5. STELLAR RADII
The luminosity radiated per unit of area of a stellar photosphere depends only on the photospheric temperature, i.e., on the spectral type. For example, we pointed out that each square centimeter of the Sun's photosphere radiates about 6500 Watts. Experiments on Earth show that the power per unit area radiated by a blackbody surface depends on the fourth power of the temperature. I.e., double the temperature and the power goes up by a factor 24 = 16. Stellar photospheres radiate almost like a blackbody. So a hot star will radiate much more per unit surface area than a cool star.
But, given equal temperatures, a large star will radiate more luminosity than a small star, simply because it has more surface area. We can put these two facts together in the following formula for the stellar luminosity:
L = 4p R2s T4, where R is the radius of the photosphere, s is a physical constant that can be measured from laboratory experiments, and T is the photospheric temperature. This equation is called the Stefan's Law . It is a powerful tool, because we can use it to infer a star's radius, a quantity that we cannot observe directly for most stars, from quantities that we can observe directly: the star's parallax (from which we can infer its distance), its brightness (from which, with distance, we can infer its luminosity L), and its spectral type (from which we can infer its photospheric temperature, T). Given L and T, and the known value of s, we can use Stefan's Law to solve for the star's radius R.If a star is cool but very luminous, it must be very big. For example, the star Betelgeuse (spectral type M2 I) is cooler than the Sun but its luminosity is more than 10,000 times greater than the Sun's. That is possible because its photosphere extends to a huge radius -- about 400 times the Sun's radius. If the Sun were this big, its photosphere would extend beyond the Earth's orbit! We call such stars
red giants.This method of inferring the radii of stars based on physical laws discovered in laboratories on Earth depends on a giant principle, which we call
The Universality of Physical Laws. This principle asserts that the laws of nature are the same everywhere, in Earth laboratory and thousands, even billions of light years away. Really, it's an assumption. We shouldn't take it for granted; we must constantly test it whenever we get a chance. Today, we are able to test the universality of the Stefan-Boltzmann law because we can measure the radius of nearby red giants such as Betelgeuse directly by stellar interferometry. We find that the radius measured by interferometry is, within the uncertainty of the measurement, the same as the radius inferred by the Stefan-Boltzmann law. So, at least in this case, the principle works.Arthur S. Eddington understood all this in 1920. You can find his discussion of the radii of stars in
The Internal Constitution of the Stars, which you should have read already while studying about the Sun.Because they can't travel to stars, astronomers are always using the Principle of Universality of Physical Laws. For example, they use when they infer the temperatures and atomic abundances of stellar photospheres by comparing stellar spectra with laboratory spectra.
So far, nobody has found any deviation from the Principle of the Universality of Physical Laws. You might then ask: why not just take it for granted? But that attitude goes against the whole philosophy by which science progresses.
There's a saying I like about the difference between religion and science: "Religion is based on faith; science is based on doubt." Most religions require that their followers take a leap of faith, accepting some tenets that cannot be proved by scientific experiments (and perhaps cannot be disproved either). Such acts of faith may greatly enhance one's religious experience.
But scientists make the greatest advances when they can overturn some commonly held belief. For example, with his Theory of Relativity, Einstein revised Newton's laws of motion and overturned commonly held ideas about the absolute nature of time.
Some great scientists have proposed theories in which the laws of nature change over great distances or times. If anyone could devise a test to verify such a theory, he or she would certainly win the Nobel Prize. But so far, nobody has found any violation of the Principle of Universality of Physical Laws.
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Last modified September 17, 2000
Copyright by Richard McCray