Limitations on the existence of a tenth planet

The orbits of 14 known Trans-Neptunian Objects were numerically integrated for time intervals of up to 107 yr in the presence of an imaginary tenth planet with a mass equal to that of Neptune as well as the known outer planets. This “planet” was placed on various hypothetical orbits. The results of these integrations allow the determination of whether or not the Trans-Neptunian Objects remained stable for the given interval. On the assumption that the real objects have ramained stable, limits can be placed on possible orbits for this hypothetical tenth planet. It transpires that if all investigated Trans-Neptunian Objects are to be stable, then the orbit of the tenth planet, if near the ecliptic, must have a semi-major axis in excess of 70 AU. Increasing the eccentricity of the orbit of this hypothetical tenth planet increases the level of instability while in contrast, increasing the inclination aids stability and for highly inclined orbits of the hypothetical tenth planet, no realistic restriction on the semimajor axis of the orbit exists through the existence of Trans-Neptunian Objects.