11institutetext: NAOC / Yunnan Observatory, Chinese Academy of Sciences,
Kunming 650011, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: Graduate School of the Chinese Academy of Sciences, Beijing
100049, China.
Received 2009 Sep. 18; accepted 2009 Nov. 12

# On the DB gap of white dwarf evolution: effects of hydrogen mass fraction
and convective overshooting

Jie Su 11 2 2 Yan Li 11

###### Abstract

We investigate the spectral evolution of white dwarfs by considering the
effects of hydrogen mass in the atmosphere and convective overshooting above
the convection zone. Our numerical results show that white dwarfs with $M_{\rm
H}\sim 10^{-16}~{}M_{\odot}$ show DA spectral type between $46,000\lesssim
T_{\rm eff}\lesssim 26,000~{}{\rm K}$ and DO or DB spectral type may appears
on either side of this temperature range. White dwarfs with $M_{\rm H}\sim
10^{-15}~{}M_{\odot}$ appear as DA stars until they cool to $T_{\rm eff}\sim
31,000~{}{\rm K}$, from then on they will evolve into DB white dwarfs as a
result of convective mixing. If $M_{\rm H}$ in the white dwarfs more than
$10^{-14}~{}M_{\odot}$, the convective mixing will not occur when $T_{\rm
eff}>20,000~{}{\rm K}$, thus these white dwarfs always appear as DA stars.
White dwarfs within the temperature range $46,000\lesssim T_{\rm eff}\lesssim
31,000~{}{\rm K}$ always show DA spectral type, which coincides with the DB
gap. We notice the importance of the convective overshooting and suggest that
the overshooting length should be proportional to the thickness of the
convection zone to better fit the observations.

###### keywords:

convection — stars: evolution — stars: white dwarfs

##  1 Introduction

White dwarfs can be classified into several spectral types in terms of the
spectral characteristics. The current classification system was introduced by
Sion et al. ([1983]) and has been modified several times. In this system, the
spectral type of a white dwarf is denoted by a letter D plus another letter
indicating its spectral characteristics. Sometimes, a suffix is added to
indicate some other features (polarization, magnetic field, pulse, etc.).
Table 1 lists a spectral classification scheme from McCook & Sion ([1999]).

Table 1: White dwarf spectral types Spectral Type | Characteristics
---|---
DA |  | Only Balmer lines; no He I or metals present
DB |  | He I lines; no H or metals present
DC |  | Continuous spectrum, no lines deeper than $5\%$ in any part of the electromagnetic spectrum
DO |  | He II strong; He I or H present
DZ |  | Metal lines only; no H or He lines
DQ |  | Carbon features, either atomic or molecular in any part of the electromagnetic spectrum
P | (suffix) | Magnetic white dwarfs with detectable polarization
H | (suffix) | Magnetic white dwarfs without detectable polarization
X | (suffix) | Peculiar or unclassifiable spectrum
E | (suffix) | Emission lines are present
? | (suffix) | Uncertain assigned classification; a colon (:) may also be used
V | (suffix) | Optional symbol to denote variability

Most of white dwarfs are of DA type which have hydrogen-dominated atmospheres.
They are found at all effective temperatures from $170,000~{}{\rm K}$ down to
about $4,500~{}{\rm K}$ (Kurtz et al. [2008]). DA white dwarfs occupy the vast
majority (about $75\%$) of all known white dwarfs.

About $25\%$ of the observed white dwarfs have helium-dominated atmospheres
which can be divided further into two spectral types. The DO white dwarfs are
found between approximately $100,000$ to $45,000~{}{\rm K}$, their outer
atmospheres being dominated by singly ionized helium (He II). The DB white
dwarfs are found between approximately $30,000$ to $12,000~{}{\rm K}$, their
outer atmospheres being dominated by neutral helium (He I).

There is an interesting fact that few white dwarfs with helium-dominated
atmospheres (DO or DB type) are found in the effective temperature range of
$45,000\lesssim T_{\rm eff}\lesssim 30,000~{}{\rm K}$. This is the so-called
DB gap. The reason for this phenomenon is not clear. On one side, there is
strong gravitational field in the white dwarf, which may cause a stratified
atmosphere. The so-called gravitational settling effect lets the light element
(hydrogen) floating to the stellar surface and the heavy element (helium)
sinking to the bottom of the stellar envelope. The typical mass fractions of
hydrogen and helium in the white dwarfs are $M_{\rm H}/M_{\rm tot}\lesssim
10^{-4}$ and $M_{\rm He}/M_{\rm tot}\lesssim 10^{-2}$, which are the mass
threshold for residual nuclear burning (Tremblay & Bergeron [2008]). If there
is no mixing between hydrogen and helium, we will expect all white dwarfs to
be DA stars. On the other side, however, many observational data show that
$M_{\rm H}$ may be significantly lower than the typical value. The existence
of a large number of non-DA white dwarfs indicates that some physical
mechanisms are competing with the gravitational settling and make the spectral
type of some DA white dwarfs changed. The DB gap is suspected to be due to the
competition between the gravitational settling and convective mixing.

Fontaine & Wesemael ([1987]) proposed that when a white dwarf starts cooling
from the hot PG $1159$ type star, hydrogen is mixed within the outer helium
envelope and the white dwarf shows the DO spectral type. After that, hydrogen
gradually floats to the stellar surface in the strong gravitational field.
When the DO star cools to $T_{\rm eff}\sim 45,000~{}{\rm K}$, hydrogen has
accumulated enough at the surface, and the white dwarf is turned into a DA
star. They supposed further that as soon as the DA star cools down to $T_{\rm
eff}\sim 30,000~{}{\rm K}$, the He I/II ionization zone in the stellar
envelope becomes convective. The convective motion may penetrate into the top
hydrogen layer and mix the hydrogen atmosphere into the helium envelope, the
less abundant hydrogen being overwhelmed by the more abundant helium and
becoming undetectable. So the DA star then appears as a DB white dwarf.

Shibahashi ([2005], [2007]) proposed a different scenario for the DB gap.
During the early stage of a white dwarf’s evolution, the convection in the He
II/III ionization zone mixes the hydrogen layer and results in a helium-
dominated atmosphere, and the white dwarf appears as a DO star. When the star
cools down to around $T_{\rm eff}\sim 45,000~{}{\rm K}$, the He II/III
ionization zone becomes deep enough that convection disappears in the DO
star’s atmosphere, so hydrogen floating to the surface and then the white
dwarf being transformed into a DA star. When the white dwarf cools further
down to $T_{\rm eff}\sim 30,000~{}{\rm K}$, the He I/II ionization zone
generates a convection zone again, as Fontaine & Wesemael’s proposal, a
similar mixing between H and He occurs and the white dwarf appears as a DB
star.

The difference between the two scenarios of the DB gap is the time scale of
the gravitational settling. In Fontaine & Wesemael’s assumption, the settling
process happens slowly as the cooling of the white dwarf. But in Shibahashi’s
assumption, it happens quickly as soon as the convection is turned off.
However, the recent data of the Sloan Digital Sky Survey (SDSS) suggest that
several DB white dwarfs do appear in the DB gap (Eisenstein et al. [2006]).
These facts imply that the formation mechanism of the DB gap are not clear and
more works should be done.

In the present paper, we calculate a series of white dwarf evolutionary models
to investigate the spectral evolution of white dwarfs caused by the convective
mixing. The details of model calculations and input physics are presented in
Section 2. In Section 3 we discuss the results of our numerical models.
Conclusions are summarized in Section 4.

##  2 model details and input physics

We have used a modified version of the White Dwarf Evolution Code (WDEC),
which was originally described by Martin Schwarzschild to simulate the
evolution of the white dwarf. Some details of the WDEC has been described in
Lamb & Van Horn ([1975]) and Wood ([1990]). Here, we only present some
summaries of the input physics in our models.

The equation of state (EOS) used in the present calculations is composed of
two parts which apply to different regions. The first part of the EOS is used
for the degenerate, completely ionized interior of the white dwarf. In this
region, we use the EOS tables provided by Lamb ([1974]). For a given chemical
composition the needed values are obtained by two-dimension, four-point
Aitken-Lagrange interpolation in terms of variables $\lg P$ and $\lg T$. For a
specific C/O mixture, they are obtained by interpolation between the carbon
and oxygen tables using the additive volume technique of Fontaine et al.
([1977]). The second part of the EOS is used for the partial ionized envelope
where non-ideal effect is important. We use the Saumon et al. ([1995]) EOS for
hydrogen and helium mixtures. The new EOS include some new physical treatments
of partial ionization caused by pressure and temperature. Mixtures of hydrogen
and helium are also obtained by the additive volume technique.

The total opacity ($\kappa$) is given by

$\dfrac{1}{\kappa}=\dfrac{1}{\kappa_{\rm r}}+\dfrac{1}{\kappa_{\rm c}}~{},$
(1)

where $\kappa_{\rm r}$ is the radiative opacity and $\kappa_{\rm c}$ is the
conductive opacity. We use the OPAL radiative opacities in our calculations.
The new tables include some new physical factors, e.g. the L-S coupling effect
of iron atoms. The conductive opacities consist of two parts which are from
Itoh et al. ([1983], [1984]) and Hubbard & Lampe ([1969]). In the actual
calculations, we use Itoh et al. opacities only in $\lg\rho\geq 1.8$ region
and Hubbard & Lampe opacities in $\lg\rho\leq 1.5$ region, in the range of
$1.5<\lg\rho<1.8$ linear interpolation being performed.

The rates of neutrino energy loss used in our calculations are provided by
Itoh and his collaborators. The rates of neutrino energy loss due to pair,
photo, plasma and bremsstrahlung processes are from Itoh et al. ([1989]), and
the rate of recombination neutrino is from Kohyama et al. ([1993]).

The high surface gravity of the white dwarf leads to gravitational segregation
of the elements in the stellar envelope, and thus models of white dwarf must
include compositionally stratified envelopes. In our calculations, we adopt
approximations of the equilibrium diffusion profiles introduced by Wood
([1990]) (see Figure 1). Our calculations do not include the impact of
convective mixing on the H/He profile. Although this may not be correct in
details, we still expect it to be a reasonable approximation, because the
abundance of hydrogen in the mixing region is several orders of magnitude less
than the helium abundance.

Figure 1: Approximations to the diffusive equilibrium profiles.

We use the standard mixing-length theory of Böhm-Vitense ([1958]) to deal with
the convection. We set the mixing-length $l$ to be equal to one local pressure
scale height, i.e.,

$l=H_{\rm P}=-\cfrac{dr}{d\ln P}=\cfrac{P}{\rho g}~{}.$ (2)

Boundaries of the convection zones are determined by the Schwarzschild
criterion. And we set the integration step to be equal to $H_{\rm P}/8$ in our
calculations.

##  3 evolutionary results

We have computed a series of white dwarf evolutionary models with mass
$M=0.6~{}M_{\odot}$, which is the typical mass of white dwarf. In order for a
DA white dwarf to change its surface chemical composition as a result of the
convective mixing, it ought to have a very thin hydrogen layer, so the
hydrogen mass of the model is supposed to vary between $10^{-16}$ to
$10^{-14}~{}M_{\odot}$. The helium mass in the computed model envelope is
fixed to be $5.0\times 10^{-5}~{}M_{\odot}$. We assume that the heavy elements
have sunk during the early phase of the cooling process to the white dwarf’s
interior due to the so-called gravitational settling effect. As a result the
envelope of the white dwarf is only composed of hydrogen and helium, and the
metallicity in the envelope is assumed to be $Z=0$. All models are evolved
from $T_{\rm eff}\sim 90,000~{}{\rm K}$ down to $T_{\rm eff}=10,000~{}{\rm
K}$. In this section we present the results of our calculations.

###  3.1 Importance of the convective overshooting

We have examined the development of convection zone in our white dwarf models.
Figure 2 shows the convection zone of a DA model with $M_{\rm
H}=10^{-15}~{}M_{\odot}$ evolving with the decreasing effective temperature.
The horizontal axis is the effective temperature which denotes the
evolutionary sequence, and the vertical axis denotes the location within the
envelope of the model. We use $1-r/R$ as the vertical axis scale in Figure 2
(a) and use $\lg(1-M_{r}/M)$ in Figure 2 (b). The solid line corresponds to
the location of the model’s photosphere ($\tau=1$) and the dashed lines
correspond to the Schwarzschild boundaries of the convection zone.

(a)  (b)

Figure 2: Location of the convection zone.

It can be found that convection occurs completely within the helium layer
which is located under the photosphere which can be regarded as the innermost
point visible to us in the white dwarf. Just above the He convection zone
there is a thin layer with a mean molecular weight gradient (the so-called
$\mu$-barrier). In the case of very thin hydrogen envelope ($M_{\rm
H}<10^{-15}~{}M_{\odot}$), Vauclair & Reisse ([1977]) have argued that the
$\mu$-barrier provides extra buoyancy to restrict the upper boundary of the He
convection zone just below it. However, at the upper boundary which is
determined by the convective stability criterion (at the $\mu$-barrier, the
Ledoux criterion $\nabla_{\rm rad}\geq\nabla_{\rm ad}+d\ln\mu/d\ln P$ is
adopted), the convective motion does not stop but moves upward further due to
inertia. In this sense, the fluid parcels with kinetic energy may penetrate
into the $\mu$-barrier until their velocity drop to zero. We perform a rough
calculation, supposing that the velocities of the fluid parcels is equal to
the mean flow velocity ($\bar{v}$) in the convection zone. When the fluid
parcels enter the $\mu$-barrier, a force $F$ (the resultant force of gravity
and buoyancy, neglecting the viscous force) acts on them and reduces the
velocity. The kinetic energy of the fluid parcels $E_{\rm
K}\sim\bar{v}^{2}/2$. When the fluid parcels deplete all the kinetic energy,
they can move beyond the convective boundary a length $L\sim E_{\rm
K}/F\sim\bar{v}^{2}/2F$. As shown in Fig. 3, the thickness of the
$\mu$-barrier $d_{\rm\mu}$ is around $3\times 10^{3}~{}{\rm cm}$ and
$L>d_{\rm\mu}$ when $T_{\rm eff}<40,000~{}{\rm K}$, and the maximum value of
$L$ is about $3\times 10^{4}~{}{\rm cm}$ which is one order of magnitude
greater than $d_{\rm\mu}$. Therefore, it seems reasonable to believe that the
convective overshooting can go cross the $\mu$-barrier. Once He penetrates
into the $\mu$-barrier, an effective mixing will flatten the composition
gradient and weaken the $\mu$-barrier. As a result, the upper boundary of the
He convection zone will thus be able to extend outward. We expect that the
convective motion can extend upward to the stellar surface, or at least, to
the photosphere in order to mix the upper hydrogen layer and to ensure that
helium can be observed. Thus, the role of convective overshooting appears to
be decisive. We suppose that the convective overshooting is able to reach the
photosphere. This suggestion let us set the distance from the photosphere down
to the top of the Schwarzschild boundary as the minimum length of the
overshooting region ($l_{\rm ovs}$). The geometrical length between the two
Schwarzschild boundaries is regarded as the length of the convection zone
($l_{\rm con}$). These two lengths vary with the evolution of the effective
temperature as shown in Figure 4. We also compare $l_{\rm ovs}$ with $l_{\rm
con}$, the ratio given in Figure 5.

Figure 3: $L$ and $d_{\rm\mu}$ versus $T_{\rm eff}$. Figure 4: The evolution
of the length of overshooting region and that of the convection zone. $l_{\rm
ovs}$ is the length of the overshooting region and $l_{\rm con}$ the length of
the convection zone. Figure 5: The ratio of $l_{\rm ovs}$ to $l_{\rm con}$
varies with $T_{\rm eff}$.

As shown in the Figure 4, $l_{\rm ovs}$ keeps almost a constant (about
$2~{}{\rm km}$), while $l_{\rm con}$ increases slowly to a few hundred meters
during a long evolutionary time scale. That is to say, the convective motion
must overshoot to a distance which is several times thicker than the
convection zone itself. Figure 6 is similar to Figure 4 but $l_{\rm ovs}$ and
$l_{\rm con}$ are expressed in unit of the local pressure scale height
($H_{\rm P}$). It is shown that $l_{\rm ovs}$ is no more than $4H_{\rm P}$.

Figure 6: Similar to Figure 4, but the two lengths are expressed in unit of
$H_{\rm P}$.

We discuss from another side of view the extension of convective overshooting
that concerns the masses in the convection and overshooting regions. We denote
$M_{\rm ovs}$ the mass within the overshooting region and $M_{\rm con}$ the
mass of the convection zone. Figure 7 shows $\lg M_{\rm ovs}$ and $\lg M_{\rm
con}$ vary with the effective temperature, respectively. Figure 8 shows the
variation of the ratio of $M_{\rm ovs}$ to $M_{\rm con}$. It can be seen that
$M_{\rm ovs}/M_{\rm con}$ decreases rapidly with $T_{\rm eff}$ and $M_{\rm
ovs}$ accounts only for a small fraction of $M_{\rm con}$ for the models with
relatively low $T_{\rm eff}$. The convection zone is thickening in the
evolutionary process and causes a rapid increase in $M_{\rm con}$. At $T_{\rm
eff}\sim 38,000~{}{\rm K}$, $M_{\rm con}$ is greater than $M_{\rm ovs}$. It is
interesting to note, that although $l_{\rm ovs}$ is considerably large than
$l_{\rm con}$, the matter in the overshooting region has a low density
comparing with the dense, turbulent convection zone. For example, when the
white dwarf model cools down to $T_{\rm eff}\sim 30,000~{}{\rm K}$, the ratio
of $l_{\rm ovs}$ to $l_{\rm con}$ is less than $3.8$ and $M_{\rm ovs}$ is only
about $1/3$ of $M_{\rm con}$. Therefore, we may reasonably believe that the
convective motion can extend to the photosphere by force of the convective
overshooting.

Figure 7: Masses of overshooting region and convection zone vary with $T_{\rm
eff}$. $M_{\rm ovs}$ is the mass of the overshooting region and $M_{\rm con}$
is the mass of the convection zone. Figure 8: The ratio of $M_{\rm ovs}$ to
$M_{\rm con}$ varies with $T_{\rm eff}$.

###  3.2 Determination of the transition temperature

If the convection zone in the helium envelope of a DA white dwarf can extend
upward to the photosphere due to the convective overshooting, the hydrogen
atmosphere will be mixed into the convective helium layer and the white dwarf
will have the opportunity to change its apparent chemical composition, in
other words, to transform into a DB star. In this section, we will discuss
when such transformation occurs.

We assume that the convective mixing region includes the overshooting region
and the convection zone, in which hydrogen and helium are homogeneously mixed.
The mass of the mixing zone is denoted as $M_{\rm mix}$, and the mass of
hydrogen in the mixing zone is denoted as $M_{\rm Hmix}$. The remainder is
helium whose mass is equal to $M_{\rm mix}-M_{\rm Hmix}$. When the mass of
helium exceeds the mass of hydrogen in the mixing zone, i.e.

$M_{\rm mix}-M_{\rm Hmix}>M_{\rm Hmix}~{},$ (3)

that is,

$M_{\rm mix}>2M_{\rm Hmix}~{},$ (4)

hydrogen will be overwhelmed by helium and we assume that the transformation
of the spectral type will occur. We use $M_{\rm mix}=2M_{\rm Hmix}$ as a
critical condition for a DA star evolving into a DB one. The effective
temperature at this critical point is called the transition temperature.

###  3.3 Discussions of the calculation results

We have computed a series model of a DA white dwarf with $M_{\rm H}=1.0\times
10^{-15}~{}M_{\odot}$. The convection zone varies with the effective
temperature as shown in Figure 9, in which we use $\lg(1-M_{r}/M)$ to indicate
the location of the convection zone. The convective motion appears just below
the H/He interface ($\lg(1-M_{r}/M)\approx-15$) and extends into the stellar
interior when the white dwarf model cools down. When the convective mixing
zone becomes thick enough, Eq. (4) is satisfied (indicated by the thick dashed
line). It can be found that the transition temperature ($T_{\rm tr}$) of this
model is about $31,000~{}{\rm K}$. If the convective overshooting can reach to
the photosphere at this temperature, we will observe that the white dwarf
evolves into a DB star. According to the discussions in Section 3.1, the
required length of the convective overshooting is about $3H_{\rm P}$.

Figure 9: A schematic representation of the spectral evolution of a white
dwarf model with $M_{\rm H}=1.0\times 10^{-15}~{}M_{\odot}$.

Other series of white dwarf evolution models we have computed show that the
thicker the hydrogen layer is, the lower the transition temperature will be.
Table 2 lists the value of $T_{\rm tr}$ of our models with different $M_{\rm
H}$. It can be noticed that the transition temperature of the model with
$M_{\rm H}=1.0\times 10^{-14}~{}M_{\odot}$ is below $20,000~{}{\rm K}$. It is
expected therefore that models with the hydrogen layers heavier than
$10^{-14}~{}M_{\odot}$ must have $T_{\rm tr}$ lower than $18,000~{}{\rm K}$.
So a DA white dwarf with $M_{\rm H}>10^{-15}~{}M_{\odot}$ may have the
opportunity to change its spectral type when it cools much below $T_{\rm
eff}\sim 30,000~{}{\rm K}$. Moreover, greater $M_{\rm H}$ also requires
stronger convective overshooting to bring helium to the stellar surface.

Table 2: Transition temperatures for different white dwarf models $M_{\rm H}~{}(M_{\odot})$ | $T_{\rm tr}~{}(~{}{\rm K})$ | $M_{\rm H}~{}(M_{\odot})$ | $T_{\rm tr}~{}(~{}{\rm K})$
---|---|---|---
$l=H_{\rm P}$ | $l=2H_{\rm P}$ | $l=H_{\rm P}$ | $l=2H_{\rm P}$
$1.0\times 10^{-15}$ | $31,084$ | $31,084$ | $1.8\times 10^{-15}$ | $25,191$ | $26,222$
$1.1\times 10^{-15}$ | $29,349$ | $29,769$ | $2.0\times 10^{-15}$ | $24,001$ | $26,209$
$1.2\times 10^{-15}$ | $27,920$ | $29,006$ | $3.0\times 10^{-15}$ | $22,112$ | $24,004$
$1.4\times 10^{-15}$ | $27,509$ | $28,456$ | $4.0\times 10^{-15}$ | $20,785$ | $22,799$
$1.5\times 10^{-15}$ | $27,257$ | $28,240$ | $5.0\times 10^{-15}$ | $19,966$ | $22,477$
$1.6\times 10^{-15}$ | $26,360$ | $27,466$ | $1.0\times 10^{-14}$ | $18,460$ | $19,802$

Furthermore, a more efficient convection can change the deepening of the inner
boundary of the He convection during the white dwarf’s evolution, and thus
change the transition temperature. In the MLT (mixing-length theory), the
mixing-length $l$ represents the efficiency of convective heat transfer. We
considered a series of models, in which the mixing-length is set to $2H_{\rm
P}$, which is $2$ times larger than before. Our calculations show that the
deepening of the convection zone do occur at a higher effective temperature.
As shown in Fig. 10, the convection zone of a model with $M_{\rm
H}=10^{-15}~{}M_{\odot}$ deepening at $T_{\rm eff}\sim 30,000~{}{\rm K}$
(compared with Figure 2). For this reason Eq. (4) will be satisfied earlier
and it will lead to a change of $T_{\rm tr}$ (see Table 2). However, the
variation of $T_{\rm tr}$ is relatively small, especially around
$30,000~{}{\rm K}$. We therefore believe that the efficiency of the convection
will not significantly affect the results.

(a)  (b)

Figure 10: Similar to Figure 2, but $l=2H_{\rm P}$.

The above discussions imply that DB white dwarfs are likely born from DA white
dwarfs because of the convective mixing. Is it possible that a DO white dwarf
can evolve into a DA star? Our calculations show that if a DA white dwarf with
a sufficiently thin hydrogen layer of $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$,
this transformation is possible. As shown in Figure 11, during the early time
of the evolution, the model’s $T_{\rm eff}$ is very high and hydrogen is
completely ionized in its atmosphere. The photosphere lies deeply in the
helium layer and thus helium is visible, resulting in the white dwarf
appearing as a DO star. When the white dwarf model cools down to $T_{\rm
eff}\sim 63,000~{}{\rm K}$, the photosphere rises to the hydrogen layer, but
at the same time convection appears in the He II/III ionization zone. Because
of the thin hydrogen layer, the convective overshooting can possibly reach to
the photosphere and Eq. (4) can easily be satisfied. The convective motion in
the helium layer will dilute the hydrogen layer, so the white dwarf is still a
DO star.

Figure 11: A schematic representation of the formation of a DO white dwarf
from a DA star with $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$.

When the white dwarf cools down below $60,000~{}{\rm K}$, as shown in Figure
12, the location of the photosphere will quickly move up toward the stellar
surface. The extension of the convective motion can not go so far to reach the
photosphere. Therefore the convective mixing of helium is invisible and
hydrogen will re-accumulate in the atmosphere, making the DO white dwarf
transforming into a DA star. This result is similar to the Shibahashi’s
assumption (Shibahashi [2005], [2007]). According to the observational data,
there are no DO white dwarfs below $T_{\rm eff}\sim 45,000~{}{\rm K}$. This
fact allows us to adjust the overshooting length to let our model change its
spectral type at $T_{\rm eff}\sim 45,000~{}{\rm K}$. We can assume reasonably
that the overshooting length is proportional to the dimension of the
convection zone. In practice, we choose the overshooting length to be the
length of convection zone (being approximately equal to $H_{\rm P}$) plus
$0.375H_{\rm P}$. Our numerical result shows that at an effective temperature
($T_{\rm eff}\sim 26,000~{}{\rm K}$) the convection zone has developed thick
enough that the convective overshooting can reach again above the photosphere.
Then the hydrogen layer will be mixed with the helium layer and the white
dwarf will become a DB star.

Figure 12: A schematic representation of the spectral evolution of a whte
dwarf model with $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$. The dash-dotted line
corresponds to the boundary of the overshooting region.

There is another possibility for DO white dwarfs transforming into DA stars in
the literature. It is probably that the progenitors of DO white dwarfs have
undergone a born-again phase and burnt most of the hydrogen envelope (Althaus
et al. [2005]). The mass loss due to stellar wind during the hot white dwarf
stage is likely to throw away the superficial hydrogen and prevents the
gravitational settling (Unglaub & Bues [2000]). When the white dwarfs cool
down to $T_{\rm eff}\sim 45,000~{}{\rm K}$, hydrogen previously left in the
internal layer is able to float to the stellar surface and DA white dwarfs are
formed.

##  4 Conclusions

From the above investigations we have found that the DB gap could be explained
as a consequence of the convective mixing in white dwarfs. DA white dwarfs
with $M_{\rm H}/M_{\odot}\sim 10^{-16}$ have opportunities to transform into
DO white dwarfs at $T_{\rm eff}\gtrsim 46,000~{}{\rm K}$ or DB white dwarfs at
$T_{\rm eff}\lesssim 26,000~{}{\rm K}$, respectively. DA white dwarfs with
$M_{\rm H}/M_{\odot}\sim 10^{-15}$ will can transform into DB stars below
$T_{\rm eff}\sim 31,000~{}{\rm K}$. White dwarfs with $M_{\rm H}$ greater than
$10^{-14}~{}M_{\odot}$ always appear as DA stars at $T_{\rm eff}\gtrsim
18,000~{}{\rm K}$. It is obvious that in the effective temperature range
between $46,000\lesssim T_{\rm eff}\lesssim 31,000~{}{\rm K}$ almost all of
white dwarfs have the DA spectral type, as shown in Figure 13. This scenario
substantially coincides with the observation. We can also estimate that the
hydrogen mass is $M_{\rm H}/M_{\odot}\sim 10^{-16}$ for the DO white dwarfs
and is $M_{\rm H}/M_{\odot}\sim 10^{-15}$ for the hot DB white dwarfs ($T_{\rm
eff}>20,000~{}{\rm K}$).

Figure 13: A schematic representation of the DB gap. $q_{H}=M_{H}/M_{\odot}$.

Based on our numerical results, the convective overshooting plays a crucial
role in the formation of the so-called DB gap, through the convective mixing
effect. It allows the convective motion penetrating into the hydrogen layer
and makes helium in the deep stellar interior being observable on the stellar
photosphere. The overshooting length is an important parameter of the model.
According to our results, the overshooting length should be proportional to
the thickness of the convection zone, which gives better agreement between the
model results and observations.

The hydrogen mass $M_{\rm H}$ is another important parameter, which is used as
a criterion for deciding when helium is dominant in the atmosphere of the
white dwarf. It determine decisively the critical effective temperature for
the white dwarf changing its spectral type.

###### Acknowledgements.

We thank Q. S. Zhang for many valuable discussions. This work is supported by
the National Key Fundamental Research Project through grant 2007CB815406.

## References

  * [2005] Althaus, L. G., Serenelli, A. M., Panei, J. A., Córsico, A. H., García-Berro, E., Scóccola, C. G. 2005, A&A, 435, 631
  * [1958] Böhm-Vitense, E. 1958, ZsAp, 46, 108
  * [2006] Eisenstein, D. J., et al. 2006, AJ, 132, 676
  * [1977] Fontaine, G., Graboske, H. C., Jr., Van Horn, H. M. 1977, ApJS, 35, 293
  * [1987] Fontaine G., Wesemael F., 1987, In: Philip A. G. D., Hayes D. S., Liebert J., eds, IAU Colloq. 95, Second Conference on Faint Blue Stars. Davis Press, Schenectady, NY, p. 319
  * [1969] Hubbard, W. B., Lampe, M., 1969, ApJS, 18, 297
  * [1983] Itoh, N., Mitake, S., Iyetomi, H., Ichimaru, S. 1983, ApJ, 273, 774
  * [1989] Itoh, N., Adachi, T., Nakagawa, M., Kohyama, Y., Munakata, H. 1989, ApJ, 339, 354
  * [1993] Kohyama, Y., Itoh, N., Obama, A., Mutoh, H. 1993, ApJ, 415, 267
  * [2008] Kurtz, D. W., Shibahashi, H., Dhillon, V. S., Marsh, T. R., Littlefair, S. P. 2008, MNRAS, 389, 1771
  * [1974] Lamb, D. Q. 1974, PhD thesis, The University of Rochester
  * [1975] Lamb, D. Q., Van Horn, H. M. 1975, ApJ, 200, 306
  * [1999] McCook, G. P., Sion, E. M. 1999, ApJS, 121, 1
  * [1984] Mitake, S., Ichimaru, S., Itoh, N. 1984, ApJ, 277, 375
  * [1995] Saumon, D., Chabrier, G., Van Horn, H. M. 1995, ApJS, 99, 713
  * [2005] Shibahashi H., 2005, In: Alecian G., Richard O., Vauclair S., eds, EAS Publ. Ser. Vol. 17, Element Stratification in Stars: 40 Years of Atomic Diffusion. EDP Sciences, Paris, p. 143
  * [2007] Shibahashi H., 2007, AIPC, 948, 35
  * [1983] Sion, E. M., Greenstein, J. L., Landstreet, J. D., Liebert, J., Shipman, H. L., Wegner, G. A. 1983, ApJ, 269, 253
  * [2008] Tremblay, P. -E., Bergeron, P. 2008, ApJ, 672, 1144
  * [2000] Unglaub, K., Bues, I. 2000, A&A, 359, 1042
  * [1977] Vauclair, G., Reisse, C. 1977, A&A, 61, 415
  * [1990] Wood, M. A. 1990, PhD thesis, The University of Texas at Austin