"In 1995, the majestic spiral galaxy NGC 4414 was imaged by the Hubble Space Telescope as part of the HST Key Project on the Extragalactic Distance Scale. [The galaxy was] observed ... on 13 different occasions over the course of two months."[1]

This is a Hubble Space Telescope Image of NGC 4414. Credit: Hubble Heritage Team (AURA/STScI/NASA).
This is an image of Messier 31, the Andromeda galaxy from 1899. Credit: Isaac Roberts.

"Images were obtained with Hubble's Wide Field Planetary Camera 2 (WFPC2) through three different color filters."[1]

"Based on [...] careful brightness measurements of variable stars in NGC 4414, [...] an accurate determination of the distance to the galaxy [was made]."[1]

"The resulting distance to NGC 4414, 19.1 megaparsecs or about 60 million light-years, along with similarly determined distances to other nearby galaxies, contributes to astronomers' overall knowledge of the rate of expansion of the universe. The Hubble constant (H0) is the ratio of how fast galaxies are moving away from us to their distance from us. This astronomical value is used to determine distances, sizes, and the intrinsic luminosities for many objects in our universe, and the age of the universe itself."[1]

"Due to the large size of the galaxy compared to the WFPC2 detectors, only half of the galaxy observed was visible in the datasets collected by the Key Project astronomers in 1995. In 1999, the Hubble Heritage Team revisited NGC 4414 and completed its portrait by observing the other half with the same filters as were used in 1995. The end result is a stunning full-color look at the entire dusty spiral galaxy. The new Hubble picture shows that the central regions of this galaxy, as is typical of most spirals, contain primarily older, yellow and red stars. The outer spiral arms are considerably bluer due to ongoing formation of young, blue stars, the brightest of which can be seen individually at the high resolution provided by the Hubble camera. The arms are also very rich in clouds of interstellar dust, seen as dark patches and streaks silhouetted against the starlight."[1]

Standard-candles astronomy is the astronomical effort to find, study and develop standard-candle candidates for use as standard candles.

Standard candles are stars in visual astronomy that may be used to calculate distances because their characteristics are, or appear to be, distance independent.

## Colors

These graphs show B-, R-, I-, and H-band Tully-Fisher relations. Credit: Michael J. Pierce.

"The empirical relationship between the luminosity of a spiral/irregular galaxy and its rotational velocity, known as the luminosity-line-width or Tully-Fisher relation (Tully and Fisher 1977, TF), has become one of the most widely used extragalactic distance indicators."[2]

"Spiral and irregular galaxies are common in both the field and in all but the richest clusters. Consequently, the TF relation is an extremely valuable technique for extensive mapping of local large scale structure, the Hubble flow, and any associated peculiar velocities. In addition, since the local population is dominated by spirals and irregulars, the TF relations can, at least in principle, be calibrated directly by Cepheids. The TF relations provide an opportunity to go from local calibrating galaxies to the "smooth" Hubble flow in a single step."[2]

Differences "in samples, extinction, and particularly line-width corrections can lead to quite different results."[2]

"The necessary data consist of apparent magnitudes, usually corrected for Galactic and internal extinction, and some measure of rotation velocities, corrected for projection effects, for a sample of galaxies. Usually, the rotational velocity is obtained via the doppler broadening of the HI 21-cm line, although Fabry-Perot imaging and long slit rotation curves (both obtained via H) are useful as well."[2]

"As originally proposed by [TF], the observational parameters of the TF relation are: (1) the photographic magnitude corrected for extinction, used as a measure of the luminosity of a galaxy, and (2) the global HI 21-cm line-width corrected for projection, used as a measure of the galaxy's rotational velocity. The raw measurements must be corrected for inclination effects, with the corrections for extinction and projection going in the opposite sense; when extinction corrections to the magnitudes are small (i.e., the galaxy is nearly face-on) the projection corrections for the rotational velocity are large, and vice versa. Consequently, considerable care must be taken to determine the proper corrections to the fundamental observables, and [there are] several different prescriptions for these corrections [...]. Nevertheless, any systematics introduced by the use of different correction schemes should in principle scale out, provided that the "local calibrators" and the sample galaxies are treated in the same way."[3]

"The publication of the Second Reference Catalog of Bright Galaxies by de Vaucouleurs et al. (1976) greatly increased the size of the existing data base of total magnitudes for a wide variety of galaxies. In addition, the completion of extensive HI 21-cm surveys [...] significantly improved the status of the line-width data. These data were subsequently used by several groups to examine both the utility of the TF relation as a distance indicator and the character of the local Hubble flow [...]. Similar photoelectric data were obtained [...] in the V, r, and IV0 ~ 1 µm) bands."[3]

"A significant improvement in the application of the TF relations occurred with the introduction of infrared H-band magnitudes (λ0 = 1.6 µm) [... The] infrared offered two distinct advantages over visible wavelengths: first, that the extinction at H is only 10% that in B, and second, that the infrared light is dominated by late-type giants. Thus, over the entire range of morphological types, the infrared is presumably a better tracer of the stellar mass than the B luminosity. Subsequent observations proved that the infrared Tully-Fisher relation (hereafter IRTF relation) has a significantly smaller dispersion than does the B-band relation, implying that uncertainties in the extinction corrections for the blue light (which were still rather crude [...]), and/or variations in mean stellar population, were significant. [Restricting] the sample of bright spirals to those with inclinations greater than 45° and with no morphological peculiarities, following [TF produced] a distance modulus for the Virgo Cluster of 30.98 ± 0.09 mag (internal errors only), with the updated Hyades distance."[4]

"For luminosity, the choices are: (i) the total magnitude for the galaxy, or (ii) a suitably defined aperture magnitude. The former offers the advantage of having a relatively simple definition, while the latter has been a necessity for infrared photometry where until recently single element detectors were state of the art. For the rotational velocity parameter the choices are: (i) the maximum of the rotation curve (Vmax), or (ii) the rotational velocity at a suitably chosen radius, perhaps corresponding to a particular aperture magnitude definition [...]."[5]

"Total magnitudes have been the choice for the luminosity parameter over visible wavelengths, even when derived through multiaperture photometry and template growth curves [...]. The use of total magnitudes was motivated by the availability of the data, the desire for a uniform magnitude system, and the expectation that the mass-to-light ratio of galaxies is constant, or at least a smooth function of mass. The latter is of particular concern at optical wavelengths where the large color difference between the bulge and disk components of spiral galaxies suggests differences in the mass-to-light ratio. This results in a significant morphological type dependence for the TF relations at bluer wavelengths. Consequently, a TF relation based on blue photometry through small apertures is of limited use [...]. [There are] numerous problems associated with multi-aperture photometry (e.g., centering errors, contaminating stars, the use of circular apertures and mean growth curves, etc.) [...]. With the introduction of CCDs, full growth curves using elliptical annuli can be easily constructed with contaminating regions (e.g., stars, cosmetic defects) suitably dealt with. The sky level for each image can also be easily estimated, provided the field of view is sufficiently large. Together, these advantages of CCD photometry can result in total magnitudes with a precision of ~ 5% [...]."[5]

Dispersions occur "for the TF relations of 0.37, 0.31, and 0.28 mag for the B, R and I bands, respectively, from CCD photometry of a complete sample of galaxies in the Ursa Major Cluster. There is likely significant line-of-sight depth for this sample, since it spans about 8° on the sky, suggesting an intrinsic dispersion for the longer wavelength TF relations of ≤ 0.25 mag, or 12% in distance. Much of the improvement over previous work (typically with ~ 0.5 mag) was attributed to an increased accuracy of inclination estimates obtained via ellipses fitted to galaxy isophotes. For example, they found a significant improvement in the dispersion for the IRTF relation using photometry from the literature, provided that the line-widths were corrected with the improved inclinations. Apparently, the redder wavelengths are preferable, though the B-band is still useful for comparative purposes."[5]

"Although the infrared aperture magnitudes have been very successful, they are not free of problems. The H-band magnitude (H-0.5) is defined to be the apparent H magnitude through an aperture size (A) given by log(A / D25b,i) = -0.5, where D25b,i is the B-band diameter of the galaxy at the 25 mag arcsec-2 isophote, corrected for internal and Galactic extinction, as well as projection. This definition allows the use of small apertures that are more tractable with traditional infrared photometers, and also reduces the effect of the sky signal to a manageable level. (The H-band sky is very bright: µH ~ 13.4 mag arcsec-2. Compare this with a mean H-band surface brightness, over D25b,i of µH ~ 18 mag arcsec-2 for a moderate luminosity spiral and ~ 21 mag arcsec-2 for a typical dwarf irregular galaxy.) There are well-known observational techniques used in infrared astronomy to combat these high background levels. Nevertheless, nearby systems having low surface brightness and large angular size pose a special problem due to uncertainties in the sky level. In addition to the general problems associated with aperture photometry mentioned above, there is an additional difficulty in tying the infrared aperture size to the optical diameter [...]. Significant systematic errors have been introduced into some of the IRTF distance estimates from biased B-band isophotal diameters, a consequence of using different catalogs of B-band diameters for different distance ranges [...]. CCD photometry is almost a necessity to provide a uniform and unbiased aperture definition, at which point aperture photometry itself becomes of limited value. The application of infrared imaging technology over the next few years should alleviate most of these concerns, provided the data reach a sufficiently faint surface brightness level for total magnitudes to be estimated."[5]

"For the TF relations to be successfully applied, the galaxies must have detectable gas in essentially circular motion; otherwise, the estimate of Vmax will be inaccurate. Although there is occasionally detectable HI in lenticular systems [...], defining a TF relation for E/S0 galaxies involves several complications: the origin of the gas is uncertain, the gas is usually patchy, its detection is difficult, and its motion may not be circular. Because of these difficulties (and the potential for significant M/L variations among early-type galaxies [...], it is best to limit samples of galaxies to types Sb - Irr having no signs of peculiar morphology, due, for example, to recent interactions. In addition, the galaxies must have enough gas in their exterior regions such that the observable rotation curve reaches a peak and Vmax can be adequately sampled. It is now clear that disk galaxies within the cores of even moderately poor clusters like Virgo suffer significant stripping of their outer envelope of HI gas [...]. Fortunately, only a few systems in Virgo are gas-stripped to radii within the turn-over in their rotation curves. The most serious cases also have morphological peculiarities and would likely have been excluded on this basis alone (e.g., NGC 4438). However, this remains an uncertainty for galaxies in the cores of more distant clusters; for these galaxies a full rotation curve may be necessary to assure that the rotation curves do indeed turn over."[6]

"The modest morphological type dependence mentioned [...] (i.e., earlier types having low luminosities for their line-widths [...]) is a consequence of late-type galaxies being systematically bluer than those of earlier type, coupled with the historical use of the B-band TF relation. Because the effect decreases dramatically toward longer wavelengths, a likely cause is the smaller bulge-to-disk ratio and the larger fraction of young stars in the low mass, gas-rich systems. Some of the effect may also be the result of a decrease in dust content for the later morphological types [...]. If so, the assumption of a mean optical depth for all morphological types could contribute to a morphological type dependence, especially at the shorter wavelengths where extinction estimates would be significantly overestimated."[6]

"As mentioned above, the kinematics of lower luminosity irregular galaxies become progressively more dominated by turbulence and less by rotational motion. Consequently, the TF relations must progressively break down below some luminosity. [... The] TF relations for systems fainter than MB ~ -15.0 become rather poorly defined as the corrections to the line-widths become large and the inclinations become more uncertain. With these factors in mind it is recommended that samples be limited to galaxies brighter than MB ~ -16.0 for which the kinematics are dominated by rotation."[6]

Disk "galaxies have flat rotation curves over a broad range of luminosity and mass, with dark matter dominating at moderately large radii. At the same time, it is clear that the TF relation implies a "disk-halo conspiracy", such that the mass within some characteristic optical radius is coupled with the mass [...]. [Gravitational] interplay between the luminous and non-luminous matter in luminous disk galaxies is sufficient to redistribute the total mass in such a way as to produce the flat rotation curves which are observed. [... Low] mass galaxies are more susceptible to significant sweeping of gas via supernovae-driven winds during protogalactic collapse than are more massive systems due to a lower binding energy. In such a situation, star formation can be radically slowed and, in some cases, essentially quenched. As a result, these systems today would have a larger dark-to-luminous matter ratio than more massive systems since the dark matter (assumed to be non-baryonic) would be unaffected by radiation and gas dynamics."[7]

"B-, R-, I-, and H-band Tully-Fisher relations for the Local Calibrators [in the set of graphs at the right] (top), Ursa Major cluster members (middle), and Virgo cluster members (bottom). It is apparent from the figures that the slope of the relations increases going to longer wavelengths and the dispersion decreases. The variation in slope is thought to arise from the differing contributions to the observed bandpass made by greater fraction of young stars found in the lower-luminosity systems. The smaller dispersion at longer wavelengths is likely due to a reduction in the sensitivity of these effects, as well as those expected from extinction variations. Nothe the much larger dispersion found for the Virgo cluster data."[8]

"There are now reliable distance estimates to three systems in or near the Local Group (M31, M33, NGC 3109), two systems in the M81 Group (M81 and NGC 2403), and one galaxy in the Sculptor Group (NGC 300). If, in addition to these systems, we could make use of the other members of the M81 and Sculptor groups with MB ≤ -16.0, the total number of calibrators would be 15."[8]

The "absolute calibration of the TF relations [...] are shown as the upper sequence of panels in [the set of graphs at the right]."[8]

"The solid points represent those systems with individual distance estimates provided by Cepheids and/or RR Lyrae variables, while the open symbols represent additional members of the M81 and Sculptor groups (square and triangles, respectively). These are assumed to be at a mean distance given by those systems with individual distance estimates. The distribution of the additional members of the M81 Group is consistent with a small dispersion, implying a small line-of-sight depth for the group. The large dispersion for members of the Sculptor Group is consistent with previous suggestions of significant line-of-sight depth for the group [...]. The dispersions are ~ 0.20 mag for the systems with individual distance estimates, implying a precision in distance estimates from the TF relations of ~ 10% for an individual galaxy."[8]

"From the six local calibrating galaxies currently with individual distances the zero point of the calibration is established to an uncertainty of 0.08 mag [...]. The distances assumed for the local calibrators are all estimated relative to the distance of the LMC, which is tied to the Galactic calibration of Cepheids and RR Lyraes. The distance of the LMC is uncertain to ~ 7% [...] and so the primary source of error in the TF calibration lies with the Galactic calibration of the Cepheid P-L and P-L-C relations."[8]

A "significant systematic color difference between field (i.e., the "local calibrators") and cluster galaxies [...]. This results in a systematic variation in the estimated distances with bandpass. A small (~ 0.25 mag) correction is necessary in the B-band in order to produce consistent distance estimates over all bandpasses and environments. The resulting calibration is given by:"[8]

${\displaystyle M_{B}^{b,i}=-7.48(log(W_{R}^{i})-2.50)-19.55+\Delta _{B},}$

with an error on the ${\displaystyle \Delta _{B}}$  of ± 0.14,

${\displaystyle M_{R}^{b,i}=-8.23(log(W_{R}^{i})-2.50)-20.46+\Delta _{R},}$

with an error on the ${\displaystyle \Delta _{R}}$  of ± 0.10,

${\displaystyle M_{I}^{b,i}=-8.72(log(W_{R}^{i})-2.50)-20.94,}$

with an error on the last term of ± 0.10,

${\displaystyle M_{H}^{b,i}=-9.50(log(W_{R}^{i})-2.50)-21.67,}$

with an error on the last term of ± 0.08,

"where the correction factors δB = 0.25 and δR = 0.06 are required for statistically consistent distances between the different bands for "cluster galaxies". The corrections for "field-galaxies" should be zero. The existence of a "color correction" diminishes the utility of the B-band relation and implies that longer wavelength bandpasses produce more reliable distance estimates. Note that these calibrations apply only to the [...] prescriptions for extinction and line-width corrections."[8]

## Minerals

This is a schematic galaxy to show brightness fluctuations. Credit: John L. Tonry.

This diagram shows a schematic galaxy up close. Credit: John L. Tonry.

"An image of an elliptical galaxy with milli-arcsec resolution would look like an enormous globular cluster, but even when the resolution is a thousand times worse, the discreteness of the stars causes measurable bumpiness in its surface brightness. This phenomenon, dubbed "surface brightness fluctuations", has been recognized for many years and is sometimes referred to as "incipient resolution". (When there are only a few stars per seeing resolution element, the eye perceives this characteristic mottling as barely resolved stars.)"[9]

"Observations of external galaxies from which fluxes could be determined for individual stars, drawn from a significant part of the stellar luminosity function, would provide us a wealth of information about the distances of galaxies, their stellar populations, and their formation history. The first steps in this direction [involved resolving] individual Pop II stars in Local Group galaxies, and [observing] the top of the luminosity function in the bulge of M31 and NGC 205. [The] resolution required to observe any but the most luminous stars in galaxies significantly beyond the Local Group will not be achieved in the near future."[9]

"Although we cannot determine the fluxes of individual stars without resolving them, we can nevertheless measure a very useful flux that is characteristic of the stellar population. [The] average profiles of globular clusters could best be determined at the bluest wavelengths where these fluctuations are minimized."[9]

"The first measurement of surface brightness fluctuations [used] a method by which the fluctuations could be quantified. [A] completely empirical [approach used] calibration of fluctuation absolute magnitudes, observations of four galaxy clusters, and the application of fluctuations to the distance scale."[9]

"The schematic galaxy at the right-hand side is twice as distant as the one on the left-hand side. Large dots symbolize giant stars, small ones main-sequence stars, and the grid represents the pixels of a CCD. Although the mean surface brightness collected in a CCD pixel is the same for the two galaxies, since fd-2 and n ∝ d2, the rms fluctuation from pixel to pixel relative to the mean varies as d-1. The image at the left is about as mottled as an I-band image of a real galaxy would be at its effective radius if the galaxy were 200 kpc distant and the scale of CCD pixels were 1"."[9]

"Surface brightness fluctuations are fundamentally a very simple effect, and [as each schematic galaxy] illustrates this with a cartoon of two galaxies, one twice as distant as the other. A grid representing a CCD's pixels is superposed on the images, and we must imagine that we can only measure the total flux within each pixel. We do not resolve individual stars in this case, but we can measure both the mean flux per pixel and the rms variation in flux from pixel to pixel. The two galaxies cannot be distinguished on the basis of mean flux per pixel (surface brightness) because the number of stars projected into a pixel of fixed angular size increases as distance squared (d2) and the flux per star decreases as d-2. If N is the mean number of stars, the mean flux is ${\displaystyle N{\bar {f}}}$ , and the variance in flux is ${\displaystyle N{\bar {f}}^{2}}$ , where ${\displaystyle {\bar {f}}}$  is a mean flux per star. In schematic galaxies, N scales as d2, ${\displaystyle {\bar {f}}}$  scales as d-2, and so the variance scales as d-2 and the rms scales as d-1. The galaxy which is twice as distant appears twice as smooth as the nearer galaxy. We can now determine the mean flux ${\displaystyle {\bar {f}}}$  as the ratio of the variance and the mean."[9]

"The mean luminosity, ${\displaystyle {\bar {L}}}$ is simply related to the moments of the luminosity function of the stellar population we are sampling. If ni is the expectation of the number of stars of luminosity Li, the average luminosity is the ratio of the second and first moments of the luminosity function:"[9]

${\displaystyle {\bar {L}}=\Sigma n_{i}L_{i}^{2}/\Sigma n_{i}L_{i}.}$

"This luminosity is roughly that of a giant star, and it is the first ratio of moments which has the dimensions of luminosity and is not dominated by the poorly known faint end of the luminosity function."[9]

"There are two aspects to determining a distance from surface brightness fluctuations: (1) measurement of a fluctuation flux, and (2) conversion to a distance by assumption of a fluctuation luminosity. The two are coupled, but it is worth discussing them separately because they involve very different problems. It is useful to bear in mind that for typical old, metal rich stellar populations the absolute fluctuation magnitudes are roughly ${\displaystyle {\bar {M}}_{B}}$  = +2.5, ${\displaystyle {\bar {M}}_{V}}$  = +1.0, ${\displaystyle {\bar {M}}_{R}}$  = +0.0, and ${\displaystyle {\bar {M}}_{I}}$  = -1.5, with sensitivity to metallicity and age ranging from very high in B to fairly low in I. [To summarize]:"[10]

(1) "This method is applicable to relatively dust-free systems such as E and S0 galaxies, spiral bulges, or globular clusters."[10]

(2) "Observation times must be long enough to collect more than 5-10 photons per source of apparent magnitude barm, and the point spread function must be well sampled and uniform. The formal measurement errors in barm can be as small as 3% times the square of the product of psf FWHM (in arcsec) and distance (in 1000 km s-1)."[10]

(3) "The I band is preferred for distance measurement because ${\displaystyle {\bar {M}}}$  is so bright at redder wavelengths that it overcomes the disadvantages of brighter sky background. Use of the I band also minimizes dust absorption. Observations of cluster galaxies shows us how barMI depends on mean color: redder populations have brighter ${\displaystyle {\bar {M}}_{I}}$ . The intrinsic scatter of ${\displaystyle {\bar {M}}_{I}}$  about this relation appears to be smaller than 0.08 magnitude."[10]

(4) "Calibration of the zero point of ${\displaystyle {\bar {M}}_{I}}$  can be based on theoretical stellar populations, galactic globular clusters, or Local Group galaxies. The latter calibration has been made, and preliminary indications are that the first two will be consistent at the 0.10 magnitude level."[10]

"The basic procedure in measuring a fluctuation flux is to perform the usual observations and reductions so as to obtain an image which is as flat and uniform as possible and for which a photometric calibration is known."[10]

"Fluctuation fluxes are subject to the usual photometric calibration errors. Fundamentally, we need to transfer the known magnitude of a standard star to the power spectrum of the psf that we fit to a data power spectrum, since zero-wavenumber of a power spectrum is just the squared integral of the flux of the psf. Applying the same photometric reduction to the standard star and the psf star accomplishes this transfer with an accuracy of roughly 1-2%. Uncertainties in atmospheric extinction, galactic extinction, and extinction within the galaxy vary, but can also contribute an error of 1-2% (or more if there is considerable intervening dust)."[11]

Asteroid 65 Cybele and 2 stars with their apparent magnitudes are labeled. Credit: Kevin Heider.
This movie is an artist's impression of a Type Ia supernova. Credit: ESO/M. Kornmesser.

Def. any "astronomical object of known diameter whose distance can then be calculated from the angle it subtends"[12] is called a standard ruler.

Def. "a numerical measure of the brightness of a star, planet etc.; a decrease of 1 unit represents an increase in the light received by a factor of 2.512"[13] is called an apparent magnitude (${\displaystyle m\!\,}$ ).

Def. "the apparent magnitude [intrinsic luminosity][14] that a star etc. [celestial body][14] would have if viewed from a distance of 10 parsecs [or about 32.6 light years][15]"[16] is called an absolute magnitude (${\displaystyle M\!\,}$ ).

Def. the "magnitude of a star in terms of the total amount of radiation received at all wavelengths"[17] is called a bolometric magnitude.

Def. a "difference between the bolometric magnitude [${\displaystyle M_{bol}}$ ] and and visual magnitude [${\displaystyle M_{V}}$ ] of a star"[18] is called a bolometric correction.

${\displaystyle M_{bol}=M_{V}+BC.}$

Def. any "astronomical object of known absolute magnitude"[19] is called a standard candle.

Its luminosity distance [${\displaystyle D_{L}\!\,}$  in parsecs] can then be calculated from its apparent magnitude.

${\displaystyle M=m-5((\log _{10}{D_{L}})-1)\!\,}$

An object's absolute magnitude may be calculated from its distance modulus (${\displaystyle \mu \!\,}$ ) and apparent magnitude using

${\displaystyle M=m-{\mu }.\!\,}$
${\displaystyle m=M+{\mu }.\!\,}$

Using the Sun (${\displaystyle \odot }$ ) as a standard candle:

${\displaystyle M_{bol_{\rm {star}}}-M_{bol_{\rm {Sun}}}=-2.5\log _{10}{\frac {L_{\rm {star}}}{L_{\odot }}},}$

Or,

${\displaystyle {\frac {L_{\rm {star}}}{L_{\odot }}}=10^{((M_{bol_{\rm {Sun}}}-M_{bol_{\rm {star}}})/2.5)},}$

where

${\displaystyle L_{\odot }}$  is the Sun's (sol) luminosity (bolometric luminosity)
${\displaystyle L_{\rm {star}}}$  is the star's luminosity (bolometric luminosity)
${\displaystyle M_{bol_{\rm {Sun}}}}$  is the bolometric magnitude of the Sun
${\displaystyle M_{bol_{\rm {star}}}}$  is the bolometric magnitude of the star.
${\displaystyle m=H+2.5\log _{10}{\left({\frac {d_{OS}^{2}d_{OOb}^{2}}{p(\chi )d_{0}^{4}}}\right)}\!\,}$

where ${\displaystyle H}$  is the absolute magnitude for a solar system object, the observer is Ob, the Sun (S), and the object is O.

${\displaystyle d_{0}\!\,=1AU}$
${\displaystyle \chi \!\,=phaseangle}$

The phase angle lies between the Sun-object and object-observer lines.

${\displaystyle \cos {\chi }={\frac {d_{OOb}^{2}+d_{OS}^{2}-d_{ObS}^{2}}{2d_{OOb}d_{OS}}}.\!\,}$

where ${\displaystyle p(\chi )\!\,}$  is between 0 and 1 for the integration of reflected light.

${\displaystyle p(\chi )={\frac {2}{3}}\left(\left(1-{\frac {\chi }{\pi }}\right)\cos {\chi }+{\frac {1}{\pi }}\sin {\chi }\right).\!\,}$
${\displaystyle d_{OOb}\!\,}$

where the distance (d) is between the observer (O) and the object (O).

${\displaystyle d_{OS}\!\,}$

where S is the Sun.

${\displaystyle d_{ObS}\!\,}$

"A supernova [in the movie at right] is one way that a star can end its life, exploding in a display of grandiose fireworks. One family of supernovae, called Type Ia supernovae, are of particular interest in cosmology as they can be used as standard candles to measure distances in the Universe and so can be used to calibrate the accelerating expansion that is driven by dark energy. One defining characteristic of Type Ia supernovae is the lack of hydrogen in their spectrum. Yet hydrogen is the most common chemical element in the Universe. Such supernovae most likely arise in systems composed of two stars, one of them being the end product of the life of sun-like stars, or white dwarfs. When such white dwarfs, acting as stellar vampires that suck matter from their companion, become heavier than a given limit, they become unstable and explode."[20]

## Entities

The image is a graph of absolute magnitude versus frequency for globular clusters. Credit: William E. Harris.

"The basic ideas [of using globular cluster luminosity functions (or GCLFs)] are illustrated in [the image at the right], which is a plot of the GCLF (the relative number Φ (m) of globular clusters as a function of magnitude m) for the Virgo giant ellipticals. The functional shape of Φ (m) is characterized by two simple parameters: the magnitude level or turnover point (m0) where the population of clusters reaches a maximum; and the dispersion (σ (m) or standard deviation of the distribution. As an empirical curve, the observations show that a Gaussian or log-normal function"[21]

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

"(where A is a normalization factor representing the peak population of globular clusters in the galaxy) does an excellent job of describing the GCLF in all galaxies studied to date. The log-normal fitting function gives a convenient way to parameterize the data and to intercompare the GCLFs from different galaxies. For the nearest rich systems of galaxies containing giant ellipticals (the Virgo and Fornax clusters) the deepest limits achieved at present reach to just about 1 σ fainter than the turnover (Harris et al. 1991)."[21]

"The actual use of the GCLF method, in its simplest form, goes as follows. The 'standard candle' is essentially the magnitude of the turnover, m0. The goal in observing any distant galaxy is then to obtain photometry of its halo clusters which reaches sufficiently faint to estimate m0 accurately. In practice, the GCLF is normally found as the residual excess of starlike objects [...], after a background [luminosity function] LF for the field has been subtracted and photometric detection incompleteness at the faintest levels has been accounted for. By fitting an analytic interpolation function (such as the Gaussian model above) to the observed data, or else by using a maximum-likelihood fit to the total observed LF (GCLF plus background), the curve parameters (A, m0, σ) and their internal uncertainties can be estimated [...]. Adopting an absolute magnitude M0(turnover), and any necessary correction for foreground reddening, immediately yields the distance modulus."[21]

"If the limiting magnitude of the photometry goes clearly past the turnover as in the example of [the image at right], both m0 and σ can be solved for simultaneously. Ideally, the uncertainty e (m0) would be equal to σ / √N, which for a sample size N of several hundred or more clusters should be ± 0.1 mag or smaller. In practice, the extra noise introduced by the background LF subtraction and completeness corrections, and the uncertainty in σ itself, lead to best-case results near e (m0) ≃ ± 0.2 mag for a single galaxy. However, if the limit of the data falls at or a bit short of the turnover, then the fitted solutions for m0 and σ become strongly correlated (Hanes and Whittaker 1987). It is still possible to estimate m0 to a typical precision of ± 0.3 mag, but only by performing a restricted function fit with an assumed, externally-known value of σ (hence the importance of the basic premise that the GCLF parameters are the same for galaxies of the same type [...]). An alternative is to adopt the well-defined Virgo GCLF ([shown in the image at the right]) as a fiducial function and to match any other galaxy to it by adjusting its distance relative to Virgo until a best fit is achieved."[21]

"The range of applicability of the GCLF method, and its basic strengths, are readily summarized."[21]

(1) "The GCLF works by far the best in giant E galaxies, which contain the largest globular cluster populations (N ≥ 1000 is observationally feasible in such galaxies). On the other hand, the GCLF method is immensely more difficult to employ in late-type galaxies (Sc, Sd, Irr), and so in most cases it is difficult to match the GCLF distance scale directly with any of the Population I techniques."[21]

(2) "Because the clusters are nonvariable objects, they do not require repeated or carefully scheduled time-series observations. Telescope time is thus employed efficiently."[21]

(3) "Because it applies to objects in the halos of large galaxies, the method is free of a host of observational problems that affect all Population I standard candles, such as internal reddening differences, crowding and source confusion, and inclination-angle corrections."[21]

(4) "The distance range of the primary GCLF method is set by the limiting magnitude at which the turnover M0 is still detectable. With current ground-based capabilities, the limit should be near d ≃ 50 Mpc (Harris 1988a)."[21]

"So far, the motivation for the GCLF as a standard candle is almost totally empirical rather than theoretical. The astrophysical basis for its similarity from one galaxy to another is a challenging problem, and is probably less well understood than for any other standard candle currently in use. Because globular clusters are old-halo objects that probably predate the formation of most of the other stellar populations in galaxies [...], to first order it is not surprising that they look far more similar from place to place than their parent galaxies do. Methods for allowing clusters to form with average masses that are nearly independent of galaxy size or type [...]. Other constraints arising from cluster metallicity distributions and the early chemical evolution of the galaxies [may occur]. None of these yet serve as more than general guidelines for understanding why the early cluster formation process should be so nearly invariant in the early universe."[21]

"After the initial formation epoch, dynamical effects on the clusters including tidal shocking and dynamical friction, and evaporation of stars driven by internal relaxation and the surrounding tidal field, must also affect the GCLF within a galaxy over many Gyr, and these mechanisms might well behave rather similarly in large galaxies of many different types. [...] their importance decreases dramatically for distances ≳ 2-3 kpc from the galaxy nucleus, and for the more massive, compact clusters like present-day globulars. [...] close to the centers of the Virgo ellipticals has shown no detectable GCLF differences with radius. The implication is therefore that today's GCLF resembles the original mass formation spectrum of at least the brighter clusters, perhaps only slightly modified by dynamical processes. Many qualitative arguments can be constructed as to why the GCLFs should, or should not, resemble each other in different galaxies, but at the present time these must take a distant second place to the actual data."[21]

## Emissions

This graph depicts using AGNs to estimate distances. Credit: Darach Watson (et al).

“Accurate distances to celestial objects are key to establishing the age and energy density of the Universe and the nature of dark energy.”[22]

“A distance measure using active galactic nuclei (AGN) has been sought for more than forty years, as they are extremely luminous and can be observed at very large distances.”[22]

Active "galactic nuclei are home to supermassive black holes which unleash powerful radiation. When this radiation ionizes nearby gas clouds, they also emit their own light signature. With both emissions in range of data gathering telescopes, all that’s needed is a way to measure the time it takes between the radiation signal and the ionization point. The process is called reverberation mapping."[23]

“We use the tight relationship between the luminosity of an AGN and the radius of its broad line region established via reverberation mapping to determine the luminosity distances to a sample of 38 AGN.”[22]

“All reliable distance measures up to now have been limited to moderate redshift — AGN will, for the first time, allow distances to be estimated to z~4, where variations of dark energy and alternate gravity theories can be probed.”[22]

"The AGN Hubble diagram [is at the right]. The luminosity distance indicator τ is plotted as a function of redshift for 38 AGN with H lag measurements. On the right axis the luminosity distance and distance modulus (m-M) are shown using the surface brightness fluctuations distance to NGC3227 as a calibrator. The current best cosmology is plotted as a solid line. The line is not fit to the data but clearly follows the data well. Cosmologies with no dark energy components are plotted as dashed and dotted lines. The lower panel shows the logarithm of the ratio of the data compared to the current cosmology on the left axis, with the same values but in magnitudes on the right. The red arrow indicates the correction for internal extinction for NGC3516. The green arrow shows where NGC7469 would lie using the revised lag estimate. NGC7469 is our largest outlier and is believed to be an example of an object with a misidentified lag."[23]

“The scatter due to observational uncertainty can be reduced significantly. A major advantage held by AGN is that they can be observed repeatedly and the distance to any given object substantially refined.”[22]

“The ultimate limit of the accuracy of the method will rely on how the BLR (broad-line emitting region) responds to changes in the luminosity of the central source. The current tight radius-luminosity relationship indicates that the ionisation parameter and the gas density are both close to constant across our sample.”[22]

## X-rays

Shown here is the explosion in optical, ultraviolet and X-ray wavelengths. Credit: NASA/Swift/S. Immler.

Supernova 2005ke, which was detected in 2005, is a Type Ia supernova, an important "standard candle" explosion used by astronomers to measure distances in the universe. Shown here is the explosion in optical, ultraviolet and X-ray wavelengths. This is the first X-ray image of a Type Ia, and it has provided observational evidence that Type Ia are the explosion of a white dwarf orbiting a red giant star.

An explosion called SN 2005ke is the first Type Ia supernova detected in X-ray wavelengths, and it is much brighter in the ultraviolet than expected. A Type Ia is an explosion of a white dwarf in orbit around either another white dwarf or a red giant star. The dense white dwarf can accumulate gas donated from the companion. When the dwarf reaches the critical mass of 1.4 solar masses, a thermonuclear explosion ensues.

Type Ia are called "standard candles" and are used by astronomers to measure distances in the universe, because each Type Ia shines with a known luminosity. Immler's team says it has the first observational evidence to support one theory about the origin of these supernovae.

Immler's group has found direct evidence in the X-ray and ultraviolet light of SN 2005ke that a white dwarf, now obliterated, was indeed orbiting a red giant. The scientists detected shock waves from the explosion ramming into gas from a red giant and found no evidence of a second white dwarf. This observation may help astronomers understand the birthplaces and evolution of these supernovae, so crucial to the field of cosmology and dark energy.

## Opticals

Spiral galaxy Messier 82 has long been known for its remarkable starburst activity. Credit:

"Spiral galaxy Messier 82 has long been known for its remarkable starburst activity, caused by interactions with its near neighbour Messier 81, and has been the subject of intense study for many years. On 21 January 2014, astronomers at the University of London Observatory in London, UK, pointed their telescope at the galaxy and spied something peculiar… an intensely bright spot seemed to have suddenly appeared within the galaxy [1]."[24]

"This bright spot is actually a new supernova known as SN 2014J — the closest supernova to Earth in recent decades! Since its discovery, SN 2014J has been confirmed as a type Ia supernova, making it the closest of its type to Earth in over 40 years (since SN 1972E) [2]. This new NASA/ESA Hubble Space Telescope image is set against a previous mosaic of Messier 82 from 2006 (heic0604a), and shows the supernova as an intensely bright spot towards the bottom right of the frame."[24]

"Type Ia supernovae are even more exciting for astronomers, as they have particular properties that we can use to probe the distant Universe. They are used as standard candles to measure distances and help us understand the scale of the cosmos. Catching such a supernova so soon after its explosion is very unusual; this early discovery will enable astronomers to explore its evolution in great detail, and to potentially infer the properties of its progenitor star."[24]

"Messier 82 is several times more luminous than our Milky Way. Because it is only 12 million light-years away, it is one of the brighter galaxies in the northern sky. It can be found in the constellation of Ursa Major (The Great Bear). The supernova is currently visible through a modest amateur telescope, so why not see if you can spot it from your back garden?"[24]

"The image shown here was taken on 31 January 2014 with Hubble’s Wide Field Camera 3. This image is inset into a photo mosaic of the entire galaxy taken in 2006 with Hubble’s Advanced Camera for Surveys."[24]

## Visuals

In this diagram, the photon spectral density of a blackbody at 6000 K (solid line) and 5000 K (dashed line) is illustrated. Credit: Douglas L. Welch.

"Classical Cepheid variable stars are potentially the most accurate distance indicators available for the near-field (0.5 kpc - 10 Mpc). Their suitability as distance indicators arises from a number of factors including luminosity (MV = -2 to -7 mag), ease of detection through variability, precision, permanence, and understanding of the pulsation phenomenon. The classical (or Type I) Cepheids are young, disk objects and as such are only found in galaxies with recent star formation (primarily spiral or irregular galaxies)."[25]

"The traditional path to a Cepheid-based distance to a galaxy has involved the following steps: (1) acquisition of plate material at several epochs; (2) discovery of variables; (3) magnitude estimates; (4) determination of periods; (5) estimation of mean magnitudes and colors on a standard system; (6) correction for absorption and distance estimate (given an absolute calibration)."[26]

"The differences between observations in the optical and near-infrared are sufficiently great that they warrant discussion. In this section, we will discuss the relative merits of optical (0.4 - 0.9 µm) and near-infrared (1.0 - 2.5 µm) observations of Cepheids. To accomplish this, we will consider, in turn, the wavelength dependence of observables, factors affecting the detected signal, and factors contributing to the noise."[27]

"Panoramic detectors used in the optical (0.4 - 0.9 µm) and the near-infrared (1.0 - 2.5 µm) are unity gain devices which produce a single electron for every detected photon. A blackbody at an effective temperature of 5500 K - typical of longer period Cepheids - has a slowly varying photon spectral density from the V bandpass (0.5 µm) out to K (2.2 µm). [...] However, the near-infrared spectra of stars of this effective temperature are strongly distorted by the H- opacity minimum at 1.6 µm, centered on the photometric H bandpass. The effect of this opacity minimum is to allow radiation characteristic of a hotter source function to escape, and consequently a Cepheid will appear approximately 20% brighter at 1.6 µm than predicted by a blackbody spectrum scaled to, say, a V magnitude. (A second and less important effect which works in the same direction is the smaller amount of limb-darkening in the near-infrared.)"[27]

"One consequence of the H- opacity feature is the near constant H - K color of Cepheids (and for that matter, RR Lyrae stars) through their pulsation cycle. Since the H - K index varies extremely slowly in this temperature regime, a useful value for the total absorption for bandpasses in the 1 - 2.5 µm regime may be estimated on the basis of the H - K index at one phase point."[27]

"In [the diagram at right], the photon spectral density of a blackbody at 6000 K (solid line) and 5000 K (dashed line) is illustrated. These effective temperatures represent the range encountered during the pulsation of a typical 10-day Cepheid. The positions of the B, V, I, J, H, and K bandpasses are indicated. The number of detected photons from sources of these effective temperatures is proportional to the area under the curve for each bandpass. Wider bandpasses and higher quantum-efficiency detectors in the near infrared are particularly advantageous for the cooler and brighter long-period Cepheids. The increased near-infrared flux due to the lower H- opacity is not shown in this diagram."[27]

"During pulsation, both the effective temperature and radius of a Cepheid vary. Since the relative change in radius rarely exceeds 5% in Cepheids (except for the longest period stars), the areal change is only of order 10%. Therefore, the large lightcurve amplitudes observed in the optical are primarily a reflection of the changes in surface brightness during the pulsation cycle, due to the changes in effective temperature. This variation is illustrated in [the diagram at the right], where the relative number of photons at minimum and maximum effective temperature are easily compared. For detection of variation, a short wavelength bandpass such as B or V is indicated, whereas for measurement of mean lightcurve characteristics, a long wavelength bandpass such as I, J, H, or K is advantageous."[27]

"Other well-known advantages of longer wavelength observation are the smaller total absorption, reduced sensitivity to flux redistribution due to metallicity, and reduced sensitivity to possible photometric contamination by an upper main-sequence companion."[27]

"The bandwidths of the J, H, and K filters are typically 0.3 - 0.35 µm, compared to 0.2 µm at I and 0.08-0.10 µm at B or V. Furthermore, the quantum efficiency of thinned CCDs at I is typically 30-35%, whereas quantum efficiencies greater than 60% over the J, H, and K bandpasses can be realized with HgCdTe detectors. The reflectivity of aluminum is 93-96% in the 1.0-2.5 µm region compared to 86% at I."[27]

"The difference between long-wavelength optical and infrared photometry is not substantial with respect to crowding. The K-giant background against which Cepheids will be detected acts as a background having statistical fluctuations due to variations in the number of giant stars per pixel, but is significantly redder (V - K = 3.3) than the variables (V - K = 2.0). The effects of both crowding and sky noise are obviously reduced when the [full width at half maximum] FWHM of the stellar images is small (i.e., good seeing conditions)."[27]

## Blues

This is a graph of the maximum magnitude-rate of decline for Galactic novae. Credit: Cohen (1985).

"Novae possess a number of attributes that make them potentially valuable standard candles. They are luminous (approaching MV = -10) and easy to recognize. Because they belong to an old stellar population, they are found predominantly in ellipticals and the bulges of spirals [...]; such environments are relatively dust-free and photometrically smooth, so that observations of novae beyond the Local Group are simpler and easier to interpret than observations of Cepheids. The available evidence suggests that observations of novae are not strongly affected by metallicity effects [...]. Finally, the calibration of novae as standard candles possesses relatively low intrinsic scatter [...], and is well understood theoretically [...]."[28]

"All [or nearly all early] observations of extragalactic novae were made in continuum blue light."[28]

"The relationship between the maximum luminosity (magnitude) of a nova and its rate of decline (MMRD) is the usual starting point for deriving extragalactic distances. [L]uminous novae decay more rapidly than intrinsically faint novae [...]. The MMRD correlation for Galactic novae [has been confirmed. In addition there are] expansion parallaxes for a number of (previously undetected) spatially resolved shells around old Galactic novae. The physical basis for the MMRD relation [is that] more massive white dwarfs require less accreted matter to produce a thermonuclear runaway, and these lower mass envelopes can be ejected more rapidly; thus the most luminous novae are also the fastest."[29]

"To measure the distance to an external galaxy using the MMRD relation, it is necessary to determine the apparent magnitudes of novae at maximum light, and a mean rate of decline over two magnitudes. At present the calibration of the MMRD relation is in the B or mpg bands [...], so that observations through these bandpasses are preferred. It is essential that the observations sample the light curves of novae frequently enough near maximum light that mmax can be estimated for the fastest (i.e., brightest) novae. The signal to noise ratio of the photometry should be high enough that novae discovered near maximum light can be followed at least 2 magnitudes below this level."[29]

"The calibration can be effected in two ways: (1) using Galactic novae (in which case the use of the MMRD relation is a primary distance indicator, calibrated using geometrical techniques); or (2) using novae in M31 (in which case the distance scale is tied to the distance of M31)."[30]

At right is a graph of the maximum "magnitude-rate of decline relation for Galactic novae [...]. Closed symbols represent the novae designated high quality [...]; the solid line [from the equation below] is a least-squares fit to the high-quality data."[30]

An "MMRD relation for Galactic novae is given by:"[30]

${\displaystyle M_{V}^{max}=-9.96-2.31\times log{\dot {m}},}$

"where ${\displaystyle {\dot {m}}}$  is the mean rate of decline (in mag d-1) over the first 2 magnitudes. The mean scatter around this relation is ± 0.52 mag (1 ) for the high quality subset [data]. The data and fit are shown in [the graph at the right]. A slightly different result is obtained if the Galactic data are corrected for the constancy of MB 15 days after maximum light [...]. In principle such a correction removes systematic errors in the absolute magnitudes, and results in a tighter MMRD correlation (σ ≃ 0.47 mag for all objects)."[30]

"An alternate calibration of the MMRD relation is obtained by studying novae in the nearby spiral galaxy M31. [...] Only about 1/3 of the known novae in M31 have sufficient information in their light curves to be useful in determining the MMRD relation, and only about 1/4 of these possess good quality light curves with a well-observed maximum and rate of decay. [...] The 1 σ scatter around the mean relation depends on the subset of data chosen, and is in the range 0.20 - 0.28 mag [...]."[30]

"To compare the M31 MMRD data with the mean Galactic MMRD, we assume (m - M)B ≃ 24.6 for M31 [...], (B - V)max ≃ 0.23 [...], and (mpg - B) ≃ -0.17 [...]. With these assumptions, [...] the agreement between the Galactic and M31 MMRD relations is not good: the flattening observed in the M31 MMRD relation for bright and faint novae is not seen for Galactic novae. In addition, there appears to be a systematic offset of about 0.3 mag between the two MMRD relations, in the sense that Galactic novae are fainter than M31 novae. (This offset would increase to ~ 0.5 mag if the mean internal absorption for M31 novae were 0.2 mag [...]. However, we note that [there may not be] any systematic difference in the MMRD relation for novae close to and far from obvious dust patches in the bulge of M31.)"[30]

"The flattening at faint magnitudes in the M31 MMRD relation may be due to Malmquist bias: in the presence of a magnitude limit, only the brightest novae will be detected. Whether or not this flattening is real has little effect on distance determinations outside the Local Group, because it is predominantly the most luminous novae that are detected at large distances. The flattening of the M31 MMRD relation for luminous novae is a more difficult problem. If one aligns the MMRD relations for Galactic and M31 novae in the linear (-1.3 ≲ log mdot ≲ -0.7) regime, then the luminous (log mdot ≲ -0.6) Galactic novae lie an average of ~ 0.8 mag above the M31 MMRD relation. One possible explanation for this is that maximum light for M31 novae is not as well sampled as it is for Galactic novae; this is particularly apparent in the light curves of Arp 1 and Arp 2 [...]."[30]

"The shift of the Galactic and M31 MMRD relations relative to each other seems to imply that the true distance modulus of M31 is 0.3 mag less than the value obtained with quality distance indicators (e.g., RR Lyrae stars, IR observations of Cepheids). However, [...] this discrepancy vanishes if a different sample of objects is chosen to define the Galactic MMRD, and if uncorrected MVmax values are used for the Galactic nova sample (instead of MV values corrected for the Buscombe - de Vaucouleurs effect). It is also worth noting that the theoretical MMRD [...] possesses a flatter slope than that observed for [...] data on Galactic novae, and hence provides a better overall fit to the S-shaped MMRD relation observed for M31 novae."[30]

It "should be noted that the Galactic MMRD relation is defined with far fewer objects than is the case for M31; furthermore, the overall quality of the Galactic data (as demonstrated by the scatter in the MMRD relation) is considerably lower than for M31, probably due to such effects as uncertain Galactic absorption, and due to the assumption of spherical symmetry that is inherent in Cohen's application of the expansion parallax technique [...]. In fact, the offset between the Galactic and M31 MMRD relations is almost exactly what would be expected if the prolate geometry of nova shells is not taken into account when applying the expansion parallax technique [...]."[30]

"In view of all of the above, it seems somewhat safer to employ the M31 MMRD relation as the calibrator for the extragalactic distance scale. This makes the distance scale dependent on an assumed distance to M31. However, [...] there is concordance in most distance estimates for M31 (except those derived using novae!); using the M31 calibration is therefore a more prudent approach at the present time."[30]

"Several other methods for using novae as distance indicators have been proposed in the literature. Here we give a brief discussion of some of these methods, and their limitations."[31]

"<M15>: [The] mean magnitude of an ensemble of novae 15 days after maximum light was a constant; from the most recent data on Galactic novae, [...] <M15> ≃ -5.60 ± 0.14, where the quoted error is the 1 σ error in the mean. The rough constancy of <M15> is a consequence of the MMRD relation [...]. For Virgo cluster novae, <B15> ≃ +26 if (m - M) = 31.5; hence the <M15> distance indicator will be strongly affected by Malmquist bias at the distance of the Virgo cluster unless the observational completeness limit goes somewhat fainter than B > 26. Note that there are exceptional objects (e.g., M31 novae Arp 1, 2, and 3) that suggest this method be used with caution."[31]

"Luminosity Function of Novae: The luminosity function of novae at maximum light is approximately Gaussian [with] the mean magnitude of this Gaussian to determine the distance to M31. [...] The most recent compilation of M31 nova data [...] appears to show a double-peaked luminosity function; [using] the magnitude of the minimum between the two peaks as a distance indicator. Yet another method is to use the integral luminosity function of novae at maximum light; this function is linear over a wide range of magnitude, and possesses a well-defined intercept [...]."[31]

"The use of the luminosity function of novae at maximum light as a distance indicator demands large samples of novae that are essentially complete at the faintest magnitudes; it is therefore unlikely that this method will be useful for any but the nearest luminous galaxies (i.e., those with high nova rates). [Using] the dip between two peaks in the luminosity function is not completely reliable, because, [...] different samples of M31 novae have luminosity functions with very different structure. (The very existence of this dip is in some question [...]"[31]

The "luminosity function of all M31 nova observations (i.e., random phases) possesses no useful information that can be used in determining extragalactic distances [...]."[31]

"Period of Visibility: [There] exists a strong correlation between the mean period of visibility of novae (down to some limiting magnitude mlim, and the absolute magnitude that this mlim corresponds to. Application of this correlation (calibrated using M31 data) to the Virgo elliptical observations [...] yields a distance modulus that is similar to that obtained using the MMRD relation. This method needs complete samples of novae down to some chosen mlim, but the samples do not have to be large. (With a large sample of novae, it would in principle be possible to apply this technique at several different mlim values.)"[31]

## Oranges

The image shows orange and red stars in NGC 416. Credit: Andrzej Udalski, Peter Garnavich and Krzysztof Stanek.

This is the Hipparcos color-magnitude diagram. Credit: Hipparcos.

This image shows the variance in the I-band. Credit: Stanek and Garnavich (1998).

"Stellar cluster NGC 416 [in the image at the right is] located in the nearby Small Magellanic Cloud galaxy."[32]

"The cluster [in the image at the right] contains many red clump stars, allowing for accurate distance measurement to the host galaxy. This photograph was made with the 1.3-meter (51-inch) Warsaw University Telescope at Las Campanas Observatory, near La Sarena, Chile."[32]

"The ideal distance indicator would be a standard candle abundant enough to provide many examples within reach of parallax measurements and sufficiently bright to be seen out to local group galaxies."[32]

Red "clump stars (shown in the Hipparcos color-magnitude diagram [second image at the right]) precisely fit this description. These stars are the metal rich equivalent of the better known horizontal branch stars, and theoretical models predict that their absolute luminosity fairly weakly depends on their age and chemical composition. Indeed the absolute magnitude-color diagram of Hipparcos clearly shows how compact the red clump is."[32]

The "variance in the I-band is only about 0.15 mag (see the [third figure at the right])."[32]

"Red clump giants can provide a very precise estimate of the distances to metal-rich Galactic globular clusters and nearby galaxies."[32]

## Reds

"Supernovae [especially Type Ia (SNe Ia)], as extremely luminous (MB ~ -19.5) point sources, offer an attractive route to extragalactic distances. [...] Type II supernovae have a wide range in peak absolute magnitude and can not be treated as standard candles. Distances to individual SNe II can be estimated by means of the expanding photosphere (Baade-Wesselink) and the expanding radiosphere methods, but only elementary applications based on simplifying assumptions have been made to SNe II beyond the Local Group [...]. [Applications] of the method to SN 1987A in the Large Magellanic Cloud [have been] based on detailed calculations [...]. Supernovae of Type Ib, Type Ic, or Type II-L may turn out to be good standard candles but the present samples are small and all three subtypes have the disadvantage of being less luminous than Type Ia."[33]

"Supernovae of Type Ia lack hydrogen lines and helium lines in their optical spectra; during the first month after maximum light they do have a strong absorption feature produced by the red doublet (λ6347, λ6371 Å) of singly ionized silicon. [... One model is that] Type Ia supernovae are the result of the nuclear detonation of a white dwarf which is at or near the Chandrasekhar mass limit [...]. Since such stars are present in the old stellar populations of all galaxies (but see Foss et al. 1991), there is good reason to believe that Type Ia supernovae behave as standard candles."[33]

"Numerous analyses of unrestricted samples of SNe Ia, involving various assumptions about relative distances and interstellar extinction, have produced values of the dispersion in peak Mpg or MB that are generally consistent with [early results having a] determined σ = 0.6 mag. Smaller dispersions of 0.3-0.5 mag have been obtained by restricting the SN Ia samples to those beyond the Local Supercluster [...], in elliptical galaxies [...], in the Virgo cluster [...], and in the Coma cluster [...]. The restriction to remote samples lowers the dispersion by a combination of two effects: (1) the avoidance of the problem of uncertain relative distances for the nearby galaxies, and (2) the tendency to select against SNe Ia that are observationally subluminous (whether they are intrinsically subluminous or are highly extinguished)."[33]

A "small intrinsic dispersion for ordinary SNe Ia that may be ≲ 0.3 mag. Most SNe Ia that are observationally subluminous tend to be red and in inclined disk galaxies, and probably just suffer high interstellar extinction. The peculiar, intrinsically subluminous SN 1991bg also was red. If observationally faint events enter into samples of remote SNe Ia, in spite of the selection against them, they can be recognized by their colors, and, in the case of those that are intrinsically abnormal, by their spectra. There is not yet any solid evidence for anomalously bright SNe Ia."[33]

"From the Hubble diagram for [a] sample of 35 SNe Ia"[33]

${\displaystyle M_{B}=-18.13+5logh,}$

with an error on the first constant of ± 0.08, "where h is the Hubble constant in units of 100 km s-1 Mpc-1."[33]

"From a sample of 40 SNe"[33]

${\displaystyle M_{B}=-18.36+5logh,}$

with an error on the first constant of ± 0.04.

"The difference is primarily due to the fact that [for the first equation no] corrections for parent-galaxy extinction, while [for the second] an inclination-dependent correction to those SNe Ia in spirals that appear to be subluminous [has been applied]. Perhaps the most accurate available estimate for the intrinsic absolute magnitude is [for] for nine SNe Ia in ellipticals:"[33]

${\displaystyle M_{B}=-18.33+5logh,}$

with an error on the first constant of ± 0.11.

"The standard model for a Type Ia supernova is the thermonuclear disruption of a carbon-oxygen white dwarf that has accreted enough mass from a companion star to approach the Chandrasekhar mass [...]. The nuclear energy released in the explosion unbinds the white dwarf and provides the kinetic energy of the ejected matter, but adiabatic expansion quickly degrades the initial internal energy and the observable light curve is powered by delayed energy input from the radioactive decay of 56Ni and 56Co. This model brings with it a self-calibration of the peak luminosity. Arnett (1982a) predicted on the basis of an analytical model that the SN Ia peak luminosity would be equal to the instantaneous decay luminosity of the nickel and cobalt, in which case the peak luminosity follows directly from the ejected nickel mass and the rise time to maximum light. The rise time can be inferred from observation but owing to uncertainties in the physics of the nuclear burning front [...] the amount of synthesized and ejected 56Ni cannot yet be accurately predicted by theory. [The] nickel mass can be estimated indirectly from spectra and light curves. The more nuclear burning, the more 56Ni and kinetic energy, and the greater the blueshifts in the spectrum and the faster the decay of the light curve. [From] the blueshifts in the spectra [...] the nickel mass must be in the range 0.4 to 1.4 M [with] a value of 0.6 M (as in the particular carbon deflagration model W7 [...]). Adopting a rise time to maximum of 17 ± 3 days and distributing the luminosity according to the observed ultraviolet-deficient flux distribution of SNe Ia, [provides an] estimated MB = -19.5+0.4-0.9) at bolometric maximum, which corresponds to MB = -19.6 with limits of -19.2 and -20.5 at the time of maximum blue light a few days earlier."[34]

## Infrareds

The images show a standard candle. Credit: NASA/JPL-Caltech/M. Marengo (Iowa State).

"This image layout [at right] illustrates how NASA's Spitzer Space Telescope was able to show that a "standard candle" used to measure cosmological distances is shrinking -- a finding that affects precise measurements of the age, size and expansion rate of our universe. The image on the left, taken by Spitzer in infrared light, shows Delta Cephei, a type of standard candle used to measure the distances to galaxies that are relatively close to us. Cepheids like this one are the first rungs on the so-called cosmological distance ladder -- a tool needed to measure farther and farther distances."[35]

"Spitzer showed that the star has a bow shock in front of it. This can be seen as the red arc shape to the left of the star, which is depicted in blue-green (the colors have been assigned to specific infrared wavelengths we can't see with our eyes). The presence of the bow shock told astronomers that Delta Cephei must have a wind that is forming the shock. This wind is made up of gas and dust blowing off the star. Before this finding, there was no direct proof that Cepheid stars could lose mass, or shrink."[35]

"The finding is important because the loss of mass around a Cepheid can obscure the star's light, making it appear brighter in infrared observations, and dimmer in visible light, than it really is. This, in turn, affects calculations of how far away the star is. Even tiny inaccuracies in such distant measurements can cause the whole cosmological distance ladder to come unhinged."[35]

"The diagram on the right illustrates how Delta Cephei's bow shock was formed. As the star speeds along through space, its wind hits interstellar gas and dust, causing it to pile up in the bow shock. A companion star to Delta Cephei, seen just below it, is lighting up the region, allowing Spitzer to better see the region. By examining the structure of the bow shock, astronomers were able to calculate how fast the star is losing mass."[35]

"In this image, infrared light captured by the infrared array camera is blue and blue-green (3.6- and 4.5-micron light is blue and 8.0-micron light is blue-green). Infrared light captured by the multiband imaging photometer is colored green and red (24-micron light is green and 70-micron light is red)."[35]

## Oxygens

The diagram is a schematic H-R showing the evolutionary tracks of planetary nebula central stars. Credit: Jacoby et al. 1989.

At right is a "schematic H-R diagram showing the evolutionary tracks of planetary nebula [PN] central stars. The tracks illustrate the strong dependence of magnitude and lifetime with core mass: the luminosity of a helium burning central star goes approximately as L ~ M3.5, but the lifetime of the object goes as τ ~ M-9.5. It is these two relationships, coupled with a sharply peaked core-mass distribution, which cause the abrupt truncation in planetary-nebula luminosity function."[36]

"It is common to think of using the brightest stars to determine extragalactic distances. Indeed, over 50 years ago, Hubble (1936a) first attempted to resolve stars in other galaxies to quantify the expansion of the universe, and since then, the use of blue and red supergiants for extragalactic distance determinations has been explored several times (e.g., Sandage and Tammann 1974, Humphreys 1983). However, only recently has it been appreciated that young planetary nebulae also fall into the "brightest stars" category and are therefore potentially useful as standard candles. As can be seen in the H-R diagram [at right], the central stars of these objects are almost as luminous as the brightest red supergiants - the fact that their continuum emission emerges in the far ultraviolet, instead of the optical or near infrared, does not affect their detectability. On the contrary, since their surrounding nebulae reprocess the EUV radiation into discrete emission lines, planetaries can be viewed through interference filters which suppress the starlight from the host galaxy. As a result, observations made through a narrow band [oxygen III 500.7 nm] 5007 filter can detect ~ 15% of the energy emitted from these extremely luminous objects, with little contamination and confusion from continuum sources."[37]

"Planetary nebulae have several advantages over other extragalactic distance indicators. Because PN are not associated with any one stellar population, they can be found in galaxies of all Hubble types, and hence are particularly valuable for probing the [elliptical] E and [intermediate] S0 galaxies which define the cores of large groups and clusters. Likewise, internal extinction is usually not a problem in extragalactic PN observations: unlike blue supergiants or Cepheids, PN can be found far away from star forming regions, in areas of the galaxy which are relatively dust free. Since PN are observed through narrow band filters which suppress the continuum, the identification and measurement of these objects does not require complex, crowded field photometric procedures, and, unlike variable star standard candles, PN observations are required only once. Perhaps most importantly, in a large galaxy there may be several hundred planetaries populating the brightest two magnitudes of the planetary nebula luminosity function (PNLF). With the luminosity function so well defined, distance derivations are straightforward, and the internal errors can be as small as 3% (cf. Jacoby et al. 1989)."[37]

"Despite these facts, the use of PN to determine extragalactic distances is a relatively new phenomenon. The first suggestion that PN might be a useful standard candle can be found in the book Galaxies and Cosmology by Hodge (1966), where it is listed in Table 12.1 along with such well-known distance indicators as Cepheids, RR Lyrae, and novae. Actual PN distance measurements, however, were [not] made until the 1970s, when Ford and Jenner (1978) used a 50 Å wide λ 5007 filter and the SIT Video Camera on the Kitt Peak 4-m telescope to find and measure the brightest PN in the bulge of M81. Ford (1978) had noticed that the absolute [O III] λ 5007 flux of the brightest PN in each of seven Local Group galaxies varied by less than 25%. Thus, by comparing the [O III] λ 5007 fluxes of M81's brightest PN with those of the brightest PN found in M31, Ford and Jenner (1978) estimated the distance ratio between these two galaxies to be ~ 4. A few years later, Jacoby and Lesser (1981) used a similar argument to place limits on the distances to five Local Group dwarfs, and Lawrie and Graham (1983) estimated the distance modulus of NGC 300."[37]

"None of the above results was exceptionally persuasive, however. Some of this skepticism arose from the analysis method which excluded all but the brightest objects from consideration. However, the main concern at the time was that little was known about the luminosity function of planetaries; hence the uncertainties associated with these distances were completely unknown. It is an irony of the subject that distances to Galactic planetary nebulae are extremely difficult to obtain, and that a single PN is definitely not a standard candle. (For instance, NGC 7027, one of the best studied Galactic PN has recent distance estimates that range from 178 pc [Daub 1982] to 1500 pc [Pottasch et al. 1982].) However, while a single PN may not be a standard candle, an ensemble of these objects can yield a well determined distance. The reason for this is the invariance of the [O III] λ 5007 planetary nebula luminosity function."[37]

## Materials

The images show necessary adjustments to Type Ia supernovae brightness. Credit: Supernova Factory.

"Type Ia supernovae result from the explosions of white dwarf stars. These supernovae vary widely in peak brightness, how long they stay bright, and how they fade away, as the lower graph shows. Theoretical models (dashed black lines) seek to account for the differences, for example why faint supernovae fade quickly and bright supernovae fade slowly. A new analysis by the Nearby Supernova Factory indicates that when peak brightnesses are accounted for, as shown in the upper graph, the late-time behaviors of faint and bright supernovae provide solid evidence that the white dwarfs that caused the explosions had different masses, even though the resulting blasts are all “standard candles.”"[38]

"Sixteen years ago two teams of supernova hunters, one led by Saul Perlmutter of the U.S. Department of Energy’s Lawrence Berkeley National Laboratory (Berkeley Lab), the other by Brian Schmidt of the Australian National University, declared that the expansion of the universe is accelerating – a Nobel Prize-winning discovery tantamount to the discovery of dark energy. Both teams measured how fast the universe was expanding at different times in its history by comparing the brightnesses and redshifts of Type Ia supernovae, the best cosmological “standard candles.”"[38]

"These dazzling supernovae are remarkably similar in brightness, given that they are the massive thermonuclear explosions of white dwarf stars, which pack roughly the mass of our sun into a ball the size of Earth. Based on their colors and how fast they brighten and fade away, the brightnesses of different Type Ia supernovae can be standardized to within about 10 percent, yielding accurate gauges for measuring cosmic distances."[38]

"Until recently, scientists thought they knew why Type Ia supernovae are all so much alike. But their favorite scenario was wrong."[38]

"The assumption was that carbon-oxygen white dwarf stars, the progenitors of the supernovae, capture additional mass by stripping it from a companion star or by merging with another white dwarf; when they approach the Chandrasekhar limit (40 percent more massive than our sun) they experience thermonuclear runaway. Type Ia brightnesses were so similar, scientists thought, because the amounts of fuel and the explosion mechanisms were always the same."[38]

“The Chandrasekhar mass limit has long been put forward by cosmologists as the most likely reason why Type Ia supernovae brightnesses are so uniform, and more importantly, why they are not expected to change systematically at higher redshifts.”[39]

“The Chandrasekhar limit is set by quantum mechanics and must apply equally, even for the most distant supernovae.”[39]

"But a new analysis of normal Type Ia supernovae, led by SNfactory member Richard Scalzo of the Australian National University, a former Berkeley Lab postdoc, shows that in fact they have a range of masses. Most are near or slightly below the Chandrasekhar mass, and about one percent somehow manage to exceed it."[38]

"While white dwarf stars are common, it’s hard to get a Chandrasekhar mass of material together in a natural way.”[40]

"A Type Ia starts in a two-star (or perhaps a three-star) system, because there has to be something from which the white dwarf accumulates enough mass to explode."[38]

"Some models picture a single white dwarf borrowing mass from a giant companion."[38]

“The most massive newly formed carbon-oxygen white dwarfs are expected to be around 1.2 solar masses, and to approach the Chandrasekhar limit a lot of factors would have to line up just right even for these to accrete the remaining 0.2 solar masses.”[40]

"If two white dwarfs are orbiting each other they somehow have to get close enough to either collide or gently merge, what Scalzo calls “a tortuously slow process.” Because achieving a Chandrasekhar mass seems so unlikely, and because sub-Chandrasekhar white dwarfs are so much more numerous, many recent models have explored how a Type Ia explosion could result from a sub-Chandrasekhar mass – so many, in fact, that Scalzo was motivated to find a simple way to eliminate models that couldn’t work."[38]

"He and his SNfactory colleagues determined the total energy of the spectra of 19 normal supernovae, 13 discovered by the SNfactory and six discovered by others. All were observed by the SNfactory’s unique SNIFS spectrograph (SuperNova Integral Field Spectrograph) on the University of Hawaii’s 2.2-meter telescope on Mauna Kea, corrected for ultraviolet and infrared light not observed by SNIFS."[38]

"A supernova eruption thoroughly trashes its white dwarf progenitor, so the most practical way to tell how much stuff was in the progenitor is by spectrographically “weighing” the leftover debris, the ejected mass. To do this Scalzo took advantage of a supernova’s layered composition."[38]

"A Type Ia’s visible light is powered by radioactivity from nickel-56, made by burning carbon near the white dwarf’s center. Just after the explosion this radiation, in the form of gamma rays, is absorbed by the outer layers – including iron and lighter elements like silicon and sulfur, which consequently heat up and glow in visible wavelengths."[38]

"But a month or two later, as the outer layers expand and dissipate, the gamma rays can leak out. The supernova’s maximum brightness compared to its brightness at late times depends on how much gamma radiation is absorbed and converted to visible light – which is determined both by the mass of nickel-56 and the mass of the other material piled on top of it."[38]

"The SNfactory team compared masses and other factors with light curves: the shape of the graph, whether narrow or wide, that maps how swiftly a supernova achieves its brightest point, how bright it is, and how hastily or languorously it fades away. The typical method of “standardizing” Type Ia supernovae is to compare their light curves and spectra."[38]

“The conventional wisdom holds that the light curve width is determined primarily or exclusively by the nickel-56 mass, whereas our results show that there must also be a deep connection with the ejected mass, or between the ejected mass and the amount of nickel-56 created in a particular supernova.”[40]

“The white dwarfs exploding as Type Ia supernovae have a range of masses, and the resulting light-curve width is directly proportional to the total mass involved in the explosion.”[39]

"For a supernova whose light falls off quickly, the progenitor is a lot less massive than the Chandrasekhar mass – yet it’s still a normal Type Ia, whose luminosity can be confidently standardized to match other normal Type Ia supernovae."[38]

"The same is true for a Type Ia that starts from a “classic” progenitor with Chandrasekhar mass, or even more. For the heavyweights, however, the pathway to supernova detonation must be significantly different than for lighter progenitors. These considerations alone were enough to eliminate a number of theoretical models for Type Ia explosions."[38]

"Carbon-oxygen white dwarfs are still key. They can’t explode on their own, so another star must provide the trigger. For super-Chandrasekhar masses, two C-O white dwarfs could collide violently, or one could accrete mass from a companion star in a way that causes it to spin so fast that angular momentum supports it beyond the Chandrasekhar limit."[38]

"More relevant for cosmolology, because more numerous, are models for sub-Chandrasekhar mass. From a companion star, a C-O white dwarf could accumulate helium, which detonates more readily than carbon – the result is a double detonation. Or two white dwarfs could merge. There are other surviving models, but the psychological “safety net” that the Chandrasekhar limit once provided cosmologists has been lost. Still, says Scalzo, the new analysis narrows the possibilities enough for theorists to match their models to observations."[38]

“This is a significant advance in furthering Type Ia supernovae as cosmological probes for the study of dark energy, likely to lead to further improvements in measuring distances. For instance, light-curve widths provide a measure of the range of the star masses that are producing Type Ia supernovae at each slice in time, well back into the history of the universe."[39]

## Sun

${\displaystyle M_{V\odot }=4.83.}$
${\displaystyle M_{bol\odot }=4.75.}$ [41]

## Betelgeuse

The star Betelgeuse may still be too far away for visual trigonometric parallax. Standard candles have probably been used to estimate its distance from the Sun. Estimates from visual trigonometric parallax may be available to evaluate the historical accuracy of standard stellar candles.

In 1977 the first direct angular-diameter observations of 119 Tauri were made.[42]

As a spectral type M2.2 Iab information is inferred about Betelgeuse (a type M2.2 Iab) from the occultation measurements of 119 Tauri.[42] The occultations were on January 31, and April 23, 1977.

The spectral type of 119 Tau has been constant since 1940.[42]

In 1977, 119 Tauri (CE Tau) was a spectral type M2.2 Iab, with a spectral range of M2.0-M2.4 Iab-Ib.[42]

In 1977, Betelgeuse was a spectral type M2.0 Iab-, with a range of M1.3-M2.8 Iab-Ib.[42]

For α Sco in 1980 it was M1.1 Iab with a range of M0.7-M1.5 Iab-Ib

As of 2014, 119 Tauri is an M2Iab according to SIMBAD.

As of 2014, Betelgeuse (alf Ori) is an M2Iab according to SIMBAD.

"The spectral type of α Ori varies roughly with the 5.8 yr period and epoch [...] for brightness, radial velocity, and possible angular-diameter variations. Recently, α Ori has shown the latest spectral type between 1973 and 1975 and again in 1980 January-February with a spectral type of M2.8 Iab-. Its spectral type was about M1.5 around 1969-1971 and again around 1977-1978. By 1982 or 1983, α Ori should again have a spectral type of about M1.5."[42]

In 1977, apparently α Scorpii (Antares) was an M2.2 Iab, but in 2014 it is an M1.5Iab-b.

The standard candle being used in 1977 for spectral region K5-M4 is the CN (cyanide) index from the CN absorption in selected bands.[42]

The apparent magnitude needed for calculating an object's distance in pc is obtained using a photospheric magnitude received for the 1.04 µm flux peak, I(104) in early-M stars.[42]

Near-infrared "photometry on the narrow-band eight-color system [...] has been obtained for these stars. The mean [CN] indices and spectral types derived from photometry of the three supergiants are"[42]

1. 119 Tau, CN index = 18 ± 2, I(104) mag = +0.84 ± 0.03, MV = -5.2 (-4.8 to -5.6), distance = 417 pc,
2. α Ori, CN index = 18 ± 3, I(104) mag = -2.68 ± 0.03, MV = -5.2, (-4.5 to -5.8), distance = 96 pc, and
3. α Sco, CN index = 19 ± 3, I(104) mag = -2.28 ± 0.02, MV = -5.5, (-4.5 to -5.9), distance = 107 pc.

"The distance to α Ori is about half the value, 200 pc, that is almost universally used in the literature."[42]

"The direct evidence for a distance of 200 pc [to Betelgeuse] is a trigonometric parallax of 0.005, which is 10 times smaller than the expected error of measurement [0.005 ± 0.05 mas]."[42]

In 1977 using the absorption spectrum of cyanide believed to be applicable for the spectral region K5-M4 to produce a CN index and the relationship between apparent magnitude and absolute magnitude, a distance of 96 pc was estimated for Betelgeuse. Parallax measurements at that time estimated a distance of 200 pc, but the error was 10 times greater than the value derived.[42]

A parallax measurement by the satellite Hipparcos indicated a distance of 197 ± 45 pc published in 2008.

While post 1980 adjustments were made to increase the estimated distance of Betelgeuse, the initial discrepancy is quite large, at least a factor of 2 using a standard candle.

In 1977, the distance to Betelgeuse estimated by various standard candles suggested 200 pc, "almost universally used in the literature."[42]

## SN 2005ke

Supernova 2005ke, which was detected in 2005, is a Type Ia supernova, an important "standard candle" explosion used by astronomers to measure distances in the universe. Shown here is the explosion in optical, ultraviolet and X-ray wavelengths. This is the first X-ray image of a Type Ia, and it has provided observational evidence that Type Ia are the explosion of a white dwarf orbiting a red giant star. Credit: NASA/Swift/S. Immler.

A Type Ia supernova is an explosion of a white dwarf in orbit around either another white dwarf or a red giant star. The dense white dwarf can accumulate gas donated from the companion. When the dwarf reaches the critical mass of 1.4 M, a thermonuclear explosion ensues. As each Type Ia shines with a known luminosity, Type Ia are called "standard candles" and are used by astronomers to measure distances in the universe.

SN 2005ke is the first Type Ia supernova detected in X-ray wavelengths, and it is much brighter in the ultraviolet than expected.

## Andromeda galaxy

This image is a mosaic of XMM-Newton's European Photon Imaging Camera’s (EPIC) observations of the central region of M31 as seen from 2000 to 2004. Credit: W. Pietsch (MPE Garching, Germany), ESA/XMM Newton.

"This image [on the right] is a mosaic of XMM-Newton's European Photon Imaging Camera’s (EPIC) observations of the central region of M31 as seen from 2000 to 2004."[43]

"The color coding is such that red displays X-ray photons received in the energy band 0.2-0.5 keV, green in the 0.5-1 keV band and blue in the 1-2 keV band. The positions of ten counterparts of optical novae detected in these images are indicated with circles and nova names. Nova names are given omitting the M31N prefix."[43]

"It was detected that eleven out of the 34 novae that had exploded in the galaxy during the previous year were shining X-rays into space. An additional seven novae remained detectable in X-rays up to 10 years after outburst."[43]

## Distances

For nearby objects:

${\displaystyle M=m+5(1+\log _{10}{p})\!\,}$
${\displaystyle m=M-5(1+\log _{10}p).\!\,}$

where p is the parallax in arcseconds.

"The use of elliptical galaxies as distance indicators developed from studies aimed at determining their physical properties. The discovery of the relationship between luminosity and central velocity dispersion, L ∝ σ4, by Faber and Jackson (1976) marks the beginning of the potential use of normal elliptical galaxies as standard candles. Following that discovery, automatic, quantitative techniques based on Fourier or cross correlation methods for measuring velocity dispersions came into frequent use [...]. The use of these quantitative methods led to a growth of work on the properties and distances of elliptical galaxies, notably the suggestion [...] of a second parameter at work in the Faber-Jackson relation and the use of that relation [...] to measure the infall of the Local Group towards the Virgo cluster. The uncertainty in the distance to a single galaxy in [a] use of the Faber-Jackson relation was 32%."[44]

"The distribution and physical properties of elliptical galaxies make them attractive as potential distance indicators because:"[44]

(1) They are luminous galaxies so that their global properties can be measured accurately at large distances. The Dn- method can be used to measure distances both within the Local Supercluster and out to distances more than twice that of the Coma cluster.

(2) "They are strongly clustered so that many galaxies can contribute to the determination of the distance to an aggregate of galaxies. This increases the precision of the distance estimate and reduces systematic "Malmquist-like" biases."[44]

(3) "They have a single dominant old stellar population and are free of the obscuring effects of dust."[44]

"The method is, however, potentially sensitive to variations in the physical properties of ellipticals: residual star formation, the unknown distribution of intrinsic shapes, variations in the degree of rotational support, the presence of central velocity dispersion anomalies, and/or the presence of a weak disk."[44]

"Unfortunately there are no nearby examples of luminous elliptical galaxies that can be used for absolute calibration using primary indicators. Furthermore, there is no theoretical basis for an absolute calibration of the Dn-σ relation. Thus, the method is best suited to measuring relative distances that can then be calibrated by some of the other methods discussed elsewhere in this review. For this reason the method has been applied primarily to measure the peculiar motions of galaxies and aggregates of galaxies, rather than to derive absolute distances for individual galaxies."[44]

"The implementation of the Dn-σ method requires the measurement of the central velocity dispersion and luminosity profile (or curve of growth) of each galaxy to sufficient precision that the measurement errors do not contribute to the intrinsic uncertainty of the distance estimator (σ1.2 / Dn). [A] typical scatter of 23% in Dn at fixed σ [occurs]. [The] single measurements of Dn range in precision from 5-12% and for σ from 9-14% (these estimates were made from repeat measurements which were used to improve the precision of their measurements). Single measurements of this quality would contribute significantly to the scatter in Dn in the best observed clusters. A reasonable aim for future work is to determine Dn to 6%, and to 5%, which in the absence of cosmological scatter would generate a distance determination with an uncertainty ≤ 9%."[45]

"If the peculiar motions of aggregates of galaxies (hypothesised to be at the same distance) are to be determined, then a measurement of recession velocity is required in addition to the parameters which enter into the distance estimator. (Of course, this can be obtained from the same observational data that was used to derive the velocity dispersions.) The measured galaxies must then be assigned to groups on the basis of position and recession velocity using an algorithm [...]."[45]

## Hypotheses

1. Many of the standard candles have such a wide range of sizes that distance estimates are at best only good to an order of magnitude.
2. Photons reaching Earth from any source say nothing about its distance.

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