Appendix A: Some More Background
279
Fig. A.3, where the angle E has been defined. This angle is called ‘eccentric
anomaly’. If you know E, ν is easy to find from the equations for the ellipse.
It does depend of course on the eccentricity of the orbit (0 for a circle, 1 for a
line). But E also does not increase uniformly with time.
Therefore, Kepler defined an imaginary ‘mean anomaly’, M, which does
run uniformly with time, but cannot be constructed in the figure. If you know
the time t, M is easy to calculate. But then you have to find E and for that you
have to use Kepler’s equation, which connects M and E. This, however, is not
straightforward to solve and had been a problem for studies of eccentric orbits
of asteroids and particularly comets. This is a mathematical problem and ways
had been found tot solve this, but it remained time consuming and tedious.
There were two general ways of tackling this problem. The first is by iteration,
i.e. repeatedly calculating a better approximation until the result is sufficiently
accurate. In that case you start with an initial guess of what M is for a particular
value of E and insert it into the equation in such a way that an improved guess
of M results. Insert that new value also until a sufficiently accurate value is
obtained. If e is significantly different from 0, this may take a long time.
The second method is by series expansion, in which M is written as an infinite
sum of terms with increasing powers of E, that decrease in value for increasing
E. One then only needs to calculate a limited number of terms. Even for an
asteroid in a fairly eccentric orbit, that may take a fair number of terms for the
series expansions used.
The solution proposed by Kapteyn falls into the second category and consisted of a new series. However, the actual improvement was marginal.
A.5 Stellar Evolution
The fundamental diagram in astrophysics is named after the Danish astronomer
Ejnar Hertzsprung and American Henri Norris Russell (see Fig. A.4). It is a
diagram that relates the temperature at the surface of stars to the luminosity.
The vertical axis is the luminosity, but for that also the absolute magnitude can
be used. On the horizontal axis we have the temperature at the surface of the
star, but one can also use for this the color index, for example the difference
between the magnitudes of the star in a blue (B) and a visual (V ) band. If the
star is relatively bright in blue, then the star is relatively hot on the surface.
But the spectral type can also be used for this, because the absorption lines in
the spectrum of the star are created by atoms or ions in the outer parts, which
absorb light at specific wavelengths; which atoms or ions are present and which
lines are prominent, is strongly influenced by the temperature. Spectral types
279
Fig. A.3, where the angle E has been defined. This angle is called ‘eccentric
anomaly’. If you know E, ν is easy to find from the equations for the ellipse.
It does depend of course on the eccentricity of the orbit (0 for a circle, 1 for a
line). But E also does not increase uniformly with time.
Therefore, Kepler defined an imaginary ‘mean anomaly’, M, which does
run uniformly with time, but cannot be constructed in the figure. If you know
the time t, M is easy to calculate. But then you have to find E and for that you
have to use Kepler’s equation, which connects M and E. This, however, is not
straightforward to solve and had been a problem for studies of eccentric orbits
of asteroids and particularly comets. This is a mathematical problem and ways
had been found tot solve this, but it remained time consuming and tedious.
There were two general ways of tackling this problem. The first is by iteration,
i.e. repeatedly calculating a better approximation until the result is sufficiently
accurate. In that case you start with an initial guess of what M is for a particular
value of E and insert it into the equation in such a way that an improved guess
of M results. Insert that new value also until a sufficiently accurate value is
obtained. If e is significantly different from 0, this may take a long time.
The second method is by series expansion, in which M is written as an infinite
sum of terms with increasing powers of E, that decrease in value for increasing
E. One then only needs to calculate a limited number of terms. Even for an
asteroid in a fairly eccentric orbit, that may take a fair number of terms for the
series expansions used.
The solution proposed by Kapteyn falls into the second category and consisted of a new series. However, the actual improvement was marginal.
A.5 Stellar Evolution
The fundamental diagram in astrophysics is named after the Danish astronomer
Ejnar Hertzsprung and American Henri Norris Russell (see Fig. A.4). It is a
diagram that relates the temperature at the surface of stars to the luminosity.
The vertical axis is the luminosity, but for that also the absolute magnitude can
be used. On the horizontal axis we have the temperature at the surface of the
star, but one can also use for this the color index, for example the difference
between the magnitudes of the star in a blue (B) and a visual (V ) band. If the
star is relatively bright in blue, then the star is relatively hot on the surface.
But the spectral type can also be used for this, because the absorption lines in
the spectrum of the star are created by atoms or ions in the outer parts, which
absorb light at specific wavelengths; which atoms or ions are present and which
lines are prominent, is strongly influenced by the temperature. Spectral types