Impact of the Galactic Disk and Large Magellanic Cloud on the Trajectories of Hypervelocity Stars Ejected from the Galactic Center

As in Kenyon et al. (2014), we work in coordinate systems with an origin at the GC (see Table 1). Stars have cartesian 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 GC longitude); the angle relative to the x–y plane is ϕ (the GC latitude). With ${\varrho }^{2}={x}^{2}+{y}^{2}$, we also specify stellar positions and velocities in spherical (r, θ, ϕ) or cylindrical (ϱ, θ, z) systems.

Table 1.  Summary of Selected HVS Variables

Description	Variable(s)
Gravitational potential	Φ
Cartesian system centered on GC, disk midplane has z = 0	(x, y, z)
Spherical system centered on GC	(r, θ, ϕ)
Cylindrical system centered on GC (ϱ2 = x2 + y2)	(ϱ, θ, z)
GC longitude ($x=\varrho \,\cos \,\theta$, $y=\varrho \,\sin \,\theta$)	θ
GC latitude ($z=r\,\sin \,\phi$)	ϕ
Velocity in the GC frame	v
Heliocentric distance	d
Heliocentric Galactic longitude	l
Heliocentric Galactic latitude	b
Galactic rest-frame radial velocity	vr
Galactic rest-frame tangential velocity	vt
Heliocentric radial velocity	${v}_{r,\odot }$
Heliocentric tangential velocity	${v}_{t,\odot }$
Initial GC distance of HVS	r0
Final GC distance of HVS	rf
Initial GC longitude and latitude of HVS	$({\theta }_{0},{\phi }_{0})$
Final GC longitude and latitude of HVS	$({\theta }_{f},{\phi }_{f})$
Initial heliocentric longitude and latitude of HVS	(l0, b0)
Final heliocentric longitude and latitude of HVS	$({l}_{f},{b}_{f})$
Initial velocity of HVS	v0
Final velocity of HVS	vf
Main-sequence lifetime	tms
Time of ejection from GC	tej
Time of observation	tobs

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To measure dynamical properties in heliocentric coordinates, we adopt a cartesian position (−R_⊙, 0, 0) and velocity (0, ${v}_{\odot }$, 0) for the Sun, where R_⊙ = 8 kpc is the distance of the Sun from the GC (e.g., Bovy et al. 2012) and ${v}_{\odot }$ = 235 $\mathrm{km}\,{{\rm{s}}}^{-1}$ is the space velocity of the Sun relative to the GC (e.g., Hogg et al. 2005; Bovy et al. 2012; Reid et al. 2014; Reid & Dame 2016; Russeil et al. 2017). Stars have distances $d={({(x+{R}_{\odot })}^{2}+{y}^{2}+{z}^{2})}^{1/2}$ and relative velocities ${v}_{\mathrm{rel}}\,={({v}_{x}^{2}+{({v}_{y}-{v}_{\odot })}^{2}+{v}_{z})}^{1/2}$. The Galactic longitude l of the star is the angle—measured counterclockwise in the x–y plane—from a line connecting the Sun to the GC, $l={\tan }^{-1}(x\,\tan \,\theta /(x+{R}_{\odot }))$. The Galactic latitude measures the height of the star above the Galactic plane, $b={\sin }^{-1}(z/d)\,={\sin }^{-1}(r\,\sin \,\phi /d)$. For $r\gg {R}_{\odot }$, θ ≈ l and ϕ ≈ b.

In this heliocentric system, the radial velocity of an HVS is

$$\begin{eqnarray}&&{v}_{r,\odot }={v}_{x}\cos l\cos b+({v}_{y}-{v}_{\odot })\sin l\cos b+{v}_{z}\sin b.\end{eqnarray} \tag{ 1 }$$

The tangential velocity follows from the relative velocity, ${v}_{t,\odot }^{2}={v}_{\mathrm{rel}}^{2}-{v}_{r,\odot }^{2}$. In the GC frame, the radial velocity is

$$\begin{eqnarray}&&{v}_{r}={v}_{r,\odot }+{v}_{\odot }\,\sin \,l\,\cos \,b.\end{eqnarray} \tag{ 2 }$$

The tangential velocity is then ${v}_{t}^{2}={v}^{2}-{v}_{r}^{2}$. For our discussion, we consider velocities in the GC frame. In an observational program, v_t requires accurate measurements of both the proper motion and distance.

We adopt a three-component model for the Galactic potential Φ_G (Kenyon et al. 2008, 2014) with parameters listed in Table 2:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{G}={{\rm{\Phi }}}_{b}+{{\rm{\Phi }}}_{d}+{{\rm{\Phi }}}_{h},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{b}(r)=-{{GM}}_{b}/(r+{r}_{b})\end{eqnarray} \tag{ 4 }$$

is the potential of the bulge,

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{d}(\varrho ,z)=-G{M}_{d}/\sqrt{{\varrho }^{2}+{[{a}_{d}+{({z}^{2}+{b}_{d}^{2})}^{1/2}]}^{2}}\end{eqnarray} \tag{ 5 }$$

is the potential of the disk, and

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{h}(r)=-{{GM}}_{h}\mathrm{ln}(1+r/{r}_{h})/r\end{eqnarray} \tag{ 6 }$$

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

Table 2.  Summary of Parameters

Parameter	Symbol	Value
Mass of central MW black hole	Mbh	3.5 × 106 M⊙
Mass of MW bulge	Mb	3.75 × 109 M⊙
Mass of MW disk	Md	6 × 1010 M⊙
Mass of MW halo	Mh	1 × 1012 M⊙
Mass of LMC	ML	1 × 1011 M⊙
Virial mass for MW	Mvir	1.7 Mh
Scale length of MW bulge	rb	105 pc
Radial scale length of MW disk	ad	2750 pc
Vertical scale length of MW disk	bd	300 pc
Scale length of MW halo	rb	20 kpc
Scale length of LMC halo	rL	15 kpc
Concentration parameter for MW	c	12.5
Distance of Sun from GC	${r}_{\odot }$	8 kpc
Orbital velocity of Sun around GC	v⊙	235 $\mathrm{km}\,{{\rm{s}}}^{-1}$
Distance of LMC from Sun	dL	49.66 kpc

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For the bulge and halo, we set ${M}_{b}=3.75\times {10}^{9}\,{M}_{\odot }$, ${M}_{h}={10}^{12}\,{M}_{\odot }$, r_b = 105 pc, and r_h = 20 kpc (Table 2). 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, 2014) and are consistent with various independent measures of the mass of the Galaxy (e.g., Gnedin et al. 2010; Watkins et al. 2010, 2018; Boylan-Kolchin et al. 2013; Piffl et al. 2014; Peñarrubia et al. 2016; McMillan 2017; Patel et al. 2017b, 2018; Gaia Collaboration et al. 2018; Monari et al. 2018; Posti & Helmi 2018).

In some applications, the potential of the halo is expressed in terms of the virial mass M_vir and the concentration parameter c (e.g., Navarro et al. 1997; Zentner & Bullock 2003; Gómez et al. 2015; Bullock & Boylan-Kolchin 2017, and references therein). For a virial radius ${r}_{\mathrm{vir}}={{cr}}_{h}$, ${M}_{\mathrm{vir}}={M}_{h}[\mathrm{ln}\,(1+c)\,-c/(1+c)]$. Specifying c and M_vir is then equivalent to setting r_h and M_h. Mass models for the MW typically have c ≈ 10–15 (e.g., Dehnen et al. 2006; Boylan-Kolchin et al. 2013; Patel et al. 2017a; Monari et al. 2018), yielding M_vir ≈ 1.5–1.8M_h.

To match the adopted circular velocity of the Sun of 235 $\mathrm{km}\,{{\rm{s}}}^{-1}$, we adopt parameters for the disk potential: ${M}_{d}=6\times {10}^{10}\,{M}_{\odot }$, a_d = 2750 pc, and b_d = 300 pc. The complete set of parameters for the bulge, disk, and halo yields a flat rotation curve from 3 to 50 kpc.

Although the formal escape velocity for the halo is unbounded, we adopt a convenient definition based on the outward velocity required for a star to reach r = 250 kpc with zero velocity. To place this reference point in context, a halo potential with a concentration parameter c = 12.5 has a virial radius r_vir = 250 kpc for the adopted r_h = 20 kpc. We derive v_esc(r) numerically by tracking the position and speed of particles dropped into the MW from rest at r = 250 kpc. For the adopted MW potential, the escape velocity in the x–y plane is roughly 1 $\mathrm{km}\,{{\rm{s}}}^{-1}$ larger than the escape velocity along the z-axis. We ignore this difference.

Following Gómez et al. (2015), we assume a spherical potential for the LMC:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{L}=-G{M}_{L}/\sqrt{{r}_{s}^{2}+{r}_{L}^{2}},\end{eqnarray} \tag{ 7 }$$

where ${M}_{L}={10}^{11}\,{M}_{\odot }$, r_L = 15 kpc, and r_s is the distance from the center of the LMC. The adopted mass and scale length are roughly in the middle of the range measured/proposed in the literature (e.g., van der Marel et al. 2002; Gómez et al. 2015; Peñarrubia et al. 2016; Patel et al. 2017b). Viewed from the GC, the LMC scale length subtends an angle θ_L ≈ 167 at a distance d_L = 50 kpc.

Adding in the central SMBH, the total potential is

$$\begin{eqnarray}&&{\rm{\Phi }}={{\rm{\Phi }}}_{G}+{{\rm{\Phi }}}_{L}-{{GM}}_{\mathrm{bh}}/r,\end{eqnarray} \tag{ 8 }$$

where ${M}_{\mathrm{bh}}=3.6\times {10}^{6}$ M_⊙ is the mass of the central black hole. Although 10%–20% lower than current best values (e.g., Boehle et al. 2016; Eckart et al. 2017; Gillessen et al. 2017), this value maintains consistency with previous studies (Bromley et al. 2006; Kenyon et al. 2008, 2014). Adopting a larger value has little impact on the results.

With our adopted $({M}_{h},{r}_{h})$ and $({M}_{L},{r}_{L})$, the total acceleration from the Galaxy always dominates the acceleration from the LMC. Thus, there is no equivalent to a "Hill sphere," a volume where the gravity of the LMC overcomes the gravity of the MW.³ For stars ejected at an angle θ₀ relative to the line of centers, however, the LMC produces an acceleration tangential to the velocity vector. When d_L = 50 kpc, 0° < θ₀ ≲ 25°, and r ≈ 35–65 kpc, the LMC acceleration is a significant fraction of the deceleration from the MW. In this regime, trajectories are gravitationally focused and bend around the LMC. In Section 4.2, we quantify this gravitational focusing.

When stars pass "through" the LMC, we ignore the possibility that ejected stars collide with gas or stars within the LMC. The trajectories of HVSs follow paths dictated solely by the potential of the MW and the potential of the LMC.

To select ${{\boldsymbol{r}}}_{0}$ and ${{\boldsymbol{v}}}_{0}$ for HVSs, we rely on published calculations. In our approach, 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; Rossi et al. 2014). 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. To select stars capable of reaching ≳10 kpc from the GC, we also set a minimum ejection velocity ${v}_{\mathrm{ej},\min }$ = 750 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (Kenyon et al. 2008). Choosing smaller a_max and larger ${v}_{\mathrm{ej},\min }$ precludes moderate velocity ejections that barely reach the Galactic halo and remain bound to the Galaxy.

These choices for a_max and ${v}_{\mathrm{ej},\min }$ are consistent with expectations for ensembles of close binaries within 1–2 pc of the SMBH. For the selection procedure outlined below, results with ${v}_{\mathrm{ej},\min }$ = 750 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and a somewhat smaller a_max between 0.6 au and 4 au are fairly similar to those with ${v}_{\mathrm{ej},\min }$ = 750 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and a_max = 4 au. In the dense stellar system within a few parsecs of the GC (e.g., Tremaine et al. 2002; Genzel et al. 2003), binaries with components of equal mass and a_max = 0.6 au (4 au) evaporate in roughly 2 Gyr (250 Myr) (Perets 2009). Defining $\zeta ={Gm}/2a{\sigma }^{2}$ where m is the mass of the binary and σ is the stellar velocity dispersion, Fragione & Sari (2018) divide binaries into "soft" (ζ ≪ 1) and "hard" (ζ ≫ 1). For the measured σ ≈ 60 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at a distance of 1–2 pc from the GC (Tremaine et al. 2002), $\zeta \approx 1.1\,(m/6\,{M}_{\odot })\,(1\,\mathrm{au}/a)$. As implied by the calculations of Perets (2009), the adopted upper limit on a_max coupled with the lower limit on ${v}_{\mathrm{ej},\min }$ is consistent with binaries that are hard enough to survive for up to 1 Gyr in the vicinity of the GC.

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

$$\begin{eqnarray}&&{p}_{H}({v}_{\mathrm{ej}})\propto {e}^{(-{({v}_{\mathrm{ej}}-{v}_{\mathrm{ej},H})}^{2}/{\sigma }_{v}^{2})},\end{eqnarray} \tag{ 9 }$$

where the average ejection velocity is

$$\begin{eqnarray}{v}_{\mathrm{ej},H} & = & 1760{\left(\displaystyle \frac{{a}_{\mathrm{bin}}}{0.1\mathrm{au}}\right)}^{-1/2}{\left(\displaystyle \frac{{M}_{1}+{M}_{2}}{2{M}_{\odot }}\right)}^{1/3}\\ & & \times \,{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{3.5\times {10}^{6}{M}_{\odot }}\right)}^{1/6}{f}_{R}\ \ \mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 10 }$$

and σ_v ≈ 0.2 ${v}_{\mathrm{ej},H}$ (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:

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

where

$$\begin{eqnarray}&&D={D}_{0}\left(\displaystyle \frac{{r}_{\mathrm{close}}}{{a}_{\mathrm{bin}}}\right)\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{D}_{0}={\left[\displaystyle \frac{2{M}_{\mathrm{bh}}}{{10}^{6}({M}_{1}+{M}_{2})}\right]}^{-1/3}.\end{eqnarray} \tag{ 13 }$$

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

$$\begin{eqnarray}&&{P}_{\mathrm{ej}}\approx 1-D/175\end{eqnarray} \tag{ 14 }$$

for 0 ≤ D ≤ 175. For D > 175, ${r}_{\mathrm{close}}\gg {a}_{\mathrm{bin}};$ 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; Ducati et al. 2011; dos Santos et al. 2017). For binaries with a = a_max, the maximum distance of closest approach is ${r}_{\mathrm{close},\max }\,=175\,{a}_{\max }/{D}_{0}$. We adopt a minimum distance of closest approach ${r}_{\mathrm{close},\min }=1$ 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; ${v}_{\mathrm{ej},H}$, D, and P_ej follow from Equations (10)–(14). 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. When P_ej ≥ P, ${v}_{\mathrm{ej}}\geqslant {v}_{\mathrm{ej},\min }$, and ${t}_{\mathrm{ej}}\lt {t}_{\mathrm{obs}}$, the star is ejected from the GC. Otherwise, we select new random numbers. We place each ejected 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. These stars have initial Galactic longitude l₀, Galactic latitude b₀, GC longitude ${\theta }_{0}$, and GC latitude ${\phi }_{0}$.

To integrate the motion of each ejected star through the MW+LMC 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 three-dimensional orbit through the Galaxy, allowing us to track position and velocity as a function of time. We integrate the orbit for a time ${t}_{f}={t}_{\mathrm{obs}}-{t}_{\mathrm{ej}}$, shorter than the main-sequence lifetime of the ejected star. This procedure allows us to integrate millions of orbits fairly rapidly. Several tests demonstrate that our approach yields typical errors of 0.01% in position and velocity after 1–10 Gyr of evolution. At t_f, stars have positions $({x}_{f},{y}_{f},{z}_{f})$, equivalently $({r}_{f},{\theta }_{f},{\phi }_{f})$ or $({\varrho }_{f},{\theta }_{f},{z}_{f})$, and velocities $({v}_{{xf}},{v}_{{yf}},{v}_{{zf}})$.

To enable comparisons with other studies, we quote results for several simple calculations (Table 3). In these tests, massless particles released at rest fall toward the GC from several locations along the x-axis (HVx models) or the z-axis (HVz models). The table lists the time t(r) to reach several distances r between the starting point r₀ and the GC. Timescales to fall in along the x-axis are 0.5–1.5 yr shorter than those along the z-axis. Velocities at r = 1 pc and r = 100 kpc are independent of the initial position. At 8 kpc, the velocity of particles falling through the disk is somewhat larger than v(r) for infall perpendicular to the disk. At the ±1 $\mathrm{km}\,{{\rm{s}}}^{-1}$ level, velocities from the numerical calculations agree with the analytic result $v=\sqrt{2({{\rm{\Phi }}}_{G,0}-{{\rm{\Phi }}}_{G})}$, where ${{\rm{\Phi }}}_{G,0}$ is the gravitational potential at r = r₀ and Φ_G is the potential at r. Calculations with these starting velocities at z₀ ≈ 1 pc (or x₀ ≈ 1 pc) achieve the appropriate maximum distance from the GC on the listed infall timescales.

Table 3.  Results for Test Calculations

		t(r) (Myr)			v(r) ($\mathrm{km}\,{{\rm{s}}}^{-1}$)
Model	r0 (kpc)	100 kpc	8 kpc	1 pc	100 kpc	8 kpc	1 pc
HVz	250	1415	1668	1680	261	568	918
HVz	500	3786	4007	4019	320	598	937
HVz	1000	9726	9932	9944	354	616	949
HVx	250	1415	1667	1679	261	579	918
HVx	500	3785	4006	4017	320	608	938
HVx	1000	9725	9931	9942	354	616	949

Note. Within a pure MW potential, particles are released at rest from a distance r₀ and fall toward the GC. The columns list the time t(r) to reach a distance r and the velocity v(r) at r for models where infall is along the z-axis (HVz) or the x-axis (HVx).

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To illustrate the deflection of HVS trajectories by the Galactic disk, we consider stars with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ traveling in the x−z plane for t_f = 1 Gyr (for other examples, see Gnedin & Gould 2005; Yu & Madau 2007). When the initial angle relative to the x–y plane is ${\phi }_{0}=1^\circ$, stars reach a maximum height above the x-axis of z ≈ 60 pc at x ≈ 8400 pc (t ≈ 13 Myr). Acceleration from the disk then pulls the star through the midplane (z = 0) at t ≈ 52–53 Myr when x ≈ 27–28 kpc. Although v_z is then only −2 $\mathrm{km}\,{{\rm{s}}}^{-1}$, the disk potential is too weak to pull the star back toward the disk (see Figure 1 of Kenyon et al. 2008). The star continues to move farther below the disk midplane, reaching z ≈ −750 pc (ϕ ≈ 025) when x ≈ 175 kpc and t ≈ 1 Gyr.

Stars ejected at somewhat larger angles end up farther below the disk midplane after 1 Gyr (Figure 1, blue and green curves). When ϕ₀ ≈ 3°–5°, stars feel a larger gravitational force from the disk. Despite their larger initial v_z, stars with larger ϕ₀ decelerate more rapidly and have v_z ≈ −4 $\mathrm{km}\,{{\rm{s}}}^{-1}$ as they pass through the midplane. After 1 Gyr, these stars almost reach the halo, z = −2200 pc (ϕ ≈ 065).

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As the initial angle of ejection ϕ₀ grows, it is harder and harder for the disk to pull the star across the disk midplane (Figure 1, light green and orange curves). In these examples, the initial v_z is too large for the disk gravity to overcome completely. Although the stars reach a peak z distance and then begin to fall back toward the disk, they remain above the midplane at t = 1 Gyr.

Among all HVSs, larger ejection velocities lead to smaller deflections (Figure 2). When v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and t_f = 600 Myr (purple curve), the difference between the initial and final values for the GC latitude, $\delta \phi =| {\phi }_{0}-{\phi }_{f}|$, grows from zero at ϕ₀ = 0° to nearly 8° at ϕ₀ ≈ 15°. Although the deflection then decreases at larger ϕ₀, it is still significant—roughly 1°—for ejections toward the Galactic pole (ϕ₀ ≈ 89°). For stars with larger v₀, the maximum deflection decreases to roughly 4° for v₀ = 1050 $\mathrm{km}\,{{\rm{s}}}^{-1}$. It is 25 for v₀ = 1200 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and 19 for v₀ = 1350 $\mathrm{km}\,{{\rm{s}}}^{-1}$. Despite the variation in the maximum deflection with initial velocity v₀, the form of the δϕ–ϕ₀ relation is independent of v₀, with the peak deflection always at ϕ₀ ≈ 15°.

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The magnitude of the deflection is sensitive to the disk parameters, M_d, a_d, and b_d. For modest changes in M_d, the maximum δϕ scales approximately with disk mass. Larger (smaller) disk masses result in larger (smaller) deflections, with no shift in the ${\phi }_{0}$ for maximum deflection. The two scale factors—a_d and b_d—control the amplitude and shape of the δϕ–ϕ₀ curve. Smaller (larger) a_d and b_d enable larger (smaller) δϕ. Larger a_d (b_d) shifts the maximum δϕ to smaller (larger) ϕ₀.

For fixed M_d, it is easier to generate larger deflections with modest changes in a_d and b_d than it is to produce smaller deflections. With M_d = 6 × 10¹⁰ M_⊙, for example, setting a_d = 2250 pc (3250 pc) results in a maximum deflection of 10° (7°) instead of the δϕ ≈ 8° for our nominal a_d = 2750 pc. Similarly, adopting b_d = 450 pc (150 pc) yields a maximum δϕ ≈ 7° (10°) instead of δϕ ≈ 8° for b_d = 300 pc.

Systematic deflection of HVS trajectories by the disk has a clear observational consequence. Without deflection, the fraction of all HVSs detected in a survey is proportional to the sky coverage; e.g., surveying 50% of the sky should yield 50% of all HVSs. Because HVSs with ${\phi }_{0}$ somewhat larger than 30° end up with ${\phi }_{0}$ somewhat smaller than 30°, deflection reduces the ability of halo surveys to recover HVSs. This reduction depends on the initial ejection velocity. In these examples, the fraction of stars with ϕ_f ≤ 30° ranges from 54% for v₀ = 1200 $\mathrm{km}\,{{\rm{s}}}^{-1}$ to 57% for v₀ = 1050 $\mathrm{km}\,{{\rm{s}}}^{-1}$ to 60% for v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$. The impact for unbound HVSs with v₀ ≥ 925–950 $\mathrm{km}\,{{\rm{s}}}^{-1}$ is smaller than for bound HVSs with v₀ ≤ 900–925 $\mathrm{km}\,{{\rm{s}}}^{-1}$.

To explore these issues in more detail, we consider 10⁷ intermediate-mass stars (3 M_⊙, B spectral type, t_ms = 350 Myr) with random ejection parameters as outlined in Section 3.2. The sample includes bound stars that barely make it out of the bulge and unbound stars ejected from the Galaxy. Aside from the larger lower velocity limit, v₀ = 750 $\mathrm{km}\,{{\rm{s}}}^{-1}$ instead of v₀ = 600 $\mathrm{km}\,{{\rm{s}}}^{-1}$, this set of calculations is identical to those in Kenyon et al. (2014). Outcomes are also similar. Although the higher minimum v₀ precludes bound stars with maximum r = 1–8 kpc, statistics for stars with r ≳ 10 kpc are nearly identical to those in Kenyon et al. (2014).

To establish the importance of bound and unbound stars in this sample, we derive the variation of the space density with distance from the GC (see also Bromley et al. 2006). For a set of radial bins, r_i, we define a space density, ${\rho }_{i}\propto {r}_{i}^{2}{N}_{i}$, where N_i is the number of stars in a bin extending from r_i − 0.5δr to r_i + 0.5δr. Setting δr = 5 kpc yields a reasonable number of stars per bin. Within the full sample, "halo-like" stars have a space velocity v smaller than 75% of the local escape velocity v_esc; "bound outliers" have 0.75 v_esc ≤ v ≤ v_esc. "Unbound" stars have velocities relative to the GC that exceed the local escape velocity. Table 4 summarizes the fraction of these three types of stars as a function of r.

Our choice for the boundary between halo-like stars and bound outliers is motivated by radial velocity surveys of the halo (e.g., Battaglia et al. 2005; Smith et al. 2007; Brown et al. 2008, 2010, 2014; Xue et al. 2008; Kafle et al. 2012; Loebman et al. 2014; King et al. 2015; Cohen et al. 2017). Within these surveys, the radial velocity distribution consists of a Gaussian component and a small set of outliers. The outliers have velocities roughly 2–3 times larger than the half-width of the Gaussian component. For typical surveys, unambiguous outliers have velocities exceeding roughly 75% of the local escape velocity.

At r ≤ 70 kpc, bound stars dominate the population (Figure 3). Nearly all of the bound stars have v ≤ 0.75v_esc; these stars have positions and velocities similar to those of the indigenous population in the Galactic halo (e.g., Brown et al. 2006a, 2006b, 2007; Kenyon et al. 2014). Roughly 20% have space velocities large enough (v > 0.75v_esc) to be identified as outliers in a halo radial velocity survey but they are still bound to the Galaxy. Only a small fraction of the ejected stars at these distances (2% at 10–20 kpc, 6% at 20–40 kpc, and 21% at 40–80 kpc; Table 4) are unbound.

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The variation of ρ_i with r for the bound stars depends on the stellar lifetime and the initial ejection velocity from the GC (Bromley et al. 2006; Kenyon et al. 2008, 2014). Bound HVSs ejected with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ take 100 Myr to reach r = 45 kpc and another 100 Myr to approach r = 77 kpc. In an ensemble of HVSs ejected from the GC at random times, a travel time of 100 Myr is a modest fraction of the main-sequence lifetime, t_ms = 350 Myr. Nearly all bound HVSs can travel to 50 kpc; the density is then roughly constant with r. Beyond 50 kpc, the travel time becomes a larger and larger fraction of t_ms; the density then begins to drop because the stars die. For HVSs ejected with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ as zero-age main-sequence stars, the maximum distance from the GC is roughly 110 kpc. At this point, the density of bound stars is zero.

When r ≳ 80 kpc (r ≳ 125 kpc), most (all) stars are unbound. Unbound stars ejected at high velocities, v₀ ≈ 1100 $\mathrm{km}\,{{\rm{s}}}^{-1}$, reach 100 kpc (125 kpc) on timescales of ∼100 Myr (175 Myr). On these timescales, it is fairly easy for a B-type main-sequence star to travel 100–110 kpc from the GC; however, it is much more challenging to reach distances much beyond 150 kpc. Thus, the density gradually rises until 100–120 kpc and then begins to fall due to the finite stellar lifetime. Although stars ejected at the largest velocities, 1500 $\mathrm{km}\,{{\rm{s}}}^{-1}$, are still on the main sequence at r ≈ 200–300 kpc, these stars are rare. At these distances, the density of B-type HVSs is negligible.

Despite the smaller set of bound stars in a sample at r ≈ 80–160 kpc, there is a dramatic variation in δϕ with r and ${\phi }_{0}$ (Figures 4 and 5; see also Yu & Madau 2007). At these distances, the highest-velocity stars with v₀ = 1400–1500 $\mathrm{km}\,{{\rm{s}}}^{-1}$ feel a modest deceleration from the disk and have δϕ ≲ 1°. For lower-velocity stars, typical deflections range from 5° (ϕ₀ ≲ 30°) to 05 (ϕ₀ ≳ 70°; Figure 4, lower panel). Maximum deflections are roughly twice the typical deflections. Within the full set of stars, 57% have ϕ_f ≲ 30°, illustrating the dramatic impact of gravitational focusing by the disk.

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Among less distant stars with r = 40–80 kpc, the range of δϕ is roughly 50% larger (Figure 4, upper panel). Compared to the 80–160 kpc group, unbound stars at these distances have somewhat smaller initial velocities and therefore experience somewhat larger overall deflections. However, most stars with the largest δϕ are bound; with v₀ ≈ 850–900 $\mathrm{km}\,{{\rm{s}}}^{-1}$, they spend more time at smaller r and undergo much larger deflections. As a result, more stars in this sample have ϕ_f ≲ 30° (60%).

For stars with r = 10–40 kpc, the variation of δϕ with ϕ₀ is more complicated (Figure 5). In this distance range, nearly all of the stars are bound (94%; Table 4 and Figure 3). Among the bound stars, roughly 90% have halo-like space velocities. On their first pass out through the Galaxy, these stars endure somewhat larger deflections than higher-velocity stars at larger r. The maximum δϕ is roughly 20° for stars with r = 20–40 kpc and ϕ₀ ≈ 15°; stars with r = 10–20 kpc experience a maximum δϕ of 30° at ϕ₀ ≈ 20°–30°. As in Figure 4, the maximum δϕ is smaller for stars with ϕ₀ ≲ 10° and ϕ₀ ≳ 30°; the typical δϕ is half the maximum.

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Some bound stars with ϕ₀ ≳ 30° and r = 10–40 kpc travel out from the GC, reach apogalacticon, and head back toward the Galactic disk. If they live long enough to pass through the midplane of the disk, they end up on the opposite side of the disk relative to their starting point and have δϕ ≳ ϕ₀. This group produces the concentrations of stars extending from (ϕ₀, δϕ) ≈ (35°, 30°) to (ϕ₀, δϕ) ≈ (90°, 150°) in each panel of Figure 5. Stars with smaller v₀ that reach smaller maximum r are more likely to live long enough to pass through the disk plane than higher-velocity stars at larger r. Thus, there are more stars with very large δϕ at 10–20 kpc than at 20–40 kpc.

When bound stars follow purely radial orbits, they simply retrace their path after reaching apogalacticon. Within a real MW, however, the disk deflects trajectories for stars on their way out of the GC and continues to deflect them as they try to return to the GC (Figure 1). In this situation, stars follow very non-radial orbits where the total deflection is roughly proportional to ϕ₀: δϕ ≈ αϕ₀ + β with α ≈ 2.5 and β = −70° for ϕ₀ = 40°–90°.

We next consider the impact of the LMC on the trajectories of HVSs. With a mass almost twice the mass of the Galactic disk, the LMC should generate larger deflections than the disk. To quantify changes to HVS trajectories, we consider simple models with a stationary LMC and then examine results for an LMC on a more realistic orbit relative to the GC.

As a first exploration of the impact of the LMC on HVSs ejected from the GC, we consider a simple potential model where an LMC analog lies along the +z-axis at a distance of 49.01 kpc from the GC and 49.66 kpc from the Sun. Compared to a system with no LMC, the extra mass in the MW+LMC potential changes r(t) and v(t) for HVSs ejected from the GC. After showing how the LMC modifies v(r) for HVSs ejected along the +z-axis, we follow the structure of the previous subsection and quantify how the gravity of the LMC modifies the radial trajectories of individual stars ejected from the GC. We then examine the range of possible deflections for ensembles of 10⁷ HVSs selected with the standard prescription outlined in Section 3.2.

Figure 6 compares v(r) for calculations with (dashed lines) and without (solid lines) an LMC on the +z-axis. In a pure MW potential, stars ejected with v₀ ≲ 775 $\mathrm{km}\,{{\rm{s}}}^{-1}$ toward the LMC have maximum r ≲ 10 kpc (see also Kenyon et al. 2008). As v₀ grows, ejected stars reach larger r. Although the LMC produces negligible changes to v(r) for low-velocity ejections that never reach the LMC, there are clear changes in v(r) for high-velocity ejections. When v₀ ≈ 800 $\mathrm{km}\,{{\rm{s}}}^{-1}$, stars traveling toward the LMC achieve greater distances (r ≈ 40 kpc) than those trying to escape from a pure MW potential (r ≈ 30 kpc). When the ejection velocity is larger (v₀ ≈ 900–1000 $\mathrm{km}\,{{\rm{s}}}^{-1}$), the LMC acceleration produces a clear "bump" in the v(r) track centered on the distance of the LMC from the GC (r ≈ 49 kpc).

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In this example, HVSs ejected toward the LMC reach larger distances than HVSs ejected into a potential with no LMC. When v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$, HVSs ejected along the +z-axis reach r = 105 kpc after 300 Myr of travel time. With no LMC, stars reach only 101 kpc. The smaller deceleration before the star reaches the LMC compensates for the larger deceleration after the star passes through the LMC. Despite the larger r, HVSs traveling through the LMC have v ≈ 200 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at 300 Myr compared to 212 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for HVSs ejected into a pure MW potential. Although the LMC helps HVSs ejected along the +z-axis reach larger distances, these HVSs have 10% smaller velocities.

Although changing r_L results in negligible differences in $v(r,t)$, the amplitude of the bump in the v(r) track responds to the adopted M_L. More (less) massive LMC analogs yield larger (smaller) bumps. In a system where the LMC has twice (half) the nominal mass, an HVS ejected with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ along the +z-axis reaches a distance of 107 kpc (103 kpc) with a velocity of 185 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (205 $\mathrm{km}\,{{\rm{s}}}^{-1}$) after a travel time of 300 Myr. For ejections along the −z-axis, v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ yields (r, v) = (100 kpc, 202 $\mathrm{km}\,{{\rm{s}}}^{-1}$) for the light LMC analog, (97 kpc, 192 $\mathrm{km}\,{{\rm{s}}}^{-1}$) for the nominal LMC analog, and (94 kpc, 172 $\mathrm{km}\,{{\rm{s}}}^{-1}$) for the heavy LMC analog,

Figure 7 illustrates trajectories for HVSs ejected with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at various angles ϕ₀ relative to the Galactic plane. When ϕ₀ = 90°, the axisymmetric potential of the disk and the LMC simply speed up or slow down an HVS without changing its overall path (Figure 6). For smaller ejection angles, however, the LMC deflects stars more than the disk does (Figure 1). Over a travel time of 600 Myr, stars with ϕ₀ ≈ 89° reach a maximum x ≈ 910 pc at z ≈ 64 kpc and then bend back toward the z-axis, reaching x ≈ −200 pc at z ≈ 146 kpc. Stars with smaller ϕ₀ achieve larger maximum x distances from the z-axis before bending around the LMC. The maximum x distance and the z distance for this maximum increase with decreasing ϕ₀.

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Doubling (halving) the LMC mass increases (decreases) deflections (Figure 7). Over the first 90–100 Myr, HVS trajectories are fairly independent of the mass of the LMC. After 100 Myr, the heavier LMC analog sharply bends the path of an HVS toward the x-axis (Figure 7, dark green dashed line). With the lighter LMC analog, the trajectory is much more radial (Figure 7, dark green dotted–dashed line). Despite the different magnitude of the deflections in this example, the overall speed of an HVS at 200 Myr is nearly identical: 267 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (light LMC), 270 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (nominal LMC), 268 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (heavy LMC). Somewhat counterintuitively, an HVS traveling past the heavy LMC travels a larger distance (85 kpc) than in the gravity well of the nominal LMC (81 kpc) or the light LMC (80 kpc).

The variation of δϕ with ejection angle and LMC mass is a signature of gravitational focusing by the LMC. All stars ejected at some angle ϕ' (= 90° − ϕ) relative to the +z-axis feel an acceleration toward the +z-axis. When ϕ' is small, the acceleration in the x–y plane is also small. At large ϕ', the Galactic potential dominates. In both regimes, focusing is negligible. For ϕ' ≈ 15°–25°, acceleration from the LMC at r ≈ 35–65 kpc is large enough to bend trajectories by 5°–10°. At these angles, the acceleration at r ≈ 35–50 kpc exceeds the deceleration at r ≈ 50–65 kpc. Compared to stars with ϕ' ≲ 15° or ϕ' ≳ 25°, these stars end up with slightly larger velocities after passing by the LMC. Because the deceleration from the Galaxy is independent of ϕ', the stars maintain their larger velocities as they continue to speed through the halo.

The amplitude of the "bump" in v in Figure 6 and the deflections of trajectories in Figure 7 are also functions of the ejection velocity from the GC. With the gravitational focusing factor ${f}_{g}\propto {({v}_{\mathrm{esc}}/{v}_{0})}^{2}$, the magnitude of the bump or deflection responds more to changes in v₀ than to changes in M_L. For our nominal M_L, ensembles of stars with v₀ = 1200 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and various ejection angles ${\phi }_{0}$ have nearly identical v_f after a travel time of 300 Myr. Deflections from purely radial trajectories are minimal. Halving or doubling the mass of the LMC has a fairly minimal impact on the trajectories for these high-velocity stars. However, halving (doubling) v₀ leads to much smaller (larger) deflections (Figure 2).

Aside from these obvious gravitational focusing effects, the LMC potential also bends the trajectories of HVSs ejected into the Galactic plane (Figure 8). For stars with ϕ₀ ≈ 1°–5°, the disk gravity works to deflect stars back toward the midplane as in Figure 1. By the time stars reach x = 40 kpc, the gravity of the more distant LMC overcomes the weaker disk gravity and pulls stars away from the plane and into the halo.

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When stars are ejected with ϕ₀ = −1° to −5°, the impact of the LMC is more pronounced. At small x, the disk gravity pulls stars toward the midplane; the z-component of the velocity changes sign from negative to positive. After stars cross the midplane, the gravity of the disk is too weak compared to that of the LMC to pull them back. The gravity from the LMC continues to pull stars farther and farther above the midplane. Because these stars already have a positive v_z, they overtake HVSs ejected with ϕ₀ > 0. The trajectories of stars with ϕ₀ < 0° are therefore bent more than those of stars with ϕ₀ > 0°.

For HVSs ejected with a range of ${\phi }_{0}$, the distance reached after a fixed time depends on v₀ and ${\phi }_{0}$. Stars ejected toward the fixed LMC (${\phi }_{0}$ ≳ 60°) with v₀ = 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ achieve 1%–5% larger distances after a travel time of 300 Myr. Stars ejected away from the LMC (${\phi }_{0}$ ≲ −60°) end up at 4% smaller distances. Stars ejected approximately into the plane (−60° ≲ ${\phi }_{0}$ ≲ 60°) are mainly slowed by the extra gravity from the LMC and have 2%–3% smaller maximum distances than stars ejected into a pure MW potential.

Independent of ${\phi }_{0}$, HVSs in an MW+LMC potential have smaller space velocities. For v₀ = 900 km s, speeds after 300 Myr range from 95% (${\phi }_{0}$ ≈ 75°) to 90% (${\phi }_{0}$ ≈ −90°) of HVS speeds in the pure MW potential. The maximum final speeds occur for HVSs that pass within 1–2 LMC scale lengths of the LMC center.

HVSs ejected with larger (smaller) v₀ have a smaller (larger) impact on their distances and speeds after 300 Myr traveling through the Galaxy. Stars with high (low) velocity spend less (more) time near the LMC and thus experience smaller (larger) overall deceleration. For HVSs capable of escaping the Galaxy (v₀ ≳ 1000 $\mathrm{km}\,{{\rm{s}}}^{-1}$), final distances and speeds are only somewhat less with the LMC than without it. Bound stars with v₀ ≲ 900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ have much smaller distances and velocities, with trajectories modified significantly by the LMC (Figures 7 and 8).

To quantify the impact of the LMC on HVS trajectories in more detail, we consider a second sample of 10⁷ stars ejected from the GC with a stationary LMC analog positioned on the +z-axis at a distance of 49.66 kpc from the Sun. Once again, the predicted δϕ is a strong function of ϕ₀ and v₀ (Figure 9). When an LMC analog lies along the +z-axis, there are three peaks in the ϕ₀–δϕ relation: (i) at ϕ₀ ≈ −15°, where stars ejected with a negative v_z are pulled toward the midplane by the LMC and the Galactic disk, (ii) at ϕ₀ ≈ 15°, where the gravity from the disk counters the gravity from the LMC, and (iii) at ϕ₀ ≈ 75°, where the LMC gravity focuses stars around it.

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With our adopted LMC mass, the three peaks have very different maximum deflections. In a pure MW potential, the two peaks at ϕ₀ = ±15° are symmetric: stars ejected with positive or negative ϕ₀ are equally drawn to the disk midplane. Adding in the LMC potential creates an asymmetry. Stars ejected with negative ϕ₀ are pulled toward the disk midplane by the LMC and the disk. For stars with positive ϕ₀, the gravity of the LMC counters the gravity of the disk. When $| {\phi }_{0}| \,\lesssim 60^\circ$, stars with negative ϕ₀ have larger δϕ than those with positive ϕ₀.

Stars ejected along the +z-axis (ϕ₀ ≳ 60°) experience much larger deflections from the gravity of the LMC than from the gravity of the disk. Among these stars, the disk gravity decelerates stars and produces a modest deflection (Figure 2). All of these stars, however, pass within 2r_L of the LMC and are focused toward the +z-axis. The amount of focusing depends on v₀: stars with large (small) v₀ spend less (more) time near the LMC and have smaller (larger) δϕ.

As in examples for the pure MW potential, the typical δϕ is a strong function of v₀. Stars with the largest v₀ reach the largest r and experience the smallest deflections (Figure 9, lower panel). Compared with HVSs in a pure MW potential, HVSs in the MW+LMC potential with r ≳ 80 kpc and ϕ₀ ≈ −15° (+15°) have larger (smaller) δϕ. Most of these stars are unbound; aside from deflecting stars above the plane, the LMC has little impact on their escape from the Galaxy.

High-velocity HVSs ejected toward the LMC (ϕ₀ ≳ 60°) fall into two groups. Nearly all of these stars are unbound (Figure 3). The LMC deflects these stars by a few degrees, as indicated by the red contour in the lower right corner of the panel. A few stars are bound; before falling back toward the GC, the LMC bends their trajectories by as much as 10°.

Lower-velocity stars that reach r = 40–80 kpc have systematically larger δϕ (Figure 9, upper panel). For nearly all of these stars, the typical δϕ is roughly 50% larger than for higher-velocity stars at 80–160 kpc. However, the shape of the δϕ–${\phi }_{0}$ relation is mostly unchanged, with two peaks at ${\phi }_{0}$ = −15° and ${\phi }_{0}\approx +15^\circ$, and a third peak at a somewhat larger ${\phi }_{0}$ ≈ +75° instead of +65°.

Stars at much smaller r experience a variety of deflections (Figure 10). Compared to results for a pure MW potential (Figure 5), the overall shape of the ${\phi }_{0}$–δϕ relation is fairly similar: (i) most stars have modest deflections, (ii) there are clear peaks in δϕ at ${\phi }_{0}$ ≈ ±15°, and (iii) some bound stars with large ϕ₀ undergo very large δϕ as they try to return to the GC.

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Among stars at 20–40 kpc, the LMC generates several new features in the ${\phi }_{0}$–δϕ relation (Figure 10, lower panel). Although a substantial "tail" of bound stars with large δϕ at large ${\phi }_{0}$ remains for ${\phi }_{0}$ ≲ −30°, there is a much weaker feature at ${\phi }_{0}$ ≳ 30°. Stars with large ${\phi }_{0}$ accelerate toward the LMC and are focused toward it. With somewhat larger velocities (due to their smaller deceleration), fewer stars return toward the GC with the same maximum deflections as their counterparts with ${\phi }_{0}$ ≲ −30°. Instead, the trajectories of these stars bend toward the LMC by 20°–30°, producing an additional peak in the ${\phi }_{0}$–δϕ relation at ${\phi }_{0}$ ≈ 60°.

Because the gravity of the LMC deflects HVSs with small ${\phi }_{0}$ (Figure 8), there is another group of stars with ${\phi }_{0}$ ≈ 0° and δϕ ≈ 20°–30°. In a pure MW potential, stars ejected into the plane feel little gravity from the disk and are undeflected. With an LMC along the +z-axis, these stars are deflected toward the LMC and make it into the halo at r ≈ 20–40 kpc.

Overall, the LMC has a modest impact on bound stars at 10–20 kpc (Figure 10, upper panel). Most stars have modest deflections. The ${\phi }_{0}$–δϕ relation has (i) the standard peaks of δϕ ≈ 30° at ${\phi }_{0}$ ≈ −20° and δϕ ≈ 20° at ${\phi }_{0}$ ≈ +20°, (ii) tails at $| {\phi }_{0}| \gtrsim 30^\circ$ with large δϕ, and (iii) small subsets of bound stars with δϕ ≈ 20°–30° at ${\phi }_{0}$ ≈ 0° and at ${\phi }_{0}$ ≈ 60°. Compared to the pure MW model, the tail at ${\phi }_{0}$ ≳ 30° has a much broader morphology and is more chaotic. With the LMC along the +z-axis, bound stars traveling toward the Galactic pole are pulled toward the LMC, producing a different set of deflections compared to the pure MW potential.

Ejecting a sample of 10⁷ HVSs into a potential with a fixed LMC at its current position yields fairly similar ${\phi }_{0}$–δϕ relations. Among unbound stars at 80–160 kpc, an LMC with d = 49.66 kpc at (l, b) = (+2805, −329) = (−795, −329) creates somewhat larger extreme deflections, with peaks at ${\phi }_{0}$ ≈ −50°, ${\phi }_{0}$ ≈ −15°, and ${\phi }_{0}$ ≈ +10° (Figure 11, lower panel). Stars injected into the Galactic plane typically have δϕ ≈ 4°–8°. Compared to a calculation with the LMC on the +z-axis, these deflections are either a degree or two smaller (${\phi }_{0}$ ≳ −30°) or a degree or two larger (${\phi }_{0}$ ≲ +30°). Stars ejected toward the Galactic pole have either small δϕ (${\phi }_{0}$ ≳ 60°) or large δϕ (${\phi }_{0}$ ≲ −60°).

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An LMC in the southern hemisphere has a more dramatic impact on stars at smaller distances, r ≈ 40–80 kpc (Figure 11, upper panel). Bound stars ejected just above the disk midplane (ϕ₀ ≈ 10°) and into the southern Galactic halo (${\phi }_{0}$ ≈ −60°) then have large δϕ ≈ 30°. These deflections are roughly 50% larger than in a system with the LMC along the +z-axis. Unbound stars injected with −30° ≲ ${\phi }_{0}$ ≲ +30° typically have modest δϕ ≈ 5°–10°. HVSs traveling above the plane (z ≳ 0) are deflected more than HVSs below the plane. Those with much larger ${\phi }_{0}$ have much smaller δϕ ≈ 1°–2°.

Bound stars with lower v₀ have even larger deflections. At 20–40 kpc (Figure 12, lower panel), stars typically have δϕ ≈ 10°–20° at low ${\phi }_{0}$ and a few degrees at large ${\phi }_{0}$. However, groups of bound stars have large δϕ for all ${\phi }_{0}$. As in previous examples, bound stars that travel out through the Galaxy, turn around, and head back toward the GC often have δϕ ≈ 90°–150°. This group grows considerably among stars with r = 10–20 kpc.

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Although the toy models illustrate how a stationary LMC impacts the trajectories of HVSs, the real LMC travels many kiloparsecs during the 100+ Myr flight time for stars ejected from the GC into the Galactic halo. To consider the impact of a more realistic LMC, we add a moving LMC into our potential model. Starting with an LMC at its current position, we calculate the acceleration of the LMC (MW) due to the MW (LMC). Relative to a fixed center of mass, we then integrate the orbits of the LMC and the MW backwards in time using the same procedure as for HVSs ejected from the GC. After an evolutionary time of 10 Gyr, we adopt the endpoint of the "backwards" integration as the starting point for a second integration, where we allow the LMC and the MW to fall back toward the center of mass. The differences between the endpoint of this second integration and the current LMC position are smaller than ±0.05 kpc in each cartesian position coordinate and less than ±0.05 $\mathrm{km}\,{{\rm{s}}}^{-1}$ in each cartesian velocity. Although we could achieve higher accuracy with shorter time steps, this agreement is satisfactory.

In this exercise, the masses, gravitational potentials, and other structural parameters of the LMC and MW are fixed in time.⁴ To treat dynamical friction by the MW on the LMC, we follow previous studies and adopt a simple formula (e.g., Besla et al. 2007; Gómez et al. 2015; Jethwa et al. 2016):

$$\begin{eqnarray}&&\displaystyle \frac{d{{\boldsymbol{v}}}_{L}}{{dt}}=-4\pi {G}^{2}\,{M}_{L}\,{\rho }_{\mathrm{MW}}\,\mathrm{ln}{\rm{\Lambda }}\,\left|{\int }_{0}^{{v}_{L}}{v}^{2}{f}_{\mathrm{MW}}{dv}\right|\,\displaystyle \frac{{{\boldsymbol{v}}}_{L}}{{v}_{L}^{3}},\end{eqnarray} \tag{ 15 }$$

where ρ_MW is the mass density of the MW at the position of the LMC, Λ = r/4800 is the Coulomb logarithm, f(v) is the velocity distribution function, and v_L is the velocity of the LMC relative to the GC.

Typically, this acceleration from dynamical friction is 10% to 15% of the acceleration from the MW on the LMC. Our approach ignores the acceleration from dynamical friction on the MW by the LMC that is a factor of 100 smaller.

It is standard to approximate the integral in Equation (15) by

$$\begin{eqnarray}&&{\int }_{0}^{{v}_{L}}{v}^{2}{f}_{\mathrm{MW}}{dv}=\mathrm{erf}(\xi )-\displaystyle \frac{2\xi }{\sqrt{\pi }}{e}^{-{\xi }^{2}},\end{eqnarray} \tag{ 16 }$$

where $\xi ={v}_{L}/(\sqrt{2}\sigma )$ and σ is the one-dimensional velocity dispersion of the MW halo (Gómez et al. 2015; Jethwa et al. 2016). For a Navarro–Frenk–White profile, ρ_MW and σ can be expressed as

$$\begin{eqnarray}&&{\rho }_{\mathrm{MW}}=\displaystyle \frac{{\rho }_{h}}{{x}_{h}{(1+{x}_{h})}^{2}}\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&\sigma =1.4393\,{v}_{\max }\left(\displaystyle \frac{{x}_{c}^{0.354}}{1+1.1756\,{x}_{c}^{0.725}}\right)\,\end{eqnarray} \tag{ 18 }$$

where ${\rho }_{h}={M}_{h}/4\pi {r}_{h}^{3}$, ${x}_{h}=r/{r}_{h}$, ${x}_{c}={r}_{c,\max }/{r}_{h}$, ${r}_{c,\max }\,=2.16258\,{r}_{h}$ is the radius of maximum circular velocity, ${v}_{\max }\,={({GM}({r}_{c,\max })/{r}_{c,\max })}^{1/2}$ is the maximum circular velocity at $r={r}_{c,\max }$, and $M({r}_{c,\max })$ is the mass contained within ${r}_{c,\max }$ (e.g., Zentner & Bullock 2003; Gómez et al. 2015; Jethwa et al. 2016). Setting

$$\begin{eqnarray}&&M(r)={M}_{h}\left[\mathrm{ln}\left(\displaystyle \frac{r+{r}_{h}}{{r}_{h}}\right)-\left(\displaystyle \frac{r}{r+{r}_{h}}\right)\right]\,\end{eqnarray} \tag{ 19 }$$

yields the mass contained inside r for an adopted M_h and r_h (Navarro et al. 1996, 1997).

To test our algorithm, we conducted a series of tests designed to reproduce published results. Our solutions for the separation of the LMC and MW, r_LMC(t), as a function of the masses and structural parameters follow the trends in Figure 1 of Gómez et al. (2015), who derive the motion of the LMC for a broad range of MW and LMC masses. For our choice of M_h and M_L, the trajectory of the LMC across the sky in (l, b) matches the trajectory in Figure 1 of Boubert & Evans (2016), who adopt an LMC orbit from the calculations of Jethwa et al. (2016).

For an ensemble of 10⁷ HVSs traveling through a time-varying MW+LMC potential, we select stars using our standard prescription. Once a star is placed at r = 1.4 pc with velocity v₀ and ejection angles ${\theta }_{0}$ and ${\phi }_{0}$, we use a look-up table to place the LMC at its expected position at a time t = t_ej. The LMC is then at a position $\delta t={t}_{\mathrm{obs}}-{t}_{\mathrm{ej}}$ earlier than its current position relative to the GC. As we integrate the orbit of the star through the MW+LMC potential, we update the LMC position every time step, until the LMC reaches its current position at t = t_obs. In this way, every star travels through a time-varying potential, with the initial LMC position set by t_ej.

In these calculations, we ignore the changing velocity of the MW relative to the center of mass. Over the 350 Myr main-sequence lifetime of the B-type stars in our model, the velocity of the MW relative to the center of mass changes by roughly 20 $\mathrm{km}\,{{\rm{s}}}^{-1}$. Compared to typical ejection velocities of 700 $\mathrm{km}\,{{\rm{s}}}^{-1}$ to more than 1000 $\mathrm{km}\,{{\rm{s}}}^{-1}$, this difference is small. Among known HVSs, travel times from the GC to the halo are 50–250 Myr (Brown et al. 2014). During this time frame, the velocity of the MW changes by only 10 $\mathrm{km}\,{{\rm{s}}}^{-1}$. This velocity is comparable to typical errors in the radial velocity and much smaller than typical errors in the tangential velocity (e.g., Brown et al. 2009, 2012, 2014, 2015; Hattori et al. 2018a; Marchetti et al. 2018b).

Overall, the moving LMC has little impact on distributions of HVS deflections. At 80–160 kpc, the magnitude of typical and extreme deflections as a function of ${\phi }_{0}$ is similar to that in models with a fixed LMC at its current position (Figure 13, lower panel). At any ${\phi }_{0}$, the largest δϕ is roughly 1° smaller; typical deflections are nearly identical. Among closer stars with r = 40–80 kpc, maximum deflections are nearly 2° smaller; typical deflections are less than 1° different from those with a stationary LMC (Figure 13, upper panel). Within both samples, a moving LMC fills a larger volume throughout the simulation and therefore deflects a larger percentage of stars. Compared to calculations with a stationary LMC, more HVSs have typical deflections in this simulation. Correspondingly fewer have the minimum deflection.

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This situation repeats for stars at 10–40 kpc (Figure 14). Although the maximum and typical deflections are smaller, more HVSs experience significant deflections. Among bound and unbound HVSs, there is a larger percentage of clearly non-radial orbits. Despite these clear differences, a moving LMC has little impact on the vast majority of stars in these samples (within the red contours in the figure). Most stars have modest deflections that range from a few degrees for stars ejected into the Galactic plane or toward the Galactic pole to 10°–30° for stars ejected with ${\phi }_{0}$ ≈ 10°–40°. While the gravity of the LMC deflects these stars, its motion has little impact.

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Within an MW+LMC potential, HVSs ejected from the GC deviate significantly from radial orbits. On its own, the gravity of the disk deflects the trajectories of HVSs moving near the Galactic plane (Figures 1 and 2). For HVSs with ${\phi }_{0}$ ≲ 30°, typical deflections range from δϕ ≈ 5° for unbound stars to δϕ ≈ 20°–30° for bound stars (Figures 4 and 5). Although stars ejected into higher Galactic latitudes experience a δϕ that is a factor of 2–3 smaller, all HVSs are deflected toward the disk. Within a large ensemble of HVSs, more than 60% have final Galactic latitude b_f ≲ 30°.

The LMC adds another source of asymmetry to the potential. With a nominal mass larger than the MW disk mass, the LMC slows down HVSs faster than the disk and generates larger deflections with respect to the initial, purely radial motion. Changes to the radial trajectories are fairly independent of the initial angle of ejection relative to the x–z plane: most HVSs have δϕ ≈ 2°–5°, a factor of 2–3 larger than in a system with no LMC.

In these examples, the position and large mass of the LMC are responsible for the shape of the ϕ₀–δϕ relation. Large deflections always occur for stars ejected with small ${\phi }_{0}$; moving the LMC closer to the disk tends to make these deflections larger. Stars with larger ${\phi }_{0}$ develop significant non-radial motions when the LMC lies somewhere near their nominal trajectory. An LMC along the +z-axis deflects stars with ${\phi }_{0}$ ≳ 60°, but not those with ${\phi }_{0}$ ≲ −60°. Similarly, placing the LMC at its current position changes the trajectories of stars with ${\phi }_{0}$ ≈ 0° to −60° and ${\theta }_{0}$ ≈ −135° to −25° much more than the trajectories of stars ejected into other quadrants of the Galaxy.

For any LMC position, reducing (increasing) the mass of the LMC leads to smaller (larger) peaks at ϕ₀ ≈ −15° and at ϕ₀ ≈ 75°. Although changing the LMC scale length r_L changes v_f and r_f for HVSs in the outer halo, r_L has little impact on the magnitude or the shape of the ϕ₀–δϕ relation. For peaks at −15° and +15°, the distance of the LMC is large compared to the scale length. The magnitude of the deflections then depends only on the LMC mass. When stars are ejected along the +z-axis, the scale length has a modest impact on the height and shape of the peak at +75°. However, changes in r_L by a factor of two produce much smaller variations in the peak than changes in the LMC mass by a factor of two.