This study employs a range of AI-based bibliometric methods to examine trends among astronomical research publications. Astronomy is a field with a long history of research and a wide variety of fields, so there are many areas in which quantitative bibliometric studies can be used to categorize topics, summarize research trends, and explore future research directions. For our first attempt we chose the oldest astronomical instrument, the sundial. We collected a total of 172 sundial and gnomon research papers from 1909 to 2024 from Web of Science and Scopus databases. A bibliometric analysis of the astronomical research papers was performed using the bibliometrix package in R. Topics were categorized and discussed using the Structural Topic Model (STM) method. Productivity, citation counts, and other metrics were compared across countries and journals and the global network of researchers engaged in the study of sundials was analyzed. Results emphasize the need for greater international collaboration and interdisciplinary integration. Current trends in sundial and gnomon research were reviewed, identifying eight research topics through the use of STM, demonstrating the evolution of this field into various applications. The article concludes by discussing future research directions for sundials and gnomons, demonstrating the applicability of AI-assisted bibliometric analysis in various fields of astronomy research.