PREDICTED SPACE MOTIONS FOR HYPERVELOCITY AND RUNAWAY STARS: PROPER MOTIONS AND RADIAL VELOCITIES FOR THE GAIA Era

To establish a framework for analyzing numerical simulations, we consider an analytic model for the proper motions of stars with simple trajectories in the Galaxy. In a Cartesian coordinate system with an origin at the GC, stars have positions (x, y, z) and velocities (v_x, v_y, v_z). The distance from the GC to the star is r; the space velocity of the star relative to the GC is v. The angle of the position vector of the star relative to the x axis is θ; the angle relative to the x–y plane is ϕ. To distinguish these angles from standard galactic longitude and latitude, we call θ (ϕ) the GC longitude (GC latitude).

In this convention, we specify coordinates in both the Cartesian (x, y, z) and spherical (r, θ, ϕ) systems (see also Binney & Tremaine 2008). These systems are appropriate for stars in the Galactic bulge or halo, where the potential is roughly spherically symmetric. To make a clear link with the cylindrical coordinate system more appropriate for the Galactic disk, we define the cylindrical radius ρ² = x² + y². In this system, we specify coordinates with (ρ, θ, z).

To connect these coordinates to the heliocentric galactic system, we assign the Sun a position (− R_☉, 0, 0) and a velocity (0, v_☉, 0), where R_☉ = 8 kpc is the distance of the Sun from the GC (e.g., Bovy et al. 2012) and v_☉ is the space velocity of the Sun relative to the GC (Figure 1). Each star then has a distance d = ((x + R_☉)² + y² + z²)^1/2 from the Sun and a relative velocity $v_{{\rm rel}} = (v_x^2 + (v_y - v_\odot)^2 + v_z)^{1/2}$. In this system, the galactic longitude l of the star is the angle—measured counter-clockwise in the x–y plane—from a line connecting the Sun to the GC, l = tan⁻¹(x tan θ/(x + R_☉)). The galactic latitude measures the height of the star above the galactic plane, b = sin⁻¹(z/d) = sin⁻¹(r sin ϕ/d). For r ≫ R_☉, θ ≈ l and ϕ ≈ b.

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Although these coordinate systems are clearly defined, angles in the heliocentric galactic system span a smaller range than in the pure GC system (Figure 2). For stars with positions r < R_☉, the range of θ (−π to π) is larger than the range of l (−l_max to l_max), where

$$\begin{equation} l_{{\rm max}} = {\rm sin^{-1}} (\rho / R_\odot). \end{equation} \tag{ 1 }$$

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When l = l_max, θ_l = cos⁻¹(− ρ/R_☉). For each l < l_max, there are two values⁴ of θ.

We derive the radial velocity v_r, the tangential velocity v_t, and the proper motion μ in the heliocentric frame. For all stars, $v_{{\rm rel}}^2 = v_r^2 + v_t^2$. The radial velocity is

$$\begin{equation} v_r = v_x {\rm \,cos\,} l {\rm \,cos\,} b + (v_y - v_\odot)\, {\rm sin\,} l {\rm \,cos\,} b + v_z {\rm \,sin\,} b. \end{equation} \tag{ 2 }$$

We separate the tangential velocity into two components, v_l and v_b, where $v_t^2 = v_l^2 + v_b^2$. The component along the direction of galactic longitude is

$$\begin{equation} v_l = -v_x {\rm \,sin\,} l + (v_y - v_\odot)\, {\rm cos\,} l. \end{equation} \tag{ 3 }$$

The latitude component is

$$\begin{equation} v_b = -(v_x {\rm \,cos\,} l + (v_y - v_\odot)\, {\rm sin\,} l)\, {\rm sin\,} b + v_z {\rm \,cos} \,b. \end{equation} \tag{ 4 }$$

For stars with b ≡ 0 and no motion in the z-direction, v_b = 0. Although each component of the tangential velocity has a clearly defined sign convention, we plot the absolute magnitude when we combine the two components into the tangential velocity, $v_t = | v_t | = (v_l^2 + v_b^2)^{1/2}$.

The standard definition for the proper motion is

$$\begin{equation} \mu = { v_t \over {4.74 d} }, \end{equation} \tag{ 5 }$$

where v_t is measured in km s⁻¹ and d is in pc. To set the proper motion in the heliocentric galactic frame, μ_l = v_l/(4.74d) and μ_b = v_b/(4.74d), where the velocities are in km s⁻¹. Positive (negative) proper motions are in the direction of increasing (decreasing) l or b.

Within this framework, we consider several simple stellar motions to explore the variation of v_r and v_t with position in the Galaxy. Most motions are composed of both a circular component and a radial component. Starting with stars following circular orbits around the GC inside and outside the solar circle (Section 3.2.1), we derive the behavior of v_r and v_t with θ, l, and b for stars with total velocity v. In this simple example, we set z = 0 and work in a coordinate system where r = ρ. The maximum tangential velocity is then fixed at v_{t, max} = v + v_☉; the maximum radial velocity falls with r inside the solar circle and then grows with r outside the solar circle. At large r, v_{r, max} ≈ v_☉. Continuing with stars on purely radial orbits (Section 3.2.2), we explore motions in the spherical coordinate system appropriate for the bulge and the halo. For stars inside the solar circle, the maximum radial and tangential velocities are v_{r, max} ≈ v and v_{t, max} ≈ v + v_☉. Extrema in v_r lie at l ≈ 0; stars have maximum v_t at l ≈ l_max. Well outside the solar circle, the maximum radial velocity (v_{r, max} ≈ v + v_☉) is much larger than the maximum tangential velocity (v_{t, max} ≈ v_☉). At intermediate r ≈ 8–20 kpc, there is a smooth transition from small v_{r, max} and large v_{t, max} to large v_{r, max} and small v_{t, max}.

3.2.1. Circular Orbital Motion

For stars following simple circular orbits around the GC, v_x = vsin θ, v_y = −v cos θ, and v_z = 0 (Figure 2). Although stars in the thin and thick disks have finite vertical distances from the Galactic plane and non-zero motion out of the Galactic plane, we set z ≡ 0 and ignore any out-of-plane motion here. Thus, our radial coordinate r is identical to the standard cylindrical coordinate ρ. With ϕ = b = 0, the heliocentric radial and tangential velocities are

$$\begin{equation} v_r = v {\rm \,sin\,} \theta \, {\rm cos\,} l - (v\, {\rm cos\,} \theta + v_\odot)\, {\rm sin\,} l \end{equation} \tag{ 6 }$$

and

$$\begin{equation} v_t = -v\, {\rm sin\,} \theta {\rm \,sin\,} l - (v {\rm \,cos\,} \theta + v_\odot)\, {\rm cos\,} l. \end{equation} \tag{ 7 }$$

In this system, v_b = 0 and v_t = v_l. Using trigonometric identities, we can simplify these to

$$\begin{equation} v_r = v {\rm \,sin} (\theta - l) - v_\odot {\rm \,sin\,} l \end{equation} \tag{ 8 }$$

and

$$\begin{equation} v_t = -v {\rm \,cos} (\theta - l) - v_\odot {\rm \,cos\,} l. \end{equation} \tag{ 9 }$$

For convenience, we can eliminate θ in the expression for the radial velocity,

$$\begin{equation} v_r = \left({R_\odot \ \over r} v - v_\odot \right) {\rm sin\,} l. \end{equation} \tag{ 10 }$$

Figure 3 illustrates the variation of v_r (dashed curves) and v_t (solid curves) with GC longitude for stars with v = 250 km s⁻¹ and r = 5 kpc (cyan lines) and r = 50 kpc (magenta lines). In this configuration, stars on the opposite side of the Galaxy from the Sun (θ = 0) have no net radial velocity and a maximum tangential velocity of v + v_☉. For v_☉ = 250 km s⁻¹, v_{t, max} = 500 km s⁻¹. Stars on the near side of the Galaxy (θ = ±π) have no net radial or tangential velocity. Thus, the minimum tangential velocity is v_{t, min} = 0.

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The behavior of v_r depends on r. For all r, v_r = 0 at θ = 0 and ±π. When r < R_☉, maximum positive v_r is at θ = −θ_l (l = −l_max). Maximum negative v_r is at θ = +θ_l. With a maximum radial velocity, v_{r, max} = ±(v − v_☉r/R_☉), the amplitude of the radial velocity variation declines from roughly 250 km s⁻¹ at r ≈ 0 to roughly zero at r ≈ R_☉. Outside the solar circle, the extrema in v_r lie at l = ±π/2 (θ = cos⁻¹(− R_☉/r)). With v_{r, max} = vR_☉/r − v_☉, the amplitude of the v_r variation grows from zero at r ≈ R_☉ to v_☉ at r ≫ R_☉. Thus,

$$\begin{equation} v_{r,{\rm max}} = \left\lbrace \begin{array}{@{}lll}\pm \left(v - {r \over R_\odot } v_\odot \right) & & r < R_\odot \\ \\ \pm \left({R_\odot \over r} v - v_\odot \right) & & r \ge R_\odot. \\ \end{array} \right. \end{equation} \tag{ 11 }$$

In the heliocentric galactic frame, the variation of v_r and v_t with l is somewhat different (Figure 4). For stars inside the solar circle, v_t follows an egg-shaped loop with minimum and maximum velocity at l = 0. Here, l_max sets the maximum extent of the loop in galactic longitude. Thus, the "egg" widens at larger r, reaching l_max ≈ ±π/2 at r ≈ R_☉. Outside the solar circle, v_t varies sinusoidally with l, with maxima of v + v_☉ = 500 km s⁻¹ at l = ±π and a minimum of zero at l = 0.

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The radial velocity also follows simple trajectories. At small r, v_r varies along a curved line with extrema of v_{r, max} (Equation (11)) at l_max. As r grows, the curves extend to larger l but have smaller maxima. At large r, v_r follows a simple sinusoid, with extreme values set by v_☉ at l = ±π/2 and zero-crossings at l = 0 and l = ±π.

3.2.2. Radial Motion

Stars moving radially away from the GC have subtly different behavior. To infer conclusions appropriate for stars in the bulge or the halo, we consider stars with a broad range of GC latitude. In the GC frame, outflowing stars have constant ϕ for all r (see Figure 5). In the heliocentric frame, nearby stars have larger b than more distant stars. For stars with ϕ > 0, r > ρ. Thus, distant stars with large b may lie inside the solar circle.

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With v_x = v cos θ cos ϕ, v_y = v sin θ cos ϕ, and v_z = v sin ϕ, the heliocentric radial and tangential velocities are

$$\begin{equation} v_r = v {\rm \,cos} (\theta - l)\, {\rm cos\,} \phi \, {\rm cos\,} b - v_\odot {\rm \,sin} \,l {\rm \,cos\,} b + v {\rm \,sin\,} \phi \, {\rm sin\,} b, \end{equation} \tag{ 12 }$$

$$\begin{equation} v_l = v\, {\rm sin} (\theta - l)\, {\rm cos\,} \phi - v_\odot {\rm \,cos\,} l, \end{equation} \tag{ 13 }$$

and

$$\begin{equation} v_b = -v {\rm \,cos} (\theta - l) {\rm \,cos} \,\phi \, {\rm sin\,} b - v_\odot {\rm \,sin\,} l\, {\rm sin\,} b + v {\rm \,sin\,} \phi \, {\rm cos\,} b. \end{equation} \tag{ 14 }$$

When stars move radially outward through the Galactic plane, ϕ = b = 0. With v_b = 0, the equations for radial and tangential velocity are then very simple: v_r = v cos(θ − l) − v_☉sin l and v_t = v_l = v sin(θ − l) − v_☉ cos l. For stars inside the solar circle, the maximum galactic longitude is l = l_max.

When ϕ > 0, the variations of v_r and v_t with θ and l are more complex. Aside from having a non-zero v_b, the amplitude of both velocity components declines with cos ϕ. For stars inside the solar circle, the maximum l scales with ϕ: R_☉ sin l_max = r cos ϕ. Thus, stars inside the solar circle at large ϕ have a smaller range in l than stars with small ϕ.

Figure 6 shows the variation of v_r (dashed curves) and v_t (solid curves) as a function of l for stars with r = 5 kpc, v = 500 km s⁻¹, and ϕ = 0^o (violet curves), 30^o (blue curves), 60^o (cyan curves), and 75^o (magenta curves). With l_max∝ϕ, curves at larger ϕ have a smaller extent in l. For stars at r = 5 kpc, the maximum galactic latitude is b ≈ 30^o–40^o. Both sets of curves follow loops in the l, v plane. For ϕ = 0, the v_t curve folds back on itself.

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When stars lie inside the solar circle, the radial velocity varies symmetrically about an average velocity v sin ϕ sin b. This average increases with ϕ, reaching v_{r, avg} = v_☉ cos b + v sin b at ϕ = 90^o. With b ≈ 30^o, v_{r, avg} ≈ v_☉. The amplitude of the v_r variation scales with cosϕ and thus declines markedly from Δv_r ≈ v to Δv_r ≈ 0 among the sequence of four curves.

The variation of v_t with l is not symmetric. At ϕ = 0^o, the tangential velocity ranges from a minimum of 0 to a maximum close to v + v_☉, roughly 700 km s⁻¹. At large ϕ, v_t approaches a constant value of roughly vcosb + v_☉sin b ≈v for v ≫ v_☉ and b ≈ 30^o–40^o.

Well outside the solar circle (r = 50 kpc), the motions are much simpler (Figure 7). At large r, v_r varies roughly sinusoidally with l about an average velocity of v_{r, avg} ≈ v. The amplitude of this variation decreases with b, reaching a constant v_r ≈ v when r ≫ R_☉ and b ≈ 90^o. Stars reach a minimum (maximum) v_r at l = ±π/2.

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The tangential velocity has a smaller amplitude and different phasing with l. At large b, the tangential velocity is roughly v_☉, a result of reflex solar motion. For stars close to the plane ϕ ≈ 0 and b ≈ 0, the tangential velocity consists of two sinusoids with amplitudes of v and v_☉ (v_t ≈ v_l = vsin (θ − l) − v_☉ cos l; Equation (13)). Thus, the minimum v_t is small and approaches v_t = 0 at b = 0. Because the Sun is offset from the GC, the phase of minimum v_t is offset from ±π/2. The solar motion and position in the galaxy produce an offset of roughly −20^o in longitude.

For stars with 5 kpc ≲ r ≲ 50 kpc, there is a smooth transition between the behavior shown in Figures 6 and 7. Stars with in-plane distances less than R_☉ (ρ < R_☉) follow the trajectories in Figure 6. Stars outside this limit follow the trajectories in Figure 7.

For an ensemble of stars with r ≈ 16 kpc, as an example, stars with ϕ ≲ π/3 have ρ ≳ R_☉ and follow the trajectories in Figure 7. Stars at larger ϕ have the closed loop trajectories in Figure 6.

To illustrate this transition in more detail, Figure 8 shows the variation of the maximum and minimum radial velocity (lower panel) and tangential velocity (upper panel) as a function of r and ϕ for stars on radial orbits with v = +500 km s⁻¹ relative to the GC. At small ϕ, the minimum v_t is close to zero for all r. This minimum increases with ϕ until v_t = v_☉. The maximum v_t is roughly constant at 500–750 km s⁻¹ at small r and then decreases smoothly to v_☉ at large r.

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The extrema in v_r have similar trends. Inside the solar circle, stars on the near side of the GC all move toward the Sun and have large negative v_r. On the far side of the GC, all stars move away from the Sun. Thus, the range in v_r is largest for stars inside the solar circle. Because v_r scales with cos ϕ, stars at small (large) ϕ have the largest (smallest) range in v_r.

Outside the solar circle, all stars in this example move away from the Sun. The minimum radial velocity is then always larger than zero, producing the large increase in minimum v_r at r = R_☉. Somewhat counterintuitively, the maximum v_r also grows with r. Stars with r ≳ R_☉ have the largest velocity with respect to the Sun when they move radially outward roughly along the y-axis. As r increases, the angle between the y-axis and the line-of-sight from the Sun to the star decreases. Thus, the radial component of the relative velocity grows with r, reaching v + v_☉ when r ≫ R_☉.

Figure 9 repeats Figure 8 for heliocentric distance d. Aside from a clear discontinuity in the minimum v_r for ϕ = 0 at d = R_☉, the behavior in v_r is almost identical to Figure 8. The minimum v_r crosses from negative to positive v_r at d = R_☉. The maximum v slowly increases to v + v_☉ at large r. The variation in the minimum v_t is also similar, a slowly decreasing function of increasing r.

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The maximum in the tangential velocity, however, exhibits a clear maximum for stars with ϕ ≲ 30^o at d = 8 kpc. Stars close to the GC produce this maximum. When GC stars move radially outward in the y–z plane, their tangential velocity is at a maximum. For ϕ = 0, this peak in v_t is a sharp feature. Although still visible for ϕ ≈ 5^o to 30^o, the feature vanishes for larger ϕ.

Despite the simple stellar motions in these examples, the behavior of v_r and v_t with d, l, and b is amazingly rich. For stars inside the solar circle, circular orbital motion and radial outflow produce large maximum tangential velocities, v_{t, max} ≈ v + v_☉. These maxima occur at distinct galactic longitudes: l ≈ 0 for stars orbiting the GC and l ≈ −l_max for stars moving radially away from the GC. Thus, proper motion measurements offer some promise for distinguishing high velocity stars ejected from the GC from stars on circular orbits around the GC.

Outside the solar circle, circular orbital motion is also distinct from purely radial motion. For stars orbiting the GC, the maximum v_t is independent of r. However, the maximum tangential velocity of radially outflowing stars gradually declines with r until v_{t, max} ≈ v_☉. This maximum v_t changes little with l. For distant stars, orbital motion yields a larger v_{t, max} than radial motion.

Trends of v_r with r and d are opposite those of v_t. Inside the solar circle, stars on circular orbits have smaller and smaller v_r at larger and larger r. For stars moving radially away from the GC, v_r has clear minima and maxima of ±v at l ≈ ±l_max. Outside the solar circle, stars on circular orbits have maximum radial velocity v_{r, max} ≈ v_☉ at large r. Stars moving radially away from the GC have much larger maximum v_r, with v_{r, max} ≈ v + v_☉. Thus, radial velocity measurements excel at separating distant stars on roughly circular orbits from high velocity stars moving radially outward from the GC.

To conclude this section, we derive predicted proper motions for stars moving radially away from the GC (Figure 10). Close to the Sun, proper motions are large, roughly 100 mas yr⁻¹. The range in proper motions is small (large) for stars at high (small) galactic latitude. At large distances (d ≈ 50–100 kpc), the maximum proper motion of roughly 1 mas yr⁻¹ results from solar reflex motion.

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At intermediate distances, there is a small "peak" at d ≈ 8 kpc in the trend of μ∝d⁻¹. Stars inside the solar circle with l ≈ l_max produce this peak. For b ≈ 0^o–10^o, the peak has the largest contrast with the general trend in proper motion (see Figure 9). At the largest galactic latitudes (b ≳ 60^o), the peak fades considerably.

These results demonstrate that radial velocities can isolate high velocity stars from the space motions of typical stars in the Galaxy. Radial velocity measurements succeed at large d, where observations can easily separate HVSs or runaways with v_r ≳ +300 km s⁻¹ from normal halo stars with |v_r| ≲ 100 km s⁻¹ (e.g., Brown et al. 2014 and references therein).

Inside the solar circle, proper motion measurements provide a clear path for isolating high velocity stars from bulge and disk stars orbiting the GC. Among B-type stars with d ≲ 1 kpc, typical proper motions are 10–40 mas yr⁻¹ (e.g., de Zeeuw et al. 1999). This motion is a factor of 3–10 times smaller than the predicted motion for nearby HVSs and runaways with space velocities of 500 km s⁻¹ (Figure 10). The observed velocity dispersion (σ_r ≈ 100 km s⁻¹) of stars in the Galactic bulge implies typical proper motions of 1–5 mas yr⁻¹ (Tremaine et al. 2002; Soto et al. 2014), smaller than the 10–20 mas yr⁻¹ predicted for high velocity HVSs and runaways escaping the inner Galaxy.

For all distances, proper motions alone cannot easily separate ejected stars from indigenous halo stars. The maximum proper motions of typical halo stars with d ≈ 1–10 kpc, ∼30–50 mas yr⁻¹ (Kinman et al. 2007, 2012; Bond et al. 2010), are comparable to the likely proper motions of typical ejected stars. We return to this issue in Section 8.5 with a direct comparison between observations of halo stars and predictions from our numerical simulations.

As in Kenyon et al. (2008), we adopt a three-component model for the Galactic potential Φ_G (for other approaches, see Gnedin et al. 2005; Dehnen et al. 2006; Yu & Madau 2007):

$$\begin{equation} \Phi _G = \Phi _b + \Phi _d + \Phi _h, \end{equation} \tag{ 15 }$$

where

$$\begin{equation} \Phi _b(r) = - G M_b / (r + a_b) \end{equation} \tag{ 16 }$$

is the potential of the bulge,

$$\begin{equation} \Phi _d(\rho,z) = - G M_d /\sqrt{\rho ^2 + \left[a_d + \left(z^2+b_d^2\right)^{1/2}\right]^2} \end{equation} \tag{ 17 }$$

is the potential of the disk, and

$$\begin{equation} \Phi _h(r) = - G M_h \ln (1+r/r_h) / r \end{equation} \tag{ 18 }$$

is the potential of the halo (e.g., Hernquist 1990; Miyamoto & Nagai 1975; Navarro et al. 1997).

For the bulge and halo, we set M_b = 3.76 × 10⁹ M_☉, M_h = 10¹² M_☉, r_b = 0.1 kpc, and r_h = 20 kpc. These parameters match measurements of the mass and velocity dispersion inside 1 kpc and outside 50 kpc (see Section 2.2 of Kenyon et al. 2008).

To match a circular velocity of 235 km s⁻¹ at the position of the Sun (e.g., Hogg et al. 2005; Bovy et al. 2012; Reid et al. 2014), we adopt parameters for the disk potential M_d = 6 × 10¹⁰ M_☉, a_d = 2750 kpc, and b_d = 0.3 kpc. The complete set of parameters for the bulge, disk, and halo yields a flat rotation curve from 3–50 kpc.

To select $\boldsymbol {r}_0$ and $\boldsymbol {v}_0$, we rely on published calculations for HVSs and runaways. For HVSs, we consider a model where a single SMBH at the GC disrupts close binary systems with semimajor axes a_bin between a_min and a_max (Hills 1988; Kenyon et al. 2008; Sari et al. 2010). Our choice of the minimum semimajor axis a_min minimizes the probability of a collision between the two binary components during the encounter with the black hole (Ginsburg & Loeb 2007; Kenyon et al. 2008). Setting the maximum semimajor axis a_max ≈ 4 AU limits the number of low velocity ejections that cannot travel more than 10–100 pc from the GC and use a substantial amount of computer time.

4.2.1. Hypervelocity Stars

Numerical simulations of binary encounters with a single black hole demonstrate that the probability of an ejection velocity v_ej is a Gaussian,

$$\begin{equation} p_{\rm H}(v_{\rm ej}) \propto e^{(-(v_{\rm ej}- v_{\rm ej,H})^2 / \sigma _v^2)}, \end{equation} \tag{ 19 }$$

where the average ejection velocity is

$$\begin{eqnarray} v_{\rm ej,H} &=& 1760 \left(\frac{a_{{\rm bin}}}{\rm 0.1\ AU}\right)^{-1/2} \left(\frac{M_1+ M_2}{2\ M_\odot }\right)^{1/3} \nonumber\\ && \times \left(\frac{M_{\rm bh}}{3.5 \times 10^6\ M_\odot }\right)^{1/6} f_{R}\ \ {\rm km\ s^{-1}}, \end{eqnarray} \tag{ 20 }$$

and σ_v ≈ 0.2 v_{ej, H} (Bromley et al. 2006). Here M₁ (M₂) is the mass of the primary (secondary) star and M_bh is the mass of the central black hole. The normalization factor f_R depends on r_close, the distance of closest approach to the black hole:

$$\begin{eqnarray} \nonumber f_{R}& = & 0.774+(0.0204+(-6.23\times 10^{-4} +(7.62\times 10^{-6} \\ && + (-4.24\times 10^{-8} +8.62\times 10^{-11}D)D)D)D)D, \end{eqnarray} \tag{ 21 }$$

where

$$\begin{equation} D = D_0 \left(\frac{r_{{\rm close}}}{a_{{\rm bin}}}\right) \end{equation} \tag{ 22 }$$

and

$$\begin{equation} D_0 = \left[\frac{2\ M_{\rm bh}}{10^6 (M_1+ M_2)}\right]^{-1/3}. \end{equation} \tag{ 23 }$$

This factor also sets the probability for an ejection, P_ej:

$$\begin{equation} P_{{\rm ej}} \approx 1 - D/175 \end{equation} \tag{ 24 }$$

for 0 ⩽ D ⩽ 175. For D > 175, r_close ≫ a_bin; the binary does not get close enough to the black hole for an ejection and P_ej ≡ 0.

To establish initial conditions, we select each HVS from a random distribution of a_bin, r_close, and v_ej. The binaries have semimajor axes uniformly distributed in log a_bin (e.g., Abt 1983; Duquennoy & Mayor 1991; Heacox 1998). For binaries with a = a_max, the maximum distance of closest approach is r_{close, max} = 175a_max /D₀. We adopt a minimum distance of closest approach r_{close, min} = 1 AU. Within this range, the probability of any r_close grows linearly with r. Choosing two random deviates thus yields a_bin and r_close; v_{ej, H}, D, and P_ej follow from Equations (20)–(24). Selecting a third random deviate from a Gaussian distribution yields the ejection velocity. Two additional random deviates drawn from a uniform distribution spanning the main sequence lifetime of the star fix t_ej and t_obs. To see whether this combination of parameters results in an ejection, we select a sixth random deviate, P, and adopt a minimum ejection velocity v_{ej, min} = 600 km s⁻¹. Stars with smaller ejection velocities cannot escape the GC (Kenyon et al. 2008). When P_ej ⩾ P, v_ej ⩾ v_{ej, min}, and t_ej < t_obs, we place the star at a random location on a sphere with a radius of 1.4 pc centered on the GC and assign velocity components appropriate for a radial trajectory from the GC. Failure to satisfy the three inequalities results in a new selection of random numbers.

4.2.2. Runaway Stars

For runaway stars, we consider two analytic models for the ejection velocity. Following Bromley et al. (2009), we assume runaway companions of a supernova have an exponential velocity distribution:

$$\begin{equation} p_S(v_{\rm ej}) \propto e^{-v_{\rm ej}/ v_{\rm ej,S}}, \end{equation} \tag{ 25 }$$

where v_{ej, S} ≈ 150 km s⁻¹. For a minimum (maximum) velocity of ejected stars of 20 km s⁻¹ (400 km s⁻¹), this distribution roughly matches simulations of binary supernova ejections (Portegies Zwart 2000).

Predicted velocity distributions for stars ejected dynamically are much steeper (Perets & Subr 2012). To allow a reasonable number of high velocity ejections, we adopt

$$\begin{equation} p_D(v_{\rm ej}) \propto \left\lbrace \begin{array}{@{}lll}v_{{\rm ej}}^{-3/2} & & v_{{\rm ej}} \le v_{\rm ej,D}\\ \\ v_{{\rm ej}}^{-8/3} & & v_{{\rm ej}} \ge v_{\rm ej,D}, \\ \end{array} \right. \end{equation} \tag{ 26 }$$

where v_{ej, D}= 150 km s⁻¹ (Perets & Subr 2012). In our standard calculations, we set a minimum ejection velocity of 20 km s⁻¹ and a maximum ejection velocity of 800 km s⁻¹. To improve the accuracy of the statistics for the highest velocity runaways, we perform a second set of simulations with a minimum velocity of 50 km s⁻¹. Together, these simulations yield a robust picture for the frequency and observable parameters for runaways produced by the dynamical ejection mechanism.

Both of these models yield small production rates for high velocity runaways. To enable more robust comparisons with simulations of HVSs, we consider a "toy" model where the ejection velocity is uniformly distributed between 400 km s⁻¹ and 600 km s⁻¹. Thus, we use Equations (25) and (26) to derive rates for high velocity runaways and the toy model to understand the galactic distribution of the highest velocity runaways.

Establishing the initial conditions for runaways also requires a set of random deviates. We assume the initial space density of runaways follows the space density of stars in the Galactic disk. Thus, the probability of ejecting a runaway from a cylindrical radius ρ₀ is

$$\begin{equation} p(\rho _0) \propto \rho _0 e^{-\rho _0 / \rho _s}, \end{equation} \tag{ 27 }$$

where the scale length is ρ_s = 2.4 kpc (Siegel et al. 2002; Bovy & Rix 2013). We adopt a range for the initial radius, ρ₀ = 3–30 kpc (Brand & Wouterloot 2007). Setting the position of the runaway requires two random deviates, one for ρ₀ and another for the initial longitude in the GC frame. In this approach, the initial height above the Galactic plane is z = 0.

Once ρ₀ is known, we choose a random deviate for v_ej and two random deviates for the ejection angles (spherical θ and ϕ). Adding the velocity from Galactic rotation yields three velocity components. We then choose a final random deviate for t_obs. In these simulations, ejections occur on timescales much shorter than the lifetime of the ejected star. Thus, t_ej = 0.

To integrate the motion of each ejected star through the Galactic potential, we use an adaptive fourth-order integrator with Richardson extrapolation (e.g., Press et al. 1992; Bromley & Kenyon 2006; Bromley et al. 2009). Starting from an initial position $\boldsymbol {r}_0$ with velocity $\boldsymbol {v}_0$, the code integrates the full 3D orbit through the Galaxy, allowing us to track position and velocity as a function of time. We integrate the orbit for a time t_f = t_obs − t_ej, which is smaller than the main sequence lifetime of the ejected star. This procedure allows us to integrate millions of orbits fairly rapidly. Several tests demonstrate our approach yields typical errors of 0.01% in position and velocity after 1–10 Gyr of evolution time.

To investigate the distribution of stars as a function of μ and d, we construct a density diagram. For stars with |b| ⩾ 30^o, we (1) divide the log d–log μ plane into bins spaced by 0.01 in log d and log μ, (2) count the number of stars in each bin, and (3) plot the relative number in a contour diagram. In each diagram, bright red represents the largest density; dark blue the smallest density. The full range in relative density varies from a factor of 5–10 for 1 M_☉ runaways to a factor of 50–500 for 1–3 M_☉ HVSs.

Figure 12 shows predicted density distributions for 3 M_☉ HVSs and runaways. Figure 13 plots predictions for 1 M_☉ stars. The HVS results assume stars ejected from equal mass binaries (1 M_☉: a_bin = 0.032–4 AU, t_ms = 10 Gyr; 3 M_☉: a_bin = 0.115–4 AU, t_ms = 350 Myr). For the runaway simulations, we adopt minimum ejection velocities of 20 km s⁻¹ (supernova ejections) or 50 km s⁻¹ (dynamical ejections). Eliminating the lower velocity dynamical ejections artificially enhances the density at large proper motions relative to small proper motions. This enhancement provides a clearer picture of the relative frequency of the highest velocity runaways.

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These results demonstrate that nearly all of the proper motions for 3 M_☉ HVSs result from reflex solar motion (Figure 12, upper panel). Most HVSs fall close to the line

$$\begin{equation} \mu = 49.6 \left({ d \over {\rm 1\ kpc} } \right) {\rm mas\, yr}^{-1}. \end{equation} \tag{ 29 }$$

At fixed d, stars with smaller μ have smaller b (see also Figure 10). Along the locus, the number of 3 M_☉ HVSs peaks at d ≈ 50 kpc.

Above the μ(d) locus, there is a cloud of stars with d ≲ 10–20 kpc and μ ≲ 100 mas yr⁻¹. This group of mostly bound HVSs lies at all b in the direction of the GC. Stars ejected along the z-axis produce this clump of high proper motion stars (see Figure 10).

Although runaways generally follow the μ(d) relation expected for reflex solar motion, the distribution about this relation is much more diffuse than for HVSs (Figure 12, middle and lower panels). Galactic rotation produces this fuzziness. In HVS ejections, the distribution of ejection velocities is Gaussian; the position of a star along the μ(d) relation is a simple function of this ejection velocity and the flight time. In runaway ejections, the ejection velocity consists of Galactic rotation plus a random velocity with a random angle relative to Galactic rotation. This randomness creates a much larger dispersion of space velocities and much larger dispersion about the simple μ(d) relation.

For d ≈ 20–100 kpc, galactic rotation also produces twin peaks in the density at fixed distance. Separated by roughly 0.3 in log μ, these twin density maxima are very prominent in the ensemble of supernova-induced runaways (middle panel) and less prominent among the dynamically generated runaways (lower panel). For runaways in the Galactic anti-center, the rotational component of their motion is parallel to the Sun's motion. These stars lie in the low proper motion peak. Distant runaways in the direction of the GC are beyond the GC; the rotational component of their space motion is anti-parallel to the Sun's motion. These stars produce the high proper motion peak. Nearby indigenous disk stars in the direction of the GC have rotational motions parallel to the Sun, eliminating the double-peaked aspect of the proper motion distribution.

The larger maximum velocities from dynamical ejections blur the double-peaked distributions of proper motions identified in runaways from supernovae (Figure 12, lower panel). Despite the diffuse nature of the contour diagram, Galactic rotation is clearly visible at 20–50 kpc. As with HVSs and supernova-induced runaways, the width of the proper motion distribution narrows with increasing distance.

Results for 1 M_☉ HVSs and runaways are similar (Figure 13). The HVSs closely follow the linear μ(d) relation out to d = 30 Mpc (Figure 13, upper panel). The density of 1 M_☉ HVSs has two clear maxima at d ≈ 10 kpc and d ≈ 2–3 Mpc. Stars at high b closely follow the line (red contour); stars at b ≈ 30^o occupy the blue contour below the line. At d ≈ 10 kpc, there is a group of stars with large μ above the red contour. High velocity ejections along the Galactic poles produce this collection of large proper motion stars (see also Figure 10).

Despite their lower frequency, 1 M_☉ runaways also clearly follow the linear μ(d) relation expected for solar reflex motion. Aside from having a shape similar to the contours for the 3 M_☉ runaways, the contours for 1 M_☉ runaways extend to slightly larger distances due to their longer main sequence lifetimes.

The number and location of density peaks for HVSs in Figures 12 and 13 depend solely on stellar lifetime (see also Bromley et al. 2006; Kenyon et al. 2008). For ejected stars with infinite lifetimes, the space density n is a simple power-law with distance from the GC, n∝r⁻². Thus, the total number of HVSs grows monotonically with distance. However, real HVSs have finite lifetimes. The total number falls at distances where the travel time exceeds the main sequence lifetime. For 1 M_☉ (3 M_☉) HVSs with ejection velocities drawn from Equation (20), lifetimes of 10 Gyr (350 Myr) result in peaks at 2–3 Mpc (50 kpc). The second peak in the density of 1 M_☉ stars results from bound stars with orbital periods smaller than the main sequence lifetime. Continuous ejection of relatively low velocity HVSs over 10 Gyr produces a large concentration of bound 1 M_☉ HVSs with d ≲ 20 kpc. The short lifetimes of 3 M_☉ HVSs preclude a significant concentration of nearby HVSs.

The space density of stars in the disk sets the density of runaways in these diagrams. For stars with an exponential distribution of ejection velocities, unbound stars are very rare (Section 5). Thus, most stars in the diagram are bound to the Galaxy. For bound stars at |b| > 30^o, the final in-plane distance from the GC, ρ_f, is similar to the initial distance, ρ₀. The space density of these stars follows the initial density, which is concentrated toward the GC. As a result, most stars have d ≲ 10–20 kpc.

In both diagrams, the dynamical and supernova ejection scenarios produce an ensemble of stars at d ≈ 10 kpc with smaller proper motions than the locus of stars with solar reflex motion. Within this group, stars with the smallest μ are concentrated toward small b at a variety of Galactic longitudes l ≈ ±100^o–280^o where the tangential velocity reaches a minimum. Some of these stars have large ejection velocities parallel to the Sun's trajectory (see Figure 7). Others have modest ejection velocities perpendicular the plane, which enable them to reach large b but not escape the Galaxy (see Figure 4).

To explore the variation of radial velocity with distance, we examine another density diagram. As in the previous section, we (1) select stars with |b| ⩾ 30^o, (2) divide the log d–v_r plane into bins spaced by 0.01 in log d and 20 km s⁻¹ in v_r, and (3) count the number of stars in each bin. In the diagrams, bright red represents the largest density; dark blue the smallest density. The range in density varies from a factor of 50 for 1 M_☉ runaways to a factor of 300 for 3 M_☉ HVSs and runaways.

HVSs and runaways from supernovae show a remarkable diversity in the relative density of 1 M_☉ stars as a function of v_r and d (Figure 14). Close to the Sun (d ≲ 1–3 kpc), the relatively few HVSs and runaways have fairly symmetric velocity distributions around a median v_r ≈ 0 km s⁻¹. At moderate distances (d ≈ 3–20 kpc), the spread in radial velocity grows smoothly with distance. Although HVSs have a much larger spread in radial velocity (see also Table 1), both groups have a clear peak in the relative number of stars at d ≈ 10 kpc and v_r ≈ 0–100 km s⁻¹.

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For stars at large distances (d ≳ 20 kpc), the velocity distributions of HVSs and supernova-induced runaways differ dramatically. With their modest maximum ejection velocities, few runaways reach the outer Galaxy (Section 5; Figure 11). For d ≳ 20 kpc, the density of runaways drops significantly and falls very close to zero at d ≈ 100 kpc. Despite the steep fall in relative density, the median velocity and the spread in radial velocity are roughly constant with distance (Table 1).

The properties of distant HVSs provide a clear contrast with distant runaways. For HVSs, the median radial velocity and the spread in the radial velocity grow with distance. The maximum radial velocity increases from roughly 1000 km s⁻¹ at d ≈ 10 kpc to 3000 km s⁻¹ at d ≈ 10–20 Mpc. Although the relative density of HVSs falls from 20 kpc to 300 kpc, the relative density displays a clear secondary peak at d ≈ 2 Mpc and v_r ≈ 400–500 km s⁻¹. Beyond 5 Mpc, the density slowly falls and reaches roughly zero at d ≈ 20 Mpc.

The distributions of 3 M_☉ runaways are similar to those of 1 M_☉ runaways (Figure 15). In the middle panel, supernova-induced runaways show a clear increase in relative density from d ≈ 300 pc to d ≈ 10 kpc. Within this range of distances, the spread in the radial velocity grows smoothly with distance; the median v_r is close to zero. Beyond this peak, the relative density drops to zero at d ≈ 100 kpc. Among the more distant stars, the maximum v_r is roughly constant with distance; the minimum v_r grows slowly with distance.

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Dynamically generated runaways yield similar results. When we adopt a minimum v_ej = 20 km s⁻¹ for dynamically generated runaways, the distribution is nearly indistinguishable from the middle panel of Figure 15. Increasing the minimum v_ej to 50 km s⁻¹ removes the ensemble of low velocity stars from the diagram, reducing the density and increasing the spread in v_r for nearby stars (Figure 15, lower panel). Despite this difference, dynamically generated runaways still display a clear peak in relative density at d ≈ 10 kpc. Around this peak, stars have a median v_r close to zero and a spread of ±500 km s⁻¹ (Table 1).

Because the dynamical model yields a maximum v_ej ≈ 800 km s⁻¹, a few runaways reach larger distances and have larger radial velocities than supernova-induced runaways. Despite this difference, very few runaways are unbound (Section 5).

Although 3 M_☉ HVSs have a nearly identical distribution of ejection velocities as 1 M_☉ HVSs, the density distributions in the d–v_r plane show several clear differences. The primary peak in the density lies at somewhat smaller distances, at ∼8 kpc instead of ∼10 kpc. The secondary peak falls at much smaller distances, ∼50 kpc instead of a few Mpc. The drop in density at large distances is much more rapid, falling to zero just inside 1 Mpc instead of reaching to 30 Mpc.

These trends have simple physical explanations (Bromley et al. 2006; Kenyon et al. 2008). For our adopted MW potential and HVS parameters, roughly 10% of ejected stars have velocities large enough to reach 10–20 kpc but too small to travel beyond 60–100 kpc. Typical travel times of 100–400 Myr for these bound stars are a significant fraction of the main sequence lifetime of a 3 M_☉ star, but are much smaller than the lifetime of a 1 M_☉ star. Bound stars with long lifetimes have median velocities close to zero and modest velocity dispersions of 100–200 km s⁻¹. However, many bound stars with short lifetimes reach d ≈ 30–60 kpc and evolve off the main sequence before returning to the solar circle. Thus, there is a large deficit of bound, massive stars with negative radial velocity. For 3 M_☉ stars, this deficit leads to a median v_r larger than zero and a larger velocity dispersion than 1 M_☉ HVSs (see also Table 1).

Among unbound stars, finite stellar lifetimes are also responsible for trends in v_r with distance. Stars with larger ejection velocities reach larger distances; the median v_r and dispersion in v_r thus increase with d. Longer lifetimes also enable stars to reach larger d. With a factor of 30 longer lifetime, the 1 M_☉ stars reach 30 times larger distances than 3 M_☉ stars (30 Mpc instead of 1 Mpc).

The initial velocity distributions of HVSs and runaways produce the stark differences in Figures 14 and 15. Among runaways ejected from 3–30 kpc in the Galactic disk, nearly all have modest ejection velocities and remain bound to the Galaxy (Section 5). Bound stars with modest ejection velocities reach maximum distances of roughly 100 kpc before falling back into the Galaxy. The small fraction of runaways that reach the halo have typical d ≈ 10 kpc and v_r ≲ 100–150 km s⁻¹.

The density plots in Figures 12–15 demonstrate the rich behavior in the predicted μ(d) and v_r(d) as a function of initial conditions and stellar properties. To focus on predictions for large ensembles of HVSs and runaways, we now consider the frequency distributions of v_r and μ for Galactic halo stars (|b| ⩾ 30^o) in a discrete set of distance bins. In these diagrams, color encodes distance (violet: d ⩽ 10 kpc, blue: 10 kpc <d ⩽ 20 kpc, green: 20 kpc <d ⩽ 40 kpc, and orange: 40 kpc <d ⩽ 80 kpc). Tables 1–3 summarize statistics for μ, v_r, and v_t in each distance bin.

Figure 16 shows the distributions of v_r (left panels) and μ (right panels) for 1 M_☉ (upper panels) and 3 M_☉ (lower panels) HVSs. The trends of radial velocity with stellar mass and distance follow the correlations in Section 6.2 (see also Bromley et al. 2006; Kenyon et al. 2008). For 3 M_☉ stars, the median radial velocity grows with increasing distance. Among the more distant stars, there is a large tail of very high velocity stars with v_r ≳ 1000 km s⁻¹. As distance decreases, a smaller and smaller fraction of stars have high velocities. In the nearby sample with d ⩽ 10 kpc, nearly all stars have v_r ≲ 500 km s⁻¹.

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For 1 M_☉ stars, the trend of increasing median velocity with increasing distance is much weaker (Table 1). For all d, the velocity dispersion and inter-quartile range are smaller. Although the typical maximum velocity is similar, a much smaller fraction of stars has v_r ≳ 1000 km s⁻¹.

The distributions of proper motion and tangential velocity reverse the trends of the radial velocity (Tables 2 and 3; Figure 16, right panels). Distant HVSs moving radially away from the GC have small transverse components of their space motion, leading to small tangential velocities and small proper motions. As the distance decreases, the angle between the line-of-sight and the velocity vector for an HVS grows, leading to larger and larger tangential velocities. With μ∝d⁻¹, nearby HVSs have much larger proper motions than more distant HVSs (see also Figure 9). Although geometry requires that the maximum v_r exceed the maximum v_t, some nearby HVSs have v_t ≈ 400–600 km s⁻¹ and μ ≈ 30 mas yr⁻¹.

Trends in the radial velocity distributions for supernova-induced runaways follow those of the HVSs (Figure 17; Table 1). More distant runaways have a larger median radial velocity and a larger tail to very large radial velocity (see also Bromley et al. 2009). At fixed distance, however, the average and median velocities of runaways are much smaller than those of HVSs. Typically, runaways are 100–500 km s⁻¹ slower than HVSs, with velocity dispersions less than half the dispersions of HVSs.

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Differences between the radial velocity distributions for HVSs and runaways in the snapshots reflect the initial distribution of ejection velocities. As summarized in Section 5, more than 20% of the HVSs ejected with initial velocities exceeding 600 km s⁻¹ reach the halo. Most HVSs that reach the halo are unbound. The fastest runaways receive a boost from Galactic rotation (Equation (28)); they all lie in the Galactic plane (see also Bromley et al. 2009). Many fewer runaways reach the halo; nearly all of these have much smaller space velocities than HVSs. As a result, runaways in the halo have smaller median radial velocities and a larger fraction of bound stars than HVSs.

At similar distances, the proper motion distribution of both types of runaways is broader than that of HVSs (Figure 17, right panels). As with HVSs, nearby runaways have larger tangential velocities than more distant runaways. However, galactic rotation produces a double-peaked distribution of proper motion for runaways at fixed distance (Figures 12 and 13). In an ensemble of stars with a broad range of distances, the double-peaked character of the distribution smears out into a single broad peak. Among stars with a smaller range of distances, the double-peaked proper motion distribution is prominent.

The distributions of v_r and μ for runaways produced from dynamical ejections have the same features as supernova-induced runaways. The median radial velocity grows with distance (Figure 18, left panels). Although dynamical ejections produce a smaller fraction of high velocity runaways, the largest ejection velocities exceed those produced from the supernova mechanism (Table 1). For calculations with a minimum ejection velocity of 20 km s⁻¹, dynamical ejections yield average and median velocities 5%–10% smaller than supernova-induced runaways. In simulations with a minimum ejection velocity of 50 km s⁻¹, the inter-quartile ranges and standard deviations for dynamical ejections lie between those of HVSs and runaways produced in supernovae.

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The larger maximum velocities from dynamical ejections shift the peaks of the proper motion distributions to larger values (Figure 18, right panels; see also Table 3). These peaks are also somewhat broader than those for other ejected stars. As with HVSs and supernova-induced runaways, the width of the proper motion distribution narrows with increasing distance.

For 3 M_☉ HVSs and runaways, Gaia can detect the typical proper motion in the d = 40–80 kpc bins (dashed lines in Figures 16–18). To establish this conclusion, we use the predicted rms errors of roughly 0.16 mas yr⁻¹ for stars with g ≈ 20 (Lindegren 2010). Observed proper motions of 0.50 mas yr⁻¹ should then be detectable at the 3σ level. Although solar-type stars with g ≈ 20 have d ≈ 10 kpc, 3 M_☉ B-type stars with g ≈ 20 have d ≈ 100 kpc (see also Section 7). Thus, reliable distances and proper motions from Gaia can test these predicted proper motion distributions.

As we described in Section 6.1, proper motion distributions for HVSs and runaways are very sensitive to stellar lifetime. Long-lived unbound stars travel great distances from the Galaxy; shorter-lived stars evolve off the main sequence before leaving the Galaxy. Long-lived bound stars generate fairly symmetric distributions around the GC; shorter-lived stars have more asymmetric distributions.

At high galactic latitude (|b| ⩾ 30^o), HVSs provide the most extreme examples of this behavior (Figure 19, top panels). With lifetimes of 10 Gyr, unbound 1 M_☉ stars reach maximum distances of roughly 30 Mpc from the GC. These unbound stars produce the prominent peak at μ ≈ 0.01–0.05 mas yr⁻¹ in the upper left panel of Figure 19. Bound 1 M_☉ stars have maximum distances of roughly 60 kpc; they orbit the GC with periods of 700 Myr or less. Smaller distances result in much larger proper motions. These stars comprise the smaller peak in the upper left histogram at μ ≈ 1–10 mas yr⁻¹.

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Among all HVS ejections, bound stars outnumber unbound stars by roughly 4:1 (Section 5). Outside the Galactic plane (|b| ⩾ 30^o), however, unbound stars dominate. Thus, the peak of unbound stars at small μ is larger than the peak of bound stars at large μ.

Despite having similar ejection velocities as low mass HVSs, massive unbound HVSs do not live long enough to reach large distances from the GC. With typical maximum distances of roughly 1 Mpc, the smallest proper motions of unbound 3 M_☉ HVSs are roughly a factor of 30 larger than those of unbound 1 M_☉ HVSs. Although there are a few massive HVSs with μ ≈ 0.03–0.10 mas yr⁻¹, most have μ ≳ 0.1 mas yr⁻¹. These stars lie within the peak at μ ≈ 1 mas yr⁻¹ in the upper right panel of Figure 19.

At |b| ⩾ 30^o, bound 3 M_☉ HVSs have a much larger range in proper motion. Marginally bound stars reach large distances from the GC, r ≈ 40–60 kpc. Before they turn around and return to the GC, these stars evolve off the main sequence. This group has fairly small proper motion μ ≈ 1 mas yr⁻¹. Among the much larger group of bound stars that reach small distances, r ≲ 10–20 kpc, some have |b| ≳ 30^o. These have large proper motions, μ ≳ 10 mas yr⁻¹. In between, stars with d ≈ 20–40 kpc fill in the histogram at μ ≈ 1–10 mas yr⁻¹.

These general conclusions apply to both types of runaways. Among stars ejected by dynamical processes, a few have ejection velocities of 600–900 km s⁻¹ and can reach large distances from the Galaxy. Low mass stars in this group are still on the main sequence at d ≳ 1 Mpc; these stars produce a long tail in the proper motion distribution at μ ≈ 0.01–0.5 mas yr⁻¹ (Figure 19, lower left panel). More massive stars cannot reach these distances while on the main sequence; 3 M_☉ unbound stars produce a small shoulder in the distribution at μ ≈ 0.1–0.3 mas yr⁻¹ (Figure 19, upper right panel).

For all masses, the dynamical ejection process yields a large population of bound, relatively nearby stars. Typical proper motions are μ ≈ 1–20 mas yr⁻¹. Nearly all stars in the peaks of both histograms are bound stars.

With much lower maximum ejection velocities, stars ejected during a supernova are almost always bound to Galaxy. Among 1–3 M_☉ stars, most are nearby. Few have μ ≲ 0.5 mas yr⁻¹. These stars simply produce a single peak in the histogram at μ ≈ 3–4 mas yr⁻¹.

At large μ, the shapes of all of the histograms are fairly similar. For all of our snapshots, nearby stars with high velocity are rare. Most runaways with d ≲ 1 kpc have small velocities and modest proper motions. HVSs fill a much larger volume of the Galaxy and have much smaller space densities. Thus, the frequency of ejected stars with μ ≳ 10 mas yr⁻¹ falls sharply and reaches zero at μ ≈ 100 mas yr⁻¹.

To conclude our analysis of complete snapshots of ejected stars, we focus on the variation of proper motion with l and b. After separating stars into four galactic latitude bins equally spaced in |sin b|, we divide the l–μ plane into a set of bins spaced by 0.01 in log μ and 1^o in l. As with the d–μ and d–v_r diagrams in Sections 6.1 and 6.2, we plot the relative density of stars in each bin as a contour diagram where bright red represents the largest density and dark blue represents the smallest density. The range in density varies from a factor of 5–10 for 1 M_☉ runaways to a factor of 50–500 for 1–3 M_☉ HVSs.

Figure 20 shows a set of four contour diagrams for 3 M_☉ HVSs. At all b, HVSs have a broad range of proper motion between 0.1 mas yr⁻¹ and 30 mas yr⁻¹. Outside the GC region, the typical proper motion is μ ≈ 1 mas yr⁻¹. Toward the GC, there is a strong concentration of stars with large proper motion, μ ≈ 10–100 mas yr⁻¹. This concentration is strongest at low Galactic latitude and weakens considerably at larger b.

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In the Galactic plane (Figure 20, lowermost panel), the variation of μ with l for HVSs shows a clear signature from the Sun's orbit around the GC (see Equation (13); compare with Figure 7). The plot shows clear minima in μ at l = −100^o and at l = +80^o. The Sun's (1) spatial offset from the GC and (2) orbit around the GC produces the lack of mirror symmetry in the minima (see also Figure 7). At somewhat larger b (middle two panels), the amplitude of the variation is visible but suppressed. Toward the Galactic poles (uppermost panel), the Sun's position and orbital velocity have no impact on the tangential velocity. Thus, the variation disappears.

Contour diagrams for supernova-induced runaways display identical features (Figure 21). Runaways have somewhat larger proper motions than HVSs, with a typical μ ≈ 3 mas yr⁻¹ and a typical range of 0.3–30 mas yr⁻¹. Although HVSs have larger space velocities, their much larger distances result in smaller proper motions.

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As with HVSs, the contour diagrams change systematically with Galactic latitude. In the Galactic plane (Figure 21, lowermost panel), runaways display a large concentration of high proper motion stars toward the GC. At larger b, this concentration weakens and spreads to a broader range of Galactic longitude. High velocity runaways outside the solar circle but close to the Galactic plane also show clear minima in μ at l ≈ −110^o and +100^o. As noted for high velocity HVS in Figure 20, these minima are a clear signature of solar rotation around the GC and the solar offset from the GC (see Figure 7). This signal gradually diminishes with increasing b.

A clear minimum in μ at l ≈ 0 and |sin b| < 0.25 (Figure 21, lowermost panel) distinguishes supernova-induced runaways from HVSs. For stars inside the solar circle, this feature is the signature of stars rotating in the Galactic disk (see Figure 4). Inside the solar circle, the tangential velocities of stars orbiting the GC lie in an egg-shaped locus with v_{t, max} ≈ 2 v_☉ and v_{t, min} ≈ 0. The lower edge of this egg produces the distinct minimum in μ at l ≈ 0.

Dynamical runaways with a minimum velocity of 20 km s⁻¹ produce distributions of μ(l, b) nearly identical to the distributions for supernova-induced runaways in Figure 21. Calculations with a minimum velocity of 50 km s⁻¹ yield dramatically different results (Figure 22). Although (1) the typical range in μ(b), (2) the heavy concentration of stars toward the GC, and (3) the clear minima in μ at l ≈ ± 100^o observed for dynamical runaways are similar to results for supernova-induced runaways, there is (1) a clear lack of stars with very small μ at l ≈ 0^o and (2) a broad minimum of stars with very small μ at l ≈ ±150^o–180^o.

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The higher minimum ejection velocity in these calculations produces both features. Stars on unperturbed orbits around the GC produce the distinct minimum in μ at l ≈ 0^o. Setting a high minimum ejection velocity in our calculations produces ensembles of stars with modest tangential velocities and proper motions at l ≈ 0^o, eliminating the pronounced minimum in μ at l = 0^o in Figure 21. This high minimum ejection velocity also tends to place stars onto orbits with modest eccentricity. Stars originating inside the solar circle—where the stellar density is large—then spend some time outside the solar circle—where the stellar density is small. This behavior increases the density of stars with small proper motion in the direction of the Galactic anti-center, where the tangential velocity is very small (see Figure 3).

Analyzing the complete sample of stars in simulations of HVSs and runaways leads to several clear results.

• 1.  
  Ejections produce a broad range in v_r (0–1000 km s⁻¹), v_t (0–600 km s⁻¹), and μ (0.01–100 mas yr⁻¹). HVSs have the largest velocities and proper motions (Figures 12–15).
• 2.  
  For distant stars, radial velocities separate unbound stars from bound stars (Figures 14–15 and Figures 16–18).
• 3.  
  For nearby stars, proper motions distinguish between unbound stars and bound stars on radial or circular orbits. However, nearby unbound stars are relatively rare compared to nearby bound stars (Figures 12–13 and Figure 19).
• 4.  
  Unbound runaways retain memory of their original orbital velocity around the GC. At large distances and high Galactic latitudes, the double-peaked proper motion distribution of runaways distinguishes them from stars on radial orbits (Figures 12 and 13).
• 5.  
  In the Galactic plane, HVSs and runaways have clear, distinctive minima in μ(l) at l ≈ ±100^o (Figures 20–22).
• 6.  
  Concentrations of high proper motion stars toward the GC are a unique signature of HVSs ejected from the GC or runaways ejected from the inner galaxy (Figures 20–22).
• 7.  
  Gaia can detect predicted proper motions of B-type HVSs and runaways with d ≲ 100 kpc (Figures 16–18).

These results clearly demonstrate the ability of radial velocity measurements to distinguish the highest velocity HVSs and runaways from indigenous halo stars (see also Brown et al. 2005, 2006, 2014; Bromley et al. 2006; Kenyon et al. 2008; Silva & Napiwotzki 2011 and references therein). For HVSs and runaways with d ≳ 50–100 kpc, the median radial velocity, 500–600 km s⁻¹, and radial velocity dispersion, σ_r ≈ 200–400 km s⁻¹, are very different from the population of halo stars with median v_r close to zero and σ_r ≈ 100–110 km s⁻¹. Thus, radial velocity surveys easily separate the highest velocity HVSs and runaways from indigenous halo stars.

Among stars with intermediate distances, d ≈ 20–40 kpc, proper motions can distinguish runaways from indigenous halo stars. For runaway stars with a small range of distances, galactic rotation produces a clearly double-peaked distribution of proper motions. With little or no rotation about the GC (e.g., Bond et al. 2010), halo stars should have a broad, single-peaked distribution. Because HVSs are ejected on purely radial orbits, their distribution of proper motions should resemble the halo distribution.

Among nearby halo stars with d ≲ 10 kpc and |b| ≳ 30^o, kinematic data do not offer a simple path for identifying ejected stars. The velocity dispersions, σ_r ≈ 100–175 km s⁻¹, of nearby HVSs and runaways are comparable to the typical velocity dispersion of halo stars (e.g., Brown et al. 2014 and references therein). Thus, it is not possible to use v_r to distinguish nearby HVSs and runaways from halo stars. Although the dispersions in v_t for nearby HVSs and runaways are small, the median v_t ≈ 250 km s⁻¹ for HVSs and dynamically generated runaways is much larger than the roughly 150 km s⁻¹ dispersion in v_t expected for indigenous halo stars. Because nearby HVSs and runaways are rare, their proper motions are fairly similar to the proper motions of nearby halo stars.

As with radial velocities, obvious outliers in proper motion are promising candidates for ejected stars. Our simulations yield maximum proper motions of 100 mas yr⁻¹ for stars with d ≲ 1–2 kpc. Along any line-of-sight, however, such high proper motion stars comprise only 0.01%–0.1% of the complete population of ejected stars. Thus, high proper motion outliers should be very rare.

Figure 25 compares the HVS candidates from Palladino et al. (2014) with predictions from the HVS and supernova-induced runaway models. Solid lines show contours of constant stellar density that contain 50% (inner) and 90% (outer) of the stars in the complete samples of simulated stars. Contours for dynamically generated runaways are nearly identical to those for the supernova-induced runaways. Filled circles indicate the measured (d, v_r) for the HVS candidates.

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Based on this comparison, the HVS model and both runaway models are wildly inconsistent with the observations. Although the observed radial velocities fall within the bounds predicted for all three models, the observed proper motions lie well above model predictions. None of the survey stars fall within the 50% contours of any model; only three fall within the 90% contours. Using the 95% contours (not shown on the figure), the runaway models fare a little better than the HVS model, with five (instead of three) stars lying within the contours. Despite this marginally better success, all models are excluded at better than 3σ confidence.

There are two possible origins of the mismatch between the models and the observations in Figure 25. Reducing the proper motions by a factor of five would place the data within the 50% contours of all models. Although Palladino et al. (2014) derive high reliabilities for their proper motion measurements, the ratio of the transverse to radial velocity for this sample is much larger than expected for a random selection of high velocity stars. Thus, there is a reasonable probability that at least some of the large proper motions are spurious (Palladino et al. 2014). Testing this hypothesis with another epoch of imaging data from the ground or with Gaia is straightforward.

The only alternative to explain the large proper motions of these stars is to develop another model that produces high velocity stars. Modifying the existing HVS or runaway models is unlikely to produce a better match: the geometry of ejections from the GC or the Galactic disk simply precludes a large population of nearby 1 M_☉ stars with modest radial velocity and very large proper motions. Supernova-induced runaways from binaries in the halo might allow a better match; however, supernova rates for halo binaries are probably much smaller than those in the disk. The Abadi et al. (2009) proposal of ejections from disrupted dwarf galaxies requires a somewhat massive Milky Way, ∼1.5–2 × 10¹² M_☉, but seems otherwise plausible. Numerical simulations of the velocity distribution of stars from disrupted dwarfs are required to test this interesting idea in more detail but are beyond the scope of this paper.

Figure 26 compares the HVS candidates from Brown et al. (2014) with predictions from the HVS and runaway models. As in Figure 25, solid lines show contours of stellar density that contain 50% (inner) and 90% (outer) of the stars in each simulation. Filled circles indicate the measured (d, v_r) for the HVS candidates. Unlike the 1 M_☉ targets, these candidates lie close to or within the 90% contours for all three models.

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This comparison strongly favors the HVS model for the origin of these high velocity stars. Only one (four) of the candidates lies within the 90% contour for the supernova-induced (dynamically generated) runaway model. Nearly all lie beyond the 95% contours (not shown on the figure). In contrast, more than half of the candidates lie within the 50% contour for the HVS model; all lie within the 90% contour. Taking the results at face value, this comparison rules out the runaway models at better than 3σ confidence.

To test the models in another way, we attempt to match the observed d and v_r of the candidates from the complete ensemble of stars in the HVS and runaway models. For each HVS candidate with Galactic coordinates (l, b), we select all model stars within a 20^o × 20^o window centered on the measured coordinates. Among this group, we count the number of model stars with distances and radial velocities within 10% of the measurements and tabulate the number of "matches" N_i for each candidate.

Figure 27 summarizes the results of this matching exercise. For each survey star, the bars show N_i for the HVS (violet) and dynamically generated runaway (cyan) models. With only one match among all HVS candidates, the supernova-induced runaway model fails the exercise. The HVS and dynamical runaway models fare much better, with N_i ≈ 30–100 for the HVS model and N_i ≈ 1–10 for the runaway model.

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Based on this approach, we conclude with strong confidence that the candidate HVSs from the Brown et al. (2014) survey are much more likely to be HVSs ejected from the GC than runaway stars ejected from the Galactic disk. For every survey star, the HVS model yields a larger N_i with N_i(HVS) ≈ 10–100 N_i(runaway). Factor of two changes in the size of the windows for the distance, galactic coordinates, and radial velocity in the matching algorithm yield indistinguishable results. Among all HVS candidates, the supernova-induced runaway model yields 0–1 matches. Analyzing each candidate separately, an ensemble of HVSs ejected from the GC always produces a factor of 10–100 more matches than an ensemble of runaways generated from dynamical interactions.

Modest changes to the probability distribution for the ejection velocity of runaways cannot change this conclusion. Matching the observed radial velocities of the 3 M_☉ HVS candidates requires much larger maximum ejection velocities (e.g., 800 km s⁻¹ instead of our adopted 400 km s⁻¹) for supernova-induced ejections or a much shallower power-law distribution of ejection velocities (e.g., $p_D(v_{{\rm ej}}) \propto v_{{\rm ej}}^{-n}$, with n ≲ 2 instead of our adopted 8/3) for dynamically generated runaways. Although doubling the maximum velocity from a supernova ejection is possible in rare circumstances (e.g., Portegies Zwart 2000), observations suggest a maximum ejection velocity of 400–450 km s⁻¹, close to our adopted value (e.g., Silva & Napiwotzki 2011). Observations also appear to preclude placing a larger fraction of runaways at the highest velocities (Silva & Napiwotzki 2011).

Figure 28 compares the runaways from Silva & Napiwotzki (2011) with predictions from the HVS and runaway models. As in Figure 25, solid lines show contours of stellar density that contain 50% (inner) and 90% (outer) of the stars in each simulation. Filled circles indicate the measured (d, v_r) for the runaway candidates; these stars lie close to or within the 90% contours for all three models.

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This comparison strongly favors the runaway model. Only two candidates lie within the 90% contour for the HVSs model; none lie within the 50% contour. Many of the candidates are close to or within contours from dynamically generated runaways with a minimum ejection velocity of 50 km s⁻¹. More than half of the candidates fall within the 90% contours of the supernova-induced runaway model. Contours generated from a set of dynamically generated runaways with a minimum ejection velocity of 20 km s⁻¹ show a similarly good agreement with the observations.

Our matching algorithm also favors the runaway model for these stars. Only one star—EC 09452-1403 with v_r = 236 km s⁻¹—yields any matches (three) to the HVS model. The supernova-induced runaway model typically yields 10–100 matches for each target.

To add a little more realism to the matching algorithm, we add the measured proper motions. Requiring the proper motion in l and b to match within ±10% eliminates all matches for the HVS model to data for EC 09452-1403. Although including proper motion data also reduces the number of matches for the supernova-induced runaway model, the typical number of matches is still 3–20 per star.

These tests demonstrate our ability to explain observations of bona-fide runaways from a set of numerical simulations and to discriminate between HVSs and runaways. The positions of these stars in the d − v_r diagram and the number of matches in (d, μ, v_r) space clearly favor an identification as runaways rather than HVSs.

We now consider five miscellaneous high velocity stars identified as possible HVSs or "hyper-runaways." Here, we focus on two basic predictions from ejected star models: (1) HVSs should have radial orbits from the GC and (2) runaways should have a significant non-radial component of motion consistent with ejection from the disk.

From the observed l, b, and v_r, we derive the radial velocity in the GC frame and the predicted proper motion for a purely radial orbit (e.g., Equation (5) and Equations (12)–(14)). For nearby HVS candidates, the tangential component of the velocity dominates the space motion; for more distant HVSs, the radial component dominates. Among runaways, the tangential component of the motion is smaller than HVS for nearby stars and larger than HVS for more distant stars. If a nearby (distant) candidate is a runaway, we expect the analytic proper motion to exceed (fall below) the observed proper motion. In addition to yielding a reasonably simple way to distinguish between HVSs and runaways, this approach avoids cpu-intensive calculations for stars with masses of 2–11 M_☉, which are not included in our suite of simulations.

To test this approach, we derive predicted proper motions for HVS candidates from Brown et al. (2014). Our analysis yields predictions of 0.3–1.2 mas yr⁻¹. For each HVS candidate, we then compare this analytic proper motion with the average proper motion from the simulated stars selected with the matching algorithm outlined in Section 8.3. The typical difference between the analytic and numerical results is small, ∼0.1–0.2 mas yr⁻¹, with a dispersion of 0.5 mas yr⁻¹. Thus, the analytic approach works reasonably well.

This analysis strongly favors runaway models for SDSS J013655.91+242546.0, HIP 60350, and HD 271791. For SDSS J013655.91+242546.0 and HD 271791, the observed magnitude of the proper motion is much larger than predicted from the simple HVS model. Coupled with the observed v_r, proper motions in l and b strongly favor ejection from the disk instead of the GC (see also Heber et al. 2008; Tillich et al. 2009). For HIP 60350, the HVS model predicts much larger proper motion than observed. Proper motions in l and b also favor a disk ejection (Irrgang et al. 2010).

For HE0437-5439 and J091206.52+091621.8, the small proper motions favor HVS models. HE0437-5439 has a distance and radial velocity similar to HVS-1 (Brown et al. 2005; Edelmann et al. 2005). Hubble Space Telescope proper motion data suggest origin in the GC (Brown et al. 2010a); ejection from the LMC is also possible (Edelmann et al. 2005; Brown et al. 2010a). For J091206.52+091621.8, the observed proper motion is comparable to or less than the predicted proper motion of roughly 3 mas yr⁻¹ (Zheng et al. 2014). However, the relatively large errors in the proper motion prevent isolating the ejection solely from the GC (Zheng et al. 2014).

To consider whether observations can distinguish HVSs and runaways from halo stars, we examine kinematic measurements from Kinman et al. (2007, 2012). Figure 29 compares data for BHB stars ("BHB"; cyan points) and RR Lyr stars ("RR"; orange points) with model contours for distance-limited samples of 3 M_☉ HVSs ("HV3"; violet curves) and runaways ("RS3"; green curves). The contours enclose 50% (inner) and 95% (outer) of the stars in each simulation. To provide an approximate match to the Galactic coordinates for halo stars in the upper (lower) panel, we select HVSs and runaways with b ⩾ 75^o (l = 160^o to l = 200^o and b = 25^o to 50^o).

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This comparison confirms our previous conclusion that radial velocity measurements directly discriminate between HVSs and indigenous halo stars (e.g., Brown et al. 2006, 2007, 2009, 2014; Bromley et al. 2006, 2009; Kenyon et al. 2008). For v_r ≈ −200 km s⁻¹ to +300 km s⁻¹, the distributions of halo stars and HVSs overlap. Beyond v_r ≈ 300 km s⁻¹, however, HVSs dominate. This separation has a simple origin: HVSs and halo stars with v_r ≲ 300 km s⁻¹ are bound to the galaxy. HVSs with larger v_r are unbound.

Although the radial velocity distributions for halo stars and runaways overlap in the direction of the NGP, the highest velocity runaways are easily distinguished from halo stars toward the Galactic anti-center. At the NGP, nearly all runaways are bound to the Galaxy. Thus, they have radial velocities similar to indigenous halo stars. Toward the anti-center, the velocity from Galactic rotation enables a significant population of unbound runaways on outbound trajectories. Bound halo stars never reach the large v_r of these unbound runaways.

Proper motion data alone do not easily discriminate HVSs and runaways from indigenous halo stars. The maximum proper motion of halo stars in Kinman et al. (2007, 2012), ∼30–50 mas yr⁻¹, exceeds the proper motions of 99% (∼100%) of HVSs and runaways toward the NGP (anti-center). Toward the anti-center, the number of high proper motion outliers, μ ≳ 100 mas yr⁻¹, is nearly zero. Toward the NGP, however, 0.1–1% of HVSs and runaways have proper motions larger than 100 mas yr⁻¹. Thus, occasional HVSs and runaways can be identified as proper motion outliers (e.g., Heber et al. 2008; Tillich et al. 2009; Irrgang et al. 2010).

For large samples of halo stars, accurate distances might provide a way to isolate runaways from indigenous stars (Figure 12). Because runaways share the rotation of stars in the disk, they should exhibit a double-peaked distribution of proper motion. With little or no rotation (e.g., Bond et al. 2010), indigenous halo stars should have a single-peaked distribution. Although synthesizing the expected distribution for indigenous halo stars is beyond our scope, isolating the runaways probably requires a significant population of ejected stars within the halo.

To complete our comparisons between the models and available data, we now estimate the relative production rates for HVSs and runaways. Accurate rate estimates allow us to predict the relative space density of ejected stars as a function of distance and Galactic latitude (e.g., Brown et al. 2009; Bromley et al. 2009). Because the matching algorithm and the contour maps draw from equal numbers of simulated HVSs and runaways, production rates allow us to normalize the number of matches to the expected space density.

Predictions for HVSs depend on the rate binaries encounter the black hole at the GC (e.g., Yu & Tremaine 2003). The time variation of the population of binaries within the "loss cone"—the ensemble of orbits that pass within the black hole's tidal radius—is an important issue in these derivations. Binary encounters with the black hole empty the loss cone; encounters between binaries and molecular clouds or other field stars fill the loss cone (e.g., Yu & Tremaine 2003; Perets & Gualandris 2010; Zhang et al. 2013; Vasiliev & Merritt 2013; Madigan et al. 2014). Rates with a "full" loss cone are larger than those with an "empty" loss cone; typical estimates are 10⁻⁵ to 10⁻³ yr⁻¹ for binaries of all types (e.g., Yu & Tremaine 2003; Bromley et al. 2012; Zhang et al. 2013). For reasonable initial mass functions, limiting the binary population to 2.5–3.5 M_☉ B-type stars (0.8–1.2 M_☉ G-type stars) implies rates 40 times (10 times) smaller, 2.5–250 × 10⁻⁷ yr⁻¹ for B-type stars and 1–100 × 10⁻⁶ yr⁻¹ for G-type stars.

To infer empirical HVS rates, we focus on the S stars at the GC (Eckart & Genzel 1997; Ghez et al. 1998) and HVSs in the Galactic halo (Brown et al. 2014). The S stars are luminous B-type stars orbiting the GC, which are the plausible captured partners of HVSs ejected into the outer Galaxy (Gould & Quillen 2003; O'Leary & Loeb 2008; Perets 2009; Madigan et al. 2014). The population of S stars implies a capture rate of roughly 2 × 10⁻⁷ yr⁻¹ for stars with masses exceeding 5 M_☉ (see also Bromley et al. 2012). Using their complete spectroscopic sample of B-type stars in the outer halo, Brown et al. (2014) estimate a production rate of 1.5 × 10⁻⁶ yr⁻¹ for unbound HVSs with masses of 2.5–4 M_☉. For an ensemble of stars selected from a Salpeter IMF, these rates agree with theoretical predictions and with each other to within a factor of two.

Predictions rates for runaways from supernovae depend on the local star formation rate, the mass range adopted for B-type stars, the fraction of stars in binaries, and the fraction of binaries with mass ratios and orbital periods capable of ejecting a star with a velocity exceeding 10–20 km s⁻¹ (e.g., Brown et al. 2009; Bromley et al. 2009). For a star formation rate of 0.5 M_☉ yr⁻¹ (Lada & Lada 2003) and for the observed properties of binaries composed of B-type stars with masses of 2.5–4 M_☉ (Kobulnicky & Fryer 2007), supernovae produce runaways with minimum ejection velocities of 10 km s⁻¹ at a rate of roughly 3 × 10⁻⁶ yr⁻¹ (see also Brown et al. 2009; Bromley et al. 2009).

Predicting ejection rates for the dynamical runaway mechanism is more challenging. From numerical simulations of dense clusters, Perets & Subr (2012) estimate an ejection rate for hyper-runaway B-type stars with v_esc ≳ 450 km s⁻¹ of 1–2 × 10⁻⁸ yr⁻¹. For a power-law probability of the ejection velocity (Equation (26)), the total ejection rate for B-type runaway stars with v_esc ≳ 10 km s⁻¹, is roughly 1–2 × 10⁻⁵ yr⁻¹. This rate suggests that dynamically generated runaways are somewhat more common than supernova-induced runaways.

Empirical rates for runaways generally agree with these estimates. To derive these rates, we estimate the formation rate of 2.5–3.5 M_☉ stars from the observed star formation rates and a Salpeter initial mass function. For the Lada & Lada (2003) star formation rate of 0.5 M_☉ yr⁻¹, one star with a mass of 2.5–3.5 M_☉ is born every 300–350 yr. To infer the fraction of runaway stars in this group, we rely on the observed frequencies of less than 1% for A-type stars (Stetson 1981; Bromley et al. 2009), 5% for B-type stars (Gies & Bolton 1986), and more than 20% for O-type stars (Tetzlaff et al. 2011). Adopting a 1–2% runaway frequency among 2.5–3.5 M_☉ stars yields a production rate of 3–6 × 10⁻⁵ yr⁻¹. This rate is nearly identical to our purely theoretical rate estimate.

Altogether, these estimates suggest our matching algorithm selects stars from ensembles with similar formation rates. If we assume that bound HVSs have a comparable frequency to unbound HVSs in the Brown et al. (2014) sample, the combined HVS production rate of 3 × 10⁻⁶ yr⁻¹ is identical to the production rate of supernova-induced runaways. For the dynamically generated runaways, we correct the complete ensemble for the fraction of runaways with v_ej ≳ 50 km s⁻¹. This correction yields an expected rate of 3–6 × 10⁻⁶ yr⁻¹, very close to the rates for HVSs and supernova-induced runaways.

Although these estimates yield roughly similar production rates for HVSs and both types of runaways, we expect unbound runaways to be much less frequent than unbound HVSs. More than 25% of HVSs ejected from the GC to distances ⩾10 kpc are unbound (Section 5). In contrast, ⩽1% of runaways have ejection velocities larger than the local escape velocity. Thus, unbound HVSs should greatly outnumber unbound runaways in the halo.

Overall, observations confirm the expectation that the number of unbound runaways is smaller then the number of unbound HVSs (see also Brown et al. 2009; Bromley et al. 2009; Perets & Subr 2012). For every unbound runaway, theory predicts 10–30 HVSs. Among known unbound stars, the vast majority are HVSs (Section 8.3; see also Brown et al. 2009).

Observations of HVS candidates and known runaways yield good tests of the numerical simulations. The comparisons in the preceding subsections lead to several clear conclusions.

• 1.  
  Runaway models provide an excellent match to observations of known runaway stars with modest radial velocities, v_r ≲ 250 km s⁻¹ (Silva & Napiwotzki 2011).
• 2.  
  The HVS models match observations of HVS candidates from Brown et al. (2014).
• 3.  
  HVS and runaway star models fail to match observations of HVS candidates from SEGUE (Palladino et al. 2014). If the proper motions are correct, some other model for ejected stars is required to match the observations.
• 4.  
  Among a few miscellaneous high velocity stars, at least three are runaways (Heber et al. 2008; Tillich et al. 2009; Irrgang et al. 2010). Two others are more likely HVSs (Edelmann et al. 2005; Brown et al. 2010a; Zheng et al. 2014).
• 5.  
  Cleanly isolating unbound ejected stars from indigenous halo stars requires radial velocities. If the halo contains a large population of runaways ejected from the disk, these stars can be identified as high proper motion outliers or by their rotational motion about the GC.
• 6.  
  Observed and theoretical formation rates for HVSs and runaways suggest that most unbound stars in the halo are HVSs.