Astronomy 162: Professor Barbara Ryden

Tuesday, February 25

THE COSMIC DISTANCE SCALE


``It is much easier to make measurements than to know exactly what you are measuring.'' - J. W. N. Sullivan

Key Concepts


(1) The distances to galaxies can be found using standard candles.

Finding the distances to celestial objects is a long-standing problem in astronomy. Different techniques are used on different length scales.

Within our Solar System, distances to planets are found, with great accuracy, by using radar (radio signals bounced off planets). Unfortunately, radar is only useful out to a distance of approximately 10 AU; beyond that distance, the radio `echo' is too faint to detect.

Within our galaxy, distances to nearby planets are found using stellar parallax (described in the lecture for Monday, January 13. Unfortunately, stellar parallax is only useful out to a distance of approximately 500 parsecs; beyond that distance, a star's shift in position is too small to measure.

Distances between galaxies are typically measured in megaparsecs. One megaparsec (abbreviated Mpc) is equal to 1,000,000 parsecs, or 3,260,000 light years. At these distances, neither radar nor stellar parallax is useful. The distance of a galaxy (which may be many megaparsecs away) can be found if the galaxy contains a standard candle. A standard candle is an object whose luminosity L is known. The known luminosity, combined with the measured apparent brightness b for the object, gives us the distance. (The relation among luminosity, apparent brightness, and distance is given in the lecture for Tuesday, January 14, if you would like a review.)

The fundamental problem with standard candles is determining their luminosity in the first place. (After all, stars aren't labeled like lightbulbs!) If you knew the distance d as well as the apparent brightness b, you could compute the luminosity L of the standard candle. BUT the distance is just what we were trying to find! Thus, we seem to be stuck in a circular argument: ``In order to determine the distance, we need to know the distance in the first place.'' To avoid this circular argument, we need to build a ``distance ladder'', starting with nearby standard candles, with distances known from their parallax, and building outward to standard candles of unknown parallax. Steps in building a distance ladder:

(NOTE: this argument assumes that the two standard candles have the same luminosity. If this assumption is wrong, your computed distance will also be wrong.)

(2) Cepheid variable stars and Type Ia supernovae are the most useful standard candles.

Cepheid variable stars are good standard candles. First, their luminosity is quite high (the most luminous Cepheids are 40,000 times more luminous than the Sun), so they can be seen to large distances. Second, their luminosities can be computed from the Period-Luminosity Relation (for a review of this relation, go to the lecture notes for Wednesday, January 29.) For instance, the star Delta Cephei (the first Cepheid variable star to be discovered) has a period of P = 5.4 days. Any other Cepheid in the universe with the same period of variability will have the same average luminosity as Delta Cephei. The distance to Delta Cephei, computed from its parallax, is 300 parsecs. A Cepheid variable star with the same period as Delta Cephei, but with an apparent brightness 1/1,000,000 as big, will be 1000 times farther away than Delta Cephei (that is, at a distance of 300,000 parsecs from us).

Beyond 30 Mpc, however, Cepheids are too dim to be detected. At larger distances, we need brighter standard candles. Type Ia supernovae are superb standard candles. They are all basically the same (if you've seen one carbon-oxygen white dwarf explode, you've seen them all!) Thus, all Type Ia supernovae have about the same luminosity: L = 4 billion Lsun. Supernovae are 100,000 times more luminous than even the brightest Cepheid stars, and can be seen at distances of thousands of megaparsecs. The biggest problem with Type Ia supernovae is that they are infrequent. Even a large galaxy only has one supernova per century, on average.


(3) The Hubble Law states that the radial speed of a galaxy is proportional to its distance.

Measure the Doppler shift of the stars in a galaxy. You will find that most galaxies are rotating (the disks of spiral galaxies rotate rapidly; elliptical galaxies rotate in a more leisurely manner). In addition, however, the galaxy as a whole will be moving towards us or away from us. The galaxy's radial speed v is found from its Doppler shift:
v = c z
where v = radial speed of galaxy
c = speed of light (300,000 km/sec)
z = fractional change in the wavelength of light.

A few nearby galaxies are found to be moving towards us. The Andromeda Galaxy (M31), for instance, is blueshifted. That is, the absorption lines in its spectrum are shifted to shorter wavelengths, by an amount z = -0.001. We deduce, therefore, that our galaxy and the Andromeda Galaxy are moving toward each other, with a relative speed
v = c z = (300,000 km/sec) (-0.001) = -300 km/sec.
(The negative speed implies that the distance between the two galaxies is decreasing with time.) A plausible explanation for this observation is that our galaxy and the Andromeda Galaxy form a binary system, on elongated orbits around their center of mass. The galaxies have recently passed apocenter (the point of maximum separation) and are now moving toward each other.

A VERY SURPRISING discovery was made in the early decades of this century. Edwin Hubble discovered that galaxies more distant than the Andromeda Galaxy are all redshifted. That is, they are all receding from us, moving steadily further away. Moreover, the recession speed of a galaxy is proportional to its distance: the farther a galaxy is from us, the faster it's moving away:

The relation between the radial speed and distance of a galaxy is known as the Hubble Law, in honor of Edwin Hubble. In mathematical form, it can be written as:

v = H0 d

where v is the radial speed of a galaxy (equal to its redshift times the speed of light), d is the distance to the galaxy, and H0 is a constant known as the Hubble constant. Ironically, Hubble himself badly overestimated the value of the Hubble constant (he thought that galaxies were closer than they actually are). The best current value of the Hubble constant is H0 = 70 km/sec/Mpc.

The Hubble Law is a surprising result. Why are galaxies flying away from us? Is it something we said? In fact, our galaxy does not possess a special repulsive force. Galaxies are receding, from our point of view, because the universe is EXPANDING. If we were located in ANY galaxy in the universe, distant galaxies would appear to be fleeing from us, due to the universal expansion (More on this subject later, when we talk about the Big Bang and the Expanding Universe.)

For our purposes at the moment, it suffices to point out that the Hubble Law permits us to find distances to far-away galaxies. Radial velocities are relatively easy to measure. Once we measure v for a galaxy, we can compute
d = v / H0 .
For instance, suppose a galaxy is moving away from us at 14,000 km/sec. Its distance, using the Hubble Law, is thus
d = (14,000 km/sec) / (70 km/sec/Mpc) = 200 Mpc.
The galaxy is 200 megaparsecs away (652 million light years), beyond the region where Cepheids can be used to compute distances. The major drawback to using the Hubble Law is that the value of H0 (the Hubble Constant) is poorly known.


Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 25 Feb 2003

Copyright 2003, Barbara Ryden