What is the length of the apogalacticon and perigalacticon of the Sun and Milky Way? The general terms seem to be apoapsis and periapsis.
My greatest efforts at Googling have failed miserably. If you can provide references as well, please do!
According to this website we need another 15 million years to the perigalacticon.
We are now about 27,000 lightyears away from the galactic center, close to the perigalacticon. The difference between apogalacticon and perigalacticon is a little more than 4,000 lightyears (15% of 27,000 lightyears). The 15% number is calculated from the 7% eccentricity of the orbit by $1.07/0.93=1.15~~$ according to the definition of eccentricity, and Kepler's 1st law.
This question was once valid, but as of 1994 we found out that our Solar System is part of the Sagittarius Dwarf Elliptical Galaxy, or Sag-DEG (M54), which is the largest of the 30 or so Dwarf galaxies orbiting the Milky Way. We are in a 500 million year long circum-polar orbit. The two apsides are very far above the equatorial plane of the Milky Way. We just passed through the disc and are now about 50 light years above it, and rising towards our northern apsis (north with respect to the Galactic Coordinate System). Our journey will take us 125 million years to reach the top, and 125 million years to reach the disc of the Milky Way again. Ironically, if the Milky Way experiences a 250 million year rotation with respect to our current position in the Orion Arm (remembering that every location of the MW disc experiences a differential rotation with different orbital speeds), then by the time we reach the disc again 250 million years from now, we will basically pierce through the Orion arm very near to where we are right now! Isn't that amazing?
As for your question, and the answer provided by Gerald, we know that the stars orbiting the MW's core do oscillate up and down. This was predicted in 1978 and is due to the Galactic Current Sheet, which is why all of the stars follow each other in a sine wave pattern as they bob up and down. Because of this, we can predict the apsides for most of these stars, meaning the high and low point above the disc. However, these are not "apogalacticon" or "perigalacticon", which technically represent the furthest and closest and points in a satellites orbit, not as it oscillates through the disc, because Satellites don't do that, but around the Galactic core in an elliptical orbit much like a Sun-Grazing comet gets close to and then very far from the Sun. Think the Large Magellanic Cloud, or the 30 known Dwarf Galaxies, or many of the 158 known Globular Clusters.