Astronomy 162: Introduction to Stars, Galaxies, & the Universe Prof. Richard Pogge, MTWThF 9:30

Notes to Unit 2

The Internal Structure of Stars

[11.1] The data used in the graph were taken from a compilation by D.M. Popper 1980, ARA&A, 18, 115. I have adopted a simple approxmation of L=M⁴ for numerical convenience in these lectures. A detailed fit to the data for M>0.4M_sun gives an exponent of 3.92+/-0.04, close enough to 4 for our purposes.

[11.2] In reality, stars are denser in their cores and become less dense as you move outwards to the surface. The mean stellar density is just a simple way to quantify how tightly packed the matter is within a given star, and has the virtue that is easy to compute (by assuming stars are spherical and the matter is uniformly distributed throughout). For the numbers used in this section, I have taken typical masses and radii for stars from Astrophysical Quantities. The mean density of air is for standard temperature and pressure at sea level, taken from the Handbook of Chemistry & Physics.

[11.3] The classical ideal gas law is more conventionally written as:

   PV = NkT

where P is the gas pressure, V is the volume occupied by the gas, N is the number of gas particles, T is the gas temperature, and k is the Boltzmann Constant. This can be rewritten as

   P = nkT

where I have substituted in n=N/V, which is the number density of gas particles (the number of particles per unit volume). This latter form lets me state the pressure in terms of density and temperature, as done in the notes.

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As Long as the Sun Shines

[12.1] William Thomson, First Lord Kelvin (1824-1907), an Irish-Scots mathematician and physicist educated at Cambridge (UK), and one of the pre-eminent scientists of the 19th century. An excellent biographical sketch is available on the Wikipedia.

[12.2] Hermann von Helmholtz (1821-1894), German physicist. An brief biographical sketch is available on the Wikipedia.

[12.3] A modern estimate based on a detailed numerical calculation of the Kelvin-Helmholtz timescale for the current reference solar interior structure model would give an age of ~30Myr, a little bit larger than Helmholtz' 22Myr. The original 19th-century K-H timescale calculation was an order-of-magnitude estimate of how long it would take the Sun to radiate away its gravitational potential energy assuming (a) that the Sun is a uniform-density sphere, and (b) that its luminosity stays the same throughout the contraction process. Both assumptions (a) and (b) are incorrect in detail, but such a calculation nonetheless yields an timescale of approximately 22Myr. The longer timescale of 30Myr quoted here addresses both these issues, and is a round number coming from modern numerical models of stellar interiors. Note that a similar, slightly smaller estimate, was also done about this time by American astronomer Simon Newcomb, using similar techniques.

Kelvin's age estimate was based on a calculation of how long it would take for a molten Earth to cool to its present temperature, which gave a range of ages between 24 and 400Myr. Because of the general agreement between the solar contraction age (Helmholtz) and earth cooling age (Kelvin), the initial response of many scientists of the late 19th century was that these two essentially independent estimates being in agreement was confirmation of this age of 20-30Myr for the Sun. Both estimates are flawed (Helmholtz and others didn't know about nuclear fusion as a stellar energy source, and Kelvin's estimates did not take into account then then-unknown additional heating due to natural radioactivity), so their apparent agreement is coincidental. Both estimates were at great odds with even conservative estimates of the age of the Earth being done by geologists, and so were not much accepted outside of physics and astrophysics circles.

[12.4] The proton-proton (PP) chain shown in the notes for this and the next lecture is more formally known as "PPI", one of 3 possible nuclear reaction chains involving proton-proton reactions. The two alternative PP chains involve reactions with existing Helium nuclei. The second chain, "PPII", becomes important between temperatures of between 14 and 23 million K and involves fusion with existing ⁴He and one of the ³He nuclei from a step in the PPI chain:

  3He + 4He -> 7Be + photon
  7Be + e- -> 7Li + neutrino
  7Li + 1H -> 4He + 4He

Note that the ⁴He nucleus that enters at the top of the PPII chain exits at the bottom (the net result is production of a new ⁴He from 4 protons).

The third chain, called "PPIII", becomes important at temperatures above 23 million K. It starts the same as PPII, but instead of an electron capture by ⁷Be in the second step, ⁷Be captures a proton to make unstable ⁸B (boron), which sends it into the PPIII chain:

  3He + 4He -> 7Be + photon
  7Be + 1H -> 8B + photon
  8B -> 8Be + e+ + neutrino
  8Be -> 4He + 4He

As with PPII, the ⁴He nucleus that enters at the top of the chain exits at the bottom with a new ⁴He produced from 4 protons inserted in the previous steps. The so-called "Boron-8 neutrino" produced in the 3rd line above is important for studies of solar neutrino production.

All three reactions run simultaneously within a hydrogen-burning star. In the Sun, PPI is responsible for ~86% of the total energy generated by PP fusion, PPII produces ~14%, and PPIII produces only about 0.11%.

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Energy Generation & Transport in Stars

[13.1] The CNO cycle shown here is the main part of the so-called "CNO bi-cycle", which includes a secondary branch:

   15N + 1H -> 16O + photon
   16O + 1H -> 17F + photon
   17F -> 17O + e+ + neutrino
   17O + 1H -> 14N + 4He

Note that the first step above is an alternative branch of the last step in the main CNO cycle:

   15N + 1H -> 12C + 4He

The ¹⁵N proton capture reaction that produces ¹⁶O and starts the second subcycle is very rare: it only occurs ~0.04% of the time, or on average once every 2500 reactions. The ¹⁴N created by the last step of the second cycle feeds back into the main CNO cycle in the middle. The net result is that over time C and N are slowly turned into ¹⁶O, which itself gets transformed into ¹⁴N and injected back into the main CNO cycle. Yeah, it is complicated.

[13.2] The PP and CNO reaction cycles run simultaneously in hydrogen-burning stars. At low temperatures, PP is more efficient than CNO. For example, CNO fusion accounts for only about 2% of the energy generated in the Sun's core when averaged over the entire nuclear burning region. Since progressively more massive stars have higher core temperatures, the overall fusion rate from both processes increases with increasing stellar mass. However, because CNO is more sensitive to temperature than PP, while both grow in strength as temperature increases, the CNO reaction rate grows stronger faster, and eventually surpasses PP as the main contributor to the total fusion energy output of the core. The cross-over between PP-dominated and CNO-dominated to occurs when the stellar mass is above about 1.1-times the mass of the Sun.

[13.3] This quantity is known as the "Photon Diffusion Timescale". The value of 200,000 years given in the notes was rounded up from an estimate of 170,000 years computed by Mitalas & Sills, 1992, ApJ, 401, 759.

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Star Formation

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The Main Sequence

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The Evolution of Low-Mass Stars

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The Evolution of High-Mass Stars