Northern Lights and Magnetic Storms.

By Academician A. N. Krylov.

(A speech delivered by the chairman at the January 1917 general meeting of the members of the Russian Physico-Chemical Society.)

§ 1. The northern lights have long attracted the gaze of observers, and scholars have long sought to unravel the nature of these phenomena, which gave rise to a whole series of legends and superstitions.

More than 200 years ago Galileo drew attention to the similarity between the phenomena of the aurora borealis and the glow that is observed when electricity flows from a strongly electrified body; he also noted that, when the aurora has the form of an arch, the summit of this arch lies in the magnetic meridian, while the inclination of the rays, or poles, is close to the inclination of the magnetic needle.

Polar navigators of the first half of the last century observed deviations in compass readings and disturbances of the magnetic needle during northern lights, and Franklin was perhaps the first to carry out detailed investigations on this question.

The glow of rarefied gases under the action of electrical discharges presented a new analogy in the realm of electrical phenomena and the northern lights. Theories were even proposed in which the northern lights were attributed to quiet discharges of terrestrial electricity through the upper rarefied layers of the atmosphere.

After magnetic observatories had been established on Gauss’s initiative, Franklin’s observations received systematic confirmation, and the connection between magnetic storms and the northern lights was established with complete certainty.

Finally, observations of the state of the sun’s surface and of sunspots, properly conducted from the beginning of the 1700s, already in the 1850s made it possible to note a connection between sunspots and the northern lights: years with developed and strong activity of the sun, accompanied also by an abundance of spots, were abundant in northern lights as well, and moreover in those years they attained the greatest intensity and visibility even in middle latitudes.

As early as the 1790s Dalton, using the position of the vertex of the arc or vault of an aurora relative to the stars, observed from two points 83 English miles (125 versts) apart, determined that this vertex was at an altitude of 100 English miles, somewhat to the south of one of these points.

In R. Potter’s article, “Calculation of the Heights of the Aurorae Boreales of 17th Sept. and 12th Oct. 1853,” printed in volume 8 of the Philosoph. Transactions of the Cambridge Royal Society, a number of similar observations are given, which likewise yielded heights from 65 to 85 English miles.

Spectral observations revealed in auroras a number of lines that were found in known terrestrial bodies.

§ 2. In approximately this state the question stood some twenty-five years ago, i.e. the phenomenon had been described and studied from the outward, mainly qualitative, side, but had not been subjected to systematic investigation by means of precise observations which would then be compared with one another and would serve as the basis of a coherent theory.

But exactly twenty years ago the Norwegian scholar Prof. Birkeland equipped his first expedition for the study of auroras.

In this expedition he took part personally, with two of his students, Helland-Hansen and Laws, accompanied by the old Finn “postvagnus,” i.e. the postman Hætta.

The aim of the expedition was to reach the summit of a mountain about 3000 feet high in Northern Norway, not far from Hammerfest, and to make use of a log hut near the summit for their camp and observations.

But this expedition was unsuccessful—less than two versts short of the hut, the expedition was caught in a snowstorm with a strong north wind and frost down to \(25^\circ\) C. All the members of the expedition suffered frostbite; they had to abandon their baggage and instruments, turn back, and only the experience of the old “postvagnus” helped them reach again the station of Gargia, which they had left 31 hours earlier.

Skillful thawing of the frostbitten hands in icy water and timely medical assistance saved the members of the expedition from gangrene and mutilation.

§ 3. The failure of the first attempt did not weaken Birkeland’s energy; on the contrary, it made him continue the work he had begun with greater caution and foresight.

In the summers of 1897 and 1898 he visited Northern Norway, climbed the summits of its highest mountains, choosing the one most suitable for the establishment of an observatory.

After such study he settled his choice upon the summits of Sukkertop and Talviktop, situated at a distance of about 3.4 kilometers from one another, at 70° N. lat. and 22° 30′ E. long. from Greenwich. The height of each summit is about 3000 ft. above sea level.

On both summits solid stone observatory buildings were constructed, connected both with each other and with the general network by telephones.

At these observatories, during the winter of 1899–1900, observations of the aurora borealis, magnetic, and meteorological observations were carried out under the direction of Prof. Birkeland himself. Without going into the details of the results obtained, since they are most curious, we shall mention only those severe storms and blizzards which the members of the expedition had to observe. The wind speed on these summits reached as much as 46 meters per second, and there were storms with speeds up to 38 meters at a frost of 16°. Birkeland says—it is difficult to imagine what this is and how such a storm acts upon a person—one of his assistants returned after several minutes with a frostbitten hand because he had not put fur mittens over his woolen ones while making observations with anemometers.

Nevertheless, a sturdy Finn of small stature regularly delivered the mail to the observatory once or twice a week, despite all storms and snowstorms. Once, when he arrived all iced over, so that he could scarcely be recognized, the professor asked him whether he was not afraid to go about in such bad weather. The Finn at first did not answer and sat quietly until the ice had thawed from him, and then said: “I am too stupid to be afraid.”

§ 4. The principal conclusion obtained on the basis of the observations made by the second expedition was that, in order to solve the question of the causes of the aurora borealis and magnetic storms, it is necessary to have at one’s disposal simultaneous records of magnetic instruments and observations at different stations, correspondingly situated at a distance of about 1000 kilometers from one another, and the same kind of records for as large a number of stations as possible, distributed over the entire surface of the earth. It proved possible to demonstrate by calculations that certain magnetic disturbances, embracing considerable regions, can be ascribed to the action of an electric current parallel to the earth’s surface in its polar parts, at an altitude of several hundred kilometers and with a strength of up to a million amperes, if only the strength of the stream of electric particles is measured in the same way as galvanic current is measured, by its magnetic action. In the polar countries these currents were well delineated and concentrated,

being located, in terms of time, between two stations—Jan Mayen and Bossekop.

The Norwegian government allocated 20,000 kroner for further investigations; five private individuals contributed 6,000 kroner each, and Prof. Birkeland himself added 30,000 kroner to this sum and equipped his third expedition for the study of the northern lights. These expeditions belong to the years 1902–1903; the processing of all the results obtained has not yet been completed, but Prof. Birkeland has already published two enormous books, constituting the first and second parts of the first volume of his treatment of these results.

The stations were established at the following four points (see Fig. 1):

Kaafjord (Northern Norway) (lat. \(69^\circ 56' N\), long. \(22^\circ 58'\) Ost.).
Dyrafjord (Iceland) (lat. \(66^\circ 15' N\), long. \(22^\circ 30' West.\)).
Axelöen (Spitsbergen) (lat. \(77^\circ 41' N\), long. \(14^\circ 50'\) Ost.).
Matochkin Shar (Novaya Zemlya) (lat. \(73^\circ 17' N\), long. \(53^\circ 57'\) Ost.).

Fig. 1.

All the stations were supplied with a complete series of self-recording variation magnetic instruments, an absolute magnetometer, inclinometers, a full set of meteorological instruments, a theodolite for astronomical observations, and chronometers,

At each station there was to be built a dwelling house, a magnetic observatory for self-recording instruments, an observatory for absolute magnetic observations, a meteorological booth, and a booth for astronomical observations. At each station there was a director and two assistants; moreover, Prof. Birkeland himself directed the station at Kaafjord, in the north of Norway, and the construction of all the stations.

On Novaya Zemlya a house built for the artist Borisov was placed at the disposal of the expedition, and the expedition was transported to Matochkin Shar and back on the government steamship “Vladimir,” by order of the then Governor of Arkhangelsk, Rear Admiral N. A. Rimsky-Korsakov.

In addition to these specially built and equipped stations, at all the principal magnetic observatories of the whole world (23 in number) records were made at the appointed hours with a more rapid motion of the registering instruments; the results of the observations were communicated to Prof. Birkeland.

It was this enormous material that he subjected to the systematic processing of which I intend now to give a brief account.

§ 5. Proceeding from the idea that the cause of the northern lights and magnetic storms is not terrestrial but cosmic in origin, and that it must be sought in the motion of electrified particles (or particles of electricity) ejected by the sun, on which spots may be sources of cathode rays, Birkeland adopted the following method of processing.

First of all, for each of the observed magnetic disturbances, occurring, as has been known since the time of Gauss, simultaneously over the whole earth, he calculated the magnitude and direction of the disturbing force for each of the places of observation and thus obtained a representation of the disturbing magnetic field on the earth’s surface. He made such a calculation for each disturbance for a whole series of successive moments, in order to be able to follow the very course of its development. To represent the above-mentioned field he adopted a peculiar method: what was plotted on the map was not the projection of the disturbing force, but, for each place of observation, a vector was constructed whose direction coincided with the direction of the horizontal electric current capable, by flowing above the given place, of producing the observed horizontal disturbing magnetic force; the length of the vector was laid off proportional to the strength of the current, and therefore also proportional to the horizontal component of the disturbing force.

To represent its vertical component, a vector was constructed perpendicular to the first and directed away from it to the left when this force was directed downward, and to the right when upward. With such a

designation, if the disturbance were actually produced by horizontal currents located above the earth, then the arrow representing the vertical force would be directed toward the place of greatest current density.

Examination of the maps constructed in this way led Birkeland to the following classification of magnetic disturbances, or storms:

1) Positive equatorial.
2) Negative equatorial.
3) Positive polar.
4) Negative polar.
5) Vortical of middle latitudes (cyclo-median).

A positive equatorial disturbance is characterized by the following features: a) in middle and low latitudes there is observed a positive (i.e., increasing the horizontal component) disturbing force lying in the plane of the magnetic meridian, i.e., producing no changes either in declination or in inclination, or only very small ones. The greatest magnitude of the force is observed in low latitudes; toward the pole its magnitude decreases.

The map (Fig. 2) presents an example of disturbances of this type.

Fig. 2.

A negative equatorial disturbance occurs comparatively rarely; an example of it may be the map (Fig. 3), on which the arrangement of the arrows is similar to the preceding one, but their direction is opposite.

A polar disturbance is characterized by a very large magnitude of the disturbing force in polar countries, but over a comparatively small region; with distance from it the magnitude of the force

rapidly decreases in middle and low latitudes, becoming very small.

Such a perturbation (negative), i.e. one in which the horizontal component in the region of greatest perturbation decreases, is presented on the map (Fig. 4); moreover, the vectors for Iceland and Spitsbergen are shown at five times shorter length than would follow if the same scale were used for them as for Central Europe.

Fig. 3.

Fig. 4.

A positive polar perturbation differs from the preceding one only in the direction of the forces.

An example of a vortical or cyclical disturbance is presented on the map (Fig. 5), on which the directions of the lines of force have a vortical character.

§ 6. Before setting forth the explanations that Birkeland gave for the observed phenomena, it is necessary to say a few words about the mathematical theory of the northern lights, given by the Norwegian mathematician Carl Störmer.

Fig. 5.

Störmer’s theory is set forth in the abstract placed in the 42nd volume of the Journal of the Russian Physico-Chemical Society, and I shall therefore confine myself only to the most essential points.

Störmer undertook the investigation of the question of under what conditions a particle carrying an electric charge, ejected by the sun, will enter the sphere of action of the earth’s magnetic field and, under its influence, reach its surface or pass very close to it.

To solve this problem, he assumes that the sun is a non-magnetic body, so that it does not form a magnetic field, and that the entire motion of the particle is determined by the earth’s field, which Störmer replaces by an elementary small magnet, whose axis is directed along the magnetic axis of the earth and whose magnetic moment is equal to its magnetic moment, i.e. \(8.52 \cdot 10^{25}\ C.\ G.\ S.\)

The differential equations of motion of a particle carrying an electric charge in a magnetic field will be written if one likens an element of the trajectory described by the particle at the moment under consideration to a current element and applies the rule for the action of a magnetic field on such a current, the current strength being proportional to the velocity of motion of the particle and to its charge.

Denoting by \(H_x, H_y, H_z\) the components of the field intensity along the coordinate axes, by \(m\) the mass of the particle, by \(e\) its charge, and by \(\alpha\) the Newtonian constant\(^1\), we shall have the equations:

\[ \left. \begin{aligned} \frac{d^2x}{dt^2} &= \alpha\,\frac{e}{m}\,[\,H_yv_z-H_zv_y\,],\\ \frac{d^2y}{dt^2} &= \alpha\,\frac{e}{m}\,[\,H_zv_x-H_xv_z\,],\\ \frac{d^2z}{dt^2} &= \alpha\,\frac{e}{m}\,[\,H_xv_y-H_yv_x\,]. \end{aligned} \right\} \qquad .\ .\ .\ .\ .\ .\ .\ (1) \]

where by \(v_x, v_y, v_z\) are denoted the projections of the velocity \(v\) of the particle on the coordinate axes. Since the direction of the force acting on the particle is perpendicular to the direction of its velocity, the velocity \(v\) is constant, and in the preceding equations Störmer introduces, instead of the time \(t\), the variable \(s\)—the length of the arc of the trajectory—with the equation \(ds=vdt\). Then directing the \(Z\)-axis along the magnetic axis of the earth and denoting its potential by \(V\), he obtains:

\[ V=M\frac{z}{r^3} \qquad .\ .\ .\ .\ .\ .\ .\ (2) \]

where \(M\) is the magnetic moment of the earth and \(r^2=x^2+y^2+z^2\); consequently,

\[ \left. \begin{aligned} H_x&=\frac{\partial V}{\partial x}=-3M\frac{xz}{r^5},\\ H_y&=\frac{\partial V}{\partial y}=-3M\frac{yz}{r^5},\\ H_z&=\frac{\partial V}{\partial z}=-M\frac{3z^2-r^2}{r^5}. \end{aligned} \right\} \qquad .\ .\ .\ (3) \]

Moreover:

\[ \frac{d^2x}{dt^2}=\frac{d^2x}{ds^2}v^2,\qquad \frac{d^2y}{dt^2}=\frac{d^2y}{ds^2}v^2;\qquad \frac{d^2z}{dt^2}=\frac{d^2z}{ds^2}v^2 \]

and, denoting by primes the derivatives taken with respect to the variable \(s\), we shall have the system

\[ \left. \begin{aligned} x''&=\frac{c_1^{\,2}}{r^5}\left[\,3yz\cdot z'-(3z^2-r^2)y'\,\right],\\ y''&=\frac{c_1^{\,2}}{r^5}\left[\,(3z^2-r^2)x'-3xz\cdot z'\,\right],\\ z''&=\frac{c_1^{\,2}}{r^5}\left[\,3xz\cdot y'-3yz\cdot x'\,\right]. \end{aligned} \right\} \qquad .\ .\ .\ (4) \]

\(^1\) The constant \(\alpha\) is the ratio of the electrostatic unit of electric mass to the electromagnetic one and is equal to \(\frac{1}{C}\), where \(C\) is the speed of light.

Here \(c_1\) denotes a certain constant depending on the ratio of the charge of the particle to its mass, on the velocity of its motion \(v\), i.e. depending on the kind of particle, and on the magnetic moment \(M\). From equations (4) it is seen that \(c_1\) represents a certain length, whose magnitude, for the above-adopted value of \(M\), is as follows:

for cathode rays \(c_1\) is equal to \(4.0\)—\(8.5\) million km.
for \(\beta\) rays of radium, \(c_1\) is equal to \(1.4\)—\(2.2\) million km.
for \(\alpha\) rays of radium, \(c_1\) is equal to \(150000\)—\(170000\) km.

It is clear that, by the homogeneity of eq. (4), any length may be taken as the unit of length.

The equations become simpler if one takes \(c_1=1\), whereby the scale of the trajectory will be determined.

In what follows we shall assume \(c_1=1\).

It turns out that system (4) has an integral similar to the integral of areas. In fact, forming the expression \(xy''-yx''\), we obtain:

\[ xy''-yx''=\frac{1}{r^5}\left[(3z^2-r^2)(xx'+yy')-3(x^2+y^2)zz'\right] \]

or, putting

\[ x^2+y^2=R^2, \]

\[ xy''-yx''=\frac{3z^2-r^2}{r^5}RR'-\frac{3R^2}{r^5}zz'\ . . . . . (5). \]

The expression standing on the right-hand side of eq. (5) is the total derivative with respect to \(s\) of the quantity \(\dfrac{R^2}{r^3}\), for

\[ \frac{\partial}{\partial R}\left(\frac{R^2}{r^3}\right)=\frac{3z^2-r^2}{r^5}R \quad \text{and} \quad \frac{\partial}{\partial z}\left(\frac{R^2}{r^3}\right)=-\frac{3R^2}{r^5}z. \]

Introducing then cylindrical coordinates

\[ x=R\cos\varphi \qquad y=R\sin\varphi \]

and keeping the \(z\)-axis unchanged, instead of equation (5) we obtain

\[ \frac{d}{ds}\left[R^2\frac{d\varphi}{ds}\right] = \frac{d}{ds}\left[\frac{R^2}{r^3}\right] \ . . . . \ (5') \]

while the other two equations of the system, after solving with respect to \(\dfrac{d^2R}{ds^2}\) and \(\dfrac{d^2z}{ds^2}\), are replaced by the following:

\[ \begin{aligned} \frac{d^2R}{ds^2} &= R\left(\frac{d\varphi}{ds}\right)^2 +\frac{r^2-3z^2}{r^5}R\frac{d\varphi}{ds} \\[6pt] \frac{d^2z}{ds^2} &= \frac{3z}{r^5}R^2\frac{d\varphi}{ds} \end{aligned} \ . . . . \ (6) \]

instead of equation (5′), the corresponding integral is taken:

\[ R^2\frac{d\varphi}{ds}=2\gamma+\frac{R^2}{r^3}\ .\ .\ .\ .\ .\ (7) \]

in which an arbitrary constant is denoted by \(2\gamma\).

Moreover, since \(s\) is the arc of the trajectory, we shall have:

\[ R^2\left(\frac{d\varphi}{ds}\right)^2+\left(\frac{dR}{ds}\right)^2+\left(\frac{dz}{ds}\right)^2=1\ .\ .\ .\ .\ ,\ (8) \]

The equation (7), corresponding to the integral of areas, will be of very great importance for what follows.

Let us write this equation in the form:

\[ R\frac{d\varphi}{ds}=\frac{2\gamma}{R}+\frac{R}{r^3}\ .\ .\ .\ .\ .\ (9) \]

and, noting that the quantity \(R\dfrac{d\varphi}{ds}\) is the sine of the angle \(\theta\) formed by the tangent to the trajectory with the plane drawn through the point of tangency and the \(z\)-axis, we obtain the inequality:

\[ -1\leqq \frac{2\gamma}{R}+\frac{R}{r^3}\leqq 1\ .\ .\ .\ .\ .\ (10) \]

which bounds those regions of space in which, for a given value of the arbitrary constant \(2\gamma\), trajectories may be found, whatever the initial conditions, from those where trajectories cannot be found. It is clear that the bounding surfaces are obtained when equality signs are taken in formula (10).

In Figs. 6, 7, 8, and 9 is shown the form of the meridians of these surfaces, which are all surfaces of revolution about the \(z\)-axis, the regions where a trajectory cannot exist being covered in black.

The values of the constant \(\gamma\) for which these figures have been constructed are the following:

Let us note with what a very narrow sector each of these surfaces approaches the origin of coordinates, in particular—

... for values of $\gamma$ close to $-1$, and consequently all trajectories, whatever the initial conditions may be, for such a value of $\gamma$ will reach the origin within this very narrow region.

Examination of these surfaces shows that they are simply connected, i.e., consist of one piece, and extend to infinity, enclosing also the origin of coordinates, if

$$ -1 \leq \gamma \leq 0. $$

Fig. 6a.

Fig. 6b.

For very small positive values of $\gamma$, the trajectories do not reach the origin of coordinates, but if $\Delta$ is a small quantity and

$$ \gamma \leq \left(\frac{2c_1}{\Delta}\right)^3, $$

then the trajectory approaches the origin to a distance less than $\Delta$.

Finally, taking for cathode rays the quantity $c_1$ equal to

5.2 million kilometers, so that the distance from the earth to the sun is equal to \(28.8 c_1\), or simply 28.8, when the length \(c_1\) is taken as the unit of length. Störmer obtained a further series of inequalities limiting the magnitudes of the possible values of \(\gamma\), as well as those initial conditions under which a particle flying out from the sun can reach the earth.

Fig. 7a.

Fig. 7b.

Consideration of the forms of the spaces containing trajectories in the vicinity of the origin of coordinates, for the above-mentioned values of the parameter \(2\gamma\), led Störmer to the following conclusions:

1°) All trajectories reach the limits of the atmosphere within belts situated around the magnetic poles at distances bounded as follows:

For cathode rays between \(2^\circ,3\) and \(3^\circ,4\).
“ \(\beta\)-rays of radium ” \(4^\circ,6\) “ \(5^\circ,8\).
“ \(\alpha\) ” “ ” \(16^\circ,6\) “ \(18^\circ,1\).

As can be seen, these belts are already zones of aurorae, but one must also keep in mind the series of simplifying assumptions that had to be made for the calculations, and that the “rigidity” of the rays emitted by sunspots, i.e. under conditions completely inaccessible to our experiments, may also differ from the “rigidity” of the rays known to us.

Fig. 8a.

Fig. 8b.

According to equation (9), the angle \(\varphi\) is found by quadrature:

\[ \varphi=\int_{s_0}^{s}\left(\frac{2r}{R^2}+\frac{1}{r^3}\right)\,ds+\varphi_0 \]

after which \(x\) and \(y\) will be found, and the trajectory can be constructed.

Thus, the whole matter reduces to integration, in other words, to compiling a table of values of the functions \(R\) and \(z\) for given \(s\).

Suppose that a series of successive equidistant values of the independent variable \(s\) is taken with tabular interval \(\Delta s=h\), sufficiently small, so that these values are

\[ s_\lambda=s_0+\lambda h \quad (\lambda=1,2,\ldots n-2,\ n-1,\ n,\ n+1,\ldots) \]

and denote the corresponding values of \(R\) and \(z\) by

\[ R_\lambda=R(s_\lambda) \quad \text{and} \quad z_\lambda=z(s_\lambda). \]

Fig. 9.

Further, introduce the quantities \(\rho_\lambda\) and \(\zeta_\lambda\) by the equalities

\[ \rho_\lambda=R''(s_\lambda)\cdot h^2 \quad \text{and} \quad \zeta=z''(s_\lambda)\cdot h^2. \]

The process of computation consists in successively adjoining, one row at a time in the tables, the quantities \(R_n,\rho_n,z_n,\zeta_n\) and their differences, whereby for finding \(\rho_n\) and \(z_n\), when \(R_n\) and \(z_n\) are known, the first two differential equations of the system serve.\(^{1}\)

Thus, let the table be filled in as shown:

\(^{1}\) This method is set forth in detail in the article by A. N. Krylov, Archive of the Physical Sciences, Nos. 1, 2, p. 68, 1918.

and, in exactly the same way, two tables for \(z\) and \(l\); it is required to compute \(R_{n+1}\) and \(z_{n+1}\).

For this purpose, from the equations

\[ R''=\left(\frac{2\gamma}{R}+\frac{R}{r^3}\right) \left(\frac{2\gamma}{R^2}+\frac{3R^2}{r^5}-\frac{1}{r^3}\right), \qquad z''=\left(\frac{2\gamma}{R}+\frac{R}{r^3}\right)\cdot\frac{3Rz}{r^5} \]

the quantities

\[ R''_{(n)} \text{ and } z''_n,\quad \text{and from them } \rho_n=R''_n h^2 \text{ and } l_n=z''_n h^2 \]

are computed directly; after this, to the table of values of the quantity \(\rho\) the numbers

\[ \rho_n,\ \Delta \rho_{n-1},\ \Delta^2\rho_{n-2},\ \Delta^3\rho_{n-3},\ \Delta^4\rho_{n-4} \]

are added, and, in exactly the same way, to the table of values of \(l\) the numbers

\[ l_n,\ \Delta l_{n-1},\ \Delta^2 l_{n-2},\ \Delta^3 l_{n-3},\ \Delta^4 l_{n-4}. \]

Using the Stirling formula, the following relation is obtained, neglecting [[unclear: damaged words]]:

\[ \Delta^2 R_{n-1} = \rho_n+\frac{1}{12} \left[ \Delta^2\rho_{n-1} +\Delta^3\rho_{n-2} +\Delta^4\rho_{n-3} -\frac{1}{20}\Delta^4\rho_{n-4} \right], \]

using which, by iteration [[unclear: damaged words]], having found \(\Delta^2R_{n-1}\), we obtain

\[ \Delta R_n=\Delta R_{n-1}+\Delta^2R_{n-1}, \]

and

\[ R_{n+1}=R_n+\Delta R_n. \]

In exactly the same way \(z_{n+1}\) is found.

Applying this process, Schwarzschild, using the services of his assistant and two computers, calculated about 120 trajectories for the following 27 values \(\gamma_1=-\gamma\):

\[ \gamma_1=0.1;\ 0.2;\ 0.3;\ 0.4;\ 0.5;\ 0.6;\ 0.7;\ 0.8; \]

\[ 0.85;\ 0.90;\ 0.92;\ 0.926;\ 0.9285;\ 0.93;\ 0.9335;\ 0.939; \]

\[ 0.94;\ 0.95;\ 0.956;\ 0.957;\ 0.97;\ 0.999;\ 1;\ 1.2;\ 1.5;\ 2;\ 5. \]

Fig. 10

For each trajectory, about 100–120 points were computed, which required more than 5000 working hours.

For all these trajectories the initial conditions were chosen so that the trajectory passed through the origin of coordinates and went off to infinity; moreover, Schwarzschild, on the basis of a study of the equations of system (4), proved that for every value of the constant \(\gamma\) between 0 and \(-1\) there are two and only two such trajectories, and at the same time he showed how to find the initial values of the variables for these trajectories.

On the basis of the computations performed, models have been constructed, photographs of which are presented in Figs. (10, 11, and 12).

But besides trajectories going off to infinity, there also exist closed or periodic trajectories, for which approximately the same work has been carried out as for those going off to infinity.

Fig. 11.

These periodic trajectories sometimes represent very complex spirals, such as, for example, the one presented in Fig. (12), and it is not difficult to imagine how much labor is required to compute a sufficient number of points for constructing such a spiral.

§ 7. Thus, Størmer’s investigations undoubtedly show that particles of electricity corresponding to cathode, β, or α rays of radium can, under the assumptions made, under the influence of the Earth’s magnetic field, reach the Earth and, moreover, in the very limited vicinity of the polar region; their trajectories will all be contained, as they approach the Earth, in an extremely narrow sector, whose section at an altitude

Fig. 12.

about 100 kilometers above the earth’s surface ranges from a few meters to several kilometers, depending on the “hardness” of the rays. It is clear that a stream of particles ejected by the sun simultaneously, even in places considerably distant from one another and moving along such trajectories, will produce the phenomenon of the northern lights with its draperies and rays.

Fig. 13.

§ 8. Birkeland approached the solution of the question somewhat differently: he began to study the motion of particles of electricity in a magnetic field experimentally. For this purpose he constructed, so to speak, a model of the terrestrial globe, which he called terrella (little earth).

In Fig. (13) his original model is shown; in Fig. (14), the second, of larger dimensions.

The essence of the arrangement of both is the same: in a glass vessel or in a box with glass walls, in which a rarefaction down to several thousandths of a millimeter of mercury is maintained, there is placed, raised in the middle of the box, an electromagnet to which the shape of a sphere has been given. This electromagnet is covered with a shell painted with a solution of barium platinocyanide. On one of the walls of the box is placed a source of cathode rays; the spherical electromagnet (terrella) can be set so that its magnetic axis takes any position relative to the straight line joining its center with the cathode; moreover, it can be given a rotation corresponding to the daily rotation of the earth.

Fig. 14.

By changing the strength of the magnetizing current, one can vary, within very wide limits, the magnetic moment of the magnet, i.e. the constant $C_1$ in Störmer’s equations, which for the earth changes depending on the kind of rays, and thus, with the same cathode rays, study also the path of rays of a different “hardness.”

Cathode rays produce luminescence in a rarefied gas; at the place where they fall upon the surface of the terrella, a bright glow is observed, owing to the coating with barium platinocyanide.

Birkeland also equipped his terrella with various screens coated with the same substance, in order to trace the paths of the rays with still greater clarity; in addition, he placed other screens so as to isolate a narrow stream of rays in the desired direction.

Fig. 15.

Figs. 14, 15, and 16 show some of the results obtained by Birkeland; here one can see both the equatorial ring and the polar luminosities concentrated in a narrow region at a distance of about 20′ from the pole.

Fig. 16.

The forms of these curves and the places where they adjoin the sphere generally correspond to Størmer’s mathematical theory.

§ 9. Under certain initial conditions, among the trajectories found by Störmer there are also such trajectories as consist, as it were, of a vertical branch approaching the earth, a small horizontal segment near its surface, and again a vertical branch receding from the earth. Such trajectories approach the earth in near-polar regions.

Birkeland obtained similar paths under certain conditions of magnetization of his terrella and of the position of its magnetic axis relative to the stream of rays.

This circumstance, together with the comparative simplicity of the field obtained during a polar magnetic storm, led him to investigate the following problem: to determine the length of the horizontal part, its height above the surface of the earth, and the strength of such a current as would produce an exciting field similar to that observed.

It turned out that the height should be taken to be about 200 kilometers, the length about 1600, and the current strength about 1,000,000 amperes; then a field close to one of those observed was obtained. For other observations, quantities of the same order were obtained.

§ 10. Störmer’s trajectories approach the earth’s surface for the most part from the side not illuminated by the sun; the same was shown to Birkeland by experiments with the terrella. He therefore set himself the task of studying the distribution of magnetic storms over the time of day, for his four polar stations. A simple count of disturbances, or the accounting of their duration at different hours of the day, is insufficient; it is also necessary to take into account the intensity of the disturbance itself.

Let, for example, \(P_h\) represent the magnitude of the horizontal component of the disturbing force in the plane of the magnetic meridian; then the integral

\[ \left| S_H \right| = S_H^a = \frac{1}{T}\int_0^T \left| P_h \right|\,dt \]

of the absolute value of this component represents the absolute mean intensity of the horizontal disturbing force along the meridian. In a similar manner, if the positive and negative values are denoted respectively by \(P_h^p\) and \(P_h^n\), then the integrals

\[ S_H^p = \frac{1}{T}\int_0^T P_h^p\,dt \]

\[ S_H^n = \frac{1}{T}\int_0^T P_h^n\,dt \]

represent the mean intensity of the positive and negative disturbance of the above-mentioned component.

It is clear that similar quantities can also be composed for the transverse component (the disturbance of declination) and for the vertical one.

Finally,

\[ S^T=\sqrt{|S_H|^2+|S_D|^2+|S_V|^2} \]

will represent the value of the mean magnitude of the total intensity, where \(|S_D|\) and \(|S_V|\) represent the corresponding quantities for the transverse and vertical components.

Such calculations were also carried out by Birkeland for each of the two-hour intervals of each day, then for the same hours over five-day periods, then by months, and, finally, for the entire time of observation.

It turns out that the mean intensity of the disturbances near the belt of the auroras follows the diurnal motion of the sun, and if these disturbances were represented by vectors of the current directions, then for each station two principal systems could be observed:

1°) the first system, having a maximum at about 6 o’clock in the evening, local time, with the arrow directed eastward along the auroral belt.

2°) the second system, with a maximum at about midnight and with the arrow directed westward.

At about 9–10 o’clock in the morning, local time, there occurs, so to speak, a magnetic lull, i.e. an absence of disturbances.

These two principal systems correspond to those disturbances which above were called positive and negative polar disturbances.

Examination of the diagrams themselves showed Birkeland that, for these disturbances, the center was located between Axoloen and Kaafiord; but, besides these principal ones, there is also encountered a certain number of local storms of lesser strength in the vicinity of the earth’s magnetic pole and north of the auroral belt.

The diurnal period, the position of the zone of polar disturbances, and the time of day when they reach their greatest intensity correspond, both according to Störmer’s theory and according to Birkeland’s experiments with the terrella, by which he discovered, by varying the rigidity of the rays, that two types of electric precipitation can be obtained on the evening side of the earth; in one of these types the motion of the particles proceeds eastward, in the other westward. (This was detected by Birkeland with the aid of screens arranged in the form of an asterisk near the poles of the terrella, observing their luminous and shadow side.) These two types can also serve as an explanation of positive and negative polar disturbances.

The source of the electric radiation is the sun.

§ 11. Birkeland’s experiments with the terrella are also curious, when he made the electromagnet itself or the terrella itself the source of cathode rays, while at the same time magnetizing it.

He obtained phenomena resembling the solar corona and the rings of Saturn, as can be seen in Fig. 17.

§ 12. It is difficult, in a few quarters of an hour, to give a complete idea of two enormous volumes, so rich in content—observational, experimental, and, finally, theoretical—as Birkeland’s work, the brief outline of which I have given.

I now turn to Störmer’s most recent works, in which he reveals his talent as an observer and experimental investigator, whereas in his preceding works he displayed his mathematical powers.

Fig. 17.

I have already mentioned at the beginning Dalton’s calculation, by which he determined the position of the apex of the arc or vault of the northern lights, and Potter’s work, dating from the year 1833.

At that time there was as yet neither photography nor the telegraph and, of course, no thought of the possibility of the telephone; it is therefore understandable that observations from two points of such a changeable phenomenon as the northern lights, with the naked eye or with the simplest angle-measuring instruments for determining the position of the prominent points of the aurora relative to the stars, could not, by their very difficulty, be distinguished by great accuracy.

Störmer decided to apply modern scientific means to the systematic investigation of this question. To this end he set up in northern Norway two observatories, at Bossekop and at Store Korsnes, the former lying at 70° north latitude and 24° east longitude from Greenwich, the latter on the same meridian, but 27 kilometers farther north.

At each of these observatories there were specially adapted photographic cameras, completely identical with one another. The two stations were connected by telephone and, consequently, it was easy to obtain fully simultaneous photographs. The cameras were направи—

...were pointed at that region of the sky where the aurora was so bright that an image of some bright star or planet was also obtained on the plate, to which the corresponding images of the very same point of the aurora could then be referred.

In short, photogrammetric photography of the aurora was carried out simultaneously with two cameras, the image of the stars on the plate giving the orientation of the camera.

It is clear that from such two simultaneous photographs it is not difficult to determine the position of any distinctive point obtained in the photographs with respect to the base, both in altitude and in azimuth, and, consequently, to transfer the projection of this point onto a map and mark on it its elevation above the surface of the earth. Störmer determined chiefly the lower edge of the aurora when it had the appearance of a curtain, and, having examined several hundred photographs, found from them the positions of many thousands of points.

Fig. 18.

Fig. 19.

In Figs. (18 and 19) a pair of photographs is presented, in which the images of the aurora and of Venus are visible.

Fig. (20) represents a summary table of observations; it shows the elevations of all the computed points, and one can see how they all group into a layer at a height of about 100 kilometers above the earth’s surface.

Störmer’s work is not yet finished; simultaneously with the photogrammetric survey he also carried out magnetic observations by means of self-recording variational instruments.

Fig. 20.

It is clear that the processing of these joint observations will yield accurate data for calculations similar to those made conjecturally by Birkeland, i.e., concerning the strength of that stream of electric particles which, penetrating into the atmosphere, produces both the aurora borealis and magnetic storms; but already now it may be said that, through the work of Birkeland and Störmer, the essence of the matter has been clarified, and in the future only the study of its details will remain.