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11 The Kapteyn Universe
attempt at ‘a general theory of the distribution of mass, forces and velocities’
in the Galaxy.
So how did Kapteyn work out this theory? First of all, he had to be able to
calculate the gravitational forces of all matter in the system at any location. To
do this, he first reduced it to an elliptical distribution, see Fig. 11.10. There
are mathematical formulas with which Kapteyn could now relatively easily
calculate these forces at any location, except for a factor that corresponded to
the average mass of the stars. But that factor had to be the same everywhere.
He then started with the vertical direction. It is clear that there must be a
certain average velocity to keep everything in equilibrium. If the stars have
a lower velocity then that, then they cannot come up and down far enough
from the plane to explain the observed thickness. Kapteyn concluded from
measurements in the literature that the average velocity of stars in this direction
is 10 km/s. He then calculated that there would be an equilibrium if each star
on average has a mass 1.4–2.2 times that of the Sun. And it was believed on the
basis of the orbits in binary stars that the components of such systems together
are on average 1.6 times heavier than the Sun. That lead to the conclusion that
Kapteyn’s system would be in equilibrium in the vertical direction if most of
the stars were binaries and Kapteyn felt that this was not unreasonable. Then
there was no need for the presence of any unseen matter in the form the ‘dark
bodies’.
This is a remarkable result: the distribution of the stars and their velocities
are precisely balanced. In 1932, Jan Hendrik Oort developed a model of this
vertical distribution in more detail and in modern times we can do the same for
other galaxies—work in which I myself have been involved. The question of a
need to postulate unseen matter of unknown nature is still open. By the way,
the vertical distribution that Kapteyn presented in his model in Fig. 11.10,
and the average velocity in that direction, correspond remarkably well with
Fig. 11.10 The distribution of the stars in space according to Kapteyn approximated by
ellipsoids. Compare Fig. 11.9, to which this should be an approximation
11 The Kapteyn Universe
attempt at ‘a general theory of the distribution of mass, forces and velocities’
in the Galaxy.
So how did Kapteyn work out this theory? First of all, he had to be able to
calculate the gravitational forces of all matter in the system at any location. To
do this, he first reduced it to an elliptical distribution, see Fig. 11.10. There
are mathematical formulas with which Kapteyn could now relatively easily
calculate these forces at any location, except for a factor that corresponded to
the average mass of the stars. But that factor had to be the same everywhere.
He then started with the vertical direction. It is clear that there must be a
certain average velocity to keep everything in equilibrium. If the stars have
a lower velocity then that, then they cannot come up and down far enough
from the plane to explain the observed thickness. Kapteyn concluded from
measurements in the literature that the average velocity of stars in this direction
is 10 km/s. He then calculated that there would be an equilibrium if each star
on average has a mass 1.4–2.2 times that of the Sun. And it was believed on the
basis of the orbits in binary stars that the components of such systems together
are on average 1.6 times heavier than the Sun. That lead to the conclusion that
Kapteyn’s system would be in equilibrium in the vertical direction if most of
the stars were binaries and Kapteyn felt that this was not unreasonable. Then
there was no need for the presence of any unseen matter in the form the ‘dark
bodies’.
This is a remarkable result: the distribution of the stars and their velocities
are precisely balanced. In 1932, Jan Hendrik Oort developed a model of this
vertical distribution in more detail and in modern times we can do the same for
other galaxies—work in which I myself have been involved. The question of a
need to postulate unseen matter of unknown nature is still open. By the way,
the vertical distribution that Kapteyn presented in his model in Fig. 11.10,
and the average velocity in that direction, correspond remarkably well with
Fig. 11.10 The distribution of the stars in space according to Kapteyn approximated by
ellipsoids. Compare Fig. 11.9, to which this should be an approximation