Cosmic distance ladder

Direct distance measurements are based upon the astronomical unit (AU), which is defined as the mean distance between the Earth and the Sun. Kepler's laws provide precise ratios of the sizes of the orbits of objects orbiting the Sun, but provide 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 is calculated. The Earth's orbit is known with an absolute precision of a few meters and a relative precision of a few parts in 100 billion (1×10⁻¹¹).

Historically, observations of Venus transits 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,[3] and by tracking interplanetary spacecraft in their orbits around the Sun through the Solar System.

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 to allow their true absolute magnitude to be determined with enough accuracy. The second problem lies in recognizing members of the class, and not mistakenly using a standard candle calibration on 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.[16]

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 when corrected, this had the effect of doubling the estimates of distances to the globular clusters, the nearby galaxies, and the diameter of the Milky Way.

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.

Discovered in 1956 by Olin Wilson and M.K. Vainu Bappu, the Wilson–Bappu effect utilizes the effect known as spectroscopic parallax. Many stars have features in their spectra, such as the calcium K-line, that indicate their absolute magnitude. The distance to the star can then be calculated from its apparent magnitude using the distance modulus.

There are major limitations to this method for finding stellar distances. The calibration of the spectral line strengths has limited accuracy and it requires a correction for interstellar extinction. Though in theory this method has the ability to provide reliable distance calculations to stars up to 7 megaparsecs (Mpc), it is generally only used for stars at hundreds of kiloparsecs (kpc).

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:

     ${\displaystyle 5\log _{10}{d}=V+(3.34)\log _{10}{P}-(2.45)(V-I)+7.52\,.}$[33]
     ${\displaystyle 5\log _{10}{d}=V+(3.37)\log _{10}{P}-(2.55)(V-I)+7.48\,.}$[34]

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.[35][36][37][38][39][40][41][42][43]

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 some cosmological parameters of the Universe may be constrained significantly better by supplying a precise value of the Hubble constant.[44][45]

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.

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.

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

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

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

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

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 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 ${\displaystyle 1.4M_{\odot }}$.

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

$${\displaystyle \ M_{B}\approx M_{V}\approx -19.3\pm 0.3\,.}$$

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.[46]

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 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:

$${\displaystyle \ M_{V}^{\max }=-9.96-2.31\log _{10}{\dot {x}}\,.}$$

Where ${\displaystyle {\dot {x}}}$ 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

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:

$${\displaystyle \ \Phi (m)=Ae^{(m-m_{0})^{2}/2\sigma ^{2}}\,}$$

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

It is 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.

Like the GCLF method, a similar numerical analysis can be used for planetary nebulae 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:

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

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

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.[47]

The Sigma-D relation (or Σ-D 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

$${\displaystyle \log(D)=1.333\log(\Sigma )+C}$$

where C is a constant which depends on the distance to the galaxy clusters.[48]

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 are not 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.