where p is its density, G is the gravitation constant, and M is the mass of the sun. If light repulsion is n times gravitation pull, GM for 62 is the acceleration of the earth in its orbit, and is equal to o ̊59 cm./sec.2; We see that a would be inversely proportional to ʼn if the repulsion continued according to the same law. But, as stated in the text, diffraction comes into play when a is very small, and the ratio of light pressure to gravitation pull diminishes when a is reduced below a certain value. The maximum ratio is probably when a is about NOTE 6, p. 73. THE AMOUNT OF MATTER WHICH CAN BE PUSHED OUT BY THE PRESSURE OF SUNLIGHT Consider a square centimetre area perpendicular to the rays from the sun. The momentum streaming through it per second is where S is the solar constant in ergs per sec. per sq. cm. at the earth's distance b from the sun and r is the distance of the square centimetre from the sun. Let the square centimetre be the origin of matter which experiences light-pressure equal to n times the gravitation pull. Let the matter be supposed to be all within a short distance behind the square centimetre. The maximum amount which can be repelled is that which absorbs all the sunlight and all its momentum. Let it be m. The gravitation pull on m is where G is the gravitation constant and M is the mass of centimetre. If the sunlight is partly reflected then something less than twice this amount of matter can have push balancing pull. If the absorbing matter is scattered at different distances behind the square centimetre, the amount of matter which can be balanced will be increased. For suppose it at double the distance. The cross section of the cone, with vertex at the sun and the square centimetre as base, will at the double distance have four times the area, and four times the matter can thus be balanced. But we confine the investigation to the case in which the matter is not far behind the square centimetre in comparison with the distance r, the case to which the tails of comets, at any rate, roughly correspond; and the correspondence is the closer in that the density must decrease rapidly as we recede from its head. With constant acceleration outwards, half the matter is in the first quarter of the tail. We see at once that no gas can be repelled. For there is no gas known in which the absorption of a layer of mass 10-4 gm. per sq. cm. at all approaches completeness. Let us suppose that the matter consists of opaque absorbing particles, and, for the sake of illustration, suppose n = 10, a value probably existing in some comets' tails. 1 if the whole momentum is absorbed. But it is an exceedingly high estimate to suppose that of the sunlight is stopped. The mass repelled even on this high estimate of absorption cannot exceed Behind a square kilometre or 1010 sq. cm. there cannot be more than 1010 = 103 grammes or 1 kilogramme of matter. We thus obtain a superior limit to the amount of matter in a comet's tail on the light-pressure theory. If we suppose that the tail is 107 kilometres long, and that it consists of absorbing spheres of density 1 and radius 10-5 cm. each, and if we suppose that n = 10, the maximum number of particles behind a square centimetre is given by As the number of c.c. behind 1 sq. cm. in a length of 107 kilometres is 1012, the number of particles per c.c. NOTE 7, p. 79. THE RESISTING FORCE ON A SPHERE MOVING ROUND THE SUN AND RADIATING R PER SQ. CM. PER SECOND AND THE REDUCTION OF ITS ORBIT ROUND THE SUN THE DOPPLER EMISSION EFFECT The investigation 1 is divided into three parts: (1) The pressure on a very small area moving along its own normal; (2) the tangential stress on a very small area moving in its own plane; (3) the application of (1) and (2) to a moving sphere The principle of the investigation for (1) and (2) consists in finding the momentum in the space between two hemispherical surfaces containing the disturbance emitted in a short time which we may without loss of precision take as the unit time. The resultant of this momentum reversed is the pressure on the area at the time of emission. The stream of energy is normal to the hemispherical surfaces, and we may therefore apply the normal stream method of Note 2, which gives the momentum density 1 An investigation of the Doppler emission effect was given by the author in Phil. Trans., 202, 1903, p. 546, but, as pointed out in a note on p. 550, the result obtained was double the true value. The method here given appears to be more direct and satisfactory. |