REDSHIFT-INDEPENDENT DISTANCES IN THE NASA/IPAC EXTRAGALACTIC DATABASE: METHODOLOGY, CONTENT, AND USE OF NED-D

The number of distance estimates published based on primary indicators is doubling approximately every four years, as shown in Figure 2. Growth by a factor of 100× in primary estimates over the last 30 years means that, currently, more than 20,000 such estimates are available compared to ∼200 published from 1980 to 1985. Indeed, growth in primary estimates is occurring at a remarkably constant rate. Estimates available have doubled every four years for the last three decades. Growth in secondary estimates has been broadly parallel to, but less steady than, that in primary estimates. Updates to NED-D are made regularly, and the cadence has increased recently from four to six releases per year. In general, the full tabulation appearing online is current to within the last six months.

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Growth in data is attributable to corresponding growth in extragalactic distance scale research activity. Growth in the number of references published with primary distances, the number of estimates published per reference, and the number of authors per reference is shown in Figure 3. Specifically, growth in primary distances estimates of 100× over the past three decades is due in small part to growth in the number of references published per year (5×), and in large part due to growth in the number of estimates per reference (20×). For the period 2012 to 2014 inclusive, primary distance estimates per reference have averaged 60. That is as many primary distances per reference today as were published in the NASA HST Key Project final report, which gave 62 Cepheids-based distances for 31 galaxies (Freedman et al. 2001).

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A graphical presentation that facilitates comparison of the distance indicators is shown in Figure 4. Each indicator applies to a range of distances, as indicated by boxes showing the 25th percentile (left side), 50th percentile (median, inner solid line), and 75th percentile (right side), along with "whiskers" spanning the minimum and maximum values. Figure 4 reveals some indicators with distance estimates as far as, or even beyond, the ∼14 Gpc radius of the observable universe based on the standard cosmological model (e.g., Lineweaver & Davis 2005, noting that radius is ∼3× greater than the simplistic Hubble radius of 4.3 Gpc based on r_H = c/H₀ and assuming H₀ = 70 km s⁻¹ Mpc⁻¹). Some extremely large distances involve indicators with very high errors, including GRBs, gravitational lenses (G Lens), and H ii luminosity function (H ii LF), which can be over-estimated (and in other cases under-estimated) by factors of 2.5 or more. Some are based on Type Ia supernovae, and evidently represent statistical outliers or tentative identifications (candidates rather than confirmed) (SNIa, and SNIa SDSS). Rather than censoring such outliers, they are included in the database with their corresponding errors to accurately reflect what has been collated from the peer-reviewed literature to date. Clearly much work remains to decrease error in the techniques, and to adjust expectations regarding realistic ranges of applicability of these specific distance indicators.

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Indicators applicable to cosmological distances, such as Type Ia supernovae and GRBs, are severely limited in the minimum distance they apply to. The nearest Type Ia supernova in three decades, SN 2014J, occurred at 3.5 Mpc. The primary concern of extragalactic distance scale research is to use nearer redshift-independent indicators to calibrate more distant, cosmological indicators, in order to fix the extragalactic distance scale more precisely, and to obtain cosmological parameters with more precision.

Specific distance indicators have a perceived quality relative to each other that is subject to interpretation. The activity researchers have devoted to each indicator however, is a matter of record. Research activity in terms of the number of references and number of authors devoted to each indicator, and the number of citations received by each since 1980, are shown in Figure 5. When the indicators are shown in increasing order of the number of references that have applied each indicator to date, the research activity in terms of numbers of authors contributing, estimates published, and citations received are clearly correlated. Specific indicators appearing toward the top of Figure 5 are in general therefore, more "tried and true," compared to indicators appearing toward the bottom. At the time of writing, the top five most applied standard candle indicators by number of references are: Cepheids, TRGB, RR Lyrae, color–magnitude diagrams (CMDs), and Type Ia supernovae. The top three most applied standard rulers are: masers, eclipsing binaries, and Type II supernovae (optical). The top three most applied secondary indicators are: Tully–Fisher, fundamental plane, and diameter-rotational velocity (D_n-sigma). Note that some indicators have little research activity but many citations, because although rarely applied, they were published for comparison with more widely used indicators in articles that garner more citations.

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Usage of NED-D has been cited in nearly 250 astronomical articles as of 2016 January 1. References in the year 2012 totaled 46, up from only two or three references per year in NED-D's first two years. Citations in 2014, the most recent full year considered, are close to 60. NED-D has been cited in more than 200 papers published in the top seven astronomical and science publications, as measured by impact factors. Examples include: McCommas et al. (2009), Freedman & Madore (2010), Burns et al. (2011), Freedman et al. (2011), Smith et al. (2012), Hunter et al. (2012), Jarrett et al. (2013), Pietrzynski et al. (2013), Petty et al. (2014), and White et al. (2015).

Two core activities for NED-D, as for NED in general, involve improving the content and search capabilities, and updating to maintain the most complete data possible. In addition to keeping pace with new data appearing in the literature, NED-D completeness will improve by incorporating ∼9000 secondary distances from 50 references, which is in our near-term work plan. The challenge is that most of these data sources involve large lists and tables of galaxies that are not currently machine-readable because they are found in older, non-digitized articles published prior to 2000.

The percentage of references being published with eligible distance estimates but without keywords "galaxies–distances" in their abstracts is a concern, and worth attempting to correct. To do so, we are planning to add an update regarding publication of redshift-independent distances to our recently published Best Practices for Data Publication (Schmitz et al. 2014).⁷ Researchers publishing distance estimates for Type Ia supernovae, GRBs, and other extragalactic objects are encouraged to include the keywords "galaxies–distances," as well as the object descriptors in their abstracts, so that NED and others interested in such estimates are able to easily locate them.

Greater inclusion, rather than exclusion of relevant data, is important, because of the relative rarity and high value of redshift-independent distances data. Whether to include data in the main NED database from non peer-reviewed references available in NED-D, however, remains an open issue. Therefore, we invite user input on this issue.⁸ In addition, the current practice of providing mean NED distances based on estimates as published, uncorrected for assumed distance scale, will be upgraded to include mean estimates accounting for author differences in scale.

The NED team is in the process of developing a number of new interactive data visualizations to facilitate understanding the database content, and to simplify new types of database queries. For example, the figures in this paper are static snapshots of interactive visualizations (allowing zooming, panning, display of data attributes while hovering over markers, etc.) that are being configured on the NED website and will be updated as new content is added to the database.

Further plans include enhancing the NED Galaxy Environment service.⁹ Introduced in 2013, this feature allows researchers to quickly ascertain and graphically display the 3D neighbors of galaxies with available redshifts. A future version will include redshift-independent distances. We are also exploring techniques for generating interactive 3D maps of galaxy distributions using distances derived both from redshifts and redshift-independent indicators, for example using the World Wide Telescope (see Roberts & Fay 2014).

A related work in progress at NED involves identifying galaxy neighbors by their hierarchy, by recording and reporting on which galaxies have been identified in the literature as being members of pairs, groups, clusters, and superclusters. For NED-D, hierarchy information could be used as a force multiplier, multiplying by three or more times the number of galaxies with effective redshift-independent distances, albeit with cautionary flags to distinguish hierarchy-based inferred distances from original distance estimates.

Our ultimate plan to ensure the most complete coverage of data relevant to NED-D, as for other data types in NED over all, is to apply, test, and put into operation text data-mining algorithms to locate, classify, tag, and simplify extraction of relevant data. Simply put, effective application of modern machine learning algorithms may be the only practical way to keep the database as comprehensive as possible amidst the rapid growth of data published annually in the literature. Initial steps in this area have begun.

Active galactic nucleus (AGN) timelag

Based on the time lag between variations in magnitude observed at short wavelengths compared to those observed at longer wavelengths in AGNs. For example, using a quantitative physical model that relates the time lag to the absolute luminosity of an AGN, Yoshii et al. (2014) obtain a distance to the AGN host galaxy MRK 0335 of 146 Mpc.

AGB stars

Based on the maximum absolute visual magnitude for these stars of M_V = −2.8 (Davidge & Pritchet 1990). Thus, the brightest AGB stars in the galaxy NGC 0253, with a maximum apparent visual magnitude of m_V = 24.0, have a distance modulus of (m-M)_V = 26.8, for a distance of 2.3 Mpc.

B-type stars (B stars)

Based on the relation between absolute magnitude and beta-index in these stars, where beta-index measures the strength of the star's emission at the wavelength of hydrogen Balmer or H-beta emission. Applied to the LMC by Shobbrook & Visvanathan (1987), to obtain a distance modulus of (m-M) = 18.30, for a distance of 46 kpc, with a statistical error of 0.20 mag or 4 kpc (10%).

BL Lac object luminosity (BL Lac luminosity)

Based on the mean absolute magnitude of the giant elliptical host galaxies of these AGNs. Applied to BL Lacertae host galaxy MS 0122.1+0903 by Sbarufatti et al. (2005), to obtain a distance of 1530 Mpc.

Black hole

Based on super-Eddington accreting massive black holes, as found the host galaxies of certain AGNs at high redshift, and a unique relationship between their bolometric luminosity and central black hole mass. Based on a method to estimate black hole masses (Wang et al. 2014), the black hole mass–luminosity relation is used to estimate the distance to 16 AGN host galaxies, including for example galaxy MRK 0335, to obtain a distance of 85.9 Mpc, with a statistical error of 26.3 Mpc (31%).

Blue supergiant

Based on the absolute magnitude and the equivalent widths of the hydrogen Balmer lines of these stars. Applied to the SMC by Bresolin (2003) to obtain a distance modulus of (m-M) = 19.00, for a distance of 64 kpc, with a statistical error of 0.50 mag or 16 kpc (25%).

Brightest stars

Based on the mean absolute visual magnitude for red supergiant stars, M_V = −8.0, Davidge et al. (1991) present an application to NGC 0253 where red supergiant stars have apparent visual magnitude m_V = 19.0, leading to a distance modulus of (m-M)_V = 27.0, for a distance of 2.5 Mpc.

Carbon stars

Based on the mean absolute near-infrared magnitude of these stars M_I = −4.75 (Pritchet et al. 1987). Thus, carbon stars in galaxy NGC 0055 with a maximum apparent infrared magnitude m_I = 21.02, including a correction of −0.11 mag for reddening, have a distance modulus of (m-M)_I = 25.66, for a distance of 1.34 Mpc, with a statistical error of 0.13 mag or 0.08 Mpc (6%).

Cepheids

Based on the mean luminosity of Cepheid variable stars, which depends on their pulsation period, P. For example, a Cepheid with a period of P = 54.4 days has an absolute mean visual magnitude of M_V = −6.25, based on the period–luminosity (PL) relation adopted by the HST Key Project on the Extragalactic Distance Scale (Freedman et al. 2001). Thus, a Cepheid with a period of P = 54.4 days in the galaxy NGC 1637 (Leonard et al. 2003) with an apparent mean visual magnitude m_V = 24.19, has an apparent visual distance modulus of (m-M)_V = 30.44, for a distance of 12.2 Mpc. Averaging the apparent visual distance moduli for the 18 Cepheids known in this galaxy (including corrections of 0.10 mag for reddening and metallicity) gives a corrected distance modulus of (m-M)_V = 30.34, for a distance of 11.7 Mpc, with a statistical error of 0.07 mag or 0.4 Mpc (3.5%).

CMD

Based on the absolute magnitude of a galaxy's various stellar populations, discernable in a CMD. Applied to the LMC by Andersen et al. (1984), to obtain a distance modulus of (m-M) = 18.40, for a distance of 47.9 kpc.

Delta Scuti

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheid and Mira variables, a PL relation gives their absolute magnitude. Applied to the LMC by McNamara et al. (2007), to obtain a distance modulus of (m-M) = 19.46, for a distance of 49 kpc, with a statistical error of 0.19 mag or 4.5 kpc (9%).

Flux-weighted gravity–luminosity relation (FGLR)

Based on the absolute bolometric magnitude of A-type supergiant stars, determined by the FGLR (Kudritzki et al. 2008). Applied to galaxy Messier 31, to obtain a distance of 0.783 Mpc.

GRB

Based on six correlations of observed properties of GRBs with their luminosities or collimation-corrected energies. A Bayesian fitting procedure then leads to the best combination of these correlations for a given data set and cosmological model. Applied to GRB 021004 by Cardone et al. (2009), to obtain a luminosity distance modulus of (m-M) = 46.60 for a luminosity distance of 20,900 Mpc. With the GRB's redshift of z = 2.3, this leads to a linear distance of 6330 Mpc, with a statistical error of 0.48 mag or 1570 Mpc (25%).

Globular cluster luminosity function (GCLF)

Based on an absolute visual magnitude of M_V = −7.6, which is the location of the peak in the luminosity function of old, blue, low-metallicity globular clusters (Larsen et al. 2001). So, for example, the galaxy NGC 0524 with an apparent visual magnitude m_V = 24.36 for the peak in the luminosity function of its globular clusters, has a distance modulus of (m-M)_V = 31.99, for a distance of 25 Mpc, with a statistical error of 0.14 mag or 1.8 Mpc (7%).

Globular cluster surface brightness fluctuations (GC SBF)

Based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of the cluster due to individual stars (Ajhar et al. 1996). Thus, the implied apparent magnitude of the stars leading to these fluctuations gives the distance modulus in magnitudes. Applied to galaxy Messier 31, to obtain a distance modulus of (m-M) = 24.56, for a distance of 0.817 Mpc, with a statistical error of 0.12 mag or 0.046 Mpc (6%).

H ii luminosity function (H ii LF)

Based on a relation between velocity dispersion, metallicity, and the luminosity of the H-beta line in H ii regions and H ii galaxies (e.g., Siegel et al. 2005, and references therein). Applied to high-redshift galaxy CDFa C01, to obtain a luminosity distance modulus of (m-M) = 45.77, for a luminosity distance of 14,260 Mpc. With a redshift for the galaxy of z = 3.11, this leads to a linear distance of 3470 Mpc, with a statistical error of 1.58 mag or 3,710 Mpc (93%).

Horizontal branch

Based on the absolute visual magnitude of horizontal branch stars, which is close to M_V = +0.50, but depends on metallicity (Da Costa et al. 2002). Thus, horizontal branch stars in the galaxy Andromeda III with an apparent visual magnitude m_V = 25.06, including a reddening correction of −0.18 mag, have a distance modulus of (m-M)_V = 24.38, for a distance of 750 kpc, with a statistical error of 0.06 mag or 20 kpc (3%).

M stars luminosity (M stars)

Based on the relationship between absolute magnitude and temperature-independent spectral index for normal M Stars. Applied to the LMC by Schmidt-Kaler & Oestreicher (1998), to obtain a distance modulus of (m-M) = 18.34, for a distance of 46.6 kpc, with a statistical error of 0.09 mag or 2.0 kpc (4%).

Miras

Based on the mean absolute magnitude of Mira variable stars, which depends on their pulsation period. As with Cepheid variables, a PL relation gives their absolute magnitude. Applied to the LMC by Feast et al. (2002), to obtain a distance modulus of (m-M) = 18.60, for a distance of 52.5 kpc, with a statistical error of 0.10 mag or 2.5 kpc (5%).

Novae

Based on the maximum absolute visual magnitude reached by these explosions, which is M_V = −8.77 (Ferrarese et al. 1996). So, a nova in galaxy Messier 100 with a maximum apparent visual magnitude of m_V = 22.27, has a distance modulus of (m-M)_V = 31.0, for a distance of 15.8 Mpc, with a statistical error of 0.3 mag or 2.4 Mpc (15%).

O- and B-type supergiants (OB stars)

Based on the relationship between spectral type, luminosity class, and absolute magnitude for these stars. Applied to 30 Doradus in the LMC by Walborn & Blades (1997), to obtain a distance of 53 kpc.

Planetary nebula luminosity function (PNLF)

Based on the maximum absolute visual magnitude for planetary nebulae of M_V = −4.48 (Ciardullo et al. 2002). So, planetary nebulae in the galaxy NGC 2403 with a maximum apparent visual magnitude of m_V = 23.17 have a distance modulus of (m-M)_V = 27.65, for a distance of 3.4 Mpc, with a statistical error of 0.17 mag or 0.29 Mpc (8.5%).

Post-asymptotic giant branch stars (PAGB Stars)

Based on the maximum absolute visual magnitude for these stars of M_V = −3.3 (Bond & Alves 2001). Thus, PAGB stars in Messier 31 with a maximum apparent visual magnitude of m_V = 20.88 have a distance modulus of (m-M)_V = 24.2, for a distance of 690 kpc, with a statistical error of 0.06 mag or 20 kpc (3%).

Quasar spectrum

Based on the observed apparent spectrum of a quasar, compared with the absolute spectrum of comparable quasars as determined based on HST spectra taken of 101 quasars. Applied to 11 quasars by de Bruijne et al. (2002), including quasar [HB89] 0000–263, to obtain a distance of 3.97 Gpc.

RR Lyrae stars

Based on the mean absolute visual magnitude of these variable stars, which depends on metallicity: M_V = F/H × 0.17 + 0.82 mag (Pritzl et al. 2005). So, RR Lyrae stars with metallicity F/H = −1.88 in the galaxy Andromeda III have an apparent mean visual magnitude of m_V = 24.84, including a 0.17 mag correction for reddening. Thus, they have a distance modulus of (m-M)_V = 24.34, for a distance of 740 kpc, with a statistical error of 0.06 mag or 22 kpc (3.0%).

Red clump

Based on the maximum absolute infrared magnitude for red clump stars of M_I = −0.67 (Dolphin et al. 2003). So, red clump stars in the galaxy Sextans A with a maximum apparent infrared magnitude of m_I = 24.84, including a 0.07 mag correction for reddening, have a distance modulus of (m-M)_I = 25.51, for a distance of 1.26 Mpc, with a statistical error of 0.15 mag or 0.09 Mpc (7.5%).

Red supergiant variables (RSV stars)

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period (Jurcevic 1998). As with Cepheid and Mira variables, a PL relation gives their absolute magnitude. Applied to galaxy NGC 2366, to obtain a distance modulus of (m-M) = 27.86, for a distance of 3.73 Mpc, with a statistical error of 0.20 mag or 0.36 Mpc (10%).

Red variable stars (RV stars)

Based on the mean absolute magnitude of RV stars, which depends on their pulsation period (Kiss & Bedding 2004). As with Cepheid variables, a PL relation gives their absolute magnitude. Applied to the SMC to obtain a distance modulus of (m-M) = 18.94, for a distance of 61.4 kpc, with a statistical error of 0.05 mag or 1.4 kpc (2.3%)

S Doradus stars

Based on the mean absolute magnitude of these stars, which is derived based on their amplitude-luminosity relation. Applied to galaxy Messier 31 by Wolf (1989), to obtain a distance modulus of (m-M) = 24.40, for a distance of 0.759 Mpc.

SNIa SDSS

Based on SNIa (Type Ia supernovae). It is distinguished from normal SNIa however, because it has been applied to candidate SNIa obtained in the SDSS Supernova Survey that have not yet been confirmed as bona fide SNIa (Sako et al. 2014). Applied to Type Ia supernova SDSS-II SN 13651, to obtain a luminosity distance modulus of (m-M) = 41.64 for a luminosity distance of 2130 Mpc. With a redshift for the supernova of z = 0.25, this leads to a linear distance of 1700 Mpc.

SX Phoenicis stars

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheid and Mira variables, a PL relation gives their absolute magnitude (e.g., McNamara 1995). Applied to the Carina Dwarf Spheroidal galaxy, to obtain a distance modulus of (m-M) = 20.01, for a distance of 0.100 Mpc, with a statistical error of 0.05 mag or 0.002 Mpc (2.3%).

Short gamma-ray bursts (SGRBs)

Similar to but distinct from the GRB standard candle, because it employs only GRBs of short, less than 2 s duration (Rhoads 2010). SGRBs are conjectured to be a distinct subclass of GRBs, differing from the majority of normal or "long" GRBs, which have durations of greater than 2 s. Applied to SGRB GRB 070724A, to obtain a linear distance of 557 Mpc.

Statistical

Based on the mean distance obtained from multiple distance estimates, based on at least several to as many as a dozen or more different standard candle indicators, although standard ruler indicators may also be included. For example, Freedman & Madore (2010) analyzed 180 estimates of the distance to the LMC, based on two dozen indicators not including Cepheids, to obtain a mean distance modulus of (m-M) = 18.44, for a distance of 48.8 kpc, with a statistical error of 0.18 mag or 4.2 kpc (9%).

Subdwarf fitting

Gives an improved calibration of the distances and ages of globular clusters. Applied to the LMC by Carretta et al. (2000), to obtain a distance modulus of (m-M) = 18.64, for a linear distance of 53.5 kpc, with a statistical error of 0.12 mag or 3.0 kpc (6%).

Sunyaev–Zeldovich effect (SZ effect)

Based on the predicted Compton scattering between the photons of the cosmic microwave background radiation and electrons in galaxy clusters, and the observed scattering, giving an estimate of the distance. For galaxy cluster CL 0016+1609, Bonamente et al. (2006) obtain a linear distance of 1300 Mpc, assuming an isothermal distribution.

SBF

Based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of the galaxy due to individual stars, primarily red giants with maximum absolute K-band magnitudes of M_K = −5.6 (Jensen et al. 1998). So, the galaxy NGC 1399, for example, with brightest stars at an implied maximum apparent K-band magnitude m_K = 25.98, has a distance modulus of (m-M)_K = 31.59, for a distance of 20.8 Mpc, with a statistical error of 0.16 mag or 1.7 Mpc (8%).

TRGB

Based on the maximum absolute infrared magnitude for TRGB stars of M_I = −4.1 (Sakai et al. 2000). So, the LMC, with a maximum apparent infrared magnitude for these stars of m_I = 14.54, has a distance modulus of (m-M)_I = 18.59, for a distance of 52 kpc, with a statistical error of 0.09 mag or 2 kpc (4.5%).

Type II Cepheids

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with normal Cepheids and Miras, a PL relation gives their absolute magnitude. Applied to galaxy NGC 4603 by Majaess et al. (2009), to obtain a distance modulus of (m-M) = 32.46, for a linear distance of 31.0 Mpc, with a statistical error of 0.44 mag or 7.0 Mpc (22%).

Type II supernovae, radio (SNII radio)

Based on the maximum absolute radio magnitude reached by these explosions, which is 5.5 × 10²³ ergs s⁻¹ Hz⁻¹ (Clocchiatti et al. 1995). So, the type-II SN 1993J in galaxy Messier 81 (NGC 3031), based on its maximum apparent radio magnitude, has a distance of 2.4 Mpc.

Type Ia supernovae (SNIa)

Based on the maximum absolute blue magnitude reached by these explosions, which is M_B = −19.3 (Astier et al. 2006). Thus, for example, SN 1990O (in the galaxy MCG +03-44-003) with a maximum apparent blue magnitude of m_B = 16.20, has a luminosity distance modulus of (m-M)_B = 35.54 (including a 0.03 mag correction for color and redshift), or a luminosity distance of 128 Mpc. With a redshift for the galaxy of z = 0.0307, this leads to a linear distance of 124 Mpc, with a statistical error of 0.09 mag or 6 Mpc (4.5%).

White dwarfs

Based on the absolute magnitudes of white dwarf stars, which depends on their age. Applied to the LMC by Carretta et al. (2000), to obtain a distance modulus of (m-M) = 18.40, for a linear distance of 47.9 kpc, with a statistical error of 0.15 mag or 3.4 kpc (7%).

Wolf-Rayet

Based on the mean absolute magnitude of these massive stars. Applied to galaxy IC 0010, by Massey & Armandroff (1995), to obtain a distance of 0.95 Mpc.

CO ring diameter

Based on the mean absolute radius of a galaxy's inner carbon monoxide (CO) ring, with compact rings of r = ∼200 pc and broad rings of r = ∼750 pc. So, a CO compact ring in the galaxy Messier 82 with an apparent radius of 130 arcsec, has a distance of 3.2 Mpc (Sofue 1991).

Dwarf galaxy diameter

Based on the absolute radii of certain kinds of dwarf galaxies surrounding giant elliptical galaxies such as Messier 87. Specifically, dwarf elliptical (dE) and dwarf spheroidal (dSph) galaxies have an effective absolute radius of ∼1.0 kpc that barely varies in such galaxies over several orders of magnitude in mass. So, the apparent angular radii of these dwarf galaxies around Messier 87 at 11.46 arcsec, gives a distance for the main galaxy of 18.0 ± 3.1 Mpc (Misgeld & Hilker 2011).

Eclipsing binary

A hybrid method between standard rulers and standard candles, using stellar pairs orbiting one another fortuitously such that their individual masses and radii can be measured, allowing the system's absolute magnitude to be derived. Thus, the absolute visual magnitude of an eclipsing binary in the galaxy Messier 31 is M_V = −5.77 (Ribas et al. 2005). So, this eclipsing binary, with an apparent visual magnitude of m_V = 18.67, has a distance modulus of (m-M)_V = 24.44, for a distance of 772 kpc, with a statistical error of 0.12 mag or 44 kpc (6%).

Globular cluster radii (GC radius)

Based on the mean absolute radii of globular clusters, r = 2.7 pc (Jordan et al. 2005). So, globular clusters in the galaxy Messier 87 with a mean apparent radius of r = 0.032 arcsec, have a distance of 16.4 Mpc.

Grav. stability gas. Disk

Based on the absolute diameter at which a galaxy reaches the critical density for gravitational stability of the gaseous disk (Zasov & Bizyaev 1996). A distance to galaxy Messier 74 is obtained of 9.40 Mpc.

Gravitational lenses (G Lens)

Based on the absolute distance between the multiple images of a single background galaxy that surround a gravitational lens galaxy, determined by time-delays measured between images. Thus, the apparent distance between images gives the lensing galaxy's distance. Applied to the galaxy 87GB[BWE91] 1600+4325 ABS01 by Burud et al. (2000), to obtain a distance of 1,920 Mpc.

H ii region diameters (H ii)

Based on the mean absolute diameter of H ii regions, d = 14.9 pc (Ismail et al. 2005). So, H ii regions in the galaxy Messier 101 with a mean apparent diameter of r = 4.45 arcsec, have a distance of 6.9 Mpc.

Jet proper motion

Based on the apparent motion of individual components in parsec-scale radio jets, obtained by observation, compared with their absolute motion, obtained by Doppler measurements and corrected for the jet's angle to the line of sight. Applied to the quasar 3C 279 by Homan & Wardle (2000), to obtain an angular size distance of 1.8 ± 0.5 Gpc.

Masers

Based on the absolute motion of masers orbiting at great speeds within parsecs of supermassive black holes in galaxy cores, relative to their apparent or proper motion. The absolute motion of masers orbiting within the galaxy NGC 4258 is V_t = 1,075 km s⁻¹, or 0.001100 pc yr⁻¹ (Humphreys et al. 2004). So, the maser's apparent proper motion of 31.5 × 10⁻⁶ arcsec yr⁻¹, gives a distance of 7.2 Mpc, with a statistical error of 0.2 Mpc (3.0%).

Orbital mechanics (Orbital mech.)

Based on the predicted orbital or absolute motion of a galaxy around another galaxy, and its observed apparent motion, giving a measure of distance. Applied by Howley et al. (2008) to the Messier 31 satellite galaxy Messier 110, to obtain a linear distance of 0.794 Mpc.

Proper motion

Based on the absolute motion of a galaxy, relative to its apparent or proper motion. Applied to galaxy Leo B by Lepine et al. (2011), to obtain a linear distance of 0.215 Mpc.

Ring diameter

Based on the apparent angular ring diameter of certain spiral galaxies with inner rings, compared to their absolute ring diameter, as determined based on other apparent properties, including morphological stage and luminosity class (Pedreros & Madore 1981). For galaxy UGC 12914, a distance modulus is obtained of (m-M) = 32.30, for a linear distance of 29.0 Mpc, with a statistical error of 0.84 mag or 13.6 Mpc (47%), assuming H = 100 km s⁻¹ Mpc⁻¹.

Type II supernovae, optical (SNII optical)

Based on the absolute motion of the explosion's outward velocity, in units of intrinsic transverse velocity, V_t (usually km s⁻¹), relative to the explosion's apparent or proper motion (usually arcseconds year⁻¹) (e.g., Eastman et al. 1996). So, the absolute motion of Type II SN 1979C observed in the galaxy Messier 100, based on the Expanding Photosphere Method (EPM), gives a distance of 15 Mpc, with a statistical error of 4.3 Mpc (29%). An alternative SNII Optical indicator uses the Standardized Candle Method (SCM) of Hamuy & Pinto (2002). Applied to Type II SN 2003gd in galaxy Messier 74, by Hendry et al. (2005), to obtain a distance of 9.6 Mpc, with a statistical error of 2.8 Mpc (29%).

Brightest cluster galaxy (BCG)

Based on the fairly uniform absolute visual magnitudes of M_V = −22.68 ± 0.35 found among the brightest galaxies in galaxy clusters (see Hoessel 1980). So, for example, for the brightest galaxy in the galaxy cluster Abell 0021, which is the galaxy 2MASX J00203715+2839334 and which has an apparent visual magnitude of m_V = 15.13, the luminosity distance modulus can be calculated, as done by Hoessel et al. (1980). The result is a luminosity distance modulus of (m-M) = 37.81, or a luminosity distance of 365 Mpc. With a redshift for the BCG in Abell 0021 of z = 0.0945, this leads to a linear distance of 333 Mpc, with a statistical error of 0.35 mag or 59 Mpc (18%).

D_n-sigma

Provides standard candles based on the absolute magnitudes of elliptical and early-type galaxies, determined from the relation between the galaxy's apparent magnitude and apparent diameter (e.g., Willick et al. 1997). Applied to galaxy ESO 409- G 012, to obtain a distance modulus of (m-M) = 33.9, for a linear distance of 61 Mpc, with a statistical error of 0.40 mag or 12 Mpc (20%).

Diameter

Certain galaxy's major diameters may provide secondary standard rulers based on the absolute diameter for example of only the largest, or "supergiant" spiral galaxies, estimated to be ∼52 kpc (van der Kruit 1986). So, from the mean apparent diameter found for supergiant spiral galaxies in the Virgo cluster of ∼9 arcmin, the Virgo cluster distance is estimated to be 20 Mpc, with a statistical error of 3 Mpc (15%).

Dwarf ellipticals

Based on the absolute magnitude of dwarf elliptical galaxies, derived from a surface-brightness/luminosity relation, and the observed apparent magnitude of these galaxies (Caldwell & Bothun 1987). Applied to dwarf elliptical galaxies around galaxy NGC 1316 in the Fornax galaxy cluster, to obtain a distance of 12 Mpc.

Faber–Jackson

Based on the absolute magnitudes of elliptical and early-type galaxies, determined from a relation between a galaxy's apparent magnitude and velocity dispersion (Lucey 1986). Applied to galaxy NGC 4874, to obtain a distance modulus of (m-M) = 34.76, for a linear distance of 89.5 Mpc, with a statistical error of 0.12 mag or 5.1 Mpc (6%).

Fundamental plane (FP)

Based on the absolute magnitudes of early-type galaxies, which depend on effective visual radius r_e, velocity dispersion sigma, and mean surface brightness within the effective radius I_e: log D = log r_e–1.24 log sigma + 0.82 log I_e + 0.173 (e.g., Kelson et al. 2000). The galaxy NGC 1399 has an effective radius r_e = 55.4 arcsec, a rotational velocity sigma = 301 km s⁻¹, and surface brightness, I_e = 428.5 L_Sun pc⁻². So, from the FP relation, its distance is 20.6 Mpc.

GC K versus (J − K)

The globular cluster K-band magnitude versus J-band minus K-band CMD secondary standard candle is similar to the CMD standard candle, but applied specifically to globular clusters within a galaxy, rather than entire galaxies (Sitko 1984). Applied to galaxy Messier 31, to obtain a linear distance of 0.689 Mpc.

GeV–TeV ratio

Based on the absolute magnitude at which this ratio equals one, which compares energy emitted at two wavelengths, giga-electron volt and tera-electron volt (Prandini et al. 2010). Applied to galaxy 3C66A, to obtain a linear distance of 794 Mpc.

Globular cluster fundamental plane (GC FP)

Based on the relationship among velocity dispersion, radius, and mean surface brightness for globular clusters, similar to the fundamental plane for early-type galaxies (Strader et al. 2009). Applied to globular clusters in galaxy Messier 31, to obtain a distance modulus of (m-M) = 24.57, for a linear distance of 0.820 Mpc, with a statistical error of 0.05 mag or 0.019 Mpc (2.3%).

H I + optical distribution

Based on neutral hydrogen I mass versus optical distribution or virial mass provides a secondary standard ruler that applies to extreme H I-rich galaxies, such as Michigan 160, based on the assumption that the distance-dependent ratio of neutral gas to total (virial) mass should equal one (Staveley-Smith et al. 1990). Applied to galaxy UGC 12578, to obtain a distance modulus of (m-M) = 33.11, for a linear distance of 41.8 Mpc, with a statistical error of 0.20 mag or 4.0 Mpc (10%).

Infra-Red Astronomical Satellite (IRAS)

Based on a reconstruction of the local galaxy density field using a model derived from the 1.2 Jy IRAS survey with peculiar velocities accounted for using linear theory (e.g., Willick et al. 1997). Applied to galaxy UGC 12897, to obtain a distance modulus of (m-M) = 35.30, for a linear distance of 115 Mpc, with a statistical error of 0.80 mag or 51 Mpc (44%).

LSB galaxies

Based on the SBF standard candle, which is based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of a galaxy due to individual stars, but applied specifically to low surface brightness (LSB) galaxies (Bothun et al. 1991). Applied to LSB galaxies around galaxy NGC 1316 in the Fornax galaxy cluster, to obtain a distance modulus of (m-M) = 31.25, for a linear distance of 17.8 Mpc, with a statistical error of 0.28 mag or 2.4 Mpc (14%).

Magnetic energy

Based on an extragalactic object's magnetic energy and particle energy, and calculations assuming certain relations between the two. It has been applied so far to only one gamma-ray source, HESS J1507-622 (Domainko 2014). Depending on which theoretical possibilities are assumed, the distance is estimated to range from 0.18 Mpc to 100 Mpc, indicating that HESS J1507-622 is extragalactic.

Magnitude

Based on the apparent magnitudes of certain galaxies, which may provide a secondary standard candle based on the mean absolute magnitude determined from a sample of similar galaxies with known distances. Assuming a mean absolute blue magnitude for dwarf galaxies of M_B = −10.70, the dwarf galaxy DDO 155 with an apparent blue magnitude of m_B = 14.5, has a distance modulus of (m-M)_B = 25.2, for a distance of 1.1 Mpc (Moss & de Vaucouleurs 1986).

Mass model

Based on the absolute radii of galaxy halos, estimated from the galaxy plus halo mass as derived from rotation curves and from the expected mass density derived theoretically (Gentile et al. 2010). Applied to galaxy NGC 1560, to obtain a linear distance of 3.16 Mpc.

Radio brightness

Based on the absolute radio brightness assumed versus the apparent radio brightness observed in a galaxy (Wiklind & Henkel 1990). Applied to galaxy NGC 0404, to obtain a distance of 10 Mpc.

Sosies

"Look Alike," or in French "Sosies," galaxies provide standard candles based on a mean absolute visual magnitude of M_V = −21.3 found for spiral galaxies with similar Hubble stages, inclination angle, and light concentrations (Terry et al. 2002). So, the galaxy NGC 1365, with an apparent visual magnitude of m_V = 9.63, has a distance modulus of (m-M)_V = 30.96, for a distance of 15.6 Mpc. Galaxy NGC 1024, with an apparent visual magnitude of m_V = 12.07 that is 2.44 mag fainter and apparently farther than NGC 1365, is also estimated to be 0.06 mag less luminous than NGC 1365, leading to a distance modulus of (m-M)_V = 33.34, for a distance of 46.6 Mpc.

Tertiary

A catch-all term for various distance indicators employed by de Vaucouleurs et al. in the 1970s and 1980s, including galaxy luminosity index and rotational velocity (e.g., McCall 1989). Applied to galaxy IC 0342, to obtain a distance modulus of (m-M) = 26.32, for a linear distance of 1.84 Mpc, with a statistical error of 0.15 mag or 0.13 Mpc (7%).

Tully estimate (Tully est)

Based on various parameters, including galaxy magnitudes, diameters, and group membership (Tully, NGC, 1988). For galaxy ESO 012- G 014, the estimated distance is 23.4 Mpc.

Tully–Fisher

Introduced by Tully & Fisher (1977), based on the absolute blue magnitudes of spiral galaxies, which depend on their apparent blue magnitude, m_B, and their maximum rotational velocity, sigma: M_B = −7.0 log sigma—1.8 (e.g., Karachentsev et al. 2003). So, the galaxy NGC 0247 has an absolute blue magnitude of M_B = −18.2, based on its rotational velocity, sigma = 222 km s⁻¹. With an apparent blue magnitude of m_B = 9.86, NGC 0247 has a distance modulus of (m-M)_B = 28.1, for a distance of 4.1 Mpc.

Here are some notes relating to Cepheids distances in particular, and to standard candle indicators in general, regarding different luminosity relations, apparent versus reddening-corrected distance, and corrections related to age or metallicity.

Cepheid variable stars have absolute visual magnitudes related to the log of their periods in days

$$\begin{eqnarray*}&&{M}_{{\rm{V}}}=-2.76\,\mathrm{log}\,P-1.46\end{eqnarray*}$$

This is the PL relation adopted by NASA's HST Key Project On the Extragalactic Distance Scale (Freedman et al. 2001).

In the galaxy NGC 1637, the longest-period Cepheid of 18 observed has a period of 54.42 days, yielding a mean absolute visual magnitude of M_V = −6.25 (Leonard et al. 2003). With the star's apparent mean visual magnitude of m_V = 24.19, its apparent visual distance modulus of is (m-M)_V = 30.44, corresponding to a distance of 12.2 Mpc.

NGC 1637's shortest-period Cepheid, with a period of 23.15 days, has a mean absolute visual magnitude of M_V = −5.23. The shorter period variable's mean apparent visual magnitude is m_V = 25.22, giving an apparent visual distance modulus of (m-M)_V = 30.45, for a distance of 12.3 Mpc. This is in excellent agreement with the distance found from the longest-period Cepheid in the same galaxy.

Nevertheless, there is in practice a significant scatter in the individual Cepheid distance moduli within a single galaxy. In the galaxy NGC 1637, for example, the average of the apparent distance moduli for all 18 Cepheids is (m-M)_V = 30.76, corresponding to a distance of 14.2 Mpc. This is ∼0.3 mag fainter than the distance moduli obtained from either the longest- or shortest-period Cepheids, and corresponds to a 15% greater distance.

Scatter in individual Cepheid distance moduli is caused primarily by differential "reddening" or dimming due to differing patches of dust within target galaxies, and to a lesser extent by reddening due to foreground dust within the Milky Way, as well as differences in the intervening intergalactic medium. Because reddening is wavelength-dependent (greater at shorter wavelengths) the difference between distance moduli measured at two or more wavelengths can be used to estimate the extinction at any wavelength, E_V-I = (m-M)_V - (m-M)_I. For NGC 1637, with (m-M)_V-I = 30.76–30.54, the extinction between V and I is E_V-I = 0.22. Extinction, when multiplied by the ratio of total-to-selective absorption and assuming that ratio to be R_V = 2.45, equals the total absorption, or dimming in magnitudes of the visual distance modulus due to dust, A_V = R_V × E_V-I = 0.54 in the case of NGC 1637. Note different total-to-selective absorption ratios are assumed by different authors. The correction for dimming due to dust obtained by Leonard et al. (2003) is deducted from the apparent visual distance modulus of (m-M)_V = 30.76 to obtain the true, reddening-corrected, "Wesenheit" distance modulus of (m-M)_W = 30.23, corresponding to a distance of 11.1 Mpc.

Cepheids formed in galaxies with higher "metal" abundance ratios (represented here by measured oxygen/hydrogen ratios), are comparatively less luminous than Cepheids formed in "younger" less evolved galaxies.

Leonard et al. (2003) apply a metallicity correction of Z = 0.12 mag, based on the difference in metal abundance between galaxy NGC 1637 and the LMC. Their final, metallicity- and reddening-corrected distance modulus is (m-M)_Z = 30.34, corresponding to a distance of 11.7 Mpc.

Different corrections for reddening and age or metallicity are applied by different authors. For a review see Freedman & Madore (2010).

Differences affecting distance estimates, whether based on Cepheid variables or other methods, include corrections for:

• 1.  
  line-of-sight extinction due to foreground, target, and IGM obscuration,
• 2.  
  age/metallicity/colors,
• 3.  
  distance scale zero point,
• 4.  
  distance scale formula (PL or other relation),
• 5.  
  photometric zero point,
• 6.  
  other biases; for example, the well-known Malmquist bias,
• 7.  
  cosmological priors; for example, the Hubble constant.

All these involve systematic and statistical errors; Freedman et al. (2001) have a discussion of the errors involved in many of the methods discussed here.