OFFICIAL ABSTRACT and CERTIFICATION Category A Mathematically driven physical analysis and confirmation study of exoplanet HD 189733 b and others in close-by star systems using principles of Machine Learning/Linear Algebra, Bayesian Statistical tests, and Mathematics Computational Python Programming

Pratham Babaria and Ethan Chandra Harmony School of Endeavor H S (Austin), Austin, Texas, US

In this study, we examined the orbital periods and photometry of one exoplanet in the HD star system: 189733 b. We constructed a high caliber exoplanet transit detection tracker that acts as a means to analyze the data constituted of the Raw Science images that we obtained from a DSLR camera. We used the Lightkurve and BATMAN Python programming library to convert our data to light curves. The transit data was taken from multiple high precision research studies, such as the NASA exoplanet database, which were then converted to a graph portraying the dip in the host star's luminosity with respect to time. Linear Algebra-based Machine learning models were developed alongside Chi-square tests to examine the likelihood that observation was due to mere chance. We hypothesized that the creation of a DSLR camera exoplanet detector would produce results that support other results. The results of our studies were statistically significant and supported our hypothesis.

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This stamp or embossed seal attests that his project is in compliance with all federal and state laws and regulations and that all appropriate reviews and approvals have been obtained including the final clearance by the Scientific Review Committee. A Mathematically driven Physical Analysis and Confirmation study of exoplanet HD 189733 b and others in close-by star systems using principles of machine learning/linear algebra, Bayesian statistical tests, and computational Python programming Scienteer Project ID: 152305 Introduction Machine Learning Photometric Analysis Flux Light curve Analysis is a method of To Study the transit light curve graphs of IMPORTANT: OUR RESEARCH WAS In this project we used Machine Learning asserting the existence of an unseen HD 189733b in terms of relative light RECENTLY PUBLISHED IN HARVARD’S JEI principles to train and analyze the data object orbiting a host star by examining flux, light curve graphs were normalized SCIENTIFIC JOURNAL. FOLLOWING IS fed to our model. We included elements the dip in normalized flux of the star. If to a baseline value of one. We calculated THE LINK TO THE PUBLICATION: of linear algebra based TensorFlow in the dip happens periodically a curve of the normalized radii of the planets using https://www.emerginginvestigators.org/ order to perform numerical computation best fit can be applied to the scatterplot the following equation: articles/analysis-of-the-exoplanet-hd- and large scale machine learning. and a trend can be observed. 189733b-to-confirm-its-existence/pdf The star orbits the center of mass because of the " gravitational attraction Δ� �! between the planet and the The Light curves in our project were = " Exoplanets are planets that orbit a star in star. generated using the Lightkurve library of � �# a distant solar system other than our the Python Programming Language own. Studying Exoplanets has enhanced our capabilities to understand our place in the cosmos. In our project, we used the method of transit photometry to Periodic Dip in the discover exoplanets orbiting various star normalized flux of the star systems. We also explain the process of with Unit of days. using tensor flow guided machine learning to discover exoplanets around

star systems. In this project, we analyzed The above table was analyzed and machine learning analysis was performed, our the light curves of the planet HD model was 96% accurate in detecting/confirming exoplanets. Measuring Planetary Properties 189733b. We created a confusion matrix to Uniform Source If the planet has a circular orbit, evaluate and allow for the its physical properties can be HD 189733 b is a “Hot Jupiter” indicating that If the star is a uniform source, it is a gas giant with an orbital period less than visualization of our algorithm and evaluated based on planetary 10 days. We formulated Machine Learning its performance as tested by our we can distinguish the true transit. Code using Python that allowed us to examine anomaly from the mean v model. � the light curves and develop Statistical Tests. anomaly using the following $ = Δ� � Hypothesis Variables formula # Moderator variable: Stars close to We hypothesize that the optimization of earth without exoplanets, test data A python code was written for machine learning algorithms coupled with us to compute the eccentricity. mathematical/statistical analysis will yield from NASA archive a significantly more accurate way to detect extrasolar planets compared to raw science Fluctuating variable: flux as a models that are frequently used in the search for exoplanets. function of time. Research Workflow Experimental Question/Purpose Constant: time Linear Algebra Observation Through the use of data-driven Chi Square Tests We observed that the star system Astronomy, we aimed to develop a star Instead of using a rudimentary tracking system that would allow us to A chi square test is used to measure how scalar, vector, or matrix approach of that we were observing would have observe exoplanets orbiting nearby star likely it is that an observed distribution is analyzing the data, we used a more a noticeable flux change that would systems. This was done through the use due to chance. In this research, the indicate the existence of an distribution was the light curve which robust approach: tensors. We of pictures obtained via a DSLR survey of created and stored multiple exoplanet with the same physical the star system using a custom-made measured the normalized flux of the star per unit of time (days). matrices of data that were then characteristics as previous literature exoplanet detector. We obtained these values. light curves using the Python analyzed and maintained through programming language and analyzed We used the Kepler Data Processing linear-algebra based TensorFlow. them through machine-learning Pipeline to convert our Data into transit Data Acquisition algorithms. graphs and Chi-Squared maps: . We used a standard celestron The “dip” in flux https://docs.google.com/spreadsheets/d caused by the telescope to observe the star transit of an /1sAas92qpUS-1jCTvuPVQplH- unnamed systems. We accompanied this with a exoplanet about pacL0vtYhu-vOWYFbWg/edit?usp=sharing an unnamed star. custom made, engineered exoplanet detector. Mathematical Tools Confirmation Process for Calibration The Table shown below examines the extent of the difference between the values that Discovery Some images that were produced we calculated and previously established Exoplanet detection can be hard at were biased images and therefore literature values. times because of regular lacked much credibility. The bias interferences from astronomical and images that were observed most meteorological phenomena. We often were Dark images, and devised a checklist in order to domeflat images ensure that exoplanet detection can be easily achieved. This checklist Using these bias, dark, and domeflat served to ensure that we would get images, we created Masterbias, the least biased data. Masterdark, and Masterflat images. To create these images, the Python Measured Results and the corresponding literature values of our high precision exoplanet research survey for the test exoplanet Checklist Programming language was used under observation. The literature values differ considerably from our obtained results thereby showing the inaccuracy of The Motion of an exoplanet around 1. Ensure meteorological along with a CCD detector. previous scientific research studies. a star is characterized by a vector phenomena does not obstruct Bibliography of �̂. We also know that the the collection of data. (Raw Science – MasterDark) gravitational force between the star 2. Known circumbinary star systems (Masterflat – Masterbias) We used a myriad of sources in our and the planet is the following: should be marked as “false research project. Hence, we alarms.” decided to compile the sources on a ��� 3. Ensure that gears on the star Google Doc. The following link �% = " ⋅ �̂ � tracking device are properly fit contains all sources used: The distance between the two and are thoroughly inspected for centers of mass (the planet and the finest precision. Raw Science Image Iimage without Bias Frame https://docs.google.com/document star) is r which can be resolved into /d/172kYhKy1eUxcTWO65qMVTi988 two components: x and y. qyP5wVKYybiOiI_8k8/edit?usp=shari Therefore, the x component is ng ; � cos � and the y component is � sin �. The equation of the horizontal velocity is: Possible false light curves that may be as a result of clouds passing by Conclusion

Master Dark image without bias Star Tracker used to gather One of the most significant parts of � = � cos � Methods frame. images the research conducted was the d �̇ = � cos � We used the same data analysis Photometry/Light confirmation of an exoplanet using d� techniques that the Kepler mission technology far less sophisticated than d d Curve Analysis �̇ = � cos � + cos � � used in its mission except with our that typically used in conventional d� d� own steps of Photometry and light In order to obtain a photometric exoplanetary research. The usage of curve modeling: We took raw light curve, we used two software Multivariate Calculus, Linear Algebra, Expanding the derivative gives us: images from the telescope lens and graphical interfaces: Sharp Cap and Statistically driven error analysis, � �� Machine Learning, and Tensor flow �̇ = � cos � − sin � ⋅ ⋅ � converted them to chi-square maps Astro Image J. These light curves helped us streamline the process of �� �� and transit light curves. were compared to the ones obtained through our Python Code, obtaining authentic light curves. Similarly, for y we get: and a Chi-Squared map was &' We used similar steps as the Kepler �̇ = cos � ⋅ ⋅ � + �̇ ⋅ sin �x developed Additionally, we used Python &( Mission used for data Analysis: Data Programming libraries such as light Chi-Square Map for our Acquisition, Calibration, exoplanet with x-axis labelled kurve to convert pixelated images of Computations for this were as hours from mid transit and Photometry, Light curve modeling, and the y-axis as the host stars to light curves. This normalized planetary radius. aggregated and compiled using the and Light curve Analysis. process proved to be very beneficial python programming language. Data Acquisition Results in our final result: confirmation of Our Hypothesis was supported because exoplanet HD 189733b. In the future, The True Anomaly was Isolated from We used a custom-built star tracking we hope to use these techniques to device in order to collect our data on the values that we found by utilizing the the mean anomaly using the Newton continue searching for the, nearly, 92 several cloudless nights We accompanied light curve transit graphs were similar – Raphson method to get closer and million exoplanets still undiscovered by these techniques using remote observing to the result outputed from our code. TESS data. closer approximations to the on the computer. All planets that we expected values. observed were “Hot Jupiters.” Project Forms

Abstract