The influence of winds on the time-dependent behavior of self-gravitating accretion discs
aa r X i v : . [ a s t r o - ph ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 1 January 2019 (MN L A TEX style file v1.4)
The influence of winds on the time-dependent behaviour ofself-gravitating accretion discs
Mohsen Shadmehri ⋆ Department of Mathematical Physics, National University Ireland Maynooth, IrelandSchool of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
ABSTRACT
We study effects of winds on the time evolution of isothermal, self-gravitatingaccretion discs by adopting a radius dependent mass loss rate because of the existenceof the wind. Our similarity and semi-analytical solution describes time evolution ofthe system in the slow accretion limit. The disc structure is distinct in the inner andouter parts, irrespective of the existence of the wind. We show that existence of windwill lead to a reduction of the surface density in the inner and outer parts of the discin comparison to a no-wind solution. Also, the radial velocity significantly increasesin the outer part of the disc, however, the accretion rate decreases due to the reducedsurface density in comparison to the no-wind solution. In the inner part of the disc,mass loss due to the wind is negligible according to our solution. But the radial sizeof this no-wind inner region becomes smaller for stronger winds.
Key words: accretion discs - quasars: general - stars: formation
The existence of outflow or wind in many accreting systemsis supported by strong observational evidences. Outflowsfrom Active Galactic Nuclei (AGN) are much more com-mon than previously thought: the overal fraction of AGNswith outflows is fairly constant, approximately 60%, overmany order of magnitude in luminosity (Ganguly & Broth-erton 2008). These mass-loss mechanisms are also observedin microquasars, Young Stellar Object (YSO) and even frombrown dwarfs (Ferrari 1998; Bally, Reipurth & Davis 2007;Whelan et al 2005). About 30% of T Tauri stars presentbipolar ejection and this percentage increases to 100% forclass 0 objects, the earliest stage of star formation. It is nowwidely accepted that winds or outflows have their origin inaccretion flows (e.g., Blandford & Payne 1982). They mayprovide an additional important sink of mass, angular mo-mentum and energy which their dynamical influence to theaccretion flow can not be neglected. However, it remainsunclear as how some part of the accreting gas can be trans-ferred into winds or outflows.Nevertheless, deviations from Keplerian rotation insome AGNs and the flat infrared spectrum of some T Tauristars can both be described by self-gravitating discs. Ver- ⋆ E-mail: [email protected] (MS); tical structure of an accretion disc under the influence ofself-gravity was studied by Paczy´nski (1978). Some authorsstudied the effects related to self-gravity of the disc in theradial direction (e.g., Bodo & Curir 1992). New class of self-gravitating discs has also been proposed, in which the energyequation is replaced by a self-regulation condition (Bertin &Lodato 1999; Lodato & Bertin 2001). Mineshige & Umemura(1997) extended the previous steady state solutions to thetime-dependent case while the effect of the self-gravity of thedisc was taken into account. They used an isothermal equa-tion of state, and so their solutions describe a viscous ac-cretion disc in the slow accretion limit. Also, Tsuribe (1999)studied the self-similar collapse of an isothermal viscous ac-cretion disc.For simplicity, outflow is neglected in most of thesteady-state or time-dependent theoretical models of theself-gravitating discs. However, some authors studied theeffect of wind or outflow on the radial structure of non-self-gravitating accretion discs (e.g., Knigge 1999). Recently,Combet & Ferreira (2008) studied the structure of YSO ac-cretion discs in an approach that takes into account the pres-ence of the protostellar jets. They showed that discs with jetpresents structure different from the standard accretion discbecause of the influences of jets on the radial structure ofthe disc. Ruden (2004) studied the physics of protoplanetarydisc evolution in the presence of a photoevaporative wind.In this paper, we generalize time-dependent solutions c (cid:13) M. Shadmehri of Mineshige & Umemura (1997) for a self-gravitating ac-cretion disc to include winds or outflows. Our paramet-ric model describes mass and angular momentum loss bywind in an isothermal, self-gravitating disc, yet applicableto many types of dynamical disc-wind models. Basic equa-tions are presented in the next section. Properties of oursemi-analytical are discussed in section 3. A summary ofimplications of the results are given in section 4.
We consider an accretion disc that is axisymmetric and ge-ometrically thin, i.e.
H/R <
1. Here R and H are, respec-tively, the disk radius and the half-thickness. The disc issupposed to be turbulent and possesses an effective turbu-lent viscosity. In our model, a central object has not yet beenformed and the radial component of the gravitational forceis provided by the self-gravity of the disc. The continuityequations reads ∂ Σ ∂t + 1 r ∂∂r ( r Σ v r ) + 12 πr ∂ ˙ M w ∂r = 0 , (1)where v r is the accretion velocity ( v r <
0) and Σ = 2 ρH is the surface density at a cylindrical radius r . Also, ρ isthe midplane density of the disc and the mass loss rate byoutflow/wind is represented by ˙ M w . So,˙ M w ( R ) = Z πR ′ ˙ m w ( R ′ ) dR ′ , (2)where ˙ m w ( R ) = ρv +z is mass loss rate per unit area fromeach disc face. Here, v +z is a mean vertical velocity at thedisc surface, i.e., at the base of a wind.The radial momentum equation is ∂v r ∂t + v r ∂v r ∂r = − c Σ ∂ Σ ∂r − GM r r + v ϕ r , (3)where v ϕ is the rotational velocity. As in Mineshige &Umemura (1997), we adopt the monopole approximation forthe radial gravitational force due to the self-gravity of thedisc, which considerably simplify the calculations and arenot expected to introduce any significant errors as long asthe surface density profile is steeper than 1 /r (e.g., Li & Shu1997; Saigo & Hanawa 1998; Tsuribe 1999; Krasnopolsky &K¨onigl 2002). Also, we assume that the disc is vertically self-gravitating and so, the half thickness of the disc, H , becomes H = c / (2 πG Σ).Similarly, integration over z of the azimuthal equationof motion gives (e.g., Knigge 1999) ∂∂t ( rv ϕ ) + v r ∂∂r ( rv ϕ ) = 1 r Σ ∂∂r ( r ν Σ ∂ Ω ∂r ) − ( lr ) Ω2 πr Σ ∂ ˙ M w ∂r , (4)where the last term of right hand side represents angularmomentum carried by the outflowing material. Here, l = 0corresponds to a non-rotating wind and l = 1 to outflowingmaterial that carries away the specific angular momentumit had at the point of ejection and it should be most ap-propriate for radiation-driven outflows (Knigge 1999). Cen-trifugally driven MHD winds are corresponding to l > l = R A /R where R A is Alfven radius (e.g., Knigge 1999).Also, ν is a kinematic viscosity coefficient and we assume ν = αc s H = α ( H/r ) c s r , where c s is the sound speed. As inMineshige & Umemura (1997), we assume α ′ = α ( H/r ) isconstant in space. Thus, in our model the viscosity coeffi-cient is in proportion to the radius which has also been usedby some other authors (e.g., Hartmann et al 1998).We introduce similarity variable x ≡ r/ ( c s t ) and thephysical quantities asΣ( r, t ) = c s πGt σ ( x ) , (5) v r ( r, t ) = c s u ( x ) , (6) v ϕ ( r, t ) = c s v ( x ) , (7) j = xv ( x ) , (8)˙ m w = c s πGt σ ( x )Γ( x ) . (9)Thus, the accretion rate becomes ˙ M acc = − πr Σ v r =( c /G ) ˙ m , where ˙ m = x ( − u ) σ is the non-dimensional simi-larity accretion rate. Now, we can write equations (1), (3)and (4) as − σ − x dσdx + 1 x ddx ( xσu ) + σ Γ = 0 , (10)2 σ dσdx + ( u − x ) dudx − σ u − xx − v x = 0 , (11)( l Γ + 1) j + ( u − x ) djdx = α ′ σx ddx [ σx ( − j + x djdx )] . (12)We can solve the above differential equations numeri-cally subject to appropriate asymptotic behaviors as bound-ary conditions. But we restrict to solution in slow accretionlimit which implies v ≫ σ ≫ | u | ≪
1. Then, equa-tion (11) gives v = σ / ( x − u ) / or j = σ / x ( x − u ) / . (13)On the other hand, equation (10) can be rewritten as d ln σd ln x = 1 x − u dud ln x − − x Γ u − x . (14)Also, after mathematical manipulation, from equation(13) we have d ln jd ln x = 1 + 12 1 x − u ( x − dud ln x ) + 12 d ln σd ln x . (15)Substituting equation (14) into equation (15), we obtain d ln jd ln x = 2 x − u + x Γ2( x − u ) . (16)Using equation (16) the similarity angular momentumequation (12) is written as ux + (2 l − α ′ σxj ddx [ σxj ( 3 u − x + x Γ x − u )] . (17)Equations (14) and (16) give d ln( σxj ) d ln x = 1 x − u dud ln x + 2 x − u + 3 x Γ2( x − u ) . (18)The above relation helps us to simplify equation (17) as c (cid:13) , 000–000 he influence of winds on the time-dependent behaviour of self-gravitating accretion discs dudx = − Ax + Bux + 3 u x − u − x Γ) x − ( x − u ) [ u + (2 l − x Γ)] α ′ ( x − u − x Γ) x , (19)where A = 4 + 6Γ − ddx ( x Γ) − ,B = − − ddx ( x Γ)] . First order differential equation (19) is the main equation ofour analysis which can be solved numerically. Having pro-file of u ( x ) from equation (19), the similarity surface den-sity variable is calculated using equation (14) numerically.Clearly, the effect of wind or outflows appears by the term Γ.If we set this parameter equal to zero, equation (19) reducesto equation (18) of Mineshige & Umemura (1997) which de-scribes no-wind solution.Behavior of the solutions with winds highly depends onthe profile of the mean vertical velocity at the disc surface,i.e. Γ. However, asymptotic behavior of the solutions nearto the central part of the disc, i.e. x →
0, is similar to a casewithout wind/outflow according to equations (14) and (19).When x tends to zero, we have | u | ∝ x, σ ∝ x − / , v ∝ x − / . (20)Appropriate boundary condition at x = 0 is determined us-ing the above asymptotic behavior. For starting the integra-tion of equation (19), we assume u = 0 at x = 0. But inorder to determine surface density profile, we can integrateequation (14) from outer boundary of the disc, i.e. x = 1, to-wards the center for a given σ ( x = 1) = σ . But we considera series of models where the accretion rate at the outer edge(i.e., the inflow rate from the parent cloud) is kept constant,which appears a natural requirement. Thus, our boundaryconditions are u = 0 at x = 0, and, ˙ m = ˙ m at x = 1. So,there is another input parameter for our model, i.e. ˙ m . Forthe mass loss profile, we consider a simple power-law formfor Γ as Γ = Γ x s . We find two different regimes accordingto our similarity solution: inner no-wind part and an outerregion with the surface density profile in proportion to x − .Thus, our prescription for v +z /c s = Γ /σ gives us a power lawmass loss rate for the wind as ˙ M w ∝ x s +1 . In steady statecase, this prescription for mass loss by wind has been usedwidely by many authors (e.g., Quataert & Narayan 1999;Beckert 2000; Turolla & Dullemond 2000; Misra & Taam2001; Fukue 2004). Note that we are doing a purely parametric approach to takeinto account effects of wind on a self-gravitating accretiondisc. So, this does not ensure that a self-consistent solutionexists for any given set of the specified parameters. For solv-ing the equations, we check out that solutions do not violatetwo important constraints: (a) the mass loss rate by the windmust be less than the accretion rate, (b) the slow accretionlimit is satisfied. We restrict our study to positive values forthe exponent s , because for s < M w ∝ x s +1 and x ≤ s . Our parameterized approach is nevertheless useful, because it illustrates the possible effects of winds onthe time-dependent structure of a self-gravitating disc.Figure 1 shows the change in the profiles of the phys-ical variables with the mass loss power-law index s . Eachcurve is labeled by corresponding index s . Also, we adopt α ′ = 0 .
1, ˙ m = 0 .
2, Γ = 0 . l = 1 (i.e., rotating wind).The surface density and the rotational velocity for the no-wind solution are represented by Σ and v ,ϕ . We find thatthe structure is represented by an inner region with a densityprofile in proportion to x − / and an outer part with densityprofile proportional to x − , irrespective of the existence ofwind or outflow. The transition radius at x tr ∼ α ′ separatesthe inner and outer parts. However, the surface density de-creases at all parts of the disc because of the wind. In orderto make an easier comparison, the ratio (Σ − Σ ) / Σ ver-sus the similarity variable x is shown in Figure 1 (top, left).The surface density reduction is more significant for smallervalues of the exponent s . As the wind becomes stronger andmore mass is extracted from the disc, reduction to the sur-face density is more significant.Profile of the rotational velocity versus the similarityvariable is also shown in Figure 1 (top, right). Generally,solutions with winds will rotate slower than those withoutwinds. So, the viscous dissipation per unit mass in the flowis expected to be smaller in the presence of a wind. Thereduced rotational velocity of the disc is sensitive to thevariations of the exponent s . Also, profile of the radial ve-locity in Figure 1 (middle, left) shows significant deviationsfrom no-wind solution because of the presence of winds. Theradial velocity is approximately uniform in the outer partsof a disc without winds. But when the wind carries away theangular momentum appropriate to the radius from which itis launched, the remaining gas in the outer parts of the dischas much larger radial velocity in comparison to the no-windsolution. As wind becomes stronger, deviations of the radialvelocity from the no-wind solution is occurring over a largerregion of the disc. Velocity of the wind at the surface of thedisc is shown in Figure 1 (middle, right). We can simplyshow that v + z /c s ∝ x s +1 at the outer part of the disc. Whenthe exponent s increases, velocity of the wind at the surfaceof the disc decreases.Profile of the accretion rate, ˙ M acc , is shown in Figure 1(bottom, left). Accretion rate for the no-wind solution is rep-resented by the dotted lines. We can see that the accretionrate decreases at all parts of the disc due to the existence ofthe wind. However, the accretion rate is not very sensitiveto the variations of the exponent s . Ratio of the mass lossrate due to the wind to the accretion rate, i.e. ˙ M w / ˙ M acc , isrepresented in Figure 1 (bottom, right). In the inner partof the disc, the mass loss by wind is negligible except for astrong wind corresponding to s = 0 .
1. We adopted the inputparameters so that the ratio is less than one at all radii ofthe disc. But most of the mass loss by wind is occurring atlarge radii, i.e. outer part of the disc. For a stronger wind,a larger fraction of the mass carries away by the wind.We explore possible effects of extraction of angular mo-mentum in Figure 2 by changing the parameter l . In thisfigure, we assume α ′ = 0 .
1, ˙ m = 0 . l = 0 .
0, 0 . .
0, 1 . = 0 . s = 0 .
7. Obviously, angular mo-mentum is not extracted by the wind when we have l = 0. c (cid:13) , 000–000 M. Shadmehri -0.32-0.30-0.28 -0.16-0.15 , (cid:77) -0.38-0.36-0.34 ( (cid:54) - (cid:54) ) / (cid:54) -0.18-0.17 ( v (cid:77) - v , (cid:77) ) / v -4 -3 -2 -1 0-0.40 log (x) -4 -3 -2 -1 0-0.19 log (x) | v r | / c s v z + / c s -4 -3 -2 -1 00.00.1 log (x) -4 -3 -2 -1 00.000.020.04 log (x) . M a cc ( G - c s ) .. M W / M a cc -4 -3 -2 -1 00.00 log (x) -4 -3 -2 -1 00.00 log (x) Figure 1.
The profiles of the physical variables for α ′ = 0 .
1, ˙ m = 0 . s = 0 .
1, 0 .
3, 0 .
5, 0 . = 0 . l = 1 (i.e,rotating wind). Surface density and the rotational velocity for no-wind solution are represented by Σ and v ,ϕ . Each curve is labeledby corresponding s . No-wind solution is shown by dotted curves. c (cid:13) , 000–000 he influence of winds on the time-dependent behaviour of self-gravitating accretion discs , (cid:77) -0.4-0.20.0 ( (cid:54) - (cid:54) ) / (cid:54) -0.3-0.2-0.10.0 ( v (cid:77) - v , (cid:77) ) / v -4 -3 -2 -1 0-0.6 log (x) -4 -3 -2 -1 0 log (x) | v r | / c s v z + / c s -4 -3 -2 -1 00.00.1 log (x) -4 -3 -2 -1 00.000.05 log (x) . M a cc ( G - c s ) .. M W / M a cc -4 -3 -2 -1 00.00 log (x) -4 -3 -2 -1 00.0 log (x) Figure 2.
The profiles of the physical variables for α ′ = 0 .
1, ˙ m = 0 . l = 0 .
0, 0 .
5, 1 .
0, 1 . = 0 . s = 0 .
7. Surfacedensity and the rotational velocity for no-wind solution are represented by Σ and v ,ϕ . Each curve is labeled by corresponding l . No-windsolution is shown by dotted curves.c (cid:13) , 000–000 M. Shadmehri -0.12-0.08-0.04 -0.06-0.03 , (cid:77) -0.32-0.28-0.24-0.20-0.16 ( (cid:54) - (cid:54) ) / (cid:54) -0.15-0.12-0.09 ( v (cid:77) - v , (cid:77) ) / v -4 -3 -2 -1 0 log (x) -4 -3 -2 -1 0-0.18 log (x) | v r | / c s v z + / c s -4 -3 -2 -1 00.00.1 log (x) -4 -3 -2 -1 00.000.04 log (x) . M a cc ( G - c s ) .. M W / M a cc -4 -3 -2 -1 00.00 log (x) -4 -3 -2 -1 00.00 log (x) Figure 3.
The profiles of the physical variables for α ′ = 0 .
1, ˙ m = 0 . = 0 .
1, 0 .
05, 0 .
01 with l = 1 . s = 0 .
7. Surface densityand the rotational velocity for no-wind solution are represented by Σ and v ,ϕ . Each curve is labeled by corresponding Γ . No-windsolution is shown by dotted curves. c (cid:13) , 000–000 he influence of winds on the time-dependent behaviour of self-gravitating accretion discs Actually, this case corresponds to a non-rotating wind andthe disc losses only mass because of the wind. However, wefound that for l < / v ≫ l = 0 and 0 . v +z is alsosensitive to the amount of the extracted angular momentum(middle, right). Profile of the accretion rate (bottom, left)shows that the accretion rate decreases due to the existenceof a rotating wind. However, as more angular momentum iscarried away by the wind (i.e. larger l ), not only the surfacedensity and the rotational velocity are reduced at all regionsof the disc, but the radial velocity is increases significantlyat the outer part of the disc.Another input parameter of our model is Γ that itspossible effects are explored in Figure 3. We assume that α ′ = 0 .
1, ˙ m = 0 . = 0 .
1, 0 .
05, 0 .
01 with l = 1 . s = 0 .
7. Again, the surface density and the rotationaland radial velocities are significantly decreasing with Γ .The velocity of the wind at the surface of the disc is highlyaffected by the parameter Γ (middle, right). Thus, the ac-cretion rate and the mass loss by the wind are decreasedwith the parameter Γ .The effect of the disc self-gravity in this paper is limitedto provide the radial gravitational field to keep the disc incentrifugal equilibrium. On the other hand, it is well knownthat self-gravitating discs can be unstable if they are toocold. We can study gravitational stability of the solutionsusing the Toomre criteria (Toomre 1964), Q = c s κπG Σ , (21)where κ = p v ϕ /r ) d ( rv ϕ ) /dr is the epicyclic frequency.We calculate Toomre parameter Q using our solutions andthe disc is gravitationally stable if Q >
1. We can rewriteToomre parameter in terms of the similarity quantities as Q = 2 √ xσ r jx djdx . (22)Figure 4 shows profile of Toomre parameter for the so-lutions which are presented in Figures 1, 2 and 3. Here, weassumed that the nondimensional similarity accretion rateat the outer boundary is 0 .
2, i.e. ˙ m = 0 .
2. To make aneasier comparison, Toomre parameter for a case withoutwind/outflow is represented in Figure 4. This plot showsthat Toomre parameter increases due to the existence of no-wind solution Q -4 -3 -2 -1 001 log (x)log (x) Figure 4.
The profile of Toomre parameter versus similarity vari-able for the solutions which are represented in Figures 1, 2 and3. It is assumed that ˙ m = 0 . Red curves are corresponding tothe solutions for α ′ = 0 . s = 0 .
1, 0 .
3, 0 .
5, 0 . = 0 . l = 1. Toomre parameter for the solutions with α ′ = 0 . l = 0 .
0, 0 .
5, 1 . = 0 . s = 0 . green curves . Also, blue curves are showing Toomre parameterwhen α ′ = 0 . = 0 .
01, 0 .
05 with l = 0 . s = 0 . winds or outflows, except for the cases with l = 0 and 0 . m ), the disc becomes more massive and the sizeof the gravitationally unstable inner region increases due tothe existence of wind/outflow. Our similarity solution show that the transition radius in-creases linearly with time, i.e. r tr ≈ α ′ c s t . Assuming thatthe radius r tr is very close to the central part of the disc ini-tially, there is not ”no-wind” region initially and wind existsat all radii of the disc. At early times of evolution, wind doesnot modify the surface density significantly. As gas accretestoward the central parts, the surface density profile changesfrom inside. However, the accretion rate is reduced initiallybecause of the wind, in comparison to the no-wind solution.Some fraction of the accreted mass and the angular mo-mentum can be carried away to infinity by the wind, whileleaving the remaining flow with smaller density at all partsand enhanced radial velocity at the outer part of the disc.We can apply our similarity solutions to a self-gravitating disc just before forming a central star, namely, inthe runaway collapse phase. This phase of evolution wouldcorrespond to the very early (class 0) stage. A molecu-lar cloud core can be approximated by an isothermal gas c (cid:13) , 000–000 M. Shadmehri
Table 1.
Integrals I and J for the solutions with different input parameters. In the absence of winds/outflows, wehave I = 1 .
28 and J = 0.(Γ , l ) = (0 . ,
1) (Γ , s ) = (0 . , .
7) ( l, s ) = (1 . , . s I J l I J Γ I J . .
83 7 .
19 0 . .
84 9 .
81 0 .
01 0 .
88 4 . . .
87 5 .
08 1 . .
88 4 .
49 0 .
05 1 .
04 2 . . .
88 4 .
49 1 . .
58 2 .
86 0 . .
22 0 . with sound speed c s ∼ . − (e.g., Hayashi & Nakano1965). Typical age is considered to be 5 × yr. The radius r = 0 .
01 pc at t = 5 × yr corresponds to x = 1 in oursimilarity variable. Then, the radius of the disc r d becomes r d = 0 . x c s . − )( t × yr ) . (23)Now, we can calculate the mass of the disc as follows M disc = Z r d π Σ rdr, (24)or M disc = (9 . × − M ⊙ )( t × yr ) I , (25)where I = R xσdx . Having our similarity solutions, we cansimply calculate this integral for different set of input pa-rameters. When there is not a wind or outflow, the integralbecomes I = 1 .
28. But existence of a wind or outflow de-creases the integral by a factor up to 2 depending on theinput parameters (see Table 1). In other words, the mass ofa self-gravitating disc with wind is reduced comparing to asimilar disc but without wind. So, reduction to the mass ofa disc with wind is mainly due to the decreased surface den-sity at all regions of the disc. This implies that the centralmass object is forming up to two times slower than a sys-tem without wind. Of course, this factor may increase if weconsider cases, in which more angular momentum is carriedaway by wind (i.e., larger l ). We can determine how long isneeded to increase the mass of the disc by one solar mass.According to equation (25), we can write τ ≃ . × I yr . (26)For a disc without wind, the mass of the disc increases byone solar mass within approximately τ ≃ . × yr. Butthe integral I decreases by a factor up to 2 according toTable 1. This allows the formation of a central core withone solar mass in 2 times slower than a similar system with-out wind, i.e. τ ≃ × yr. This result is consistent withnumerical simulations of early stages of massive discs withwinds/outflows (e.g., Banerjee & Pudritz 2007). Numericalstudy of Banerjee & Pudritz (2007) is concentrated towardsanalyzing massive star formation and inseparable links be-tween gravitational collapse and early wind-driven outflows.They showed that the disc, in the early stages of formation ofa high mass star, is more massive than the protostar that isforming within it (see also Banerjee & Pudritz 2006). Domi-nance of mass of the disc is kept within 7 × yr according tothe simulations of Banerjee & Pudritz (2006). In our paper,we neglected mass of the central protostar at early stages evolution which is a good approximation at early stages offormation. Also, the other analytical studies show that thecentral core grows to one solar mass in 1 . × yr if othereffects are ignored (e.g., Tsuribe 1999)Although we presented the ratio of the total mass lossrate by wind to the accretion rate at each radius of thedisc in Figures 1, 2 and 3, it is desirable to calculate howmuch mass is lost by wind during the early stages. Actually,observations of different systems show that the ratio of massloss rate by wind to the accretion rate is around 0 . M w = (1 . × − M ⊙ / yr) J , (27)where J = R xσ Γ dx . Obviously, exact value of this integraldepends on the profiles of the physical quantities of the disc,i.e. σ and Γ. Table 1 shows this integral for the solutionswith different sets of the input parameters. We can simplyshow that during formation of a core with one solar mass theamount of mass loss by the wind is J / I times the accretedmass. This ratio varies approximately from 0 .
001 to 0 . x tends to infinity (Mineshige & Umemura 1997). But werestricted our solutions within x ≤
1, and so, the total massof the disc is finite. Once the outer part of the disc is depletedwith gas and a central mass is formed, we can not apply oursimilarity solution with wind. However, the effects of wind onthe early evolution of self-gravitating discs are remarkablyworth to be considered, in particular reduced surface densityand the accretion rate at all parts of the disc.
ACKNOWLEDGMENTS
I thank an anonymous referee for his/her constructive com-ments.
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