When will the Sagittarius Dwarf Galaxy collide with the Milky Way?

Based on its current trajectory, the Sag DEG main cluster is about to pass through the galactic disc of the Milky Way within the next hundred million years, while the extended loop-shaped ellipse is already extended around and through our local space and on through the Milky Way galactic disc, and in the process of slowly being absorbed into the larger galaxy, calculated at 10,000 times the mass of Sag DEG.

At first, many astronomers thought that Sag DEG had already reached an advanced state of destruction, so that a large part of its original matter was already mixed with that of the Milky Way. However, Sag DEG still has coherence as a dispersed elongated ellipse, and appears to move in a roughly polar orbit around the Milky Way as close as 50,000 light-years from the galactic core. Although it may have begun as a ball of stars before falling towards the Milky Way, Sag DEG is now being torn apart by immense tidal forces over hundreds of millions of years. Numerical simulations suggest that stars ripped out from the dwarf would be spread out in a long stellar stream along its path, which were subsequently detected.

However, some astronomers contend that Sag DEG has been in orbit around the Milky Way for some billions of years, and has already orbited it approximately ten times. Its ability to retain some coherence despite such strains would indicate an unusually high concentration of dark matter within that galaxy.

In 1999, Johnston et al. concluded that SagDEG has orbited the Milky Way for at least one Gya and that during that time its mass has decreased by a factor of two or three. Its orbit is found to have galactocentric distances that oscillate between ~13 and ~41 kpc with a period of 550 to 750 million years. The last perigalacticon was approximately fifty million years ago. Also in 1999, Jiang & Binney found that it may have started its infall into the Milky Way at a point more than 200 kpc away if its starting mass was as large as ~1011 M☉.

The models of both its orbit and the Milky Way's potential field could be improved by proper-motion observations of SagDEG's stellar debris. This issue is under intense investigation, with computational support by the MilkyWay@Home project.

Based on its current trajectory, the Sag DEG main cluster is about to pass through the galactic disc of the Milky Way within the next hundred million years, while the extended loop-shaped ellipse is already extended around and through our local space and on through the Milky Way galactic disc, and in the process of slowly being absorbed into our galaxy.

I don't think it will have any effect on Earth at all. The Milky Way has absorbed several dwarf galaxies in the past.

The Sagittarius Dwarf Elliptical Galaxy, presently in a close dance with our home Galaxy, may be small but there must be some effects of this merger. Even if it is slowly absorbed by the MWG, the two will gravitationally act and react with each other.