Les Comptes Rendus de l'Academie des Sciences (C.R.A.S.), 1846, Semester 1, Vol. 22, No. 22, 907-918
ASTRONOMY
Research on the movements of Uranus
by Mr. U.-J. Le Verrier
[Translated by V.C. Tsai with help from Z. Duputel]
I propose, in this memoire of which I have the honor to submit an extract to the Academy, to
study the nature of the irregularities of the movement of Uranus, to go back to their cause,
seeking to discover, from the path that is affected, the direction and magnitude of the force from
which it is produced.
The theory of Uranus concerns astronomers today. It gave rise to much speculation more or less
plausible, but which, devoid of any geometrical consideration, could have no real value. Several
Societies have even proposed this theory as a subject of competition. So I think, because of the
importance of the question, I should quickly summarize its history: so that the Academy may
better judge the purpose of my work, of the path I traveled and the results to which I arrived.
We had, in 1820, forty years of regular meridian observations of Uranus. The planet had also
been observed seventeen times from 1690 until 1771 by Flamsteed, Bradley, Mayer and
Lemonnier. These astronomers had recorded [Uranus] as a star of sixth magnitude. On the other
hand, the analytical expressions for the perturbations that Jupiter and Saturn have on Uranus
were developed in Volume III of the "Celestial Mechanics". It was hoped that by using all these
data, we could successfully build accurate Tables of the motion of the planet, which is what Mr.
Bouvard, member of the Academy of Sciences, tried to accomplish. However, he encountered
unexpected difficulties.
When one bases Tables of a planet on a small number of observations, it may happen that these
Tables, in the course of time, cannot predict the exact positions of the celestial object or, at least,
that the predicted positions can only be known as rigorously as the observations used: you can
even say that it is much easier to satisfy [the Tables], that which employs fewer [observations]. It
was not so in the construction of tables of Uranus. There was an inability to represent both 17
ancient observations and many modern observations. In this embarrassing situation, the learned
academician [Mr. Bouvard] cast doubts on the accuracy of past observations; he completely
dismissed and had respect solely for modern observations. But we must say that even if the
observations of Flamsteed, Bradley, Mayer and Lemonnier are not as accurate as those of the
astronomers of our time, we cannot, in all likelihood, regard them as highly erroneous as accused
by the current Tables [of Bouvard]. The author of these Tables even indicated that this was his
opinion, since he added, reporting the difficulties he had encountered:
"This is the alternative that is provided in [my] Tables of planet Uranus, since if the old
observations are combined with modern, the first will be fairly represented, while the latter will
not be with the precision that they contain, and if one rejects [the old] to keep only the other
[new ones], the resulting Tables that have any relation to the desirable accuracy of the modern
observations cannot adequately satisfy the old observations. One must decide between these two
options; I choose the second, as the one that more likely meets the truth, and time will tell if this
is the right choice: if the difficulty of reconciling the two systems actually stems from the
inaccuracy of old observations, or if it depends on any foreign and unnoticed disturbance acting
upon the planet."
Twenty five years since then, we have learned that [the predictions of] the current Tables, which
do not represent the old places, agree no better with the observed positions in 1845. Should we
attribute this disagreement to the theory not being precise enough? Or that this theory has not
been compared with observations with sufficient accuracy, in the work that has formed the basis
of the existing Tables? Finally, could it be that Uranus was subject to influences other than those
arising from actions of the Sun, Jupiter and Saturn? And, in this case, may we, by a careful study
of the disturbed motion of the planet, successfully determine the cause of these unexpected
inequalities? Could we come to fix the point in the sky where the observational astronomers
should recognize the foreign body, the source of so much trouble?
These are the questions that the history of Uranus raises today. It must be said that there had
been no satisfactory answer to any of them, when last year I undertook to probe carefully all
points of this theory to illuminate the details as far as the principle features of the attraction of
matter.
I have made known to the Academy, in the meeting of 10 November 1845, the result of the first
part of my research. I have proven this time that which had previously been neglected in
calculating the interference caused by Jupiter and Saturn, the many very significant terms, the
omission which would result in the impossibility of infallibly representing exactly the motion of
Uranus. So, previously, we had the impossibility of knowing whether to believe, whether [the
discrepancy] was real or if it was apparent.
I had to ask myself if these corrections, brought into the current Tables [of Bouvard] would
remove the huge mistakes that affect them. Taking into account this purpose, alterations that
disturbances produce that had been neglected in the orbital elements, I recognized that even if the
discrepancies of the Tables, in 1845, actually significantly decreased by the use of [my] new
expressions, they were still very considerable and higher than the errors of the observations. The
consequence thereby would have been very clear if I could count, in an absolute manner, the
accuracy of the path that had been followed in the construction of Tables published in 1821. I
could say, in the month of November, that there was a need to look elsewhere in the imperfection
in the orbital elements for the cause of the strange inequality of Uranus. Unfortunately,
examining with great attention the preamble to [Bouvard's] very concise tables of Uranus, I
discovered several causes of errors, which it was impossible to assess with accuracy, and without
correction we could not draw any immediate and accurate inference regarding the tables
themselves.
Without wishing to dwell on this point, I must, however, point out briefly some of the mistakes,
the existence of which have a great influence on our way forward.
The coefficients of the equations of condition are given in four significant digits. However, of
these four figures, three are mostly inaccurate.
Secondly, the author calculated the quadratures as oppositions, without considering the possible
error of the radius vector. If there had been an inaccuracy in this, it ought to be corrected by a
change in the heliocentric longitude.
I omit several other causes of uncertainty. Those that I have just reported are sufficient, indeed,
to stop us from making conclusions based on the immediate use of the existing Tables. We lack
completely data that would be needed to assess the ultimate influence of these errors. Is this
influence comparable in size to the variances of the Tables? It is impossible to judge, and we
understand that one would need to follow those [errors] back to provide new foundations in its
entirety, to make a comparison of the theory with observations. That is what I will do now.
Due to the importance of the topic, I made it a rule to review everything, and check it all myself.
With regard to the former observations, I again reduced those of Flamsteed, Bradley, Mayer and
Lemonnier, and among the new, I chose two hundred and sixty two, made mainly at times of
oppositions and quadratures. For the first twenty years, from 1781 until 1800, I used the
publications of the Observatory of Greenwich. Observations published by the Observatory of
Paris, in the Connaissance des Temps and in two folio volumes, served me from 1801 until 1828.
In 1829 and 1830, I took English observations. Finally from 1835 until 1845, I was able to enjoy
the new series of, as yet unpublished, excellent observations made in Paris, which M. Arago
kindly provided to me.
Then, based on the orbital elements of Uranus, already known with great approximation, I
calculated the heliocentric position of the planet at the times corresponding to the observations,
and I added the expressions of disturbances as a result of part of my work. Heliocentric positions
thus obtained, and combined with the places of the Sun, I derived the most accurate tables, which
gave me the geocentric positions of the planet. Finally, subtracting the calculated coordinates
observed, I obtained variations that affect the theory with respect to the observations, when
adopting elliptic elements in use and when it is assumed that the planet, obeying the main action
of the Sun, is not subject to secondary forces other than those resulting from actions of known
planets. Recognize that this assumption is correct: since the disturbances produced by the planets
were established with accuracy, the deviations of the theory with respect to the observations can
only come from the errors of the orbital elements taken as a starting point, and that in suitably
modifying these elements, we will bring the calculated positions to have a lower error compared
with the observations. So by examining whether it is possible to eliminate errors by theoretical
changes in the orbital elements, and seeking to give our conclusion rigorous geometric
demonstration, we can know definitively whether Uranus obeys only the actions of the Sun and
other [known] planets.
Take four exact longitudes of the planet, the determination of each of which we have made with
several concordant observations, and calculate the orbital elements so that they comply strictly
with these four longitudes then compare the determined positions using these elements, with the
series of observations we have, and carefully examine the causes which may delay the result of
the calculation, relative to the observation. The [causes] are three in number, namely: (1) error
characteristic of the new comparative observation; (2) uncertainty that can affect the calculated
result from the mistakes of longitudes which served as the basis for determining the orbital
elements position; (3) finally the theoretical error due to the planet actually obeying some
unknown secondary force. If we can prove that the first two causes are not sufficient to explain
the difference between the calculation and observation, we are forced to admit serious influence
of the third. I apply this method of reasoning to the issue before us.
The orbital elements determined by four longitudes, taken at times very distant from each other,
leaving in 1838 for example, results in 124.98" error in the theory. The three [above] possibilities
could explain this difference; the first, which is due to the error in location observed, can be
considered insensible, as the position was deduced from several meridian observations consistent
with each other: it cannot inspire the slightest doubt. The second part of the total error is more
difficult to estimate, as it is necessary to obtain the expressions of the four fundamental changes
in longitudes introduced by the orbital elements, and then deduce the corresponding corrected
positions calculated using the elements. Suppose that fundamental errors influence all longitudes
in the same direction as the error of longitude in 1838; suppose further that each of these
fundamental longitudes is as erroneous as the uncertainties included in the observations; despite
the unlikely accumulation errors, we will not be able to explain, in this case, more than 30" of the
125" error, found in 1838. The rest, that is to say nearly one hundred seconds, will of necessity
be attributed to the third question, a hitherto unknown foreign influence, acting on Uranus. What
we are saying of Uranus' position in 1838 also applies to the position of the planet at other times.
In 1831, for example, the calculated location away from the observed location of 188", about
140" cannot be explained if one does not admit another influence than the Sun, Jupiter and
Saturn.
To clearly establish the meaning of the result which I have just reached, I ask permission to
emphasize two points. I relied on accurate formulas, an advantage over my predecessors who did
not start with a deeper theory; this negligence would always make suspect the accuracy of their
conclusions. It should be noted, secondly, that I am not merely trying combinations of more or
fewer equations, to declare that I had failed to represent the motion of the planet; in this case, we
would not have failed to object that I might have missed the true combination, that another could
happily discover. We would be found in the same uncertainty as before: but this is not the path
that I followed. I demonstrated, if I am not mistaken, that there was inconsistency between the
formal observations of Uranus and the assumption that this planet would not be subject to the
actions of the Sun and other planets, acting in accordance with the principles of universal
gravitation. We will never succeed in this case, to represent the observed motions.
No sooner had we started, a few years ago, to suspect that the motion of Uranus was modified by
some unknown cause, already all possible hypotheses were hazarded on the nature of the cause.
Everyone, it is true, simply followed the bent of his imagination, without giving any
consideration to support his assertion. We thought of the resistance of the ether; a large satellite
to accompany Uranus; or a yet unknown planet; or the disruptive force should be taken into
consideration; and some even went so far as to assume that at this enormous distance from the
Sun, the law of gravitation could lose something of its strength. Finally, a comet would she not
have disturbed Uranus suddenly through her march?
I repeat, all these opinions were issued in the form of hypotheses, and without having sought to
substantiate any of them with positive considerations. One should not be surprised. The problem
of the movement of Uranus had not been treated with such rigor; it was demonstrated that we
could not manage to solve it by the consideration of the forces currently known. In this
uncertainty, it was without doubt allowable to hazard a guess. But no one could bring themselves
to undertake considerable work on inconsistencies whose existence was still problematic. Today
it is quite different. We can no longer doubt these inconsistencies, and the time has come to try to
disentangle the direction and magnitude of the force that produces them.
I do not conceal from myself the pitfalls which are sown on the road I will go now. More than
once, unexpected obstacles would have made me give up my business if I had not had the deep
conviction of its usefulness. How, indeed, will observational astronomers discover in the vast
expanse of the sky, the physical cause of the disturbances of Uranus, if we are unable to stake
their work to a small enclosed part of the sky? And who of them would resolve to seek a
telescopic object successively in the twelve signs of the zodiac? So we must begin by proving
that the research should be concentrated in a small number of degrees. We can then expect that
the vigil of the observers will not fail, and that before long, physical astronomy will be enriched
with the celestial object which theoretical astronomy will have revealed the existence and fixed
position of.
I will not dwell on the idea that the laws of gravitation could cease to be rigorous, to the great
distance that Uranus is located from the Sun. This is not the first time we have had to explain
inconsistencies that do not understand, and that we have attacked the principle of universal
gravitation. But we also know that these [previous] assumptions have been wiped out by a closer
examination of the facts. The alteration of the laws of gravitation would be a last resort to which
one could be allowed to use only after having exhausted the review of other causes, after having
recognized these to be powerless to produce the observed effects.
I cannot believe more in the resistance of the ether, resistance which we barely glimpsed traces
of in the movement of lower density bodies, that is to say in the circumstances that would be
most likely to demonstrate the action of the fluid.
Could these specific inequalities of Uranus be due to a large satellite accompanying the planet?
Oscillations that would occur in the motion of Uranus would affect a very short period, and it is
precisely the opposite result of the observations. The inconsistencies that concern us develop
very very slowly. It is therefore impossible to use the current assumption, especially as the
satellite should actually be very large, and could not escape observers.
Could it be a comet falling on Uranus, at one time, that suddenly changed the magnitude and
direction of its movement? I have already said we pretty much satisfactorily predicted the
movement of the planet between 1781 and 1820, without the aid of any extraordinary strength.
This remark, which seems to prove that the disturbing force has not exercised significant
influence during this period, would be quite consistent with the current hypothesis of a sudden
alteration of the motion of the planet. But then the period 1781 to 1820 could be bound naturally
to a series of previous observations, or the series of subsequent observations, and would be
inconsistent with one of them. Now this did not happen. We can prove that the intermediate
series cannot agree on the one hand, with older observations, and on the other, with the new.
There thus remains another hypothesis to try, that of a body acting in a continuous manner on
Uranus, changing its movement in a very slow manner. This body, according to what we know
of the constitution of our solar system, could only be a planet, so far ignored. But is this
hypothesis more plausible than the previous? Is it inconsistent with the inconsistencies? Is it
possible to assign the place that should occupy this planet in the sky?
First of all, it cannot be placed below Saturn, as it does bother Uranus, and we know that its
influence on Saturn is insensitive.
Can we assume it between Saturn and Uranus? It would require placement inherently much
closer to the orbit of Uranus than that of Saturn, and its mass should be small enough to only
produce on Uranus disturbances which are ultimately small. It is easy to conclude that the
interfering action will be exercised only when it happens to be in the vicinity of Uranus, and the
little difference there would be between the periods of revolution of two stars in this
circumstance would not have been encountered only once in the period covered by the
observations of the planet. This result is contrary to what is inferred from observations.
The disturbing planet will [therefore] be located beyond Uranus. We should not assume that it is
close, because then its mass would be very small, and thus would fall back into the same
impossibilities as before. It will be far beyond Uranus, that we can hope to discover this new
body whose mass is considerable. We know from the mean distances of the [known] planets to
the Sun, that the most distant planets are located at distances from the center which are very
nearly double of each other; it [therefore] would be natural to assume that the new body is twice
as far from the Sun as Uranus, even if the following considerations do not make it certain. I said
that the planet sought could not be located within a short distance of Uranus. However, it is not
possible to place it at a great distance, such as triple that of the Uranus Sun distance since, in this
case, it would give this planet a very considerable mass and its great distance from both Saturn
and Uranus would make its actions on these two planets comparable, and it would not be
possible to explain inconsistencies in Uranus without Saturn also developing highly sensitive
disturbances, and of which there is no trace.
Adding to this that the orbits of Jupiter, Saturn and Uranus are little inclined to the ecliptic, it can
be assumed, in a first approximation, that it is the same for the planet sought; observations of the
latitudes of Uranus prove without replica, since these latitudes have little other sensitivity than
those due to the actions of Jupiter and Saturn. We are thus led to ask the following question:
"Is it possible that the inconsistencies of Uranus are due to the action of a planet located in the
ecliptic at an average distance twice that of Uranus? And if so, where is this planet currently
located and what is its mass, and what are the elements of the orbit it travels?"
The problem stated in these terms is that which I rigorously solve.
If we could determine, for each time period, the variation in disturbances due to the action of the
unknown mass, one could deduce the direction in which Uranus falls, owing to the incessant
action of the perturbing body: thus we would know the position of this body. But the problem is
presently far from being that simple. Numerical expressions for the perturbations could be
determined immediately if rigorous values were known for the orbital elements described by
Uranus around the sun, but these, in turn, we cannot determine exactly if we do not know the
amount of perturbation. Seen this way, it is impossible to split into two distinct parts research of
Uranus' elements and the elements of the body that perturbs it. We could hope in vain, forming
empirical equations, to find, a priori, the law of disturbances; the risk of error would be high,
since we would thus have obtained a proper expression to represent the excess disturbances on
errors from inaccuracies of the orbital elements, and not the disturbances themselves. There is
only one way forward: we must form expressions for the disturbances due to the new body, as
functions of its mass and unknown orbital elements it describes, and we will introduce these
disturbances in Uranus' coordinates calculated using the unknown orbital elements of this planet
that travels around the Sun. Matching the coordinates obtained with the observed coordinates, we
will solve for the unknowns in the equations that will result, not only the orbital elements
described by Uranus, but the orbital elements described by the disturbing planet, for which we
seek a position.
We can rigorously eliminate the orbital elements of Uranus, thus obtaining the relationship
between the mass of the planet sought, the eccentricity of its elliptical orbit and the value of the
mean longitude at the origin time. The following discussion requires special attention.
The new relationship is sufficient to determine, with certainty, expressions for the eccentricity of
the orbit and the longitude of perihelion, as functions of the mass and the longitude at the epoch.
Imagine that the calculation has been made, and that we have eliminated the different
relationships between eccentricity and longitude of perihelion. It will fall out of equations that
are no longer arbitrary containing the mass of the planet and the average longitude at the origin
time, and must all be satisfied by a suitable choice of the unknowns.
It is quite remarkable that the mass very nearly disappears from these same equations. The
elimination of the eccentricity and longitude of perihelion causes, not the complete
disappearance of the mass dependent terms, but reduces them to such a degree of smallness, that
this mass cannot be accurately determined, and it will be allowed to assume a large range of
possibilities. In any case, we can neglect, very nearly, the terms that depend on the mass in the
final equations that we attain, and we will only require the longitude at the epoch to solve the
[equations].
However, I show that one can choose this longitude to satisfy both the final equations as well as
all observations of the planet with the accuracy that we have. I prove that there is a possible
solution, and that the further away from the solution, the deviations of the theory relative to the
observations become considerable, from which I conclude that we can effectively represent the
irregularities of the movement of Uranus by the action of a new planet located at a distance twice
the distance of Uranus to the Sun; and, what is very important, that this is achieved only one way.
In saying that the problem likely has one solution, I mean that there are not two regions of the
sky that we can choose at will to place the planet at a given time, on 1 January 1847 for example.
One should understand that, in this unique area, there are certain limits to the prediction of
astronomical position, smaller if the observations are accurate and of a suitable number, and
extended if the observations are insufficient. Let us at last discuss the position of the planet in the
sky.
To not make this summary too abstract, I shall confine myself to describe the expression of
longitude at 1 January 1847. This is the most important goal of my work, and is the result that
should serve as a starting point for observers to discover the new celestial object. m being the
mass of the planet relative to the ten-thousandth part of the Sun taken as unity, and α an
unknown, I found for the expression of the heliocentric longitude of the planet , expressed in
sexagesimal degrees, on 1 January 1847
ν = 314.5° + 12.25° α + 1 / m {20.82° - 10.79° α - 1.14° α2}
The discussion of this formula, as regards to the limits within which m and α must remain, so
that we do not cease to satisfy the observations, shows that assigning heliocentric longitude 325
degrees to the planet on 1 January 1847 has an error of no greater than 10 degrees.
This is the important result at which I have arrived. I will not try to produce smaller limits today.
This work which I have presented as an abstract to the Academy should be considered as a draft
of the beginnings of a theory. When in full ignorance of the position of the planet sought, I found
it necessary to extend the discussion of formulas and comparisons with observations in all
regions of the ecliptic, I had necessarily to simplify my work so as to not make it impossible, and
to only consider a number of selected positions of Uranus; but now that the orbital elements
described by the planet are determined with approximation, it becomes possible to use all
observations we have in constraining the solution. All these data will, without doubt, assign to
the current position of the planet a much smaller error than what I have outlined above. You can
even correct the timing of the periodic revolution.
I will plan to bring to the new theory the improvements of which it is capable: despite the
documents that I have collected on this subject, I do not know if I will finish this before the next
opposition. I will try to get, by that time, all information capable of leading us to the goal with
greater certainty.
We see, in short, that to reconcile the theory with observations, I needed to successively consider
the following:
Recalculate the perturbations that Jupiter exerts on Uranus; determine those produced by Saturn,
pushing the approximations to the square and the products of the masses, which introduced
significant changes in the accepted theories;
Reduce the number of meridian observations of Uranus to close to 300;
Calculate the corresponding heliocentric positions of the planet, assuming that it obeys only the
combined action of the Sun, and Saturn and Jupiter; deduce the geocentric coordinates with the
help of the Tables of the Sun, and prove conclusively that there is inconsistency between the so
calculated places and observed places.
The existence of an unknown planet, being thus placed beyond doubt, I reversed a problem that
has been proposed in the calculation of perturbations. Instead of having to measure the action of
a specific planet, I had to leave in the irregularities recognized in Uranus, to deduce the orbital
elements of the perturbing planet to give the position of the planet in the sky, and show that its
action was perfectly aware of the apparent irregularities of Uranus.
There will be, without doubt, people who want to reduce the solar system to its known narrow
limits, and draw a conclusion against the existence of a new celestial object. In this case,
however, I would say we have had the same reasons to affirm, on March 12, 1781, that Saturn
was the last of the planets, except to be contradicted the next day by the discovery of Uranus. Is
it new this assumption that there are more distant planets from the Sun than we know? In the
year 1758, the famous mathematician Clairaut stated in a public meeting of the Academy of
Science, during the disturbances of Halley's Comet, that a body that comes from far away could
be subject to totally unknown forces, such as the action of planets, too distant to ever be
perceived.
Let us just hope that the stars [celestial objects] of which Clairaut spoke will not all be invisible;
that although chance had discovered Uranus, that one will succeed in seeing the planet of which I
have shared the position.
C. R., 1846 1st Semester. (T. XXII , No. 22 )