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Orbital evolution of planetesimals in gaseous disks

Planets are formed from collisional growth of small bodies in a protoplanetary disk. Bodies much larger than approximately $1$\,m are mainly controlled by the gravity of the host star and experience weak gas drag; their orbits are mainly expressed by orbital elements: semimajor axes $a$, eccentricities $e$, and inclinations $i$, which are modulated by gas drag. In a previous study, $\dot a$, $\dot e$, and $\dot i$ were analytically derived for $e \ll 1$ and $i \ll H/a$, where $H$ is the scale height of the disk. Their formulae are valid in the early stage of planet formation. However, once massive planets are formed, $e$ and $i$ increase greatly. Indeed, some small bodies in the solar system have very large $e$ and $i$. Therefore, in this paper, I analytically derive formulae for $\dot a$, $\dot e$, and $\dot i$ for $1-e^2 \ll 1$ and $i \ll H/a$ and for $i \gg H/a$. The formulae combined from these limited equations will represent the results of orbital integration unless $e \geq 1$ or $i > π- H/a$. Since the derived formulae are applicable for bodies not only in a protoplanetary disk but also in a circumplanetary disk, I discuss the possibility of the capture of satellites in a circumplanetary disk using the formulae.