Z<EMAIL_ADDRESS>XXXXXXX/PAR.20XX.00000 1 Fundamentals of Stars:
Critical Looks at Mass-Luminosity Relations and Beyond–Fundamentals of Stars:
Critical Looks at Mass-Luminosity Relations and Beyond

# Fundamentals of Stars: Critical Looks at Mass-Luminosity Relations and
Beyond

Z. Eker${}^{1\cc}$0000-0003-1883-6255  F. Soydugan2,30000-0002-5141-7645 and
S. Bilir40000-0003-3510-1509
1Akdeniz University  Faculty of Sciences  Department of Space Sciences and
Technologies  07058  Antalya  Türkiye
2Çanakkale Onsekiz Mart University  Faculty of Sciences  Department of Physics
17100  Çanakkale  Türkiye
3Çanakkale Onsekiz Mart University  Astrophysics Research Center and Ulupınar
Observatory  17100  Çanakkale  Türkiye
4 Istanbul University  Faculty of Science  Department of Astronomy and Space
Sciences  34119  Beyazıt  Istanbul  Türkiye

(2023; XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX )

###### Abstract

Developments on various relations among stellar variables such as the main
sequence empirical mass-luminosity (MLR), mass-radius (MRR) and mass-effective
temperature (MTR) relations were reviewed. Conceptual changes on their
understanding and usages were discussed. After its discovery, MLR was treated
as one of the fundamental secrets of the cosmos, at least until questioned by
the end of the 20th century. Differences between fundamental laws and
statistical relations were used to understand long-term developments of MLR,
together with improvements observed in MRR and MTR. Developments show a break
point, initiated by Andersen (1991), in the line of progress. Before the break
when reliable data were limited, MLR and MRR were calibrated using $M$, $L$,
and $R$ of binary components of all kinds visual, spectroscopic, and eclipsing
for two purposes: i) obtaining mean mass ($\langle M\rangle$), mean luminosity
($\langle L\rangle$), and mean radius ($\langle R\rangle$), ii) to estimate
$M$ and $R$ of single stars. By the time of the break, the number of solutions
from detached double-lined eclipsing binaries (DDEB) giving accurate $M$ and
$R$ within a few percent levels are increased. Parameters from very close,
semi-detached, and contact binaries were excluded for the sake of refinement,
however, MLR and MRR diagrams were found insufficient to derive MLR and MRR
functions because the dispersions are not only due to random observational
errors but also due to chemical composition and age differences. Then, the new
trend was adopted by replacing classical MLR and MRR with empirical $M$ and
$R$ predicting relations.Thus, the purpose one was suppressed also because the
new trend found a fruitful application in determining $M$ and $R$ of exoplanet
hosting single stars. Corrections on misnames and devising new classical MLR,
MRR, and MTR, giving mean values ($\langle M\rangle$, $\langle L\rangle$,
$\langle R\rangle$) are encouraged since they are still useful and needed by
astrophysical models requiring such mean values, not only beneficial to
astrophysics, but also beneficial to galactic, extragalactic search, even
cosmological models.

###### keywords:

Stars: fundamental parameters, Stars: luminosity function, mass function,
Galaxies: luminosity function, mass function, Cosmology: miscellaneous

††volume: 0

##  1 Introduction

Accuracy and precision of observational parameters of stars are crucial not
only for improving stellar structure and evolution theories but also for
fundamental astrophysics as well as Galactic and extragalactic studies, and
ultimately even for the cosmological models because stars and galaxies are
primary building blocks of the universe. Fundamental and statistical relations
are essential for understanding physical events occurring in various parts of
the universe. The fundamental relations are the relations like Stefan-
Boltzmann law ($L=4\pi R^{2}\sigma T_{\rm eff}^{4}$) which are characterised
by at least two properties; i) how various properties of stars are related,
e.g. how the luminosity of a star is related to its radius and effective
temperature; ii) validity is not limited by certain conditions, that is, it is
applicable to all stars and radiating surfaces as long as the source of
radiation is thermal. The statistical relations, on the other hand, may not
work in all possible cases; appropriate statistical conditions are demanded.

A good example is the kinetic temperature. A temperature is a physical
quantity that is measured in the macro world but does not exist in the micro
world. This is because temperature is a statistically defined quantity
implying an average kinetic energy per particle in a substance which could be
a solid, liquid, or gas. The definition of kinetic temperature tells us that a
single particle or insignificant number of particles cannot be associated with
any temperature. For example, when we talk about air temperature, it indicates
the average kinetic energy per air particle, which could be written as:

$\frac{1}{2}m\langle v^{2}\rangle=\frac{3}{2}kT$ (1)

where $m$ and $\langle v^{2}\rangle$ are the mean mass and root-mean-square
(rms) speed of particles. They are not physical quantities but just average
values. Increasing the speed of the substance itself by moving it faster, e.g.
putting it on a very fast airplane, does not change its temperature.
Therefore, equation (1) cannot be written for a randomly chosen particle (or a
few particles), that is, the right-hand side does not exist except for a group
of particles satisfying the implied statistic.

Similarly, the mass–luminosity relation (MLR) in the form ($L\propto
M^{\alpha}$), which is defined so far for main-sequence stars, is a
statistical relation like equation (1) which indicates how main-sequence
luminosities are related to main-sequence masses. As it is not valid for non-
main sequence stars, depending upon expected accuracy, it may also not be
valid for an individual star. So, it is the relation devised for estimating a
typical mass from a typical luminosity, or vice versa.

Consequently, inter-related mass-luminosity (MLR), mass-radius (MRR), and
mass-effective temperature relations (MTR) of Eker et al. (2018) all are
statistical relations, which should not be treated like fundamental relations.
While, bolometric correction – effective temperature relation (BC-$T_{\rm
eff}$) of Flower (1996); BC-$T_{\rm eff}$ and BC-mass relations of Eker et al.
(2020, 2021b); Bakış & Eker (2022); Eker & Bakış (2023) are all also
statistical relations which are not to be confused by fundamental relations,
otherwise unexpected results or upsetting errors in the computed quantities
becomes unavoidable.

One cannot say “statistical relations are less valuable than the fundamental
relations”. On the contrary, a statistical relation could be more valuable;
even more practical, or easier to use, e.g. calculating the total energy of a
gas is equal to the mean energy per particle multiplied by the number of
particles in the gas. Otherwise, the probability of each particle having a
certain kinetic energy, and the number of particles having this energy is
required before integrating them over all possible kinetic energies.
Similarly, using an MLR is more practical to determine the masses and
luminosity of galaxies, which are the key parameters to determine dark matter
in galaxies. Otherwise, summing individual masses and luminosities of the
stars in a galaxy, which are impractical observationally, are needed to obtain
the total mass and the total luminosity of the galaxy in question.

Moreover, there could be various astrophysical research demanding MLR, MTR,
and MTR besides stellar astrophysics. Live examples are such as cometary
research (Wysoczańska et al., 2020a, b), Oort clouds (Baxter et al., 2018),
heliophysics and planetary habitability (Schrijver et al., 2019), exoplanet
investigations (Berger et al., 2020; Arora & Hasegawa, 2021; Burt et al.,
2021; Caballero et al., 2022; Dattilo et al., 2023), planetary nebula (Munday
et al., 2020; Aller et al., 2020), open clusters (Ilin et al., 2021; Akbulut
et al., 2021; Yontan et al., 2021, 2023; Yontan, 2023), dark matter searches
(Garani & Palomares-Ruiz, 2022; Peled & Volansky, 2022) and quasars (Albert et
al., 2021), neutron stars (Yuan et al., 2022), black holes (Gomel et al.,
2021a, b), general relativity (Lalremruati & Kalita, 2021), gravitational
lensing (Ramesh et al., 2022; Pietroni & Bozza, 2022) and even search for
extra-terrestrial intelligence (SETI, Kerins, 2021; Kerins et al., 2023),
which are all used at least one of the statistical relations MLR, MRR and MTR
of Eker et al. (2018).

Apparently, differentiation between fundamental and statistical relations is
important from an astrophysical point of view. Recognising statistical
relations is even more important, otherwise using them as fundamental
relations would be misleading. Unfortunately, there are authors such Malkov
(2003, 2007), Henry (2004) and Gafeira et al. (2012) including Eker et al.
(2015) did not hesitate to call MLR “fundamental law”, “sufficiently
fundamental to be applicable to many areas of astronomy”, “one of the most
famous empirical law” and “one of the fundamental secrets of the cosmos”. On
the contrary, Andersen (1991) and Torres et al. (2010) preferred not to define
a MLR and preferred displaying the $\log M-\log L$ diagram without a function
(MLR) fitting to the data because the scatter on the diagram is not due to
observational random errors but most likely abundance and evolutionary
effects. Andersen (1991) claims “….departures from a unique relation is real”.
If there is no unique function to represent the data on the diagram, why
bother to define one?

Therefore, this review article is dedicated to investigating the evolution of
the statistical functions MLR, MRR, and MTR starting from their discovery
until today and trying to explain why such quarrels occurred and whether it is
possible to resolve such quarrels by identifying the nature of the relation.
Moreover, the presentation of their conceptual advances and realizing whether
they were perceived as fundamental laws or statistical relations will benefit
to astronomical community. Developing such a conscious analysis would result
in a better understanding of their previous usages as well as future studies.

##  2 Overview

Calibrations of MLR, MRR, and MTR require pre-determined accurate stellar
parameters such as mass ($M$), luminosity ($L$), radius ($R$), and effective
temperature ($T_{\rm eff}$). The most critical parameter among them is $L$
because it is not an observable quantity, that is, there is no telescope nor a
detector to be able to observe the total radiation radiated from a star at all
frequencies (Bakış & Eker, 2022; Eker & Bakış, 2023). Fortunately, there are
only one direct and two indirect methods to calculate $L$ of a star (Eker et
al., 2021c). The first method is a direct one because it uses independently
determined observational $R$ and $T_{\rm eff}$ of a star to calculate its $L$
directly from its radiating surface area ($4\pi R^{2}$) and bolometric flux
($\sigma T_{\rm eff}^{4}$), which is commonly known as the Stefan-Boltzmann
law ($L=4\pi R^{2}\sigma T_{\rm eff}^{4}$). The other two methods are indirect
because the first one providing $L$ of a star from its $M$ requires a pre-
determined classical MLR in the form of $L\propto M^{\alpha}$, while the other
supplying $L$ of the star from its apparent brightness and distance requires a
pre-determined BC-$T_{\rm eff}$ relation. Unfortunately, the indirect methods
are useless without their pre-determined relations, and thus, one has no other
choice but to use the Stefan-Boltzmann law to produce earliest sample of $L$
values for the first calibration of MLR, MRR, and MTR.

Assuming that the sample $L$ values are ready, the next most critical
parameter is stellar $M$ to establish not only MLR but also MRR and MTR since
sequentially required $R$ and $T_{\rm eff}$ should have been already used in
computing $L$ of the sample stars. However, like $L$ of a star, $M$ of a star
is also not observable. This fact makes the parameter $M$ even more critical
since only the masses of binaries (or multiple systems) could be calculated
using Kepler’s Third Law from the observed orbital semi-major axis of the
components and the orbital period which could be deduced from the observed
orbits of visual binaries (or multiple systems) or from the radial velocity
curves of double-lined eclipsing spectroscopic binaries. Stellar spectra do
not provide orbital inclinations, thus spectroscopic binaries without eclipses
cannot provide component masses unless orbital inclinations are available
independently.

Eclipsing binaries are ideal objects to collect observed radii of stars while
the single stars are null, except the ones that are close in distance and
large enough where interferometry could be useful. Although binarity adds
complications to estimating component effective temperatures, the eclipsing
binaries are still advantageous to revealing the most accurate effective
temperatures and temperature ratios from the depths of minima if the effective
temperature for one of the components is estimated correctly.

Visual binaries (multiple systems too) and double-lined eclipsing binary
systems are the only objects provide $M$ from Kepler’s third law, with $R$
from eclipses. On the other hand, there is almost no other way to obtain
accurate masses and radii of single stars except seismic analysis obtained
mass and radii data for especially solar-like pulsating stars (e.g. Gaulme et
al., 2016; Bellinger et al., 2019). Considering some basic information and
data knowledge about this research topic, we must now start reviewing the
statistical relations starting from the most prominent one: MLR.

###  2.1 Revisiting MLR

The famous stellar mass-luminosity relation (MLR) was discovered empirically
in the middle of the first half of the 20th century by Hertzsprung (1923) and
Russell et al. (1923) independently from the masses and absolute brightness of
a very limited number of visual binaries. Eclipsing binaries were included
later in the statistics. In his MLR, Eddington (1926) was able to use 13
eclipsing binaries together with 29 visual binaries and five cepheids, which
was available to him at that time, while McLaughlin (1927) increased the
number of eclipsing systems to 41 in his plots.

Almost immediately after discovery, Eddington (1926, page 154) fit a curve to
the data on a diagram $\log M-M_{\rm Bol}$ ($\log$ mass versus absolute
bolometric magnitude) and call it “the theoretical mass-luminosity law”.
Examining masses and bolometric magnitudes of 18 visual binaries, Gabovits
(1938) agreed Eddington (1926) after twelve years by re-examining the
mass-$M_{\rm Bol}$ data again and declared “We conclude that, as revealed by
our selected first-class data, the stars (chiefly of the main-sequence)
probably follow a strict mass-luminosity law”. On the other hand, having
larger data sets including visual binaries, spectroscopic binaries, Hyades and
Trumpler stars and white dwarfs, Kuiper (1938) was rather suspicious to accept
the mass-luminosity relation as a law because he commented “It is doubtful
whether this mean relation has any physical significance” after discussing
them on a $\log M-M_{\rm Bol}$ diagram.

MLR has been updated and revised many times, until a major break occurred at
the very beginning of the last decade of the 20th century on the issue of
whether it is a statistical relation or a fundamental law. Looking at the
developments before this break would be useful to understand it better. Petrie
(1950a, b) used 93 spectroscopic binary systems, Strand & Hall (1954) studied
23 visual binaries, Eggen (1956) investigated 34 visual binaries, McCluskey &
Kondo (1972) considered 40 visual binaries and 35 eclipsing systems, Cester et
al. (1983) gathered 45 visual and 40 spectroscopic binaries, Griffiths et al.
(1988) analyzed 72 detached main-sequence binaries, 25 detached OB, six
resolved binaries and 23 visual binaries when revising the MLR relations.
Demircan & Kahraman (1991) preferred to study masses and luminosities of 70
eclipsing binaries (140 stars) only, including the main sequence components of
detached and semi-detached binaries as well as the components of OB-Type
contact and near contact binaries. Karetnikov (1991) used 303 eclipsing
systems of different types. At this point, we must keep in mind that the
observational data was very limited, therefore, authors combined different
kinds of binaries whether they are eclipsing or not without differentiating
between detached, semi-detached, and contact systems. This was done to
increase the statistical reliability of MLR’s calibrated.

Those early generation relations, including the very earliest ones, were
demonstrated as mostly mass-absolute magnitude diagrams, some are with the
best fitting function, and some are without a fitting curve. Considering the
classical form $L\propto M^{\alpha}$ of MLR, first Eggen (1956) intended to
define the power of mass ($\alpha$) so he expressed $L=\mu^{3.1}$, where $\mu$
is the total mass of a double-star system defined as $\mu=a^{3}/P^{2}\pi^{3}$
from the Kepler’s Harmonic Law. Then, McCluskey & Kondo (1972) preferred to
use a relation in the form $M\propto L^{\beta}$, where $M$ and $L$ are the
masses and luminosities of the components while $\alpha$ and $\beta$ are the
constants to be determined by the data on various mass-absolute bolometric
magnitude diagrams. Accordingly, Cester et al. (1983); Griffiths et al. (1988)
and Demircan & Kahraman (1991) preferred to study MLR on a mass-luminosity
diagram for defining unknown constants on the classical form ($L\propto
M^{\alpha}$) of MLR either by fitting a curve to all data or dividing the mass
range into two regions as the low and the high mass stars, or three regions as
the high, intermediate or Solar, and low mass, in order to determine the
inclinations (power of $M$) and it is zero point constants of the linear MLRs
on the $\log M-\log L$ diagram.

####  2.1.1 Is MLR a Fundamental Relation?

The major brake on the concept and splitting practices on the purpose of
calibrating a MLR is triggered by Andersen (1991) who collected 45 detached
double-lined eclipsing binary (DDEB) systems (90 stars) having both masses and
radii accurate within 2%, which were the most accurate stellar data of the
time. Preferring only DDEB stars was not intended to reject the reliability of
the other sources for determining masses and radii; only because $M$ and $R$
from DDEB were considered as the most reliable.

Andersen (1991) rejected calibrating any form of MLR function to fit data on
the $\log M-\log L$ diagram. The diagram displayed by him is without a fitting
curve (MLR) because the scatter from the curve is certainly not only due to
random observational errors but also due to abundance and evolutionary
effects. He exclaimed: “At first glance, the mass-luminosity diagram shows a
tight, well-defined mean relation. Closer inspection, taking the individual
uncertainties into account, reveals that the departures from a unique relation
are real”. Then, if there is no unique curve expressible by a function to
represent data, why bother to draw one?

In very early times, especially after its discovery, MLR was claimed to be one
of the most prominent empirical laws of nature by Eddington (1926) and
Gabovits (1938) and commonly used by researchers either reckoning $L$ of a
star from its $M$ or estimating $M$ of a star from its $L$ at least until the
middle of the 20th century, and perhaps until Andersen (1991) objecting it.
This practise was extremely useful for single stars because there was no other
observational way to access their masses directly, but the absolute bolometric
magnitudes of the ones with known parallaxes were rather easy to get by using
a proper bolometric correction (BC), and then estimate their masses from a
pre-determined bolometric magnitude-mass relation. The method was even helpful
to double-check trigonometric parallaxes of single stars and even eclipsing
binaries with light curve solutions giving $R$ and $T_{\rm eff}$ of the
components, thus $L$ values are compatible by the Stefan-Boltzmann law. Those
very early times were a period in astrophysics when nuclear reactions were not
fully established and understood correctly. The discovery of nuclear reactions
in the cores of stars are attributed to Sir Arthur Eddington (Bahcall, 2000) a
few years before the discovery of MLR by Hertzsprung (1923) and Russell et al.
(1923).

Astronomers waited until 1932 for the discovery of neutrons (Chadwick, 1933)
to study and fully understand hydrogen fusion as a main source of stellar
energy. Only after the CNO cycle was established by Hans Bethe and Von
Weizsacher (Clayton, 1968), and only a year later the p-p chain reactions were
suggested by Bethe (1939), were the solutions of stellar structure equations
with the nuclear energy which placed our theoretical understanding of the
evolution of stars with evolutionary tracks on a solid ground (Clayton, 1968).
Empirically discovered MLR was confirmed later theoretically in a sense that
mass ($M$) is the prime parameter that determines the internal structure, size
($R$), and luminosity ($L$) of a star not only for the time span of the main
sequence but also throughout the star’s lifetime until its death, where its
initial chemical composition can cause little variations. So, the scatter on a
$\log M-\log L$ diagram for field main-sequence stars is not only due to
observational errors but also due to various ages and chemical compositions.
Additionally, at least until the middle of the 20th century, maybe until about
the middle of its second half, but certainly not until Andersen (1991), the
observational accuracy was not high enough to differentiate theoretical (true)
$L$ and $M$ of stars which are marked on an H-R diagram to form evolutionary
tracks. As long as observational accuracies of $L$ and $M$ are very low that
the error bars covering the thickness of the main sequence, then astronomers
did not suspect inconsistency between the predicted and the observed
quantities using a MLR. It was normal for them to be satisfied as in the case
of fundamental law.

By the time of Andersen (1991), who collected the most accurate $M$ and $L$
data from DDEB, there were sufficiently accurate masses and luminosities thus
one can deduce the scatter on the $\log M-\log L$ diagram are not only due to
observational errors of $M$ and $L$ but also due to abundance and evolutionary
effects. Andersen (1991) felt something was not right, that is, there must be
a problem in treating classical MLR in the form $L\propto M^{\alpha}$ as a
fundamental relation. This was because, Andersen thought, such a fundamental
relation must not only fit the data uniquely but also must contain the other
parameters involving chemical composition and evolution of the stars on the
diagram.

Being influenced by the common usage, a kind of paradigm not easy to be free
off, Andersen (1991) was expecting, like the others before him, to obtain the
mass of a single star from its luminosity within acceptable accuracy [ideally
$\pm$5% see Andersen (1991) page 93]. However, with a mean MLR in the form
$L\propto M^{\alpha}$ for the main-sequence stars, this was not possible
anymore due to improved observational techniques with higher accuracy in both
$L$ and $M$ values available to him. Nevertheless, he succeeded in this aim
about two decades later by replacing $\log L$ with a function including
observational parameters $T_{\rm eff}$, surface gravity ($\log g$), and
relative iron abundance [Fe/H] to incorporate evolution (age) and chemical
composition in a new review paper lead by Torres et al. (2010) as

$\log M=a_{1}+a_{2}X+a_{3}X^{2}+a_{4}X^{3}+a_{5}(\log g)^{2}+a_{6}(\log
g)^{3}+a_{7}{\rm[Fe/H]},$ (2)

where $X\equiv\log T_{\rm eff}-4.1$, which is the parameter contains the most
effective variable, $T_{\rm eff}$, directly related to the luminosity with its
fourth power in the Stefan-Boltzmann law, and $a$’s which are the calibrated
coefficients $a_{1}=1.5689\pm 0.058$, $a_{2}=1.3787\pm 0.029$,
$a_{3}=0.4243\pm 0.029$, $a_{4}=1.139\pm 0.240$, $a_{5}=-0.1425\pm 0.011$,
$a_{6}=0.01969\pm 0.0019$, and $a_{7}=0.1010\pm 0.058$. This is a function
providing 6.4% accuracy for main-sequence and evolved stars above
$0.6M_{\odot}$. It is obvious that such a function cannot be drawn on a $\log
M-\log L$ diagrams but provides stellar $M$ as the classical MLR but much more
accurate not only for main-sequence stars but also giants and sub-giants thus,
both Andersen (1991) and Torres et al. (2010) presented their $\log M-\log L$
diagrams without a fitting function.

Incorrect diagnoses are natural to be continued by incorrect treatments. First
of all, classical MLR in the form of $L\propto M^{\alpha}$ is a statistical
relation devised for estimating a typical mass from a typical luminosity, or
vice versa, not for estimating $M$ or $R$ of a single star, even if it was
once used to estimate $M$ of a star from its $L$. Therefore, It is not right
to diagnose the classical MLR as one of the fundamental relations to calculate
the mass of a star from its luminosity. It is not right to look for a relation
giving the mass of a star from other stellar observational parameters for
replacing MLR. It is not right to call this relation MLR if it provides $M$ of
a star from the other parameters even if $L$ is included in the right-hand
side of the equal sign except if the right-hand side contains only $L$ as a
variable. It is not right to claim that there is no uniquely fitting function
to the data on $\log M-\log L$ the diagram despite a tight, mean relation
between $M$ and $L$ is obvious as if the least squares method would fail to
produce one.

Nevertheless, Andersen (1991)’s exclamation had a noticeable consequence in
stellar astrophysics. It appears as if a main cause of deviation on
understanding, definition, and usage of newly defined MLR functions from the
main path which was continued by Ibanoǧlu et al. (2006) and Eker et al. (2015,
2018) where statistical relation between masses and luminosities of main-
sequence stars was kept in the form $L\propto M^{\alpha}$.

####  2.1.2 Mass Predicting Relations of the Deviated Path

Andersen (1991)’s objection was very effective in the literature that some
authors (Malkov, 2003, 2007; Torres et al., 2010) also presented their
empirical mass-luminosity diagrams without a curve fitting to the data.
Gafeira et al. (2012) just gave mass-luminosity relations for main sequence
FGK stars without displaying them on a mass-luminosity diagram. Fernandes et
al. (2021), who compared their results to Torres et al. (2010), could be
considered an improved version of Gafeira et al. (2012), thus also did not
display mass-luminosity relations on the mass-luminosity diagram.

Various kinds of mass-predicting empirical relations were calibrated and most
of them were erroneously called MLR. 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. Henry & McCarthy (1993), who
preferred $\log M=aM_{\xi}+b$ for infrared colors, where $M_{\xi}$ indicates
absolute magnitudes at the $J$, $H$, and $K$ bands, and $\log M=aM_{\rm
V}^{2}+bM_{\rm V}+c$ for the $V$ band to express various MLR with unknown
coefficients $a$, $b$, and $c$ to be determined by the data on various
diagrams. The former relation in the form $M_{\rm Bol}=a+b\log M$ could be
justified to be named mass-luminosity relation since $M_{\rm Bol}$ of a star
is directly related to its luminosity, but a relation in the form $\log
M=aM_{\xi}+b$, is definitely not a mass-luminosity relation. Such relations
should better be called mass-absolute brightness relations to avoid confusion.

Style of expressing $V$-band mass-absolute brightness relation as second-
degree polynomials covering masses $0.6~{}M_{\odot}$ to 22.89 $M_{\odot}$ is
continued by Xia & Fu (2010) who calibrated it within two regions one for
$M_{\rm V}<1.05$ mag ($2.31<M/M_{\odot}<22.89$) and other for $M_{\rm V}>1.05$
mag ($0.60<M/M_{\odot}<2.31$) using the dynamical masses and $V$-band absolute
magnitudes of 203 main-sequence stars, but still the relation is called MLR.
The accuracy of predicted masses is estimated to be within 5%. The style of
displaying mass ($M/M_{\odot}$) versus absolute magnitude data, where the
absolute magnitudes are in the visual and the $K$-bands is continued by
Benedict et al. (2016) for the low end of the main sequence ($M<0.6M_{\odot}$)
but in different functional forms predicting masses as accurate as $\pm
0.035M_{\odot}$ in the region $M=0.2M_{\odot}$. Similarly, both direct
(absolute magnitude for a given mass) and inverse (mass for a given absolute
magnitude) relations of Benedict et al. (2016) are better to be called
empirical mass-brightness relations rather than mass-luminosity relations. The
empirical data of Benedict et al. (2016) are also compared to the theoretical
mass-absolute brightness curves of age 1 Gyr of Baraffe et al. (2015) and
Dotter (2016). $K$-band data was found to fit better than the $V$-band data.

Mass-Absolute visual magnitude $[M_{\rm V}-(\log m)]$, mass-luminosity $[\log
L-(\log m)]$, mass-temperature $[\log T_{\rm eff}-(\log m)]$, and mass-radius
[$\log R-(\log m)]$ relations and corresponding inverse relations are
calibrated for the intermediate-mass stars in the mass range
$1.4<M/M_{\odot}<12$ by Malkov (2007) in forms of polynomials with empirically
determined coefficients from the fundamental parameters of stars collected
from DDEB and visual binaries. Similar empirical relations were studied later
in a wider perspective by Moya et al. (2018) using a set of observational
parameters $M$, $R$, $T_{\rm eff}$, $g$ (surface gravity), $\rho$ (mass
density), and [Fe/H] of 934 stars of eclipsing binaries and single stars
observed by asteroseismology and interferometry where the two-thirds of the
stars are on the main sequence. A total of 576 linear combinations of $T_{\rm
eff}$, $L$, $g$, $\rho$, and [Fe/H] (and logarithms) were used as independent
variables to estimate $M$ and $R$, but Moya et al. (2018) presented only 38 of
them with regression statistics adj-$R^{2}$ higher than 0.85. Accuracy better
than 10% was achieved in almost all cases of 38 equations. The term “empirical
relations” used by Moya et al. (2018), and the names of relations given by
Malkov (2007) are proper and consistent in comparison to the names used for
the mass predicting relations in the previous paragraph.

Using $\log L/L_{\odot}$, [Fe/H] and star age/age⊙ as free variables, Gafeira
et al. (2012) suggested three different equations so that one can choose one
according to the availability of data. The simplest is the first one which is
a third-degree polynomial without a constant term, having $X$, where
$X\equiv\log L/L_{\odot}$. The second equation adds [Fe/H] as the new variable
in another third-degree polynomial into the equation in addition to the first.
The third equation combines three third-degree polynomials, where the third
uses the relative age (age/age⊙) of the star as a variable. With all
parameters, the third relation looks like equation (2) above by Torres et al.
(2010). That is, the part $a_{1}+a_{2}X+a_{3}X^{2}+a_{4}X^{3}$ is replaced by
$0.0219(\pm 0.023)\log L/L_{\odot}+0.063(\pm 0.060)(\log
L/L_{\odot})^{2}-0.119(\pm 0.112)(\log L/L_{\odot})^{3}$ while the part
$a_{7}$[Fe/H] is replaced by $+0.079(\pm 0.031){\rm[Fe/H]}-0.122(\pm
0.119){\rm[Fe/H]}^{2}-0.145(\pm 0.234){\rm[Fe/H]}^{3}$ at last the part
$+a_{5}(\log g)^{2}+a_{6}(\log g)^{3}$ is replaced by $+0.144(\pm 0.062)({\rm
age/age_{\odot}})-0.224(\pm 0.104)({\rm age/age_{\odot}})^{2}-0.076(\pm
0.045)({\rm age/age_{\odot}})^{3}$. Adding age and metallicity improved the
mass estimation (15% to 5%) for FGK stars (Gafeira et al., 2012). However,
only the first equation could be called MLR, but not the other two, which are
better to be called mass-luminosity-metallicity relation and mass-luminosity-
metallicity-age equation respectively. Nevertheless, Fernandes et al. (2021)
corrected this by changing the name “the mass and radius-luminosity-
metallicity-age relations” in the new version which was calibrated by 56 stars
with metallicity and mass in the ranges ${\rm-0.34<[Fe/H](dex)<0.27}$ and
$0.66<M/M_{\odot}<1.8$ from the DEBCat
catalogue111https://www.astro.keele.ac.uk/jkt/debcat/ . Estimated accuracy in
the new version is 3.5% and 5.9% in predicting $M$ and $R$ of single stars,
respectively. Serenelli et al. (2021) commented “Gafeira et al. (2012)
provided three relations for the stellar mass, but only two of them can be
easily applied”.

Serenelli et al. (2021) studied mass determination methods from a wider
perspective including all existing techniques older and newer; involving
spectroscopy and/or photometry; theoretical or observational and at last
summarized them in a figure (Fig. 16 of Serenelli et al., 2021) graphically
where one can see their applicable ranges of stellar mass as well as the
accuracy and precision of the predicted masses. Despite deceiving examples as
summarized in the three paragraphs above, Moya et al. (2018) were careful not
to use the word “MLR” for naming their “empirical relations”, which are
established to estimate $M$ and $R$ only. While Fernandes et al. (2021)
constantly called their simiar functions MLR and claimed “for single nearby
Solar-type stars, the luminosity can be obtained observationally, but not the
mass” opposing the fact that the luminosity of a star is actually not an
observable parameter because there is no telescope/detector to observe at all
wavelengths. Calling these $M$ or $R$ predicting relations ML/MR would be a
main source of confusion among readers. For example, “empirical ML/MR
relations” marked in Figure 16 of Serenelli et al. (2021) at an
accuracy/precision in between 10% and 15% for stars $M<2.5M_{\odot}$, where in
the caption “ML/MR for mass-luminosity and mass-radius relations” written
clearly, could be indeed confusing to a careful reader.

This is because: Is it possible Serenelli et al. (2021) to use ML/MR to
indicate both of the empirical relations of Moya et al. (2018) and the
classical mass-luminosity/mass-radius relations (MLR/MRR) of Eker et al.
(2018)? According to Figure 1 (Figure 16 of Serenelli et al., 2021) the answer
would be “yes” because the relations of Moya et al. (2018) are not marked as
“empirical relations” on the figure, and it is not possible to forget them
because they cover one of the very important sections of the review (Section
4.4) where the $M$ predicting relations of other authors are summarized and
compared to their counterparts in Moya et al. (2018) (see Table 1). However,
Serenelli et al. (2021) declared “All the relations except two (those with the
largest number of dimensions) have precisions better than 5%”. If the answer
is "yes", then, another problem arises: “If ML/MR in Figure 1 implies both
classical MLR/MRR and the new M/R predicting relations, why the precision for
the ML/MR is marked to be from 10% to 15% while M/R predicting relations are
said to have precisions better than 5%?”.

Figure 1:  Mass ladder summarizing the capacity of various methods to obtain
stellar masses (credit to Serenelli et al., 2021).

It is possible ML/MR in Figure 1 were actually marked from 3% to 15%, but this
region on the figure is shadowed by other various methods of predicting
stellar mass. Then another conceptual problem comes out: Only similar
quantities could be compared. it is not right to compare the
accuracy/precision of the empirical relations of Moya et al. (2018), which are
solely devised for predicting masses of single stars, to the
accuracy/precision of the classical MLR, which is primarily devised to
establish as a statistical relation between typical masses and luminosities of
main-sequence stars. It is inconsistent and meaningless to compare E18 to M18
in Table 9 of Serenelli et al. (2021) (see Table 1). Similarly, it is
scientifically inconsistent if empirical ML/MR in Figure 1 indicates classical
MLR/MRR as well, rather than the empirical relations of Moya et al. (2018).
This is because Eker et al. (2018) declared “(classical) MRR and MTR
functions, as well as the MLR functions, are needed by the astronomical
community for practical purposes. Those include the need to be able to
estimate a typical luminosity, radius, and $T_{\rm eff}$ for main-sequence
stars of a given mass”. Which is definitely different than the purpose of
predicting $M$ or $R$ of single stars all over the H-R diagram?

After having $M$ and $R$ predicting relations, further calculations are needed
for a meaningful comparison. First calculate $M$, $R$, and then $L$ (if not
used as input variable) for main-sequence stars only. Then, plot $\log M-\log
L$ and $\log M-\log R$ diagrams, and fit preferred functions (ML or MR) using
the least squares method. At last ML/MR becomes comparable to MLR/MRR of Eker
et al. (2018). Why bother to do that, if there are many reliable direct
methods of obtaining $M$, $R$, and $L$ of main-sequence stars within accuracy
1% and a few % respectfully from radial velocity and light curves of DDEB
stars which are marked in the bottom of Figure 1 covering the full ranges of
stellar masses. Thus, ML/MR is incompatible with classical MLR/MRR (Figure 1).

Table 1:  Mass predicting empirical relations of other authors are compared to
counterparts in Moya et al. (2018) (credit to Serenelli et al., 2021).

Ref. | Relation | Acc/Prec | Ref. | Corresponding relation | Acc/Prec
---|---|---|---|---|---
T10 | $M=f(X,X^{2},X^{3},{\rm log}^{2}g,{\rm log}^{3}g,{\rm[Fe/H]})$ | 7.4/52.9 | M18 | $M=f(T_{\rm eff},{\rm log}g,{\rm[Fe/H]})$ | 7.5/3.4
G12 |  $M=f({\rm log}L,{\rm log}^{2}L,{\rm log}^{3}L$) | 14.0/0.6 | M18 | ${\rm log}M=f({\rm log}L)$ | 10.1/0.1
G12 | $M=f({\rm log}L,{\rm log}^{2}L,{\rm log}^{3}L,{\rm[Fe/H]},{\rm[Fe/H]}^{2},{\rm[Fe/H]}^{3})$ | 8.9/0.8 | M18 | ${\rm log}M=f({\rm log}L,{\rm[Fe/H]})$ | 9.9/0.9
M07 | $M=f({\rm log}L,{\rm log}^{2}L)$ | 11.2/— | M18 | ${\rm log}M=f({\rm log}L)$ | 10.08/0.13
E18 | ${\rm log}L=f({\rm log}M)$ | 33.3/6.9 | M18 | ${\rm log}L=f({\rm log}M)$ | 31.9/0.6

References: T10 (Torres et al., 2010), G12 (Gafeira et al., 2012), M07
(Malkov, 2007), E18 (Eker et al., 2018), M18 (Moya et al., 2018).

####  2.1.3 Recognition of statistical MLRs

The classical form of MLR ($L\propto M^{\alpha}$) appreciated by Cester et al.
(1983), Griffiths et al. (1988), Karetnikov (1991) and Demircan & Kahraman
(1991) before the break initiated by Andersen (1991). This classical form
continued by Ibanoǧlu et al. (2006) when comparing mass-luminosity relations
for detached and semidetached Algols despite Andersen (1991)’s exclamation.
The most recent examples are by Eker et al. (2015, 2018). The classical or any
other form reducible to the classical ($L\propto M^{\alpha}$) has the
advantage of being easy to interpret with a value given to the power of $M$,
which is known to change by energy generation rate per star mass at cores of
stars. Thus, the derivative of MLR function ($dL/dM$), the inclination of a
line, on the $\log M-\log L$ diagram is the value of alpha. On the other hand,
the real advantage is not only that it works in both directions ($M$ from $L$,
or $L$ from $M$), but also because it permits one to relate typical masses and
luminosities of main-sequence stars in general.

Obviously, the statistically determined relation between $M$ and $L$ for the
main-sequence stars is not true for a single star. This is because $L$, $R$,
and $T_{\rm eff}$ of a star change by time, while $M$ of the star stays
constant (an evolutionary effect) since mass loss of main-sequence stars
(especially for the ones cooler than B spectral types) are too small thus mass
loss, is usually ignored (Daszyńska-Daszkiewicz & Miszuda, 2019; Bressan et
al., 2012). Moreover, there is a metallicity effect that also changes $L$,
$R$, and $T_{\rm eff}$ slightly, therefore, stellar structure and evolution
models use chemical composition and $M$ as the two basic free parameters for
computing $L$, $R$, and $T_{\rm eff}$, which stands for an output of an
internal structure model to be confirmed externally. Finally, it can be
concluded that any kind of relation between $M$ and $L$ of a single star
cannot easily be deduced from evolutionary tracks or from isochrones. However,
the distribution of available stellar luminosities on H-R diagrams or on $\log
M-\log L$ diagrams clearly shows that there must exist, at least a
statistical, one-to-one relation between a typical mass and typical luminosity
of main-sequence stars. But, on the other hand, if $L$ and $M$ were totally
independent, identification of main-sequence stars on the H-R diagram would
not be possible. In fact, the main-sequence stars were first recognized just
according to their extraordinary positional appearance even on very primitive
H-R diagrams. Because of their distinct positions, they were named main-
sequence stars which is still actively used. After all, it is obvious to
everyone now, that as soon as main-sequence luminosities are placed on a $\log
M-\log L$ diagram, a one-to-one relation between $M$ and $L$ shows itself
clearly with a high-level statistical significance.

What actually happened so far is that; the statistical relation between
stellar $M$ and $L$ for main-sequence stars is so strong and obvious that it
was discovered even before the stellar structure and evolution theory was
fully established. That was the first reason why it had been evaluated as one
of the fundamental secrets of the cosmos like the H-R diagram itself (why are
stars not distributed evenly on the surface of the H-R diagram but mostly
gathered on the main sequence?). The second reason was that the observational
accuracy of those early years of astrophysics was not sufficient to
distinguish it as a statistical relation, so it was treated as a fundamental
relation even though some serious objections mentioned as above have occurred.
Even so, no one has yet clearly declared that MLR is just a statistical
relation.

Andersen (1991) totally rejected the existence of a real relation between $M$
and $L$ of the main-sequence stars because according to him a unique relation
(like a Planck law, keeping $T_{\rm eff}$ constant, the deviations from the
Plank curve are only due to random errors of observed intensities) does not
exist since deviations from a fitting curve (MLR) is not just only due to
random observational errors of $L$ and $M$, but also due to chemical
composition and age differences. Eker et al. (2015, 2018) defended existence
of a MLR relation by claiming any sample of data could be expressed by a
unique relation because a unique fit of a curve on sample data is guaranteed
by the least squares method.

However, this defence occurred intuitively because the distinction between MLR
to be a fundamental or a statistical relation was not yet fully founded. Even
Eker et al. (2015) said “One of the fundamental secrets of the cosmos, the
famous stellar mass–luminosity relation (MLR), was discovered empirically …”.
Only later, Eker et al. (2018) declared “The main-sequence MLR is one of the
fundamentally confirmed and universally recognized astronomical relations”.

How to solve non-uniqueness problems attributed to the light curves of spotted
stars, which is a kind of problem Andersen (1991) pointed out against MLR, was
discussed first by Eker (1999). The same principles adopted by Eker et al.
(2018) for defending the uniqueness of MLR, MRR, and MTR functions from the
most accurate masses and luminosities of 509 main-sequence stars as the
components of DDEB in the Solar neighbourhood of the Milky Way. There could be
three types of non-uniqueness problems. According to Eker et al. (2018), there
is no non-uniqueness problem of type I because the preferred function
expressing MLR is a power law ($L\propto M^{\alpha}$). There should not be a
non-uniqueness problem of type II also because there exist many methods like
the least squares for one to achieve a unique fit and determine its
coefficients uniquely. Finally, the problem of type III does not exist either
because the parameter space of MLR is so simple that there is always one
correlation between $L$ and $M$, that is, there is only one $L$ value for a
given $M$ and vice versa in the case of inverse MLR. The most general approach
to non-uniqueness problems recently applied to main sequence BC-$T_{\rm eff}$
relations by Eker et al. (2021b) when discussing the chronic zero-point
problems of the BC scale, which is important for obtaining accurate $L$ of
single stars from apparent magnitudes if distances are known.

Apparently, there is no non-uniqueness problem associated with classical MLR,
but still one may run into problems of obtaining accurate $L$ from a given $M$
or vice versa. Such problems are inevitable since classical MLR is not devised
solely for obtaining accurate $L$ from an accurate $M$ or vice versa. The main
obstacle for obtaining accurate $M$ of a single star from its $L$ using
classical MLR is not because of the absence of a unique function (Andersen,
1991), but because of the degeneracy induced by the stellar structure and
evolution theory (Eker et al., 2018). This is because, theoretically, there
are an infinite number of $L$ values for a main-sequence star of given $M$
depending upon its chemical composition and age. Thus, varying chemical
composition and ages are not there to cause non-uniqueness to an existing MLR.
On the contrary, the chemical composition and age of a star is there to break
inferred degeneracy for obtaining accurate $L$, $M$, and $R$, which appear on
classical MLR and MRR functions as only variables. For this, one must conduct
further investigation on evolutionary tracks. Knowing the mass and the
chemical composition of a star, the correct track will be chosen. By knowing
the age, both $L$ and $R$ will be read on the track. If the age is not known,
then either $L$ or $R$ must be known to read the age of the star on the track.
Stellar structure and evolution theory is even useful to determine the
accurate mass of a single star from its accurately determined [Fe/H] and $L$,
where the value of $R$ could also be obtained if $T_{\rm eff}$ of the star is
known accurately.

Without distinguishing them from a fundamental relation, Eker et al. (2015)
calibrated four linear MLR functions covering mass range
$0.38<M/M_{\odot}<32$, using masses and luminosities of 268 main-sequence
stars selected from 514 stars as the components of 257 DDEB from the catalogue
of Eker et al. (2014). Three distinct break points separating the four
distinct mass domains on a $\log M-\log(L/M)$ diagram were identified. The
mass domains were named as low mass ($0.38<M/M_{\odot}\leq 1.05$),
intermediate mass ($1.05<M/M_{\odot}\leq 2.4$), high mass
($2.4<M/M_{\odot}\leq 7$) and very high mass ($7<M/M_{\odot}\leq 32$). The
linear MLRs of the four mass domains were compared to linear and quadratic MLR
of the full range $0.38<M/M_{\odot}<32$ and the four-piece linear MLRs were
found best to represent the data on a $\log M-\log L$ diagram. The break
points separating the mass domains were interpreted as abrupt changes in the
power of $M$, most probably due to changing the type of efficient nuclear
reaction operating in the cores of main-sequence stars.

The statistical nature of MLR was sensed but not fully grasped yet by Eker et
al. (2018). The referees were against calibrating new MLRs which are useless
in predicting $M$ and $R$ of single stars. The new trend in determining MLRs
was to use them to obtain $M$ and $R$ of single stars as done by Andersen
(1991), Henry (2004), Malkov (2007), Torres et al. (2010), and Gafeira et al.
(2012). Therefore, the justifications for re-calibrating them again after only
three years were explained by Eker et al. (2018) as: There are two tables in
the handbook of astronomers, commonly known as “Allen’s Astrophysical
Quantities” (Cox, 2000). The first of the two tables list calibration of MK
spectral types with seven columns spectral type, $M(V)$, $B-V$, $U-B$, $V-R$,
$R-I$, $T_{\rm eff}$ and BC (Table 15.7 on page 388). The second table with
six columns Sp type, mass, radii, surface gravity, mean mass density and
rotational speed (Table 15.8 on page 389) is actually the continuation of the
first table. The columns of the two tables, thus, were connected by spectral
types. The second table has a notification mark to say “columns containing
uncertain values”, which actually indicates statistically determined typical
masses, radii, surface gravity and mean mass for a given spectral type. The
rationale of the new paper was obvious that interrelated MLR, MRR, and MTR
would be very useful in supplying demanded reliable statistical information to
the astronomical community.

After three years, Eker et al. (2018), this time, calibrated a six-piece MLR
covering the mass range $0.179<M/M_{\odot}<31$, using 509 main-sequence stars
selected from 639 stars as the components of 318 pair (DDEB) and one detached
spectroscopic triple. One should not be surprised by the higher increasing
rate of the calibrating stars, which appears to be 90% while the number of
newly added DDEB binaries is only 24%. This is because stars with less
accurate $M$ and $R$ up to 15% were used in the calibrations rather than up to
3% as in the previous study. Thus, Eker et al. (2018) concluded that it is not
always good to eliminate less accurate data for a better study because it may
mean loss of information rather than gain. Lowering limiting relative accuracy
to 15% was a real contribution to extending the low mass limit down to
$0.179M_{\odot}$ from $0.38M_{\odot}$ and discovery of two more break points
on the $\log M-\log L$ diagram. That is, adding new DDEB stars (24%) to the
list was not effective as much as lowering the limiting accuracy.

The distribution of the luminosities of 509 main-sequence stars on a $\log
M-\log L$ diagram is shown in Figure 2, where the vertical lines mark the
positions of the break points. Between the break points there is a linear MLR
luminosities of that domain, thus there exists six-piece MLR to cover the full
range $0.179<M/M_{\odot}\leq 31$. A six-degree polynomial, shown by a blue
dotted line to fit the full range, was found producing a best fit better than
the higher and the lower degree polynomials to explain the full range data by
a single function. It is obvious in Figure 2 that the six-piece MLRs are even
better at representing stellar luminosities than any polynomial of any degree.

Figure 2: The dotted (blue) line is a sixth-degree polynomial, solid (red)
lines are classical MLRs, and the vertical lines are the break points
separating mass domains where the linear lines were fitted (credit to Eker et
al., 2018). Figure 3: Mass domains and classical MLRs representing stellar
masses and luminosities in each domain. 6th degree polynomial representing all
data (same as the dotted blue line in Figure 2) is not as successful as linear
MLRs in low-mass domains ($M<1.05$) and very high-mass domain ($M>7$). Data
accuracy is: (o) very accurate $<3\%$, ($+$) accurate (3-6%), and ($\times$)
less accurate (6-15%) (credit to Eker et al., 2018).

The six-mass domains, which are named ultra-low mass ($0.179<M/M_{\odot}\leq
0.45$), very-low mass ($0.45<M/M_{\odot}\leq 0.72$), low mass
($0.72<M/M_{\odot}<1.05$), intermediate mass ($1.05<M/M_{\odot}\leq 2.4$),
high mass ($2.4<M/M_{\odot}\leq 7$) and very-high mass ($7<M/M_{\odot}\leq
31$), and their best fitting MLR functions are shown in Figure 3, where they
were also compared to a single MLR function of sixth degree covering the whole
mass range $0.179<M/M_{\odot}\leq 31$. Analytical expressions, and statistical
parameters $N$ (number of stars in the domain), $R^{2}$ (correlation
coefficient), $\sigma$ (standard deviation), and $\alpha$ (the power of $M$)
as the inclination of a linear MLR on a $\log M-\log L$ diagram is listed in
Table 2.

Table 2:  Classical MLRs for main-sequence stars at various mass domains (credit to Eker et al., 2018). Domain | $N$ | Mass Range | Classical MLR | $R^{2}$ | $\sigma$ | $\alpha$
---|---|---|---|---|---|---
Ultra low-mass | 22 | $0.179<M/M_{\odot}\leq 0.45$ | $\log L=2.028(135)\times\log M-0.976(070)$ | 0.919 | 0.076 | 2.028
Very low-mass | 35 | $0.45<M/M_{\odot}\leq 0.72$ | $\log L=4.572(319)\times\log M-0.102(076)$ | 0.857 | 0.109 | 4.572
Low mass | 53 | $0.72<M/M_{\odot}\leq 1.05$ | $\log L=5.743(413)\times\log M-0.007(026)$ | 0.787 | 0.129 | 5.743
Intermediate mass | 275 | $1.05<M/M_{\odot}\leq 2.40$ | $\log L=4.329(087)\times\log M+0.010(019)$ | 0.901 | 0.140 | 4.329
High mass | 80 | $2.4<M/M_{\odot}\leq 7$ | $\log L=3.967(143)\times\log M+0.093(083)$ | 0.907 | 0.165 | 3.967
Very highmass | 44 | $7<M/M_{\odot}\leq 31$ | $\log L=2.865(155)\times\log M+1.105(176)$ | 0.888 | 0.152 | 2.865

###  2.2 Revisiting MRR

Although it appears to be natural to sense a relation between $M$ and $R$ of
main-sequence stars right after the discovery of MLR by Hertzsprung (1923) and
Russell et al. (1923), the empirical MRR of main-sequence stars did not appear
in the literature for another one and a half decades. Kuiper (1938), who was
suspicious about MLR to be one of the fundamental relations, also plotted a
$\log M-\log R$ diagram but had no discussion about it. Only after mid of the
20th century, the studies discussing empirical stellar mass-radius relation
(MRR) begin to appear in the literature (McCrea, 1950; Plaut, 1953; Huang &
Struve, 1956; Lacy, 1977, 1979; Kopal, 1978; Patterson, 1984; Gimenez &
Zamorano, 1985; Harmanec, 1988; Demircan & Kahraman, 1991).

If $M$ and $L$ are related, why not $M$ and $R$ since $L$ is already known to
be related to the square of $R$ according to the Stefan-Boltzmann law? The
main obstacle seems to be the difficulty of accessing reliable stellar radii.
McCrea (1950) appears to be one of the first founders of the idea for two
reasons: i) Using the light curve solutions of eclipsing binaries, not
necessarily all being detached, that is, including W UMa binaries too, Plaut
(1953) has drawn a diagram in the manner proposed by McCrea (1950). ii)
Readers were referred to McCrea (1950) for theoretical interpretation of the
relation in the form $\log(M/R)=a+b\log M$ suggested by Plaut (1953). Using
the parameters of 30 systems with both components on the main sequence chosen
among 130 eclipsing binaries of all kinds compiled by Plaut (1953), the
coefficients of the relation were rectified as $a=-0.058\pm 0.026$,
$b=0.335\pm 0.29$ with a dispersion of $\pm 0.11$ according to the least
squares.

The masses and radii of the eclipsing stars compiled by Plaut (1953) had been
studied by Huang & Struve (1956), who plotted them on the two separate
spectral type-$\log R$ diagrams, on which both components smaller thus not
touching, both components are touching, primary is touching but secondary is
not, and the secondary is touching but the primary is not touching the inner
contact surfaces were marked by special symbols for each. The two separate
diagrams made Huang & Struve (1956) believe that the sub-giant components in
such systems could have evolved from early main-sequence stars.

An empirical MRR in the form $\log R(M)=8.495-0.2M_{\rm Bol}(M)-2\log T_{\rm
eff}(M)$ suggested by Hoxie (1973) for the low mass stars ($M<1M_{\odot}$) in
the Solar neighbourhood. Comparing this MRR with low mass model calculations
of stellar $R$, Hoxie (1973) announced a discordance implying theoretical
radii are 30% or smaller than the observationally derived radii in the mass-
radius plane for stars of mass less than $0.5M_{\odot}$. Afterward, a method
of estimating $R$ of nearby stars was declared by Lacy (1977). Being based on
the Barnes-Evans relation and being free of assumptions of spectral types,
luminosity class, effective temperature, or bolometric correction, this method
is applied to nearby single and double stars with accurate parallaxes and
$V-R$ photometry. The double stars (visual binaries) were useful to supply
components $M$ and $R$, where $R$ needed to be compared with predicted $R$ by
the new method with Barnes-Evans relation. Additional comparison was also
possible to Lacy (1977) using $R$ from non-contact eclipsing binaries with
well-determined dimensions selected from Batten (1967) and Koch et al. (1970).
Having a sufficient number of accurate masses and radii (both 5% or better)
from the visual systems and non-contact eclipsing binaries, Lacy (1977) was
also able to produce the most reliable $\log M-\log R$ diagram so far, where
the empirical relation of Hoxie (1973) was plotted with a special symbol
different from the symbols of visual binaries, eclipsing systems undergone
mass exchange and without any mass exchange.

Therefore, Lacy (1977) suggested $\log R=0.640\log M+0.011$ for the region
$0.12\leq\log M\leq 1.30$ and $\log R=0.917\log M-0.020$ for the region
$-1.00\leq\log M\leq 1.30$ as two MRR functions to indicate zero age main
sequence (ZAMS), where $R$ and $M$ are in Solar units. Lacy (1977) interpreted
the break point as a signal crossing over from the region of p-p chain to the
C-N-O cycle. The terminal age main sequence (TAMS) line was deduced from the
models of Iben (1967) and Paczyński (1970) marked on the $\log M-\log R$ as a
dashed line, while ZAMS is shown by a continuous line with a break at $1.3\pm
1M_{\odot}$. Theory and observation were found in good agreement for the stars
$M\gtrsim 1M_{\odot}$, but the models of $M$ dwarfs having 25% smaller radii
than real stars. Another line of development was using the Catalogue of the
Elements of Eclipsing Binaries (Kopal & Shapley, 1956), where Kopal (1959)
obtained a good statistical definition of the MRR. He found a linear relation,
in logarithmic scales, for the values of individual stars irrespectively being
primary or secondary, but the slope is different for massive and less massive
stars with a transition at $\sim 2M_{\odot}$ (Kopal & Shapley, 1956; Kopal,
1978). Popovici & Dumitrescu (1974), who compiled data from Kopal & Shapley
(1956) and Svetchnikoff (1969)222Catalog Orbitalnii Elementov, Mass i
Svetimostii Tesnii Dvornik Svezd, Sverdlovsk, were mainly interested in the
radius-luminosity diagram. Five mean empirical mass-radius relations had been
constructed by Habets & Heintze (1981). The number of MRR is five because it
is calibrated not only for main-sequence stars but also for luminosity class
IV (subgiants), III (giants), $V_{\odot}$ (ZAMS) and $EV_{\odot}$ (EZAMS)
using visual and eclipsing binaries which are at the same time double-lined
spectroscopic systems collected by themselves. Patterson (1984) was
unsatisfied by the MRRs suggested before him, thus he proposed the empirical
ZAMS mass-radius relation in the form $R=\alpha M^{\beta}$, where $M$ and $R$
in Solar units and $\alpha=1$, $\beta=0.88\pm 0.02$ for the region $0.1\leq
M/M_{\odot}\leq 0.8$ and $\alpha=0.98$, $\beta=1.00$ for the region $0.8\leq
M/M_{\odot}\leq 1.4$ by a quotation “poorly defined for $M\leq 0.4M_{\odot}$”
for himself to apply it to cataclysmic variables. Gimenez & Zamorano (1985)
preferred using the Catalogue of Stellar Masses and Radii published by Popper
(1980) for studying the classical MRR on $\log M-\log R$ diagram. Using
reliable $M$ and $R$ from OB and B6-M detached eclipsing and visual binaries
and resolved spectroscopic binaries, Gimenez & Zamorano (1985) determined
$a=0.041\pm 0.011$ and $b=0.749\pm 0.011$ for an MRR in the form $\log
R=a+b\log M$ with a correlation coefficient 0.96.

It must be noted that Lacy (1977, 1979), Gimenez & Zamorano (1985) and
Demircan & Kahraman (1991) presented their $\log M-\log R$ diagrams without
fitting a MRR function to data, but with a ZAMS line as the lower limit, and
with a TAMS line as the upper limit of main-sequence stars which are estimated
by the help of theoretical stellar structure and evolution models. The most
recent example is by Eker et al. (2018) is shown in Figure 4, where it is
clear that the statistical MRR does not appear as strong as classical MLR
expressed on $\log M-\log L$ diagram (Figure 3). Looking at the difference of
data distribution on both diagrams $\log M-\log L$ and $\log M-\log R$,
therefore, it is not difficult to understand why $\log M-\log R$ diagram is
presented without a fitting curve like Figure 3.

Figure 4: Main sequence mass-radius diagram of DDEB stars. ZAMS and TAMS lines
are from PARSEC models of Bressan et al. (2012) (credit to Eker et al., 2018).
The vertical line is a division at $M=1.5M_{\odot}$ ($\log M=0.176$).

Because of the very large scatter caused by the evolution of stars $M\gtrsim
1.3M_{\odot}$ ($\log M>0.15$), data of main-sequence stars on a $\log M-\log
R$ diagram are not suitable to be expressed by a function or a curve.
Obviously, plotting a ZAMS MLR on a $\log M-\log R$ diagram permits one to see
the evolution of stellar radii for main-sequence stars $M\gtrsim
1.3M_{\odot}$.

Despite its hassle-free appearance, tight and narrow distribution of stellar
$L$ on a $\log M-\log L$ diagram for the full mass range of main-sequence
stars, different forms of MLR functions causing confusion were suggested,
calibrated, and shown together with data. Looking at the already suggested
main sequence MLR and MRR functions so far, one should see another noticeable
difference. Being contrary to different forms of MLR suggested by different
study groups, main sequence MRR in the form $\log R=a+b\log M$ happened to be
common. The form $R\propto M^{\beta}$ suggest by Patterson (1984), Demircan &
Kahraman (1991) and Karetnikov (1991), appears different, but it is not. The
same form of function is expressed once in logarithmic form while the other is
a power law.

Despite such obvious differences (appearance on a diagram, formal expression),
similarities between MLR and MRR studies also exist. Andersen (1991)’s
exclamation against classical MLR also affected MRR studies, thus modern MRR
also appears to be deviated from the main path. Older MRR studies aimed to
calibrate an MRR relation for two practical purposes: i) to get an answer to
the question: how mean (or typical) $M$ and $R$ of main-sequence stars are
related? ii) to estimate $R$ of a star for a given $M$, or vice versa. The
former is useful for making models that require mean $M$ and $R$, the latter
is useful for guessing the $R$ of a star from its $M$, or vice versa. Models
using mean $M$ and $R$ as well as mean $L$, have larger application area not
only in stellar astrophysics but also Galactic and extra-galactic studies.

The exclamation of Andersen (1991), unfortunately, caused the first aim to be
neglected. As explained above, in the new trend of calibrating MLR, there is
only one aim: to obtain the most accurate $M$ and $R$ together from the other
observed parameters of the star. Thus, Malkov (2007), Torres et al. (2010),
Moya et al. (2018), and Fernandes et al. (2021) have not only calibrated a
single relation for predicting $R$ of a single star, but also at least one
more relation useful to deliver $M$ of the same star. Thus, the new trend of
MRR cannot be considered independent of the new trend of MLR appearing after
Andersen (1991).

###  2.3 Revisiting MTR

If $M$ and $L$ are related, why not $M$ and $T_{\rm eff}$ since $L$ is known
to equal surface area times surface flux, where the flux is proportional to
the fourth power of the effective temperature. A relation even stronger than
mass-radius is expected because the power of $T_{\rm eff}$ is two times bigger
than the power of $R$ in the Stefan-Boltzmann law. Its calibration and usage
also appear easier than the calibration and usage of MRR relation. Despite
all, if the spectral type-effective temperature (horizontal axis of H-R
diagram) relation is not counted, it took almost six decades for an MTR to be
seen in the literature.

First empirical mass-$T_{\rm eff}$ relation is studied by Habets & Heintze
(1981) and shown on a $\log M-\log T_{\rm eff}$ diagram for main sequence-
stars after studying newly calibrated mass-spectral type and spectral
type-$T_{\rm eff}$ relations from the main-sequence components of eclipsing
binaries. Empirical relation was shown together with two theoretical
mass-$T_{\rm eff}$ relations of Stothers (1974, for a metallicity $Z=0.02$
with $Y=0.49$ and 0.25) and the one by Demarque & Gisler (1975, for $Z=0.02$
and $Y=0.25$). The form of the relation was not given. Karetnikov (1991) later
determined coefficients of mass-$T_{\rm eff}$ relation in the form $\log
T_{\rm eff}=a+b\log M$ for six different kinds of eclipsing systems using
absolute parameters of 303 eclipsing binaries of different types with varying
and constant orbital periods without showing them on diagrams. The relations
were determined for primaries and secondaries separately, thus coefficients of
24 relations were determined and listed without showing them on $\log M-\log
T_{\rm eff}$ diagrams.

Apparently, MTR relation had also been affected by Andersen (1991)’s objection
to MLR because Malkov (2007) announced a direct $[\log T_{\rm eff}-(\log M)]$
and an inverse $[\log M-(\log T_{\rm eff)}]$ mass-effective temperature
relations together with his $M$ and $R$ predicting relations and their inverse
functions. Like $\log M-\log L$ diagram without a fitting curve, his $M$
versus $\log T_{\rm eff}$ diagram too presented without a fitted curve. Being
not interested in calibrating an MTR, Moya et al. (2018), on the other hand,
used $T_{\rm eff}$ as one of the free parameters for predicting $M$ and $R$ of
single stars. Moya et al. (2018), calibrated 38 relations: 18 for $M$ and 20
for $R$. This is very much similar to Torres et al. (2010) who suggested only
two relations using $T_{eff}$, $\log g$ and [Fe/H] as free parameters, one for
obtaining $M$ one for obtaining $R$, to replace classical MLR claimed
inadequate to provide stellar $M$ by two reasons: i) It is a mean relation,
thus $M$ from $L$ is very inaccurate. ii) Scatter on $\log M-\log L$ diagram
is not only due to random observational errors of $M$ and $L$ but also due to
stellar age and chemical composition differences. As if, a reliable (or true)
MLR function must contain all of the parameters which introduce scattering.
Such a MLR function, however, cannot be drawn on a $\log M-\log L$ diagram.

On the other hand, following the old tradition, that is, looking for mean
relations for MRR and MTR, Eker et al. (2015) have compared $L$, $R$, and
$T_{\rm eff}$ distributions on $\log M-\log L$, $\log M-\log R$ and $\log
M-\log T_{\rm eff}$ diagrams. The first comparison between $\log M-\log L$,
$\log M-\log R$ has been commented as “the appearance of data on the $\log
M-\log R$ diagram is very different than the appearance on the $\log M-\log L$
diagram (compare Figures 2 and 4), which rather looks like a band of data
expressible by a function; however, with a very narrow distribution of radii
for masses $M<1M_{\odot}$ and a broad band of radii for stars with
$M>1M_{\odot}$, a single function to express a MRR would be odd and
meaningless”. Then, comparison between $\log M-\log R$, $\log M-\log T_{\rm
eff}$ were commented as “the temperature evolution within the main-sequence
band is not that obvious on the $M-T_{\rm eff}$ diagram. At first look, it
resembles the MLR”. It is indeed not like MRR, where the main-sequence
evolution of $R$ is obvious for $M>1M_{\odot}$ (compare Figures 2, 4, and 5).

Despite the distribution of data on $\log M-\log T_{\rm eff}$ diagram (Figure
5) resembles the distribution of data on $\log M-\log L$ diagram, Eker et al.
(2015) preferred not to calibrate a mean MTR since it would have been odd or
inappropriate that time to oppose literature where MLR and MRR are many, but
MTR is almost absent. Later, Eker et al. (2018) noted “Stefan–Boltzmann law
clearly indicates stellar luminosities are related to stellar radii and
effective temperatures. Having empirically determined the MLR and MRR
available, one is not free to determine another independent mass–effective
temperature relation (MTR)”. The three independently calibrated MLR, MRR, MTR
functions are not guaranteed to give consistent $L$, $R$ and $T_{\rm eff}$ for
a given $M$. The solution to the problem is in the next subsection.

###  2.4 Revisiting interrelated MLR, MRR, MTR

It has been noticed that the distribution of $R$ on the $\log M-\log R$
diagram (Figure 4) on the left of the vertical line ($M\leq 1.5M_{\odot}$) is
smooth and tight, that is, it is expressible by a simple function which is to
be called MRR. But the high mass region $M>1.5M_{\odot}$ due to faster
evolution, $R$ values are scattered very much, and thus, its band-like
appearance does not seem expressible by a curve of a function.

Despite its overall appearance resembling an MLR, the $T_{\rm eff}$
distribution on the $\log M-\log T_{\rm eff}$ diagram shows almost opposite
characteristics for one to choose the same mass domain to calibrate a MTR. One
can easily notice that the low-mass region $M\leq 1.5M_{\odot}$ (left of the
vertical line) in Figure 5 has a tight but wavy distribution, which is rather
not possible to fit a simple function. On the contrary, one can easily spot
the domain of high-mass stars $M>1.5M_{\odot}$ (right of the vertical line) in
Figure 5 with a sufficiently tight but smoothly varying distribution, which is
easily expressible by a function to be called MTR.

Figure 5:  Main sequence mass-effective temperature diagram of DDEB stars
(credit to Eker et al., 2018). The vertical line is a division at
$M=1.5M_{\odot}$ ($\log M=0.176$).

Having six linear MLRs already calibrated (Table 2, Figures 2 and 3) for the
main-sequence stars in the full range of observed stellar masses, probable
inconsistencies among the mean $L$, $R$, and $T_{\rm eff}$ values for a given
$M$ will be eliminated if the MRR was calibrated for the low-mass region
$M\leq 1.5M_{\odot}$ only and the MTR was calibrated for the high-mass region
$M>1.5M_{\odot}$ only. This is because the vertical line is a dimensionless
border between the low and the high mass regions in both $R$ and $T_{\rm eff}$
distributions as shown in Figures 4 and 5. Otherwise, with independently
calibrated MLR, MRR, and MTR functions, one will have a mean luminosity
($\langle L\rangle$), a mean radius ($\langle R\rangle$), and a mean effective
temperature ($\langle T_{\rm eff}\rangle$) for a given $M$. No one would know
which ones of the three mean values are wrong because the mean $\langle
L\rangle$ will not be equal to a mean surface area $\langle 4\pi
R^{2}\rangle=4\pi(\langle R^{2}\rangle)$ multiplied by a mean surface flux
$\langle\sigma T^{4}\rangle=\sigma(\langle T_{\rm eff}^{4}\rangle)$ for a
typical main-sequence star of given $M$. Choosing the most eligible regions on
$\log M-\log R$ and $\log M-\log T_{\rm eff}$ diagrams as compensating mass
domains for covering the full mass range, not only removes a probable
inconsistency but also guarantees the most trustable MLR and MRR for the low-
mass stars ($M\leq 1.5M_{\odot}$) and the most trustable MLR and MTR for the
high-mass stars ($M>1.5M_{\odot}$). Then, one could calculate consistent
$\langle L\rangle$, $\langle R\rangle$ and $\langle T_{\rm eff}\rangle$ for
the full range of main-sequence stars, which happens in two steps for both of
the mass domains. For the low-mass stars: i) Use MLR and MRR to calculate
$\langle L\rangle$ and $\langle R\rangle$ for a given mass, ii) use Stefan-
Boltzmann law to calculate $\langle T_{\rm eff}\rangle$ for the same mass from
its already computed $\langle L\rangle$ and $\langle R\rangle$. For the high-
mass stars: i) Use MLR and MTR to calculate $\langle L\rangle$ and $\langle
T_{\rm eff}\rangle$ for a given mass, ii) use Stefan-Boltzmann law to
calculate $\langle R\rangle$ for the same mass from its already computed
$\langle L\rangle$ and $\langle T_{\rm eff}\rangle$.

Utilizing the least squares, Eker et al. (2018) determined an empirical MRR
directly from $M$ and $R$ of 233 main-sequence stars for low-mass stars within
range $0.179\leq M/M_{\odot}\leq 1.5$, and an empirical MTR from $\log M$ and
$\log T_{\rm eff}$ of 276 main-sequence stars for high-mass stars within range
$1.5<M/M_{\odot}\leq 31$. Because the table giving the empirical MRR and MTR
by Eker et al. (2018) has misprints, the open forms of the functions,
correlation coefficients ($R^{2}$), and standard deviations ($\sigma$) from
Eker et al. (2021a) are given here. The empirical MRR in the form of quadratic
equation, where $R$ and $M$ are Solar units, is

$R=0.438(0.098)\times M^{2}+0.479(0.180)\times M+0.137(0.075),$ (3)

has $R^{2}=0.867$ and $\sigma=0.176$. The empirical MRR in the form of a
quadratic equation, where $T_{\rm eff}$ and $M$ are Kelvin and Solar units, is

$\log T_{\rm eff}=-0.170(0.026)\times(\log M)^{2}+0.888(0.037)\times\log
M+3.671(0.010),$ (4)

has $R^{2}=0.961$ and $\sigma=0.042$.

The empirical MRR and MTR by Eker et al. (2018) are shown together with data
in Figure 6. The division between the low and high mass stars at
$1.5M_{\odot}$ is a nice coincidence that plots of MRR and MTR appear with the
same numbers in the horizontal axis, thus one must be careful MTR is shown in
logarithmic scale while MRR is not.

Figure 6: Empirical MRR and MTR calibrated from 509 main-sequence stars for
the mass ranges $0.179\leq M/M_{\odot}\leq 1.5$ and $1.5\leq M/M_{\odot}\leq
31$ of DDEB stars. Note MTR is in logarithmic scale, but MRR is not (credit to
Eker et al., 2018). Figure 7: The mean values ($\langle L\rangle$, $\langle
R\rangle$, $\langle T_{\rm eff}\rangle$) computed from empirical MLT, MRR and
MTR are compared to $L$, $R$ and $T_{\rm eff}$ of 509 main-sequence stars. The
mean values directly from MLR, MRR or MLR, MTR are solid, The mean values from
the Stefan-Boltzmann law are dotted. Dashed vertical lines are the break
points of MLR (credit to Eker et al., 2018).

The mean values ($\langle L\rangle$, $\langle R\rangle$, $\langle T_{\rm
eff}\rangle$) computed from the empirical MLR, MRR and MTR of Eker et al.
(2018) are compared to $L$, $R$ and $T_{\rm eff}$ of 509 main-sequence stars
chosen from 318 DDEB and one detached eclipsing triple are shown in Figure 7.
The solid lines mark the mean values directly from MLR, MRR or MLR, MTR, while
the dotted lines mark the mean values $\langle T_{\rm eff}\rangle$ calculated
from $\langle L\rangle$, $\langle R\rangle$ for the low-mass stars and
$\langle R\rangle$ calculated from $\langle L\rangle$ and $\langle T_{\rm
eff}\rangle$ for the high-mass stars.

Calculating the mean values ($\langle L\rangle$, $\langle R\rangle$, $\langle
T_{\rm eff}\rangle$) in two steps as described above has an advantage further
checking them up and confirming whether the mean values of the first step is
consistent or not. Note that, only if $\langle L\rangle$ and $\langle
R\rangle$ values of the low mass stars are consistent, then in the second step
consistent $\langle T_{\rm eff}\rangle$ values could be produced from $\langle
L\rangle$ and $\langle R\rangle$; similarly, only if $\langle L\rangle$ and
$\langle T_{\rm eff}\rangle$ values of the high-mass stars are consistent,
then in the second step, consistent $\langle R\rangle$ values could be
produced according to the Stefan-Boltzmann law. The three break points and
varying inclinations of MLR before, between and after the break points in the
mass range $M\leq 1.5M_{\odot}$, apparently, causing the wavy look of the MTR
of low-mass stars. What simple function would have produced the current
successful appearance of the fit displayed in Figure 7c by the dotted line?
Similar influence of the break points and effect of varying inclinations of
MLR before, between and after the break points for the high mass stars
$M>1.5M_{\odot}$ are there in the middle panel (Figure 7b) but appears to be
lost within the scatter caused by the faster evolution of more massive stars.
Therefore, it could be said that it is a bull’s eye to choose the mass region
$M\leq 1.5M_{\odot}$ for devising MRR and to choose the mass region
$M>1.5M_{\odot}$ for devising MTR.

The two-step procedure in determining the mean values of $\langle L\rangle$,
$\langle R\rangle$, $\langle T_{\rm eff}\rangle$ for a given $M$ may appear
problematic to introduce extra errors in the error propagation. Users should
not be deceived from this illusion. First of all, this is because, consistent
mean values of higher uncertainty is better than inconsistent more accurate
ones. To avoid inconsistency, one may use two-step procedure as described
above or will use MLR and MRR only or MLR and MTR only for the full mass range
of main-sequence stars. Thus, in either case, the two-step procedure is
inevitable for consistent results. For the high mass stars $M>1.5M_{\odot}$
the scatter on $R$ is too big thus authors usually prefer not to determine
MRR. Then for the former case, MRR and MTR would be missing for the high mass
stars while interrelated MLR, MRR and MTR would be possible only for low mass
stars. For the low mass stars $M<1.5M_{\odot}$, on the other hand, it is not
possible to find a simple function to fit in determining MTR. Then,
interrelated MLR, MRR and MTR becomes possible only for the high mass stars
since MRR and MTR would be missing for the low mass stars $M<1.5M_{\odot}$. if
one prefers to fit a MRR together with MLR in the full mass region, then
he/she will end up having very unreliable $\langle R\rangle$ for the stars
$M>1.5M_{\odot}$. Error propagation, then, would produce similarly unreliable,
even worse $\langle T_{\rm eff}\rangle$ since independent MTR function is not
there. if one prefers to calibrate MLR and MTR in the full range of masses,
similar unreliable or erroneous $\langle T_{\rm eff}\rangle$, $\langle
R\rangle$ values emerge for the low mass stars $M<1.5M_{\odot}$. Therefore,
the two-step procedure as described in the paragraph before this one in
determining the mean values of $\langle L\rangle$, $\langle R\rangle$,
$\langle T_{\rm eff}\rangle$ for a given $M$ always better and more reliable
than the independently determined MLR, MRR and MTR for sure.

##  3 Conclusions

Fundamental relations are handy tools to explain simple natural phenomena or
starting points for one to understand more complex happenings. The main
difference between statistical and fundamental relations is that a statistical
relation is valid under certain conditions implied by the data from which it
was formulated while a fundamental relation is rather independent of data,
that is, data is not there to constrain it but only to confirm or to falsify
it. All empirical relations derived from observational or experimental data
are statistical in spirit. This does not, however, mean a fundamental relation
can not be derived from observational and/or experimental data. A statistical
study could reproduce and even may find a fundamental relation.

Physical laws are indifferentiable from fundamental relations. Their scope is
wider; for example, the Stefan-Boltzmann law ($L=4\pi R^{2}\sigma T_{\rm
eff}^{4}$) is valid for all stars radiating thermally, also for a hypothetical
star representing $\langle L\rangle$, $\langle R\rangle$, and $\langle T_{\rm
eff}\rangle$ implied by main sequence MLR, MRR, and MTR. On the contrary, MLR,
MRR, and MTR are valid to give mean values for a main-sequence star of a given
$M$ within a valid mass range, which was suggested after their calibrations.
As they are not valid for non-main sequence stars, they are also not valid to
estimate $L$, $R$, or $T_{\rm eff}$ of a main-sequence star with a known $M$
only. This is because, nowadays many stellar structure and evolution models
exist for one to look for $L$, $R$, and $T_{\rm eff}$ of a star according to
its $M$, chemical composition ($X,Y,Z$) and age.

Because of their simplicity, some statistical relations could be mistaken or
misused as a fundamental relation. A good example of this is the classical
MLR. This is because, right after its discovery by Hertzsprung (1923) and
Russell et al. (1923), for a while, there were no other methods to obtain
masses of single stars but estimate it from $L$ or absolute bolometric
magnitude ($M_{\rm Bol}$) by using a MLR in the forms of $L\propto M^{\alpha}$
or $M_{\rm Bol}$ \- mass diagrams.

Having no alternative is another property of fundamental relations. Indeed,
the Stefan-Boltzmann law or the Planck law (spectral energy distribution of a
black body with specific temperature) do not have alternatives. If there is no
alternative to estimate $M$ of a single star, but using a classical MLR,
convinced early astronomers that it could be used as a fundamental relation.
Today, there are alternatives to getting $M$ and $R$ of single stars, thus
there is no excuse to assume any form of MLR as a fundamental relation. One
might still want to use MLR, MRR, and MTR, as a fundamental law, that is,
he/she is more interested in knowing $M$ of star from its $L$. This is
permissible with a larger uncertainty covering with of the main-sequence at
the value of $M$ on a $\log M-\log L$ diagram, simply $\pm\sigma$, which is
given in Table 2.

On the other hand, statistical relations could be devised to serve a very
specific purpose; for example, to obtain $M$ and $R$ of a single star from its
other observables. In this respect, a statistical relation may appear to be
operating like a fundamental relation, but this is an illusion. $M$ and $R$
predicting relations are still statistical relations if their validity depends
on the data from which they were calibrated. Also, because there are many
alternatives to provide $R$ or $M$ of a star from its $T_{\rm eff}$, $L$, $g$,
$\rho$, and [Fe/H] as expressed by Moya et al. (2018) and as summarized by
Serenelli et al. (2021). Malkov (2007), Torres et al. (2010), and Fernandes et
al. (2021) have other alternatives.

Naming a newly devised or renaming a re-calibrated empirical relation is very
important; as important as classifying it as another statistical relation or
one of the fundamental laws. Names must be unique to avoid confusion and guide
users towards the original purpose of the relation. Repeating the same
improper and non-unique names, should not be advocated by saying “Let us keep
the same name in the past”. This will be nothing but insisting on the same
error. There are many miscalling or non-unique naming examples in the past,
which are identified with possible correct names in the previous section.
Especially after Andersen (1991)’s exclamation that some $R$ and $M$
predicting relations were non-uniquely and incorrectly called MLR. Let us
hope, Chevalier et al. (2023), and Malkov et al. (2022) would be the last
examples who erroneously named their mass-$M_{\rm G}$ diagrams, where $M_{\rm
G}$ is absolute brightness in Gaia $G$ band, “mass-luminosity” diagram, and
their mass – $M_{\rm G}$ relations “mass-luminosity” relation. Possible unique
names for them are the “mass-absolute brightness” diagram or the “mass-
absolute brightness” relation at Gaia $G$ band.

The empirical relations for predicting $M$ and $R$ of single stars developed
as alternatives to classical MLR and MRR after the split, initiated after
Andersen (1991)’s exclamation, occurred on the classical path aiming to get
mean values $\langle L\rangle$, $\langle M\rangle$, and $\langle R\rangle$,
and individual $L$, $R$, and $M$ of single stars. In this respect, it is not
possible for one to say, “the empirical relations giving $M$ and $R$ of single
stars are more valuable than the classical MLR and MRR”. Claiming the opposite
is also not correct. Both schools of thought have useful applications,
valuable with respect to their own aims. Empirical $M$ and $R$ predicting
relations had found a fruitful application to explore exoplanets hosting
single stars. Estimating $M$ and $R$ of planet-hosting stars are the starting
point for exploring hosted exoplanets (Stassun et al., 2017, 2018). Classical
MLR, MRR, and MTR, on the other hand, are practical for constructing
astrophysical models that need mean values, not only beneficial to stellar
astrophysics but also Galactic, extragalactic search, and even cosmological
models.

Further improvements in the predicting accuracy of single stars parameters
depend mainly on the quantity and the quality of the radial velocity and light
curve solutions of DDEB. Advances on already established relations or new
forms are possible due to increasing demand for exoplanet investigations.
Developments on the classical MLR, MRR, and MRR, on the other hand, are
encouraged to include the metallicity effect, as well as to extend them
further towards high and low mass limits, not only for the main sequence but
also for other luminosity classes.

## Acknowledgements

Authors thank the two anonymous referees who both gave valuable comments in
improving the quality of discussions in the manuscript.

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