Description of the distance determination methods
Direct methods determine distances straight from the measurement data and do not depend on external calibrations.
They are the basis for construction of the cosmic distance ladder.
The most important distance estimates use trigonometric parallax of individual stars.
The methods of statistical parallax and moving cluster parallax allow us to derive distances for groups of stars.
It is very useful for the calibration of methods based on luminosity of Cepheids and RR Lyrae.
Unfortunately, these methods are usually restricted to our Galaxy or to its nearby satellites.
A notable exception is NGC 4258 whose precise Maser distance of 7.6 Mpc is precious to calibrate the other methods.
The Detached Eclipsing Binaries (DEB) provide an accurate geometric method for distance determination.
The fundamental parameters of the stars (the radii, effective temperatures, masses, and luminosities)
can be determined from the light and radial velocity curves of an eclipsing binary.
This method is independent of any intermediate calibration steps.
The Expanding Photosphere Method (EPM) and the Expanding Shock front Method (ESM)
are a geometric distance determination technique based on comparing radial velocities with proper motion
of an expanding shell after supernova explosion.
The method is based on study of kinematics of an accretion disk around a supermassive black hole by a radio maser emission.
It gives a direct geometric estimate of an absolute distance.
Humphreys et al. (2013, ApJ, 775, 13) measured the distance of 7.6 Mpc with 3% uncertainty to the Seyfert II galaxy NGC 4258
using 10 years observations of the H2O maser.
This class contains some of the most precise and important distance indicators for extragalactic astronomy:
the Cepheids and RR Lyrae variable stars,
the tip of the red giant branch (TRGB) and
the horizontal branch (HB) stars.
These distance indicators can be calibrated using direct methods.
Except SN Ia, all these methods are effective only for the nearby Universe on scale from several to few dozen Mpc.
All these methods use the luminosity of the brightest stars in galaxies as standard candles.
BBSLF and BRSLF consider the luminosity function of the brightest, respectively blue and red, stars.
BS3B and BS3R take the mean absolute magnitude of the three brightest blue or red stars, respectively.
The luminosity of brightest blue and red supergiants depends on the magnitude of the parent galaxy
(Rozanski & Rowan-Robinson, 1994, MNRAS, 271, 530).
This method is based on a fit of the magnitude of the He-burning Blue Loop stars with theoretical isochrones.
Because of the large uncertainties in the theoretical models, this method cannot be considered as a reliable distance indicator.
This is one of the most important standard candles.
The method is based on the period-luminosity (PL) relation for Cepheid variable stars.
There are many calibrations of the relation in different pass-bands using
the Galactic or Large Magellanic Cloud (LMC) PL relation, for example,
the Hubble Space Telescope Key Project On the Extragalactic Distance Scale (Freedman et al., 2001, ApJ, 553, 47),
the HIPPARCOS trigonometric parallaxes (Feast & Catchpole, 1997, MNRAS, 286, L1),
or the Baade-Wesselink methods (Storm et al., 2011, A&A, 534, A95).
It is foreseen that the calibration will be dramatically improved in the coming years
thanks to the GAIA astrometric satellite which starts to operate now.
It uses various features of the composite colour-magnitude diagram (CMD) of a galaxy resolved into individual stars
to estimate the distance by comparison with template CMD or with theoretical isochrones.
For instance, Dolphin (2000, ApJ, 531, 804) develops the software which fits the observed CMD with synthetic data,
to estimate in the same time the distance and the star formation history of a galaxy.
The carbon-rich Stars (CS) in the TP-AGB phase form the horizontal red tail on CMD, at about 0.5 mag brighter than the TRGB.
Battinelli & Demers (2005, A&A, 442, 159) find the absolute I-band magnitude of CS as a function of the metallicity of the parent galaxy:
〈MI〉 = -4.33+0.28[Fe/H].
The Flux-weighted Gravity-Luminosity Relationship (FGLR)
is a technique to derive the distance from a spectral analysis of the B and A supergiant stars (Kudritzki et al., 2008, ApJ, 681, 269).
It is based on a tight correlation between the absolute bolometric magnitude and the flux-weighted gravity,
The old globular cluster luminosity function (GCLF) method uses
the peak (or turnover, TO) of the GCLF as a standard candle.
For instance, Di Criscienzo et al. (2006, MNRAS, 365, 1357) derive MV,TO=-7.66±0.09
with adopted the LMC distance modulus of 18.50.
The median of the Globular Cluster half-light Radii (GCR) of 2.7±0.3 pc (Jordan et al., 2005, ApJ, 634, 1002)
can be used as a standard ruler for distance estimate.
The half-light radius of individual GC should be corrected for colour, surface brightness, and host galaxy colour.
These methods use the horizontal branch (HB) or the blue horizontal branch (BHB) stars as standard candles.
Carretta et al. (2000, ApJ, 533, 215) give the relation between absolute magnitude and metallicity of HB:
MV(HB) = (0.13±0.09)([Fe/H]+1.5) + (0.54±0.07).
This method fits the position of the Main Sequence below the turn-off with a theoretical isochrones or with template CMD.
It relates to the CMD distance determination method.
Mira Ceti stars are long period variable stars in the asymptotic giant branch phase.
Ita & Matsunaga (2011, MNRAS, 412, 2345), among others, derive the period-magnitude relations for Mira-like variables in the LMC
using bolometric, near- and mid-infrared magnitudes.
This method uses the sharp exponential truncation of the planetary nebulae luminosity function (PNLF) as a standard candle.
The zero-point, M*=-4.48, is based on the M 31 distance of 710 kpc (Ciardullo et al., 2002, ApJ, 577, 31).
The Red Clump (RC) is populated by core helium-burning stars of intermediate age.
Their mean absolute magnitude provides a standard candle for distance determination.
Girardi & Salaris (2001, MNRAS, 323, 109) find important non-linear dependences
on both the age and the metallicity of the stellar population.
The method is based on the mean absolute magnitude for RR Lyrae variable stars, which depends on metallicity:
MV(HB) = (0.18±0.09)([Fe/H]+1.5) + (0.57±0.07) (Carretta et al. 2000, ApJ, 533, 215).
This method uses the period-luminosity relation for the Red Supergiant Variable (RSV) stars.
The calibration of the PL relation by Pierce et al. (2000, MNRAS, 313, 271) adopts the distance modulus of 18.50 mag for LMC.
The RSVs as well as the Miras are long period variable stars.
The Surface Brightness Fluctuations (SBF) method relies on the luminosity fluctuations
that arise from the counting statistics of stars contributing the flux in each pixel of an image (Tonry & Schneider 1988, AJ, 96, 807).
The absolute fluctuation magnitude depends on the stellar populations and, consequently, on the colour of the galaxy.
It can only be applied to old stellar populations.
Because of their extremely high luminosity and regular behaviour the type Ia supernovae (SN Ia) provide
a powerful tool for measuring cosmological distances.
The method uses the relationship between the light-curve shape and the maximum luminosity of a SN Ia.
The Tip of the Red Giant Branch (TRGB) is an excellent distance indicator for nearby galaxies resolved into individual stars.
The method, relying on the old stellar population, can be used for galaxies of any morphological types.
Thanks to the shallow colour-dependence of the magnitude of the TRGB in the I-band,
the method is one of the most precise distance indicators.
For example, Rizzi et al. (2007, ApJ, 661, 815) calibrates the zero-point of the TRGB method using HB stars:
MIJC = -4.05(±0.02) + 0.22(±0.01)[(V-I)-1.6].
Empirical relationships between
the intrinsic luminosity of a galaxy and its properties such as kinematics, surface brightness, and so on.
The most important ones are the Tully-Fisher (TF) relation for spirals and the fundamental plane (FP) for early-type galaxies.
Because the methods use the total luminosity of a galaxy as a standard candle, they can be applied on scales up to hundreds Mpc.
These methods provide low precisions for individual measurements, but they give good results in a statistical sense with huge sets of data.
This is especially true for the Tully-Fisher relation, where obtaining observational data is relatively inexpensive.
The TF and FP methods allow us to investigate the cosmic flows in the Universe on scale of hundred Mpc.
The Faber-Jackson (FB) relation provides a standard candle for elliptical and early-type galaxies
based on the relationship between absolute magnitude and central velocity dispersion.
The Fundamental Plane (FP) is a distance determination method for early-type galaxies
based on relation between the absolute magnitude, effective radius, velocity dispersion, and mean surface brightness.
log D = logre - 1.24 logσ + 0.82 log〈I〉e +0.173
(Kelson et al. 2000, ApJ, 529, 768).
The methods using the Surface Brightness-total Magnitude relation (SB-M) or the Sersic index-total Magnitude relation (Sersic-M)
can be considered as rough distance estimate for small mass elliptical galaxies.
The method of `look alike' (sosie in French) is proposed by Paturel (1984, ApJ, 282, 382).
It is based on the idea that galaxies with the same morphological type, the same inclination, and the same HI line width
must to have the same absolute luminosity according to the TF relation.
The Tully-Fisher (TF) is a standard candle based on empirical relationship between the absolute magnitude of a spiral galaxy
and its maximum rotational velocity, estimated by a HI line width
The recent calibration I-band TF relation gives
MIb,i,k = -21.39-8.81(log Wimx-2.5)
(Tully & Courtois 2012, ApJ, 749, 78).
The Baryonic Tully-Fisher (BTF) relation uses relationship between the amplitude of rotation and the baryonic mass of the galaxy.
This relation takes into account not only the stellar light from optical data as in the original TF relation,
but also the mass of gas in neutral and molecular forms.
The BTF relation is comparable to the TF one for giant spiral galaxies,
and it represents an improvement for dwarf galaxies with circular velocities below 90 km s-1 (McGaugh et al 2000, ApJ, 533, L99)
where the cold gas represents an important and variable dynamical component.
The BTF can also be applied to gas rich dwarf elliptical galaxies (De Rijcke et al. 2007, ApJ, 659, 1172).