- Email: [email protected]
The Effect of Radiative Efficiency on the Growth of the Black Hole Mass
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 31 (2007) 109–116
The Effect of Radiative Efficiency on the Growth of the Black Hole Mass† HU Chen3,4 YANG Fang1,2,4 WANG Jian-min1,2 ZHANG En-peng1,2,4 WU Mei1,2 1 2
Key Laboratory for Particle Astrophysics, Chinese Academy of Sciences, Beijing 100049 Center for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 3 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012 4 Graduate School of Chinese Academy of Sciences, Beijing 100039
Abstract The effect of variation of radiative efficiency on the growth of the black hole mass is discussed. In the process of accretion, the black hole’s angular momentum varies, resulting in a variation of the radiative efficiency, and the growth of the black hole mass is thereby affected. For the exponential growth model of back holes and taking into account the effect of a varying radiative efficiency, the equation for the growth of the black hole mass is solved numerically. Compared to the model that assumes a constant radiative efficiency, the result indicates that variation of radiative efficiency has a marked, retarding effect on the growth of the black hole mass. This model can explain quantitatively some recent observational results. Key words: black hole physics—accretion—accretion disks—galaxies: bulges— galaxies: evolution
1. INTRODUCTION It is generally believed that black holes exist in the nuclei of most galaxies, and that the black hole mass is closely related to the host galaxy, as shown by the relations of the black hole mass with the absolute luminosity of the bulge[1] and with the velocity dispersion of the stars †
Supported by National Natural Science Foundation Received 2005–09–05; revised version 2006–04–14 A translation of Acta Astron. Sin. Vol. 47, No. 4, pp. 355–361, 2006
c 2007 Elsevier B . V. All rights reserved. 0275-1062/07/$-see front matter doi:10.1016/j.chinastron.2007.04.003
110
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
in the galaxy[2,3]. It led to the conclusion that the black holes probably co-evolve with their host galaxies[4]. With the luminosity function of quasars and assuming that the black hole mass comes entirely from accretion, Soltan[5] ingeniously calculated the mass density of black holes in the local universe. Later, by assuming that the back hole mass in active galactic nuclei (AGNs) comes from accretion, Salucci et al.[6] derived the mass density of black holes that have ever been active, and found the same mass density as in the nearby galaxies. From the data of the 2dF Redshift Survey and Sloan Digital Sky Survey (SDSS), Yu[7] found that by using Soltan’s assumption and Boyle’s luminosity function, the absorption efficiency of black holes can be derived to be η =0.1. All these authors used the relation between the black hole mass and the galaxy velocity dispersion. However, in these studies a constant radiative efficiency in the course of accretion was assumed. Clearly, it is not a very reasonable assumption[8] . From the X-ray background, η >0.15 is obtained[9] , and from the mass distribution function of black holes, η 0.3-0.4 can be derived[10] . Compared with these results, η =0.1 is obviously too small. When we study the growth of the black hole mass, models of two extreme kinds are generally considered: (1) The linear growth model: it assumes the accretion rate M˙ to be a constant. Then the equation for the growth of the black hole mass is dM /d t = (1 − η)M˙ , η being the radiative efficiency, and the solution of the equation is M = M0 + (1 − η)M˙ t, M0 being the initial black hole mass; (2) The exponential growth model: this model assumes m ˙ = M˙ / M˙ Edd =constant, and this leads to the following equation for the growth of the black hole mass: dM 1−η = (1 − η)m ˙ M˙ Edd = k Mm ˙ . dt η
(1)
Here k = 4πGmp /(cσ) = 7.04×10−17s−1 , G the gravitational constant, mp the proton mass, σ the Thomson scattering cross-section, and c the light velocity. The solution of Eq.(1) is: 1−η M = M0 exp k mt ˙ . (2) η In the above two models, η is always taken as a constant. However, η is a function of the dimensionless angular momentum a, and as the black hole mass grows, the magnitude of a will vary. In the course of accretion, the radiative efficiency η will vary, and so will affect the solution of the equation of the black hole mass growth. For orientation, we can make first the following estimation: when a = 0, η =0.057, (1 − η) /η =16.5; and when a =1, η =0.42, (1−η) /η =1.38. We thus find that varying η has a relatively small effect in the linear growth model, but probably exerts a very large effect in the exponential growth model. Recently, Peng et al.[11] found that for quasars with z > 2, the mass ratio M /Mbulge between the black hole and the bulge is 3-6 times the value of the quasars in the local universe. This implies that the growth of the black hole mass and the galaxy’s evolution are not strictly in step. This may be caused in two ways: (1) A feedback effect has suppressed the growth of the black hole mass[12] ; (2) An enhancement of the galaxy evolution, resulting in a decrease of the M /Mbulge ratio. At present, the available observational data are not yet sufficient to make a decision between the two possibilities. This paper will discuss mainly the effect of a varying η on the evolution of M /Mbulge in the exponential growth mode.
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
111
2. MODEL OF THE GROWTH OF THE BLACK HOLE MASS TAKING INTO CONSIDERATION VARIATION OF RADIATIVE EFFICIENCY This paper adopts the geometrized units: c = G =1, the dimensionless angular momentum a = J /M 2 (with values of a in the range from -1 to 1), and the dimensionless radius r = R /M . The discussion is made for the thin-disk accretion (Rin = Rms ) and the case of M˙ = M˙ Edd. Considering the variation of η, the equation for the growth of the black hole mass can be written as: dM =k dt
M in which, η( M ) expresses η as a function of 0
d(
M ) M0 M, M η( ) M0
1 − η(
M M0 .
(3)
Eq.(3) can be further expressed as:
M M 1 − η( ) ) M0 M0 M =k . M dt M0 η( ) M0
(4)
M In order to solve Eq.(4), we need the expression of η( M ). 0 The radiative efficiency η is a function of rms : Rg 2 =1− 1− . η = 1− 1− 3Rms 3rms
(5)
And rms is in turn a function of a: 1
in which,
rms = 3 + Z2 ∓ [(3 − Z1 )(3 + Z1 + 2Z2 )] 2 ,
(6)
1 1 1 Z1 = 1 + (1 − a2 ) 3 (1 + a) 3 + (1 − a) 3 ,
(7)
1
Z2 = (3a2 + Z12 ) 2 ,
(8)
“−” means direct accretion, and “+” means retrograde accretion. As the accretion proceeds, the dimensionless angular momentum of the black hole will change. Taking no account of the angular momentum carried away by radiation, a will eventually reach 1. The accreted materials on the accretion disk may gradually lose their angular momentum because of viscosity, until they arrive at a minimum stable orbit. Once inside the minimum stable orbit, the material will rapidly fall into the black hole. Thus, the variations of the the black hole mass and angular momentum caused by the accreted materials can be calculated by the energy and angular momentum per unit mass carried by the material on the minimum stable orbit, multiplied by the total mass of the accreted materials. Let Ems and Lms denote, respectively, the energy and angular momentum carried by unit mass of the material on the minimum stable orbit. Then the following relation holds[13] :
112
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
Ems =
1−
2 3rms
12
,
1 2M Lms = √ 1 + 2(3rms − 2) 2 . 3 3 And the variations of the black hole mass M and angular momentum J satisfy:
(9) (10)
dM = Ems dm ,
(11)
dJ = Lms dm ,
(12)
in which dm is the mass of accreted matterial. Combining Eqs.(9-12), we obtain J d 2 1 Lms da M − 2a . = = d ln M d ln M M Ems In addition, a can be expressed as the function of rms :
(13)
1 1 13 rms 4 − (3rms − 2) 2 . 3 By this relation, Bardeen has obtained the solution of Eq.(13):
(14)
a=
rms M 2 = rms0 M02 .
(15)
Inserting Eq.(15) into Eq.(14), the evolutionary of a with the black hole mass can be √ M ≤ rms0 , obtained[14] . When M 0 ⎧ 12 ⎫
1 2 ⎨ ⎬ 2 1 rms0 M0 M0 , (16) a= −2 4 − 3rms0 ⎭ 3 M ⎩ M and when
M M0
≥
√ rms0 , a = 1,
(17)
in which rms0 is the dimensionless minimum stable orbit at t =0, and it can be calculated from the initial angular momentum a. Thus, the functional relation of η with respect to M /M0 is given by Eqs.(5-17). In the following we will further obtain the solution of Eq.(4) numerically.
3. RESULT OF NUMERICAL CALCULATIONS AND DISCUSSION The discussion will be made separately for the two cases of direct and retrograde accretion, neglecting the angular momentum carried away by radiation. 3.1 Direct Accretion In the case of direct accretion, the rotational direction of the black hole is the same as that of the accretion disk, the accretion speeds up the rotation of the black hole, the value of a
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
113
increases monotonically until reaching the steady value 1. We have selected two sets of initial conditions in solving the equation: (1) for M = M0 , a =0; (2) for M = M0 , a =0.5. From the formulae (16) and (17), we find that as √ the accretion proceeds, a keeps on increasing, √ until it reaches 1 when M /M0 = rms0 = 6, and this value is held thereafter. The results of the numerical calculations under the two sets of initial conditions are shown separately in Fig. 1 and Fig.2. The solid lines without dots are the results with tSalp =0.45 Gyr.
Fig. 1 Growth of the black hole mass under direct accretion (initial angular momentum a =0). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
Fig. 2 Growth of the black hole mass under direct accretion (initial angular momentum a =0.5). The constant radiative efficiencies for comparisons are shown at the right corner.
In Fig.1, ingly, and the t / tSalp ≥0.21, and the black
when t / tSalp ≤0.21, a increases uninterruptedly, and η increases accordgrowth of the black hole mass deviates from the exponential law. When a reaches 1, from then on the radiative efficiency is a constant, η =0.42, hole mass grows according to an exponential law. For comparisons, Fig.1
114
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
shows also the growth of the black hole mass when the radiative efficiency takes a few different constant values (0.057, 0.42, and 0.1). 0.057 and 0.42 are the radiative efficiencies when a =0 and a =1, respectively, and 0.1 is the radiative efficiency generally adopted in the calculations. At t / tSlap =0.21, i.e., when a reaches 1, our calculations give the result √ M /M0 = 6 ≈2.5, and the values of M /M0 calculated with η =0.057 and η =0.1 are 31.1 and 6.5, respectively. In the comparison, when the variation of radiative efficiency is taken into account, the growth rate of black hole mass becomes obviously slower. The explanation is as follows: from Eq.(1) we find that for the same value of M , the greater the value of η, the slower the growth of the black hole mass. The formulae (5-8) show η to be an increasing function of a (when 0< a <1), and the formula (16) shows a to be an increasing function of M /M0 , so η is an increasing function of M /M0 . As the black hole mass increases, a increases, and η increases as well, this slows down the growth of the black hole mass. Fig. 2 shows a similar process as Fig.1, apart from some numeral differences. When t / tSalp ≈0.2, a reaches 1, then the growth is in accordance with an exponential law. And from Figs. 1 and 2 we can find that the growth of the black hole mass is somewhat slower for the initial condition a =0.5, than for a =0. 3.2 Retrograde Accretion In the case of retrograde accretion, the black hole and the accretion disk rotate in opposite directions (a is negative). As the accretion proceeds, the angular momentum brought in by the accreted matter slows down the rotation of the black hole. After a reaches 0, the continued accretion will make the black hole rotate in the same direction as the accretion disk, at an ever increasing speed. After a reaches 1, it remains constant. From formulae (5),(6) and (16), we can see that in the process of accretion, as a increases continuously from its initial value to 1, η also increases continuously: when a =-1, η =0.038; when a =0, η =0.057; and when a =1, η =0.42. We have obtained solutions for the two sets of initial conditions with a = −1 and a = −0.5, and the results are shown in Fig.3 and Fig.4, respectively, in which the solid lines without dots are the results of numerical calculations. For comparisons, the growth curves for three constant η values (0.057, 0.42 and 0.1) are also given in the figures . The calculation under the initial condition “M = M0 , a = −1” indicates that when t / tSalp =0.22, a increases to 1, M /M0 = 3.0. And for the initial condition “M = M0 , a = −0.5”, when t / tSalp = 0.21, a increases to 1, M /M0 = 2.75. From Fig.3 and Fig.4 we can see that because of the variation of η, the growth of the black hole mass deviates from the exponential law. And that in the case of retrograde accretion, the variation of η with time also slows down the growth of the black hole mass relative to the case of η = 0.1. 3.3 Discussion From Figs.1-4 we can see clearly that when the variation of radiative efficiency is taken into consideration, the growth of the black hole is delayed. The observations demonstrate that the ratio M/ Mbulge is 3—6 times greater for quasars of z >2, than for quasars in the local universe. From Figs.1—4 we can make a quantitative estimation. By comparing the situation when the variation of radiative efficiency has been taking into consideration (the solid lines in the figures) with the situation of the constant radiative efficiency η =0.1 (the dotted lines in the figures), we see that when t / tSalp ≥0.25, the growth of the black hole
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
115
mass has decreased by a factor of 3. This theoretical result is in general agreement with the observations made by Peng et al.[11] . This means that the deceleration of the mass growth caused by accretion is not a negligible physical process, and this indicates that black holes probably do not evolve in step with their host galaxies.
Fig. 3 Growth of the black hole mass under retrograde accretion (initial angular momentum a =-1). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
Fig. 4 Growth of the black hole mass under retrograde accretion (initial angular momentum a =-0.5). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
4. CONCLUSIONS Our calculated results demonstrate that in the process of the accretion-caused growth of black holes, variation of radiative efficiency has a rather marked effect on the growth of the black hole mass. In the process of accretion, a varying radiative efficiency delays the growth of the black hole, resulting in a smaller M / Mbulge ratio. We find that theoretical estimation
116
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
is generally consistent with the observational results. This means that black holes probably do not co-evolve with their host galaxies. Further studies are awaiting the accumulation of more observational data. ACKNOWLEDGEMENT work.
We thank Dr. Jin Jing for participating in a part of this
References 1
Magorrian J., Tremaine S., Richstone D., et al., AJ, 1998, 115, 2285
2
Gebhardt K., Bender R., Bower G., et al., ApJ, 2000, 539, L13
3
Ferrarese L., Merritt D., ApJ, 2000, 539, L9
4
Ho L.C., ed. Carneigie Observatories Astrophysics Series Vol.1: Coevolution of Black Holes and
5
Soltan A., MNRAS, 1982, 200, 115
6
Salucci P., Szuszkiewicz E., Monaco P., et al., MNRAS, 1999, 307, 637
Galaxies, Cambridge: Cambridge Univ. Press, 2004, p.1
7
Yu Q., Tremaine S., MNRAS, 2002, 335, 965
8
Shapiro S.L., ApJ, 2005, 620, 59
9
Elvis M., Risaliti G., Zamorani G., ApJ, 2002, 565, L75
10
Wang J.M., Chen Y.M., Ho L.C., et al., ApJ, 2006, 642, L111
11
Peng C.Y., Impey C.D., Ho L.C., et al., ApJ, 2006, 640, 114
12
Wang J.M., Chen Y.M., Hu C., ApJ, 2006, 637, L85
13
Bardeen J.M., Nature, 1970, 226, 64
14
Moderski R., Sikora M., A&AS, 1996, 120, 591
ELSEVIER
Chinese Astronomy and Astrophysics 31 (2007) 109–116
The Effect of Radiative Efficiency on the Growth of the Black Hole Mass† HU Chen3,4 YANG Fang1,2,4 WANG Jian-min1,2 ZHANG En-peng1,2,4 WU Mei1,2 1 2
Key Laboratory for Particle Astrophysics, Chinese Academy of Sciences, Beijing 100049 Center for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 3 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012 4 Graduate School of Chinese Academy of Sciences, Beijing 100039
Abstract The effect of variation of radiative efficiency on the growth of the black hole mass is discussed. In the process of accretion, the black hole’s angular momentum varies, resulting in a variation of the radiative efficiency, and the growth of the black hole mass is thereby affected. For the exponential growth model of back holes and taking into account the effect of a varying radiative efficiency, the equation for the growth of the black hole mass is solved numerically. Compared to the model that assumes a constant radiative efficiency, the result indicates that variation of radiative efficiency has a marked, retarding effect on the growth of the black hole mass. This model can explain quantitatively some recent observational results. Key words: black hole physics—accretion—accretion disks—galaxies: bulges— galaxies: evolution
1. INTRODUCTION It is generally believed that black holes exist in the nuclei of most galaxies, and that the black hole mass is closely related to the host galaxy, as shown by the relations of the black hole mass with the absolute luminosity of the bulge[1] and with the velocity dispersion of the stars †
Supported by National Natural Science Foundation Received 2005–09–05; revised version 2006–04–14 A translation of Acta Astron. Sin. Vol. 47, No. 4, pp. 355–361, 2006
c 2007 Elsevier B . V. All rights reserved. 0275-1062/07/$-see front matter doi:10.1016/j.chinastron.2007.04.003
110
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
in the galaxy[2,3]. It led to the conclusion that the black holes probably co-evolve with their host galaxies[4]. With the luminosity function of quasars and assuming that the black hole mass comes entirely from accretion, Soltan[5] ingeniously calculated the mass density of black holes in the local universe. Later, by assuming that the back hole mass in active galactic nuclei (AGNs) comes from accretion, Salucci et al.[6] derived the mass density of black holes that have ever been active, and found the same mass density as in the nearby galaxies. From the data of the 2dF Redshift Survey and Sloan Digital Sky Survey (SDSS), Yu[7] found that by using Soltan’s assumption and Boyle’s luminosity function, the absorption efficiency of black holes can be derived to be η =0.1. All these authors used the relation between the black hole mass and the galaxy velocity dispersion. However, in these studies a constant radiative efficiency in the course of accretion was assumed. Clearly, it is not a very reasonable assumption[8] . From the X-ray background, η >0.15 is obtained[9] , and from the mass distribution function of black holes, η 0.3-0.4 can be derived[10] . Compared with these results, η =0.1 is obviously too small. When we study the growth of the black hole mass, models of two extreme kinds are generally considered: (1) The linear growth model: it assumes the accretion rate M˙ to be a constant. Then the equation for the growth of the black hole mass is dM /d t = (1 − η)M˙ , η being the radiative efficiency, and the solution of the equation is M = M0 + (1 − η)M˙ t, M0 being the initial black hole mass; (2) The exponential growth model: this model assumes m ˙ = M˙ / M˙ Edd =constant, and this leads to the following equation for the growth of the black hole mass: dM 1−η = (1 − η)m ˙ M˙ Edd = k Mm ˙ . dt η
(1)
Here k = 4πGmp /(cσ) = 7.04×10−17s−1 , G the gravitational constant, mp the proton mass, σ the Thomson scattering cross-section, and c the light velocity. The solution of Eq.(1) is: 1−η M = M0 exp k mt ˙ . (2) η In the above two models, η is always taken as a constant. However, η is a function of the dimensionless angular momentum a, and as the black hole mass grows, the magnitude of a will vary. In the course of accretion, the radiative efficiency η will vary, and so will affect the solution of the equation of the black hole mass growth. For orientation, we can make first the following estimation: when a = 0, η =0.057, (1 − η) /η =16.5; and when a =1, η =0.42, (1−η) /η =1.38. We thus find that varying η has a relatively small effect in the linear growth model, but probably exerts a very large effect in the exponential growth model. Recently, Peng et al.[11] found that for quasars with z > 2, the mass ratio M /Mbulge between the black hole and the bulge is 3-6 times the value of the quasars in the local universe. This implies that the growth of the black hole mass and the galaxy’s evolution are not strictly in step. This may be caused in two ways: (1) A feedback effect has suppressed the growth of the black hole mass[12] ; (2) An enhancement of the galaxy evolution, resulting in a decrease of the M /Mbulge ratio. At present, the available observational data are not yet sufficient to make a decision between the two possibilities. This paper will discuss mainly the effect of a varying η on the evolution of M /Mbulge in the exponential growth mode.
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
111
2. MODEL OF THE GROWTH OF THE BLACK HOLE MASS TAKING INTO CONSIDERATION VARIATION OF RADIATIVE EFFICIENCY This paper adopts the geometrized units: c = G =1, the dimensionless angular momentum a = J /M 2 (with values of a in the range from -1 to 1), and the dimensionless radius r = R /M . The discussion is made for the thin-disk accretion (Rin = Rms ) and the case of M˙ = M˙ Edd. Considering the variation of η, the equation for the growth of the black hole mass can be written as: dM =k dt
M in which, η( M ) expresses η as a function of 0
d(
M ) M0 M, M η( ) M0
1 − η(
M M0 .
(3)
Eq.(3) can be further expressed as:
M M 1 − η( ) ) M0 M0 M =k . M dt M0 η( ) M0
(4)
M In order to solve Eq.(4), we need the expression of η( M ). 0 The radiative efficiency η is a function of rms : Rg 2 =1− 1− . η = 1− 1− 3Rms 3rms
(5)
And rms is in turn a function of a: 1
in which,
rms = 3 + Z2 ∓ [(3 − Z1 )(3 + Z1 + 2Z2 )] 2 ,
(6)
1 1 1 Z1 = 1 + (1 − a2 ) 3 (1 + a) 3 + (1 − a) 3 ,
(7)
1
Z2 = (3a2 + Z12 ) 2 ,
(8)
“−” means direct accretion, and “+” means retrograde accretion. As the accretion proceeds, the dimensionless angular momentum of the black hole will change. Taking no account of the angular momentum carried away by radiation, a will eventually reach 1. The accreted materials on the accretion disk may gradually lose their angular momentum because of viscosity, until they arrive at a minimum stable orbit. Once inside the minimum stable orbit, the material will rapidly fall into the black hole. Thus, the variations of the the black hole mass and angular momentum caused by the accreted materials can be calculated by the energy and angular momentum per unit mass carried by the material on the minimum stable orbit, multiplied by the total mass of the accreted materials. Let Ems and Lms denote, respectively, the energy and angular momentum carried by unit mass of the material on the minimum stable orbit. Then the following relation holds[13] :
112
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
Ems =
1−
2 3rms
12
,
1 2M Lms = √ 1 + 2(3rms − 2) 2 . 3 3 And the variations of the black hole mass M and angular momentum J satisfy:
(9) (10)
dM = Ems dm ,
(11)
dJ = Lms dm ,
(12)
in which dm is the mass of accreted matterial. Combining Eqs.(9-12), we obtain J d 2 1 Lms da M − 2a . = = d ln M d ln M M Ems In addition, a can be expressed as the function of rms :
(13)
1 1 13 rms 4 − (3rms − 2) 2 . 3 By this relation, Bardeen has obtained the solution of Eq.(13):
(14)
a=
rms M 2 = rms0 M02 .
(15)
Inserting Eq.(15) into Eq.(14), the evolutionary of a with the black hole mass can be √ M ≤ rms0 , obtained[14] . When M 0 ⎧ 12 ⎫
1 2 ⎨ ⎬ 2 1 rms0 M0 M0 , (16) a= −2 4 − 3rms0 ⎭ 3 M ⎩ M and when
M M0
≥
√ rms0 , a = 1,
(17)
in which rms0 is the dimensionless minimum stable orbit at t =0, and it can be calculated from the initial angular momentum a. Thus, the functional relation of η with respect to M /M0 is given by Eqs.(5-17). In the following we will further obtain the solution of Eq.(4) numerically.
3. RESULT OF NUMERICAL CALCULATIONS AND DISCUSSION The discussion will be made separately for the two cases of direct and retrograde accretion, neglecting the angular momentum carried away by radiation. 3.1 Direct Accretion In the case of direct accretion, the rotational direction of the black hole is the same as that of the accretion disk, the accretion speeds up the rotation of the black hole, the value of a
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
113
increases monotonically until reaching the steady value 1. We have selected two sets of initial conditions in solving the equation: (1) for M = M0 , a =0; (2) for M = M0 , a =0.5. From the formulae (16) and (17), we find that as √ the accretion proceeds, a keeps on increasing, √ until it reaches 1 when M /M0 = rms0 = 6, and this value is held thereafter. The results of the numerical calculations under the two sets of initial conditions are shown separately in Fig. 1 and Fig.2. The solid lines without dots are the results with tSalp =0.45 Gyr.
Fig. 1 Growth of the black hole mass under direct accretion (initial angular momentum a =0). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
Fig. 2 Growth of the black hole mass under direct accretion (initial angular momentum a =0.5). The constant radiative efficiencies for comparisons are shown at the right corner.
In Fig.1, ingly, and the t / tSalp ≥0.21, and the black
when t / tSalp ≤0.21, a increases uninterruptedly, and η increases accordgrowth of the black hole mass deviates from the exponential law. When a reaches 1, from then on the radiative efficiency is a constant, η =0.42, hole mass grows according to an exponential law. For comparisons, Fig.1
114
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
shows also the growth of the black hole mass when the radiative efficiency takes a few different constant values (0.057, 0.42, and 0.1). 0.057 and 0.42 are the radiative efficiencies when a =0 and a =1, respectively, and 0.1 is the radiative efficiency generally adopted in the calculations. At t / tSlap =0.21, i.e., when a reaches 1, our calculations give the result √ M /M0 = 6 ≈2.5, and the values of M /M0 calculated with η =0.057 and η =0.1 are 31.1 and 6.5, respectively. In the comparison, when the variation of radiative efficiency is taken into account, the growth rate of black hole mass becomes obviously slower. The explanation is as follows: from Eq.(1) we find that for the same value of M , the greater the value of η, the slower the growth of the black hole mass. The formulae (5-8) show η to be an increasing function of a (when 0< a <1), and the formula (16) shows a to be an increasing function of M /M0 , so η is an increasing function of M /M0 . As the black hole mass increases, a increases, and η increases as well, this slows down the growth of the black hole mass. Fig. 2 shows a similar process as Fig.1, apart from some numeral differences. When t / tSalp ≈0.2, a reaches 1, then the growth is in accordance with an exponential law. And from Figs. 1 and 2 we can find that the growth of the black hole mass is somewhat slower for the initial condition a =0.5, than for a =0. 3.2 Retrograde Accretion In the case of retrograde accretion, the black hole and the accretion disk rotate in opposite directions (a is negative). As the accretion proceeds, the angular momentum brought in by the accreted matter slows down the rotation of the black hole. After a reaches 0, the continued accretion will make the black hole rotate in the same direction as the accretion disk, at an ever increasing speed. After a reaches 1, it remains constant. From formulae (5),(6) and (16), we can see that in the process of accretion, as a increases continuously from its initial value to 1, η also increases continuously: when a =-1, η =0.038; when a =0, η =0.057; and when a =1, η =0.42. We have obtained solutions for the two sets of initial conditions with a = −1 and a = −0.5, and the results are shown in Fig.3 and Fig.4, respectively, in which the solid lines without dots are the results of numerical calculations. For comparisons, the growth curves for three constant η values (0.057, 0.42 and 0.1) are also given in the figures . The calculation under the initial condition “M = M0 , a = −1” indicates that when t / tSalp =0.22, a increases to 1, M /M0 = 3.0. And for the initial condition “M = M0 , a = −0.5”, when t / tSalp = 0.21, a increases to 1, M /M0 = 2.75. From Fig.3 and Fig.4 we can see that because of the variation of η, the growth of the black hole mass deviates from the exponential law. And that in the case of retrograde accretion, the variation of η with time also slows down the growth of the black hole mass relative to the case of η = 0.1. 3.3 Discussion From Figs.1-4 we can see clearly that when the variation of radiative efficiency is taken into consideration, the growth of the black hole is delayed. The observations demonstrate that the ratio M/ Mbulge is 3—6 times greater for quasars of z >2, than for quasars in the local universe. From Figs.1—4 we can make a quantitative estimation. By comparing the situation when the variation of radiative efficiency has been taking into consideration (the solid lines in the figures) with the situation of the constant radiative efficiency η =0.1 (the dotted lines in the figures), we see that when t / tSalp ≥0.25, the growth of the black hole
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
115
mass has decreased by a factor of 3. This theoretical result is in general agreement with the observations made by Peng et al.[11] . This means that the deceleration of the mass growth caused by accretion is not a negligible physical process, and this indicates that black holes probably do not evolve in step with their host galaxies.
Fig. 3 Growth of the black hole mass under retrograde accretion (initial angular momentum a =-1). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
Fig. 4 Growth of the black hole mass under retrograde accretion (initial angular momentum a =-0.5). The constant radiative efficiencies adopted for comparisons are shown at the right corner.
4. CONCLUSIONS Our calculated results demonstrate that in the process of the accretion-caused growth of black holes, variation of radiative efficiency has a rather marked effect on the growth of the black hole mass. In the process of accretion, a varying radiative efficiency delays the growth of the black hole, resulting in a smaller M / Mbulge ratio. We find that theoretical estimation
116
WANG Jian-min et al. / Chinese Astronomy and Astrophysics 31 (2007) 109–116
is generally consistent with the observational results. This means that black holes probably do not co-evolve with their host galaxies. Further studies are awaiting the accumulation of more observational data. ACKNOWLEDGEMENT work.
We thank Dr. Jin Jing for participating in a part of this
References 1
Magorrian J., Tremaine S., Richstone D., et al., AJ, 1998, 115, 2285
2
Gebhardt K., Bender R., Bower G., et al., ApJ, 2000, 539, L13
3
Ferrarese L., Merritt D., ApJ, 2000, 539, L9
4
Ho L.C., ed. Carneigie Observatories Astrophysics Series Vol.1: Coevolution of Black Holes and
5
Soltan A., MNRAS, 1982, 200, 115
6
Salucci P., Szuszkiewicz E., Monaco P., et al., MNRAS, 1999, 307, 637
Galaxies, Cambridge: Cambridge Univ. Press, 2004, p.1
7
Yu Q., Tremaine S., MNRAS, 2002, 335, 965
8
Shapiro S.L., ApJ, 2005, 620, 59
9
Elvis M., Risaliti G., Zamorani G., ApJ, 2002, 565, L75
10
Wang J.M., Chen Y.M., Ho L.C., et al., ApJ, 2006, 642, L111
11
Peng C.Y., Impey C.D., Ho L.C., et al., ApJ, 2006, 640, 114
12
Wang J.M., Chen Y.M., Hu C., ApJ, 2006, 637, L85
13
Bardeen J.M., Nature, 1970, 226, 64
14
Moderski R., Sikora M., A&AS, 1996, 120, 591