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# Vertically Self-Gravitating ADAFs in the Presence of Toroidal Magnetic Field

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## Abstract

Force due to the self-gravity of the disc in the vertical direction is considered to study its possible effects on the structure of a magnetized advection-dominated accretion disc. We present steady-sate self similar solutions for the dynamical structure of such a type of the accretion flows. Our solutions imply reduced thickness of the disc because of the self-gravity. It also imply that the thickness of the disc will increase by adding the magnetic field strength.
arXiv:1110.2941v1 [astro-ph.HE] 13 Oct 2011
Vertically Self-Gravitating ADAFs in the Presence of
Toroidal Magnetic Field
Abstract Force due to the self-gravity of the disc in
the vertical direction is considered to study its possi-
ble eﬀects on the structure of a magnetized advection-
dominated accretion disc. We present steady-sate self
similar solutions for the dynamical structure of such a
type of the accretion ﬂows. Our solutions imply re-
duced thickness of the disc because of the self-gravity.
It also imply that the thickness of the disc will increase
by adding the magnetic ﬁeld strength.
Keywords accretion, accretion ﬂow, self-gravity
1 INTRODUCTION
There has been rapidly progress over the past three
decades towards a better understanding of the accretion
processes in astrophysics, in particular accretion discs
around compact objects or even black holes (see re-
views by Narayan, Mahadevan & Quataert 1998; Kato,
Fukue & Mineshige 2008). Black hole accretion is one
S. Abbassi
J. Ghanbari
ghanbari@ferdowsi.um.ac.ir
1School of Physics, Damghan University of Basic Sciences,
Damghan, 36715-364, Iran
2School of Astronomy, Institute for Research in Fundamental Sci-
ences (IPM), Tehran, 19395-5531, Iran
3Department of Physics, Faculty of Science, Golestan University,
Basij Square, Gorgan, Iran
4Department of Physics, School of Sciences, Ferdowsi University
5Khayam Institute of Higher Education, P.O.Box 918974-7178,
of the most important ingredient when considering as-
trophysical scenarios of galaxy/quasar formation. Cur-
rent belief is that there are two sites where an accretion
disc around a black hole can be found: In close binary
systems called X-ray binaries (XBs), and at the cen-
ter of the galaxies. The birth of modern accretion disc
theory is traditionally attributed to the original model
presented by Shakura & Sanyev (1973). This standard
geometrically thin, optically thick accretion disc model
(SSD) can successfully explain most of observational
features in active galactic nuclei (AGNs) and X-ray bi-
naries. In the standard thin disc model, the motion
of matter in the accretion disc is nearly Keplerian, and
the viscous heat in the disc is radiated away locally. An
alternative accretion disc model, namely, the advection-
dominated accretion ﬂows (ADAFs), was suggested for
the black holes accreting at very low rates (Ichimaru
1977, Narayan & Yi 1994).
In some of the scenarios of structure formation in as-
trophysics, in particular those related to the formation
of the stars or galaxies, self-gravity of the system may
play a vital role. Formation of an accretion disc, as one
stage of the structure formation, is an important part of
any theory in this ﬁeld. However, in the standard accre-
tion disc model, the eﬀect of self-gravity in the vertical
or radial direction of the disc is neglected for simplicity
and the disc is supported in the vertical direction only
by the thermal pressure. Although in some of the ac-
creting systems it is a reasonable assumption, there are
situations, in which one can hardly neglect self-gravity
of the disc itself. According to the current theories,
self-gravity of the disc not only can contribute in gen-
erating turbulence inside the disc but its force in the
radial and the vertical directions would modify the an-
gular velocity and and the vertical scale-hight, respec-
tively Duschl et al. 2000. Therefore, it is not an easy
task to include self-gravity of a disc in a self-consistent
approach. In fact, the theory of self-gravitating ac-
2
cretion discs is less developed. Early numerical work
of self-gravitating accretion discs began with N-body
modeling (Cassen & Moosman 1981; Toomley; Cassen
& Stein-Cameron 1991). Although great progress has
been made in recent years in increasingly sophisticated
numerical accretion disc simulations, simple analytic
disc models still are the only accessible way of mak-
ing direct link between the theory and observations.
The theoretical treatment can estimate the spectra and
other observational features of the accretion powered
objects. To solve the nonlinear equations of the self-
gravitating accretion discs, sometimes the technique of
self-similarity is useful. Some of the astrophysical sys-
tems often attain self-similar limits for a wide range of
the initial conditions. Moreover, the self-similar prop-
erties allows us to investigate properties of the solu-
tions in arbitrary details, without any of the associated
diﬃculties of numerical magneto-hydrodynamics. Sev-
eral classes of the self-similar solutions for the structure
of self-gravitating accretion discs have been studied by
now (e.g., Paczynski 1978; Mineshige & Umemura 1996;
Shadmehri 2004). Some of them are presented for the
standard accretion discs (e.g., Paczynski 1978), while
the others are describing ADAFs including self-gravity
(e.g., Mineshige & Umemura 1996; Shadmehri 2004).
Abramowicz et al. 1984 examined a number of issues re-
lated disc’s self-gravity in a steady state accretion disc,
especially in regard to the eﬀect of disc self-gravity on
the disc topology and disc dimensions. Bu, Yuan &
Xie (2009) have proposed a self-similar solution of a
magnetized ADAFs without self-gravity. Mineshige &
Umemura (1996) presented a set of self-similar solu-
gravitational force of the disc in the radial direction.
However, such solutions are hardly applicable because
the gravitational force of the disc due to its self-gravity
the central object. However, the gravitational force of
the disc in the vertical direction can be a fraction of the
vertical component of gravitational force of the central
object, at least at the outer part of ADAFs. Then, as
we will show, the structure of the disc would modify be-
cause of the self-gravity of the disc itself. We think, our
solutions are applicable to the outer parts of ADAFs in
X-ray binaries.
Although we did not apply our solution to a speciﬁc
astronomical system, we believe the solutions are appli-
cable to the discs with an inner hot part surrounded by
an outer cold part. In the most cases, one can hardly
recognize the transition region. For example, the outer
regions of the X-ray binaries are SSD, though their in-
ner parts resemble to the hot accretion ﬂows (e.g., Esin
et al. 1997). Our solutions are suitable for a region
where the disc is self-gravitating but the advection of
the dissipated energy is not negligible. Also, the discs
of the low-luminosity AGNs (e.g., Ho 2008) may have a
similar situation and they do not show clear transition
from SSD to ADAF. The solutions are also applicable
to Slim discs, and thus have its potential applications
in ULXs and NLS1s (Mineshige et al. 2000; Watarai
et al. 2000). The next step for future studies is to
relax our simplifying assumptions, in particular simi-
larity method, and solve the relevant equations for one
of the mentioned systems and in doing so, our similarity
solutions will guide us.
2 The Basic Equations
We are interested in analyzing the dynamical behav-
ior of a magnetized ADAF when the force due to the
self-gravity of the disc in the vertical direction is con-
sidered. But self-gravitational force in the radial di-
rection is neglected because ADAFs do not extend ra-
dially so much. We consider only toroidal component
of the magnetic ﬁeld. We suppose that the gaseous
disc is rotating around a compact object of mass M.
Thus, for a steady axisymmetric accretion ﬂow, i.e.
∂/∂ t =/∂φ = 0, we can write the standard equations
in the cylindrical coordinates (r, φ, z). We vertically
integrate the ﬂow equations and so, all the physical
variables become only functions of the radial distance
r. We also neglect the relativistic eﬀects and Newto-
nian gravity in radial direction is considered. The disc is
supposed to be turbulent and possesses an eﬀective tur-
bulent viscosity ν. As for the energy conservation, we
assume the generated energy due to viscosity dissipa-
(e.g., Narayan & Yi 1994).
The equation of continuity gives,
1
r
d
dr (rΣvr) = 2 ˙ρH (1)
where vris the accretion velocity (i.e., vr<0) and ˙ρ
denotes the mass-lose rate per unit volume due to the
wind or outﬂow, His the disc half-thickness, and Σ is
the surface density. We can also write Σ = 2ρH .
The equation of motion in the radial direction is
vr
dvr
dr =v2
ϕ
rGM
r21
Σ
d
dr c2
s)c2
A
r1
d
dr c2
A),
(2)
where vϕ, csand cAare the rotational velocity of the
disc, sound and Alfven velocities of the gas, respec-
tively. Sound speed and the Alfven velocity are deﬁned
as c2
s=pgasand c2
A=B2
ϕ/4πρ = 2pmag where
3
pgas and pmag are the gas and the magnetic pressures,
respectively.
The vertically integrated angular momentum equa-
tion becomes
rΣvr
d
dr (rvϕ) = d
dr (r3νΣd
dr ) (3)
where Ω = vφ/r is the angular velocity. Also νrepre-
sents the kinematic viscosity coeﬃcient and we assume
(Shakura & Sunyaev 1973),
ν=αcsH(4)
where αis a nondimensional parameter less than unity.
The goal of this study is to investigate the eﬀect
of self-gravity of the disc on its structure. One can
estimate the importance of self-gravity of the disc by
comparing the contributions to the local gravitational
acceleration in the vertical direction by both the central
object and the disc itself. The vertical acceleration at
the disc surface due to its self-gravity is 2πGΣ and by
the central object is GMh/r3. Thus, force due to the
self-gravity of the disc in vertical direction is dominated
if:
Md
Mπr2Σ
M
>1
2
h
r(5)
where Mdis the mass enclose the disc within a radius
r. For ADAFs the typical values of h/r is around 1.
Since the enclosed mass Mdis an increasing function of
the radial distance r, the eﬀect of self-gravity becomes
signiﬁcant when the disc is thick. Here we consider ke-
plerian selfgravitating (KSG) disks in which selfgravity
is signiﬁcant only in the vertical direction and which
satisfy the constraint (1/2)(h/r)MMd(r)M.
Compared to the standard ADAF solutions (Narayan
& Yi 1994; Akizuki & Fukue 2006), the KSG disks re-
quire modiﬁcation of the equation of hydrostatic sup-
port in the direction perpendicular to the disk. Thus,
while in the standard ADAF solutions the local vertical
pressure gradient is balanced by the z component of the
gravitational force due to the central object, in the SG
case we have balance between two local forces, namely
the pressure force and the gravitational force due to the
disks local mass. Also in the KSG case, in the radial
direction centrifugal forces are still balanced by gravity
from a central mass (Keplerian approximation) .
For a self-gravitating disc, the hydrostatic equilib-
rium in the vertical direction yields (e.g., Paczyncki
1978, Duschl et al. 2000)
Pc=πGΣ2(6)
where Pcis the pressure in the central plane (z= 0)
and Gis the gravitational constant. Since details of
the thermodynamics in the vertical direction are not
considered in our model, we shall assume the disc to be
isothermal in the vertical direction.
For a magnetized Keplerian self-gravitating disc, we
have
Pc
¯ρ=c2
s[1 + 1
2(c2
A
c2
s
)2] = c2
s(1 + β) (7)
where ¯ρ=Σ
2His a vertically averaged mass density and
β=Pmag
Pgas =1
2(cA
cs)2, where this parameter shows the
important of magnetic ﬁeld pressure in comparison to
the gas pressure. We will take βas an input parameter
of our model and its eﬀect on the structure of the disc
is studied for diﬀerent values of β. Having Equations 6
and 7, we can write
H=(1 + β)c2
s
2πGΣ.(8)
Also, the energy equation becomes
Σvr
γ1
dc2
s
dr +c2
s
r
d
dr (rvr) = f νr2(d
dr )2(9)
where γis the ratio of speciﬁc heats. The advection
parameter fmeasures the degree to which the ﬂow is
advected-dominated (Narayan & Yi 1994), and it is sup-
posed to be constant.
Induction equation with the ﬁeld scape can be writ-
ten as (Akizuki & Fukue 2006)
d
dr (VrBϕ) = ˙
Bϕ,(10)
where ˙
Bϕis the ﬁeld scaping/creating rate due to the
magnetic instability or the dynamo eﬀect. We can
rewrite this equation as
vr
dc2
A
dr +c2
A
dvr
dr c2
Avr
r= 2c2
A
˙
Bφ
Bφc2
A
2 ˙ρH
Σ.(11)
Now, we have a set of ordinary diﬀerential equations
that describe the dynamical behavior of a magnetized
ADAF ﬂow including self-gravity of the disc. Solu-
tions of these equations give us the dynamical behavior
of the disc, which depends on the viscosity, magnetic
ﬁeld strength, self-gravity and advection rate of energy
transport.
3 Self-Similar Solutions
Global behavior of the disc can not be described us-
ing similarity solutions, because boundary conditions
4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
D
c1
f = 0.1
f = 0.5
f = 0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.95
0.96
0.97
0.98
0.99
1
D
c2
f = 0.1
f = 0.5
f = 0.7
α = 0.1 , β = 0.1 , γ = 1.0
α = 0.1 , β = 0.1 , γ = 1.0
Fig. 1 These plots show behavior of the coeﬃcients
C1(
vr
αvk) and C2(vϕ
vk) versus self-gravitating parameter D
for diﬀerent values of the advection parameter f. Here, the
input parameters are β= 0.1, α= 0.1 and γ= 1.
are not considered in this method. But, as long as we
are not interested in the solutions near the boundaries,
similarity solutions describe correctly, true and useful
asymptotically behavior of the ﬂow in the intermediate
regions.
We assume that the physical quantities can be ex-
pressed as a power law of the radial distance,i.e. rν,
where νis determined by substituting the similarity
solutions into the main equations and solving the re-
sulting algebraic equations. Therefore, we can write
similarity solutions as
vr(r) = C1αvk(r),(12)
vϕ(r) = C2vk(r),(13)
c2
s(r) = C3v2
k(r),(14)
c2
A(r) = B2
ϕ(r)
4πρ(r)= 2βC3v2
k(r),(15)
where
vk(r) = rGM
r,(16)
and the parameters C1,C2and C3will be determined
from the main equations.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
D
c1
β = 0.1
β = 0.5
β = 1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.95
0.96
0.97
0.98
0.99
D
c2
β = 0.1
β = 0.5
β = 1.0
α = 0.1 , f = 0.7 , γ = 1.0
α = 0.1 , f = 0.7 , γ = 1.0
Fig. 2 These plots show behavior of the coeﬃcients
C1(
vr
αvk) and C2(vϕ
vk) versus self-gravitating parameter D
for diﬀerent values of the magnetic ﬁeld parameter β. Here,
the input parameters are f= 0.7, α= 0.1 and γ= 1.
In addition, the surface density Σ is assumed to be
a form of
Σ = Σ0rs,(17)
where Σ0and sare constants. Then, in order for the
self similar treatment to be valid, the mass-loss rate
per unit volume and the escaping rate must have the
following form,
˙ρ= ˙ρ0rs5/2,(18)
˙
Bφ=˙
B0r(s5)/2,(19)
where ˙ρ0and ˙
B0are constants.
Considering hydrostatic equation, we obtain the disc
half-thickness Has
H
r=C3(1 + β)
2Md
M
=C3(1 + β)
D,(20)
where Mdis the mass enclosed in the disc within a
radius rand is given approximately by Md=πr2Σ.
Also, we deﬁne parameter D(= 2 Md
M) as self-gravitating
coeﬃcient.
By substituting the above similarity solutions into
the continuity and the motion equations and also the
energy and the induction equations, we obtain the fol-
5
lowing system of dimensionless algebraic equations,
˙ρ0=(s+1
2)C1αΣ0GMD
2(1 + β)C3
,(21)
1
2α2C2
1+C2
21s1 + β(s+ 1)C3= 0,(22)
C1=3(s+ 1)(1 + β)
DC3/2
3,(23)
H
r=(1 + β)C3
D,(24)
C2
2=2D
9f(1 + β)(3γ
γ1)C1C1/2
3,(25)
˙
B0=(3 s)
2αC1(GM)s4πDΣ0β
(1 + β),(26)
As is easily seen from the above equations, for s=1/2
there is no mass loss, while there exist mass loss for
s > 1/2. On the other hands, the escape and cre-
ation of magnetic ﬁelds are balanced each other for
s= 3. Although outﬂow is one of the most important
progresses in accretion theory,(see Narayan & Yi 1995;
Blandford & Begelman 1999; Stone & Pringle & Begel-
man 1999 and also some recent work like Xie & Yuan
2008; Ohsuga & Mineshige 2011), but in out model we
chouse self-similar solution in a way that ˙ρ= 0 and
˙
Bϕr11/4,(s=1/2). So we ignored the eﬀect of
wind and outﬂow on the structure of the disks.
After some algebraic manipulations, we can reduce
the above equations to a sixth order algebraic equation
for C1,
A3C6
1+ 3A2EC 4
1+ (3AE2+B3)C2
1+E3= 0,(27)
where the coeﬃcients depend on the input parameter
as
A=1
2α2,(28)
B=ǫ+1
2(3 β)( 2D
3(1 + β))2/3,(29)
E=1,(30)
ǫ=1
35/3f(2D
1 + β)2/3(3γ
γ1).(31)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
D
h/r
f = 0.3
f = 0.5
f = 0.7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
D
h/r
β = 0.1
β = 0.5
β = 1.0
α = 0.1 , β = 0.1 , γ = 1.0
α = 0.1 , f = 0.7 , γ = 1.0
Fig. 3 These plots show behavior of the h/r versus self-
gravitating parameter Dfor diﬀerent values of the advection
parameter fupper panel and magnetic ﬁeld parameter, β,
lower panel. Here, the input parameters are f= 0.7, α=
0.1, β= 0.1 and γ= 1.
Having C1from this algebraic equation, the other vari-
ables (i.e. C2and C3) can be determine easily.
Now, we can do a parameter study considering our
input parameters. Figure 1 shows how the coeﬃcients
C1and C2depend on the self-gravitation parameter D
velocity is determined by C1which is shown in the up-
less than free fall velocity on a point mass, but it be-
comes larger if the advection parameter fis increased
and such a behavior is consistent with the previous an-
alytical solutions (e.g., Akizuki & Fukue 2006; Abbassi
et al. 2008, 2010). Obviously, larger parameter D im-
plies a more self-gravitating disc and top plot of Figure
1 shows that the accretion velocity decreases as param-
eter Dincreases. Compare to a non-self-gravitating
disc, low D, the velocity of the ﬂow will decrease some
order of magnetude. Moreover, as more dissipated en-
velocity is more sensitive to the variations of the self-
gravitating parameter within the range D5. But
when the disc becomes strongly self-gravitating (i.e.,
large D), the accretion velocity is more or less indepen-
dent of the advected energy. Bottom plot of Figure 1
shows coeﬃcient C2(i.e., rotational velocity) versus pa-
rameter D. Here, the rotational velocity is not aﬀected
by the variation of the parameter D, but it decreases
6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
f
c1
D = 0.1
D = 0.2
D = 0.3
D = 0.5
D = 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.94
0.95
0.96
0.97
0.98
0.99
1
f
c2
D = 0.1
D = 0.5
D = 1.0
α = 0.1 , β = 0.1 , γ = 1.0
α = 0.1 , β = 0.1 , γ = 1.0
Fig. 4 These plots show behavior of the coeﬃcients
C1(
vr
αvk) and C2(vϕ
vk) versus advection parameter ffor dif-
ferent values of the self-gravitating parameter D. Here, the
input parameters are β= 0.1, α= 0.1 and γ= 1.
In Figure 2, we assume the fraction of the advected
energy is f= 0.7 and then behavior of the coeﬃcients
C1and C2versus parameter Dare shown for diﬀer-
ent values of magnetic ﬁeld parameter β. For a given
D, both the radial and rotational velocities increase as
the disc becomes more magnetized (i.e., larger β) A
magnetized disc must rotate faster than a case without
magnetic ﬁeld because of the eﬀect of magnetic tension.
The radial velocity is more sensitive to the variations
of Dwhen the disc is more magnetized. The rotational
behavior is the same for diﬀerent values of Dbut it
slightly shifts up when parameter βincreases.
In Figure 3, we show vertical thickness of the disc in
terms of the self-gravitating parameter Dfor diﬀerent
values of fand β. The disc becomes thinner as the
parameter Dincreases. If we neglect force due to the
self-gravity of the disc in the vertical direction, the disc
becomes thicker as more energy advected with the ﬂow.
But our analysis shows for a strongly advected disc even
not very large values of Dlead to a reduction of the disc
thickness by a factor of two. For a given D, by increas-
ing the parameters fand βthe vertical thickness in-
creases which means that advection and magnetic ﬁeld
will cause the disc becomes thicker.
Figure 4 shows how the coeﬃcients C1and C2de-
pend on the advection parameter ffor diﬀerent val-
ues of self-gravitating parameter D. Radial velocity is
determined by C1which is shown in the upper panel.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
f
c1
β = 0.1
β = 0.5
β = 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.94
0.95
0.96
0.97
0.98
0.99
1
f
c2
β = 0.1
β = 0.5
β =1.0
α = 0.1 , D =0.2 , γ = 1.0
α = 0.1 , D =0.2 , γ = 1.0
Fig. 5 These plots show behavior of the coeﬃcients
C1(
vr
αvk) and C2(vϕ
vk) versus advection parameter ffor dif-
ferent values of the magnetic ﬁeld parameter β. Here, the
input parameters are D= 0.2, α= 0.1 and γ= 1.
parameter radial velocity will increases while the rota-
tional velocity decrease ( Akizuki & Fukue 2006). In-
ﬂuence of the self-gravitation of the disc will lead to
decrease of radial velocity while the rotational velocity,
c2, is not aﬀected. In Figures 5 and 6 the inﬂuence of
magnetic ﬁeld strength, β, on the behavior of Cisand
vertical thickness of the ﬂow were plotted respectively.
By adding β, which indicates the role of magnetic ﬁeld
in the dynamic of accretion discs, we will see that the
radial ﬂow increase as well as rotational velocities and
vertical thickness increase. On the other hand radial
and toroidal velocity increase when the toroidal mag-
netic ﬁeld becomes large. This is due to the magnetic
tension term, which dominates the magnetic pressure
term in the radial momentum equation that assist the
4 Conclusions
In this paper, we studied an accretion disc in the ad-
vection dominated regime considering purely toroidal
magnetic ﬁeld and the force due to the self-gravity of
the disc in the vertical direction. A set of similarity so-
lutions was presented for such a conﬁguration. Our so-
lutions reduce to the previous analytical solutions (e.g.,
Akizuki & Fukue 2006; Fukue 2004; and Abbassi et al
2008) when self-gravity is neglected. Some approxima-
tions were made in order to simplify the main equations.
7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
f
h/r
D = 0.1
D = 0.2
D = 0.3
D = 0.5
D = 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
f
h/r
β = 0.1
β = 0.5
β = 1.0
α = 0.1 , β = 0.1 , γ = 1.0
α = 0.1 , D = 0.2 , γ = 1.0
Fig. 6 These plots show behavior of the h/r versus self-
gravitating parameter Dfor diﬀerent values of the self-
gravitating parameter Dupper panel and magnetic ﬁeld
parameter, β, lower panel. Here, the input parameters are
β= 0.1, α= 0.1, D= 0.2 and γ= 1.
We assume an axially symmetric and static disc with
α-prescription for the viscosity. We also ignored the
relativistic eﬀects.
Generally, it is believed that the ADAFs are hot and
thick because of their ineﬃciency in radiating out the
dissipated energy. In particular, as more energy is ad-
vected, the disc becomes thicker. But our analysis may
slightly change this picture in particular regarding to
the thickness of the disc if the force due to the self-
gravity of the disc in the vertical direction is considered.
Our disc still rotates with nearly Keplerian proﬁle, but
its thickness is reduced signiﬁcantly because of the self-
gravity. In other words, we may have a fully advected
disc, but its thickness is reduced in comparison to a
case without self-gravity. Also, as the disc becomes
more advective, the eﬀect of self-gravity becomes more
evident. We think the present solutions are suitable for
the outer parts of an ADAF not in inner parts.
It is diﬃcult the evaluated the precise picture of Ad-
vection Dominated Accretion Flows (ADAFs) in the
presence of B-ﬁeld and self-gravity with self-similar
method. However, this method can reproduce overall
dynamical structure of the disc whit a set of given phys-
ical parameter. Although our preliminary self-similar
solutions are too simpliﬁed, they clearly improve our
understanding of physics of ADAFs around a black
hole.
we believe the solutions are are presented, are appli-
cable to the discs with an inner hot part surrounded
by an outer cold part where the self-gravity plays an
important role and and advection of the energy is not
negligible. The discs of the low-luminosity AGNs may
have a similar situation and they do not show clear
transition from SSD to ADAF. The solutions are also
applicable to Slim discs, and thus have its potential
applications in ULXs and NLS1s.
It would be interesting if we add the eﬀect of the
wind and outﬂow in our future investigation to ﬁnd
out how it would change solutions. The next step for
future studies is to relax our simplifying assumptions,
in particular similarity method, and solve the relevant
equations for one of the mentioned systems and in doing
so, our similarity solutions will guide us.
Acknowledgements We are grateful to the referee
for a very careful reading of the manuscript and for
his/her suggestions, which have helped us improve the
presentation of our results.
8
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