The previous post touched on some ways in which the orbits of certain outer Solar System objects are similar. These may be quickly summarized in the following way: both the arguments and longitudes of the objects' perihelia are unusually clustered around certain values.
The above image shows numerous relevant parameters concerning the position of an orbit. In the case of orbits in the Solar System, the plane of reference is the plane of the Earth's orbit and the Sun, also known as the ecliptic. The reference direction often used for heliocentric objects is called the First Point of Aries, defined as the position of Earth's vernal equinox and so named for its location within the constellation Aries. The ones with which we are concerned here are the argument of periapsis ω (this is the general name for argument of perihelion to include non-heliocentric objects) and the longitude of the ascending node Ω. The sum of these two angles is called the longitude of perihelion because it measures the angle between the perihelion and the reference direction. In summary, the similarity in the arguments of perihelion indicates that the members of the relevant population of objects have similar orientations with respect to the plane of the Solar System, while the similarity in the longitudes indicates a clustering of these orbits in space.
A 2016 paper by Konstantin Batygin and Michael E. Brown ran a statistical analysis of these parameters for the six most extreme known trans-Neptunian (beyond Neptune) objects. Since they were discovered by a number of distinct observational surveys, the possibility of observational bias was dismissed. The analysis found that the clustering of the objects had only a 0.007% probability of occurring by chance. This suggested that another explanation was in fact required for the phenomenon. Further simulations suggested that a Planet Nine could account for the observations, provided that it have the required heft: at least around 10 Earth masses (or, equivalently, 5000 Pluto masses). In comparison, all the previously known trans-Neptunian objects put together weighed much less than a single Earth mass.
Shortly afterward, more evidence for Planet Nine was discovered, using data from a surprising source: the Cassini space probe. Launched in 1997, this Saturn orbiter allowed the calculation of the position of Saturn over time to unprecedented precision. These were compared to an extremely precise gravitational model of the Solar System known as INPOP, which accounts for the gravitational influence of the Sun, the planets, and many asteroids. The model then outputs planetary ephemerides, namely positions of the planets at given times. A paper published in February 2016 by Agnès Fienga et al. experimented with adding a Planet Nine at different positions to the INPOP. If the residuals (differences in Saturn's position between the predictions of INPOP and the real measurements from Cassini) are increased, this rules out the existence of Planet Nine in this position. However, if they are decreased, then this is evidence in support of Planet Nine, since it would partially explain the observed discrepancy.
The results of the paper are summarized in the diagram above. They showed that Planet Nine of 10 Earth masses and a semi-major axis of 700 AU was ruled out by Cassini's data to be in the red zones (this increased the residuals). The pink zones correspond to areas that would be ruled out by further inclusion of Cassini's data (the paper only used the measurements through 2014). The green zone, however, is where a Planet Nine would decrease residuals, making the INPOP model a more accurate picture of the Solar System. Therefore, the paper found this to be the most likely zone to find Planet Nine (with the single most likely position indicated). The addition of a Planet Nine in the farther regions of its orbit would not produce significant perturbations, and thus this is labeled "uncertainty zone".
Further analysis fine-tuned the estimates of mass, eccentricity, semi-major axis, and other parameters for the supposed Planet Nine. With an array of increasingly large telescopes at their disposal, astronomers will soon be able to settle the Planet Nine hypothesis one way or the other, bringing new insight into the current structure and the formation of our Solar System.
Sources: https://arxiv.org/pdf/1601.05438v1.pdf, https://en.wikipedia.org/wiki/Argument_of_periapsis, http://arxiv.org/pdf/1602.06116v3.pdf, http://arxiv.org/pdf/1603.05712.pdf
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