The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement to an astronomical object is only possible for those objects that are "close enough" (within about a thousand parsecs) to earth. The techniques for determining distances to more distant objects are all bases on various measured correlations between methods that work at close distances with methods that work at larger distances. The ladder analogy arises because no one technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distances at the next higher rung.

Direct methods of distance determination

At the base of the ladder are fundamental distance measurements, in which distances are determined directly, with no physical assumptions about the nature of the object in question. These direct methods are:

• parallax (or triangulation) based upon trigonometry, using precise measurements of angles, similar to what is used in surveying.
• Light travel time (that is, the constancy of the speed of light), as in radar.

Radar can (for practical reasons) only be used within the Solar System.

The precise measurement of stellar positions is part of the discipline of astrometry.

Parallax based methods

The most important fundamental distance measurements come from parallax. The Earth's motion around the sun causes small shifts in stellar positions. These shifts are angles in a right triangle, with 1 AU making the short leg of the triangle and the distance to the star being the long leg. One parsec is the distance of a star whose parallax is one arc second. Astronomers usually express distances in units of parsecs; light-years are used in popular media, but almost invariably values in light-years have been converted from numbers tabulated in parsecs in the original source.

Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars whose parallax is larger than the precision of the measurement. In the 1990s, the Hipparcos mission obtained parallaxes for over a hundred thousand stars with a precision of about a milliarcsecond, providing useful distances for stars out to a few hundred parsecs.

Another fundamental distance method is statistical and secular parallax. This technique combines measurements of the motions and brightnesses of members of a selected, homogeneous group of stars in a statistical way to deduce an average distance to the group. It remains an important technique for the Cepheids and the RR Lyrae variables.

Moving cluster parallax is a technique where the motions of individual stars in a nearby star cluster (only open clusters are near enough for this technique to be useful) can be used to find the distance to the cluster. In particular the distance obtained for the Hyades has been an important step in the distance ladder.

Other individual objects can have fundamental distance estimates made for them under special circumstances. If the expansion of a gas cloud, like a supernova remnant or planetary nebula, can be observed over time, then an expansion parallax distance to that cloud can be estimated. Binary stars which are both visual and spectroscopic binaries also can have their distance estimated by similar means. The common characteristic to these is that a measurement of angular motion is combined with a measurement of the absolute velocity (usually obtained via the Doppler effect). The distance estimate comes from computing how far away the object must be to make its observed absolute velocity appear with the observed angular motion.

Expansion parallaxes in particular can give fundamental distance estimates for objects that are very far away, because supernova ejecta have large expansion velocities and large sizes (compared to stars). Further, they can be observed with radio interferometers which can measure very small angular motions. These combine to mean that some supernovae in other galaxies have fundamental distance estimates^[1]. Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.

Astronomical unit

The use of the parallax method usually requires precise determination of the distance between the Earth and the Sun (the radius of the Earth's orbit), called the Astronomical Unit (AU).

Other astronomical distance measures build outward from this.

Historically, observations of transits of Venus were crucial in determining the AU; in the first half of the 20th Century, observations of asteroids were also important. Presently the AU is determined with high precision using radar measurements of Venus and other nearby planets and asteroids,^[2] and by tracking interplanetary spacecraft in their orbits around the Sun through the Solar System. Kepler's Laws provide precise ratios of the sizes of the orbits of objects revolving around the Sun, but not a real measure of the orbits themselves. Radar provides a value in kilometers for the difference in two orbits' sizes, and from that and the ratio of the two orbit sizes, the size of Earth's orbit comes directly.

Distance determination based on physical assumptions

With few exceptions, distances based on direct measurements are available only out to about a thousand parsecs, which is a modest portion of our own Galaxy. For distances beyond that, measures depend upon physical assumptions, that is, the assertion that one recognizes the object in question, and the class of objects is homogeneous enough that its members can be used for meaningful estimation of distance.

Almost all of these physical distance indicators are standard candles. These rely upon recognizing an object as belonging to some class, which has some known absolute magnitude, measuring its apparent magnitude, and using the inverse square law to infer the distance needed to make the "candle" appear at its observed brightness. Some means of accounting for interstellar extinction, which also makes objects appear fainter, is also needed. The difference between absolute and apparent magnitudes is called the distance modulus, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.

Physical distance indicators, used on progressively larger distance scales, include:

• Main sequence fitting, usually for open clusters of stars
• Cepheids and novae
• Individual galaxies in clusters of galaxies
• The Tully-Fisher relation
• Type Ia supernovae
• Redshifts and Hubble's Law

Two problems exist for any class of standard candle. The principal one is calibration, determining exactly what the absolute magnitude of the candle is. This includes defining the class well enough that members can be recognized, and finding enough members with well-known distances that their true absolute magnitude can be determined with enough accuracy. The second lies in recognizing members of the class, and not mistakenly using the standard candle calibration upon an object which does not belong to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.

(Another class of physical distance indicator is the standard ruler, but few of these are used at this time.)

A succession of distance indicators, which is the distance ladder, is needed for determining distances to other galaxies. The reason is that objects bright enough to be recognized and measured at such distances are so rare that few or none are present nearby, so there are too few examples close enough with reliable trigonometric parallax to calibrate the indicator. For example, Cepheid variables, one of the best indicators for nearby spiral galaxies, cannot be satisfactorily calibrated by parallax alone. The situation is further complicated by the fact that different stellar populations generally do not have all types of stars in them. Cepheids in particular are massive stars, with short lifetimes, so they will only be found in places where stars have very recently been formed. Consequently, because elliptical galaxies usually have long ceased to have large-scale star formation, they will not have Cepheids. Instead, distance indicators whose origins are in an older stellar population (like novae and RR Lyrae variables) must be used. However, RR Lyrae variables are less luminous than Cepheids (so they cannot be seen as far away as Cepheids can), and novae are unpredictable and an intensive monitoring program — and luck during that program — is needed to gather enough novae in the target galaxy for a good distance estimate.

Because the more distant steps of the cosmic distance ladder depend upon the nearer ones, the more distant steps include the effects of errors in the nearer steps, both systematic and statistical ones. The result of these propagating errors means that distances in astronomy are rarely known to the same level of precision as measurements in the other sciences, and that the precision necessarily is poorer for more distant types of object.

Another concern, especially for the very brightest standard candles, is their "standardness": how homogeneous the objects are in their true absolute magnitude. For some of these different standard candles, the homogeneity is based on theories about the formation and evolution of stars and galaxies, and is thus also subject to uncertainties in those aspects. For the most luminous of distance indicators, the Type Ia supernovae, this homogeneity is known to be poor; however, no other class of object is bright enough to be detected at such large distances, so the class is useful simply because there is no real alternative.

The observational result of Hubble's Law, the proportional relationship between distance and the speed with which a galaxy is moving away from us (usually referred to as redshift) is a product of the cosmic distance ladder. Hubble observed that fainter galaxies are more redshifted. Finding the value of the Hubble constant was the result of decades of work by many astronomers, both in amassing the measurements of galaxy redshifts and in calibrating the steps of the distance ladder. Hubble's Law is the primary means we have for estimating the distances of quasars and distant galaxies in which individual distance indicators cannot be seen.

External links

• The ABC's of distances (UCLA)
• The Extragalactic Distance Scale by Bill Keel
• The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
• The Hubble Constant, a historical discussion
• The Cepheid Distance scale by N. J. Allen

Notes