae reach \(10\ km/sec\). Between the clou–

INTERSTELLAR POLARIZATION OF LIGHT

...contains a rarefied gas with concentration \(n \simeq 0.1\) atom \(\cdot \mathrm{cm}^{-3}\). The dispersion of its velocities is of the order of \(70\text{--}100\ \mathrm{km/sec}\), and the semithickness is about 10,000 parsecs (the diameter of the Galaxy is about 20,000 parsecs).

The principal element in interstellar gas, as in stars, is hydrogen (about 80% by mass). Next comes helium (about 20%) and other elements (about 1%), among which the most abundant are O, N, C. Metals relative to these elements amount to about 1%, and among them iron is the most abundant.

In its physical properties the gas is divided into sharply distinct regions: zones of ionized and nonionized hydrogen \(^{22}\). The former extend 50–100 parsecs around hot stars. Most clouds are H I regions. The gas between clouds is ionized near the plane of the Galaxy and nonionized far from it, where there are few hot stars. In H II regions the electron concentration is of the order of the gas concentration (10–20 in clouds and 0.1 between clouds), and the kinetic temperature is about \(10\,000^\circ\). In nonionized clouds \(n_e \simeq 0.01\ \mathrm{cm}^{-3}\) (the electrons are produced mainly by the ionization of carbon); the kinetic temperature is of the order of \(60\text{--}100^\circ\ \mathrm{K}\). Part of the gas forms molecules \(\mathrm{H}_2\), \(\mathrm{CH}\), \(\mathrm{CN}\), etc. Data on temperature have been obtained from the relative intensities of lines in emission nebulae, from the radio emission of ionized gas, from the radiation of the radio line of neutral hydrogen (\(\lambda = 21\ \mathrm{cm}\)), and from theoretical calculations \(^{23}\).

Besides gas, there is dust in interstellar space. Its presence was first established from dark nebulae visible directly in photographs when they are projected onto a luminous nebula or onto the bright background of the Milky Way. If there is a bright star near a dust nebula, the nebula scatters its light and therefore appears bright. It was later shown that the light of practically all stars located near the plane of the Galaxy and not too close to us is weakened in passing through interstellar space. The absorption is selective and in the visible region of the spectrum is proportional to \(\lambda^{-1}\). Because of absorption the stars appear redder, and this makes it possible to determine the amount of absorption with reasonable confidence. From the amount of absorption the mass of the dust was estimated (on the assumption that the mean size of the particles is of the order of several tenths of a micron; see below), which near the galactic plane amounts to approximately 1% of the mass of the gas. Later it was shown that this ratio is also valid for the Orion Nebula \(^{24}\).

The dust forms a layer of approximately the same thickness as the gas clouds (about 100 parsecs). A study of fluctuations in absorption showed that the dust is distributed nonuniformly, but is concentrated in individual clouds \(^{25}\), with one cloud absorbing on average about 0.25 stellar magnitude \(^{26,27}\) (one stellar magnitude corresponds to a change in brightness by a factor of 2.512, or to a change of the logarithm...

magnitude of brightness by 0.4). Naturally, the question arose of the identity of clouds of gas and dust. Studies by several authors have shown that there is a certain correlation between the intensity of interstellar absorption lines and the reddening of stars, although more complicated secondary effects (chiefly the nonproportionality of the gas column density and the line intensity) complicate the relation. It was pointed out[^28] that, since gas in its motion must carry dust along with it, a concentration of gas in a cloud must also cause a concentration of dust, although the relative motion of dust and gas under the action of radiation pressure and other forces may lead to different relative concentrations of dust and gas. A comparison of the radiation of diffuse nebulae in the $H_\alpha$ line and in the continuous spectrum[^29] showed that in brighter nebulae it is impossible to detect the light of a star reflected by dust, owing to the strong radiation of an atomic nature. In less bright nebulae, radiation due to dust appears. Therefore one may think that the presence of dust is characteristic of the majority of diffuse nebulae. Recently the connection between the distribution of gas and dust was confirmed[^30] by comparing absorption with the intensity of radio emission at $\lambda = 21\ \mathrm{cm}$; these showed a close correlation.

Observations of radio emission at a wavelength of $21\ \mathrm{cm}$ and other methods, as well as the study of light absorption in other galaxies, have shown that both dust and gaseous clouds are located, for the most part, along the spiral arms of galaxies together with hot stars. The very existence of spiral structure remains puzzling: it is not a purely mechanical effect. It is unclear why clouds do not spread out from the arms, what gives them such stability, and what processes led to the formation of the spirals that are so typical of galaxies. It is interesting that the presence of dust in different galaxies is closely connected with the presence and degree of development of spiral structure, increasing with the latter.

The growth of dust particles apparently occurs by adsorption of atoms and molecules of interstellar gas on their surfaces[^31],[^32]. Almost all atoms can be adsorbed except H and He, which evaporate. Proceeding from this, van de Hulst[^33] considers that dust particles consist of “icy” crystals with the most probable composition: $\mathrm{H_2O}$, $\mathrm{NH_3}$, $\mathrm{CH_4}$, and other hydrogen compounds, with a small admixture of heavier elements, including metals. For growth, a comparatively low temperature and high density are necessary; these conditions are fulfilled in non-ionized gaseous clouds. It is possible that this circumstance is also responsible for the connection between the distribution of gas and dust. The growth of dust particles is limited by their evaporation during high-velocity collisions (collisions of clouds). If there is appreciable partial evaporation of dust particles, then less volatile fractions, in particular metals, may remain. However, this process is unlikely. A quantitative theory of the formation of interstellar dust has not yet been ...

does not yet exist. The question of the composition of the dust is of great importance. Observations can provide some information here, since the spectral law of absorption, the reflecting power (the ratio of scattered light to the light removed by the dust grain from the main beam), and the distribution of scattered light by directions (the scattering indicatrix) depend on the composition and size of the particles.

The theory of the attenuation of light by homogeneous spherical particles of radius \(r\) with refractive index \(m\) (in the general case complex, with the imaginary part characterizing absorption) was created by Mie \({}^{34}\). This theory, based on Maxwell’s equations, gives the quantity \(Q\), equal to the ratio of the effective scattering cross section to the geometrical cross section of the sphere \((\pi r^2)\), as a function of the quantity

\[ \alpha=\frac{2\pi r}{\lambda}. \]

Numerical calculations according to this theory have been carried out by a number of authors \({}^{35,36}\) for various \(m\) at \(\alpha<12\). A detailed analysis of Mie theory for various cases, with physical interpretation and methods of numerical calculation, was given by van de Hulst \({}^{37}\). An example of the dependence of \(Q\) on \(\alpha\) for real \(m=1.33\) is given in Fig. 6. It is evident from the figure that the absorption is approximately proportional to \(\alpha\) (i.e. \(\lambda^{-1}\)) for \(1.5<\alpha<4\). Such inequalities determine, for each value of \(m\), the interval of particle sizes that will give the observed law of absorption. If one plots the dependence of \(Q\) on the quantity \(2\alpha(m-1)\), equal to the phase shift of the light that has passed through the particle, then the curves practically coincide for all real \(m\) from 1 to 2 \({}^{37}\). This circumstance makes it possible to use a single calculation for the entire indicated interval. For particles that strongly absorb light (\(m\) has a large imaginary component), the maximum lies between \(\alpha=1\) and 2, and the fluctuations after the maximum are weakly expressed. It should be kept in mind that in this case \(m\) depends on \(\lambda\), and the curve of \(Q\) versus \(\alpha\) does not directly represent the dependence of \(Q\) on \(\lambda\).

Fig. 6.

Analysis of solutions based on Mie theory has shown \({}^{33}\) that the observed law of interstellar absorption can be explained either by icy \((m=1.25—1.33)\) particles with an average diameter

$\sim 0.8\,\mu$ or metallic (iron) particles with a diameter of $\sim 0.08\,\mu$. In the first case the reflectivity is $\sim 1$ (dielectric particles scatter, rather than absorb, light), in the second—from 0.10 to 0.15 (the light is attenuated mainly through absorption). The dependence of the intensity of the scattered (unabsorbed) light on wavelength follows Rayleigh’s law ($\lambda^{-4}$), since the sizes of the metallic particles are smaller than the wavelength (the total attenuation of light is proportional to $\lambda^{-1}$).

The scattering indicatrix in the first case is strongly elongated in the direction of the incident light (diffraction); in the second case it is symmetric with respect to the particle and is determined by the law $1 + \cos^2 \theta$ (Rayleigh scattering).

A comparison[^33] of these properties with the results of observations of the brightness of scattered light in the Galaxy in the region of the constellation Cygnus[^38] led to the conclusion that the particles are predominantly dielectric, with a small admixture of metals. However, Güttler[^39] believes that the observational data are insufficient for an unambiguous solution of the question and that, under certain conditions, iron particles of smaller sizes can also explain the observations. He points to the influence of the large[^40] dispersion of the refractive indices of the absorbing material and other effects that can exert a strong influence on the calculation. Mathematical difficulties did not allow him to solve the problem exactly as applied to the conditions of interstellar dust grains, but from approximate calculations he concluded that the observations cannot rule out the possibility of a significant admixture of metals in the dust grains. It should be noted that from the calculation of the brightness of reflection nebulae and its comparison with observations[^29],[^41] one can draw a conclusion in favor of a high reflectivity of dust, i.e. in favor of its dielectric nature. Apparently the latter is much more probable than a metallic composition.

To refine the physical conditions in which the substance of the dust grains is found, it is necessary to know their temperature. The temperature of dust grains is determined by the condition of stationarity: the amount of absorbed light energy is equal to the energy emitted by the dust grain (in the far infrared region). Since the mean density of radiant energy in the Galaxy is well known, it is easy to calculate that a black body should have a temperature of about $3^\circ\mathrm{K}$. If the absorption coefficient differs from unity but is constant over the spectrum, the value of the temperature will not change. However, dust grains absorb light with wavelengths considerably larger than their sizes only very weakly (diffraction); consequently, their emissivity in the far infrared region is small. Therefore the equilibrium value of the temperature of dust grains with diameters from $0.2$ to $2\,\mu$ is higher than that of a black body, and lies within the limits $10$–$40^\circ\mathrm{K}$[^33]. We now turn to theories that directly consider the polarization of starlight.

4. POLARIZATION OF LIGHT BY ORIENTED ELONGATED PARTICLES

Without dwelling for the moment on the mechanism that orients the particles, let us consider what effect elongated, identically oriented dust grains will have on transmitted light. Unfortunately, mathematical difficulties have not allowed the absorption by such dust grains to be calculated when their dimensions are comparable with the wavelength, as is the case in reality. An exact solution was obtained by Gans for ellipsoids whose dimensions are much smaller than the wavelength,^42 and by Schaefer and Grossmann for infinitely long dielectric circular cylinders.^43 These theories were applied to explain the polarization of light, respectively, by Davis and Greenstein^44 and by van de Hulst.^45

Gans calculates the induced dipole moment under the assumption that the field \(E\) is homogeneous in the particle. This means that the size of the particles is much smaller than the wavelength, that the material of the particle is homogeneous, and that the magnetic field of the wave is negligible. The particle is a spheroid with an axis of symmetry of length \(2a_A\) and a perpendicular axis \(2a_T\). The electric field inside an ellipsoid surrounded by a medium with dielectric constant equal to unity is determined by the equations

\[ E_A=\frac{E_{A0}}{1+\frac{m^2-1}{4\pi}P}; \qquad E_T=\frac{E_{T0}}{1+\frac{m^2-1}{4\pi}P'}, \tag{4.1} \]

where \(E_{A0}\) and \(E_{T0}\) are the components of the external electric field along the axis of symmetry and along one of the perpendicular axes, \(m\) is the complex refractive index, and \(P\) is a certain factor depending only on the geometric configuration of the ellipsoid. The value of this factor is expressed in terms of the spheroid axis ratio \(x=\dfrac{a_A}{a_T}\) as follows (in ^42 the eccentricity is used instead of \(x\)):

\[ \begin{aligned} P&=\frac{4\pi}{x^2-1} \left[ \frac{x}{(x^2-1)^{1/2}}\operatorname{arcch}x-1 \right], \qquad (x>1),\\ P&=\frac{4\pi}{1-x^2} \left[ 1-\frac{x}{(1-x^2)^{1/2}}\arccos x \right], \qquad (x<1),\\ P'&=2\pi-\frac{1}{2}P. \end{aligned} \tag{4.2} \]

Let the effective cross section of a spheroid attenuating light, with electric vector \(\mathbf E\) parallel to the axis of symmetry \(OA\), be denoted by \(\sigma_A\), and for light with \(\mathbf E \perp OA\) by \(\sigma_T\). Then, in the case of pure scattering (\(m=n\) real),

\[ \sigma_A= \frac{128\pi^5 a^6}{3\lambda^4} \left[ \frac{m^2-1}{3+(m^2-1)\dfrac{3P}{4\pi}} \right]^2 = \frac{128\pi^5 a^6}{3\lambda^4}G_A, \tag{4.3} \]

and for metallic particles ($m$ has a large imaginary part)

\[ \sigma_A=\frac{8\pi^2 a^3}{\lambda}\operatorname{Im} \left[ \frac{1-m^3}{3+(m^2-1)\dfrac{3P}{4\pi}} \right] = \frac{8\pi^2 a^3}{\lambda}\,H_A . \tag{4.4} \]

Here $a$ is the mean radius of the spheroid, equal to $(a_Aa_T^2)^{1/3}$. The quantities $\sigma_T$, $G_T$, and $H_T$ are obtained by replacing $P$ by $P'$. Polarization of the transmitted light will occur if $P\ne P'$. For dielectric particles the cross section is proportional to $\lambda^{-4}$ (Rayleigh’s law); for metallic particles it is $\sim\lambda^{-1}$, if the dependence of $m$ on $\lambda$ is neglected. The results of calculations of $\sigma_A$ and $\sigma_T$ for elongated spheroids, both dielectric and metallic, are given in Table I, taken from $^{44}$. The last three

Table I

Effective cross sections and inertial properties of small particles

	$x=a_A/a_T$	1	1.091	1.667	2.294	3.203	5.025
$G_A$	$n=\sqrt{2}$	0.06250	0.06487	0.07590	0.08416	0.09175	0.09973
$G_T$	$n=\sqrt{2}$	0.06250	0.06140	0.05707	0.05471	0.05285	0.05126
$\sigma_A/\sigma_T$	$n=\sqrt{2}$	1.000	1.053	1.328	1.538	1.738	1.945
$H_A$	$m=\sqrt{2}(1-i)$	0.6000	0.6612	1.0105	1.2352	1.3739	1.4163
$H_T$	$m=\sqrt{2}(1-i)$	0.6000	0.5714	0.4640	0.4135	0.3764	0.3465
$\sigma_A/\sigma_T$	$m=\sqrt{2}(1-i)$	1.000	1.158	2.18	2.99	3.65	4.08
$\gamma$		1	1.095	1.889	3.132	5.628	13.126
$100\,\Gamma$		0	0.303	1.816	1.888	1.800	1.358
$100\,\Gamma_{pg}va^2/I$		0	0.804	6.38	8.16	9.76	9.97

rows of the table contain quantities that will be needed by us below.

If $m=\sqrt{2}(1-0.5i)$, then the polarization is approximately half as large as in the table. In general, the polarization is large even for weakly elongated particles. Let us consider the dependence of the polarization on wavelength. Calculation shows that $\dfrac{\sigma_A}{\sigma_T}=\dfrac{G_A}{G_T}$ and $\dfrac{H_A}{H_T}$ depend only weakly on $\lambda$. The ratio of polarization to absorption $p/A$ is characterized by the quantity $\dfrac{\sigma_A-\sigma_T}{\sigma_A+\sigma_T}$ and, consequently, also depends only weakly on $\lambda$. Since $A\sim\lambda^{-1}$, $p$ should be proportional to $\lambda^{-1}$, which contradicts the observed independence of polarization from $\lambda$. It has been suggested that the presence of particles of different sizes can reduce this contradiction, since spheroids of size of order $\lambda$ will give a smaller dependence of $p$ on $\lambda$, but at the same time also a smaller polarization. Van de Hulst’s estimates $^{45}$ show-

they show that the assumption of the existence in interstellar space of two classes of particles, one of which produces polarization (small particles), while the other is responsible for absorption, creates great difficulties. Indeed, in order for \(p\) to be independent of \(\lambda\), the absorption by large particles must be 10 times greater than the absorption by small particles, and this greatly reduces the polarization and requires a very large elongation of the small particles (\(x\) from 2 to 6 for \(m\) from 2 to 1.4, respectively).

Nevertheless, we shall consider the application of Gans’ theory to the calculation of the polarization of light, since, first, the objections are not decisive and, second, the apparatus developed in \(^{44}\) can be used also in more general cases.

Let the axis of the spheroid make an angle \(\alpha\) with the electric vector of the quasistatic homogeneous field of the light wave. Considering the incident polarized light as the sum of two waves polarized parallel and perpendicular to the axis of the particle, it is easy to obtain:

\[ \sigma(\alpha)=\sigma_A \cos^2 \alpha+\sigma_T \sin^2 \alpha =\sigma_T+(\sigma_A-\sigma_T)\cos^2 \alpha . \tag{4.5} \]

Let now the dust grain rotate in a homogeneous magnetic field \(\mathbf{B}\). The angular momentum is represented by the vector \(\mathbf{H}\) (Fig. 7); the axis of symmetry \(OA\) describes, about \(\mathbf{H}\), a cone with angle \(\theta\) (nutation). The light propagates in the direction \(S\), which lies in the plane \(zOy\) and forms an angle \(\nu+\dfrac{\pi}{2}\) with \(\mathbf{B}\); \(\mathbf{H}\) is localized by the angles \(\beta\) and \(\Phi\), while the axis \(OA\) is localized by the angles \(\theta\) and \(\psi\) about \(\mathbf{H}\), as about a polar axis. If there is no interaction between the dust grain and the magnetic field, then \(\Phi\), \(\beta\), and \(\theta\) are constant, while \(\psi\) increases uniformly. The effect of the dust grain on the light must be averaged over \(\psi\) and over \(\Phi\), since all values of \(\Phi\) are equally probable.

Fig. 7.

From (4.5) it is seen that the average is to be taken of the square of the cosine of the angle between the axis of symmetry and the direction of the electric vector \(\mathbf{E}\). We decompose \(\mathbf{E}\) into \(E_\pi\) (in the plane \(BS\)) and \(E_\sigma\), corresponding to intensities \(\mathcal{J}_\pi\) and \(\mathcal{J}_\sigma\). The mean energy removed from the light beams by the dust grain is respectively equal to

\[ \begin{aligned} \mathcal{J}_\pi\left[\sigma_T+(\sigma_A-\sigma_T)\overline{\cos^2\alpha_\pi}\right] &=\mathcal{J}_\pi \Sigma_\pi(\beta,\theta,\nu,a),\\ \mathcal{J}_\sigma\left[\sigma_T+(\sigma_A-\sigma_T)\overline{\cos^2\alpha_\sigma}\right] &=\mathcal{J}_\sigma \Sigma_\sigma(\beta,\theta,\nu,a). \end{aligned} \tag{4.6} \]

The averages are computed by means of spherical trigonometry. Fina-

the final expressions have the form

\[ \left. \begin{aligned} \Sigma_{\pi} &= \sigma_T+(\sigma_A-\sigma_T)\frac{1}{2} \left(1-\cos^2\beta\cos^2\theta-\frac{1}{2}\sin^2\beta\sin^2\theta\right),\\ \Sigma_{\sigma} &= \left[\sigma_T+(\sigma_A-\sigma_T) \left(\cos^2\beta\cos^2\theta+ \frac{1}{2}\sin^2\beta\sin^2\theta\right)\right]\cos^2\nu +\Sigma_{\delta}\sin^2\nu . \end{aligned} \right\} \tag{4.7} \]

Let us now pass from one dust particle to an aggregate of them. Let \(y\) be the coordinate along the line of sight, \(n_d(y)\) the number of dust particles in \(1\ \mathrm{cm}^3\), \(\rho(a)\,da\) the fraction of dust particles having size \(a\), and \(f(\beta,\theta,a)\) the function of their distribution over the angles \(\beta\) and \(\theta\). The number of dust particles characterized by the quantities \(\beta,\theta\), and \(a\) in \(1\ \mathrm{cm}^3\) is equal to \(f(\beta,\theta,a)\rho(a)n_d(y)\,d\beta\,d\theta\,da\). If no magnetic field acts on the dust particle, then \(f(\beta,\theta,a)\) is determined by the condition of equipartition of the energy acquired by the particle in collisions with atoms, and has the form

\[ f_e(\beta,\theta)=\frac{1}{2}\gamma^{\frac{1}{2}}\sin\beta\sin\theta \left(\gamma\cos^2\theta+\sin^2\theta\right)^{-\frac{3}{2}}, \tag{4.8} \]

where \(\gamma\) is the ratio of the moment of inertia about the short axis to the moment of inertia \(I\) about the axis of symmetry (Table I). \(f_e(\beta,\theta)\) does not depend on \(a\) and \(I\). In reality, if the dust particle possesses paramagnetic properties, a weak braking torque will act on it in a magnetic field; the action of this torque will be considered in the following paragraph. This torque changes the distribution, which will now have the form

\[ f(\beta,\theta,a)=f_e(\beta,\theta)+f_1(\beta,\theta,a). \tag{4.9} \]

To calculate the absorption, we introduce, instead of \(y\), the variable \(N\), defined by the condition \(dN=n_d(y)\,dy\), whose physical meaning is obvious. It can be shown that

\[ d\mathcal{J}_{\pi}=-S_{\pi}\mathcal{J}_{\pi}\,dN, \tag{4.10} \]

where

\[ S_{\pi}=\int_{0}^{\infty}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\pi} \Sigma_{\pi}(\beta,\theta,\nu,a)f(\beta,\theta,a)\rho(a)\,d\beta\,d\theta\,da . \tag{4.11} \]

The end of the axis \(OA\) is chosen so that \(\theta\leq \dfrac{\pi}{2}\); therefore the integration with respect to \(\theta\) is carried out up to \(\dfrac{\pi}{2}\). Similarly,

\[ d\mathcal{J}_{\sigma}=-S_{\sigma}\mathcal{J}_{\sigma}\,dN, \tag{4.12} \]

INTERSTELLAR POLARIZATION OF LIGHT

where

\[ S_\sigma=\int_{0}^{\infty}\int_{0}^{\pi/2}\int_{0}^{\pi} \Sigma_\sigma(\beta,\theta,\nu,a) f(\beta,\theta,a)\rho(a)\,d\beta\,d\theta\,da . \tag{4.13} \]

The result of the integration can be expressed in terms of three functions:

\[ F(a)=-\int_{0}^{\pi/2}\int_{0}^{\pi} \left(\cos^2\beta\cos^2\theta+\frac{1}{2}\sin^2\beta\sin^2\theta\right) f_1(\beta,\theta,a)\,d\beta\,d\theta , \tag{4.14} \]

\[ S_t=\frac{1}{3}\int_{0}^{\infty}(2\sigma_T+\sigma_A)\rho(a)\,da, \tag{4.15} \]

\[ S_p=\frac{3}{2}\int_{0}^{\infty}(\sigma_A-\sigma_T)F(a)\rho(a)\,da, \tag{4.16} \]

where \(F(a)\) is a certain characteristic of the distribution of particles over angles, equal to zero for \(f=f_e\) and attaining the value \(1/3\) in the case of the greatest ordering of elongated paramagnetic particles (see below), when all the long axes of the particles are perpendicular to \(\mathbf B\); \(S_tN\) is the optical thickness for the beam as a whole, \(S_\pi N\) and \(S_\sigma N\) are the optical thicknesses for \(\mathcal J_\pi\) and \(\mathcal J_\sigma\); \(S_pN\cos^2\nu\) is the “optical thickness” for polarization. It can be shown that

\[ S_\pi=S_t-S_p\left(\cos^2\nu-\frac{1}{3}\right);\qquad S_\sigma=S_t+\frac{1}{3}S_p . \tag{4.17} \]

We now pass from optical thicknesses to the total absorption, color excess, and polarization. The equation \(d\mathcal J=-S\mathcal J\,dN\), in the case of constant \(S\), has the solution \(\mathcal J=\mathcal J_0\exp\{-SN\}\), i.e. the attenuation of light in stellar magnitudes is

\[ \Delta m=-2.5\lg\frac{\mathcal J}{\mathcal J_0}=1.086SN, \]

where \(SN\) is the optical thickness. In the case when \(\mathbf B\) is homogeneous along the entire path from the star, and \(\rho(a)\) and \(f\) are constant, then

\[ \Delta m_\pi=1.086S_\pi N \quad\text{and}\quad \Delta m_\sigma=1.086S_\sigma N . \]

The difference in brightness for the two positions of the polaroid is

\[ \Delta m_p=1.086NS_p\cos^2\nu . \]

The degree of polarization is

\[ p=\frac{\mathcal J_\pi-\mathcal J_\sigma}{\mathcal J_\pi+\mathcal J_\sigma} =\frac{1-\exp(-S_pN\cos^2\nu)}{1+\exp(-S_pN\cos^2\nu)} \simeq \frac{1}{2}S_pN\cos^2\nu =0.4605\Delta m_p . \tag{4.18} \]

if \(S_p N \ll 1\). Absorption by dust is \(A_{pg}=1.086 S_l N\), whence

\[ \frac{p}{A_{pg}}=0.46\,\frac{S_p}{S_l}\cos^2\nu . \tag{4.19} \]

For grains of a single size one can find that

\[ \frac{p}{A_{pg}}=2.07F\cos^2\nu\, \frac{\sigma_{A/s_T}-1}{\sigma_{A/s_T}+2}. \tag{4.20} \]

For comparison with Hogg’s data (§ 2), it should be borne in mind that \(A_{pg}=4.7E=9E_1\), where \(E_1\) is the color excess on the Stebbins scale, which is used in many works. The observed value for the maximum, \(p/A_{pg}\simeq 0.02\), can be explained by the theory for \(\nu=0\), \(F=\frac{1}{3}\), and \(\sigma_{A/s_T}=1.086\), or, for example, for \(F=\frac{1}{15}\) and \(\sigma_{A/s_T}=1.52\) \((x\simeq 2)\). The dependences of absorption and polarization on wavelength are not thereby explained.

Further, in \({}^{44}\) the quantity \(F\) is calculated in the presence of a magnetic field. Before presenting this calculation, let us consider the polarization and absorption produced by infinitely long circular dielectric cylinders. The general formulas for this case \({}^{43}\) are similar to the formulas of Mie theory. The quantity \(Q\) denotes the ratio of the effective cross section to the geometrical one; the subscripts 1 and 2 correspond, respectively, to light with the electric vector parallel and perpendicular to the axis of the cylinder. Then \(Q\) is expressed through \(\alpha=2\pi r/\lambda\) (\(r\) is the radius of the cylinder) and the refractive index \(m\) as follows:

\[ Q_1(\alpha,m)=\frac{2}{\alpha}\sum_{n=-\infty}^{\infty}\Re(b_n); \qquad Q_2(\alpha,m)=\frac{2}{\alpha}\sum_{n=-\infty}^{\infty}\Re(a_n), \tag{4.21} \]

where

\[ b_n(\alpha,m)= \frac{mJ'_n(y)J_n(\alpha)-J_n(y)J'_n(\alpha)} {mJ'_n(y)H_n(\alpha)-J_n(y)H'_n(\alpha)}, \tag{4.22} \]

\[ a_n(\alpha,m)= \frac{J'_n(y)J_n(\alpha)-mJ_n(y)J'_n(\alpha)} {J'_n(y)H_n(\alpha)-mJ_n(y)H'_n(\alpha)}, \tag{4.23} \]

where \(y=m\alpha\), \(J_n(\alpha)\) are Bessel functions, \(H_n(\alpha)\) are Hankel functions of the second kind, the prime denotes differentiation, the symbol \(\Re\) indicates that the real part is taken, and \(b_{-n}=b_n\).

The results of calculations \({}^{45}\) for various \(m\) are presented in Fig. 8, where \(Q_1\) and \(Q_2\) are plotted as functions of the quantity \(\rho=2\alpha(m-1)\). Up to 10 terms of the expansion were computed. The lower curve corresponds

with a refractive index close to unity, when \(Q_1 = Q_2 = \pi \Sigma_1(\rho)\), where \(\Sigma_1\) is the function of Struve \(^{46}\). The first terms of the expansion of \(Q_1\) and \(Q_2\) are shown in the figure by a dotted line. For any real values of \(m\) they have the form

\[ \left. \begin{aligned} Q_1 &= \frac{\pi^2(m^2-1)^2}{8}\,a^3;\\ Q_2 &= \frac{\pi^2(m^2-1)^2}{4(m^2+1)^2}\,a^3;\\ \frac{Q_1}{Q_2} &= \frac{(m^2+1)^2}{2}. \end{aligned} \right\} \tag{4.24} \]

However, these values cannot be used for calculating the polarization, since it is seen from the figure that the first terms give a sufficiently accurate approximation only for \(\rho < 1\) (especially for \(m = 1.25\)).

Fig. 8.

Meanwhile, to explain interstellar absorption proportional to \(\lambda^{-1}\) in the visible and near-infrared regions, it is necessary to assume that, for real cylinders, \(\rho\) varies from 1 to 3 in the indicated spectral interval. One may take \(\rho = \lambda^{-1}\),

if \(\lambda\) is expressed in microns. In this case the ratio \(Q_1/Q_2\) is considerably smaller than for the first terms.

Table II\(^{33}\) gives, for \(\lambda = 5000\) Å, the values of \(Q_1\), \(Q_2\), and their ratios; here \(r\) has been chosen so that the spectral law of absorption is satisfied.

Table II

\[ \lambda = 5000\ \text{Å} \]

Cylinder material	\(m\)	\(\dfrac{2r}{\mu}\)	\(\dfrac{2\pi r}{\lambda}\)	\(Q_1\)	\(Q_2\)	\(\dfrac{Q_1}{Q_2}\)
Ice	1,25	0,70	4,4	2,62	2,40	1,10
Glass	1,50	0,35	2,2	2,90	2,26	1,28
The same, with an admixture of metals	\(1,50 - 0,1 i\)	0,32	2,0	2,55	1,90	1,34
The same, with an admixture of metals	\(1,41 - 0,41 i\)	0,07	0,44	1,54	0,92	1,68
Observations	—	—	—	—	—	1,07

In the last row the observed value is given under the condition of complete orientation of the particles, obtained from the maximum value of \(p/A_{pg}\) with the aid of the relations:

\[ p = 0{,}46\Delta m_p,\qquad \Delta m_p = A_1 - A_2, \]

\[ A_{pg} = \frac{500}{410} A_{5000} = \frac{500}{410}\,\frac{1}{2}(A_1 + A_2); \qquad \frac{Q_1}{Q_2} = \frac{A_1}{A_2}, \]

where \(A_1\) and \(A_2\) are the absorption in stellar magnitudes for polarized light with different orientations. If one uses Hoag’s data for \(p/E\), then

\[ \frac{Q_1}{Q_2} = 1{,}06. \]

It is seen from the table that fairly well oriented ice needles can explain the observed polarization. For glass needles the margin is still greater, but their presence is less probable. A rough estimate, the details of which are not given in \(^{33,45}\), shows that \(Q_1 - Q_2\) decreases by a factor of two if the needles are replaced by elongated spheroids with an axial ratio of \(2:1\), and by a factor of four if the axial ratio is \(1{,}3:1\). In this case ice ellipsoids can no longer explain the observed polarization, whereas those with an admixture of metals can, but only with sufficiently great ordering. This will be discussed in more detail below. It is very important that, although \(Q_1\) and \(Q_2\) are approximately proportional to \(\lambda^{-1}\), their difference in the same spectral interval does not depend on \(\lambda\) (Fig. 8), if one disregards small oscillations that are smoothed out when particles of different sizes are present. This corresponds to the observed fact that the polarization is independent of wavelength and serves as an additional argument in favor of the correctness of the proposed explanation.

5. MECHANISM OF PARTICLE ORIENTATION

Hiltner already pointed out,^11 that the orientation of particles may be determined by a magnetic field. The first quantitative hypothesis, proposed by Spitzer and Tukey^47 to explain the ordering, likewise proceeded from the existence of an interstellar magnetic field. It was assumed that the particles contain a large amount of iron, nickel, etc., and possess ferromagnetic properties and a permanent magnetic moment. The interaction of this moment with the external field tends to turn the particle along the field, like a compass needle. Since the particle rotates, the gyroscopic effect prevents the turn, and the axis of the particle will precess, describing a cone about the direction of the magnetic field. Orientation can occur only if the moment is very large, and also if the rotation of the dust grain in the magnetic field is braked during the time between two successive collisions because of thermal losses associated with the phenomenon of hysteresis. However, this process is inefficient. It was therefore necessary to assume,^47 that the strength of the interstellar field is of order \(10^{-4}\), that the ferromagnetic properties are very strong, and that the gas temperature is lower than usual (\(10^{\circ} \mathrm{K}\)). In this case the direction of the field must be perpendicular to the galactic plane, which corresponds to the general direction of polarization, but does not explain the dependence of polarization on longitude. This circumstance, as well as the requirement of almost unrealistic properties of the dust and physical conditions, make the hypothesis^47 unlikely, and we shall not consider it in detail. A recently published paper^48 also proceeds from the ordering of small ferromagnetic dust grains by a magnetic field and encounters the same difficulties.

More interesting is the hypothesis of Davis and Greenstein,^44 who proceed from the assumption that ferromagnetic atoms are contained in interstellar dust in small quantities and give the dust grain as a whole not ferromagnetic but paramagnetic properties. The rotation of a dust grain in a magnetic field causes its continuous remagnetization; in this process part of the energy is converted into heat, the source of which is the kinetic energy of rotation, as a result of which a small moment acts on the dust grain, slowing its rotation. Before presenting the calculations in detail, let us consider roughly the general physical picture of the phenomenon. In Fig. 9, \(\boldsymbol{\omega}\) denotes the vector of the angular velocity of the particle, \(\mathbf{B}\) the vector of magnetic-field strength. The magnetic moment per unit volume of the particle \(\mathbf{M}\) (magnetization) does not coincide in direction with \(\mathbf{B}\); because of the finite relaxation time associated with the dissipation of energy, it is, as it were, carried along by the dust grain. The interaction of \(\mathbf{M}\) and \(\mathbf{B}\) gives a torque

\[ \mathbf{L} = V\mathbf{M} \times \mathbf{B}, \]

having the direction \((\boldsymbol{\omega} \times \mathbf{B}) \times \mathbf{B}\) (\(V\) is the volume of the particle). In fact,

in fact the picture is somewhat more complicated, since the axis of rotation is not fixed relative to the particle; the latter undergoes nutation, and its axis \(OA\) and the vector \(\omega\) describe cones about the angular-momentum vector \(\mathbf H\). Since the interaction is small and the nutation takes place with a high frequency, so that during one nutation the changes are insignificant, here and below it is assumed that the process may be averaged, regarding \(\omega\) as constant and coincident with \(\mathbf H\).

Fig. 9.

The action of the moment \(\mathbf L\) on the rotating body causes aperiodic precession: the angular momentum gradually approaches the direction of the field \(\mathbf B\). Mathematically this follows from the equation \(d\mathbf H/dt=\mathbf L\), which shows that, in the course of this approach, the projection of \(\mathbf H\) on \(\mathbf B\) remains constant, since \(\mathbf L\perp \mathbf B\). Consequently, as a result of this process the particle will rotate about an axis coinciding with \(\mathbf B\). It is shown below that the axis of rotation will be the least axis of the ellipsoid of inertia, i.e., the particle will turn in such a way that its long axis is set perpendicular to the magnetic field. The difference between the process described and the mechanism of Spitzer and Tukey is that for the latter it is necessary to bring the dust grain to a complete stop between collisions, whereas here the rotation is accomplished with only a slight loss of angular velocity. Therefore paramagnetic properties and a weaker field are sufficient for the Davis-Greenstein mechanism. Collisions with atoms interfere with the process of orientation of the particles. Ultimately, there must be established some partially ordered distribution of axes corresponding to the dynamical equilibrium of the two processes. We shall now consider all this in more detail, following \(^{44}\).

Moment arising from paramagnetic braking. Let the field in the substance vary sinusoidally,

\[ \mathbf B=\mathbf e B_0 \cos(\omega t+\delta), \]

where \(\mathbf e\) is a unit vector. As with an alternating electric field in a dielectric, the interaction of \(\mathbf B\) with the substance may be described by means of a complex magnetic susceptibility, whose imaginary part characterizes the losses.

The magnetization ( \(^{49}\), p. 20) is

\[ \mathbf M=\mathbf e B_0\,[\chi'\cos(\omega t+\delta)+\chi''\sin(\omega t+\delta)], \tag{5.1} \]

where \(\chi'\) and \(\chi''\) are the real and imaginary parts of the magnetic suscepti-

tivity. If the field frequency \(\omega=0\), then \(\chi''=0\) and \(\chi'=\chi_0\), the ordinary static susceptibility.

Suppose that \(\mathbf B=\mathrm{const}\) and that the grain rotates with constant angular velocity \(\boldsymbol\omega\) about an axis fixed both in the grain and in space. Let \(\mathbf e_x,\mathbf e_y,\mathbf e_z\) be unit vectors in a fixed coordinate system, which we choose so that \(B_y=0\), and let \(\mathbf i,\mathbf j,\mathbf k\) be those in the rotating system. The origin of time is chosen so that

\[ \mathbf e_x=\mathbf i\cos\omega t-\mathbf j\sin\omega t;\qquad \mathbf e_y=\mathbf i\sin\omega t+\mathbf j\cos\omega t;\qquad \mathbf e_z=\mathbf k=\frac{\boldsymbol\omega}{\omega}. \tag{5.2} \]

Then, in the rotating system,

\[ \mathbf B=B_z\mathbf k+B_x\left[\mathbf i\cos\omega t+\mathbf j\cos\left(\omega t+\frac{\pi}{2}\right)\right], \tag{5.3} \]

and if the body is isotropic, then the magnetization is

\[ \mathbf M=\chi_0B_z\mathbf k+B_x\{\mathbf i(\chi'\cos\omega t+\chi''\sin\omega t)+ \]

\[ +\mathbf j(-\chi'\sin\omega t+\chi''\cos\omega t)\}= \chi_0B_z\mathbf e_z+\chi'B_x\mathbf e_x+\chi''B_x\mathbf e_y. \tag{5.4} \]

Consequently, in the rotating body the component of \(\mathbf M\) along the \(x\)-axis is changed, in comparison with the static case, by a factor \(\chi'/\chi_0\), while the “drag” angle of \(\mathbf M\) is equal to \(\operatorname{arctg}(\chi''/\chi')\). The mechanical torque is

\[ \mathbf L=V\mathbf M\times\mathbf B =V(\chi_0-\chi')\omega^{-2}(\mathbf B\cdot\boldsymbol\omega)(\boldsymbol\omega\times\mathbf B)+ \]

\[ +V\chi''\omega^{-1}(\boldsymbol\omega\times\mathbf B)\times\mathbf B. \tag{5.5} \]

The first term acts perpendicular to \(\boldsymbol\omega\) and does not change the rotational energy; it produces only precession about \(\mathbf B\), which does not affect the distribution, since \(VB^2(\chi_0-\chi')\ll kT\). For the same reason one may neglect the action of the additional very small torque \(\chi_0^2B^2V\), which tends to turn the long axis of the paramagnetic ellipsoid parallel to \(\mathbf B\). Only the term with \(\chi''\) produces a cumulative effect.

The quantity \(\chi''\), in general form \((^{49},\ \text{p. }97)\), for a frequency of the order of \(\omega_e\), corresponding to equipartition of energy, has the form

\[ \chi''=\frac{\chi_0\omega}{\omega_0}\left(\frac{\pi}{2}\right)^{1/2} \exp\left(-\frac{\omega^2}{2\omega_0^2}\right), \]

where

\[ h\frac{\omega_0}{2\pi} = 2\beta^2\left[8S(S+1)\sum_{p\ne q} r_{pq}^{-6}\right]^{1/2}. \tag{5.6} \]

Here \(\beta=eh/4\pi mc=0.927\cdot10^{-20}\ \mathrm{erg/gauss}\) is the Bohr magneton, \(S\) is the spin quantum number of the ferromagnetic ions, and \(\sum r_{pq}^{-6}=7\cdot2n_c^2\), where \(n_c\) is the concentration of Fe ions (cf. \(^{49}\), pp. 13–15), \(2[S(S+1)]^{1/2}=5.92\)

for Fe IV; for other ions it is of the same order or somewhat smaller. With these values we obtain \(\omega_0=3.66\cdot 10^{12} n\), where \(n=10^{-24}n_c\) is the number of Fe atoms per cubic ångström. The static magnetic susceptibility (\(^{49}\), p. 6)

\[ \chi_0=\frac{n_c g^2 J(J+1)\beta^2}{3kT_g}=\frac{7.28n}{T_g}, \tag{5.7} \]

where \(T_g\) is the temperature of the dust grain, and it is assumed that

\[ g[J(J+1)]^{1/2}=2[S(S+1)]^{1/2}=5.92. \]

Since the practically interesting values are \(\omega<0.7\omega_0\), the exponential is close to unity, and after substitution of the constants

\[ \chi''=2.5\cdot 10^{-12}\frac{\omega}{T_g} \]

provided that

\[ 10^{-2}>n>6\cdot 10^{-21}\left(\frac{T}{a^5\rho_g}\right)^{1/2}=10^{-6}-10^{-7}. \]

The lower limit follows from the condition \(\omega<0.7\omega_0\), where the rotation frequency

\[ \omega\approx\omega_e=\left(\frac{2\cdot \frac{3}{2}kT}{I}\right)^{1/2} = \]

\[ =1.57\cdot 10^{-8}\left(\frac{T}{a^5\rho_g}\right)^{1/2}\ \text{rad/sec}\approx 5\cdot 10^5-6\cdot 10^6 \tag{5.8} \]

in the H I and H II regions. The upper limit follows from the condition of applicability of Curie’s law: the iron atoms must not be so close together that a strong interaction appears. It may be calculated that \(n\) in interstellar dust satisfies the stated conditions. It is interesting that in this case \(\chi''\) does not depend on the concentration of Fe atoms, since the change of \(\chi_0\) with changing \(n\) is compensated by the displacement of \(\omega\) from the resonant frequency \(\omega_0\), which also depends on \(n\). This circumstance means that, within the indicated limits, the calculation does not depend on the composition of the dust (on the fraction of ferromagnetic atoms).

The influence of a weak retarding moment on the orientation of a particle. Fig. 10, \(a\) depicts the plane passing through the angular-momentum vector \(\mathbf H\) and the axis of symmetry \(\mathbf e_A\). By symmetry, the vector of angular velocity \(\boldsymbol\omega\) also lies in this plane. The angles between the vectors—\(\theta\), \(\theta_H\), and \(\theta_\omega\)—are indicated in the drawing. \(I\) and \(\gamma I\) are the moments of inertia about \(\mathbf e_A\) and \(\mathbf e_T\). We may write:

\[ \boldsymbol\omega=\mathbf e_A\omega\cos\theta_\omega+\mathbf e_T\omega\sin\theta_\omega, \tag{5.9} \]

\[ \mathbf H=\mathbf e_A H\cos\theta+\mathbf e_T H\sin\theta =\mathbf e_A I\omega\cos\theta_\omega+\mathbf e_T I\gamma\omega\sin\theta_\omega. \tag{5.10} \]

From these equations

\[ \cos\theta_\omega=\frac{H}{I\omega}\cos\theta;\qquad \sin\theta_\omega=\frac{H}{I\gamma\omega}\sin\theta. \tag{5.11} \]

INTERSTELLAR POLARIZATION OF LIGHT

Since \(\theta_H=\theta-\theta_\omega\), we have

\[ \cos\theta_H=\frac{H}{I\gamma\omega}\left(\gamma\cos^2\theta+\sin^2\theta\right); \]

\[ \sin\theta_H=\frac{H}{I\gamma\omega}(\gamma-1)\sin\theta\cos\theta. \tag{5.12} \]

The kinetic energy of rotation is

\[ R=\frac{1}{2}\left[I(\omega\cos\theta_\omega)^2+I\gamma(\omega\sin\theta_\omega)^2\right]. \tag{5.13} \]

Substituting (5.11), we obtain:

\[ \frac{2I\gamma R}{H^2}=\gamma\cos^2\theta+\sin^2\theta, \tag{5.14} \]

whence

\[ \frac{2R}{H\omega}=\cos\theta_H. \tag{5.15} \]

The basic equations of motion have the form

\[ \dot{\mathbf H}=\mathbf L =V(\chi_0-\chi')\omega^{-2}(\mathbf B\cdot\boldsymbol\omega)(\boldsymbol\omega\times\mathbf B) +V\frac{\chi''}{\omega}(\boldsymbol\omega\times\mathbf B)\times\mathbf B, \tag{5.16} \]

\[ \dot R=\boldsymbol\omega\cdot\mathbf L =-V\frac{\chi''}{\omega}\left[\omega^2B^2-(\boldsymbol\omega\cdot\mathbf B)^2\right], \tag{5.17} \]

where a dot denotes differentiation with respect to time. In order to estimate the cumulative effect, it is necessary to average \(\dot{\mathbf H}\) and \(\dot R\) over the nutation. For this we consider Fig. 10, b. As a result of the nutation the plane,

Fig. 10.

containing \(OT\), \(\mathbf H\), \(\boldsymbol\omega\), and \(OA\), rotates with constant angular velocity \(\dot\psi\) about \(\mathbf H\), so that \(\theta\), \(\theta_\omega\), and \(\theta_H\) remain constant. It is easy to see that \(\boldsymbol\omega \times \mathbf B\) is directed along \(OQ\), i.e. along \(\mathbf H \times \mathbf B\), and \((\boldsymbol\omega \times \mathbf B)\times \mathbf B\) along \(OR\), i.e. in the plane of \(\mathbf B\) and \(\mathbf H\). Hence it follows that the first term on the right-hand side of (5.16) causes a precession of \(\mathbf H\) about \(\mathbf B\), changing neither \(\beta\) nor \(H\). Therefore it may be disregarded in calculating the orientation. The mean value of the second term is equal to

\[ V\frac{\gamma''}{\omega}\overline{(\boldsymbol\omega\times\mathbf B)\times\mathbf B} = V\frac{\gamma''}{\omega}(\boldsymbol\omega\times\mathbf B)\times\mathbf B = \]

\[ = -\,V\frac{\gamma''}{\omega}\,\omega B^2\sin\beta\cos\theta_H\,\mathbf e_\sigma . \tag{5.18} \]

Consequently, \(\mathbf H\) approaches \(\mathbf B\) along a spiral, and the projection of \(\mathbf H\) on \(\mathbf B\) is constant. In projections this may be written as:

\[ \overline{\dot H_\pi}=0;\qquad \frac{\overline{dH_\sigma}}{dt} = \frac{\overline{d(H_\pi \operatorname{tg}\beta)}}{dt} = -\,V\frac{\gamma''}{\omega}\,\omega B^2\sin\beta\cos\theta_H . \tag{5.19} \]

From this one can determine the rate of change of \(\beta\)—one of the basic parameters determining the orientation,

\[ \dot\beta = -\,\frac{\gamma''}{\omega}\, \frac{VB^2}{I\gamma}\, \sin\beta\cos\beta(\gamma\cos^2\theta+\sin\theta). \tag{5.20} \]

In the limit \(\beta=0\), \(\mathbf H\parallel \mathbf B\).

The rate of change of the other orientation parameter \(\theta\) is determined by the equations:

\[ \frac{\overline{dH^2}}{dt} = \frac{\overline d}{dt}\left(H_\pi^2+H_\sigma^2\right) = -\,2V\frac{\gamma''}{\omega}\,\omega B^2H\sin^2\beta\cos\theta_H, \tag{5.21} \]

\[ \dot{\overline R} = -\,V\frac{\gamma''}{\omega}\,\omega^2B^2(1-\overline{\cos^2\alpha}) = \]

\[ = -\,V\frac{\gamma''}{\omega}\,\omega^2B^2 \left( 1-\cos^2\theta_H\cos^2\beta-\frac12\sin^2\theta_H\sin^2\beta \right), \tag{5.22} \]

where \(\overline{\cos^2\alpha}\) has been transformed with the aid of spherical trigonometry.

We differentiate (5.14), solve for \(\dot\theta\), and average. Using (5.15) and (5.12), we obtain:

\[ \dot{\overline\theta} = -\frac{I_\gamma R}{H^2(\gamma-1)\sin\theta\cos\theta} \left\{ \frac{\dot{\overline R}}{R} - \frac{1}{H^2}\frac{\overline{dH^2}}{dt} \right\} = \]

\[ = \frac{I_\gamma V\gamma''B^2\omega}{H^2(\gamma-1)\sin\theta\cos\theta} \sin^2\theta_H \left(1-\frac12\sin^2\beta\right) = \]

\[ = \frac{\gamma''}{\omega}\, \frac{VB^2}{I\gamma}\, (\gamma-1)\sin\theta\cos\theta \left(1-\frac12\sin^2\beta\right). \tag{5.23} \]

The physical meaning of (5.23) is that the angle between the axis of symmetry and \(\mathbf H\) increases continuously until it reaches \(90^\circ\), after which \(\dot\theta=0\) \((\cos\theta=0)\); the axis of symmetry of the particle is perpendicular to the field, and the particle rotates about the minor axis oriented along the field.

Calculation of the distribution integral \(F(a)\). The complete ordering described above may, under real conditions, fail to occur because of collisions of dust grains with atoms, chiefly hydrogen. The time between two collisions is

\[ t_{H,d}=\frac{1}{\pi a^2 v_H n_H} =\frac{2\cdot 10^{-5}}{a^2 n_H T^{1/2}} \simeq 2\cdot 10^3\ \text{sec.}, \tag{5.24} \]

and the relaxation time of the distribution is

\[ \tau=\frac{t_{H,d}m_d}{m_H} =\frac{5\cdot 10^{10}\rho_a a}{n_H T^{1/2}}. \tag{5.25} \]

Let us first consider the case in which during the time \(\tau\) the change in \(\beta\) and \(\theta\) is small, so that the braking torque plays the role of a small perturbation. The change in the distribution function of the particles with respect to the angles \(\beta\) and \(\theta\) per unit time due to collisions may be roughly estimated by

\[ \left.\frac{dF}{dt}\right|_c=-\frac{F}{\tau}. \tag{5.26} \]

In the case of equipartition \(F=0\), while in the case of complete ordering of the minor axes, which simultaneously serve as axes of rotation,

\[ F=\frac{1}{3}. \]

The braking torque changes \(F\) in 1 sec by

\[ \left.\frac{dF}{dt}\right|_m, \]

and the stationarity condition has the form

\[ F=\tau\left.\frac{dF}{dt}\right|_m. \tag{5.27} \]

It is necessary to calculate

\[ \left.\frac{dF}{dt}\right|_m \]

from \(\dot\beta\) and \(\dot\theta\). Each dust grain may be characterized by a point in the space \(\beta,\theta\); the density of these points is \(f(\beta,\theta,a)\) (4.9), and the points move with velocity

\[ \mathbf v=\dot\beta\,\mathbf e_\beta+\dot\theta\,\mathbf e_\theta. \tag{5.28} \]

The continuity equation for these points has the form

\[ \frac{df}{dt}=-\operatorname{div}(\mathbf v f). \tag{5.29} \]

We estimate the magnitude of the derivatives, assuming that the system passes through a state close to the equipartition state \(f_e\):

\[ \frac{df_1}{dt} =-\frac{\partial}{\partial\beta}(\dot\beta f_e) -\frac{\partial}{\partial\theta}(\dot\theta f_e). \tag{5.30} \]

Differentiating (4.14) under the integral sign, substituting into (5.27) and using (5.30), we obtain:

\[ F_l(a)=\tau \int\limits_0^{\frac{\pi}{2}}\int\limits_0^\pi \left(\cos^2\beta \cos^2\vartheta+\frac{1}{2}\sin^2\beta \sin^2\vartheta\right)\times \]

\[ \times\left[ \frac{\partial}{\partial \beta}\overline{(\dot{\beta}f_e)} + \frac{\partial}{\partial \vartheta}\overline{(\dot{\vartheta}f_e)} \right]\,d\beta\,d\vartheta, \tag{5.31} \]

where the subscript \(l\) on \(F\) means that the linear approximation is being considered (the perturbation is weak), i.e. \(F_l \ll \frac{1}{3}\). Using (4.8), (5.20), and (5.23) and carrying out the integration, we obtain:

\[ F=F_l=\frac{\Gamma(\gamma)V_{\chi}^{\prime\prime}B^2\tau}{I\omega_e}, \tag{5.32} \]

where \(\Gamma(\gamma)\) is a constant depending on the elongation of the spheroid and equal, for elongated particles \((\gamma>1)\), to

\[ \Gamma(\gamma)= \frac{2}{15\gamma^{1/2}(\gamma-1)^{1/2}} \left[ \frac{2\gamma+1}{\gamma-1}\operatorname{arcsh}(\gamma-1)^{1/2} - 3\left(\frac{\gamma}{\gamma-1}\right)^{1/2} \right]. \tag{5.33} \]

The values of \(\Gamma(\gamma)\) and other quantities for various \(x=a_A/a_T\) are given in Table I. The calculation of \(\gamma\) was performed by the formula

\[ \gamma=\frac{I\gamma}{I} = \frac{\rho_z V(a_A^2+a_T^2)/5}{\rho_g V(a_T^2+a_T^2)/5} = \frac{1}{2}\left(\frac{a_A^2}{a_T^2}+1\right). \tag{5.34} \]

After some simplifications, for the interval of \(x\) of interest to us we have approximately

\[ F_l=\frac{6.31\cdot 10^6 B^2}{aT^{1/2}n_H T_g}. \tag{5.35} \]

Let us now consider another limiting case, when the relaxation time of the distribution is large in comparison with the time of rotation of the axes under the action of paramagnetic braking. In this case the particles rotate about their small axes, oriented along \(\mathbf{B}\), and

\[ F=\frac{1}{3}. \]

Consequently, if \(F_l \ll \frac{1}{3}\), then \(F=F_l\); if \(F_l \gg \frac{1}{3}\), then \(F=\frac{1}{3}\); in the intermediate interval a rough interpolation is possible. If

\[ B=10^{-5},\quad a=2\cdot 10^{-5},\quad T=70^\circ,\quad n_H=10,\quad T_g=10^\circ, \]

then \(F=0.04\), i.e. the ordering is comparatively small, but it increases rapidly with increasing field strength. With the aid of (4.20) one can calculate that, in order to explain the observed maxim—

...the value \(p/A_{pg}'=0.025\) for \(F=0.04\) and \(\cos\gamma=1\), the ratio \(\sigma_A/\sigma_T(=Q_1/Q_2)\) must be equal to 2.3. This value is too large (see Tables I and II). Therefore, for the indicated values of the parameters the polarization will be smaller than the observed one. If one takes \(B=2\cdot10^{-5}\), then \(F=0.15\) and \(\sigma_A/\sigma_T=1.26\), which is close to a possible value, but still large.

Taking into account the approximate character of the calculation, one may say that a field of the order of several units in \(10^{-5}\) can explain the observed polarization. The required value of the field strength is somewhat large (see below), and this constitutes a difficulty for the theory, though not a decisive one. Besides theories based on the assumption of the presence of a magnetic field, other mechanisms of orientation have been proposed, for example the effect of the relative motion of dust and gas\(^{50,51}\), or radiation pressure, whose field in the Galaxy is not isotropic\(^{52}\). The latter effect proved to be insignificant and may be disregarded. The first effect was considered for idealized conditions. It was assumed that collisions of gas atoms with dust particles are completely inelastic, that their relative velocity is greater than the thermal velocity of the atoms, and that the dust particles have the form of needles of very small thickness. The mechanism of orientation reduces to the following: if an atom imparts some momentum to a needle-shaped dust particle, applied at any point, and the direction of the momentum coincides with the direction of the atom’s velocity (the impacts are inelastic), then the rotational moment arising in this case will be perpendicular to the atom’s velocity. The dust particle will begin to rotate in a plane passing through its long axis and the atom’s velocity vector. An ensemble of dust particles will rotate in different planes, but all these planes will pass through the vector of the relative velocity, like meridians through a pole. Repeated impacts will not change the picture, since all new moments lie in the same plane, perpendicular to the velocity. The probability that the long axis of a dust particle will at some instant be directed toward the “pole” is determined not by the fraction of the area of the sphere near the pole, as in chaotic rotation, but by the fraction of the length of a meridian near the pole, which is relatively larger. The ordering factor, defined as the ratio of the sums of the projections of the long axes of the particles onto the direction of the relative velocity and onto the perpendicular direction, is, in the ideal case,

\[ \frac{ \displaystyle \int_{0}^{\pi/2} d\varphi \int_{0}^{\pi/2} \cos\theta\,d\theta }{ \displaystyle \int_{0}^{\pi/2} d\varphi \int_{0}^{\pi/2} \sin\theta\cos\varphi\,d\theta } =\frac{\pi}{2}. \tag{5.36} \]

Under real conditions, when the partial elasticity of the collision, the finite thickness of the dust grain, the thermal velocities of the atoms, etc., are taken into account, the effect is reduced, but some preferential orientation is preserved. It is difficult to carry out quantitative calculations here, since the properties of the dust grains are insufficiently known; moreover, it is not clear how long the motion of a dust grain in a gas at supersonic velocity can continue.

It is true that in paper ⁵¹ it is indicated that the supersonic velocity is not very important, and that the effect will also occur at subsonic velocities, but it will undoubtedly be considerably smaller. Gold assumes that ordering by motion may be important, in particular at the edges of gas clouds, where the expansion of the gas under the action of a pressure gradient creates a relative velocity. However, this mechanism cannot explain the general polarization of starlight in the Galaxy, since for it one must assume that there are rapid relative motions of gas and dust perpendicular to the plane of the Galaxy, which is unlikely. Most importantly—and this applies to any mechanism that assumes orientation in some direction—it is impossible in this way to explain the dependence of the polarization on Galactic longitude (this circumstance for some reason has not been noted in the literature).

Concerning the connection of the orientation of particles with the polarization of light, Gold remarks without proof that the van de Hulst analysis (§ 4), which requires a strong ordering of the particles in order to explain the observed polarization, does not cover all cases, and that thin needles, with a certain distribution of their lengths, can apparently give an absorption law \(\lambda^{-1}\) and a stronger polarization.

6. PROPERTIES OF THE MAGNETIC FIELD

In order to obtain appreciable polarization, the dust grains must be identically oriented over almost the entire path from the star to the observer, especially if one takes into account the data of § 4 on the degree of ordering necessary for explaining the observations. Therefore the presence of only chaotic fields produced by turbulent motions cannot explain the polarization. The field must be almost homogeneous over distances of the order of 1000 parsecs and directed along the arm. However, the very fact of the interstellar polarization of light does not yet prove that the field throughout the arm has the same direction. Since the dust is concentrated in clouds, it is sufficient that the field have the same direction only in the clouds, while between the clouds it may have an arbitrary direction ⁵³. An identical orientation of the field in the clouds could apparently arise from their systematic motion relative to the rarefied gas, for example because of motion along the arm. But the presence of elongated nebulae ¹¹ shows that the field in the arms, apparently,

INTERSTELLAR POLARIZATION OF LIGHT

... homogeneous and outside the clouds, that this is an external field, and not an internal field induced by the motion of the cloud itself. At the same time, the identical orientation of dust particles inside the cloud means that the field penetrates into the cloud as well. Since the field cannot penetrate into it from outside during the lifetime of the cloud (the relaxation time of the magnetic flux is more than \(10^{10}\) years), it must apparently be assumed that the clouds are always in a field; when they disperse, the magnetic flux is distributed over a larger volume, and when they condense it is gathered again. It is still premature to consider these questions in greater detail.

Fig. 11.

Of interest is the comparison of data obtained from the polarization of light and from the elongation of dark and bright nebulae, carried out by G. A. Shajn\(^{54}\). In Fig. 11, borrowed from this work, the outlines of a number of dark nebulae and of one emission nebula (NGC 1499) are shown schematically, together with the polarization of the light of stars (the direction of the electric vector), according to the data of Hiltner\(^{19}\) and Behr\(^{58}\), in the region of the constellations Perseus and Taurus. In the middle of the lower part of the figure one can see a group of hot stars whose polarization strongly deviates from the Galactic equator and exhibits a systematic change with the change in the position of the star. It is significant that all these stars are at a close, approximately identical, distance, so that their light apparently passes through one cloud (or group of clouds). The agreement is clearly noticeable...

orientation of the plane of oscillation of the electric vectors and of elongated dark nebulae, and also of fibers in the emission nebula NGC 1499. The nebula located below the stars is also elongated in this direction, although the field inside it is directed differently.

In the lower part of the figure, dotted lines mark very thin, long, bright fibers in the Pleiades, apparently not associated with the stars. Their direction deviates strongly from the galactic equator and is parallel to the elongated dark nebulae. Apparently, here there is a large-scale fluctuation of the magnetic field, encompassing a region several hundred parsecs in size in the constellations Perseus and Taurus. A similar correspondence has been noted in the constellation Cygnus. In the region \(\alpha = 20^h5^m,\ \delta = +35^\circ\) there is a field of nebulae showing a predominant orientation in a direction almost perpendicular to the galactic equator. The polarization of stars in this region is also predominantly close to this direction. In a second region, \(\alpha = 20^h—22^h,\ \delta = 40—50^\circ\), between the nebulae “America” and “Pelican,” there stretches a band of dark matter projected onto the bright nebula and producing its apparent division. This band breaks up into many thinner parallel bands. In the same region there are several stars whose polarization is large and is directed along the band. An important independent indication is Baer’s \({}^{56}\), that at the eastern edge of the “Pelican” the polarization is large and oriented parallel to the dark fibers. In the bright dusty nebulae in the Pleiades star cluster (constellation Taurus) there are curved fibers. In the work it was noted that the stars in this region exhibit polarization approximately parallel to the fibers*).

Each of the facts cited could be considered accidental, although some of them encompass tens of stars, but the whole set of facts, especially in the Perseus–Taurus region, seems to testify quite reliably to the presence of large-scale magnetic fields, which determine both the orientation of elongated nebulae and fibers and the polarization of light.

Of interest is the character of the field around the remnants of outbursts of novae and supernovae, which usually have the form of filamentary nebulae. It turned out that in IC 443 in Gemini, identified as a source of radio emission, in addition to the ring \({}^{54}\), there is a large number of fibers oriented in the same way, at a large angle to the galactic equator. In NGC 6960-92 there is also a sharply expressed tendency toward identical orientation of the fibers \({}^{57}\). These and several other cases indicate that in the surroundings of a supernova there exists a local magnetic field which, possibly,

*) By polarization here and throughout is meant the direction of the electric vector; in work \({}^{56}\) the magnetic vector is used, and there it is said that the polarization is perpendicular to the direction of the fibers.

is the frozen field of the star itself, carried away from its surface by the envelope.^54 Further, in that work a number of arguments are given in favor of the view that the fibers characteristic of most nebulae also arise with the participation of local magnetic fields. However, this question already lies beyond the scope of the present review.

A very important question is that of the strength of the magnetic field in the arms. For the orientation of dust grains and the explanation of polarization, the strength must be from \(10^{-5}\) to \(10^{-4}\). To restrain nebulae from expanding in the direction perpendicular to the field,^11 the latter must have a strength greater than, or of the order of, \(10^{-5}\) oersted. Chandrasekhar and Fermi^17 made an attempt to determine the strength by two independent methods.

The first is based on the observed dispersion of the directions of the plane of polarization in individual regions, which indicates a dispersion of the directions of the lines of force. The curvature of the lines of force occurs, in the authors’ opinion, because of chaotic motions of the clouds across the general field. This motion may be regarded as a magnetohydrodynamic wave,^3 whose propagation velocity is \(V=(4\pi\rho)^{-1/2}H\). The deviation of a line of force from its initial position is determined by the equation

\[ y=a\cos k(x - Vt). \tag{6.1} \]

whence

\[ \begin{aligned} \frac{dy}{dx} &= -ak\sin(x - Vt); \qquad \dot y = akV\sin(x - Vt);\\ V^2\left(\frac{dy}{dx}\right)^2 &= \dot y^{\,2}. \end{aligned} \tag{6.2} \]

The displacement velocity of a line of force \(\dot y\) is equal to the velocity of motion of the gas in the given direction \(v/\sqrt{3}\); \(\frac{dy}{dx}\) is equal to the mean inclination of the line \(\alpha\). Hence

\[ H=\sqrt{\frac{4}{3}\,\pi\rho}\,\frac{v}{\alpha}. \tag{6.3} \]

Putting \(\rho=2\cdot10^{-24}\), \(v=5\ \text{km/sec}\) and \(\alpha=0.2\) radian, Chandrasekhar and Fermi obtained \(H=7\cdot10^{-6}\). In fact \(\alpha\), apparently, is somewhat smaller,^18 and is rather equal to \(0.12\) radian; the density in the clouds which interest us in the present case, since they contain mainly dust, is \(2\text{–}4\cdot10^{-23}\), and the space velocity of the clouds is of the order of \(10\text{–}12\ \text{km/sec}\). Taking these data into account, \(H\) is approximately equal to \(10^{-4}\). This value, quite sufficient for ordering by the Davis and Greenstein mechanism, is possibly overestimated, since the deviation of the plane of polarization is produced by the motion of the cloud perpendicular to the galactic plane, whereas we have taken the dispersion of velocities from data on interstellar absorption lines,

which characterize the motion in the plane of the Galaxy. It is possible that the velocity dispersion in the direction perpendicular to the plane of the Galaxy is smaller; then \(H\) is correspondingly reduced as well. G. A. Shain applied (6.3) to calculate \(H\) from the dispersion of the directions of elongated features in the constellation Cygnus (angle \(\alpha = \pm 15^\circ\)). Taking \(v = 5\ \mathrm{km/sec}\) and \(\rho = 5 \cdot 10^{-23}\) (comparatively dense formations visible in emission are considered), he obtained the value \(H \simeq 2 \cdot 10^{-5}\), and for dark nebulae \(H \simeq 10^{-5}\).

The second method of estimating the strength of the magnetic field is based on calculating the stability of a spiral arm. The latter is considered as a circular cylinder with radius 250 parsecs. The equilibrium of the gas under the action of gas and magnetic pressure forces and of gravity is considered. The latter is determined mainly by the stars. This method gives \(H \simeq 6 \cdot 10^{-6}\). To summarize, one may say that different methods give, for the strength of a uniform field, values of the order of \(10^{-5}—3 \cdot 10^{-5}\). Similar estimates are given by the condition for confining cosmic rays within the Galaxy and by radio-astronomical data, but for them the main role is apparently played not by a uniform field, but by chaotic fields forming a spherical subsystem. This follows from the nearly spherical shape of the radio-emitting region in our Galaxy and in the Andromeda Nebula, which means that cosmic rays are confined not in the arms, but in a spherical gas subsystem.

The relationship between the chaotic and the regular field is still unclear. Apparently, in the spiral arms, which occupy less than \(1\%\) of the volume of the spherical Galaxy, the regular field predominates; chaotic motions affect only the curvature of the lines of force. Outside the arms there is apparently no regular field; the principal role is played by chaotic fields induced by turbulent motion. The question of the origin of the regular field is still completely unclear. The possibility of its formation as a result of the differential rotation of the Galaxy, in which the inner parts move with a greater angular velocity than the outer ones, has been considered.\(^{8,58}\) In this case the lines of force should be stretched in the direction of the displacement, and some preferential direction of the magnetic field should be formed. However, in the process of such stretching much still remains unclear. Moreover, since during the lifetime of the Galaxy it has made less than one hundred revolutions, the indicated process could have amplified the initial field of large scale by about 100 times (a chaotic field is not amplified in this way). It is almost as difficult to imagine the formation of a regular field with strength \(10^{-7}—10^{-6}\) as one 100 times stronger, if one takes into account the relaxation time of the field in the Galaxy and the maximum current density (under stationary conditions) for mean values of the conductivity and of the gradients of temperature and density.\(^{8}\) Perhaps this initial regular

...field is a remnant of the field that existed in the metagalactic medium from which the Galaxy was formed. There this field could have been induced, for example, by large-scale motion, as in a magnetohydrodynamic wave. It is still premature to discuss all these questions.

A regular magnetic field is apparently the main factor contributing to the preservation of spiral structure. As was noted^11, the gas of the spiral arms would quickly disperse in the Galaxy if it were not held by magnetic forces. In addition to gas, the spiral arms contain hot stars, apparently recently formed. Stars are not held by the magnetic field; therefore, the formation of stars in the spiral branches shows that the material from which they formed had rather the form of diffuse matter than of compact heavy bodies. Thus, the problem of the regular magnetic field of the Galaxy is closely connected with questions concerning the nature of spiral structure, the formation of stars and nebulae, the relation between stars and diffuse matter, and, finally, with the question of the formation of the Galaxy as a whole, and is one of the most important problems of modern astrophysics.

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