C / 1882 R1 (September Great Comet)

Opening

By September 1, 1882, the first records of the observation of a comet in the Gulf of Guinea and on the Cape of Good Hope relate. In Auckland (New Zealand), it was first seen on September 3 (16:48 UT). In Cordoba (Argentina), on September 5, at 9:36 a comet became visible in the sky at dawn, and its brightness was comparable with Venus, as the American astronomer B. A. Gould reported ^{[1] [2]} . The next day, September 6, there is evidence of observations from Panama ^[1] , and on the same day she was seen from the ship “Caraki” ^{[1] [3]} .

The first astronomer to observe the comet was - this happened on September 7 (around 5 p.m. local time) at the observatory on the Cape of Good Hope , where he conducted observations together with , who registered the comet on October 8th. They were able to distinguish a large comet’s head, a stellar core of −3 ^m and a diameter of 10-15 ’in the southern region of the head, with a thickness of 40-50 ″ in the vicinity of the nucleus and a 1-2.5 ° tail, the southern border of which was more distinct, bright and long ^{[1] [2] [4] [5] [6] [7] [8]} . On the same day at the also observed a comet , who received a message the day before from the ship “Caraki.” Using a spectroscope, he recorded a very bright continuous spectrum, where there were no lines typical of comets, but there was a bright yellow line that he identified l as a sodium line ^[3] . In the morning of September 9, the comet was first registered by R. Ellery at another Australian observatory, this time in ^[9] . Also on September 9 was the Australian astronomer J. Tebbutt in Windsor ( New South Wales) ), received on the eve of a comet report from Melbourne and other places, made its independent discovery, noting that the nucleus was large and bright, and the tail had a length of 3-4 ° ^{[1] [2] [10]} . A day later, J. Reed did the same on the HMS Triumph ship south of the Cape Verde Islands . And on September 12, at the in Rio de Janeiro, also discovered a comet ^{[2] [8] [11]} - he noted that it was visible with the naked eye, and suggested its identity with Comet Pons Brooks 1812 ^{[1] [12]} .

Before perihelion

In the first days after its appearance in the sky, the comet, moving to the perihelion, was visible only before dawn. So, J. Reid reported that he saw her on September 12 and 13 only a few minutes (due to bad weather conditions during this period), but managed to fix a pronounced undistorted tail 2-2.5 ° long and a bright coma - a core surrounded by brighter ring ^{[1] [13]} . British amateur astronomer L.E. Eddie describes the comet observed by him at his Greyamstown Observatory on September 13 as a narrow brightly luminous strip of red-brown color ending in a nucleus comparable in brightness and size to Jupiter, and in color to Venus but with a slightly golden hue; the coma was dense, its borders clear; the tail seemed straight and had a length of 12 ° ^{[1] [14]} . Another Briton, E. Марк. Markwick, who first registered the comet on September 14 in Durban, noted its reddish tint and vague head structure ^[15] . The core was equivalent in brightness to Jupiter on September 15, and the next day, Eddie observed through a telescope that the core became less pronounced, the coma had a greater width on the north side than on the south side, and the tail lengthened and became darker in the middle, as if bifurcated ^{[ 1] [14] [16]} . Then the comet, as B. A. Gould testifies in his reports, was visible through the telescope-finder during the day ^{[1] [16]} .

Perihelion passage

During perihelion, the brightness of the comet increased so much that it was visible even in the daytime for more than two days ^{[1] [9]} .

So, on September 16 (early morning of September 17, local time), J. Tebbutt observed her 4 ° west of the Sun, rapidly moving in its direction, and her head and tail were clearly distinguishable with a length of about 1/3 ° ^{[1] [ 16]} . L.E. Eddie testified that a comet appeared in the sky (September 17 at 5:44 local time) 14 minutes before sunrise, its size and brightness then increased so much that it became easily visible near the Sun with the naked eye, even without the slightest efforts; first, the tail had a length of about 8 ′, then during the day its total length was about 1 ° ^{[1] [14] [16]} . A little later (at 8:10) the inhabitants of Reus (Spain) observed the comet only 1 ° from the Sun; they were amazed at how bright it was: it was visible even through thin clouds, and the tail could be distinguished in ordinary binoculars with tinted glass ^{[1] [8]} . At 10:45 a comet was registered by the English amateur astronomer E. Ком. Common at his observatory in Ealing , where he conducted a daily survey of the sky in search of near-solar comets using a helioscope ^{[1] [16] [8] [11] [12]} . He noted a very bright tail and a large core with a diameter of 45 "and an almost round shape ^{[1] [17]} . The comet was rapidly moving in the direction of the Sun, and according to Eddie’s reports, by 2 o’clock in the afternoon (local time) it was already difficult to locate its location , and at 16:30 (also local time) he saw it for the last time 14 ′ from the edge of the solar disk.According to the results of his observations with a telescope, the astronomer described the nucleus as a solid white luminous ball, superior in brightness to Venus and located almost at the tip tail, coma was quite heaven of a larger size, and its borders were a denser luminous flux than the inner region, while the northern part was narrower and brighter than the south ^{[1] [14]} . Eddie noted that as the comet approached the Sun there were no signs of change its physical structure, which distinguished it from other comets ^[14] . Gould wrote that at 14:53 the comet was still visible, moreover, clearly and distinctly, although for this it was necessary to use a “protective glass”; at 15:21, according to his reports, she was already in the same field of view with the Sun ^[1] . G. C. Russell, who observed the comet almost continuously throughout the day, wrote that it looked like a bright star, and with the telescope one could clearly distinguish the core and concentric layers of the coma, of which the outermost was the sharpest ^[3] . W. G. Finlay, who also conducted observations throughout the day with W. L. Elkin, noted that at noon the comet's disk size was 4 " ^{[1] [4] [6] [16]} , and in one field of view with the solar limb, it was at 16:40 (local time), while the silver color of the comet was clearly distinguished from the red-yellow color of the sun ^{[16] [6]} . Around 16:51 (local time), according to these astronomers , the comet disappeared from the visibility ^{[18] [6] [7] [8] [12]} ; at 16:19 Gould could not see it - it began to pass through the disk of the Sun. It lasted 1 hour 17 minutes, until 16:34 ^{[1] [18]} . Elkin compared l this process with what the moon coverage of a 4th magnitude star looks like ^[18] .

Perihelion occurred at 17:17, at 17:46 the comet reached a maximum elongation of 27 ′ and began to move beyond the Sun - its coverage with the solar disk occurred from 18:57 to 20:53; until the next day she was no longer visible in the sky ^{[1] [18]} .

According to estimates, the brightness of the comet during and immediately after perihelion was no less than −9 ^m , and at the time of observation against the background of the solar limb it was even brighter than it ^[18] .

After perihelion

After passing the perihelion, the comet was the first to register Tebbutt on September 18 at 01:26 UTC, less than 1 ° from the western edge of the solar disk ^[18] . On this day, the comet was 3-4 ° away from the Sun and became so clearly visible even in the daytime that its presence in the sky was noted by numerous observers around the world, measurements of its position and characteristics were made at several observatories ^{[8] [12]} . So, when observing at the Cape of Good Hope Observatory, its director D. Gill estimated the size of the nucleus at 4 ″ and compared its brightness with a star of the first visible magnitude ^{[4] [5]} . He noted that it was enough to protect the eyes from the Sun by covering it with outstretched hand to clearly see the radiant-white core and a clearly defined tail 0.5 ° long ^{[18] [5]} . On the same day, the French physicist estimated the diameter of the nucleus at 15 ″, and the total size of the coma and part of the tail visible to the naked eye at 20 ′, he described their contour as half an ellipse with an eccentricity of about 4, so that the nucleus, very large and bright, was located between the apex and the focus of this ellipse ^[8] . In addition, this scientist obtained a spectrum of the nucleus and neighboring elements - narrow and bright, in it there were double sodium lines and a number of other bright lines coinciding with the iron lines, while a typical line spectrum with carbon bands was not observed ^{[8] [11] [12]} . The redshift of the sodium lines (almost 1/4 of the distance between them) indicated that the comet was rapidly moving away from the Earth, and the absence of the dark Fraunhofer lines meant that reflected sunlight was not the main source of its glow ^{[8] [12]} . In the next 2 days, the comet was also clearly visible to the naked eye, information about it spread everywhere, and all people interested in astronomy were absorbed in its observations ^[12] . So, according to Eddie’s September 19 reports, the comet continued to move in the same direction as before the perihelion, and the next day the head became less bright, while the tail increased in size: its length was 5 °, at a distance of 2 ° from the nucleus it expanded sharply to 1 °, then more smoothly to 1.5 ° ^[14] . C.O. Young astronomer at Princeton University , who observed the comet on the same day for the first time, reported that the nucleus had a diameter of 4 ″ and was round and bright, but fuzzy ^[12] . Also on September 19, shortly after noon, the comet and astronomers from the US Naval Observatory were first registered; and W.K. Winlock compared her outlines with spread wings of a bird ^[19] .

Since September 21, the comet has been visible with the naked eye only where the sky was clear of clouds ^{[8] [12]} . So, on September 22, an astronomer from Nashville, E. E. Barnard, reported that he could see it without a telescope for 15 minutes after sunrise - such a long period of observation of the comet with the naked eye in the daytime had taken place only once before in 1402. In addition, in the predawn dusk, an American scientist recorded a tail length of 12 °. On the same day, telescopic observations were successfully carried out immediately before sunrise by J. M. Scheberle in Ann Arbor and immediately after, by E. Millosevich in Rome ^[20] . It is noteworthy that at the same time, on September 22, in Paris, attempted to rise in a balloon belonging to the famous balloonist V. de Fonviel (on his behalf), above the clouds - the comet was indeed much better visible, but to register precisely her position in such conditions, of course, was impossible ^{[8] [12]} . According to Gill's reports, on September 22, the core was comparable to a third-magnitude star, while the remaining part of the head was less distinguishable ^[5] . Finally, on the same day, , who first observed the comet on that day at the Palermo Observatory, received a spectrum of nucleus and coma, which was narrow and continuous and contained, in addition to sodium D-lines, several others that he could not accurately identify ^{[8] [12]} .

Eddie noted that the comet reached its greatest brightness and size on September 24: the tail was 25 ° long, its outer borders were lighter, and it was as if divided in two by a dark stripe (from the nucleus almost to the end of the wide part of the tail), with the northern part being more bright, it was somewhat convex near the head, the latter consisted of a still clearly pronounced small nucleus and a dense coma also of small thickness ^[14] . Shortly afterwards, on September 24–25, Ellery, who observed the comet at dawn at the observatory in Melbourne, recorded that the tail was 15 ° long and about 1 ° wide at the end ^[9] . At the same time, on September 25, E.E. Markvik, seeing the comet for the first time after perihelion in Pietermaritzburg , noted that the nucleus was white, bright and star-like, and the tail was very long, located parallel to the ecliptic, and inside it was visible a dark strip closer to the southern border, which was more clearly defined than the northern one ^[15] . Until September 27, with the naked eye, one could see after the extinction of stars of the first magnitude with sunrise the portion of the tail 12 ° long ^[20] . An exceptionally large tail length (up to 30 °), approximately 10 days after the perihelion, was also observed by Kruls ^[8] . From September 27 to October 1, Rikko and from Johns Hopkins University in Baltimore also observed a dark strip passing through the entire tail, and the American physicist noted, in addition to it, another shorter parallel strip ^{[ 12]} .

During this period (end of September – beginning of October), the comet remained exceptionally bright, almost zero in magnitude, its tail was clearly distinguishable — rather long (20 °) and narrow (less than 1 °) ^[20] . Thus, observers on the ship Earnock, proceeding from Glasgow to Melbourne, reported that the comet was very bright, of the order of 1 ^{m [21]} . From another ship, “Superb,” on September 28, observers determined the tail length at 9.5 °, and on October 1, about 11 ° ^[22] . Also on September 28, from the US Naval Observatory recorded a tail length of 15 °, a maximum width of 1.5 °, its slight curvature, concavity on the north side and a dark strip passing through it; the nucleus was very clear, and the entire comet was an exceptionally bright object. The next day, his colleague Fribsey also described the core - an irregular shape, elongated in the direction of the tail, it had a length of 15 ″ and a maximum width of 3 ″ ^[19] . An important observation concerning the comet spectrum belongs to September 29: doublet lines of sodium became insoluble, iron lines almost disappeared, and on the whole it acquired the form typical of comets ^{[8] [11]} . Marcwick also noted on September 30 that the comet's head was comparable in brightness to a star of the first magnitude, although its shape was rather fuzzy compared to Jupiter; the core took the form of a line deviating at a small angle from the axis of the tail, while the tail itself became noticeably curved. The next day, a British astronomer estimated its length at 12.5 °, and its maximum width at 1.75 ° ^[15] . According to the observations of C. O. Young, on October 2, the tail had a length of about 14 °, was quite bright and clear, especially along the edges, slightly curved and convex towards the horizon. An astronomer from Princeton also noted a well-defined dark band beginning behind the core, and made various assumptions about its nature. In addition, he obtained an intense continuous spectrum of the nucleus and tail, on which carbon bands were superimposed ^[12] . Another American, Winlock, recorded that day, the tail, rather narrow, had a length of 19 ° ^{[approx. 1]} , and the core was round and rather vague ^[19] . By October 3, according to Eddie’s reports, the length of the tail was reduced (relative to his previous observation) to 15 °, the width was 3 °, its brightness became more uniform, the shape more blurred; the southern part became brighter, denser and slightly rounded, the dark strip starting from the comet's head began to disappear, but another appeared - coming from the end of the tail about 1/4 of its length and slightly curved to the north. The next day, the main dark band became even less pronounced ^[14] . The British astronomer and meteorologist C. L. Prince, who saw a comet for the first time (due to weather conditions) on his Crowborough Observatory only on October 4, recorded a tail length of 25 ° on this day ^[23] , while the American Frisbee - at 17-18 ° ^[19] . Young on this day noted that the sodium lines in the spectrum, which ceased to be pronounced as early as October 2, became almost indistinguishable ^[12] . Finally, Markwick reported on October 4 that the outlines of the nucleus became blurry, the tail, compared with previous observations, became slightly longer but less bright, the dark band became barely noticeable, and the next day was already indistinguishable; the end of the tail on October 5 became uneven ^[15] . Later, on October 6, Winlock testified that the tail, having a total length of 17 ° and a width at the end of 3 °, was slightly concave and discontinuous on the north side, its brightest area was 8 ° from the head closer to the southern border, and north of it an almost black strip 4–5 ° long was visible; the head was located at a slight angle to the tail and was slightly wider on the south side than on the north, where it seemed somewhat flattened. The next day, an astronomer from Washington noted that the tail with the same length had a width at the end of 5 °, and 3 ° from the head - 2 °; its southern part was clearly expressed, while the northern was vague and intermittent ^[19] . Then, on October 7, according to evidence from Vishakhapatnam (India), the length of the brightest part of the tail was reduced to 7-8 ° ^[24] .

In the first half of October, the comet began to fade, became visible again before sunrise and moved southwest ^{[1] [12]} . On October 1, Barker estimated its brightness at 0.5 ^m , and Markwick at 1 ^{m [1]} . Then on October 4, Ch. L. Prince - at 2 ^{m [23]} , on October 6, the Austro-Hungarian astronomer , who worked at the Golis-Leipzig Observatory - at 1.8 ^{m [1]} , and the New Zealander J. T. Stevenson - at 1 ^{m [25]} . Thus, the comet in this period was an exceptionally bright (comparable to Jupiter) object in the sky ^[12] . Most observers ^[15] noted a pronounced white color of the comet at the beginning of this period, but Barnard mentioned the “pearl shade” ^[1] , and Prince wrote that the core was orange (October 4), and the tail was shining silver and bizarrely curved, resembling tangled rope (October 10) ^[23] . Eddie also noted the S-shaped curve of the southern part of the tail in his reports of October 8 ^[14] , and on the same day Frisbee described his fan-shaped form. And on October 9, Winlock recorded a tail length of 14–16 ° and a branch at its end ^[19] . Then Markwick reported that the southern part of the tail became sharper and at the end had the shape of a diverging beam, while the northern and the end of the tail became more blurry; the core remained elongated and located at an angle relative to the tail. The latter had a length the next day, October 10, 18 °, and October 11 - 17 ° ^[15] . Also on this day (October 10), Winlock again estimated the tail length at 16 ° ^[19] , and in addition, according to Young's data, the sodium lines finally disappeared from the spectrum ^[12] . Ricco also confirmed that at the same time, carbon bands appeared more distinctly in these first days of October - the Italian astronomer recorded that their source was the nucleus and the region about 5 ′ around it ^[8] .

Thus, the tail remained quite long during this period - 15–20 °, narrow, bright and clearly defined ^{[1] [12] [14]} . Observers from the Superb ship noted that during October it became increasingly pale, but increased in length ^[22] . There was also a noticeable dark strip starting from the comet's head and dividing the tail lengthwise into two parts ^{[1] [26]} ; JT Stevenson, who conducted observations in Auckland (New Zealand), reported that the north lying was much wider ^[25] , while according to Prince's reports, they were approximately the same width, but the southern one was much brighter ^[23] , Eddie he wrote that the southern one was longer, and both of them diverged in different directions, like horns ^[14] . Ricco reported that during this period the southern part of the tail was (at some points - significantly) brighter than the northern one, and the tail as a whole was almost straight and only slightly deviated south ^[8] . In addition to the dark band located at an angle of 8-10 ° relative to the nucleus, Young on October 10 also recorded another bright 2–3 ′ length, starting from the same point at the beginning of the tail, but aligned with it. A few days later, on October 15, according to a report by an astronomer from Princeton, only a bright band was visible, starting from the end of the nucleus, and from the place of its bend; the tail was 18 ° long, that is, about 100 million km ^[12] . Winlock reported on October 14 that the previously recorded dark band was still visible in the middle of the tail, which diverged 3-4 ° from the end, where it had a width of 3.5 °, its total length was 17 °. The next day, another astronomer from Washington, Sampson, established in the spectrum of the nucleus 3 bands typical of comets (the central one was the brightest), and the spectrum of the tail was described as continuous, as bright as possible in the green part ^[19] . The characteristic dark band and the separation of the tail was shown in his sketches from October 18–20 by an astronomer from Germany, E. V. L. Tempel , who worked at the Arcetri observatory ^[26] . At the same time, through the tail, one could see the stars behind him ^{[14] [15] [19] [23] [25]} . Later, on October 23, Prince estimated the brightness of the nucleus at 5 ^m , and the tail length at 20 ° ^[23] . On this day, Markwick reported that the tail length was 18 °, and the brightness of the nucleus was 1 ^m , although he noted that the comet, albeit very slightly, faded ^[15] . The fact that the comet became less bright was also confirmed on October 24 by Young ^[12] and Winlock. The latter also noted that the southern part of the tail was brighter, its length was 12 °, and its head was fuzzy and slightly flattened from the northern side ^[19] . In addition, at the end of October, the comet was removed from the Sun by the same distance as the Earth, and its lines completely disappeared in its spectrum, but there were 3 hydrocarbon bands ^[11] .

The Hungarian astronomer , in his Gurbanovo observatory, was only able to observe the comet on November 1 (weather had previously prevented this), noted that on this day the tail was bent upward and its border directed towards the horizon , was much brighter and better delineated. The nucleus had a pronounced yellow color, while the coma was a greenish tint; its edges were fuzzy. At the same time, rays diverging from the nucleus, typical of comets, were not observed, and the whole head was more like a candle flame in a fog. The spectrum of the nucleus was very bright, especially in the red part, and continuous, no sodium lines were visible in it. The coma had a rather bright spectrum with bands typical of comets inherent in hydrocarbons, with maxima at wavelengths of the order of 600 nm, 560 nm, 514 nm, 470 nm, and 430 nm ^[27] . On November 2, Markwick estimated the tail length at 19.5 °, the thickness at the end at 3.5 °, and also noted that the comet noticeably faded ^[15] . The latter was confirmed on the same day by Winlock, who also reported that the head remained somewhat flattened on the north side, and the tail had a length of 10-12 ° ^[19] .

Earl Crawford J. Linsdy, who was informed by B.J. Hopkins of London about his observations, noted that on November 4 the core had a yellowish-white color and an ellipse shape with an axis in the direction of the tail; the latter had a length of 20 ° and a width of up to 1.5 °, slightly bent upward, 2/3 of its length was divided by a dark stripe, and at the end it was quite blurred ^[28] . On the same day, Young estimated the tail length at 16 °, its maximum width at 4 °, and the comet's head brightness at 4 ^m - when observed with the naked eye, it was unusually bright for an object located at the same distance from the Sun as the Earth ^[12] . Later, on November 8, Winlock recorded a tail length of 10 °, noting that its northern side was much duller, while in general the comet remained quite bright ^[19] . On the same day, according to Hopkins, the core ^{[approx. 2]} was comparable to a star of the second magnitude ^[29] , the tail length was 19 °, it was straight by 4/5 of its length, and its end, sharply bending upwards, had the shape of a fan 4 ° wide, the dark strip became less noticeable, although the southern part of the tail was still somewhat brighter. By the time of the next observation, November 14, the nucleus was very elongated, the length of the tail increased to 30 °, it bifurcated at the end, and the northern part bent sharply upward, separated from the southern semicircular region, so that the entire tail resembled the shape of the letter γ ^[28] .

According to the evidence from the Earnock and Superb, at this time (November 7-8), the tail was still quite long - 17.5 ° ^{[21] [22]} . Markwick reported on November 9 that the tail became wider and had a length of 20 ° and a width at the end of 3.5 °, the dark strip almost disappeared, and the core of the bright line became dull and spherical, resembling a woolen ball ^[15] . Eddie recorded on November 10 that the nucleus became even brighter and when observed with the naked eye began to resemble a star more, and the visible movement of the comet was very small ^[14] . There is evidence that on November 12 the tail on the side of the head was comparable in brightness to a third-magnitude star, and the head itself, apparently, was not much brighter ^[29] . On the same day, Markwick estimated the core brightness at 5 ^m , as well as a tail length of 19 ° and a width at the end of 5 ° ^[15] . Winlock on November 13 and 15 estimated the length of the tail to be approximately 10 °, and also noted a dark strip running down to about its middle. The same length of the tail was established by him on November 18, when the tail, according to his observations, became less curved, and the head - much less distinct ^[19] . From Gill's photographs, which he regularly made during November, it can be concluded by comparison with the nearest stars of known magnitude that in the middle of the month the comet brightness was about 3.5 ^{m [29]} . By November 20, Markwick testified that the comet as a whole really became brighter ^[15] , specifying the next day that it was brighter than fifth-magnitude stars ^{[approx. 3] [29]} . Also on November 20, Young recorded that although the comet had faded compared to its previous observation at the beginning of the month, the tail was still quite long and visible without the help of a telescope ^[12] . Winlock reported on November 20 that the tail had a length of 15 ° and a width at the end of 3 °, its bifurcation was clearly noticeable for more than half its length, and the head, especially from the north, was clearly outlined. The next day, an astronomer from the US Naval Observatory recorded a tail length of 10 ° and a similar view of the head of a comet. Also on November 22, the head did not have a clear border on the south side; the tail that day, when the moonlight did not interfere with the observations, had a length of up to 12 ° ^[19] . In addition, on November 22, as Tebbutt noted, the angle between the main axis of the ellipse, the shape of which had a core, and the celestial parallel became maximum and reached 45 ° ^[30] . This situation was also recorded upon observation from the Greenwich Observatory on November 26; then it was reported that in general the comet was very pale, and the brightest part was 1/3 of its length from the northern end ^[31] .

By the end of November, according to Eddie’s notes, the tail had shrunk to only 12 ° in length, but its width at the end was 6 °, and it changed, curving downward (eastward), so that it looked like a shepherd’s staff or ostrich feather with lowered end ^[14] . Markwick also noted that the length of the tail was reduced to 15 °, the width at the end was 4.5 °, the whole tail, like the head, became more vague; the head had a brightness of 5 ^{m [15] [32]} . A few days later, on December 2, and Winlock reported a decrease in the length of the tail to 6-7 ° and the pallor of the comet as a whole, and on December 3, an astronomer from Washington described the shape of the tail: it became almost straight, brighter on the south side and fuzzy at the end ^{[ 19]} . And on December 4, Eddie recorded that the length of the bright part of the tail decreased to 8 °, and the width 4 ° from the head was 2 °, the curvature at the end became almost invisible ^[14] . The same length of the tail, which is still clearly distinguishable, was established on December 7 (and then December 11) and Winlock, who also noted then the clearly defined border of the head of the comet, which was still slightly flattened from the north side ^[19] . Markwick reported on December 8 that the tail was pale but wide, its length was 8 °, the head when viewed through a telescope looked like a blurry nebula, the core was already indistinguishable. After 2 days, according to his reports, the tail was straight and had a length of 12 ° ^[15] . Eddie testified that by December 9 the comet had already become very dim, and after 5 days he completed the observations ^[14] . However, Markwick was able to accomplish several more: on December 15, the tail was still visible, although its shape was so fuzzy that it was impossible to determine where it ended, the approximate length was 15 °, the width at the end was 5 °; December 22 is the last observation from Pietermaritzburg ^[15] . According to Winlock, at the end of December, the comet was very pale and not visible to the naked eye ^[19] . Tebbutt reported that in December and January the comet resembled a vaguely defined nebula of approximately elliptical shape, slightly brighter in the center ^[30] .

Kernel Change

Since the end of September, after the passage of perihelion, the nucleus began to change ^{[1] [33]} . So, on September 27, Barnard noticed that the nucleus extended in the direction of the tail ^[1] ; by September 30, elongation along the celestial parallel was also recorded by Tebbutt ^[30] , O.K. Wendell from the Harvard Observatory ^[1] , Finlay and Elkin on the Cape of Good Hope ^[4] , as well as A. Ricco. A significant change occurred on October 2: on this day, according to the notes of the American astronomer , the nucleus took an ovoid shape, with its wide end oriented toward the Sun ^[1] , and Tebbutt reported that the major axis of the elongated core became 3 -4 times exceed the small ^[30] ; another American, C. O. Young, also noted significant traction ^[12] . The very next day, L.E. Eddie and the Belgian , observing at his observatory in Louvain , independently registered the division of the nucleus into two parts, both clearly elliptical ^{[1] [20] [34]} ; Eddie noted that each of them resembled a candle flame, and the first, the head one, lay a little south relative to the axis of the tail ^[14] . And on October 4, he wrote that one was larger and brighter than the other, and they looked like rice grains lying one after the other ^{[1] [14]} . At that time, Young recorded an even stronger core extension compared to his previous observation, so that it began to resemble an Indian mace in shape ^[12] . Finally, on the same day, E. Frisbee at the US Naval Observatory discovered that the core lengthened and became denser in the region of 1/4 of its length from the end (from the side closest to the tail) ^[19] . The next day, October 5, Barnard and his compatriot G.K. Wilson , who worked at , independently reported 3 cores already, and Wilson characterized them as being arranged in a row parallel to the right (northern) border of the tail. Pritchett described approximately the same picture on October 6, adding that they “seemed to be floating in a cloud of yellow dust” ^[1] . On the same day, the division into 3 parts was also registered by the German astronomer A. Kruger ^[35] , and Stevenson noted that the entire head of the comet, slightly deviating from the axis of the tail, resembles a double star, and when viewed through a telescope, it is elongated in the direction of the comet's motion - from east to west ^[25] . Finally, on October 6, 3 pronounced light clots were also observed by specialists at the US Naval Observatory - Frisbee and Winlock, in particular, the latter measured the sizes of the brightest clot - approximately 12 × 4 ″ - and the total core length - 25-30 ″ ^{[ 19]} .

Over the next few days, observers using small telescopes continued to see one, but very elongated core, while large telescopes could see 2-3 parts ^[1] . So, Tebbutt observed in the structure of the nucleus the brightest light clot about 1/4 of the eastern end of the nucleus ^[30] . And on 9 October Finlay recorded several clots, of which two were the brightest ^[4] . The distance between the two brightest fragments recorded on October 8 by Krueger was 13 ″, and Finlay on October 10 was 22 ″ ^[1] ; similar measurements were made these days by J. M. Scheberle in Ann Arbor ^[35] . There is also evidence from Washington, DC: on October 8, Frisbee observed at least 3 seals in the structure of the nucleus, more pronounced than the day before, with the brightest central (although it was located closer to the tail, away from the head), and Winlock on October 9 recorded that there were already 4 clots, and the entire core extended even more in a line (slightly convex to the southerly direction), so that its width became 5 times less than the length that that day, according to the measurement of their colleague J. R. Eastman , was 31.8 ″ ^[19] . In addition, on October 10, Finlay also noted that the central bright points lay somewhat south of the others, located in a line, and 2 days later he estimated the length of this line to be 39 ″ ^[4] . And on the same day, Young recorded significant changes in the structure of the nucleus relative to the previous observation a week ago: the line 40 ″ in length, into which it turned, resembled a spindle in shape and consisted of 6 luminous points similar to stars. The brightest, which, according to the astronomer, apparently was the real core, was at a distance of 1/3 from its beginning, and the next 2-3 ″ behind it was the second brightest. With the onset of dawn, the paler fragments gradually disappeared, so that at some point the comet seemed to be divided into only two parts ^[12] . Finally, at the same time, Winlock estimated the maximum width of the core (still resembling a string of 3 beads, the central of which became more round and large, though already not curved) at 9 ″ ^[19] . On October 11, Eddie also noted significant changes in relation to his previous observation of three days ago, when the part in front was an elongated oval, and in the far part one could already see 2 new round and dimmer condensation centers ^[14] : the fragment located in front decreased significantly and looked very bright , a star-like dot, while the other stretched out a lot and acquired a dumbbell shape, that is, it was two seals connected by a jumper (moreover, one is more spherical in shape, ugoe located closer to the tail - more elongated), thus showing a clear tendency to further divide. And on October 15, he made an observation that one core and a design of two luminous spots were clearly distinguished, in each of which with a 100-fold increase it was possible to consider, in turn, 2 more weakly expressed parts ^{[1] [14] [34]} . On October 13, the British Prince noted that the core, which earlier (on October 4) had a pear-shaped shape, strongly extended into a flickering line ^[23] . Eastman October 14 estimated the length of the nucleus at 43 ″, meanwhile, Winlock - his colleague from Washington - no longer observed separate clearly identified fragments, only the part closest to the tail seemed denser ^[19] . On October 15, Young reported that the largest fragment - now the third from the beginning (from the side of the Sun) - had a diameter of 6-7 ″, and the whole core was 48.5 ″ in length and from the straight line it became slightly curved, like the tail, forming a single arc with it ^[12] .

In the second half of October, many continued to observe one very elongated core, and through large telescopes it was possible to see up to 6 separate small nuclei, which, for example, J. M. Scheberle described as a chain of beads strung on a thread, and this line was oriented according to the records Russian astronomer I. Cortazzi , in the direction of a point with a position angle of 286 ° October 19 and 293 ° October 31 ^[1] . Gill recorded on October 17 that the core was cigar-shaped and consisted of 5 parts with a diameter of 2-3 ″, with a total length of about 1 ′ ^[4] . Prince saw 3 parts on October 20, and on October 23, together with his colleague 4 - 4-5 (a sketch is given in the report of an astronomer from Crowborough), which they also compared with a string of beads and noted their frequent flickering; by the end of October, these effects, according to Prince's notes, became much less distinguishable ^[23] . Eddie, who carefully monitored the dynamics of the structure of the nucleus every day, noted that some fragments were brighter than others, and their position in the sequence from brightest to faintest changed each time ^[1] . So, according to his reports, starting from October 21, another bright fragment appeared before the first bright fragment, so that it was possible to distinguish groups of 2 and 4 fragments surrounded by a light nebula and located in a line perpendicular to the line of sight; its total length was October 23, 1′15 ″ ^[14] . On October 24, Winlock observed 4–5 “beads” in the core, the second being the brightest from the end ^[19] . Other astronomers provided similar descriptions. On the side of the core lying closer to the Sun, one could see 5-6 "caps", and the inner one, according to the reports of K. G. F. Peters , actually merged with the luminous line of the core region ^[1] . The distance between the fragments, in addition to Scheberle, was regularly recorded by Wilson; there were also separate observations from Cordoba ( ), Rio de Janeiro (L. Kruls), Palermo ( ) and the Dudley Observatory in Albany ( ) ^[35] .

E.E. Common, who, due to weather conditions, was not able to conduct high-quality observations until October 30, recorded on that day that the line into which the core had turned had a length of 58 ″ and a width of 10 ″, and the very next day, when the observation conditions improved, at the ends this line became much brighter, in the middle of it there was a gap of 11 ″ wide, and its length grew to 110 ″. And on November 2, the part closest to the tail became even brighter and acquired the appearance of an independent nucleus ^[17] . Tebbutt reported that the core, which continued to stretch during October, became fainter in the middle than at the end of this month and at the beginning of the next month, moreover, on the one closest to the tail - brighter than on the other; by the end of November, this effect disappeared ^[30] .

The length of the luminous strip into which the nucleus turned gradually increased further throughout the entire period when accurate observations were possible ^[33] . So, on November 1, Finlay estimated it at 3.5 ′ along the right ascension arc (the shape became much less clear) ^[4] . The next day, Winlock also recorded that the core extended even more and faded; according to his measurements, the distance between 1 m and 2 m fragment was 12.4 ″, between 1 m and 3 m - 22.7 ″, total length - 55.9 ″, width - 5 ″. His colleague at the U.S. Naval Observatory that day, W. Sampson , according to the results of his measurements, received the distance between 1 m and 2 m fragment - 10.1 ″, between 2 m and 3 m - 9.9 ″, as well as the length of the first fragment - 7.2 ″. After 3 days, he testified that these distances increased ^[19] . B. J. Hopkins, comparing the data of his observations in London of November 8 with the previous of November 4, recorded the separation of the nucleus into 2 parts, the line connecting them formed a small angle with the axis of the tail; by November 14, they were extended even more, and the distance between them was reduced ^[28] . Until November 8, the American Wilson regularly recorded the distance between the fragments ^[35] . His compatriot Winlock on November 13 continued to see 2 obvious light clots and behind them 2 less clear, so that the distance between the 1st and 3rd was 23.3 ″. And on November 15, an astronomer from Washington could no longer distinguish more than 2 even with high resolution, but he measured the distances between them and the blurry borders of the entire core, its total length is 92.1 ″ and its maximum width is 9.9 ″, and determined the position angle of the line connecting the fragments - 309.4 °. He got almost the same value on November 18, when the distance between the two fragments observed by him reached, according to his measurements, a little more than 20 ″ - his colleague Sampson, who also saw the third fragment and estimated the distance to him (from the 1st) to 30 ″, and in addition, a day earlier he reported that the total core length was 80.6 ″ ^[19] .

According to Eddie’s testimony, the last of the 6 fragments he saw earlier in November gradually faded, and by the end of this month it was difficult to see only 2-3 parts, nevertheless, the granular structure was clearly visible until December 5 ^[14] . In late November and early December, the distance between the fragments of the nucleus was also recorded by G. Bigurdan on the island of Martinique, I. Palis in Vienna ^[35] and specialists from Washington ^[35] . So, on November 20, Winlock reported that it was impossible to distinguish individual parts in the composition of the nucleus — it was only slightly brighter from the side of the tail — however, the next day, I again observed 2 light clusters almost identical in brightness, and the position angle of the segment formed by them was 313.8 °. Then, on December 3, Sampson determined the distance between the 1st and 2nd fragments - 23.1 ″ and the total length of the nucleus - 102.1 ″. And his colleague Winlock received for the lengths of these segments, respectively, the values ​​of 34.4 ″ and 105.6 ″; he registered the 3rd fragment, but very close (about 11 ″) to the 2nd; the position angle, according to his measurements, was 333.3 ° that day. After 2 days, Sampson testified that between the 1st and 2nd fragments there were 29.6 ″ ^[19] . The angular size of the core, according to Finlay's reports, peaked in December, then began to gradually decrease ^[4] . At the end of December, measurements were made by Tom in Cordoba, Wendell in Cambridge and C. Trepier in Algeria ^[35] .

On January 27, Common described the core as a line of 5 points: the 2nd and 3rd were the brightest (11 ^m ), the 1st and 4th were about 3 times less bright and the last was the paler; the distance between the 2nd and 3rd parts was 31.5 ″, and between the first and last - 135.5 ″. This situation continued until February 24; the distance between the 2nd and 3rd fragments on this day was 33.5 ″, and the entire line was slightly curved to the north ^[17] . There was evidence that even in February it was possible to distinguish brighter points in the structure of core 2 at a distance of about 35 ″ from each other and on both sides of them two paler points at a distance of about 50 ″ and 22 ″ respectively ^[36] . Meanwhile, on February 1, Winlock captured a picture that did not differ much from what he observed almost 2 months ago — 3 light clots — except that the entire core had faded somewhat. The distance from the central, which was the brightest, to the head ^{[approx. 4]} , was 35 ″, to another - 42 ″, and the total core length - 80 ″. A similar situation was registered by him and independently by his colleague Frisbee on February 23; for the position angle on that day, Winlock received a value of 76.5 °. A. Hall , another astronomer from the US Naval Observatory, on February 26 also determined the distances between these points - 34.5 ″ and 47.14 ″ respectively, as well as the position angle - 77.2 °, but he also observed 4- th point. The next day, an American scientist received for these values, respectively, the values ​​of 34.58 ″, 48.99 ″ and 78.8 °, as well as for the distance from the 3rd to 4th point - 22.27 ″. Then, on February 27, Winlock recorded that he had observed the 5th at about 50 ″ from the 4th, and the total core length on that day was 106 ″. The appearance of a very pale 5th point was recorded on February 28 and Eastman ^[19] . In addition to experts from Washington, in February Bigurdan (already at the Paris Observatory), Wendell, Common, W. Chur (in Strasbourg ) and E. B. Bayot (in Toulouse ) received quite a lot of data ^[35] .

At the beginning of March 1883, when the comet had already substantially faded, the core, according to observations at the Cape of Good Hope Observatory, was an ellipse with semi-axes of 3.5 ′ and 1.5 ′ ^[4] , and the distance between its two largest fragments was estimated at 35 ″ ^[33] . Meanwhile, on March 3, Winlock reported that 3 points of approximately the same brightness (of the order of 12 ^m ) and the 4th more pale one are still distinguishable. A month later, he registered only one clearly distinguishable point approximately in the middle of a very pale and vague strip 20-30 ″ long, into which the core turned. His colleague Frisbee, however, still saw 3 separate light clusters on April 4, of which the closest to the tail was the brightest ^[19] . Bigurdan and Wendell managed to make some measurements of the distance between the fragments in March and April ^[35] .

Separation of the nucleus, as noted by the American astronomer Young, was the most unusual effect, largely due to what happened after the passage of perihelion ^[12] . In addition, this was a source of inaccuracy in position measurements, since it was not possible to unambiguously establish the focus point of the observation tool ^{[12] [30]} .

Various Unusual Effects

Starting in September and especially throughout October, another unusual effect - various objects near the comet and around it - could be observed both through a telescope and without its help in those places where the conditions were quite favorable (i.e. a fairly dark sky at night). Around the middle of September, some observers noted a faint luminous halo almost completely surrounding the comet; it was most pronounced from October 6 to 17, and then by the beginning of November it gradually faded away. The first to notice this effect was September 14, Charles Gruver, a member of a British expedition sent to Brisbane to observe the passage of Venus across the disk of the Sun - in his report he described a cylindrical surface enveloping the comet ^[1] . Young reported on September 19 that the comet was like a white bird flying fast toward the Sun, whose “wings” were formed by elongated arcs connecting the envelopes of the core of the surface; they were located 30 ”and 2-3 ′ in front of the head, respectively. The next day these arcs were no longer visible ^[12] . Markvik on September 30 and October 1 reported that a slight haze could be observed around the head and part of the comet’s tail ^[15] . On October 2, Young also noted a bright “cap” 30 ″ from the core. In his notes dated October 3, Eddie wrote that the head and a quarter of the northern half of the tail were circled by a strip of weakly glowing loose substance about 1/4 ° thick, and the next day its size increased in places to 1 °, and it seemed to merge with the northern part tail w ^[14] . Then, October 4 Markwick besides haze near comet heads found in 1,5 ° to the south of it two small luminous clot resembling nebula ^{[1] [15] [34]} . The same day, Yang reported that the northern border of the tail became more blurred and covered with a light nebula starting from the head, and the halo near the nucleus registered by him 2 days earlier turned into a weakly expressed nebula. Also, an American physicist noted that one can see one bright the comet inside another more ble Noah, but not in the middle, and a little to the south ^[12] . On October 6, Winlock from the US Naval Observatory also began to observe a pale envelope 30-45 ′ thick at 65 ′ from the head of the comet: on the northern part of the head, at a distance of 0.5 ° from it, it was almost straight and stretched for another 1 ° beyond its borders, and to the south lay much closer and did not protrude beyond its borders; the inner (located on the side of the head) region was more dull and vague than the outer one. The next day, according to the reports of the American astronomer, the line became even more pronounced, especially from the south side. During both (and subsequent) observations, he determined the exact position of this object relative to the nearest stars and made detailed sketches ^[19] .

Regardless of the above observers, A. V. Nursing Row in his observatory in Vishakhapatnam on October 7 recorded a bright limb on the sunny side of the comet's head, rounded on each side of the tail, as well as narrow streaks of light above the tail more than 12 ° long, bending southward ^{[24 ]} . On the next day, October 8, according to Eddie’s reports, this object completely covered the comet’s head, as if with a hood, forming a comet-shaped cone in front of it with a poorly distinguishable apex. The British astronomer reported that the effect was clearly distinguishable with the naked eye and with the telescope-finder and in the following days of observation - on October 9 and 11, and the “hood” extended to 1.5 ° in front of the comet's head ^[14] . Markwick also noted a weak halo around the comet's head: on October 9, it had a thickness of 1 °, and on October 10 and 11, it was 2 ° ^[15] . Also on October 9, Young reported that he had previously noticed a nebula in the region of the northern border of the tail became more pronounced ^[12] . Winlock made a similar observation that day, and the next day, according to his reports, the envelope line became even brighter and clearer. A few days later, on October 14, an American astronomer recorded that it almost closed around the head of a comet ^[19] .

Also on October 9, the German astronomer and geophysicist I.F. Yu. Schmidt , who made observations at the Athens National Observatory , noticed a certain object moving parallel to it 4 ° south-west of the comet, which he described in his article in the journal Astronomische on October 12 Nachrichten ”as a new comet ^{[1] [8] [12] [34]} . According to his data on the position of this object on October 10 and 11, even the parameters of its hypothetical orbit were calculated, in particular, the perihelion transit time on September 24 or 25. However, later it was no longer observed. The German astronomer also registered at the Strasbourg Observatory on October 10, to the southwest of the comet, a large nebula similar, as he said, to a comet with a bright nucleus and a fan-shaped tail, but on October 13 could not see it again. This unusual phenomenon is also mentioned in Barnard’s reports: south of the comet he found a comet-like object with a diameter of 15 ′, immediately behind it - the same, but less bright, and on the opposite side of the first - one more and more dim, so all three were located almost in a row; and 6 ° southeast of the head of the main comet, he observed a group of 6–8 more such objects ^{[1] [34]} . In addition, on October 21, U. R. Brooks of New York, noticed a certain pale comet-like object 2 ° long and 8 ° southeast of the main comet ^{[1] [12]} .

A number of observers noted a rather pronounced cloud of light located outside the sunflower side of the coma and looking like an unusual tail , straight and directed towards the Sun ^[1] . So, on October 6, J.T. Stevenson wrote that, appearing some time ago first to the north of the comet’s head, it was a continuation of the main brighter tail (although, upon closer examination, it deviated from this axis by a small angle) and had length about 2 °, and by October 10 - up to 4 ° ^[25] . On October 9, he also registered a faint light strip at the University of Kansas with almost parallel widths of the same width as the tail and directed toward the sun. By October 15, it became even more noticeable, located from the border between the tail and head 1-2 ° above the core down 2-3 ° below the head, had distinct edges, a width of up to 30 ′, and also uniform brightness, however, according to the latter the parameter was significantly inferior even to the faintest parts of the tail ^[12] . The length of this anomalous object, measured by Barnard and the German astronomer A. Auvers on October 16 and 17, was 4-6 °, while Barnard estimated its width at 1 °. Another astronomer from Germany, E. V. L. Tempel, described it on October 18, accompanied by sketches, like a cylinder with bright but fuzzy borders and a dark region along the axis ^{[1] [26]} . British astronomer confirmed that he received evidence of this unusual tail from Constantinople, and Athens, as well as from J. W. Schiaparelli and D. Gill ^[25] .

Basically, observers note that the described effect gradually became less noticeable by October 20 and completely disappeared by October 27. It was unique in its kind, since nothing similar was observed when observing other comets. According to amateur astronomer and comet researcher , the “comets” recorded by Schmidt, Hartwig and Barnard were the brightest clots of a luminous halo seen by other observers, which consisted of dust ejected by or a large September comet during its previous passage of perihelion about 8 centuries before, or by another comet from a large family of near-solar comets ^{[1] [29]} .

Meanwhile, similar unusual effects were noted during further observations. So, E.E. Common on January 27 reported that a light haze surrounded the head of the comet, the southern part of which was more pronounced, and the borders were fuzzy ^[17] .

Completion of Observations

The comet remained visible with the naked eye until February 1883, even then its tail remained quite long - 4-6 °. After that, in March, the light of the moon no longer allowed to see it ^[1] .

E.E. Common, in particular, noted that on January 27 the comet resembled a rather bright extended nebula, the tail was already almost indistinguishable to the naked eye. The British astronomer wrote in his report that the comet was bright enough for observations until February 24, and he could continue them further, but it already ceased to be visible above the horizon ^[17] . Tebbutt recorded the position and brightness of the comet until March 2 ^[30] .

Astronomers from the US Naval Observatory continued to observe using the meridian circle until March 3, and using telescopes for another month, until April 4 ^[19] .

Gould testified that the last observation with the naked eye was carried out on March 8 ^[32] .

The last time (through a telescope) the comet was observed on June 1, 1883 in the form of an extremely pale, according to the reports of , light spot ^{[1] [32]} . Gould, like Common, noted that the comet was no longer visible, but not because it became too dim, but because of the too low position in the west above the horizon with the setting of the Sun ^[1] .

Some astronomers made the first calculations of the parameters of shaving based on the results of observations of their colleagues and their own shortly after their completion. For example, , using data from three different observation points in Europe and the USA, using the graphical constructions, simulated the parabolic orbit of the comet ^[37] . S.K. Chandler from Harvard came to the conclusion that for a parabolic orbit there would be a strong discrepancy with the recorded trajectory, and proposed (after several corrections and taking into account a larger amount of data) elliptic with a period of 8.5 years ^[8] . J. Tatlock at ( ) under the direction of from a series of observations in October and November 1882 and January 1883, calculated the parameters of the orbit, also arriving at the conclusion about its ellipticity ^[38] .

Based on the results of more than 60 observations made, in particular, by U. L. Elkin and U. G. Finlay between September 7, 1882 and June 1, 1883, Heinrich Kreutz calculated the perihelion distance a few years later - of the order of 0.003 a. e. ^[33] and the periods of the orbit for each of the 4 separate fragments - 664.3, 769.2, 875.2 and 959.4 years, respectively, and for their center of mass - 817.3 years ^[35] . He came to the conclusion about the connection (but not full identity) of it with the , and also about the common origin of the comets of 1882, 1880 and 1843 ^[33] . Finlay and Elkin themselves ^[39] , as well as other astronomers, calculating the distance at the perihelion, the longitude of the ascending node, and the argument of the perihelion , also noted this similarity and made similar conclusions ^{[9] [11] [12] [8]} . In favor of the theory of the origin of the comets of 1882, 1880 and 1843 from , the destruction of the nucleus of the great September comet of 1882, which was recorded by many observers, also spoke.   , and objects marked by astronomers near it   , which with high probability were also fragments separated from it ^[11] .

Later, already in the 20th century, a particularly strong similarity between the comet and the comet Ikea - Seki of 1965 was revealed. It has been suggested that they correlated with each other and with the progenitor comet in the same way as the fragments of the nucleus of the great September comet of 1882 with the original whole nucleus after its destruction. It is believed that these two comets are members of the second subgroup of fragments of the original comet of 1106 ^{[35] [40] [41]} .

Notes

1. ↑ This is the maximum value recorded by astronomers from Washington.
2. ↑ Although he probably meant the head
3. ↑ But apparently, did not exceed stars of the fourth magnitude
4. ↑ Which was previously on the tail side when the comet was observed before sunrise, while during the February observations it culminated before midnight
5. ↑ With a focus not in the center of the sun, but in the center of mass of the entire solar system
6. ↑ ^{1 2} Based on the formula${\displaystyle \mathrm{<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">e</span></span></span>}<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span></span>\mathrm{<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">one</span></span></span>}<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span></span>\mathrm{<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">q</span></span></span>}<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span></span>\frac{\mathrm{<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">one</span></span></span>}}{\mathrm{<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span></span>}}<span\; class="reference-text"><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span></span>}$ {\ displaystyle e = 1-q ({1 \ over a})}   Error in the footnotes ^? : Invalid <ref> : name “MarsdenNote1” defined several times for different content

Footnotes