1,519 research outputs found
Qualitative analysis of dynamic equations on time scales
In this article, we establish the Picard-Lindelof theorem and approximating
results for dynamic equations on time scale. We present a simple proof for the
existence and uniqueness of the solution. The proof is produced by using
convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz
condition is not necessary for uniqueness. The existence of epsilon-approximate
solution is established under suitable assumptions. Moreover, we study the
approximate solution of the dynamic equation with delay by studying the
solution of the corresponding dynamic equation with piecewise constant
argument. We show that the exponential stability is preserved in such
approximations.Comment: 13 page
On the Formation of Planetesimals via Secular Gravitational Instabilities with Turbulent Stirring
We study the gravitational instability (GI) of small solids in a gas disk as
a mechanism to form planetesimals. Dissipation from gas drag introduces secular
GI, which proceeds even when standard GI criteria for a critical density or
Toomre's predict stability. We include the stabilizing effects of turbulent
diffusion, which suppresses small scale GI. The radially wide rings that do
collapse contain up to Earth masses of solids. Subsequent
fragmentation of the ring (not modeled here) would produce a clan of chemically
homogenous planetesimals. Particle radial drift time scales (and, to a lesser
extent, disk lifetimes and sizes) restrict the viability of secular GI to disks
with weak turbulent diffusion, characterized by . Thus
midplane dead zones are a preferred environment. Large solids with radii
cm collapse most rapidly because they partially decouple from the
gas disk. Smaller solids, even below mm-sizes could collapse if
particle-driven turbulence is weakened by either localized pressure maxima or
super-Solar metallicity. Comparison with simulations that include particle
clumping by the streaming instability shows that our linear model underpredicts
rapid, small scale gravitational collapse. Thus the inclusion of more detailed
gas dynamics promotes the formation of planetesimals. We discuss relevant
constraints from Solar System and accretion disk observations.Comment: Accepted for publication in the Astrophysical Journal; 20 pages, 10
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