I wanted a short animation of the Big Dipper from 100,000 years ago to 100,000 years ahead. The idea is old. Richard Pogge at Ohio State has such a movie on his course page, last updated in 2007. Before I computed anything I pictured the Dipper in 100,000 years as 7 stars going in 7 directions. Which star goes where, I had no idea. So I computed it and then measured what is left of the shape.
Gaia did not have the stars
It is 2026 and Gaia is what I would take for anything about stars, so I started there. Python 3.12, astropy 8.0.1, astroquery 0.4.11, a query to Gaia DR3 around each of the 7 positions. Dubhe was not there at all. The brightest source within 3 arcminutes of it has G = 14.0, while Dubhe in Hipparcos has Hp = 1.95. Phecda and Alioth are in the catalog with a position only, without parallax or proper motion. Alkaid too. The other 3 have full astrometry with RUWE between 2.6 and 6.0. Above 1.4 such a solution is usually not trusted. I widened the search radius to 3 arcminutes before I believed it. Then I accepted that for these 7 stars Gaia is not the source.
So the data is older. Positions and motions on the sky come from the Hipparcos new reduction by van Leeuwen (2007), epoch 1991.25. Radial velocities come from SIMBAD, for 5 stars from Gontcharov 2006. Megrez has its value from Gaia DR3 and Mizar from Pourbaix 2004. Each star gets a position and a velocity in 3D and moves on a straight line. Every 500 years I project it back onto the sky. That is 401 frames from minus 100,000 to plus 100,000 years. At 24 frames per second the video is 16.7 seconds. The frames are drawn with numpy 2.5.3 and matplotlib 3.11.2, then ffmpeg 6.1.1 glues them into the video.
The middle of the Dipper moves as one piece
In 100,000 years every star of the Dipper moves 2.6 to 3.9 degrees across the sky, which is 5 to 8 Moon widths. This agrees with Sky & Telescope, which gives one Moon diameter per 16,000 years for the group. But the video does not look like 7 stars scattering. The handle bends at Mizar and most of the figure travels as one piece.
To put a number on “one piece” I took the 5 middle stars, from Merak to Mizar. I fitted a shift, a rotation and a scale that take them from today to their places in 100,000 years. Then I looked at how far each of the 7 stars is from where this fit puts it. For the 5 stars the residual is at most 0.17 degrees. Dubhe is off by 6.9 degrees and Alkaid by 7.3. The rotation is 0.13 degrees. The scale is 1.044 because the 5 stars come toward us, Merak from 24.4 to 23.1 parsecs. Going back 100,000 years gives the same picture: the 5 hold, Dubhe is off by 6.4 and Alkaid by 6.8.
The velocities explain it. Relative to the mean of the 5, each of them moves at 0.7 to 2.9 km/s. Dubhe moves at 36 km/s and Alkaid at 33. The 5 middle stars belong to the Ursa Major moving group and the 2 end stars do not. This is old knowledge. S&T says it in the same article. Still, before the fit I did not expect that the whole “falling apart” is only these 2 stars.
Alkaid leaves the fitted shape by a Moon width in about 7,100 years and Dubhe in 7,600. The first of the 5 to do the same is Phecda, after about 300,000 years. That number I trust less. Phecda moves at 0.9 km/s relative to the group and at such speeds the catalog errors matter. Pushed through them, its residual at 300,000 years lands anywhere from 0.34 to 0.66 degrees.
How much the straight line lies
The model is simple, so I checked what it ignores. Everything below is in degrees at plus 100,000 years and it is all in the chart after this section.
The easy way to do this is to add the proper motion to today’s coordinates and forget the distance. Against my 3D lines this differs by 0.16 to 0.26 degrees per star. 2 things make this difference and they are of the same size. Without the radial velocity the stars land 0.08 to 0.17 degrees from where they land with it. Coming closer speeds up their motion on the sky. Adding degrees to coordinates near the pole is the second error, 0.09 to 0.26 degrees, largest for Dubhe at 62 degrees north.
The catalog errors I pushed through a Monte Carlo with 4,000 draws per star. For 6 stars the 95th percentile of the shift is at most 0.03 degrees. Mizar has 0.11, because its Hipparcos proper motion error is about 1.5 mas per year against 0.1 to 0.4 for the others. Mizar is a multiple star and its radial velocity depends on the source. With Gaia’s value for Mizar B, minus 10.3 km/s instead of minus 6.3, it ends up 0.06 degrees away. The bend of the orbit in the Galaxy I estimated from above with the vertical tide of the disk. It gives 6 arcseconds.
None of this comes close to the 7 degrees of Dubhe and Alkaid. The straight line is fine for this horizon. The 0.17 degrees of Phecda is also not noise from the catalog: with the errors pushed through the fit, it stays between 0.15 and 0.20. So within this data the 5 stars really do change shape a little.
What I did not check
The brightness in the video is only the Hipparcos magnitude corrected for distance. Megrez gets brighter by 0.11 because it comes closer. Other stars of the constellation are not in the video, only the 7. I did not model the orbits inside Mizar and Alcor. The 300,000 years for Phecda depend on a velocity difference of about 1 km/s. I did not test how they change with other radial velocity catalogs.