2.3.1. The First Rung: Calibrating the Main Sequence Luminosities of Stars via Stellar Parallax

Galaxies emit optical radiation by the combined luminosity of their stellar populations. Therefore we must know the absolute brightnesses of different types of stars if we are to make any progress in determining the extragalactic distance scale. Hence the first rung of the distance scale ladder is firmly grounded in trigonometric parallaxes of nearby stars that anchor the lower end of the main sequence. The phenomenon of stellar parallax is a consequence of the Earth's revolution around the sun. Relative to the baseline defined by the earth-sun distance, a nearby star traces out a small ellipse in its apparent position as measured with respect to more distant stars. To accurately determine the distance to that nearby star requires two precise measurements:

The earth-Sun distance in physical units

the semi-major axis of the ellipse in angular units

The earth-sun distance can now be measured with high accuracy by first determining the distance to Venus via radar reflection timing. That distance, combined with the Sun-Venus-Earth angle, directly determines the distance to the sun. The second measurement is more problematic. In general, atmospheric aberration limits the angular resolution of a single ground-based image (photographic or digital) to 0.5 - 1.0 arcseconds. The distance a star must have in order to have a parallax of 1 arc-second is 3.26 light years (3.086 x 10¹⁸ cm). This distance is known as a parsec. The closest star to the sun has a distance of 4 light years. A star at a distance of 20 parsecs from the sun would exhibit a parallactic angle of only 0.05 seconds or arc, approximately a factor of 10 below the best angular resolution that can be achieved in a single image. While the centroid of a stellar image can be determined to a positional accuracy that is many times smaller than the seeing disk, the limiting factor for parallax measurements is systematic error associated with the determination of absolute position on the sky. These errors are best eliminated or reduced by repeated observation of the same star. Thus, determining the absolute luminosity scale of stars, based on distances to nearby stars, is a time consuming process that requires many years of measurement in order to reduce the random error in the distance determination. In general, random errors as low as +/- 0.01 arcseconds have been obtained using photographic plates as the imaging system. Still, at a distance of 20 parsecs, this represents a 20% error in distance and, worse still, a 45% error in intrinsic stellar luminosity. Although there are some 8000 stars whose parallax has been determined, most of these are relatively low quality and have formal errors that are not well determined. We thus regard 20 parsecs as a limiting volume out to which stellar distances can be directly determined with reasonable accuracy, from photographically based measurements.

If we consider the disk of our galaxy as having a radius of 10 kiloparsecs (kpc), and a thickness of 1 kpc, then the volume that we sample with trigonometric parallax is only 10^-7 of the total volume. How can we be sure that this tiny volume contains a representative sample of stars? Fortunately, a theoretical argument involving stellar lifetimes serves as a consistency check. In general, star formation events in our galaxy make many more low mass stars than high mass ones. Since high mass stars have very short lifetimes, we would not expect to find many in a volume limited sample, nor would we expect to find many stars that are in short-lived evolutionary phases. Hence, a representative sample of stars should be heavily weighted towards low-mass main sequence stars (plus a few white dwarfs) and this is generally what is observed in the trigonometric parallax sample.

A long term program underway at the United States Naval Observatory employs CCD measurements to improve the accuracy of stellar parallax determinations. To date, measurements of approximately 200 stars indicate a median error of +/- 1.1 milliarcseconds, which is an order of magnitude improvement over what is available in the current Yale Parallax catalog. In principle then, distance measurements for stars with distances up to 200 parsecs can now be made. In addition, the HIPPARCOS astrometric satellite made measurements (to be released in spring of 1997) of about 100,000 stars down to 8th magnitude with a median error of 1.8 milli-arcseconds. These new data bases should greatly improve the precision for which we know lower main sequence luminosities. More importantly, the larger volume accessed by these two surveys should allow for direct determinations of distances to Population II subdwarfs. As will be seen later, this can provide a very important and independent check of the distance scale(s) which are ultimately derived from the positions of stars in the HR diagram.

With the HIPPARCOS data and the CCD data of the United States Naval Observatory a check on the reliability of the old photographic parallax scale can be done. Any systematic error in the distances to the nearest stars changes the luminosity of the main sequence and hence affects all distances that are subsequently derived. A preliminary analysis of the HIPPARCOS parallax data by Perryman et al. (1995) shows the following:

The median standard error for all parallax measurements is 1.5 milliarcseconds.

Any systematic errors are less than 0.1 milliarcseconds.

There is no significant difference in the position of the lower main sequence as defined by previous parallax measurements of stars located within 10-15 pc.

There are some serious disagreements for individual stars with distances nominally within 25 pc of the sun between the HIPPARCOS data and specific ground-based observations. This will cause a re-evaluation of the masses and luminosities of these stars but, overall, will not have a profound effect on the placement of the main sequence (many of the discrepant stars are not main sequence stars but or Red or Blue giants).

In sum, the parallax measurements do allow for a reliable determination of stellar luminosity for main sequence stars. As shown below, this provides a means to calibrate the distance to a stellar cluster in which the main sequence can be identified.