History of Astronomy Example Exam Question Hints and Suggestions

• 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 19^th century technology).
  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 19^th 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 19^th 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, 19^th century physicists and astronomers generally accepted Kelvin's idea (especially as Kelvin himself had immense prestige in British physics).

The models:

• Zöllner's cooling hypothesis, in which stars start out white and turn red as they age;
• 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.

• discovery of radioactivity in late 19^th 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 in stars.
• 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).