Jeans Instability of Palomar 5's Tidal Tail

Tidal tails composed of stars should be unstable to the Jeans instability and this can cause them to look like beads on a string. The Jeans wavelength and tail diameter determine the wavelength and growth rate of the fastest growing unstable mode. Consequently the distance along the tail to the first clump and spacing between clumps can be used to estimate the mass density in the tail and its longitudinal velocity dispersion. Clumps in the tidal tails of the globular cluster Palomar 5 could be due to Jeans instability. We find that their spacing is consistent with the fastest growing mode if the velocity dispersion in the tail is similar to that in the cluster itself. While all tidal tails should exhibit gravitational instability, we find that clusters or galaxies with low concentration parameters are most likely to exhibit short wavelength rapidly growing Jeans modes in their tidal tails.Comment: sumbmitted to MNRA

Stability Boundaries for Resonant Migrating Planet Pairs

Convergent migration allows pairs of planet to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to an equilibrium value that depends on the ratio of migration time scale to the eccentricity damping timescale, $K=\tau_a/\tau_e$, with higher values of equilibrium eccentricity for lower values of $K$. For low equilibrium eccentricities, $e_{eq}\propto K^{-1/2}$. The stability of a planet pair depends on eccentricity so the system can become unstable before it reaches its equilibrium eccentricity. Using a resonant overlap criterion that takes into account the role of first and second order resonances and depends on eccentricity, we find a function $K_{min}(\mu_p, j)$ that defines the lowest value for $K$, as a function of the ratio of total planet mass to stellar mass ($\mu_p$) and the period ratio of the resonance defined as $P_1/P_2=j/(j+k)$, that allows two convergently migrating planets to remain stable in resonance at their equilibrium eccentricities. We scaled the functions $K_{min}$ for each resonance of the same order into a single function $K_c$. The function $K_{c}$ for planet pairs in first order resonances is linear with increasing planet mass and quadratic for pairs in second order resonances with a coefficient depending on the relative migration rate and strongly on the planet to planet mass ratio. The linear relation continues until the mass approaches a critical mass defined by the 2/7 resonance overlap instability law and $K_c \to \infty$. We compared our analytic boundary with an observed sample of resonant two planet systems. All but one of the first order resonant planet pair systems found by radial velocity measurements are well inside the stability region estimated by this model. We calculated $K_c$ for Kepler systems without well-constrained eccentricities and found only weak constraints on $K$.Comment: 11 pages, 7 figure