MacTutor
KARL SCHWARZSCHILD was born at Frankfort 1873 October 9. He was educated successively at the public school at Frankfort and at Strasburg and Munich universities. He gave early evidence of remarkable mathematical ability, and when barely sixteen contributed two excellent papers to the Astronomische Nachrichten on the problem of determining an orbit from three observations. At Munich he came under the inspiration of Seeliger's teaching, which had a very lasting influence on his mind. His first astronomical post was at the Von Kuffner Observatory, Vienna, where he was assistant from 1896 to 1899.
After a short interval as privatdocent at Munich, he was appointed Director of the Observatory of Göttingen in succession to Schur. Here he joined a brilliant circle of scholars who have made Göttingen world-famous. To a man of his wide interests in all branches of mathematics and physics the surroundings must have been very congenial; and to his growing reputation as an astronomer he added important achievements in other fields of research. Finally, in 1909 he was chosen at the age of thirty-six to succeed Vogel as Director of the Astrophysical Observatory, Potsdam, a position which he filled with distinguished success.
In reviewing his contributions to astronomy we may start with his work on photographic magnitudes, a long and sustained investigation, which required much perfection of detail as well as brilliance of conception. His first method, carried out at the Von Kuffner Observatory, was to measure the greyness of star-images taken out of focus. He set out to determine the variation of photographic effect with the length of exposure, as a step towards forming an absolute scale of magnitudes. It was already known that an increase of exposure-time has not the same effect as an increase of light-intensity-doubling the exposure is not so effective as doubling the brightness. Schwarzschild proposed the now well-known law that the photographic effect is proportional to , where is the intensity of the light, and the exposure-time. For the constant he found the value 0.675; but this appears to have been too low, and in the Göttingen Aktinometrie he redetermined the constant as 0.76 – which means practically that an increase of exposure which might be expected to give a gain of four magnitudes in reality only gains three.
The fruit of this preliminary work was reaped when he came to Göttingen and, with the aid of his assistants, carried out the "Aktinometrie." This was a large undertaking, involving the determination of the magnitudes of all the stars brighter than between 0º and +20° dec., and their reduction to an absolute scale. The great area of the sky covered by the survey necessitated the use of plates covering a large field 20° square. Naturally, in the outer parts of the field the extra-focal images were distorted and unsuitable for measurement, so Schwarzschild adopted the plan of displacing the image in a regular manner during the exposure so as to build up a square of uniform greyness; the device is known as the "Schraffierkassette." The immediate results of this work were of great interest; the visual magnitudes of all these stars had been measured at Potsdam, and the differences, photographic minus visual, gave the colour-indices of some 3500 stars. (The use of this quantity as a measure of colour seems to have been first suggested by him some years earlier.) In conjunction with similar work by King and Parkhurst, the Aktinometrie showed that spectral type is very accurately a function of the colour-index. It also brought out the fact that it is the stars of intermediate colour (yellow) which possess the largest proper motions, and that after a maximum about type G, the proper motions diminish very rapidly for the redder types. This feature of the red stars was just beginning to be recognised; but the change of proper motion appeared particularly striking when exhibited as a function of a continuous variable, colour-index, instead of the broad divisions of spectral class.
In 1895, acting on a suggestion by Michelson, Schwarzschild introduced the plan of placing a coarse grating of parallel strips in front of the object-glass of a telescope so as to give diffraction images at a small distance from the principal image of each star. This device has since been used extensively both for measuring the colour, or effective wave-length, of stellar light and for establishing an absolute scale of magnitudes. Schwarzschild's first application was not to these purposes but to the measurement of double stars, the arrangement acting as a kind of double-image micrometer. The separation of the diffraction-images could be varied in a known manner by tilting the grating at different angles to the line of vision. Successful results were obtained with a 10-inch refractor; but no doubt there would be difficulties in working the arrangement at the end of a long telescope..
The task of determining accurate data for large numbers of stars inevitably leads the mind to consider the great problems of the structure of the stellar universe. Somewhat in the way that the Cape Durchmusterung led Kapteyn to his cosmical researches, so the Göttingen Aktinometrie may have turned Schwarzschild's thoughts to this wide field of discovery. But in his case there was already the influence of his teacher Seeliger, the pioneer in the mathematical theory of the distribution of stars. Schwarzschild's best-known work in this subject is his ellipsoidal hypothesis of stellar motions. It had been shown by Kapteyn that the individual motions of stars are not haphazard, but are especially directed in two favoured directions that there are, in fact, two streams of stars. The directions of these streams, when referred not to the sun but to the stars as a whole, automatically become opposite to one another. In an approximate mathematical representation it was perhaps natural to take the streams as though they were two independent systems of stars passing through one another; but the theory proposed by Schwarzschild showed clearly that this division into two systems was not an essential part of Kapteyn's discovery. The important point to preserve in the formulæ was the colotropic distribution of the motions a greater mobility to and fro along one axis in space than in perpendicular directions. He noticed that this property could be represented by a simple modification of Maxwell's law of velocities; by setting the frequency of a velocity proportional to the distribution stands in the same kind of relation to Maxwell's random distribution as an ellipsoid does to a sphere. The apparent antagonism between the two-drift and ellipsoidal hypotheses disappears if we remember that the purpose of both is descriptive. In a letter communicating his results to the writer, Schwarzschild said, "The formal comparison of our two hypotheses of distribution shows very well how nearly the same they are.... Within our visible system the gravitation of the whole is certainly much stronger than the attraction of single stars at casual approaches. We could represent the stellar system as a homogeneous ellipsoid, and assume that the initial velocities of the stars are so distributed that this homogeneous ellipsoid always remains similar; then the velocities parallel to the greatest axis must be on the average greater than those parallel to the least. In that way one can try to render my hypothesis more plausible. But I freely admit that it has difficulties like yours. You will certainly continue your researches on the two-drift hypothesis, as I shall on the ellipsoidal; and by this competition I hope the solution of the whole marvellous problem may be accelerated." The ellipsoidal theory has great mathematical elegance both in its conception and its application. Whilst the two-drift theory has often been preferred in the ordinary proper-motion investigations, on account of an additional constant in the formulæ which gives it a somewhat greater flexibility, the ellipsoidal theory has been found the more suitable for discussions of radial velocities and the dynamical theory of the stellar system.
The study of proper motions leads naturally to the question of the distribution of the stars in space. In an elementary conception of this problem we may suppose that each part of space contains stars of varying degrees of luminosity in proportions determined by some standard law, and with various speeds also following a uniform law, but there will probably be a thinning out in the number of stars per unit volume as we recede from the centre of the stellar system. As knowledge has progressed, much complexity has been added to this simple scheme; we have had to step aside to investigate star-streaming or the varying behaviour of different spectral types. But the luminosity law and the density law, involving the specification of the shape and dimensions of the stellar system, must remain a fundamental aim of research. It is theoretically possible to deduce the two laws from counts of stars between different limits of magnitude together with the mean parallax (or parallactic motion) for each magnitude. The general solution of this problem was given by Schwarzschild; it involved the solution of certain integral equations by an original method of considerable mathematical interest. In practice it is usually more desirable to use a less general method, assuming particular forms of the laws which give a sufficient approximation; here Schwarzschild followed more closely the lines laid down by Seeliger, but by his keen insight he brought the mathematical discussion into an admirably clear form. His latest and fullest discussion of the observational data is given in the Sitzungsberichte of the Berlin Academy, 1914, where the different spectral types are treated separately. Workers in this kind of statistical research owe a grudge to the discovery of "giant" and "dwarf" stars, which has played havoc with some of the hoped-for simplicity of the problem, and seems to have placed definitive results for the moment out of reach; but step by step the knowledge of stellar distribution is improving, and Schwarzschild's mathematical treatment has notably contributed to the advance.
In solar physics Schwarzschild has left his mark by his theory of radiative equilibrium of the sun's atmosphere, published in 1906. Three possible modes of transfer are recognised by which the heat squandered into space from the outer layers may be replaced from the stores of energy within. Conduction can scarcely play a significant part, and it has usually been considered that convection currents are the main agents of transfer, as in the terrestrial atmosphere. But at the high temperature of the sun it is more likely that radiation outweighs all other causes – certainly in the interior, and probably also in the atmosphere proper. In this case the state of equilibrium must be determined by the condition that each element radiates just as much energy as it gains by absorption of the radiation passing through in all directions. In proposing this state of radiative equilibrium, Schwarzschild was unwittingly following the theory put forward by Sampson twelve years earlier; but by introducing Stefan's law, that the density of radiation varies as the fourth power of the temperature, he was able to give definite shape to the formule. He showed also that the radiative hypothesis appears to account satisfactorily for the observed degree of darkening of the sun's limb.
An account of Schwarzschild's achievements necessarily has a disjointed appearance, so varied were the subjects he pursued. We can only refer very briefly to some other scientific contributions. Among his theoretical researches with a definitely astronomical application may be noted his calculation of the light-pressure on small solid particles. He showed that the increase in the ratio of light-pressure to gravitation with diminishing size ceases to hold when the dimensions of the particle are comparable with a wave-length of light, so that there is a maximum value of this ratio (for solid particles), which in the case of the sun is about 20. Since the repulsion observed in comets' tails is believed frequently to exceed the limit of twenty times the gravitation, this conclusion is of considerable practical interest. Schwarzschild also made an important step in the discussion of the equilibrium of the pear-shaped figure of rotating fluid in connection with Poincaré's theory of exchange of stabilities. In geometrical optics he devised a reflecting telescope with two concave mirrors, which should give a large field of good definition (equivalent to that of the astrographic telescopes) for a focal ratio of 1 in 3. A preliminary memoir, in which, however, he expressly disclaims originality, contains what is probably the most concise and clear account to be found anywhere of the use of the characteristic function (or eikonal) in practical optics. Other branches of research were so diverse as the electron theory and the navigation of airships; for the latter he published suitable tables in 1909, and later devised a special form of sextant. The recent upheaval of the fundamental ideas of physics interested him deeply; a paper by him on the relativity explanation of the motion of the perihelion of Mercury is not yet accessible in this country, and during his last illness he corrected the proofs of a paper on the quantum theory for the Berlin Academy.
The wide range of his contributions to knowledge suggests a comparison with Poincaré; but Schwarzschild's bent was more practical, and he delighted as much in the design of instrumental methods as in the triumphs of analysis. It is interesting, though sometimes misleading, to trace a unity in the varied researches of a man's lifetime, for it often happens that a thread of association binds the whole together. But Schwarzschild's versatility baffles any such summary; his joy was to range unrestricted over the pastures of knowledge, and, like a guerilla leader, his attacks fell where they were least expected. He sought no unity save the one bond which unites together all branches of scientific endeavour.
"Mathematics, physics, chemistry, astronomy, march in one front. Whichever lags behind is drawn after. Whichever hastens ahead helps on the others. The closest solidarity exists between astronomy and the whole circle of exact science. From this aspect I may count it well that my interest has never been limited to the things beyond the moon, but has followed the threads which spin themselves from there to our sublunar knowledge; I have often been untrue to the heavens. That is an impulse to the universal which was strengthened unwittingly by my teacher Seeliger, and afterwards was further nourished by Felix Klein and the whole scientific circle at Göttingen. There the motto runs that mathematics, physics, and astronomy constitute one knowledge, which, like the Greek culture, is only to be comprehended as a perfect whole."
Soon after the outbreak of war Schwarzschild was placed in charge of a meteorological station for the German army at Namur. He was afterwards attached to the artillery staff, at first in France and later on the Eastern front. He received the order of Knight of the Iron Cross. As might be expected, the German army organisation took care to give scope to his scientific ability, and in 1915 November he contributed to the Berlin Academy a paper on "The Effect of Wind and Air-density on the Path of a Projectile": publication has, of course, been postponed. On military service he contracted a disease which ultimately proved fatal. After a long illness, attended with much suffering, he died on 1916 May 11. His lively and attractive character and his readiness to co-operate with others brought him into cordial relationship with astronomers in many parts of the world, and his early death is felt as a deep personal loss. He was elected an Associate on 1909 June 11.
A. S. E.
After a short interval as privatdocent at Munich, he was appointed Director of the Observatory of Göttingen in succession to Schur. Here he joined a brilliant circle of scholars who have made Göttingen world-famous. To a man of his wide interests in all branches of mathematics and physics the surroundings must have been very congenial; and to his growing reputation as an astronomer he added important achievements in other fields of research. Finally, in 1909 he was chosen at the age of thirty-six to succeed Vogel as Director of the Astrophysical Observatory, Potsdam, a position which he filled with distinguished success.
In reviewing his contributions to astronomy we may start with his work on photographic magnitudes, a long and sustained investigation, which required much perfection of detail as well as brilliance of conception. His first method, carried out at the Von Kuffner Observatory, was to measure the greyness of star-images taken out of focus. He set out to determine the variation of photographic effect with the length of exposure, as a step towards forming an absolute scale of magnitudes. It was already known that an increase of exposure-time has not the same effect as an increase of light-intensity-doubling the exposure is not so effective as doubling the brightness. Schwarzschild proposed the now well-known law that the photographic effect is proportional to , where is the intensity of the light, and the exposure-time. For the constant he found the value 0.675; but this appears to have been too low, and in the Göttingen Aktinometrie he redetermined the constant as 0.76 – which means practically that an increase of exposure which might be expected to give a gain of four magnitudes in reality only gains three.
The fruit of this preliminary work was reaped when he came to Göttingen and, with the aid of his assistants, carried out the "Aktinometrie." This was a large undertaking, involving the determination of the magnitudes of all the stars brighter than between 0º and +20° dec., and their reduction to an absolute scale. The great area of the sky covered by the survey necessitated the use of plates covering a large field 20° square. Naturally, in the outer parts of the field the extra-focal images were distorted and unsuitable for measurement, so Schwarzschild adopted the plan of displacing the image in a regular manner during the exposure so as to build up a square of uniform greyness; the device is known as the "Schraffierkassette." The immediate results of this work were of great interest; the visual magnitudes of all these stars had been measured at Potsdam, and the differences, photographic minus visual, gave the colour-indices of some 3500 stars. (The use of this quantity as a measure of colour seems to have been first suggested by him some years earlier.) In conjunction with similar work by King and Parkhurst, the Aktinometrie showed that spectral type is very accurately a function of the colour-index. It also brought out the fact that it is the stars of intermediate colour (yellow) which possess the largest proper motions, and that after a maximum about type G, the proper motions diminish very rapidly for the redder types. This feature of the red stars was just beginning to be recognised; but the change of proper motion appeared particularly striking when exhibited as a function of a continuous variable, colour-index, instead of the broad divisions of spectral class.
In 1895, acting on a suggestion by Michelson, Schwarzschild introduced the plan of placing a coarse grating of parallel strips in front of the object-glass of a telescope so as to give diffraction images at a small distance from the principal image of each star. This device has since been used extensively both for measuring the colour, or effective wave-length, of stellar light and for establishing an absolute scale of magnitudes. Schwarzschild's first application was not to these purposes but to the measurement of double stars, the arrangement acting as a kind of double-image micrometer. The separation of the diffraction-images could be varied in a known manner by tilting the grating at different angles to the line of vision. Successful results were obtained with a 10-inch refractor; but no doubt there would be difficulties in working the arrangement at the end of a long telescope..
The task of determining accurate data for large numbers of stars inevitably leads the mind to consider the great problems of the structure of the stellar universe. Somewhat in the way that the Cape Durchmusterung led Kapteyn to his cosmical researches, so the Göttingen Aktinometrie may have turned Schwarzschild's thoughts to this wide field of discovery. But in his case there was already the influence of his teacher Seeliger, the pioneer in the mathematical theory of the distribution of stars. Schwarzschild's best-known work in this subject is his ellipsoidal hypothesis of stellar motions. It had been shown by Kapteyn that the individual motions of stars are not haphazard, but are especially directed in two favoured directions that there are, in fact, two streams of stars. The directions of these streams, when referred not to the sun but to the stars as a whole, automatically become opposite to one another. In an approximate mathematical representation it was perhaps natural to take the streams as though they were two independent systems of stars passing through one another; but the theory proposed by Schwarzschild showed clearly that this division into two systems was not an essential part of Kapteyn's discovery. The important point to preserve in the formulæ was the colotropic distribution of the motions a greater mobility to and fro along one axis in space than in perpendicular directions. He noticed that this property could be represented by a simple modification of Maxwell's law of velocities; by setting the frequency of a velocity proportional to the distribution stands in the same kind of relation to Maxwell's random distribution as an ellipsoid does to a sphere. The apparent antagonism between the two-drift and ellipsoidal hypotheses disappears if we remember that the purpose of both is descriptive. In a letter communicating his results to the writer, Schwarzschild said, "The formal comparison of our two hypotheses of distribution shows very well how nearly the same they are.... Within our visible system the gravitation of the whole is certainly much stronger than the attraction of single stars at casual approaches. We could represent the stellar system as a homogeneous ellipsoid, and assume that the initial velocities of the stars are so distributed that this homogeneous ellipsoid always remains similar; then the velocities parallel to the greatest axis must be on the average greater than those parallel to the least. In that way one can try to render my hypothesis more plausible. But I freely admit that it has difficulties like yours. You will certainly continue your researches on the two-drift hypothesis, as I shall on the ellipsoidal; and by this competition I hope the solution of the whole marvellous problem may be accelerated." The ellipsoidal theory has great mathematical elegance both in its conception and its application. Whilst the two-drift theory has often been preferred in the ordinary proper-motion investigations, on account of an additional constant in the formulæ which gives it a somewhat greater flexibility, the ellipsoidal theory has been found the more suitable for discussions of radial velocities and the dynamical theory of the stellar system.
The study of proper motions leads naturally to the question of the distribution of the stars in space. In an elementary conception of this problem we may suppose that each part of space contains stars of varying degrees of luminosity in proportions determined by some standard law, and with various speeds also following a uniform law, but there will probably be a thinning out in the number of stars per unit volume as we recede from the centre of the stellar system. As knowledge has progressed, much complexity has been added to this simple scheme; we have had to step aside to investigate star-streaming or the varying behaviour of different spectral types. But the luminosity law and the density law, involving the specification of the shape and dimensions of the stellar system, must remain a fundamental aim of research. It is theoretically possible to deduce the two laws from counts of stars between different limits of magnitude together with the mean parallax (or parallactic motion) for each magnitude. The general solution of this problem was given by Schwarzschild; it involved the solution of certain integral equations by an original method of considerable mathematical interest. In practice it is usually more desirable to use a less general method, assuming particular forms of the laws which give a sufficient approximation; here Schwarzschild followed more closely the lines laid down by Seeliger, but by his keen insight he brought the mathematical discussion into an admirably clear form. His latest and fullest discussion of the observational data is given in the Sitzungsberichte of the Berlin Academy, 1914, where the different spectral types are treated separately. Workers in this kind of statistical research owe a grudge to the discovery of "giant" and "dwarf" stars, which has played havoc with some of the hoped-for simplicity of the problem, and seems to have placed definitive results for the moment out of reach; but step by step the knowledge of stellar distribution is improving, and Schwarzschild's mathematical treatment has notably contributed to the advance.
In solar physics Schwarzschild has left his mark by his theory of radiative equilibrium of the sun's atmosphere, published in 1906. Three possible modes of transfer are recognised by which the heat squandered into space from the outer layers may be replaced from the stores of energy within. Conduction can scarcely play a significant part, and it has usually been considered that convection currents are the main agents of transfer, as in the terrestrial atmosphere. But at the high temperature of the sun it is more likely that radiation outweighs all other causes – certainly in the interior, and probably also in the atmosphere proper. In this case the state of equilibrium must be determined by the condition that each element radiates just as much energy as it gains by absorption of the radiation passing through in all directions. In proposing this state of radiative equilibrium, Schwarzschild was unwittingly following the theory put forward by Sampson twelve years earlier; but by introducing Stefan's law, that the density of radiation varies as the fourth power of the temperature, he was able to give definite shape to the formule. He showed also that the radiative hypothesis appears to account satisfactorily for the observed degree of darkening of the sun's limb.
An account of Schwarzschild's achievements necessarily has a disjointed appearance, so varied were the subjects he pursued. We can only refer very briefly to some other scientific contributions. Among his theoretical researches with a definitely astronomical application may be noted his calculation of the light-pressure on small solid particles. He showed that the increase in the ratio of light-pressure to gravitation with diminishing size ceases to hold when the dimensions of the particle are comparable with a wave-length of light, so that there is a maximum value of this ratio (for solid particles), which in the case of the sun is about 20. Since the repulsion observed in comets' tails is believed frequently to exceed the limit of twenty times the gravitation, this conclusion is of considerable practical interest. Schwarzschild also made an important step in the discussion of the equilibrium of the pear-shaped figure of rotating fluid in connection with Poincaré's theory of exchange of stabilities. In geometrical optics he devised a reflecting telescope with two concave mirrors, which should give a large field of good definition (equivalent to that of the astrographic telescopes) for a focal ratio of 1 in 3. A preliminary memoir, in which, however, he expressly disclaims originality, contains what is probably the most concise and clear account to be found anywhere of the use of the characteristic function (or eikonal) in practical optics. Other branches of research were so diverse as the electron theory and the navigation of airships; for the latter he published suitable tables in 1909, and later devised a special form of sextant. The recent upheaval of the fundamental ideas of physics interested him deeply; a paper by him on the relativity explanation of the motion of the perihelion of Mercury is not yet accessible in this country, and during his last illness he corrected the proofs of a paper on the quantum theory for the Berlin Academy.
The wide range of his contributions to knowledge suggests a comparison with Poincaré; but Schwarzschild's bent was more practical, and he delighted as much in the design of instrumental methods as in the triumphs of analysis. It is interesting, though sometimes misleading, to trace a unity in the varied researches of a man's lifetime, for it often happens that a thread of association binds the whole together. But Schwarzschild's versatility baffles any such summary; his joy was to range unrestricted over the pastures of knowledge, and, like a guerilla leader, his attacks fell where they were least expected. He sought no unity save the one bond which unites together all branches of scientific endeavour.
"Mathematics, physics, chemistry, astronomy, march in one front. Whichever lags behind is drawn after. Whichever hastens ahead helps on the others. The closest solidarity exists between astronomy and the whole circle of exact science. From this aspect I may count it well that my interest has never been limited to the things beyond the moon, but has followed the threads which spin themselves from there to our sublunar knowledge; I have often been untrue to the heavens. That is an impulse to the universal which was strengthened unwittingly by my teacher Seeliger, and afterwards was further nourished by Felix Klein and the whole scientific circle at Göttingen. There the motto runs that mathematics, physics, and astronomy constitute one knowledge, which, like the Greek culture, is only to be comprehended as a perfect whole."
Soon after the outbreak of war Schwarzschild was placed in charge of a meteorological station for the German army at Namur. He was afterwards attached to the artillery staff, at first in France and later on the Eastern front. He received the order of Knight of the Iron Cross. As might be expected, the German army organisation took care to give scope to his scientific ability, and in 1915 November he contributed to the Berlin Academy a paper on "The Effect of Wind and Air-density on the Path of a Projectile": publication has, of course, been postponed. On military service he contracted a disease which ultimately proved fatal. After a long illness, attended with much suffering, he died on 1916 May 11. His lively and attractive character and his readiness to co-operate with others brought him into cordial relationship with astronomers in many parts of the world, and his early death is felt as a deep personal loss. He was elected an Associate on 1909 June 11.
A. S. E.
Karl Schwarzschild's obituary appeared in Journal of the Royal Astronomical Society 77:4 (1917), 314-319.