## 5 Juno Trajectory Geometry
### Statistical Properties of Position, Velocity and Acceleration Components
Abrupt variations in acceleration may serve as indications of certain occurrences, such as the activation of thrusters or the unexpected collision with external forces. Examining the rate of change of acceleration, known as the jerk, aids in identifying these occurrences and comprehending their influence on the spacecraft's movement. Essentially, acceleration informs you about the rate at which the velocity is changing, whereas jerk indicates the speed at which that change is occurring. Here's what a boxplot of the acceleration derivative (Jerk) would typically show:  
* The center line of the box represents the median, which is the middle value when the data are ordered from least to greatest. * The box itself encompasses the middle 50% of the data, also known as the interquartile range (IQR). The lower boundary of the box corresponds to the first quartile (Q1), while the upper boundary indicates the third quartile (Q3). * Lines extending from the box, called whiskers, typically reach up to 1.5 times the IQR from the quartiles. These whiskers capture most of the remaining data points. * Any data points falling outside the whiskers are considered outliers and are plotted individually as circles or asterisks. This visual summary allows you to quickly assess the following:  
* Center: The typical value of the acceleration derivative. * Spread: How much the data vary around the center. * Skewness: If the distribution leans toward positive or negative values. * Outliers: Any extreme values in the dataset. While the acceleration (the first derivative of velocity) and jerk (the first derivative of acceleration) are crucial for spacecraft control, analyzing the snap, the second derivative of acceleration, provides even deeper insight. Figure 8 shows the second derivative of the velocity (the snap). The center line of the boxplot shows the median, which is the middle number when the data are ranked from lowest to highest. This is also called the interquartile range (IQR) and it shows the middle 50% of the data. The first quartile (Q1) is shown at the bottom of the box, and the third quartile (Q3) is shown at the top. The lines that go out from the box are called whiskers, and they usually go up to 1.5 times the IQR from the quartiles. Most of the last few data points are picked up by these edges. Although the impacts of snap may not be as apparent as those of acceleration or jerk, it offers a more profound comprehension of the subtleties of a spacecraft's movement. Figure 7: Boxplot of the jerk component (for the time interval between 7 August 2011 and 4 January 2023). Through the analysis of the snap, engineers may attain remarkable fluidity, improve control systems for maximum effectiveness, and guarantee the comfort and welfare of astronauts or the integrity of delicate cargo. It is crucial to emphasize that the analysis of snap becomes more significant for high-precision operations or spacecraft carrying very delicate payloads. When it comes to regular missions, it may be enough to prioritize acceleration and jerk. Nevertheless, when the limits of accuracy and comfort are tested in space exploration, snap analysis emerges as a significant technique for attaining improved control over spacecraft. While this study focuses on the Juno mission, the methods and findings have broad applicability to other spacecraft missions. The analytical approaches developed here are designed to be adaptable to various mission profiles and orbital characteristics. For example, our anomaly detection method can be applied to future outer planet missions, helping to identify potential navigation issues early. The spatial-temporal clustering approach can be used in mission planning for future asteroid or comet rendezvous missions, optimizing trajectory design. The insights gained from analyzing Juno's trajectory variations can inform the design of future missions to the outer solar system, improving our understanding of the challenges in navigating these environments.