Tuesday, February 26, 2019


The history of Le Verrier’s mathematical discovery of Neptune is my favorite story, period. It’s literally got everything you’d want in a good novella - differential equations, integrals, telescopes, intrigue, you name it. Rather than try to rehash it here without doing it justice, I’ll point the interested reader to an excellent 2016 article by Davor Krajnović called “The contrivance of Neptune.” Here, I only want to call attention to what Le Verrier (and Adams) got right and what they got less right.

It’s widely known that Neptune was discovered “with the tip of a pen.” Indeed, Le Verrier was able to derive Neptune’s location on the sky from orbital anomalies of Uranus with exquisite accuracy, such that Galle and D’Arrest’s observational campaign to discover this elusive planet took less than a single night. What is somewhat less well known is that Le Verrier and Adams’ calculations of Neptune’s orbit and mass were not as precise. The figure below shows the true orbits and locations of Uranus (gray) and Neptune (black) between 1830 and 1860, as well as the predicted orbits of Neptune (in purple).
Notice that the inferred semi-major axis of Neptune was about 40 (rather than 30) AU and the derived mass (36 and 50 Earth masses for Le Verrier and Adams, respectively) also significantly exceeded that of Neptune. In light of the fact that the discovery of Neptune represents the only successful mathematical prediction of a planet to date, this level of uncertainty sets the gold standard for dynamically motivated planetary predictions. In other words, if we get Planet Nine to a similar level of precision, I’ll be satisfied. It is also useful to point out that the most significant quantity in perturbing the orbit of Uranus was the anomalous acceleration in the radial direction produced by the new body - GM/r^2 - a ratio that was calculated to higher accuracy than the individual values of mass and semi-major axis. As I will highlight later, the general framework of the Planet Nine hypothesis is characterized by comparable degeneracies between P9’s mass and orbital parameters.

Following Le Verrier’s mathematical discovery of Neptune, the planet prediction business didn’t simply get a boost - it exploded. Jacques Babinet (1848), David P. Todd (1877), George Forbes (1880), Camille Flammarion (1884), William Pickering (1909-1932) all took turns predicting trans-Neptunian planets that later turned out to not be there. But no planetary prediction is quite as emblematic as Percival Lowell’s hypothesized “Planet X.” Briefly, the story goes as follows: despite the addition of Neptune to the solar system’s ledger of planets, small apparent discrepancies in the orbits of the giant planets remained, and pointed to the existence of a ~7 Earth mass planet beyond Neptune. The search continued well past Lowell’s death, and in 1930, a bright moving object was discovered by Clyde Tombaugh in the approximate location on the sky where Planet X was envisioned to be. Because Planet X was the object of the original search, the newly found body was initially considered to be the long-sought-after Planet X.

Immediately, however, there was a problem. Planet X was supposed to be like Neptune, but Tombaugh’s new planet appeared dim and point-like, and therefore much much smaller. It soon became clear that the new member of the solar system could not be THE Planet X. The object was subsequently named Pluto, and its estimated mass steadily declined for the next 5 decades.
Although Planet X - as originally formulated by Lowell - does not exist, the discovery of Pluto turned out to be the tip of an extraordinary trans-Neptunian iceberg called the Kuiper belt. The mapping and subsequent characterization of the Kuiper belt in the ‘90s and the ‘00s, generated a new wave of planetary proposals — check out Brunini & Melita (2002), Gladman & Chan (2006), Gomes et al. (2006), Lykawka and Mukai (2008), Trujillo and Sheppard (2014), Volk and Malhotra (2017) and many others that are referenced therein. All of these hypothetical planets were invoked to explain different observational puzzles, and attempt to do so through individual dynamical mechanisms. Stepping away from specific predictions, however, it is worthwhile to examine the question of where still-undetected planets can hide in the solar system, from a completely model-independent perspective. As it turns out, the combination of ephemerides, orbital stability, and definition of a planet alone leave only a limited parameter space where additional solar system planets can hide (shown as the shaded region on the plot below):
Remarkably, Planet Nine falls right in the center of that region.


  1. I don't really understand that line of dwarf planets in the picture. The classification of a dwarf planet doesn't have with mass to do

  2. the classification of a dwarf planet very much is a function of mass... See criterion by Jean-Luc Margot

    1. The definition of mass fits well with the stability condition of a rotating system (at the upper limit). Let me remind you: the stability condition is a consequence of the law of conservation of momentum. It is necessary to take into account: the mass of all known planets of the Solar system, the mass of an unknown celestial body + the amplitude of the Sun's sinusoid.
      (x, y, z) Σmi·ri(t)= 0 or (x, y, z) Σ mi·ri(t) = Const
      Let me remind you: The rotating system is
      stable if the center of mass, the center of gravity, the center of rotation of this rotate system are coincide. In our situation there is an additional condition
      -this point coincides with the central body. That generally corresponds to all the Three Generalized Laws of Kepler.

    2. As for the rest, you must answer the following questions:

      1. The inclinations of the orbital planes of asteroids;
      2. The Kirkwood's gaps or dips in the asteroid belt. Orbital characteristics of "The fifth giant planet " and perturbing effect for objects of main asteroid belt;
      3. Anomaly of the Pioneers, causes;
      4. Conditions for the visibility of the "Fifth Giant Planet" ;
      5. What could make Venus rotate in the opposite direction?
      6. Why does Uranus lie on its side?
      7. Why do several planets have a slope of the diurnal axes to
      the plane of the orbit more than 20 degrees?
      8. Non-symmetry of the heliosphere.
      9.What is the mechanical law or mechanical reason of statistical rule resonance of Kozai -Lidov?

    3. I think you become a hostage to the wrong theory. 3 years of experiment from you - the result is negative. Maybe you should reconsider and accept the fallacy of the theory. I have a positive result of the experiment. My offer of friendship and cooperation remains valid. Your task: the identification of the object and its satellite, the refinement of the orbital characteristics. It will be honest and decent. I wait to your reply. More in my blog.

  3. Le Verrier, Adams had foretold mass of Neptune cca 36-50 Earth masses. Neptune mass is but aprox. 17Earths. Foretold was but little more distant and more prolongated orbit. There is quite big difference 19-33Earths,...So, why such big difference? If P9,P10 are 50-500AU from Sun,..so mainly if the first heavy body next to Pluto is 200 or more AU from us? There is 100x smaller gravity influence, pull,...from 300Au, than from 30AU,..so P9,10 if so distant must be 1000Earth masses,...or more,... Pavel Smutny

    1. At first glance, your question may seem stupid. But if you carefully examine, the difference in the 19-33 Earth mass may indicate an additional effect (perturbation) which coming from the inside part of the solar system.

    2. maybe yes,...but they (Le Verrier, Adams) located possible position,...mass of Neptune for 30 years,..so the whole orbit of Saturn, 3 orbits of Jupiter,..so I dont think that disproportion of 19-30 Earths was due to perturbation,...from the inside part of our solar system,.. Also Harrington,...had proposal for X quite close to orbit of Neptune with 5-7Earths Pavel Smutny,...

    3. Dear Pavel, this is a separate topic. Therefore, the comment may be with the provision of calculations 2-3 pages, may be more...I will not pay attention to this topic.

    4. The disproportion of 19-30 Earths arose precisely because of the disturbance from the inner part of the solar system ( I'm based on real calculation in dynamics). The main criterion of any theory is the result of the experiment. I have a positive experimental result. I discovered the massive celestial body of the solar system. Any astronomer can observe this planet. But because of biggest cynicism and simple unwise, specialists are too lazy to direct a telescope at the coordinates that are published on my blog. Brown and Batygin are no exception. My articles provide calculations on mechanics, optics and thermodynamics: the mechanical reasons for Kozai-Lidov resonance (Sixth exact solution "Three bodies problem"), visibility conditions of the Fifth Giant Planet (or if you want Planet Nine) and much more are explained . You can see the evolution of content in the form of corrections. The main thing in this situation is do not lazy to test the ephemeris. It is very simple. I want to express the hope that Brown and Batygin will find the strength to recognize the fallacy of their theory and accept my point of view. Acknowledging a mistake is a big step forward, progress. I do not blame B & B, their work is the apotheosis of statistical theories. A big and difficult job is planned for the future. B & B in this work will play a very large positive role.

  4. What more can be done to calculate P9's present position, rather than only its orbit?

    1. Seems P9 is simply too far away from the other large planets to have a noticeable effect on their orbital speeds. Can't use anomalies in Neptune's orbital speed to find P9, same as Neptune was found by calculations on the anomalies of Uranus's orbital speed. Or, can we? Is the greater precision of our measurements today enough to discern P9's miniscule effects on Neptune? Why did anyone think that P9 could cause a noticeable anomaly in the Cassini orbiter's paths about the Saturn system?

    2. Elizabeth Bailey has a negative result that mean-motion resonances have too many degrees of freedom to narrow down P9's position.

    3. Orbital speed of any body is fastest when near perigee, therefore, at any given time, P9 is more likely to be closer to its apogee. That fact only helps for guiding the necessarily slow campaign of trying to find the planet by a plain old search, looking everywhere it could be, but looking near its apogee first, so that it might be found in 5 years, rather than, say, 20 years.

    4. Various surveys would have found it already it was near perigee, is that right? Another reason to search near the presumed apogee.

    5. And then there's the area where the Milky Way is in the background.

    6. I suppose astrometry on the sun (the wobble method) is no good. Even though we have the advantage of being much, much, much closer to Sol than to any other star in which we've detected a wobble from an exoplanet, P9 is just too far away and too slow. Maybe if we had several centuries of very precise measurements of solar wobble?

    7. Maybe someone gets really lucky and just happens to see P9 while looking at other stuff, and realizes that they are seeing a planet, not a star. Odds of that happening in the next 3 years must be, what, much less than 0.1%?

    Is that about it?

    1. Regarding your point 6:
      Being closer to a star would usually be a positive, but we're TOO close to the Sun - any acceleration from P9 affecting the Sun would affect the Earth at almost equal strength.

      (Also, the wobble would become noticable if we have the star change its motion over time due to the exoplanet. Relative to the star's average movement it would sometimes go more towards Earth and sometimes away from it - but we can only really notice that if we see both of those, and for P9 those would be thousands of years apart)

  5. Yes, I wondered about the problem of us being wobbled right along with the sun, and whether there was any place we could park a space telescope to avoid that. Perhaps a polar orbit about the sun? But even if we could, thousands of years to get enough wobble data is just too long to wait.

    How about this one?

    8. Transit method, with the Milky Way. Watch the stars of the Milky Way for objects passing in front of them.

    1. With the transit method, you would only know how long the background star was occulted, but you wouldn't know whether it was by a small close object or a large distant one.

      At least one group announced the observation of a large outer Solar System body in this way, but their observation in the given region of the sky could not be reproduced. The occultation was real, but most likely caused by a normal small main belt asteroid.

      For a good discovery, you'd need at least three observations of the object in question. With the first two you wouldn't know if it was one slowly moving object or two different fast moving objects, and the third observation can be used to test whether it appears at the spot where an object that just followed its orbit would be.

    2. Certainly you are right. Several transits (eclipses) are required. That is exactly what was done. In my articles you will see 2 transit, which we published. But there were more observations. It is on the basis of all observations that the orbital parameters were determined.