11institutetext: Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004,
18080 Granada, Spain 22institutetext: Dept. Física Teórica y del Cosmos,
Universidad de Granada, Campus de Fuentenueva s/n, 10871, Granada, Spain

# Rotationally and tidally distorted compact stars

A theoretical approach to the gravity-darkening exponents for white dwarfs

A. Claret 11 2 2

(Received ; accepted; )

###### Abstract

Context. To the best of our knowledge, there are no specific calculations of
gravity-darkening exponents for white dwarfs in the literature. On the other
hand, the number of known eclipsing binaries whose components are tidally
and/or rotationally distorted white dwarfs is increasing year on year.

Aims. Our main objective is to present the first theoretical approaches to the
problem of the distribution of temperatures on the surfaces of compact stars
distorted by rotation and/or tides in order to compare with relevant
observational data.

Methods. We used two methods to calculate the gravity-darkening exponents: (a)
a variant of our numerical method based on the triangles strategy and (b) an
analytical approach consisting in a generalisation of the von Zeipel theorem
for hot white dwarfs.

Results.  We find discrepancies between the gravity-darkening exponents
calculated with our methods and the predictions of the von Zeipel theorem,
particularly in the cases of cold white dwarfs; although the discrepancy also
applies to higher effective temperatures under determined physical conditions.
We find physical connections between the gravity-darkening exponents
calculated using our modified method of triangles strategy with the convective
efficiency (defined here as the ratio of the convective to the total flux). A
connection between the entropy and the gravity-darkening coefficients is also
found: variations of the former cause changes in the way the temperature is
distributed on distorted stellar surfaces. On the other hand, we have
generalised the von Zeipel theorem for the case of hot white dwarfs. Such a
generalisation allows us to predict that, under certain circumstances, the
value of the gravity-darkening exponent may be smaller than 1.0, even in the
case of high effective temperatures.

Conclusions.  To constrain the gravity-darkening exponent values
observationally it would be necessary to find and investigate eclipsing
binaries constituted by white dwarfs showing tidal and/or rotational
distortions that were double-lined and that were bright enough to obtain good
radial-velocity semi-amplitudes for both components. It would be very
interesting and useful if observer were to focus their attention on this kind
of system to check our theoretical results.

###### Key Words.:

stars: binaries: close; stars: evolution; stars: rotation; stars: white
dwarfs; stars: atmospheres

††offprints: A. Claret<EMAIL_ADDRESS>
##  1 Introduction

Gravity-darkening is an important piece of the stellar structure and evolution
that has been studied for almost a hundred years (von Zeipel 1924). Gravity-
darkening exponents (GDE) are key tools for analysing light curves of
eclipsing binaries or in isolated rotating stars through long-baseline optical
and infrared interferometry. Considering stars in strict radiative equilibrium
(pseudo-barotrope), in 1924 von Zeipel showed that the variation of brightness
over the surface is proportional to the effective gravity, that is,

${F}=-{4acT^{3}\over{3\kappa\rho}}{dT\over{d\psi}}{g^{\beta_{1}}},$ (1)

or equivalently

${{T_{\rm eff}}^{4}}\propto{g^{\beta_{1}}},$ (2)

where $\psi$ is the potential, $g$ the local gravity, T the local temperature,
$\kappa$ the opacity, $\rho$ the local density, $a$ the radiation pressure
constant, $c$ the velocity of light in vacuum, Teff the effective temperature,
and $\beta_{1}$=1.0 is the GDE, which is a bolometric quantity. Although we
are still far from fully understanding the gravity-darkening phenomenon, in
the last 22 years several theoretical papers related to the GDE have appeared
in the literature that have shed some light on the scenario, such as Claret
(1998, 2000, 2012), Espinosa Lara and Rieutord (2011, 2012), and McGill,
Sigut, and Jones (2013). For a historical summary of the theoretical research
on the GDE see Claret (1998, 2015). From now on we also designate the GDE as
$\beta_{1}$.

Another important and complementary ingredient in the analysis of systems with
non-spherical configurations are the so-called gravity-darkening coefficients
(GDCs). As we can only observe determined band-limited stellar flux (not the
bolometric), the introduction of these coefficients is necessary in order to
model a distorted star. Such a coefficient can be written as (Claret& Bloemen
2011):

$\displaystyle y(\lambda,T_{\rm eff},\log[A/H],\log g,V_{\xi})=$
$\displaystyle\left(\frac{d\ln T_{\rm eff}}{{d\ln
g}}\right)\left(\frac{\partial{\ln I_{o}(\lambda)}}{{\partial{\ln T_{\rm
eff}}}}\right)_{g}+\left(\frac{\partial{\ln I_{o}(\lambda)}}{\partial{\ln
g}}\right)_{T_{\rm eff}},$ (3)

where $\lambda$ is the wavelength, $I_{o}(\lambda)$ the intensity at a given
wavelength at the centre of the stellar disc, and $V_{\xi}$ is the
microturbulent velocity. We note that the expression $\left(\frac{d\ln T_{\rm
eff}}{{d\ln g}}\right)$ can be written as $\beta_{1}/4$. In order to progress
in our investigation of stellar configurations distorted by rotation and/or
tides, we recently studied the GDCs for the case of compact stars (Claret et
al. 2020). In that study we computed GDC for DA, DB, and DBA white dwarf
models, covering the transmission curves of the Sloan, UBVRI, Kepler, TESS,
Gaia, and HiPERCAM photometric systems. These computations are available for
log [H/He] = -10.0 (DB), -2.0 (DBA), and He/H = 0 (DA). The covered range of
log g was 5.0-9.5, while for the effective temperatures the respective range
was 3750 K-100000 K.

From an observational point of view, discoveries of binary systems whose
components are tidally and/or rotationally distorted white dwarfs are ongoing
(see e.g. Burdge et al. (2019a, 2019b, 2020), Kupfer et al. (2020) and
references therein). However, as far as we know, there are no specific GDE
calculations for white dwarf sequences or even for an individual model. In
this short paper we present, for the first time, GDEs for three white dwarf
cooling sequences. We also introduce some improvements to our methods for
calculating the GDE: $\beta_{1}$ is computed as a function of the optical
depth, that is, $\beta_{1}=\beta_{1}(\tau)$. In addition, we introduce an
extra condition in our calculations, namely the relationship between the local
gravities and the corresponding optical depths for a given equipotential
surface. In the following, sections we discuss these points in more detail.

The paper is organised as follows: Sect. 2 is dedicated to describing our
methods and some aspects of the distorted stellar configurations. In Sect. 3
we discuss both observational and theoretical evidence for deviations from von
Zeipel’s theorem (1924) and present our results and conclusions.

##  2  The numerical method

Our numerical method is based on the triangles strategy introduced by
Kippenhahn et al. (1967). A complete description of our method can be found in
Claret (1998), but for the sake of clarity, we summarise it below. To save
computing time, Kippenhahn et al. (1967) introduced an ingenious method: when
an evolutionary sequence is being calculated, if the external boundary
conditions are unchanged at the fitting point (envelope-interior), the outer
layer integrations must be the same as the previous ones. Three envelopes
corresponding to three points in the Hertzsprung-Russell (HR) diagram around
the current values of luminosity and effective temperature are computed. As
the model evolves, its properties are checked to see if they remain within
this triangle. If so, the boundary conditions are unchanged. It is important
to highlight that this strategy is valid only if the triangle is sufficiently
small. This warning is particularly valid for convective envelopes. To
simulate a distorted star, we are interested in several triangles showing
different effective temperature distributions over the surface. Therefore, we
increase the number of triangles in the HR diagram, that is, we compute
several envelopes with different temperature distributions but imposing the
same physical conditions at a given interior point. Figure 2 in Claret (2000)
shows a simplified scheme of our method where only three points are shown in
the HR diagram. Once the triangles have been computed for each point of the
evolutionary track we can derive $\beta_{1}$ by differentiating the
neighbouring envelopes. Such a procedure is performed for the next
evolutionary track point and so on.

Our method has some advantages: (a) it can be applied to convective and/or
radiative envelopes; (b) one can investigate the influence of the optical
depth in the GDE by changing the fitting point to impose the boundary
conditions, without loss of generality; (c) the GDE can be computed as a
function of initial mass, chemical composition, evolutionary stage (in the
present paper up to the white dwarf phase), and other ingredients of the input
physics, and (d) more realistic atmosphere models can easily be incorporated
as external boundary conditions, as done in Claret (2012).

We introduce an extra condition in our procedure: the relationship between
local gravity and optical depth over a given equipotential surface. In
reality, solving the hydrostatic equilibrium equation for two points on an
equipotential characterised by [$g(\mu),\tau(\mu,\psi)$] , we obtain
$g(\mu)\tau(\mu,\psi)=g(\mu_{o})\tau(\mu_{o},\psi)$. In this relationship, $g$
is the local gravity, $\psi$ is the total potential (rotational one included),
$\tau$ is the optical depth, and $\mu$ is given by $\cos(\phi)$ where $\phi$
is the angle between the radiation field and the $z$ axis. The subscript $o$
indicates the reference point, for example the pole. To guarantee that the
triangle technique represents an equipotential, we have taken as reference the
absolute dimensions for each point of the evolutionary models, that is,
[$g(\mu_{o}),\tau(\mu_{o},\psi)$]. The equipotentials are then configurated
introducing additional triangles from this point.

As outlined in Sect. 1, for stars in strict radiative equilibrium, von
Zeipel’s theorem predicts an exponent $\beta$ = 1.0. However, if we inspect a
typical HR diagram in the log g $\times$ Teff plane for stars with different
initial masses, for example, 1.0 M⊙ (convection predominates in the envelope)
and 10.0 M⊙ (envelope predominantly in radiative equilibrium), it can be
verified that both models have different average slopes. If we make a simple
analogy between these two slopes and the GDE given by Eq. 2, the respective
$\beta_{1}$ would be different, with the one corresponding to the star with
1.0 M⊙ being smaller than that of the model with 10.0 M⊙; see Fig. 3 by Claret
(1998) for a graphic example. This figure seems to indicate that stars with
convective envelopes do not strictly obey von Zeipel’s theorem. Additional and
more elaborate evidence has also been found to support these statements, such
as that outlined in Claret (2012, 2015) for example. In these latter studies,
significant deviations were found when the GDEs are computed for the upper
layers of a distorted star.

On the other hand, Kopal (1959) derived the following equation for the stellar
distortions:

$\displaystyle{{g-g_{o}}\over{g_{0}}}={\sum_{j}}\left(1-{5\over{\Delta_{j}}}\right)\left({r\over{a_{1}}}-1\right),$
(4)

where go is the reference local gravity, $\Delta_{j}$ = 1 + 2 kj , and kj is
the apsidal motion constant of order $j$. Therefore, the radius of an
equipotential $r$ (order 2) can be written as

$r=a_{1}\left(1-f_{2}P_{2}(\theta,\phi)\right),$ (5)

with

$\displaystyle f_{2}={5\omega^{2}a_{1}^{3}\over{3GM_{\psi}(2+\eta_{2})}},$ (6)

where $\omega$ is the angular velocity, P${}_{2}(\theta,\phi)$ is the second
surface harmonic, $a_{1}$ is the mean radius of the level surface, $\eta_{2}$
is the logarithmic derivative of the amplitudes of the surface distortions
defined through Radau’s differential equation, and $M_{\psi}$ is the mass
enclosed by an equipotential. Claret (2000) has shown that there is a close
connection between the GDE and the shape of the distorted stellar
configuration, its internal structure, and the details of the rotation law. In
fact, for stellar masses around 1.5 M⊙ there is a change in the predominant
source of thermonuclear energy from the proton–proton chain to the CNO cycle.
This change causes a readjustment of the mass concentration through the
parameter $\eta_{2}$. We reiterate the fact that $\eta_{2}$ is connected to k2
through a simple equation: $k_{2}=(3-\eta_{2}(R))/(2(2+\eta_{2}(R)))$, where
$\eta_{2}(R)$ is evaluated at the stellar surface. This readjustment will
affect how a star reacts under distortions and will consequently effect the
parameter $\beta_{1}$ (see Eq. 3 and also Fig. 1 by Claret (2000) where log k2
and $\beta_{1}$ are shown for ZAMS models with masses varying from 0.075 to
40.0 M⊙). In addition, convection also begins to contribute significantly to
the total flux in this range of effective temperatures. For masses greater
than 1.5 M⊙ the mass concentration decreases almost linearly with the stellar
mass.

##  3 Discussion of the results and final remarks

Some years ago, Claret (2016, see the corresponding Eqs. 8 and 9) adopted an
expression for the convective efficiency (ratio of the convective to the total
flux, denoted by the symbol $f$) to investigate the GDEs. Here we generalise
that result for a range of opacities (through parametrised formulae) as
indicated by Eq. 7:

$\displaystyle{f}\approx A\gamma\left({r\over
R}\right)^{2}\left({3\Gamma_{1}\over{5\mu_{1}\beta}}\right)^{1/2}\left({2c_{P}\mu_{1}\beta\over{5}}\right)T^{1/2}\left[g\over{T_{\rm
eff}^{{(4n+4+\left|n+s\right|)}}}\right].$ (7)

In the above equation, $A$ is a constant, $R$ the star radius, $r$ the radial
coordinate, $\mu_{1}$ the mean molecular weight, $\beta$ the ratio of gas to
total pressure, $c_{P}$ the specific heat at constant pressure, and
$\Gamma_{1}$ is the adiabatic exponent. The parameter $\gamma$ is given by

$\displaystyle{\gamma}={1\over 2}{\overline{v}\over{v_{s}}}{\Delta T\over T},$
(8)

with $\overline{v}$ being the mean convective velocity, $\Delta$T the excess
of temperature of a rising element over the mean temperature of the
surrounding environment, and $v_{s}$ is the velocity of sound. Here, we assume
that for the derivation of Eq. 7 the opacity can be written as
$\kappa\approx\kappa^{\prime}P^{n}T^{-s}$ or
$\kappa\approx\kappa^{\prime\prime}\rho^{n}T^{-n-s}$, where $\kappa^{\prime}$
and $\kappa^{\prime\prime}$ are constants, and $n>$ 0 and $s<$ 0, assuming a
perfect gas equation. Although we will not use this expression directly
because it is only an approximation, it is useful in order to analyse the
correlation between $f$ —through the role of the opacities— and $\beta_{1}$,
as we see in the following paragraphs.

On the other hand, the evolutionary track from the pre-main sequence (PMS) up
to the white dwarf stage was computed using the MESA module (Paxton et al.
2011, 2013 and 2015), version 7385. The basic input physics of the MESA code
is given in the above references. Another set of DA- and DB-type white dwarf
models was computed with LPCODE (Althaus et al. 2001a, 2001b). The models were
followed from the zero age main sequence (ZAMS) up to the white dwarf stage
(Renedo et al. 2010). This code considers modern input physics such as a
detailed network for thermonuclear reactions, OPAL radiative opacities, full-
spectrum turbulence theory of convection, detailed equations of state, and
neutrino emission rates.

We first discuss the cases of ‘pure’ DA- and DB-type white dwarfs, without
considering the previous evolutionary phases. We analyse models with 1.0 M⊙
although the results are similar if we consider models with different initial
masses. Figure 1 illustrates the resulting GDE computed at optical depth
$\tau$= 100.0 (thick continuous line) using the modified triangles strategy
method. The log g during the evolution of this cooling sequence vary
approximately between 8.3 and 8.7. As expected, for high effective
temperatures where radiative transfer predominates and at large optical depth,
the results are compatible with the equation of diffusion, that is, with those
resulting from the von Zeipel (1924) approach. However, as the model evolves,
the influence of convection begins to appear which translates into a drop of
$\beta_{1}$. The drop-off threshold is located at logTeff $\approx$ 4.12 and
we have found large deviations from the von Zeipel theorem for effective
temperatures smaller than 10000K. We note that this transition temperature is
slightly different from that typical of main sequence stars (logTeff $\approx$
3.90). This is due to differences in chemical compositions and degree of
compactness (log g). The dashed (log g = 8.5) and thin continuous (log g=8.0)
lines are useful to establish a more direct connection between the
contribution of the convection to the flux and $\beta_{1}$. Such a
contribution can be set in terms of the ratio F${}_{c}(\tau)$/F${}_{r}(\tau)$
where F${}_{c}(\tau)$ is the convective flux and F${}_{r}(\tau)$ is the
radiative one for a given optical depth (these data were kindly provided by E.
Cukanovaite 2020; see also Cukanovaite et al. 2019). The
F${}_{c}(\tau)$/F${}_{r}(\tau)$ ratio is slightly different from that given by
Eq. 7 but shows the same appearance, taking into account the different scales:
the F${}_{c}(\tau)$/F${}_{T}(\tau)$ ratio varies from 0 to 1.0,
F${}_{T}(\tau)$ being the total flux.

We can use Eq. 7 to help us visualise the behaviour of $\beta_{1}$ and its
relation to opacity. The connection between $\beta_{1}$ and the convective
contribution to the flux is evident; it is especially clear near the maxima
and minima. The behaviour of the GDE shown in Fig. 1 (also that of the
following figures) is the result of the combination of the different sources
of opacity but we can gain some insight into its influence on the GDE by
adopting some opacity laws in their parameterised form. Firstly, we analyse
the effects of opacity in the colder models. For this range of effective
temperatures one of the predominant sources of opacity is the negative
hydrogen ion whose dependence on $\rho$ and T is given by
$\kappa\approx\kappa_{1}\rho^{1/2}T^{7.7}$, where $\kappa_{1}$ is a constant.
We note the high dependence of negative hydrogen ion opacity with temperature.
The expression $g\over{T_{\rm eff}^{{(4n+4+\left|n+s\right|)}}}$ in Eq. 7
drives the behaviour of the gravity-darkening exponent, and, introducing the
relevant values of $n$ and $s$ for such models at a given convective
efficiency, we get $\beta_{1}\approx$ 0.30. In addition, the opacities $ff$,
$bf,$ and $bb,$ because of electronic transitions, are given by Kramers law:
$\kappa\approx\kappa_{2}\rho T^{-7/2}$, where $\kappa_{2}$ is a constant. The
approximate corresponding value of $\beta_{1}$ in this case is $\approx$ 0.35.
Considering that Eq. 7 is just a rough approximation, these results are
surprising because they predict deviations from the von Zeipel’s theorem and,
in addition, they are in reasonable agreement with the values of $\beta_{1}$
found in the literature for late-type stars (semi-empirical and theoretical
ones; see below). On the other hand, one of the main sources of continuum
opacity in hot star atmospheres is the so-called Thomson scattering. As known,
this opacity is ‘grey’ because there is no dependence on temperature and
density and it can be written as follows for the case of complete ionisation:
$\kappa\approx 0.2(1.0+X)$, where $X$ is the hydrogen content. Using the same
treatment as for the negative hydrogen ion case and Kramers law, we obtain
$\beta_{1}\approx 1.00$, which is in good agreement with the predictions of
the von Zeipel’s theorem for hot stars. Again it is gratifying that Eq. 7 is
capable of predicting the typical values of $\beta_{1}$, even considering its
limitations.

In addition to the analysis of the influence of convective flux on
$\beta_{1}$, there are other ways to correlate $\beta_{1}$ with some physical
magnitudes. Considering the additive properties of the specific entropy in the
nonrelativistic case we have

$\displaystyle{s}=B+{N_{o}k\over{\mu_{i}}}ln{T^{3/2}\over{\rho}}+{{N_{o}k\over{\mu_{e}}}}\left[{5\over{3}}{F_{3/2}(\alpha)\over{F_{1/2}(\alpha)}}+\alpha\right]+{4a\over{3}}{T^{3}\over{\rho}},$
(9)

where the symbols have the following meaning: B is a constant, $\alpha$ the
degeneracy parameter, No is Avogadro’s number, $k$ the Boltzmann constant,
$\mu_{i}$ the mean molecular weight per ion, and $\mu_{e}$ is the mean
molecular weight per electron. The functions $F_{1/2}(\alpha)$ and
$F_{3/2}(\alpha)$ are auxiliary functions that can be written as

$\displaystyle
F_{1/2}(\alpha)=\int_{0}^{\infty}{u^{1/2}du\over{e^{\alpha+u}+1}},$ (10)

and

$\displaystyle
F_{3/2}(\alpha)=\int_{0}^{\infty}{u^{3/2}du\over{e^{\alpha+u}+1}},$ (11)

where $u=p^{2}/(2mkT)$, with $p$ being the particle momentum and $m$ its mass.
The three components in Eq. 9 above can be easily identified: the second term
is the entropy due to ions, the third is connected to electrons, and the
fourth to radiation. We can see in Fig. 2 how the GDE is related to the
entropy for the same models and conditions shown in Fig. 1. As mentioned, the
convection onset is in log Teff $\approx$ 4.12 (log T${}_{\rm
eff\leavevmode\nobreak\ onset}$) where a sudden variation of the entropy is
indicated by a vertical arrow. This effective temperature is in agreement with
those by Tremblay (2020). There are three other inflexion points (also marked
with vertical arrows). These four points shown in Fig. 2 indicate that the
entropy varies with the effective temperature (also with log g), which in turn
drives the behaviour of the GDE. The changes in $\beta_{1}$ with entropy are
not surprising. Indeed, the differential of the entropy is given by
$dS=c_{p}(\nabla-\nabla_{adia})$, where $\nabla=dlnT/dlnP$ and
$\nabla_{adia}=(dlnT/dnlP)_{adia}$. On the other hand, it can be shown that,
alternatively, the convective flux
F${}_{c}(\tau)\propto(\nabla-\nabla_{adia})^{x}$, where x = 3 (small
convective efficiency) or x = 3/2 for large convective efficiency. Thus, under
determined physical conditions, the entropy can also be used as a convective
stability criterion.

Another point to note is that the entropy does not significantly depend on the
optical depth for effective temperatures $\leq$ T${}_{\rm
eff\leavevmode\nobreak\ onset}$. Indeed, the curves for $\tau$ = 100.0 and
500.0 coincide for Teff $\leq$ T${}_{\rm eff\leavevmode\nobreak\ onset}$. This
implies that the behaviour of the resulting GDEs should not vary
significantly, at least within the range of optical depths, log g, and
effective temperatures explored here. We note that, for hot models, because
the entropy hardly varies with Teff, the corresponding GDE values are almost
constant and equal to 1.0, which restores von Zeipel’s theorem. These results
are more general than those obtained decades ago, whose GDE value was constant
and approximately equal to 0.32 for envelopes in convective equilibrium.

Figure 3 shows a comparison between the GDE for DB and DA models at $\tau$ =
100.0. The profiles of $\beta_{1}$ are very similar, with the exception of the
transition zone where there is a shift due to the difference in the chemical
composition of the models and its influence on pressure and temperature.

An interesting comparison that can be made is related to the masses of white
dwarfs. In Fig. 4 we show the evolution of the GDE for DA models with masses 1
M⊙ (continuous line) and 0.52 M⊙ (dashed line). The calculations were also
performed for $\tau$ = 100.0. Because of the dependence of the onset of
convection on the local gravity, the GDEs for the model with 0.52 M⊙ are
shifted towards lower temperatures, in reasonable agreement with other studies
(Cunningham et al. 2019, Tremblay (2020)).

As indicated, another set of evolutionary models was generated with the MESA
module with an initial mass of 2.0 M⊙, X = 0.703, and Y = 0.277. For stars
with convective envelopes, we employed the standard mixing-length formalism
(Böhm-Vitense 1958) with $\alpha_{MLT}$ = 1.80. The opacity tables adopted are
those given by Grevesse and Sauval (1998). The models were followed from the
PMS up to the cooling white dwarf stage. Figure 5 shows the complete
evolutionary track in the HR diagram. The final mass in the cooling stage is
0.56 M⊙. The results concerning gravity-darkening are shown in Fig. 6 where we
add the profile of $\beta_{1}$ shown in Fig. 1 for comparison. Within our
present level of approximation, the differences are small, being more
appreciable only in the interval around logTeff $\approx$ 3.90.

As we can see in Figures 1-4 and 6, there are clear indications of deviations
from the approach by von Zeipel. Deviations from von Zeipel’s theorem were
previously found using suitable evolutionary models in stars evolving towards
the branch of red giants and/or in low-mass main sequence stars (low effective
temperatures) where the flux is predominantly convective in their envelopes
(Claret 1998, 2000). Additional evidence for deviations from von Zeipel’s
theorem comes from 3D simulations of cold stars (Ludwig, Freytag&Steffen
1999). A more complete historical review about this subject can be found in
above references.

On the other hand, there is an analytical approach by which we can explore the
behaviour of GDEs in compact stars whose outermost layers are in radiative
equilibrium. Such an analysis was carried out in Claret (2012) for main
sequence stars. Here we adapt the physical conditions for the case of white
dwarfs. Such an equation can be written as (see Appendix A)

$\displaystyle\beta_{1}\approx
1+\left[{{2+N}-\chi(4+N)\over{2+N}}-{{2+N+\chi(-4+8\alpha_{1}-5N)}\over{2+N+\chi(\kappa_{\rho}-\kappa_{T})(N-2\alpha_{1})}}\right]$
$\displaystyle{\tau_{p}\over{\tau_{e}}}.$ (12)

Equation 12 opens up some possibilities on the investigation of the
distribution of rotational velocities through the parameter $\alpha_{1}$ and
on geometry through $N$. We note that this equation is valid only for hot
stars. We note that the ratio $\tau_{p}/\tau_{e}\leq 1.0$. For example, we
would restore the predictions of von Zeipel’s theorem for the cases
$\alpha_{1}=N/2$ or for uniform rotation and no $\theta$ dependence. An
interesting feature of Eq. 12 can be explored and is related to
$\kappa_{\rho}$ and $\kappa_{T}$. Depending on the behaviour of these two
variables, we could have values of $\beta_{1}$ significantly smaller than
1.0.111We have also found significant deviations from von Zeipel’s theorem
using our modified numerical method, at the upper layers of hot white dwarfs.
For example, for $\alpha$=-1.0, N = 1, $\chi$ = 0.2, $\tau_{p}$/$\tau_{e}$=
0.71, and $\beta_{1}$ = 0.80, and the resulting condition would be

$\displaystyle-6.0\lessapprox\left[\left({{\partial ln\kappa}\over{\partial
ln\rho}}\right)_{T}-\left({{\partial ln\kappa}\over{\partial
lnT}}\right)_{\rho}\right]\lessapprox-5.0.$ (13)

Such a condition is approximately fulfilled for envelopes with effective
temperatures in the interval 6000 K $\lessapprox$ Teff $\lessapprox$ 13000 K.
There are no semi-empirical data yet for white dwarfs in this effective
temperature range for comparison, but it is interesting to note that values of
$\beta_{1}$ smaller than 1.0 were detected using long-baseline
optical/infrared interferometry in isolated fast rotators. Some of these
systems show effective temperatures within the mentioned range. A summary of
their properties can be found in Table 1 of Claret (2016).

Values of the GDE smaller than 1.0 were also observationally detected in main
sequence and/or subgiants stars in binary systems. However, a direct
connection between the results from long-baseline interferometry and from
eclipsing binaries (mostly main sequence stars) and those provided by Eq. 12
(compact ones) is not straightforward. However, it can give us some clue for
future research. A turning point concerning semi-empirical $\beta_{1}$ was the
pioneering paper by Rafert&Twigg (1980) using eclipsing binaries. Such
research was followed by others, such as Pantazis and Niarchos (1998),
Niarchos (2000), and Djurasevic et al. (2003, 2006), for example, who explored
a wide range of effective temperatures. The semi-empirical GDEs derived by
these latter authors for hot stars are scattered around the classical von
Zeipel value. Some of these systems show $\beta_{1}$ as low as 0.60. Another
important result of these investigations is that the derived values for
systems with cooler components also contradict the predictions of von Zeipel’s
theorem. These semi-empirical GDE values are quite significant but do not yet
constitute a critical test of the theory of temperature distribution on a
distorted stellar surface. However, despite the fact that these observations
are not conclusive, they seem to indicate a transition zone for the GDE. That
zone approximately coincides with the change in the prevalence of the process
of energy transport in the envelopes, that is, radiative to convective, as
indicated by the Fig. 3 in Claret (2003). Such a transition zone is also
predicted in the present paper for compact stars.

Finally, it would be very interesting and useful if observers were to focus
their attention on close binary systems constituted by white dwarfs distorted
by rotation and tides, so that the validity of the preliminary calculations
presented here can be verified. To constrain the GDE values observationally,
it would be necessary to investigate eclipsing binary white dwarfs that are
double-lined and bright enough to obtain good radial-velocity semi-amplitudes
for both components. We hope that such systems will be found in the not too
distant future.

Figure 1: DA-type white dwarf models (1.0 M⊙). The thick solid line represents
the GDE as a function of effective temperature. The continuous thin line
indicates the ratio F${}_{c}(\tau)$/F${}_{r}(\tau)$ for log g = 8.5 and the
dashed one denotes the same but for log g = 8.0. All calculations were
performed at $\tau$ = 100.0. We note that the Teff range of the atmosphere
models by Cukanovaite (2020) is limited to Teff $\leq$ 60000 K. The arrow
indicates the direction of time evolution. Figure 2: Entropy for the same
models and conditions shown in Fig. 1 (red lines). The vertical arrows
indicate the points where there is a marked variation of the entropy with
effective temperature. The solid line represents the entropy for $\tau$ = 100
while the dashed-dotted line denotes $\tau$ = 500; both for log g = 8.5. The
horizontal arrow indicates the direction of time evolution. Figure 3:
Comparison between the GDE for a DB model (solid line, 1.0 M⊙) and DA (dashed
line, 0.5 M⊙). All calculations were performed at $\tau$ = 100.0. The
horizontal arrow indicates the direction of time evolution. Figure 4: Effect
of log g on the onset of convection for DA models. The continuous line
represents a sequence of DA model (1.0 M⊙) while the dashed one also denotes
DA models but with 0.52 M⊙. The horizontal arrow indicates the direction of
time evolution. Figure 5: HR diagram for an initial mass of 2.0 M⊙ from the
PMS to cooling white dwarf stage. Initial chemical composition X = 0.703, Y =
0.277, $\alpha_{MLT}$ = 1.80. Figure 6: GDE for the white dwarf cooling
sequence for the models shown Fig. 5 (continuous line). The DA model shown in
Fig. 1 is represented by a dashed line. The horizontal arrow indicates the
direction of time evolution.

###### Acknowledgements.

I would like to thank E. Cukanovaite for providing models of the structure of
white dwarfs atmospheres and an anonymous referee for his/her helpful
suggestions. The Spanish MEC (ESP2017-87676-C5-2-R, PID2019-107061GB-C64, and
PID2019-109522GB-C52) is gratefully acknowledged for its support during the
development of this work. A.C. also acknowledges financial support from the
State Agency for Research of the Spanish MCIU through the “Center of
Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía
(SEV-2017-0709). This research has made use of the SIMBAD database, operated
at the CDS, Strasbourg, France, and of NASA’s Astrophysics Data System
Abstract Service.

## References

  * (1) Althaus, L. G., & Serenelli, A. M., & Benvenuto, O. G. 2001a, MNRAS, 323, 471
  * (2) Althaus, L. G., & Serenelli, A. M., & Benvenuto, O. G. 2001b, MNRAS, 324, 617
  * (3) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * (4) Burdge, K. B., Coughlin, M. W., Fuller, J., et al. 2019a, Nat, 571, 528
  * (5) Burdge, K., Fuller, J., Phinney, E. S. et al. 2019b, ApJL, 886, L12
  * (6) Burdge, K. B., Prince, T. A., Fuller, J. et al., 2020, 2020arXiv200902567B
  * (7) Claret, A. 1998, A&AS, 131, 395
  * (8) Claret, A. 2000, A&A, 359, 289
  * (9) Claret, A. 2003, A&A, 406, 623
  * (10) Claret, A. 2012, A&A, 538, A3
  * (11) Claret, A. 2016, A&A, 588, A15
  * (12) Claret, A., Bloemen, S. 2011, A&A, 529A, 75C
  * (13) Claret, A., Cukanovaite, E., Burdge, K. et al. 2020, A&A, 634A, 93C
  * (14) Cukanovaite, E. 2020 (private communication)
  * (15) Cukanovaite, E., Tremblay, P.-E., Freytag, B., et al. 2019, MNRAS, 490, 1010
  * (16) Cunningham, T., Tremblay, P-E., Freytag, B. et al. 2019, MNRAS, 488, 2503C
  * (17) Djurasevic, G., Rovithis-Livaniou, H., Rovithis, P., Georgiades, N., Erkapic, S., Pavlovic, R. 2003, A&A, 402, 667D
  * (18) Djurasevic, G., Rovithis-Livaniou, H., Rovithis, P., Georgiades, N., Erkapic, S., Pavlovic, R. 2006, A&A, 445, 291D
  * (19) Espinosa Lara, F., Rieutord, M. 2011, A&A, 545, 130D
  * (20) Espinosa Lara, F., Rieutord, M. 2012, A&A, 547, 32E
  * (21) Grevesse, N., Sauval, A. J. 1998, Space Sci. Rev., 85, 161
  * (22) Kippenhahn, R. 1977, A&A, 58, 267
  * (23) Kippenhahn, R., Weigert, A., Hofmeister, E., 1967 in Computational Physics, Academic Press, New York, Vol. 7, p. 129
  * (24) Kopal, Z. 1959, Close Binary Systems, Chapman & Hall
  * (25) Kupfer, T., Bauer, E. B., Burdge, K. B. et al., 2020, ApJL, 898, L25
  * (26) Ludwig, H-G., Freytag, B., Steffen, M. 1999, A&A, 346, 111
  * (27) McGill, M. A., Sigut, T. A. A., Jones, C. E. 2013, ApJS, 204, 2
  * (28) Pantazis, G., Niarchos, P. G. 1998, A&A, 335, 199
  * (29) Paxton, B., Bildsten, L., Dotter, A. et al. 2011, ApJS, 192, 3
  * (30) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * (31) Paxton, B., Marchant, P.,Schwab, J., et al. 2015, ApJS, 220, 15
  * (32) Rafert, J. B., Twigg, L. W. 1980, MNRAS, 193, 79
  * (33) Renedo, I., & Althaus, L. G., & Miller Bertolami, M. M. et al. 2010, ApJ, 717, 183
  * (34) Tremblay, P-E. 2020 (private comunication)
  * (35) von Zeipel, H. 1924, MNRAS, 84, 665

##  Appendix A Brief description of the derivation of Equation 12

For completeness, we give here a brief summary of the derivation of Eq. 12
which is a generalisation of von Zeipel’s theorem. For more details, see
Kippenhahn (1977) and Claret (2012).

Let us assume that the angular velocity has the following general form

${\omega^{2}}={h(r)\sin^{N}\theta},$ (14)

where $h(r)$ is a generic function and $\theta$ is the colatitude angle. In
our approach we assume that the effect of the centrifugal force on the stellar
structure is small. The total potential $\psi$ can be written as

$\psi={-GM\over{r}}-{r^{2}\omega^{2}\sin^{2}\theta\over{N+2}}.$ (15)

The integrating factor was chosen in such a way that $\Lambda\@vec{g}$ can be
derived from a potential. As a consequence, it can be shown that the pressure
is constant on equipotential surfaces. If the effects of rotation are small,
we obtain for the integrating factor:

${\Lambda}=1+{1\over{N+2}}\chi(N-2\alpha_{1})\sin^{2}\theta.$ (16)

In the above equation we define $\alpha_{1}={\partial ln\omega\over{\partial
lnr}}$ and $\chi={r^{3}\omega^{2}}/(GM_{\psi}$).

On the other hand, the equation of the radiative transport can be written as

${\<EMAIL_ADDRESS>(17)

Assuming that the density and temperature are functions of the structural form
of the potential we have ${\rho}=d(\Psi)\Lambda$ and $T={t(\Psi)/{\Lambda}}$
and

${\@vec{F}}={4act^{3}\over{3\kappa
d}}\left({dt\over{d\Psi}}{\@vec{g}\over{\Lambda^{4}}}+{t\over{\Lambda^{6}}}{\@vec{\nabla}}\Lambda\right).$
(18)

As an extra condition we adopt the relationship
$g(\mu_{e})\tau(\mu_{e},\psi)=g(\mu_{p})\tau(\mu_{p},\psi)$, where the
subscripts $e$ and $p$ denote values at the equator and at the poles.
Calculating the values of local gravity, of $\Lambda$ and of Eq. A.5 at the
pole and at the equator we finally obtain the GDE through the logarithmic
derivatives of the flux and effective gravity

$\beta_{1}\approx{{\left({F_{e}-F_{p}}\over{F_{p}}\right)}\over\left({{g_{e}-g_{p}}\over{g_{p}}}\right)}.$
(19)

After some algebra and incorporating the effects of variable opacity we
finally get

$\displaystyle\beta_{1}\approx
1+\left[{{2+N}-\chi(4+N)\over{2+N}}-{{2+N+\chi(-4+8\alpha_{1}-5N)}\over{2+N+\chi(\kappa_{\rho}-\kappa_{T})(N-2\alpha_{1})}}\right]$
$\displaystyle{\tau_{p}\over{\tau_{e}}},$ (20)

where ${\kappa_{\rho}}\equiv\left({{\partial ln\kappa}\over{\partial
ln\rho}}\right)_{T}$ and ${\kappa_{T}}\equiv\left({{\partial
ln\kappa}\over{\partial lnT}}\right)_{\rho}$.