# How Far are the Stars? (part 2)

A few months ago, I shared with you how astronomers measure distances to the nearest stars using simple geometry. I also pointed out, however, that we can measure only our small neighborhood using the geometric method we call parallax.

How then, can we possibly know the distances of stars even farther out?

Well, we all know that a light gets dimmer the farther away it is.  Therefore, we can estimate a star’s distance if we can measure how bright it appears to us and then compare that to how bright it’s ‘supposed to be.’

 Hipparchus

Astronomers describe the brightness of any celestial object as its magnitude.  The term goes back to antiquity when the Greek astronomer Hipparchus put stars into six classes of brightness.  The brightest stars he could see were called first magnitude, and the dimmest stars he could barely make out were sixth magnitude.  For one thing, this means that lower magnitudes describe brighter objects, while higher magnitudes describe dimmer objects–the reverse of what most people would expect.  Also, this means that the scale is logarithmic rather than linear, as the human eye does not detect brightness linearly.

A star’s brightness as it appears to us is its apparent magnitude.  Astronomers also define a star’s absolute magnitude as the brightness it would have if it were 10 parsecs (about 32.6 light years) away.  The difference between a star’s apparent and absolute magnitudes is the distance modulus, a direct measure of the star’s distance.  A star’s absolute magnitude is related to its luminosity (the amount of light it emits).  Objects of known luminosity, enabling us to measure their distance, are called ‘standard candles.’

Standard Candles

Among the more important standard candles for determining distances are stars called ‘Cepheid variables.’  These are stars that vary in brightness over a period of several days as they pulsate (expand and shrink again).  The period over which Cepheids vary in brightness indicates their luminosity.

Cepheids are one of several types of variable stars in the instability strip of the Hertzsprung-Russell diagram.  These stars pulsate because in these stars, a layer of helium is subjected to enough heat and pressure that helium atoms lose their electrons and become ionized.  Doubly-ionized helium or He III (with both electrons gone) readily absorbs light that a normal helium atom transmits.  Therefore, He III makes stars slightly dimmer.  However, all heated gases expand and then cool as a result of the expansion.  Thus, the Cepheid pulses outward, and in the cooler environment of the expanded star, electrons recombine with helium ions.  No longer ionized, the helium no longer absorbs light, and the star brightens again.  When the star has expanded too far, its gravity causes all the stellar material to fall back towards the center of the star.  In the heated environment of the compressed star, helium atoms lose their electrons again, the star dims, and the process repeats itself.  In 1917, Arthur Stanley Eddington suggested that Cepheids were types of heat engines; Sergei A. Zhevakin in 1953 correctly identified helium as the particular gas involved.

From Pulsation to Mass, Mass to Luminosity

Since a star’s mass determines how fast and how far it will expand before collapsing under its own gravity, the period of a Cepheid’s pulsation is related to its mass.  A star’s mass, in turn, is related to its luminosity.  As a result, we when we measure how much time it takes for a Cepheid variable to brighten, get dimmer, and brighten again, we have information about its luminosity.  Comparing this to the star’s observed apparent magnitude tells us its distance.  Once enough Cepheids have been observed, it becomes possible to establish a relation that lets us measure distances to any Cepheids, even those in nearby galaxies.

In 1784, English amateur astronomer John Goodricke discovered that the star Delta Cephei varied in brightness over a period of about six days.  Since most stars known at the time to change their brightness were novae or supernovae, Delta Cephei became the prototype of a new type of variable star.  (It turns out that a few months earlier, Goodricke’s friend Edward Pigott had discovered that the star Eta Aquila varies in the same way as Delta Cephei.  Nevertheless, the name ‘Cepheid’ remains).

 Lesser Magellanic Cloud Photo courtesy of NASA

In 1908, Henrietta Swan Leavitt was studying photographic plates that Edward Charles Pickering had taken of the Magellanic Clouds when she noticed a strong relationship between Cepheids’ brightness and the log of their pulsation period.  Leavitt assumed (correctly) that the Magellanic Clouds were much, much smaller than their distance from us; all the stars she was measuring on her photographic plates were thus at about the same distance away.  Thus Leavitt’s period-luminosity relation was a way to determine the luminosity of a Cepheid independently of its distance.

Edwin Hubble & the Andromeda Galaxy

In 1924, Edwin Hubble used Leavitt’s relation to show that the Andromeda Galaxy was indeed a different galaxy and not a nebula in our own Milky Way as many believed at the time.  In 1929, Hubble and Milton Humason used distances to galaxies calculated using Cepheids to establish that more distant galaxies recede from us faster than nearby ones, thus formulating Hubble’s law.

It turns out that there are a variety of stars in the Cepheids’ instability strip.  Walter Baade in the 1940s discovered a second type of Cepheids now called W Virginis variables, after their prototype star in Virgo, the Virgin.  Less massive and dimmer on average than the classical Cepheids, these are older population II stars with fewer heavy elements.  Conflating the two types of Cepheids had introduced errors in distances to nearby galaxies.  For example, Baade’s corrections increased the known distance to the Andromeda Galaxy by a factor of four.  Still smaller and dimmer, with shorter periods of pulsation, are the RR Lyrae variables, named after their prototype star in Lyra, the Lyre.  Astronomers use RR Lyrae stars to measure distances in our own galaxy, but their dimness makes them hard to detect in other galaxies.

The use of Cepheids as standard candles continues into recent decades as well.  The Key Project of the Hubble Space Telescope was to determine the Hubble constant (the rate at which a galaxy at a given distance from us is receding from us)  by measuring the distances to 18 different galaxies using Cepheids.

With modern methods, we are able to detect Cepheid variables in galaxies up to 29 million parsecs (94.6 million light-years) away.  With Cepheids, then, we can measure much more of the universe than with parallax alone.  However, much of the observable universe is so far from us  that it still remains out of reach.  To measure even greater distances, we will need other standard candles, which we shall discuss at a later time.