Kinematic Models

 In order to understand the intrinsic velocity structure underlying the observed kinematics, we have undertaken two separate sets of three-dimensional Monte-Carlo wind simulations: a rounded expanding bubble and a pair of cones arranged as a funnel. Although primarily geometric in nature, these models allow us to estimate the three-dimensional morphology of the outflow, as well as intrinsic gas velocities, both of which are important constraints for physical models of the wind emission mechanisms and other observational phenomena.

 

Figure 11: Y-Z cuts along the axes of two Monte-Carlo models of the southern outflow in M82: a. a single truncated bubble and b. a pair of inclined cones. Dimensions are given in parsecs; arrows denote the direction toward the observer.  

For our initial model, we used rounded bubble geometries, given by the spherical functions,

$$\rho = A\ \cos{m\theta} \qquad -\frac{\pi}{2m} < \theta < +\frac{\pi}{2m}$$ (4)

or

$$\rho = A\ \cos^m \theta \qquad -\frac{\pi}{2} < \theta < +\frac{\pi}{2}$$ (5)

(see Fig. 11a). Both functions produce a parabolic leading surface becoming conical at small radii. The parameter m can be related to an opening angle and a Mach number in cases where the leading surface is produced in a bow shock. Truncating the bubble at inner and outer radii allows for the selection of a outflow structure with specific opening angle and radial curvature. We superimposed two alternate velocity laws upon this spatial geometry: a radial velocity vector, which corresponds physically to the case in which each gas parcel is accelerated by the flow originating at the bubble apex, or a velocity vector perpendicular to the surface of the bubble, which may be more appropriate for an expansion due to increasing heat and pressure inside the bubble, such as an inflating balloon.

We then executed a series of Monte-Carlo simulations for each velocity law, varying bubble parameters such as opening angle, inclination angle, and inner and outer truncation radii. The conclusion reached from this set of simulations is that the rapid divergence of the velocity components in the southern outflow cannot be reproduced by a single bubble, at least not without invoking highly contrived velocity profiles for the wind. Studies of the minor axis x-ray distribution are similarly unable to model the emission with a single bubble or cone (e.g., [Suchkov et al. 1996]).

Morphologically, one can divide the southern region of split lines into two separate velocity regimes: the region within approximately 200 pc of the nucleus, where the split line components are separated by $\sim$50 km s^-1, and the region beyond 500 pc radius, in which the components are separated by a much larger, but still relatively constant, projected value of $\sim$300 km s^-1. The inner component is not observed north of the galaxy, presumably because the split lines cannot be resolved at a sufficiently small radius due to the intervening inclined disk of the galaxy. Note that the H$\alpha$ and [NII] flux maps (Fig. 1) also suggest the presence of two distinct regions, as the line flux drops sharply at the same radius at which the velocity components rapidly separate ($\sim$500 pc).

We therefore performed another set of Monte-Carlo simulations, this time using a double cone geometry, given by the cylindrical function  

$$r = \cases{\kappa_1 (z - z_{01}), & 0 $<$\space z $<$\space ... ... \kappa_2 (z - z_{02}), & 350 $<$\space z $<$\space 800~pc\cr}$$ (6)

(see Fig. 11b). The $\kappa$ parameters determine the opening angles of the cones, while the z₀ parameters control the radial extent of the cones through truncation. Again, we superimposed both radial and normal (i.e., perpendicular) velocity laws, but determined in the end that a velocity vector tangential to the cone surfaces provided the best match to the observations and was most easily understood physically, in terms of material entrained by the high-velocity wind. An additional parameter was used to smooth the abrupt projected velocity transition where the two cones meet (at z=350 pc).

 

Figure 12: Comparison of the observed H$\alpha$ emission and a two-cone Monte-Carlo simulation of the southern outflow in M82. Panel a compares the model with the spatial distribution of H$\alpha$ flux (contour map) and the regions of split H$\alpha$ lines (grayscale). Panels b-d are two-dimensional spectra extracted along the axes drawn in panel a, and illustrate the multiple velocity components. The spectrum in panel b has been extracted along the axis of the cones, while panels c and d have been extracted perpendicular to the cone axis, at distances of 650 and 910 pc from the nucleus. Panel b is oriented with North at the top; panels c and d are oriented with West at the top.  

 

Inner cone Outer cone

inner height z₁ [pc] 350

outer height z₂ [pc] 350 800

minimum diameter D₁ [pc] 375 390

maximum diameter D₂ [pc] 400 590

$$\alpha$$ opening angle 5^o 25^o

$$\iota$$ inclination angle 5^o 15^o

position angle pa 150^o-165^o 165^o

radial velocity v(r) [km s^-1] ([pc]) 525 + 0.13 r

After performing simulations over a range of cone sizes, inclination angles, opening angles, and velocity laws, we derived the final model shown in Figure 12. Parameters of the cones which best fit the observations are given in Table 3. The inner ``cone'' is almost a cylinder of radius equal to the injection zone ($r\sim200$ pc), with a small, but non-zero, opening angle. The outer cone has an opening angle of approximately 25<sup>o</sup>, in relative agreement with the ``small'' opening angle models of [Bland & Tully 1988] and [McKeith et al. 1995]. Models in the ``large'' opening angle regime (e.g., 60^o; [Heckman, Armus, & Miley 1990]) do not match the observations, requiring excessively low intrinsic velocities and a larger projected spatial extent for the outflow region (see Fig. 12a). A large opening angle for the primary outflow cone also produces substantially skewed Doppler ellipses in spectra perpendicular to the outflow axis. This effect is due to the slit cutting through the back and front of the cone at different nuclear radii, and is not observed in our synthetic spectra (see Figs. 12c and 12d).

The inner and outer cones are inclined toward the observer by $\sim$5^o and $\sim$15^o, respectively, roughly aligning the back sides of the two cones (see Fig. 11b). This is required to explain the lack of a sharp velocity gradient in the low-velocity component (LVC) at a radius of 350 pc, as is observed in the high-velocity component (HVC). In addition, since the LVC exhibits small but non-zero projected velocities, the back sides of the cones must be at a slight angle to the plane of the sky. These inclination angles agree with previous estimates (e.g., [Burbidge, Burbidge, & Rubin 1964]; [Hennessy 1996]). While the observed velocities of the HVC could be duplicated with smaller cone opening angles and a larger inclination, the low velocities of the LVC require small inclination angles.

Initial attempts to model the kinematics of the outflow with a constant velocity law, i.e., using only the simple double-cone geometry to reproduce the observed velocity structure, were not successful. Figure 10 illustrates that both velocity components in the south and north exhibit non-zero slopes in the position-velocity plot. This can be understood as either a continuous change in the intrinsic gas velocity or as a change in the projected velocity through a continuous change in the outflow cone geometry. The latter case implies that both sides of the outer cone are constantly bending toward the observer, producing a slowly increasing projected velocity with radius. A gradually increasing intrinsic wind velocity is probably necessary as well, and can be understood from a physical standpoint. Buoyancy effects in the hot wind from the decreasing disk density with scale height, decreasing wind densities from the lack of collimation at larger radii, and other effects contribute to produce wind velocities that increase with radius in standard galactic wind models (e.g., [Chevalier & Clegg 1985]; [Suchkov et al. 1994]). After testing a number of stronger power-law expressions for the gas velocity dependence on radius, we finally chose the simple linear model given in Table 3. Together with a constant cone opening angle, this intrinsic velocity structure produces linear projected velocity gradients that correspond well to those observed in Figure 10. The intrinsic velocities of the gas range from 525 km s^-1 near the nucleus to 655 km s^-1 at a radius of 1 kpc. These velocities are comparable to the escape velocity for M82 (see Table 1), implying that the most distant entrained filaments are not bound to the galaxy. This conclusion is also supported by the large radial extent of the fast wind itself, as seen in soft X-rays, along the minor axis.

Just as the outflow cones are not aligned with the minor axis of the galaxy along the line of sight, neither are they aligned in the plane of the sky. The region of split H$\alpha$ lines constitutes a cone on the sky with the expected opening angle of $\sim$25^o, but with a position angle of $\sim$165^o, approximately 15^o greater than what the literature (e.g., [McKeith et al. 1995]) had previously defined as the ``outflow axis.'' The cone axis is rotated $\sim$100^o from the major axis of the galaxy, and places the eastern edge of the cone almost directly parallel to the minor axis. In fact, Figure 12a illustrates that this eastern edge is quite pronounced, both in the split emission lines and in the H$\alpha$ flux observed from the inner collimated zone. This suggests that the tilting of the outflow cones in the plane of the sky has been produced by a relative density enhancement in the eastern lower halo which maintains collimation of the wind even as it fans out toward the west and toward the observer.

While the large outflow cones appear to originate east of the galaxy's minor axis, a small region of split lines is also observed on the western edge of the collimated zone, approximately 300 pc from the nucleus. The morphology of the H$\alpha$ and [OIII] flux maps (Figs. 1 and 2) indicates that this region constitutes a small bubble on the side of the larger outflow structure. We clearly see an enhanced rim of H$\alpha$ emission around the bubble, and split lines within it.

Recalling again the identification of two outflow ``streams'' in the [OIII] flux map and the [NII]/H$\alpha$ ratio map (Figs. 2 and 4), one might expect a more substantial outflow from the western half of the nucleus, or at least a more centrally-positioned outflow cone. However, it appears that gas densities on the western side of the lower halo are substantial enough to keep the western stream from expanding into a cone. The stream appears to bend toward the west in the [OIII] flux map (Fig. 2), and the only expanding structure that we observe is the one small bubble.

But at larger radii, the ambient density toward the west must drop relative to the eastern side. While the outflow remains tightly confined to the east, even beyond the collimated zone, certain Fabry-Perot maps show evidence of variations in the western side of the outer outflow cone, suggesting a more azimuthally-extended morphology there, e.g., break-out into a less dense region. For example, the H$\alpha$ velocity map (Fig. 3) reveals high velocity clumps at radii of $\sim$1 kpc that are distributed westward from the sharp eastern edge over almost 90^o in azimuth. These clumps are clearly associated with the outflow, and can also be seen as the blue-shifted emission just west of the split line region in our most distant outflow cut, Figure 12d. The fact that this gas exhibits smaller velocities in panel c of this figure illustrates that kinematical effects from disk rotation are greater closer to the disk. In contrast, the emission immediately east of the split line region lies at effectively the same velocity in both panels c and d of Figure 12, as this coincides with the sharp collimating edge on that side of the outflow. The spatial structure of the H$\alpha$ velocity map (like that published in [Heckathorn 1972]) also suggests the presence of rotating disk material that has been entrained and gradually diverted to the outflow.

To summarize, the inner 350 pc of the outflow constitutes a flow down a pipe. The outflow is collimated, presumably by ambient and entrained disk material, and highly inclined to our line of sight, such that the observed radial component of the flow velocity is only $\sim$50 km s^-1. Beyond 350 pc, however, the collimation weakens and the outflow expands rapidly as a cone of emission with an opening angle of 25^o and a projected front-to-back velocity separation of approximately 300 km s^-1. This expansion is preferentially toward the west and toward the observer. A linearly increasing intrinsic gas velocity with an initial value of 525 km s^-1 and a gradient of 0.13 km s^-1 pc^-1 matches the observations well out to a radius of a kiloparsec.

The line-splitting phenomenon indicates that the H$\alpha$-emitting filaments are produced on the surface of the outflow cones, at the interface between the wind and the ambient halo material. In addition to the minor axis velocity structure, this model explains the increased flux within the collimated region (see Fig. 12a) as a result of an elevated density with respect to the outer expanding cone (${\cal F}\propto n_e^2$). An increased density in the innermost regions has also been indicated observationally by the ``filling in'' of the line profiles at those radii (e.g., [McKeith et al. 1995]).

While the inner bubble has an extremely small opening angle, we find that an outer cone of opening angle $\sim$20^o-30^o fits the data most closely. This range agrees well with the ``small'' opening angle values found by other authors (e.g., $\sim$30<sup>o</sup>; [McKeith et al. 1995]). Our observations do not support the ``large'' opening angle regime (e.g., $\sim$60^o; e.g., [Heckman, Armus, & Miley 1990]).