# Standard stellar luminosities; what are typical and limiting accuracies in
the era after Gaia?

Z. Eker,1 F. Soydugan,2,3 S. Bilir,4 and V. Bakış1
1Akdeniz University, Faculty of Sciences, Department of Space Sciences and
Technologies, 07058, Antalya, Turkey
2Department of Physics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart
University, 17100 Çanakkale, Turkey
3Astrophysics Research Center and Ulupınar Observatory, Çanakkale Onsekiz Mart
University, 17100, Çanakkale, Turkey
4Istanbul University, Faculty of Science, Department of Astronomy and Space
Sciences, 34119, Istanbul, Turkey
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Methods of obtaining stellar luminosities ($L$) have been revised and a new
concept, standard stellar luminosity, has been defined. Among the three
methods (direct method from radii and effective temperatures, method using a
mass-luminosity relation (MLR) and method requiring a bolometric correction),
the third method, which uses the unique bolometric correction (BC) of a star
extracted from a flux ratio ($f_{\rm V}/f_{\rm Bol}$) obtained from the
observed spectrum with sufficient spectral coverage and resolution, is
estimated to provide an uncertainty ($\Delta L/L$) typically at a low
percentage, which could be as accurate as 1% perhaps more. The typical and
limiting uncertainties of the predicted $L$ of the three methods were
compared. The secondary methods requiring either a pre-determined non-unique
BC or MLR were found to provide less accurate luminosities than the direct
method, which could provide stellar luminosities with a typical accuracy of
8.2% - 12.2% while its estimated limiting accuracy is 2.5%.

###### keywords:

stars: fundamental parameters – stars:general

††pubyear: 2015††pagerange: Standard stellar luminosities; what are typical
and limiting accuracies in the era after Gaia?–References

##  1 Introduction

Luminosity of a star is not a directly observable parameter. It is, rather, an
empirical parameter to be computed from observable parameters; a radius ($R$)
and an effective temperature ($T_{\rm eff}$). Therefore, the most reliable
stellar luminosities so far are the ones which are calculated directly from
the Stefan-Boltzmann law ($L=4\pi R^{2}\sigma T_{\rm eff}^{4}$) from observed
radii and effective temperatures of Detached Double-lined Eclipsing Binaries
(DDEB) (Andersen, 1991; Torres, Andersen, & Giménez, 2010; Eker et al., 2014,
2015, 2018, 2020) obtained from simultaneous solutions of their radial
velocity and eclipsing light curves and/or analysis of disentangled spectra of
the components (Hadrava, 1995; Bakış et al., 2007). Except for a very limited
number of nearby single stars which have radii available by interferometry and
any kind of observed effective temperatures, the direct method, alas, has a
serious defect that it is not applicable to single stars, visual and
spectroscopic binaries (or multiple systems) because their stellar radii are
not observable directly. Moreover, effective temperatures implied by intrinsic
colors, in most cases, are also not observable due to interstellar reddening.
The direct method even faces difficulties for contact, semi-detached and even
close binaries because of proximity effects which deform star shapes, thus the
Stefan-Boltzmann law is not applicable directly.

In addition to this limited primary source of stellar luminosities (the direct
method), there are two other secondary sources, which indirectly provide
stellar luminosities. They are indirect because both rely upon a prior
relation determined by the most accurate stellar parameters mostly from DDEB.

###  1.1 Mass-Luminosity relations for predicting L

From the Historical point of view, the first of the secondary sources is the
main-sequence mass-luminosity relation ($L\propto M^{\alpha}$). Despite being
applicable to only main-sequence stars, this method has the power to enlarge
the availability of stellar luminosities towards single stars (if their masses
are known or estimated somehow) and visual binaries (or multiple systems) with
orbital parameters. If orbital parameters could be extracted from a visual
orbit, then component masses would be known according to Kepler’s third law.
Note that, unlike a visual binary with an orbital inclination deduced from its
visual orbit, a spectroscopic binary is without an orbital inclination because
it is not possible to deduce the orbital inclination from the observed
spectra. Therefore, spectroscopic binaries do not provide the true masses of
components except for a mass ratio or a mass function, hence this method is
not applicable to spectroscopic binaries unless orbital inclinations are
somehow available.

The main-sequence mass-luminosity relation (MLR) was discovered independently
by Hertzsprung (1923) and Russell, Adams & Joy (1923) empirically in the
middle of the first half of the 20th century. As newer and more accurate data
came along, it has been revised, updated and improved (Eddington, 1926;
McLaughlin, 1927; Kuiper, 1938a; Petrie, 1950a, b; Strand & Hall, 1954; Eggen,
1956; McCluskey & Kondo, 1972; Cester, Ferluga & Boehm, 1983; Griffiths,
Hicks, & Milone, 1988; Demircan & Kahraman, 1991) many times. Early relations
were demonstrated as mass-absolute bolometric magnitude diagrams, some with
the best fitting curve and some without. First, Eggen (1956) attempted to
define the power of mass (alpha) so the relation is expressed as
$L=\mu^{3.1}$, where $\mu$ is defined as $\mu=a^{3}/P^{2}\varpi^{3}$ from
Kepler’s Harmonic Law in which $a$ and $\varpi$ are semimajor axis and
parallax of the double star in units of arcseconds, $P$ is orbital period in
years, and $\mu$ mass of the system in solar masses. While McCluskey & Kondo
(1972) tried to establish the relation as $M=\alpha L^{\beta}$, where $M$ and
$L$ are the masses and luminosities of components in which $\alpha$ and
$\beta$ are the constants determined on various mass-absolute bolometric
magnitude diagrams, Cester et al. (1983), Griffiths et al. (1988) and Demircan
& Kahraman (1991) preferred to study on a mass-luminosity diagram in order to
define unknown constants on the classical form of mass-luminosity ($L\propto
M^{\alpha}$) relation either by fitting a curve to all data or dividing the
mass range into two (low mass stars, high mass stars) or three (high mass,
intermediate mass or solar, low mass) in order to define the inclination of
the linear MLR (power of $M$) and its zero point constant on $\log M-\log L$
diagrams.

Andersen (1991) objected to defining any form of MLR and preferred to display
the $\log M-\log L$ diagram without a curve fit because the scatter from the
curve is not only due to observational errors but also due to abundance and
evolutionary effects. He claimed: “… departures from a unique relation are
real”, so if there is no unique function to represent data, why bother to
define one? Because of this objection, Henry (2004) and Torres et al. (2010)
also displayed their diagrams without a curve. Except for Gorda & Svechnikov
(1998), who preferred the form $M_{\rm Bol}=a+b\log M$, where $M_{\rm Bol}$ is
the absolute bolometric magnitude and $M$ is the mass, and Henry & McCarthy
(1993), who preferred $\log M=aM_{\xi}+b$ for infrared colours, where
$M_{\xi}$ stands for absolute magnitude at the $J$, $H$ and $K$ bands, and
$\log M=aM_{V}^{2}+bM_{V}+c$ for the $V$ band to express various MLR with
unknown coefficients $a$, $b$, and $c$ to be determined by data on the
diagram, and Malkov (2007), who defined MLR and inverse MLR functions, authors
such as Gafeira, Patacas, & Fernandes (2012), Torres et al. (2010), Benedict
et al. (2016), Moya et al. (2018) and recently, Fernandes, Gafeira & Andersen
(2021) deviated from the tradition of defining MLR, which could be used on
both directions ($L$ computed from $M$ or $M$ computed from $L$), with the
idea that the luminosity of solar type single stars could be obtained from
observations with fair accuracy but not the mass (Fernandes et al., 2021).
Therefore, MLR should be established only for estimating the mass of single
stars from other astrophysical stellar parameters such as its luminosity,
metallicity ($Z$) and age, not for estimating luminosities from masses
(Fernandes et al., 2021). Nevertheless, the predicted relation is still called
MLR despite it not being a relation solely between mass and luminosity but
also including metallicity and age as the observable parameters. Devised for
estimating mass rather than luminosity, MLR of this kind is not suitable for
this study.

Classical MLR in the form $L\propto M^{\alpha}$ was appreciated by Ibanoǧlu et
al. (2006) when they were comparing mass-luminosity relations for detached and
semidetached Algols. The tradition of MLR in the form $L\propto M^{\alpha}$
(or reducible to it) has been continued by Eker et al. (2015, 2018). Note that
any curve or a polynomial of any degree fitting data on a $\log M-\log L$
diagram is reducible to the form $L\propto M^{\alpha}$ because the derivative
of the fitting function at a given mass gives the value of alpha. The
advantage of such MLR is not only that it works both ways ($L$ from $M$, or
$M$ from $L$), but also because it permits one to relate typical masses and
luminosities of main sequence stars in general. Since this study is primarily
interested in estimating typical accuracies of mass and luminosities, the six-
piece classical MLRs of Eker et al. (2018), as the most recent determined
MLRs, are more suitable for this study than any of the other MLR forms.

Although $L$ obtained from a classical MLR would be a unique value for a given
mass ($M$), it is akin to the mean value of all luminosities from the Zero Age
Main-Sequence (ZAMS) to Terminal Age Main-Sequence (TAMS) of stars with the
same mass but with different age and various chemical composition. Thus, the
uncertainty of obtained $L$ is expected to be very large for those who are
looking for an accurate $L$ of a star in question. Still, this was the only
method for producing the luminosities of single stars with known masses and
visual binary or multiple systems with visual orbits in earlier times, when
the third source using bolometric corrections (BC) was not available yet or
the BC values were not as accurate as today.

###  1.2 Bolometric Corrections for predicting L

Perhaps the most powerful secondary method for obtaining $L$ is the method
using bolometric correction (BC). It appears to be even more powerful than the
direct method not only because it is applicable to all stars, single or binary
(or multiple), but also because it is more practical and easier to use. The
method can even work with a single observation at a preferred filter, let us
say $V$ filter, which nowadays it could be accurate at a milimag level,
perhaps more accurate, if the star is bright enough, though not a binary or a
multiple system. For binaries and multiple systems, however, the method
requires the relative light contributions of the components. Only
disadvantage, compared to the direct method, is that a trigonometric parallax
($\varpi$) and reddening (or extinction) of the stars must be provided.
Nevertheless, despite these advantages, the method is still secondary because
it does not work if a pre-determined BC value is not available, which could be
read from any of the tabulated BC tables in the literature, or from $BC-T_{\rm
eff}$, $BC-M/M_{\odot}$ (Flower, 1996; Eker et al., 2020, 2021), or similar
relations if available.

Analytical $BC-T_{\rm eff}$ relations, however, are determined only by two
authors (Flower, 1996; Eker et al., 2020), the rest of the available BCs are
all in tabular format (Kuiper, 1938b; Popper, 1959; McDonald & Underhill,
1952; Johnson, 1964, 1966; Heintze, 1973; Code et al., 1976; Malagnini et al.,
1985; Cayrel et al., 1997; Bessell, Castelli & Plez, 1998; Girardi et al.,
2008; Sung et al., 2013; Chen et al., 2019) where the variation of BC values
with the other stellar parameters such as spectral type, intrinsic colour,
luminosity class, metallicity and surface gravity may also be given.

It was Torres (2010) who first noticed inconsistencies in the use of $BC_{\rm
V}$ values that may lead to errors of up to 10% or more in the derived
luminosity equivalent and about 0.1 mag or more uncertainty in the bolometric
magnitudes. According to Torres (2010), the problems arise from the
arbitrariness attributed to the zero point of the BC scale. Recently, Eker et
al. (2021) revised the zero points of the BC scales on the tabulated tables
and confirmed Torres (2010) independently. According to the results of Eker et
al. (2021) there could be up to 0.1 mag systematic shifts of BC values
corresponding up to 10% systematic errors in predicted stellar luminosities,
which are intolerable in the era after Gaia.

In this paper, we must first re-emphasize IAU 2015 General Assembly Resolution
B2, which Eker et al. (2021) relied upon in solving the problems originating
from the arbitrariness attributed to the BC scale. Fixing the zero point of
the BC according to IAU 2015 General Assembly Resolution would actually mean
the standardization of BC. Briefly, the standardization of BC values was a
solution suggested by Eker et al. (2021) to remove uncertainties both on
stellar absolute bolometric magnitudes and predicted stellar luminosities
caused by the arbitrariness attributed to the zero point constants of BC
values appearing in the literature.

Since stellar luminosities are used by Galactic and extragalactic astronomers
to estimate Galactic and extragalactic structures and luminosities, which
could then be used to estimate Galactic and extragalactic distances as well as
the luminous mass contained in galaxies and universe, the standardization
apparently is expected to have a widespread effect on dark matter research,
Hubble law and cosmological models. For this reason, one must be careful about
uncertainties and errors reflected in BC values, which naturally propagate to
stellar luminosities. Not only the luminosities of stars, but also for deep
space and cosmology researches, are affected. This study intends to explain
(1) how these negative propagating effects could be minimized and (2) how to
estimate and compare accuracies of the predicted stellar luminosities by the
three methods, as well as (3) the need to emphasize the differences between
standard and non-standard luminosities.

##  2 Data

Preliminary data of this study originates with to the “Catalog of Stellar
parameters from the Detached Double-Lined Eclipsing Binaries in the Milky Way”
containing 514 stars (257 systems) by Eker et al. (2014). Although the updated
catalog (Eker et al., 2018) was expanded to include 639 stars (318 binaries
and one eclipsing spectroscopic triple), after removing the stars having
errors greater than 15% both on mass and radius, and eliminating the stars
belonging to globular clusters, then finally choosing main-sequence stars with
a mass of $0.179\leq M/M_{\odot}\leq 31$ and metal abundance $0.008\leq Z\leq
0.040$ ranges within the limits of theoretical ZAMS (Zero Age Main Sequence)
and TAMS (Terminal Age Main Sequence) according to PARSEC models (Bressan et
al., 2012), 509 stars were retained by Eker et al. (2018) to study the MLR of
the sample representing the nearby stars in the Galactic disc in the solar
neighborhood.

The most accurate stellar luminosities and propagated uncertainties were
calculated according to the direct method (Method 1) using the most accurate
radii and effective temperatures and the associated observed uncertainties of
the 509 main-sequence stars, which are the “components of DDEB” in the updated
catalog. The six-piece MLRs in the form of $\log L=a\log M+b$ calibrated by
the data on the $\log M-\log L$ diagram covering the mass $0.179\leq
M/M_{\odot}\leq 31$ and metal abundance $0.008\leq Z\leq 0.040$ ranges for the
main-sequence stars in the Galactic disc in the solar neighborhood by Eker et
al. (2018) are taken for granted in this study.

##  3 Relative accuracies According to three methods

###  3.1 Relative uncertainty of L according to Method 1

Uncertainty of luminosity by the direct method using the Stefan-Boltzmann law
($L=4\pi R^{2}\sigma T_{\rm eff}^{4}$) could be calculated by:

${\frac{\Delta L}{L}}=\sqrt{\left(2\frac{\Delta
R}{R}\right)^{2}+\left(4\frac{\Delta T_{\rm eff}}{T_{\rm eff}}\right)^{2}.}$
(1)

where, $\Delta R/R$ and $\Delta T_{\rm eff}/T_{\rm eff}$ are relative
observational random errors of a star’s radius and effective temperature,
which usually comes from simultaneous solutions of the light and radial
velocity curves of DDEB and spectral analysis of the disentangling spectra of
the system’s components.

###  3.2 Relative uncertainty of L according to Method 2

In order to estimate relative uncertainty of $L$ using the method of error
propagation, the form of adopted MLRs ($L\propto M^{\alpha}$) indicates

${\frac{\Delta L}{L}}=\alpha\frac{\Delta M}{M}$ (2)

where $\Delta M/M$ is the relative uncertainty of the observed mass of a star,
$\alpha$ is the power of $M$ and $\Delta L/L$ is the relative uncertainty of
the predicted luminosity. According to Eker et al. (2015, 2018), the equation
(2) is invalid because the dispersions on the $\log M-\log L$ diagram are not
only due to observational uncertainties of $M$, but also due to the age and
chemical composition differences of the stars in the sample (Andersen, 1991;
Torres et al., 2010; Eker et al., 2015, 2018). It is better to use:

${\frac{\Delta L}{L}}=\frac{SD}{0.4343}$ (3)

where $SD$ is the standard deviation of data from the MLR function, which
should be chosen according to the mass of the star in question. There are six
MLR functions with standard deviations and inclinations, already computed by
Eker et al. (2018), which will be used in this study. Only if the cases are
$\alpha\frac{\Delta M}{M}>\frac{SD}{0.4343}$ would equation (2) then be valid.
Because the typical relative uncertainties of $M$ are in the order of 1-2%
(Eker et al., 2014), equation (2) is not valid. It would be valid for stars
with relative uncertainties bigger than about 6%, for low mass stars
($M<2.4M_{\odot}$), and relative uncertainties bigger than about 10% for high
mass stars ($M>2.4M_{\odot}$) (Eker et al., 2015).

###  3.3 Relative uncertainty of L according to Method 3

The same data set of 509 main-sequence stars used for method 2 above is also
used for method 3 here, which uses a pre-determined bolometric correction (BC)
for predicting the luminosity of a star. Unfortunately, many of the binary
systems containing the 509 stars as components of DDEB had to be eliminated
because some of the binaries have not been observed in standard $V$ magnitudes
or otherwise, the light contributions of components in the $V$ band could not
be achieved, or a reliable trigonometric parallax for the star ($\varpi$) did
not exist, or reliable interstellar reddening could not be found.

Eker et al. (2020) could find only 206 binaries which have at least one
component in the main-sequence (194 systems with components, eight systems
with primaries, four systems with secondaries on the main-sequence), leaving a
total of 400 main-sequence stars which are eligible to compute bolometric
correction (BC) coefficients in the $V$ band. A standard $BC_{\rm V}-T_{\rm
eff}$ curve (a fourth degree polynomial) which is valid in the range $3100\leq
T_{\rm eff}\leq 36000$ K was calibrated and the coefficients of the
polynomial, errors of the coefficients, standard deviation ($SD=0.215$) and
correlation coefficient ($R^{2}=0.941$) have already been announced by Eker et
al. (2020). Containing the necessary statistics Table 5 of Eker et al. (2020)
was adopted for this study as the basic data to compute the luminosity of a
star ($L$) according to Method 3.

In this method, the $L$ of a star could be achieved according to the following
relation:

$M_{\rm Bol}=-2.5\log L+C_{\rm Bol},$ (4)

where $M_{\rm Bol}$ is the absolute bolometric magnitude of the star and
$C_{\rm Bol}=71.197~{}425~{}...$ or if $L$ is in SI units, and/or $C_{\rm
Bol}=88.697~{}425~{}...$ or if $L$ is in cgs units (see IAU 2015 General
Assembly Resolution B2 and Eker et al., 2021). The only requirement is that
the $M_{\rm Bol}$ of the star must be known beforehand then $L$ can be
extracted. $M_{\rm Bol}$ is available according to following relation:

$M_{\rm Bol}=M_{\rm Filter}+BC_{\rm Filter}=M_{\rm V}+BC_{\rm V},$ (5)

where $M_{\rm V}$ and $BC_{\rm V}$ are the absolute visual magnitude and
associated bolometric correction for the star in question. Notice that the
absolute bolometric magnitude of a star is independent of the filter used in
observations. Since $V$ filter observations are the oldest and most available,
the $V$ filter was chosen here to symbolize all BC values of many photometric
bands. If the $BC_{\rm V}$ value of the star is available, then an additional
step must be taken to obtain $M_{\rm V}$ directly from observable parameters
according to the following relation,

$M_{\rm V}=V+5\log\varpi+5-A_{\rm V},$ (6)

where $V$ is the apparent visual magnitude, $\varpi$ is the trigonometric
parallax of the star, which is nowadays available up to 21st magnitude (Gaia
Collaboration et al., 2020) and $A_{\rm V}$ is the extinction in the $V$ band,
which can be ignorable if the star is in the local bubble (Leroy, 1993;
Lallement et al., 2019) or could be estimated using galactic dust maps (e.g.
Schlafly & Finkbeiner, 2011; Green et al., 2019).

It is clear in this method that the only uncertainty to propagate up to $L$
comes from $M_{\rm Bol}$. It can be considered that a well-defined constant
(IAU 2015 GAR B2) $C_{\rm Bol}$ does not make any contribution, thus:

${\frac{\Delta L}{L}}={\frac{\Delta M_{\rm Bol}}{2.5\log e}=0.921\times\Delta
M_{\rm Bol}}.$ (7)

On the other hand, equation (5) indicates

$\Delta M_{\rm Bol}=\sqrt{\Delta M_{\rm V}^{2}+\Delta BC_{\rm V}^{2}+ZPE_{\rm
V}^{2}}.$ (8)

which means that there are three possible error contributions to $\Delta
M_{\rm Bol}$. These are: 1) random observational errors associated with the
absolute visual magnitude ($\Delta M_{\rm V}$), 2) the error of the BC value
itself ($\Delta BC_{\rm V}$), and 3) the zero point uncertainty of the BC
scale ($ZPE_{\rm V}$). This study primarily aims to estimate the amount of the
zero point error of the BC scale if the BC value comes from non-standard
bolometric corrections, which are tabulated or any other source. Consequently,
we may assume, just for now, that the first two contributions are zero. Then,
the equation (7) changes to:

${\frac{\Delta L}{L}}=0.921\times ZPE_{\rm V}.$ (9)

from which one could obtain relative uncertainty of $L$ caused by the
uncertainty of the zero point of the BC scale alone; that is, if the absolute
visual magnitude and bolometric correction are errorless. Unfortunately, this
is not the case nowadays because there are many BC sources giving non-standard
BC values. If one of them is used, it is better not to omit $ZPE_{V}$ in
equation (8). Only if one uses a standard BC in equation (5), could $ZPE_{V}$
in equation (8) be omitted, which corresponds to $ZPE_{V}$ being equal to
zero, then, there is no need for equation (9). Now the question is how
standard and non-standard BC values can be recognized. This is explained in
the next section.

##  4 Definition and Recognition of standard stellar luminosities

###  4.1 Definition of standard luminosities

It is assured that the fixed zero point constant in equation (4) does not
cause any uncertainty. Thus, the relative uncertainty $\Delta L/L$ in equation
(7), should not include a term implying an uncertainty coming from $C_{\rm
Bol}$ because the derivative of a constant is zero by definition.

If and only if the $BC_{\rm V}$ value used in equation (5) comes from a
standard source of BC coefficients, it is not necessary to include $ZPE_{V}$
in equation (8). Then, it converts to

$\Delta M_{\rm Bol}=\sqrt{\Delta M_{\rm V}^{2}+\Delta BC_{\rm V}^{2}}.$ (10)

which means there are only two error contributions to $\Delta L/L$; the
observational random errors of absolute visual magnitude and the error of the
BC value itself. Consequently, the $L$ obtained by equation (4), righteously,
would be called standard luminosity. The concept of standard BC was originally
is suggested by Eker et al. (2021).

The term “standard” or “non-standard” before the word “luminosity”, would be
meaningful if the $L$ of any star is calculated by the method requiring a
bolometric correction coefficient. If a standard BC which is already defined
by Eker et al. (2021) is used, the computed $L$ is called standard. To the
contrary, a non-standard BC value makes the computed $L$ non-standard.
Standardization of BC values, therefore, is equivalent to the standardization
of stellar luminosities. According to the definition of Eker et al. (2021),
the tabulated values of $BC_{\rm V}$ could be considered as standard sources
for the $BC_{\rm V}$ values if the nominal value of solar absolute bolometric
magnitude $M_{\rm Bol,\odot}=4.74$ mag, and the nominal solar luminosity
$L_{\odot}=3.828\times 10^{26}$ W as used in:

$M_{\rm Bol}=M_{\rm Bol,\odot}-2.5\log{\frac{L}{L_{\odot}}},$ (11)

when computing $M_{\rm Bol}$, where $L$ requires observational $R$ and $T_{\rm
eff}$ from DDEB stars (Eker et al., 2014, 2018). Then, a standard $BC_{\rm V}$
is obtained according to the basic definition of bolometric correction
$BC_{\rm V}=M_{\rm Bol}-M_{\rm V}$.

###  4.2 Recognizing Non-standard Luminosities

The equations (4) and (11) are both valid for calculating $L$ of a star from
its absolute bolometric magnitude. The validity is assured by:

$C_{\rm Bol}=M_{\rm Bol,\odot}+2.5\log L_{\odot},$ (12)

according to Eker et al. (2021). Since equation (4) and the nominal values
$M_{\rm Bol,\odot}=4.74$ mag, and $L_{\odot}=3.828\times 10^{26}$ W were
introduced only recently, any other non-standard values of $M_{\rm Bol,\odot}$
and $L_{\odot}$ which were used in computing a BC source (tabulated BC values
or $BC_{\rm V}-T_{\rm eff}$ relation) most likely would not produce a $C_{\rm
Bol}=71.197~{}425~{}...$ mag if $L$ is in SI units, and/or $C_{\rm
Bol}=88.697~{}425~{}...$ mag if $L$ is in cgs units according to equation
(12). This is the first and clear indication that there is a zero point error
in the BC value used, which certifies that it is not a standard BC .

It is unlikely but still under a possibility that the non-standard $M_{\rm
Bol,\odot}$ and $L_{\odot}$ values would produce a $C_{\rm
Bol}=71.197~{}425~{}...$ mag if $L$ is in SI units, and/or $C_{\rm
Bol}=88.697~{}425~{}...$ mag if $L$ is in cgs units according to equation
(12). This mathematical possibility is unavoidable because there could be an
infinite number of $M_{\rm Bol,\odot}$, $L_{\odot}$ pairs to produce the same
$C_{\rm Bol}$. This is the second type (an unseen indication) of a zero point
error in BC values used. How to treat these two different types of zero point
errors will be discussed later in the Discussion section.

For now, let us review some of the contemporary $BC_{\rm V}$ sources in the
near past with different $C_{\rm Bol}$ and corresponding non-standard $M_{\rm
Bol,\odot}$ and $L_{\odot}$ values. Bessell et al. (1998) gave a table for
comparing the estimated $BC_{\rm V}$ of the Sun by various authors, who
preferred adopting $M_{\rm Bol,\odot}$ rather than adopting $BC_{\rm V}$ of
the Sun. Relying on the observed apparent visual magnitude of the Sun (-26.76
mag Bessell et al., 1998; Torres, 2010), (-26.74 mag Allen, 1976; Schmidt-
Kaler, 1982), (-26.79 mag Durrant, 1981) and adopting either one of the
quantities $M_{\rm Bol,\odot}$ or $BC_{\rm V,\odot}$ was inevitable in those
years because the zero point of the $BC_{\rm V}$ scale was not yet fixed, thus
it was assumed arbitrary. Adopting one of these quantities meant defining a
zero point for tabulated $BC_{\rm V}$ values. Here we have reconstructed a
similar table (Table 1), which enables us to compare various $C_{\rm Bol}$
values as well as the $M_{\rm Bol,\odot}$ and $L_{\odot}$ values defining it.

Table 1: Comparing $BC_{\rm V}$ values attributed to the Sun and zero point constants of bolometric magnitude scale ($C_{\rm Bol}$) and corresponding $M_{\rm Bol,\odot}$ and $L_{\odot}$ according to various authors in the near past. | $V$ | $M_{\rm V}$ | $M_{\rm Bol}$ | $BC_{\rm v}$ | $f(\times 10^{6})$ | $F(\times 10^{10})$ | $L(\times 10^{33})$ | $C_{\rm Bol}$ | Reference
---|---|---|---|---|---|---|---|---|---
Order | (mag) | (mag) | (mag) | (mag) | (erg cm${}^{-2}\rm s^{-1}$) | (erg cm${}^{-2}\rm s^{-1}$) | (erg s-1) | (mag) |
1 | -26.74 | 4.83 | 4.75 | -0.08 | 1.36 | 6.284 | 3.826 | 88.70686 | Allen (1976)
2 | -26.79 | 4.87 | 4.74 | -0.13 | 1.37 | 6.329 | 3.853 | 88.70450 | Durrant (1981)
3 | -26.74 | 4.83 | 4.64 | -0.19 | 1.37 | 6.330 | 3.850 | 88.60365 | Schmidt-Kaler (1982)
4 | -26.76 | 4.81 | 4.74 | -0.07 | 1.371 | 6.334 | 3.856 | 88.70534 | Bessell et al. (1998)
5 | -26.75 | 4.82 | 4.74 | -0.08 | 1.367 | 6.322 | 3.845 | 88.70224 | Cox (2000)
6 | -26.76 | 4.81 | 4.75 | -0.06 | 1.368 | 6.324 | 3.846 | 88.71252 | Torres (2010)
7 | -26.76 | 4.81 | 4.75 | -0.06 | 1.361 | 6.294 | 3.828 | 88.70729 | Casagrande & VandenBerg (2018)
8 | -26.76 | 4.81 | 4.74 | -0.07 | 1.361 | 6.294 | 3.828 | 88.69743 | Eker et al. (2020)
9 | -26.76 | 4.81 | 4.645 | -0.165 | 1.361 | 6.294 | 3.828 | 88.60229 | This study

The columns of Table 1 are self-explanatory. The sequence number, apparent
visual (a measured quantity), absolute visual and the adopted absolute
bolometric magnitudes of the Sun are given in the first four columns. The
$BC_{\rm V}$ of the Sun as the difference between absolute bolometric and
visual magnitudes is in the fifth column. Corresponding solar fluxes just
outside the Earth’s atmosphere and on the solar surface are given in columns
six and seven, while the corresponding solar luminosity is in column eight.
Column nine displays the standard (row number 8) and other figures non-
standard zero point constants ($C_{\rm Bol}$) according to various $BC_{\rm
V}$ sources. The references are in column ten in chronological order.

Figure 1 compares the nominated bolometric zero point constant and the nominal
$M_{\rm Bol,\odot}$ or $BC_{\rm V,\odot}$ values of the Sun to the
corresponding values of non-standard sources. A horizontal solid line on
Figure 1a marks the nominated zero point constant ($C_{\rm
Bol}=88.697~{}425~{}~{}...$ mag) of IAU 2015 GAR B2.

Figure 1: Comparing nominated bolometric zero point constant and nominal
$M_{\rm Bol,\odot}$ and $L_{\odot}$ values to the corresponding values of non-
standard sources: 1) Allen (1976), 2) Durrant (1981), 3) Schmidt-Kaler (1982),
4) Bessell et al. (1998), 5) Cox (2000), 6) Torres (2010), 7) Casagrande &
VandenBerg (2018), 8) Eker et al. (2020), 9) In order to shift $BC_{\rm
V}-T_{\rm eff}$ curve to make all $BC_{\rm V}\leq 0$ mag (This study).

Except for (3) (Schmidt-Kaler, 1982) and (9), which is the test point of this
study on Figure 1a with a special $M_{\rm Bol,\odot}$ (4.645 mag) to make all
BC values less than zero, the other $C_{\rm Bol}$ values in the uppermost box
(a) lie close to the horizontal solid line with various values which are all
greater than $C_{\rm Bol}=88.697~{}425~{}...$ mag (see Table 1). The largest
$C_{\rm Bol}$ (88.712523) is with Torres (2010). According to Table 1 and
Figure 1a, the difference between the largest and smallest $C_{\rm Bol}$ is
0.110 mag, which is equivalent to 10.13% uncertainty on the predicted stellar
luminosities even if $M_{\rm V}$ and $BC_{\rm V}$ are errorless. The least-
deviated $C_{\rm Bol}$ is associated with Cox (2000), which has a $C_{\rm
Bol}$ value 0.005 mag bigger than the nominal value, corresponding to a 0.46%
error on $L$.

Figure 1b compares the nominal value of $L_{\odot}$ (horizontal solid line) to
the adopted $L_{\odot}$ values (data) of the other $BC_{\rm V}$ sources. The
two horizontal dashed lines marks the random observational error limits
associated with the nominal solar luminosity ($L_{\odot}=3.8275\pm
0.0014\times 10^{33}$ erg s-1) (IAU General Assembly Resolution B3). This
means that in an ideal case (no error contributions from $M_{\rm V}$ and
$BC_{\rm V}$ and $ZPE$), a standard stellar luminosity could be as accurate as
0.036% (4 out of 10000).

Please notice that, using a non-standard $L_{\odot}$ in equations (4) and (11)
would produce a systematic error on $L$ even if $M_{\rm V}$ and $BC_{\rm V}$
are errorless and no zero point error exists. The largest of such systematic
errors appear as an overestimation of stellar luminosities of about 0.74% if
one uses the $BC_{\rm V}$ values of Bessell et al. (1998), who has the biggest
$L_{\odot}$ in Table 1. The zero point errors caused only by non-standard
$L_{\odot}$ are apparently are less than 0.74% according to Table 1 and Figure
1b. The value of $L_{\odot}$ by Cox (2000) in Table 1 implies a 0.46% zero
point error on Figure 1b, which confirms the same amount of error caused by
his non-standard $C_{\rm Bol}$ according to Figure 1a. According to Figure 1c,
because Cox (2000) uses $M_{\rm Bol,\odot}=4.74$ mag, which is the nominal
value, one assumes no zero point error contribution from it. Despite an error
of approximately about 0.46% error on $L$ according to Table 1 and Figure 1,
the BC values of Cox (2000) all appear to have be shifted 0.095 mag towards
smaller (more negative) values compared to the BC values of Eker et al.
(2020).

The 0.095 mag systematic shift, creating a difference between the $BC_{\rm V}$
(max) of Eker et al. (2020) and Cox (2000), on the other hand, means an error
of about 8.85% on stellar luminosities. In other words, the $BC_{\rm V}$
values of Cox (2000) dominates the zero point errors and propagate within
$BC_{\rm V}$ systematically. Thus, anyone who uses $BC_{\rm V}$ values of Cox
(2000) will overestimate the stellar luminosities by about 9%, without even
including the observational errors and the error of the $BC_{\rm V}$ value
used.

Figure 1c compares the nominal value 4.74 mag (horizontal solid line) to the
other adopted $M_{\rm Bol,\odot}$ (data) values of the other $BC_{\rm V}$
sources. The nominal value is that used by Durrant (1981), Bessell et al.
(1998), Cox (2000) and Eker et al. (2020). The non-standard value 4.75 mag is
that used by Allen (1976), Torres (2010) and Casagrande & VandenBerg (2018).
Another non-standard value $M_{\rm Bol,\odot}=4.64$ mag is used by Schmidt-
Kaler (1982), which is very close to our non-standard trial value of this
study (Table 1, Figure 1c).

It can be concluded here that a non-standard stellar $L$ is easy to recognize.
Standard $L$ could be obtained only if standard bolometric correction
coefficients were used. One can recognize non-standard bolometric corrections
by checking whether nominal $M_{\rm Bol,\odot}$ and nominal $L_{\odot}$ were
used or not. Although the zero point of absolute bolometric magnitudes is
given as $C_{\rm Bol}=M_{\rm Bol,\odot}+2.5\log L_{\odot}$, the nominal
$C_{\rm Bol}$ does not necessarily guarantee standardization of the pre-
computed $BC_{\rm V}$ values. On the contrary, a non-standard $C_{\rm Bol}$
implies non-standard $BC_{\rm V}$ values. $BC_{\rm V}$ values could be
considered standard if and only if nominal $M_{\rm Bol,\odot}$ and nominal
$L_{\odot}$ are used and if the zero point of the BC scale ($C_{\rm 2}$) is
calculated as Eker et al. (2021) describes. Authors such as Cox (2000), who
assume it is arbitrary, thus $C_{\rm 2}=0$, in order to make all $BC_{\rm V}$
less than zero (see Eker et al., 2021, and references therein). This is the
case when the zero point error of the BC scale cannot be estimated from pre-
assumed $M_{\rm Bol,\odot}$ and $L_{\odot}$ but shows itself directly on the
$BC_{\rm V}$ value itself.

##  5 Discussion

Choosing a non-standard $L_{\odot}$ from Table 1 contributes very little (less
than 1%) to the uncertainty of a computed $L$. Therefore, the largest error
contribution, definitely, comes from the choice of $M_{\rm Bol,\odot}$. This
is because there are an infinite number of $M_{\rm Bol}$ and $M_{\rm
Bol,\odot}$ pairs to indicate a single value of $L/L_{\odot}$ according to
equation (11). On the process of computing $BC_{\rm V}$ values, $L/L_{\odot}$
is a quantity most probably coming from DDEB, thus it is a fixed value. Since
$BC_{\rm V}=M_{\rm Bol}-M_{\rm V}$, where $M_{\rm V}$ is also a fixed value.
Therefore, pre-computed $BC_{\rm V}$ values are effected directly by the
choice of $M_{\rm Bol,\odot}$ with the classical method using equation (11).
As in the case of Casagrande & VandenBerg (2018), who preferred to use $M_{\rm
Bol,\odot}=4.75$ rather than the nominal 4.74 mag, all pre computed $BC_{\rm
V}$ values would be 0.01 mag smaller compared to the standard $BC_{\rm V}$.
Although Casagrande & VandenBerg (2018) do not state clearly which $L_{\odot}$
value used, we have assumed they are using the nominal $L_{\odot}$ since they
cite IAU 2015 GAR B3 for the value of $L_{\odot}$ used in their equation (3)
for obtaining the bolometric flux received from a star.

In another aspect, $M_{\rm Bol,\odot}$ acts as the arbitrary zero point for
bolometric magnitudes, as Casagrande & VandenBerg (2018) propose that “any
value is equally legitimate on the condition that once chosen, all bolometric
corrections are scale accordingly”. That is, different authors using different
$M_{\rm Bol,\odot}$ end up calculating different $BC_{\rm V}$ values for the
same star. Because $BC_{\rm V}=M_{\rm Bol}-M_{\rm V}$, the zero point problem
shows itself in the produced $BC_{\rm V}$ values. Using the nominal values of
$M_{\rm Bol,\odot}$ and $L_{\odot}$ Eker et al. (2020) produced a standard
$BC_{\rm V}-T_{\rm eff}$ relation, which has a maximum $BC_{\rm V}$ value of
$BC_{\rm V}$ (max) = 0.095 mag at $T_{\rm eff}=6897$ K. For this study we have
searched for the value of $M_{\rm Bol,\odot}$, without changing the nominal
value of $L_{\odot}$, that would obtain a $BC_{\rm V}-T_{\rm eff}$ relation
with $BC_{\rm V}$ (max) = 0.00. We concluded that the answer is $M_{\rm
Bol,\odot}=4.645$ mag; then, all computed BC values will be reduced (more
negative) to be less than zero, as listed in Cox (2000). However, such a
relation is cannot be definitely considered standard.

The problem with the $BC_{\rm V}$ values of Cox (2000) is not the same as
adopting $M_{\rm Bol,\odot}=4.645$ mag in order to make all $BC_{\rm V}$
values negative. His $BC_{\rm V}$ values are all negative, despite his use of
$M_{\rm Bol,\odot}=4.74$ mag. In fact, his $C_{\rm Bol}$ value is the nearest
$C_{\rm Bol}$ among the other sources, which wrongly imply that if one uses a
$BC_{\rm V}$ value of Cox (2000), he/she would obtain a very accurate
luminosity (0.46%) despite, in reality, the computed $L$ having a 9%
systematic zero point error. Most probably, the $BC_{\rm V}$ values of Cox
(2000) were calculated using the following definition of $BC_{\rm V}$:

$BC_{\rm V}=2.5\log\frac{f_{\rm V}}{f_{\rm Bol}}+(C_{\rm Bol}-C_{\rm
V})=2.5\log\left(\frac{\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm
d\lambda}{\int_{0}^{\infty}\\!f_{\lambda}\rm d\lambda}\right)+C_{2}.$ (13)

where $C_{2}$, the zero point constant of the BC scale, was assumed arbitrary,
thus, $C_{2}=0$ value was taken arbitrarily. If $C_{2}$ is assumed zero, all
$BC_{\rm V}$ values becomes unquestionably be less than zero. This is because
visual flux ($f_{\rm V}$) is never zero but less than the bolometric flux
($f_{\rm Bol}$). The logarithm of numbers between zero and one ($f_{\rm
V}/f_{\rm Bol}$) is always negative, which requires a positive $C_{2}$
otherwise (if $C_{2}=0$ or $C_{2}<0$) the BC scale does not have a zero point;
a negative number plus zero or adding two negative numbers does not produce a
number zero. The BC producers, who assumed that the zero point constant of the
BC scale is arbitrary, believed that they had the right to make it zero. This
way, they unknowingly carried the zero point error into the BC value itself.

Obviously there are two approaches to remove the zero point errors. The first
approach is that suggested by Casagrande & VandenBerg (2018) and Torres
(2010). This entails being cautious when calculating the $M_{\rm Bol}$ of a
star from its $M_{\rm V}$ and $BC_{\rm V}$. Before using the $BC_{\rm V}$ on
the formula $M_{\rm Bol}=M_{\rm V}+BC_{\rm V}$, check it out and first learn
which $M_{\rm Bol,\odot}$ and $L_{\odot}$ values were used before producing
the tabulated $BC_{\rm V}$ values or $BC_{\rm V}-T_{\rm eff}$ relation from
which the $BC_{\rm V}$ value was taken. Then, it is safe to proceed in
calculating $M_{\rm Bol}=M_{\rm V}+BC_{\rm V}$ in the first step. However, do
not use equation (4); instead, use equation (11) when calculating $L$. Do not
forget to use the same $M_{\rm Bol,\odot}$ and $L_{\odot}$ values, which you
have searched for in order to ensure that they are consistent with the
$BC_{\rm V}$ value you are using.

The second approach is suggested in this study. Do not use any value of
$BC_{\rm V}$ haphazardly. Use only standard $BC_{\rm V}$ values from standard
sources. You can use either one of the equations (4) or (11). It does not
matter which, since both are valid for producing the standard $L$ of stars.

Notice that, the first approach fails to produce a standard $L$ if one uses
$BC_{\rm V}$ values from the sources such as Cox (2000) which may appear to be
using nominal $M_{\rm Bol}$ and $L_{\odot}$ but their $BC_{\rm V}$ values are
not standard because they were produced by assuming $C_{2}=0$. Concentrating
only on the difference between the $M_{\rm Bol}$ of a star and $M_{\rm Bol}$
of the Sun, the first approach has no answer as to which $L_{\odot}$ was used
if the stellar $L$ is taken from published papers where the published $L$ are
usually expressed in solar units.

There are no such problems in the second approach.

###  5.1 Typical Accuracy of L in Method 1 (direct method)

The peak of the histogram distribution of relative radius errors of DDEB is 2%
according to Eker et al. (2014). For today’s accuracy we may take it as 1%,
since many newly-published papers give stellar radii even more accurate than
$\sim 1\%$. According to Eker et al. (2015), typical temperature accuracy is
2-3%. The acceptable stellar effective temperature uncertainty for single
stars in general is 1-2.5% according to Masana, Jordi & Ribas (2006). Since
the direct method uses the effective temperatures of DDEB, we may find a
typical temperature uncertainty of 2-3%. Consequently, equation (1) indicates
that the typical uncertainty range of stellar $L$ is 8.2% - 12.2%. On the more
accurate side, there could be stars like the primary component of V505 Per
having $\Delta R/R=1.09\%$ and $\Delta T_{\rm eff}/T_{\rm eff}=0.32\%$ with
corresponding $\Delta L/L=2.53\%$ (Tomasella et al., 2008). That is, the
accuracy of a few percentage levels has been already achieved by the direct
method.

###  5.2 Typical Accuracy of L in Method 2 (using MLR)

Among the three methods, the method using a MLR is the least accurate. This is
because the relative uncertainty of $L(\Delta L/L)$ is determined by the
standard deviation ($SD$) of stellar luminosities on a $\log M-\log L$ diagram
according to equation (3), where $SD$ could be different at different mass
ranges (Eker et al., 2015, 2018). The most accurate mass range, which was
named the ultra-low mass domain covering stellar masses in the range
$0.179<M/M_{\odot}<0.45$, has $SD=0.076$ mag which corresponds to $\Delta
L/L=17.5\%$. The most dispersed range, which is called the high mass domain
covering stellar masses in the range $2.4<M/M_{\odot}\leq 7$, has $SD=0.165$
mag which corresponds to $\Delta L/L=37.99\%$ (Eker et al., 2018).

###  5.3 Typical Accuracy of L in Method 3

####  5.3.1 Typical Accuracy of L using a standard BC

Standard stellar luminosities are those calculated by the method using a pre-
determined standard $BC_{\rm V}$. The most important problem with this method
is that the systematic zero point errors of non-standard $BC_{\rm V}$ could be
as big as 0.11 mag, corresponding to 10.13% errors in the predicted stellar
luminosities. Such systematic errors could be removed if and only if one uses
standard $BC_{\rm V}$ sources. To make the comparison meaningful, here we
assume the zero point error has already been removed or taken to be zero.

A typical accuracy of standard $L$, therefore, could be calculated by
equations (7) and (8) but taking $ZPE_{\rm V}$ as zero in the equation (8).

There could be three contributions to the uncertainty of $\Delta M_{V}$
according to equation (6), Eker et al. (2020) gives:

$\Delta M_{V}=\sqrt{\left(\Delta m_{V}\right)^{2}+\left(5\log
e\frac{\sigma_{\varpi}}{\varpi}\right)^{2}+\left(\Delta A_{V}\right)^{2},}$
(14)

where, the first term in the square root stands for the error contribution
from the apparent brightness, the second term represents the contribution from
stellar parallaxes, and the last term is for the contribution from
interstellar extinctions. Nowadays, apparent magnitude uncertainties are in
the order of milli magnitudes. If we assume extinction errors ($\Delta A_{\rm
V}$) are about 0.01 mag, definitely the parallax errors would definitely
dominate the others; If there is a 10% error on the parallax, $\Delta M_{\rm
V}=0.217$ mag. For stars with a 5% error in their parallaxes, $\Delta M_{\rm
V}=0.109$ mag. Histogram distribution of the parallaxes for 206 DDEB (400
stars), from which the standard $BC_{\rm V}$ was extracted (Eker et al.,
2020), has a peak (median) of 2%. If we take this as a typical parallax error,
then typically $\Delta M_{\rm V}=0.044$ mag.

The typical standard error of a $BC_{\rm V}$ value, if it comes from a
standard $BC_{\rm V}-T_{\rm eff}$ curve, would be in the orders of 0.011 mag
(Table 2) for the range of temperatures (3000-36000 K) considered for main-
sequence stars by Eker et al. (2020). The re-arranged data indicates a
standard error of $BC_{\rm V}-T_{\rm eff}$ being reduced to 0.009 mag for
medium temperatures (5000-10000 K) while it is 0.59 mag for lower
temperatures, and 0.028 mag for the higher temperatures. Inserting typical
$\Delta M_{\rm V}=0.044$ mag and a typical standard error of a BC (0.009 mag)
into equation (10), a typical $\Delta M_{\rm Bol}=0.045$ mag is obtained for
the middle temperatures. Inserting this into equation (7), a typical $\Delta
L/L$ becomes 4.14%.

Table 2: Standard deviation of data from $BC_{\rm V}-T_{\rm eff}$ curve of Eker et al. (2020) and standard errors of $BC_{\rm V}$ values. $T_{\rm eff}$ range (K) | $N$ | $SD$ | $SD/\sqrt{N}$
---|---|---|---
3000-5000 | 35 | 0.349 | 0.059
5000-10000 | 261 | 0.142 | 0.009
10000-36000 | 104 | 0.290 | 0.028
3000-36000 | 400 | 0.215 | 0.011

Now, let us assume the extreme case of a star (CM Dra) with a relative
parallax error of 0.050% (Gaia Collaboration et al., 2020). We may assume no
extinction because it is only 14.86 pc away according to Gaia eDR3 data. Then,
the milli-magnitude accuracy of its apparent brightness would imply $\Delta
M_{\rm V}=0.0019$ mag. Assuming its $BC_{\rm V}$ has a standard error of 0.011
mag ($4^{\rm th}$ line in Table 2), its standard luminosity would have an
uncertainty of 1.03%. Accuracy in standard luminosity even increases to 0.82%
for the middle temperatures.

Here, we run into an unexpected case: the secondary method, using a BC,
provides more accurate stellar luminosities than the direct method. It is too
good to be true. Apparently, the problem must be taking the standard error of
a standard $BC_{\rm V}-T_{\rm eff}$ curve as the standard error of a the BC
value before computing $\Delta M_{\rm Bol}$.

As Eker et al. (2021) stated, a standard $BC_{\rm V}-T_{\rm eff}$ curve
obtained from $BC_{\rm V}$ and $T_{\rm eff}$ values is similar to mass-
luminosity relations obtained from masses and luminosities, but not like the
Planck curve representing the spectral energy distribution (SED) of a star.
Thus, the standard error of a mass-luminosity relation (curve) cannot be used
as the standard error of an $L$ for a given $M$. This is because the
scattering of data from the relation is not only due to observational errors
but also due to the different ages and chemical compositions of the stars on
the $M-L$ diagram. Therefore, $\Delta L/L$ should be calculated directly from
the standard deviations, as described in Section 2.2

Similarly, one must not use standard errors (column 3 of Table 2) but use the
standard deviations (column 2 of Table 2) when computing typical errors of
$M_{\rm Bol}$. If this is done, 0.215 mag standard deviation for the total
range indicates $\Delta L/L=0.198$, and 0.142 mag standard deviation for the
middle temperatures indicates $\Delta L/L=0.131$ for the star CM Dra which has
$\Delta M_{\rm V}=0.0019$ mag. For a typical $\Delta M_{\rm V}=0.044$ mag,
$\Delta L/L$ becomes 20.2% for all main-sequence temperatures in general, and
13.7% for the middle temperature only. Therefore, the indirect method of using
standard BC when computing standard stellar luminosities could be considered
better than using MLR, but not as good as the direct method.

####  5.3.2 Typical Accuracy of $L$ using a Unique BC

Another advantage of the method of computing the standard $L$ of a star using
its BC is when one does not need a pre-computed $BC_{\rm V}-T_{\rm eff}$ curve
or a table giving standard BC values. The unique BC of a star may be obtained
directly from its SED according to equation (13). Using this equation,
however, requires spectroscopic observation of stars covering a sufficient
spectral range at least more than the photometric filter which is used in
determining its absolute magnitude in equation (6). Interstellar reddening is
still there to spoil SED, but restoring of the SED is always possible as done
by Stassun & Torres (2016). This may even not be necessary if the star is in
the local bubble where interstellar extinction could be ignored.

Assuming that $C_{2}$ is a well-defined quantity (Eker et al., 2021), the
error of BC in this method depends on how accurate $f_{\rm V}/f_{\rm Bol}$
could be determined. If the quantity $f_{\rm V}/f_{\rm Bol}$ is determined in
a few percentage level, since there is no error contribution from $C_{2}$,
then according to equation (5) the accuracy of $M_{\rm Bol}$ depends on the
accuracy of two quantities, $M_{\rm V}$ and $BC_{\rm V}$, which means that the
$M_{\rm Bol}$ of a star could be obtained at a low percentage. For example,
using typical $\Delta M_{\rm V}=0.044$ mag as pointed out in Section 5.3.1,
assuming $\Delta BC_{\rm V}=0.03$ mag, $\Delta M_{\rm Bol}$ is 0.053 mag which
means $\Delta L/L$ is 4.9%. If a star has a very accurate parallax like CM
Dra, then its absolute visual magnitude could be as accurate as 0.0019 mag,
and if its BC is calculated with an accuracy of 1%, then its luminosity would
be 0.9%.

We estimate here that the typical accuracy of a standard stellar luminosity
could be in the order of a few percent levels if the method uses the star’s
unique BC, which could be computed from its SED. For ideal cases it could be
as accurate as 1% or even more, depending upon the accuracies of $M_{V}$ and
the unique BC itself. Therefore, using a unique BC which is computed from its
SED is a real advantage over all other methods including the direct method.

##  6 Conclusions

The accuracy of predicted stellar luminosities using the direct and two
secondary methods has been revised and a new concept, “standard stellar
luminosity”, has been defined. The luminosities which were calculated by the
direct method from observational radii and effective temperatures are more
accurate than the luminosities estimated by the secondary methods which
require a MLR or a pre-computed standard BC. The luminosities produced by the
direct method have been shown to be accurate from about 8.2% to 12.2%, and
they could be more accurate by 2-3%. Stellar luminosities obtained by the
direct method are standard by definition because there could be only one
luminosity for a star of given radius and effective temperature.

The stellar luminosities by the method using MLR are the least accurate.
Depending upon the mass range where the classical MLR operates in the form
$L\propto M^{\alpha}$, they could be most accurate at about 17.5% (for low
mass stars; $0.179\leq M/M_{\odot}\leq 0.45$) and least accurate at about 38%
(for high mass stars; $2.4\leq M/M_{\odot}\leq 7$). Luminosities calculated by
a method with MLR cannot be called standard because, first of all, a standard
MLR has not been defined yet, and perhaps it never will. This is because a MLR
in the form $L\propto M^{\alpha}$ is the relation between the typical mass and
typical luminosity of a main-sequence star belonging to a certain region
(e.g., Galactic disc stars in the solar neighborhood). Moreover, there is no
relation or process providing the true luminosity of a main-sequence star from
its mass except a standard stellar structure and evolutionary model (not yet
defined or agreed upon), which naturally would include the chemical
composition and age, as with the other independent parameters.

The stellar luminosities produced by the method using a pre-determined BC are
more accurate than the luminosities computed by the method of using MLR, but
less accurate than the luminosities produced by the direct method. It has been
shown that the method which uses a standard BC would provide 13.7% accuracy in
predicted luminosities if the typical accuracy of the absolute visual
magnitude is about $\pm 0.044$ mag for main-sequence stars with middle
temperatures of $5000\leq T_{\rm eff}\leq 10000$ K. Depending upon other
factors, such as the accuracy of its parallax, interstellar extinction, and
accuracy of the standard BC, the accuracy would be improved or becomes worse.
It has been said that such luminosities are called standard only if the BC
value used in computing it is a standard BC. Non-standard BCs produce non-
standard luminosities.

Note that if a BC value is from a tabulated standard BC table and/or from a
standard $BC_{\rm V}-T_{\rm eff}$ relation, the BC value cannot be called
unique. Tabulated standard BC tables and/or $BC_{\rm V}-T_{\rm eff}$ relations
in the literature do not provide unique BC values because they have already
been produced from pre-computed unique BC values; thus, they provide only a
mean value to represent stars of a given effective temperature at different
age and chemical composition.

Therefore, a unique BC is a BC computed according to equation (13) for a star
from its spectrum with sufficient spectral coverage and resolution. Unlike
tabulated tables and/or $BC_{\rm V}-T_{\rm eff}$ relations where chemical
composition and age information has been lost, a computed $f_{\rm V}/f_{\rm
Bol}$ from the well-observed spectrum of a star retains this information;
therefore, such computed BC values of stars are unique. Using unique BC when
computing standard luminosity has the advantage of providing an even more
accurate standard $L$ than the other methods, including the direct method.

Unfortunately, this new method, described in the present study, has no
application yet because it requires $C_{2}$ (equation 13) for the $V$ band and
other bands, which are not available in the literature (Eker et al., 2021).
Therefore, we encourage determining $C_{2}$ values for various bands first,
and then the determinations of $f_{\rm V}/f_{\rm Bol}$ from observed stellar
spectra in order to compute a unique $BC_{\rm V}$ value for each star. Then,
unique stellar luminosities could be computed using equations (4), (5) and
(6).

## Acknowledgements

We are grateful to anonymous referee whose comments was useful improving the
revised manuscript. Thanks to Mr. Graham H. Lee for careful proofreading of
the text and correcting its English grammar and linguistics. This work has
been supported in part by the Scientific and Technological Research Council
(TÜBİTAK) under the grant number 114R072. Thanks are also given to the BAP
office of Akdeniz University for providing partial support for this research.

## Data Availability

The data underlying this article are available at
https://dx.doi.org/10.1093/mnras/staa1659

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