This is bookwork, though you need to remember to answer both parts of the question!
The main 19th century theories both relied on conversion of gravitational potential
energy, but differed in the assumed source of the potential energy:
- Lockyer's meteoritic hypothesis assumed the potential energy came from the
infall of small solar system bodies (comets and meteors). An infalling object of mass
m loses potential energy GMm/R, where M and R are the Sun's mass
and radius, so generating the Sun's current luminosity of ~4×1026 W requires
an infall of about 6×1022 kg/year.
This is inadequate because it changes the Sun's mass by ~3×10-6% per
year. This would produce a small but measurable change in the Earth's orbit
(about half a second per year, quite within the capabilities of 19th century
Therefore we do not want to add mass to the Sun. The alternative is
- the Kelvin-Helmholtz contraction hypothesis, in which the Sun as a whole
shrinks under gravity. The potential energy of the Sun is proportional to
GM2/R, with a constant of proportionality which depends on
how the Sun's density varies with radius (it's 0.6 for a uniform Sun); the Sun's
current luminosity requires a shrinkage rate of around 100 m/yr, which is not
detectable with 19th century technology.
This is inadequate because it implies a solar lifetime of ~10 million years
(this is the time taken for the Sun to shrink from infinite radius to the present measured
radius while maintaining the same luminosity), which is beginning to look rather small
compared to the requirements of 19th century geology and evolutionary biology.
Note that the objection to Lockyer's hypothesis is much stronger than the objection to
Kelvin's (at this time, geologists cannot prove their timescale for the age of the
Earth). Therefore, 19th century physicists and astronomers generally accepted
Kelvin's idea (especially as Kelvin himself had immense prestige in British physics).
Note that there are four parts to this question: you need to describe the models,
relate them to the theories, and discuss how they stand up to both the HR
diagrams and the mass-luminosity diagram. (These are really the wrong way round
in the question: the HR diagram is earlier than the mass-luminosity relation.)
- Zöllner's cooling hypothesis, in which stars start out white and turn red as
- Lockyer's hypothesis, in which stars start out large and red, contract and heat up
to become small and white, and then cool to small and red.
Their relationship to the theories:
- Zöllner's model would match Lockyer's meteoritic hypothesis, if we assume that
the amount of infalling debris decreases with time (which seems reasonable). It doesn't
match the contraction model, because if we differentiate GM2/R we
get something proportional to R -2: it would be natural for stars
to get brighter as they contracted, not fainter.
- In contrast, Lockyer's model matches the contraction hypothesis beautifully: the
large, low-density red giants contract until their density is too great to allow further
contraction; they then cool. It doesn't match the meteoritic hypothesis particularly
well: there seems no motivation in this theory for the change in direction of evolution
(red to white to red).
Their relationship to the HR diagram:
- Zöllner's model desn't explain why there are two classes of red star: the
large, luminous red giants and the small, faint red "dwarfs" (main sequence stars). In
Zöllner's model, there should be no red giants.
- Lockyer's model matches the HR diagram very well: the red giant branch is the
contraction phase, and the main sequence is the cooling phase.
Their relationship to Eddington's mass-luminosity relation
In both these models, all stars go through the same process: the mass
determines how fast they do it, not what they do (and there is no unanimity about whether
higher mass stars should evolve faster or slower, though thermodynamics suggests the former).
In contrast, Eddington's results show
that, on the main sequence at least, luminosity appears to be determined by mass. This
would be understandable in a star cluster (though the direction of the relation implies
that low mass stars must evolve faster, which is unexpected), but is much more difficult
to explain given that Eddington's results are not for stars in the same cluster.
The "correct explanation" we are looking for is hydrogen fusion (the stages involving
fusion of heavier elements were mostly worked out after 1940).
The points we need to make are:
You need all these points, with names and (at least approximate) dates, for full marks -
4 marks represent 16 minutes' work, so a fair amount of detail is expected.
- discovery of radioactivity in late 19th century offers prospect of new
energy source capable of powering stars for long periods;
- understanding that heavier atoms generally weigh less than the equivalent
number of hydrogen atoms (Morley 1895; more precise values by Aston in the 1920s), plus
Einstein's E = mc2 (1905), suggests that fusion of light elements might
generate energy (hydrogen fusion first suggested by Eddington in 1920);
- the Bohgr atom (1913) and its further development led to a quantitative understanding
of spectra. As a result, Saha's equation (1921) applied by Payne (1924)
showed that stars are mostly hydrogen,
further supporting the idea of hydrogen fusion.
- Initially, hydrogen fusion looked unlikely because stellar interiors are not hot
enough to allow the protons to overcome their mutual Coulomb barrier and fuse. However,
the uncertainty principle (Heisenberg) led to the idea of quantum mechanical tunnelling
(Gamow; Atkinson, Houtermans, 1929) which showed that it was in principle possible to fuse hydrogen
- Finally, the exact routes by which hydrogen can be fused to helium (the CNO cycle
and the pp chain) were worked out by Hans Bethe and collaborators (1939).