DISKS AROUND BROWN DWARFS IN THE EJECTION SCENARIO. I. DISK COLLISIONS IN TRIPLE SYSTEMS

Figure 7 shows a typical example of a recircularized radial surface density disk profile after a close triple encounter, together with its initial profile. Many features of the radial disk profile seen in this diagram are similar to the results for parabolic two-body encounters investigated by Hall (1997), e.g., the disks are severely truncated and the surface density drops by orders of magnitudes in the outer disk regions. Also as in Hall (1997), the inclusion of the approximate viscous evolution of the strongly perturbed disk by recircularizing the disk material, further depletes the outer disk regions, because at larger radii the disk material loses much angular momentum and, therefore, ends up at smaller radii (Hall 1997; Hall et al. 1996). This has also the effect that in the inner regions of the disk the surface density is enhanced relative to the initial density. We also find, as in Hall (1997), that the surface density of the outer parts of the disk can be described by an exponential function. However, Hall (1997) gave no detailed description of the radial structure of the inner disk, where most of the disk mass resides.

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From our simulations we found that most of our disk profiles can be divided into three distinct regions, a power-law region with Σ∝r⁻¹ for disk radii below 0.2r_min and two regions, located between 0.2 and 0.7r_min and above 0.7r_min, where the surface density decreases exponentially as Σ∝exp (log (1/2) r/τ), with the outer region being flatter, i.e., having a larger full-width-half-value τ, than the inner one. At 0.2r_min Kobayashi & Ida (2001) also found that the structure of the disk is changing by investigating the eccentricity change of the disk material after a prograde encounter. Inside that radius the disk is only weakly perturbed, whereas further out, resonances lead to strong perturbations. Although only strictly valid for parabolic encounters, they also found that this boundary is only weakly dependent on the eccentricity of the perturber. In our simulations the eccentricity of the perturber with the closest encounter distance was always below 2.5 at the time of closest approach and, therefore, the region was at most shifted to 0.24r_min which can be neglected given the uncertainties of our fits.

As already mentioned, inside the inner power-law region the surface density is enhanced relative to the initial surface density. As can bee seen from Figure 7 it is increased by a constant factor, which varied in our simulations between 1.4 and 2. This factor is similar to the surface density enhancement for the parabolic, retrograde encounters of Hall (1997) at r = 0.4r_min, although in all our runs we observed this increase at somewhat lower radii. This might be partly due to the influence of the prograde encounter, as Hall (1997) observes the enhancement at 0.25r_min for this case, but also because of the, on average, higher eccentricity at the time of closest approach. Another consequence of the higher eccentricity of the orbits might be that the density enhancement never exceeded a factor of two, in contrast to the prograde, parabolic encounters in Hall (1997), where the surface density was five times higher than its initial value. However, as we will see in Section 3.4, even if the eccentricities are close to the parabolic case, the disks' surface densities are not as strongly enhanced in the inner region as in that case, which indicates that the higher eccentricities have a rather limited influence on the disk structure. The reason for the lower surface density enhancement might have rather something to do with the dynamics of the triple encounters that cannot be sufficiently described in terms of two-body encounters, as we will discuss in Section 3.4.

In order to find out how the surface density profile is related to the encounter parameters r_min and r₁/r₂, we fitted for each of our obtained post-encounter disk profiles two exponential functions to the outer two regions and determined the full-width-half values τ₁, for the region between 0.2 and 0.7r_min, and τ₂, for the region outside 0.7r_min. We then plotted these values over r_min for each r₁/r₂ separately. We found that, while the uncertainties of the fitted values for τ₁ were mostly between 2% and 10%, they were much larger for τ₂, mainly because of the much lower number of particles left in the outer regions of the disk. In those regions many particles are stripped off the disk and the remaining ones are recircularized to smaller radii, due to their much smaller angular momentum. Therefore, we can give for most of the r₁/r₂ values we investigated only upper limits for τ₂.

Figure 8 shows τ₁ as a function of r_min for different values of r₁/r₂. In all cases τ₁ increases with increasing r_min which means that for larger r_min the radial surface density disk profile gets flatter in this region. Furthermore, τ₁ seems to depend linearly on r_min. However, we find that the values of τ₁ scatter much more around this linear relation than one would expect from their individual errors, which come from the quality of the fit. As already mentioned in Sections 2.2 and 3.1 the reason for the large scatter is mainly intrinsic to our parameterization. Furthermore, we find no significant difference between triple collisions, where the disk suffered a retrograde or a prograde encounter at the time the ejected body reached the closest encounter distance r_min. This indicates that the orientation of the disk plays only a minor role and most of the large scatter comes from the rather large variety of initial conditions at the chosen maximum distance between the bodies. Also, we did not find that the deviations of τ₁ correlate with any other parameter we investigated, like the eccentricity of the perturber orbit and the escape velocity of the ejected body. We therefore come to the conclusion that the value of τ₁ must depend on many parameters in a rather complex way, which, in order to investigate further, would require to investigate a much larger parameter space than we can currently study.

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For each of the values of r₁/r₂ we determined the slope of the linear fit of the τ₁ values with the corresponding error and plotted these values over r₁/r₂ in Figure 9.

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As can be readily seen, for values of r₁/r₂ greater than 0.5 there is only a weak increase in the slope of τ₁(r_min) and, considering the errors, it is not absolutely certain if the slope at r₁/r₂ = 0.5 is really larger than the one at r₁/r₂ = 0.95 or if this is an effect of the limited number of points available for the fit. For encounters with r₁/r₂ < 0.5 we find that the slope of τ₁(r_min) increases much stronger and nearly all values of τ₁ for r₁/r₂ = 0.2 were larger than the corresponding values for r₁/r₂ ⩾ 0.5. This clearly shows that at least in this regime the radial surface density distribution depends strongly on the parameter r₁/r₂ and leads to disk profiles which are generally much flatter for lower values of r₁/r₂.

As mentioned earlier, the uncertainties in the values of τ₂ are much larger than they are for τ₁ because of the lower number of particles left in the outer disk regions for the fit. Only for the cases r₁/r₂ = 0.2 and r₁/r₂ = 0.3 we could determine τ₂ with comparable accuracy as τ₁, and found a value of ≈0.2 with an uncertainty of 15% for both. In all other cases we can only say that a₂ must be between 0.1 and 0.2. As we cannot constrain how a₂ changes with r₁/r₂ we, therefore, assume for simplicity a constant value of a₂ = 0.2 for our study. Although this value is an overestimate for most of our disks after triple encounters with r₁/r₂ ⩾ 0.5, the error we introduce into our model is only minor. This is because, as we will also show later, the mass contained in the outermost part of the disk is, for these cases, always less than 6%.

From our results we are now able to construct a scale-free model for the recircularized post-encounter surface density disk profile as a function of the encounter parameters r_min and r₁/r₂. For its general form we can write

$${\fontsize{9.6}{11.6}\selectfont{ \begin{eqnarray} &&\Sigma (r_{{\rm min}},\, r_{1}/r_{2})=\Sigma _{0}\, k\,\nonumber\\ &&\times\left\lbrace \begin{array}{@{}ll@{}}\frac{r_{0}}{r} ;& r\le 0.2r_{{\rm min}}\\ \frac{r_{0}}{0.2r_{{\rm min}}}\,\tilde{a}\,\exp \left(\log (1/2)\,\frac{r}{a_{1}(r_{1}/r_{2})\, r_{{\rm min}}}\right) ;&0.2r_{{\rm min}}<r<0.7r_{{\rm min}}\\ \frac{r_{0}}{0.2r_{{\rm min}}}\,\tilde{a}\,\tilde{b}\,\exp \left(\log (1/2)\,\frac{r}{a_{2}r_{{\rm min}}}\right) ;& r\ge 0.7r_{{\rm min}}\end{array}\right.\nonumber\\ \end{eqnarray}}} \tag{ 1 }$$

with

$$\begin{eqnarray} \tilde{a} & = & \exp \left(\log (1/2)\,\frac{0.2}{a_{1}}\right)^{-1}\nonumber\\ \tilde{b} & = & \exp \left(\log (1/2)\,0.7\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)\right), \end{eqnarray} \tag{ 2 }$$

where we replaced the exponential slopes τ_{1, 2} with a_{1, 2} r_min.

In this equation, the density enhancement for the region below 0.2r_min is accounted for by a constant factor k while Σ₀ is the initial value of the surface density at radius r₀. The constants $\tilde{a}$ and $\tilde{b}$ are coefficients to ensure continuity for Σ. The parameter a₁ is given by our fits in Section 3.3.2, while a₂ = 0.2, as we mentioned earlier. However, as a₁ is only given for a limited number of r₁/r₂ values, we need to fit these data points with a simple function in order to be able to apply our results to different encounter parameters. Given the error bars, it is certainly legitimate to represent a₁(r₁/r₂) by two linear fits, one for r₁/r₂ ⩾ 0.5 and another one for r₁/r₂ ⩽ 0.5, modifying the a₁ intercept of each of them slightly to guarantee continuity at r₁/r₂ = 0.5. The parameters a₁ and a₂ are then given by

$$\begin{eqnarray} a_{1}(r_{1}/r_{2}) & = & \left\lbrace \begin{array}{@{}ll@{}}-0.046\, r_{1}/r_{2}+0.108 ;& r_{1}/r_{2}>0.5\\ -0.25\, r_{1}/r_{2}+0.21 ;& r_{1}/r_{2}\le 0.5\end{array}\right.\\ a_{2} & = & 0.2.\nonumber \end{eqnarray} \tag{ 3 }$$

Thus, Equations (1)–(3) completely define the resulting post-encounter disk profile in terms of r_min and r₁/r₂, apart from the constants k and Σ₀(r₀). While the latter is related to the initial disk and, therefore, to the specific problem one wants to apply our model to, the constant k can be determined by considering the relative mass loss of the disk, which will be done in the next section.

From our model surface density profile we can now determine how the total disk mass changes with r_min for a given value of r₁/r₂. In addition, it allows us to measure the extent of the disks based on their mass distribution. There is, however, no unique definition of disk radius. In the literature the half-mass radius is frequently used for that purpose, which has the advantage that this radius is roughly consistent with the apparent 2.7 mm flux sizes of disks around T Tauri stars (Hartmann et al. 1998). However, the apparent disk size depends crucially on the optical depth of the disk and hence on the absolute value of the surface density. Therefore, it is not clear, a priori, that this relation will hold for disks around brown dwarfs, because if we assume roughly the same ratio of disk to central mass for the observed brown dwarfs as for T Tauri stars, which is also supported by observations of Klein et al. (2003) and Scholz et al. (2006), brown dwarf disks will have much lower surface densities than T Tauri disk and consequently very different optical depths. In absence of detailed radiative transfer modeling of low surface density disks there is no practical disk size definition which can be directly compared to observations.

For our investigation, we use the radius $r_{90\%}$, within which 90% of the disk mass is contained, as a measure of the size of the disk. We can then compare it with the result from parabolic two-body disk encounters of Hall et al. (1996), who found $r_{90\%}=0.5r_{{\rm min}}$.⁷

In order to obtain the disk mass M_D from our model we integrate Equation (1) over the radius r:

$$\begin{equation} M_{D}=2\pi \int _{0}^{\infty }r\,\Sigma (r)\, dr. \end{equation} \tag{ 4 }$$

The general solution can then be written in the form

$$\begin{equation} M_{D}=2\pi k\Sigma _{0}r_{0}\big(0.2+M_{{\rm out}}^{1}(a_{1})+M_{{\rm out}}^{2}(a_{1},a_{2})\big)\, r_{{\rm min}}, \end{equation} \tag{ 5 }$$

where M^{1, 2}_out are coefficients which are proportional to the mass of the disk contained in the corresponding region. As can be seen, in our model the disk mass is directly proportional to r_min for given values of a_{1, 2}, which is also consistent with the disk masses we get from our simulations, shown in Figure 10.

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In these figures, we plotted for each value of r₁/r₂ the masses of the disks in units of the initial disk mass over r_min and found that the values are linearly correlated with a correlation factor larger than 95%. The values of M^{1, 2}_out together with $r_{90\%}$ for each value of r₁/r₂ we investigated are shown in Table 1. Also shown is the fraction of the total post-encounter disk mass that is contained within the three different disk regions.

Table 1. Disk Mass Coefficients M_inn, M^{1, 2}_out and the 90%-mass-radii $r_{90\%}$ in Units of r_min for Our Disk Models with Different Values of r₁/r₂

r1/r2	Minn	M1out	M2out	$r_{90\%}$ (rmin)
0.95	0.2 (60%)	0.13 (38%)	<0.006 (2%)	0.38
0.8	0.2 (52%)	0.17 (44%)	<0.015 (4%)	0.49
0.6	0.2 (49%)	0.19 (46%)	<0.02 (5%)	0.52
0.5	0.2 (49%)	0.19 (46%)	<0.02 (5%)	0.52
0.3	0.2 (29%)	0.35 (51%)	0.14 (20%)	0.9
0.2	0.2 (27%)	0.37 (51%)	0.16 (22%)	1.0

Notes. M_inn and M^{1, 2}_out are proportional to the mass contained in the respective region. In parenthesis the mass fraction with respect to the total post-encounter disk mass is given.

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This table clearly shows that for r₁/r₂ ⩾ 0.5 there is only a very small amount of mass contained in the outer region of the disk with r ⩾ 0.7r_min and more than 95% of the disk mass is within 0.7r_min, making the post-encounter disk very compact. Therefore, the rather large uncertainty of the disk profile and mass in the outermost region, which is of the order of 50%, does not have a significant influence on the overall mass distribution of the disk. Furthermore, we find only a weak increase in mass in the region between 0.2r_min and 0.7r_min for decreasing r₁/r₂ ⩾ 0.5, as it was to be expected from the results of τ₁(r_min). Comparing this mass with the mass that is contained within 0.2r_min we find that they are nearly equal, given the errors of about 10% for M¹_out. Only in the case with r₁/r₂ = 0.95 there seems to be more mass contained in the innermost region, with about 60% of the total disk mass.

Because of the strong increase of τ_{1, 2} for r₁/r₂ ⩽ 0.5, as shown in Section 3.3, we also have a strong increase in mass contained in the outer disk regions for this case. As a consequence, these disks are no longer strongly concentrated and appear to be much flatter, as now there are only about 78% of the total disk mass contained within 0.7r_min as opposed to more than 95% for r₁/r₂ ⩾ 0.5. In contrast to disk encounters with r₁/r₂ ⩾ 0.5 the outermost part of the disk now contains almost a quarter of the total mass of the disk and most of its mass is contained in the region between 0.2r_min and 0.7r_min. This comes close to the corresponding values for a disk with Σ∝1/r truncated at a radius 1r_min.

So far, we completely specified the disk profile as shown in Equations (1)–(3) apart from a constant k, which specifies how much the surface density in the inner disk region below 0.2r_min is enhanced relative to the initial surface density. As in that region we are severely limited by noise, fitting each single profile for that region in order to determine k is rather problematic. This is because the power-law indices of the fits will also scatter around the mean value of −1, which in turn adds much to the error of k. We, therefore, determined k by assuming that the inner profile follows strictly a power law with index −1 and use the linearly fitted disk masses from Figure 10 together with Equation (5). This gives a statistically more robust value for k because the determination of the masses are based on a larger number of particles than that of the inner disk profile. For the determination of k, we expressed Equation (5) in terms of the initial disk mass M_init = 2π Σ₀r₀(r_out − r_in), where r_out is the outer and r_in is the inner initial disk radius. We then inserted the values from Table 1 for each r₁/r₂ and set the numeric value of the slope of M(r_min) equal to the corresponding slope of the fits, thus obtaining k. In all cases, we found that k = 1.7 ± 0.2, which lies also in the range we roughly expected from the disk profiles in Section 3.3. Since this factor does not seem to depend on r₁/r₂, it follows that a flatter disk profile implies a larger total disk mass. From this it follows further that in general the masses of the post-encounter disks increase with decreasing r₁/r₂, which also agrees with our results shown in Figure 10.

Since we noticed that disks after triple encounters are qualitatively similar to the ones after two-body encounters, we now want to find out to what extent this similarity holds, and compare these two cases in a more quantitative manner. Here we are only concerned with a comparison of disk masses and, in Section 3.5.2, of disk radii, as the only study that was concerned with recircularized surface density profiles (Hall 1997) did not have a sufficient resolution for the inner disk to compare our results with quantitatively.

For two-body disk collisions, Hall et al. (1996) found that the post-encounter disk has only one-third of the initial mass if the disk has a radius of 4r_min, which has also been found in a more recent study of Olczak et al. (2006). As our disk radius is always 20 AU this means that, in order to compare to our results, the corresponding disk mass in our case is the one at r_min = 5 AU for each r₁/r₂. From our calculations we find values ranging from 15% to 18% of the initial disk mass for r₁/r₂ ⩾ 0.5, while for r₁/r₂ < 0.5 we find values of up to ≈33%. This reflects the trend we intuitively would expect, as for larger values of r₁/r₂ we have roughly speaking two two-body encounters with similar strength and consequently more stripping of disk material, whereas for lower values of r₁/r₂ we are getting closer to a single two-body encounter. In addition, we also find that this trend is strongly nonlinear and not a simple function of r₁/r₂. The fact that in a triple encounter we only find the same disk mass for r₁/r₂ ⩽ 0.2 has the direct consequence, together with the distribution of r₁/r₂ shown in Figure 5, that the majority of close triple encounters is more destructive than the corresponding two-body ones and only in one-third of the cases we should get at least comparable disk masses.

However, the convergence of the disk masses for smaller r₁/r₂ is not as trivial as it might sound, because we also find that the influence of the third body in the case of r₁/r₂ = 0.2 is still significant. In order to demonstrate this effect, we carried out a calculation for one particular triple encounter where the disk is only affected by the escaper and the body with the closest encounter distance r_min, but otherwise leave the orbits of the bodies unchanged. In that example, we find that the post-encounter disk mass is increased by almost 10% of the initial disk mass which makes the post-encounter disk mass 33% larger compared to the case, where the disk is affected by all three bodies. As we find that for the r₁/r₂ = 0.2 case the disk masses are comparable to the two-body case, it directly follows that the particular motion of the escaping body in a triple encounter tend to increase the mass of the disk while the presence of the third body reduces it. We therefore expect that for ratios r₁/r₂ ⩽ 0.2 the trend for higher disk masses compared to two-body collisions for a given r_min is continued.

From our simulation, we suggest that the trend of higher disk masses for lower r₁/r₂ is closely related to the formation of the final binary after the triple encounter. To demonstrate this, we compared in Figure 11 the distance between the escaper and the body of the binary with the smallest encounter distance r_min and the distance these bodies would have if they moved on a hyperbolic orbit with the same eccentricity at r = r_min. As can be clearly seen, the distance between the bodies in the triple system is almost always lower than in the two-body case, leading to an enhanced interaction time which is much larger than five dynamical times found for the parabolic two-body case by Hall et al. (1996; one dynamical time is the ratio of r_min over the velocity v_min at that point). Before the encounter, the distances are lower because the bodies do not approach each other from an infinite distance, but are part of a bound system with finite size. The lower distances after the encounter are due to the formation of a binary where one of the two bodies is slowed down relative to the escaper because of the presence of the third body. For the disk around the escaper this means, on the one hand, that the disk starts losing material earlier in the encounter than it would do in a hyperbolic encounter. On the other hand, due to the lower distances after the encounter, particles that became unbound after the closest collision are slowed down again and could be re-captured by the escaper or the other two bodies depending on the direction of their ejection. Since much of the disk material is initially on smaller radii around the ejected body than after the encounter, the initial deviation from the hyperbolic encounter should only play a minor role, while the deviations after the closest encounter are more effective, because much of the disk material is distributed over a much larger area. Figure 12 shows a snapshot of a disk after the final binary has formed and where the disk is only affected by the perturber with the closest encounter distance r_min. Although, at this time, after more than 20 dynamical times, the particle states, i.e., whether they are gravitationally bound or unbound to any of the three bodies, would have been long evident in a two-body encounter (Hall et al. 1996), in a triple encounter there is a substantial amount of unbound material that gets captured later because of the perturber getting drastically slower when it forms a binary. The increased effect of the perturber on the disk can bee seen by looking on the strong spiral arm that develops during the encounter, which, in a two-body encounter, is always pointing away from the perturber. In this figure, however, it is bent toward the binary because of the much lower velocity of the perturber after binary formation relative to the escaping body, making it possible that this otherwise unbound material gets captured by the escaper or by the perturber. Here we want to stress that the effect is solely because of the deceleration of the closest perturber and not because of the mass of the third body, which has been artificially set to zero while retaining the orbit of the ejected body in this particular simulation. The re-capture of disk material can also be seen in Figure 13 where we plotted the change of mass of the disk around the ejected body. Here it is clearly shown that first, the time required for the particle states to settle is much longer than in the two-body case and, second, that just after the encounter and after the formation of a binary much of the disk material lost is re-captured by the ejected body. From this figure it is also evident that the change of disk mass in an orbit of a triple system is no longer a simple function over time and a substantial amount of disk material changes its gravitational state more than once. Without the rather abrupt deceleration the disks would have masses very close to the corresponding minimum shown in Figure 13. Therefore, the loss caused by the presence of the third body is in that case compensated by approximately 10% of the initial disk mass, leading to the mass expected for only two masses in a nearly parabolic two-body encounter.

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In contrast to the behavior of the disk masses for lower r₁/r₂, the case r₁/r₂ = 0.2 does not converge toward the expected two-body result for the 90%-mass-radii $r_{90\%}$. Here we find that the disks after triple encounters are comparable or much larger than after slightly parabolic two-body ones. As shown in Table 1 we also find the tendency for the post-encounter disks to become flatter with decreasing r₁/r₂ in the value of $r_{90\%}$. The dependence on r₁/r₂ is nearly the same as for the slope of τ₁(r_min), i.e., only a weak increase for r₁/r₂ ⩾ 0.5 and a much stronger one for values below. As a consequence of the high-mass concentration in the inner disk regions for r₁/r₂ ⩾ 0.5, the 90%-mass-radius has rather low values. However, compared to the results of Hall (1997), where $r_{90\%}=0.5r_{{\rm min}}$, our values are not much different in the cases with r₁/r₂ ⩾ 0.5, although here we would expect to have a much stronger deviation. As already mentioned, for r₁/r₂ < 0.5 we do not get similar 90%-mass-radii and find that $r_{90\%}$ is around 1r_min, that is, twice as large as in the parabolic two-body case. Considering that for the r₁/r₂  =  0.2 case we have about the same disk mass as for the corresponding two-body case, we come to the conclusion that parabolic two-body collisions produce disks with a much higher mass concentration in the inner regions and, therefore, with a significantly increased average surface density.

The generally flatter mass distribution of disks after triple encounters is presumably caused by the capture process of unbound material after the final binary has formed, as we described in the previous few paragraphs. Since it is known that the unbound material carries most of the angular momentum away from the disk (Hall et al. 1996), the re-capture process brings some of this high-angular-momentum material back to the disk, which recircularizes after the collision to larger radii, producing a flatter disk profile. In our example (Figure 13), one-third of the final disk material was re-captured for the case where all three bodies affect the disk, and should have a larger angular momentum thus ending up at larger radii.