Cosmic distance ladder

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 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.

This article's lead section may be too short to adequately summarize the key points. (January 2023)

The ladder analogy arises because no single 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.

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. Early fundamental distances—such as the radii of the earth, moon and sun, and the distances between them—were well estimated with very low technology by the ancient Greeks.^[2]

Parallax

This section is an excerpt from Parallax in astronomy.[edit]

The most important fundamental distance measurements in astronomy come from trigonometric parallax, as applied in the stellar parallax method. 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, even for the nearest stars, measuring 1 arcsecond for an object at 1 parsec's distance (3.26 light-years), 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 which are near enough to have a parallax larger than a few times the precision of the measurement. In the 1990s, for example, the Hipparcos mission obtained parallaxes for over a hundred thousand stars with a precision of about a milliarcsecond,^[4] providing useful distances for stars out to a few hundred parsecs. The Hubble Space Telescope's Wide Field Camera 3 has the potential to provide a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to 5,000 parsecs (16,000 ly) for small numbers of stars.^[5][6] The Gaia space mission provided similarly accurate distances to most stars brighter than 15th magnitude.^[7]

Distances can be measured within 10% as far as the Galactic Center, about 30,000 light years away. 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 of 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.^[8]

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 observed 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.^[11]

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 do not 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.^[12] Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.

Almost all astronomical objects used as physical distance indicators belong to a class that has 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, coined by Henrietta Swan Leavitt.^[13]

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, 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 parsecs^[14] as follows: $${\displaystyle 5\cdot \log _{10}d=m-M+5}$$ or $${\displaystyle d=10^{(m-M+5)/5}}$$ where m is 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 correcting for interstellar extinction, which also makes objects appear fainter and more red, is needed, especially if the object lies within a dusty or gaseous region.^[15] The difference between an object's absolute and apparent magnitudes is called its distance modulus, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.

Gravitational waves originating from the inspiral phase of compact binary systems, such as neutron stars or black holes, have the useful property that energy emitted as gravitational radiation comes exclusively from the orbital energy of the pair, and the resultant shrinking of their orbits is directly observable as an increase in the frequency of the emitted gravitational waves. To leading order, the rate of change of frequency ${\displaystyle f}$ is given by^{[19][20]: 38} $${\displaystyle {\frac {df}{dt}}={\frac {96\pi ^{8/3}(G{\mathcal {M}})^{\frac {5}{3}}f^{\frac {11}{3}}}{5\,c^{5}}},}$$ where ${\displaystyle G}$ is the gravitational constant, ${\displaystyle c}$ is the speed of light, and ${\displaystyle {\mathcal {M}}}$ is a single (therefore computable^[a]) number called the chirp mass of the system, a combination of the masses ${\displaystyle (m_{1},m_{2})}$ of the two objects^[22] $${\displaystyle {\mathcal {M}}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}.}$$ By observing the waveform, the chirp mass can be computed and thence the power (rate of energy emission) of the gravitational waves. Thus, such a gravitational wave source is a standard siren of known loudness.^[23][20]

Just as with standard candles, given the emitted and received amplitudes, the inverse-square law determines the distance to the source. There are some differences with standard candles, however. Gravitational waves are not emitted isotropically, but measuring the polarisation of the wave provides enough information to determine the angle of emission. Gravitational wave detectors also have anisotropic antenna patterns, so the position of the source on the sky relative to the detectors is needed to determine the angle of reception.

Generally, if a wave is detected by a network of three detectors at different locations, the network will measure enough information to make these corrections and obtain the distance. Also unlike standard candles, gravitational waves need no calibration against other distance measures. The measurement of distance does of course require the calibration of the gravitational wave detectors, but then the distance is fundamentally given as a multiple of the wavelength of the laser light being used in the gravitational wave interferometer.

There are other considerations that limit the accuracy of this distance, besides detector calibration. Fortunately, gravitational waves are not subject to extinction due to an intervening absorbing medium. But they are subject to gravitational lensing, in the same way as light. If a signal is strongly lensed, then it might be received as multiple events, separated in time, the analogue of multiple images of a quasar, for example. Less easy to discern and control for is the effect of weak lensing, where the signal's path through space is affected by many small magnification and demagnification events. This will be important for signals originating at cosmological redshifts greater than 1. It is difficult for detector networks to measure the polarization of a signal accurately if the binary system is observed nearly face-on.^[24] Such signals suffer significantly larger errors in the distance measurement. Unfortunately, binaries radiate most strongly perpendicular to the orbital plane, so face-on signals are intrinsically stronger and the most commonly observed.

If the binary consists of a pair of neutron stars, their merger will be accompanied by a kilonova/hypernova explosion that may allow the position to be accurately identified by electromagnetic telescopes. In such cases, the redshift of the host galaxy allows a determination of the Hubble constant ${\displaystyle H_{0}}$.^[22] This was the case for GW170817, which was used to make the first such measurement.^[25] Even if no electromagnetic counterpart can be identified for an ensemble of signals, it is possible to use a statistical method to infer the value of ${\displaystyle H_{0}}$.^[22]

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.^[26] More recently the physical scale imprinted by baryon acoustic oscillations (BAO) in the early universe has been used. In the early universe (before recombination) the baryons and photons scatter off each other, and form a tightly coupled fluid that can support sound waves. The waves are sourced by primordial density perturbations, and travel at speed that can be predicted from the baryon density and other cosmological parameters.

The total distance that these sound waves can travel before recombination determines a fixed scale, which simply expands with the universe after recombination. BAO therefore provide a standard ruler that can be measured in galaxy surveys from the effect of baryons on the clustering of galaxies. The method requires an extensive galaxy survey in order to make this scale visible, but has been measured with percent-level precision (see baryon acoustic oscillations). The scale does depend on cosmological parameters like the baryon and matter densities, and the number of neutrinos, so distances based on BAO are more dependent on cosmological model than those based on local measurements.

Light echos can be also used as standard rulers,^[27][28] although it is challenging to correctly measure the source geometry.^[29][30]

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, uses orbital parameters of visual binaries to measure the mass of the system, and hence use 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 Large Magellanic Cloud (LMC), Small Magellanic Cloud (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 to a distance of around 3 Mpc (3 million parsecs).^[31]
• 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):^[32]
  • 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
• 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.^[33] A notable exception is SN 2003fg, the "Champagne Supernova", a Type Ia supernova of unusual nature.
• Redshifts and Hubble's law

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 use 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.

Supernovae

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

$${\displaystyle \omega ={\frac {\Delta \theta }{\Delta t}}\,,}$$ 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 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 ${\displaystyle 1.4M_{\odot }}$.

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.^[49]

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:

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

Globular cluster luminosity function

This section does not cite any sources. (July 2022)

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.

Planetary nebula luminosity function

This section does not cite any sources. (July 2022)

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.

Surface brightness fluctuation method

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. 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.^[50]

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, though the Gaia space mission can now weigh in on that specific problem. 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. 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.^{[52][clarification needed]} 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. Edwin 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.

• Araucaria Project
• Distance measure
• Orders of magnitude (length)#Astronomical
• Standard ruler