NED-D Distance Methods Explanation and Examples

Standard Candles

• AGN time lag

  The AGN time lag standard candle is based on the time lag between observed variations in the AGN’s magnitude at short wavelengths compared to those same variations observed at longer wavelengths. For example, based on a quantitative physical model that relates the time lag to the absolute luminosity of an AGN (see Yoshii et al., 2014ApJ...784L..11Y) find a distance to the AGN host galaxy MRK 0335 of 146 Mpc.

• Asymptotic Giant Branch Stars (AGB)

  The Brightest AGB Stars standard candle is based on the maximum absolute visual magnitude for these stars M_V = -2.8 (see Davidge & Pritchet, 1990AJ....100..102D). 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, or a distance of 2.3 Mpc.

• B-type Stars (B Stars)

  The B Stars standard candle is based on the beta-index/absolute-magnitude relation of these stars (see e.g. Shobbrook & Visvanathan, 1987MNRAS.225..947S).

• BL Lac Object Luminosity (BL Lac Luminosity)

  The BL Lac standard candle is based on the mean absolute magnitude of the giant elliptical host galaxies of these active nuclei (see Sbarufatti, et al. 2005ApJ...635..173S).

• Black Hole

  Black Holes of a specific kind known as super-Eddington accreting massive black holes, as found in certain Active Galactic Nuclei hosting galaxies at high redshift, have a unique relationship between their bolometric luminosity and their black hole mass. Using the reverberation mapping technique to estimate black hole masses (see Wang et al., 2014ApJ...793..108W), the mass-luminosity relation is used to estimate the distance to 16 AGN hosting galaxies, including for example for the galaxy MRK 0335, finding D = 85.9 Mpc, with a statistical uncertainty of 26.3 Mpc (31%).

• Blue Supergiant

  The Blue Supergiant standard candle is based on the absolute magnitude and the equivalent widths of the Balmer lines of these stars (see Bresolin, 2003LNP...635..149B).

• Brightest Cluster Galaxy (BCG)

  The brightest galaxies in galaxy clusters provide standard candles suitable for use as a secondary distance indicator, because their absolute visual magnitudes tend to be very similar at approximately M(V) = -22.68 mag ± 0.35 mag (see Hoessel, 1980ApJ...241..493H). 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 mag, the luminosity distance modulus can be calculated as m-M(L) = m(V) - M(V), as done by Hoessel, Gunn & Thuan (see Hoessel et al., 1980ApJ...241..486H). The result is a luminosity distance modulus of m-M(L) = 37.81 mag, which gives a luminosity distance of 365 Mpc. Since the luminosity distance equals the linear distance divided by 1 + z, where z is the object’s redshift, and the redshift of the BCG in Abell 0021 is z = 0.094509, the linear distance of that galaxy is estimated to be 333 Mpc, with a statistical uncertainty of 0.35 mag or 59 Mpc (18%).

• Brightest Stars

  The Brightest Stars standard candle is based on the mean absolute visual magnitude for the brightest red giant halo stars, M_V= -2.8. Davidge & Pritchet, 1990AJ....100..102D, present an application to NGC 0253 where the brightest red giants become obvious at apparent visual magnitude m_V= 24.0, leading to a distance modulus (m-M)_V= 26.8, or a distance of 2.3 Mpc.

• Carbon Stars

  The Carbon Stars standard candle is based on the mean absolute near-infrared magnitude of these stars M_I = -4.75 (see Pritchet et al., 1987ApJ...323...79P). 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 (m-M)_I = 25.66, or a distance of 1.34 Mpc, with a statistical uncertainty of 0.13 mag or 0.08 Mpc (6%).

• Cepheids

  The Cepheid standard candle is 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 (see Freedman et al. 2001ApJ...553...47F).
  
  Thus, a Cepheid with a period of P = 54.4 days in the galaxy NGC 1637 (see Leonard et al. 2003ApJ...594..247L) with an apparent mean visual magnitude m_V = 24.19, has an apparent visual distance modulus (m-M)_V = 30.44, or a distance of 12.2 Mpc.
  
  Averaging the apparent visual distance moduli for the eighteen Cepheids known in this galaxy (including corrections of 0.10 mag for reddening and metallicity) gives a corrected distance modulus (m-M)_V = 30.34, or a distance of 11.7 Mpc, with a statistical uncertainty of 0.07 mag or 0.4 Mpc (3.5%).
  
  Here are some notes relating to Cepheid distances:
  
  
  • Period-Luminosity Relation
    
    Cepheid variable stars have absolute visual magnitudes related to the log of their periods in days

    M_V = -2.76 log P - 1.46

    This is the PL relation adopted by NASA’s Hubble Space Telescope Key Project On the Extragalactic Distance Scale (see Freedman et al. 2001ApJ...553...47F).
    
    In the galaxy NGC 1637, the longest period Cepheid of 18 observed has a period of 54.42 days, yeilding a mean absolute visual magnitude of M_V = -6.25 (again, see Leonard et al. 2003ApJ...594..247L). With the star’s apparent mean visual magnitude of m_V = 24.19, its apparent visual distance modulus 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, or a distance of 12.3 Mpc. This is in excellent agreement with the distance found from the longest-period Cepheid in the same galaxy.
  
  
  • Apparent distance
    
    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 and 15% larger than the distance derived based on just the longest or shortest period Cepheids.
  
  
  • Reddening-corrected 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, R_V = 2.45, equals the total absorption, or dimming in magnitudes of the visual distance modulus due to dust, A_V = R_V x E_V-I = 0.54 in the case of NGC 1637. This correction for dimming due to dust must be deducted from the visual distance modulus of (m-M)_V = 30.76 to find the true, reddening-corrected, "Wesenheit" distance modulus of (m-M)_W= 30.23, corresponding to a distance of 11.1 Mpc.
  
  
  • Metallicity-corrected distance
    
    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. (2003ApJ...594..247L) apply a metallicity correction of Z = 0.12 mag, based on the different metal abundances for our example galaxy NGC 1637 and for the Large Magellanic Cloud. Their final, metallicity- and reddening-corrected distance modulus is (m-M)_Z= 30.34, corresponding to a distance of 11.7 Mpc.
  
  
  • Distance uncertainty
    
    Differences affecting distance estimates, whether based on Cepheid variables or other methods, include corrections for:
    1. dimming/reddening/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. (2001ApJ...553...47F) have a discussion of the errors involved in many of the methods discussed here.

• Color-Magnitude Diagrams (CMD)

  The CMD standard candle is based on the absolute magnitude of a galaxy’s various stellar populations, discernable in a color-magnitude diagram, including Tip of the Red Giant Branch, Red Clump, and Horizontal Branch stars, and others. This method has been largely supplanted by these alternatives, which have been shown to be more accurate.

• Delta Scuti

  The Delta Scuti standard candle is based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheids and Mira variables, a period-luminosity (PL) relation gives their absolute magnitude (see, for example, McNamara et al. 2007AJ....133.2752M).

• Flux-Weighted Gravity-Luminosity Relation (FGLR)

  The FGLR standard candle is based on the absolute bolometric magnitude of A-type supergiant stars, determined by the Flux-weighted Gravity- Luminosity Relation (see Kudritzki et al., 2008ApJ...681..269K).

• Gamma-Ray Burst (GRB)

  The GRB standard candle is based on six correlations of observed properties of GRBs with their luminosities or collimation-corrected energies. A Baysian fitting procedure then leads to the best combination of these correlations for a given data set and cosmological model. (see Cardone et al. 2009MNRAS.400..775C).

• Globular Cluster Luminosity Function (GCLF)

  The Globular Cluster Luminosity Function (GCLF) standard candle is based on the maximum absolute visual magnitude of M_V = -7.6 of old, blue, low-metallicity globular clusters (see Larsen et al. 2001AJ....121.2974L). So, for example, the galaxy NGC 0524 with a maximum apparent visual magnitude for its globular clusters m_V = 24.36, has a distance modulus of (m-M)_V = 31.99, or a distance of 25 Mpc, with a statistical uncertainty of 0.14 mag or 1.8 Mpc (7%).

• Globular Cluster Surface Brightness Fluctuations (GC SBF)

  The GC SBF standard candle is based on the fluctations in surface brightness arising from the mottling of the otherwise smooth light of the cluster due to individual stars (see Ajhar et al. 1996AJ....111.1110 for an application to a cluster in M31). Thus, the implied apparent magnitude of the stars leading to these fluctuations gives the distance modulus in magnitudes, and distance in Mpc.

• HII Luminosity Function (HII LF)

  The HII LF standard candle is based on a relation between velocity dispersion, metallicity, and the luminosity of the H-beta line in HII regions and HII galaxies (see e.g. Siegel et al., 2005MNRAS.356.1117S for references and an application to high-redshift starburst galaxies).

• Horizontal Branch

  The Horizontal Branch standard candle is based on the absolute visual magnitude of horizontal branch stars, which is close to M_V = +0.50, but depends on metallicity (see Da Costa, et al. 2002AJ....124..332D). Thus, horizontal branch stars of this metallicity in the galaxy Andromeda III with an apparent visual magnitude m_V= 25.06, including a redding correction of -0.18 mag, have a distance modulus (m-M)_V= 24.38, or a distance of 750 kpc, with a statistical uncertainty of 0.06 mag or 20 kpc (3%).

• M Stars luminosity (M Stars)

  A relationship exists between absolute magnitude and a temperature-independent spectral index for normal M Stars (see Schmidt-Kaler & Oestreicher, 1998AN....319..375S).

• Miras

  The Mira standard candle is based on the mean absolute magnitude of Mira variable stars, which depends on their pulsation period. As with Cepheid variables, a period-luminosity (PL) relation gives their absolute magnitude. As with any other distance indicator, their apparent magnitude gives their distance modulus in magnitudes, and the distance in Mpc is calculated from that.

• Novae

  The novae standard candle is based on the maximum absolute visual magnitude reached by these explosions, which is M_V = -8.77 (see Ferrarese et al. 1996ApJ...468L..95F). 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, or a distance of 15.8 Mpc, with a statistical uncertainty of 0.3 mag or 2.4 Mpc (15%).

• O- and B-type Supergiants (OB Stars)

  The OB Stars standard candle is based on the relationship between spectral type, luminosity class, and absolute magnitude for these stars (see Walborn & Blades, 1997ApJS..112..457W for an application to 30 Doradus in the LMC).

• Planetary Nebula Luminosity Function (PNLF)

  The PNLF standard candle is based on the maximum absolute visual magnitude for planetary nebulae of M_V = -4.48 (see Ciardullo et al. 2002ApJ...577...31C). So, the planetary nebulae in the galaxy NGC 2403 with a maximum apparent visual magnitude m_V = 23.17 have a distance modulus (m-M)_V = 27.65, or a distance of 3.4 Mpc, with a statistical uncertainty of 0.17 mag or 0.29 Mpc (8.5%).

• Post-Asymptotic Giant Branch Stars (PAGB Stars)

  The PAGB standard candle is based on the maximum absolute visual magnitude of these stars M_V= -3.3 (see Bond & Alves, 2001ASSL..265...77B). Thus, PAGB Stars in Messier 031 (Andromeda) with a maximum apparent visual magnitude m_V= 20.88 have a distance modulus (m-M)_V= 24.2, or a distance of 690 kpc, with a statistical uncertainty of 0.06 mag or 20 kpc (3%).

• Quasar spectrum

  The quasar spectrum standard candle is based on the observed apparent spectrum of a quasar, compared with the absolute spectrum the quasar ought to have based on Hubble Space Telescope spectra taken of 101 quasars. For example, de Bruijne et al. (see 2002A&A...381L..57D), find a distance to the quasar [HB89] 0000-263 of 3.97 Gpc.

• RR Lyrae Stars

  The RR Lyrae standard candle is based on the mean absolute visual magnitude of these variable stars, which depends on metallicity:

  M_V = F/H x 0.17 + 0.82 mag

  (see Armandroff et al. 2005AJ....129.2232P). So, RR Lyrae stars for example, with metallicity -1.88 in the nearby 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 (m-M)_V = 24.34, or a distance of 740 kpc, with a statistical uncertainty of 0.06 mag or 22 kpc (3.0%).

• Red Clump

  The red clump standard candle is based on the maximum absolute infrared magnitude for Red Clump stars of M_I = -0.67 (see Dolphin et al. 2003AJ....125.1261D). 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, or a distance of 1.26 Mpc, with a statistical uncertainty of 0.15 mag or 0.09 Mpc (7.5%).

• Red Supergiant Variables (RSV Stars)

  The RSV standard candle is based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheid and Mira variables, a period-luminosity (PL) relation gives their absolute magnitude (see for example, Jurcevic Thesis, 1998.

• Red Variable Stars (RV Stars)

  The Red Variable stars standard candle is based on the mean absolute magnitude of RV Stars, which depends on their pulsation period. As with Cepheid variables, a period-luminosity (PL) relation gives their absolute magnitude. As with other distance indicators, their apparent magnitude then gives their distance modulus in magnitudes, and from that the distance in Mpc is calculated (see Kiss & Bedding, 2004MNRAS.347L..83K).

• S Doradus Stars

  The S Doradus Stars standard candle is based on the mean absolute magnitude of these stars, which is derived based on their amplitude-luminosity relation. As with other distance indicators, their apparent magnitude then gives their distance modulus in magnitudes, and from that the distance in Mpc is calculated (see Wolf, 1989A&A...217...87W).

• SNIa SDSS

  The SNe Ia SDSS standard candle is based on SNIa (see 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.

• SX Phoenicis Stars

  The SX Phoenicis Stars standard candle is based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheids and Mira variables, a period-luminosity (PL) relation gives their absolute magnitude (see, for example, McNamara et al. 1995AJ....109.1751M).

• Short Gamma-Ray Bursts (SGRB)

  The SGRB standard candle is similar to the GRB standard candle but employs only GRBs of short, less than 2 second duration, as opposed to GRBs of long, more than 2 second duration (see Rhoads, 2010ApJ...709..664R).

• Statistical

  The statistical method is 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 (see 2010ARA&A..48..673F), analyzed 180 estimates of the distance to the Large Magellanic Cloud, based on two dozen indicators not including Cepheids, to obtain a mean distance modulus of (m-M) = 18.44 ± 0.18, or a distance of 48.8 kpc.

• Subdwarf Fitting

  The Subdwarf fitting standard candle leads to an improved calibration of the distances and ages of globular clusters (see Carretta et al., 2000ApJ...533..215C, Section 4).

• Sunyaev-Zeldovich Effect (SZ effect)

  The SZ effect standard candle is 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 in Mpc (see for example Bonamente et al. 2006ApJ...647...25B).

• Surface Brightness Fluctuations (SBF)

  The Surface Brightness Fluctuations (SBF) standard candle is based on the fluctations 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 M_K = -5.6 (see Jensen et al. 1998ApJ...505..111J). So, the galaxy NGC 1399, for example, with brightest stars at an implied maximum apparent K-band magnitude m_K = 25.98, has distance modulus of (m-M)_K= 31.59, or a distance of 20.8 Mpc, with a statistical uncertainty of 0.16 mag or 1.7 Mpc (8%).

• Tip of the Red Giant Branch (TRGB)

  The Tip of the Red Giant Branch (TRGB) standard candle is based on the maximum absolute infrared magnitude for TRGB Stars with M_I= -4.1 (see Sakai et al. 2000AJ....119.1197S). So, the Large Magellanic Cloud (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, or a distance of 52 kpc, with a statistical uncertainty of 0.09 mag or 2 kpc (4.5%).

• Type II Cepheids

  The Type II Cepheids standard candle is based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with normal Cepheids and Miras, a period-luminosity (PL) relation gives their absolute magnitude (see, for example, Majaess et al. 2009AcA....59..403M).

• Type II Supernovae, Radio (SNII radio)

  The type-II supernova (SNII) radio standard candle is based on the maximum absolute radio magnitude reached by these explosions, which is 5.5 x 10²³ ergs/s/Hz (see Clocchiatti et al. 1995ApJ...446..167C). 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)

  The SNe Ia standard candle is based on the maximum absolute blue magnitude reached by these explosions, which is M_B = -19.3 (see Astier et al. 2006A&A...447...31A). 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 the redshift for the galaxy z = 0.0307, this leads to a linear distance of 124 Mpc, with a statistical uncertainty of 0.09 mag or 6 Mpc (4.5%).

• White Dwarfs

  The white dwarf cooling sequence standard candle is based on the absolute magnitudes of white dwarf stars, which depends on their age (see Carretta et al., 2000ApJ...533..215C, Section 3.1).

• Wolf-Rayet

  The Wolf-Rayet standard candle is based on the mean absolute magnitude of these massive stars (see for example, Massey & Armandroff, 1995AJ....109.2470M).

Standard Rulers

• CO ring diameter

  The carbon monoxide (CO) ring diameter standard ruler is based on the mean absolute radius of a galaxy’s inner CO ring, with compact rings R ~ 200 pc and broad rings 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 (see Sofue, 1991PASJ...43..671S).

• Dwarf Galaxy Diameter

  The Dwarf Galaxy Diameter ruler is 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 arcseconds, gives a distance for the main galaxy of 18.0 +/- 3.1 Mpc (see Misgeld & Hilker 2011MNRAS.414.3699M).

• Eclipsing Binary

  Eclipsing Binary stars provide 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 (see Ribas et al. 2005ApJ...635L..37R). So, this eclipsing binary, with an apparent visual magnitude of m_V = 18.67, has a distance modulus of (m-M)_V = 24.44, or a distance of 772 kpc with a statistical uncertainty of 0.12 mag or 44 kpc (6%).

• Globular Cluster Radii (GC radius)

  Globular Cluster rulers are based on the mean absolute radii of globular clusters, r = 2.7 parsecs (see Jordan et al. 2005ApJ...634.1002J). So, globular clusters in the galaxy Messier 87 with a mean apparent radii of r = 0.032 arcsec, have a distance of 16.4 Mpc.

• Grav. Stability Gas. Disk

  The Gravitational Stability Gaseous Disk standard ruler is based on the absolute diameter at which a galaxy reaches the critical density for gravitational stability of the gaseous disk (see Zasov & Bizyaev, 1996AstL...22...71Z).

• Gravitational Lenses (G Lens)

  The gravitational lens standard ruler is 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 in Mpc.

• HII Region Diameters (HII)

  The HII region diameter ruler is based on the mean absolute diameter of HII regions, d = 14.9 parsecs (see Ismail et al. 2005JKAS...38....7I). So, HII 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

  The Jet Proper Motion standard ruler is 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. For example, Homan (see 2000ApJ...535..575H) find an angular size distance to the quasar 3C 279 of 1.8 ± 0.5 Gpc.

• Masers

  The maser ruler is based on the absolute motion of masers orbiting at great speeds within mere 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, or 0.001100 parsecs/yr (see Humphreys et al. 2004AAS...205.7301H). So, the masers’ apparent proper motion 31.5 x 10^-6 arcsec/yr, gives a distance of 7.2 Mpc with a statistical uncertainty of 0.2 Mpc (3.0%).

• Orbital Mechanics (Orbital Mech.)

  The orbital mechanics standard ruler is based on the predicted orbital or absolute motion of a galaxy around another galaxy, and its observed apparent motion, giving a measure of distance in Mpc.

• Proper Motion

  The Proper Motion ruler is based on the absolute motion of a galaxy, relative to its apparent or proper motion (see, for example, Lepine et al. 2011ApJ...741..100L).

• Ring Diameter

  The Ring Diameter ruler is 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 (see Pedreros & Madore 1981ApJS...45..541P).

• Type II Supernovae, Optical (SNII Optical)

  Supernova Type II (SNII) optical rulers are based on the absolute motion of the explosion’s outward velocity, in units of intrinsic transverse velocity, V_t (usually km/s), relative to the explosion’s apparent or proper motion (usually arcseconds/year) (see for example, Eastman, Schmidt and Kirshner, 1996ApJ...466..911E). 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.

Secondary Methods

• D-Sigma

  The D-Sigma relation 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 (again, see Willick et al. 1997ApJS..109..333W).

• Diameter

  Certain galaxy’s major diameters may provide secondary standard rulers based on the absolute diameter for example of only the largest, or "giant" spiral galaxies, estimated to be ~52 kpc. So, from the mean apparent diameter found for giant spiral galaxies in the Virgo cluster of ~9 arcmin, the Virgo cluster distance is estimated to be 20 Mpc (see van der Kruit, 1986A&A...157..230V).

• Dwarf Ellipticals

  Dwarf Ellipticals provide secondary standard candles based on the absolute magnitude of dwarf elliptical galaxies, derived from a surface-brightness/luminosity relation, and the observed apparent magnitude of these galaxies (see Caldwell & Bothun, 1987AJ.....94.1126C).

• Faber-Jackson

  The F-J relation provides secondary standard candlea based on the absolute magnitudes of elliptical and early-type galaxies, determined from a relation between a galaxy’s apparent magnitude and velocity dispersion (see Lucey, 1986MNRAS.222..417L).

• Fundamental Plane (FP)

  The Fundamental Plane (FP) relation provides standard candles 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

  (see, for example, Kelson et al. 2000ApJ...529..768K). The galaxy NGC 1399 has an effective radius r_e = 55.4 arcsec, a rotational velocity sigma = 301 km/s, and surface brightness, I_e = 428.5 L_Sun/pc². So, from the FP relation, its distance is 20.6 Mpc.

• GC K vs. (J-K)

  The globular cluster K-band magnitude vs. J-band minus K-band Color-Magnitude Diagram secondary standard candle is similar to the Color-Magnitude Diagram standard candle, but applied specifically to globular clusters within a galaxy, rather than entire galaxies (see Sitko, 1984ApJ...286..209S).

• GeV TeV ratio

  The GeV TeV ratio secondary provides standard candles based on the absolute magnitude at which this ratio equals one (see Prandini et al., 2010MNRAS.405L..76P).

• Globular Cluster Fundamental Plane (GC FP)

  The GCFP relation provides standard candles based on the relationship among velocity dispersions, radii, and mean surface brightnesses for globular clusters, similar to the fundamental plane for early-type galaxies. Strader et al. 2009AJ....138..547S have an application for globular clusters in M31.

• H I + optical distribution

  The 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 (see Staveley-Smith et al., 1990ApJ...364...23S).

• Infra-Red Astronomical Satellite (IRAS)

  The IRAS standard candle is 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 (see Willick et al. 1997ApJS..109..333W for references and details).

• L(H{beta})-{sigma}

  The L(H{beta})-{sigma} relation is used as a secondary standard candle based on a correlation between the Hβ luminosity and the Hβ line width (dispersion), as introduced by Terlevich (2015MNRAS.451.3001T). It is similar to the Faber-Jackson relation.

• LSB galaxies

  Low Surface Brightness galaxies provide secondary standard candles, based on the Surface Brightness Fluctuations (SBF) standard candle. It 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 LSB galaxies (see Bothun et al., 1991ApJ...376..404B).

• Magnitude

  Certain galaxy’s apparent magnitudes may provide a secondary standard candle based on the mean absolute magnitude previously estimated for a sample of similar galaxies with known distances. Assuming a mean absolute blue magnitude for dwarf galaxies of MB = -10.70, the dwarf galaxy DDO 155 with an apparent blue magnitude of mB = 14.5, has a distance modulus (m-M)B = 25.2, or a distance of 1.1 Mpc (see Moss & de Vaucouleurs, 1986PASP...98.1282M).

• Mass Model

  The Mass Model provides secondary standard rulers 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 (see Gentile et al., 2010MNRAS.406.2493G).

• Radio Brightness

  The Radio Brightness secondary standard candle is based on the absolute radio brightness assumed versus the apparent radio brightness observed in a galaxy (see Wiklind & Henkel, 1990A&A...227..394W).

• Sosies

  "Look Alike", or in French "sosie", galaxies provide standard candles based on the similar absolute magnitudes of spiral galaxies with similar Hubble stages, inclination angle, and light concentrations. Out of thousands of non-similar galaxies within 100 Mpc, only 283 sosie galaxies were found (see Terry et al. 2002A&A...393...57T). Of these, ten with accurate, Cepheid-based distances determined the average absolute visual magnitude for sosie galaxies, M_V = -21.3. 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, or a distance of 15.6 Mpc.
  
  In practice, slight differences in absolute magnitude between sosie galaxies due to their Hubble type, inclination, and/or light concentration can be accounted for. Thus, the 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 estimated to be 0.06 V-mag less luminous than NGC 1365, leading to a distance modulus of (m-M)_V = 33.34, or a distance of 46.6 Mpc.

• Tertiary

  The Tertiary secondary method is 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 (see, for example, McCall 1989AJ.....97.1341M).

• Tully Estimate (Tully est)

  The Tully estimate distances are based on various parameters, including galaxy magnitudes, diameters, and estimates by Tully based on group membership (see Tully, Nearby Galaxies Catalog, 1988).

• Tully-Fisher

  The Tully-Fisher relation provides standard candles 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 = -7.0 log sigma - 1.8

  (see, for example, Karachentsev et al. 2003A&A...404...93K). So, the galaxy NGC 0247 has an absolute blue magnitude of M_B = -18.2, based on its rotational velocity, sigma = 222 km/s. With an apparent blue magnitude of m_B = 9.86, NGC 0247 has a distance modulus of (m-M)_B = 28.1, or a distance of 4.1 Mpc.