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In the early 1900s astronomers could measure the distance to only a small number of nearby stars. Jacobus Kapteyn, a Dutch astronomer at the University of Groningen, still wanted the shape of the whole Milky Way. So he counted. His picture was wrong in important ways, but his method still teaches how to model from incomplete data.
Who was Jacobus Kapteyn?
Kapteyn (1851-1922) was appointed professor of astronomy and theoretical mechanics at Groningen in 1878, according to the Kapteyn Astronomical Institute's history page. In 1896 he set up an astronomical laboratory, which the institute calls an "Observatory without telescopes". He measured photographic plates taken by David Gill at the Cape of Good Hope; the result, the Cape Photographic Durchmusterung (1896-1900), listed the positions and magnitudes of 454,875 stars.
How did Kapteyn use star counts to model the Milky Way?
Direct distances come from parallax, the small yearly shift of a nearby star, which we explain in our post on Friedrich Bessel and the first measured star distance. Astronomer P.C. van der Kruit's review of the Kapteyn Universe explains that Kapteyn soon realised most stars were too far away for this. Instead he worked from counts of stars sorted by apparent brightness and by proper motion (how fast they creep across the sky). He made three assumptions:
The spread of true brightnesses among stars is the same everywhere.
Star motions have no preferred direction.
No starlight is absorbed on its way to us.
Counting every star was impossible, so in 1906 Kapteyn published the Plan of Selected Areas (University of Groningen): 206 small patches spread regularly over the sky, each studied in as much detail as possible by many observatories (van der Kruit).
The result came in Kapteyn's 1922 paper in the Astrophysical Journal. He described the stars as filling nested, flattened ellipsoids of equal density, with an axial ratio of 5.1. Star density fell to one-hundredth of its value near the Sun at 8,465 parsecs along the plane of the Milky Way, but at only 1,660 parsecs toward the galactic pole. For the calculation he assumed the Sun sat at the centre, though he noted the true centre was probably about 650 parsecs away.
What were Kapteyn's two star streams?
Around 1902 Kapteyn noticed that stars did not move at random. Their proper motions pointed toward two preferred directions about 100 degrees apart on the sky. Once the Sun's own motion was removed, the two directions were roughly opposite and lay in the Milky Way (van der Kruit). He announced the two streams in 1904 at St Louis and at the British Association meeting in Cape Town in 1905. Radial velocities showed that the streams moved relative to each other at about 40 km/s.
Karl Schwarzschild soon offered the reading accepted today: one set of stars whose random speeds are larger along one axis. Kapteyn kept two streams, and in 1922 treated them as two groups of stars rotating in opposite directions, with a relative velocity of 39 km/s (Kapteyn 1922). In 1928 Jan Oort, who always considered himself Kapteyn's student, showed that the galaxy rotates in a different way, which we cover in Jan Oort and the spinning galaxy.
Why was the Kapteyn universe wrong?
Assumption 3 failed. Dust between the stars dims distant stars, so counts thin out too soon and the system looks smaller, centred near us. In 1909 he estimated 0.001 magnitude per parsec, but Harlow Shapley's 1916 work on star colours in the globular cluster M13 seemed to show that space was transparent, and this must have convinced Kapteyn and others (van der Kruit). Shapley then put the centre of the globular clusters toward Sagittarius, about 20 kiloparsecs away. That story is in Harlow Shapley and how we found the Sun is not at the centre.
Van der Kruit adds that in the vertical direction Kapteyn's results are "substantially correct". The model failed in the plane of the galaxy, where the dust is.
Try Kapteyn's reasoning: a worked example
One magnitude step is a brightness ratio of about 2.512, and five steps make exactly 100 (apparent magnitude). Now imagine identical stars spread evenly through clear space.
Brightness falls as 1/d squared. To see stars one magnitude fainter, you can see 2.512 to the power 0.5, about 1.585 times farther.
Volume grows as d cubed, so you take in 1.585 cubed, about 3.98 times as much space.
So each extra magnitude should roughly quadruple the count. If a patch has 100 stars brighter than magnitude 10, expect about 398 brighter than 11.
Suppose you count only 250. Either the stars thin out with distance, or dust is dimming them. Counts alone cannot tell you which, and that is exactly the trap Kapteyn fell into.
Ask three questions of any model:
What am I sampling, and is the sample fair?
What am I assuming, in plain words?
What result would prove an assumption wrong?
If inverse-square scaling still feels slippery, the Fermi Concept Library gives each physics concept its own lesson, and the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
