ADVERSARIOS Y SEGUIDORES DE KANT Adversarios

1.1.1 Stellar Radii

The stellar radius is a fundamental physical characteristic of a star. Unfortunately, this property of a star is not well known due to the difficulty to measuring it directly. In eclipsing binary systems, the radii are measured by the combination of the binary’s spectroscopic and photometric data, and absolute dimensions of their radii can be determined without the distance to the star being known. Although this is a straightforward approach to determining stellar radii, there are a limited amount of eclipsing binaries (52 individual components in the A, F, and G star range from the sample in Andersen 1991) to study in this manner.

Radii measurements of single stars are more challenging. They require special observing techniques to measure directly their small angular size,θ. The combination ofθwith trigono- metric parallax Π allows the linear radius to be determined. Thanks to the HIPPARCOS

mission (ESA 1997; van Leeuwen 2007), we now know accurate parallaxes (out to a certain distance) to most of the bright stars in the sky. However, because stars are at such great distances from us, they are typically unresolved point sources of light, so their angular sizes can only be determined with clever techniques in astronomy such as using lunar or Jovian occultation (LO, JO) events, speckle interferometry, and long-baseline optical interferometry (LBOI).

The largest stars to be resolved in our sky are evolved stars (e.g. supergiants and giants), where although they reside at large distances from the Sun, their big intrinsic radii provide angular sizes that are large enough to be easily resolved by lunar occultation observations and by interferometers with modest baselines. Stars that have not evolved off the main sequence far outnumber the evolved stars is our sky, because ≈ 90% of a star’s life is spent on the main sequence. However, the radius of a main sequence star is typically one to three orders of magnitude (or 10−1000) smaller than that of an evolved star, making it much smaller in angular size, despite its close vicinity to the Sun. These main sequence stars are also intrinsically several magnitudes dimmer than giants, due to their smaller radii.

The resolution limits to measuring the size of a single star using occultations (lunar or Jovian) or speckle interferometry depend on the size or diffraction limit of the telescope, and thus only the largest of stars may be observed with these techniques. Intensity inter- ferometery can measure the size of a star to great accuracy (dependent on the baseline), but is limited to bright stars as in the case of the Narrabri Stellar Intensity Interferometer (Hanbury Brown et al. 1974), which only observed stars brighter than B=2.5 mag). The CHARA Array, an amplitude (Michelson-type) interferometer, has the highest resolution of any interferometer in the world due to its long baselines, and, although the telescopes are only 1-meter in diameter, the sensitivity of the CHARA Array depends on the beam combiner and wavelength used for observation.

1.1.2 Stellar Effective Temperatures

In addition to measuring the linear radius of a star, we may determine another fundamental property of a star, the effective temperature, TEFF. This property provides the link be-

tween the theory of stellar structure and evolution and model atmospheres. The effective temperature of a star is defined through the Stephan-Boltzmann law:

F =σT_EFF4 (1.1)

where F is the total emergent flux of the star and σ is the Stefan-Boltzmann constant. Transforming this equation to observables at Earth, we arrive at the expression:

FBOL=

1 4θ

2_σT4

EFF (1.2)

where FBOL is the bolometric flux received at Earth, and θ is the angular diameter of the

star in radians. This is the only empirical method of determining a star’s temperature, and it mostly depends on the tricky task of measuring the angular diameter of the star. Fortunately, the error in the effective temperature is relatively insensitive to errors in θ or FBOL. For instance, because TEFF ∝ θ−2 then σ(TEFF) ∝ 1₂σ(θ), and because TEFF ∝ FBOL4

then σ(TEFF)∝ 1₄σ(FBOL) (Booth 1997).

The renowned results from the survey of angular diameters of 32 stars conducted by the Narrabri Stellar Intensity Interferometer (Hanbury Brown et al. 1974; Code et al. 1976) extended from O to F type stars, eleven of which were on the main sequence. The average accuracy of these angular diameter determinations depended primarily on the brightness of the object, and was≈ 6.5% for the 32 stars measured. Distance errors at the time were not of high accuracy, and only eleven of the stars had less than a 20% error in parallax, limiting the results of the linear radius derived from the angular diameter measurement as well. This survey (conducted more than three decades ago), has been a key resource in calibrating several less direct relationships to stellar properties.

One such relation was first established by Barnes & Evans (1976), with the use of angu- lar diameters of stars from lunar occultation (LO) measurements with other forms of direct measurements having been added to the calibration since this work was first published. It provides a relationship between the surface brightness of a star and its color index to the angular diameter of the star. Another technique, the Infrared Flux Method (IRFM), was first established by Blackwell & Shallis (1977). The IRFM embraces the idea that one can determine the angular diameter and temperature of a star simultaneously. A monochro- matic version of the method was developed by Gray (1967), where the observed spectral energy distribution is compared to a model spectral energy distribution of a star, so that by conservation of energy:

4πR2F = 4πd2FBOL (1.3)

where R is the radius of the star,F is the total flux emitted at the surface of the star, and d is the distance to Earth. Because θ= 2R/d, then we have the relation:

F FBOL

=θ2/4. (1.4)

The IRFM performs this same task, but assumes that the flux in the ratio of F/FBOL holds

for monochromatic wavelengths, in particular in the IR. In their work, Blackwell & Shallis (1977) justify this relation by arguing that there is a weak influence in the IR due to the temperature of the star versus the flux distribution (i.e., the monochromatic flux in the IR depends only on temperature to the first power, whereas the full integrated flux depends on the temperature to the fourth power). Smaller effects due to line-blanketing and opacity sources are more well known in this region as well. This method has developed sophistication

over the years to take these issues into account (see Gonz´alez Hern´andez & Bonifacio 2009, and references therein) and boasts a 1% accuracy on effective temperature determinations.

These relationships are extremely useful in extending our knowledge to a larger number of stars, at distances too far to resolve accurately their sizes. However, it has been noted over the years that in the absence of a more complete sample of stars, these relationships are only as good as the data upon which the calibrations were based (McAlister 1985).

1.1.3 Angular Diameters of Main Sequence Stars

As mentioned before, the Narrabri Stellar Intensity Interferometer (Hanbury Brown et al. 1974; Code et al. 1976) measured the angular diameters of eleven main sequence stars, pro- viding the means to calibrate properties of stars on the hot, massive end of the main sequence. For several decades, luminosity class I, II, and III stars were observed with interferometry, but no main sequence star earlier than A7 was observed (Davis 1997). As an update, the CHARM2 Catalogue1 _{(Richichi et al. 2005) is a compilation of stellar diameters by means}

of direct measurements by high angular resolution methods, as well as indirect estimates. The CHARM2 Catalogue includes all results as of July 2004, a total of 8231 entries, for 3238 unique sources. Of these 8231 entries, 905 are from direct measurements, and 458 of these are unique sources. Of the latter sample, 242 have errors in the angular diameter measurements of <5%, and only 24 of these reside on the main sequence (luminosity class V or IV). In a recent work by Holmberg et al. (2008), they remark that measurements of the angular diameters of main sequence F and G stars need to be better than 2%, yielding temperatures to 1%, in order for offsets in the color-temperature calibrations to be minimal.

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At that time, only nine stars met this criterion. This accuracy limit reiterates the target accuracy proposed by Blackwell et al. (1979) for the limits to the Infrared Flux Method, that a good TEFF determination goal should be 1% to match the best atomic data available

for abundance determinations and logg estimates (Davis 1985; Booth 1997).

The determination of accurate temperatures also becomes an important issue when de- termining stellar ages. Holmberg et al. (2007) give several good examples of how an offset in effective temperature will, in turn, offset the metallicity [Fe/H] measurements, and that these effects double up when determining the ages of the stars, thereby producing false age- metallicity relations. With 1% errors in the effective temperature scale, it is also possible to challenge stellar models to achieve greater accuracy than now attainable, by constraining mixing length theory and convective overshooting, to name a few issues at hand. The long baselines of the CHARA Array are uniquely suited for observing diameters of main-sequence stars to great accuracy.