The Influence of the Secular Perturbation of an Intermediate-mass Companion. II. Ejection of Hypervelocity Stars from the Galactic Center

For different components of the Galactic potential, we adopt the conventional prescription for the SMBH, IMC, and Galactic bulge, disk, and halo, such that

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{\mathrm{SMBH}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{SMBH}}}{r}+\mathrm{post-Newtonian}\ \mathrm{terms},\\ {{\rm{\Phi }}}_{\mathrm{IMC}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{IMC}}}{\sqrt{| \vec{r}-{\vec{r}}_{\mathrm{IMC}}{| }^{2}+{s}_{\mathrm{IMC}}^{{\prime} 2}}},\\ {{\rm{\Phi }}}_{\mathrm{bulge}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{bulge}}}{r+{a}_{\mathrm{bulge}}},\\ {{\rm{\Phi }}}_{\mathrm{disk}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{disk}}}{\sqrt{{R}^{2}+{\left({a}_{\mathrm{disk}}+\sqrt{{z}^{2}+{d}_{\mathrm{disk}}^{2}}\right)}^{2}}},\\ {{\rm{\Phi }}}_{\mathrm{halo}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{halo}}}{r}\mathrm{ln}\left(1+\displaystyle \frac{r}{{a}_{\mathrm{halo}}}\right).\end{array}\end{eqnarray} \tag{ 1 }$$

In the above expression, r, R, and z are the spherical radius, cylindrical radius, and distance above the disk, and r_IMC is the distance of the IMC from the GC. In Table 1 of Paper I, we specify that a_bulge (=0.6 kpc), a_disk (=5 kpc), d_disk (=0.3 kpc), and a_halo (=20 kpc) are the scaling lengths for the bulge, disk, and halo, respectively; M_SMBH (=4 × 10⁶ M_⊙) and M_IMC (=10⁴ M_⊙) are the mass of the SMBH and IMC; and M_bulge (=10¹⁰ M_⊙), M_disk (4 × 10¹⁰ M_⊙), and M_halo (=10¹² M_⊙) are the mass scaling factor for the bulge, disk, and halo, respectively (Miyamoto & Nagai 1975; Hernquist 1990; Navarro et al. 1997). For the central SMBH, we include the post-Newtonian corrections (first and second orders) near the Schwarzchild radius, same as the formulae in Kidder (1995). As it has negligible effects on most orbits, the second-post-Newtonian term is only considered for those highly eccentric orbits where the closest distance to the central SMBH is smaller than ∼10⁻⁵ pc to avoid large energy changes during close encounters with the SMBH, as discussed in Rodriguez et al. (2018).

In one of the models (SOFT), we consider the possibility of an IMC being a dense cluster with a Plummer potential and a softening parameter ${s}_{\mathrm{IMC}}^{{\prime} }(=\,5\times {10}^{3}$ au). In all models, the IMC's initial semimajor axis and eccentricity are a_IMC = 0.35 pc and e_IMC = 0.3, respectively.

We neglect the departure from spherical symmetry for the Galactic bulge. Although a triaxial mass distribution in the Galactic bulge can lead to the IMC's precession, which modifies the secular evolution of single and binary stars' orbits (Petrovich & Antonini 2017; Bub & Petrovich 2020), it is negligible in comparison with the IMC's secular perturbation on the relatively massive main-sequence stars within the central parsec from the GC during their life span (i.e., over a few to tens of megayears). However, a nonspherical component in the mass distribution in the Galactic halo can significantly modify the trajectories of HVSs on a dynamical timescale. We consider this possibility in model I60-TRI with a triaxial potential (Yu & Madau 2007) such that

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{\mathrm{halo}} & = & -\displaystyle \frac{{{GM}}_{\mathrm{halo}}}{{r}^{{\prime} }}\mathrm{ln}\left(1+\displaystyle \frac{{r}^{{\prime} }}{{a}_{\mathrm{halo}}}\right),\\ \mathrm{with}\ {r}^{{\prime} } & = & {p}_{\mathrm{halo}}^{1/3}{q}_{\mathrm{halo}}^{1/3}{\left({x}^{2}+\displaystyle \frac{{y}^{2}}{{p}_{\mathrm{halo}}^{2}}+\displaystyle \frac{{z}^{2}}{{q}_{\mathrm{halo}}^{2}}\right)}^{1/2},\end{array}\end{eqnarray} \tag{ 2 }$$

where p_halo and q_halo describe the asphericity of the halo potential. If p_halo = q_halo = 1, the halo potential is spherical. In the I60-TRI model, we set p_halo = 0.8 and q_halo = 0.7, with a z-axis normal to the Galactic disk, the same as the assumption in Yu & Madau (2007).

For the gaseous disk, we adopt a prescription similar to our previous models of protostellar disks. In the absence of a gap, the precession of the embedded stars is driven by the gaseous disks' central gravity,

$$\begin{eqnarray}&&{f}_{\mathrm{emb},\mathrm{gas}}=-4\pi G{\rm{\Sigma }}{Z}_{k},\end{eqnarray} \tag{ 3 }$$

where Z_k =1.094 is the same as in Ward (1981), Nagasawa et al. (2005), and Zheng et al. (2017a). This force is also applied to an IMC with an inclined orbit or a counterorbiting IMC. For a co-orbiting IMC with a semimajor axis a_IMC (=0.3–0.4 pc) and eccentricity e_IMC (=0.2–0.4), we assume that it opens up a gap in the disk with a width

$$\begin{eqnarray}&&{a}_{\mathrm{in}/\mathrm{out}}={a}_{\mathrm{IMC}}(1\mp {e}_{\mathrm{IMC}})\left[1\mp {\left(\displaystyle \frac{{M}_{\mathrm{IMC}}}{3{M}_{\mathrm{SMBH}}}\right)}^{1/3}\right]\end{eqnarray} \tag{ 4 }$$

such that the central gravity from the gaseous disk on it becomes

$$\begin{eqnarray}\begin{array}{rcl}{f}_{\mathrm{gap},\mathrm{gas}} & = & 2\pi G{\rm{\Sigma }}\sum _{l=0}^{\infty }{\left[\displaystyle \frac{(2l)!}{{2}^{2l}{\left(l!\right)}^{2}}\right]}^{2}\left(\displaystyle \frac{2}{4l+1}\right)\\ & & \times \left[(2l){\left(\displaystyle \frac{r}{{a}_{\mathrm{out}}}\right)}^{2l+1/2}-(2l+1){\left(\displaystyle \frac{{a}_{\mathrm{in}}}{r}\right)}^{2l+1/2}\right].\end{array}\end{eqnarray} \tag{ 5 }$$

For illustrative purposes, we assume a power-law surface density (Σ) distribution constructed for the minimum-mass solar nebula (Hayashi et al. 1985) such that

$$\begin{eqnarray}&&{\rm{\Sigma }}={{\rm{\Sigma }}}_{0}{\left(\displaystyle \frac{r}{{R}_{0}}\right)}^{-3/2}{e}^{-t/{\tau }_{\mathrm{dep}}}\,{\rm{g}}\,{\mathrm{cm}}^{-2},\end{eqnarray} \tag{ 6 }$$

where τ_dep(=3.16 Myr) is the depletion timescale of the gas nebula, and Σ₀(=600 g cm⁻²) and R₀(=10³ au) are the fiducial surface density and radii.

We assume that the stars emerge from the gaseous disk and that the gaseous disk preserves its spin orientation. In one model (I60-IS60), we assume that the gaseous disk is inclined to the Galactic disk by 60°. In all other models, the gaseous and Galactic disks are assumed to be in the same plane. For models where the orbital planes of the stars, IMC, and/or Galaxy are inclined, the inclination of the stars relative to the gaseous disk (i_*) evolves with time. The force in the (z) direction normal to the gaseous disk plane is approximated with

$$\begin{eqnarray}&&{f}_{{\text{}}z,{\rm{gas}}}=-2\pi G{\rm{\Sigma }}(R)\displaystyle \frac{\mathrm{sign}(z){\left(z/H\right)}^{2}}{1+{\left(z/H\right)}^{2}},\end{eqnarray} \tag{ 7 }$$

where the gaseous disk's aspect ratio $H/R=({H}_{0}/{R}_{0}){\left(R/{R}_{0}\right)}^{1/4}$ with H₀/R₀ = 10⁻².

In order to investigate the effects of sweeping secular resonance on the evolution of GC stars and their parameter dependence, we test a series of numerical models with various setups. Table 1 lists all of those models, including simulations with three different gas disk depletion timescales. In all cases, the stars are assumed to reside in the gaseous disk with initially circular orbits.

Table 1. Parameters for Various Numerical Simulations

Models	${s}_{\mathrm{IMC}}^{{\prime} }(\mathrm{AU})$ a	iIMC,⋆(deg) b	i⋆(deg) c	Gas Disk Potential d	Gas Disk Gap e	Halo Potential f
NGD	0	0	0	−	−	Spherical
Fiducial	0	0	0	+	+	Spherical
SOFT	5000	0	0	+	+	Spherical
CCW	0	180	0	+	−	Spherical
CCW-NGD	0	180	0	−	−	Spherical
I30	0	30	0	+	−	Spherical
I30-NGD	0	30	0	−	−	Spherical
I60	0	60	0	+	−	Spherical
I60-NGD	0	60	0	−	−	Spherical
I60-IS60	0	60	60	+	−	Spherical
I60-TRI	0	60	0	+	−	Triaxial
I120	0	120	0	+	−	Spherical
I120-NGD	0	120	0	−	−	Spherical

Notes.

^a The softening parameter of the IMC's potential. ^b The initial mutual inclination between the orbits of the IMC and disk stars. ^c The initial inclination of the orbit of disk stars (gas disk) relative to the Galactic disk (reference plane). ^d The presence of the gas disk potential. ^e The presence of a gas-free gap around the IMC's orbit. ^f The form of the halo potential.

Download table as:  ASCIITypeset image

We assess the influence of the sweeping secular resonance by turning off the gravity due to the gaseous disk in a comparative model no gas disk (NGD). We also simulated a SOFT model in which the IMC's gravity is approximated by a Plummer potential. In the fiducial, NGD, and SOFT models, we assume that Sgr A^⋆ and the companion, stars, and gaseous disk are all in the same plane, and their orbital angular momentum vectors are aligned. In the counter-clockwise (CCW) model, we consider the possibility of a counterorbiting IMC, as inferred from some observations of IRS 13E (Genzel et al. 2003). In a series of 3D cases, we assume a 30°, 60°, and 120° misalignment angle between the IMC's and the star's initial orbital plane.

To examine this possibility, we consider several models that are variations of the fiducial model (Figure 2). In all of these models, we use Σ₀ = 600 g cm⁻², τ_dep = 3.16 Myr, M_IMC = 10⁴ M_⊙, a_IMC = 0.35 pc, and e_IMC = 0.3. For each model, we randomly place a population of 5000 disk stars with an initial semimajor axis a₀ randomly distributed between 0.05 pc and a_in (∼0.22 pc; Equation (4)). We first analyze the results of the NGD model, which neglect the effect of the sweeping secular resonance. In this case, large eccentricity excitation occurs only for stars with orbits that are initially around 2:1 and 5:2 MMRs with the orbit of the IMC (at ∼0.22 and ∼0.19 pc, respectively). Very few stars are ejected into hyperbolic orbits with an asymptotic speed v > 500 km s⁻¹ after their ${r}_{\min }$ is reduced to ∼1 au (top panel in Figure 2).

Zoom In Zoom Out Reset image size

In the fiducial model (middle panel in Figure 2), the evolving gaseous disk introduces the effect of sweeping secular resonance. After 6 Myr (∼2 τ_dep), a population of disk stars in the region 0.15–0.22 pc from the SMBH attains hyperbolic orbits (e_⋆ > 1) and reaches a distance ≳100 pc with asymptotic speed with a magnitude v > 500 km s⁻¹ mostly in the direction away from the SMBH. We refer to these stars as the candidate HVSs. In our models, we have not included compact binary systems that are the progenitors of HVSs under the Hills hypothesis. This ejection channel for HVSs through sweeping secular resonance occurs rapidly during the epoch of disk depletion. It can account for the HVSs' young ages and high masses. Moreover, a minute fraction of stars attain ${r}_{\min }$ comparable to the tidal disruption radius (Frank & Rees 1976), ${r}_{\mathrm{tide}}\simeq {({M}_{\mathrm{SMBH}}/{m}_{\star })}^{1/3}{s}_{* }(\approx 3.5\times {10}^{-6}$ pc). For all of those very close approaching events, we recheck them with higher accuracy. These rare close encounters raise the possibility of infrequent tidal disruption events.

The HVSs' asymptotic speed in the fiducial model is ∼1% of the escape speed for hypothetical stars on circular orbits at ${r}_{\min }$. However, it is slightly larger than the IMC's orbital speed at its periastron. This "slingshot" is accomplished in three steps (see discussions in Section 3 and illustrative Figure 1). (1) Most of the "heavy lifting" is accomplished through the IMC's secular perturbation, which induces some stars to attain nearly parabolic orbits (with very little orbital energy). (2) These marginally bound stars receive some impulsive "gravitational assist" from the IMC during their close encounters. Although the fractional change on the stellar orbits (∼${({M}_{\mathrm{IMC}}{a}_{\mathrm{IMC}}/{M}_{\mathrm{SMBH}}{r}_{\mathrm{IMC},* })}^{1/2}$, where r_IMC,* = ∣r_IMC − r∣ is the stars' distance from the IMC) is modest, the IMC and SMBH's gravity together may significantly modify some stars' energy and impact parameters relative to the SMBH. (3) Some of the scattered stars undergo an additional close encounter with the SMBH and escape from its gravitational confinement with e_⋆ ≫ 1.

We select a group of HVSs and plot their distance r_IMC,* and angular separation Δθ_*,IMC = ∣θ_IMC − ω_*∣ from the IMC during or immediately after the instant of their closest encounters (Figure 3). Dispersion of Δθ_*,IMC from π is due to the scattering during the closest encounters, and it can attain nonnegligible values for stars with orbits marginally overlapping that of the IMC. The ejected HVSs' distances from Sgr A^⋆ at the end of the calculation are indicated by color. The lack of correlation between the minimum star–IMC distance and the HVSs' distances from Sgr A^⋆ is an indication that the IMC is not the primary culprit of hypervelocity ejection. In contrast to the correlation between ${r}_{\min }$ and the HVSs' final distances at the end of the simulation, Figures 2 and 4 provide supporting evidence to suggest that their scattering by the SMBH after their impulsive close encounters with the IMC was the event that led to their escape with hypervelocities.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the model SOFT, we adopt a Plummer model for the companion's gravitational potential. The softening parameter ${s}_{\mathrm{IMC}}^{{\prime} }\sim 0.07\,{a}_{\mathrm{IMC}}$ is marginally smaller than its Hills radius, ${r}_{{\rm{H}}}={({M}_{\mathrm{IMC}}/3{M}_{\mathrm{SMBH}})}^{1/3}{a}_{\mathrm{IMC}}\sim 0.09\,{a}_{\mathrm{IMC}}$. This modified potential weakens the impact of star–IMC close encounters without sacrificing the IMC's distant secular perturbation on the stars. In addition, we turn off the merging possibility between the IMC and disk stars only in this SOFT model. The bottom panel of Figure 2 indicates that e_* for a large fraction of stars is excited to ∼1. With their nearly zero total orbital energy, close encounters with the IMC's relatively shallow potential are adequate to perturb these stars from parabolic to hyperbolic orbits. The result of the SOFT model suggests that the IMC does not need to be an intermediate-mass black hole (IMBH), and the distributed potential of a compact stellar cluster is a sufficient agitator for the HVSs.

We also simulate a CCW model with the identical gas and stellar disks as the fiducial model. The IMC's M_IMC, a_IMC, and e_IMC are also identical, but its orbit is in the counterclockwise direction (i.e., its orbital angular momentum is antiparallel to that of the stars and gas disk). A wealth of disk stars in more extended regions are significantly excited (with e_⋆ ≫ 1) by this counterorbiting IMC, albeit in the 2D limit. However, those disk stars initially distributed within ∼0.13 pc are less perturbed compared to the fiducial and SOFT models in Figure 2. Moreover, the MMRs are less pronounced in this model, as shown in Figure 4. The CCW model can produce many more ejections and HVSs than the fiducial/SOFT models. The frequency of HVSs is close to 1% among all disk stars (shown in Figure 5). This result is consistent with the analytic calculation in Paper I of this series, where we showed that a counterorbiting IMC could indeed induce secular resonances over a wide region. Although the presence of a gas disk introduces additional precession that initially reduces the effectiveness of the IMC's secular perturbation, the influence of a counterorbiting IMC is reinstalled after the disk depletion. In another series of CCW-NGD models, a counterorbiting IMC is imposed without the gas disk potential (i.e., with Σ₀ = 0). The results in Figure 4 show that it is just as capable of exciting e_⋆ and ejecting HVSs.

Zoom In Zoom Out Reset image size

In most of the above models, we assume that the simulated disk stars formed in the midplane of a gas disk, and they are coplanar to a co-orbiting or counterorbiting IMC. Due to the uncertainties in the observed orbit of IRS 13E, a more general analysis is warranted. Although we can generalize the analytic calculation in Paper I, many degrees of freedom expand its complexity. Instead, we adopt additional illustrative series of numerical simulations. For models with evolving gas disks, we use the same value of the Σ distribution (but without a gap), Σ₀, and τ_dep as the fiducial model.

In I30, the orbits of all of the disk stars are initially set to be in the same plane with i_* = 0°. We also adopt a mutual initial inclination of i_IMC − i_⋆ = 30° for the IMC's orbit. The IMC's mass, semimajor axis, and eccentricity (M_IMC, a_IMC, and e_IMC) in this and other models with inclined stellar orbits are identical to those in the fiducial model. Similarly, models I60 and I120 are set with an initial mutual inclination between the disk stars and the IMC of 60° and 120°, respectively. For all of these inclined models, disk stars are initially embedding in the gaseous disk and coplanar to the Galactic disk (referred to as the reference plane), but the IMC is in an inclined orbit. Accordingly, the related NGD models, I30-NGD, I60-NGD, and I120-NGD, describe the same misalignment between the stars and IMC but without considering the existence of a gas disk.

Compared to the 2D cases, the sweeping secular resonance of an inclined IMC can still significantly excite disk stars, albeit less effectively, especially for a highly inclined orbit, e.g., the I60 model. Even though there is no gas disk under the coplanar circumstance, the IMC's secular perturbation and MMRs can stimulate some hypervelocity escapees in the NGD model. However, in the slightly inclined case without a gas disk, model I30-NGD, only a few disk stars can be incredibly excited with a periapse distance smaller than 10⁻⁴ pc. As a result, no HVS was ejected. The situation is improved in the I60-NGD model. In the absence of a gaseous disk, the sweeping secular resonance effect does not contribute to any HVS population in this case. The mutual inclination between the IMC and the disk stars is larger than the threshold value of the Lidov–Kozai secular resonance around a point-mass potential, which is generally set as 39°. Even though the Galactic bulge potential introduces additional precession, more disk stars are excited and ejected in the I60-NGD model than the I30-NGD model.

In Figure 6, we focus on the inclination distribution of disk stars under each inclined model after ∼6 Myr evolution. For a clearer contrast, the left panels of Figure 6 show the results of inclined cases with the gas disk, while the right panels are the corresponding models without the gas disk. As expected, the presence of a gas disk not only introduces sweeping secular resonance, which leads to efficient e_* excitation and ejection of HVSs, but also damps the stars' excited inclination relative to the disk and preserves their initial i_*, especially for those residuals at r_* < 0.1 pc (see also Paper I).

Zoom In Zoom Out Reset image size

In Paper I, we show that the characteristic timescales for secular evolution for the GC stars depend weakly on a_*. Since they are comparable to the simulated duration (∼6 Myr) for all of the models, typical stars, including those with a_* ≲ 0.1 pc, went through approximately one to a few secular cycles in both e_* and (i_* − i_IMC). This pattern is evident in the relatively low e_* and the gradient of (i_* − i_IMC) after ∼6 Myr. The range of e_* modulation is small for a_* < 0.1 pc. However, the eccentricities of some more distant stars (with a_* > 0.1 pc) are significantly excited. These stars also endure large changes in i_* from their initial values. Due to their contribution to the potential, changes in i_* in models with a gas disk are generally smaller than those without a gas disk (i.e., the NGD models).

The potential for the gas disk modifies the frequency of both longitudinal and nodal precessions. The disk potential diminishes on a timescale τ_dep, which is also comparable to the duration of the simulation and the eccentricity excitation timescale in models I30, I60, and I120. Direct comparisons between these models with their diskless counterparts (models I30-NGD, I60-NGD, and I120-NGD) indicate that changes in both e_* and i_* are reduced for the close-in stars (with a_* < 0.1 pc) by the disk potential (Figures 7 and 6). Outside ∼0.1 pc, changes in the disk potential during its depletion significantly enlarge the radial extent of the IMC's secular resonance (cf. equivalent models without the evolving gas disks). Significant eccentricity excitation (leading to ${r}_{\min }\lt {10}^{-3}$ pc) and inclination changes occur for stars with a₀ ∼ 0.15–0.22 pc, similar to the fiducial and SOFT models. This region is populated by stars with a wide range of inclination and relatively high eccentricity. This result may be linked to the observed inclination diversity of the young stars around Sgr A^⋆ (Lu et al. 2006, 2009; Paumard et al. 2006). While the eccentricity of many stars is excited to values comparable to that observed among the CCWs, few stars acquire an ${r}_{\min }$ small enough (≲10⁻⁴ pc) to undergo a close encounter with the SMBH and then be ejected as HVSs. Although the frequency of HVSs is increased by the presence of the disk, it is also reduced by the IMC's inclined orbit (Figure 5).

Zoom In Zoom Out Reset image size

At a distance greater than ∼1 kpc, the HVSs' orbital trajectories are under the influence of the Galactic disk and halo. In all of the above inclined models, we assume the initial orbits of all of the stars (including those eventually ejected as HVSs) are in the same plane as the Galactic disk. We carried out another model in which we considered the possibility that the stellar disk is inclined 60° to the Galactic disk (Paumard et al. 2006) in Figures 8 and 9. It shows that the HVSs' asymptotic trajectories are indeed randomly distributed rather than clustering around the Galactic disk's pole.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In all but one of the simulations, we adopt a spherically symmetric potential for the Galactic halo. Triaxial mass distribution in the Galactic halo (Wegg et al. 2019) can lead to the randomization of the HVSs' orbits relative to the Galactic plane and the reorientation of their trajectories from a radially outward direction relative to the GC (Yu & Madau 2007; Brown 2015; Du et al. 2019). In Figure 8, we plot the stars' orbital inclination relative to their natal disk and galactocentric distance. Their position in the Galactic longitude and latitude viewed from the GC is shown in Figure 9, in a similar way as Lu et al. (2010).

In model I60-TRI, we use Equation (2) to approximate a triaxial mass distribution and potential. We plot the distribution of the angles

$$\begin{eqnarray}&&\begin{array}{l}\theta ({\boldsymbol{r}},{\boldsymbol{v}})={\cos }^{-1}\left(\displaystyle \frac{{\boldsymbol{r}}\cdot {\boldsymbol{v}}}{| {\boldsymbol{r}}| | {\boldsymbol{v}}| }\right)\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \ \\ \phi ({\boldsymbol{z}},{\boldsymbol{v}})={\cos }^{-1}\left(\displaystyle \frac{{\boldsymbol{z}}\cdot {\boldsymbol{v}}}{| {\boldsymbol{z}}| | {\boldsymbol{v}}| }\right)\end{array}\end{eqnarray} \tag{ 9 }$$

for the HVSs after 60 Myr (left panel of Figure 10). This sample of 14 HVSs is chosen with r > 10² pc and v > 500 km s⁻¹ as in Figure 2. Six HVSs that are escaping the Galactic halo reach 30–60 kpc with relatively small θ( r , v ); i.e., their trajectories have not deviated significantly from the radial direction. Seven HVS distances remain within ∼10 kpc. Their θ( r , v ) are widely distributed, as shown by Yu & Madau (2007). Although the triaxial potential has randomized their trajectory, there is no obvious indication of clustering along the minor axis of the halo potential.

Zoom In Zoom Out Reset image size

We also plot the evolution of θ( r , v ) (right panel of Figure 10) for the same HVS sample. The most distant stars are also the fastest-moving stars. Their trajectories have not deviated significantly from radial motion away from the GC. However, the orientation of the closer HVS appears to evolve under the influence of nonspherical halo potential. Thus, we do not expect the observed proper motion of all of the HVSs to be directed away from the GC.

We also find that in the CCW model (not plotted here), most of the escaping stars retain their initial retrograde orbits (with i_⋆ = 180°), though the magnitude of their e_⋆ is extensively excited. However, some escaping disk stars flip their orbital angular momentum vector to follow the orbital direction of the IMC. Since both the disk stars and the IMC are in the same orbital plane in the CCW model, no star attains an inclined orbit. Nevertheless, misalignment between the orbit of the disk stars with respect to the Galactic plane and a triaxial mass distribution in the halo can also change the spatial distribution of HVSs in this model.