Eddington Solar Model

where is the total mass contained within a sphere of radius , centered on the origin, and the mass density at radius . Now, as is well known, the gravitational acceleration at some radius in a spherically symmetric mass distribution is the same as would be obtained were all the mass located within this radius concentrated at the center, and the remainder of the mass neglected (Fitzpatrick 2012). In other words,

(13.22) |

where is the gravitational potential energy per unit mass, and the radial gravitational acceleration. The force balance criterion (13.1) yields

(13.23) |

where is the pressure. The previous three equations can be combined to give

In order to make any further progress, we need to determine the relationship between the Sun's internal pressure and density. Unfortunately, this relationship is ultimately controlled by energy transport, which is a very complicated process in a star. In fact, a star's energy is ultimately derived from nuclear reactions occurring deep within its core, the details of which are extremely complicated. This energy is then transported from the core to the outer boundary via a combination of convection and radiation. (Conduction plays a much less important role in this process.) Unfortunately, an exact calculation of radiative transport requires an understanding of the opacity of stellar material, which is an exceptionally difficult subject. Finally, once the energy reaches the boundary of the star, it is radiated away. The following ingenious model, due to Eddington (Eddington 1926), is appropriate to a star whose internal energy transport is dominated by radiation. This turns out to be a fairly good approximation for the Sun. The main advantage of Eddington's model is that it does not require us to know anything about stellar nuclear reactions or opacity.

Now, the temperature inside the Sun is sufficiently high that radiation pressure cannot be completely neglected with respect to conventional gas pressure. In other words, we must write the solar equation of state in the form

(13.25) |

where

is the gas pressure (modeling the plasma within the Sun as an ideal gas of free electrons and ions), and (Chandrasekhar 1967)

(13.27) |

the radiation pressure (assuming that the radiation within the Sun is everywhere in local thermodynamic equilibrium with the plasma). Here, is the Sun's internal temperature, the Boltzmann constant, the mass of a proton, and the relative molecular mass (i.e., the ratio of the mean mass of the free particles making up the solar plasma to that of a proton). Note that the electron mass has been neglected with respect to that of a proton. Furthermore, , where is the Stefan-Boltzmann constant, and the velocity of light in a vacuum. Incidentally, in writing Equation (13.26), we have expressed in the equivalent form .

Let

where the parameter is assumed to be uniform. In other words, the ratio of the radiation pressure to the gas pressure is assumed to be the same everywhere inside the Sun. This fairly drastic assumption turns out--perhaps, somewhat fortuitously (Mestel 1999)--to lead to approximately the correct internal pressure-density relation for the Sun. In fact, Equations (13.26)-(13.29) can be combined to give

where

It can be seen, by comparison with Equation (13.4), that the previous pressure-density relation takes the form of an adiabatic gas law with an effective ratio of specific heats . Note, however, that the actual ratio of specific heats for a fully ionized hydrogen plasma, in the absence of radiation, is . Hence, the exponent, appearing in Equation (13.30), is entirely due to the non-negligible radiation pressure within the Sun.

Let , , and , be the Sun's central temperature, density, and pressure, respectively. It follows from Equation (13.30) that

and from Equations (13.26) and (13.28) that

Suppose that

(13.34) |

where is a dimensionless function. According to Equations (13.26), (13.28), and (13.30),

(13.35) | ||

(13.36) |

Moreover, it is clear, from the previous expressions, that at the center of the Sun, , and at the edge, (say), where the temperature, density, and pressure are all assumed to fall to zero. Suppose, finally, that

where is a dimensionless radial coordinate, and

Thus, the center of the Sun corresponds to , and the edge to (say), where , and

Equations (13.35)-(13.38) can be used to transform the equilibrium relation (13.24) into the non-dimensional form

Moreover, Equation (13.21) can be integrated, with the aid of Equations (13.35), (13.37), and (13.40), and the physical boundary condition , to give

where

(13.42) |

Equation (13.40) is known as the

According to Equation (13.41), the solar mass, , can be written

which reduces, with the aid of Equations (13.31) and (13.38), to

where

and

(13.46) |

Moreover, it is easily demonstrated that

According to Equations (13.44) and (13.45), the ratio, , of the radiation pressure to the gas pressure in a radiative star is a strongly increasing function of the stellar mass, , and mean molecular weight, . In the case of the Sun, for which , Equation (13.44) can be inverted to give the approximate solution

(13.48) |

Using the observed solar mass , and the value (which represents the best fit to the Standard Solar Model mentioned in the following), we find that . In other words, the radiation pressure inside the Sun is only a very small fraction of the gas pressure. This immediately implies that radiative energy transport is much less efficient than convective energy transport. Indeed, in regions of the Sun in which convection occurs, we would expect the convective transport to overwhelm the radiative transport, and so to drive the local pressure-density relation toward an adiabatic law with an exponent . Fortunately, convection only takes place in the Sun's outer regions, which contain a minuscule fraction of its mass.

Equations (13.32), (13.33), (13.39), (13.43), and (13.47) yield

(13.49) | ||

(13.50) | ||

(13.51) |

where the solar radius has been given the observed value . The actual values of the Sun's central temperature, density, and pressure, as determined by the so-called