In Chapter 11 we derived an analytic approximation for the pressure, density and temperature variation sufficiently near the centre of a solar mass star on the main sequence, the Clayton model. In Chapter 17 we gave a purely descriptive account of the evolution of stars of various masses both during and beyond the main sequence. In this Chapter we return to the detailed mathematical structure of main sequence stars, but with the objective of covering stars of arbitrary mass.
Initially we consider an alternative approximate model based on the assumption of a polytropic equation of state, i.e. that pressure is proportional to a certain fixed power of density, P = K.density^gamma. Together with hydrostatic equilibrium this leads to a second order, non-linear differential equation for the density.
Integrating this hydrostatic-polytropic equation provides a fairly good representation of the Sun. This model contains four arbitrary constants: the two coefficients in the polytropic equation, plus two integration constants. One constant (K) is fixed by specifying the mass of the star, whilst the requirement that the density is maximum at the centre determines one of the integration constants. The other two constants, taken to be gamma and the density at the centre of the star, are not determined by the model and are adjusted to give a reasonable fit to the known results for the Sun. Specifically we find that a central density of 90,000 kg/m3 together with gamma = 1.33 give a good representation of the Sun. These reproduce:-
An expression is derived for the radiative heat flux in terms of the temperature, the density, the temperature gradient and the opacity.
Simple power-law expressions in temperature and density are given for the power density resulting from the pp and CN reaction sequences. They are compared with power densities obtained from published reaction rates. The CN power density is proportional to the density of nitrogen-14, and hence the CN sequence does not apply early in the life of first generation stars. The pp power density is roughly proportional to T^4 whereas the CN power density is roughly proportional to T^16. Consequently, the CN sequence dominates at higher temperatures. We find that the pp sequence dominates below 19 million K, and the CN sequence dominates above this temperature.
Whilst the hydrostatic-polytropic model is quite successful in terms of the above results, it was necessary to provide it with the values of two parameters from a more detailed model. Consequently, the hydrostatic-polytropic model is not fundamental. The ingredient which is missing from the models considered so far is the requirement that the rates of heat transport balance against the rate of heat production at every point (without which we would not have a quasi-static solution). The correct, complete, set of fundamental equations is presented here for the case of radiation dominated heat transport. The polytropic equation is not required, but rather the pressure and the density will both follow from solving the complete set of equations.
We do not attempt to solve the complete equation set here. This would have to be by numerical integration. Instead we note that very informative scaling relations may be obtained in the case that the opacity and the power density can both be approximated by power laws in density and temperature. In this case it is shown that the solution is essentially 'the same' for stars of different masses, when suitably scaled. The scaling required is that the dependent variables pressure, density, temperature and luminosity are scaled by (M/M0)^x and their distributions are plotted against the fractional mass parameter (m/M). The stellar equation set is used to determine the powers x for each field quantity for this so-called 'homologous model'.
Assuming the (constant) Thompson opacity and perfect gas behaviour we deduce that luminosity is proportional to the cube of the star's mass, and that the stellar lifetime is inversely proportional to the square of the mass. These results obtain independently of the nuclear reaction rates. Thus, contrary to naive expectation, more massive stars burn out far more quickly than lighter stars. More detailed models suggest an even greater sensitivity to mass, the exponents being closer to 3.5 and -2.5 respectively.
Two sets of scaling behaviours for the pressure, density, central temperature and stellar radius are presented: one for CN dominance and one for pp dominance. The latter applies for stars of mass below M0 ~ 1.8 solar masses, and the former for heavier stars. Some results are unsurprising, e.g. that the radius of the star and its central temperature increase monotonically with increasing mass. However, we find, contrary to naive expectation, that the density reduces monotonically with increasing mass. More massive stars are disproportionately larger and hence have lower densities. More surprisingly still, we find that stars with masses either less than M0 or greater than M0 have lower pressures! Thus, of all main sequence (hydrogen burning) stars, those of 1.8 solar masses have the largest pressure.
The homologous model result for luminosity implies a surface temperature proportional to mass to a power ~0.34 for more massive stars, or to power ~0.64 for less massive stars. Hence, more massive stars are hotter - but note that this statement relates solely to main sequence stars. [Red giants might be very massive, but with low surface temperature].
Having deduced both the luminosity and the surface temperature we can finally derive an expected slope for the Hertzsprung-Russell diagram, which is a logarithmic plot of luminosity versus surface temperature (colour). Our estimates derived from the homologous model are 5.6 to 8.8, which compare reasonably well with the observed value of ~7.
Finally, in an Annex to this Chapter, we derive from first principles the temperature dependence of the rate-controlling CN reaction. The purpose is to explain the extreme temperature sensitivity of this reaction (proportional to temperature to a power of around 16). The explanation lies in the usual Gamow peak, which provides an exponential dependence on temperature. The coefficient appearing in the exponent depends upon the product of the reactants' atomic numbers and their atomic masses. Thus, the exponent is a larger (negative) number for the CN sequence because the rate controlling step involves nitrogen-14, a heavier nucleus than those involved in the pp sequences. The reason why these truely exponential temperature dependencies are often approximated as power laws should now be apparent - they facilitate approximate scaling results as discussed above for the homologous.
An advantage of the polytropic model over the Clayton model is that it leads to a clearly defined surface where the pressure, density and temperature drop to zero. Using the known nuclear reaction rates for a given temperature and density, the total luminosity of the star can be found from the above model results. We estimate a surface temperature of 5915 K (cf. the accepted value of ~5885 K) for a luminosity of 4.21 x 10^26 W.
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The Cat's Eye Nebula (NGC6543) in Draco: a planetary nebula. The central star has ejected most of its initial 5 solar masses during its red giant phase. Current mass about one solar mass with a very high surface temperature of ~80,000K. One of William Herschel's many discoveries.