Chapter 6: Stellar Kinematics

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176
Chapter 6: Stellar Kinematics
6.1 Introduction
The velocity of a star in three dimensional space includes radial and transverse
components. The radial component is observed directly as the star’s photospheric
radial velocity which is measured from the Doppler shift of its spectral features. The
transverse component is measured as the angular displacement of a star in the plane of
the sky over time, otherwise known as proper motion. Distance is used to convert the
angular measurements of the proper motion into the transverse velocity. Eggen (1961)
outlines how to convert the proper motion, distance, and radial velocity of a star into a
full space velocity in the heliocentric rectangular coordinate system (UVW velocities).
The space velocity can provide insight into a star’s past and present association with a
given kinematic population and when considered in a model of the galactic potential,
can be used to trace the past trajectory. In this way the kinematics of a star can provide
valuable information about its nature and origin.

6.1.1 Galactocentric Cylindrical Coordinate System
The UVW velocities calculated using the method of Eggen(1961) are subjected
to an additional coordinate rotation to transform each star’s space velocity into the
galactocentric non-rotating cylindrical frame (V
π
, V
θ
, V
z
). In this frame of reference,
V
π
is the radial motion of the star relative to the galactic center. A positive V
π
is
directed away from the galactic center. V
θ
is the rotational velocity component,
orthogonal to V
π
in the galactic plane. V
θ
is positive in the direction of galactic
177
rotation. Lastly, V
z
is the velocity of the star perpendicular to the galactic plane, where
a positive V
z
is going towards the North Galactic Pole (NGP) . The coordinate
transformation assumes that the Sun is 8kpc from the galactic center and moving
relative to that point with the velocity (V
π
, V
θ
, V
z
) = (-9 km/s, 232 km/s, 7 km/s). The
solar motion is a combination of the motion of the Local Standard of Rest (LSR, V
θ
=
220 km/s) and the Sun’s own peculiar motion (Mihalas & Binney 1981).

6.2 Diagnostics from Stellar Kinematics
Full space velocities are useful for the purpose of identifying a star as a member
of a given kinematic population. The Milky Way has three main stellar kinematic
components: the bulge/bar, the disk, and the halo. The bulge/bar population is largely
confined to within a few kiloparsecs of the galactic center so it is of little concern to this
study.
The disk is a rotationally supported population with an exponential distribution
on the z-axis that can be divided into a thin disk that has a scale height of 300 parsecs
and a thick disk that is kinematically hotter and has a scale height of 1.35 kiloparsecs
(Binney & Merrifield 1998). The key property of the stars, in both the thin and thick
disk populations, is a strong rotational velocity (V
θ
) on the order of 200 km/s. The
velocity dispersions of the stars in the thick disk are slightly larger than the thin disk
(Table 6.1). However, both disk populations have smaller velocity dispersions than the
halo population.
178
Table 6.1 Kinematic Parameters of Galactic Stellar Populations

Population
<V
θ
>
(km/s)
σ
π

(km/s)
σ
θ

(km/s)
σ
Z

(km/s)
Thin Disk
a
214 34 21 18
Thick Disk
a
184 61 58 39
Halo
b
25 135 105 90
a Edvardsson (1993)
b Binney & Merrifield (1998)

The stellar halo occupies a roughly spherical volume surrounding the disk and
bulge. Stars in the halo population are characterized by a roughly isotropic velocity
distribution without any strong signature of rotation (Table 6.1). Many of the stars in
the halo have velocities with a much stronger radial component (V
π
) than stars in the
disk population.

6.2.1 Expected Kinematics of Sub-Luminous Stars
Stars in the sample which are on the blue horizontal branch (BHB) are in most
cases significantly less luminous than main sequence stars with the same effective
temperature. If the intrinsic luminosity of a main sequence star is mistakenly applied to
a sub-luminous star in order to calculate its photometric distance, then that distance will
be overestimated. A distance that is too large will inflate the calculated full space
velocity of that star. This can cause a member of either the thick or thin disk population
to look kinematically more like a member of the halo. In extreme cases, an
overestimated distance can even result in velocities which appear to exceed the local
escape velocity of the Milky Way (at least 500 km/s; Carney, Latham, & Laird 1988).
If one uses a more appropriate luminosity, many of these evolved stars are not
several kiloparsecs; but only a few hundred parsecs above the plane. This means that
179
they are more likely to be members of the galactic thick disk than the halo. Thus by
applying proper intrinsic luminosities to these stars, it may be possible to sort them
kinematically into the halo and thick disk populations.

6.2.2 Expected Kinematics of Runaway Stars
Stars which have been ejected from the disk, regardless of ejection mechanism,
should have a unique kinematic signature which is similar to the disk population with
some kinematic heating of the velocity components. This assumes that the ejection
velocities are not so large that the residual kinematic signature of the parent disk
population is completely obliterated by kinematic heating. Since these stars formed in
the thin disk, they started out with V
θ
on the order of 200 km/s and small V
π
and V
z

velocities (Table 6.1). The mechanisms which eject them from the disk alter their space
velocity, probably without favoring any component. However, there may be an
observational bias in favor of stars that received a large kick in V
z
since the focus here
is on stars that manage to travel far out of the galactic plane in a relatively short time.
The remainder of the ejection velocity that does not go into the V
z
component is added
to the V
π
and V
θ
components.
Without knowing the distribution of ejection velocities imparted to runaways, it
is not possible to model how those velocities might be partitioned into each of the
components. The maximum ejection velocity is about 300 km/s for either the binary
supernova scenario (BSS) or the dynamic ejection scenario (DES) (Iben & Tutukov
1997; Leonard 1991). Assuming that most of the kick gets put into the V
z
component
(due to the bias in favor of stars further from the galactic plane), then at most a few
180
hundred km/s is left to be distributed between the V
π
and V
θ
components. Therefore, on
a plot of V
π
versus V
θ
a sample of stars that have been ejected from the disk should look
like a kinematically heated disk population and fall in a circular area within a few
hundred km/s of (V
θ
, V
π
)=(220 km/s, 0 km/s).
The kinematics of runaway stars, halo stars, and sub-luminous stars with
overestimated distances should have differences with respect each other. In a sample
with velocities calculated from photometric distances assuming main sequence
luminosities, this would leave only the runaway stars having disk-like kinematics in this
part of the V
θ
and V
π
plane. Generally speaking it can be difficult to assign individual
stars to populations on the basis of kinematics alone because their distributions overlap
in velocity space. Some stars that are not runaways may still randomly fall inside the
region of kinematically heated disk-like velocities but they should be distinguishable
from ejected Population I B stars by their metal poor photospheric elemental
abundances.

6.2.3 Past Stellar Trajectories
The full space velocity of a star can be integrated with a model of the Milky
Way’s gravitational potential to calculate the galactic orbit of that star. A solution
integrated backwards in time yields the time of travel from the disk to the present
location of a suspected runaway. A star that has a travel time less than or equal to its
main sequence lifetime could have originated in the disk. However, it is important to
keep in mind the uncertainties associated with the calculation of both the flight times
and the main sequence lifetimes (Section 6.4.3 and 6.5). The past trajectory of a star
181
can also reveal where the star may have originated in the disk and the velocity with
which it was ejected. The range of ejection velocities is helpful as a constraint on the
theoretical ejection mechanisms.

6.3 Input Data
The algorithm of Eggen (1961) computes full space velocities as a function of
distance, proper motion, and radial velocity. The errors in the velocities were estimated
from the errors of the input parameters using a Monte Carlo technique. Normally the
largest contribution to the errors is from the proper motions because they are small
angular measures which usually have errors that are a significant fraction of their value
and they are multiplied by the distance. To ensure the best results, attention must be
paid to the error contributions from each input parameter.

6.3.1 Proper Motions
Every star in the study has proper motions from the Hipparcos Catalog (HIP)
(Turon et al. 1997), the Tycho-2 Catalog (Hog et al. 2000), and at least one other
catalog including: the ACT (Urban et al. 1997), the Carlsberg Meridian Catalogs
(CMC) (Fabricius 1993), and the First U.S. Naval Observatory CCD Astrograph
Catalog (UCAC1) (Zacharias et al. 2000). Three or more independent proper motion
measures for each star were combined by weighted average as described in Martin &
Morrison (1998). In most cases, the errors of the average proper motions are a factor of
1.5 to 2.0 smaller than the errors in the Hipparcos catalog. The average proper motions
and their errors are listed in Table 6.2.
182
Table 6.2 Proper Motions
Average Proper Motion
Star
µ
α
(mas/yr) µ
δ
(mas/yr)
Source
a

HIP 1214 57.0±0.5 10.6±0.5 ACT,CMC
HIP 1511 -3.8±1.4 -34.4±1.1 CMC
HIP 3013 8.0±1.0 0.1±0.6 ACT,UCAC1
HIP 6419 -4.9±0.6 -17.7±0.4 ACT,UCAC1
HIP 6727 13.2±0.6 -1.2±0.8 ACT,UCAC1
HIP 11809 -17.4±1.5 -12.9±1.5 CMC
HIP 11844 4.8±0.9 -2.4±0.9 ACT,UCAC1
HIP 12320 4.9±1.7 -2.4±1.6 CMC
HIP 15967 -1.4±0.7 2.1±0.8 ACT,UCAC1
HIP 16130 8.6±0.7 3.2±0.5 ACT,UCAC1,CMC
HIP 28132 -4.0±0.7 8.3±0.6 ACT,UCAC1,CMC
HIP 37903 2.0±1.0 -5.9±0.8 ACT
HIP 41979 11.7±0.8 -23.4±0.7 ACT
HIP 45904 2.1±0.7 -9.8±0.7 ACT
HIP 48394 3.5±1.0 -5.5±0.6 ACT
HIP 50750 -12.3±0.7 4.3±0.6 ACT
HIP 52906 -6.7±1.3 0.1±0.8 ACT,CMC
HIP 55461 -1.5±0.7 -6.1±0.7 ACT,CMC
HIP 56322 3.6±1.1 11.4±0.9 ACT
HIP 58046 8.7±1.0 5.3±0.7 ACT
HIP 59067 -8.3±0.9 -6.0±0.7 ACT,CMC
HIP 59955 -10.4±0.7 -2.5±0.7 ACT
HIP 60578 -12.1±0.9 4.8±0.7 ACT
HIP 60615 -1.3±1.2 1.4±0.9 ACT
HIP 61800 0.0±0.6 -5.0±0.6 ACT,CMC
HIP 63591 -4.2±0.8 -8.4±1.0 ACT
HIP 65388 -26.4±1.5 10.2±1.4 CMC
HIP 66291 -7.6±1.1 -8.7±0.8 ACT
HIP 68297 1.4±0.9 17.8±0.7 ACT
HIP 69247 -0.4±0.8 -2.6±0.7 ACT,UCAC1
HIP 70275 4.6±0.9 -11.7±0.7 ACT,CMC
HIP 71667 -9.9±0.7 5.7±0.6 ACT
HIP 75577 8.5±0.5 -10.5±0.4 ACT,CMC
HIP 76161 1.8±0.7 -1.3±0.8 ACT,UCAC1
HIP 77131 -3.4±0.7 -2.4±0.6 ACT,UCAC1
HIP 77716 -12.7±1.1 1.6±1.0 ACT
HIP 79649 0.2±0.7 -3.8±0.6 ACT
HIP 81153 -7.6±0.9 -12.8±0.6 ACT
HIP 82236 -4.6±1.0 3.3±0.6 ACT
HIP 96130 1.7±0.7 -0.2±0.6 ACT,UCAC1,CMC
HIP 98136 -3.4±0.7 3.5±0.4 ACT,UCAC1
HIP 104931 4.2±0.8 -7.3±0.7 ACT
HIP 107027 -6.4±0.8 -11.0±0.6 ACT,UCAC1,CMC
HIP 109051 0.7±0.7 -0.5±0.5 ACT,UCAC1
HIP 111396 -0.5±0.6 3.6±0.5 ACT,UCAC1,CMC
HIP 112790 -18.5±0.7 -13.8±0.6 ACT,UCAC1
HIP 114569 46.5±0.8 33.4±0.7 ACT,UCAC1,CMC
HIP 115729 3.5±0.7 -1.4±0.5 ACT,UCAC1,CMC
HIP 116560 23.6±1.2 -26.9±0.9 ACT,UCAC1
a Sources averaged into the proper motion in addition to HIP and Tycho-2.
183
6.3.2 Distances
The distances to the stars in the sample of study were determined by
photometric parallax. Until proven otherwise, all the stars in the sample are assumed to
have main sequence luminosities. After the stars which are sub-luminous have been
positively identified, their space velocities will be revisited assuming a more
appropriate distance for their diminished luminosity (see Section 8.3.3).
The absolute Johnson V magnitudes were calculated from Geneva Photometry
for the stars using the relation of Cramer (1999). The Cramer (1999) relation was
calibrated using Hipparcos parallaxes to obtain the distances and intrinsic luminosities
of stars and then fitting a function to the Geneva reddening free X and Y color indices
(Cramer & Maeder 1979). The standard deviation of the fit to the Cramer (1999})
relation is 0.44 magnitudes. Since this method uses the X and Y indices it is only valid
for stars with effective temperatures hotter than 11000 K.
Stars which fall outside the scope of the Cramer (1999) relation or have no
Geneva photometry were assigned absolute Johnson V magnitudes from the tables of
Wegner (2000) according to their effective temperature and surface gravity. The
Wegner (2000) relation is also calibrated using the Hipparcos parallaxes of nearby OB
stars by averaging the absolute magnitude for each spectral type and fitting a smoothed
relation to the data.
184
Figure 6.1 Comparison of Absolute Magnitude Relations

A comparison of the absolute magnitudes calculated using the Cramer (1999) and
Wegner (2000) relations.

Comparisons of the results from the Wegner (2000) and Cramer (1999)
calibrations are plotted in Figure 6.1 and Figure 6.2. Figure 6.1 shows that for stars
with absolute magnitudes fainter than -1.0, Cramer (1999) estimates significantly lower
absolute magnitudes than Wegner (2000). Figure 6.2 shows that there may be some
trend to the difference between the two relations with respect to effective temperature
for stars hotter than about 15000 K. However, that trend appears to be minimal
compared to the 0.44 magnitude accuracy of the Cramer (1999) relation. Below an
effective temperature of 15000 K (log(T
eff
) = 4.18) it is difficult to tell which relation is
correct.
185

Figure 6.2 Difference in Absolute Magnitude Versus Effective Temperature

The difference in the absolute magnitudes calculated using the Cramer (1999) and
Wegner (2000) relations as a function of effective temperature.

The differences between the results of the Cramer (1999) and Wegner (2000)
relations highlight the difficulty with photometric distances. In addition to uncertainty
in the absolute magnitude, uncertainty in the apparent magnitude and the interstellar
extinction contribute to the distance error. The 0.44 magnitude accuracy of the Cramer
(1999) relation corresponds to roughly a factor of 20% in distance. Considering the
Wegner (2000) relation and other sources of error, the photometric distances are
assigned a ±30% error (about ±0.57 magnitudes).
Distance moduli were calculated from the apparent Johnson V magnitudes using
extinction corrections which were calculated from the E(B-V) values, assuming an
186
interstellar reddening coefficient (R = A
v
/E(B-V)) of 3.1 for all lines of sight. The
photometric distances derived by this method are listed in Table 6.3.
The distance to HIP 76161 has been independently determined. It is an
eclipsing binary composed of two main sequence B stars that was analyzed by Martin
(2003). From that work, a distance of 2.4 kpc is adopted with an error of ±30%. This
distance compares favorably with the distance for this star from the Wegner (2000)
relation (2.3 ±0.7 kpc).

187
Table 6.3 Photometric Distance Derived From Main Sequence Luminosity
Apparent Absolute Distance
Star V mag A
V
V mag
a
(kpc)
HIP 1214 8.98 0.12 0.42 0.49
HIP 1511 11.61 0.20 0.05
b
1.87
HIP 3013 10.87 0.06 -2.09 3.79
HIP 6419 9.52 0.13 -0.57 0.98
HIP 6727 11.14 0.11 -0.03
b
1.63
HIP 11809 11.08 0.09 -1.15
b
2.68
HIP 11844 10.10 0.07 0.50 0.80
HIP 12320 12.04 0.27 -1.00
b
3.59
HIP 15967 10.35 0.03 -1.56 2.37
HIP 16130 9.94 0.00 -0.14 1.04
HIP 28132 9.83 0.13 -1.49 1.72
HIP 37903 9.90 0.15 -2.85
b
3.32
HIP 41979 11.48 0.11 -2.30
b
5.43
HIP 45904 10.01 0.05 -3.15 4.19
HIP 48394 10.11 0.02 -2.15
b
2.80
HIP 50750 9.98 0.14 -0.86 1.38
HIP 52906 11.25 0.06 -2.75
b
6.15
HIP 55461 11.19 0.13 -1.45
b
3.18
HIP 56322 10.12 0.08 -2.69 3.51
HIP 58046 10.22 0.03 -0.85
b
1.61
HIP 59067 11.07 0.12 -1.30 2.82
HIP 59955 9.84 0.07 0.16 0.84
HIP 60578 10.66 0.04 -1.14 2.25
HIP 60615 10.48 0.03 -2.15
b
3.31
HIP 61800 8.79 0.04 -1.88 1.34
HIP 63591 10.64 0.08 -0.03
b
1.31
HIP 65388 12.45 0.03 -2.15
b
8.19
HIP 66291 11.86 0.06 -1.70
b
5.01
HIP 68297 10.26 0.20 -2.55 3.32
HIP 69247 9.36 0.27 -2.84 2.44
HIP 70275 9.68 0.15 -2.29 2.31
HIP 71667 10.15 0.08 -1.85
b
2.42
HIP 75577 7.95 0.12 -2.91 1.40
HIP 76161 9.10 0.70 -3.40
c
2.29
HIP 77131 8.88 0.84 -4.02 2.59
HIP 77716 10.84 0.04 -3.52 7.31
HIP 79649 9.06 0.03 -2.40 1.93
HIP 81153 7.79 0.87 -3.49 1.21
HIP 82236 10.53 0.18 -3.13 4.96
HIP 96130 9.80 0.56 -3.22 3.11
HIP 98136 9.38 0.79 -3.28 2.37
HIP 104931 10.51 0.21 -2.85
b
4.26
HIP 107027 9.33 0.24 -3.00 2.61
HIP 109051 9.86 0.12 -1.95 2.18
HIP 111396 9.04 0.19 -1.07 0.97
HIP 112790 10.13 0.14 -0.85 1.47
HIP 114569 9.75 0.08 -1.29 1.55
HIP 115729 8.30 0.09 -1.81 1.01
HIP 116560 10.87 0.08 1.00 1.44
a Main sequence absolute magnitude from Cramer (1999) relation unless noted otherwise.
b Main sequence absolute magnitude from Wegner (2000) relation.
c Main sequence absolute magnitude from Martin (2003)
188
6.3.3 Radial Velocities
Radial velocities were measured for forty (41) of the stars in the sample (Section
2.5). Of the remaining eight stars, no spectra were obtained for six and it is not possible
to measure the radial velocity from the spectra of two other stars due to very short
period variability and irregularly shaped line profiles which made it difficult to measure
the line centers from the long exposures taken in this study (see Section 2.5.2). As a
result, the radial velocities for these stars were taken from the literature (Table 6.4).

Table 6.4 Stars With Radial Velocities from Other Sources
Heliocentric
Star V
R
(km/s) Source
HIP 6419 -7±2.4 Greenstein & Sargent 1974
HIP 50750 49±2.4 Greenstein & Sargent 1974
HIP 60578 115±10 Evans (1979)
HIP 68297 28±2.4 Greenstein & Sargent 1974
HIP 70275 239±2.4 Greenstein & Sargent 1974
HIP 71667 27±10 Evans (1979)
HIP 77131 -9±5 Evans (1979)
HIP 82236 2.42±0.53 Lynas-Gray et al.(1984)

6.4 Analysis
The full space velocities calculated from the photometric distances, heliocentric
radial velocities, and proper motions by the method of Eggen (1961), transformed to the
galactocentric cylindrical reference frame are listed in Table 6.5. The errors for the
velocity components are calculated by a Monte-Carlo method using the estimated errors
of the input parameters.
189
Table 6.5 Positions & Space Velocities in the Galactocentric Cylindrical Frame
Position
a
Velocity
b

Star R (kpc)
θ (deg)
c
Z (kpc)
d
V
π
V
θ

V
z

HIP 1214 8.2 3.2 -0.3 124.4±20.1 179.3± 8.2 5.9± 2.9
HIP 1511 8.9 11.0 -1.0 -241.0±23.4 -17.3±18.7 -122.8±53.8
HIP 3013 9.9 21.4 -3.7 139.6±22.0 168.3±14.2 -88.9± 5.0
HIP 6419 8.8 3.7 -0.9 -68.9±10.0 184.9± 8.3 -17.0± 6.7
HIP 6727 9.4 5.5 -1.5 74.7±12.7 165.7±11.6 19.7± 3.1
HIP 11809 10.6 3.5 -2.1 -233.5±38.9 287.0±12.9 -158.1±36.9
HIP 11844 8.8 2.0 -0.7 -23.5± 2.5 223.8± 3.8 53.6± 4.6
HIP 12320 11.5 4.5 -2.7 55.5±15.5 163.9±23.0 0.4±16.0
HIP 15967 10.1 6.8 -1.9 19.6± 5.3 253.7± 7.3 -57.3± 5.3
HIP 16130 8.9 3.6 -0.8 29.8± 5.8 213.3± 3.7 10.9± 7.1
HIP 28132 9.2 7.9 -0.6 49.3± 9.6 269.4± 9.3 -17.3± 3.9
HIP 37903 11.1 7.1 1.7 113.4± 9.2 177.2±18.0 72.0±12.0
HIP 41979 13.1 9.8 2.8 -296.8±58.3 -461.8±106.5 66.8±18.7
HIP 45904 12.1 4.1 2.9 7.2± 8.9 50.7±35.0 85.4±15.3
HIP 48394 10.7 5.5 2.0 -19.7± 9.2 186.1±12.6 65.2±13.7
HIP 50750 9.1 6.0 1.1 76.4±12.8 233.9± 3.1 9.9± 8.2
HIP 52906 14.2 0.9 5.4 217.8±36.4 204.2±19.3 -7.4±24.3
HIP 55461 9.8 17.2 2.9 -36.3± 7.5 117.8±15.2 37.3± 9.0
HIP 56322 9.3 21.6 3.1 -4.7±12.9 296.3±29.2 319.9±21.1
HIP 58046 8.8 9.3 1.6 -59.4± 7.5 291.3±11.0 51.2± 4.8
HIP 59067 8.6 19.0 2.7 37.1±11.2 108.6±19.6 -1.0± 9.9
HIP 59955 8.0 6.0 0.8 14.6± 5.2 199.1± 4.7 14.5± 3.3
HIP 60578 9.7 9.6 2.1 177.9±22.4 238.1± 6.6 81.0±10.2
HIP 60615 11.1 7.7 3.3 32.7±11.1 249.0± 9.0 30.4± 3.1
HIP 61800 9.0 5.9 1.3 -27.6± 3.6 201.6± 5.1 -35.1±10.3
HIP 63591 7.9 9.4 0.9 4.4± 3.4 173.0± 9.7 5.8± 2.8
HIP 65388 13.2 35.8 7.5 1015.6±170.8 -424.2±83.6 28.9±17.0
HIP 66291 3.6 25.7 4.8 -20.2±12.2 -49.8±40.5 129.4± 9.8
HIP 68297 5.2 16.2 2.8 28.4±17.8 392.4±35.3 164.8±30.5
HIP 69247 6.1 12.2 1.6 -48.6± 3.8 172.6± 9.3 -1.6± 6.7
HIP 70275 5.9 8.3 1.7 -223.2±14.7 82.4±10.2 95.4±19.8
HIP 71667 5.8 8.8 2.2 99.0±17.5 189.3± 7.0 88.2±14.2
HIP 75577 6.7 4.5 1.1 -39.5±11.2 197.8± 3.9 -86.1±10.9
HIP 76161 5.8 6.5 1.0 -0.5± 4.0 208.6±12.7 -35.3± 7.6
HIP 77131 5.5 5.7 1.1 -2.8± 3.4 148.2±13.1 8.9± 5.8
HIP 77716 6.8 58.4 5.7 132.6±41.9 -164.7±36.8 215.1±61.8
HIP 79649 8.1 13.8 1.3 -1.2± 6.2 240.1± 4.2 26.0± 4.7
HIP 81153 6.8 1.7 0.5 -138.3± 5.4 161.4±12.8 66.9± 4.1
HIP 82236 4.6 34.4 2.7 104.4±12.1 115.0±25.7 117.6±26.9
HIP 96130 5.0 8.1 -1.1 56.0± 4.8 160.7±18.6 2.0± 8.7
HIP 98136 5.9 9.2 -0.9 -2.6± 4.2 228.0±14.4 43.2±11.5
HIP 104931 6.5 32.1 -2.1 28.7± 9.9 116.8±18.0 -134.4±30.7
HIP 107027 6.0 14.3 -1.9 -133.9±16.7 166.1±17.9 -58.3± 7.6
HIP 109051 6.6 13.2 -1.6 -2.3± 3.8 242.5± 6.7 -52.6± 5.7
HIP 111396 7.4 5.8 -0.8 20.6± 2.0 226.0± 3.4 39.7± 4.2
HIP 112790 7.2 9.4 -1.2 -143.3±23.3 196.6± 8.2 26.7± 7.6
HIP 114569 6.9 8.7 -1.4 418.3±64.5 279.6± 5.2 4.6±11.8
HIP 115729 7.7 7.0 -0.9 10.2± 2.5 228.4± 2.5 -24.0± 2.6
HIP 116560 7.1 8.8 -1.4 36.0±10.5 21.6±35.5 -133.4±14.7
HIP 116560 7.1 8.8 -1.4 36.0±10.5 21.6±35.5 -133.4±14.7
a Assumes that the Sun is 8.0 kpc from the galactic center and 0.0 kpc from the galactic plane.
b The velocity of the Sun in this frame is: (V
π
, V
θ
, V
z
) = (-9 km/s, 232 km/s, 7 km/s)
c The angle of separation between the Sun and the star as viewed from the galactic center.
Positive angles are measured in the direction of galactic rotation.
d The distance from the galactic plane to the star in the direction of the north galactic pole
190
A number of stars, HIP 65388 in particular, stand out as having unusually large
velocities. These velocities might be the result of overestimating the luminosity of a
sub-luminous star. The V
z
for twenty two (22) of the stars is under 50 km/s. This is not
unexpected because it is more likely to observe a star at the apex of its trajectory (where
V
z
is small) since stars spend more time near the apex of their orbit than traveling to or
from that point. Fourteen (14) stars are actually moving back toward the galactic plane.
If they are runaways, these stars have already passed through the peak of their
trajectory.

6.4.1 Identifying Potential Runaways Using Space Velocities
A plot of V
θ
versus V
π
(Figure 6.3) shows a clustering of stars around (V
θ
, V
π
) =
(220 km/s , 0 km/s) as expected for normal disk stars and predicted for a kinematically
heated population of runaway stars ejected from the disk. The space velocities of the
stars in the abundance control sample are plotted for comparison since that sample is
representative of the parent population for stars ejected from the disk.
As discussed in Section 6.3.2, runaway stars plotted on Figure 6.3 should appear
as a population with a disk-like mean rotational velocity, kinematically heated so that
the total velocity dispersion in V
θ
and V
π
is on the order of a few hundred km/s. A
circle is drawn on the plot to enclose a region within 100 km/s of the LSR (Local
Standard of Rest), (V
θ
, V
π
) = (220 km/s , 0 km/s). A circle/oval with this radius
includes a clump of stars that is evenly distributed around the LSR. The circle could
have been drawn at 150 km/s to include some of the stars just outside the right side of
191
Figure 6.3 V
θ
versus V
π
For Sample Stars (Assuming Main Sequence Distances)

A plot of V
θ
versus V
π
calculated assuming that the stars in the sample all have main-sequence
luminosities. The circle with radius 100 km/s centered on the LSR ((V
θ
, V
π
) = (220 km/s , 0 km/s))
denotes the region were most stars ejected from the galactic disk may be expected to fall. The velocities
of the nearby B stars in the abundance control sample are plotted as triangles that are densely cluster
around the LSR.

the 100 km/s boundary. However, then the enclosed stars are not evenly distributed
about the LSR. Assuming that there is no preferred direction in which stars are ejected,
the distribution should fall symmetrically around this point. There is a slight hint of
some structure in the kinematic distribution of stars close to the LSR in Figure 6.3. This
apparent structure is discussed in detail in Section 8.4.3.
The region within 100 km/s of the LSR, which includes the most likely
runaways, contains twenty five (25) of the sample stars. That is not to say that all
twenty five of the stars in that region are definitely runaways but rather that this is one
piece of information in favor of that classification for these stars. Likewise some of the
192
stars which lie outside of the 100 km/s circle, particularly those that lie within 150 km/s
of the LSR, are not completely excluded from being runaway stars.

6.4.2 Past Trajectories From Velocities
Stars ejected from the disk are required to have travel times from the disk to
their present location that do not exceed their projected main sequence lifetimes. Past
trajectory and travel time from the disk are calculated from the full space velocities by
integrating them in a model of the galactic potential. The orbital integrator and galactic
potential model are described in detail by Harding et al (2001). A brief description of
the model potential is given below. The errors in a star’s distance and velocity are
propagated through the integration by a Monte-Carlo simulation.
A three component model is used by Harding (2001) to simulate the Milky
Way’s potential following the prescription of Johnston, Spergel, & Hernquist (1995).
The three components are: (1) a Miyamoto & Nagai (1975) disk potential, (2) a
Hernquist (1990) spheroid potential, and (3) a logarithmic dark halo potential. The
parameters for each of the three components are selected so that together they reproduce
the observed rotation curve of the Milky Way’s disk (Table 6.6). These parameters are
identical to those used by other contemporary studies (Johnston, Spergel, & Haydn
2002; Helmi & White 2001) to model the motions of stars in the Milky Way’s halo.
Table 6.7 and Figures 6.4 & 6.5 show that this integrator and galactic potential model
are also able to reasonably reproduce the flight times and ejection velocities of Magee et
al. (2001).
193
Table 6.6 Components and Parameters of Milky Way Potential Model
Miyamoto & Nagai (1975) Disk Potential
2222
disk
disk
)bza(R
GM
+++
−=Φ

M
disk
= 1.0 x 10
11
M
￿

a = 6.5 kpc
b = 0.26 kpc
Hernquist (1990) Bulge/Spheroid Potential
c
r
GM
elgbu
spheroid
+
−=Φ

M
bulge
= 3.4 x 10
10
M
￿

c = 0.7 kpc
22
zRr +=

Dark Halo Logarithmic Potential
)drln(v
222
halohalo
+×−=Φ

v
halo
= 128 km/s
d = 12.0 kpc


Figure 6.4 Comparison of Ejection Velocities with Magee et al. (2001)

A plot of ejection velocities calculated by the method used in this study versus the
ejection velocities from Magee et al. (2001).
194
Table 6.7 A Comparison With The Results of Magee et al. (2001)
Flight Time (Myr)
Ejection Velocity (km/s

Star This Work Magee et al.This Work Magee et al.
EC 00321-6320 45.2±2.9 30 57.3±2.2 58
EC 00358-1516 28.5±1.1 32 225.7±17.7 278
EC 00468-5622 38.6±2.2 33 96.3±3.0 93
EC 03342-5243 18.0±0.9 17 89.0±2.5 145
EC 05515-6231 35.9±4.2 21 38.9±4.4 37
EC 19071-7634 6.1±1.3 6 177.6±30.9 266
EC 19337-6734 13.7±1.8 12 39.3±2.7 48
EC 19476-4109 35.1±5.8 52 80.9±8.8 79
EC 20089-5659 21.4±3.4 35 38.7±2.2 60
Figure 6.5 Comparison of Travel Times with Magee et al. (2001)

A plot of flight times as calculated by the method used in this study versus the flight
times from Magee et al. (2001).

195
6.4.3 Travel Time Versus Main Sequence Lifetime
Flight times of stars from the disk were calculated assuming that each star
originated in the galactic plane with Z = 0. The star forming regions of the thin disk
define the plane of the galaxy but a few, like the Orion complex, are on the order of 100
pc from the galactic plane. When stars on ballistic trajectories are close to the galactic
plane they are moving with velocities at or close to their ejection velocity (about 100
km/s or more). A star moving at 100 km/s crosses 100 pc in roughly 1 Myr. Therefore,
there is no significant difference in travel time whether it is figured from Z = 0 or Z =
100 pc. Table 6.8 contains the flight times for each star from Z = 0 to its present
location.
The main sequence lifetime was determined from the effective temperature of
each star by a two step process. First the tables of Lang (1992) were used to estimate
the main sequence mass of the star from its effective temperature. Then the main
sequence lifetime was estimated from the mass using the tables of Iben (1967). The
estimated main sequence lifetimes are listed in Table 6.8. The errors of main sequence
lifetimes are difficult to estimate. Heger & Langer (2000) show that stellar rotation and
other mixing can change the main-sequence lifetime of a massive star by as much as
30%. In fact, many of the processes involved are only roughly modeled (Heger,
Langer, & Woosley 2000) which affords additional uncertainty.
Figure 6.6 compares the estimated main sequence lifetime to the calculated
flight time for each star in the sample. The stars having velocities withing 100 km/s of
the LSR (Figure 6.3) are highlighted. Stars in the upper left corner of the plot that have
196
Table 6.8 Flight Times and Main Sequence Lifetimes

Star
MS Lifetime
(Myr)
T
Flight
(Myr)

Z
ej
a
(km/s)

HIP 1214 246 20.1± 1.2 -36.2± 1.8
HIP 1511 2014 9.4± 6.5 -136.8±46.3
HIP 3013 21 26.2± 0.7 -193.8± 6.9
HIP 6419 105 22.7± 2.2 -58.5± 2.4
HIP 6727 1356 36.8± 1.4 -106.2± 5.0
HIP 11809 103 13.0± 2.9 -176.8±32.9
HIP 11844 309 55.1± 2.9 -68.6± 3.1
HIP 12320 115 46.2± 5.9 -122.8±11.9
HIP 15967 30 23.0± 1.1 -104.6± 3.7
HIP 16130 100 30.7± 2.9 -60.3± 2.3
HIP 28132 43 18.5± 1.4 -47.0± 1.9
HIP 37903 36 18.7± 2.1 +105.3± 8.1
HIP 41979 16 35.8± 7.8 +90.7±16.2
HIP 45904 19 26.5± 2.9 +132.8±10.4
HIP 48394 21 23.2± 2.8 +107.5± 8.5
HIP 50750 42 25.7± 2.4 +72.9± 2.1
HIP 52906 28 59.4± 8.4 +191.8±17.2
HIP 55461 55 34.2± 2.4 +122.6± 3.9
HIP 56322 12 9.4± 0.5 +342.3±19.0
HIP 58046 117 20.5± 1.0 +92.9± 3.0
HIP 59067 60 38.4± 2.4 +149.0± 9.3
HIP 59955 105 20.1± 0.9 +59.2± 0.8
HIP 60578 56 18.7± 1.3 +139.1± 5.7
HIP 60615 25 37.7± 1.0 +130.4± 3.8
HIP 61800 163 47.6± 5.0 +81.0± 4.4
HIP 63591 1085 23.4± 0.9 +61.9± 0.6
HIP 65388 27 76.4±15.1 +122.0±25.1
HIP 66291 39 26.0± 1.1 +232.2± 8.8
HIP 68297 16 14.5± 2.1 +213.6±24.6
HIP 69247 295 30.7± 2.4 +94.0± 4.6
HIP 70275 17 14.9± 2.1 +132.2±15.5
HIP 71667 86 17.6± 1.5 +156.5± 7.9
HIP 75577 174 67.3± 7.1 +101.8± 7.8
HIP 76161 20 33.1± 3.7 +79.4± 4.5
HIP 77131 10 20.7± 1.8 +91.0± 6.2
HIP 77716 19 21.5± 4.3 +330.5±46.4
HIP 79649 20 23.7± 1.3 +81.0± 1.9
HIP 81153 10 7.8± 0.4 +81.0± 3.4
HIP 82236 10 15.8± 2.0 +232.2±29.8
HIP 96130 17 20.2± 2.4 -101.4± 9.1
HIP 98136 16 36.8± 5.9 -74.3± 5.3
HIP 104931 61 13.4± 2.2 -183.8±21.8
HIP 107027 18 20.4± 1.4 -112.5± 6.2
HIP 109051 20 19.5± 1.1 -109.6± 3.9
HIP 111396 88 39.9± 2.1 -70.1± 2.2
HIP 112790 56 49.5± 6.3 -58.0± 5.7
HIP 114569 36 24.4± 2.5 -115.7± 5.6
HIP 115729 36 19.1± 0.7 -66.9± 1.0
HIP 116560 2814 9.4± 0.9 -157.5±11.9
a Component of ejection velocity in the Z direction.
197
flight times which are longer than their lifetimes could be stars that formed in situ in the
halo or sub luminous stars whose distances, velocities, and flight times have been over
estimated. A number of stars identified in Figure 6.3 as runaway star candidates appear
to be unable to travel from the disk to their present location within their estimated main
sequence lifetimes.





Figure 6.6 Main Sequence Lifetime Versus Travel Time

The flight time of a star from the disk plotted against the estimate of its main sequence
lifetime. Stars inside the circled region from Figure 6.3 are highlighted with a filled in
diamond. Stars on the upper left side of the diagram take longer than their projected
lifetime to return to the disk while stars on the lower left have no such conflict.


198
6.5 The Effects of A Lower Disk Mass Surface Density
A decrease in the disk potential will cause an increase in flight times which
could create a conflict with the main sequence lifetimes for several stars in Figure 6.6.
The disk potential is the dominant component of the model as far as most stars in this
study are concerned. The parameters adopted for the Miyamoto & Nagai (1975) disk
potential translate to a local disk mass surface density (Σ
disk
) of about 95 M
￿
/pc
2
. This
is significantly larger than the local disk mass surface density indicated by dynamical
studies of the solar neighborhood for the disk alone (Σ
disk
= 48 ± 9 M
￿
/pc
2
) (Kuijken &
Gilmore 1989) and for the total integrated mass within 1.1 kpc of the galactic plane (Σ
1.1
kpc
=71 ± 6 M
￿
/pc
2
) (Kuijken & Gilmore 1991).
In simple terms a decrease in Σ
disk
should produce a proportional increase in
flight times for stars near the disk which are most influenced by its potential. A smaller
Σ
disk
leads to a decreased restoring force for stars displaced from the galactic plane,
allowing runaway stars to travel further from the disk for a given ejection velocity. This
means stars would travel at lower velocities to reach the same positions, lengthening
their time of flight from the disk. The disk potential is embedded in the larger dark halo
potential, so stars that journey farther may pass beyond the immediate influence of the
disk and be less effected by changes in Σ
disk
.
Simply changing Σ
disk
in the model is a non-trivial procedure because this
potential model relies on a balance between its three components in order to reproduce
the observed rotation curve of the galactic disk. A complete reformulation of the model
is beyond the scope of this study. However, a simple reassessment of the model can
give an idea of the effects which a lower Σ
disk
will have on the flight times.
199
The model of the galactic potential is constrained by observations of the galactic
rotation curve. The galactic rotation curve produced by the baseline model used in the
analysis is flat at about 210 km/s between 5 kpc and 20 kpc from the galactic center (see
Figure 6.7A). This is in good agreement with the observed rotation of the Milky Way
disk (Clemens 1985) and is able to reproduce the motion of the LSR in simulations.
Altering Σ
disk
, by either changing the radial scale length of the disk or the total disk
mass (M
disk
), will cause the rotation curve to take a different shape. The rotation curve
can be returned to its generally accepted form by altering the parameters of the halo to
rescale and flatten it appropriately.
The flight times calculated using a series of models with different Σ
disk
(Table
6.9) were compared to the baseline model used in the analysis (Table 6.6). The dark
halo potential in each of the models is adjusted to flatten the rotation curves (Figure
6.7). The fractional difference in the flight times calculated from each model with
respect to flight times calculated from the baseline model are shown in Figure 6.8. The
No Bulge model illustrates that the bulge does not have a significant effect on the flight
times as long as the rotation curve has been flattened by altering the halo parameters
(Figure 6.8A).

Table 6.9 Alternative Potential Models
Σ
disk
M
disk
a M
bulge
c V
halo
d
Model (M
￿
/pc
2
) (M
￿
) (kpc) (M
￿
) (kpc) (km/s) (kpc)
Baseline
a
95 1.00e11 6.5 3.4e10 0.7 128 12.0
No Bulge 95 1.00e11 6.5 0.0e10 0.0 103 1.0
Disk 0.5 47 0.50e11 6.5 3.4e10 0.7 113 4.0
Disk 0.6 56 0.60e11 6.5 3.4e10 0.7 115 5.0
Disk 0.7 66 0.70e11 6.5 3.4e10 0.7 115 6.0
Short Disk 47 0.67e11 2.6 0.0e10 0.0 132 6.5

* See Table 6.6 for the form of the model and definition of the parameters
a The baseline model is the model of Harding et al. (2001) used in the analysis

200
The Disk 0.5, Disk 0.6, and Disk 0.7 models explore a range of Σ
disk
values by
changing M
disk
and maintaining the scale length used by the baseline model. As
expected, reducing Σ
disk
lengthens the time of flight for a star from the disk to its present
location (Figures 6.8B-D). The expected transition between stars dominated by the disk
potential and the dark halo occurs at about 1.5 kpc from the galactic plane. The flight
times of stars more than 1.5 kpc from the plane, which are less influenced by the disk
potential, are at most ten to fifteen percent larger than the flight times calculated using
the baseline model. Meanwhile, stars which are within 1.5 kpc of the disk have larger
differences in their flight times which depend on Σ
disk
.

201
Figure 6.7 Rotation Curves of Galactic Potential Models


Panels A through F show the galactic rotation curves for the series of models with parameters listed in
Table 6.9. The solid line is the total rotation curve for the model. The dotted line is the rotation curve for
the Baseline model. The dash-dot line is the contribution from the bulge potential. The dashed line is the
contribution from the disk. And the dash-dot-dot-dot line is the dark halo contribution.
202
Figure 6.8 Effects of Different Models On Flight Times With Respect to Z


Panels A through E show the fractional difference of the flight times from the model
named on the panel relative to those from the baseline model as a function of a star’s
distance from the galactic plane.

203
Recent studies have favored a significantly shorter radial disk scale length than
employed in the baseline model ( Kent, Dame, & Fazio 1991; Spergel, Malhotra, &
Blitz 1996; Drimmel, & Spergel 2001;and other references therein). The Short Disk
model has the same Σ
disk
as the 0.5 Disk model with a radial scale length of 2.6 kpc
(versus 6.5 kpc in the 0.5 Disk model), which is consistent with more recently accepted
values. The bulge has been left out of the Short Disk model because including it causes
difficulties flattening the rotation curve and omitting it should not have any significant
effect (see the No Bulge model, Figure 6.8A). The fractional differences in the flight
times calculated using the Short Disk model (Figure 6.8E) closely mirror the
distribution of flight time differences calculated using the 0.5 Disk model (Figure 6.8B).
This demonstrates that the flight times are not really dependent on the scale length and
M
disk
except through their influence on Σ
disk
.
In summary, changes to Σ
disk
do not cause significant changes to the flight times
for stars that are more than 1.5 kpc from the galactic plane. Reducing Σ
disk
by up to a
factor of two lengthens calculated flight times for those stars by less than ten to fifteen
percent. This is much smaller than the uncertainty of at least 30% in their main
sequence lifetimes. However, the flight times of stars which are 1.5 kpc or closer to the
plane can be more sensitive to variations in Σ
disk
. This may lead to a consequential
conflict between the flight time and main sequence lifetime of a star. In light of this
information the travel times, presented for a number of stars in Table 6.8 and Figure 6.6
should be considered conservative lower limits. Figure 6.9 shows how the differences
in travel time between the model used in the analysis and the 0.5 Disk model affect
204
Figure 6.6. These issues and their implications will be discussed in more detail in
section 8.4.2.

Figure 6.9 The Effect of the 0.5 Disk Model on Flight Times in Relation to
Main Sequence Lifetimes

Like Figure 6.6, this is a plot of the flight time of a star versus its estimated main
sequence lifetime. An * denotes the flight time calculated with the model described in
Table 6.6. A triangle denotes the flight time calculated using the 0.5 Disk model (see
Table 6.9). A solid line connects the flight times calculated for each star with each
model.