THE MAGNETIC FIELD OF ROTATING MASSIVE BODIES*

P. M. S. Blackett

FROM THE EDITORS

In printing below a translation of an interesting article by Prof. P. Blackett, we consider it necessary to supplement the historical survey given by the author with a reference to the well-known work of P. N. Lebedev, “Magnetometric Investigation of Rotating Bodies. First Communication” (Journal of the Russian Physico-Chemical Society. Physical Section, vol. 45, p. 484, 1911; see also Collected Works, p. 207. Moscow, 1913**). The published first communication represents only the beginning of a large work, interrupted by the premature death of P. N. Lebedev. In this work Lebedev set himself the task of “directly investigating, by magnetometric means, the magnetic phenomena caused by the rotation of bodies.” The impetus for the work was the magnetic field of sunspots discovered by Hale (in 1909). In the first part of the work two hypotheses were subjected to experimental verification: 1. Sutherland’s hypothesis (see p. 56 of Blackett’s article); 2. the hypothesis of centrifugal displacements, according to which a rotating body acquires a negative surface charge as a result of centrifugal accelerations, so that negative charges, according to this hypothesis, experience greater accelerations than positive ones. To test these hypotheses Lebedev constructed an apparatus in which the body under investigation (a ring 6 cm in diameter) was set into rotation at a speed of 30,000–35,000 revolutions per minute, and the magnetic field was measured directly with a sensitive magnetometer.

The experiments gave a negative result, although the calculation made on the basis of both hypotheses showed that the sensitivity of the magnetometer would have been more than sufficient to detect the magnetic field if these special hypotheses had been correct.

* Reported on May 15, 1947, at the Royal Society in London. Nature, May 17, 1947, vol. 159, pp. 658–666, No. 4046. Translated by K. E. Willer.
** P. N. Lebedev’s work was also published in German: Ann. d. Physik, 39, 840 (1912).

In concluding his work, Lebedev writes: “As the experiments have shown, the above-stated hypotheses concerning the formation of magnetic fields around rotating bodies have not withstood direct experimental verification. The two hypotheses indicated by no means exhaust the possible connection between the motion of matter and the formation of magnetic fields, which we observe in the case of sunspots and the phenomena of normal geomagnetism. Other hypotheses that may be advanced concerning this connection, and that are sufficient to explain the magnetic forces of very large moving masses, lead one to expect that, under the conditions and dimensions of the experiments described above, only very weak magnetic fields can arise, which cannot be detected magnetometrically; to test these hypotheses, the very scheme of the experiments must be changed, so as to obtain sufficient sensitivity of measurement, many times greater than that which could be used in the preliminary investigations described above.”

These words, once again testifying to P. N. Lebedev’s remarkable physical intuition, should be compared with the results of the calculations given in § 5.3 of Blackett’s article.

Brief Summary

It had long been known, especially from the work of Schuster, Sutherland, and Wilson—though, it is true, little attention has recently been paid to this fact—that the magnetic moments \(P\) of the Earth and the Sun are almost proportional to their angular momenta \(U\), and that the coefficient of proportionality is approximately equal to the square root of the gravitational constant \(G\), divided by the speed of light \(c\). We may therefore write:

\[ P = \beta \frac{G^{1/2}}{2c} U, \]

where \(\beta\) is a constant of the order of unity.

Recently the magnetic field of the star 78 Virginis was measured for the first time (Babcock, 1947). Calculated by means of this equation from the most accurate known values of its mass, radius, and speed of rotation, the magnetic field of this star agrees with the measured value. We thus have a rough confirmation of the validity of this equation for three bodies: the Earth, the Sun, and 78 Virginis, with the values of \(P\) and \(U\) varying within the limits \(10^{10}:1\), whereas the measured values of the magnetic field differ only in the ratio \(2000:1\). The author therefore considers that the above equation deserves serious attention, since it expresses a possible general law of nature for all rotating massive bodies.

If white dwarf stars are formed by contraction from stars similar to the Sun, then they must have the same angular momentum and, consequently, according to the equation given above, the same magnetic moment. In view of their small dimensions, the magnetic field at their surfaces should be of the order of \(10^6\) gauss. The idea is expressed that magnetic fields of this order may be the cause of the large width of the Balmer lines in the majority of white dwarf stars, as well as of the complete absence of any lines in certain very dense stars. Various mutually exclusive theories of the magnetic field of the Earth and the Sun are considered.

A preliminary proposal is put forward: to consider that, according to the available observational evidence, with increasing values of the expression given below, certain new basic properties of rotating matter become expressed. It is possible that this relation will yield the long-sought connection between electromagnetic and gravitational phenomena.

§ 1. THE MAGNETIC FIELD OF THE EARTH, THE SUN, AND A STAR

In investigating the possible influence of the magnetic fields of the stars of the Galaxy on cosmic radiation, I drew attention to two facts which seemed to me extremely interesting from the physical point of view. The magnetic moments \(P\) of the Earth and the Sun are almost proportional to their angular momenta \(U\), which are calculated by the formula

\[ U=\frac{2}{5}\omega MR^{2} \tag{1} \]

for a uniformly dense sphere with mass \(M\), radius \(R\), and angular velocity \(\omega\). This proportionality may be verified from the data of Table 1.

Table 1

The measured values of the magnetic fields at the poles are also given here. They are related to the magnetic moments by the expression:

\[ H_p=2P/R^{3}. \tag{2} \]

It has long been known that the directions of rotation and of the magnetic moments of both bodies are also related to one another; indeed, the south magnetic poles of both bodies are situated close to their northern geographic poles.

It was natural to compare the mean ratio \((P/U)_1\) of magnetic moment to angular momentum for these two astronomical bodies with the analogous ratio for the Bohr magneton

\[ (P/U)_2=e/2mc=0.88\cdot10^{7}\ \mathrm{cm}^{1/2}\mathrm{g}^{-1/2}. \tag{3} \]

We obtain

\[ (P/U)_1/(P/U)_2=1.08\cdot10^{-22}. \tag{4} \]

and we immediately see that this numerical value is close to the value of the well-known dimensionless ratio

\[ G^{1/2}m/e = 4.90 \cdot 10^{-22} \tag{5} \]

of the gravitational mass of the electron to its charge, expressed in electrostatic units. We may therefore write:

\[ (P/U)_1=\beta\,\frac{G^{1/2}m}{e}\,(P/U)_2, \tag{6} \]

where \(\beta\) is a small numerical constant of order \(1/4\); whence, using (3), we obtain the following relation between the magnetic and angular momenta of both astronomical bodies:

\[ P=\beta\,\frac{G^{1/2}}{2c}\,U. \tag{7} \]

The simplicity of this result, whether in the form (6) or (7), suggested the idea that it must have a deep physical meaning.

This prompted me to undertake a study of the extensive literature concerning the origin of the magnetic fields of the Earth and the Sun, and, to my surprise, I found that the essence of these facts had been known for many years, but for various reasons interest in these questions has recently ceased. In the following paragraph a brief review of earlier work will be given.

In searching for further ways of testing equation (7), I immediately thought of the possibility of measuring the magnetic field of some rapidly rotating star. I am indebted to Professor S. Chandrasekhar, who informed me of the first measurement made of the magnetic field of a star. The measurements by Babcock¹ over the star 78 Virginis, already published at the present time, gave a field at its pole of 1500 gauss. In order to calculate \(P\) and \(U\), we must know the mass of the star, its radius, and its angular velocity. Statistical data for stars of this type give, for these quantities expressed in terms of the corresponding solar quantities, the following values: \(M=2.3,\ R=2.0,\ \omega=25\) (see § 4). Table 2 gives the required data for the star.

Table 2

We again obtain approximately the same ratio for \(P/U\), showing that equation (7) is valid not only for the Earth and

of the Sun, but also for the star 78 Virginis. The relation between the directions of rotation and of the magnetic field is unknown.

This further confirmation of the validity of equation (7) says, on the one hand, that it deserves serious attention as expressing a possible new general law of nature, and, on the other hand, that a more rigorous check of its lawfulness must be made, taking into account the variation of density and, possibly, also the rotation inside bodies.

§ 2. THE FIRST THEORIES OF THE MAGNETIC FIELD OF THE EARTH AND THE SUN

Although it is known that the principal dipole field of the Earth is of internal origin, it is still unclear how it is excited. Chapman and Bartels² (1940) critically examined the extensive literature on this question. The earlier reviews by Schuster³ (1912), Brunt⁴ (1913), and Swann⁵ (1923) are still of great value. In one of the more recent papers, Cowling⁶ (1945) considers possible theories of the Sun’s field.

Theories of the principal field of the Earth or the Sun may conveniently be divided into two classes, which may be called particular and general theories.

A particular theory ascribes the magnetic properties of a rotating body to the specific properties of the matter of which it consists; according to this theory, the magnetic moment of a rotating massive body will depend on the specific electrical, magnetic, thermal, and mechanical properties of the material and, in all probability, will have entirely different values for different physical states of the body.

The first theory of this kind, according to which the Earth was regarded as ferromagnetic, was long ago rejected, since the internal temperature of the Earth is almost certainly far above the Curie points for any material that could be present there. Another possible variant is, evidently, the assumption of the existence in the Earth of an electric conduction current. The difficulty here lies in finding a mechanism that maintains it. In order to justify the sign of the field, it is necessary that the positive current should flow from east to west, or that the negative current should flow from west to east.

Elsasser⁷ recently (1939) attempted to ascribe the current to thermoelectric electromotive forces arising as a consequence of temperature differences in the liquid core, caused by convective motions. The considerable asymmetry required to create circulation around the axis is attributed by him to the action of Coriolis forces on masses in convective motion. In a subsequent paper⁸ (1941) Elsasser considers the relation between the strong main dipole field of the Earth and its irregular part, amounting to several pro—

centers from the principal field. He assumes that both these fields are produced by a central core of molten metal, whose radius is about \(0.55 R\).

A somewhat similar theory of Frenkel\({}^{9}\) (1945) additionally introduces a self-excitation mechanism, the same as that which, according to the supposition of Gurevich and Lebedinsky\({}^{10}\), accounts for the magnetic field of sunspots. The first theory to suppose the presence of self-excitation, or of a dynamo mechanism, was advanced by Larmor\({}^{11}\) (1919) and then developed in detail by Cowling\({}^{12}\) (1933), but it turned out that it could explain neither the magnetic field of sunspots nor the principal fields of the Earth and the Sun. Cowling\({}^{6}\) (1945) also showed that the thermal effects caused by convection in the rotating Sun give only the correct sign of the field, and in magnitude only \(10^{-7}\) of its measured value. He also showed that the time of electromagnetic decay of the Sun’s general magnetic field is \(10^{10}\) years, so that it might be that the field we observe at present represents the remnants of an entirely different primordial state. However, Lamb (see, for example,\({}^{2}\) p. 704) long ago showed that the decay of currents in the Earth would occur too rapidly for it to be admissible that the field observed by us at present is a relic.

The general theory ascribes the existence of a dipole field to some general property of rotating matter. As early as 1891 Schuster\({}^{13}\), suspecting, on the basis of the appearance of the solar corona, that the Sun may have a magnetic field, raised the question whether “every rotating mass is not a magnet.”

The only obvious atomic effect is magnetization upon rotation, i.e. the gyro-magnetic effect. This gives the correct sign of the field, but a magnetic moment \(10^{10}\) times smaller (\({}^{2}\) p. 705). However, as Schuster\({}^{3}\) showed, with such uniform magnetization the total moment is proportional to the volume of the sphere and, consequently, the field at the surface does not depend on the radius. Thus, any such theory can be excluded.

In a whole series of papers Sutherland\({}^{14}\) (1900—1908) assumed that the Earth’s magnetic field is due to the Earth possessing a positive volume charge, which is compensated by a negative surface charge. He calculated the required charge density and showed that this entails the presence within the Earth of electric fields of the order of \(10^{8}\ \mathrm{V/cm}\). He also drew attention to the fact that the separation of charges, calculated from the measured magnetic field of the Earth, has almost the same magnitude as would be obtained if one assumed, on the basis of the Lorentzian relativistic theory (see, e.g.,\({}^{3}\)), that the separation is explained by a small difference in the electric forces acting between two protons, two electrons, and an electron and a proton. This is essentially the first confirmation of a numerical result established later in explicit

in Wilson’s form^15, which gives a somewhat different expression for the result we cited above in paragraph 1.

In 1912 Schuster made a survey^3 of possible theories and their difficulties, and investigated in detail one of them, a theory similar to Sutherland’s, based on the assumption of unequal forces acting between like and unlike particles of the atom. Brent^4 (1913) considered many different theories and also showed that the separation of the positive and negative charges of the Sun cannot account for more than \(10^{-15}\) of the Sun’s magnetic field.

Following Schuster’s idea, T. A. Wilson^15 (1923) showed that the correct order of magnitude of the fields of the Earth and the Sun is obtained if it is assumed that a moving element of mass \(M\), measured in gravitational units, produces the same magnetic effect as a moving electric charge \(Q\), measured in electrostatic units. The measure \(M\) of the mass of a body in gravitational units is determined by the equation \(F=M_1M_2/r^2\) for the force of attraction between two masses. Comparing this with the expression \(F=Gm_1m_2/r^2\), where \(m_1\) and \(m_2\) are measured in grams, we obtain \(M=G^{1/2}m\), whence the gravitational unit of matter is equal to \(G^{-1/2}=3.870\) g.

Expressed formally, Wilson’s hypothesis is equivalent to the assumption that an element of mass of \(m\) grams, or of \(M\) gravitational units, when moving with velocity \(\mathbf{v}\), gives at a distance \(r\) the magnetic field

\[ \mathbf{H}=-\frac{M}{cr^3}[\mathbf{v}\cdot\mathbf{r}] =-\frac{G^{1/2}m}{cr^3}[\mathbf{v}\cdot\mathbf{r}], \tag{8} \]

by analogy with the nonrelativistic expression for the magnetic field of a moving charge of \(Q\) electrostatic units, which is equal to

\[ \mathbf{H}=\frac{Q}{cr^3}[\mathbf{v}\cdot\mathbf{r}]. \tag{9} \]

From (8) we obtain by integration that the magnetic moment \(P\) of a sphere of mass \(M\), uniform density \(\rho\), radius \(R\), and angular velocity \(\omega\) is given by the expression

\[ P=\frac{1}{5}\frac{G^{1/2}}{c}\omega MR^2. \tag{10} \]

Let us note that (1) and (10) together give (7), but without the constant \(\beta\). According to (2), the magnetic field at the surface at the pole is equal to

\[ H_p=\frac{2}{5}\frac{G^{1/2}}{c}\frac{\omega M}{R} \tag{11} \]

or, expressed in terms of the density \(\rho\):

\[ H_p=\frac{8}{15}\pi\frac{G^{1/2}}{c}\omega\rho R^2. \tag{11a} \]

Wilson showed that (11) gives the correct ratio of the fields of the Sun and the Earth, but the numerical values come out three times too large.

This result may also be expressed in another way: the Earth’s magnetic field is such as if the Earth possessed a negative volume charge of density \(\sigma\), given in terms of the mass density \(\rho\) by the expression

\[ \sigma \simeq G^{1/2}\rho . \tag{12} \]

Angenheister \(^{16}\) (1925) also drew attention to the circumstance that the assumption of proportionality between the density of electric charge and the density of mass gives approximately the correct ratio of the magnetic fields of the Sun and the Earth. He clearly recognized the physical difficulties connected with the assumption of the existence of real charges of such magnitude and of the strong electric fields which must accompany them in such an electrically conducting material as the Earth’s core.

Wilson \(^{15}\) (1928) carried out a laboratory experiment with a swinging iron rod in order to test the hypothesis that a moving mass creates a magnetic field. He showed experimentally that the field determined by (8), under the assumption that \(\dot m\) is the mass of the Earth and \(v\) the relative velocity of the iron rod, does not exist. Other arguments against this hypothesis are given in paragraph 5.

Schlomka \(^{17}\) (1933) examined in detail a theory essentially based on Schuster’s hypothesis of unequal Coulomb forces acting at large distances between identical and unlike particles.

Haalk \(^{18}\) (1937—1938) attributes the required separation of charges not to a difference in the Coulomb forces acting between electrons and nuclei, but to forces acting at small distances between electrons and molecules, i.e. to forces determining the elastic properties of the material. He assumes that such forces must lead to a separation of charges proportional to some power of the radius and having a maximum at the center of the Earth.

Suyun \(^{19}\) (1927) developed a complicated theory based on a small arbitrary modification of the equations of electrodynamics, chosen in such a way that they give the required magnetic field without an undesirable electric field. However, Suyun’s basic hypothesis was that the magnetic field of a rotating body is proportional to \(\rho \omega^{4} R^{4}\), and not to \(\rho \omega R^{2}\) according to (11a). This was adopted in order to avoid the difficulty connected with translational motion, and is equivalent to the assumption that the magnetic field created by an element of mass depends on certain time derivatives of its velocity and thus is equal to zero in uniform translational motion. Suyun chose this special form because it gives the correct ratio of the fields of the Sun and the Earth. However, this theory also gives relatively large values for the fields of rotating bodies of laboratory dimensions, the existence of which was

later refuted by the experiments of Suonna and Longmire \(^{20}\) (1918), than this theory too was refuted, at least for small bodies. Suonna’s theory also encounters other objections, namely that it appears improbable that the vector quantity \(\mathbf P\) could depend on an even power of the vector \(\omega\), and, moreover, this theory does not give the correct numerical value of the magnetic moment, except in cases where arbitrary parameters are introduced.

Cases in which the general theory is justified

Chalmers and Bartels \(^{2}\) (1940, vol. 2, p. 701) consider all the various theories in considerable detail, but they do not attach special importance to the separate parts of the work of Sutherland, Schuster, Wilson, and Einstein, which together establish the results presented above. In general they come to a conclusion against the general theory and in favor of a private theory of some type, but they do not consider that any of the private theories existing at that time can be used with confidence. Their principal arguments against the general theory consist in the presence of an inclination of the magnetic axis to the axis of rotation, which is equal to \(4^\circ\) for the Sun and \(12^\circ\) for the Earth, and also in the existence of secular variation. For the Earth this means that if the general theory could even explain the existence of its principal symmetric magnetic field, it would still remain to explain the presence of a field amounting to as much as \(10\%\) of the principal one. It seems to me, however, that if it were possible to find a general explanation for the existence of the principal field, then it might easily turn out that the changes introduced into it by convection motions, etc., which must exist inside the Earth, would be sufficient to give the observed field.

Undoubtedly, it is much more difficult to find such a private theory that would lead to the following two striking results: (a) that the magnetic moments of the Sun and the Earth are almost proportional to their angular momenta, while these vary within limits differing by more than \(10^{8}:1\); (b) that the coefficient of proportionality is approximately equal to \(G^{1/2}/c\), than to explain the deviation of the directions of the two axes and the existence of secular variation.

It appears exceptionally improbable that the approximate validity of equation (7) could be accidental. The simplicity of this equation, containing only two macroscopic constants \(G\) and \(c\), is in sharp contrast with the complex and arbitrary character of the special theories advanced hitherto, which have failed to give a qualitative explanation either for the existence of the field of the Earth or of the Sun, still less of both.

The presence of only a single constant \(G^{1/2}/c\), multiplied by the angular momentum, apparently rules out—apart from the case of an astonishing number of numerical coincidences—the possibility of determining the magnetic

of the moment of a rotating body with any of its particular properties, except its size, mass, and rotation.

If \(P\) in fact depends only on the macroscopic quantities \(M, R, \omega, G, c\), then from dimensional considerations one can find functional dependences of the corresponding form. Restricting ourselves to consideration of cases where \(P\) is proportional to \(M\), we find that \(P\) can be expressed in the form:

\[ P \sim G^{1/2}c^{-1}\omega MR^2 f(\omega R/c). \tag{13} \]

For \(f(\omega R/c)=\mathrm{constant}\) this gives our original expression (7). Since \(P\) must depend on an odd power of \(\omega\), the most probable expression for it is obtained if one puts \(f(\omega R/c)=(\omega R/c)^2\), which gives

\[ P \sim G^{1/2}c^{-3}\omega^3 MR^4 . \tag{14} \]

Whereas our original expression gives both the observed dependence of \(P\) on \(M, \omega\), and \(R\), and also its roughly correct numerical magnitude, this expression gives neither. It is easy to see that the latter expression differs from the former only by the factor \((\omega R/c)=v/c\), if \(v\) is the peripheral velocity. It is, of course, possible that at very large peripheral velocities it will be necessary to introduce a relativistic correction containing \((v/c)^2\).

§ 3. THE ANGULAR MOMENTUM OF A STAR

As was already explained above, the approximate validity of equation (7) was first noticed in comparing the magnetic moments of the Earth and the Sun under the assumption of uniform density for both. In reality, however, this is far from the case, especially for the Sun, and therefore in calculating angular momenta it is necessary to take into account the condensation of these bodies at the center.

It is convenient to introduce the ratio

\[ k = I/I_0 \tag{15} \]

of the moment of inertia of a given sphere, condensed at its center, to the moment of inertia of a sphere of the same radius and rotating with the same angular velocity, but with uniform density equal to the mean density of the sphere condensed at its center. Then instead of (1) we obtain

\[ U=\frac{2}{5}k\omega MR^2 . \tag{16} \]

For the Earth and most planets, \(k\) is known from the work of Jeffreys\(^{21}\) (1924, 1937). For the Sun and other stars it is necessary to calculate its value by adopting some stellar model. We have, first of all, Emden polytropic gas spheres,

for which Eddington^22 (1926) gives the density distribution for three different values of the polytropic index \(n\). There also exists a still later point-convection model by Cowling^23 (1935), for which new calculations were made by Blanch, Lowan, Marshak, and Bethe^24 (1941) and by Marshak and Blanch^25 (1946).

From the density distribution, by graphical integration, the values of \(k\) were roughly calculated; these are given in Table 3 together with the ratio of the density at the center of the body \(\rho_0\) to the mean density \(\bar{\rho}\).

Table 3

Density at the center \(\rho_0\) and relative moment of inertia of stellar bodies

Thus, for the Sun the value of \(k\) is subject to considerable uncertainty. Below we shall adopt for it the value 0.16, as the mean of the point-convection model and the polytrope with index \(n = 3\).

Non-uniform rotation

From considerations of stability it appears probable that the angular velocity of a gaseous stellar body will not be uniform, but should increase toward its center (see Milne^26, 1930, p. 241). On the other hand, it is known that the rotational velocity on the surface of the Sun at the poles is 40% less than at the equator. Although there still apparently exists no reliable method for estimating this effect, it is convenient to introduce the quantity:

\[ \eta = U/U_0, \]

where \(U\) is the true moment and \(U_0\) is the angular moment calculated from the measured peripheral velocity at the equator, allowing for the variation of density but not for the variation of angular velocity. Then we know that \(\eta\) will be of order unity, but in all probability greater than unity.

I am indebted to Prof. Cowling for communicating that the available data on the mean period of rotation of the entire mass of the Sun are very uncertain, but that this period is probably shorter than ten days, although in

in reality it may also be more than twenty-five days. This gives for \(\eta\) limits between 2.5 and, say, 0.7.

Thus we have the following expression for the angular momentum of a rotating celestial body:

\[ U=\frac{2}{5}k\eta\omega MR^2, \tag{17} \]

which, together with (2) and (7), gives:

\[ H_p=\frac{2}{5}\beta G^{1/2}k\eta\omega M/Rc, \tag{18} \]

where \(\omega\) is the measured angular velocity at the surface at the equator. Only for a few stars are \(M\) and \(R\) accurately known; in general they have to be derived statistically. The values of \(k\) and \(\eta\) can, of course, be obtained only from a definite theoretical model of the internal structure of the star.

§ 4. THE MAGNETIC FIELD OF A ROTATING STAR

4.1. Babcock’s measurements of the star 78 Virginis

It had long been known that many stars rotate much more rapidly than the Sun and, consequently, according to our hypothesis that equation (7) expresses a general law, these stars should have very strong magnetic fields. However, only quite recently was the magnetic field of a star other than the Sun measured for the first time. This was done by Babcock¹ (1947) for the star 78 Virginis (spectral type A2), and, as has already been stated above, it was gratifying to find that, within the limits of the rather large error in measuring the magnetic field and the statistical uncertainty in determining the mass, radius, and angular velocity, the results agree with what we expected, although the relation between the directions of the magnetic field and of rotation is unknown.

The rotational velocity of stars is measured from the broadening, due to the Doppler effect, of the spectral lines of light coming from different parts of the disk. A general description of the method is given by Elvey²⁷ (1930), Rosseland²⁸ (1936, p. 262), and Becker²⁹ (1942, p. 66). Stars of early spectral types O, B, and A and early F are usually in rapid rotation, most often with peripheral velocity about \(100\ \text{km/sec}\), compared with the velocity of the Sun (type G0) of \(2\ \text{km/sec}\). A small fraction of the stars rotate with velocities greater than \(200\ \text{km/sec}\) (see Table 4). Rotation seems to disappear abruptly in stars belonging to the types between F2 and F5, and no observable rotation whatever, i.e. peripheral velocities greater than, say, \(20\ \text{km/sec}\), has been observed for stars of later spectral types. However, a few stars of early types give narrow spectral lines, and this is interpreted as indicating that their axes of rotation are directed parallel to the line of sight.

line of sight. Only on such stars can the Zeeman effect be measured at the present time.

Babcock’s measurements for 78 Virginis give, for its field at the pole, 1500 gauss. He takes its equatorial velocity to be \(60\ \mathrm{km/sec}\) and, comparing the figures he obtained with the field of 53 gauss and the equatorial velocity of \(2\ \mathrm{km/sec}\) for the Sun, comes to the conclusion that the field is proportional to the equatorial velocity. However, from the physical point of view this is not convincing.

To check equation (7) we need to know \(M\), \(R\), and \(\omega\). From Becker’s\(^ {29}\) data (1942, p. 64) in tables for different spectral classes, we find for a star of type A2 that \(M=2.3\) and \(R=2.0\), relative to the values of these same quantities for the Sun. From Becker’s data (p. 69) on rotational velocities, given in Table 4, it follows that the mean value of the equatorial velocity of stars of type A is rather about \(100\ \mathrm{km/sec}\) than the \(60\ \mathrm{km/sec}\) adopted by Babcock. The angular velocity \(\omega\) should therefore be 25 times greater than the Sun’s velocity, i.e. almost the same as for the Earth.

Table 4

Distribution of equatorial rotational velocities for stars of early spectral types (Becker 1942)

If it is assumed that \(k\) and \(\eta\) have the same values for the Sun and 78 Virginis, then according to (11) the ratio of their fields at the poles should be proportional to \(\omega M/R\), and consequently it should be equal to 29, whereas the measured value is 28. Taking into account the statistical character of the data for the star, which is clearly seen from Table 4, such a close agreement must be regarded as entirely accidental. The direction of the magnetic field was determined, but the direction of rotation, of course, was not. Thus, for this star it is impossible to compare the signs of \(P\) and \(U\).

4.2. Comparison of the Data for the Earth, the Sun, and the Star 78 Virginis

Table 5 collects the data for all three bodies. For the Earth, Chapman’s\(^ {30}\) (1943) data are given for the component of the dipole moment in the direction of the axis of rotation.

MAGNETIC FIELD OF ROTATING MASSIVE BODIES

Quite recently one uncertainty was clarified, namely, concerning the nature and magnitude of the Sun’s dipole field. Hale’s early measurements gave a much more rapid decrease of \(H\) with height in the Sun’s atmosphere than should have been the case for a dipole field. However, Tissein’s work\(^{31}\) (1946) apparently

Table 5

The ratio \(P/U\), allowing for condensation at the center

definitively established the existence of the Sun’s dipole field and gave for the field at the pole the value \(53\pm12\) gauss.

In the last column of Table 5 are given the values of \(\beta\), calculated with the aid of equation (7) from the measured values of \(P\) and \(U\) for
\[ G^{1/2}/2c=4.31\cdot10^{-15}\ \mathrm{cm}^{1/2}\,\mathrm{g}^{-1/2}. \]
It is easy to see that \(\beta\) is nearly equal to unity for the Sun and 78 Virginis, but only about \(0.3\) for the Earth; this latter circumstance was first noted by Wilson\(^{15}\) (1923). \(\beta\) may also be obtained directly from the expression:

\[ \beta=\frac{5}{2}\,\frac{H_pRc}{k\omega MG^{1/2}}. \tag{18a} \]

This equation is convenient to use, since from it one can see how errors in the determination of the various quantities affect the value of \(\beta\).

It is necessary to note that the almost exact proportionality between \(P\) and \(U\) for the Earth and the Sun, found under the assumption of their uniform density, is violated if condensation at the center of the Sun and the Earth is taken into account.

In Table 6, for comparison, are given the measured relative values of \(H_p\) for the three bodies, and also those calculated from equation (7); moreover, for the case of uniform density the angular momenta were calculated with the aid of equation (1), and for spheres condensed at the center, with the aid of equation (16).

Table 6

Measured and calculated values of the field at the pole

4.3. Possibility of further testing the main conclusion on stars

It is clear that, in order to test the theory, it is extremely important to carry out further measurements of the magnitude and direction of the magnetic field of rapidly rotating stars. Babcock[^1] (1947) asserts that the relation between the directions of \(H'\) and \(\omega\) could be determined on visual binary stars. As is seen from Table 4, some stars belonging to the early spectral classes and resembling 78 Virginis to a certain degree have rotational velocities almost 2.5 times greater [see also Struve[^32] (1930), Uestett[^33] (1933, 1934)]. Therefore the field of such stars at the poles should be of the order of 4000 gauss.

The maximum magnetic field is determined by Roche’s limit*), which indicates the maximum angular velocity \(\omega_m\) for stability. This maximum angular velocity is expressed as

\[ \omega_m = 1.52G^{1/2}\bar{\rho}^{1/2}, \tag{19} \]

where \(\bar{\rho}\) is the mean density (Jeans[^34], 1928, p. 246). This indicates that angular velocities exceeding the velocity of 78 Virginis by a factor of 4 are possible, and consequently the maximum field of a star of this type may reach 6000 gauss. If in (11a) we substitute \(\omega_m\) from (19), then we obtain that the maximum possible field \(H_m\) of a star is proportional to \(\bar{\rho}^{1/2}R^2\). The variation of \(\bar{\rho}\) and \(R\) as functions of the spectral type of the star (p. 62) shows that, for the main sequence, \(H_m\) changes comparatively slowly in passing from one type to another. We find the relative values of \(H_m\) for the Sun: \(H_m = 1.2\) for stars of type (M0) and \(H_m = 0.5\) for type (B0). Stars of the latter types, although apparently they normally do not rotate rapidly, are capable dynamically of producing stronger fields than stars of earlier types.

) See Jeans, Astronomy and Cosmogony*, 1928, p. 246.

It would be best to test the theory on a star for which one could directly measure both the magnitude and the direction of the magnetic field, as well as the angular velocity, rather than relying, in determining the angular velocity, on statistical data, as in the case of 78 Virginis. In principle this appears feasible by simultaneous measurement of the Zeeman effect and the Doppler effect of light coming from the nearly eclipsed edge of the rear component of an eclipsing binary star. Here, however, the experimental difficulties will probably be very great.

4.4. White Dwarfs

White dwarfs, if they rotate with angular velocities comparable with that of the Sun, should also have a strong field, because of the large value of the ratio \(M/R\), on which the field strength depends [equation (11)]. For many white dwarfs the value of this ratio, which also determines the red shift, exceeds its value for the Sun by a factor of 20–30 (Russell \(^{35}\), 1945, p. 760), and consequently their field should be of the order of 1300 gauss. If, however, for example, the dark companion of Sirius has the same angular velocity of rotation as Sirius itself, i.e. if its period of revolution is 8 days, then its magnetic field will be of the order of 5000 gauss.

Kuiper \(^{36}\) (1941) expressed the idea that white dwarfs may possess angular momenta comparable with the angular momentum of the Sun, which is what one would expect if the dwarfs were formed as a result of the contraction of stars such as the Sun. Sirius B has the same mass as the Sun, while its radius is only \(1/50\) of the solar radius. If its angular momentum is equal to that of the Sun, then according to our hypothesis their magnetic moments will likewise be equal. The magnetic field of Sirius B will therefore be \(50^3\) times greater than the field of the Sun, i.e. the field at its equator will be equal to \(3 \cdot 10^6\) gauss.

The most conspicuous, and often the only, lines in the spectra of dwarfs are the Balmer hydrogen lines. These lines, however, are broader than in normal main-sequence stars. Such a breadth is usually attributed to broadening of the lines under the action of pressure, i.e. to the Stark effect caused by atomic collisions, which in these stars must be exceptionally intense because of the high pressure associated with very strong gravitational fields. From Kuiper’s \(^{36}\) (1941) arguments it follows that a rough quantitative explanation of these lines can be given.

It is interesting, however, to note that a magnetic field of several million gauss will cause the same broadening of spectral lines. Since the double Zeeman splitting of the normal triplet of the wavelength \(H\gamma\) is approximately equal to \(2 \cdot 10^{-5}\) Å/gauss, the width of the Zeeman splitting should be of the order of 60 Å. It must be remembered that, owing to the Paschen–Back effect, the Balmer li-

will give an almost normal triplet. Since the field at the poles is twice the field at the equator, the calculated value of the width will give approximately twice the width for half the light intensity from the star as a whole. Sirius B itself apparently gives comparatively thin Balmer lines, with a width of less than \(60\,\text{\AA}\), and also some metallic lines. However, a whole series of dwarfs—for example, 40 Eridani B and Wolf 1346—give Balmer lines with a width of about \(50\,\text{\AA}\) (Kuiper1, 1941). Thus one cannot regard as implausible the supposition that the lines of white dwarfs are noticeably broader than those of main-sequence stars not only because of the strong Stark effect, but partly also because of a strong Zeeman effect. Kuiper’s comparison of the spectra of 40 Eridani B and the comparison star shows that the width of the latter’s lines, apparently connected with pressure broadening, would be sufficient to obscure the Zeeman-splitting structure typical of a white dwarf, in the form of a double triplet.

Many of these stars give broad Balmer lines and a comparatively narrow central part of the spectrum, which is difficult to explain by the Stark effect; it is possible that it owes its existence to the undisplaced components of the Zeeman splitting, which remain unchanged over the whole disk.

Of great importance, perhaps, is the circumstance that most dwarfs whose sizes are assumed to be still smaller than that of Sirius B either give no Balmer lines at all (Wolf 489, 457, 219), or give very weak ones (AC 70, 8247). One star (Ross 627) is anomalous in this respect, since it gives comparatively narrow lines; nevertheless, it is assumed to have a high density.

In the spectrum of the star (AC 70, 8247) there are two weak lines, with widths greater than \(100\,\text{\AA}\), centered at \(4480\,\text{\AA}\) and \(4140\,\text{\AA}\). If these lines are taken to correspond to \(H\gamma\) (4341) and \(H\delta\) (4102), and their width is attributed to the Zeeman effect, then it turns out that the magnetic field required for this must be of the order of \(10^7\) gauss. If Kuiper’s estimate of the radius as \(0.004\) of the Sun’s radius is used, then the magnetic moment, and also the angular momentum of the star relative to the corresponding moments of the Sun, will be

\[ (10^7/25)\cdot(0.004)^3 = 0.024. \]

It is possible, however, that the observed weak lines are not complete absorption lines, but only their thinner central part. The true width in this case will be considerably greater than \(100\,\text{\AA}\), and, consequently, the angular momentum will also be greater and closer to the Sun’s moment.

It is clear that it is urgently necessary to measure the polarization in the spectra of white dwarfs, and also to compute the line profiles that result from the Zeeman effect in strong fields with simultaneous pressure broadening.

I am indebted to Prof. H. H. Plaskett for pointing out the existence of one white dwarf, Wolf 489 (see\(^{36}\) p. 211), which confirms the correctness of explaining the broadening of the lines by the Zeeman effect rather than by the Stark effect. This star is unusual in that it belongs to a later color type (G8) and, consequently, should have intense lines of ionized calcium. In reality, however, it shows no lines at all. Since the Ca\(^+\) lines are much less subject to the action of the Stark effect than the hydrogen lines, their absence is difficult to explain by the Stark effect. Consequently, the Zeeman effect, to which the H and Ca\(^+\) lines are equally sensitive, gives a more plausible explanation.

Since the width of the Zeeman splitting \(\Delta\lambda_Z\) is proportional to the magnetic field at the surface and, consequently, according to (2) and (7), also to \(U/R^3\), while the width of the Doppler displacement \(\Delta\lambda_D\) is proportional to the peripheral velocity, i.e. \(U/MR\), the ratio \(\Delta\lambda_Z/\Delta\lambda_D\) is proportional to \(M/R^2\) and, consequently, does not depend on the velocity of rotation. For very small and dense stars the width of the Zeeman splitting becomes greater than the width of the Doppler displacement. For Sirius B, \(\Delta\lambda_Z\) is almost 20 times greater than \(\Delta\lambda_D\). For a star whose mass is equal to the mass of the Sun and whose radius is 5 times greater than the radius of Sirius B, these two quantities will be equal. If the angular momentum of Sirius B is the same as that of the Sun, then its period of rotation is 15 minutes, and its peripheral velocity is 100 km/sec. In view of its great density, the limiting period of this star, determined by Roche’s limit, is about 40 seconds.

4.5. The Largest Planets

It is also of interest that Jupiter may possess a magnetic field capable of being measured. If one takes for \(k\) the value given by Jeffreys\(^{21}\), 0.66, with \(R=7.0\cdot10^9\) cm, \(\bar{\rho}=1.35\), \(T=9.8\) hours, then from (18a) we obtain, for the field at the pole, 30 gauss for \(\beta\) the same as for the Earth, and 120 gauss for \(\beta\) the same as for the Sun. Such a weak field apparently cannot be measured, since the only lines available to us will be the absorption bands of methane and ammonia; however, the low temperature of about 150° abs. may help, by causing these lines to narrow.

§ 5. THEORETICAL DISCUSSION

5.1

As was already said above, in the first attempts to explain the existence of the Earth’s magnetic field on the basis of a general theory, it was provisionally assumed that the Earth is in some way electrically charged. Since the Earth has a very weak external electric

field, it proved necessary to assume that its resultant charge is zero, and thus one postulated that a separation of positive and negative charges takes place inside the Earth. To explain the sign of the magnetic field it proved necessary to suppose that the negative charge is distributed closer to the surface of the Earth. A special case of such a distribution (Sutherland[^14], 1904) is a uniform distribution of positive charge throughout the volume, with a uniform distribution over the surface of negative charge of the same total magnitude.

A theory based on the assumption of this kind of charge distribution has the merit that it very simply explains both the existence of a magnetic field in a rotating body and its absence in purely translational motion. However, if one regards the ordinary laws of electromagnetism as valid here, it is quite impossible to believe in the existence of an actual separation of charges of the required magnitude. It is given by expression (12), and for the Earth the required charge density is \(10^{-3}\) electrostatic units, or \(10^6\) electrons per unit volume. But such a distribution of charges means that inside the Earth there must be an electric field of the order of \(10^8\ \mathrm{V/cm}\), exceeding the dielectric strength of any known materials. The impossibility of the existence of such a field inside the Sun is still more obvious. Since the conductivity both of the central core of the Earth and of the ionized matter of the Sun is relatively large, all such charges, if they were produced by any static electromotive forces, would rapidly be carried away. According to the calculations of Chapman[^37] (1928) and Cowling[^6] (1945), the conductivity of matter at the center of the Sun is of the same order as the conductivity of copper at normal temperature.

In reality, in an ionized gas situated in a gravitational field there will, of course, be some separation of positive and negative charges owing to the large difference in the masses of electrons and positive ions. Pannekoek[^38] (1922) and Rosseland[^39] (1924) considered this question and, with the aid of the Maxwell–Boltzmann equation, proved that a charge density of magnitude

\[ \sigma_e = g\rho\mu/e, \tag{20} \]

where \(\mu\) is the mass of the positive ions and \(e\) the electron charge, is caused by the tendency of the electrons to move into regions with higher gravitational potential.

The mean molecular weight of the Sun relative to the hydrogen atom is about 2.2. Comparing (20) with (12), we see that the ratio of the charge density—and consequently also of the electric field—caused by the separation of electrons and ions in a gravitational field, to the charge density and electric field that could explain the existence of the Earth’s magnetic field, is about \(10^{-17}\). Essentially the same result was obtained earlier by Bren-

volume 4 (1913), in a less general form. The sign of the charge distribution is the same as is required for the explanation of the terrestrial magnetic field, namely the negative charge is the outermost one.

Cowling^40 (1929) pointed out that when an ionized medium moves in a gravitational field at right angles to the magnetic field, the electric field caused by the separation of charges is considerably greater than that given by expression (20). However, it is still too small for the observed magnetic moments to be explained by it.

If one assumes that the normal laws of electromagnetism are applicable, then it is clear that no adequate real separation of charges can exist. A change in the fundamental equations then becomes inevitable. It is difficult, however, to believe that the equations of electrodynamics can be changed in such a way that they would permit the existence in the Earth and the Sun of a real charge density of the required magnitude. Nevertheless, it is possible that the theories of Schuster, Schlomka, and others concerning charge separation, connected with the hypothesis of the absence of exact equality between electrons and protons, etc., should usefully be subjected to further study.

Another kind of modification of the equations of electromagnetism was proposed by Wilson, who postulated that even an electrically neutral mass, moving with velocity $v$, creates a magnetic field determined by expression (8).

However, this hypothesis is, of course, incorrect if $v$ is interpreted as a relative translational velocity. Thus, for example, one may regard an observer moving in an airplane at a speed of 500 km/hour relative to the Earth as being at rest, while the Earth passes by him with the very same speed. Then, according to (8), he should observe a magnetic field of the order of 1 gauss, whose direction would be quite different from that of the real terrestrial field. This relation was also refuted by Wilson’s laboratory experiments.

There is, of course, also a theoretical objection to the existence of any magnetic field connected with the translational motion of a neutral mass. The point is that the normal Lorentz transformations for free space show that the magnetic field determined by expression (8) can exist only when the mass taken as “neutral” has a charge $Q \sim G^{1/2}m$, and this contradicts the assumption that this mass is uncharged.

We must, therefore, arrive at the conclusion that a neutral mass moving with a purely translational velocity does not create a magnetic field corresponding to (8). However, in order to confirm the correctness of equation (7), it may also not be necessary to assume that expression (8) is valid for purely translational motion. Thus, for example, if the velocity of the elemen-

of the mass \(\delta m\) as a velocity relative to the observer, caused by the absolute rotation of the body, then one may use expression (8) to confirm the validity of equation (7), without necessarily drawing the conclusion that a neutral mass, in purely translational motion, creates a magnetic field. We thus arrive at the conclusion that, upon the introduction of the additional factor \(\beta\), expression (8) may be used, as a test, to confirm the validity of equation (7), provided only that \(v\) is defined as the relative velocity caused by rotation, measured with respect to the inertial frame of reference of the universe, i.e. by the equation \(\mathbf{v}=[\boldsymbol{\omega}\cdot\mathbf{R}]\), where \(\boldsymbol{\omega}\) is the measured absolute angular velocity of rotation and \(\mathbf{R}\) is the radius vector from the center of gravity.

For me, however, it is unclear whether it is possible to preserve expression (8) for rotational motion and to regard it as unsuitable for translational motion without introducing far-reaching changes into the equations of electromagnetism and dynamics. No difficulties arise in using this expression in the case of a single rigid rotating body, but how to avoid the difficulties arising in connection with the orbital motion, say, of planets or of a double star is still not so clear. As was already pointed out above, in the case of the hypothesis of a real separation of charges no difficulties arise. The point is that here we have two equations of type (9): one for positive \(Q\) and \(H\), and the other for negative ones. The mutual annihilation of the magnetic fields produced by opposite charges does in consequence give a magnetic field for rotational, but not for translational, motion. The question arises whether, with the help of equation (8) alone, the very same result can be obtained. If this proves impossible, then it may happen that we shall be forced to return again to some theory equivalent to the assumption of virtual electric charges, i.e. charges that produce a magnetic field without an electric one. Consideration of the neutron field may suggest how this could be done.

At the time when the first attempts were made to explain the existence of the earth’s magnetic field, no magnetic fields were known other than those connected with the motion of electric charges or with a change in time of electric intensities. At present, however, it is well known that the neutron possesses a magnetic moment. It amounts to \(1.80\) Bohr nuclear magnetons, and the vectors of the magnetic moment and of the spin (angular momentum) are directed oppositely to one another, just as for the Earth and the Sun. In some modern theories of the meson field, the difference between the magnetic moments of the neutron and the proton is ascribed to the existence of virtual mesons, i.e. virtual, and not real, electric charges are introduced. It is possible that this could be taken as a hint that the magnetic field of a rotating massive body

can be attributed to such a virtual separation of charges. The observed linear dependence of \(H\) on \(\omega\) shows that such a virtual separation of charges cannot be caused by rotation, but must be a property of the massive body at rest.

Since the ratio \(P/U\) is about \(10^{-3}\) of this ratio for the Bohr atomic magneton, then, as is seen from (6), there is no evident numerical dependence between \(P/U\) for a massive body and \(P/U\) for the neutron, unless one introduces arbitrarily such a dimensionless ratio as \(e^2/hc = 1.16\cdot 10^{-3}\). At present it seems impossible to find an explanation for the fact that \(\beta \simeq 0.3\) for the Earth and about unity for the Sun and 78 Virginis. The effect of the shielding action of ferromagnetic materials of the Earth’s crust can hardly amount to 1 percent. It is possible, of course, that the error lies in the values of \(k\) and \(\eta\) adopted by us for the Sun and 78 Virginis. If it turned out that in reality the products \(k\eta\) are three times larger than we assumed, then the discrepancy would be removed. The same would be true if \(\beta\) had for the Sun and 78 Virginis the value, attractive to us, of about unity.

Another possible explanation is that this difference is caused by some weak action of the particular properties of terrestrial and stellar matter, which is superposed on the general mechanism responsible for the existence and order of magnitude of the main field. One could, of course, attribute the difference in the values of \(\beta\), say, to the content of free electrons in the matter, or to the ratio of the number of protons to the number of neutrons.

It is clear that the experimental dependence (7) and its hypothetical explanation (8) are suitable only for the nonrelativistic case. At peripheral velocities comparable with \(c\), changes must naturally be expected.

Generally speaking, it appears very probable that a satisfactory explanation of equation (7) can be found only on the basis of a unified field theory. In this connection equation (6) may acquire special significance in view of the important role which the dimensionless ratio \(G^{1/2}m/e\) must play in any such theory.

5.2. Magnetic field of an asymmetric body

Up to now we have tacitly assumed that the static magnetic field, given by equation (7) for a sphere, must ultimately be expressible as an integral taken over all parts of the body. Although equation (8) represents a differential form which gives a result agreeing with the observations for a sphere, if the coefficient \(\beta\) is not counted, it is still necessary to determine whether solutions of other types are possible. For the reasons set forth below this appears unlikely. First, the circumstance that, according to experimental data, \(H\) varies proportionally to \(\rho\) [see equa-

... (11a)], means that \(H\) must be expressed as the vector sum of all \(\delta H\) from each mass element separately. If, for example, \(\delta H\) from each given mass element depended not only on \(\delta m\), but, say, also on the gravitational potential at this element, then \(H\) for the whole body as a whole would not be a linear function of \(\rho\). Therefore, if one assumes that each mass element makes an independent contribution to the total field, then equation (8) is the only form that gives the experimentally observed linear dependence of \(H\) on \(\rho\).

It is possible, of course, that the correct solution will have not the form (8), but rather (9), if by \(\delta Q\) one understands a virtual charge of some kind, taking both positive and negative values. If this is so, then \(\delta Q\) for any volume element may depend not only on \(\delta m\), but on the entire configuration of the body, as in the case of a real separation of charges.

Until this question is resolved, it will be impossible to calculate the magnetic field of an asymmetric body, for example, of a massive ellipsoid rotating about its own shortest axis. This is of great importance in connection with the possible performance of laboratory experiments, since technically it may prove easier to measure the variable magnetic field that could be expected from such an asymmetric body than the static field of a symmetric body.

Then the general question arises whether it is in fact possible to extrapolate (7) and (8) to laboratory scales. Although, roughly speaking, equation (7) is experimentally confirmed for three astronomical bodies within a variation of \(U\) of \(10^{10}:1\), this still does not justify the required extrapolation by \(10^{28}:1\) from the case of the Earth to the case, say, of a bronze sphere 1 meter in diameter rotating at a speed of 100 rev/sec, which should give about \(10^{-8}\) gauss (see the next paragraph).

The physical conditions of the material of such an experimental sphere differ so sharply from the conditions for the Earth not only in size, but also in temperature, pressure, stress, the ratio of gravitational forces to centrifugal forces, etc., that until a reliable theory of equation (7) is found, only very limited conclusions are possible.

It may be noted that for the three astronomical bodies mentioned above the quantities \(R\), \(\omega\), and \(\rho\) vary within the following limits: \(20:1\), \(25:1\), and \(14:1\). Extrapolation to an experimental sphere entails an extrapolation in the ratio \(1:10^{-7}\) for \(R\) and \(1:10^{7}\) for \(\omega\).

5.3. Possibility of carrying out laboratory experiments

If it is assumed that extrapolation is permissible, then we can calculate the field of a sphere of a given material. Assigning \(\beta\) the value 0.30, as for the Earth, we obtain from (11a) for the field at the po...

the pole of a rotating sphere:

\[ H_p=1.07\cdot 10^{-1}\rho\omega R^2 . \tag{21} \]

In this expression we must substitute the maximum possible value of \(\omega_m\), rad/sec, corresponding to the maximum permissible tensile stress \(S\), g/cm\(^2\); using Swedberg’s data (1940) for a disk, the expression for \(\omega_m\) can be written in the form

\[ \omega_m=50S^{1/2}\rho^{-1/2}R^{-1}, \tag{22} \]

we obtain:

\[ H_p=5.4\cdot 10^{-11}S^{1/2}\rho^{1/2}R . \tag{23} \]

We see, therefore, that \(H_p\sim R\), i.e., that it is advantageous to take a large body rotating correspondingly slowly. If, for example, we take \(\rho=8\) and \(S=3.0\cdot 10^6\) g/cm\(^2\), or 43,000 pounds/sq. inch, then we obtain:

\[ H_p=2.6\cdot 10^{-10}R . \tag{24} \]

For a sphere 1 meter in diameter (\(R=50\) cm) the maximum speed will be 100 rev/sec, and \(H_p=1.3\cdot 10^{-8}\) gauss. For a sphere 10 meters in diameter the speed will be only 10 rev/sec, and the field \(1.3\times 10^{-7}\) gauss. If it should turn out that the correct value of \(\beta\) is the same as for the Sun, and not for the Earth, then the field values obtained will have to be multiplied by three.

If one takes an asymmetric body with the same linear dimensions, its field would be smaller than for a sphere; however, the circumstance that this field would be variable would make it easier to detect than a stronger static field.

The only attempt to detect the magnetic field of a rotating body was the experiments of Suon and Longacre (1928)*), who rotated a sphere 10 cm in diameter at a speed of 200 rev/sec. According to (20), the field of this sphere should have been about \(10^{-9}\) gauss. However, the sensitivity of their method made it possible only to establish that no field of order \(10^{-4}\) gauss was detected.

CITED LITERATURE

1. Babcock, Astrophys. J., 105, 105 (1947).
2. Chapman and Bartels, “Geomagnetism” (Oxford, 1940).
3. Schuster, Proc. Phys. Soc., London, 24, 121 (1912).
4. Brunt, Astr. Nachr., 196, 169 (1913).

*) And also the experiments of P. N. Lebedev, apparently unknown to the author. See the introduction “From the Editors” at the beginning of this article — Ed.

1. Swann, J. Franklin Inst. (1932).
2. Cowling, Mon. Not. Roy. Ast. Soc., 105, 166 (1946).
3. Elsasser, Phys. Rev., 55, 489 (1939).
4. Elsasser, Phys. Rev., 60, 876 (1941).
5. Frenkel, C. R. Ac. Sci. U. S. S. R., 49, 98 (1945).
6. Gurevich and Lebedinsky, C. R. Acad. Sc. U. S. S. R., 49, 92 (1945).
7. Larmor, Brit. Assoc. Rept., 159 (1919).
8. Cowling, Mon. Not. Roy. Ast. Soc., 94, 40 (1933).
9. Schuster, Brit. Assoc. Rept. (1891).
10. Sutherland, Terr. Magn., 5, 73 (1900); 8, 49 (1903); 9, 167 (1904); 13, 155 (1908).
11. Wilson, Proc. Roy. Soc., A, 104, 451 (1923).
12. Angenheister, Phys. Zeits., 26, 367 (1925).
13. Schlomka, Gerlands Beitr. Geophys., 38, 357 (1933).
14. Haalck, Zeits. f. Phys., 105, 81 (1937); Gerlands Beitr. Geophys., 52, 25 (1938).
15. Swann, Phil. Mag., 3, 1088 (1927).
16. Swann and Longacre, J. Frankl. Inst., 206, 421 (1928).
17. Jeffries, Mon. Not. Roy. Ast. Soc., 84, 534 (1924); Mon. Not. Roy. Ast. Soc., Geophys. Suppl., 4, 63 (1937).
18. Eddington, Internal Constitution of the Stars (Cambridge, 1936).
19. Cowling, Mon. Not. Roy. Ast. Soc., 96, 57 (1935).
20. Blanch, Lowan, Marshak and Bethe, Astrophys. J., 94, 37 (1941).
21. Marshak and Blanch, Astrophys. J., 104, 82 (1946).
22. Milne, Thermodynamics of Stars (Berlin, 1930).
23. Elvey, Astrophys. J., 71, 771 (1930).
24. Rosseland, Theoretical Astrophysics (Oxford, 1936), 202—215.
25. Becker, Sterne und Sternsysteme (Leipzig, 1912).
26. Chapman, Mon. Not. Roy. Ast. Soc., 103, 116 (1943).
27. Thiessen, Ann. Astrophys., 9, 101 (1946).
28. Struve, Astrophys. J., 72, 1 (1930).
29. Westgate, Astrophys. J., 78, 46 (1933); 79, 357 (1934).
30. Jeans, Astronomy and Cosmogony (1926).
31. Russel, Dugan, Stewart, Astronomy (Ginn, 1945).
32. Kuiper, Novae and White Dwarfs (Herman, Paris, 1941).
33. Chapman, Mon. Not. Roy. Ast. Soc., 89, 57 (1928).
34. Pannekoek, Bull. Ast. Inst. Netherlands, No. 19 (1922).
35. Rosseland, Mon. Not. Roy. Ast. Soc., 84, 720 (1924).
36. Cowling, Mon. Not. Roy. Ast. Soc., 90, 40 (1929).

1. Kuiper. ↩