The effective mass problem for the Landau-Pekar equations

We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar.

On a variant of Newton-Coulomb's law

A variant of the usual formulas for gravitational and electrostatic fields, differing from those by a logarithmic term, is introduced in order to solve some questions connected with a possible atomic contraction phenomenon at large time scale and the so-called hidden mass problem in cosmology. This approach is conceptuallly related to the MOND theory of M. Milgrom but allows a reversal of attracting forces when the distance becomes very small. The asymptotic behavior of solutions of a related dissipative ODE is studied, we obtain that all bounded trajectories converge to a point where the field vanishes.

Solution to the debris disc mass problem: planetesimals are born small?

ABSTRACT Debris belts on the periphery of planetary systems, encompassing the region occupied by planetary orbits, are massive analogues of the Solar system’s Kuiper belt. They are detected by thermal emission of dust released in collisions amongst directly unobservable larger bodies that carry most of the debris disc mass. We estimate the total mass of the discs by extrapolating up the mass of emitting dust with the help of collisional cascade models. The resulting mass of bright debris discs appears to be unrealistically large, exceeding the mass of solids available in the systems at the preceding protoplanetary stage. We discuss this ‘mass problem’ in detail and investigate possible solutions to it. These include uncertainties in the dust opacity and planetesimal strength, variation of the bulk density with size, steepening of the size distribution by damping processes, the role of the unknown ‘collisional age’ of the discs, and dust production in recent giant impacts. While we cannot rule out the possibility that a combination of these might help, we argue that the easiest solution would be to assume that planetesimals in systems with bright debris discs were ‘born small’, with sizes in the kilometre range, especially at large distances from the stars. This conclusion would necessitate revisions to the existing planetesimal formation models, and may have a range of implications for planet formation. We also discuss potential tests to constrain the largest planetesimal sizes and debris disc masses.

Positive multipeak solutions to a zero mass problem in exterior domains

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text], and the nonlinearity [Formula: see text] is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as [Formula: see text].