# Multiple mobility edges in a 1D Aubry chain with Hubbard interaction in
presence of electric field: Controlled electron transport

Srilekha Saha Saha  Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India  S. N. Karmakar  Condensed Matter Physics Division, Saha Institute
of Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India

###### Abstract

Electronic behavior of a 1D Aubry chain with Hubbard interaction is critically
analyzed in presence of electric field. Multiple energy bands are generated as
a result of Hubbard correlation and Aubry potential, and, within these bands
localized states are developed under the application of electric field. Within
a tight-binding framework we compute electronic transmission probability and
average density of states using Green’s function approach where the
interaction parameter is treated under Hartree-Fock mean field scheme. From
our analysis we find that selective transmission can be obtained by tuning
injecting electron energy, and thus, the present model can be utilized as a
controlled switching device.

###### pacs:

72.20.Ee, 71.27.+a, 71.30.+h, 73.23.-b

##  I Introduction

Electron transport in low-dimensional system has created a lot of interest
among researchers due to its immense applicability in the field of
nanoscience. Transport in low-dimensional systems led to interesting quantum
effects. In one-dimension ($1$D) in presence of random disorder all the
eigenstates are exponentially localized irrespective of however weak is the
strength of disorder, this is the well-known phenomenon of Anderson
localization ander . Based on this fact it is a common belief that no mobility
edge, energy eigenvalues separating localized states from the extended states,
can exist in 1D. In addition to Anderson localization there exists another
kind of localization which is Wannier Stark localization wann  that occurs due
to application of bias voltage. It has also drawn much attention like the case
of Anderson localization. Here localization is obtained even in absence of
disorder and only due to the resulting electric field. Many theoretical wt1 ;
wt2 ; wt3 ; wt4 ; wt5  and experimental wae  analysis are available on Stark
localization just like Anderson localization. Even in this case mobility edge
could not be detected.

However, it has been pointed out that in correlated disordered systems all
eigenstates are not localized angel , rather some states are of extended in
nature as well. In a work Dunlap et al. considered a random dimer model dunlap
and showed that the system supports extended eigenstates at certain discrete
eigenvalues. Similarly, a number of works have also appeared in the literature
l1 ; l2 ; ssa2 ; sn1  to establish the presence of delocalized states along
with the localized ones thereby exhibiting metal to insulator transition. With
all these special classes of lattice models, Aubry-Andre (AA) model abam
always gives a a classic signature in transport phenomena. The on-site
potential in the AA 1D chain has the form of a cosine function mag ; sharma ;
eco :

$\epsilon_{n}=\lambda\cos(Qna)$ (1)

where $\lambda$ is the modulation amplitude, $Q$ is an irrational multiple of
$\pi$ and $a$ is the lattice spacing. It is a quasiperiodic lattice something
intermediate between periodic and random disordered systems. The parameter
$\lambda$ has an important role on the localization behavior of the
eigenstates. Using Thouless formula thf  Aubry and Andre demonstrated that
this model exhibits energy independent metal insulator transition in the
parameter space of the Hamiltonian at $\lambda=2t$, where $t$ represents the
nearest-neighbor hopping integral. For $\lambda<2t$ all eigenstates are
extended and in case of $\lambda>2t$ all are localized, the equality relation
is the point of duality with exotic critical eigenstates which are neither
extended nor localized. This interesting feature of the AA model aroused
immense

Figure 1: (Color online). Schematic diagram of a $1$D tight-binding chain
coupled to two $1$D semi-infinite electrodes, viz, source and drain.

curiosity in the minds of researchers to advance in this field. Large number
of articles dealt with this model both for $1D$ chain as well as in case of
ladder aubry79 ; ssa1 . However to the best of our knowledge, the study of the
phenomenon of Stark localization in an AA chain in presence of Hubbard
interaction is absent in literature. In the present manuscript we have
elaborately studied the phenomena of localization in a 1D AA chain in absence
and presence of electron-electron (e-e) interaction. First we analyze the
effect of applied bias voltage on the localization behavior of AA chain in
absence of any Hubbard interaction and finally we approach to the case in
presence of interaction. Thus we see how an interesting interplay of
localization as well as mobility edge phenomena occurring.

Rest of the article is arranged as follows. In Section II we present the model
and theory based on which the results have been derived and discussed in
Section III. Lastly we conclude in Section IV.

##  II Model and Theory

Figure 1 depicts a one-dimensional AA chain coupled to two semi-infinite
leads. The chain comprising $N$ atomic sites is subjected to an incommensurate
Aubry potential. We describe the model embracing the tight-binding formalism
and Hamiltonian for the entire system can be expressed as,

$H=H_{S}+H_{chain}+H_{tun}+H_{D}$ (2)

where the different sub-Hamiltonians are described as follows. The
Hamiltonians

Figure 2: (Color online). Variation of voltage dependent on-site potentials in
a $1$D chain with $200$ atomic sites when the bias voltage is fixed at
$0.2\,$V. Three different electrostatic potential profiles, one linear and two
non-linear, are taken into account those are represented by three different
colored curves.

for the semi-infinite source and drain electrodes are $H_{S(D)}$ and they can
be written explicitly as,

$H_{S(D)}=\sum\limits_{p}\epsilon_{0}d_{p}^{{\dagger}}d_{p}+\sum\limits_{p}t_{0}[d_{p+1}^{{\dagger}}d_{p}+h.c.]$
(3)

where $\epsilon_{0}$ and $t_{0}$ correspond to the on-site energy and nearest-
neighbor hopping integral, respectively, in the electrodes. Creation and
annihilation operators of electron inside the electrodes in the $n$th Wannier
state are respectively denoted by $d_{n}^{{\dagger}}$ and $d_{n}$.

The second term in Eq. 2 describes the Hamiltonian for the 1D AA chain. In
absence of electron-electron interaction, the AA chain has on-site energy of
the form $\epsilon_{i}=\lambda\cos(Qia)$ where, $a$ represent the lattice
constant, $Q$ is an irrational multiple of $\pi$ and $i$ corresponds to
positions of atomic sites. Therefore Hamiltonian for a non-interacting AA
chain is of the form

$H_{chain}=\sum\limits_{i}\epsilon_{i}c_{i}^{{\dagger}}c_{i}+\sum\limits_{i}t[c_{i+1}^{{\dagger}}c_{i}+h.c.]$
(4)

$t$ being nearest-neighbor hopping integral and $c_{i}^{{\dagger}}$ ($c_{i}$ )
denotes the creation (annihilation) operator in the AA chain. When all site
energies are identical, the AA chain maps to a perfect ordered chain and in
that case we set $\epsilon_{i}=0$ $\forall\;i$, without loss of generality. As
we apply bias voltage $V$ across the chain, it results an electric field to
develop across it and the on-site energy gets modified to
$\epsilon_{i}^{\prime}=\epsilon_{i}+\epsilon_{i}(V)$, where $\epsilon_{i}(V)$
is the voltage dependent part of on-site energy. Now, the question naturally
comes how on-site energy depends on the voltage. Several attempts pr1 ; pr2 ;
pr3 ; pr4 ; pr5 ; pr6  have been done along this line to find potential
profile along a linear conductor coupled to source and drain electrodes.
Mostly ab initio or semiempirical models are used, but few groups have also
done it by solving Poisson’s equations. From these studies it has been
suggested that depending on electron screening, potential profile along a
conductor sandwiched between source and drain can be of linear type or non-
linear one. For very large screening length linear voltage drop is expected,
while entire voltage drops at the interfaces between chain and electrode for
very short screening length pr5 . Considering symmetric potential drop at the
two interfaces we can write voltage dependent on-site energy for linear
potential profile as pr7  $\epsilon_{i}(V)$=$V/2-iV/(N+1)$, where $V$ is the
applied bias voltage. While, for non-linear potential profile there is no such
specific functional form. Keeping in mind the effect of electron screening, in
our model calculations we choose two different functional forms for two non-
linear curves as shown in Fig. 2. For pink curve we set
$V/2\;\mbox{Exp}[-iV/5]$, while for blue curve it gets the form
$V/2\;\mbox{Exp}[-iV/1.5]$. Running the variable $i$ (viz, site index) from
$1$ to $N/2$ we generate the datas for fixed voltage $V$ and then for $N/2+1$
to $N$ values of $i$ we take the negative of these values in reverse order to
make the profile symmetric. One can also take other functional forms to
generate these identical curves and the entire physics will remain same as
electron transport depends only on the values of voltage dependent on-site
energies, not on the actual functional form. Here two non-linear profiles
correspond to two different electron screening as suggested by earlier studies
pr5 ; pr6 . Here it is important to note that, the variation of electrostatic
potential profile depends on the material itself. But for our model
calculations we consider these three different profiles, and, we believe that
with our results general features of electric field on electron transmission
across a junction can be clearly analyzed.

The third term represents coupling Hamiltonian due to the coupling of the AA
chain and side-attached leads, and it reads as

$H_{tunn}=\tau_{S}[c_{1}^{\dagger}d_{0}+h.c.]+\tau_{D}[c_{N}^{\dagger}d_{N+1}+h.c.]$
(5)

where $\tau_{S}$ and $\tau_{D}$ give the strengths by which the system is
coupled to source and drain, respectively.

Now if we incorporate on-site Coulomb interaction in the AA chain through
Hubbard term along with the effect of bias voltage, the Hamiltonian takes the
form,

$\displaystyle H_{chain}$ $\displaystyle=$
$\displaystyle\sum\limits_{i,\sigma}\epsilon_{i,\sigma}^{\prime}c_{i,\sigma}^{\dagger}c_{i,\sigma}+\sum\limits_{\langle
ij\rangle,\sigma}t[c_{i,\sigma}^{\dagger}c_{j,\sigma}+c_{j,\sigma}^{\dagger}c_{i,\sigma}]$
(6) $\displaystyle+$
$\displaystyle\sum\limits_{i}Uc_{i\uparrow}^{\dagger}c_{i\uparrow}c_{i\downarrow}^{\dagger}c_{i\downarrow}$

where $U$ is the Coulomb interaction strength.

To study electronic behavior of such an interacting system we use Hartree-Fock
mean field kato ; kam ; new1  theory. In this approach, Eq. 6 can be written
as

$\displaystyle H_{chain}$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i\uparrow}^{\prime\prime}n_{i\uparrow}+\sum_{\langle
ij\rangle}t\left[c_{i\uparrow}^{\dagger}c_{j\uparrow}+c_{j\uparrow}^{\dagger}c_{i\uparrow}\right]$
(7) $\displaystyle+$
$\displaystyle\sum_{i}\epsilon_{i\downarrow}^{\prime\prime}n_{i\downarrow}+\sum_{\langle
ij\rangle}t\left[c_{i\downarrow}^{\dagger}c_{j\downarrow}+c_{j\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle-$ $\displaystyle\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ $\displaystyle=$
$\displaystyle H_{c,\uparrow}+H_{c,\downarrow}-\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$

where, $H_{c,\uparrow}$ and $H_{c,\downarrow}$ are Hamiltonians for the up and
down spin electrons, respectively, similar to Eq. 4 with modified on-site
energies as
$\epsilon_{i,\uparrow}^{\prime\prime}=\lambda\cos(Qia)+\epsilon_{i}(V)+U\langle
n_{i,\downarrow}\rangle$ and
$\epsilon_{i,\downarrow}^{\prime\prime}=\lambda\cos(Qia)+\epsilon_{i}(V)+U\langle
n_{i,\uparrow}\rangle$, $n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i,\sigma}$
being the number operator. The last term in the above equation (Eq. 7)
represents a shift in the total energy and it depends on the average number of
up and down spin electrons. Once we get the decoupled Hamiltonians for up and
down spin electrons, we find the eigenvalues self-consistently considering
some initial guess values of $\langle n_{i,\uparrow}\rangle$ and $\langle
n_{i,\downarrow}\rangle$. With the starting guess values of $\langle
n_{i,\sigma}\rangle$ we diagonalize the Hamiltonians $H_{c,\uparrow}$ and
$H_{c,\downarrow}$, and compute a new set of values for $\langle
n_{i,\sigma}\rangle$. Next we replace the initial guess values with the new
set of values of $\langle n_{i,\sigma}\rangle$ and repeat the process until
the values of all $\langle n_{i,\sigma}\rangle$ converges. Substituting the
final set of values in the Hamiltonians we calculate the two-terminal
transmission probability using the Landauer formula. Transmission probability
for up or down spin evaluated separately in terms of Green’s function from the
relation datta1

$T_{\sigma}(E)=Tr[\Gamma_{S}^{\sigma}G_{chain,\sigma}^{r}\Gamma_{D}^{\sigma}G_{chain,\sigma}^{a}]$
(8)

where $\Gamma_{S(D)}^{\sigma}$ is the coupling matrix bearing the imaginary
part of the self-energy $\Sigma_{S(D)}^{\sigma}$, arising due to the coupling
between chain and the semi-infinite leads. $G_{chain,\sigma}^{r}$ and
$G_{chain,\sigma}^{a}$ are the retarded and advanced Green’s functions of the
chain which include the effect of electrodes. Thus we can write datta1
$G_{chain,\sigma}^{r}=(E-H_{c,\sigma}-\Sigma_{S}^{\sigma}-\Sigma_{D}^{\sigma})^{-1}$.
For complete derivation of self-energies and effective Green’s function
($G_{chain,\sigma}$) of the chain, see Appendix A and Ref. datta1 .
Determining transmission probabilities of up and down spin electrons we can
calculate the total transmission probability from the relation
$T(E)=\sum\limits_{\sigma}T_{\sigma}(E)$. Finally, we evaluate the average
density of states (ADOS) using the relation
$\rho(E)=-(1/N\pi)Tr[Im[G_{chain}^{r}]]$.

During the computation we use $\lambda=1$ in unit of $t$ (eV), except for Fig.
5 where $\lambda$ is varying, and set $Q=(1+\sqrt{5})/2$. The other common
parameters are as follows: $\tau_{S}=\tau_{D}=1\;$eV, $t=1\;$eV,
$\epsilon_{0}=0$ and $t_{0}=2\;$eV. The on-site energy
($\epsilon_{i,\uparrow}^{\prime\prime}$ or
$\epsilon_{i,\downarrow}^{\prime\prime}$) in the chain, a sum of tree terms
($\lambda\cos(Qia)$, $\epsilon_{i}(V)$ and $U\langle n_{i,\downarrow}\rangle$
or $U\langle n_{i,\uparrow}\rangle$), on the other hand is not a constant, and
therefore, we cannot assign a common value of it. Throughout the calculations
we measure all energies in unit of $t$ and bias voltage in unit of Volt ($V$)
and set $c=e=h=1$, for simplification.

##  III Numerical Results and Discussion

In this section we present the results based on the above theoretical
formulation.

Figure 3: (Color online). Total transmission probability (pink color) and
average DOS (green color) vs. energy $E$ for a non-interacting chain
considering $N=200$ and $U=0$, where (a) and (b) correspond to the chain with
identical site potentials, while (c) and (d) represent the Aubry chain. For
the left column we set $V=0$, and, it is $0.2$ for the right column. The
results are computed for the linear bias drop across the chain.

First we study the case of electron transmission through a non-interacting AA
chain in presence of bias voltage and then we consider effect of Hubbard
interaction into it.

Figures 3(a) and (b) present the results for the case of a non-interacting 1D
chain in absence of incommensurate potential i.e., the chain becomes a perfect
$1$D lattice. For such a case we choose the bare site potentials to zero,
without loss of generality. In Fig. 3(a) we set the bias voltage $V$ to zero,
while it is fixed at $0.2$ in Fig. 3(b). In absence of external bias voltage
all the energy eigenstates are extended in nature, and therefore, the
transmission probability is finite at all

Figure 4: (Color online). Total transmission probability and average DOS as a
function of energy $E$ for an interacting chain with $U=2$, where (a)-(d)
correspond to the identical meaning as in Fig. 3. All the other parameters are
same as considered in Fig. 3.

Figure 5: (Color online). $T$-$E$ and $\rho(E)$-$E$ curves for a $200$ site
interacting chain ($U=2$) for different values of $\lambda$, where (a), (b),
(c) and (d) correspond to $\lambda=0.5$, $1$, $2$ and $2.5$, respectively. For
all these cases we set $V=0$.

energy eigenvalues. In presence of finite bias eigenstates at the band edges
are no longer extended as it is evident from Fig. 3(b) that the transmission
probability is zero while inside the band eigenstates are extended as we have
finite transmission probability. It indicates that the choice of Fermi energy
is quite important. If it is chosen to lie well inside the energy band the
chain will be of conducting nature, while if it lies near the band edges the
chain behaves as an insulator. Such sharp transitions of the conducting
behavior gives an idea of existence of mobility edges in presence of a finite
bias. In the same figure, cases (c) and (d) represent the transmission
probability and ADOS of a non-interacting AA chain in absence and presence of
a linear bias drop. Presence of incommensurate potential leads to splitting of
band. We see that there are two gaps embedded inside three bands. In presence
of bias voltage, say $V=0.2$, the energy eigenstates belonging to the
outermost two bands have negligible contribution to transmission unlike those
of the middle band, and hence extended energy eigenstates as well as the
localized ones are present leading to metal-insulator transition. The more we
increase the bias voltage, more number of localized states will appear and at
one stage all the states will be localized.

Next we study the interplay between electric field and the Aubry ordering in
1D Hubbard chain. The transmission characteristics together with ADOS

Figure 6: (Color online). $T$-$E$ and $\rho(E)$-$E$ characteristics for an
interacting AA chain considering $300$ atomic sites with $U=2$ and
$\lambda=1$. The results are shown for a linear bias drop when $V=0.3$.

Figure 7: (Color online). $T$-$E$ and $\rho(E)$-$E$ characteristics for an
interacting AA chain with $N=200$, $U=2$ and $\lambda=1$ considering non-
linear bias drop across the chain. In (a) set the electrostatic potential
profile as given by the pink line in Fig. 2, while in (b) we choose it
according to the blue line of Fig. 2. Here we fix $V=0.2$.

of an AA interacting chain are shown in Fig. 4. First row represents $T(E)$
and $\rho(E)$ as functions of $E$ for an interacting chain in absence of Aubry
potential. In the half-filled case we have a Mott insulator (Fig. 4(a)) with a
single gap at the band centre. For the $1$D AA Hubbard chain, we have a Mott
gap at the center of the band, but additional gaps appear due to Aubry
potential as clearly seen from Fig. 4(c). In absence of bias voltage, all the
eigenstates of the periodic as well as the AA Hubbard chains have finite
electron transmission probability. On the other hand, in presence of finite
bias localized states appear at the band edges both in case of a $1$D periodic
Hubbard chain (Fig. 4(b)) and $1$D AA Hubbard chain (Fig. 4(d)).

To investigate the precise role of $\lambda$ on transmission and ADOS we
present results for different values of $\lambda$ in Fig. 5 setting $V=0$. We
see that for $\lambda<2t$, all the eigenstates are of extended in nature even
in presence of $U$. The fact that for $\lambda<2t$ all the eigenstates of $1$D
AA chain behave like extended states and their behavior remains unaltered even
for $U\neq 0$ which can be noticed from Figs. 5(a) and (b). It is well known
that for $\lambda=2t$ states are critical and for $\lambda>2t$ all the
eigenstates are localized, and for both these cases we have zero transmission
probability. Quite interestingly from Figs. 5(c) and (d) we see that few
conducting states appear in the middle of the inner two bands when $U\neq 0$.
Physically it implies that electron-electron interaction changes the behavior
of the AA chain.

To test the invariant nature of the above discussed phenomena with respect to
the parameter values, in Fig. 6 we present the characteristics features of
transmission probability together with average density of states considering a
chain with different set of parameter values where $N=300$ and $V=0.3$. From
the spectra it is clear that all the physical pictures remain unchanged and
certainly it strengthens the invariant character of our analysis and can be
verified experimentally.

Till now we have shown all the cases in presence of linear voltage drop across
the chain. To get an idea regarding the behavior of transmission and average
density of states in presence of non-linear bias drop let us focus on the
results given in Fig. 7. Two different non-linear profiles are taken into
account following the curves (pink and blue lines) shown in Fig. 2. For these
two cases we also get similar kind of band splitting and localization
phenomenon, but a careful observation suggests that the transmission
probability becomes higher for the flatter profile (blue line of Fig. 2)
compared to the other (pink line of Fig. 2) one. With increasing the flatness
the localization effect due to electric field decreases, and, for the limiting
case i.e., when the bias drop takes place only at the two edges of the chain,
transmission probability will be maximum when all the other parameters are
kept unchanged.

##  IV Conclusion

In the present work we critically investigate the role of electric field,
developed due to external bias, in an interacting $1$D Aubry chain. The
interaction parameter is described within a Hartree-Fock mean field level
under tight-binding framework where transmission probability and ADOS are
evaluated from Green’s function approach. The interplay between Aubry lattice,
Coulomb correlation and electric field provides multiple mobility edges at
different energies. Under this situation if we scan throughout the energy band
window then electrons can allow to pass from source to drain via the selective
conducting energy channels providing finite electron transmission, while for
all other cases we get the insulating phase since then no electron can
transmit through the localized channels. This phenomenon clearly emphasizes
that the present model can be utilized as a selective switching device.

##  Appendix A Evaluation of self-energies and effective Green’s function of
the chain coupled to source and drain electrodes

Since there is no spin flip mechanism (viz, from up spin to down spin or vice
versa) in our problem, the net transmission probability is obtained from the
relation $T(E)=\sum_{\sigma}T_{\sigma}(E)=T_{\uparrow}+T_{\downarrow}$. To
find $T_{\sigma}$ we need to evaluate $H_{c,\sigma}$, $\Sigma_{S}^{\sigma}$,
$\Sigma_{D}^{\sigma}$ and the effective Green’s function $G_{chain,\sigma}$.
The Hamiltonian $H_{c,\sigma}$ is determined from the mean field scheme which
is clearly described in Sec. II, and therefore, here we discuss elaborately
how to calculate other factors i.e., self energies and effective Green’s
function. For this, we can now ignore the spin index, as a matter of
simplification, because the prescription is same for both the two spin cases.

Following the definition we can write the Green’s function of the full system,

$G^{r}=\left[(E+i\eta)I-H\right]^{-1}.$ (9)

where, $I$ is the identity matrix and $\eta\rightarrow 0^{+}$. But there is a
problem with Eq. 9. Here we are working with an open system i.e., a conductor
connected with two semi-infinite electrodes. Therefore, $H$ has infinite
dimension and so also $G^{r}$. Now it is not possible to do any calculation
with a matrix whose dimension is infinity. So, to get rid of this situation we
apply the partitioning technique which maps the Green’s function matrix in the
reduced Hilbert space of the conductor itself, and the effects of the side
attached leads are incorporated there.

Suppose we are considering a conductor attached to electrode $p$. Hence, the
total Hamiltonian of the system can be written in a matrix form as,

$H=\left(\begin{array}[]{cc}H_{p}&\tau_{p}\\\
\tau_{p}^{\dagger}&H_{c}\end{array}\right).$ (10)

Here, $H_{c}$ and $H_{p}$ are the Hamiltonian matrices describing the
conductor and the side attached electrode. $\tau_{p}$ corresponds to the
coupling matrix due to coupling of the conductor to the side attached
electrode.

Hence, the Green’s function is,

$\displaystyle G^{r}$ $\displaystyle=$
$\displaystyle\left((E+i\eta)I-H\right)^{-1}$ (13) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}(E+i\eta)I-H_{p}&-\tau_{p}\\\
-\tau_{p}^{\dagger}&EI-H_{pc}\end{array}\right)^{-1}$

Now, partitioning the Green’s function matrix in the same way like the
Hamiltonian matrix, we can rewrite the above equation as,

$\displaystyle\left(\begin{array}[]{cc}G_{p}&G_{pc}\\\
G_{cp}&G_{c}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}(E+i\eta)I-H_{p}&-\tau_{p}\\\
-\tau_{p}^{\dagger}&EI-H_{c}\end{array}\right)^{-1}$ (18)

i.e.,

$\displaystyle\left(\begin{array}[]{cc}G_{p}&G_{pc}\\\
G_{cp}&G_{c}\end{array}\right)\left(\begin{array}[]{cc}(E+i\eta)I-H_{p}&-\tau_{p}\\\
-\tau_{p}^{\dagger}&EI-H_{c}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ (25)

We obtain two decoupled equations from the above equation, which are as
follows,

$G_{cp}\left[(E+i\eta)I-H_{p}\right]-G_{c}\tau_{p}^{\dagger}=0$ (27)

and,

$-G_{cp}\tau_{p}+G_{c}(EI-H_{c})=1$ (28)

We define, $(E+i\eta)I-H_{p}=(g_{p}^{r})^{-1}$. So, from Eq. (27) we get,

$\displaystyle G_{cp}(g_{p}^{r})^{-1}$ $\displaystyle=$ $\displaystyle
G_{c}\tau_{p}^{\dagger}$ $\displaystyle\Rightarrow G_{cp}$ $\displaystyle=$
$\displaystyle G_{c}\tau_{p}^{\dagger}g_{p}^{r}$ (29)

Using the expression of $G_{cp}$ from Eq. (29) in Eq. (28), we get,

$\displaystyle-G_{c}\tau_{p}^{\dagger}g^{r}_{p}\tau_{p}+G_{c}(EI-H_{c})=1$
(30) $\displaystyle\Rightarrow$ $\displaystyle G_{c}\left[EI-
H_{c}-\tau_{p}^{\dagger}g^{r}_{p}\tau_{p}\right]=1$ $\displaystyle\Rightarrow$
$\displaystyle G_{c}\left[EI-H_{c}-\Sigma_{p}\right]=1$
$\displaystyle\Rightarrow$ $\displaystyle G_{c}=\left[EI-
H_{c}-\Sigma_{p}\right]^{-1}$

where,

$\Sigma_{p}=\tau_{p}^{\dagger}g^{r}_{p}\tau_{p}$ (31)

Here, $G_{c}(=\mathcal{G})$ is the effective Green’s function which
incorporates the effect of the electrode attached with the conductor.

In Eq. 30, all matrices have the same dimension ($C$ $\times$ $C$), where, $C$
is the dimension of the conductor. But at first sight it might appear that the
problem of inverting an infinite dimensional matrix still remains, while
evaluating $g^{r}_{p}$ to obtain the expression for $\Sigma_{p}$. But
fortunately for an isolated infinite lead $g^{r}_{p}$ can be calculated
analytically.

Now for $ij$-th element of the self-energy matrix,

$\displaystyle\Sigma_{ij}$ $\displaystyle=$ $\displaystyle\langle
i|\tau_{p}^{\dagger}g_{p}^{r}\tau_{p}|j\rangle$ (32) $\displaystyle=$
$\displaystyle\sum_{m,n}\langle i|\tau_{p}^{\dagger}|m\rangle\langle
m|g_{p}^{r}|n\rangle\langle n|\tau_{p}|j\rangle$ $\displaystyle=$
$\displaystyle t^{2}g_{p}^{r}(p_{i},p_{j}).$

Here, $g_{p}^{r}(p_{i},p_{j})$ denotes the ($p_{i},p_{j}$)-th element of the
matrix $g_{p}^{r}$. To work with Eq. 30 we have to reconstruct the
$\Sigma_{p}$ matrix in $C$ $\times$ $C$ dimension. Here, all elements of
$\Sigma_{p}$ matrix would be zero, except at the points ($i$,$j$) inside the
conductor, which are adjacent to points ($p_{i}$,$p_{j}$) inside the
electrode.

For more than one electrodes we have to add the effects of individual
electrodes. It means if we have $p$ number of side-attached electrodes, then
the effective Green’s function will be,

$\mathcal{G}=[EI-H_{c}-\sum\limits_{p}\Sigma_{p}]^{-1}$ (33)

From this expression we can easily write the desired effective Green’s
function for our two-terminal system as
$\mathcal{G}=G_{chain}=(E-H_{c}-\Sigma_{S}-\Sigma_{D})^{-1}$ which includes
the effects of both source and drain electrodes. For two different spin sub-
spaces it can be generalized as
$G_{chain,\sigma}=(E-H_{c,\sigma}-\Sigma_{S}^{\sigma}-\Sigma_{D}^{\sigma})^{-1}$.

## References

  * (1) P. W. Anderson, Phys. Rev. 109, 1492 (1958).
  * (2) G.H. Wannier, Phys. Rev. 117 432 (1960).
  * (3) H. M. James, Phys. Rev. 76, 1611 (1949).
  * (4) D. Emin and C. F. Hart, Phys. Rev. B 36, 7353 (1987).
  * (5) R. Ouasti, N. Jekri, A. Brezini, and C. Depollier, J. Phys.: Condens. Matter. 7, 811 (1995).
  * (6) N. Zekri, M. Schreiber, R. Ouasti, R. Bouamrane, and A. Brezini, Z. Phys. B 99, 381 (1996).
  * (7) J. R. Borysowicz, Phys. Lett. A. 231, 240 (1997).
  * (8) C. Hamaguchi, M. Yamaguchi, H. Nagasawa, M. Morifuji, A. Di Carlo, P. Vogl, G. Bohm, G. Trankle, G. Weimann, Y. Nishikawa and S. Muto, Jpn. J. Appl. Phys. 34, 4519 (1995).
  * (9) A. Sanchez, Phys. Rev. B 49, 147 (1994).
  * (10) D. H. Dunlap, H.-L. Wu, and P. Phillips, Phys. Rev. Lett. 65, 88 (1990).
  * (11) X. F. Hu, Z. H. Peng, R. W. Peng, Y. M. Liu, F. Qiu, X. Q. Huang, A. Hu, and S. S. Jiang, J. Appl. Phys. 95, 7545 (2004).
  * (12) R. L. Zhang, R. W. Peng, X. F. Hu, L. S. Cao, X. F. Zhang, M. Wang, A. Hu, and S. S. Jiang, J. Appl. Phys. 99, 08F710 (2006).
  * (13) S. Sil, S. K. Maiti, and A. Chakrabarti, Phys. Rev. B 78, 113103 (2008).
  * (14) S. K. Maiti and A. Nitzan, Phys. Lett. A 377, 1205 (2013).
  * (15) S. Aubry and G. Andre, in: L. Horwitz, Y. Neeman (Eds.), Group Theoretical Methods in Physics, Annals of the Israel Physical Society, Vol. 3, American Institute of Physics, New York, 1980, p. 133.
  * (16) M. Johansson and R. Riklund, Phys. Rev. B 42, 8244 (1990).
  * (17) S. Das Sarma, S. He, and X.C. Xie, Phys. Rev. Lett. 61, 2144 (1988).
  * (18) C. M. Soukoulis and E. N. Economou, Phys. Rev. Lett. 48, 1043 (1982).
  * (19) D. J. Thouless, J. Phys. C 5, 77 (1972).
  * (20) S. Aubry and G. Andre, Ann. Isr. Phys. Soc. 3, 133 (1979).
  * (21) S. Sil, S. K. Maiti, and A. Chakrabarti, Phys. Rev. Lett. 101, 076803 (2008).
  * (22) S. Datta, W. Tian, S. Hong, R. Reifenberger, J. I. Henderson, and C. P. Kubiak, Phys. Rev. Lett. 79, 2530 (1997).
  * (23) W. Tian, S. Datta, S. Hong, R. Reifenberger, J. I. Henderson, and C. P. Kubiak, J. Chem. Phys. 109, 2874 (1998).
  * (24) V. Mujika, A. E. Roitberg, and M. A. Ratner, J. Chem. Phys. 112, 6834 (2000).
  * (25) N. D. Lang and P. Avouris, Phys. Rev. Lett. 84, 358 (2000).
  * (26) A. Nitzan, M. Galperin, G.-L. Ingold, and H. Grabert, J. Chem. Phys. 117, 10837 (2002).
  * (27) S. Pleutin, H. Grabert, G.-L. Ingold, and A. Nitzan, J. Chem. Phys. 118, 3756 (2003).
  * (28) V. Mujika, M. Kemp, A. Roitberg, and M. A. Ratner, J. Chem. Phys. 104, 7296 (1996).
  * (29) H. Kato and D. Yoshioka, Phys. Rev. B 50, 4943 (1994).
  * (30) A. Kambili, C. J. Lambert, and J. H. Jefferson, Phys. Rev. B 60, 7684 (1999).
  * (31) S. K. Maiti and A. Chakrabarti, Phys. Rev. B 82, 184201 (2010).
  * (32) S. Datta, Electronic transport in mesoscopic systems (Cambridge University Press, Cambridge, 1995).


# On the Zero Point Constant of the Bolometric Correction Scale

Z. Eker 1, V. Bakış1, F. Soydugan2,3 and S. Bilir4
1Akdeniz University, Faculty of Sciences, Department of Space Sciences and
Technologies, 07058, Antalya, Turkey
2Department of Physics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart
University, 17100 Çanakkale, Turkey
3Astrophysics Research Center and Ulupınar Observatory, Çanakkale Onsekiz Mart
University, 17100, Çanakkale, Turkey
4Istanbul University, Faculty of Science, Department of Astronomy and Space
Sciences, 34119, Istanbul, Turkey
E-mail<EMAIL_ADDRESS>
###### Abstract

Arbitrariness attributed to the zero point constant of the $V$ band bolometric
corrections ($BC_{V}$) and its relation to “bolometric magnitude of a star
ought to be brighter than its visual magnitude” and “bolometric corrections
must always be negative” was investigated. The falsehood of the second
assertion became noticeable to us after IAU 2015 General Assembly Resolution
B2, where the zero point constant of bolometric magnitude scale was decided to
have a definite value $C_{Bol}(W)=71.197~{}425~{}...$ . Since the zero point
constant of the $BC_{V}$ scale could be written as $C_{2}=C_{Bol}-C_{V}$,
where $C_{V}$ is the zero point constant of the visual magnitudes in the basic
definition $BC_{V}=M_{Bol}-M_{V}=m_{bol}-m_{V}$, and $C_{Bol}>C_{V}$, the zero
point constant ($C_{2}$) of the $BC_{V}$ scale cannot be arbitrary anymore;
rather, it must be a definite positive number obtained from the two definite
positive numbers. The two conditions $C_{2}>0$ and $0<BC_{V}<C_{2}$ are also
sufficient for $L_{V}<L$, a similar case to negative $BC_{V}$ numbers, which
means that “bolometric corrections are not always negative”. In sum it becomes
apparent that the first assertion is misleading causing one to understand
bolometric corrections must always be negative, which is not necessarily true.

###### keywords:

stars: fundamental parameters - stars: general - Sun: fundamental parameters -
Sun: general

††pagerange: On the Zero Point Constant of the Bolometric Correction
Scale–Data Availability

##  1 Introduction

One of the most very basic definitions in stellar astrophysics is bolometric
correction (BC) known as the difference between the bolometric and visual
magnitudes of a star ($BC_{V}=M_{Bol}-M_{V}=m_{Bol}-m_{V}$). Preference of the
$V$ filter is simply a convention since most (including the oldest)
photometric data is in the visual.

After giving its basic definition as $BC_{V}=m_{Bol}-V$ and stating “… this
definition is usually interpreted to imply that the bolometric corrections
must always be negative, although many of the currently used tables of
empirical $BC_{V}$ values violate this condition”, Torres (2010) paid
attention to another definition of $BC_{V}$,

$BC_{V}=2.5\log\left(\frac{\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm
d\lambda}{\int_{0}^{\infty}\\!f_{\lambda}\rm d\lambda}\right)+C_{2},$ (1)

where $S_{\lambda}(V)$ is the sensitivity function of the $V$ magnitude
system, $f_{\lambda}$ is the monochromatic flux from a star, and $C_{2}$ is an
assumed arbitrary constant of integration. Consequently, another problem was
pointed out: “The constant $C_{2}$ contains an arbitrary zero point that has
been a common source of confusion”. This is because “When making use of
tabulations of empirical bolometric corrections for stars ($BC_{V}$), a
commonly overlooked fact is that while the zero point is arbitrary, the
bolometric magnitude of the Sun ($M_{Bol,\odot}$) that is used in combination
with such tables cannot be chosen arbitrarily. It must be consistent with the
zero point of $BC_{V}$ so that the apparent brightness of the Sun is
reproduced” as reported by Torres (2010).

Confirming Torres (2010), Casagrande & VandenBerg (2018) stated that despite
the solar luminosity being a measured quantity, $M_{Bol,\odot}$ is an
arbitrary zero point and any value is equally legitimate on the following
equation

$M_{Bol}-M_{Bol,\odot}=-2.5\log(L/L_{\odot}),$ (2)

on the condition that once chosen, all bolometric corrections are scaled
accordingly. Therefore, equation (2) indicates the existence of an infinite
number of $M_{Bol}$ and $M_{Bol,\odot}$ pairs having the same magnitude
difference ($M_{Bol}-M_{Bol,\odot}$) to compute the single stellar luminosity
($L/L_{\odot}$) in solar units. Consequently, an infinite number of $BC_{V}$
is inevitable for a single star because there is no similar mechanism to
change its $M_{V}$ parallel to its $M_{Bol}$ in order to keep $BC_{V}$
constant in the basic definition ($BC_{V}=M_{Bol}-M_{V}$). Therefore,
according to Casagrande & VandenBerg (2018), an infinite number of zero points
for the bolometric magnitudes and for BC scales must exist and all must be
valid.

Arbitrariness attributed to the zero point of the $BC_{V}$ scale was claimed
long before by Hayes (1978) because all the $BC_{V}$ values tabulated by Hayes
(1978) are less than zero and his statement “I have taken the zero of B.C. to
be at its minimum absolute value, with the result that $B.C._{\odot}=-0^{\rm
m}.14$” clearly indicates arbitrariness of the BC scale, where the smallest of
the corrections, $BC_{V}=-0.01$, are associated with main sequence stars of
the A7-F7 spectral types. Similarly, a tabulated empirical $BC_{V}$ table by
Habets & Heintze (1981) does not contain any positive value where $BC_{V}$ for
the sun is -0.34 mag. The smallest correction, $BC_{V}=-0.07$, is associated
with F2 spectral type. $BC_{V}$ by Kuiper (1938), who preferred to take the
zero point at 6600 K, are the same; all are less than zero. McDonald &
Underhill (1952) adjusted their corrections to a scale giving $BC_{V}=-0.11$
mag for the Sun with $T_{\odot}=5713$ K. Wildey (1963) evaluated all $BC_{V}$
determinations and presented them in a two-dimensional diagram $(B-V)_{\rm
0}-BC_{V}$, where all BC values appear less than zero. Another author, who
gave $BC_{V}$ values less than zero, is Popper (1959). The $BC_{V}$ values of
pre-main-sequence stars are all given in the negative by Pecaut & Mamajek
(2013). The handbook of astronomers, Allen’s Astrophysical Quantities,
contains $BC_{V}$ numbers for main-sequence, giants, and supergiants which are
all negative, where the smallest of the corrections, $BC_{V}=-0.00$ mag, is
associated with F2 supergiants, and $T_{eff}=7030$ K (Cox, 2000). Considering
his note on page 381, where the definition of BC is given by Cox (2000), the
negative sign appears to be intentional to emphasize that all $BC_{V}$ must be
negative.

However, the picture is totally different among the other half of researchers,
who use the same basic definition, and the alternative definition used by
Torres (2010).

According to Heintze (1973), $f_{\lambda}$ for the Sun is a measurable
quantity over a sufficiently large wavelength region, thus $BC_{V,\odot}$ is
known from the basic definition, $M_{Bol,\odot}$ and the zero point was
adopted for the bolometric magnitude scale. So, $C_{2}$ is known (Heintze,
1973). Confirming this, Code et al. (1976) say that “$C_{2}$ is uniquely
defined if we adopt a value for the bolometric correction for a specified
spectral distribution of $f_{\lambda}$”. Using not only the Sun’s spectral
distribution, but also the spectral distribution of 21 non-reddened single
stars with well-established $\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm
d\lambda$, Code et al. (1976) determined the value of $C_{2}=0.958$. Heintze
(1973) did not give any table, but Code et al. (1976) gave the tabulated
$BC_{V}$ as being less than zero ($BC_{V}<0$) for main-sequence stars in the
effective temperature range 5780-34000 K, except for a limited portion in the
7000-8000 K range with $BC_{V}=0.01$.

The table of Code et al. (1976) is not the only example to include a limited
number of positive $BC_{V}$ values. The tables presented by Johnson (1964,
1966), Flower (1977, 1996), Bessell, Castelli & Plez (1998), Sung et al.
(2013) and recently Casagrande & VandenBerg (2018) and Eker et al. (2020) all
contain a limited number of positive $BC_{V}$ values.

Clearly, there are two schools of thought who treat arbitrariness of the
$BC_{V}$ scale differently. One school intentionally shifts computed $BC_{V}$
values to make them all negative while the other school prefers retaining them
as they are, even if there are positive $BC_{V}$ values. The first group
appears self consistent according to the principle that a positive $BC_{V}$
would violate the fact that the bolometric magnitude of a star ought to be
brighter than its $V$ magnitude; therefore, this group’s members feel free to
shift the zero point of the $BC_{V}$ scale to make sure that all $BC_{V}$ are
negative. However, the second group, which retains the $BC_{V}$ as they are,
even if there are positive values, either ignores the note “$BC=m_{Bol}-V$
(always negative)” (page 381 in Cox, 2000) or a positive $BC_{V}$ is not seen
as a violation. If the zero point of the BC scale is indeed arbitrary, why not
take advantage and use it to make all BC values negative in order to avoid a
conceptual problem?

Here we should stress the difference between absolute and limited discrete
arbitrariness. Absolute arbitrariness is the arbitrariness expressed by Torres
(2010) on equation (1) where $C_{2}$ could be any number from minus infinity
to plus infinity, which would all be legitimate, and satisfy the solution.
However, for the limited discrete arbitrariness, a researcher does not have
the freedom to choose any value except a limited number of choices. The former
is problematic while, the latter option, is acceptable. One step after limited
discrete arbitrariness is the nominal choice (or standard choice). That is,
standardization; if all users have a single option. Although the arbitrariness
by Casagrande & VandenBerg (2018) gives the impression of absolute
arbitrariness, after reading why they choose $M_{Bol,\odot}=4.74$ mag, it
becomes clear why this is not an absolute but a limited discrete
arbitrariness.

Motivated to try to solve the dilemma between these two schools of thought, in
this study we investigated the problem of arbitrariness on the $BC_{V}$ scale
and its relationship to the statements “the bolometric magnitude of a star
ought to be brighter than its visual magnitude” and “bolometric corrections
must always be negative”.

##  2 Groundwork

###  2.1 Non-Uniqueness (Arbitraryness) Anaysis

One of the most distinguished philosophers of science in the 20th century, Sir
Karl Popper (1972), says that if a theory explains everything, it has no
scientific value. Similarly, if a scientific problem has an infinite number of
solutions (absolute arbitrariness), such as solutions of an indefinite
integral with an arbitrary constant, it would be equivalent to no solutions at
all. Therefore, before discussing the problems associated with a positive BC,
it is necessary to analyze the non-uniqueness (arbitrariness) attributed to
the BC scale. Unfortunately, the word “unique” does not have a unique meaning.
Thus, one must first be aware of already identified types of non-uniqueness
(arbitrariness) too. Similar analysis had been done for the synthetic light
curves of spotted stars (Eker, 1999) and main-sequence mass-luminosity (MLR),
mass-radius (MRR) and mass-effective temperature (MTR) curves (Eker et al.,
2018), where three types of non-uniqueness (Type 1, Type 2, Type 3) were
investigated.

Type I applies to a scientist who has already collected data (measurements,
observations) and is at the point of trying to find a best-fitting curve
(function), which could be picked from a list of many mathematical functions.
For most cases, however, mathematical functions are known; e.g., a Planck
function is for explaining the spectral energy distribution (SED) of a star. A
problem with Type 1 exists if scientists do not have any notion about which
function to apply and there could be more than one, perhaps many different
mathematical functions that equally explains the data.

If a Type 1 problem does not exist, the next problem is called Type 2, which
concerns whether a unique fit is possible or not. Nevertheless, there are
methods how to deal with Type 2 problems; e.g., the least squares method
guarantees a unique fit for a given function; free parameters and coefficients
can be determined even with their associated errors. In the case of SED, the
free parameter is effective temperature of the star, which is a unique (non-
arbitrary) value.

Lastly, the problem of Type 3, involving parameter space, needs to be
investigated if there are no problems of Type 1 and Type 2. This is because
there could be more than one set of parameters (discrete arbitrariness), even
an infinite number of sets (unscientific case), that may produce the same
curve. A good example of limited arbitrariness would be the light curve of a
single circular spot on a rotating spherical star. Here, there is a trade off
between the inclination of the rotation axis and the spot latitude; that is, a
circular spot at latitudes 10 degrees (spot position with respect to the
equator) on a spherical rotating star with 45 degrees of inclination produces
exactly the same light curve as another spherical star with 80 degrees of
inclination having the same size circular spot at 45 degrees latitudes (Eker,
1996). A spot modeler cannot differentiate between these two models from the
light curves unless the inclination of the rotation axis is pre-assumed or
known. On the other hand, since each of the MLR, MRR and MTR functions (Eker
et al., 2018) has one to one correspondence (one luminosity, one radius, one
effective temperature for a main-sequence star of a given mass), there is no
Type 3 problem for the interrelated MLR, MRR and MTR functions.

When discussing these types of arbitrariness, Eker (1999) and Eker et al.
(2018) assumed that the scientific data (observed light curves, stellar masses
and parameters respectively) did not involve any kinds of zero point
complexities. The arbitrariness attributed to the zero point constant of the
BC scale, if BC numbers are treated as measurements (data), is another kind of
non-uniqueness, which is a totally new case in addition to the non-uniqueness
(arbitrariness) problems discussed above. Since this new problem is directly
connected with the measurements (data), it should be called Type 0 and should
be tested or investigated before the other types. One may imagine the mass of
stars being expressed by unknown units in such a way that it is impossible to
convert them into a known unit, because the scales used by different observers
have different zero points. The result is that the data would be unusable.

Science has overcome these problematic issues so that today’s scientist enters
a world where all units and unit systems (SI, cgs, etc) in science are ready
and rules how to convert from one system to another already exist. In one way
or another, the determination of a standard zero point constant of the BC
measurements was completed only very recently. Another peculiarity is that the
definition of BC in equation (1) involves definite integrals. The definite
integrals, however, never take an arbitrary constant. Why such an arbitrary
constant $C_{2}$ required in equation (1) is one of our basic questions, but
let us leave this issue aside for the moment and start investigating whether
there is a way out of the absolute arbitrariness implied by the arbitrary zero
point constant $C_{2}$. It is clear to us since there is neither a boundary
condition to apply nor could it be invented to set up its own value, that the
only way out of the absolute arbitrariness is a consensus, as was done
historically for the fundamental units of length, mass and time. Here, it
would be useful to review earlier attempts to avoid absolute arbitrariness
before arriving at a final solution.

####  2.1.1 Early Attempts to Avoid Absolute Arbitrariness

There are two possible ways of breaking off the absolute arbitrariness of the
BC scale using a star, the Sun, despite the difficulty of measuring its
apparent visual magnitude, which is about 26 mag brighter than the brightest
star in the sky, and unlike stars the Sun is not a point source.

1- Assume a bolometric correction ($BC_{V,\odot}$) for the Sun, and then
calculate the solar bolometric magnitudes from the observationally determined
solar visual magnitudes $V_{\odot}$ and $M_{V,\odot}$ using
$BC_{V}=M_{Bol}-M_{V}=m_{Bol}-V$.

2- Assume an absolute bolometric magnitude ($M_{Bol,\odot}$) for the Sun, and
then calculate its bolometric correction ($BC_{V,\odot}$) from its absolute
visual magnitude ($M_{V,\odot}$).

After choosing one of the two methods above, equation (2) could be used for
calculating the $M_{Bol}$ of a star if its $L$ is known. Consequently, the
star’s $BC_{V}$ could be calculated from its visual magnitudes if its distance
and interstellar extinction are available. After collecting a sufficient
number of $BC_{V}$ from all types of stars, a $BC_{V}-T_{eff}$ relation could
be established; this is a useful relation for estimating the $BC_{V}$ of
single stars which could be used for calculating the $L$ of other stars from
their observed $V$ magnitudes if their distances and extinctions are known.
Most $BC_{V}-T_{eff}$ relations are in tabular format; only Flower (1996) and
Eker et al. (2020) gave it as a function of $\log T_{eff}$ in a polynomial
format.

Although the word “assume” in the beginning of both methods implies absolute
arbitrariness, as soon as a decision is made ($BC_{V,\odot}$ value in the case
1, or $M_{Bol,\odot}$ value for the case 2), the problem reduces to limited
discrete arbitrariness for users who access tabulated tables of $BC_{V}$ from
several producers. For a producer, however, as long as the assumed value is
treated as a unique quantity, all the other parameters ($BC_{V}$ and
$M_{Bol}$) of other stars could be defined uniquely, fulfilling the fact that
stars must have unique $BC_{V}$ and $M_{Bol}$.

Historically, both methods are equally popular. Flower (1996) and Cox (2000)
could be considered among the latest who assumed $BC_{V,\odot}=-0.08$ mag
first, then calculated $M_{Bol,\odot}=4.74$ mag from $M_{V,\odot}=4.82$ mag,
itself calculated from $V_{\odot}=-26.75$ mag, which is the primary
observational quantity. Bessell et al. (1998) gives a table (Table A4) where
various authors (Allen, 1976; Durrant, 1981; Schmidt-Kaler, 1982) who assumed
$M_{Bol,\odot}$ first, then $BC_{V,\odot}$ calculated from $M_{V,\odot}$ which
is itself obtained from $V_{\odot}$ together with corresponding solar fluxes
and luminosities. Torres (2010) and Casagrande & VandenBerg (2018) are the
most recent examples where stellar $BC_{V}$ was obtained by adopting
$M_{Bol,\odot}=4.75$ mag, while Eker et al. (2020) preferred to adopt
$M_{Bol,\odot}=4.74$ mag since it is a nominal value suggested by IAU 2015
General Assembly.

Both methods are quite well acceptable and eligible to produce a consistent
unique $BC-T_{eff}$ curve. In addition to the discrete arbitrariness (users
have several choices), what is more problematic is that users must be alerted
when choosing BC corrections from literature. They must first search what
values of $M_{Bol,\odot}$ and $L_{\odot}$ were adopted at the very beginning
because the same adopted values are needed when calculating the $L$ of other
stars via equation (2). Even a slight change on any one or both of those
adopted values will be reflected in as systematic errors on the computed $L$.
The risk of error has already been point out by Torres (2010) “When making use
of tabulations of empirical bolometric corrections for stars ($BC_{V}$), a
commonly overlooked fact is that while the zero point is arbitrary, the
bolometric magnitude of the Sun ($M_{Bol,\odot}$) that is used in combination
with such tables cannot be chosen arbitrarily”.

In order to avoid such problems and also remove discrete arbitrariness,
consensus on a method and its assumed parameter is needed. Such a consensus
has recently been achieved by IAU 2015 General Assembly Resolution B2.

####  2.1.2 IAU 2015 GAR B2

It is not possible to say that the IAU has not issued a formal resolution on
the matter of $BC_{V}$ zero points because two of its commissions did agree
only on a zero point of the bolometric magnitude scale by adopting a value for
$M_{Bol,\odot}=4.7$5 mag at the Kyoto meeting of 1997 (Andersen, 1999, pp. 141
and 181). On one occasion, the bolometric magnitude scale was set by defining
a star with $M_{Bol}=0.00$ mag for an absolute radiative luminosity
$L=3.055\times 10^{28}$ W (see also Cayrel, 2002). However, as Casagrande &
VandenBerg (2018) commented, since $L_{\odot}$ is a measured quantity while
$M_{Bol,\odot}$ is an arbitrary zero point, any value of $M_{Bol,\odot}$ is
legitimate on the condition that once chosen, all bolometric corrections are
scaled accordingly. That is, no extra consensus is needed to pin down the
value of the zero point constant of the $BC_{V}$ scale. Indeed, fixing the
zero point of the bolometric magnitude scale has the consequence of fixing the
zero points of the all BC scales of all other bands, not only $BC_{V}$.

Arbitrariness of various BC scales, apparently, was an issue discussed first
by IAU commissions. Nevertheless, the decisions on the subject were finalized
by IAU 2015 General Assembly Resolution B2 (after IAU 2015 GAR B2). The zero
point constant of bolometric magnitudes was set according to:

$M_{Bol}=-2.5\log(L)+C_{Bol}.$ (3)

with the clear statement “a radiation source with absolute bolometric
magnitude $M_{Bol}=0$ has a radiative luminosity of exactly
$L_{0}=3.0128\times 10^{28}$ W and the absolute bolometric magnitude $M_{Bol}$
for a source of luminosity $L$ (in W) is:

$M_{Bol}=-2.5\log(L/L_{0})=-2.5\log L+71.197~{}425~{}...~{}.$ (4)

The zero point was selected so that the nominal solar luminosity corresponds
closely to absolute bolometric magnitude $M_{Bol,\odot}=4.74$ mag, the value
most commonly adopted in the recent literature (e.g. Bessell, Castelli & Plez,
1998; Cox, 2000; Torres,
2010)”111https://www.iau.org/static/resolutions/IAU2015 English.pdf together
with a reminder: the notations of $M_{Bol}$ and $m_{Bol}$ were adopted by
Commission 3 (Notations) at the 6th IAU General Assembly in Stockholm in
1938222https://www.iau.org/static/resolutions/IAU1938 French.pdf. It is
clearly declared that $M_{Bol}$ and $m_{Bol}$ refer specifically to absolute
and apparent bolometric magnitudes respectively. Thus, the zero point constant
of the bolometric magnitude scale is $C_{Bol}=71.197~{}425~{}...$ if $L$ is in
SI units, and $C_{Bol}=88.697~{}425~{}...$ if $L$ is in cgs units.

Equation (3) is more fundamental than equation (2) because if equation (3) is
written for the Sun:

$M_{Bol,\odot}=-2.5\log L_{\odot}+C_{Bol}.$ (5)

Subtracting this equation side by side from equation (3), the equation (2) is
obtained. Contrary to equation (2), which is the source of arguments favouring
arbitrariness, the more fundamental relation equation (3) leaves no space for
any kind of arbitrariness if the value of $C_{Bol}$ is known. In equation (2),
the term $C_{Bol}$ cancels out during the subtraction. From equation (5) it
can be understood why the Sun was chosen for setting the zero point constant
($C_{Bol}$) of the bolometric magnitudes. This is because the most accurate
stellar luminosity ever determined is solar luminosity.

The best estimate of solar luminosity is $L_{\odot}=4\pi({\rm
1~{}au})^{2}S_{\odot}=3.8275(\pm 0.0014)\times 10^{26}$ W, which is a value
computed from the recently estimated best value of the solar constant
$S_{\odot}=1361(\pm 1)$ Wm-2, obtained from the total solar irradiance (TSI)
data of 35 years of space-born observations during the last three solar cycles
(Kopp, 2014). The adopted values of $C_{Bol}$ and $L_{\odot}$ indicate the
nominal value of $M_{Bol,\odot}=4.739~{}996~{}...$ mag, which is indeed very
close to $M_{Bol,\odot}=4.74$ mag as specifically pointed out by IAU 20015 GAR
B2.

####  2.1.3 Other Types of Arbitrariness?

The discrete arbitrariness of the zero point of the $BC_{V}$ scale is removed
after assigning a definite value to the zero point constant for bolometric
magnitudes. If $M_{Bol}$ and $L$ have one-to-one correspondence, both could be
considered as unique properties of a star. Since visual absolute magnitude
$M_{V}$ is also a unique property of this star, the basic definition,
$BC_{V}=M_{Bol}-M_{V}$ is there to produce unique $BC_{V}$. This fact,
however, became noticeable to us after IAU 2015 GAR B2; that is, the
arbitrariness of type 0 attributed to the zero point of the $BC_{V}$ scale had
been removed.

The tabulated tables of $BC_{V}$ actually define a relationship between
$BC_{V}$ coefficients and other stellar parameters, even though there is no
definite form for a function. The primary parameter influencing $BC_{V}$ is
effective temperature. Flower (1996) and Eker et al. (2020) give the
$BC_{V}-T_{eff}$ relation as polynomials, using the logarithm of $T_{eff}$ as
a single variable. Therefore, it can be said that there is no arbitrariness of
Type 1 associated with $BC_{V}$ values as well as $BC_{V}-T_{eff}$ curves.
Moreover, there should not be an arbitrariness problem with Type 2 either
because the least squares method guarantees a unique fitting curve
($BC_{V}-T_{eff}$ function) with arbitrary coefficients determined, including
internal random errors. The parameter space of the $BC_{V}-T_{eff}$ function
is also unique; hence there is no space for a Type 3 problem because for a
given $T_{eff}$ there is only one value of $BC_{V}$. Thus, the arbitrariness
problems of Type 0, Type 1, Type 2 and Type 3 do not exist for $BC_{V}$ and
$BC_{V}-T_{eff}$ relations in reality.

###  2.2 Apparent Magnitudes and Zero Point Constants

A comprehensive description of the magnitude system is given concisely by Cox
(2000) and various textbooks. There are various filters to isolate certain
spectral features or wavelength ranges. The fluxes received through these
filters are transformed to magnitude values. For monochromatic radiation, a
flux density per unit wavelength of $3.631\times 10^{-9}$ erg s-1cm-2 Å-1
defined as m = 0.0 (Casagrande & VandenBerg, 2014), equivalent to -21.10 mag,
which is known as the zero point constant of ST magnitudes. If the
monochromatic magnitudes use fluxes of per unit frequency, the system is
called the AB system. For AB magnitudes, a flux of $3.631\times 10^{-20}$ erg
s-1cm-2 Hz-1 is set up for $m=0.0$ magnitude, where the zero point constant
becomes -48.60 mag (Casagrande & VandenBerg, 2014). Bessell et al. (1998)
provide them to three digits accuracy: $C_{\lambda}=-21.100$, and
$C_{\nu}=-48.598$. The fluxes which make $m=0.0$ were chosen so that, for
convenience, $\alpha$ Lyr (Vega) has very similar magnitudes in all systems:
ST, AB and VEGA (Casagrande & VandenBerg, 2014).

In reality, the magnitudes of various photometric systems are heterochromatic.
Using $\alpha$ Lyr (Vega) as the primary calibrating star, the VEGA system is
the most well known and deliberated for heterochromatic measurements. Among
these systems, the most famous is the Johnson-Cousins system. 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 et al. (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. (2012), Bessell &
Murphy (2012) and Casagrande & VandenBerg (2014). For the other bands, Vega is
found to be just slightly positive ($\sim 0.02$ mag) at most bands (Rieke et
al., 2008).

For the heterochromatic bands of various photometric systems, the zero point
constant $C_{\xi}$ needs to be derived for each bandpass $\xi$ using a star of
known absolute flux, usually Vega (Casagrande & VandenBerg, 2014), or Sirius
and Vega (Bohlin, 2014). Although $C_{\xi}$ are usually not given for the
photometric systems in the literature, where only monochromatic fluxes at
effective wavelengths of the filters are listed, all zero point constants are
well defined quantities (Bessell et al., 1998; Cox, 2000; Girardi et al.,
2012; Casagrande & VandenBerg, 2014).

The advantage of the magnitude system is that it works even if $C_{\xi}$ are
unknown. Since zero point constants cancels out, only ratio of fluxes can be
determined by subtracting apparent magnitudes; then knowing the absolute flux
of one star is sufficient to calculate the absolute flux of the other star
from their magnitude differences. On the other hand, unique (unchanging)
$C_{\xi}$ for each band is there to define the intrinsic colour indexes (if
there are no interstellar and atmospheric effects). Therefore, photometric
bands and their special zero points (definite fluxes unique to each band) are
natural for the astronomical photometry to work.

Apparent visual magnitude of a star could be expressed as:

$V=-2.5\log
f_{V}+C_{V}=-2.5\log\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm
d\lambda+C_{V},$ (6)

where $f_{V}=\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm d\lambda$ is the
$V$ flux reaching to the telescope (no interstellar and atmospheric effects),
$S_{\lambda}(V)$ is the $V$ band transparency function, $f_{\lambda}$ is the
monochromatic flux coming from the star and $C_{V}$ is the zero point constant
for the $V$ band. Similarly, it can be adopted for bolometric apparent
magnitudes as:

$m_{Bol}=-2.5\log f_{Bol}+C_{Bol}=-2.5\log\int_{0}^{\infty}\\!f_{\lambda}\rm
d\lambda+C_{Bol},$ (7)

where $f_{Bol}=\int_{0}^{\infty}\\!f_{\lambda}\rm d\lambda$ is the bolometric
flux reaching to the telescope (again, no interstellar and atmospheric
effects) and $C_{Bol}$ is the zero point constant for the bolometric apparent
magnitudes. Using the nominal solar values, $m_{Bol,\odot}=26.832$ mag (Cox,
2000), and $f_{Bol,\odot}=1361$ Wm-2 (IAU 2015 GAR B2), one can calculate the
value of the zero point constant $C_{Bol}=-18.997~{}351~{}...$ mag for the
apparent bolometric magnitudes. Consequently,
$f_{Bol}=2.518~{}022~{}...10^{-8}$ Wm-2 could be calculated using the same
$C_{Bol}=-18.997~{}351~{}...$ mag for making $m_{Bol}=0$.

Bolometric apparent magnitude appears as if there is brightness at another
band, in addition to the other bands of the Johnson photometry. This
hypothetical brightness, however, represents a band of radiation so broad that
it covers all wavelengths; thus, it represents the total radiation of a star.
As is calculated for obtaining the intrinsic colour of an unreddened star,
subtracting side by side equation (6) from equation (7), according to its
basic definition, $BC_{V}$ becomes:

$\displaystyle BC_{V}=m_{Bol}-V=2.5\log\frac{f_{V}}{f_{Bol}}+(C_{Bol}-C_{V})$
(8)
$\displaystyle=2.5\log\left(\frac{\int_{0}^{\infty}\\!S_{\lambda}(V)f_{\lambda}\rm
d\lambda}{\int_{0}^{\infty}\\!f_{\lambda}\rm d\lambda}\right)+C_{2}.$

Here is the answer to the question of why there is an arbitrary constant in
equation (1), despite the definite integrals not taking any constant. $C_{2}$
is certainly not a constant of integration, even though it appears as if it is
one. It is a natural result of absolute photometry. Also, it is not correct to
call it “arbitrary” anymore, because $C_{2}=C_{Bol}-C_{V}$, where $C_{Bol}$
and $C_{V}$ are the definite zero point constants of the bolometric and visual
magnitudes according to equations (7) and (6).

###  2.3 Absolute Magnitudes and Zero Point Constants

Absolute magnitudes are also expressed by the same notation as apparent
magnitudes. Thus, the absolute magnitude of a star at the photometric band
$\xi$ is expressed as:

$M_{\xi}=-2.5\log F_{\xi}+C_{\xi}.$ (9)

and the bolometric absolute magnitudes as:

$M_{Bol}=-2.5\log F_{Bol}+C_{Bol}.$ (10)

then, $M$ and $F$ stand for absolute magnitudes and fluxes if a star is moved
to a distance of 10 pc. Now the question is: What happens to the zero point
constants $C_{\xi}$ and $C_{Bol}$? Are they the same constants used for
apparent magnitudes?

It is logical to assume the zero point constants of absolute magnitudes are
the same as the zero point constants of apparent magnitudes because the zero
point flux for a photometric band, which marks zero magnitude on the magnitude
scale, is independent of the distance. But still this can be tested by taking
the $C_{Bol}$ of the apparent bolometric magnitudes given above, and using
$M_{Bol,\odot}=4.74$ mag,

$4.74=-2.5\log F_{Bol,\odot}-18.997~{}351~{}...~{},$ (11)

then the bolometric flux of the Sun, if it is at a distance of 10 pc
$F_{Bol,\odot}=3.199~{}334~{}...\times 10^{-10}$ W m-2 is obtained. If this
flux is multiplied by the surface area of a sphere with a radius of 10 pc, it
gives $L_{\odot}=3.828\times 10^{26}$ W. This is the nominal solar luminosity
announced by IAU GAR B2, which was calculated from the solar constant
$S_{\odot}=1361(\pm 1)$ Wm-2. Therefore, we can claim confidently that the
zero point constants of apparent (equation 7) and absolute bolometric
magnitudes (equation 10) are the same
$C_{Bol}{\rm(fluxes)}=-18.997~{}351~{}...$ . Moreover, the same conclusion
could be obtained by $BC_{\xi}=M_{Bol}-M_{\xi}=m_{Bol}-m_{\xi}$.

Using the part

$M_{Bol}-M_{\xi}=m_{Bol}-m_{\xi}$ (12)

and moving $m_{\xi}$ to the left, then moving $M_{Bol}$ to the right
algebraically, the equation becomes:

$m_{\xi}-M_{\xi}=m_{Bol}-M_{Bol}.$ (13)

Both equations (12 and 13) are valid. The former is the basic definition of
$BC_{\xi}$ while the latter is the distance modulus, which is equal to ($5\log
d-5$), where $d$ is in parsecs. Since the wavelength dependent zero point
constants are cancelled between the equal signs separating various photometric
bands, equation (13) tells us that the distance modulus is independent of
wavelengths. This could happen only if the zero point constant for apparent
and absolute magnitudes have the same value (Table 1). While the distance
modulus is independent of wavelengths, conversely, the bolometric corrections
defined by equation (12) are dependent on wavelengths (or bands). Thus,
bolometric correction is a unique feature of a star, as with its intrinsic
colours.

Table 1: Zero point constants for calculating stellar luminosities or fluxes. $C_{Bol}$ for $L$ | $L_{0}$ | Unit |  $C_{Bol}$ for $f_{Bol}$ |  $C_{Bol}$ for $F_{Bol}$ | $f_{0}$ | Unit
---|---|---|---|---|---|---
71.197 425 … | 3.0128E+28 | W | -18.997 351 … | -18.997 351 … | 2.5180E-08 | W m-2
88.697 425 … | 3.0128E+35 | erg s-1 | -11.497 351 … | -11.497 351 … | 2.5180E-05 | erg s-1 cm-2

###  2.4 From a Flux Ratio to a Luminosity Ratio

Taking one of the $M_{\xi}$ (equation 9), let it be $M_{V}$, and by
subtracting it side by side from $M_{Bol}$ (equation 10), it can be written:

$M_{Bol}-M_{V}=2.5\log\frac{F_{V}}{F_{Bol}}+(C_{Bol}-C_{V}).$ (14)

where, $F_{V}/F_{Bol}$ is the ratio of visual flux to bolometric flux reaching
to Earth (no atmospheric and interstellar effects) if the star is moved to a
distance of $d=10$ pc. According to the basic definition of $BC_{V}$, this
equation and equation (8) are equivalent. Hence, it can be written:

$\frac{F_{V}}{F_{Bol}}=\frac{f_{V}}{f_{Bol}}.$ (15)

which means that the ratio of visual flux to bolometric flux for a star is
independent of its distances. Therefore, some more equal signs could be added:

$\frac{F_{V}}{F_{Bol}}=\frac{f_{V}}{f_{Bol}}=\frac{f_{V}(S)}{f_{Bol}(S)}=\frac{4\pi
R^{2}f_{V}(S)}{4\pi R^{2}f_{Bol}(S)}=\frac{L_{V}}{L}=\frac{f_{V}}{\sigma
T^{4}}.$ (16)

The third term is the flux ratio on the surface of the star. Multiplying the
surface fluxes (visual and bolometric) by $4\pi R^{2}$ (surface area of a
star), then its equivalent, the luminosity ratio ($L_{V}/L$), could be added.
Finally, by replacing the surface bolometric flux by $\sigma T^{4}$, one can
see the primary parameter (effective temperature) to determine the BC values
of stars according to the rightmost ratio. After replacing $F_{V}/F_{Bol}$ in
equation ( 14) by $L_{V}/L$, one can deduce:

$M_{Bol}=-2.5\log L+C_{Bol},$ (17)

and,

$M_{V}=-2.5\log L_{V}+C_{V}.$ (18)

We must be careful here; the $C_{Bol}$ providing $M_{Bol}$ of stars from their
$L$ is different ($C_{Bol}=71.197~{}425~{}...$ mag) from the $C_{Bol}$
providing the same $M_{Bol}$ from its bolometric flux if it is at 10 pc
[$C_{Bol}{\rm(fluxes)}=-18.997~{}351~{}...$ mag]. Similar case must be true
for the apparent magnitudes. Subtracting equation (18) from equation (17), the
following could also be written for $BC_{V}$

$BC_{V}=2.5\log\frac{L_{V}}{L}+C_{2}.$ (19)

where $C_{2}=C_{Bol}-C_{V}$. Here, one must be aware of the fact that the zero
point constants proper to the luminosities (equations 17 and 18), which are
different from the zero point constants for fluxes (equations 6, 7 and 9, 10).

###  2.5 Analytical Evidence for Limited Arbitrariness and Positive BC

According to equations (1), (8) and (19), the arbitrariness attributed to the
zero point constant ($C_{2}$) of the $BC_{V}$ scale cannot be called absolute
arbitrariness but it could be called limited arbitrariness. In other words; no
value of $C_{2}$ from minus infinity to plus infinity (absolute arbitrariness)
satisfies these equations. This is because the visual to bolometric flux
ratio, or ratio of luminosities $L_{V}$ to $L$, is a unitless number
[$(0<f_{V}/f_{Bol}<1)$ or $(0<L_{V}/L<1)$] between zero and one. Logarithms of
numbers between zero and one are all negative. Only a positive $C_{2}$ would
satisfy equations (1), (8) and (19) if the $BC_{V}$ of a star is zero, which
occurs at $M_{Bol}=M_{V}$. Therefore, from equations (1) and (14):

$C_{2}(V)=-2.5\left|\log\frac{f_{V}}{f_{Bol}}\right|_{BC_{V}=0}.$ (20)

This equation could easily be adapted for $C_{2}(\xi)$ by replacing $f_{V}$ by
$f_{\xi}$. Thus, each band has to be evaluated at the point $BC_{\xi}=0$,
which occurs at $M_{Bol}=M_{\xi}$. Since the flux of each band, not only the
visual flux, is less than the bolometric flux, the right hand side of equation
(20) is a positive number. The ratio in the logarithmic term has no chance to
be either zero or one, therefore, $C_{2}(\xi)$ is strictly a number greater
than zero.

The $C_{2}(\xi)$ is still arbitrary depending upon which star is chosen for
$BC=0$ mag, but the arbitrariness implied by equations (1), (8) and (19) is
not an absolute arbitrariness, which would also include zero and negative
numbers. Since only positive numbers are possible for $C_{2}(\xi)$, the type
of the arbitrariness is limited.

After knowing the fact that $C_{2}(\xi)>0$, it becomes easy to investigate
whether a positive BC exists or not. From equation (20) the following could be
deduced: if the absolute values of the logarithmic terms in equations (1),
(8), and (19)

$\left|2.5\log\frac{f_{\xi}}{f_{Bol}}\right|<C_{2}(\xi).$ (21)

is less than the value of the zero point constant $C_{2}(\xi)$, the $BC_{\xi}$
values are positive, if

$\left|2.5\log\frac{f_{\xi}}{f_{Bol}}\right|>C_{2}(\xi).$ (22)

the $BC_{\xi}$ values are negative. It can therefore be concluded here that a
limited number of positive BC values is inevitable.

A limited number of positive BC may also be deduced from following:

$L_{V}=L\times 10^{\frac{BC_{V}-C_{2}}{2.5}}.$ (23)

which is another form of equation (19). All of the $BC_{V}$ values smaller
than $C_{2}$, including zero and limited range of positive numbers ($0\leq
BC_{V}<C_{2}$), are valid to produce a $L_{V}$ which is less than $L$.
$BC_{V}=C_{2}$ would mean $L_{V}=L$ and $BC_{V}>C_{2}$ would produce
$L_{V}>L$, which are invalid cases because $L_{V}\geq L$ is unphysical.

##  3 Discussion

###  3.1 Positive Bolometric Corrections; Problematic or not Problematic?

There should not be any other option in between; a positive BC is either a
conceptual problem, or not. No one should say “whether the BC values are
positive or negative is subject to the caprice of astronomers” and then ask
the question “Does anybody really care whether BCs “have to be” positive or
not?”

Positive science does not operate on the caprice of scientists and care must
be given not to stray from scientific consistency. A positive BC was already
said to contradict the fact that “the bolometric magnitude of a star ought to
be brighter than its $V$ magnitude” (Torres, 2010). Moreover, to emphasize the
importance for BC numbers to be negative, Cox (2000) listed the smallest BC of
stars (main-sequence, giants and supergiants) as “-0.00” for F2 supergiants in
the astronomers’ handbook Allen’s Astrophysical Quantities, 4th edition, on
page 389. The negative sign appearing as “-0.00” indicates that BCs are not
even neutral (zero) but strictly negative numbers. “-0.00” implies that the
$BC_{V}$ of F2 supergiants could contain a numeral other than zero, with at
least two zeros after the decimal point.

A positive BC is also related to the arbitrariness problem attributed to the
zero point of the BC scale. Arbitrariness, especially absolute arbitrariness,
provides BC producers the freedom to make a systematic shift of
$BC_{V}-T_{eff}$ diagram or on the tabulated BC numbers to make them all
negative in order to avoid a dilemma. Two schools of thought have emerged. One
group of astronomers intentionally apply a systematic shift to avoid even a
small single positive BC (because this solves the problem of contradicting the
fact that “the bolometric magnitude of a star ought to be brighter than its
$V$ magnitude”), while the other group of astronomers keep positive BC in
their list and serve them to their users. The group keeping positive BC
appears problematic if the arbitrariness problem is not solved; the group
could be called agnostic or uncaring about the problems of positive BC. If
indeed there is a solution to the arbitrariness problems, users of
$BC_{V}-T_{eff}$ relations or tabulated tables of BC do have a right to hear
what the solution is. If there is a solution, then the former group becomes
inconsistent. There is no way in between, either one of the groups must give
up its practice for consistent science, even if their practice is several
decades or centuries old.

###  3.2 Theoretical and Observational Support for a Limited Number of
Positive BC

The analytical evidence for the limited arbitrariness and positive BC
presented in Section 2.5 could be considered as theoretical support for a
limited number of positive BC values, which appear from time to time in the
literature. As discussed in Section 2.5, although a positive BC is inevitable
analytically, there is no indication where (or at what temperature) the zero
point of the BC scale for different photometrical bands would occur. This is
the kind of problem where one should be able to see the existence of a limited
number of positive BC values, but leave the astronomer with the freedom to
define the zero point optionally, at best conventionally. The applicability
and necessity of the decision by IAU 20015 GAR B2 is founded firmly in this
freedom. In other words, the zero point of the BC scale suggested by IAU 20015
GAR B2 would be consistent with the analytical prediction if the zero point is
not at the peak (or above the peak) of the $BC-T_{eff}$ curve. It has already
been shown in Section 2.5 that unphysical results such as $L_{V}=L$ would
occur if the zero point of the BC scale is placed right on the peak of the
curve, which means $BC_{V}=C_{2}$. This case, however, is impossible according
to equation (20). The unphysical result $L_{V}>L$ occurs if the zero point is
placed above the peak, that is, if $BC_{V}>C_{2}$, where $C_{2}$ stands for
the zero point constant for the $BC_{V}$ scale. Physically consistent results
require $BC_{V}<C_{2}$, which includes both positive ($0<BC_{V}<C_{2}$) and
negative ($BC_{V}<0$) bolometric corrections.

All empirical $BC_{V}$ vales computed from the observed stellar parameters
(Code et al., 1976; Johnson, 1964, 1966; Flower, 1977, 1996; Bessell, Castelli
& Plez, 1998; Sung et al., 2013; Casagrande & VandenBerg, 2018; Eker et al.,
2020) could be considered as observational support for the existence of a
limited number of positive $BC_{V}$. The $V$ band empirical bolometric
correction coefficients of main-sequence stars computed most recently by Eker
et al. (2020) are of special importance not only because they were computed
from the most updated and/or most accurate stellar parameters, but also
because the nominal values $M_{Bol,\odot}=4.74$ mag, and
$L_{\odot}=3.828\times 10^{26}$ W were used in computing the $BC_{V}$ values
of 400 stars, as suggested by IAU 2015 General Assembly Resolution B2.

According to the results of Eker et al. (2020), the peak of the
$BC_{V}-T_{eff}$ curve occurs at $T_{eff}=6897$ K, at about a spectral type
$\sim$ F1V with a peak value $BC_{V,max}=0.95$ mag and the two zero points
($BC_{V}=0$ mag) occur at effective temperatures $T_{eff,1}=5859$ K, and
$T_{eff,2}=8226$ K corresponding to the spectral types $\sim$ G2V and $\sim$
A5V, respectively. Thus, according to the $BC_{V}-T_{eff}$ curve of Eker et
al. (2020), main-sequence stars within the temperature limits
$5859<T_{eff}<8226$ K have positive $BC_{V}$ and the rest, that is, the
earlier and later spectral types, have negative $BC_{V}$.

Therefore, here we can go beyond the norm to say “bolometric correction of a
star is not always negative”. Considering the widespread effect of the well-
known axiom “the bolometric magnitude of a star ought to be brighter than its
$V$ magnitude” among practitioners for about a century, this conclusion is not
just a new result but a paradigm change, which is supported not only by
theoretical and observational evidence but also according to the convention
announced by IAU 2015 GAR B2.

###  3.3 Comparing $M_{Bol}$ and $M_{V}$ rather than $L$ and $L_{V}$ is
misleading

Comparing the bolometric magnitude of a star to its $V$ magnitude, and then to
say which one is brighter, could be misleading because the human senses are
not absolute. Thus, the perceptions of the eye could be deceiving. Even though
there is one-to-one correspondence between luminosity and corresponding
brightness in magnitude units, as they are displayed by equations (3), (17)
and (18), the same conclusion cannot be drawn by comparing total luminosity
(whole spectrum) to one of its parts ($L_{V}$ formed by visual photons).

First of all, luminosities are physical but magnitudes are not physical
quantities. Comparing physical quantities is definitely more meaningful and
definite than comparing non-physical, which is most probably subjective rather
than being objective. Then, magnitude is the measure of human perception of a
luminosity, which was first suggested by Hipparchus of Nicaea (190-120 BC) and
reformulated by Norman Robert Pogson in 1854 by defining a first magnitude
star as a star that is 100 times brighter than a sixth magnitude star (Cox,
2008), which is still used today. Thus, eye comparisons of magnitudes are
meaningful only if they are conducted in the same wavelength range within the
visible part of the electromagnetic spectrum; otherwise, meaningless to
indicate which one of two stars is more luminous due to different eye
sensitivity at different wavelengths.

Since the magnitude scale is in reverse order, a brighter magnitude is shown
by a smaller number. Consequently, the phrase that “bolometric magnitude of a
star ought to be brighter than its $V$ magnitude” indicates that the
bolometric magnitude of a star has a smaller numerical value than its $V$
magnitude. Then, subtracting a bigger number ($M_{V}$ or $V$) from a smaller
number ($M_{Bol}$ or $m_{Bol}$), one can deduce a negative number for the
bolometric correction of a star according to the basic definition
($BC_{V}=M_{Bol}-M_{V}=m_{Bol}-V$), which is confirmed only for stars of the
earliest and latest spectral types but not for main-sequence stars of spectral
types $\sim$ (G2-A5). Drawing attention to comparing $M_{Bol}$ and $M_{V}$
rather than $L$ and $L_{V}$, the phrase “bolometric magnitude of a star ought
to be brighter than its $V$ magnitude” could also be considered misleading
because the conclusion drawn by this comparison is not firmly established,
that is, bolometric corrections are not always negative according to the
observational and theoretical evidences discussed above.

###  3.4 Visualizing a Fictitious Problem: Arbitrariness

Assuming the absolute bolometric magnitude of the Sun ($M_{Bol,\odot}$) is
arbitrary (Casagrande & VandenBerg, 2018), when computing the absolute
bolometric magnitude ($M_{Bol}$) for a star according to equation (2) means
that the computed $M_{Bol}$ of the star is also arbitrary. This way, there
could be an infinite number of $M_{Bol}$ values computed from a single
luminosity. This situation is visualized in Figure 1 by three preferred
$M_{Bol,\odot}$ values using the luminosities of 400 stars which were selected
by Eker et al. (2020) for computing their $BC_{V}$ values, from which the
$BC_{V}-T_{eff}$ relation for nearby main-sequence stars is predicted.

Figure 1: Filled circles ($\bullet$) are $M_{Bol}$ of main-sequence stars
(Eker et al., 2020) calculated according to equation (2) using
$M_{Bol,\odot}=-0.26$ mag (upper), $M_{Bol,\odot}=4.74$ mag (middle),
$M_{Bol,\odot}=9.74$ mag (lower). Empty circles (o) are absolute visual
magnitudes $M_{V}$, (o) for the same stars.

The filled and empty circles in the middle of Figure 1 show $M_{Bol}$ and
$M_{V}$ values of the 400 main-sequence stars plotted as a function of stellar
mass in solar units. Despite the stellar luminosities staying constant,
because of the enforced arbitrariness (Casagrande & VandenBerg, 2018), the
$M_{Bol}$ values are shifted up or down according to the adopted values of
$M_{Bol,\odot}$ as indicated on the figure. Only two examples of all positive
BC or all negative BC were indicated. That is, if one prefers to use
$M_{Bol,\odot}=9.74$ mag, then all computed $BC_{V}$ values would be positive,
while choosing $M_{Bol,\odot}=-0.26$ mag makes all $BC_{V}$ values negative
(See Figure 2).

Figure 2: All positive (upper $M_{Bol,\odot}=9.74$ mag), standard (middle
$M_{Bol,\odot}=4.74$ mag) and all negative (below $M_{Bol,\odot}=-0.26$ mag)
$BC_{V}$ coefficients of the main-sequence stars (Eker et al., 2020) as a
function of $\log M/M_{\odot}$.

If the absolute visual magnitudes ($M_{V}$) of stars in Figure 1 are
subtracted from the corresponding $M_{Bol}$, then the corresponding $BC_{V}$
are obtained as a function of stellar mass (Figure 2). The same $BC_{V}$ data
could be re-arranged according to stellar effective temperatures (Figure 3),
which is the same $BC_{V}$ data from which Eker et al. (2020) derived their
$BC_{V}-T_{eff}$ relation. The similar appearance of $BC_{V}-M/M_{\odot}$ and
$BC_{V}-T_{eff}$ is not surprising.

Figure 3: All positive (upper $M_{Bol,\odot}=9.74$ mag), standard (middle
$M_{Bol,\odot}=4.74$ mag) and all negative (below $M_{Bol,\odot}=-0.26$ mag)
$BC_{V}$ coefficients of the main sequence stars (Eker et al., 2020) as a
function of $\log T_{eff}$.

It is by definition, that a star has a unique $M_{Bol}$, thus a unique
$BC_{V}$, coming from its unique $L$ and $M_{V}$. Only after both absolute and
limited discrete arbitrariness attributed to the zero point of BC scales is
removed by IAU 2015 GAR B2 can such a consistency be achieved. The
arbitrariness attributed to $M_{Bol,\odot}$, (Casagrande & VandenBerg, 2018)
violates this fact as displayed in Figures 1, 2, and 3, so the infinite number
of $BC_{V}$ becomes inevitable. At this point, it can be said that the
arbitrariness attributed to $M_{Bol,\odot}$ and the zero point of the BC scale
is a fictitious problem caused by the method of computing according to the
basic definition ($BC_{V}=M_{Bol}-M_{V}=m_{Bol}-V$), where the true value of
$M_{Bol,\odot}$ was not known, thus assumed to be arbitrary before IAU 2015
GAR B2.

It may also be noticed here that the positive and negative occurrences of
standard $BC_{V}$ values are also marked on Figure 2 and Figure 3 where the
values above the horizontal line are positive. The very limited number of
positive occurrences is clearly apparent.

Stressing the uniqueness of a BC for a star at any band is no different than
the uniqueness of its $M_{Bol}$, $M_{V}$, $L$, $R$ or $T_{eff}$; there should
not be a function such as $BC-T_{eff}$ and/or $BC-M/M_{\odot}$, which could
relate a BC value directly to a star’s $T_{eff}$ or mass. Similar to the MLR,
MRR and MTR functions of Eker et al. (2018), which are not the functions to
provide the $L$, $R$ or $T_{eff}$ of a star given mass but to provide mean
values of $L$, $R$ and $T_{eff}$ of main-sequence stars of given mass.
According to stellar structure and evolution theories Clayton (1968) true $L$,
$R$, and $T_{eff}$ are unique for a star which depend on its mass, chemical
composition and age. In the same way of thinking, $BC-T_{eff}$ or
$BC-M/M_{\odot}$ should provide typical (a kind of mean value) BC value for
the typical effective temperature of main-sequence stars. Thus, the typical
mass, typical effective temperatures and typical BC are related. The unique BC
of a single star is provided only by the basic definition of BC
($BC_{\xi}=M_{Bol}-M_{\xi}=m_{Bol}-m_{\xi}$). Scatter of data on a
$BC_{V}-T_{eff}$ relation are not really due only to observational errors, as
in the case of SED represented by a Planck function, but also due to
differences of mass, chemical compositions and age. Thus, the $BC_{V}-T_{eff}$
relation derived by Eker et al. (2020) cannot be used to calculate the
$BC_{V}$ of a star with a $T_{eff}$. It rather provides a mean value of
$BC_{V}$ for main-sequence stars having a typical $T_{eff}$ value, which could
be related to a typical mass according to the MTR of Eker et al. (2018), or
could be derived by directly from the masses of stars, as shown in Figure 2 in
this study. The tabulated BC tables are also not free from this problem. They
too provide a mean $BC_{V}$ for a given mean value of a stellar parameter.

###  3.5 Importance of Standard BC

There are two ways to obtain a standard $BC_{\xi}$ for a star from its
luminosity ($L$):

  * 1)

Obtain its $M_{Bol}$ from its $L$ according to equation (3) using
$C_{Bol}=71.197~{}425~{}...$ if $L$ is in SI units,
$C_{Bol}=88.697~{}425~{}...$ if $L$ is in cgs units.

  * 2)

Obtain its $M_{Bol}$ from its $L$ according to equation (2) using nominal
solar bolometric absolute magnitude $M_{Bol,\odot}=4.74$ mag, and the nominal
solar luminosity $L_{\odot}=3.8275\times 10^{26}$ W.

After having its unique (standard) $M_{Bol}$ from its unique L according to
one of the two methods above, its absolute magnitude ($M_{\xi}$) in the band
$\xi$ should be calculated properly from its de-reddened apparent magnitude
($m_{\xi}$) using its most accurate stellar parallax. Then, the difference
$M_{Bol}-M_{\xi}$ could be called standard $BC_{\xi}$. Using any other
$C_{Bol}$ in the method (1) and any other non-standard $M_{Bol,\odot}$ and/or
$L_{\odot}$ values in the method (2) is sufficient to make it a non-standard
($BC_{\xi}$). Finally, the standard BC of many stars, e.g. main-sequence
stars, could be used to calculate the standard $BC_{V}-T_{eff}$ curve, as done
by Eker et al. (2020).

Although both methods are equivalent, the method (1) is primarily advised
since it is more straightforward, and does not demand additional knowledge
other than $C_{Bol}$. Standard BC tables or standard $BC-T_{eff}$ curves are
very important for standardizing stellar luminosities. It also saves time for
users trying to find out from inspecting which value of $M_{Bol,\odot}$ were
used at the beginning, and prevents them from any kind of making mistakes by
using the improper $M_{Bol,\odot}$ and $L_{\odot}$ when re-computing unknown
$L$.

Considering the non-standard tabulated tables of $BC_{V}$, especially where
all $BC_{V}$ are less than zero, our estimate is that there could be about
$\sim$0.1 mag differences between standard and non-standard BC of a star. An
approximate $\sim$0.1 mag difference, however, is equivalent to a $\sim$5%
systematic error in the distances (or trigonometric parallaxes) and about 10%
systematic error in the stellar luminosities. Such uncertainty was tolerable
in the era before Gaia. However now, after Gaia, it is no longer tolerable.
Among 206 binaries used for establishing the standard $BC_{V}-T_{eff}$
relation of nearby main-sequence stars by Eker et al. (2020), who relied on
Gaia Data Release 2 distances (Gaia Collaboration, 2018), 120 systems ($\sim
60\%$) have 2% or better accuracy in the trigonometric parallaxes.

Reducing the uncertainties of stellar $L$ by standard BC is not only important
for internal structure and evolution theories, which demand not only the most
accurate observational parameters (mass, radius, temperature, luminosity), but
are also very important for galactic and extra-galactic astronomy, even
cosmology and Hubble law; the luminosity functions of galaxies all require
better stellar luminosities.

##  4 Conclusions

The zero point constant of any BC scale [$C_{2}(\xi)$] must be a positive
number according to equations (1), (8), (14) and (19) due to the fact that the
apparent flux ($0<f_{\xi}/f_{Bol}<1$) or absolute flux ($0<F_{\xi}/F_{Bol}<1$)
or luminosity ($0<L_{\xi}/L<1$) of a star, which is limited by a filter, is
less than the apparent bolometric, or absolute bolometric fluxes or total
luminosity of the star. A positive $C_{2}(\xi)$, however, require at least
some of the $BC_{\xi}$ values to be positive.

Consequently, shifting BC values to make them all less than zero is not
necessary, despite any arbitrariness attributed to it. In this study, it
became clear to us that the zero point constant $C_{2}$ in equation (1) is not
a constant of integration, even though it looks like one. This is because
definite integrals never take a constant of integration. In reality, it is a
zero point constant imposed by definitions of bolometric and visual magnitudes
appearing in the basic definition according to absolute photometry [see
equations (8), (14)]. It is shown in this study that $C_{2}=C_{Bol}-C_{V}$.
Since the zero points of existing photometric systems are all well-defined
(Casagrande & VandenBerg, 2014), but only very recently has IAU 2015 GAR B2
defined the zero point of bolometric magnitudes, then earlier astronomers had
no option but to assume it arbitrary. Only, after IAU 2015 GAR B2 is it
understood to be well defined quantity because the definite values of
$C_{Bol}$ and $C_{V}$ leave no space for it to be an arbitrary number, even
though a definite value of $C_{V}$ cannot be found easily in the literature
[for each bandpass $C_{\xi}$ could be derived using a star of known absolute
flux, usually Vega (Casagrande & VandenBerg, 2014), or Sirius and Vega
(Bohlin, 2014)] where usually monochromatic fluxes at effective wavelengths of
filters for an apparent magnitude zero are given (Bessell et al., 1998; Cox,
2000; Girardi et al., 2012; Casagrande & VandenBerg, 2014).

A positive $C_{2}$ implies $C_{Bol}>C_{V}$. However one must be careful here.
$C_{Bol}=71.197~{}425~{}...$ if $L$ is in SI units,
$C_{Bol}=88.697~{}425~{}...$ if $L$ is in cgs units are for calculating $L$
from $M_{Bol}$, or vice versa, according to equation (3) or (17). The zero
point constant for equations (7) and (10), where apparent and absolute
bolometric fluxes reaching Earth (if no atmospheric and interstellar effects)
are related to apparent and absolute magnitudes, is different. Perhaps, it is
more proper to indicate it as $C_{Bol}({\rm fluxes})=-18.997~{}351~{}...$ mag
(IAU 2015 GAR B2) corresponding to a flux of $2.518\times 10^{-8}$ Wm-2 to
make the absolute and apparent magnitudes zero. Table 1 indicates which of the
zero point constants are good for calculating stellar luminosities or fluxes.

Arbitrariness attributed to the zero point of the BC scale does not exist in
reality. It is a fictitious arbitrariness caused by the method of computing or
because the zero point of bolometric magnitudes was not yet fixed. Since
$C_{2}=C_{Bol}-C_{V}$ and if $C_{Bol}$ is not fixed, then $C_{2}$ become
arbitrary. It is irrational to assume a star having more than one BC or
$M_{Bol}$. Absolute arbitrariness attributed to the zero point of BC scale,
however, contradicts this fact because it implies a star can have an infinite
number of BC and $M_{Bol}$, which is not true. Comments such as “… while the
solar luminosity $L_{\odot}$ is a measured quantity, $M_{Bol,\odot}$ is an
arbitrary zero-point and any value is equally legitimate on the condition that
once chosen, all bolometric corrections are scaled accordingly” are indeed
incorrect because stars can have only one $M_{Bol}$, and the Sun cannot be the
only exception to having an infinite number of absolute bolometric magnitudes.

## Acknowledgements

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

## References

  * Allen (1976) Allen, C. W., 1976, Astrophysical Quantities (3rd ed.; London: Athlone Press)
  * Andersen (1999) Andersen, J., 1999, Proc. Twenty-Third General Assembly, Transactions of the IAU, Vol. XXIIIB (Dordrecht: Kluwer)
  * Bessell et al. (1998) Bessell, M. S., Castelli, F., Plez, B., 1998, A&A, 333, 231
  * Bessell & Murphy (2012) Bessell, M., Murphy, S., 2012, PASP, 124, 140
  * Bohlin (2014) Bohlin, R. C., 2014, AJ, 147, 127
  * Casagrande & VandenBerg (2014) Casagrande, L., VandenBerg, D. A., 2014, MNRAS, 444, 392
  * Casagrande & VandenBerg (2018) Casagrande, L., VandenBerg, D. A., 2018, MNRAS, 479, L102
  * Cayrel (2002) Cayrel, R. 2002, in ASP Conf. Ser. 274, Observed HR Diagrams and Stellar Evolution, ed. T. Lejeune & J. Fernandes (San Francisco, CA: ASP), 133
  * Clayton (1968) Clayton, D. D., 1968, Principles of Stellar Evolution and Nucleosynthesis. McGraw-Hill Book Company, New York
  * Code et al. (1976) Code, A. D., Bless, R. C., Davis, J., Brown, R. H., 1976, ApJ, 203, 417
  * Cox (2000) Cox, A. N., 2000, Allen’s astrophysical quantities, New York: AIP Press; Springer, Edited by Arthur N. Cox. ISBN: 0387987460
  * Cox (2008) Cox, M., 2008, Antiquarian Astronomer, 4, 23
  * Durrant (1981) Durrant, C. J., 1981, Landolt Bornstein vol VI/2A (Neue Series: Springer-Verlag), p. 82
  * Eker (1996) Eker, Z., 1996, ApJ, 473, 388
  * Eker (1999) Eker, Z., 1999, NewA, 4, 365
  * Eker et al. (2018) Eker, Z., Bakıs, V., Bilir, S., et al., 2018, MNRAS, 479, 5491
  * Eker et al. (2020) Eker, Z., Soydugan, F., Bilir, S., et al., 2020, MNRAS, 496, 3887
  * Flower (1977) Flower, P. J., 1977, A&A, 54, 31
  * Flower (1996) Flower, P. J., 1996, ApJ, 469, 355
  * Gaia Collaboration (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T. et al., 2018, ApJ, 616, id.A1, 22
  * Girardi et al. (2012) Girardi, L., Bertelli, G., Bressan, A., Chiosi, C., Groenewegen, M. A. T., Marigo, P., Salasnich, B., Weiss, A., 2012, A&A, 391, 195.
  * Habets & Heintze (1981) Habets, G. M. H. J., Heintze, J. R. W., 1981, A&AS, 46, 193
  * Hayes (1978) Hayes, D. S., 1978, In: The HR diagram-The 100th anniversary of Henry Norris Russell; Proceedings of the Symposium, Washington, D.C., November 2-5, 1977. Dordrecht: Reidel, p. 76
  * Heintze (1973) Heintze, J. R. W., 1973, Proceedings of IAU Symposium no. 54 Held in Geneva, Switzerland, Sept. 12-15, 1972. Ed by B. Hauck and Bengt E. Westerlund. Dordrecht, Boston, Reidel, p. 231
  * Johnson (1964) Johnson, H. L., 1964, Boletin de los Observatorios de Tonantzintla y Tacubaya, 3, 305
  * Johnson (1966) Johnson, H. L., 1966, ARA&A, 4, 193
  * Johnson et al. (1966) Johnson, H. L., Mitchell, R. I., Iriarte, B., Wisniewski, W. Z., 1966, ColPL, 4, 99
  * Kopp (2014) Kopp, G., 2014, Journal of Space Weather and Space Climate, 4, A14
  * Kuiper (1938) Kuiper, G. P., 1938, ApJ, 88, 472
  * McDonald & Underhill (1952) McDonald, J. K., Underhill, A. B., 1952, ApJ, 115, 577
  * Pecaut & Mamajek (2013) Pecaut, M. J., Mamajek, E. E., 2013, ApJS, 208, article id. 9, 22
  * Popper (1959) Popper, D. M., 1959, ApJ, 129, 647
  * Popper (1972) Popper, K. R., 1972, in: Objective knowledge, Oxford University press, 1972, Oxford
  * Rieke et al. (2008) Rieke, G. H., Blaylock, M., Decin, L. et al., 2008, AJ, 135, 2245
  * Schmidt-Kaler (1982) Schmidt-Kaler, Th., 1982, Landolt-Bornstein: Numerical Data and Functional Relationships in Science and Technology, edited by K. Schaifers and H.H. Voigt (Springer-Verlag, Berlin), VI/2b
  * Sung et al. (2013) Sung, H., Lim, B., Bessell, M. S., Kim, J. S., Hur, H., Chun, M., Park, B., 2013, JKAS, 46, 103
  * Torres (2010) Torres, G., 2010, AJ, 140, 1158
  * Wildey (1963) Wildey, R. L., 1963, Nature, 199, 988

## Data Availability

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