The cosmic distance ladder (also known as the extragalactic distance scale) is the way astronomers measure the distance of objects in space. No one method works for all objects and distances, so astronomers use a number of methods.

A real direct distance measurement of an astronomical object is possible only for those objects that are close enough to Earth (within about a thousand parsecs). It is the larger distances which are the problem. Several methods rely on a standard candle, which is an astronomical object that has a known standard luminosity.

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 measures

Astronomical Unit

The astronomical unit is the mean (average) distance of the Earth from the Sun. This we know quite accurately. Radar bounce off artificial satellites and inner planets provide the data, then the result is worked out by computer.

Parallax

Parallax is the use of trigonometry to discover the distances of objects near to the solar system.

As the Earth orbits around the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in a right triangle, with 2 AU making the short leg of the triangle and the distance to the star being the long leg. The amount of shift is quite small, measuring 1 arcsecond for an object at a distance of 1 parsec (3.26 light-years)

This method works for distances up to a few hundred parsecs.

Standard candles

Objects of known brightness are called standard candles. Most physical distance indicators are standard candles. These are objects which belong to a class that has a known brightness. By comparing the known luminosity of the latter to its observed brightness, the distance to the object can be computed using the inverse square law.

In astronomy, the brightness of an object is given in terms of its absolute magnitude. This quantity is derived from the logarithm of its luminosity as seen from a distance of 10 parsecs. The apparent magnitude is the magnitude as seen by the observer. It can be used to determine the distance D to the object in kiloparsecs (kiloparsec = 1,000 parsecs) as follows:

$\begin{smallmatrix}5 \cdot \log_{10} \frac{D}{\mathrm{kpc}}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}$

where m the apparent magnitude and M the absolute magnitude. For this to be accurate, both magnitudes must be in the same frequency band and there can be no relative motion in the radial direction.

Some means of accounting for interstellar extinction, which also makes objects appear fainter and more red, 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.

Problems

Two problems exist for any class of standard candle. The principal one is calibration, finding out exactly what the absolute magnitude of the candle is.

The second lies in recognizing members of the class. The standard candle calibration does not work unless the object belongs to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.

A significant issue with standard candles is the question of how standard they are. For example, all observations seem to indicate that Type Ia supernovae that are of known distance have the same brightness. The possibility that the distant Type Ia supernovae have different properties than nearby Type Ia supernovae exists.[1]

Galactic distance indicators

Nova Eridani 2009 (apparent magnitude ~8.4) during a full moon

With few exceptions, distances based on direct measurements are available only up 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.

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

• Eclipsing binaries — In the last decade, measurement of eclipsing binaries offers a way to gauge the distance to galaxies. Accuracy at the 5% level up to a distance of around 3 million parsecs.[2]
• RR Lyrae variables — are periodic variable stars, commonly found in globular clusters, and often used as standard candles to measure galactic distances. These red giants are used for measuring distances within the galaxy and in nearby globular clusters.
• In galactic astronomy, X-ray bursts (thermonuclear flashes on the surface of a neutron star) are used as standard candles. Observations of X-ray burst sometimes show X-ray spectra indicating radius expansion. Therefore, the X-ray flux at the peak of the burst should correspond to Eddington luminosity,[3] which can be calculated once the mass of the neutron star is known (1.5 solar masses is a commonly used assumption).
• Cepheid variables and novae
• Cepheids are a class of very luminous variable stars. The strong direct relationship between a Cepheid variable's luminosity and pulsation period, secures for Cepheids their status as important standard candles for establishing the Galactic and extragalactic distance scales.[4]
• Novae have some promise for use as standard candles. For instance, the distribution of their absolute magnitude is bimodal, with a main peak at magnitude −8.8, and a lesser one at −7.5. Novae also have roughly the same absolute magnitude 15 days after their peak (−5.5). This method is about as accurate as the Cepheid variable stars method.[5]
• White dwarfs. Because the white dwarf stars which become supernovae have a uniform mass, Type Ia supernovae produce consistent peak luminosity. The stability of this value allows these explosions to be used as standard candles to measure the distance to their host galaxies, because the visual magnitude of the supernovae depends primarily on the distance.[6]
• Redshifts and Hubble's Law By using Hubble's law, which relates redshift to distance, one can estimate the distance of any particular galaxy.

Main sequence fitting

In a Hertzsprung-Russell diagram the absolute magnitude for a group of stars is plotted against the spectral classification of the stars. Evolutionary patterns are found that relate to the mass, age and composition of the star. In particular, during their hydrogen burning period, stars lie along a curve in the diagram called the main sequence.

By measuring the properties from a star's spectrum, the position of a main sequence star on the H-R diagram can be found. From this the star's absolute magnitude is estimated. A comparison of this value with the apparent magnitude allows the approximate distance to be determined, after correcting for interstellar extinction of the luminosity because of gas and dust.

In a gravitationally-bound star cluster such as the Hyades, the stars formed at approximately the same age and lie at the same distance. This allows relatively accurate main sequence fitting, providing both age and distance determination.

This is not a complete list of methods, but it does show the ways astronomers go about estimating the distance of astronomical objects.

References

1. Linden, Sebastian; Virey, Jean-Marc; Tilquin, André (2009). "Cosmological parameter extraction and biases from type Ia supernova magnitude evolution". Astronomy and Astrophysics 506 (3): 1095–1105. doi:10.1051/0004-6361/200912811., and references therein.
2. Bonanos, Alceste Z. (2006). "Eclipsing binaries: tools for calibrating the extragalactic distance scale". Binary stars as critical tools and tests in contemporary astrophysics: International Astronomical Union. Symposium no. 240, held 22–25 August 2006 in Prague, Czech Republic, S240, #008 2: 79–87. doi:10.1017/S1743921307003845.
3. In a star, the point at which gravitational force inwards equals the radiation force outwards.
4. Freedman, Wendy L.; Madore, Barry F. (2010). "The Hubble Constant". Annual Review of Astronomy and Astrophysics 48: 673. doi:10.1146/annurev-astro-082708-101829.
5. Robert, Gilmozzi; Della Valle, Massimo (2003). "Novae as distance indicators". In Alloin, D.; Gieren, W.. Stellar candles for the extragalactic distance scale. Springer. pp. 229–241. ISBN 3-540-20128-9.
6. Colgate S.A. (1979). "Supernovae as a standard candle for cosmology". Astrophysical Journal 232 (1): 404–408. doi:10.1086/157300.