NON-GAUSSIANITY OF ERROR
DISTRIBUTIONS
FOR ROTATION VELOCITY
MEASUREMENTS
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By: Tia Camarillo
Supervisor: Dr. Bharat Ratra
Kansas State University | Physics Department | REU Program
Project Overview:
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Figure 1: Matthew Newby, milkyway.cs.rpi.edu |
The amount of luminous mass in galaxies do not agree
with the expected total masses of the galaxies required for objects in their
outer rims to maintain their rotational velocities about the galaxy’s center.
Figure 1 shows the observed rotation curve of the Milky Way Galaxy in green,
with the blue line being what we would expect from Keplerian
motion. The hypothesized “Dark Matter” was then thought to be the missing link
in the Missing Mass Problem. The rotation curve of a galaxy can yield plenty of
understanding as to the galaxy’s properties, notably the density and
distribution of both luminous and dark matter. We cannot however contrive a
well-fitting rotation curve for specifically the Milky Way if the data
providing such a curve is ill-suited for statistical analysis. My project was
to conduct this analysis.
Data on the Milky Way has long existed, but has been
scattered across publications. Until recently, analyses have not been
all-inclusive regarding rotation curve measurements. Miguel Pato
and Fabio Iocco created a comprehensive compilation
of rotation curve data and a tool, named "galkin”
that allowed us to access and analyze the measurements individually by tracer
type or all-together. This new tool is significant to the study of our galactic
mass distribution and, since it is open source code, extremely convenient to
extracting values on which to perform qualitative data analysis.
I worked with applying various statistical techniques
(mainly weighted means and median statistics) to this new compilation of Milky
Way rotation curve measurements. Raw data is often sliced into sections, or
bins, and the data specific to each individual bin can be analyzed to your
choosing. The results from each bin are brought together to the new analyzed,
binned data. My objective was to compare the original data to the results from
multiple analyzed, binned data sets in search of new physical (meaningful)
conclusions about constraints on luminous and dark matter in the galactic mass
budget.
Research Description and Results:
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Figure 2: Crandall, S.,
Camarillo, T., & Ratra, B. “Non-Gaussian Error
Distributions of Milky Way Rotation Curve Measurements.” |
The data we used was a compilation of 2700+ rotation velocity
measurements of different “kinematic tracers”, or, objects in space that we can
trace and observe the kinematics of. It is conventional to bin large
astronomical data sets in order to better see like groups, and so I binned them
in the following ways: by the square root of the total number of measurements
(to maximize bin number and counts per bin) 
, by kinematic tracer (e.g. star, gas, and maser
objects); by galactic angle (done by using the provided position data). Figure
2 shows the color-coordinated unbinned
distribution of tracer rotational velocity (V) and angular rotational velocity
(W). In our error distribution analysis, we use the latter.
Upon binning the data in these three ways I was
concerned with the Gaussianity of the error distributions.
We would expect, especially from such a large set of data, to have a Gaussian
distribution of errors should the measurements themselves be independent of
each other (ref). If they show to be non-Gaussian, then
we might assume that there is some statistical dependence between some of the
sources’ measurements and so we wouldn’t be able to use the entirety of the
data in future endeavors. For a Gaussian distribution centered about a central
estimate, there exists 68.3% (95.5%) of the data within 1σ (2σ). This
means that for the distribution of Nσ,
or the number of standard deviations a single value deviates from the chosen
central estimate (I did this with both a median and weighted mean), we should
expect a Gaussian distribution to have 
and 
for 1σ and 2σ respectively where the center
is at 0. By and large this is not what I found.
In fact, I calculated that on average the data was best
fit by sharper peaked distributions such as the Lorentzian and Laplacian
distributions, which have sharper peaks than a standard Gaussian distribution.
Figure 3 shows an example set of histograms pulled by the method of binning by where I have
kept the bin’s distribution unsymmetrized on the left
hand side and symmetrized the Nσ
distribution on the right hand side. This particular bin was chosen because of
the International Astronomical Union’s recommendation (ref) for R0 so it should be
reasonable to expect this particular region to have uncorrelated measurements
and independently quoted errors, given the abundance of data centered on our
Sun.
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Figure 3: Signed (left) and symmetrized (right) error distributions
for galactocentric range |
We find that for the bin in Figure 3 the range for
1σ and 2σ respectively is 
and 
. This means that for and we find that the percentages of values within these
ranges are 78.9% and 84.6% respectively, instead of the expected percentages of
68.3% and 95.5% (also respectively).
My colleague Sara Crandall is due to present this
research at the upcoming Mid-American Regional Astrophysics Conference next
spring and our results are in due time to be submitted for publication and so I
will not put the entirety of our results up for the sake of length. However, we
hope that knowledge of the non-Gaussianity of the
error distributions for these measurements as a whole does not greatly impact
the quality of further statistically analyzing it, as this independence is an
important assumption in many cases.
To download my final presentation for the summer click
here. We received this data exactly a week before the close of the NSF REU so I
will be finishing this project after I put up this webpage. I will update it
after our paper is submitted and accepted.
To Prospective REU Students!
The NSF REU at Kansas State University provided me the
tools to begin research in my intended field: theoretical cosmology. The chance
to work with Dr. Ratra was a momentous opportunity
and I would advise any prospective undergraduate interested in cosmology or
astrophysics to consider applying for this REU at Kansas State. I made
wonderful, lifelong friends during this summer and looked forward to our
adventures together each weekend just as much as I looked forward to work each
Monday. We were provided lectures by various professors in our department over
their fields of research. To name just a few from the Cosmology/High Energy
groups: Dr. Ratra spoke of the expansion of the
universe, Dr. Horton-Smith lectured us on neutrinos and energy conservation,
and Dr. Bolton taught on the Higgs boson.
If you have any questions whatsoever about the
experience, how we possibly managed to have fun in Manhattan KS, about my
continuation of research on this vast data set, or anything else you can think
of please do not hesitate to contact me. I can be reached via tiacamarillo@ksu.edu. Below are a few
pictures from our long list of memorable times during this summer REU of 2015!
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We couldn’t hold
this “angry” look for very long. |
We made a tent in
the dorm lobby and watched scary movies. |
We had a Fourth of
July barbeque at my off-campus apartment. |
I am going into my senior year at Kansas State
University. I will receive my Bachelor of Science in Physics in May 2016, and
intend to continue on to graduate school in astrophysics or cosmology. I just
really love space. The REU program has helped me grow as a team member, and as
a scientist contributing to the field of cosmology. This was my first
experience programming, so learning the basics of C++ and Python taught me
patience with- and commitment to- my work (but mostly patience). I am the
president of Physics Club here at K-State and work as an undergraduate teaching
assistant. I am the primary lab instructor and coordinator for our algebra
based Physics 1 course and enjoy tutoring outside of class. I am an artist and
I unapologetically eat a lot of peanut butter. At the time of creating this
webpage, I am in ongoing health battles one-year post having an organ removed
that was causing me a lot of grief- so if you’re reading this and also facing
chronic illnesses while pursuing physics, you can still pull through and there
is a place in science for you. We only need to have eager minds; our bodies are
trivial.
Useful Links:
American
Physical Society Statements on Ethics
This program is funded by the National Science Foundation through grant number PHYS-1461251. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.