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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 DIMINU TION 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.
(Continued from our laft, page 246.)
24. TH HERE is also another circum- of the fun exceeds that of a wax canftance, from which, perhaps, dle in no lefs a proportion than that fome little additional probability might of 8000 to 1. If therefore the brightbe derived, with regard to the real nefs of any of the fixed stars should not distance of a star, fuch as that we have exceed that of our common candles, fuppofed; but upon which however, which, as being fomething lefs lumiit must be acknowledged, that no great nous than wax, we will fuppofe in ftrefs can be laid, unless we had fome round numbers to be only one 10.0oodth better analogy to go upon than we part as bright as the fun, such a star have at prefent. The circumftance I would not be vifible at more than an mean is the greater fpecific brightness 100dth part of the distance, at which which fuch a ftar muft have, in pro- it would be vifible, if it was as bright portion as the real distance is less than as the fun. Now, because the fun that fuppofed, and vice verfa; fince, would ftill appear, I apprehend as luin order that the ftar may appear equal- minous as the ftar Sirius, when rely luminous, its fpecific brightnefs moved to 400.000 times his prefent muft be as the fourth power of its di- diftance, fuch a body, if no brighter ftance inverfely; for the diameter of than our common candles, would only the central ftar being as the cube of appear equally luminous with that star the diftance between that and the re- at 4000 times the diftance of the fun, volving ftar, and their diftance from and we might then begin to be able, the earth being in the fimple ratio of with the best telescopes, to distinguish their diftance from each other, the fome fenfible apparent diameter of it; apparent diameter of the central ftar but the apparent diameters of the stars must be as the fquare of its real di- of the lefs magnitudes would still be ftance from the earth, and confequent- too fmall to be diftinguishable even ly, the furface of a fphære being as the with our beft telescopes, unless they fquare of its diameter, the area of the were yet a good deal lefs luminous, apparent dife of fuch a ftar must be as which may poffibly however be the the fourth power of its diftance from cafe with fome of them; for, though the earth; but in whatever ratio the we have indeed very flight grounds to apparent difc of the ftar is greater or go upon with regard to the fpecific lefs, in the fame ratio inverfely muft brightnefs of the fixed ftars compare be the intensity of its light, in order with that of the fun at prefent, and to make it appear equally luminous. can therefore only form very uncertain Hence, if its real diftance fhould be and random conjectures concerning it, greater or less than that fuppofed in yet from the infinite variety which we the proportion of 2 or 3 to 1, the in- find in the works of the creation, it is tenfity of its light must be lefs or not unreasonable to fufpect, that very greater, in the firft cafe, in the pro poffibly fome of the fixed ftars may portion of 16, or, in the latter of 81 have fo little natural brightness in proportion to their magnitude, as to admit of their diameters having fome fenfible apparent fize, when they shall
25. According to Monf. Bouguer (fee his Traité d'Optique) the brightnefs LOND. MAG. May 1785.
come to be more carefully examined, and with larger and better telefcopes than have been hitherto in common ufe. 26. With regard to the fun, we know that his whole furface is extremely luminous, a very fmall and temporary interruption fometimes from á few fpots only excepted. This univerfal and exceffive brightnefs of the whole furface is probably owing to an atmosphere, which being luminous throughout, and in fome measure alfo tranfparent, the light, proceeding from a confiderable depth of it, all arrives at the eye; in the fame manner as the light of a great number of candles would do, if they were placed one behind another, and their flames were fufficiently tranfparent to permit the light of the more diftant ones to pafs through thofe that were nearer, without any interruption.
27. How far the fame conftitution may take place in the fixed ftars we do not know; probably however it may do fo in many; but there are fome appearances with regard to a few of them, which feem to make it probable, that it does not do fo univerfally. Now, if I am right in fuppofing the light of the fun to proceed from a luminous atmosphere, which maft neceffarily diffufe itfelf equally over the whole furface, and I think there can be very little doubt that this is really the cafe, this conftitution cannot well take place in thofe ftars, which are in fome degree periodically more and lefs luminous, fuch as that in Collo Ceti, &c. It is alfo not very improbable, that there is fome difference from that of the fun, in the conftitution of thofe ftars, which have fometimes appeared and fometimes difappeared, of which that in the conftellation of Caffiopeia is a notable instance. And if thofe conjectures are well founded which have been formed by fome philofophers concerning ftars of thefe kinds, that they are not wholly luminous, or at least not conftantly fo, but that ail, or by far the greatest part of their furfaces is fubject to confiderable changes, fometimes becoming luminous, and at other times being extinguished; it is amongit the ftars of this
fort, that we are most likely to meet with inftances of a fenfible apparent diameter, their light being much more likely not to be fo great in proportion as that of the fun, which, if removed to four hundred thousand times his prefent diftance, would ftill appear, I apprehend, as bright as Sirius, as I have obferved above; whereas it is hardly to be expected, with any telefcopes whatfoever, that we should ever be able to diftinguifh a well defined dife of any body of the fame fize with the fun at much more than ten thoufand times his diftance.
28. Hence the greatest distance at which it would be poffible to diftinguish any fenfible apparent diameter of a body as denfe as the fun cannot well greatly exceed five hundred times ten thoufand, that is, five million times the diftance of the fun; for if the diameter of fuch a body was not less than five hundred times that of the fun, its light, as has been fhewn above, in art, 16, could never arrive at us.
29. If there fhould really exift in nature any bodies, whofe denfity is not lefs than that of the fun, and whofe diameters are more than 500 times the diameter of the fun, fince their light could not arrive at us; or if there fhould exift any other bodies of a fomewhat fmaller fize, which are not naturally luminous; of the exiftence of bodies under either of these circumftances, we could have no information from fight; yet, if anyother luminous bodies fhould happen to revolve about them we might ftill perhaps from the motions of these revolving bodies infer the existence of the central ones with fome degree of probability, as this might afford a clue to fome of the apparent irregularities of the revolving bodies, which would not be eafily explicable on any other hypothefis; but as the confequences of fuch a fuppofition are very obvious, and the confideration of them fomewhat befide my prefent purpose, I shall not profecute them any farther.
30. The diminution of the velocity of light, in cafe it fhould be found to take place in any of the fixed ftars, is the principal phenomenon
whence it is propofed to difcover their distance, &c. Now, the means by which we may find what this diminution amounts to, feems to be fupplied by the difference which would be occafioned in confequence of it, in the refrangibility of the light, whofe velocity fhould be fo diminished. For let us fuppofe with Sir Ifaac Newton (fee his Optics, prop. vi. paragr. 4 and 5) that the refraction of light is occafioned by a certain force impelling it towards the refracting medium, an hypothefis which perfectly accounts for all the appearances. Upon this hypothefis 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 fines of incidence and refraction will, in confequence of this, bear the fame ratio to each other with thefe velocities inverfely, Thus, according to this hypothefis, if the fines of the angles of incidence and refraction, when light paffes out of air into glafs, are in the ratio of 31 to 20, the velocity of light in the glafs muft be to its velocity in air in the fame proportion of 31 to 20. But becaufe the areas, reprefenting the forces generating thefe velocities, are as the fquares of the velocities, fee art. 5 and 6, these areas muft be to each other as 961 to 400. And if 400 reprefents the area which correfponds to the force producing the original velocity of light, 561, the difference between 961 and 400, muft reprefent the area correfponding to the additional force, by which the light was accelerated at the furface of the glaís.
31. In art. 19, we fuppofed, by way of example, the velocity of the light of fome particular ftar to be diminished in the ratio of 19 to 20, and it was there observed, that the area reprefenting the remaining force which would be neceffary to generate the velocity 19, was therefore properly reprefented by 3dth parts of the area, that fhould reprefent the force that would be neceffary to generate the whole velocity of light, when undiminifhed. If then we add 561, the area representing the force by which
the light is accelerated at the surface of the glafs, to 361, the area reprefenting the force which would have generated the diminished velocity of the ftar's light, the fquare root of 922, their fum, will reprefent the velocity of the light with the diminished velo city, after it has entered the glass. And the fquare root of 922 being 30,364, the fines of incidence and refraction of fuch light out of air into glafs will confequently be as 30,364 to 19, or what is equal to it, as 31,96 to 20 inftead of 31 to 20, the ratio of the fines of incidence and refraction, when the light enters the glafs with its velocity undiminished.
32. From hence a prifm, with a fmall refracting angle, might perhaps be found to be no very inconvenient inftrument for this purpose: for by fuch a prifm, whofe refracting angle was of one minute, for inftance, the light with its velocity undiminifhed would be turned out of its way 33", and with the diminished velocity 35, 88 nearly, the difference between which being almoft 2" 53", would be the quantity by which the light, whofe velocity was diminished, would be turned out of its way more than that whofe velocity was undiminished.
33. Let us now be fuppofed to make ufe of fuch a prifm to look at two ftars, under the fame circumstances as the two ftars in the example abovementioned, the central one of which fhould be large enough to diminish the velocity of its light one twentieth part, whilft the velocity of the light of the other, which was fuppofed to revolve about it as a fatellite, for want of fufficient magnitude in the body from whence it was emitted, should fuffer no fenfible diminution at all. Placing then the line, in which the two faces of the prifm would interfect each other, at right angles to a line joining the two ftars; if the thinner part of the prifm lay towards the fame point of the heavens with the central ftar, whofe light would be most turned out of its way, the apparent diftance of the ftars would be increased 2" 53" and confequently become 3".53"" inftead of 1", only, the ap
parent distance fuppofed above in art. 21. On the contrary, if the prifm fhould be turned half way round, and its thinner part lie towards the fame point of the heavens with the revolying ftar, their diftance must be diminifhed by a like quantity, and the central ftar therefore would appear 1". 53" diftant from the other on the oppofite fide of it, having been removed from its place near three times the whole distance between them.
34. As a prifm might be made ufe of for this purpofe, which fhould have a much larger refracting angle than that we have propofed, efpecially if it was conftructed in the achromatic way, according to Mr. Dollond's principles, not only fuch a diminution, as one part in twenty, might be made ftill more diftinguishable; but we might probably be able to difcover confiderably lefs 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 faid above, would be occafioned by fphæres, whofe diameters fhould be to that of the fun, provided they were of the fame denfity, in the feveral proportions nearly of 70, 50, 30, and 22 to 1 refpectively.
35. If fuch a diminution of the velocity of light, as that above fuppofed, fhould be found really to take place, in confequence of its gravitation towards the bodies from whence it is emitted, and there fhould be feveral of the fixed ftars large enough to make it fufficiently fenfible, a fet of obfervations upon this fubject might probably give us fome confiderable information with regard to many circumftances of that part of the univerfe, which is vifible to us. The quantity of matter contained in many of the fixed stars might from hence be judged of, with a degree of probability, within fome mederate limits; for though the exact quantity muft ftill depend upon their denfity, yet we must fuppofe the denfity moft enormously different from that of the fun, and more fo, indeed, than one can easily conceive to take place in fact, to make the
error of the fuppofed quantity of matter very wide of the truth, fince the denfity, as has been fhewn above in art. 11 and 12, which is neceffary to produce the fame diminution in the velocity of light, emitted from different bodies, is as the fquare of the quantity of matter contained in those bodies inversely.
36. But though we might poffibly from hence form fome reasonable guess at the quantity of matter contained in feveral of the fixed ftars; yet, if they have no luminous fatellites revolving about them, we shall still be at a loís to form any probable judgement of their diftance, unlefs we had fome analogy to go upon for their specific brightnefs, or had fome other means of difcovering it; there is, however, a cafe that may poffibly occur, which may tend to throw fome light upon this matter.
37- I have fhewn in my Enquiry into the probable Parallax, &c. of the Fixed Stars, published in the Philofophical Tranfactions 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 conftitute. one of thefe groups. Now of the stars which we there fee collected together, it is highly probable, as I have obferved in that paper, that there is not one in a hundred which does not belong to the group itfelf; and by far the greateft part, therefore, according to the fame idea, muft lye within a fphære, a great circle of which is of the fame fize with a circle, which appears to us to include the whole group. If we fuppofe, therefore, this circle to be about 2°. in diameter, and confequently only about a thirtieth part of the distance at which it is feen, we may conclude, with the highest degree of probability, that by far the greatest part of thefe ftars do not differ in their diftances from the fun by more than about one part in thirty, and from thence deduce a fort of fcale of the proportion of the light which is produced by different ftars of the fame group or fyftem in the Pleiades at least; and, by a fomewhat probable analogy,
we may do the fame in other fyftems likewife. But having yet no means of knowing their real diftance, or fpecific brightnefs, when compared either with the fun or with one another, we shall still want fomething more to form a farther judgement from.
38. If, however, it should be found, that amongst the Pleiades, or any other like fyftem, there are fome ftars that are double, triple, &c. of which one is a larger central body, with one or more fatellites revolving about it, and the central body should likewife be found to diminish the velocity of its light; and more efpecially, if there fhould be feveral fuch inftances met with in the fame fyftem; we fhould then begin to have a kind of meafure both of the diftance of fuch a fyftem of stars from the earth, and of their mutual diftances from each other. And if feveral instances of this kind fhould occur in different groups or fyftems of ftars, we might alfo, perhaps, begin to form fome probable conjectures concerning the specific denfity and brightnefs of the itars themfelves, especially if there fhould be found any general analogy between the quantity of the diminution of the light and the diftance of the fyftem deduced from it; as, for inftance, if those stars, which had the greateft effect in diminishing the velocity of light fhould in general give a greater diftance to the fyftem, when fuppofed to be of the fame denfity with the fun, we might then naturally conclude from thence, that they are lefs in bulk, and of greater fpecific denfity, than thofe ftars which diminish the velocity of light lefs, and wice verfa. In like manner, if the
larger ftars were to give us in general a greater or lefs quantity of light in proportion to their bulk, this would give us a kind of analogy, from whence we might perhaps form fome judgement of the specific brightness of the ftars in general; but, at all adventures we fhould have a pretty tolerable meafure of the comparative brightness of the fun and thofe ftars, upon which fuch obfervations fhould be made, if the refult of thera fhould turn out agreeable to the ideas above explained
39. Though it is not improbable, that a few years may inform us, that fome of the great number of double, triple ftars, &c. which have been obferved by Mr. Herschel, are systems of bodies revolving about each other, efpecially if a few more obfervers, equally ingenious and induftrious with himself could be found to fecond his labours; yet the very great distance at which it is not unlikely many of the fecondary ftars may be placed from their principals, and the confequently very long periods of their revolutions*, leave very little room to hope that any very great progrefs can be made in this fubject for many years, or perhaps fome ages to come; the above outlines, therefore, of the use that may be made of the obfervations upon the double ftars, &c. provided the particles of light fhould be fubject to the fame law of gravitation with other bodies, as in all probability they are, and provided alfo that fome of the stars should be large enough fenfibly to diminish their velocity, will, I hope, be an inducement to thofe, who may have it in their power, to make these obfervations for the benefit of future gene
* If the fun, when removed to 10.000 000 times his prefent diftance, would ftill appear as bright as a ftar of the fixth magnitude, which I apprehend to be pretty near the truth, any fatellite revolving round fuch aitar, provided the ftar was not either of lefs fpecific brightnefs, or of greater denfity than the fun, muft, if it appeared at its greatest elongation, at the diftance of one fecond only from its principal, be between three and four hundred years in performing one revolution; and the time of the revolution of the very fmall ftar near a Lyre, if it is a fatellite to this latter, and its principal is of the fame fpecific brightness and denfity with the fun, could hardly be lefs than eight hundred years, though 37" the distance at which it is placed from it, according to Mr. Herfchel's obfervations, thould happen to be its greatest distance. These periodical times, however, are computed from the above distances, upon the fuppofition of the ftar, that revolves as a fatellite, being very much smaller than the central one, fo as not to disturb its place fenfibly; for if the two ftars fhould contain equal, or nearly equal, quantities of matter, the periodical times might be fomewhat lefs, 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 fatellite must revolve upon the other fuppofition.