Negative Heat Capacity in Self-Gravitating Systems

1 Self-Gravitating Systems Jae Un Kim Department of Physics in Ajou University Course Title Professor Name October 27, 2025 2 Abstract Negative Heat Capacity in Self-Gravitating Systems A self-gravitating system, such as a star cluster or molecular cloud, exhibits an intriguing thermodynamic property known as negative heat capacity. This means that when the system loses energy, its internal temperature increases, which is contrary to ordinary thermodynamic intuition. Physical Origin For a system bound by gravity, the total energy is given by E=K+U,(1) where Kis the kinetic energy and Uis the gravitational potential energy, U∝−GM2 R. When the system radiates energy (dE <0), the radius Rdecreases due to gravitational contraction, making Umore negative. Since K=E−U, the decrease in Eleads to an increase in K, which corresponds to a higher temperature: dE <0⇒dT >0.(2) Thus, the heat capacity Csatisfies C=dE dT <0,(3) indicating that the system heats up as it loses energy. Implications This counterintuitive behavior underlies the thermodynamic instability of self-gravitating systems. It explains why gravitationally bound systems tend to undergo core collapse and halo expansion, a process that eventually prevents them from reaching true thermal equilibrium. In summary, negative heat capacity arises because gravitational potential energy becomes increasingly negative during contraction, converting part of this potential energy into kinetic energy, which raises the system’s temperature even as total energy decreases. 3 Self-Gravitating Systems Introductory text goes here. A wild example parenthetical reference appears (Contributor, 2023), followed by an in-text sample citation by Book Author (2021). Note that paragraphs are separated by a blank line in the editor. If you do not do this, it will continue as a single paragraph. This line is technically on a new line in the editor, but will print in the same paragraph. Method Participants Participant information goes here. Materials Words, along with a sample table (Table 1). Table 1 Sample words from this hypothetical experiment. First word Second word Yeet Yoink Hot Lit Measures Some words about the measures used. Design Some more words Procedure Description of the procedure. Results Descriptive Statistics Statistical words. A demonstrative Figure 1. 4 Figure 1 The mean vibes for each word type and counterbalanced condition order. Error bars represent one standard error of the mean. Inferential Statistics Analytical words. Discussion Discussion words. 5 References Book Author, F. P. (2021). Fancy Pants Whole Book. So & Fancy Publishing. Contributor, C. (2023). The Title of the Chapter. In F. Editor & S. Editor (Eds.), The Book Title (pp. 42–69). Other Publishing Place.