Stellar Structure and Evolution

One of the triumphs of stellar evolution theory is a detailed understanding of the preferred location of stars in the physical Hertzsprung-Russell diagram, which plots luminosity versus temperature (Figure 17). There are a number of uncertainties associated with stellar evolution models, and hence age estimates based on the models.

Probably the least understood aspect of stellar modeling is the transport process of matter, angular momentum and magnetic field at macroscopic and microscopic levels, including in particular the process of convection. Numerical simulations hold promise for the future, but at present one must view properties of stellar models which depend on the treatment of convection to be uncertain, and subject to possibly large systematic errors. Main sequence stars and red giants have surface convection zones. Hence, the surface properties of the stellar models (such as its effective temperature, or colour) are rather uncertain. Horizontal branch stars have convective cores, so the predicted luminosities and lifetimes of these stars are subject to possible systematic errors. Other domains such as the statistical physics at high density and/or low temperature or the nuclear reaction rates of heavy nuclei also require improvement.

This lack of knowledge has consequences on topics as fundamental as the chemical evolution of the Universe, the rate of formation of heavy elements and of dust in the interstellar medium, and on the measurement of the age of the Universe. Understanding the dynamics of stellar interiors remains a key challenge for astronomy.

A stellar model is constructed by solving the four basic equations of stellar structure: (1) conservation of mass; (2) conservation of energy; (3) hydrostatic equilibrium and (4) energy transport via radiation, convection and/or conduction. These four, coupled differential equations represent a two point boundary value problem. Two of the boundary conditions are specified at the centre of the star (mass and luminosity are zero), and two at the surface. In order to solve these equations, supplementary information is required. The surface boundary conditions are based on stellar atmosphere calculations. The equation of state, opacities and nuclear reaction rates must be known. The mass and initial composition of the star need to be specified. Finally, as convection can be important in a star, one must have a theory of convection which determines when a region of a star is unstable to convective motions, and if so, the efficiency of the resulting heat transport. Once all of the above information has been determined a stellar model may be constructed. The evolution of a star may be followed by computing a static stellar structure model, updating the composition profile to reflect the changes due to nuclear reactions and/or mixing due to convection, and then re-computing the stellar structure model.

  

Figure: Colour-magnitude diagram of the globular cluster M15 ([Durrell & Harris1993]).

The agreement between predicted and observed properties of stars has remained qualitative due to the modest accuracy and relative scarcity of the relevant observed quantities. The development of accurate astrometry with Hipparcos, and high-resolution, high signal-to-noise ratio spectroscopy has allowed major progress in this area, but measurement of the global stellar parameters is often insufficient for testing the internal regions of the stars, where the evolution proceeds. To date, direct information on these regions is available only for the Sun, but even there theory is unable to reconcile the observed neutrino flux with the helioseismology data.

The next decade will see the opening of the field of asteroseismology from space. It will provide for the first time direct indicators of the physical status of stellar interiors. The corresponding data, combined with measurement of global parameters, as provided by GAIA will allow a major step improvement in our understanding of stellar evolution.

Stellar oscillation frequencies can be used to constrain stellar evolution through both a direct as well as an inverse approach. In the former, given initial approximate stellar parameters and a set of stellar models, the frequency information can be used to derive the fundamental stellar parameters, i.e., the mass and radius of the oscillating stars, thus placing it on the H-R diagram and determining its evolutionary status ([Christensen-Dalsgaard et al.1995]; [Guenther & Demarque1996]; [Petersen & Christensen-Dalsgaard1996]; [North et al.1997]; [Popper1997]). In the inverse approach, the observed fundamental stellar parameters are taken as a starting point, i.e., the mass M, the stellar radius R, the effective temperature $T_{\rm eff}$, and the chemical composition. With the exception of the double-lined eclipsing binary systems, the accurate determination of fundamental stellar parameters requires knowledge of the distance. Given a stellar structure model, the oscilation frequencies are then predicted, and compared with the measurements, thus testing the underlying physics of the assumed stellar structure.

In the few cases for which asteroseismology has been used to derive accurate stellar parameters, i.e. for the nearby G sub-giant $\eta$ Boo ([Bedding et al.1998]) and for the two $\delta$ Scu stars SX Phe and AI Vel ([Høg & Petersen1997]), the resulting uncertainties on the stellar parameters, and in particular on the absolute luminosity, match very well the uncertainty deriving from the Hipparcos parallaxes. With the precision offered by GAIA parallaxes, the challenge for the models (and for asteroseismology) to match the accurately measured luminosity will be much greater.

As described in Section 2.3, the global stellar parameters used in the field of stellar structure and evolution need to be obtained by GAIA itself. The key to success is the building of a complete and homogeneous sample covering a large variety of independent parameters. The stellar absolute luminosity is derived from the parallax and the apparent magnitude, corrected for extinction, which can be deduced from the GAIA photometric and spectroscopic data. These also provide the effective temperature $T_{\rm eff}$, metallicity indicators, and the projected rotational velocity of the stars, $v\sin i$. Ages can then be inferred from the location of stars in the Hertzsprung-Russell diagram.

Unfortunately, masses can be measured directly only in special cases, where the gravitational interactions with other bodies are easily measurable. This is the case for binaries, discussed in Section 1.5. An additional possibility comes from astrometric microlensing (Section 1.4.10). The large number of systems for which the mass will be measured by GAIA will be used to validate the modeling of stars for which mass is known, providing in turn indirect estimates of the mass of other stars, for example through the mass-luminosity relation.

Single stars, which will most likely constitute the majority of targets for future asteroseismology space missions such as COROT, MOST and MONS are, from the point of view of a priori accurate stellar parameters, the worst case. Dynamical mass determinations cannot be performed on individual stars, for which the only possible mass determination will be through models. Accurate luminosities can be derived from accurate parallaxes and photometry, which, given $T_{\rm eff}$, will yield stellar radii.