282
Appendix A: Some More Background
then is so great that the degeneration pressure can no longer compensate for it;
then the electrons are, as it were, merged into the protons and form neutrons.
This process is so fast that an enormous pressure wave propagates through the
star, blowing itself up like a supernova. More chemical elements are formed
and this material is thrown out into space. The remaining central parts are then
a neutron star, which is in equilibrium under the influence of the degeneration
pressure of the neutrons, or, if the initial star mass is even larger and the
contraction forces too strong, a black hole.
The derivation of the formulae above and more background on the issues in
this section can be found among others in my introductory astronomy lectures
[119, 120].
A.6 The ‘Star Ratio’
This concept was used by Edward Pickering in 1903 to obtain information
about the distribution of stars in space. Assume that all stars have the same
intrinsic luminosity and are uniformly distributed in space. Take a certain
distance from the Sun, then all stars at that distance have the same apparent
brightness or magnitude, say m. Stars of a magnitude weaker, i.e. m + 1,
in the sky are according to the definition of the magnitude scale a factor
5
√
100 = 2.512 fainter and have to be therefore a factor
√
2.512 further away.
The stars between apparent magnitude m and m + 1 fill a shell. Now take
stars that are another magnitude fainter, so m + 2. They are a factor of 2.512
fainter than the stars of magnitude m + 1 and are a factor of
√
2.512 further
away. The shell between m + 1 and m + 2 is a factor 2.512 larger than the
one between m and m + 1 and the thickness is a factor
√
2.512 larger. So the
volume of the shell is a factor 2.512 ×
√
2.512 = 2.512 3/2 larger and so is
the number of stars in it.
This means that the number of stars between magnitude m + 1 and m + 2
should be the same factor 2.512 3/2 = 3.981 times larger than the number
between m and m +1. This ratio was called the ‘star ratio’ by Edward Pickering
and 3.981 is its reference value. In practice, he did not use the factor 3.981
itself, but actually its logarithm, which is exactly 0.6. In reality on the sky,
this ratio is smaller than this theoretical value; this may be due to changes in
the density of the stars with changing distances or to the distribution of their
intrinsic luminosity, but also to absorption of starlight by dust in interstellar
space.
Appendix A: Some More Background
then is so great that the degeneration pressure can no longer compensate for it;
then the electrons are, as it were, merged into the protons and form neutrons.
This process is so fast that an enormous pressure wave propagates through the
star, blowing itself up like a supernova. More chemical elements are formed
and this material is thrown out into space. The remaining central parts are then
a neutron star, which is in equilibrium under the influence of the degeneration
pressure of the neutrons, or, if the initial star mass is even larger and the
contraction forces too strong, a black hole.
The derivation of the formulae above and more background on the issues in
this section can be found among others in my introductory astronomy lectures
[119, 120].
A.6 The ‘Star Ratio’
This concept was used by Edward Pickering in 1903 to obtain information
about the distribution of stars in space. Assume that all stars have the same
intrinsic luminosity and are uniformly distributed in space. Take a certain
distance from the Sun, then all stars at that distance have the same apparent
brightness or magnitude, say m. Stars of a magnitude weaker, i.e. m + 1,
in the sky are according to the definition of the magnitude scale a factor
5
√
100 = 2.512 fainter and have to be therefore a factor
√
2.512 further away.
The stars between apparent magnitude m and m + 1 fill a shell. Now take
stars that are another magnitude fainter, so m + 2. They are a factor of 2.512
fainter than the stars of magnitude m + 1 and are a factor of
√
2.512 further
away. The shell between m + 1 and m + 2 is a factor 2.512 larger than the
one between m and m + 1 and the thickness is a factor
√
2.512 larger. So the
volume of the shell is a factor 2.512 ×
√
2.512 = 2.512 3/2 larger and so is
the number of stars in it.
This means that the number of stars between magnitude m + 1 and m + 2
should be the same factor 2.512 3/2 = 3.981 times larger than the number
between m and m +1. This ratio was called the ‘star ratio’ by Edward Pickering
and 3.981 is its reference value. In practice, he did not use the factor 3.981
itself, but actually its logarithm, which is exactly 0.6. In reality on the sky,
this ratio is smaller than this theoretical value; this may be due to changes in
the density of the stars with changing distances or to the distribution of their
intrinsic luminosity, but also to absorption of starlight by dust in interstellar
space.