114
6 Laboratory and Statistical Astronomy
their distribution in space. With an accurate determination of the position
of the Apex and of the velocity of the Sun relative to the nearby stars, a first
attempt could be made. Kapteyn began by formulating three assumptions that
were necessary to obtain an unambiguous result.
In the first place Kapteyn assumed that there was no dust between the stars,
at least not to a significant extent, that scatters or absorbs the light of the stars.
After all, if that were the case, the apparent brightness of a star would not only
be determined by its intrinsic luminosity and distance. Kapteyn figured out a
way to test this. This requires the concept of surface brightness. In astronomy
nowadays this is often expressed in magnitudes per square arcsecond. For a
dark night sky, in visual light (visible to the eye), that is something like 22
or 23; that means that for an area of one square second of arc (one by one
arcsecond), the amount of light that comes from it is the same as that of a star
of magnitude 22. Now look at a nebula. It also has a certain surface brightness.
Now imagine it to be twice as far away. Then the total amount of light you
receive is four times smaller. But the size on the sky (in arcseconds, for example)
is twice as small; so the area in the sky it covers is four times smaller in square
arcseconds. The surface brightness then remains the same and this property
thus is independent of the distance. Kapteyn then surmised that if there were
absorption of light in interstellar space, the surface brightness of nebulae at
greater distances would have to be reduced on average. But he could not find
any unequivocal signs of this.
Secondly, Kapteyn assumed that the motions of the stars in space were random, i.e. that there was no preferred direction and that the average velocity was
the same everywhere and in all directions. With these assumptions he was able
to estimate the average distance of stars as a function of apparent magnitude
from the available observations of proper motions. The result was a table in
which, if you look up a particular apparent magnitude and a particular proper
motion, you find the average distance.
For individual stars with a measured parallax, you can of course compare
this to the result you get from this method. And if you would have enough
of such stars, you can also determine how in general the stars are distributed
around those averages. This is what Kapteyn called the ‘frequency law’. To
see how that works, I cite how Kapteyn explained that to David Gill in his
resumé from 1907 [63]. If you are not interested in such details, skip to the
next paragraph.
Take stars of magnitude 6, so 5 m .5 to 6 m .5. There are about 4800 such stars
all over the sky. According to Auwers-Bradley [i.e., the catalog], 9
1
2 percent of
such stars, i.e., about 460, have proper motions between 0 .04 and 0 .05 per year.
According to the formula, the average parallax of such stars is almost exactly
6 Laboratory and Statistical Astronomy
their distribution in space. With an accurate determination of the position
of the Apex and of the velocity of the Sun relative to the nearby stars, a first
attempt could be made. Kapteyn began by formulating three assumptions that
were necessary to obtain an unambiguous result.
In the first place Kapteyn assumed that there was no dust between the stars,
at least not to a significant extent, that scatters or absorbs the light of the stars.
After all, if that were the case, the apparent brightness of a star would not only
be determined by its intrinsic luminosity and distance. Kapteyn figured out a
way to test this. This requires the concept of surface brightness. In astronomy
nowadays this is often expressed in magnitudes per square arcsecond. For a
dark night sky, in visual light (visible to the eye), that is something like 22
or 23; that means that for an area of one square second of arc (one by one
arcsecond), the amount of light that comes from it is the same as that of a star
of magnitude 22. Now look at a nebula. It also has a certain surface brightness.
Now imagine it to be twice as far away. Then the total amount of light you
receive is four times smaller. But the size on the sky (in arcseconds, for example)
is twice as small; so the area in the sky it covers is four times smaller in square
arcseconds. The surface brightness then remains the same and this property
thus is independent of the distance. Kapteyn then surmised that if there were
absorption of light in interstellar space, the surface brightness of nebulae at
greater distances would have to be reduced on average. But he could not find
any unequivocal signs of this.
Secondly, Kapteyn assumed that the motions of the stars in space were random, i.e. that there was no preferred direction and that the average velocity was
the same everywhere and in all directions. With these assumptions he was able
to estimate the average distance of stars as a function of apparent magnitude
from the available observations of proper motions. The result was a table in
which, if you look up a particular apparent magnitude and a particular proper
motion, you find the average distance.
For individual stars with a measured parallax, you can of course compare
this to the result you get from this method. And if you would have enough
of such stars, you can also determine how in general the stars are distributed
around those averages. This is what Kapteyn called the ‘frequency law’. To
see how that works, I cite how Kapteyn explained that to David Gill in his
resumé from 1907 [63]. If you are not interested in such details, skip to the
next paragraph.
Take stars of magnitude 6, so 5 m .5 to 6 m .5. There are about 4800 such stars
all over the sky. According to Auwers-Bradley [i.e., the catalog], 9
1
2 percent of
such stars, i.e., about 460, have proper motions between 0 .04 and 0 .05 per year.
According to the formula, the average parallax of such stars is almost exactly