Modeling the Longitudinal Asymmetry in Sunspot Emergence -- the Role of the Wilson Depression

The distributions of sunspot longitude at first appearance and at disappearance display an east-west asymmetry that results from a reduction in visibility as one moves from disk centre to the limb. To first order, this is explicable in terms of simple geometrical foreshortening. However, the centre-to-limb visibility variation is much larger than that predicted by foreshortening. Sunspot visibility is also known to be affected by the Wilson effect: the apparent dish shape of the sunspot photosphere caused by the temperature-dependent variation of the geometrical position of the tau=1 layer. In this article we investigate the role of the Wilson effect on the sunspot appearance distributions, deducing a mean depth for the umbral tau=1 layer of 500 to 1500 km. This is based on the comparison of observations of sunspot longitude distribution and Monte Carlo simulations of sunspot appearance using different models for spot growth rate, growth time and depth of Wilson depression.Comment: 18 pages, 10 figures, in press (Solar Physics

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n \gg 1 even, the probability that they have no real root on the full real axis decays like n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like \exp{(-2\theta_{\infty}} \sqrt{n}) and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is such that \theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde \phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and \tilde \phi(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent {-2}. These analytical results are confirmed by detailed numerical computations.Comment: 32 pages, 16 figure

Multiwavelength variability of BL Lacertae measured with high time resolution

In an effort to locate the sites of emission at different frequencies and physical processes causing variability in blazar jets, we have obtained high time-resolution observations of BL Lacertae over a wide wavelength range: with the Transiting Exoplanet Survey Satellite (TESS) at 6000–10000 Å with 2 minute cadence; with the Neil Gehrels Swift satellite at optical, UV, and X-ray bands; with the Nuclear Spectroscopic Telescope Array at hard X-ray bands; with the Fermi Large Area Telescope at γ-ray energies; and with the Whole Earth Blazar Telescope for measurement of the optical flux density and polarization. All light curves are correlated, with similar structure on timescales from hours to days. The shortest timescale of variability at optical frequencies observed with TESS is ~0.5 hr. The most common timescale is 13 ± 1 hr, comparable with the minimum timescale of X-ray variability, 14.5 hr. The multiwavelength variability properties cannot be explained by a change solely in the Doppler factor of the emitting plasma. The polarization behavior implies that there are both ordered and turbulent components to the magnetic field in the jet. Correlation analysis indicates that the X-ray variations lag behind the γ-ray and optical light curves by up to ~0.4 day. The timescales of variability, cross-frequency lags, and polarization properties can be explained by turbulent plasma that is energized by a shock in the jet and subsequently loses energy to synchrotron and inverse Compton radiation in a magnetic field of strength ~3 G.Accepted manuscrip