THE RADIUS VECTOR:
A ‘TRIVIAL’ QUESTION?

Following Adams’ abortive visit in late October, Airy wrote to Adams on November 5^th 1845 and put the question:

'I shall be very glad to know whether the assumed perturbation will explain the error of the radius vector of Uranus. This error is now very considerable .. ' and he referred Adams to ‘the Greenwich observations for each year.’ Why did Airy want to know this, and was his question merely ‘trivial’, as Adams in 1883 told Glaisher? Accounts have tended to endorse the latter view, by way of explaining why Adams never replied to Airy’s letter.

Each year the Nautical Almanac, published from Greenwich, gave predicted planetary positions, including the logarithm of the radius vector of Uranus (called, ‘The Georgian’) to eight-figure accuracy. This was given every day, in two columns, one heliocentric and the other geocentric: five thousand digits a year, from Greenwich about Uranus’ distance. As the value only changes slowly through the years, the purpose of giving daily computations remains opaque. One is amazed by such useless labour, baffled by the spurious orders of magnitude of these figures, and perplexed as to how the daily distance values were obtained. Later, Astronomical Observations made at the Royal Observatory noted the daily position of Uranus, together with (rather enigmatically) Airy’s computed ‘Error of tables’ indicating the perceived anomaly. Adams was being asked to comment on these observations.

In 1836, Airy had published in the Astronomische Nachrichten, a calculation showing how Uranus was then further away than it should have been, according the Bouvard’s tables. (1838, Vol.15, 217-220, using observations 1833-36). It was, Airy had found, too far by about five-thousandths of an AU, which is quite a small amount, and he believed that to be significant. Over the course of a year one obtained a value of this distance, by a parallax method which compared Uranus’ geocentric longitude on the two occasions when Earth and Uranus were in square to the Sun. Adams, wanting to engage in discussion with Airy, would be familiar with this publication, as well as the high orders of magnitude to which Airy’s workers were, every day of every year, citing Uranus’ radius-distance.

Adams notebooks show him continuing to grapple with this issue after Airy put the question, until finally on September 1^st he obtained a value similar to that published by Airy, for the 1830s. This concordance with Airy’s empirically-found data was stated both in his September 2^nd letter and at his RAS presentation.

Airy always viewed the question he had put as crucial:

‘I therefore considered that the trial, whether the error of the radius vector would be explained by the same theory which explained the longitude, would be truly an experimentum cruces,’

he explained to the RAS on November 13^th. His view was however mistaken, in the view of most commentators, eg Turner (1904):

‘The ‘error of the radius vector’ came before Airy in an entirely independent way, and as  an entirely independent phenomenon, from the ‘error of longitude’, and there was nothing unnatural in regarding it as requiring independent explanation. It is true that, as the event proved, a mere readjustment of the orbit of Uranus got rid of this error of the radius vector (this was substantially Le Verrier’s answer to Airy’s question). (Astronomical Discovery, p.58)

Uranus was in these years approaching its apogee, so that its distance or ‘radius vector’ was continually increasing. Uranus lagging behind its predicted position would thereby diminish its radius vector, whereas the reverse viz. an increase was what required explaining. The theory involved was succinctly explained in Airy’s textbook Gravitation (1834). Airy there described how, during a planetary conjunction, as between Jupiter/Saturn or Uranus/Neptune, the outer planet pulls upon the inner one:

‘The effect then of a force in the direction of a planet’s motion, which increases the planet’s velocity, is to increase the size of its orbit.’

The inner planet (Uranus) has its orbit increased due to the transfer of angular momentum, as increases its potential energy, and this process is then reversed after the conjunction. Airy thus saw the problem in terms of Newtonian physics. (The perturbation-effect worked mainly twenty years or so either side of the Uranus-Neptune conjunction, say 1800-1840. The calculations of Adams and Leverrier only became feasible in the aftermath of this event, they could not have been done earlier.) Adams’ commented to Airy (November 18^th, 1846) as to why he had not thought the matter was of much importance. Airy in his livid (and long-censored) letter of December 8^th to Sedgwick merely said:

‘I do not enter into any details about Adams’ notion that the examination of the effect of the radius vector was unimportant. It now suffices for my guidance that I thought it important and still think so. Perhaps it might be sufficient for your persuasion, to tell you that Leverrier also thought it important.’

The issue is far from resolved, and a reconstruction using modern equations would certainly help.