Looking for phase-space structures in star-forming regions: An MST-based methodology

Abstract

We present a method for analysing the phase space of star-forming regions. In particular we are searching for clumpy structures
in the 3D sub-space formed by two position coordinates and radial velocity. The aim of the method is the detection of kinematic
segregated radial velocity groups, that is, radial velocity intervals whose associated stars are spatially concentrated. To
this end we define a kinematic segregation index, Λ~\tilde{\Lambda }(RV), based on the Minimum Spanning Tree graph algorithm, which is estimated for a set of radial velocity intervals in the
region. When Λ~\tilde{\Lambda }(RV) is significantly greater than 1 we consider that this bin represents a grouping in the phase space. We split a star-forming
region into radial velocity bins and calculate the kinematic segregation index for each bin, and then we obtain the spectrum
of kinematic groupings, which enables a quick visualization of the kinematic behaviour of the region under study. We carried
out numerical models of different configurations in the sub-space of the phase space formed by the coordinates and the that
various case studies illustrate. The analysis of the test cases demonstrates the potential of the new methodology for detecting
different kind of groupings in phase space.

Full text

Mon. Not. R. Astron. Soc. 000,??–?? (2015) Printed 4 December 2015 (MN L
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Looking for phase-space structures in star-forming regions:
An MST-based methodology
Emilio J. Alfaro?and Marta Gonz´alez
Instituto de Astrof´ısica de Andaluc´ıa, CSIC, Glorieta de la Astronom´ıa, s/n, Granada 18008, Spain
Accepted year month day. Received year month day; in original form year month day
ABSTRACT
We present a method for analysing the phase space of star-forming regions. In
particular we are searching for clumpy structures in the 3D subspace formed by two
position coordinates and radial velocity. The aim of the method is the detection of
kinematic segregated radial velocity groups, that is, radial velocity intervals whose
associated stars are spatially concentrated. To this end we define a kinematic segre-
gation index, ˜
Λ(RV), based on the Minimum Spanning Tree (MST) graph algorithm,
which is estimated for a set of radial velocity intervals in the region. When ˜
Λ(RV)
is significantly greater than 1 we consider that this bin represents a grouping in the
phase space. We split a star-forming region into radial velocity bins and calculate the
kinematic segregation index for each bin, and then we obtain the spectrum of kine-
matic groupings, which enables a quick visualization of the kinematic behaviour of
the region under study. We carried out numerical models of different configurations
in the subspace of the phase space formed by the coordinates and the radial velocity
that various case studies illustrate. The analysis of the test cases demonstrates the
potential of the new methodology for detecting different kind of groupings in phase
space.
Key words: astronomical data bases: miscellaneous – stars: kinematics and dynam-
ics – stars: formation – radial velocities – Galaxy: open clusters and associations –
methods: data analysis – methods: statistical.
1 INTRODUCTION
One of the main observational objectives in the study of stel-
lar clusters and star-forming regions is the search for charac-
teristic patterns in the phase space and their evolution over
time. While there are many works published on the spa-
tial distribution of cluster members at different evolution-
ary stages (Cartwright & Whitworth 2004; Ma´ız-Apell´aniz,
P´erez, & Mas-Hesse 2004; Kumar & Schmeja 2007; Wang et
al. 2008; Schmeja, Kumar, & Ferreira 2008; Bastian et al.
2009; S´anchez & Alfaro 2009; Gregorio-Hetem et al. 2015,
among others), this number noticeably shrinks when we look
for pattern analysis of the kinematic subspace. So far only
four clusters seem to show a clumpy structure in radial ve-
locity data: NGC 2264 (F˝ur´esz et al. 2006; Tobin et al. 2015),
Orion Nebula Cluster (F˝ur´esz et al. 2008), and more recently
the kinematic analysis of two clusters, Gamma Velorum (Jef-
fries et al. 2014) and NGC 2547 (Sacco et al. 2015), based on
Gaia-ESO Survey (GES) data. The low number of studies
is due to the lack of precise and complete kinematic data
?E-mail: emilio@iaa.es
for the cluster members as well as to the absence of reliable
statistical tools specifically designed for this purpose.
The studies cited above were mainly based on hand-
made exploratory analysis of the spatial and kinematic infor-
mation, providing a qualitative description of the kinematic
patterns and raw quantitative estimates of the main vari-
ables characterizing sub-structures. In addition, this kind of
customized procedure is far from being the most suitable,
in terms of time and homogeneity, for analysing the amount
of data expected either from ground-based projects such as
APOGEE (Majewski et al. 2010), LAMOST (Zhao et al.
2012), and GES (Gilmore et al. 2012), or from the Gaia
space mission (Lindegren et al. 2008).
In this work we propose a methodology that relies on the
Minimum Spanning Tree (MST) graph algorithm (Jarn´ık
1930; Prim 1957). It can be easily implemented in any
pipeline developed to mine large databases and leads to a
quantitative description of the kinematic pattern allowing
a comparative analysis between different clusters, environ-
ments and datasets in a homogeneous way. The paper is di-
vided in four sections, the first being this introduction. The
foundation and description of the procedure are shown in
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2Emilio J. Alfaro & Marta Gonz´alez
section 2, the modelled case studies and the results of their
analysis by the proposed methodology are presented in sec-
tions 3 and 4, respectively. Finally, the main conclusions of
this work are summarised in section 5.
2 FOUNDATION AND PROCEDURE
The search for phase-space structures in stellar systems re-
quires specific tools that respond to different concepts of
what a stellar grouping is. Here we consider the existence
of a clumpy velocity pattern where there are velocity ranges
(channels) whose spatial distribution is more concentrated
than that of the whole kinematic interval.
The Minimum Spanning Tree graph algorithm has been
shown to be a useful tool for tackling a large variety of as-
tronomical problems (e.g. Cartwright & Whitworth 2004;
Cartwright, Whitworth, & Nutter 2006; Campana et al.
2008; Allison et al. 2009; Parker et al. 2011; Billot et al.
2011; Parker, Maschberger, & Alves de Oliveira 2012; Mac-
farlane, Gibson, & Flynn 2015). In the following we will
deal with the so-called Euclidean Minimum Spanning Tree,
where the edge weight is defined by the Euclidean distance
between vertices. For the sake of simplicity we call it MST
and it can be defined as follows: given a set of points on the
plane, the MST is the minimum length path connecting all
the vertices together without closed loops (Prim 1957).
Thus, the MST is the tree where the distance between
each two adjacent points, edge length (l), is minimum. This
property of the MST provides the clue for solving our prob-
lem. The edge-length distribution of an MST graph will show
a lower central value when the point distribution is more
spatially concentrated.
If we divide a set of points on the plane, sorted by radial
velocity (RV) values, into bins of equal number of objects,
and determine the MST for each bin, those with the lower
central value of the edge distribution are expected to form
spatial sub-structures in the sense of being more densely
grouped. In order to normalize the results, the central value
of the edge distribution for each bin has to be compared with
that of a set representative of the whole sample distribution
containing the same number of objects per bin. The ratio
between the central length of the random sample and that
corresponding to each bin should provide a measure of how
both sets are distributed. Values close to 1 would indicate
that objects in the selected bin share the spatial distribution
of the whole sample, and the selected RV range appears not
to be spatially segregated. On the contrary, if this value is
significantly higher than 1 we would get evidence of a spatial
grouping associated with a specific RV interval.
We base our procedure for detecting and analysing the
kinematic pattern of star-forming regions on the aforemen-
tioned arguments. The ideas discussed above are conceptu-
ally similar to those presented by other groups (see Allison
et al. 2009; Parker et al. 2011; Maschberger & Clarke 2011)
for the case of mass segregation. However, this methodolog-
ical approach should not necessarily be limited to analysing
the mass distribution and can be easily extended to other
physical variables, such as radial velocity, proper motions,
metallicity, etc., measured on the cluster stellar population.
The main constraint on this methodology is data qual-
ity. If the precision of the measured variables is not good
enough, internal errors will dominate over the data intrin-
sic variance and any kind of spatial structure would remain
hidden. Radial velocity is, for the time being, the only kine-
matic variable measured with enough precision for a sig-
nificant number of stars in a large number of clusters and
star-forming regions. Thus, in the next sub-section we will
address the analysis of this kinematic information and how
it can be used to draw the phase-space pattern of stellar
clusters and star-forming regions.
2.1 Detecting the kinematic groupings
The procedure designed to search for kinematic segregation
starts from a sample of objects in the region under study,
with accurate spatial coordinates and precise radial velocity
data.
•The first step is to sort the data by radial velocity val-
ues. Ascending or descending order is not important.
•Next, we separate the sample into bins of equal number
of data. When working with RV data that show, at first
approach, a Gaussian distribution, we chose a bin size given
by the closest integer to √N, which, henceforth, we denote
as R. We also introduce an incremental step S, so any bin
is shifted Spositions of the sorted variable with respect to
the previous one. The choice of Sdepends on the sampling
strategy, and we recommend a fine scanning of the complete
velocity spectrum which corresponds to S= 1 and would be
covered by 1 + N−Rbins.
•Then, we calculate the median for the spatial coordi-
nates as well as for the radial velocity for each data bin.
Median, as a central value estimator, is more robust than
mean against the presence of outliers, and we expect to find
some points away from the central value in most cases. A
robust estimator of the standard deviation is given by:
σmed =b×
^
|Xi−
f
Xi| ≡ b×MAD (1)
(see Gott et al. 2001, for a detailed discussion of median
statistics), where MAD is the median of the absolute devi-
ations from the data median. The error of this central value
can then be approached as σmed/√R. The bparameter is
dependent on the distribution function underlying the data.
In the following we will take b= 1.4826 as default.
•The next step is to obtain the MST graph for the Rob-
jets in each bin. The median of the edge length distribution
is the first indicator of the group compactness (equation 2).
As discussed by Maschberger & Clarke (2011), median is the
best estimator of the central value of the edge length when
we could be facing a clumpy distribution for a given velocity
range.
•The same is applied to a set of Rrandomly extracted
objects from the complete sample. We repeat this step 500
times and calculate the mean and the standard deviation of
the 500 medians. We chose 500 repetitions to get a reliable
average of the edge length median for the Rsized random
sample, as recommended by Allison et al. (2009). This way
we estimate the edge length median of the MST for a refer-
ence set of Robjects, representing the whole sample.
•Then we estimate the kinematic segregation index,
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Phase-space structures in star-forming regions 3
Figure 1. Schematic representation of the six test cases. From top to bottom and from left to right they are: 1.- Two spatially separated
clusters with the same central velocity of 1 km s−1; 2.- Two clusters separated both in space and in velocity, where the mean velocities of
the clusters are 1 and 2 km s−1, respectively; 3.- Two spatially superimposed clusters with velocities of 1 and 2 km s−1, respectively; 4.-
Two filaments that intersect at one of the ends with homogeneous spatial distribution and a velocity gradient in each one proportional
to the spatial coordinate Y; 5.- One filament with a homogeneous spatial distribution and a mean velocity distribution of 1 km s−1and
the same dispersion as that estimated for the clusters; 6.- Four clusters that partially overlap in space and velocity distributions with
means of 1, 2, 3, and 4 km s−1, respectively, and the same velocity dispersions given by its virial equilibrium. Different colours mean for
different central velocities. Case 3 is coloured by blending blue (1 km s−1) and red (2 km s−1) colours. Spatial coordinates, Xand Y
axes, are in parsecs.
˜
Λ(RVj), as
˜
Λ(RVj) = ˜
l500
R
˜
li,i+R
(2)
where jis the bin identification and iis the order of the
first element in the bin, both being connected through the
relation i= 1 + (j−1) ×S. Supra-index 500 indicates that
the average is calculated over 500 stochastic realizations.
The total number of bins is given by the integer part of
1 + ((N−R)/S).
•A plot of ˜
Λ(RVj) versus the median of the radial veloc-
ity in the RVjinterval
g
RV for all j, shows what we name
the spectrum of kinematic groupings which provides an in-
teresting and useful tool for the exploratory analysis of the
stellar system phase space.
•Those ˜
Λ(RVj) that verify the inequality
˜
l500
R
˜
li,i+R−2טσ500
R
˜
li,i+R≡˜
Λ(RVj)−2×σ˜
Λ(RVj)>1 (3)
mark the radial velocity channels, RVj, where a kinematic
segregation has been detected. Here we decided to multiply
by two the standard deviation term as a conservative crite-
rion in the detection of groupings (confidence level greater
than 95%). Note that ˜σ500
Ris not the median of the 500
standard deviations but the standard deviation of the 500
medians.
3 TEST CASES
In order to analyse the potential of the new methodology,
we have numerically generated six different case studies. The
cases analysed are:
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4Emilio J. Alfaro & Marta Gonz´alez
−3−2−101234
RV (km s−1)
0
5
10
15
20
25
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
−4−2 0 2 4
RV (km s−1)
0
5
10
15
20
25
30
35
40
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
−4−3−2−101234
RV (km s−1)
0
5
10
15
20
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
−6−4−20246
RV (km s−1)
0
5
10
15
20
25
30
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
Figure 2. Upper mosaic: Case 1 (see Fig. 1 and description in the text). The mosaic contains four panels, corresponding to: a) The
spectrum of velocity segregation, which represents ˜
Λ against the radial velocity for each one of the intervals selected. The red circles mark
the segregated velocity intervals according to the inequality defined by the equation (3) (upper left panel); b) The position of all the stars
of the sample (blue circles) superimposed on its spatial density map, and with the stars, within the segregated velocity intervals, marked
in red (upper right panel); c) Velocity histogram of the sample, onto which the velocity histogram of the stars within the segregated
intervals has been superimposed (colour red) (lower left panel); d) Density contours of the segregated stars (colour) on the density map
of the sample (greyscale). Lower mosaic: Case 2 with the same panel distribution and explanation as the upper mosaic.
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Phase-space structures in star-forming regions 5
−4−3−2−10123
RV (km s−1)
0
5
10
15
20
25
30
35
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
−8−6−4−20246
RV (km s−1)
0
5
10
15
20
25
30
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
−4−2 0 2 4
RV (km s−1)
0
2
4
6
8
10
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0008
0.0016
0.0024
0.0032
0.0040
0.0048
0.0056
0.0064
−6−4−202468
RV (km s−1)
0
5
10
15
20
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0008
0.0016
0.0024
0.0032
0.0040
0.0048
0.0056
0.0064
Figure 3. Upper and lower mosaics correspond to cases 3 and 4, respectively. Description and explanation as in Figure 2.
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6Emilio J. Alfaro & Marta Gonz´alez
−4−3−2−101234
RV (km s−1)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0008
0.0016
0.0024
0.0032
0.0040
0.0048
0.0056
0.0064
−6−4−20246
RV (km s−1)
0
5
10
15
20
25
30
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.0000
0.0008
0.0016
0.0024
0.0032
0.0040
0.0048
0.0056
0.0064
−4−3−2−1012345
RV (km s−1)
0
2
4
6
8
10
12
14
16
e
Λ
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
−6−4−20246
RV (km s−1)
0
5
10
15
20
25
30
N
-10.0 -5.0 0.0 5.0 10.0
X (pc)
−5
0
5
Y (pc)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
Figure 4. Upper and lower mosaics correspond to cases 5 and 6, respectively. Description and explanation as in Figure 2.
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Phase-space structures in star-forming regions 7
1.- Two spatially separated clusters with the same central
velocity of 1 km s−1
2.- Two clusters separated both in space and in velocity,
where the mean velocities of the clusters are 1 and 2 km s−1,
respectively
3.- Two spatially superimposed clusters with velocities of
1 and 2 km s−1, respectively
4.- Two filaments that intersect at one of the ends with
homogeneous spatial distribution and a velocity gradient in
each one proportional to the spatial coordinate Y
5.- One filament with a homogeneous spatial distribution
and a mean velocity distribution of 1 km s−1and the same
dispersion as that estimated for the clusters
6.- Four clusters that partially overlap in space and velocity
distributions with means of 1, 2, 3, and 4 km s−1, respec-
tively, and the same velocity dispersions given by its virial
equilibrium
See Fig. 1 for a schematic representation of the different
cases in the phase space. Spatial coordinates are in parsecs.
In all of them there is an underlying field of 200 stars
with homogeneous spatial distribution and Gaussian radial
velocity with zero mean and 10 ×σdispersion, σbeing the
internal dispersion of the groupings in phase space, consid-
ering that they are in virial equilibrium. When the grouping
in phase space is a cluster, we have modelled 50 stars with a
radial spatial distribution of exponent -2 and radius 1. The
velocities always follow a Gaussian distribution with differ-
ent means and dispersions corresponding to virial equilib-
rium. In the case of the filamentary distributions, we have
spatially modelled them as a narrow rectangle containing
50 randomly distributed stars, and with Gaussian velocity
distribution and the same sigma as that estimated for the
clusters. The case with two spatial filaments, showing a lin-
eal velocity gradient in each one, can be seen as a particular
challenge for this methodology, as this is designed to de-
tect groupings in phase space, and a lineal distribution of
velocities does not correspond to a phase-space central con-
centration but to a filamentary structure in itself. In other
words, this method makes it possible to detect spatial struc-
tures of almost any form as long as the velocity is concen-
trated around a central value, but, in principle, the internal
structure of the algorithm is not designed for other cases
of velocity distribution. Nonetheless, we have modelled and
analysed it, and surprisingly the results are more positive
than we might have hoped.
4 RESULTS ANALYSIS
In all of the cases the number of stars per bin, R, has been
chosen equal to √Nand the step Sequal to 1. In this way,
and following the protocol established in section 2, we have
determined the spectrum of the kinematic groupings. This
spectrum, the first result of the application of this method-
ology, has become a very efficient tool for the analysis of the
existence and detection of groupings in phase space. As a
graphical synthesis of the results we have designed a figure
that contains four panels, corresponding to: a) The spec-
trum of velocity segregation, which represents ˜
Λ against the
radial velocity for each one of the intervals selected. The
red circles mark the segregated velocity intervals according
to the inequality defined by the equation (3) (upper left
panel); b) The position of all the stars of the sample (blue
circles) superimposed on its spatial density map, and with
the stars, within the segregated velocity intervals, marked in
red (upper right panel); c) Velocity histogram of the sam-
ple, onto which the velocity histogram of the stars within
the segregated intervals has been superimposed (colour red)
(lower left panel); d) Density contours of the segregated
stars (colour) on the density map of the sample (greyscale).
These four panels make it possible to visualise directly this
method’s potential for detecting and isolating the different
groupings in phase space contained in the test cases.
The results, grouped in pairs, are visualized in figures
2, 3, and 4, corresponding to the six test cases carried out.
In all of the cases the velocity segregation spectrum detects
statistically significant groupings in phase space. Even in
simple view (upper left panel in every figure) it can be seen
how the local maxima of ˜
Λ are coincident with the mean
values of the velocity in the simulated groupings. Only in
the case of the two filaments that intersect at a vertex and
with a lineal gradient of velocity in each one of the filaments,
the method detects two groups of segregated velocities cor-
responding to the ends of the velocity gradient of each of the
filaments. We stress again that the designed procedure does
not contemplate, a priori, the detection of these types of ve-
locity distributions, since the fundamental condition is that
each velocity interval is more spatially concentrated than
any random selection of the sample with the same number
of points. Each filament contains 50 stars, and given that the
space occupied is greater in this case than for the clusters,
the spatial density is less. This density is also less in the
velocity subspace, as the 50 stars are distributed between 5
and -5 km/s. When the velocities are well separated from
that of the mean of the field, the groupings are detected,
but when the velocity of the stars in the filament are close
to that of the field mean (0 km/s), the combined effect of
the low density in both subspaces leads to non-detection
of groupings in the phase space. Below we will discuss the
general characteristics of the phase-space groups for all the
test cases except for the case of the two filaments. In ev-
ery case the groupings detected are contaminated by field
stars that share the same velocity interval but do not con-
ceal or blur the spatial localization of the density maxima
(see upper right panel in figures 2, 3, and 4). The selection
of bona fide members each of the clusters or filaments will
require a subsequent careful membership analysis by any of
the methods available in the literature (Sampedro & Alfaro
2015; Cabrera-Ca˜no & Alfaro 1990, and references in that
articles).
The structure of the velocity field of the sample is re-
flected in the segregated velocities histogram and in the spa-
tial density map of the stars belonging to these kinematic in-
tervals (left and right lower panels, respectively). In all cases
we see how the spatially segregated velocities histogram (in
red) reproduces the distributions of the sample modelled.
The spatial density of the members of the groupings de-
tected also shows the velocity spatial distribution simulated
in the models.
The parameters chosen for thetest cases correspond to
feasible cases that we can find on the large radial velocity
surveys that are currently being performed. However, Gaia
is the ultimate goal, and this methodology can easily be
expanded to the distribution of proper motions that this
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8Emilio J. Alfaro & Marta Gonz´alez
space mission will provide us with. It is evident that this
analysis can be extended to a broad set of parameters but
we believe that that task lies beyond this article’s objectives,
which are to make the method known, explain how it works
and apply it to a few concrete cases that simulate part of
the extensive types of cases that can be found in reality.
5 CONCLUSIONS
•Based on the MST graph algorithm we developed a
new methodology to identify kinematic segregated groups,
i.e. velocity ranges that show a significantly more concen-
trated spatial distribution than the average of the whole
velocity distribution.
•We have defined the kinematic segregation index,
˜
Λ(RVj) (equation 2) that forms the basis or our analysis.
The variation of this index with the radial velocity depicts
the so-called spectrum of kinematic groupings which be-
comes a singular tool for visualising, at a single glance, the
kinematic behavior of the clustered population in the stellar
system.
•The spectrum of kinematic groupings thus constitutes
the fundamental nucleus of study from which a varied quan-
tity of sub-products can subsequently be derived that enable
different visualizations of the problem in order to improve
its further analysis.
•For those cases analysed the method provides an excel-
lent description of the map of the velocities modelled and de-
tects the phase-space groupings present in the sample, with
the exception of the case of the filaments with an internal
velocity gradient.
•The algorithm is easy to implement in any pipeline
aimed at analysing the phase space of stellar systems and
has been specially designed for the study of data expected
from the Gaia mission. Although in this paper we have fo-
cused on the analysis of the radial velocity, the method can
be easily extrapolated to other velocity components, such as
the proper motions.
ACKNOWLEDGMENTS
We warmly thank Simon Goodwin, this paper’s referee, for
his enlightened and useful comments that have helped to im-
prove the content and presentation of this work. Likewise,
we also wish to thank Richard Parker for his reading of the
draft and the discussion held concerning specific points of
the article. We acknowledge support from the Spanish Min-
istry for Economy and Competitiveness and FEDER funds
through grant AYA2013-40611-P.
REFERENCES
Allison R. J., Goodwin S. P., Parker R. J., Portegies Zwart
S. F., de Grijs R., Kouwenhoven M. B. N., 2009, MNRAS,
395, 1449
Bastian N., Gieles M., Ercolano B., Gutermuth R., 2009,
MNRAS, 392, 868
Billot N., et al., 2011, ApJ, 735, 28
Cabrera-Ca˜no J., Alfaro E. J., 1990, A&A, 235, 94
Campana R., Massaro E., Gasparrini D., Cutini S., Tra-
macere A., 2008, MNRAS, 383, 1166
Cartwright A., Whitworth A. P., 2004, MNRAS, 348, 589
Cartwright A., Whitworth A. P., Nutter D., 2006, MNRAS,
369, 1411
F˝ur´esz G., Hartmann L. W., Megeath S. T., Szentgyorgyi
A. H., Hamden E. T., 2008, ApJ, 676, 1109
F˝ur´esz G., et al., 2006, ApJ, 648, 1090
Gilmore G., et al., 2012, Msngr, 147, 25
Gott J. R., III, Vogeley M. S., Podariu S., Ratra B., 2001,
ApJ, 549, 1
Gregorio-Hetem J., Hetem A., Santos-Silva T., Fernandes
B., 2015, MNRAS, 448, 2504
Gutermuth R. A., et al., 2008, ApJ, 674, 336
Higuchi A. E., Kurono Y., Saito M., Kawabe R., 2009, ApJ,
705, 468
Jarn´ık V., 1930, Pr´ace moravsk´e pˇr´ıodovˇedeck´e spoleˇcnosti,
6, 57
Jeffries R. D., et al., 2014, A&A, 563, A94
Kumar M. S. N., Schmeja S., 2007, A&A, 471, L33
Lindegren L., et al., 2008, IAUS, 248, 217
Macfarlane B. A., Gibson B. K., Flynn C. M. L., 2015,
arXiv, arXiv:1505.02059
Ma´ız-Apell´aniz J., P´erez E., Mas-Hesse J. M., 2004, AJ,
128, 1196
Majewski S. R., Wilson J. C., Hearty F., Schiavon R. R.,
Skrutskie M. F., 2010, IAUS, 265, 480
Maschberger T., Clarke C. J., 2011, MNRAS, 416, 541
Parker R. J., Bouvier J., Goodwin S. P., Moraux E., Allison
R. J., Guieu S., G¨udel M., 2011, MNRAS, 412, 2489
Parker R. J., Maschberger T., Alves de Oliveira C., 2012,
MNRAS, 426, 3079
Prim R. C., 1957, The Bell System Technical Journal, 36,
1389
Sacco G. G., et al., 2015, A&A, 574, L7
Sampedro, L., Alfaro E. J., 2015, (MNRAS, submitted)
S´anchez N., Alfaro E. J., 2009, ApJ, 696, 2086
Schmeja S., Kumar M. S. N., Ferreira B., 2008, MNRAS,
389, 1209
Teixeira P. S., et al., 2006, ApJ, 636, L45
Tobin J. J., Hartmann L., F˝ur´esz G., Hsu W.-H., Mateo
M., 2015, AJ, 149, 119
Walker M. F., 1956, ApJS, 2, 365
Wang J., Townsley L. K., Feigelson E. D., Broos P. S.,
Getman K. V., Rom´an-Z´u˜niga C. G., Lada E., 2008, ApJ,
675, 464
Zhao G., Zhao Y.-H., Chu Y.-Q., Jing Y.-P., Deng L.-C.,
2012, RAA, 12, 723
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