The First Stellar Parallaxes Revisited
RReceived xy July 2020; Revised xy July 2020; Accepted xy July 2020DOI: xxx/xxxx
ORIGINAL ARTICLE
The First Stellar Parallaxes Revisited
Mark J. Reid | Karl M. Menten Center for Astrophysics Harvard &Smithsonian, Cambridge, MA, USA Max-Planck-Institut für Radioastronomie,Bonn, Germany
Correspondence
Mark J. Reid, Center for Astrophysics Harvard & Smithsonian, 60 Garden Street,Cambridge, MA 02138, USA Email:[email protected]
We have re-analyzed the data used by Bessel, von Struve, and Henderson in the 1830sto measure the first parallax distances to stars. We can generally reproduce theirresults, although we find that von Struve and Henderson have underestimated someof their measurement errors, leading to optimistic parallax uncertainties. We findthat temperature corrections for Bessel’s measured positions are larger than antici-pated, explaining some systematics apparent in his data. It has long been a mysteryas to why von Struve first announced a parallax for Vega of . ®® , only later withmore data to revise it to double that value. We resolve this mystery by finding thatvon Struve’s early result used two dimensions of position data, which independentlygive significantly di erent parallaxes, but when combined only fortuitously give thecorrect result. With later data, von Struve excluded the “problematic” dimension,leading to the larger parallax value. Allowing for likely temperature corrections andusing his data from both dimensions, reduces von Struve’s parallax for Vega to avalue consistent with the correct value. KEYWORDS: history and philosophy of astronomy – astrometry – stars: distances – methods: data analysis
In 1838, Friedrich Wilhelm Bessel published two papersreporting what has generally been considered to be the firsthighly significant measurement of the distance to a star. Onepaper published in the
Monthly Notices of the Royal Astronom-ical Society (Bessel, 1838c) was an edited version, translatedfrom German to English by John Herschel, of a more detailedpaper in
Astronomische Nachrichten (Bessel, 1838a) titled"Determination of the distance to the 61 st star in Cygnus."These papers reported what has generally been considered tobe the first highly significant measurement of the distance toa star. Bessel used the Earth’s orbit as a surveyor’s baseline tomeasure the apparent angular shift of a nearby star as the Earthmoves around the Sun (trigonometric parallax). This achieve-ment culminated centuries of e orts to determine what wase ectively a scale size of the Universe for th century astron-omy, as well directly validating that the Earth went around theSun. Fernie (1975) and Hirshfeld (2001) place this work in a historical context, detailing the lives and e orts that lead thismomentous result.In the same time period, two other astronomers also col-lected astrometric data that would yield trigonometric parallaxdistances to stars. In Cape Town, South Africa, Thomas Hen-derson observed ↵ Centauri with a mural circle between 1832and 1833, as part of a program to measure the positions of largenumbers of stars. Years after his return to England, he ana-lyzed this data and claimed to have a parallax measurement forwhat is now known to be the nearest stellar system (Henderson,1840). In 1835, Friedrich Georg Wilhelm von Struve startedobservations designed specifically to measure a trigonomet-ric parallax for ↵ Lyrae (Vega), and, after collecting a year’sdata, he communicated a tentative result in a chapter of hismonograph “Stellarum duplicium et multiplicium mensuraemicrometricae” (von Struve, 1837).Interestingly, both von Struve and Bessel used telescopesdesigned by Joseph Fraunhofer specifically for parallax mea-surement. These were the last telescopes built by Fraunhofer
Reid & Menten and in the hands of these two astronomers produced outstand-ing results, providing a prime example for the validity ofthe paradigm that major discoveries follow major technologi-cal innovations in observational instruments as recognized byHarwit (1981).Motivated by his friend von Struve’s tentative result, Bessel,who had worked on the parallax of 61 Cygni as early as 1812,but had to a await a better telescope (Ashbrook, 1954), startedan intensive observing program to measure the parallax of thisstar and beat von Struve to the prize. Given this complicatedand interwoven sequence, attribution of priority for this trulytransformational result has been a subject of discussion to thisdate.We have been developing radio astronomical techniques inorder to directly map the spiral structure of the Milky Way.This endeavor involves measuring trigonometic parallaxes forlarge numbers of massive stars that have recently formed ingiant molecular clouds and trace spiral structure throughoutthe Galaxy. This prompted our interest in the history of thefirst parallax measurements. Fortunately, much of the raw datahas been published, and this paper presents a re-analysis, madepossible by modern computers, in order to better assess theirsignificance.
Bessel chose 61 Cygni for a parallax measurement because itcould easily be observed over a full year at the KönigsbergObservatory in Prussia and, importantly, it was known to havean extremely large motion across the sky (proper motion), sug-gesting it was nearby. He used a heliometer telescope builtspecially for high precision astrometry by Joseph Fraunhofer,the world’s premier telescope maker. This was the last tele-scope designed by Fraunhofer, and it was delivered to Besselafter Fraunhofer’s death. A heliometer has the objective lenscut in half and the two pieces can be slid along side each otherto superpose the images of two stars. Their angular separa-tion can then be determined by a precision micrometer screw,which tells how far one lens was slid with respect to the other.61 Cygni is a binary system, with the two stars havingapproximately a 700-year orbital period. Bessel gives theirseparation in 1838 as ®® along a position angle of ˝ Eastof North. He measured the angular distance from the center ofthe binary to each of two dimmer reference stars: star a sepa-rated by ®® at a position angle of ˝ and star b separatedby ®® at a position angle of ˝ . Bessel took upwards of100 measurements of the position of 61 Cygni relative to eachreference star over one year and performed a least-squares fit to estimate a parallax and a residual proper motion . His paral-lax, obtained by combining the results for both reference stars,was . ®®
314 ± 0 . ®® , where the uncertainty assumes that indi-vidual measurements have errors near ±0 . ®® and that the dimreference stars are at great distances so as to not have signif-icant parallaxes of their own. The true parallax of 61 Cyg, asrecently measured by Gaia , a European Space Agency parallaxmission, is . ®® . ®® (Gaia Collaboration, 2018).Bessel’s individual and combined parallax results are listedin Table 1. Our re-analyses, assuming, as did Bessel, a con-stant measurement uncertainty, yields nearly identical parallaxestimates (see column 4 of Table 1). Plots of data and fittedparallax and proper motion are shown in Fig. 1. Clearly Besselwas able to perform the extremely laborious calculations fora least-squares fit of ˘ measurements manually withouterror.The di erence in the parallax estimates of 61 Cyg measuredagainst reference star a and star b is . ®® . ®® . This di er-ence appears statistically significant ( . ) and, indeed, Besselnoted this tension (Bessel, 1838c):“The observations seem also to indicate that the di er-ence of the parallaxes of 61 and b is smaller than that of61 and a ; which must be the case, indeed, if b itself havea sensible parallax greater than a .”Bessel was pointing out that each result is a relative paral-lax, which measures the di erence in true parallax of 61 Cygand that of a reference star, and he was suggesting that the(unknown) parallax of reference star b could perhaps be sig-nificant. For example, were star b ’s parallax near . ®® , thatwould remove the tension between the two measures. However, Gaia DR2 parallaxes are . ®® for star a (BD+37 4173) and . ®® for star b (BD+37 4179). So both reference stars aree ectively at “great distances” and did not significantly biaseither parallax measurement.In Fig. 1, one can see that while the model (solid line) forstar a fits the data quite well, this is not the case for star b .The residual di erences between the measurements and a bestfitting parallax (and proper motion) are shown in Fig. 2 andfor star b reveal systematic problems. Specifically, one can seethat nearly all residuals in the autumn/winter period between1837.8 and 1838.3 are negative, while those in the spring/sum-mer period between 1838.3 and 1838.6 are mostly positive.This suggests a temperature related problem. Indeed, John Her-schel, who translated Bessel’s 1838 announcement paper forthe Monthly Notices of the Royal Astronomical Society , wasconcerned about possible temperature issues (Bessel, 1838b). Bessel removed the known large motion of the binary system from his angularo set measurements. eid & Menten TABLE 1
Parallax of 61 Cygni.
Star Data Used Parallax Parallax ParallaxBessel Re-analysis T-adjusted
61 Cyg Reference star a . ®®
369 ± 0 . ®®
028 0 . ®®
368 ± 0 . ®®
028 0 . ®®
61 Cyg Reference star b . ®®
260 ± 0 . ®®
028 0 . ®®
259 ± 0 . ®®
028 0 . ®®
61 Cyg Combined . ®®
314 ± 0 . ®®
020 0 . ®®
313 ± 0 . ®®
020 0 . ®® Comparison of parallax measurements of the center of the 61 Cyg binary by Friedrich Wilhelm Bessel and two modern re-analyses of his data. The “Re-analysis” values assume a constant measurement uncertainty, with the parallax uncertainty scaledto give a reduced chi-squared per degree of freedom on unity. The “T-adjusted” values are from Bessel’s extended data through1840, using his “temperature correction” formulae for parallax. We have set his temperature parameter k = 1 . to give acombined result that matches the true parallax of 61 Cyg of . ®® .Fortunately, Bessel listed the temperature corrections, T ,that he applied to his data in his followup 1838 paper inthe Astronomische Nachrichten . We investigated the e ects ofthese corrections on estimates of parallax by adding to themodel a scaled version of the corrections. The scaling wascontrolled by an adjustable parameter and optimized in theleast-squares fitting. Interestingly, we find that the fit can begreatly improved by adding . . ù T to the data forstar b . The revised parallax estimate for star b is . ®® . ®® .This fit has a = 74 . for 94 degrees of freedom, consid-erably improved compared to a = 101 . for 95 degrees offreedom for no temperature adjustments. If, instead, we use themodern value for parallax of . ®® as a strong prior, and solvefor the scaling parameter, we obtain a value of . , whichis consistent with our fitted value of 1.03 within its formaluncertainty.Clearly, temperature corrections could have had a significante ect on Bessel’s parallax estimates. Applying the same fittingprocedure to the data for star a , yields a highly uncertain scal-ing parameter of * . . (note the opposite sign comparedto star b ). If, instead, we adopt the better determined temper-ature scaling parameter (1.03) from star b , we find a parallaxof . ®®
291 ± 0 . ®® for the star a data. The excellent agreementwith the modern parallax value is, perhaps, fortuitous, but thiscorrection certainly improves the value of . ®®
369 ± 0 . ®® thatBessel published for this star.Bessel continued his observations beyond 1838 and pub-lished extended results in Bessel (1840a) and Bessel (1840b).In these papers, he explicitly confronts the issue of tempera-ture corrections, giving formulae for parallaxes that include atemperature parameter ( k ):
61 Cyg * star a ... ⇡ = 0 . ®® * . ®® k , (1)61 Cyg * star b ... ⇡ = 0 . ®® * . ®® k , (2) and a combined result
61 Cyg * stars a & b ... ⇡ = 0 . ®® * . ®® k . (3) Bessel states, k is “a small indeterminate correction depend-ing on the e ects of temperature on the micrometer-screw.” Hepoints out that “On deducing the value of k from the obser-vations, those of the first star give, therefore, k = * . ;and those of the second, k = 0 . .” Above, we noted thatwhen we solved for a temperature scaling parameter, for star a we obtained * . , essentially the same value as Besselfound. However, for star b Bessel’s scaling parameter of . is markedly di erent from our value of . .Knowing the true parallax of 61 Cyg of . ®® , we can usethe Bessel (1840a) parallax formulae, Eqs. (1), (2) and (3),to assess the consistency of the inferred parallaxes for thetwo stars, based on Bessel’s extended data set. The combinedformula given by Eq. (3), sets k = 1 . . Adopting this temper-ature correction, yields individual parallaxes for star a and star b of . ®® and . ®® . Assuming uncertainties of ±0 . ®® (seeTable 1), these are statistically consistent. Thus, in hindsight,it appears that temperature corrections were larger than Besselanticipated, and this can explain his small overestimate of theparallax of 61 Cygni. Early in 1837 and just before Bessel began his intensive obser-vations, Wilhelm von Struve from the Dorpat Observatory inRussia , announced what we might now term a “tentative”parallax for ↵ Lyrae (Vega) of . ®®
125 ± 0 . ®® (von Struve,1837). This . detection closely matches the true parallax Today, Dorpat is named Tartu and is part of Estonia
Reid & Menten
FIGURE 1
Bessel’s measurements of the angular separationof the midpoint of the 61 Cygni binary and two backgroundstars ( a and b ). The solid lines are our fit allowing for parallaxand (residual) proper motion, assuming uniform uncertaintiesfor the measurements as Bessel did. While the data for star a fit the model well, the data for star b show clear systematicproblems.of . ®® . ®® . (van Leeuwen, 2007) , although thisclose agreement is fortuitous given the large fractional uncer-tainty. Von Struve measured the position of Vega relative to abackground star with a separation of 42”. The parallax of thatstar (Gaia id = 2097892344993257344), not known at the time,has a very small value of 0.001463" +/- 0.000025" and thus did Vega it is too bright for precision
Gaia observations; this parallax is from the
Hipparcos mission.
FIGURE 2
Same as Fig. 1 but plotting residuals to the modelfit. The points are color coded in red for warm-weather periods(0.42 to 0.75 decimal years), in blue for cold-weather periods(0.92 to 0.25), and in green for moderate-temperatures peri-ods. Note that for star b the cold-weather points tend to havenegative residuals and the warm-weather points tend to havepositive residuals.not significantly a ect the (absolute) parallax measurement ofVega.The raw data used by von Struve for his 1837 announcementis available in tabular form on page CLXXI in the voluminousbook by von Struve (1837), whose th chapter is devoted tohis parallax measurement of Vega. Von Struve gives his modelterms equated to his measured data in two coordinates: “dis-tance” along the line between Vega and the background star eid & Menten FIGURE 3
Von Struve’s parallax data for Vega. His tenta-tive parallax of . ®® , announced in 1837, used data throughthe end of 1836 (indicated by the red-dashed line). He contin-ued observations through late 1838 and revised his parallax to . ®® . The solid line shows our best-fitting parallax of . ®® ,after removing an o set. The true parallax of Vega is . ®® .and “direction” perpendicular to that line. On the same page,von Struve gives a “probable error” for his position measure-ments of . ®® . When we use this to fit his data, we find similarresults (a parallax of . ®®
129 ± 0 . ®® ), but with a reduced chi-squared ( ⌫ ) of 2.17. In order to achieve a ⌫ of unity, oneneeds to inflate his position uncertainties to . ®® , giving amore realistic parallax uncertainty of ±0 . ®® . It is interesting to fit von Struve’s measurements in eachcoordinate separately. Using the “distance” measurementsonly, we find a parallax of . ®®
252 ± 0 . ®® , whereas using the“direction” measurements yields * . ®®
225 ± 0 . ®® . So, vonStruve’s tentative parallax in 1837 was essentially a weightedaverage of two results in significant tension ( . ), whichbalanced each other to give the “correct” result.In October 1839, now with 96 measurements in hand, vonStruve (1840) claimed a parallax of Vega of . ®®
261 ± 0 . ®® .This result is based on data taken between 1835 and 1838 We were able to reproduce von Struve’s model parallax factors (i.e., expectedo sets for a parallax of 1”) by rotating expected parallax o sets calculated for the( X , Y )=(East,North) coordinates by 132 ˝ East of North, yielding ( X ® , Y ® ), reversingthe signs of Y ® , and assigning X ® for the “distance” and Y ® for the “direction” model.The sign flip changes from our left- to his right-handed system. Note also that vonStruve lists residuals from his best fit that use the (now) unconventional definition“model minus data”. This is common practice today when one is uncertain of the magnitude ofnoise in a data set. In the NASA Astronomical Data System, various of F. G. W. von Struve’spublications are erroneously credited to his son Otto Wilhelm von Struve. and was published after Bessel’s 1838 paper in the
MonthlyNotices of the Royal Astronomical Society . While von Struve’sraw data starting from the beginning of 1837 are not includedin his paper, his great-grandson Otto Struve, also a prominentastronomer, wrote articles in Sky and Telescope in 1956 aboutthe first parallaxes. Struve (1956) plots the 1835–1838 separa-tion distances and position angles on the sky between Vega andthe background star versus time. We digitized these datasetsand converted both to angular o sets in arcseconds. (Since thisappears to be the only digitized record of von Struve’s historicobservations, we detail them in the Appendix.)Our re-analysis of the separation distances (Fig. 4) givesa parallax of . ®®
271 ± 0 . ®® , very close to that quoted byvon Struve. However, scaling the measurement uncertaintiesto achieve a post-fit ⌫ of unity results in a larger parallaxuncertainty of ±0 . ®® . Thus, it is likely that Struve adoptedan optimistic estimate for the noise in his data. The directiondata yield a smaller, but positive, parallax of . ®®
077 ± 0 . ®® .Combining both distance and direction data yields a parallaxof . ®®
218 ± 0 . ®® (see Table 2). FIGURE 4
Residuals for von Struve’s “distance” data forVega after removing the true parallax of . ®® and the best fit-ting o set and motion. The points are color coded in 4-monthbins in red for warm-weather periods (0.42 to 0.75 decimalyears), in blue for cold-weather periods (0.92 to 0.25), and in green for moderate-temperatures periods. Note the tendencyfor the warm-weather points to have a positive bias and thecold-weather points a negative bias. Reid & Menten
TABLE 2
Parallax of ↵ Lyrae (Vega).
Star Data Used Parallax Parallaxvon Struve Re-analysis
Vega 1835–1836 . ®®
125 ± 0 . ®®
055 0 . ®®
129 ± 0 . ®® Distances only ... . ®®
252 ± 0 . ®® Directions only ... * . ®®
225 ± 0 . ®® Vega 1835–1838 ... . ®®
218 ± 0 . ®® Distances only . ®®
261 ± 0 . ®®
025 0 . ®®
271 ± 0 . ®® Directions only ... . ®®
077 ± 0 . ®® Temperature adjusted ... . ®®
151 ± 0 . ®® Comparison of parallax measurements of Vega by Friedrich Georg Wilhelm von Struve and our re-analyses of his data. The“Re-analysis” values allow for a residual proper motion and a constant measurement uncertainty, with the parallax uncertaintyscaled to give a reduced chi-squared per degree of freedom of unity. The “Temperature adjusted” fit used both distances anddirections and found temperature scaling parameters of . ®® . ®® for distance measurements and . ®® . ®® for directionmeasurements. The true parallax of Vega is . ®® .This raises an interesting question: Why, with more obser-vations, had von Struve discarded the “direction” data and onlyused “distance” measurements? In his paper, von Struve (1840)states the following:"From my [96] measurements, the parallax could bedetermined in two di erent ways, from the observedseparations or from the measured directions of the lineconnecting the two stars against the declination circle, theso-called position angles. However, since circumstancesexist that impair the accuracy of the latter [measure-ments], these were not allowed to be used for the deter-mination of the parallax and it was necessary that theparallax was derived from the separations alone."Our re-fitting of the direction data leads to a parallax uncer-tainty that is comparable to that of the distance data, suggestingthat the direction data was not of lesser precision. We can onlyspeculate that the absense of a signficant parallax amplitude inthe direction data led von Struve by 1840 to no longer trust thatdata.Von Struve’s 1840 result, even after inflating his uncertaintyto achieve a ⌫ of unity, departs significantly (by ) fromVega’s true parallax of . ®® . Is there evidence in von Struve’sdata for a temperature e ect, as we found for Bessel’s data? InFig. 4 we plot “distance” residuals after subtracting the e ectsof the true parallax of . ®® and a best fitting constant andmotion. These residuals show some systematic e ects, with thewarmest four months biased to positive values and the coldestfour months biased to negative values. (We note that the clus-ter of points near 1838.3, which is within what normally is acool-to-moderate temperature period at Dorpat Observatory’s location, are also biased to negative values.) The average resid-ual o set in warm months is +0 . ®® while in cold months is * . ®® . This systematic di erence correlates strongly with thepeaks of the parallax sinusoid (see Fig. 3) and could lead to afalsely large fitted parallax by upwards of half their di erence: +0 . ®® .Knowing the true parallax of Vega, we could do a more rig-orous investigation of systematics in von Struve’s data owing toyearly temperature changes over the seasons. Lacking explicittemperature correction information from von Struve, we mod-eled the e ects on measured distances and directions as yearlysinusoids, peaking in early July, and scaled by adjustableparameters to be determined in the least-squares fitting pro-cedure. We held the parallax at its true value of . ®® andfound the temperature scaling parameter for distance data tobe . ®®
100 ± 0 . ®® . This value is in excellent agreement withour analysis in the previous paragraph, based on average biasesseen in post-fit residuals. For the direction data we find atemperature scaling parameter of . ®®
047 ± 0 . ®® , possiblyindicating less sensitivity of the direction measurements toseasonal variations.With stong evidence for significant temperature e ects invon Struve’s measurements, we performed a final least-squaresfitting, allowing all parameters to vary simultaneously. Thisyields a parallax for the combined data of . ®®
151 ± 0 . ®® and distance and direction temperature parameters of . ®®
082 ±0 . ®® and . ®®
044 ± 0 . ®® , respectively. This parallax estimateis consistent with the true value for Vega and has . for-mal statistical significance. The increased parallax uncertainty,compared to that with no temperature corrections, is owing toa large correlation coe cient of * . between parallax andthe distance data temperature parameter. eid & Menten Thomas Henderson observed ↵ Centauri from a newly estab-lished observatory near Cape Town, South Africa. ↵ Centauriis a triple star system which includes two bright stars, ↵ Cenand ↵ Cen, separated by ®® . He reported transit measure-ments with a mural circle telescope of the absolute positionsof both stars, calibrated by requiring that other stars observedon the same day had zero parallax. His paper, published in the Memoirs of the Royal Astronomical Society , gives tables of theright ascension and declination positions, as well as declina-tions from “reflected observations” (Henderson, 1840). Sincethe reflected observations had significantly larger uncertaintiesthan the direct declinations, we do not consider them here.
FIGURE 5
Plot of Henderson’s (more precise) declinationdata and the “Both Combined” parallax fit. A constant o sethas been removed. Open circles are for ↵ Cen and filled circlesare for ↵ Cen.Parallaxes measured from absolute positions (i.e., not rela-tive to nearby stars measured concurrently as did Bessel andvon Struve) are directly a ected by corrections for aberrationof light, which also is owing to the orbital motion of the Earthand has a yearly period. Henderson notes that his tabular data assumed a constant of aberration of . ®® , virtually identi-cal to the modern value . He also comments on the probableuncertainties for measurements of right ascension ( ±0 . ®® )and declination ( ±0 . ®® ). However, for the more accurate, andhence more important, declination data he notes that experi-ence from his previous “Catalog of declinations” (Henderson,1837) suggested a larger declination uncertainty of ±0 . ®® wasappropriate for the typical zenith distance of ˝ of ↵ Cen, andhe adopts the average of these two declination uncertainties: ±0 . ®® .Henderson’s data and our parallax fit are plotted in Fig. 5and listed in Table 3. In most cases, there are reasonably goodcorrespondences between Henderson’s parallax values and ourre-fitted ones. One exception is for the right ascension parallaxfor ↵ Cen. In some cases, Henderson’s and our parallax uncer-tainties agree within 10%. However, there are several exampleswhere our uncertainties are upwards of 50% larger than Hen-derson’s. These di erences can be accounted for by scaling ofmeasurement uncertainties to achieve a reduced ⌫ of unity.Thus, it appears that Henderson’s estimates of measurementuncertainties are likely optimistic in some cases.All in all, it is clear that Henderson’s final parallax esti-mate, combining the right ascension and declination results forboth stars was precise and formally significant at the level.However, the true parallax of ↵ Cen, as measured by the
Hip-parcos mission (van Leeuwen, 2007), is . ®® . ®® . Thus,our re-analysis of Henderson’s data, which gives a parallax of . ®® . ®® , is . from the true value. This moderate tensionwith the true parallax suggests some small residual systematicsin his data. We have performed an in-depth re-analysis of the datasetsthat Bessel, von Struve, and Henderson used to obtain thefirst stellar parallaxes, and we can reproduce their results. It isclear that von Struve and Henderson used a priori estimatesof data errors and arrived at optimistic parallax uncertainties.We have re-scaled their data errors to achieve post-fit ⌫ perdegree of freedom of unity and arrived at more realistic paral-lax uncertainties. Even based on more realistic uncertainties,we can confirm that all three astronomers detected their stellarparallaxes with reasonable statistical significance.We confirm Bessel’s 1838 parallax estimates for 61 Cygniin detail. Bessel measured parallaxes relative to two reference Henderson also presents parallax results adopting a value of . ®® for theconstant of aberration. Using this value changes his parallax results for ↵ Cen and ↵ Cen by 12% and 26% for right ascension and by 1% and 3% for declination data. These stars are too bright for precision
Gaia observations.
Reid & Menten
TABLE 3
Parallax of ↵ Centauri.
Star Data Used Parallax ParallaxHenderson Re-analysis ↵ Cen Right Ascension . ®®
82 ± 0 . ®®
35 0 . ®®
80 ± 0 . ®® ... Declination . ®®
40 ± 0 . ®®
19 1 . ®®
37 ± 0 . ®® ... Combined . ®®
27 ± 0 . ®®
17 1 . ®®
25 ± 0 . ®® ↵ Cen Right Ascension . ®®
38 ± 0 . ®®
34 0 . ®®
47 ± 0 . ®® ... Declination . ®®
02 ± 0 . ®®
18 1 . ®®
00 ± 0 . ®® ... Combined . ®®
88 ± 0 . ®®
16 0 . ®®
94 ± 0 . ®® Both Combined . ®®
06 ± 0 . ®®
12 1 . ®®
09 ± 0 . ®® Comparison of parallax measurements of ↵ Cen by Thomas Henderson and our re-analysis of his data. Henderson’s values aregiven for the constant of aberration of . ®® and assume that proper motion had been removed from the data. For comparison,the modern parallax of ↵ Cen is approximately . ®® .stars ( a and b ). These two parallax estimates were in sometension and Bessel, noting the di erence, suggested the mea-surement with the smaller parallax (star b ) could be explainedif that reference star itself had a significant parallax. How-ever, the Gaia space mission has measured the parallax ofboth reference stars and found them to be quite small, dis-counting Bessel’s suggestion. Upon closer inspection of thedata, we find that star b has residuals that display systematictrends, which correlate well with his recorded temperature cor-rections. Doubling his temperature corrections removes theproblems seen in the systematic residuals. In 1840 with moredata, Bessel published improved parallaxes, which includedprescriptions for the e ects of temperature corrections onparallax estimates. With perfect hindsight, knowing the trueparallax of 61 Cygni to be . ®® , we verify the need for largetemperature corrections, which then yield excellent agreementbetween the parallaxes measured against each reference star.It has long been a mystery as to why von Struve claimeda parallax for Vega of . ®® , which is nearly identical to itstrue value, only later with more measurements to arrive at aparallax of . ®® . His early result comes from combining twodimensions of measurements: along and perpendicular to theposition angle of the reference star. These two measurementsets yield substantially di erent parallaxes, but the di erencesfrom the true parallax fortuitously cancel when a weightedaverage is performed. Von Struve later discarded the perpen-dicular data, only using the “more precise” distances betweenVega and the reference star for a parallax fit. As we found forBessel, this dataset also shows some residual systematics thatare likely from temperature-dependent e ects. Accounting forthese e ects yields a parallax value of . ®®
151 ± 0 . ®® , whichis statistically consistent with the true parallax of . ®® . We note that Bessel and von Struve also made significantcontributions to the study of wide binaries. Since these mea-surements are still used to provide a long time-baseline tocomplement current observations, our finding of temperaturesensitivities in their astrometric data might be of importancefor studies of long period binaries.In general, we can reproduce Henderson’s analyses for ↵ Centauri, although in some cases scaling the measurementerrors to achieve a ⌫ of unity results in larger parallax uncer-tainties. Accounting for this, we find a best-fit parallax of . ®®
09 ± 0 . ®® , which is only in mild tension with the true valueof . ®® .Remarkably, after a century-long quest for the first stel-lar parallax, within the brief period of just 3 years, threeastronomers met success and obtained results that have stoodthe test of time. ACKNOWLEDGMENTS
We would like to thank Arne Hoyer, who traced a paper copyof F. G. W. von Struve’s monograph
Mensurae micrometricae to the Bavarian State Library and to that fine institution for pro-viding a digital copy of this work’s chapter 14. We are thankfulto Jens Kau mann for his expert help with L A TEX formattingissues.
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Table 4 lists Wilhelm von Struve’s measurements of the sepa-ration (“distance”) between Vega and a reference star and theperpendicular o set (“direction”). The data were hand digi-tized from plots in a paper by Otto Struve in Sky & Telescope(1956). We subtracted 5.616 from the “distance” data andscaled them by 7.75 to convert to arcsecond o sets. For the“direction” data, originally listed as position angles, we sub-tracted . ˝ , converted to radians, and multiplied by theseparation between stars of . ®® . Reid & Menten
TABLE 4
Wilhelm Struve’s data for Vega.
Date Distance Direction Date Distance Direction ( ®® ) ( ®® ) ( ®® ) ( ®® )1835.841 –0.612 –0.308 1837.736 0.248 0.3451835.843 –0.186 –0.353 1837.739 0.054 0.0381835.873 –0.085 –0.278 1837.775 0.202 0.2701836.542 0.403 –0.443 1837.778 0.070 0.3151836.624 0.171 –0.210 1837.780 0.031 0.2251836.638 0.132 –0.263 1837.783 –0.023 0.1351836.682 –0.023 0.068 1837.789 0.248 ...1836.769 0.186 –0.038 1837.791 –0.008 0.0601836.778 0.209 –0.293 1837.821 0.132 0.1351836.797 0.085 –0.248 1837.824 0.031 0.1881836.799 0.264 0.015 1837.832 0.248 0.1951836.895 0.147 0.113 1837.947 –0.357 0.1281836.988 0.093 –0.315 1837.950 –0.124 0.3601836.991 –0.264 –0.173 1837.975 0.093 –0.1201836.994 –0.031 –0.210 1837.977 0.543 –0.1501836.997 –0.333 –0.563 1837.994 0.543 –0.2101837.002 –0.457 0.083 1837.997 0.403 –0.0451837.112 –0.822 –0.338 1837.999 0.651 0.0981837.115 –0.783 –0.128 1838.003 –0.140 0.2851837.176 –0.202 –0.173 1838.041 –0.178 0.2931837.178 –0.318 0.315 1838.044 –0.225 0.3451837.180 –0.085 –0.150 1838.060 –0.395 0.2331837.372 0.147 –0.218 1838.066 –0.147 0.4431837.378 ... 0.278 1838.068 0.016 0.4131837.383 ... –0.113 1838.071 0.109 0.2631837.386 0.031 0.315 1838.189 –0.473 0.0531837.392 ... 0.098 1838.192 –0.217 0.2251837.397 0.132 0.353 1838.194 –0.279 0.3601837.405 –0.194 0.653 1838.197 ... 0.6381837.408 –0.047 0.413 1838.326 ... 0.1881837.411 0.031 –0.060 1838.329 ... 0.6611837.454 –0.101 0.015 1838.331 0.093 0.3531837.463 –0.047 –0.038 1838.334 –0.070 0.7131837.471 –0.124 0.083 1838.337 –0.395 0.4881837.591 0.791 0.285 1838.340 –0.279 0.5181837.594 0.419 0.383 1838.342 –0.434 0.5931837.602 0.302 0.188 1838.345 –0.171 0.7361837.605 0.225 0.435 1838.370 0.070 0.3981837.608 0.171 0.240 1838.375 –0.333 0.6681837.624 0.496 0.038 1838.405 ... 0.2481837.627 0.419 0.180 1838.408 0.287 0.3001837.630 0.302 0.075 1838.413 –0.085 0.4201837.632 0.403 0.390 1838.416 –0.163 0.6081837.635 0.171 0.173 1838.419 –0.217 0.3301837.687 0.403 0.075 1838.422 ... 0.1881837.693 0.496 0.135 1838.630 0.264 0.3151837.698 0.264 0.165 1838.632 0.395 0.383Columns are (twice): year, "distance" and "direction" data for ↵↵