Small-N collisional dynamics – V. From N ≲ 10 to N ≳ 103

ABSTRACT Direct collisions between finite-sized particles occur commonly in many areas of astrophysics. Such collisions are typically mediated by chaotic, bound gravitational interactions involving small numbers of particles. An important application is stellar collisions, which occur commonly in dense star clusters, and their relevance for the formation of various types of stellar exotica. In this paper, we return to our study of the collision rates and probabilities during small-number chaotic gravitational interactions ($N\, \lesssim$ 10), moving beyond the small-number particle limit and into the realm of larger particle numbers ($N\, \gtrsim$ 103) to test the extent of validity of our analytic model as a function of the particle properties and the number of interacting particles. This is done using direct N-body simulations of stellar collisions in dense star clusters, by varying the relative numbers of particles with different particle masses and radii. We compute the predicted rate of collisions using the mean free path approximation, adopting the point-particle limit and using the sticky-star approximation as our collision criterion. We evaluate its efficacy in the regime where gravitational focusing is important by comparing the theoretical rates to numerical simulations. Using the tools developed in previous papers in this series, in particular Collision Rate Diagrams, we illustrate that our predicted and simulated rates are in excellent agreement, typically consistent with each other to within 1 standard deviation.

Maritime Autonomous Surface Ship’s Path Approximation Using Bézier Curves

This work is devoted to the second order rational Bézier curve coefficients estimation. We present the methodology of unique coefficients for each type of ship computation. In the presented formulas of ship’s length, a draft and angular path combined with a drift path are used. This approach leads to the simplest and most accurate Maritime Autonomous Surface Ships (MASS) path modeling. Three rational curve control points are waypoints (WPT). Using WPTs as curve control points allows integrating a trajectory intuitive for the navigator with a path predicting model used as a reference in the control system. Research was done based on real-time data originating from the MASS autonomous trajectory tracking system. The presented mathematical modeling tool may be treated as the best way of future trajectory prediction due to low computation power required.

Studying temperature dependence of pairing gap parameter in a nucleus as a small superconducting system

In this paper, we have taken the effect of small size of nucleus and static fluctuations into account in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity calculations of [Formula: see text]Ti nucleus. Thermodynamic quantities of [Formula: see text]Ti have been extracted within the BCS model with the inclusion of the average value of the pairing gap square, extracted by the modified Ginzburg–Landau (MGL) method for small systems. Calculated values of the excitation energy and entropy within the MGL+BCS method improve the extracted results within the usual BCS model and show a smooth behavior around the critical temperature with a very good agreement with the semi-empirical values. The result of using MGL+BCS method for the heat capacity of [Formula: see text]Ti is compared with the corresponding semi-empirical values and the calculated values within the BCS, static path approximation (SPA) and Modified Pairing gap BCS (MPBCS) which is a method that was proposed in our previous publications. Both MGL+BCS and MPBCS avoid the discontinuity of the heat capacity curve, which is observed in the usual BCS method, and lead to an S-shaped curve with a good agreement with the semi-empirical results.

Critical Path Statistics of Max-Plus Linear Systems with Gaussian Noise

The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.