RUNAWAY STARS, HYPERVELOCITY STARS, AND RADIAL VELOCITY SURVEYS

First reported by Humason & Zwicky (1947), runaway stars are short-lived stars at unexpectedly large distances and velocities relative to their probable site of origin (Blaauw & Morgan 1954; Greenstein 1957). In the solar neighborhood, roughly 10%–30% of O stars and 5%–10% of B stars are runaways (Gies 1987; Stone 1991). The main population of runaways has a velocity dispersion, 30 km s⁻¹, roughly 3 times larger than the velocity dispersion of non-runaway stars in the Galactic disk. However, several runaways have velocities ≳100 km s⁻¹ relative to the local standard of rest (e.g., Gies 1987; Stone 1991; Martin 2006) and large distances, ≳500 pc, from the Galactic plane (Greenstein & Sargent 1974; Martin 2006). The large velocity dispersion and vertical scale height suggest a dynamical process that ejects runaways from the thin disk into the halo.

Runaways probably originate within star-forming regions in the disk of the Milky Way. Disruption of massive binaries is the likely source of runaways. In the binary supernova mechanism, a main-sequence runaway is ejected when its former companion star explodes as a supernova; subsequent mass loss and the kick velocity from the asymmetric explosion are sufficient to unbind the binary (see Blaauw 1961; Hills 1983). In this scenario, the runaway moves away with roughly the sum of the kick velocity and the orbital velocity of the binary, a velocity physically limited by the orbital velocity at the surface of the stars. In the dynamical ejection mechanism, runaways are ejected in dynamical three- or four-body interactions; the outcome is any combination of single stars and binaries (Poveda et al. 1967; Hoffer 1983; Hut & Bahcall 1983). The maximum ejection velocity from dynamical binary–binary encounters is formally the escape velocity of the most massive star (Leonard 1991).

Understanding whether the proposed formation mechanisms can produce the observed runaway star populations requires high-quality kinematic data and a clear understanding of the dynamics of close binaries and dense star clusters. For large ensembles of runaways, the low binary frequency of runaways favors the dynamical ejection mechanism (Gies & Bolton 1986). However, the predicted kick velocity of supernovae is uncertain (e.g., Burrows et al. 1995; Murphy et al. 2004); if it is large enough, runaways produced by the supernova ejection mechanism will also be single. For individual runaways, it is possible to identify a point of origin by measuring accurate distances, proper motions, and radial velocities (e.g., Hoogerwerf et al. 2001; de Wit et al. 2005; Martin 2006; Heber et al. 2008). For example, using Hipparcos data and accurate radial velocities, Hoogerwerf et al. (2001) (1) associate the runaway star ζ Oph and the pulsar PSR J1932+1059 with a supernova in the Sco OB2 association and (2) confirm that AE Aur, μ Col, and ι Ori were ejected from the Trapezium cluster in the Orion nebula. Thus, both runaway mechanisms occur in nature.

Existing observational and theoretical results place few constraints on the spatial and velocity distribution of runaways throughout the Milky Way. Because accurate proper motions are often crucial for identifying runaways, known runaways are observed largely in the solar neighborhood (e.g., Conlon et al. 1990; Holmgren et al. 1992; Mitchell et al. 1998; Rolleston et al. 1999; Hoogerwerf et al. 2001; Ramspeck et al. 2001; Magee et al. 2001; Lynn et al. 2004; Martin 2004, 2006). Although some runaways have large distances from the Galactic plane (e.g., Greenstein & Sargent 1974; Heber et al. 2008), few are in the Galactic halo. Numerical simulations currently provide little insight into runaways as probes of the Galactic halo structure. Davies et al. (2002) simulated the distribution of runaways, but only in the context of high-velocity white dwarfs near the Sun. Martin (2006) proposed using runaways to constrain the Galactic potential, but he concluded that known runaways provide few constraints on the large-scale properties of the Galaxy.

Recent observations of "HVSs" in the halo motivate a broader investigation of the spatial and velocity distribution of runaways. The first HVS is a 3 M_☉ main-sequence star traveling at least twice the escape velocity of the Galaxy at its distance of ∼110 kpc (Brown et al. 2005, 2009a). After two other serendipitous discoveries (Hirsch et al. 2005; Edelmann et al. 2005), subsequent targeted searches of the halo yielded a sample of 15 unbound HVSs with Galactic rest-frame velocities of 350–700 km s⁻¹ and a similar number of bound HVSs with rest-frame velocities of 275–350 km s⁻¹ (Brown et al. 2006a, 2006b, 2007a, 2007b, 2009a). Although the known HVSs are 3–4 M_☉ main-sequence stars roughly uniformly distributed in Galactic latitude, they are not isotropically distributed on the sky (Brown et al. 2009b).

The large space velocity of HVS1 suggests an origin in the Galactic center. As first predicted by Hills (1988), the tidal field of the massive black hole at the Galactic center can unbind a close binary and eject one of the stars at velocities exceeding 2000 km s⁻¹. To distinguish these high-velocity stars from traditional runaway stars, Hills coined the term "hypervelocity star." For typical ejection velocities expected from this mechanism (Hills 1988), we expect a range of velocities similar to those observed in HVSs. Although other dynamical mechanisms involving a black hole can produce HVSs (e.g., Hansen & Milosavljević 2003; Yu & Tremaine 2003; O'Leary & Loeb 2008), the Hills mechanism makes clear predictions for the expected velocity distribution of HVSs ejected from the Galactic center. Kenyon et al. (2008) use these predictions in a numerical simulation of the trajectories of HVSs through the Galaxy. They show that the relative number of bound and unbound HVSs predicted by the Hills mechanism agrees with observations of known HVSs.

Recent observations suggest that runaways can also achieve unbound velocities. The likely unbound star HD 271791 was probably ejected from the disk at 12–16 kpc from the Galactic center (Heber et al. 2008; Przybilla et al. 2008a). Justham et al. (2009) propose that the hot subdwarf US 708, with a heliocentric radial velocity of 708 km s⁻¹ (Hirsch et al. 2005), is also a runaway. The apparent overlap in the velocity and spatial distributions of runaways and HVSs suggests that several mechanisms may inject massive main-sequence stars into the halo. The relative contributions of these processes to the structure in the halo remain unknown.

The distribution of runaways and HVSs is also an important issue for large radial velocity surveys. Surveys like RAVE (Zwitter et al. 2008) and SEGUE (Adelman-McCarthy et al. 2008) measure radial velocities for hundreds of thousands of stars in the thin disk, thick disk, and halo. These data will provide fundamental constraints on the escape velocity and total mass of the Milky Way (Smith et al. 2007; Xue et al. 2008; Siebert et al. 2008). Although there are theoretical predictions for observable properties of HVSs in the halo (e.g., Kenyon et al. 2008), there are no predictions for runaway stars in the halo. Thus, the contribution of runaways to velocity outliers in these surveys is unknown.

Here, we use numerical simulations to make a first assay of the spatial and kinematic signature of runaways in large radial velocity surveys. Our focus is on intermediate mass 1.5–6 M_☉ main-sequence stars ejected from the disk into the halo. Intermediate mass stellar lifetimes are 10⁸–10⁹ yr; thus, these stars formed recently in the disk. The supernova binary disruption scenario requires that the former companion of the runaway was a more massive star with a shorter lifetime. We do not consider massive (>6 M_☉) runaways, because they do not live long enough to reach large distances in the halo. We also do not consider low-mass stars because they are intrinsically faint and unobservable at large distances (Kollmeier & Gould 2007; Kenyon et al. 2008).

We construct a model for the spatial and velocity distributions of runaways. We use predicted ejection velocities from the supernova mechanism to eject stars from the exponential disk, and then we track their orbits throughout the Galaxy. In Section 2, we describe the model and the results of our simulations. We apply the simulations to the runaway HD 271791 in Section 3, to HVS surveys in Section 4, and to halo radial velocity surveys in Section 5. We conclude in Section 6.

To generate populations of runaways in the Galaxy, we perform Monte Carlo simulations of 10⁶–10⁷ stars ejected into three-dimensional orbits from the exponential disk. Each star begins on a circular orbit with velocity $\vec{v}_{0}(r_{\rm init})$ at some distance r_init from the Galactic center in the plane of the Galaxy. The star is ejected at an angle θ_i relative to the orbital velocity vector and an angle b_i relative to the plane of the Galactic disk. For a randomly oriented ejection velocity $\vec{v}_{\rm ej}$, the initial velocity of the ejected star is $\vec{v}_{\rm init} = \vec{v}_{0} + \vec{v}_{\rm ej}$. For a star ejected in the direction of Galactic rotation (θ_i≈ 0), r_init becomes the pericenter of its orbit; for a star ejected opposite Galactic rotation (θ_i ≈ π), r_init becomes the apocenter of its orbit.

To follow the trajectories of these ejected stars through the Galaxy, we integrate the equations of motion numerically. In Bromley et al. (2006) and Kenyon et al. (2008), we used a simple leap-frog integrator to track stellar orbits through a one-dimensional Galactic potential. To provide more accurate solutions for the trajectories of runaways in a three-dimensional Galactic potential, here we use an adaptive fourth-order integrator with Richardson extrapolation (e.g., Equations (1)–(3) of Bromley & Kenyon 2006, see also Chapter 15 of Press et al. 1992). Starting at r_init with velocity $\vec{v}_{\rm init}$, the code integrates the full three-dimensional orbit through the Galactic potential to track position and velocity as a function of time. We integrate the orbit for a random time t, where t lies between 0 and t_ms, the main-sequence lifetime of the runaway star. For stars ejected during the supernova explosion of a more massive and much shorter-lived companion star, t is roughly the age of the runaway star.

To motivate choices for the initial conditions of our simulations, we derive the approximate velocities of runaways capable of reaching the outer halo of the Galaxy. Numerical simulations of binary systems disrupted by supernova explosions show that runaways have maximum ejection velocities of ∼400 km s⁻¹ (Portegies Zwart 2000; Dray et al. 2005). Realistic dynamical ejection models yield similar maximum ejection velocities (Leonard 1993; Fregeau et al. 2004). If the highest velocity runaways are ejected in the direction of Galactic rotation, they receive an additional kick of ≈200 km s⁻¹. Thus, the maximum possible ejection velocity is roughly v_ej,max≈ 600 km s⁻¹. Unbound HVSs in the outer halo have v_rad≳ 400 km s⁻¹ at r≈ 70 kpc (Brown et al. 2009a). To match these properties in our model for the gravitational potential of the Milky Way, runaways must have initial velocities v_ej = 725 km s⁻¹ at r_init = 1 kpc, v_ej = 550 km s⁻¹ at r_init = 10 kpc, and v_ej = 500 km s⁻¹ at r_init = 30 kpc (see Figure 2 of Kenyon et al. 2008). Thus, runaways ejected from r_init ≲ 5 kpc cannot reach the outer halo. This conclusion leads us to consider a fiducial set of simulations with v_ej = 400 km s⁻¹ and r_init = 10 kpc.

Long travel times through the halo constrain our choices for the initial ages and masses of main-sequence runaway stars. Stars with v_ej = 400 km s⁻¹ travel ∼40 kpc in ∼100 Myr. Thus, stars with main-sequence lifetimes t_ms≳ 100 Myr and initial masses m≲ 4.5 M_☉ can reach the outer halo (Schaller et al. 1992; Schaerer et al. 1993; Demarque et al. 2004). Stars with m≈ 4–6 M_☉ and t_ms≈ 65–160 Myr can reach the inner halo. To avoid confusion with indigenous halo stars (see Brown et al. 2006a, 2009a), we focus on main-sequence stars with m = 1.5–6 M_☉ and t_ms = 65 Myr to 2.9 Gyr (Schaller et al. 1992; Schaerer et al. 1993; Demarque et al. 2004, see also Table 1). These lifetimes are much longer than the lifetimes, t_ms≲ 10–20 Myr, of their m≳ 10 M_☉ binary companions that explode as supernovae (Schaller et al. 1992; Schaerer et al. 1993). Thus, we assume for simplicity that runaways have ages of zero when they are ejected from the binary (see also Portegies Zwart 2000).

We first investigate the Galactic distribution of runaways for the fiducial parameters r_init = 10 kpc, v_ej = 400 km s⁻¹, and m = 3 M_☉. Figure 1 shows results for 10⁶ runaways; we plot v_rad as a function of r. In the upper panel, we color code the points in four intervals of θ_i for stars with b_i< 30°. Blue (yellow) points indicate stars ejected with θ_i = −π/4 → π/4 (θ_i = 3π/4 → 5π/4). Cyan and magenta points indicate stars ejected at intermediate angles, θ_i = ±(π/4 → π/2) for cyan points and θ_i = ±(π/2 → 3π/4) for magenta points. In the lower panel, we color code the points in three intervals of b_i: b_i< 30°(cyan), 30° <b_i< 50°(magenta), and b_i > 50°(yellow).

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The distribution of runaway stars in Figure 1 has several features. In the top panel, stars with the largest initial velocities (blue points) are marginally bound to the Galaxy. Thus, they travel far into the halo, reach a maximum distance r_max, and then fall back toward the Galactic center. However, the travel time for this orbit, t ≈ πr_max/v_init ∼ 1 Gyr, is longer than the main-sequence lifetime of a 3 M_☉ star, t_ms≈ 350 Myr. These stars can only be observed on their outward path through the halo and always have positive radial velocities. Thus, the sequence of blue points from (r, v_rad) = (20, 500) to (160, 340) is an age sequence, with 20–30 Myr-old stars at small r and 300–350 Myr-old stars at the largest r.

Stars ejected opposite to the direction of Galactic rotation (yellow points) have the smallest initial velocities and thus cannot reach large r. These stars live long enough to orbit the Galaxy at least once. Thus, the group of yellow points represents a mixture of young and old stars with a velocity distribution symmetric about zero.

Stars ejected at intermediate angles (cyan and magenta points) fill the (r, v_rad) space in between the group ejected along or against Galactic rotation. For each group, the sequence from largest v_rad to largest r is an age sequence, with younger stars at small r and older stars at large r. Thus, most of the lower envelope of the complete ensemble of points, extending from (r, v_rad) = (30, −200) to (r, v_rad) = (160, 340) consists of old stars near the end of their main-sequence lifetimes.

The lower panel of Figure 1 illustrates how the maximum radial velocities decrease with increasing b_i. Stars at high latitude (yellow points; b_i ⩾ 50°) have the smallest velocities, with v_rad ≲ 400 km s⁻¹ at all r. Stars at intermediate latitudes (magenta points; 30° ⩽ b_i ⩽ 50°) reach v_rad ≲ 450 km s⁻¹ at all r; stars at low latitudes (cyan points; b_i ⩽ 30°) have the largest final velocities (v_rad ∼ 500 km s⁻¹) and reach the largest final radii, r = 150–160 kpc in 350 Myr. The form of the Galactic potential produces this variation. Stars ejected perpendicular to the Galactic disk receive a smaller kick from Galactic rotation compared to stars ejected parallel to the disk. Once stars leave the plane of the disk, they also feel the full disk potential. Thus, stars ejected perpendicular to the disk have smaller initial radial velocities and decelerate faster than stars ejected parallel to the disk.

To show how the results depend on each input parameter, Figure 2 plots the velocity distributions for simulations where we vary each of the three fiducial parameters separately and hold the other two fixed. Here, we color code the points with their final galactic latitude. Instead of illustrating relative densities, our intent in this figure is to show changes in the shape of the distributions of v_rad and r as functions of the various input parameters and b_f, the Galactic latitude of stars at the end of the simulation. The upper left panel of Figure 2 repeats the distribution of points in the lower panel of Figure 1 with a different color coding. Here, the variation of v_rad and r_max with final latitude is much more pronounced than the variation with ejection angle.

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The predicted distributions clearly depend on r_init and v_init (upper right and lower left panels of Figure 2). Stars ejected at small r have to climb out of a deeper potential well than stars ejected at large r. At fixed v_ej, runaways ejected from 30 kpc have larger velocities in the outer halo (maximum v_rad = 500 km s⁻¹) than stars ejected from 10 kpc (maximum v_rad = 400 km s⁻¹). Similarly, stars with smaller (larger) ejection velocities maintain smaller (larger) velocities throughout their passage through the Galaxy. Thus, at fixed r_init, runaways ejected at larger velocities have larger velocities in the outer halo and reach larger distances in the halo.

Finally, stellar lifetimes have a significant impact on the distributions of position and velocity (Figure 2; lower right panel). For fixed initial ejection velocities, stars with longer main-sequence lifetimes travel farther into the halo. Stars with longer lifetimes can reach large distances with smaller initial radial velocities. Thus, lower mass stars have smaller v_rad at larger r. For our adopted main-sequence lifetimes, 1.5–4 M_☉ runaway stars ejected from 10 kpc have asymptotic radial velocities

$$\begin{equation} v_{\rm rad} (r_{{\rm max},10}) \approx 250 (M_{\star } / 2\;M_{\odot })^{2/3}\; {\rm km\; s^{-1}} \end{equation} \tag{ 1 }$$

at maximum Galactocentric distances

$$\begin{equation} r_{{\rm max},10} \approx 400 (M_{\star } / 2\;M_{\odot })^{-2.4}\;{\rm kpc}. \end{equation} \tag{ 2 }$$

At 30 kpc, runaways have larger asymptotic radial velocities

$$\begin{equation} v_{\rm rad} (r_{{\rm max},30}) \approx 400 (M_{\star } / 2\;M_{\odot })^{1/6}\; {\rm km\; s^{-1}} \end{equation} \tag{ 3 }$$

at maximum Galactocentric distances

$$\begin{equation} r_{{\rm max},30} \approx 575 (M_{\star } / 2\;M_{\odot })^{-2.6}\;{\rm kpc}. \end{equation} \tag{ 4 }$$

In both equations for r_max, the large exponent follows from the mass dependence of the main-sequence lifetime (t_ms ∝ M⁻³_⋆; Schaller et al. 1992; Schaerer et al. 1993; Demarque et al. 2004).

Tables 1 and 2 quantify these generic conclusions for an expanded set of fiducial simulations. For simulations with r_init = 10 kpc, the median velocities in Table 1 show clear trends with stellar mass and r. Only the highest velocity runaways can reach large r; for all stellar masses, the median velocity increases with increasing r. Because their main-sequence lifetimes are longer, lower mass stars typically orbit the Galactic center 2–3 times. Thus, median velocities for lower mass stars are closer to zero than those for higher mass stars.

The trends of median radial velocity with b_f depend on r and stellar mass. Stars ejected into the disk have larger initial velocities and always travel farther than stars ejected perpendicular to the disk. Thus, at large r, stars at the smaller galactic latitude have a larger median v_rad. Higher mass stars show the largest variation of median v_rad. For every 30° increase in b_f, the median v_rad declines ∼25–75 km s⁻¹ for 1.5–3 M_☉ stars. At small r, 1.5–3 M_☉ stars with b_f< 30° and b_f > 50° were ejected against Galactic rotation; these stars have a range of orbital phases and thus have the median v_rad close to zero. At intermediate b_f, there is a mixture of stars ejected opposite to Galactic rotation (which have median v_rad close to zero) and stars ejected with Galactic rotation (which have larger median v_rad). Thus, the median velocities for b_f = 30°–50° are generally large.

The typical radial velocity dispersions of runaway stars also vary consistently with r, b_f, and stellar mass (Table 2). Stars observed at small r are a mix of young stars ejected into the outer Galaxy and older stars orbiting in the inner Galaxy. Thus, these stars have large velocity dispersions for all stellar masses. Because massive stars have shorter lifetimes, they do not travel far from their ejection point and tend to have smaller velocity dispersions. Stars observed at large r are older stars traveling on extended bound orbits. None of these stars live long enough to fall back into the Galaxy. Thus, all have large, positive v_rad, and small velocity dispersions.

We now consider a simulation for a complete ensemble of runaways ejected from the full Galactic disk. To make this simulation, we adopt probability functions to assign r_init and v_ej for each runaway star, integrate the orbit through the Galaxy, and derive the radial velocity v_rad and position r for each runaway at a random time t in its orbit. Our approach differs from that of Davies et al. (2002), who simulated runaways ejected uniformly from 1 < r < 10 kpc.

For the initial distribution of r_init, we assume that stars are ejected from an exponential disk with a radial scale length of 2.4 kpc (Siegel et al. 2002). We adopt an inner radius of 3 kpc and an outer radius of 30 kpc (Brand & Wouterloot 2007). Thus, the probability distribution for r_init is

$$\begin{equation} p(r_{\rm init}) \propto r_{\rm init} e^{-r_{\rm init}/2.4\,{\rm kpc}}. \end{equation} \tag{ 5 }$$

Runaways ejected from r_init < 3 kpc cannot reach the outer halo. Runaways ejected from r_init > 30 kpc are too rare to make a significant impact on the derived distribution of (r, v_rad).

For the initial distribution of v_ej, we adopt results from published analyses of runaways ejected from binary systems disrupted by supernovae. Although dynamical encounters can also produce runaways, there are no published simulations predicting p(v_ej) the probability distribution of ejection velocities. For v_ej ∼ 20–400 km s⁻¹, a simple function,

$$\begin{equation} p(v_{\rm ej}) \propto e^{-v_{\rm ej}/150\,{\rm km\, s^{-1}}}, \end{equation} \tag{ 6 }$$

provides a reasonable match to the Portegies Zwart (2000) simulations of binary supernova ejections.

We select the Portegies Zwart (2000) ejection velocity distribution function because it is physically well motivated and because a similar distribution function is not available for the dynamical ejection mechanism. Dynamical ejections can attain higher velocities (Leonard & Duncan 1988, 1990; Leonard 1991, 1993); however, the theoretical maximum ejection velocity is not realizable because compact binary interactions are more likely to merge stars than to eject runaways (Fregeau et al. 2004). Moreover, the ejection rate from binary–binary encounters is probably smaller than supernova ejections for intermediate mass stars. Dynamical ejections depend on the joint probability of colliding two binaries within the main-sequence lifetime of the stars (Brown et al. 2009a). Thus, we use the binary-supernova ejection velocity distribution as the representative of the runaway process for 1.5–6 M_☉ stars.

Full-disk simulations of runaways yield many of the same features in the v_rad–r diagram (Figure 3, top panels). As in Figure 2, the lower envelope of the set of points is defined by an ensemble of old runaways close to the end of their main-sequence lifetimes. Stars ejected into the disk (cyan points) receive the maximum ejection velocity. Thus, these stars have the largest v_rad at all r. Stars ejected into the halo (yellow points) have the smallest ejection velocity, the smallest v_rad at all r, and the smallest radial extent. More massive stars with shorter t_ms do not live long enough to reach large r. Thus, lower mass stars have more extended radial distributions than more massive stars.

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The major difference between the fiducial and the full-disk simulations is the variation of the maximum v_rad with r. Runaways ejected from the inner disk slowly decelerate as they move from their point of origin. In an ensemble of runaways ejected from a single radius in the inner disk, stars at larger r therefore have smaller v_rad than stars at smaller r (upper left panel of Figure 2). However, runaways ejected from the outer disk coast outward at roughly constant v_rad (upper right panel of Figure 2). These two features of the evolution combine to produce a roughly constant maximum v_rad with r in the top panels of Figure 3. This maximum velocity is independent of stellar mass and is roughly 100 km s⁻¹ larger than the maximum ejection velocity.

Tables 3–5 list the median velocities, radial velocity dispersions, and vertical velocity dispersions for the full-disk simulations of 1.5–6 M_☉ stars. For all stellar masses, stars ejected at larger v_rad reach larger r. Thus, the median radial velocity increases with r. Because the most distant stars must have roughly the same high ejection velocity to reach large r, these stars have smaller velocity dispersions than the mix of low- and high-velocity runaways at small r. For stars ejected with similar velocities, lower mass stars live longer and can reach larger r. Thus, the median v_rad and the velocity dispersions increase with increasing M_⋆.

The distribution of runaways in the v_rad–r diagram differs from the distribution of HVSs (Figure 3, lower right panel). The lower envelope of the HVS distribution is identical to the runaway distribution and is composed of stars with ages comparable to their main-sequence lifetimes. However, the distribution of HVSs has three features not observed in runaway stars. The HVS distribution has a core of stars at r≲ 3 kpc; these stars were ejected from the Galactic center at small velocities (≲700 km s⁻¹) and cannot reach large r (Kenyon et al. 2008). Because some HVSs are ejected at very high speeds (≳1200 km s⁻¹), these stars can reach large r with radial velocities much larger than any runaway star. Finally, the velocities of HVSs ejected isotropically from the Galactic center are independent of b. Thus, high-velocity HVSs are observable at all b. In contrast, high-velocity runaway stars are only observable at low galactic latitude.

The velocity distribution of runaways is also very different from the velocities of halo stars (Figure 3, lower left panel). Halo stars have a one-dimensional velocity dispersion of roughly 100–110 km s⁻¹ (Helmi 2008; Brown et al. 2009a) and a spatial density close to n ∝ r⁻³ (Siegel et al. 2002). Thus, the envelope of the halo velocity distribution is symmetric about zero and declines slowly with radius. At large r, there are many halo stars with v_rad ≈ 0. In contrast, there are no intermediate mass HVSs or runaways with v_rad ≈ 0 at large r.

To quantify the differences among runaways, HVSs, and halo stars, we now consider the predicted radial surface density of runaways (Figure 4). Compared to the initial surface density of the Galactic disk (dot-dashed line in Figure 4), runaways are much more radially extended. Runaways ejected against Galactic rotation populate the inner disk (r≲ 3 kpc). These runaways have a large velocity dispersion at all b (top panels of Figure 3). Runaways ejected at high velocity along Galactic rotation populate the outer disk. These ejections produce a power-law surface density profile, Σ ∝ r⁻ⁿ with n = 3.5–3.6, at intermediate r and an exponential decline at large r. Longer lived lower mass stars have the most extended power-law component. Short-lived massive stars have exponential density profiles more similar to the density profile of the Galactic disk.

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Although the density profiles of 1.5–3 M_☉ runaways are extended, they are much steeper than the density profiles of halo stars and HVSs. Fits to observations of halo stars typically yield power-law density profiles with n≈ 2.7–3.5 (Helmi 2008, and references therein). Our numerical simulations suggest that low-mass HVSs have a bound component with n = 3 and an unbound component with n = 2–2.5 (Kenyon et al. 2008; see also Hills 1988). The short, finite lifetimes of massive stars steepen the HVS density profile (n≳ 3) at r≳ 80 kpc. However, these density profiles remain shallower than the density profiles of runaway stars.

The predicted vertical density distribution also distinguishes runaway stars from halo stars and HVSs (Figure 5). This density distribution has two components. Low-velocity runaways produce a "thick disk" with a vertical scale height of 300–1000 pc. Although more distant runaways have slightly larger "thick disk" scale heights, the scale height is independent of stellar mass. High-velocity runaways lie in an extended disk-shaped halo with a vertical scale height of 2–40 kpc. More distant runaways have much larger vertical scale heights. At r≲ 50–60 kpc, the scale height of the extended halo is independent of stellar mass (Table 6). At r≳ 60 kpc, the vertical scale height depends on stellar mass. Massive stars (M_⋆ ≳ 4 M_☉) will short main-sequence lifetimes cannot reach large r and thus have no measurable scale height. For stars with longer stellar lifetimes, lower mass stars have smaller scale heights at large r. These smaller scale heights result from low-velocity ejected stars, which have time to reach large r only for the lowest mass main-sequence stars.

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The disk-shaped density distribution of the highest velocity runaways differs from the spherically symmetric density distributions of halo stars and HVSs. In our simulations, HVSs ejected from the Galactic center have a spherically symmetric, power-law density distribution with n≈ 2–2.5 (Kenyon et al. 2008). Halo stars are also distributed spherically symmetrically and have a steeper radial density profile (n≈ 2.7–3.5; Helmi 2008). Because HVSs have a shallower density profile than halo stars, it is easier to identify HVSs at large halo distances than at small halo distances (e.g., Brown et al. 2005; Bromley et al. 2006; Kenyon et al. 2008). The steeper density profiles produced in our runaway star simulations suggest that nearby runaways are easier to identify than distant runaways. We consider this possibility further in Sections 4 and 5.

To conclude this section, we examine several additional properties of simulated runaways in the outer Galaxy. Most runaways with r≳ 60 kpc are ejected from inside the solar circle (Figure 6). Although most ejected stars have r_init ≈ 3–6 kpc, small ejection velocities prevent them from reaching r≳ 60 kpc (Section 2.1). Because the depth of the Galactic potential is smaller for runaways with r_init ≈ 10–20 kpc, these stars make up a large percentage of stars with r≳ 60 kpc (see also Figure 2). In addition, most runaways at large r are low-mass stars with stellar lifetimes long enough to reach the outer Galaxy. With median radial velocities of 10–200 km s⁻¹ (Table 3), nearly all of these runaways are bound to the Galaxy. Before reaching the outer galaxy, massive stars (M_⋆ ≳ 4 M_☉) evolve off the main-sequence and are unobservable at r≳ 60 kpc.

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Our simulations yield a small fraction—0.07%—of runaways that are not bound to the Galaxy. More than half of unbound runaways are ejected from outside the solar circle (double dot-dashed line in Figure 6). Nearly all unbound runaways have low Galactic latitude (b_f< 30°; Figure 7 and Table 7). For 1.5–6 M_☉ stars, the shape of the cumulative probability function for b_f is nearly independent of stellar mass. The median of the distribution, however, varies slowly with stellar mass. Our results suggest median Galactic latitude b_f,med = 9°–10° for 1.5–3 M_☉ stars and b_f,med = 7° for 6 M_☉ stars (Table 7).

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These results for the distribution of b_f contrast with numerical simulations of HVSs, which yield a uniform distribution in b_f (Bromley et al. 2006; Kenyon et al. 2008). In the Hills (1988) ejection mechanism, HVSs are ejected isotropically from the Galactic center. For these stars, the disk is a small perturbation on the potential. Thus, simulated HVSs remain uniformly distributed in b_f throughout their path through the Galaxy. For runaway stars, Galactic rotation provides a significant kick to the ejection velocity. Stars ejected with this rotation are more likely to have unbound velocities than other stars (Figures 1–3). Thus, unbound runaways are confined to the plane of the disk.

Although unbound runaways have low b_f independent of M_⋆, the predicted distances are sensitive to M_⋆ (Figure 8 and Table 7). The results in Figure 8 show that the cumulative probability of r grows roughly linearly from r≈ 6 kpc to r≈ 0.75r_max and then asymptotically approaches 1 at r ≈ r_max. The limiting distance r_max of unbound runaways is a strong function of stellar mass. Long-lived low-mass stars travel far into the outer Galaxy (r_max≈ 500–1000 kpc for M_⋆= 1.5–2 M_☉); short-lived high-mass stars rapidly evolve off the main sequence and cannot reach large r (r_max≈ 50–70 kpc for M_⋆ = 5–6 M_☉).

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Our simulations demonstrate how stellar evolution and the Galactic potential combine to influence the dynamical properties of runaway stars ejected from the Galactic disk. Runaways that receive the maximum kick from the binary-supernova mechanism, ≈ 400 km s⁻¹, can travel from the disk into the halo. These stars produce an extended disk-shaped distribution of stars, where the radial and vertical scale lengths are much larger than those of the main stellar disk. The size of this extended disk is very sensitive to stellar mass. Massive stars with short stellar lifetimes are much less extended than long-lived low-mass stars. Because runaways ejected along the direction of Galactic rotation have higher ejection velocities and climb out of a shallower potential well than other runaways, these stars reach larger distances in the outer Galaxy. Although high-velocity runaways appear at all b_f, the fastest unbound runaways are at low galactic latitude, b_f≲ 30°.

Comparisons of our results with observations of halo stars and simulations of HVSs suggest clear differences between the three populations. Halo stars and HVSs are uniformly distributed in b; runaways are concentrated in the disk. The radial density gradients of halo stars and HVSs are shallower than those of runaways. The v_rad distributions of HVSs and runaways are clearly non-Gaussian compared to observations of halo stars; HVSs tend to have larger v_rad than runaways. These differences suggest clear observational discriminants of the populations, which we explore in the following sections.

We begin by "observing" our numerical simulations for 3 M_☉ runaways and HVSs in a heliocentric reference frame. Known HVSs come mostly from the radial velocity survey of Brown et al. (2006a, 2006b, 2007a, 2007b, 2009a), who target objects with the colors of 3–4 M_☉ stars. Three of the HVSs are confirmed ≃3 M_☉ main-sequence stars (Fuentes et al. 2006; Przybilla et al. 2008b; López-Morales & Bonanos 2008). Girardi et al. (2002, 2004) stellar evolutionary tracks show that a solar metallicity, 3 M_☉ star spends 350 Myr on the main sequence with an average luminosity of M_g = 0.0.

Thus, we calculate heliocentric distances and apparent magnitudes for the simulated runaways assuming M_g = 0.0 appropriate for a 3 M_☉ star. We assume the that Sun is located at r = 8 kpc. We shift the origin of the coordinate system to the Sun and derive heliocentric radial velocities by taking $\vec{v} \cdot \hat{r}$.

Figure 9 plots a heliocentric view of the simulations as a function of Galactic longitude, latitude, and apparent magnitude. The top panels of Figure 9 plot the number distribution of runaways and HVSs. The bottom panels of Figure 9 plot the 50th, 90th, and 99th percentile heliocentric radial velocity of runaways and HVSs.

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The spatial concentration of runaways in the disk, discussed in Section 2 above, is evident in Figure 9. Our models predict a greater fraction of runaways in the Galactic center hemisphere |l| < ±90° and at low Galactic latitudes b < 30°. The fraction of HVSs, on the other hand, is larger than the fraction of runaways in the direction of the Galactic anti-center l = 180° and the Galactic pole |b| = 90°. Compared to HVSs, runaways are also apparently bright: 80% of 3 M_☉ runaways are brighter than g = 16, whereas 85% of 3 M_☉ HVSs are fainter than g = 16.

Runaways are ejected from a rotating disk. This rotation is apparent in the distribution of heliocentric radial velocities: the median (50th percentile) runaway velocity is negative in the direction of Galactic rotation 0° < l < 180° and positive in the opposite direction of Galactic rotation 180° < l < 360°. Similarly, the latitude dependence of runaway velocities reflects the boost from Galactic rotation at low latitudes. HVSs, on the other hand, are ejected on purely radial trajectories and show no rotation. Median HVS velocities exceed the 99th percentile runaway velocity in every direction on the sky. Fainter stars are faster because only the fastest runaways and HVSs survive to reach the largest distances.

We now compare our runaway simulation of 3 M_☉ stars to observations of HVSs. Observed HVSs are significant velocity outliers with minimum radial velocities in the Galactic rest frame > +400 km s⁻¹. The well-defined survey of Brown et al. (2006a, 2006b, 2007a, 2007b, 2009a) samples stars with magnitudes 15 < g < 20.5 and latitudes 30° ≲ b < 90°.

Over the range of magnitude and latitude sampled by the Brown et al. survey, only 0.001% of the simulated 3 M_☉ runaways have velocities > +400 km s⁻¹. These runaways differ in three ways from the observed HVSs. (1) Simulated runaways with radial velocities exceeding 400 km s⁻¹ are located at low latitude with median b = 34°; observed HVSs are distributed uniformly across Galactic latitude with median b = 51° (Brown et al. 2009b). (2) The >400 km s⁻¹ simulated runaways are bright with median g = 16; observed HVSs are faint with median g = 19 (Brown et al. 2009a). (3) The fastest simulated runaway with b > 30° has velocity +450 km s⁻¹; observed HVSs have velocities up to +700 km s⁻¹. We conclude that runaways cannot significantly contaminate the HVS samples because the observed distribution of HVSs differs so markedly from that expected for runaways (Figure 9).

We now consider how many runaways with radial velocities exceeding +400 km s⁻¹ might be included in the Brown et al. HVS survey. To explore this point, we normalize the number of runaways in our simulation to the total number of 3 M_☉ stars formed in the last 350 Myr. Assuming that the star formation rate in the Galactic disk is 4 M_☉ yr⁻¹ (Diehl et al. 2006), 1.4 × 10⁹ M_☉ of stars have formed in the disk in the past 350 Myr. A Salpeter initial mass function, integrated from 0.1 to 100 M_☉ and normalized to 1.4 × 10⁹ M_☉, predicts 1.3 × 10⁷ 3–4 M_☉ stars. Assuming that ∼1% of all stars are ejected as runaways (see below), we predict ∼one 3 M_☉ runaway with +400 km s⁻¹ radial velocity in the Brown et al. HVS survey. This prediction is comparable to the analytic estimate in Brown et al. (2009a), and suggests that perhaps one of the observed HVSs may be a runaway.

We begin by comparing our runaway simulations to the observed stellar population of the Milky Way as revealed by star counts in the SDSS Data Release 6 (DR6; Adelman-McCarthy et al. 2008). For this comparison, we select simulations for 1.5 M_☉, 2 M_☉, and 3 M_☉ stars. As before, we "observe" the simulations from a heliocentric reference frame. We calculate apparent magnitudes assuming the 1.5 M_☉, 2 M_☉, and 3 M_☉ runaways have main-sequence luminosities of M_g = +2.9, +1.5, and 0.0, respectively (Girardi et al. 2002, 2004).

To compare observed number counts of stars with predictions from our numerical simulations, we consider only runaways that fall in the region of sky imaged by the SDSS DR6. For the stars in the SDSS, we count those stars with colors 0.15 < (g − r)₀ < 0.20, −0.15 < (g − r)₀ < −0.05, and −0.35 < (g − r)₀ < −0.25 appropriate for 1.5 M_☉, 2 M_☉, and 3 M_☉ stars, respectively, according to the Girardi et al. (2002, 2004) stellar evolutionary tracks for solar metallicity stars.

Figure 10 plots the resulting number fraction of runaways and SDSS stars as a function of Galactic latitude and apparent magnitude. Over the region surveyed by the SDSS, a larger fraction of runaways are found at low latitudes compared to star counts. This latitude dependence reflects the flattened distribution of runaways compared to the population of halo stars that dominate the star counts. Runaways contribute a negligible amount to the observed stellar (halo) population at faint magnitudes, g ≳ 17.

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The runaway fraction of O- and B-type stars has long been known to be ∼40% and ∼5%, respectively (Blaauw 1961; Gies & Bolton 1986). However, there are few comparable constraints on the fraction of A-type runaways. Stetson (1981) used tangential velocity cuts of bright stars with rough spectral types to estimate that at most ∼0.3% of solar neighborhood A-type stars are runaways. Here, we combine constraints from our simulations and existing spectroscopic surveys to place an independent upper limit on the fraction of 2 M_☉ stars ejected as runaways.

To make this estimate, we use the Brown et al. (2008) spectroscopic survey of A-type stars in the 2MASS (Skrutskie et al. 2006). The Brown et al. (2008) survey is complete over 4300 deg² to a magnitude limit of J₀ = 15.5, equivalent to g ≃ 15.5 for a zero-color A star. Brown et al. (2008) find that 40% of A-type stars at 15th mag are main-sequence stars (the other 60% are evolved horizontal branch stars). A third of the main-sequence stars are consistent with having solar metallicity, thus 13% of 15th mag A stars are possible runaways. In our 2 M_☉ simulation, however, the fraction of runaways is 6 times larger than the observed fraction in star counts at g = 15 (Figure 10). Thus, if the main-sequence A stars observed by Brown et al. (2008) are all runaways, the absolute fraction of A stars ejected as runaways is no larger than 0.13/6 ≃ 2%, in agreement with the upper limit estimated by Stetson (1981).

We now broaden our discussion and look at the distribution of runaways in the context of the Galactic structure.

Spatially, 80% of runaways in our simulations are located at |b| < 15° (Figure 10), a population an observer might call "thick disk." The Galactic thick disk has an observed scale height of 0.75–1.0 kpc (i.e., Siegel et al. 2002; Girard et al. 2006; Jurić et al. 2008), and it dominates the number density of stars in the region 1 kpc < |z| < 5 kpc. We note that claims of unusually large thick disk scale heights are possibly confused with the inner stellar halo (Kinman et al. 2009). By comparison, the number density of runaways in our simulations, selected with 7 kpc < r_cyl < 9 kpc and 1 kpc < |z| < 5 kpc, is well fit by an exponential distribution with a vertical scale height h_z = 0.54 ± 0.01 kpc.

Kinematically, runaways are a hot, rotating population analogous to the thick disk and inner stellar halo. In the region away from the plane |b|>15°, simulated runaways have a mean 135 km s⁻¹ component of velocity in the direction of Galactic rotation, comparable to that observed for the thick disk (Chiba & Beers 2000). Runaways also have a 130 km s⁻¹ heliocentric radial velocity dispersion, essentially identical to the velocity dispersion of inner halo stars with the same apparent magnitude. The Milky Way inner halo, as described by Carollo et al. (2007) and Morrison et al. (2009), has a small ∼25 km s⁻¹ prograde rotation and a one-dimensional velocity dispersion of ∼120 km s⁻¹. Thus, runaways have similar kinematics to the Galactic thick disk and inner halo, and are difficult to identify by radial velocity alone.

The mean metallicity of the inner halo is [Fe/H] = −1.6 (Carollo et al. 2007), but Ivezić et al. (2008) report solar metallicity stars up to 5 kpc above the plane. Perhaps runaways can contribute to the high metallicity population and to the small prograde rotation observed in the inner stellar halo.

The inner halo is too metal poor to be composed entirely of runaways, yet metal-rich, short-lived runaways should be present in the halo of the Milky Way. The deaths of massive runaways must necessarily result in metal enrichment and energy input into the halo. This conclusion applies for all star-forming galaxies at all redshifts. In particular, runaways should be more abundant early in the evolution of a galaxy when the star formation rate is larger. Thus, the distribution of runaways may have important implications for feedback processes.