
'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 Dday, i.e. September 23rd 1846. (For telescope work one used
geocentric coordinates, but calculations are done heliocentrically).
It seems that the perturbationequations 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 ‘Dday’
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 NeptuneDiscovery Date}
True
helio long. Of Neptune at DDay (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
Dday: 
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 Dday 
1°
28’ 
0°
2’ 
0°
24’ 
0°
27’ 
MHL at
Dday 
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
severalfold, 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 perturbationequations to a higher order of accuracy
and obtained a final error in celestial longitude of merely sixteen
arcminutes, as compared with the fiftytwo 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 resonanceinstability
effect appeared, due to a 2:5 ratio between their orbitperiods,
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 perturbationmaths 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 perturbationmath: Adams computations
were reconstructed by C. Brookes (Celestial Mechanics, 1970, 3,
6780) and I asked him what maths he had to learn to do this and
he replied: that of Laplace, who designed the perturbationtheory,
and of Pontecoulant who wrote the textbook. There was nothing in
English to read, he added. Adams’ notes often allude to Pontecoulant.
^{Coordinates Five different
coordinate systems were used in the Neptunepredictions.}
^{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
Coordinates }
^{(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 starmap, where one ‘Hour’
equals fifteen degrees of Right Ascension. The Berlin starmaps
had these coordinates.} 
^{ The
Original Bremiker starmap 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 TwentyDegree Swing
Adams’ preferred solutions swung over a twentydegree
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 orbitradius. We have
seen above how the perturbationmath goes awry when this happens,
due to a resonanceinstability that neither Adams nor Leverrier
were in a position to apprehend. Adams decides that he prefers a
mean orbitradius 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 Neptuneprediction, 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
skysearch ...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 closetoDday 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
meandistance 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
^{ }
