Document:Dark Stars 1783
John Michell's classic paper, "On the Means of discovering the Distance, Magnitude, etc. of the Fixed Stars ..." was written as a letter to his friend Henry Cavendish, in 1783 and published in the Royal Society's "Proceedings" in 1784. It's notable for predicting spectral shifts and calculating the critical horizon limit of a dark star or black hole, from Newtonian principles, more than a century before Einstein.
- "... if the semi-diameter of a sphaere of the same density with the sun were to exceed that of the sun in the proportion 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface a greater velocity than that of light, and consequently, supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity."
This is not a bad estimate: we currently believe the Sun's surface escape velocity to be about 617.5 km/s, so with the escape velocity for a body with a given density increasing proportionally to the radius, multiplying this by 500 gives an escape velocity of around 309,000 km/s ... and the speed of light is close to 300,000 km/s. So Michell's calculation is correct increase the radius of the Sun by 500 times, while adding enough mass to keep the density constant (500×500×500=125 million solar masses (!) ) would push the resulting star's escape velocity to just over the speed of light.
- " If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from light; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability ..."
In this section Michell suggests that we should be able to spot dark stars that are part of binary systems by the motion of their visible partner.
Michell goes on to suggest a method for measuring the shift in energy of starlight using a prism ... but he talks in terms of the velocity of light being modified rather than just its energy – at this point we did not yet have the concept of local lightspeed constancy. Michell also didn't have the luxury of knowing about spectral bands, so he was discussing the image of the entire star being shifted sideways in a prism-split image ... which would be true if stars emitted monochromatic light. However, with a star emitting a broad spread of frequencies, although an original spectral line in the image would be shifted by a prism, for the overall light, the redder end of the spectrum would be shifted out of the human visual range into the infrared, and the ultraviolet range would enter the human visual range, so the resulting smear of frequencies from the star would look (to the human eye) pretty much the same.
The paper itself
VII. On the Means of discovering the Distance, Magnitude, &c. of the Fixed Stars, in consequence of the Diminution of the Velocity of their Light, in case such a Diminution should be found to take place in any of them, and such other Data should be procured from Observations, as would be farther necessary for that Purpose.
By the Rev. John Michell, B. D. F.R.S. – In a Letter to Henry Cavendish, Esq. F.R.S. and A. S. – Read November 27, 1783. – Thornhill, May 26, 1783.
THE method, which I mentioned to you when I was last in London, by which it might perhaps be possible to find the distance, magnitude, and weight of some of the fixed stars, by means of the diminution of the velocity of their light, occurred to me soon after I wrote what is mentioned by Dr. PRIESTLEY in his History of Optics, concerning the diminution of the velocity of light in consequence of the attraction of the sun; but the extreme difficulty, and perhaps impossibility, of procuring the other data necessary for this purpose appeared to me to be such objections against the scheme, when I first thought of it, that I gave it then no further consideration. As some late observations, however, begin to give us a little more chance of procuring some at least of these data, I thought it would not be amiss, that astronomers should be apprized of the method, I propose (which as far as I know, has not been suggested by any one else) lest, for want of being aware of the use, which may be made of them, they should neglect to make the proper observations, when in their power; I shall therefore beg the favour of you to present the following paper on this subject to the Royal Society.
I am, &c.
THE very great number of stars that have been discovered to be double, triple, &c. particularly by Mr. HERSCHEL * , if we apply the doctrine of chances, as I have heretofore done in my "Enquiry into the probable Parallax, &c. of the Fixed Stars," published in the Philosophical Transactions for the year 1767, cannot leave a doubt with any one, who is properly aware of the force of those arguments, that by far the greatest part, if not all of them, are systems of stars so near to each other, as probably to be liable to be affected sensibly by their mutual gravitation; and it is therefore not unlikely, that the periods of the revolutions of some of these about their principals (the smaller ones being, upon this hypothesis, to be considered as satellites to the others) may some time or other be discovered.
See his Catalogue of Stars of this kind, published in the Philosophical Transactions for the year 1782, which is indeed a most valuable present to the astronomical world. By a happy application of very high magnifying powers to his telescopes, and by a most persevering industry in observing, he has made a very wonderful progress in this branch of astronomy, in which almost nothing of any consequence had been done by any one before him.
Now the apparent diameter of any central body, round which any other body revolves, together with their apparent distance from one another, and the periodical time of the revolving body being given, the density of the central body will be given likewise. See Sir ISAAC NEWTON's Prin. b. III. pr. VIII. cor. I.
But the density of any central body being given, and the velocity any other body would acquire by falling towards it from an infinite height, or, which is the same thing, the velocity of a comet revolving in a parabolic orbit, at its surface, being given, the quantity of matter, and consequently the real magnitude of the central body, would be given likewise.
Let us now suppose the particles of light to be attracted in the same manner as all other bodies with which we are acquainted; that is, by forces bearing the same proportion to their vis inertiae, of which there can be no reasonable doubt, gravitation being, as far as we know, or have any reason to believe, an universal law of nature. Upon this supposition then, if any one of the fixed stars, whose density was known by the above-mentioned means, should be large enough sensibly to affect the velocity of the light issuing from it, we should have the means of knowing its real magnitude, &c.
It has been demonstrated by Sir ISAAC NEWTON, in the 39th proposition of the first book of his Principia, that if a right line be drawn, in the direction of which a body is urged by any forces whatsoever, and there be erected at right angles to that line perpendiculars every where proportional to the forces at the points, at which they are erected respectively, the velocity acquired by a body beginning to move from rest, in consequence of being so urged, will always be proportional to the square root of the area described by the aforesaid perpendiculars. And hence,
If such a body, instead of beginning to move from rest, had already some velocity in the direction of the same line, when it began to be urged by the aforesaid forces, its velocity would then be always proportional to the square root of the sum or difference of the aforesaid area, and another area, whose square root would be proportional to the velocity which the body had before it began to be so urged; that is, to the square root of the sum of those areas, if the motion acquired was in the same direction as the former motion, and the square root of the difference, if it was in a contrary direction. See cor.2 to the aforesaid proposition.
In order to find, by the foregoing proposition, the velocity which a body would acquire by falling towards any other central body, according to the common law of gravity, let C in the figure (tab. III. ) represent the centre of the central body, towards which the falling point is urged, and let CA be a line drawn from the point C, extending infinitely towards A. If then the line RD be supposed to represent the force, by which the falling body would be urged at any point D, the velocity which it would have acquired by falling from an infinite height to the place D would be the same as that which it would acquire by falling from D to C with the force RD, the area of the infinitely extended hyperbolic space ADRB, where RD is always inversely proportional to the square of DC, being equal to the rectangle RC contained between the lines RD and CD. From hence we may draw the following corollaries.
8. Cor. 1.
The central body DEF remaining the same, and consequently the forces at the same distances remaining the same likewise, the areas of the rectangles RC, rC will always be inversely as the distances of the points D, d, from C, their sides RD, rd being inversely in the duplicate ratio of the sides CD, cd: and therefore, because the velocity of a body falling from an infinite height towards the point C, is always in the sub-duplicate ratio of these rectangles, it will be in the sub-duplicate ratio of the lines CD, Cd, inversely. Accordingly the velocities of comets revolving in parabolic orbits are always in the sub-duplicate ratio of their distances form the sun inversely; and the velocities of the planets, at their mean distances (being always in a given ratio to the velocity of such comets, viz. in the sub-duplicate ratio of 1 to 2) must necessarily observe the same law likewise.
9. Cor. 2.
The magnitude of the central body remaining the same, the velocity of a body falling towards it from an infinite height will always be, at the same distance from the point C, taken any where without the central body, in the sub-duplicate ratio of its density; for in this case the distance Cd will remain the same, the line rd only being increased or diminished in the proportion of the density, and the rectangle rC consequently increased or diminished in the same proportion.
10. Cor. 3.
The density of the central body remaining the same, the velocity of a body falling towards it from an infinite height will always be as its semi-diameter, when it arrives at the same proportional distance from the point C; for the weights, at the surfaces of different sphaeres of the same density are as their respective semi-diameters; and therefore the sides RD and CD, or any other sides rd and Cd, which are in a given ratio to those semi-diameters, being both increased or diminished in the same proportion, the rectangles RC or rC will be increased or diminished in the duplicate ratio of the semi-diameter CD, and consequently the velocity in the simple ratio of CD.
11. Cor. 4.
If the velocity of a body falling from an infinite height towards different central bodies is the same, when it arrives at their surfaces, the density of those central bodies must be in the duplicate ratio of their semi-diameters inversely; for by the last cor. the density of the central body remaining the same, the rectangle RC will be in the duplicate ratio of CD; in order therefore that the rectangle RC may always remain the same, the line RD must be inversely, as CD, and consequently the density inversely, as the square of CD.
12. Cor. 5.
Hence the quantity of matter contained in those bodies must be in the simple ratio of their semi-diameters directly; for the quantity of matter being always in a ratio compounded of the simple ratio of the density, and the triplicate ratio of their semi-diameters, if the density is in the inverse duplicate ratio of the semi-diameters, this will become the direct triplicate and inverse duplicate, that is, when the two are compounded together, the simple ratio of the semi-diameters.
The velocity a body would acquire by falling from an infinite height towards the sun, when it arrived at his surface, being, as has been said before in article 3d, the same with that of a comet revolving in a parabolic orbit in the same place, would be about 20.72 times greater than that of the earth in its orbit at its mean distance from the sun; for the mean distance of the earth from the sun, being about 214.64 of the sun's semidiameters, the velocity of such a comet would be greater at that distance than at the distance of the earth from the sun, in the subduplicate ratio of 214.64 to 1, and the velocity of the comet being likewise greater than that of planets, at their mean distances, in the subduplicate ratio of 2 to 1; these, when taken together, will make the full sub-duplicate ratio of 429.28 to 1, and the square root of 429.28 is 20.72, very nearly.
The same result would have been obtained by taking the line RD proportional to the force of gravity at the sun's surface, and DC equal to his semi-diameter, and from thence computing a velocity, which should be proportional to the square root of the area RC when compared with the square root of another area, one whose sides should be proportional to the force of gravity at the surface of the earth; and the other should be, for instance, equal to 16 feet, 1 inch, the space a body would fall through in one second of time, in which case it would acquire a velocity of 32 feet, 2 inches per second. The velocity thus found compared with the velocity of the earth in its orbit, when computed from the same elements, necessarily gives the same result. I have made use of this latter method of computation upon a former occasion, as may be seen in Dr. PRIESTLEY's History of Optics, p787, &c. but I have rather chosen to take the velocity from that of a comet, in the article above, on account of its greater simplicity, and its more immediate connexion with the subject of this paper.
The velocity of light, exceeding that of the earth in its orbit, when at its mean distance from the sun, in the proportion of about 10,310 to 1, if we divide 10,310 by 20.72, the quotient 497, in round numbers, will express the number of times, which the velocity of light exceeds the velocity a body could acquire by falling from an infinite height towards the sun, when it arrived at his surface; and an area whose square root should exceed the square root of the area RC, where RD is supposed to represent the force of gravity at the surface of the sun, and CD is equal to his semi-diameter, in the same proportion, must consequently exceed the area RC in the proportion of 247,009, the square of 497 to 1.
Hence, according to article 10, if the semi-diameter of a sphaere of the same density with the sun were to exceed that of the sun in the proportion 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface a greater velocity than that of light, and consequently, supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity.
But if the semi-diameter of a sphaere, of the same density with the sun, though the velocity of light emitted from such a body, would never be wholly destroyed, yet would it always suffer some diminution, more or less, according to the magnitude of the said sphaere; and the quantity of this diminution may be easily found in the following manner: Suppose S to represent the semi-diameter of the sun, and aS to represent the semi-diameter of the proposed sphaere; then, as appears from what has been shewn before, the square root of the difference between the square of 497 S and the square of aS will be always proportional to the ultimate remaining velocity, after it has suffered all the diminution, it can possibly suffer from this cause; and consequently the difference between the whole velocity of light, and the remaining velocity, as found above, will be the diminution of its velocity. And hence the diminution of the velocity of light emitted from the sun, on account of it's gravitation towards that body, will be somewhat less than a 494,000dth part of the velocity which it would have had if no such diminution had taken place; for the square of 497 being 247,009, and the square of 1 being 1, the diminution of the velocity will be the difference between the square root of 247,009, and the square root of 247,008, which amounts, as above, to somewhat less than one 494,000th part of the whole quantity.
The same effects would likewise take place, according to article 11, if the semi-diameters were different from those mentioned in the two last articles, provided the density was greater or less in the duplicate ratio of those semi-diameters inversely.
The better to illustrate this matter, it may not be amiss to take a particular example. Let us suppose then, that it should appear from observations made upon some of those double stars above alluded to, that one of the two performed its revolution about the other in 64 years, and that the central one was of the same density with the sun, which it must be, if its apparent diameter, when seen from the other body, was the same as the apparent diameter of the sun would be if seen from a planet revolving round him in the same period: let us further suppose, that the velocity of the light of the central body was found to be less than that of the sun, or other stars whose magnitude was not sufficient to affect it sensibly, in the proportion of 19 to 20. In this case then, according to article 17, the square root of 247,009 SS must be to the square root of the difference between 247,009 SS and aaSS as 20 to 19. But the squares of 20 and 19 being 400 and 361, the quantity 247,009 SS must therefore be to the difference between this quantity and aaSS in the same proportion, that is as 247,009 to 222,925.62; and aaSS must consequently be equal to 24,083.38 SS, whose square root is 155.2 S nearly, or, in round numbers, 155 times the diameter of the sun, will be the diameter of the central star sought.
As the squares of the periodical times of bodies, revolving round a central body, are always proportional to the cubes of their mean distances, the distance of the two bodies from each other must therefore, upon the foregoing suppositions, be sixteen times greater in proportion to the diameter of the central body, than the distance of the earth from the sun in proportion to his diameter; and that diameter being already found to be also greater than that of the sun in the proportion of 155.2 to 1, this distance will consequently be greater than that of the earth and sun from each other in the proportion of 16 times 155.2, that is 2483.2 to 1.
Let us farther suppose, that from the observations, the greatest distance of the two stars in question appeared to be only one second; we must then multiply the number 2483.2 by 206,264.8, the number of seconds in the radius of a circle, and the product 512,196,750 will shew the number of times which such a star's distance from us must exceed that of the sun. The quantity of matter contained in such a star would be (155.2)³ or 3,738,308 times as much as that contained in the sun; its light, supposing the sun's light to take up 8'.7 in coming to the earth, would, with its common velocity, require 7,900 years to arrive at us, and 395 years more on account of the diminution of that velocity; and supposing such a star to be equally luminous with the sun, it would still be very sufficiently visible, I apprehend, to the naked eye; notwithstanding its immense distance.
In the elements which I have employed in the above computations, I have supposed the diameter of the central star to have been observed, in order to ascertain its density, which cannot be known without it; but the diameter of such a star is much too small to be observed by any telescopes yet existing, or any that it is probably in the power of human abilities to make; for the apparent diameter of the central star, if of the same density with the sun, when seen from another body, which would revolve round it in 64 years, would be only the 1717th part of the distance of those bodies from each other, as will appear from multiplying 107.32, the number of times the sun's diameter is contained in his distance from the earth, by 16, the greater proportional distance of the revolving body, corresponding to 64 years instead of 1. Now the 1717th part of a second must be magnified 309,060 times in order to give it an apparent diameter of three minutes; and three minutes, if the telescopes were mathematically perfect, and there was no want of distinctness in the air, would be but a very small matter to judge of. *
- In Mr. HERSCHEL's Observations upon the Fixed Stars abovementioned, almost all of them are represented as appearing with a well-defined round disc. That this is not the real disc, but only an optical appearance, occasioned perhaps by the constitution of the eye, when the pencil, by which objects are seen, is so exceedingly small as those which he employed upon this occasion, is very manifest, from the observations themselves, of which indeed Mr. HERSCHEL seems to be himself sufficiently aware: if it were not so, the intensity of the light of these stars must either be exceedingly inferior indeed to that of the sun, or they must be immensely larger, otherwise they must have a very sensible parallax; for the sun, if removed to 10,000,000 times his present distance, would still, I apprehend, be of about the brightness of the stars of the sixth magnitude; in which case he must be magnified 1,000,000 times to make his apparent disc of any sensible magnitude; or, on the other hand; if he was only removed to a thousandth part of that distance, then he must be less luminous in the proportion of 1,000,000 to 1, to make him appear no brighter than a star of the sixth magnitude. Now the sun's diameter being contained nearly 215 times in the diameter of the earth's orbit, the annual parallax therefore of such a body in that case; if it was placed in the pole of the ecliptic, would be 215 times its apparent diameter; and as the bright star in Lyra appeared to Mr. HERSCHEL about a third part of a second in diameter, if this was its real disc, and it was no bigger than the sun, it would consequently have an annual parallax in the pole of the ecliptic of about 72.
But though there is not the least probability that this element, so essential to be known, in order to determine with precision the exact distance and magnitude of a star, can ever be obtained, where it is in the same circumstances, or nearly the same, with those above supposed, yet the other elements, such as perhaps may be obtained, are sufficient to determine the distance, &c. with a good deal of probability, within some moderate limits; for in whatever ratio the real distance of the two stars may be greater or less than the distance supposed, the density of the central star must be greater or less in the sixth power of that ratio inversely; for the periodic time of the revolving body being given, the quantity of matter contained in the central body must be as the cube of their distance from each other. See Sir I. Newton's Prin. b. 3d. pr. 8th. cor 3d. But the quantity of matter in different bodies, at whose surfaces the velocity acquired by falling from an infinite height is the same, must be, according to art. 12, directly as their semi-diameters; the semi-diameters therefore of such bodies must be in the triplicate ratio of the distance of the revolving body; and consequently their densities, by art. 11, being in the inverse duplicate ratio of their semi-diameters, must be in the inverse sextuplicate ratio of the distance of the revolving body. Hence if the real distance should be greater or less than that supposed, in the proportion of two or three to one, the density of the central body must be less or greater, in the first case, in the proportion of 64, or in the latter of 729 to 1.
There is also another circumstance, from which perhaps some little additional probability might be derived, with regard to the real distance of a star, such as that we have supposed; but upon which however, it must be acknowledged, that no great stress can be laid, unless we had some better analogy to go upon than we have at present. The circumstance I mean is the greater specific brightness which such a star must have, in proportion as the real distance is less than that supposed, and vice versa; since, in order that the star may appear equally luminous, its specific brightness must be as the fourth power of its distance inversely; for the diameter of the central star being as the cube of the distance between that and the revolving star, and their distance from the earth being in the simple ratio of their distance from each other, the apparent diameter of the central star must be as the square of its real distance from the earth, and consequently, the surface of a sphaere being as the square of its diameter, the area of the apparent disc of such a star must be as the fourth power of its distance from the earth; but in whatever ratio the apparent disc of the star is greater or less, in the same ratio inversely must be the intensity of its light, in order to make it appear equally luminous. Hence, if its real distance should be greater or less than that supposed in the proportion of 2 or 3 to 1, the intensity of its light must be less or greater, in the first case, in the proportion of 16, or, in the latter of 81 to 1.
According to Mons. BOUGER (see his Traité d'Optique) the brightness of the sun exceeds that of a wax candle in no less a proportion than that of 8000 to 1. If therefore the brightness of any of the fixed stars should not exceed that of our common candles, which, as being something less luminous than wax, we will suppose in round numbers to be only one 10,000dth part as bright as the sun, such a star would not be visible at more than an 100dth part of the distance, at which it would be visible, if it was as bright as the sun. Now because the sun would still appear, I apprehend, as luminous, as the star Sirius, when removed to 400,000 times his present distance, such a body, if no brighter than our common candles, would only appear equally luminous with that star at 4000 times the distance of the sun, and we might then begin to be able, with the best telescopes, to distinguish some sensible apparent diameter of it; but the apparent diameters of the stars of the less magnitudes would still be too small to be distinguishable even with our best telescopes, unless they were yet a good deal less luminous, which may possibly however be the case with some of them; for, though we have indeed very slight grounds to go upon with regards to the specific brightness of the fixed stars compared with that of the sun at present, and can therefore only form very uncertain and random conjectures concerning it, yet from the infinite variety which we find in the works of the creation, it is not unreasonable to suspect, that very possibly some of the fixed stars may have so little natural brightness in proportion to their magnitude, as to admit of their diameters having some sensible apparent size, when they shall come to be more carefully examined, and with larger and better telescopes than have been hitherto in common use.
With regard to the sun, we know that his whole surface is extremely luminous, a very small and temporary interruption sometimes from a few spots only excepted. This universal and excessive brightness of the whole surface is probably owing to an atmosphaere, which being luminous throughout, and in some measure also transparent, the light, proceeding from a considerable depth of it, all arrives at the eye; in the same manner as the light of a great number of candles would do, if they were placed one behind another, and their flames were sufficiently transparent to permit the light of the more distant ones to pass through those that were nearer, without any interruption.
How far the same constitution may take place in the fixed stars we don't know; probably however it may do so in many; but there are some appearances with regard to a few of them, which seem to make it probable, that it does not do so universally. Now, if I am right in supposing the light of the sun to proceed from a luminous atmosphaere, which must necessarily diffuse itself equally over the whole surface, and I think there can be very little doubt that this is really the case, this constitution cannot well take place in those stars, which are in some degree periodically more and less luminous, such as that in Collo Ceti, &c. It is also not very improbable, that there is some difference from that of the sun, in the constitution of those stars, which have sometimes appeared and sometimes disappeared, of which that in the constellation of Cassiopeia is a notable instance. And if those conjectures are well founded which have been formed by some philosophers concerning stars of these kinds, that they are not wholly luminous, or at least not constantly so, but that all, or by far the greatest part of their surfaces is subject to considerable changes, sometimes becoming luminous, and at other times being extinguished; it is amongst the stars of this sort, that we are most likely to meet with instances of a sensible apparent diameter, their light being much more likely not to be so great in proportion as that of the sun, which, if removed to four hundred thousand times his present distance would still appear, I apprehend, as bright as Sirius, as I have observed above; whereas it is hardly to be expected, with any telescopes whatsoever, that we should ever be able to distinguish a well-defined disc of any body of the same size with the sun at much more than ten thousand times his distance.
Hence the greatest distance at which it would be possible to distinguish any sensible apparent diameter of a body as dense as the sun cannot well greatly exceed five hundred times ten thousand, that is, five million times the distance of the sun; for if the diameter of such a body was not less than five hundred times that of the sun, its light, as has been shewn above, in art.16. could never arrive at us.
If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from light; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present purpose, I shall not prosecute them any farther.
The diminution of the velocity of light, in case it should be found to take place in any of the fixed stars, is the principal phaenomenon whence it is proposed to discover their distance, &c. Now the means by which we may find what this diminution amounts to, seems to be supplied by the difference which would be occasioned in consequence of it, in the refrangibility of the light, whose velocity should be so diminished. For let us suppose with Sir ISAAC NEWTON (see his Optics, prop. VI. paragr. 4 and 5) that the refraction of light is occasioned by a certain force impelling it towards the refracting medium, an hypothesis which perfectly accounts for all the appearances. Upon this hypothesis the velocity of light in any medium, in whatever direction it falls upon it, will always bear a given ratio to the velocity it had before it fell upon it, and the sines of incidence and refraction, in consequence of this, bear the same ratio to each other with these velocities inversely. Thus, according to this hypothesis, if the sines of the angles of incidence and refraction, when light passes out of air into glass, are in the ratio of 31 to 20, the velocity of light in the glass must be to its velocity in air in the same proportion of 31 to 20. But because the areas, representing the forces generating these velocities, are as the square of the velocities, see art. 5. and 6. these areas must be to each other as 961 to 400. And if 400 represents the area which corresponds to the force producing the original velocity of light, 561, the difference between 961 and 400 must represent the area corresponding to the additional force, by which the light is accelerated at the surface of the glass.
In art. 19. we supposed, by way of example, the velocity of the light of some particular star to be diminished in the ratio of 19 to 20, and it was there observed, that the area representing the remaining force which would be necessary to generate the velocity 19, was therefore properly represented by (361 / 400)dth parts of the area, that should represent the force that would be necessary to generate the whole velocity of light, when undiminished. If we then add 561, the area representing the force by which the light is accelerated at the surface of the glass, to 361, the area representing the force which should have generated the diminished velocity of the star's light, the square root of 922, their sum, will represent the velocity of the light with the diminished velocity, after it has entered the glass. And the square root of 922 being 30.364, the sines of incidence and refraction of such light out of air into glass will consequently be as 30.364 to 19, or what is equal to it, as 31.96 to 20 instead of 31 to 20, the ratio of the sines of incidence and refraction, when the light enters the glass with its velocity undiminished.
From hence a prism, with a small refracting angle, might perhaps be found to be no very inconvenient instrument for this purpose: for by such a prism, whose refracting angle was of one minute, for instance, the light with its velocity undiminished would be turned out of its way 33, and with the diminished velocity 35.88 nearly, the difference between which being almost 2.53' would be the quantity by which the light, whose velocity was diminished, would be turned out of its way more than that whose velocity was undiminished.
Let us now be supposed to make use of such a prism to look at two stars, under the same circumstances as the two stars in the example above-mentioned, the central one of which should be large enough to diminish the velocity of its light one twentieth part, whilst the velocity of the light of the other, which was supposed to revolve about it as a satellite, for want of sufficient magnitude in the body from whence it was emitted, should suffer no sensible diminution at all. Placing then the line, in which the two faces of the prism would intersect each other, at right angles to a line joining the two stars; if the thinner part of the prism lay towards the same point of the heavens with the central star, whose light would be most turned out of its way, the apparent distance of the stars would be increased 2.53 and consequently become 3.53 instead of 1. only, the apparent distance supposed above in art. 21. On the contrary, if the prism should be turned half way round, and its thinner part lye towards the same point of the heavens with the revolving star, their distance must be diminished by a like quantity, and the central star therefore would appear 1.53 distant from its place near three times the whole distance between them.
As a prism might be made use of for this purpose, which should have a much larger refracting angle than that we have proposed, especially if it was constructed in the achromatic way, according to Mr. DOLLOND's principles, not only such a diminution, as one part in twenty, might be made still more distinguishable; but we might probably be able to discover considerably less diminutions in the velocity of light, as perhaps a hundredth, a two-hundredth, a five-hundredth, or even a thousandth part of the whole, which, according to what has been said above, would be occasioned by sphaeres, whose diameters should be to that of the sun, provided that they were of the same density, in the several proportions nearly of 70, 50, 30, and 22 to 1 respectively.
If such a diminution of the velocity of light, as that above supposed, should be found really to take place, in consequence of its gravitation towards the bodies from whence it is emitted, and there should be several of the fixed stars large enough to make it sufficiently sensible, a set of observations upon this subject might probably give us some considerable information with regard to many circumstances of that part of the universe, which is visible to us. The quantity of matter contained in many of the fixed stars might from hence be judged of, with a great degree of probability, within some moderate limits; for though the exact quantity must still depend upon their density, yet we must suppose the density most enormously different from that of the sun, and more so, indeed, than one can easily conceive to take place in fact, to make the error of the supposed quantity of matter very wide of the truth, since the density, as has been shewn above in art. 11. and 12. which is necessary to produce the same diminution in the velocity of light, emitted from different bodies, is as the square of the quantity of matter contained in those bodies inversely.
But though we might possibly from hence form some reasonable guess at the quantity of matter contained in several of the fixed stars; yet, if they have no luminous satellites revolving about them, we shall still be at a loss to form any probable judgement of their distance, unless we had some analogy to go upon for their specific brightness, or had some other means of discovering it; there is, however, a case that may possibly occur, which may tend to throw some light upon this matter.
I have shewn in my Enquiry into the probable Parallax, &c. of the Fixed Stars, published in the Philosophical Transactions for the year 1767, the extremely great probability there is, that many of the fixed stars are collected together into groups; and that the Pleiades in particular constitute one of these groups. Now of the stars which we there see collected together, it is highly probable, as I have observed in that paper, that there is not one in a hundred which does not belong to the group itself; and by far the greatest part, therefore, according to the same idea, must lie within a sphaere, a great circle of which is of the same size with a circle, which appears to us to include the whole group. If we suppose, therefore, this circle to be about 2°. in diameter, and consequently only about a thirtieth part of the distance at which it is seen, we may conclude, with the highest degree of probability, that by far the greatest part of these stars do not differ in their distances from the sun by more than about one part in thirty, and from thence deduce a sort of scale of the proportion of the light which is produced by different stars of the same group or system in the Pleiades at least; and, by a somewhat probable analogy, we may do the same in other systems likewise. But having yet no means of knowing their real distance, or specific brightness, when compared either with the sun or with one another, we shall still want something more to form a farther judgement from.
If, however, it should be found, that amongst the Pleiades, or any other like system, there are some stars that are double, triple, &c. of which one is a larger central body, with one or more satellites revolving about it, and the central body should likewise be found to diminish the velocity of its light; and more especially, if there should be several such instances met with in the same system; we should then begin to have a kind of measure both of the distance of such a system of stars from the earth, and of their mutual distances from each other. And if several instances of this kind should occur in different groups or systems of stars, we might also, perhaps, begin to form some probable conjectures concerning the specific density and brightness of the stars themselves, especially if there should be found any general analogy between the quantity of the diminution of the light and the distance of the system deduced from it; as, for instance, if those stars, which had the greatest effect in diminishing the velocity of light should in general give a greater distance to the system, when supposed to be of the same density with the sun, we might then naturally conclude from thence, that they are less in bulk, and of greater specific density, than those stars which diminish the velocity of light less, and vice versa. In like manner, if the larger stars were to give us in general a greater of less quantity of light in proportion to their bulk, this would give us a kind of analogy, from whence perhaps we might form some judgement of the specific brightness of the stars in general; but, at all adventures, we should have a pretty tolerable measure of the comparative brightness of the sun and those stars, upon which such observations should be made, if the result of them should turn out agreeable to the ideas above explained.
Though it is not improbable, that a few years may inform us, that some of the great number of double, triple stars, &c. which have been observed by Mr. HERSHEL, are systems of bodies revolving about one another, especially if a few more observers, equally ingenious and industrious with himself could be found to second his labours; yet the very great distance at which it is not unlikely many of the secondary stars may be placed from their principals, and the consequently very long periods of their revolutions *, leave very little room to hope that any great progress can be made in this subject for many years, or perhaps some ages to come; the above outlines, therefore, of the use that may be made of the observations upon the double stars, &c. provided the particles of light should be subject to the same law of gravitation with other bodies, as in all probability they are, and provided also that some of the stars should be large enough sensibly to diminish their velocity, will, I hope, be an inducement to those, who may have it in their power, to make these observations for the benefit of future generations at least, how little advantage soever we may expect from them ourselves; and yet very possibly some observations of this sort, and such as may be made in a few years, may not only be sufficient to do something, even at present, but also to shew, that much more may be done hereafter, when these observations shall be come more numerous, and have been continued for a longer period of years.
If the sun, when removed to 10,000,000 times his present distance, would still appear as bright as a star of the sixth magnitude, which I apprehend to be pretty near the truth, any satellite revolving round such a star, provided the star was not either of less specific brightness, or of greater density than the sun, must, if it appeared at its greatest elongation, at the distance of one second only from its principal, be between three and four hundred years in performing one revolution; and the time of the revolution of the very small star near [alpha] Lyrae, if it is a satellite to this latter, and its principal is of the same specific brightness and density with the sun, could hardly be less than eight hundred years, though 37 the distance at which it is placed from it, according to Mr. HERSCHEL's observations, should happen to be its greatest distance. These periodical times, however, are computed from the above distances, upon the supposition of the star, that revolves as a satellite, being very much smaller than the central one, so as not to disturb its place sensibly; for if the two stars should contain equal, or nearly equal, quantities of matter, the periodical times might be somewhat less, on account of their revolving about their common centre of gravity, in circles of little more than half as great a diameter as that in which the satellite must revolve upon the other supposition.
— , John Michell, , Philosophical Transactions of the Royal Society, , 1784