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After Uranus was discovered and its orbit calculated, its future orbit was calculated, and its future positions as seen from Earth were calculated.

And observers of Uranus began to notice that Uranus was deviating from its calculated positions. Eventually the possibility that an undiscovered planet was perturbing the orbit of Uranus was used by Adams and Le Verrier to calculate the orbit of Neptune and led to the discovery of Neptune.

So how large in angles and/or in miles/kilometers were the deviations in the apparent position of Uranus that were used to calculate the orbit and apparent position of Neptune?

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Starting off with a crude order-of-magnitude estimate, the anomalous acceleration of Uranus ($m_1$) due to Neptune ($m_2$) should be on the order of $Gm_1m_2/a^2$, where $a$, a scale for the distance between the two planets, can be taken to equal the radius of Uranus's orbit. The secular trend probably vanishes, so let's take the time of interaction to be on the order of Uranus's orbital period, which is given by Kepler's laws. Then ignoring universal factors of order unity, the displacement of Uranus should be $\Delta x\sim (m_2/m_S)a$, where $m_S$ is the mass of the sun. This results in an angular displacement $\Delta\theta\sim\Delta x/a\sim m_2/m_S\sim 10''$.

In terms of actual observation, here is a graph of the residuals from Danjon, 1946:

graph of residual error in observations of Uranus

The simple order-of-magnitude estimate agrees reasonably well with the observations. Presumably people at the time did some kind of estimate similar to this in order to estimate $m_2$, and the fact that it was on the right order of magnitude for a planet encouraged them to hypothesize a new planet.

The relative orbital period $1/(1/T_1-1/T_2)$ is 170 years, which seems to equal, to within a factor 2, the time-scale for the observed oscillations in the residuals. This could have been estimated from Bode's law, which would further support the hypothesis of a new planet.

What seems to be a much more difficult problem is to estimate the new planet's orbital elements and the uncertainty of those elements, hence the controversy between supporters of Adams and Le Verrier.

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  • $\begingroup$ This is a great answer... even if more in the spirit of physics SE $\endgroup$ – Martin C. Jun 24 at 6:21
  • $\begingroup$ For the "purists," 10 arc-sec $\approxeq$ 48.5 microradians $\endgroup$ – Carl Witthoft Jun 24 at 11:41
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This topic is extensively discussed by astronomer, historian, and Marxist theorist Anton Pannekoek in his 1961 book A History of Astronomy on pp. 359-363 and his 1953 article "The Discovery of Neptune".

As he explains, the deviation in the calculated orbit and the observed orbit was 30" in 1835 and 70" by 1840.

LeVerrier and Adams both used these calculations and observations to predict an orbit of the hypothetical planet, but it is important to remember that these predictions contained many unknown variables that depended on the other variables, like mass of the hypothetical planet, its average distance from the sun and its eccentricity. To nevertheless compute an orbit, they used the at-the-time well-known Titius-Bode law that stated that the orbits of planets in AU followed the equation $$a = 0.4 + 0.3 \times 2^m$$ with m being -∞ for Mercury, 0 for Venus, 1 for Earth, 2 for Mars, 3 for Ceres, Pallas, Juno, and Vesta (which were considered planets), 4 for Mars, 5 for Jupiter, 6 for Saturn, and 7 for Uranus. It would therefore make sense that the new planet should have a semimajor axis of 38.8 AU (m=8).

Following LeVerrier's predictions, Galle of the Berlin Observatory indeed found a unmarked star in the region where LeVerrier had predicted the new planet would be. After tracking the star for months, however, astronomers quickly realized that the star was a planet but that it followed a different orbit than predicted. It turned to be smaller and nearer than the calculations predicted. Pannekoek illustrated this with a diagram:enter image description here

This realization that the orbit was completely different than predicted led to a whole new discussion on whether LeVerrier and Adams had indeed discovered the planet, or that this was a new planet was a different object altogether.

Pannekoek goes into this discussion from a Marxist perspective and attempts to explain it by invoking the social circumstances of the various astronomers involved. France is in the middle of the 1848 revolutions and the bourgeoisie want to illustrate the power of the natural sciences against received wisdom, which explains why LeVerrier is so adamant at prompting Galle to find the new planet and why the French resist the argument that the calculated and the observed planets are different objects. Britain, meanwhile, already had a revolution in the seventeenth century and has the bourgeoisie already entrenched in government, which explains why Adams was only interested in theoretical astronomy and not so much the observational verification. The United States on the other hand had started as a bourgeois and democratic society where debate was important, so it made sense that American astronomers would challenge the idea that the calculated and observed planets are the same.

For discussions on Pannekoek's interpretations, see:

Robert W. Smith, "The Cambridge network in action: The discovery of Neptune", Isis: A Journal of the History of Science 80 (1989): 395-422

John G. Hubbell and Robert W. Smith, "Neptune in America: Negotiating a Discovery", Journal for the History of Astronomy 23 (1992): 261-291

Bart Karstens, "Anton Pannekoek as a Pioneer in the Sociology of Knowledge" in: Chaokang Tai, Bart van der Steen, and Jeroen van Dongen (eds.), Anton Pannekoek: Ways of Viewing Science and Society (Amsterdam University Press, 2019): 197-218

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It is possible that some persons here might not understand the scale of deviations in the answers, So I decided to post an answer myself using the data from the two answers so far to explain the scale a little bit more.

The answer by Ben Crowell makes a rough calculation that the displacement should be on the order of ten arc seconds. It also has a plot of the variation in longitude of the observed and predicted positions of Uranus over time, which seem to differ by up to a few tens of arc seconds from predicted positions.

The answer by cktai states that according to his source the deviation between the calculated and the observed orbit was 30 arc seconds in 1835 and 70 arc seconds by 1840.

An arc second is one in 1,296,000, or 0.0000007, of a full circle.

At the average distance between the Sun and Uranus, about 19.22 Astronomical units or AU, a full circle is about 120.76271 AU. Since the orbital period of Uranus is 84.01 years, Uranus travels at an orbital speed of about 1.4374801 AU per year, or about 0.0039356 AU per day, or about 0.0001639 AU per hour, or about 0.0000027 AU per minute, or about 0.000000045 AU per second.

0.000000045 AU is 6.81433 kilometers, so the average orbital speed of Uranus should be about 6.81433 kilometers per second. The average orbital speed of Uranus is listed as 6.8 kilometers per second.

Earth is at an average distance of 1.0 Astronomical Units from the Sun, so the distance between Earth and Uranus varies between about 17.33 and 21.11 AU, and the average distance is about 19.22 AU.

So since there are 360 degrees in a full circle, at the average distance between Earth and Uranus a degree of arc would be about 0.33545 AU, and an arc minute would be about 0.00559 AU, and an arc second would be about 0.00009318 AU or 13,939.61 kilometers.

Being off position by 30 arc seconds would equal being off position by 418,188.38 kilometers, and being off position by 70 arc seconds would equal being off position by 975,772.88 kilometers. And that would be approximately the distance traveled by Uranus in about a day.

And I guess that gives some idea of the scale of the deviations which were considered a problem and which led to the discovery of Neptune.

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