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Astron. Astrophys. 361, 781-787 (2000)

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3. Discussion and summary

Thus, we have obtained the following corrections of the order of [FORMULA] due to the pseudo-Newtonian potential over the self-similar solution of Narayan & Yi 1994;

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where, all the coefficients with the suffix [FORMULA] can be determined from Eqs. (15-18) and they are exactly the same as self-similar solutions of Narayan & Yi (1994), while the perturbation coefficients [FORMULA]; [FORMULA] or [FORMULA] can be determined from Eqs. (34-37).

Using Eqs. (34-37) we can study the magnitude of the perturbed quantities as functions of [FORMULA] and [FORMULA]. In Fig. 1-Fig. 2 we plot ratios of the coefficients of perturbation to the coefficients of self-similar solution: [FORMULA], [FORMULA], [FORMULA] and [FORMULA] as functions of [FORMULA].

[FIGURE] Fig. 1. Ratios of magnitude of the coefficients of perturbed quantities to the coefficients of self-similar solution are plotted, for [FORMULA]1.5, as functions of [FORMULA]. The plot indicates all the ratios: velocity [FORMULA], sound speed [FORMULA], angular frequency [FORMULA] and density [FORMULA] are well behaved functions of [FORMULA].

[FIGURE] Fig. 2. Caption is same as Fig. 1 but the case is shown for [FORMULA]=1.55. The plot indicates a singularity in the ratio around [FORMULA] 0.35. This indicates a complete breakdown of the self-similarity over a very large distances from the central compact object.

Fig. 1 shows the behaviour of purturbed parameters for the case [FORMULA]=1.5. Four curves of all the perturbed quantities mentioned above are shown for the entire range of the viscosity parameter [FORMULA] i.e. [FORMULA]1. Magnitude of the coefficient of perturbation in sound velocity remain less than unity over the complete range of [FORMULA]. This in turn implies that [FORMULA] remains finite over the entire range [FORMULA]. Thus self-similarity is an excellent approximation for sound speed in ADAF. This feature had been first observed by NKH. Magnitude of the velocity perturbation remains greater than unity for all values of the viscosity parameter. However, [FORMULA] decreases with increasing values of [FORMULA]. From the above discussed validity criterion of the perturbation, one can see that for lower values of [FORMULA], the perturbation grows faster as compared to case where [FORMULA]1. In other words it seems that self-similarity is a good approximation for velocity perturbation in the proximity of the central object only for the higher values of [FORMULA]. Similar trends can be seen in [FORMULA] and [FORMULA] i.e. the perturbations are strongest in [FORMULA] 0 limit. In this regime the strongest departure from self-similarity can occur when [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA].

All the above features reported for [FORMULA] case in Fig. 1 are broadly in agreement with the global solutions of ADAFs obtained by NKH. In other words, our perturbative analysis confirms their results: (a) self-similar behavior of the sound velocity, in the presence of pseudo-Newtonian potential, remains more accurate compared to those for the radial velocity, angular velocity and density. (b) self-similarity violations are stronger in the low [FORMULA] regime. However, it must be noted that our analysis does not treat the inner boundary incorporating [FORMULA], and therefore one cannot confirm the prediction of the sonic point.

For [FORMULA] all the above perturbed quantities can be plotted as functions of [FORMULA]. However, for some values of [FORMULA] we find that the magnitude of the perturbations grows indefinitely, although both on higher and on lower side of this critical [FORMULA] value, the perturbations have controlled behavior. Top left panel in Fig. 2 depicts logarithm of the ratio of the coefficient of sound perturbation to that of the self-similar sound velocity as a function of [FORMULA]. It shows that the perturbation can grow indefinitely around [FORMULA]. All the other perturbed quantities have the similar behavior around the same value of [FORMULA], as shown in the figure. Also there is an over all increase in values of all perturbed quantities in the regions [FORMULA] and [FORMULA] as compared to Fig. 1. This may be indicative of high sensitivity of the solution depeding on [FORMULA].

In view of the above, a natural question is, whether the singular behavior is only an artifact of the perturbative approach or it is genuine? In what follows we try to answer this question.

Presence of the gravity term in Eq. (6) can introduce inhomogeneity, which can dictate the form of any power solution. In the present case, inhomogeneous term would read [FORMULA]. It is the first term on the right hand side which gives the self-similar solution as zeroth order in the expansion. In general the solution for the nth order can have the following form [FORMULA]: [FORMULA] go like [FORMULA] and [FORMULA] go like [FORMULA]. Since [FORMULA] we can ascertain that all the higher orders in perturbations are smaller than their counter parts in the lower order. Therefore, the singularity in the parameter space should be considered to be genuine. Moreover, the singularity should manifest itself in the global numerical solutions also. It appears that the system does indicate a strange behavior for higher [FORMULA] at [FORMULA]. In fact it has been noticed by others too, that the numerical global solution code seem to develop difficulties in integration, for the value [FORMULA] and higher [FORMULA] (Narayan 1999, Private communication).

This in turn would imply that the breakdown of the perturbation analysis and the problem faced in numerical integration could be indicative of a new branch of solution. Whether the new branch of solution is related with the shock singularity (Chakrabarti 1996, Chakrabarti & Titarchuk 1995) can not be answered within the framework of the perturbative approach. However, the breakdown of the self-similarity, in the present work, occurs in a rather small region of the parameter space unlike the referred work of Chakrabarti & Titarchuk (1995).

Finally we compare our perturbative solution with the global solution of NKH and the self-similar solution. In Fig. 3 we plot radial velocity v, sound speed [FORMULA] and angular frequency [FORMULA] as function of [FORMULA] for [FORMULA]=0.3 and [FORMULA]=1.5. In general our perturbative solution(SP) provides a better approximation to the exact global solution (NKH) as compared to the self-similar solution (SS) especially for the region [FORMULA] small. It also reconfirms that the self-similar profile of sound velocity is a good approximation even at the smaller values of [FORMULA] for the current values of [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Curves with solid lines show exact global numerical solution of NKH (see text), while the curves with dotted lines show our approximate solution (SP). The dashed curve represent self-similar (SS) solution of Narayan & Yi 1995a. All the results are obtained for [FORMULA]=0.3 and [FORMULA]=1.5. Top: variation of radial velocity with the radius. Middle: variation of sound velocity. Bottom: shows variation of angular frequency with radius. The plot shows that the pertubative solution provides a better approximation to the global solution at small radii.

In summary, we have studied the self-similar solution in a pseudo-Newtonian potential in a perturbative fashion. We have also shown that our approximate solution provide a better approximation to the exact global but numerical solution in proximity of the central object. We find that our approach gives a convenient representation of the parameter space of the self-similar solutions. It is also shown that for [FORMULA], the self-similarity would be broken in the region far away from the central compact object.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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