# A Silent Revolution in Fundamental Astrophysics

Zeki Eker Akdeniz University, Faculty of Sciences, Department of Space
Sciences and Technologies, 07058, Antalya, Turkey  Faruk Soydugan Department
of Physics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart University,
17100 Çanakkale, Turkey  Astrophysics Research Center and Ulupınar
Observatory, Çanakkale Onsekiz Mart University, 17100, Çanakkale, Turkey
Volkan Bakış Akdeniz University, Faculty of Sciences, Department of Space
Sciences and Technologies, 07058, Antalya, Turkey  Selçuk Bilir Istanbul
University, Faculty of Science, Department of Astronomy and Space Sciences,
34119, Beyazıt, Istanbul, Turkey  Ian Steer 176 The Esplanade, Ste. 705,
Toronto, Canada M5A 4H2

(Received XXXX; Revised XXXX; Accepted XXXX YY, 2022)

###### Abstract

Arbitrariness in the zero point of bolometric corrections is a nearly century
old paradigm leading to two more paradigms. “Bolometric corrections must
always be negative,” and “bolometric magnitude of a star ought to be brighter
than its $V$ magnitude”. Both were considered valid before IAU 2015 General
Assembly Resolution B2, a revolutionary document that supersedes all three
aforementioned paradigms. The purpose of this article is to initiate a new
insight and a new understanding of the fundamental astrophysics and present
new capabilities to obtain standard and more accurate stellar luminosities and
gain more from accurate observations in the era after Gaia. The accuracy
gained will aid in advancing stellar structure and evolution theories, and
also Galactic and extragalactic research, observational cosmology and dark
matter and dark energy searches.

stars: fundamental parameters, cosmology: miscellaneous

††journal: AJ

##  1 Introduction

According to Kuhn (1962), one of the preeminent philosophers of science in the
20th century, scientists work under paradigms between scientific revolutions.
This is the period when scientists tend to ignore anomalies opposing the
dominant paradigm. It is only when odds are too much to hide that a crisis and
then a revolution occur before the next normal science period begins and a new
paradigm guides scientists. The modern history of science attests to two major
revolutions separating three such periods. First, the Copernican revolution
advanced science from a geocentric to a heliocentric view, although both
models could be considered static. Second, Einstein’s theories of special and
general relativity advanced science from a static to a dynamical (expanding)
model. The discovery of Hubble’s law (Hubble, 1929) led to a major revolution
in observational astrophysics and cosmology and the new world picture of a
universe undergoing expansion. One must also keep in mind however, that every
branch of science has its minor as well as major revolutions. They can include
less noticed and even silent revolutions.

The discover that the expansion of the universe is apparently accelerating,
based on observations of Type Ia supernovae, was a surprise that led to a
partial revolution in cosmology (Perlmutter et al., 1998; Riess et al., 1998;
Schmidt et al., 1998). Furthermore, a crisis in cosmology known as Hubble
tension has developed. The Hubble constant ($H_{0}$) or universal expansion
rate determined from the cosmic microwave background (CMB) of $67.4\pm 0.5$ km
s-1 Mpc-1 (Planck Collaboration et al., 2020), does not agree with the rate
determined from Cepheid-calibrated Type Ia supernova, at $73.04\pm 0.04$ km
s-1 Mpc-1 (Riess et al., 2022). The $4\sigma-5\sigma$ disparity appears
potentially intractable because it continues to increase in severity as the
sensitivity of measurements gets better (Melia, 2022). Melia (2022) also
thinks perhaps the observations may not be fully consistent with an
accelerating universe. Even if it were so, the incompatibility of the Hubble
rates noted cannot be explained by simply replacing the currently accepted
cosmological model.

For us the problem appears more fundamental than that. That is, there may not
be an immediate solution but rather a marathon of revising current
extragalactic distances and distance indicators and aiming to improve various
methods of measuring $H_{0}$ in order to be consistent with the rate according
to CMB observations. One of the intolerable negative effects of the paradigm
of arbitrariness attributed to the zero point of the bolometric correction
(BC) scale is the extra uncertainty in the predicted luminosity ($L$) caused
by different zero points preferred by different authors in the literature for
more than 80 years. It was first discussed by Torres (2010) that this
uncertainty could be about 10% or larger. It is named the zero-point error
caused by arbitrariness by Eker et al. (2021a, b), who not only confirmed
Torres (2010) but also claimed that it is possible to avoid it if a standard
BC is used when computing standard $L$ for a star. Being equivalent to a 5%
error in parallax, a systematic 10% zero-point error in a predicted $L$ in
addition to its uncertainly caused by random observational errors associated
with absolute magnitude $M_{\zeta}$ and ${\rm BC}_{\zeta}$ of a star, where
$\zeta$ could be any photometric band, is intolerable nowadays, especially
after Gaia.

We suspect it is not a simple coincidence that 10% systematic error in
predicted $L$ is on par with the discrepancy involved in the Hubble tension
(Planck Collaboration et al., 2020; Riess et al., 2022). Improving the
accuracy of stellar luminosities is the starting point to improving galactic
and extragalactic luminosities, luminosity functions, and distances because
galaxies are mainly made of stars. Accurate determination of luminous mass in
galaxies and galaxy clusters would enable us to determine dark matter more
accurately. On the other hand, improvements in extragalactic luminosities
enable improved extragalactic distance estimates. Thus, cosmological models
and the value of the Hubble constant cannot be independent of current
improvement in fundamental astrophysics. One step has already been taken to
avoid the legacy problems caused by the arbitrariness of the BC scale by
defining a truly standard BC and standard $L$ (Eker et al., 2021a, b). It has
also been discussed and shown that standard stellar luminosities with
accuracies at 1% are now possible (Eker et al., 2021b; Bakış & Eker, 2022).

IAU 2015 General Assembly Resolution B2 is a revolutionary document in part
because it has the potential to start a new revolution in astrophysics and to
aid researchers in solving the current crisis in cosmology. First, it discards
the paradigm of arbitrariness in the zero point of bolometric magnitudes. It
furthermore discards two following paradigms. “Bolometric corrections must
always be negative,” and “bolometric magnitude of a star ought to be brighter
than its $V$ magnitude”.

We see IAU 2015 General Assembly Resolution B2 as a silent and so far
unnoticed revolution. We believe its full potential is far from being
realized. One may note that there are numerous articles in respected journals
that continue to ignore the resolution, which we do not reference directly so
as not to offend. Evidently, the three paradigms involved are so basic that
many authors prefer to ignore the effects as trivial anomalies, typical of
normal science. In this study, we show how and why adoption of the precepts,
advantages, and advancements in the resolution will improve fundamental
astrophysics, and as a result advance modern cosmology and extragalactic as
well as Galactic and stellar research.

##  2 Breaking Paradigms of Fundamental Astrophysics

The first paradigm broken was the arbitrariness attributed to the zero-point
constant of the BC scale. IAU 2015 General Assembly Resolution B2111${\rm
https://www.iau.org/static/resolutions/IAU2015\\_English.pdf}$ (hereafter IAU
2015 GAR B2) superseded this paradigm by issuing:

$M_{\rm Bol}=-2.5\times\log(L/L_{0})=-2.5\times\log L+71.197~{}425~{}...$ (1)

where $L_{0}=3.0128\times 10^{28}$ W is the radiative luminosity of a star
with absolute bolometric magnitude $M_{\rm Bol}=0$ mag. It corresponds to the
value of the zero-point constant $C_{\rm Bol}=71.197~{}425~{}...$ mag if the
star’s $L$ is in SI units. Then, the following relation gives absolute
bolometric magnitude $M_{\rm Bol}$ of a source of luminosity $L$ expressed in
SI unit W.

$M_{\rm Bol}=-2.5\times\log L+C_{\rm Bol}.$ (2)

If $L$ is replaced by $L_{\rm V}$, which is $V$ filtered luminosity of the
same star, it can be written for its visual absolute magnitudes as:

$M_{\rm V}=-2.5\times\log L_{\rm V}+C_{\rm V}.$ (3)

Since $L_{\rm V}$ of a star is less than its $L$, the zero-point constant for
the $V$ filter ($C_{\rm V}$) must be smaller than $C_{\rm Bol}$. Subtracting
equation (3) from (2),

${\rm BC}_{\rm V}=M_{\rm Bol}-M_{\rm V}=2.5\times\log(L_{\rm V}/L)+C_{\rm
Bol}-C_{\rm V}.$ (4)

and identifying $C_{2}=C_{\rm Bol}-C_{\rm V}$ as the zero-point constant of
the $V$ band bolometric corrections ${\rm BC}_{\rm V}$, it is obvious that
$C_{2}$ is a positive number because both $C_{\rm Bol}$ and $C_{\rm V}$ are
positive and $C_{\rm Bol}>C$.

Equation (4) indicates that the zero point of ${\rm BC}_{\rm V}$ is not
arbitrary because both $C_{\rm Bol}$ and $C_{\rm V}$ are well-defined
constants and the zero-point constant is equal to their difference. Therefore,
the first superseded paradigm is “the zero-point constant of the bolometric
correction scale is arbitrary”.

Like a chain reaction, the other two paradigms “bolometric corrections must
always be negative” and “bolometric magnitude of a star ought to be brighter
than its visual magnitude” are also superseded. Since $L_{\rm V}/L$ is a
number between zero and one ($0<L_{\rm V}/L<1$) indicating $\log(L_{\rm V}/L$)
is a negative number while $C_{2}$ is a positive number, then ${\rm BC}_{\rm
V}$ is positive if the absolute value of the logarithmic term is smaller than
$C_{2}$ or else it is negative. Note: according to (4), ${\rm BC}_{\rm
V}=C_{2}$ and ${\rm BC}_{\rm V}>C_{2}$ are not allowed. Positive BC is also
possible by the following equation derived from (4)

$L_{\rm V}=L\times 10^{(BC_{\rm V}-C_{2})/2.5}.$ (5)

Since it is unphysical to have $L_{\rm V}=L$ or $L_{\rm V}>L$, equation (5)
indicates all ${\rm BC}_{\rm V}$ values less than $C_{2}$ are valid. In other
words, not only ${\rm BC}_{\rm V}<0$, but also $0<{\rm BC}_{\rm V}<C_{2}$ are
valid to produce $L_{\rm V}<L$. Therefore, BCs cannot be limited only to
negative numbers. There could be stars with positive BC (see Eker et al.,
2020, 2021a, 2021b) satisfying the condition $L_{\rm V}<L$. Main-sequence
stars having effective temperatures between 5859 K and 8226 K with positive
${\rm BC}_{\rm V}$ have indeed been found (Eker et al., 2020).

Even a single occurrence of a positive BC (regardless of its value) breaks the
paradigm that “bolometric magnitude of a star ought to be brighter than its
$V$ magnitude” because a positive BC indicates $M_{\rm Bol}>M_{\rm V}$, which
is an obvious case sufficient to break this paradigm if the word “brighter”
mean smaller number. This paradigm is also incorrect factually and
linguistically since eye comparisons of magnitudes are meaningful only if they
are done in the same wavelength range within the visible part of the
electromagnetic spectrum; otherwise, the comparison is meaningless due to
different eye sensitivity at different wavelengths. One final comment: BC and
colors, e.g. $U-B$, $B-V$, $V-R$, $V-I$ … etc, are not defined to compare the
brightness of a star at different bands, they are rather defined as physical
entities to reveal information about effective temperature and/or Spectral
Energy Distribution (SED) of stars.

##  3 Superseding the Paradigms in Apparent Magnitudes

IAU 2015 GAR B2 did more than fix the zero point of absolute bolometric
magnitudes. It also fixed the zero point of apparent bolometric magnitudes,
because fixing the zero point of absolute bolometric magnitudes (equation 2)
automatically fixes the zero point of apparent bolometric magnitudes. It is
also written in the same document: “the zero point of the apparent bolometric
magnitude scale by specifying that $m_{\rm Bol}=0$ mag corresponds to an
irradiance or heat flux density of $f_{0}=2.518021002~{}...~{}\times 10^{-8}$
W m-2 and hence the apparent bolometric magnitude $m_{\rm Bol}$ for an
irradiance $f$ (in W m-2) is

$m_{\rm Bol}=-2.5\times\log(f/f_{0})=-2.5\times\log f-18.997~{}351~{}...$ (6)

The irradiance $f_{0}$ corresponds to that from an isotropically emitting
radiation source with absolute bolometric magnitude $M_{\rm Bol}=0$ mag
(luminosity $L_{0}$) at a standard distance of 10 parsecs” (IAU 2015 GAR B2).
Equation (6) also reveals that the zero-point constant of the apparent
bolometric magnitude scale is no longer arbitrary as well. Thus, its value is:
$c_{\rm Bol}({\rm irr})=-18.997~{}351~{}...$ mag, then

$m_{\rm Bol}=-2.5\times\log(f_{\rm Bol})+c_{\rm Bol}(\rm{irr}),$ (7)

where $f$ is replaced by $f_{\rm Bol}$ to be more specific. Equation (7) could
be adopted for $V$ apparent visual magnitudes as

$V=-2.5\times\log(f_{\rm V})+c_{\rm V}(\rm{irr}),$ (8)

where $c_{\rm V}({\rm irr})$ is the well-defined zero-point constant for $V$
magnitudes, and $f_{\rm V}$ is the $V$ filtered star flux reaching the
telescope (if no extinction). Since bolometric flux ($f_{\rm Bol}$) from a
star is bigger than its $V$ flux, $c_{\rm Bol}({\rm irr})>c_{\rm V}({\rm
irr})$. Even if $c_{\rm Bol}({\rm irr})$ is a negative number (numerical value
of $f_{\rm Bol}$ is less than one), $c_{\rm V}({\rm irr})$ must also be a
negative number but a larger absolute value, thus $c_{\rm Bol}({\rm irr})$
minus $c_{\rm V}({\rm irr})$ is always positive. Subtracting equation (8) from
(7), the following equation is written.

$BC_{\rm V}=m_{\rm Bol}-V=2.5\times\log(f_{\rm V}/f_{\rm Bol})+c_{\rm
Bol}-c_{\rm V}.$ (9)

Since both $c_{\rm Bol}$ and $c_{\rm V}$ are well-defined constants, equation
(9) requires as well that the zero point of the ${\rm BC}_{\rm V}$ scale is
also not arbitrary. Even though the zero-point constants are not the same as
in equation (4), where they were defined to give luminosities, here, they are
defined to give irradiance fluxes reaching the telescope, and the zero-point
constant $C_{2}$, which is defined before as $C_{\rm Bol}-C_{\rm V}$, has the
same value as $C_{2}=c_{\rm Bol}-c_{\rm V}$ in equation (9). That is because
the left hand (${\rm BC}_{\rm V}$) side of both equations are the same and

$\frac{f_{\rm V}}{f_{\rm Bol}}=\frac{L_{\rm V}}{L}.$ (10)

Despite the fact that the zero-point constants in (2), (3), and (4) are
different than the zero-point constants in equations (7), (8), and (9), one
ends up with the same numerical value for $C_{2}$. Taking it as $C_{2}=c_{\rm
Bol}-c_{\rm V}$ from the equation (9), and replacing $f_{\rm V}/f_{\rm Bol}$
by $L_{\rm V}/L$ according to (10), the following equation could be written:

$L_{\rm V}=L\times 10^{({\rm BC}_{\rm V}-C_{2})/2.5}.$ (11)

This is the very same equation used above (equation 5) as an argument for
superseding the paradigms of “bolometric corrections must always be negative”
and “bolometric magnitude of a star ought to be brighter than its visual
magnitude”. It is clear that fixing $c_{\rm Bol}=-18.997~{}351~{}...$ by IAU
2015 GAR B2 not only breaks the paradigm of arbitrary zero-point for apparent
bolometric magnitudes but also breaks the paradigm of arbitrary zero-point for
the $BC_{\rm V}$ scale. It also breaks the other two paradigms. Since the
value of $c_{\rm Bol}$ is derived from the value of $C_{\rm Bol}$, as
explained in IAU 2015 GAR B2, fixing $C_{\rm Bol}$ is sufficient to break all
three paradigms in a single step. That is so whether the BC value of the star
is coming from its apparent (equation 9) or its absolute (equation 4)
magnitude.

Note, that the $V$ filter here represents any filter in any photometric
system. The equations written for the $V$ filter are also valid for other
filters with proper BC and $C_{2}$ in any photometric system.

##  4 Discussions

###  4.1 Vega System of Magnitudes and BCζ

One counter argument used for advocating the arbitrariness of the BC scale is
that IAU 2015 GAR B2 did not set the bolometric *correction* scale. It defined
only the bolometric magnitude scale set to SI irradiance ($m_{\rm Bol}$,
$f_{\rm Bol}$) and luminosity ($M_{\rm Bol}$, $L_{\rm Bol}$) values, whereas
the photometric magnitudes are independently defined by standard stars (for
Vega magnitudes, or fluxes for ST or AB scale). The IAU could easily set the
former, but not the later like the Johnson $V$ system. Since IAU did not set
the BC scale, nothing new could be claimed about it, especially about the
arbitrariness attributed to the BC scale which has been used since about eight
decades.

Such an argument advocating the BC system used before 2015 since about eight
decades, is invalid anymore because: equation (9) could be generalized for any
photometric system: e.g. Johnson system of apparent magnitudes ($U$, $B$, $V$,
$R$, $I$) as

$\displaystyle m_{\rm Bol}=\zeta+{\rm BC}_{\zeta}=U+{\rm BC}_{\rm U}=B+{\rm
BC}_{\rm B}$ (12) $\displaystyle=V+{\rm BC}_{\rm V}=R+{\rm BC}_{\rm R}=I+{\rm
BC}_{\rm I}$

and equation (4) could be generalized to include the Johnson system of
absolute magnitudes ($M_{\rm U}$, $M_{\rm B}$, $M_{\rm V}$, $M_{\rm R}$,
$M_{\rm I}$) as

$\displaystyle M_{\rm Bol}=M_{\zeta}+{\rm BC}_{\zeta}=M_{\rm U}+{\rm BC}_{\rm
U}=M_{\rm B}+{\rm BC}_{\rm B}$ (13) $\displaystyle=M_{\rm V}+{\rm BC}_{\rm
V}=M_{\rm R}+{\rm BC}_{\rm R}=M_{\rm I}+{\rm BC}_{\rm I}$

where subscripts indicate a filter in a photometric system. Apparently,
whether it was intentional or not, the resolution conducted by IAU 2015 GAR B2
was the most easiest and logical one. This is because it is illogical to set
innumerable zero points for the BC of each band in various photometric systems
while there exists an easy way to resolve the problem in a single step. It is
obvious in the equations above that fixing the zero point of bolometric
magnitudes by equation (6) (or by equation 1) is sufficient to fix the zero
points of $\rm BC_{\zeta}$ at once because the Vega system of magnitudes has
already well defined zero points as expressed by equation (8). The
arbitrariness of the BC scale was removed automatically together with the
arbitrariness of the bolometric magnitude system at the moment IAU 2015 GAR B2
was issued. No extra setting was required.

IAU 2015 GAR B2 is revolutionary in part because it opens new insight into
fundamental astrophysics, resolving chronic problems of lasting legacy while
providing new opportunities towards most accurate stellar $L$. Accuracy may
even surpass the direct method (Eker et al., 2021b), which can provide typical
accuracy of 8.2-12.2% based on accurate stellar $R$ and $T_{\rm eff}$.

Indicated by equal signs, It is obvious in equations (12, 13) that
measurements at any filter are independent from the measurements at other
filters. That is, it is possible to obtain independent BC-$T_{\rm eff}$
relations at various filters using the most accurate $R$ and $T_{\rm eff}$ of
Double-lined Detached Eclipsing Binaries (DDEB) by the methods of Flower
(1996) and Eker et al. (2020). Independent relations at Gaia filters ($G$,
$G_{\rm BP}$, $G_{\rm RP}$) and Johnson $B$ and $V$ were obtained by Bakış &
Eker (2022). Coefficients and basic statistics of five independent BC-$T_{\rm
eff}$ relations (fourth degree polynomials) are given in Table 1, where the
columns are order, band, coefficients and associated errors to define

${\rm BC}=a+bX+cX^{2}+dX^{3}+eX^{4}$ (14)

where $X=\log T_{\rm eff}$, number of DDEB stars ($N$), standard deviation
(SD) and correlation coefficient ($R^{2}$).

Table 1: Coefficients of multiband $BC$-$T_{\rm eff}$ relations taken from Bakış & Eker (2022). Order | Band | $a$ | $b$ | $c$ | $d$ | $e$ | $N$ | SD | $R^{2}$
---|---|---|---|---|---|---|---|---|---
1 |  Gaia $G$ | -1407.14 | 1305.08 | -453.605 | 70.2338 | -4.1047 | 402 | 0.1108 | 0.9793
|  |  $\pm$256.7 |  $\pm$258.9 |  $\pm$97.67 |  $\pm$16.34 |  $\pm$1.023 |  |  |
2 |  Gaia $G_{\rm BP}$ | -3421.55 | 3248.19 | -1156.82 | 183.372 | -10.9305 | 402 | 0.1266 | 0.9738
|  |  $\pm$293.6 |  $\pm$296.1 |  $\pm$111.7 |  $\pm$18.68 |  $\pm$1.169 |  |  |
3 |  Gaia $G_{\rm RP}$ | -1415.67 | 1342.38 | -475.827 | 74.9702 | -4.44923 | 402 | 0.1092 | 0.9884
|  |  $\pm$253.3 |  $\pm$255.4 |  $\pm$96.34 |  $\pm$16.12 |  $\pm$1.009 |  |  |
4 | Johnson $B$ | -1272.43 | 1075.85 | -337.831 | 46.8074 | -2.42862 | 342 | 0.1363 | 0.9616
|  |  $\pm$394.2 |  $\pm$394.4 |  $\pm$147.7 |  $\pm$24.53 |  $\pm$1.152 |  |  |
5 | Johnson $V$ | -3767.98 | 3595.86 | -1286.59 | 204.764 | -12.2469 | 386 | 0.1201 | 0.9788
|  |  $\pm$288.8 |  $\pm$290.9 |  $\pm$109.6 |  $\pm$18.32 |  $\pm$1.146 |  |  |

Fig. 1 displays the curves of BC-$T_{\rm eff}$ relations in Table 1 as
functions of $\log T_{\rm eff}$. Note all five curves cross over one another,
as displayed, at $T_{\rm eff}\cong 10000$ K. This is not a coincidence. It is
a natural result of using Vega system of magnitudes.

Using $\alpha$ Lyr (Vega) as the primary calibrating star, the VEGA system is
the most well known and deliberated for heterochromatic measurements. Although
the zero points are often determined observationally from a network of
standard stars, it is formally just a single object. A hypothetical star of
the spectral type A0V with magnitude $V=0.0$ mag on the Johnson system is
given in Table 16.6 (Cox, 2000), where UBVRI bands and monochromatic fluxes at
effective wavelengths are presented. Note that Vega is used as a calibrating
star but its apparent magnitude is not exactly zero. $V=0.03$ mag has been
measured by Johnson (1966) and Bessell et al. (1998). The standard Johnson
value of $V=0.03$ mag is cited by Bohlin (2014). The same value ($V=0.03$ mag)
was adopted by Cox (2000), Girardi et al. (2002), Bessell & Murphy (2012) and
Casagrande & VandenBerg (2014). For the other bands, Vega is found to be just
slightly positive (0.02 mag) at most bands (Rieke et al., 2008). For the
heterochromatic bands of various photometric systems, equation (8) could be
written as:

$\zeta=-2.5\times\log(f_{\rm\zeta})+c_{\rm\zeta}(\rm{irr}),$ (15)

where the zero point constant $c_{\rm\zeta}(\rm{irr})$ needs to be derived for
each bandpass $\zeta$ using a star of known absolute flux, usually Vega
(Casagrande & VandenBerg, 2014), or Sirius and Vega (Bohlin, 2014). Although
$c_{\rm\zeta}(\rm{irr})$ are usually not given for the photometric systems in
the literature, where only monochromatic fluxes making apparent magnitudes
zero at effective wavelengths of the filters are listed, all of the zero point
constants associated with Vega system of magnitudes at various bands of
different photometric systems are well-defined quantities (Bessell et al.,
1998; Cox, 2000; Girardi et al., 2002; Casagrande & VandenBerg, 2014).

Figure 1 confirms usage of Vega system of magnitudes through equations (12)
and (13) because ${\rm BC}-T_{eff}$ curves cut each other at a single point
($T_{\rm eff}\cong 10000$ K) as the color curves would do the same since the
colors of the hypothetical star A0V is taken to be $U-B=B-V=V-R=V-I=0$ mag.
Curves intercepting at single point is naturally expected because the
difference of ${\rm BC}_{\zeta}$ values at two different bands is equal to the
negative value of the colors defined in the photometric system according to
the basic definition of BCs. All of these could be counted as undeniable
evidences indicating the zero point constants of ${\rm BC}_{\zeta}$ are not
arbitrary but well defined constants as the zero point constants of Vega
system of magnitudes.

Figure 1: Independent BC-$T_{\rm eff}$ relation at Gaia $G$, $G_{\rm BP}$,
$G_{\rm RP}$ and Johnson $B$ and $V$ (Bakış & Eker, 2022).

Each curve in Fig. 1 could be used to estimate a BC of a star if its effective
temperature is known. Then this BC could be used to calculate its bolometric
magnitude as $M_{\rm Bol}=M_{\zeta}+{\rm BC}_{\zeta}$ if its distance and
$E(B-V)$ color excess are known. At last, its standard luminosity ($L$) could
be calculated from $M_{\rm Bol}$ using equation (2). The accuracy of $M_{\rm
Bol}$, and accuracy of $L$ depend on the accuracies of $M_{\zeta}$ and ${\rm
BC}_{\zeta}$ if standard BC values are used, otherwise zero-point errors
($\sim 10\%$) caused by arbitrary definitions (Torres, 2010; Eker et al.,
2021a, b) must also be added. SD column in Table 1 indicates typical accuracy
of a BC. That is if $M_{\zeta}$ is errorless it is possible to obtain $M_{\rm
Bol}$ with accuracy $\pm$SD, which means $\Delta L/L\sim 0.921\times$SD (Eker
et al., 2021b). Thus according to Table 1 the most accurate $L(\sim 10\%)$
could be obtained using BC from BC-$T_{\rm eff}$ relation at the band Gaia
$G_{\rm RP}$, obviously not better than the direct method.

Most important is that one may increase the accuracy of predicted $L$ if many
independent standard BC-$T_{\rm eff}$ relations are used in. For example, the
primary of HP Aur has $M=0.9543\pm 0.0041M_{\odot}$ $R=1.0278\pm
0.0042R_{\odot}$ and $T_{\rm eff}=5810\pm 120$ K (Lacy et al., 2014). The
Stefan-Boltzmann law gives its $L=4.149\pm 0.344\times 10^{26}$ W, with an
accuracy of 8.302%. Using interstellar extinctions $A_{\rm V}=0.335$, $A_{\rm
G}=0.298$, $A_{\rm BP}=0.366$, $A_{\rm RP}=0.207$ mags and its Gaia EDR3
trigonometric parallax $5.2432\pm 0.0306$ mas (Gaia collaboration et al.,
2021), Bakış & Eker (2022) calculated its absolute magnitudes $M_{\rm
V}=4.753\pm 0.033$, $M_{\rm G}=4.583\pm 0.033$, $M_{\rm BP}=4.860\pm 0.042$,
and $M_{\rm RP}=4.152\pm 0.024$ mags. Using the BC values from Table 1, its
bolometric absolute magnitudes were found 4.824, 4.688, 4.730, and 4.713 mags,
which are then combined to a single value as $4.739\pm 0.030$ mag. At last its
luminosity of $L=3.831\pm 0.096\times 10^{26}$ W is found with an uncertainty
of 2.5%. That is about four times better than the direct method. Hitherto, no
method available could provide a stellar luminosity more accurate than the
direct method. This is the first, and became possible only after IAU 2015 GAR
B2. Accurate stellar luminosities are not only needed to test stellar
structure and evolution theories but also required for improving Galactic and
extra-galactic studies, dark matter search and even perhaps to be used to
resolve the Hubble tension at last because stars are one of the most important
primary building blocks to understand the universe.

###  4.2 Attention Required On Zero-Point Constants

The apparent magnitude of a star disregards distance information. Thus,
equation (7) is valid only for the apparent magnitudes of stars. The very same
equation could be adopted for absolute bolometric magnitudes as

$M_{\rm Bol}=-2.5\times\log(F_{\rm Bol})+c_{\rm Bol}(\rm{irr}),$ (16)

on the condition that the star is assumed to be at a fixed distance of 10 pc.
It is only under this condition that an irradiance or heat flux density of
$F_{\rm Bol}=f_{0}=2.518021002~{}...\times 10^{-8}$ W m-2 at the focal point
of the telescope, makes it possible to write $M_{\rm Bol}=0$ mag. It is
obvious that this equation and equation (2) appear different despite the left-
hand side of both equations being the same. Same $M_{\rm Bol}$ before the
equal sign implies that

$-2.5\times\log(L)+C_{\rm Bol}=-2.5\times\log(F_{\rm Bol})+c_{\rm
Bol}(\rm{irr}).$ (17)

The algebraic sum of the two quantities on the right-hand side of both
equations must be the same. Thus, the zero-point constant associated with
luminosities is different than the zero-point constant associated with
irradiances.

Subtracting equation (16) from equation (7), which has the same zero-point
constant, they must cancel in the subtraction, thus

$m_{\rm Bol}-M_{\rm Bol}=2.5\times\log(F_{\rm Bol}/f_{\rm Bol}).$ (18)

Because $F_{\rm Bol}$ and $f_{\rm Bol}$ correspond to bolometric fluxes of the
same star if it is at a distance of 10 pc and d pc away (assuming no
extinctions) and the flux of a star is inversely proportional to the square of
its distance, then $(F_{\rm Bol}/f_{\rm Bol})=(d/10)^{2}$. Inserting this into
(18),

$m_{\rm Bol}-M_{\rm Bol}=5\times\log d-5,$ (19)

the distance modulus of the star is obtained. So, one has to be careful using
absolute magnitudes. Proper usage of the zero-points is necessary. If $M_{\rm
Bol}$ is associated with $L$, then use $C_{\rm Bol}$ as in equation (2). Else,
use $c_{\rm Bol}{\rm(irr)}$ as in equation (16).

##  5 Conclusions

The results obtained in this study are listed below:

  * •

Although its primary aim was to set the zero point constant of bolometric
magnitudes, IAU 2015 GAR B2 was shown to be a revolutionary document to imply
the zero point constants of ${\rm BC_{\rm\zeta}}$, where $\zeta$ indicates
various photometric bands, are not arbitrary but well defined constants.

  * •

IAU 2015 GAR B2 is a revolutionary document not only because it solves nearly
a century old problems caused by the arbitrariness attributed to the BC scale,
but also to indicate “bolometric corrections must always be negative” and
“bolometric magnitude of a star ought to be brighter than its $V$ magnitude”
are not right anymore.

  * •

The falsehood of the three paradigms (BC scale is arbitrary, BC values must
always be negative, bolometric magnitude of a star ought to be brighter than
its $V$ magnitude) are not only proven mathematically but also confirmed from
observational point of view by multiband ${\rm BC}-T_{\rm eff}$ relations
calibrated by Bakış & Eker (2022) who used published observational data of
DDEB stars.

  * •

Increasing the number of photometric bands with standard ${\rm BC}-T_{\rm
eff}$ relations, which is to be used in calculating standard stellar
luminosity from $M_{\rm Bol}=M_{\zeta}+{\rm BC}_{\zeta}$ and $M_{\rm
Bol}=-2.5\times\log L+C_{\rm Bol}$, increases the accuracy of the standard
luminosity of a star. Using this method in predicting stellar luminosities is
also a revolution because this method is shown to be predicting stellar
luminosities much more accurate than the classical direct method using stellar
radii and effective temperatures through the Stefan Boltzmann law. Achieving
such accuracy became possible only after discarding the arbitrary zero point
errors caused by the arbitrariness attributed to the BC scale; this was
possible, however, only after understanding the potential of IAU 2015 GAR B2.

  * •

Such an accuracy in calculated stellar luminosities is not only needed by
stellar structure and evolution theories to be refined but also needed by
Galactic and extragalactic studies including observational and theoretical
cosmology, dark matter search etc to be improved further since stars are
primary building blocks of the universe.

##  6 Acknowledgements

This work has been supported in part by the Scientific and Technological
Research Council (TÜBİTAK) under grant number 114R072. This research has made
use of NASA’s Astrophysics Data System Bibliographic Services. This research
made use of VizieR and Simbad databases at CDS, Strasbourg, France. We made
use of data from the European Space Agency (ESA) mission _Gaia_
222https://www.cosmos.esa.int/gaia, processed by the _Gaia_ Data Processing
and Analysis Consortium
(DPAC)333https://www.cosmos.esa.int/web/gaia/dpac/consortium. Funding for DPAC
has been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement.

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