In the early part of 1984, astronomers Marc Davis and Richard Muller at UC Berkeley proposed that the apparent 25 - 30 million year periodicity in the cratering record of the Earth might be caused by a small, dim dwarf companion to the sun in a highly elliptical orbit which would have a size of 90,000 Astronomical Units and a period of 26 million years. Currently it would be 2.5 light years away on its return leg.
In 1983 David Raup and John Sepkoski at the University of Chicago tracked 600 families of marine animals that had vanished during the last 250 million years and claimed to have uncovered 12 distinct eradication peaks at roughly 26 million year intervals during this period. The most recent took place 11.3 million years ago. However, using the same data and a different analytical approach, Michael Rampino found a 30 million year period.
There is so much uncertainty surrounding the extinction and cratering history that few researchers take this Death Star idea seriously. The star would be so loosely held by the Sun that its interaction with other stars would likely eject it from any periodic orbit with respect to the Sun. Also, searches have been conducted for such objects and none larger than 10 times the size of jupiter have even been detected.
The basic idea is that this star would perturb the Oort Cloud and send millions of comets raining down upon the inner solar system. There would be a long period of increased cometary collisions with the Earth which would produce the periodicities in the cratering and extinction histories. Astronomers no longer take this idea seriously in light of the uncertainties in the extinction and cratering history data.
For more information, look at the May, 1984 article in Sky and Telescope magazine on page 406. Also, the 1989 book Cosmic Catastrophes by Clark Chapman and David Morrison, published by Plenum Press, New York also discusses this.