Properties of the accretion ring in an X-ray binary, and accretion and excretion two-layer flows from it

Turbulent motions are likely to be always excited in the torus by the impact of the stream on it and to efficiently transfer the angular momentum from the inner half side facing to the compact object to the outer half side, causing the spreading of the torus matter.

Let typical scale and velocity of the turbulences be $\lambda$ and $\tilde{v}$, then the angular momentum transfer rate through the turbulent mixing, $\dot{J}_{\rm mix}$, per unit arc length of the boundary between the inner and outer half sides is approximately given as

As a result of the angular momentum transfer from the inner half side to the outer half side of a turbulent element, the specific angular momentum distribution tends to change from the initial one as $j_{0}=\sqrt{r_{0}GM}$ to that for the mass spreading as $j_{\rm s}=\sqrt{rGM}$. Then, when $j$ becomes slightly flatter than $j_{\rm s}$, the mass spreading begins; the matter in the inner half side starts to flow inward to the compact object, while that in the outer half side goes outward. If we express the angular momentum flow rates from the inner side and the outer side as $\dot{J}_{\rm in}$ and $\dot{J}_{\rm out}$ respectively, they can be written as

and

where $u$ is the absolute value of the flowing velocity which is assumed to be the same for the both sides. Since the difference of the angular momentum flow rate, $\Delta\dot{J}$, between the outer side and the inner side defined as

should be the result of the angular momentum transfer through the turbulent mixing, we have an equation as

Inserting equations (24) and (27) into equation (28), and with the help of a relation as $\Omega=j_{0}/r^{2}$, we get

where we have introduced the concept of the $\alpha$ model for the turbulent viscosity as $\tilde{v}\lambda\simeq\alpha c_{\rm s,\;e}x_{\rm s,\;e}$ in terms of the sound velocity $c_{\rm s,\;e}$ and the scale thickness $x_{\rm s,\;e}$ of the envelope.

The spreading flow rate $\dot{M}_{\rm s,\;e}$ to both the inner and outer sides can be approximated as

Then, the time scale for the mass spreading, $t_{\rm s,\;e}$, can be estimated from equations (21) and (30) as

where $c_{\rm s,\;e}$ is expressed as

The typical cooling time scale of the envelope, $t_{\rm c,\;e}$, can be estimated as

where $\varepsilon$ ($\geq 1$) is a parameter expressing an effect of X-ray irradiation, which is given in appendix, and $\Lambda$ is the cooling function (e.g. Sutherland & Dopita 1993). When $t_{\rm c,\;e}\ll t_{\rm s,\;e}$, the total mass of the envelope, $M_{\rm e}$ is written as

where $\dot{M}_{0}$ is the mass inflow rate from the companion star. Since $M_{\rm e}$ relates to $n_{\rm s,\;e}$ as given in equation (21), $t_{\rm c,\;e}$ can be calculated from equations (33), (34) and (21) as

where denominator numbers in parentheses for $M$, $R$, $T_{\rm e}$ and $\dot{M}_{0}$ are in units of erg cm³ s^-1, $M_{\odot}$, cm, K and g s^-1 respectively and $\Lambda_{0}=10^{-22.6}$ erg cm³ s^-1 which is obtained at $T=10^{7}$ K in the case of the solar abundance from Sutherland & Dopita (1993). We use the same units for the parameters in parentheses and omit the respective units in the following equations.

As seen from equation (35), the cooling time $t_{\rm c,\;e}$ decreases as the mass inflow rate $\dot{M}_{0}$ increases. Then, when $t_{\rm c,\;e}$ is shorter than the matter circulation time in the ring, $t_{\rm circ}$ defined as

the thick envelope does not appear from the beginning. However, the effect of the X-ray heating does not allow such a situation as $t_{\rm c,\;e}<t_{\rm circ}$ to happen and ensures that the thick envelope should always exist, as discussed below.

Figure 2 shows that when $\xi\gtrsim 10^{4}$ erg cm s^-1, the X-ray heating makes a temperature of an illuminated matter as high as $\sim 10^{7}$ K. We can relate the X-ray luminosity from a compact object $L$ to the mass inflow rate $\dot{M}_{0}$ as

where $\eta$ is the energy conversion efficiency associated with the mass accretion onto the compact object and the factor 1/2 multiplied to $\dot{M}_{0}$ comes from an assumption that a half of the inflowing matter is accreted by the compact star. $n_{\rm s,\;e}$ is, on the other hand, given as in equation (22) and $M_{\rm e}$ relates with $\dot{M}_{0}$ as $M_{\rm e}=\dot{M}_{0}\ t_{\rm stay,\;e}$ in terms of the matter staying time $t_{\rm stay,\;e}$ in the envelope. Hence, we can get the following condition on $t_{\rm stay,\;e}$ for the sufficient X-ray heating to occur from an equation $L/(n_{\rm s,\;e}r_{0}^{2})\gtrsim\xi_{4}=10^{4}$ erg cm s^-1 as

This means that X-ray heating prevents a situation as $t_{\rm c,\;e}<t_{\rm circ}$ from appearing by enlarging $\varepsilon$ to much larger than unity for parameter values considered here.

Following the above discussions, the envelope is expected to have two internal flows. One is a flow towards the torus center as a result of the radiative cooling, and the other is a flow spreading out from the torus, bifurcating to a thick accretion flow and a thick excretion flow, as a result of the angular momentum transfer. We call the former and the latter flows respectively as the cooling flow and the mass-spreading flow hereafter.

The mass flow rates through the cooling flow and the mass-spreading flow, $\dot{M}_{\rm c,\;e}$ and $\dot{M}_{\rm s,\;e}$, are approximately calculated as

and

where $t_{\rm c,\;e}$ and $t_{\rm s,\;e}$ are given in equations (31) and (35) respectively. In the steady state, the following equation should be satisfied as

From the above three equations (39), (40) and (41), we get

and

By comparing equations (35) and (31), we see that $t_{c,\;e}\gg t_{s,\;e}$ when $\dot{M}_{0}\ll\dot{M}_{1}$. $\dot{M}_{1}$ is given by equating $t_{\rm c,\;e}$ with $t_{\rm s,\;e}$ as

where $\varepsilon$ is set to be 1, since X-ray heating is negligible here. In this case, it is expected that all the inflow from the companion star is transferred to the thick spreading flow, namely $\dot{M}_{\rm s,\;e}\simeq\dot{M}_{0}$ and $\dot{M}_{\rm c,\;e}\simeq 0$.

If $\dot{M}_{\rm s,\;e}$ is evenly divided into the accretion flow and the excretion flow, the accretion rate from the envelope, $\dot{M}_{\rm ac,\;e}$, is expressed as

and

Since $M_{\rm e}\simeq t_{\rm s,\;e}\dot{M}_{0}$ in this case, the scale number density, $n_{\rm s,\;e}$, is estimated from equation (22) as

Then, the ionization parameter, $\xi_{\ell}$, in this case, is calculated with the help of equation (31) as

This $\xi_{\ell}$ - $T_{\rm e}$ line is drawn in figure 2 and is seen to be in the range where the effect of X-ray heating is negligibly small. Even if the X-ray heating is negligible, the density is too low for the envelope to radiatively cool down and the ring tube is expected to be kept thick with the initial temperature, $T_{\rm in}$, in the low-accretion-rate case. And, a thick accretion flow and a thick excretion flow are expected to extend from the envelope.

When $\dot{M}_{0}\gtrsim\dot{M}_{1}$, $t_{\rm c,\;e}$ comes to satisfy the relation as $t_{\rm c.\;e}\lesssim t_{\rm s,\;e}$ and the mass inflow rate, $\dot{M}_{0}$, from the companion star is divided into the two components, $\dot{M}_{\rm c,\;e}$ and $\dot{M}_{\rm s,\;e}$ following equations (43) and (44) respectively. If we assume again that $M_{\rm c,\;e}$ turns evenly to the thin accretion disk and the thin excretion disk from the core in the steady state, the accretion rate from the core, $\dot{M}_{\rm ac,\;c}$, is written as

The total X-ray luminosity is reasonably expressed with the help of equations (43) and (44) as

The scale number density of the envelope in this case is calculated from equations (22) and (42) as

Then, the ionization parameter, $\xi_{\rm m}$, is given with the help of equation (50) as

In this medium accretion rate range, as $\dot{M}_{0}$ increases from $\dot{M}_{1}$, $t_{\rm c,\;e}$ decreases according to its proportionality to $\dot{M}_{0}^{-1/2}$ as in equation (35), and $\xi_{\rm m}$ increases due to the $(t_{\rm c,\;e}+t_{\rm s,\;e})/t_{\rm c,\;e}$ term in equation (54). Then, the $\xi_{\rm m}$ line moves right in figure 2 and eventually gets close to the line tagged with $\xi_{\rm t}$ which touches the theoretical $\xi$ - T curve at $\xi_{\rm t,\;0}$. Here, we define $\xi_{\rm t}$ as

If $\xi_{\rm m}$ just reaches $\xi_{\rm t,\;0}$, the envelope should be governed by the situation at the tangential point where the radiative cooling rate balances with the X-ray heating rate, and then, $t_{\rm c,\;e}$ should be regarded to be infinity as discussed in appendix. Since this situation of $t_{\rm c,\;e}\gg t_{\rm s,\;e}$ is the same as discussed in the low accretion rate case in sub-subsection 2.5.1, $\xi$ is required to return to $\xi_{\ell}$ then. This means that $\xi$ of the envelope cannot get just to $\xi_{\rm t}$, and a steady state solution exists in the vicinity of $\xi_{\rm t}$, which will be discussed in the next sub-subsection.

If we define the medium accretion rate range as in the case of $\dot{M}_{0}\gtrsim\dot{M}_{1}$ and $\xi_{\rm m}\ll\xi_{\rm t}$, the upper bound of $\dot{M}_{0}$ in this range, $\dot{M}_{2}$, is calculated by equating $\xi_{\rm m}$ in equation (54) to $\xi_{\rm t}$ in equation (55) with the help of equation (35) and setting $\varepsilon=1$ as

In the medium accretion rate range with $\dot{M}_{1}\lesssim\dot{M}_{0}\ll\dot{M}_{2}$, the cooling time becomes comparable to or shorter than the mass spreading time. Although the effect of the X-ray heating is not so significant yet, the temperature of the envelope is still kept $\sim T_{\rm in}$ during the cooling time. Then, two parallel accretion flows are expected to appear. One is a thin disk accretion flow from the core with the accretion rate, $\dot{M}_{\rm ac,\;c}$, as

and the other is a thick accretion flow from the envelope with the accretion rate, $\dot{M}_{\rm ac,\;e}$, as

Here, $F$ is defined as

where $t_{\rm c,\;e}$ is a function of $\dot{M}_{0}$ as in equation (35). In this range, $t_{\rm c,\;e}\lesssim t_{\rm s,\;e}$ and thus $F$ is in a range of $1/2\lesssim F<1$. The accretion rate through the thin accretion disk is slightly dominant to that through the thick accretion flow in this case.

When $\dot{M}_{0}\gtrsim\dot{M}_{2}$, $\xi$ of the envelope gets close to $\xi_{\rm t,\;0}$ and the effect of the X-ray irradiation on the envelope becomes significant. The ionization parameter in this $\dot{M}_{0}$ range, $\xi_{\rm h}$, is determined, utilizing equations (88) and (90) in appendix, by an equation as

to which equation (54) is modified as a $\xi$ - $T$ relation along the $T_{0}=10^{6}$ K line between $\xi_{\ell,\;0}$ and $\xi_{t,\;0}$ as revealed in figure 2. When $\dot{M}_{0}$ exceeds $\dot{M}_{2}$, $\xi_{\rm h}$ should have become larger than $\xi_{\rm t,\;0}$ unless X-ray heating exist, but the term of $[1-(\xi_{\rm h}/\xi_{\rm t,\;0})]^{1/2}$ reflecting the effect of X-ray heating suppresses $\xi_{\rm h}$ slightly lower than $\xi_{\rm t,\;0}$.

Combining equations (59) and (60) and considering $\xi_{\rm h}\sim\xi_{\rm t,\;0}$, we have

where $X$ is defined and calculated with equations (50) and (55) as

In this high accretion rate range, the accretion rate through the thick flow is considered to be approximately several % as small as that through the thin disk.

The ram pressure of the stream on the surface of the torus, $P_{\rm in}$, can be obtained as

where $\rho$ is the density of the stream and $v_{\bot}$ is its velocity component in the direction perpendicular to the torus surface when it hits the torus. Since we can approximate the mass flow rate of the stream, $\dot{M}_{0}$ as

where $v_{\rm in}$ is the stream velocity and $d$ is the radius of the stream-cross section, $P_{\rm in}$ can be rewritten as

If the orbital radius of the matter in the ring shifts $\Delta r$ from the Keplerian circular motion by the ram pressure of the stream, the repulsive force per mass, $f$, can be calculated as

Since the average column density of the ring-core in the radial direction is roughly given as $M_{\rm c}/(4\pi r_{0}x_{\rm s,\;c})$, the repulsive pressure of the ring-core, $P_{\rm rep}$, can be approximated as

By equating $P_{\rm in}$ to $P_{\rm rep}$, we have a relation

where $M_{\rm c}=\dot{M}_{0}t_{\rm s,\;c}$ is assumed and $v_{\rm K}=\sqrt{GM/r_{0}}$. Considering $t_{\rm circ}/t_{\rm s,\;c}\sim 10^{-3}$ as seen from equations (36) and (45) for appropriate parameters, and $v_{\bot}<v_{\rm K}<v_{\rm in}$, a moderate shift with $\Delta r<x_{\rm s,\;c}$ can make the ring-core stop the stream unless $d$ is much less than a few % of $r_{0}$. Since $d$ could be several % or a little more of $r_{0}$ (e.g. Lubow & Shu 1975; 1976), the stream can be considered to be well braked by the core.

Once the stream stops around the core surface, the matter in the stream should be heated up to $T_{\rm in}\sim 10^{7}$ K as discussed in subsection 2.1. Then, it is expected to expand to the same size as the scale thickness of the envelope, $x_{\rm s,\;e}$, since the temperature of the envelope is considered to be the same as $T_{\rm in}$. A key question is whether the expanded matter cools down soon or not.

The expanded matter of the stream just after the hit to the core should start flowing in the envelope, with velocity $v_{\rm K}$ and with mass flow rate $\dot{M}_{0}$ in the steady state. If we assume that the flow of the fresh matter from the stream has eventually the same cross section as the ring-envelope, the average number density of the fresh flow, $n_{\rm f}$, can be written from the continuity equation as

On the other hand, the scale number density of the envelope, $n_{\rm s,\;e}$, is expressed in equation (22) replacing $M_{\rm r}$ to $M_{\rm e}$ in equation (34).

From the above equations we get

This equation indicates that $n_{\rm f}$ should be smaller than $n_{\rm s,\;e}$ under a situation that the cooling flow time scale of the ring envelope is sufficiently longer than the rotational period of the ring. This situation is the premise of the present sturdy. Otherwise, the geometrically thick ring should not have been realized from the beginning. Thus, considering that the time scale of the radiative cooling is inversely proportional to the number density, we cay say that the cooling time of the hot matter newly added from the stream to the ring-envelope should be longer than that in the main flow, as far as the premise is held.

The impact of the stream on the ring can be considered to give no large effect on the basic frame of the present study.