Comets are cosmic snowballs of frozen gases, rock and dust that orbit the Sun. Isaac Newton suspected that comets were the origin of the life-supporting component of air and a key source for water replenishment in planetary interiors. A close-up view of comet Hartley 2 was taken by NASA's EPOXI mission during its flyby of the comet, using the spacecraft's medium resolution instrument. Comet Hartley has a novel asymmetric dumbbell-like shape. We employed mathematical models to study comet Hartley. Using calculus-based methods, we estimated various static properties, including the arc lengths (outer boundary length), surface area, and volume of Comet Hartley. Assuming a constant density, we also estimated the mass, center of mass, and moments of inertia for Comet Hartley using triple integration methods using cylindrical coordinates. The center of mass, moments of inertia, and radius of gyration form key inputs for studying the orbital mechanics of the comet in outer space. This research project provides excellent opportunities for hands-on explorations using multivariable calculus studies for engineering and space sciences applications. This research is important as studies of comets unravel secrets about the formation of the solar system.

Gliders are robotic vehicles used in the air and underwater to collect and transmit real-time data. Studies using gliders have important applications in oceanography, engineering, and remote sensing. The goal of this project was to model and identify aspects of a glider’s flight using vector-calculus and matrix-algebra based methods. We employed mathematical models to study the flightpath of a glider using vector valued functions and calculated the osculating plane of the glider. The model parameters were optimized to minimize turbulence. We studied the kinematics of underwater gliders using GPS data reported from gliders deployed by Rutgers University and the University of Washington. We analyzed the reported glider velocity data and applied vector-calculus based methods to calculate the instantaneous and average velocities and acceleration vectors. Additionally, we applied matrix-algebra based methods to translate and rotate the glider to position it at appropriate coordinates underwater for gathering data. This research provided insight into mathematical modeling of real-world data and involved applied optimization and data visualization. These studies provide novel avenues for hands on exploration and application of key mathematical concepts.

STS-121 is a NASA space shuttle mission to the International Space Station (ISS). The ISS is a habitable satellite (Space station) in a low Earth orbit. We employ calculus-based methods to analyze and study the flightpaths, altitude, velocity, and acceleration profiles of the STS121 data reported by NASA as it travelled through outer space. Our studies unravel information about the critical points, local maxima and minima, concavity, and inflection points in the altitude data. The velocity profiles were fitted to polynomial functions using least square data fitting using linear algebra-based methods. The acceleration data involve piecewise functions which is related to the time scales involving burning of the propellent and separation of the external propellant tank as the space shuttle gets ready to move into orbit. We estimated the work done in transferring a load from Earth to the International Space station. We used optimization methods to design an optimal solar panel geometry for a satellite by minimizing the surface area. This research provides novel applications of the fundamental theorems of calculus to study motion in outer space and involves mathematical modeling, optimization, curve fitting, data analysis and data visualization.

Dissolved oxygen (DO) is a vital component of marine ecosystems, providing the key life source for thousands of species of marine vertebrates and invertebrates. Oxygen’s solubility in seawater is influenced by many variables, which can make DO difficult to predict. Estuarine systems experience DO fluctuations, as DO can limit ecosystem reproduction and health. Levels below 4 mg/L induce hypoxic conditions, creating stress for marine organisms, which makes tracking DO levels over time an essential tool for monitoring marine ecosystem health. My research provides Spatial-temporal depth analysis of DO data from the years 2014 through 2021 in the Snohomish River Estuary in Everett, Washington. Temporally, I predicted DO to exhibit a seasonal trend with highs in the winter and lows in the summer and decrease yearly at all depths due to global ocean temperature increase. Spatially, I expected DO to be higher at sites closer to the Snohomish River, and slightly lower at locations further from the river, in the center of the sound. With regard to depths, I predicted DO to be higher near the surface and lower near the bottom, and the oxycline is expected to get closer to the surface over time. Data were collected using an EXO2 Sonde at five different field sites at varying distances from the Snohomish River. I analyzed data using Excel, RStudio, and ArcGIS. Results found that DO is increasing over most sites with seasonal fluctuations of higher DO in the winter, and lower in the summer. There was one hypoxic event in 2016 at Buoy, along with a yearly increase in DO that suggests hypoxic conditions in Possession Sound may not last. Spatially, DO is higher at sites closer to the mainland, contrary to my hypothesis. Continuation of research will include further analysis of Spatial-temporal data in ArcGIS and Rstudio.

Local processes in marine ecosystems, including coastal estuaries, modify ocean acidification caused by rising atmospheric CO2. Because ocean acidification poses a threat to shell-forming organisms, it is critical to understand how these processes affect acidification in specific regions. In the Snohomish River estuary, freshwater from river discharge deposits directly into Possession Sound, impacting the salinity and temperature of the area. River discharge in estuaries has been found to be slightly acidic, as well as a source of nutrients that fuel blooms of phytoplankton. Large phytoplankton blooms can lower the pH at depth because of the process of respiration, which releases CO2 and decreases dissolved oxygen levels. My research examines changes in pH with temperature, salinity, chlorophyll, and dissolved oxygen at different depths in Possession Sound, Washington, using data collected from January 2017 through January 2021 with a YSI EXO2 Sonde. I hypothesized that near-surface depths and sites located closer to the river would have lower temperatures and salinities correlating with lower pH. Additionally, lower dissolved oxygen at greater depths would correlate with greater amounts of chlorophyll and a decrease in pH at depth. Depths near the halocline were predicted to have alkaline pH values due to photosynthetic organisms. I analyzed data using Microsoft Excel and R Studio. Results found that with chlorophyll less than ~1.25 RFU, pH was greater than 8.0, while with lower dissolved oxygen, pH was less than 7.75. Temperatures less than 10°C corresponded with more pH values between 7.0 and 7.5, while salinity had no apparent trend. In most seasons, pH appeared to decrease slightly at greater depths. The exception to this was winter, when more acidic pH values were observed at near-surface depths. Overall these results indicate that local processes in the Snohomish River estuary are affecting changes in pH.

Possession Sound is a dynamic salt wedge estuary system near Everett, Washington that is fed by the Snohomish River. In salt wedge estuaries, a mix of salt and fresh water creates a salinity gradient between the two sources, with the denser, saltier water making up the lower sections of the vertical gradient, and the freshwater residing above it. Turbulence from river flow and tidal currents decreases the concentration of suspended sediment in the water column, measured as turbidity. As stratification increases, turbulence increases too, which then causes lower turbidity. This study explores influences on turbidity at the Everett Marina during the year 2020. In the Everett Marina, North flow corresponds with flood tides, and South flow during ebb tides. During these tidal exchanges, the prediction is that when the tide is slack, the horizontal velocities of the water would show an east/west flow. This east/west flow would create vertical mixing because of upwelling and higher turbidity in the water. Two-dimensional horizontal river flow velocities from 0.9 to 4.9 meters from the riverbed at half-meter and meter increments were measured using a grant-supported deployment of an Aquadopp ADCP (Acoustic Doppler Current Profiler). Turbidity was collected using a CTD deployed 1.7 meters from the surface. The volume, velocity, and sediment deposition of river water were compiled from the United States Geological Survey (USGS), and tide heights were published by the National Oceanic and Atmospheric Administration (NOAA). Preliminary results indicate an inverse relationship between tidal height and turbidity and with an emphasis on further Spatio-temporal relationships, more conclusions may be found. The Everett Marina hosts dredging of the estuary in order to maintain safe river flow to the Possession sound itself. Without this river flow, needed nutrients may not reach the saltwater, disrupting the ecosystem, and increasing flooding.