Activity and Kinematics of Low Mass Stars
by
John Sebastian Pineda
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2010
@ Massachusetts Institute of Technology 2010. All rights reserved.
Author ...... ................................
.
May 7, 2010
n
I
Certified by
D....................
Department of Physics
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....
Andrew A. West
Assistant Professor of Astronomy, BU
Thesis Supervisor
Certified by .
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*~*1~**!
7
//
Adam J. Burgasser
Associate Professor of Physics, MIT
Thesis Supervisor
Accepted by ....................................
.. .................
Professor David E. Pritchard
Senior Thesis Coordinator, Department of Physics
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ARCHNES
AUG 13 2010
LI3RAR{ES
Activity and Kinematics of Low Mass Stars
by
John Sebastian Pineda
Submitted to the Department of Physics
on May 7, 2010, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
We present an analysis of the magnetic activity, photometry and kinematics of approximately 70000 M dwarfs from the Sloan digital Sky Survey (SDSS) Data Release 7. This
new analysis explores the spatial distribution of these M dwarf properties as a function of
vertical distance from the Galactic plane (Z) and distance from the Galactic center (R). We
confirm the established trends of decreasing magnetic activity, as measured by Ha emission, with increasing distance from the mid-plane of the disk but also observe a new trend
in Galactocentric radius, apparent in the analysis of spectral types M3 and M4 of a small
increase in activity with increasing R. Examining the color indices r - z, r - i and g - r
from the SDSS ugriz photometry reveals noticeable gradients in the vertical direction but
not in the radial direction. To analyze the kinematics we develop a new technique utilizing probability distributions and a pseudo-montecarlo data fitting scheme to determine the
parameters (o- 1 , pi, 0-2, 12) and normalization of the underlying Gaussians making up the
kinematic distributions of the stellar population. We analyze each of the spatial velocities
VR, Vz , and Ve defined in a Galactocentric cylindrical coordinate system. The kinematic
analysis reproduced previous trends of increasing dispersion with increasing distance from
the mid-plane, but with much greater accuracy and reliability and to distances farther out
away from the mid-plane. The analysis did not reveal any significant kinematic trends in
the radial domain.
Thesis Supervisor: Andrew A. West
Title: Assistant Professor of Astronomy, BU
Thesis Supervisor: Adam J. Burgasser
Title: Associate Professor of Physics, MIT
4
Acknowledgments
I would like to acknowledge the Paul E. Gray fund in providing monetary support for the
Undergraduate Research Oppurtunities Program at MIT. I would also like to acknowlege
Andrew West, John Bochanski, Adam Burgasser for their support in getting this work together, as well as Sarah Schmidt for her assistance.
Thanks to all of my friends for being there for me. Also, thank you mom.
6
Contents
1 Introduction
2 Data
3 Magnetic Activity
4
3.1
Activity Fractions
3.2
Degree of Activity
Color Indices
5 Kinematics
5.1
Analysis Method
5.2
Results . . . . . .
6 Conclusion
...............................
8
List of Figures
. . . . . . . . . . . . . . . . . . . . . . . . . 20
2-1
DR7 M dwarf Position Map
3-1
Activity Fraction Map of MO . . . . . . . . . . . . . . . . . . . . . . . . . 25
3-2
Activity Fraction Map of M1 . . . . . . . . . . . . . . . . . . . . . . . . . 26
3-3
Activity Fraction Map of M2 . . . . . . . . . . . . . . . . . . . . . . . . . 27
3-4
Activity Fraction Map of M3 . . . . . . . . . . . . . . . . . . . . . . . . . 28
3-5
Activity Fraction Map of M4 . . . . . . . . . . . . . . . . . . . . . . . . . 29
3-6
Activity Fraction Map of M5 . . . . . . . . . . . . . . . . . . . . . . . . . 30
3-7
Activity Fraction Map of M6 . . . . . . . . . . . . . . . . . . . . . . . . . 31
3-8
Activity Fraction Map of M7 . . . . . . . . . . . . . . . . . . . . . . . . . 32
3-9
Activity Fraction Map of M3 - below the plane
. . . . . . . . . . . . . . . 33
3-10 Activity Fraction Map of M4 - below the plane
. . . . . . . . . . . . . . . 34
3-11 Activity Level Map of M3
. . . . . . . . . . . . . . . . . . . . . . . . . . 35
3-12 Activity Level Map of M4
. . . . . . . . . . . . . . . . . . . . . . . . . . 36
3-13 Activity Level Map of M5
. . . . . . . . . . . . . . . . . . . . . . . . . . 37
3-14 Activity Level Map of M6
. . . . . . . . . . . . . . . . . . . . . . . . . . 38
3-15 Activity Level Map of M7
. . . . . . . . . . . . . . . . . . . . . . . . . . 39
4-1
r - z Map for MO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4-2
r - i Map for MO
4-3
g - r Map for MO . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . 45
4-4
r - z Map for M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4-5
r - i Map for M1
4-6
g - r Map for M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4-7
r - z Map for M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4-8
r - i Map for M2
4-9
g - r Map for M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4-10 r - z Map for M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4-11 r - i Map for M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4-12 g - r Map for M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4-13 r - z Map for M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4-14 r - i Map for M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4-15 g - r Map for M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4-16 r - z Map for M5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4-17 r - i Map for M5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4-18 g - r Map for M5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 60
4-19 r - z Map for M6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4-20 r - i Map for M6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4-21 g - r Map for M6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4-22 r - z Map for M7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4-23 r - i Map for M7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4-24 g - r Map for M7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5-1
Example Probability Plots
5-2
Example Analysis in f 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5-3
Example Uncertainty Estimate . . . . . . . . . . . . . . . . . . . . . . . . 73
5-4
Probability Plots for Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5-5
Fraction values vs. IZI
. . . . . . . . . . . . . . . . . . . . . . . . . . 69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5-6 Dispersions vs. IZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5-7 Means vs. ZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5-8
Fraction value Map in VR . . . . . .
5-9
Fraction value Map in Vz -.
.
..
.
. ..
. . . . . . . . . . . 78
. . . . . . . . . . . . . . . . . . . . . . . . . 79
5-10 Fraction value Map in VD . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5-11 Cold Dispersion Map in VR . . . . . . . . . . . . . . . . . . . . . . . . . . 81
82
5-12 Cold Dispersion Map in Vz
5-13 Cold Dispersion Map in VD
. . . . . . . . . . . 83
5-14 Hot Dispersion Map in VR
. . . . . . . . . . . 84
5-15 Hot Dispersion Map in Vz
- - - - - - - - - . 85
5-16 Hot Dispersion Map in VD
. . . . . . . . . . . 86
5-17 Cold Mean Map in VR . . .
. . . . . . . . . . . 87
5-18 Cold Mean Map in Vz - - .
. .
5-19 Cold Mean Map in VD
. . . . . . . . . . . 89
. .
5-20 Hot Mean Map in VR - - -
- - - - - - - - 88
S- - - - - - - -
- 90
---
- - - - - - - - - - 91
5-22 Hot Mean Map in Vz ...
. . . . . . . . . . . 92
5-21 Hot Mean Map in Vz
12
Chapter 1
Introduction
Low mass stars such as M dwarfs have lifetimes that exceed the current age of the universe. Since they also constitute the majority of all stars in the Milky Way, they are ideal
for examining galactic dynamics and for tracing the evolution of the Galaxy (Bochanski et
al. 2010). There have been many studies examining the distribution and dynamics of M
dwarfs in the Solar neighborhood (Wielen 1977; Reid et al. 1995; Hawley et al. 1996; Reid
et al. 2002; Bochanski et al. 2007b). Wielen (1977) examined the kinematics of roughly
500 McCormick stars finding that the velocity dispersion increases with stellar age. Reid
et al. (2002) examined a volume complete sample of M dwarfs from the Palomar Michigan
State University survey yielding a kinematic analysis with approximately 400 stars showing the necessity of fitting several Gaussian components to the kinematic distribution. The
more recent study by Bochanski et al. (2007b) using the Sloan Digital Sky Survey (SDSS)
utilized a sample with upwards of 7000 stars in their kinematic analysis to discover the
increase in velocity dispersion with absolute distance from the Galactic plane. Larger photometric and spectroscopic samples of M dwarfs have also been assembled using data from
the SDSS (Bochanksi et al. 2010; West et al. 2008, 2010). The SDSS Data Release 5
included more than 40000 M Dwarfs (West et al. 2008) and the M Dwarf sample in preparation from the Data Release 7 will include upwards of 70000 stars (West et al. 2010). With
the advent of the SDSS, there are many more stars available for analysis then ever before,
allowing for a detailed examination of the local Galactic disk.
In addition to having exceedingly long lifetimes, M dwarfs are also known to host mag-
netic dynamos, producing large scale magnetic activity. Although the mechanism behind
magnetic field production in low mass stars remains an unsolved problem, it is thought
to be associated with stellar rotation. For solar-type stars, current models of magnetic
field production suggest that they can be produced by the rotational shear generated at the
boundary between the radiative zone and the convective zone deep within the star, known
as the tachocline (Parker 1993; Ossendrijver 2003; Thompson et al. 2003). After the magnetic fields are produced they rise to the surface as magnetic loops. Once at the surface
reconnection events can deposit energy in the upper atmosphere, leading to flares and other
quiescent emission. This interior structure of M dwarfs is also known to change across
spectral subtypes. For spectral types later than roughly M3, the stellar interior changes
from only partially convective to fully convective (Reid and Hawley 2005). This shift
likely alters the nature of the magnetic dynamo and the way large scale magnetic fields
are generated. Despite this change to the stellar interior, late-type M dwarfs show strong
activity (West et al. 2006, 2008), and strong fields (Reiners and Basri 2009). Additionally,
simulations have shown field generation in fully convective stars (Browning 2008).
This magnetic activity in M dwarfs has been observed in a variety of ways. Observations of flare activity from M dwarfs have been examined in recent photometric studies
(Kowalski et. al. 2009). In the optical, emission lines have been interpreted as originating
from excited atoms in the chromosphere. The emission lines, namely the hydrogen balmer
series and Call Hand K lines, are a direct product of chromospheric heating, and can be
considered a signature of the magnetic activity. Accordingly, the strength of Ha in emission has been used in several studies as a proxy for M dwarf magnetic activity (Hawley et
al. 1996; West et al. 2004, 2006, 2008). This aspect of M dwarfs has proven useful because of the apparent link between magnetic activity and age. Wilson and Woolley (1970)
demonstrated this link, measuring the magnetic activity using the strength of Ca II emission lines, and correlating it with apparent age. Later studies confirmed this link, using
Ha emission, for late type dwarfs (Hawley et al. 1996, 1999, 2000). Younger stars show
more magnetic activity than older stars (Gizis et al. 2002). The exact relationship between
age and magnetic field production for M dwarfs is still unknown. One hypothesis is that as
M dwarfs age their rotational periods increase due to angular momentum loss from mag-
netized stellar winds. The spindown changes the dynamics of the magnetic dynamo and
weakens the strength of the magnetic field.
Dynamical studies have shown a link between age and galactic position (Wielen 1977).
Specifically, stars passing through the Galactic plane are perturbed by large concentrations
of molecular gas and other stars. Repeated crossings have pushed the orbits of older stellar
populations away from the plane of the disk and into more elliptical orbits. This dynamical
heating effect also increases the velocity dispersions of these stars (Wielen 1977). Kinematic studies have demonstrated the presence of least two mathematically distinct stellar
populations, a younger dynamically colder component with lower dispersion, known as the
thin disk, and an older dynamically hotter component with larger dispersion known as the
thick disk (Reid et al. 2002; Bochanksi et al. 2007a). A kinematic study of M dwarfs near
the plane of the Galaxy revealed that the velocity dispersion of the thin disk is roughly 20
km s-1, whereas the velocity dispersion of the thick disk is roughly 40 km s-1 (Bochanski et al. 2007a). A recent kinematic study of L dwarfs confirmed these results (Schmidt
et al. 2010). Stellar density studies, utilizing star counts, have also revealed the need to
account for distinct components in the galactic disk; two exponentially decaying distributions are necessary to reproduce the observed density distribution (Gilmore & Reid 1983;
Reid & Majewski 1993; Buser et al. 1999; Norris 1999; Siegel et al. 2002; Bochanski et
al. 2010). The relative normalizations and scale heights for the two components are still
relatively uncertain (Norris 1999). Fitting density distributions with multiple exponentials
is a degenerate problem and thus requires additional methodology. The origins of these
components are still unclear. Detailed kinematic analysis of the DR7 M dwarfs will lead to
a better understanding of the structure of the Milky Way and help constrain formation models. In addition, the current large surveys of M dwarfs will also allow for the comparison
of these observed distributions to more recent N-body simulations of galaxies (Loebman
2008; Roskar 2010).
Recent studies have also examined how the magnetic activity in M dwarfs varies as
a function of spectral type and position in the Galaxy (West et al. 2006, 2008). Using
the presence of Ha emission as a proxy for magnetic activity, West et al. (2006, 2008)
showed that for each spectral subtype, magnetic activity decreases as a function of absolute
distance away from the Galactic plane. These results fit well with the kinematic studies
suggesting that populations further from the Galactic mid-plane are likely older; accordingly both Galactic position and magnetic activity can be used as a proxy for age. Studying
the kinematics and activity of M dwarfs can thus provide insights into the history of stellar
populations in the Milky Way.
Additional attributes to take into consideration are M dwarf colors. SDSS ugriz photometry measures the stellar flux in several bandwidths. For M dwarfs, the color indices,
r - z, r - i, and g - r can be indicative of many other intrinsic properties of the stars. For
example, the colors change as a function of spectral subtype. In particular, the color index
g - r has been shown to be very sensitive to metallicity (West et. al. 2004; Lepine 2009).
Metal poor M dwarfs can be much redder in g - r than typical M dwarfs (West et al. 2004).
The lack of metals is suggestive of an older population of stars that formed early on in the
history of the Milky Way when the molecular gas clouds were metal poor compared to the
current level of metallicity in the gas clouds. Examining how the colors are distributed
spatially will help correlate these trends with the known kinematic and magnetic trends.
The trends in activity and kinematics have been examined in one dimension as functions of the absolute vertical distance from the Galactic mid-plane. However, the domain
of Galactocentric radius (centered at Sagittarius A*) has so far remain unexplored. In this
cylindrical radial coordinate, a myriad of effects can influence the observed properties of
the Milky Way disk. The underlying density distribution of the disk and the star formation history would influence ages and observed positions. Like the vertical density profile
of the Galactic disk, the radial density profile also decreases exponentially outward from
the Galactic center (Bochanski et. al. 2010). Superimposed on this structure is also the
presence of dense spiral arms which have many star forming regions. Although the local
neighborhood is not particularly close to a spiral arm, the history of star formation in the
local neighborhood would influence the present density distributions. Recent simulations
have also shown the need to consider the effects of radial distributions because of the effect
of radial migration (Roskar 2010). Stars that were once in circular orbits closer to the center of the galaxy can scatter outward in radius through resonant interactions with the dense
spiral arms, preserving a circular orbit. These stars ride density waves drifting outward
in the disk. These various affects can alter the observed normalization between the thin
and thick disk and influence the history of dynamical interactions that have produced the
present distribution of M dwarfs. Consequently, the strictly one dimensional studies that
have been done so far have averaged over these affects, neglecting the radial differences in
the distribution of stars.
This study will examine both the magnetic activity and the kinematics of more than
70,000 M dwarfs from SDSS DR7 as functions of both the distance away from the midplane and the Galactocentric radius. In addition, the color indices of these stars will also
be examined in the same way. Section 2 presents the publicly available data sample used
in the analysis. Section 3 examines the magnetic activity as it varies through the galaxy,
while Section 4 looks for similar trends using various color indices. Section 5 covers
the kinematic analysis and the new technique developed for determining the kinematic
parameters. Lastly Section 6 summarizes the discussion.
18
Chapter 2
Data
In this study we use a spectroscopic and photometric sample of M dwarfs from the SDSS
DR7 with more than 70000 stars. The data were selected from the SDSS database using
known M dwarf colors as criterion (West et al. 2008; Kowalski et al. 2009). The resulting
star sample was then spectral typed using HAMMER and verified by eye (Covey et al.
2007). Additional cuts were applied using the SDSS photometric flags to ensure a high
quality data sample (see West et al. 2010 for full details). The total number of stars in
the sample came to 70823. The full sample was then reduced to N = 59319 when only
considering those stars with good photometry, signal-to-noise, ratio > 3, and eliminating
duplicates and stars with colors matching those of white dwarf M dwarf binaries. All of the
SDSS photometry was corrected for extinction using the Schlegel, Finkbeiner and Davis
(1998) maps.
To determine the distance to all of these stars we used the photometric parallax methods
of Bochanksi et al. (2010). Using these distances and SDSS astrometry, a spatial position in
the Galaxy for each star was computed assuming that the Sun is 15 pc above the mid-plane
of the Galactic disk and that it is 8500 pc from the center of the Galaxy. Thus, for each star
we calculated a position R and Z in Galactocentric cylindrical coordinates. In figure 2-1,
we plot the positions of all of these stars in the sample. The sample was also matched to
2MASS to provide infrared colors and to the USNO-B catalog which provided the baseline
for proper motions. This yielded N = 36208 stars with well determined proper motions,
good photometry and spectra. The radial velocities for these stars were determined using
the methods employed by Bochanski et al (2007a). Combining the radial velocities, proper
motions and the distance determinations, space motions for all of these stars were also calculated. UVW velocities were calculated, accounting for the motion of the Sun. Assuming
a local standard of rest that is moving at 220 km s-1 clockwise around the Galaxy, we convert these space motions into Galactocentric coordinates VR, Vz, and Ve in which negative
values of VD are moving with the general rotation of the Galactic disk. The sample used in
this study does not represent a complete sample of M dwarfs in the local solar neighborhood. However, due to SDSS selection effects, it spans a wide range of stellar properties
and kinematics, allowing for an accurate analysis despite any inherent selection effects.
DR7 Positions, Z vs R
2000
1000
-1000
.
~~ ~ ~
..
.I
i
-
rl
.
IVA
-2000
7000
8000
9000
10000
R (pc)
Figure 2-1: Positions for DR7 M dwarf sample plotted in cylindrical Galactocentric coordinates, R and Z. Positive Z corresponds to the northern hemisphere. The position of the
Sun is taken to be (R, Z) = (8500, 15) pc.
Chapter 3
Magnetic Activity
3.1
Activity Fractions
The spatial distribution of M dwarf activity has been examined previously as a function of
distance from the plane of the Galaxy (West et. al. 2006, 2008). Their findings showed that
for M dwarfs the fraction of active stars decreases farther away from the Galactic plane.
The fraction is defined as the number of active stars divided by the sum of the active and
non-active stars in a given bin (West et al. 2006, 2008). The previous studies however,
did not examine the possibility that the activity could also change as a function of distance
from the Galactic center. Given the large sample sizes now available from SDSS, it is
possible to examine the distribution of M dwarf activity in the radial domain. Each star in
the large data sample was categorized according to its level of magnetic activity, quantified
by LHa/Lbl (Hawley et al. 1996; Walkowicz, Hawley and West 2004). Those stars that
had an Ha emission level with an equivalent width greater than 1 A were deemed as active
stars, those stars that were sufficiently below the cut off value were deemed as inactive and
those stars in between or near the cutoff value were regarded as either weakly active or
potentially active stars (see West et al. 2008 for more details). Stars in this latter category
of M dwarf activity were excluded from the distribution analysis. Applying this criterion
reduced the available sample size to 7384 active stars and 46770 inactive stars for a total of
59309 M dwarfs. Table 3.1 breaks down the activity by spectral type. The column showing
the mean activity % is equivalent to
Nactive /(Nactive + Nnotactive).
It gives an overall measure
of activity for each spectral type in the sample.
Table 3.1: M dwarf activity by spectral type
Spectral Type
N
Active
Not-Active
'Weak'
Mean Activity%
MO
M1
M2
M3
M4
M5
M6
M7
9914
8142
9282
10067
8077
3566
5010
4454
145
173
297
502
892
1025
2015
1919
9588
7743
8655
9026
6455
1982
1885
1320
181
226
330
539
730
559
1110
1215
1.5
2.2
3.3
5.3
12.1
34.1
51.7
59.3
We explore how this activity varies spatially in the Galaxy. Examining each spectral
type individually and folding the data over the mid-plane we separate the stars into distance
bins and calculate the fraction of stars in each bin that are active and map the results. The
subsequent activity maps show how this activity level is distributed spatially. In figures
3-1 to 3-8, we show the distribution of activity for the spectral types MO - M7. The color
in the map corresponds to this activity fraction, where redder indicates a larger fraction.
All of the maps, except those for M6 and M7 use a binning of 100 pc by 100 pc (Z vs.
R) to make sure that each bin has plenty of stars and reduce the uncertainty associated
with the fraction determination. For spectral types M6 and M7 we use a binning of 50
pc by 50 pc (Z vs. R) because of the large density of late type M dwarfs near the solar
neighborhood. The roughly 3000 stars used in the maps for these types are concentrated
within 300 pc, allowing for a smaller bin sampling. Additionally, only bins with at least ten
stars are included in the plots. Below the activity maps are corresponding maps showing the
full length of the uncertainty associated with each fraction determination. Because of the
nature of the binomial distribution the uncertainty is not symmetric across the associated
data point. The maps reproduce previous results (West et. al. 2008) with regard to the
decreasing trend in activity away from the Galactic plane. The trend is evident in all but
the plots for the early type M dwarfs; because the early types are generally not active there
is not much of a pattern.
In figures 3-4 and 3-5, corresponding to M3 and M4, it is also clear that there seems
to be an increase in activity away from the Galactic center. A possible explanation for this
trend is that it is a result of the past star formation history in the disk. The stars closer to the
Galactic center represent an older population birthed in the denser gas closer to the galactic
center. Farther out the stars are younger from later episode of stellar formation. Knowing
the connection between age and activity, the observed trend in activity could be indicative
of gradient in average age for the M dwarfs in the sample. This new trend is only evident
in the plots for spectral types M3 and M4 because only for these types are there enough
active stars with sufficient numbers of stars far from the Galactic center. The stars for latter
types are only sampled close to the Sun and the stars for earlier types are not sufficiently
active for there to be a noticeable trend.
Utilizing the distributions in both R and Z, it is possible to remove some of the selection
bias in the SDSS sample when analyzing the data. One such bias is the sky coverage of the
SDSS. The SDSS takes data of wide strips of sky along particular sightlines. As a result
(as evident in figure 2-1) stars that are either closer to the Galactic center than the Sun is
or farther away from the Galactic center than the Sun is, are necessarily farther away from
the Galactic plane. Because of this selection bias, we should expect that the stars that are
farther away from the Galactic center should also be proportionally less active. However, it
is evident from figure 3-4 that the level of activity does not necessarily drop off with height
as expected from a simple one dimensional perspective for stars farther away than the Sun
is from the Galactic center. It is important to consider both these effects in order to obtain
an accurate representation of activity trends in both dimensions. A simple one dimensional
plot of the trend away from the galactic plane would fail to recognize the changes that
are apparent at different Galactocentric radii. As part of the full spatial analysis, we also
examined if there were any differences between the regions above and below the plane. As
expected there were no trends and no significant evidence that there is an asymmetry across
the plane of the Galaxy. In figures 3-9 and 3-10 the activity fractions are plotted both above
and below the Galactic plane. These plots are basically symmetric across the mid-plane
but they also show an interesting SDSS sightline in the southern hemisphere pointed away
from the Galactic center. There is no corresponding sightline in the northern hemisphere
sampling similar stars above the mid-plane.
3.2
Degree of Activity
In the activity fraction maps of the previous section it is shown how the general activity
varies spatially and by spectral type. Additionally, we can also examine how the level of
activity, of those stars which are active, varies spatially. Folding the data across the midplane and using the same binning as the analysis of the activity fractions with each spectral
type, we examine the distribution of LHa /Lbol as a measure of the level of activity. In figures
3-11 to 3-15 we map out the spatial variation in this activity level. The plots only include
those stars that are deemed as active following the criterion in section 3.1. Accordingly,
the corresponding maps for MOs, MIs, and M2s, with so few active stars are not included.
The maps plot out the median level of LHa/Lbol for each bin on a logarithmic scale. Only
bins with at least ten stars are included in the plots. The accompanying uncertainty plots
show the width of the middle 50%/ of each bin, indicating the spread of the data in each bin.
From all of these plots there does not seem to be much of any trends, however there may
be a decrease in activity level away from the Galactic plane. The stars farther away from
the mid-plane, likely older stars that have been dynamically heated away from the center
of the disk, have potentially lower activity than similar active stars closer to the center of
the disk that are younger. Such a relation would give support to the notion of aging stars
gradually decreasing their levels of activity. However, the plots are inconclusive and likely
require a larger sample size to make definitive arguments. Similarly, no radial trends can
be detected in this analysis.
......
....
....
..................
MO Activity Fractions
2000
1500
18
18
15
15
13
13
11
11
CD
ci)
9
00
Q500
0
7500
8000
8500
R (pc)
9000
9500
10000
9
9
6
6
4
4
2
2
0
0
Activity %
Fraction Uncertainty
22
22
18
18
15
15
11
11
7-7
3U3
0
0m
7500
8000
8500
9000
R (pc)
9500
10000
0
Uncertainty
Figure 3-1: Top - MO activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc.
M1 Activity Fractions
2000
C 1500
9h 9
0)
1000
CD5
-~500
0
7500
8000
8500
9000
9500
10000
R (pc)
Activity %
Fraction Uncertaintv
2000
18
18
15
15
12
12
9
9
6
6
3
3
0
0
1500
1000
500
0m
7500
8000
8500
R (pc)
9000
9500
10000
Uncertainty
Figure 3-2: Top - M1 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc.
.....................
............
.......
M2 Activity Fractions
1500
U
1000
500
0
8000
8500
9000
R (pc)
9500
17
17
14
14
11
11
8
8
5
5
2
2
0
0
Activity %
Fraction Uncertaintv
1500
CL
CD
T 1000
0
22
22
18
18
14
14
11
11
7-7
*<n 500
3H3
0
8000
8500
9000
R (pc)
9500
0
Uncertainty
Figure 3-3: Top - M2 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc.
M3 Activity Fractions
1200
1000
->
31
31
27
27
23
23
5 800
600
T>
400
200
0
8000
9000
8500
9500
R (pc)
Activity %
Fraction Uncertainty
4H4
0 0
8000
8500
9000
9500
Uncertainty
R (pc)
Figure 3-4: Top - M3 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc. Notice the trend of slightly increasing activity outward in Galactocentric radius.
. ..
....
..
.........
..........
......
M4 Activity Fractions
1000
800
43
43
38
38
32
32
27
27
21
21
16
16
10
10
5
5
0
0
600
400
200
-
0
8000
8200
8400
8800
8600
R (pc)
9000
9200
Activity %
Fraction Uncertainty
1000
0
0m
8000
0
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 3-5: Top - M4 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc. Notice the trend of slightly increasing activity outward in Galactocentric radius.
....
....----
M5 Activity Fractions
69
69
61
61
52
52
800
600
400
.0
< 200
8200
8400
8600
8800
9000
9200
R (pc)
Activity %
Fraction Uncertaintv
0 0
Uncertainty
8200
8400
8600
8800
9000
9200
R (pc)
Figure 3-6: Top - M5 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc.
......
...
...
...
.
.
. ...........
...
....
..
.....
M6 Activity Fractions
400
o.300
200
100
0
8300
8400
8500
8600
R (pc)
8700
8800
Activity %
Fraction Uncertaintv
29
29
24
24
19
19
14
14
oU 0
0m
8300
8400
8500
8600
R (pc)
8700
8800
Uncertainty
Figure 3-7: Top - M5 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 50 pc
by 50 pc.
M7 Activity Fractions
400
C 300
60 r
-C
60
C0)
. 200
)
100
0
8300
8400
8600
8500
8700
8800
R (pc)
Activity %
Fraction Uncertaintv
400
24
24
20
20
16
16
12
12
C. 300
CD
200
8
.1 00
0
8
0
0-0
0m
8300
8400
8500
8600
8700
8800
Uncertainty
R (pc)
Figure 3-8: Top - M5 activity fractions as functions of Galactocentric radius and absolute distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 50 pc
by 50 pc.
M3 Activity Fractions
31
27
500
23
-500
-1000
8000
8500
9000
R (pc)
9500
Activity %
Fraction Uncertainty
500
25
25
21
21
16
16
12
12
0
0
-500
-1000
8000
8500
9000
R (pc)
9500
Uncertainty
Figure 3-9: Top - M3 activity fractions as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the activity fraction, defined as the number of active stars divided by the
sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc. Shows the activity fractions for both above and below the plane. Note the sightline in the southern
hemisphere on the right.
M4 Activity Fractions
500
-500
44
44
36
36
29
29
22
22
14
14
7Q7
8000 8200 8400 8600 8800 9000 9200 9400
R (pc)
0
0
Activity %
Fraction Uncertainty
500
.C
0Y)
i
0
22
22
18
18
14
14
11
11
7-7
)
-500
3
3
0
0
Uncertainty
8000 8200 8400 8600 8800 9000 9200 9400
R (pc)
Figure 3-10: Top - M3 activity fractions as functions of Galactocentric radius and distance from the
Galactic plane. The color corresponds to the activity fraction, defined as the number of active stars divided by
the sum of the number of active and not active stars. Redder colors correspond to bins that are more active.
Bottom - Map of total uncertainties corresponding to the activity fraction map of the top panel. Uncertainties
determined from the binomial distribution. Lighter shades correspond to lower uncertainty. Bins are 100 pc
by 100 pc. Shows the activity fractions for both above and below the plane. Note the sightline in the southern
hemisphere on the right.
...
.....
..
.
. ..
...........
. ....
..
.....
.......
.........
.
M3 Level of Activity
600
-3.68
-3.68
-3.73
-3.73
-3.77
-3.77
-3.82
-3.82
-3.87
-3.87
200
-3.92
-3.92
< 100
-3.97
-3.97
-4.02
-4.02
-4.07
-4.07
- 500
a-2) 400
tD
2 300
(
0
8300 8400 8500 8600 8700 8800 8900
R (pc)
Uncertainty
Om
8300
8400
8500
8600
R (pc)
8700
8800
8900
Log(
LH-
3I
-3.92
-3.92
-3.99
-3.99
-4.06
-4.06
-4.13
-4.13
-4.20
-4.20
-4.27
-4.27
-4.34
-4.34
Log Spread
Figure 3-11: Top - M3 activity levels as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the level of activity, LHjL+ot. Redder colors correspond to bins that are
more active. Bottom - Map of spread of data in bin, width of middle 50%, corresponding to the activity level
map of the top panel. Lighter shades correspond to narrower distribution. Bins are 100 pc by 100 pc.
.......
..........
.. ........
....
M4 Level of Activity
800
-3.57
-3.57
-3.61
-3.61
600
d -3.65
9 400
( 200
.0
8400
8600
8800
9000
R (pc)
-3.70
-3.70
-3.74
-3.74
-3.78
-3.78
-3.82
-3.82
-3.87
-3.87
-3.91
-3.91
Log(
LHa
L
)
Uncertainty
800
8
-3.63
-3.63
-3.74
-3.74
-3.84
-3.84
-3.94
-3.94
-4.05
-4.05
-4.15 -
-- 4.15
600
400
0
200
-4.26
-4.26
Log Spread
8400
8600
R (pc)
8800
9000
Figure 3-12: Top - M4 activity levels as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the level of activity, LHu/Lboj. Redder colors correspond to bins that are
more active. Bottom - Map of spread of data in bin, width of middle 50%, corresponding to the activity level
map of the top panel. Lighter shades correspond to narrower distribution. Bins are 100 pc by 100 pc.
..
....
..
....
..
....
M5 Level of Activity
600
500
-.
-3.76
-3.76
-3.78
-3.78
-3.81
-3.81
-3.83
-3.83
-3.86
-3.86
-3.88
-3.88
-3.91
-3.91
-3.94
-3.94
9400
0
2300
1200
200
0
8300 8400 8500 8600 8700 8800 8900 9000 -3.96
R (pc)
Log(
-3.96
LHa
I
)
Uncertaintv
600
-3.59
-3.59
-3.69
-3.69
400
-3.79
-3.79
300
-3.89
-3.89
-4.00
-4.00
500
0.
2 200
.0
8300
8400
8500
8600
8700
R (pc)
8800
8900
9000
-4.10
K
-4.20
L. -4.20
-4.10
Log Spread
Figure 3-13: Top - M5 activity levels as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the level of activity, LHIL mathrmbo. Redder colors correspond to bins that
are more active. Bottom - Map of spread of data in bin, width of middle 50%, corresponding to the activity
level map of the top panel. Lighter shades correspond to narrower distribution. Bins are 100 pc by 100 pc.
M6 Level of Activity
600
-3.98
-3.98
500
-4.00
-4.00
-4.02
-4.02
-4.04
-4.04
-4.06
-4.06
-4.08
-4.08
-4.10
-4.10
-4.11
-4.11
-4.13
-4.13
Log( LHa
Lbo|)
-4.12
-4.12
-4.18
-4.18
-4.24
-4.24
-4.31
-4.31
-4.37
-4.37
-4.44
-4.44
-4.50
-4.50
C.
2 400
.9
300
200
C,)
-10
<
100
0
8200 8300 8400
8500 8600 8700 8800 8900
R (pc)
Uncertainty
Om
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
Log Spread
Figure 3-14: Top - M6 activity levels as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the level of activity, LHa,/L mathrmbo. Redder colors correspond to bins that
are more active. Bottom - Map of spread of data in bin, width of middle 50%, corresponding to the activity
level map of the top panel. Lighter shades correspond to narrower distribution. Bins are 50 pc by 50 pc.
....
..
..............
...
..
.. ....
.......
M7 Level of Activity
600
C,
-4.38
-4.38
-4.41
-4.41
-4.44
-4.44
-4.46
-4.46
-4.49
-4.49
-4.51
-4.51
-4.54
-4.54
-4.57
-4.57
8200 8300 8400 8500 8600 8700 8800 8900 -4.59
-4.59
500
C0
.9
400
300
200
CO
4
100
0
R (pc)
Log(
LHa I
)
Uncertainty
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
-4.38
-4.38
-4.47
-4.47
-4.57
-4.57
-4.66
-4.66
-4.75
-4.75
-4.85
-4.85
-4.94
-4.94
Log Spread
Figure 3-15: Top - M7 activity levels as functions of Galactocentric radius and distance from the Galactic
plane. The color corresponds to the level of activity, LH,/L mathrmbol. Redder colors correspond to bins that
are more active. Bottom - Map of spread of data in bin, width of middle 50%, corresponding to the activity
level map of the top panel. Lighter shades correspond to narrower distribution. Bins are 50 pc by 50 pc.
.. .......
1........
40
Chapter 4
Color Indices
Using SDSS ugriz photometry, we also examined the spatial variation of the color indices
r-z, r- i, and g - r. The data were split up by spectral type and divided into spatial bins in R
and Z. For spectral type MO - M5 the bins are 100 pc by 100 pc, whereas for types M6 and
M7 the bins are 50 pc by 50 pc. Only bins with more then ten stars were considered. For
each bin the median value of the color index was calculated and maps analogous to those
of the previous sections on magnetic activity were made. Redder colors indicate a larger
value of the color index. In figures 4-1 to 4-24 we plot the results of this analysis. The top
panel is the map, whereas the accompanying uncertainty plots in the bottom panel show
the width of the middle 50% of each bin, indicating the spread of the data. Whenever there
appeared to be some asymmetry in the plots, the corresponding map was plotted for both
above and below the plane. Figures 4-1 to 4-3 correspond to the maps for spectral type MO.
The plots for MO seem to be slightly bluer, with a lower color index at larger R, a result
due to stars with a smaller value for the color index in the southern hemisphere. Figures
4-4 to 4-6 correspond to the maps for spectral type M1. Figures 4-7 to 4-9 correspond to
the maps for spectral type M2. The plots for M1 and M2 in g - r, although mostly uniform
show a slight increase in the color index going from lower left to upper right in the plots.
Figures 4-10 to 4-12 correspond to the maps for spectral type M3. The M3 plot in g - r
shows a large discrepancy between the northern and southern hemispheres. Figures 4-13
to 4-15 correspond to the maps for spectral type M4. The M4 plot in g - r also shows a
large discrepancy between the northern and southern hemispheres. Since the photometry is
corrected for Galactic extinction it is possible that this difference is due to over correcting
in the southern hemisphere. All the plots for spectral types M3 and M4 also show that
the southern sightline closest to the Galactic plane looking at larger Galactocentric radii is
seeing stars that do not match well with the rest of the data sample. Figures 4-16 to 4-18
correspond to the maps for spectral type M5. Figures 4-19 to 4-21 correspond to the maps
for spectral type M6. Figures 4-22 to 4-24 correspond to the maps for spectral type M7.
In all of the plots for r - z and r - i there was a clear trend in decreasing color index as
absolute distance from the Galactic plane increased.
...
..........
..
.....
, ... ..
..
...
..........
..
........
...
. ..
MO Median Color Index: r - z
1500
N
1.03(
1.030
0.954
0.954
0.879
0.879
0.803
803
0.727
727
0.651
651
0.575
575
0.500
500
10000 0.424
424
1000
500
0
-500
-1000
-1500
7500
8000
8500 9000
R (pc)
9500
r-z
Error in Color Index
1500
0.54
0.54
1000
0.46
0.46
500
0.37
0.37
0
0.29
0.29
0.20
'0.20
0.11
-0.11
-500
-1000
0.03
-1500
7500
8000
8500
9000
9500
10000
0.03
Spread
R (pc)
Figure 4-1: Top - MO r - z median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Whiter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
1500
MO Median Color Index: r - i
0.646
0.646
0.604
0.604
0.561
0.561
0.519
0.519
0.47
0.476
0.434
0.434
0.392
0.392
0.349
0.349
10000 0.307
0.307
1000
500
0
-500
-1000
-1500
7500
8000
8500 9000
R (pc)
9500
Error in Color Index
7500
8000
8500 9000
R (pc)
9500
10000
0.37
0.37
0.32
0.32
0.26
0.26
0.21
0.21
0.15
01
0.10
0.10
0.04
0.04
Spread
Figure 4-2: Top - MO r - i median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread of
data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
.
. ............
........
...
...
...
......
..............
.
MO Median Color Index: g - r
1500
1000
U
1.424
1.424
1.354
1.354
1.284
1.284
1.214
1.214
500
1.144
-500
1.074
1.074
1.004
1.004
0.934
.934
0.864
.864
-1000
-1500
7500
8000
8500
9000
9500
10000
R (pc)
g- r
Error in Color Index
1500
0.65
1000
0.55
500
0.45
0.65
0.45
0.35
0
0.24
0.24
0.14
0.14
0.04
0.04
-500
-1000
-1500iM
7500
8000
Spread
8500
9000
9500
10000
R (pc)
Figure 4-3: Top - MO g - r median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
.
.. .
..
.......
......
.......
2000
M1 Median Color Index: r - z
cL 1500
1.289
1.289
1.226
1.226
1.164
]M
1.102
1000
1.040
.040
0.978
.978
0
0.91510.915
500
<
7500
8000
8500
9000
9500
0.853
0.853
0.791
.791
R (pc)
r -z
Error in Color Index
2000
0.55
0.55
0.47
0
CL
1500
2)
0.39
0.39
0.31
0.31
1000
0.24
0
0.16
<500
0.08
7500
8000
8500
9000
9500
0.08
Spread
R (pc)
Figure 4-4: Top - M1 r - z median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread of
data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. Note lower bound in Z is 100 pc.
...........................................
.......
........
.................
.....
M1 Median Color Index: r - i
2000
0.81610.816
0.782
0.782
c 1500
0.749
0)
$
500
7500
8000
9000
8500
9500
0.715
0.715
0.681
0.681
0.647
0.647
0.613
0.613
0.579 1
0.579
0.546
0.546
0.36
0.36
0.31
0.31
R (pc)
Error in Color Index
0.25
0.20
0.15
0.10U010
0.04
0.04
Spread
7500
8000
8500
9000
9500
R (pc)
Figure 4-5: Top - M1 r - i median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have larger r - i value. Bottom - Map of spread of
data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. . Note lower bound in Z is 100 pc.
M1 Median Color Index: g - r
2000
d. 1500
0)
'
1000
1.641
1.641
1.584
1.584
1.527
1.527
1.470
1.470
1.413
1.413
1.356
1.356
1.299
1.299
1.241
1.241
1.184
1.184
4-1
.0
500
7500
8000
8500
9000
9500
R (pc)
g- r
Error in Color Index
2000
0.87
0.87
0.74
0.74
0.60
0.60
0.46
0.46
0.32
0.32
1500
1000
0.18-
500
0.04
-0.18
0.04
Spread
7500
8000
8500
9000
9500
R (pc)
Figure 4-6: Top - Ml g - r median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. . Note lower bound in Z is 100 pc.
.....
.......
...................
... .. .....
....
....
....
....
M2 Median Color Index: r - z
1500
0
1.552
1.552
1.52:
1.522
1.492
1.492
1.463
1.463
1.433
1.433
1.403
1.403
1.373
1.373
1.343
1.343
1.313
1.313
0)
1000
0
500
8000
9000
8500
9500
R (pc)
r-z
Error in Color Index
0.37
1500
0.31
0.
CL
0.26
-
1000
0.20
0.14
0.14
0.08
0.08
0.03
0.03
50)
0
8000
9000
8500
9500
Spread
R (pc)
Figure 4-7: Top - M2 r - z median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread of
data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. . Note lower bound in Z is 100 pc.
M2 Median Color Index: r - i
1.004
1.004
0.98(
0.980
0.956 r
0.956
0.933
0.933
0.909
0.909
0.885
0.885
0.862
0.862
0.838
0.838
0.814
0.814
1500
U
1000
500
8000
8500
9000
9500
R (pc)
Error in Color Index
0.30
0.26
0.22
0.09L 0.09
0.05
8000
8500
R (pc)
9000
9500
_0.05
Spread
Figure 4-8: Top - M2 r - i median color index as function of Galactocentric radius and distance from the
Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread of
data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. . Note lower bound in Z is 100 pc.
...
......
....
........................
....
M2 Median Color Index: g - r
1500
1000
500
8000
8500
9000
1.672
1.672
1.624
1.624
1.576
1.576
1.528
1.528
1.480
1.480
1.432
1.432
1.384
1.384
1.336
1.336
1.289
9500
R (pc)
g- r
Error in Color Index
0.72
0.72
0.61
0.61
0.50
0.50
0.39
0.39
0.28
0.28
0.17
0.17
0.06
0.06
1500
1000
500
Spread
8000
8500
R (pc)
9000
9500
Figure 4-9: Top - M2 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc. . Note lower bound in Z is 100 pc.
1000
M3 Median Color Index: r - z
500
1.871
1.871
1.803
1.803
1.734
1.734
1.665
1.665
1.597
1.597
1.528
1.528
1.459
1.459
1.391
1.391
9500 1.322
1.322
i
U
-500
-1000
8000
8500
9000
R (pc)
r -z
Error in Color Index
1001
0.39
0.39
0.33
0.33
0.28E0.28
0.23
-500
-1000
8000
8500
9000
9500
0.17
0.17
0.12
0.12
0.07
0.07
Spread
R (pc)
Figure 4-10: Top - M3 r - z median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
::::::::::
..
...............
.......
.....................
.........
........
-----11111 - -- . . ......
- ----
1000
. .......
..........
M3 Median Color Index: r - i
1.224
1.224
1.184
1.184
1.144
1.144
1.104
1.104
1.064
1.064
1.024
1.024
0.984
0.984
0.943
0.943
9500 0.903
0.903
500
-500
-1000
8000
8500
R (pc)
9000
Error in Color Index
0.30
0.26
0.22
0.18
0.18
0.1410.14
0.10-
0.10
0.07-0.07
8000
8500
R (pc)
9000
9500
Spread
Figure 4-11: Top - M3 r - i median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
M3 Median Color Index: g - r
1000
500
V.
t
1.723
1.723
1.655
1.655
1.588
1.588
1.520 Iiiil1.520
1.452
1.452
1.384
1.384
1.317
1.317
1.249
1.249
9500 1.181
1.181
-500
-1000
8000
8500
9000
R (pc)
g- r
Error in Color Index
1000
0.87
0.87
0.73
0.73
500
0.60
0.47
0.47
0.34
-500
-1000
0.20
0.20
0.07
0.07
Spread
8000
8500
R (pc)
9000
9500
Figure 4-12: Top - M3 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
...
.........
-
.....
.................
.........
........
....
...
....
M4 Median Color Index: r - z
1000
500
-500
-1000
8000
8500
R (pc)
9000
9500
2.182
2.182
2.103
2.103
2.025
2.025
1.947
1.947
1.868
1.868
1.790
1.790
1.712
1.712
1.634
1.634
1.555
1.555
r-z
Error in Color Index
8000
8500
R (pc)
9000
9500
0.43
0.43
0.37
0.37
0.31
0.31
0.24
0.24
0.18
0.18
0.12
0.12
0.05
0.05
Spread
Figure 4-13: Top - M4 r - z median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
1000
M4 Median Color Index: r - i
1.418
1.418
1.370
1.370
1.322
1.322
1.274
1.274
1.226
1.226
1.179
1.179
1.131
1.131
1.083
1.083
9500 1.035
1.035
500
-500
-1000
8000
8500
9000
R (pc)
Error in Color Index
1000
0.36
0.36
0.31
500
0.26
0.26
0.20
-500
-1000
8000
8500
R (pc)
9000
0.15
0.15
0.10-
0.10
0.04
0.04
Spread
9500
Figure 4-14: Top - M4 r - i median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
..
.. ft- . -, v:.........................
' '.
: ::::::
.... .......
..
..
. .......
.
.
. ..
......
. ......
M4 Median Color Index: g - r
1000
1.776
1.776
1.703
1.703
1.630
1.630
1.557
1.557
1.485
1.485
1.412
1.412
1.339
1.339
1.266
1.266
9500 1.194
1.194
500
-500
-1000
8000
8500
9000
R (pc)
g -r
Error in Color Index
1000
0.79
0.79
0.67
0.67
0.55
0.55
0.43
0.43
0.32
0.32
500
-500
0.20
0.08
-1000
0.08
Spread
8000
8500
9000
9500
R (pc)
Figure 4-15: Top - M4 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
. .....
.
..........
2.. J. -
M5 Median Color Index: r - z
800
600
76
400
2.832
2.832
2.708
2.708
2.585
2.585
2.461
2.461
2.337
2.337
2.214
2.214
2.090
2.090
1.967
1.967
1.843
1.843
0
200
8200
8400
8600
R (pc)
8800
9000
r -z
Error in Color Index
800
0.68
0.59
0.59
0.49
0.49
0.40
0.40
S600
a)
400
0.3010.30
0
200
0.21 -
0.21
0.11 [__J0.11
8200
8400
8600
R (pc)
8800
9000
Spread
Figure 4-16: Top - M5 r - z median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
...........
...........
.........
....
..........
...........
.........
.. ..
.....
....
.......
......
...................
........
. .............
...
......
.......
.......................................
.. ...........
.......
........
..........
......
.. .......
M5 Median Color Index: r 800
.
1.810
1.810
1.736
1.736
1.662
1.662
1.588
1.588
1.514
1.514
1.440
1.440
1.366
1.366
1.292
1.292
1.218
1.218
0.43
0.43
0.37
0.37
0.31
0.31
0.25
0.25
0.19
0.19
600
CD
o 400
:2
0
200
0
8200
8400
8600
R (pc)
8800
9000
Error in Color Index
0.13-
-0.13
0.07-0.07
Spread
8200
8400
8600
8800
9000
R (pc)
Figure 4-17: Top - M5 r - i median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
Iaq
-
-
.
. ..
........
.. ....
-
, -
-
-
--
M5 Median Color Index: g - r
800
CO 600
1.563
1.563
1.527
1.527
1.491
1.491
1.455
1.455
1.418
1.418
1.382
1.382
1.346
1.346
1.310
1.310
1.274
1.274
(D
0)
9400
t
200
0
8200
8400
8600
R (pc)
8800
9000
g -r
Error in Color Index
0.53
0.53
0.46
0.46
0.38
0.31
0.31
0.24
0.17
0.09
0.09
Spread
8200
8400
8600
R (pc)
8800
9000
Figure 4-18: Top - M5 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 100 pc by 100 pc.
M. I
......
.......
.
..
.
..
.
..............
. ..........
-
M6 Median Color Index: r - z
600
500
3.104
3.104
3.066
3.066
3.027
3.027
2.989
2.989
400
S00
<a
0.
300
2.951
0
200
8300
8400
8600
R (pc)
.
8700
8800
2.913
2.875
2.875
2.836
.836
.798
8900
r-z
Error in Color Index
100
2 200
0
8500
2.913
800
0.47
0.47
0.41
0.41
0.35
0.35
0.28
0.28
0.22
0.22
0.16
0.16
0.09
0.09
100
0
8200
8300
8400
8500
8600
R (pc)
8700
8800
8900
Spread
Figure 4-19: Top - M6 r - z median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
M6 Median Color Index: r - i
600
500
0.
c 400
0a
.9 300
200
2.004
2.004
1.976
1.976
1.947
1.947
1.918
1.918
1.889
1.889
1.860
1.860
1.831
1.831
1.803
1.803
1.774
1.774
100
0
8200
8300
8400
8500
8600
8700
8800
8900
R (pc)
Error in Color Index
0.30
0.26
0.26
0.22
0.18
0.14
0.14
0.10
).10
0.06
OM
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
Li
0.06
Spread
Figure 4-20: Top - M6 r - i median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
.
.. .........
....
......
M6 Median Color Index: g - r
600
500
400
1.598
1.598
1.582
1.582
1.566
1.566
1.550|M1.550
100
1.534
1.534
1.519
1.519
1.503
1.503
1.487
1.487
0
8200
8300
8400
8500
8600
8700
8800
8900
R (pc)
g- r
Error in Color Index
-
0.32
0.32
400
0.27
0.27
300
0.22
0.22
0.18
0.18
(D)
20 200
0.13-
-0.13
100
0.08
0M
8200
8300
8400
8500
8600
R (pc)
8700
8800
8900
0.08
Spread
Figure 4-21: Top - M6 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
M7 Median Color Index: r - z
600
500
CL
400
ras
.
300
2 200
3.564
3.564
3.485
3.485
3.406
3.406
3328
3328
3.249
249
3.170
170
3.092
3.092
3.013
3.013
2.935
2.935
0
100
0
8200
8300
8400
8500
8600
8700
8800
8900
R (pc)
r-z
Error in Color Index
600
0.61
0.61
500
0.53
0.53
400
0.44
Q.
300
UD
0.36
0.36
0.28
0.28
0.19-
0.19
0.111
0.11
2 200
0
100
0M
8200
Spread
8300
8400
8500 8600
R (pc)
8700
8800
8900
Figure 4-22: Top - M7 r - z median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - z value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
.. ..
....
.....................
...............
.
..
.....
.....
.....
.
M7 Median Color Index: r - i
600
500
2.283
2.283
2.232
2.232
2.181
2.181
C.
400
130
<)
.
300
2.07
2.079
2.028
2.028
1.977
1.977
1.926
1.926
1.875
1.875
4)
A
200
0
100
0
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
Error in Color Index
0.33
0.30
0.
400
0.26
300
0.22
0.22
0.14
0.14
0.10
0.10
)
20
2 200
0
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
Spread
Figure 4-23: Top - M7 r - i median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have a larger r - i value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
M7 Median Color Index: g - r
600
500
CL
1.599
1.599
1.581
1.581
1.563
1.563
1.545
1.545
1.527
1.527
1.510
1.510
1.492
1.492
1.474
1.474
1.456
1.456
400
300
0
0
100
0
8200
8300
8400
8500
8600
8700
8800
8900
R (pc)
g -r
Error in Color Index
0.43
0.38
0.32
0.32
0.27
0.22
0.170.11
8200
8300
8400
8500 8600
R (pc)
8700
8800
8900
0.22
-0.17
-0.11
Spread
Figure 4-24: Top - M7 g - r median color index as function of Galactocentric radius and distance from
the Galactic plane. Redder colors correspond to bins that have larger g - r value. Bottom - Map of spread
of data in bin, width of middle 50%, corresponding to the color index map of the top panel. Lighter shades
correspond to a narrower distribution. Bins are 50 pc by 50 pc.
Chapter 5
Kinematics
Like magnetic activity and color, the kinematics of M dwarf show spatial variation (Bochanski 2007b). With the large sample size from the SDSS, distances out to more than 1500
pc away from the Sun are probed. Accordingly, the traditional UVW velocity system is
no longer useful. It works fine very close to the sun, but since these coordinates define a
rectilinear system in reference to the Sun, with U positive towards the Galactic center at
the Sun's position and V positive in the direction of motion of the Sun around the Galactic
center, they can correspond to arbitrary directions in space for stars that are not in the close
solar neighborhood. It is thus necessary to use a better system. In cylindrical Galactocentric coordinates, R, Z and <D, with Z positive in the northern hemisphere and with the
Galaxy rotating in the -CD direction, we define the rectilinear system of VR, Vz, V.
The analysis of the kinematics is complicated because there are at least two components of the stellar population, the thin and thick disk (Bochanksi 2007b). Because the
components cannot be explicitly separated it is difficult to determine the characteristics of
each. These components can be described by two gaussians. However, simply fitting two
Gaussians to the data is an unreliable method to determine the underlying parameters. This
difficult non-linear fit requires the data to be binned. There is no aprioriway to select the
proper binning. This would not necessarily be a problem if the fits were consistent across
all choices of bin size; however, this is not the case, meaning the fit parameters depend on
the selection of bin size. To circumvent these issues, the following method was developed
to determine the parameters o-, 0-2, 1, and P2 that define these Gaussian components of
the populations.
5.1
Analysis Method
We make use of probability distributions to measure the parameters of the two components.
Lutz and Upgren (1980) introduced the use of probability plots in astronomy. Since then,
there have been many studies advancing the use of these plots to study stellar kinematics
by characterizing and distinguishing populations thought to exhibit gaussian distributions
(Bochanski 2007; Reid et al. 1995, 2002). The method plots the data values against the
inverse of the cumulative distribution. The abscissa for these plots, in units of the standard
deviation, is thus the expected difference from the mean. For a population characterized by
a gaussian distribution, the probability plot would yield a straight line such that the slope of
the line corresponds to the dispersion of the distribution and the y-intercept corresponds to
the mean of the distribution (see figure 5-1, left panel). The advantage of this method is that
it eliminates any binning requisite in fitting a gaussian distribution to the data and makes
fitting easier since fitting lines is a much simpler task than fitting curves. Additionally,
a probability plot that deviates significantly from a straight line is an indication that the
underlying distribution is not a simple gaussian. Taking advantage of these properties and
using these probability plots, the aforementioned studies have described thin and thick disk
star populations in the local neighborhood.
Although there has been much success in using the probability plots, there are issues
with applying this technique to populations consisting of more than one component. Populations that are believed to come from a distribution that is the sum of two distinct gaussians
will not appear as a straight line in the probability plot, instead such plots will exhibit distinct characteristics depending on the nature of the underlying parameters. For example,
a distribution of two components with close means and separate dispersions could appear
as a plot that rises at constant slope, plateaus to a new slope, and then begins to rise again
at the original rate (see figure 5-1, center panel). Such a plot can be described as having a
central core and two outlying wing regions. It is this kind of plot that dominates the probability plots in Vz of our sample. A distribution of two components with sufficiently distinct
means and similar dispersion would appear as two distinct linear regions that are offset but
have similar slopes. Adding distinct dispersions to this case would yield two linear regions
with distinct slopes (see figure 5-1,right panel). It is this latter case that dominates the
probability plots in V0. Fitting lines to the various pseudo-linear portions of the probability
plots, as was done in Bochanski et al. (2007a), produces a reasonable determination of the
underlying parameters. Nevertheless, these estimates are flawed. Because of the nature of
the data transformation, the populations are mixed up, thus any linear fit cannot guarantee
that it is properly discriminating the components. Moreover, because of the mixing, a linear
fit to the core region is an overestimation of the actual parent parameter while fitting the
wing regions can be wildly variable. There is also always a question of which abscissa values make up which pseudo-linear portions of such plots; it is not explicitly clear where any
cutoffs should be made in fitting to accurately determine the parameters of the underlying
distribution. Additional difficulties arise when the dispersion of the Gaussian components
are relatively close; the probability plot, having diminished wings, can often resemble a
straight line even though the distribution has multiple components.
Example Probablity Plots
1000
21
60
2-00
-2
f
0
/
-
2
4
1000
I
-/-
-1
-
0
01
-1000
-4
-2
0
2
4
-4
-2
0
2
4
-4
-2
0
Figure 5-1: Example probability plots, N= 1000. Left plot corresponds to a single gaussian with mean
p = 0 and dispersion o- = 30. Center plot corresponds to probability plot of a two component population
with p, = 0, a-, = 20, and p2 = 0, 0-2 = 80 and f = .5. Right plot corresponds to probability plot of a two
component population with pi = -40, a-I = 25, and p2 = 40, 0-2 = 75 and
f
= .5.
To circumvent many of these issues, this work develops and presents a method to de-
termine the underlying parameters of the distribution associated with our constructed probability plots. This technique is then applied to the large M-dwarf sample to characterize
the structure of the galactic thin and thick disk in our local neighborhood. Since most
of the analysis will consist of determining the parameters of a two component gaussian
distribution, the following description of the method will focus on that type of analysis.
However, the method can be easily extended to a greater number of components and parameters. Although, this is theoretically feasible, constraints in sample size currently limit
the applicability and significance of such efforts.
Like Reid et. al. (2002), we construct models against which to compare the data samples. We expand upon this idea by allowing all of the parameters to vary freely except for
the relative normalization of the gaussian components. Because changing the relative normalization a small amount does not necessarily change the relative number composition of
the sample, we step through the normalization to produce the best fit. The fraction value,
f,
defined as the proportion of the total sample in a given bin that consists of the dynami-
cally colder component, meaning it has the lower dispersion value, is incremented by .01.
We use the Xy statistic defined in equation 5.1 to determine the best parameter fit. Where
the y, are the data points, the ye are the corresponding model points, aYis the error in the
data points, N is the total number of data points and p is the number of free parameters;
p = 4 in this study. Although a Bayesian approach may be preferable, with the underlying
Gaussians the aforementioned metric is not unreasonable.
2
XV-
1
-p
E
Yi
(
YC
r
)2
(5.1)
The models that we use as comparison are generated by randomly populating a comparison gaussian data set. The Box-Muller method for generating normally distributed random
numbers is utilized. First, we set a fraction value to consider and generate a comparison
model made of two components with the same number of stars as the data sample. The
parameters (o-1, pi, o-2, P2) used to generate the model are varied using the LevenbergMarquardt method for least-squares fitting. The result is thus a randomly populated data
set generated with the parameters that best approximate the data sample. Each comparison
model, however, is subject to random fluctuations, and the results will vary somewhat. To
eliminate the random error associated with this process, we perform this model fitting 1000
times and average the resulting parameters. Although more iteration would theoretically be
more accurate, the marginal improvements beyond this point come at a high computational
cost.
This thus defines the best fit parameters of a given value for the normalization. To
compare between fractions, f, we need a single estimate of the goodness of fit for the determined parameters. In order to apply this comparison, 1000 random models are produced
using the best fit parameters as inputs. Each model is then compared to the data and the X2
statistic is computed. The 1000 values of )( constitute estimates for the goodness of fit of
a particular set of parameters. We then compute the mean and standard deviation of these
1000 estimates and take the mean, k2 , as our goodness of fit statistic to compare between
values for the fraction. The uncertainty in this statistic is taken to be o-
=
0' 2
X: the
standard deviation divided by the square root of the number of iterations. The best fraction,
f, and set of parameters, o-1, pi,
0-2,
and P2, is thus the combination that minimizes this
average value, ,22 , stepping through all possible f-values. Figure 5-2 shows an example of
this analysis. The five panels show slices of the six dimensional space ( 2 , f, 0-1, P1, 0-2 ,
used in determining the best fit parameters. The minima in each plot correspond to the
best fit parameters of the population. The uncertainty in each parameter was determined by
/12)
the range of parameter values that yielded a value of 2 within 3max(oyP) of the minimum
value. Figure 5-3 shows an example estimate of the uncertainty. The right panel zooms
in on the minimum shown in the left panel. The uncertainty is taken to be the range of
f-values below the given line. Because of the asymmetry in the distributions, most of the
uncertainty measurements are asymmetric.
We tested the technique extensively before applying it to our data sample. For any set
fraction, the method is capable of accurately reproducing the values of the test parameters with a great number of iterations. For the single gaussian cases, the technique, at a
computational cost with a sufficient number of iterations, performs just as well as fitting
a straight line through the probability plots. The uncertainties in the estimates increase
with less iteration. The limit of our method lies in the number of data points available to
I-raCions
0.6
0.4
0.2
0.0
1.0
0.8
fvalue
ispersion
-
Hot
.1.2
01.0
.C
0 0.8
0.6
0.4
A2
0
5
10
60
40
25
20
15
a (km s-)
0.2
120
100
Hot Mean
Cold Mean
1.2
80
1
o (kms
)
-
-
1.0
O 0.8
'0
0.6
0.4
0.
2 0.2
0.0 - - - 1.0
1.5
.
3.0
2.5
2.0
Mean
Velocity
(kms")
3.5
4.0
10
20
50
40
30
Mean
Velocity
(kms")
60
2
Figure 5-2: Example plots from analysis to determine best fit parameters minimizing f . The minima
correspond to the parameter estimates. Example given is for the bin 200 <
the center panels for figures 5-5,5-6 and 5-7
IZI < 300 in Vz used in each of
0.35
0.30-
1.0
o
00.25-
0.8 -
0
0
I..
0.68-
0.2
4
(1 0.4 -
0.0
C
-0.15
0.2
0 .0 -,.
--
0
=30.20
0.6
0.10
. . . . . . . . .,..
0.2
0.4
0.6
f value
, , .
0.8
-.
1.0
0 .0 5 1 i....
i
.
.
.
.
1
.
.
.
0.75 0.80 0.85 0.90 0.95
f value
.
1.00
Figure 5-3: Example plot from analysis determining the uncertainties in the best parameter estimates. Uses
same plot for fractions as in 5-2. Zoomed in panel on the right demonstrates the uncertainty estimate. The line
corresponds to min(k 2) + 3max(yV2). The range of values below the given line corresponds to the uncertainty
in the parameter.
do the fitting. Under these circumstances it seems that the individual random fluctuations
often obscure the results and it becomes more difficult to come up with proper determinations of the parameters. From empirical tests, the method reliably separates the gaussian
components when each individual component is well populated, preferably with at least 50
members. For samples with a low number of stars (N < 500), the analysis yields large
uncertainties with a minimum value of Nmin ~ 225 for viable results. For few data points
in the one component case it is still preferable to fit a single line.
5.2
Results
We apply these methods to analyze the kinematics of the large sample of M dwarfs. The
sample was first cut to only those stars that had well determined proper motions combining
all of the spectral types together. After rotating the UVW velocities into VR, Vz, and V
the data were also cut to eliminate any stars with a spatial velocity greater than 1000 km
s- 1 . Each dimension also had additional velocity cuts. In VR, the data were cut to only
include those stars with IVRI < 300 km s- . In Vz, the data were cut to only include those
stars with IVz| < 300 km s-.
In VD, the data were cut to only include those stars with
-520 < |Vel < 80 km s- (the local standard of rest is taken to be into [VR, Vz,V ] =
[0, 0, -220] km s-1). The purpose of the velocity cuts is to minimize the errant stars with
abnormal velocities and to minimize the influence of potential Halo stars. The cuts lead to
data samples in each dimension of NR = 35177, Nz = 35356 and, NO = 34557. We plot in
figure 5-4 the probability plots of the entire sample in each of the three velocities, VR, Vz,
V. From the slopes and curves it is clearly evident that the distributions are not of single
Gaussians.
Probability Plots
300-
300-
200
200
100.
100-
0
-100-
-200-
-300-100
-100 -
-200-
-200
-300
-300
-400-
-500-
-4
-2
0
2
4
-4
-2
0
2
4
-4
-2
0
2
4
Figure 5-4: Probability plots corresponding to the entire data sample in each of the three velocities, VR
(left), Vz (center), VD (right). Large number of stars analyzed: NR = 35177. Nz = 35356. ND= 34557 (see
text for exact data cuts). Note that the distributions are clearly non-linear.
As a comparison with previous studies we first look at the changes in kinematics as a
function of absolute distance from the Galactic plane (there were no significant asymmetries between the northern and southern hemispheres). The data were folded over across
the Galactic plane so that abscissa represents absolute distance in Z and so that values of
Vz that are positive mean the stars are moving away from the center of the Galactic disk,
negative meaning towards the disk. The data were separated into 100 pc bins out to roughly
1500 pc, beyond which the number of stars in a particular bin did not permit a viable analysis as discussed in the previous section. In figures 5-5, 5-6 and 5-7, we plot the results of
this analysis. From figure 5-5 it is evident that the normalization across the various components of velocity are roughly consistent and that the f-value drops off with increasing
distance from the Galactic plane. From figure 5-6 it is evident that both dispersions in each
of the velocity components increase away from the plane. The results here reproduce the
kind of behavior that was established in the analysis by Bochanski et. al. (2007a); we,
however, provide more consistent measurements of what they call the 'wing' component.
Going farther out in the plane there also seems to be evidence of a plateau in dispersions for
all velocity components. The results shown in figure 5-7 are more inconclusive. A couple
trends however do seem evident. In Vz the hot component, which makes up a larger portion
of the data in each bin as distance from the plane increases, is generally moving towards
the Galactic plane; this is consistent with Galactic dynamics in that those stars could be
near the peak of their elliptical orbits on their way back towards the center of the Galactic
disk. In VD there is a clear separation between the two components. The kinematically
cold component moving near the velocity of the local standard of rest is separated from the
kinematically hot component that is lagging behind that local standard. It is also clear that
the mean of the kinematically cold component increases farther away from the plane (-VD
decreases in figure 5-7).
The R and Z spatial distribution of magnetic activity and colors were examined in 3.1
and 3.2, and 4. We did the same for the kinematic analysis. The data were separated into
spatial bins of 100 pc by 100 pc in R and Z, spanning a range of 8100 < R < 9200 pc and
0 < IZI < 1100 pc. As before the data were folded over the mid-plane of the Galaxy. The
results are shown in figures 5-8 to 5-22. The top panel in each figure shows the map for a
particular parameter in a given velocity component. The bottom panel is the accompanying
uncertainty in the values for the map of the top panel, it maps the total length of the error
bar (note that these uncertainties are generally asymmetric). Typically the bins that are
on the fringe of the maps, with relatively low numbers of stars in each bin, have higher
uncertainty values. Bins that did not yield conclusive results, due to insufficient number
Fractions
r T--r-r-T-v-f-
u,
1.0
1
A
0.8 -
F+
tf1f
. .
I
*+
'
I
0.6
0.4
0.4
0.4
0.2
-
02
0.2
0
I
0.8
0.8-
0.6 -i0.6-
00
. .
* I *
I
a a
1500
500
1000
Absolute Vertical Distance from the Plane (pc)
o
*
n
0
a a* I a a
00
nn
1000
1500
500
Absolute Vertical Distance from the Plane (pc)
0
a Ia a a
e a * i
500
1000
1500
Absolute Vertical Distance from the Plane (pc)
Figure 5-5: One dimensional plots of f-values as a function of absolute distance from the plane. Analysis
in VR (left), VZ (center), Ve (right). Data points are plotted at center of 100 pc bins.
Dispersions
140
140-
140-
120
120
120
100
100
10
80
80
in
S
0.>
60
a 10
:
40 -40
0
500
1000
1500
Absolute Vertical Distance from the Plane (pc)
0
80 -
-
60'
-
0
500
1000
1500
Absolute Vertical Distance from the Plane (pc)
0
500
1000
1500
Absolute Vertical Distance from the Plane (pc)
Figure 5-6: One dimensional plots of dispersions as a function of absolute distance from the plane. Analysis
in VR (left), Vz (center), Vo (right). Diamond symbol corresponds to the dispersion, o-1 of the kinematically
colder component. Square symbol corresponds to the dispersion, o-2 of the kinematically hotter component.
Data points are plotted at center of 100 pc bins.
100
0
15M
0
-10
-20
'_U
-40n . . . .
1500
50
1000
0
AbsoluteVertical Distance fromthe Plane(pc)
.
0
.
l
.
.
.
.
i
.
.
.
.
l.
500
1000
1500
AbsoluteVertical Distancefromthe Plane (pc)
0
500
1000
1500
AbsoluteVerticalDistance fromthe Plane(pc)
Figure 5-7: One dimensional plots of means as a function of absolute distance from the plane. Analysis in
VR (left), Vz (center), IVoI (right). Diamond symbol corresponds to the mean, pi of the kinematically colder
component. Square symbol corresponds to the mean, P2 of the kinematically hotter component. Data points
are plotted at center of 100 pc bins.
of stars, were excluded from the maps. In figures 5-8 to 5-10 we map the f-values. In
figures 5-11 to 5-13 we map the dispersions of the kinematically colder component. In
figures 5-14 to 5-16 we map the dispersions of the kinematically hotter component. In
figures 5-17 to 5-19 we map the means of the kinematically colder component. Finally,
in figures 5-17 to 5-19 we map the means of the kinematically hotter component. The
trends in absolute distance from mid-plane of the Galactic disk, demonstrated in the one
dimensional analysis, are shown once again in these maps. The analysis, however, did not
reveal any significant radial trends. The available data may not yet allow us to probe such
trends due to an insufficient number of stars at significantly different radii, R. Additionally,
the radial trends may not change much on the scales to which we have so far been sensitive
to. The width for reliable determinations of the kinematic parameters was at most roughly
500 pc. To probe significant radial trends, more data must become available across different
Galactocentric radii.
..........
__ ..................
......
. .. .
Fractions in VR
0.91
1000
0.84
800
600
400
200
8200
8400
8600
8800
9000
9200
0.77
0.77
0.69
0.69
0.62
0.62
0.55
0.55
0.48
0.48
0.40
0.40
0.33
0.33
R (pc)
f value
Parameter Uncertainty
1000
0.88
0.88
0.74
0.74
0.61
0.61
800
600
0.47
0.33
400
0.20
200
0.06
0n
0.06
Uncertainty
8200
8400
8600
8800
R (pc)
9000
9200
Figure 5-8: Top -The spatial distribution of f-values from analysis of VR, corresponds to fraction of data
in each bin that belongs to the kinematically colder component. Redder colors correspond to bins that have a
larger value of f. Bottom- The total uncertainty in the calculated parameter values associated with the map
in the top panel. A whiter shade corresponds to bins with lower uncertainty. Note that the error bars are
asymmetric. Bins are 100 pc by 100 pc.
.. ..........
. ..
......
...........
.................
...
Fractions in Vz
0.98
1000
800
600
400
0.93
0.93
0.89
0.89
0.84
0.84
0.80
0.80
0.75
0.75
0.70
200
0.66
8200
8400
8600
8800
R (pc)
9000
9200
0.61
f value
Parameter Uncertaintv
1000
0.65
0.65
0.55
0.55
0.44
0.44
0.34
0.34
0.24
0.24
800
600
8200
8400
8600
8800
R (pc)
9000
9200
0.13
-0.13
0.03
-__0.03
Uncertainty
Figure 5-9: Top -The spatial distribution of f-values from analysis of Vz, corresponds to fraction of data
in each bin that belongs to the kinematically colder component. Redder colors correspond to bins that have a
larger value of f. Bottom- The total uncertainty in the calculated parameter values associated with the map
in the top panel. A whiter shade corresponds to bins with lower uncertainty. Note that the error bars are
asymmetric. Bins are 100 pc by 100 pc.
Fractions in V,
0.97
1000
0.93
0.90 0.90
800
0.86
600
0.82
400
200
8200
8400
8600
R(pc)
8800
9000
9200
0.78
0.78
0.75
0.75
0.71
0.71
0.67
0.67
f value
Parameter Uncertainty
0.40
0.40
0.34
0.28
0.22
0.22
0.15
400
0.09
200
003
L
0.09
_0.03
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-10: Top -The spatial distribution of f-values from analysis of V4, corresponds to fraction of data
in each bin that belongs to the kinematically colder component. Redder colors correspond to bins that have a
larger value of f. Bottom- The total uncertainty in the calculated parameter values associated with the map
in the top panel. A whiter shade corresponds to bins with lower uncertainty. Note that the error bars are
asymmetric.. Bins are 100 pc by 100 pc.
.............
- ------------- -----
-
. ..
.......
..
......
.....
Cold Component Dispersion in VR
1000
800
600
400
200
47.80
47.80
43.51
43.51
39.22
39.22
34.94
34.94
30.65
30.65
26.36
26.36
22.07
2.07
17.79
8200
8400
8600
8800
9000
9200
R (pc)
13.51
a,
Parameter Uncertaintv
13.50
(km s-')
33.90
33.90
28.67
28.67
23.43
3.43
18.20
18.20
12.97
12.97
7.73-
-7.73
2.50-2.50
8200
8400
8600
8800
9000
9200
Uncertainty
R (pc)
Figure 5-11: Top -The spatial distribution of the dispersion of the kinematically colder component, o-1,
from analysis of VR. Redder colors correspond to bins that have a larger value of o-1. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
L
M
RRW
___
"I"I"I'll-
- -
____
-
-
__ -
-
-
" I
-
-
-
.
__-
..
Cold Component Dispersion in Vz
1000
800
31.50
m31.50
28.90
28.90
26.30
26.30
23.70 iiai23.70
600
21.10 m 21.10
400
200
8200
8400
8800
8600
R (pc)
9000
9200
18.50
18.50
15.90
15.90
13.30
13.30
10.70 M10.70
a, (kmn s-)
Parameter Uncertainty
19.60
m19.60
16.53
16.53
13.47
13.47
10.40
10.40
7.33
7.33
4.27
4.27
800
600
400
200
1.20 L _1.20
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-12: Top -The spatial distribution of the dispersion of the kinematically colder component, o-1,
from analysis of Vz. Redder colors correspond to bins that have a larger value of o-1. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
.....
...........
................
......
....
.
.....
......
..
..
.....
...........
........
....
....
.
........
Cold Component Dispersion in V,
1000
800
600
400
200
0
8200
8400
8600
8800
9000
9200
R (pc)
45.40
45.40
41.44
41.44
37.48
37.48
33.51
33.51
29.55
29.55
25.59
25.59
21.63
21.63
17.66
17.66
13.70
13.70
a,
(km s-1 )
Parameter Uncertaintv
10001
13.90
13.90
11.87
11.87
800
600
400
9.83
9.83
7.80
7.80
5.77
5.77
3.73-
-3.73
200
1.70-1.70
8200
8400
8600
R (pc)
8800
9000
9200
Uncertainty
Figure 5-13: Top -The spatial distribution of the dispersion of the kinematically colder component, o-1,
from analysis of VD. Redder colors correspond to bins that have a larger value of o-1. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
Hot Component Dispersion in VR
1000
800
134.00
134.0(
122.
122.5(
111.00
1111.0(
99.50
600
400
200
8200
8400
8600
8800
9000
9200
R (pc)
88.00
88.00
76.50
76.50
65.00
65.00
53.50
53.50
42.00
42.00
02
(km s-)
Parameter Uncertainty
1000
96.00
96.00
80.67
).67
65.33
5.33
50.00
50.00
34.67
34.67
800
0
40
200
4.00
0
4.00
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-14: Top -The spatial distribution of the dispersion of the kinematically hotter component, 0-2,
from analysis of VR. Redder colors correspond to bins that have a larger value of o-2. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
...............
...
..
..
Hot Component Dispersion in Vz
1000
800
600
400
200
8200
8400
8600
8800
9000
9200
R (pc)
114.00
114.0(
103.06
103.0(
92.12
92.12
81.1
81.19
70.25
70.25
59.31
59.31
48.38
48.38
37.44
37.44
26.50
26.50
02
Parameter Uncertaintv
101.00
1000
(km s1)
101.0(
85.37
69.-1
54.1
38.47
8200
8400
8600
8800
R (pc)
9000
9200
38.47
22.83c
-22.83
7.20 .
7.20
Uncertainty
Figure 5-15: Top -The spatial distribution of the dispersion of the kinematically hotter component, o-2,
from analysis of Vz. Redder colors correspond to bins that have a larger value of o-2. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
Hot Component Dispersion in V,
1000
800
95.00
5.00
87.00
7.00
79.00
79.00
71.00
71.00
63.00
63.00
55.00
55.00
47.00
47.00
39.00
39.00
31.00
31.00
600
400
200
8200
8400
8600
8800
R (pc)
9000
9200
02 (km s-1)
Parameter Uncertainty
46.00
100C
46.00
39.17
600
200
200
32.33
32.33
25.50
25.50
18.67
8.67
11.83
11.83
5.00
5.00
Uncertainty
0
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-16: Top -The spatial distribution of the dispersion of the kinematically hotter component, o-2,
from analysis of VD. Redder colors correspond to bins that have a larger value of 0-2. Bottom- The total
uncertainty in the calculated parameter values associated with the map in the top panel. A whiter shade
corresponds to bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100
pc.
....................
.
Cold component Mean in VR
1000
800
600
400
11.00
11.00
8.86
8.86
6.73
6.73
4.59
4.59
2.45
2.45
0.31
0.31
-1.82
-1.82
200
-3.96
8200
8400
8600
8800
9000
9200
R (pc)
f
-3.96
-6.10
[
-6.10
(km s-)
Parameter Uncertaintv
7.40
1000
7.40
6.13
4.87
4.87
3.60
2.33
2.33
1.07-a
1.07
-0.20
8200
8400
8600
8800
R (pc)
9000
9200
-0.20
Uncertainty
Figure 5-17: Top -The spatial distribution of the mean of the kinematically colder component, pl, from
analysis of VR. Redder colors correspond to bins that have a larger value of p I. Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100 pc.
.
Cold component Mean in Vz
1000
800
7.10
7.10
5.57
5.57
4.05
4.05
2.52
2.52
1.00
1.00
-0.52
-0.52
-2.05
-2.05
-3.57
-3.57
-5.10
-5.10
600
400
200
8200
8400
8600
8800
9000
9200
R (pc)
[t
(km s-1 )
Parameter Uncertainty
1000
800
5.00
5.00
4.18
4.18
3.37
3.37
600
2.55
400
1.73
0.92
200
0.10
L
0.92
0.10
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-18: Top -The spatial distribution of the mean of the kinematically colder component, p1i, from
analysis of Vz. Redder colors correspond to bins that have a larger value of pl. Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100 pc.
............
................
- ------. ......
.....
Cold component Mean in V,
1000
800
600
400
200
8200
8400
8800
8600
R (pc)
9000
9200
-189.10
-189.1
-192.08
-192.C
-195.05
-195.C
-198.03
-198.C
-201.00
-201.C
-203.97
-203.
-206.95
-206.S
-209.92
-209.c
-212.90
-212.c
g
(km s")
Parameter Uncertaintv
9.80
9.80
8.28
8.28
6.77
6.77
600
5.25
5.25
400
3.73
3.73
1000
800
2.22-
-2.22
200
0.70-0.70
8200
8400
8600
8800
R (pc)
9000
9200
Uncertainty
Figure 5-19: Top -The spatial distribution of the mean of the kinematically colder component, pi, from
analysis of VD. Redder colors correspond to bins that have a larger value of p 1 . Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. For reference, the local standard of rest
is taken to be moving with VD = -220 km s-. Bins are 100 pc by 100 pc.
-"a-
_-_ALM--AjV__z
- _.
-
_ ___
mffmwyl
-
- I-
-
___
-
- -
-
t
-
11
=
-,
Hot Component Mean in VR
1000
800
26.10
26.10
20.09
20.09
14.08
14.08
8.06
8.06
2.05
2.05
-3.96
-3.96
-9.97
-9.97
-15.99
.15.9
-22.00
-22.OC
600
400
200
0
8200
8400
8800
8600
R (pc)
9000
9200
R2
1
(km s )
Parameter Uncertainty
1000
29.00
29.00
24.23
24.23
19.47
19.47
14.7
14.70
9.93
5.17
0.40
9.93
-5.17
0.40
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-20: Top -The spatial distribution of the mean of the kinematically hotter component, P2, from
analysis of VR. Redder colors correspond to bins that have a larger value of p2. Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100 pc.
..........
....................
......
......
...
..
..
....
....
.. ... . ......
.....
.......
.
.....
...........
.
"I.........
---- --
Hot Component Mean in Vz
1000
800
600
53.00
53.00
44.50
44.50
36.00
36.00
27.50
27.50
19.00:19.00
400
10.5(
200
8200
8400
8600
8800
9000
9200
R (pc)
10.50
2.00
2.00
-6.50
-6.50
-15.00
-15.0C
p2 (km s-1)
Parameter Uncertainty
1000
56.00
56.00
46.83
46.83
37.67
28.50
19.3'
19.33
10.17 [-10.17
1.00 .- 1.00
8200
8400
8600
8800
9000
9200
Uncertainty
R (pc)
Figure 5-21: Top -The spatial distribution of the mean of the kinematically hotter component, p2, from
analysis of Vz. Redder colors correspond to bins that have a larger value of p2. Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. Bins are 100 pc by 100 pc.
...
.......
Hot Component Mean in V,
2.00
2.00
1000
800
-22.12
-22.12
-46.25
-46.2r
-70.38
-70.3E
-94.50
-94.5C
-118.62
-118.E
-142.75
-142.7
-166.88
-166.E
-191.00
-191.C
600
400
200
8200
8400
8600
8800
9000
9200
R (pc)
[2
(km s-1)
Parameter Uncertainty
143.00
143.0(
120.83
120.8'
98.67
76.50
76.50
54.33P54.33
32.17 -
32.17
10.00
10.00
Uncertainty
8200
8400
8800
8600
R (pc)
9000
9200
Figure 5-22: Top -The spatial distribution of the mean of the kinematically hotter component, P2, from
analysis of VD. Redder colors correspond to bins that have a larger value of p2. Bottom- The total uncertainty
in the calculated parameter values associated with the map in the top panel. A whiter shade corresponds to
bins with lower uncertainty. Note that the error bars are asymmetric. For reference, the local standard of rest
is taken to be moving with VD = -220 km s- 1. Bins are 100 pc by 100 pc.
Chapter 6
Conclusion
The large sample of M dwarfs from the SDSS, has allowed us to investigate the spatial distribution of M dwarf properties in the radial domain, R, in addition to the vertical domain,
Z above and below the plane. In terms of magnetic activity, as traced by Ha emission, we
reproduced and extended previous results concerning the decline in magnetic activity with
increase in absolute distance from the Galactic plane. The vertical gradient is especially
prominent for M6s and M7s where there is a significant drop in the activity fraction away
from the mid-plane on the length scale of 300 pc. However, with the exception of a small
radial activity trend in spectral types M3 and M4, there were no really significant changes
across the radial domain. We also demonstrated the presence of vertical gradients in the
color indices r - z, r - i and g - r. The indices r - z and r - i decreased with distance away
from the center of the Galactic plane. The g - r index on the other hand showed a discrepancy between the southern and northern hemispheres. In the north the index increased
with distance from the mid-plane. This trend can be explained by considering the stars farther away as being older and less metal rich. However, in the south this was not observed,
instead the index seem to be uniform across all length scales. This is especially evident
in the maps for M3 and M4 in g - r. Accordingly, we may be observing an effect due to
over correcting for extinction in the southern hemisphere. There were not significant radial
trends in the color index maps. For the most part, we may not yet be probing sufficiently
large radial distances to notice many radial trends, however from the observed trends in
spectral types M3 and M4 there is a hint of radial structure. We must also take not of the
peculiar sightline in the southern hemisphere demonstrated in several of the maps showing
higher activity and different colors. It warrants further investigation.
The other results of this study concerned the kinematics of these stars. The newly developed technique using pseudo-montecarlo data fitting allowed us to determine with good
accuracy the parameters of the underlying Gaussians that make up the kinematic distribution of the stellar populations in each of the velocities VR, Vz, and VD. We separated the
populations into two components, a kinematically colder component and a kinematcially
hotter component. Mapping out the distribution of these parameters as functions of R and
Z we revealed significant trends in the vertical domain but due to small samples at large
radii and the requisites of our computational method we were unable to observe significant
radial trends. Nevertheless, it can be noted that in terms of the radial scales to which we
were sensitive (~ 300 pc), radial differences are negligible.
In terms of Galactic dynamics for the very well populated bins we attribute the parameters characterizing the dynamically colder component to the thin disk. Whereas, parameters
for the kinematically hotter component can be attributed to the thick disk, however, we must
have some reservation in making this latter assumption. In many of the bins that were analyzed the kinematically hotter component was not well populated (this is especially true at
the map fringes where the bin sizes are small) meaning that the hotter component in these
cases constituted only a handful of stars. Under these circumstances, we should be hesitant
in attributing the means and dispersions of these particular handfuls of stars to the greater
thick disk as a whole. Nevertheless these estimates are still useful and give a reasonable
accounting of the dynamics in the solar neighborhood. Additionally, since there were no
observable trends in the radial domain, the one dimensional analysis provided in section
5.2 is currently the most accurate description of the local dynamics in the Galactic disk.
Continued study with increasingly large data samples will allow us to learn more about the
origins and evolution of the Milky Way.
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