The Aberration of LightApril 15, 2022
The explanation of the aberration is so simple.
It uses the startling hypothesis that the luminiferous aether passes freely through the sides of the telescope and through the earth itself.
The planets carry a portion of the aether with them. The aether, close to their surfaces, is at rest relatively on those surfaces.
- Its velocity changes as we recede from the surface, till, at no great distance, it is at rest in space.
According to the undulatory theory, the direction in which a heavenly body is seen is normal to the fronts of the waves which have emanated from it, and which have reached the neighbourhood of the observer. The aether near him is at rest relatively to him.
- If the aether in space were at rest, the front of a wave of light at any instant being given, its front at any future time could be found by the method explained in Airy’s Tracts.
- If the æther were in motion, and the speed of light were infinitely small, the wave’s front would be displaced as a surface of particles of the æther.
Neither of these suppositions are true because the aether moves while light goes through it.
The displacements of a wave’s front in an elementary portion of time due to the two causes just considered take place independently.
w be the resolved parts along the rectangular axes of
z of the speed of the aether-particle whose co-ordinates are
V be the speed of light supposing the æther at rest.
Because of the distance of the heavenly bodies, only the plane waves are considered, as they they are distorted by the æther’s motion.
Let the axis of
z be taken in, or nearly in the direction of the wave considered, so that the equation to the wave’s front at any time will be:
z = C + VT + ζ (1.)
Cis a constant
tis the time
ζis a small quantity, a function of
ζ are of the order of the aberration, their squares and products may be neglected.
y the angles which the normal to the wave’s front at the point (x, y, z) with the axes, we have, to the first order of approximation:
cos a = - dζ / dz, cos B = - dζ / dy, cos y = 1 (2.)
If we take a length
Vdt along this normal, the co-ordinates of its extremity will be:
x - (dζ / dx) Vdt, y - (dζ / dy) Vdt, z + Vdt
If the aether were at rest, the locus of these extremities would be the wave’s front at the time
t + dt. But since it is in motion, the co-ordinates of those extremities must be further increased by udt, vdt, wdt.
Denoting then by
x1, y1, z1, the co-ordinates of the point of the wave’s front at
, which corresponds to the point ( and eliminating and ) in its front at the time , we have from these equations and (1.), and denoting by , we have for the equation to the wave’s front at the time or, expanding, neglecting and the square of the aberration, and suppressing the accents of and , (3.) But from the definition of it follows that the equation to the wave’s front at the time be got from (1.) by putting for , and we have therefore for this equation, will (4.) Comparing the identical equations (3.) and (4.), we have This equation gives = but in the small term this comes to taking the approximate value of we may replace given by the equation by
, instead of , for the parameter of the system of surfaces formed by the wave’s front in its successive positions. Hence equation (1.) becomes Combining the value of just found with equations (2.), we get, to a first approximation,(5.) equations which might very easily be proved directly in a more geometrical manner. If random values are assigned to and equations will be a complicated one; but if , the law of aberration resulting from these and are such that is an exact differential, we have whence, denoting by the suffixes 1, 2 the values of the variables belonging to the first and second limits respectively, we obtain (6.)
If the motion of the æther be such
is an exact differential for one system of rectangular axes, it is easy to prove, by the transformation of co-ordinates, that it is an exact differential for any other system. Hence the formulae (6.) will hold good, not merely for light propagated in the direction first considered, but for light propagated in any direction, the direction of propagation being taken in each case for the axis of . If we assume that is an exact differential for that part of the motion of the æther which is due to the motions of translation of the earth and planets, it does not therefore follow that the same is true for that part which depends on their motions of rotation. Moreover, the diurnal aberration is too small to be detected by observation, or at least to be measured with any accuracy, and I shall therefore neglect it. It is not difficult to show that the formulae (6.) lead to the known law of aberration. In applying them to the case of a star, if we begin the integrations in equations (5.) at a point situated at such a distance from the earth that the motion of the æther, and consequently the resulting change in the direction of the light, is insensible, we shall have moreover, we take the plane and ; and if, to pass through the direction of the earth’s motion, we shall havethat is, the star will appear to be displaced towards the direction in which the earth is moving, through an angle equal to the ratio of the velocity of the earth to that of light, multiplied by the sine of the angle between the direction of the earth’s motion and the line joining the earth and the star.
In considering the effect of aberration on a planet, it will be convenient to divide the integrations in equation (5.) into three parts, first integrating from the point considered on the surface of the planet to a distance at which the motion of the æther may be neglected, then to a point near the earth where we may still neglect the motion of the æther, and lastly to the point of the earth’s surface at which the planet is viewed. For the first part we shall have will be the resolved parts of the planet’s velocity.
The increments of interval will be, therefore, .
For the second interval for the third their increments will be and , and and for the first will remain constant, while , just as in the case of a star, and being now the resolved parts of the earth’s velocity.
Fig. 1  Fig. 2 Fig. 1 represents what is conceived to take place. light quitted it; is the planet in the position it had when the the earth in the position it has when the light reaches it. The lines , &c. represent a small portion of a wave of light in its successive positions. The arrows represent the directions in which and may be conceived to move. The breadth is supposed to becomparable to the breadth of a telescope. In fig. 2, the surfaces , &c. ; is the point of the planet from which the light starts, the earth which it reaches. The trajectory the ends at represents an orthogonal trajectory to the point of may be considered a straight line, except near and , where it will be a little curved, as from to and from to . The curvature will have the same effect on the apparent position of the planet as it would have on that of a star in the same direction = as to the curvature at , if we draw produced, the curvature will have the effect of causing angle between the tangents at and perpendicular to to be seen as if it were at . Now the being that through which a star in the direction of is displaced by aberration to an observer at , and the distance being by hypothesis small (two or three radii of the planet suppose), it follows that the angle is extremely small, and may be neglected. Hence a planet will appear to be displaced from the position which it had when the light left it, just as a star in the same direction is displaced. But besides this, the planet has moved from while the light has been travelling to . These two considerations combined lead to the formula for aberration, which is applicable to the planets, as is shown in treatises on astronomy. The same reasoning which applies to a planet will apply equally to the sun, the moon, or a comet.
To give an idea of the sort of magnitudes neglected in neglecting
pmequal to the diameter of P
- the curvature from p to m uniform
- r be the radius of P
- v its velocity
- R the distance PE
The greatest possible value of the angle between the tangents . In this case we should have as seen from , suppose . Hence the angle
being the semidiameter must be very much greater for the moon than for any other body of the solar system; for in the case of the planets the value of is in no instance double its value for the earth or moon, while their discs are very small compared with that of the moon; and in the case of the sun, although its disc is about as large as that of the moon, its velocity round the centre of gravity of the solar system is very small.
It would indeed be more correct to suppose the sun’s centre absolutely at rest, since all our measurements are referred to it, and not to the centre of gravity of the solar system.
Taking then the case of the moon, and supposing , we find that the angle is about th of a second, an insensible quantity.
If the whole solar system were moving with a velocity comparable with that of the earth around the sun, then from the linearity of the equations employed, this motion can be considered separately.
Regarding this motion, the sun, moon, and planets will come into the positions in which they are seen just at the instant that the light from them reaches the earth.
With respect to the stars also, that part of the aberration which varies with the time of year, the only part which can be observed, will not be affected.
If we suppose the æther which fills the portion of space occupied by the solar system to be moving in a current, with a velocity comparable with that of the earth in its orbit, the result will still be the same.
For if we suppose a velocity equal and opposite to that of the æther to be impressed, both on the æther and on the bodies of the solar system, the case is reduced to that of the solar system moving through the æther supposed to be at rest.