By: Tia Camarillo

Supervisor:  Dr. Bharat Ratra



Kansas State University | Physics Department  | REU Program




Project Overview:


Figure 1: Matthew Newby,

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:



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.



Figure 3: Signed (left) and symmetrized (right) error distributions for galactocentric range  from  binning.

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 Below are a few pictures from our long list of memorable times during this summer REU of 2015!


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.



About Me:


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

American Institute of Physics

Stack Overflow




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.