Cosmic distance ladder

For various definitions of distance in cosmology, see Distance measures (cosmology).

• Light green boxes: Technique applicable to star-forming galaxies.
• Light blue boxes: Technique applicable to Population II galaxies.
• Light Purple boxes: Geometric distance technique.
• Light Red box: The planetary nebula luminosity function technique is applicable to all populations of the Virgo Supercluster.
• Solid black lines: Well calibrated ladder step.
• Dashed black lines: Uncertain calibration ladder step.

The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only 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 based on various measured correlations between methods that work at close distances and methods that work at larger distances. Several methods rely on a standard candle, which is an astronomical object that has a known luminosity.

The ladder analogy arises because no 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 measurement

Statue of an astronomer and the concept of the cosmic distance ladder by the parallax method, made from the azimuth ring and other parts of the Yale–Columbia Refractor (telescope) (c 1925) wrecked by the 2003 Canberra bushfires which burned out the Mount Stromlo Observatory; at Questacon, Canberra, Australian Capital Territory

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

Astronomical unit

Main article: Astronomical Unit

Direct distance measurements are based upon the Astronomical unit (AU), which is the distance between the Earth and the Sun. 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 orbit of Earth is determined with high precision using radar measurements of distances to Venus and other nearby planets and asteroids, 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 orbiting the Sun, but provides no measurement of the overall scale of the orbit system. Radar is used to measure the distance between the orbits of the Earth and of a second body. From that measurement and the ratio of the two orbit sizes, the size of Earth's orbit comes. The Earth's orbit is known with a precision of a few meters.

Parallax

Main article: Parallax

Stellar parallax motion from annual parallax. Half the apex angle is the parallax angle

The most important fundamental distance measurements come from trigonometric parallax. As the Earth orbits the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in an isosceles triangle, with 2 AU (the distance between the extreme positions of Earth's orbit around the Sun) making the base leg of the triangle and the distance to the star being the long equal length legs. The amount of shift is quite small, measuring 1 arcsecond for an object at the 1 parsec (3.26 light-years) distance of the nearest stars, and thereafter decreasing in angular amount as the distance increases. Astronomers usually express distances in units of parsecs (parallax arcseconds); light-years are used in popular media.

Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars whose parallax is larger than a few times the precision of the measurement. Parallax measurements typically have an accuracy measured in milliarcseconds. In the 1990s, for example, 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. The Hubble telescope WFC3 now has the potential to provide a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to for small numbers of stars. By the early 2020s, the GAIA space mission will provide similarly accurate distances to all moderately bright stars.

Stars have a velocity relative to the Sun that causes proper motion (transverse across the sky) and radial velocity (motion toward or away from the Sun). The former is determined by plotting the changing position of the stars over many years, while the latter comes from measuring the Doppler shift the star's spectrum caused by motion along the line of sight. For a group of stars with the same spectral class and a similar magnitude range, a mean parallax can be derived from statistical analysis of the proper motions relative to their radial velocities. This statistical parallax method is useful for measuring the distances of bright stars beyond 50 parsecs and giant variable stars, including Cepheids and the RR Lyrae variables.

The motion of the Sun through space provides a longer baseline that will increase the accuracy of parallax measurements, known as secular parallax. For stars in the Milky Way disk, this corresponds to a mean baseline of 4 AU per year, while for halo stars the baseline is 40 AU per year. After several decades, the baseline can be orders of magnitude greater than the Earth–Sun baseline used for traditional parallax. However, secular parallax introduces a higher level of uncertainty because the relative velocity of other stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the uncertainty is inversely proportional to the square root of the sample size.

Moving cluster parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster. Only open clusters are near enough for this technique to be useful. In particular the distance obtained for the Hyades has historically 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. Those measurements however suffer from uncertainties in the deviation of the object from sphericity. Binary stars which are both visual and spectroscopic binaries also can have their distance estimated by similar means, and don't suffer from the above geometric uncertainty. The common characteristic to these methods 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 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, 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 provide fundamental distance estimates to supernovae in other galaxies. Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.

See also: Stellar Parallax and Parsec

Standard candles

Almost all astronomical objects used as physical distance indicators belong to some class that have a known brightness. By comparing this known luminosity to an object's observed brightness, the distance to the object can be computed using the inverse square law. These objects of known brightness are termed standard candles.

The brightness of an object can be expressed 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, or the magnitude as seen by the observer (an instrument called a bolometer is used), can be measured and used with the absolute magnitude to calculate the distance D to the object in kiloparsecs (where 1 kpc equals 1000 parsecs) as follows:

$5 \cdot \log_{10} D = m - M - 10,$

or

$D = 0.2 \cdot 10^{m -M -10},$

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, especially if the object lies within a dusty or gaseous region. 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, that is the determination of 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 of that class 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 that 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.

A significant issue with standard candles is the recurring 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 (corrected by the shape of the light curve). The basis for this closeness in brightness is discussed below; however, the possibility exists that the distant Type Ia supernovae have different properties than nearby Type Ia supernovae. The use of Type Ia supernovae is crucial in determining the correct cosmological model. If indeed the properties of Type Ia supernovae are different at large distances, i.e. if the extrapolation of their calibration to arbitrary distances is not valid, ignoring this variation can dangerously bias the reconstruction of the cosmological parameters, in particular the reconstruction of the matter density parameter.

That this is not merely a philosophical issue can be seen from the history of distance measurements using Cepheid variables. In the 1950s, Walter Baade discovered that the nearby Cepheid variables used to calibrate the standard candle were of a different type than the ones used to measure distances to nearby galaxies. The nearby Cepheid variables were population I stars with much higher metal content than the distant population II stars. As a result, the population II stars were actually much brighter than believed, and this had the effect of doubling the distances to the globular clusters, the nearby galaxies, and the diameter of the Milky Way.

(Another class of physical distance indicator is the standard ruler. In 2008, galaxy diameters have been proposed as a possible standard ruler for cosmological parameter determination.)

Galactic distance indicators

See also: distance measures (cosmology)

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.

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

• Dynamical parallax, using orbital parameters of visual binaries to measure the mass of the system and the mass-luminosity relation to determine the luminosity
  • Eclipsing binaries — In the last decade, measurement of eclipsing binaries' fundamental parameters has become possible with 8—meter class telescopes. This makes it feasible to use them as indicators of distance. Recently, they have been used to give direct distance estimates to the LMC, SMC, Andromeda Galaxy and Triangulum Galaxy. Eclipsing binaries offer a direct method to gauge the distance to galaxies to a new improved 5% level of accuracy which is feasible with current technology up to a distance of around 3 Mpc.
• RR Lyrae variables — used for measuring distances within the galaxy and in nearby globular clusters.
• The following four indicators all use stars in the old stellar populations (Population II):
  • Tip of the red giant branch (TRGB) distance indicator.
  • Planetary nebula luminosity function (PNLF)
  • Globular cluster luminosity function (GCLF)
  • Surface brightness fluctuation (SBF)
• 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, which can be calculated once the mass of the neutron star is known (1.5 solar masses is a commonly used assumption). This method allows distance determination of some low-mass X-ray binaries. Low-mass X-ray binaries are very faint in the optical, making their distances extremely difficult to determine.
• Interstellar masers can be used to derive distances to galactic and some extragalactic objects that have maser emission.
• Cepheids and novae
• Individual galaxies in clusters of galaxies
• The Tully–Fisher relation
• The Faber–Jackson relation
• Type Ia supernovae that have a very well-determined maximum absolute magnitude as a function of the shape of their light curve and are useful in determining extragalactic distances up to a few hundred Mpc. A notable exception is SN 2003fg, the "Champagne Supernova", a Type Ia supernova of unusual nature.
• Redshifts and Hubble's Law

Main sequence fitting

When the absolute magnitude for a group of stars is plotted against the spectral classification of the star, in a Hertzsprung–Russell diagram, 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 these properties from a star's spectrum, the position of a main sequence star on the H–R diagram can be determined, and thereby the star's absolute magnitude 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.

Extragalactic distance scale

Extragalactic distance indicators

Method	Uncertainty for Single Galaxy (mag)	Distance to Virgo Cluster (Mpc)	Range (Mpc)
Classical Cepheids	0.16	15–25	29
Novae	0.4	21.1 ± 3.9	20
Planetary Nebula Luminosity Function	0.3	15.4 ± 1.1	50
Globular Cluster Luminosity Function	0.4	18.8 ± 3.8	50
Surface Brightness Fluctuations	0.3	15.9 ± 0.9	50
D–σ relation	0.5	16.8 ± 2.4	> 100
Type Ia Supernovae	0.10	19.4 ± 5.0	> 1000

The extragalactic distance scale is a series of techniques used today by astronomers to determine the distance of cosmological bodies beyond our own galaxy, which are not easily obtained with traditional methods. Some procedures utilize properties of these objects, such as stars, globular clusters, nebulae, and galaxies as a whole. Other methods are based more on the statistics and probabilities of things such as entire galaxy clusters.

Wilson–Bappu effect

Main article: Wilson–Bappu effect

Discovered in 1956 by Olin Wilson and M.K. Vainu Bappu, The Wilson–Bappu effect utilizes the effect known as spectroscopic parallax. Certain stars have features in their emission/absorption spectra allowing relatively easy absolute magnitude calculation. Certain spectral lines are directly related to an object's magnitude, such as the K absorption line of calcium. Distance to the star can be calculated from magnitude by the distance modulus:

$\ M - m = - 2.5 \log_{10}(F_1/F_2) \,.$

Though in theory this method has the ability to provide reliable distance calculations to stars roughly at 7 megaparsecs (Mpc), it is generally only used for stars hundreds at kiloparsecs (kpc).

This method is only valid for stars over 15 magnitudes.

Classical Cepheids

Beyond the reach of the Wilson–Bappu effect, the next method relies on the period-luminosity relation of classical Cepheid variable stars. The following relation can be used to calculate the distance to Galactic and extragalactic classical Cepheids:

$5\log_{10}{d}=V+ (3.34) \log_{10}{P} - (2.45) (V-I) + 7.52 \,.$

$5\log_{10}{d}=V+ (3.37) \log_{10}{P} - (2.55) (V-I) + 7.48 \,.$

Several problems complicate the use of Cepheids as standard candles and are actively debated, chief among them are: the nature and linearity of the period-luminosity relation in various passbands and the impact of metallicity on both the zero-point and slope of those relations, and the effects of photometric contamination (blending) and a changing (typically unknown) extinction law on Cepheid distances.

These unresolved matters have resulted in cited values for the Hubble Constant ranging between 60 km/s/Mpc and 80 km/s/Mpc. Resolving this discrepancy is one of the foremost problems in astronomy since the cosmological parameters of the Universe may be constrained by supplying a precise value of the Hubble constant.

Cepheid variable stars were the key instrument in Edwin Hubble’s 1923 conclusion that M31 (Andromeda) was an external galaxy, as opposed to a smaller nebula within the Milky Way. He was able to calculate the distance of M31 to 285 Kpc, today’s value being 770 Kpc.

As detected thus far, NGC 3370, a spiral galaxy in the constellation Leo, contains the farthest Cepheids yet found at a distance of 29 Mpc. Cepheid variable stars are in no way perfect distance markers: at nearby galaxies they have an error of about 7% and up to a 15% error for the most distant.

Supernovae

SN 1994D (bright spot on the lower left) in the NGC 4526 galaxy. Image by NASA, ESA, The Hubble Key Project Team, and The High-Z Supernova Search Team

There are several different methods for which supernovae can be used to measure extragalactic distances.

Measuring a supernova's photosphere

We can assume that a supernova expands in a spherically symmetric manner. If the supernova is close enough such that we can measure the angular extent, θ(t), of its photosphere, we can use the equation

$\omega = \frac{\Delta\theta}{\Delta t} \,,$

where ω is angular velocity, θ is angular extent. In order to get an accurate measurement, it is necessary to make two observations separated by time Δt. Subsequently, we can use

$\ d = \frac{V_{ej}}{\omega} \,,$

where d is the distance to the supernova, V_ej is the supernova's ejecta's radial velocity (it can be assumed that V_ej equals V_θ if spherically symmetric).

This method works only if the supernova is close enough to be able to measure accurately the photosphere. Similarly, the expanding shell of gas is in fact not perfectly spherical nor a perfect blackbody. Also interstellar extinction can hinder the accurate measurements of the photosphere. This problem is further exacerbated by core-collapse supernova. All of these factors contribute to the distance error of up to 25%.

Type Ia light curves

Type Ia supernovae are some of the best ways to determine extragalactic distances. Ia's occur when a binary white dwarf star begins to accrete matter from its companion star. As the white dwarf gains matter, eventually it reaches its Chandrasekhar Limit of $1.4 M_{\odot}$.

Once reached, the star becomes unstable and undergoes a runaway nuclear fusion reaction. Because all Type Ia supernovae explode at about the same mass, their absolute magnitudes are all the same. This makes them very useful as standard candles. All Type Ia supernovae have a standard blue and visual magnitude of

$\ M_B \approx M_V \approx -19.3 \pm 0.3 \,.$

Therefore, when observing a Type Ia supernova, if it is possible to determine what its peak magnitude was, then its distance can be calculated. It is not intrinsically necessary to capture the supernova directly at its peak magnitude; using the multicolor light curve shape method (MLCS), the shape of the light curve (taken at any reasonable time after the initial explosion) is compared to a family of parameterized curves that will determine the absolute magnitude at the maximum brightness. This method also takes into effect interstellar extinction/dimming from dust and gas.

Similarly, the stretch method fits the particular supernovae magnitude light curves to a template light curve. This template, as opposed to being several light curves at different wavelengths (MLCS) is just a single light curve that has been stretched (or compressed) in time. By using this Stretch Factor, the peak magnitude can be determined .

Using Type Ia supernovae is one of the most accurate methods, particularly since supernova explosions can be visible at great distances (their luminosities rival that of the galaxy in which they are situated), much farther than Cepheid Variables (500 times farther). Much time has been devoted to the refining of this method. The current uncertainty approaches a mere 5%, corresponding to an uncertainty of just 0.1 magnitudes.

Novae in distance determinations

Novae can be used in much the same way as supernovae to derive extragalactic distances. There is a direct relation between a nova's max magnitude and the time for its visible light to decline by two magnitudes. This relation is shown to be:

$\ M^\max_V = -9.96 - 2.31 \log_{10} \dot{x} \,.$

Where $\dot{x}$ is the time derivative of the nova's mag, describing the average rate of decline over the first 2 magnitudes.

After novae fade, they are about as bright as the most luminous Cepheid Variable stars, therefore both these techniques have about the same max distance: ~ 20 Mpc. The error in this method produces an uncertainty in magnitude of about ±0.4

Globular cluster luminosity function

Based on the method of comparing the luminosities of globular clusters (located in galactic halos) from distant galaxies to that of the Virgo cluster, the globular cluster luminosity function carries an uncertainty of distance of about 20% (or 0.4 magnitudes).

US astronomer William Alvin Baum first attempted to use globular clusters to measure distant elliptical galaxies. He compared the brightest globular clusters in Virgo A galaxy with those in Andromeda, assuming the luminosities of the clusters were the same in both. Knowing the distance to Andromeda, Baum has assumed a direct correlation and estimated Virgo A’s distance.

Baum used just a single globular cluster, but individual formations are often poor standard candles. Canadian astronomer René Racine assumed the use of the globular cluster luminosity function (GCLF) would lead to a better approximation. The number of globular clusters as a function of magnitude is given by:

$\ \Phi (m) = A e^{(m-m_0)^2/2\sigma^2} \,$

where m₀ is the turnover magnitude, M₀ is the magnitude of the Virgo cluster, and sigma is the dispersion ~ 1.4 mag.

It is important to remember that it is assumed that globular clusters all have roughly the same luminosities within the universe. There is no universal globular cluster luminosity function that applies to all galaxies.

Planetary nebula luminosity function

Like the GCLF method, a similar numerical analysis can be used for planetary nebulae (note the use of more than one!) within far off galaxies. The planetary nebula luminosity function (PNLF) was first proposed in the late 1970s by Holland Cole and David Jenner. They suggested that all planetary nebulae might all have similar maximum intrinsic brightness, now calculated to be M = −4.53. This would therefore make them potential standard candles for determining extragalactic distances.

Astronomer George Howard Jacoby and his colleagues later proposed that the PNLF function equaled:

$\ N (M) \propto e^{0.307 M} (1 - e^{3(M^{*} - M)} ) \,.$

Where N(M) is number of planetary nebula, having absolute magnitude M. M* is equal to the nebula with the brightest magnitude.

Surface brightness fluctuation method

Galaxy cluster

The following method deals with the overall inherent properties of galaxies. These methods, though with varying error percentages, have the ability to make distance estimates beyond 100 Mpc, though it is usually applied more locally.

The surface brightness fluctuation (SBF) method takes advantage of the use of CCD cameras on telescopes. Because of spatial fluctuations in a galaxy’s surface brightness, some pixels on these cameras will pick up more stars than others. However, as distance increases the picture will become increasingly smoother. Analysis of this describes a magnitude of the pixel-to-pixel variation, which is directly related to a galaxy’s distance.

D–σ relation

The D–σ relation, used in elliptical galaxies, relates the angular diameter (D) of the galaxy to its velocity dispersion. It is important to describe exactly what D represents, in order to understand this method. It is, more precisely, the galaxy’s angular diameter out to the surface brightness level of 20.75 B-mag arcsec⁻². This surface brightness is independent of the galaxy’s actual distance from us. Instead, D is inversely proportional to the galaxy’s distance, represented as d. Thus, this relation does not employ standard candles. Rather, D provides a standard ruler. This relation between D and σ is

$\log_{10}(D) = 1.333 \log (\sigma) + C \,.$

Where C is a constant which depends on the distance to the galaxy clusters.

This method has the potential to become one of the strongest methods of galactic distance calculators, perhaps exceeding the range of even the Tully–Fisher method. As of today, however, elliptical galaxies aren’t bright enough to provide a calibration for this method through the use of techniques such as Cepheids. Instead, calibration is done using more crude methods.

Overlap and scaling

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 yet 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, 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.

See also

• Distance measures (cosmology)
• Standard ruler
• Orders of magnitude (length)#Astronomical

Bibliography

• An Introduction to Modern Astrophysics, Carroll and Ostlie, copyright 2007.
• Measuring the Universe The Cosmological Distance Ladder, Stephen Webb, copyright 2001.
• Pasachoff, JM & Filippenko, AV, The Cosmos: Astronomy in the New Millennium, Cambridge: Cambridge University Press, 4th edition, 2013 ISBN 9781107687561.
• The Astrophysical Journal, The Globular Cluster Luminosity Function as a Distance Indicator: Dynamical Effects, Ostriker and Gnedin, May 5, 1997.
• An Introduction to Distance Measurement in Astronomy, Richard de Grijs, Chichester: John Wiley & Sons, 2011, ISBN 978-0-470-51180-0.

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
• NASA Cosmic Distance Scale
• PNLF information database
• The Astrophysical Journal