# How do astronomers use Cepheid luminosities to calculate distances?

ERNI HALAXA, DUBLIN, CALIFORNIA
| Published: January 1, 2005
Cepheid distances are determined through the so-called inverse-square law of light, which tells astronomers how a star’s apparent brightness diminishes with distance. If a star appears to have a certain brightness at some given distance, then it will appear 4 times fainter at twice that distance, 9 times fainter at triple that distance, and so on.
Conversely, if you know the intrinsic brightness of a star and measure its apparent brightness, you can determine its distance. Expressed in mathematical terms, the relationship is

log d = 0.2 (m – M + 5)

where d is the distance of the star in units called parsecs (1 parsec is 3.26 light-years), m is the measured brightness of the star in magnitudes, and M is the star’s intrinsic brightness in magnitudes, formally called its absolute magnitude.

Usually, the problem in applying this formula lies in how we determine absolute magnitude, and it is here that Cepheids shine. These pulsating stars expand and contract with periods ranging from roughly 1 day to more than 50 days. Fortunately, a few Cepheids are members of star clusters, for which astronomers can derive distances by other means. These cluster Cepheids inform us that their pulsation periods (P) and absolute magnitudes share a mathematical relathionship:

M = –2.9 log P – 1.2

A Cepheid changes its brightness throughout its pulsation cycle, so astronomers find the value of P by measuring the star’s apparent brightness through one full cycle. Values for absolute and apparent magnitudes are averages over the pulsation cycle. Plugging them into the first equation allows astronomers to determine the distance to the star.

There are, of course, a few caveats. First, the numbers in the second equation relate to magnitudes measured through a yellow filter; using blue or red magnitudes alters the numbers. Second, Cepheids are generally so far away that we view them through a considerable amount of interstellar dust, which makes them look fainter and redder than they really are, and astronomers must correct for this. Third, there are two groups of Cepheids, each with slightly different period-luminosity relationships. — J. DONALD FERNIE, UNIVERSITY OF TORONTO (ONTARIO)