'WITHIN ONE DEGREE'

Within two months of the discovery, Challis, Airy and James Glaisher, three of Britain’s top astronomers, had publicly committed themselves to the radically unsound view that Adams and Leverrier’s predictions agreed ‘within one degree.’ Nothing has more skewed the debate than this claim: they differed by about four degrees. As Challis was scouring the sky at nine arcminute intervals, this difference was substantial. The only thing within one degree was Leverrier’s prediction.

Following the example of John Herschel in his Outlines of Astronomy (1852), we compare predictions by expressing them as true heliocentric longitude at D-day, i.e. September 23rd 1846. (For telescope work one used geocentric co-ordinates, but calculations are done heliocentrically). It seems that the perturbation-equations that Adams used produced initially mean helio longitude (uniformly circular motion), for a given epoch: Adams used the date of Oct 1st 1845, while Leverrier computed for Jan 1st, 1847. We must first convert these to ‘D-day’ using a suitable value for mean motion, according to the distance assumed, then apply the Equation of Centre to get true helio longitude. (NB, four terms are required in this expansion on account of the large eccentricities involved: Solar System Dynamics Murray & Dermott, 1999, p.41)

Accuracy of Four Historic Solutions Compared, at the Neptune-Discovery Date

 True helio long. Of Neptune at D-Day (23.9.46) was 326° 57’ Adams 1 Adams 2 Leverrrier 1 Leverrier 2 Mean Helio Long 323° 34’ 323° 2’ 325° 318° 47’ Epoch 1st Oct ’45 1st Oct ’46 1st Jan ’47 1st Jan ‘47 Days to D-day: 357 -8 -100 -100 Distance AU 38.4 37.3 38.4 36.2 Orbit period (Yrs) 237.6 227.4 237.6 217.4 Motion to D-day 1° 28’ -0° 2’ -0° 24’ -0° 27’ MHL at D-day 325° 2’ 322° 59’ 324° 35’ 318° 19’ Perihelion 315° 55’ 299° 11’ 284° 45’ Anomaly 9° 7’ 23° 48’ 33° 34’ Eccentricity 0.161 0.121 0 0.108 Eqn. of Centre 3° 38’ 6° 27’ 0 7° 39’ True Helio Long. 328° 41’ 329° 27’ 324° 35’ 325° 58’ Error 1° 44’ 2° 30’ -2° 21’ -0° 58’ For comparison: Herschel (1852) 328° 42’ 329° 25’ 326° 0

Leverrier’s accuracy increased several-fold, between his first and second prediction, whereas Adams decreased his between his Hyp I (supposedly, October 1845) and Hyp. II (Sept 2nd 1846). The Table has a line giving ‘error’ in degrees and minutes. A reconstruction of Leverrier’s computations was performed by Baghdady (1980) in a PhD, who solved Leverrier’s Lagrangian perturbation-equations to a higher order of accuracy and obtained a final error in celestial longitude of merely sixteen arcminutes, as compared with the fifty-two arcminutes error of the historical prediction - sterling evidence that Leverrier’s prediction really was the brilliant feat of deductive logic which he believed it was, and not just a coincidence as is sometimes alleged (The computations by Perceval Lowell as led to the discovery of Pluto, were of this latter kind).

Distance Estimates

Adams’ Hyp II planet was at 32.8 AU in 1830 while Leverrier’s was then at 32.2 AU. The Table elucidates this remarkable synchrony:

 Mean AU P’helion AU Discovery AU Eccy. Leverrrier final 36.2 32.3 33 0.11 Adams Hyp I 38 31.9 32 0.16 Adams Hyp II 36.9 32.5 32.9 0.12

Neither could decrease their mean radius any further, because at around 35.3 AU a resonance-instability effect appeared, due to a 2:5 ratio between their orbit-periods, unknown to either of them, as caused a discontinuity making their computations go haywire. The US astronomer Benjamin Peirce of Harvard pointed out this strange instability in the maths: ‘The distance of 35.3 then is a complete barrier to any logical deduction, and the investigations with regard to the outer space cannot be extended to the interior’(p.443 of Amer. Jnl Sci & Arts 1847; Hubbell & Smith JHA 1992 p.271).But, the perturbation-maths tended to work better at a closer orbit, as Neptune is at 30 AU, defying ‘Bode’s Law’. Both mathematicians circumvented this problem by using a hugely exaggerated eccentricity with the planet near to its perihelion, thereby getting it much nearer than its mean radius.

They both used French theory of perturbation-math: Adams computations were reconstructed by C. Brookes (Celestial Mechanics, 1970, 3, 67-80) and I asked him what maths he had to learn to do this and he replied: that of Laplace, who designed the perturbation-theory, and of Pontecoulant who wrote the textbook. There was nothing in English to read, he added. Adams’ notes often allude to Pontecoulant.

Co-ordinates Five different co-ordinate systems were used in the Neptune-predictions.

 Mean Heliocentric Longitude Adams’ solutions were in this form. It is not the position in the sky, but is that of a fictitious ‘mean’ sphere having a uniform circular orbit. True Helio Longitude Leverrier’s August prediction was in this form. A circular orbit has its mean and true longitudes identical. This is the essential value for comparing accuracies. True and mean longitudes coincide twice per orbit, at aphelion and perihelion. Geocentric longitude A triangle between Earth, sun and Uranus is used to convert from helio- to geo- Equatorial Co-ordinates (RA and Declination) These are also geocentric, and are what the astronomer requires for pointing his telescope. ‘Adams’ July Ephemeris’ converted helio longitudes into this form. Greenwich Hour Angle These are required for looking at a star-map, where one ‘Hour’ equals fifteen degrees of Right Ascension. The Berlin star-maps had these co-ordinates.

 The Original Bremiker star-map from which Galle and D'Arrest found Neptune. The two positions marked are where leverrier predicted it (drawn in on the right) would be and where Neptune was found (between 52' and 48'). The distance between these positions appears on the map as four 'minutes' of the Greenwich Hour angle, which is equivalent to one degree (see 'Mapless in Cambridge').

A Twenty-Degree Swing

Adams’ preferred solutions swung over a twenty-degree range from July to September of 1846 - no wonder Challis was unsure where to point his telescope! We saw above how the July Ephemeris which he constructed for Challis advocated 336º (see Adams’ July Ephemeris). Adams’ letter of September 2nd to Airy contained parameters for his ‘Hyp I’ and ‘Hyp II’ versions, but went on to give his preferred value, always omitted in tellings of the story. His preferred value came from a further shortening of the mean orbit-radius. We have seen above how the perturbation-math goes awry when this happens, due to a resonance-instability that neither Adams nor Leverrier were in a position to apprehend. Adams decides that he prefers a mean orbit-radius of 33.6 AU. His mode of expression is a little opaque. From considering the most recent ‘errors’ i.e. perturbations of Uranus,
‘it may be inferred that the agreement of theory and observation, would be rendered very close by assuming a/a1 = 0.57, and the corresponding mean longitude on the 1st October, 1846, would be 315 20’, which I am inclined to think is not far from the truth. It is plain also that the eccentricity corresponding to this value a/a1 would be very small.’
He’s out here by eleven degrees.

One can imagine the terrible temptation to which Airy became subject. Looking at this letter - the only one Adams ever sent making a Neptune-prediction, before its discovery - after the great debacle, with Challis having spent six weeks not finding a planet which the Germans found in half an hour, and Airy’s own reputation at stake because he had had set up the clandestine sky-search ...if only, he mused, one were to ignore the last part of this letter, stating Adams third and preferred option ... Would people really notice? Could he stand up before his colleagues and brazen it out? He decided to take the risk, which proved well justified: nobody ever did notice.

Adams’ a/a1 signifies the helio distance of Uranus, divided by that of the unknown planet. Nowhere in his published work does he give a value of the Uranus mean radius, but taking a likely value will give us 33.6 AU as his final and preferred radius of the unknown planet. The three mean helio longitudes specified in this letter, for his close-to-D-day epoch are: 325, 323 and 315 degrees. As we’ve seen, the first two come out at 329 degrees true helio after applying the Equation of Center. Here however the eccentricity has become ‘very small.’ That’s all we are told, so let’s add on one degree for the Equation of centre. That gives us around 316 degrees, which errs by eleven degrees.

After the discovery, Adams wrote uneasily to Airy about this blunder: ‘In my letter of September 2nd, I inferred that the mean distance used in my first hypothesis must be greatly diminished, but I rather hastily concluded that the change in the mean Long. deduced would be nearly proportional to the change in the assumed mean distance.’ (15th October ’46). In his RAS presentation, Adams managed to give himself credit for his shrewd mean-distance reduction, but without mentioning the large error into which it led him.

Herschel's Portrait courtesy Royal Astronomical Society
LeVerrier pictures by permission of the Observatoire de Paris Archives