Content uploaded by Jamshid Ghanbari

Author content

All content in this area was uploaded by Jamshid Ghanbari

Content may be subject to copyright.

arXiv:1110.2941v1 [astro-ph.HE] 13 Oct 2011

Vertically Self-Gravitating ADAFs in the Presence of

Toroidal Magnetic Field

A. Mosallanezhad 1•S. Abbassi1,2 •

M. Shadmehri 3•J. Ghanbari4,5

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

A. Mosallanezhad

S. Abbassi

amin.mosallanezhad@gmail.com, abbassi@ipm.ir

M. Shadmehri

m.shadmehri@gu.ac.ir

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

of Mashhad, P.O.Box 91775-1436, Mashhad, Iran

5Khayam Institute of Higher Education, P.O.Box 918974-7178,

Mashhad, Iran

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-

tions for steady-state structure of ADAFs considering

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

in ADAfs is generally smaller than the radial force of

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-

tion is balanced by the radiation and advection cooling

(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

r−GM∗

r2−1

Σ

d

dr (Σc2

s)−c2

A

r−1

2Σ

d

dr (Σc2

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=pgas/ρ and c2

A=B2

ϕ/4πρ = 2pmag /ρ 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 (r3νΣdΩ

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 GM∗h/r3. Thus, force due to the

self-gravity of the disc in vertical direction is dominated

if:

Md

M∗∼πr2Σ

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)M∗≤Md(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

γ−1

dc2

s

dr +c2

s

r

d

dr (rvr) = f νr2(dΩ

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

Bφ−c2

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= ˙ρ0rs−5/2,(18)

˙

Bφ=˙

B0r(s−5)/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)C1αΣ0√GM∗D

2(1 + β)C3

,(21)

1

2α2C2

1+C2

2−1−s−1 + 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)(3−γ

γ−1)C1C−1/2

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

˙

Bϕ∝r−11/4,(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(3−γ

γ−1).(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

for diﬀerent values of advection parameter f. Radial

velocity is determined by C1which is shown in the up-

per panel. In ADAFs the radial velocity is generally

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-

ergy is advected within the ﬂow, we see that the radial

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

when the advection parameter fincreases.

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.

In the ADAFs solutions usually by adding advection

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

radial in-fall motion.

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

References

[26]Abbassi S., Ghanbari J., Najjar S., 2008, Mon. Not. R.

Astron. Soc., 388, 663

[26]Abbassi S., Ghanbari J., Ghasemnezhad M., 2010, Mon.

Not. R. Astron. Soc., 409, 1113

[26]Abramowicz M. A.; Curir A.; Schwarzenberg-Czerny, A.;

Wilson, R. E., 1984, Mon. Not. R. Astron. Soc., 208, 279

[26]Akizuki C. & Fukue J., 2006, PASJ, 58, 1073

[26]Bu De-Fu., Yuan. F., Xie Fu-Guo, 2009, Mon. Not. R.

Astron. Soc., 392, 325

[26]Blandford R. D., Begelman M. C., 1999, Mon. Not. R.

Astron. Soc., 303, L1

[26]Cassen P., Moosman A., 1981, Icarus, 48, 353

[26]Duschl W. J., Strittmatter P. A., Biermann, P. L., 2000,

A&A, 357, 1123

[26]Esin A. A.; McClintock J. E.; Narayan R., 1997, Apj,

489, 865

[26]Ichimaru S., 1977, ApJ, 214, 840

[26]Kato S., Fukue J., Mineshige S., 2008, Black Hole Ac-

cretion Discs, Kyoto University Press

[26]Kawabata R., Mineshige S., 2009, PASJ, 61, 1135

[26]Ho L. C., 2008, ARA&A, 46, 475

[26]Mineshige S., Takeuchi., M., 2000, New. Astron. Rev.,

435, 44

[26]Mineshige S., Umemura M., 1996, Astrophys. J., 469, 49

[26]Narayan R., & Yi, I. 1994, Astrophys. J., 428, L13

[26]Narayan R., & Yi, I. 1995, Astrophys. J., 444, 231

[26]Narayan R., Mahadevan R., Quataert E., 1998, The-

ory of Black Hole Accretion Discs, Cambridge University

Press, 1998., p.148

[26]Ohsuga K., Mineshige S., 2011, Astrophys. J., 736, arti-

cle id. 2

[26]Paczynski B., 1978, Acta Astronomica, 287, 91

[26]Shadmehri M. 2004, Astrophys. J., 612, 1000

[26]Shakura N. I., & Sunyaev R.A. 1973, A&A, 24, 337

[26]Stone J. M., Pringle J. E., Begelman M. C., 1999, Mon.

Not. R. Astron. Soc., 310, 1002

[26]Tomley L., Cassen P., Steinman-Camera T., 1991, As-

trophys. J., 382, 530

[26]Watarai Ken-ya., Mizuno T., Mineshige S., 2001, Astro-

phys. J., 549, L77

[26]Xie, Fu-Guo., Yuan, F., 2008, Astrophys. J., 681, 499

This manuscript was prepared with the AAS L

A

T

E

X macros v5.2.