Aphelion distances of long-period comets show a slight excess around 30 000 to 50 000 au from the Sun. Positions of cometary aphelia within these distance limits are aligned along a great circle inclined to both the ecliptic and the Galactic plane. This paper examines one of the possible explanations for this non-random clustering: that it is due to orbital perturbations by an undiscovered object orbiting within the above-mentioned distances. A model consistent with the observations gives a retrograde orbit (inclination 120°) for the object with a longitude of the ascending node at 77° ± 13°, a period of 5.8 × 106 yr and a radius of 32 000 au. The same model gives a present position for the undiscovered object of RA 20h 35m, Dec. +5°, with an error ellipse semimajor axis of 14° and a semiminor axis of 7°. The magnitude is likely to be fainter than 23. Such a distant object would almost certainly not remain bound for the age of the Solar system, and recent capture into the present orbit, although also of low probability, remains the least unlikely origin for this hypothetical planet.