Dynamics of planetary rings under thermal forces

Abstract

Planetary rings provide natural laboratories for studying the fundamental processes that govern the evolution of planetary systems. However, several key features, such as the sharp inner edges of Saturn's rings remain unresolved. In this work, we introduce and quantify the Eclipse-Yarkovsky (EY) effect, a thermal torque arising from asymmetric thermal emission of particles during planetary eclipses, which is effective for particles larger than millimeters in size. We formulate this effect within a continuum framework appropriate for collisionally coupled planetary rings and derive the continuum evolution equation that includes the EY torque and viscous diffusion (Eq.26), constraining its magnitude using ring particle spin distributions obtained from N-body simulations. We find that the EY effect systematically produces a positive angular momentum flux that could overcome the viscous torque, driving ring material outward and leading to long-term decretion. The total EY torque principally depends on the optical depth, in which we identify three dynamical regimes: dense, transitional, and tenuous regimes, each exhibiting distinct evolutionary pathways. In the dense or transition regimes, the EY torque can produce a sharp inner edge such as that of Saturn's A ring. In the tenuous regime, it can drive an entire ring outward while preserving shape. This outward transport may also facilitate satellite formation beyond the Roche limit. We also quantitatively show that planetary thermal radiation on rings exerts an opposing torque, namely planetary-Yarkovsky effect, whose importance depends on planetary emissivity and ring-particle albedo, and may lead to inward transport in Saturn's close-in rings.

Figures (10)

• Figure 1: Schematic of the planetary ring with a shadow. The planet is surrounded by a ring with a tilted angle $\sim 20^\circ$, mirroring the configurations of Mars or Saturn. The radiation from a surface element of the planet follows Lambert emission low, denoted by the arrows within the red dashed circle. The thermal force arises from the thermal emission of particles. In particular, the eclipse leads to an asymmetric thermal emission of particles and therefore produces a net thermal torque after averaging the orbit, namely the Eclipse-Yarkovsky effect. The right panel shows the simulated box space with ring particles as an example.
• Figure 2: The timescale of the EY effect for individual particles as a function of the particle size for Saturn's system. The planetocentric distance is set to be 2 planetary radii and the EY coefficient $f_\mathrm{EY} = 0.002$.
• Figure 3: The factor $\eta_{\rm p}$ due to the planetary radiation as a function of the orbital distance $R$ and $(1 - A_{\rm IR})/(1-A_{\rm v})$ (a higher value represents a higher influence of planetary radiation), considering different planet's emissivity $\Xi = 1$ (e.g. Martian case) and 1.78 (Saturn's case). The emissivity can partially come from the internal energy of the planet. For comparison among results obtained with different values of $\Xi$, we fix the planetary obliquity at $\varepsilon_{\rm p} = 25^\circ$, representative of both Mars ($25.2^\circ$) and Saturn ($26.7^\circ$). The planet's albedo is set as $A_{\rm p} = 0.342$. For Saturn's rings, previous works adopt $(1 - A_{\rm IR})/(1-A_{\rm v}) = 1.33$ for A/B rings rub2006Vokrouhlicky2007, while $A_{\rm IR}$ is highly uncertain.
• Figure 4: Illustration of the EY effect on the ring particles, taking the example of the Saturn ring. These ring particles are located at 2 Saturn radii, with the thermal inertia of $10$ tiu. The particle size ranges from 0.7 m to 2.3 m, distributed in a power law with the power index of -3. The left panel shows the rotation (i.e. the spin rate $\omega$ and the obliquity $\varepsilon$) distribution of ring particles at the $t = 0$ (black square) and $t = 5 ~t_{\rm orb}$ (green circles). The colors in the background denote the values of $f_{\rm EY}$. The right panel shows the convergence of the EY coefficient $f_{\rm EY}$ over time with the epochs in the left panel included, considering different optical depths $\tau$. This indicates that we can simply adopt a constant $\bar{f_{\rm EY}}$ over the long-term evolution.
• Figure 5: Ring evolutionary examples for the viscous evolution (a) and EY-included evolution in the order of decreasing the optical depth from (b) to (d). The simulation domain extends to $4\, r_{\rm p}$, while the panel displays to $3\, r_{\rm p}$, corresponding to the commonly adopted fluid Roche limit. The rings are assumed to orbit Saturn, with a mean particle size of 0.1 m. Note the difference in $y$-axis scales between the subfigures.