Appendix A: Some More Background
281
which depends only on the temperature T . The amount per unit surface area
is given by the law of Stefan-Boltzmann, named after Josef Stefan (1835–1893)
and Ludwig Eduard Boltzmann (1844–1906). When for a star its distance is
known, we can calculate the total energy emitted and from the distribution of
brightness with wavelength determine the temperature. This law then can be
used to calculate the total surface area of the star and thus its radius.
Most stars lie along the Main Sequence from the top left to the bottom
right. After being formed from a cloud of gas cloud, contraction and release
of potential energy will make the inside of a star hotter, and as the density
increases more energy is added than can be radiated away. At ten million
degrees, nuclear reactions start, converting hydrogen into helium. The stars
at the top are bright, heavy, and hot and for them this period is shortest, the
M-stars are faint, light and cool and they live longer than the present age of
the Universe.
Eventually all hydrogen in the central parts of a star will be used up, so that
the core, which now consists entirely of helium, will be extinguished. Initially
‘hydrogen burning’ continues for a while in a shell around the core and the
star becomes brighter. A star like the Sun then moves up along the sub- and
the giant branches (along numbers IV and III in Fig. A.4). The temperature
T at the surface drops and the star becomes redder, but the radius R will
then increase. Because energy is no longer produced there, the core contracts,
but then gets hotter. When it becomes hot enough, helium burning will start
there, converting it into carbon and oxygen. A star like the Sun is then on the
‘clump’ halfway along the red giant branch. Then the star expels its outer layers
and forms a so-called planetary nebula. The star then cools down to a white
dwarf at the bottom of the figure. The matter becomes very compact, but at a
certain moment the contraction stops because Fermi’s exclusion principle and
Heisenberg’s uncertainty relationship together forbid that the electrons come
even closer together (the so-called degeneration pressure). The uncertainty
relationship says that the position and speed of an electron (or other elementary
particle) cannot be measured infinitely accurately at the same time. Then two
different electrons can in principle not get so close together and have the
same speed so accurately that they would be indistinguishable. The Fermi
principle prohibits this and so matter cannot become more compact than
when all particles can just be distinguished in position and speed. This is
called degeneration pressure.
In a more massive star than the Sun, the pressure in the core becomes so
great that the center continues to contract; the star then becomes so hot that
even heavier chemical elements are formed. But in the long run this process
also stops and the central parts contract even further. The contraction force
281
which depends only on the temperature T . The amount per unit surface area
is given by the law of Stefan-Boltzmann, named after Josef Stefan (1835–1893)
and Ludwig Eduard Boltzmann (1844–1906). When for a star its distance is
known, we can calculate the total energy emitted and from the distribution of
brightness with wavelength determine the temperature. This law then can be
used to calculate the total surface area of the star and thus its radius.
Most stars lie along the Main Sequence from the top left to the bottom
right. After being formed from a cloud of gas cloud, contraction and release
of potential energy will make the inside of a star hotter, and as the density
increases more energy is added than can be radiated away. At ten million
degrees, nuclear reactions start, converting hydrogen into helium. The stars
at the top are bright, heavy, and hot and for them this period is shortest, the
M-stars are faint, light and cool and they live longer than the present age of
the Universe.
Eventually all hydrogen in the central parts of a star will be used up, so that
the core, which now consists entirely of helium, will be extinguished. Initially
‘hydrogen burning’ continues for a while in a shell around the core and the
star becomes brighter. A star like the Sun then moves up along the sub- and
the giant branches (along numbers IV and III in Fig. A.4). The temperature
T at the surface drops and the star becomes redder, but the radius R will
then increase. Because energy is no longer produced there, the core contracts,
but then gets hotter. When it becomes hot enough, helium burning will start
there, converting it into carbon and oxygen. A star like the Sun is then on the
‘clump’ halfway along the red giant branch. Then the star expels its outer layers
and forms a so-called planetary nebula. The star then cools down to a white
dwarf at the bottom of the figure. The matter becomes very compact, but at a
certain moment the contraction stops because Fermi’s exclusion principle and
Heisenberg’s uncertainty relationship together forbid that the electrons come
even closer together (the so-called degeneration pressure). The uncertainty
relationship says that the position and speed of an electron (or other elementary
particle) cannot be measured infinitely accurately at the same time. Then two
different electrons can in principle not get so close together and have the
same speed so accurately that they would be indistinguishable. The Fermi
principle prohibits this and so matter cannot become more compact than
when all particles can just be distinguished in position and speed. This is
called degeneration pressure.
In a more massive star than the Sun, the pressure in the core becomes so
great that the center continues to contract; the star then becomes so hot that
even heavier chemical elements are formed. But in the long run this process
also stops and the central parts contract even further. The contraction force