Phase Transition of Degeneracy in Minor-Closed Families

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1$ for any sequence $(G_n)_{n \in \mathbb{N}}$ over ${\mathcal G}$ with $\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty$, where $G_n(p(n))$ is the random subgraph of $G_n$ such that each edge remains independently with probability $p(n)$. In this paper we study the upper threshold for the family of $H$-minor free graphs and for the graph property of being $(r-1)$-degenerate, which is one fundamental graph property with many applications. Even a constant factor approximation for the upper threshold for all pairs $(r,H)$ is expected to be very difficult by its close connection to a major open question in extremal graph theory. We determine asymptotically the thresholds (up to a constant factor) for being $(r-1)$-degenerate for a large class of pairs $(r,H)$, including all graphs $H$ of minimum degree at least $r$ and all graphs $H$ with no vertex-cover of size at most $r$, and provide lower bounds for the rest of the pairs of $(r,H)$. The results generalize to arbitrary proper minor-closed families and the properties of being $r$-colorable, being $r$-choosable, or containing an $r$-regular subgraph, respectively

The [Ne III] Jet of DG Tau and its Ionization Scenarios

Forbidden neon emission from jets of low-mass young stars can be used to probe the underlying high-energy processes in these systems. We analyze spectra of the jet of DG Tau obtained with the Very Large Telescope/X-Shooter spectrograph in 2010. [Ne III] $\lambda$3869 is clearly detected in the innermost 3" microjet and the outer knot located at $\sim$6".5. The velocity structure of the inner microjet can be decomposed into the low-velocity component (LVC) at $\sim -70$ km/s and the high-velocity component (HVC) at $\sim -180$ km/s. Based on the observed [Ne III] flux and its spatial extent, we suggest the origins of the [Ne III] emission regions and their relation with known X-ray sources along the jet. The flares from the hard X-ray source close to the star may be the main ionization source of the innermost microjet. The fainter soft X-ray source at 0".2 from the star may provide sufficient heating to help to sustain the ionization fraction against the recombination in the flow. The outer knot may be reionized by shocks faster than 100 km/s such that [Ne III] emission reappears and that the soft X-ray emission at 5".5 is produced. Velocity decomposition of the archival Hubble Space Telescope spectra obtained in 1999 shows that the HVC had been faster, with a velocity centroid of $\sim -260$ km/s. Such a decrease in velocity may potentially be explained by the expansion of the stellar magnetosphere, changing the truncation radius and thus the launching speed of the jet. The energy released by magnetic reconnections during relaxation of the transition can heat the gas up to several tens of megakelvin and provide the explanation for on-source keV X-ray flares that ionize the neon microjet

Velocity-Resolved [Ne III] from X-Ray Irradiated Sz 102 Microjets

Neon emission lines are good indicators of high-excitation regions close to a young stellar system because of their high ionization potentials and large critical densities. We have discovered [Ne III]{\lambda}3869 emission from the microjets of Sz 102, a low-mass young star in Lupus III. Spectroastrometric analyses of two-dimensional [Ne III] spectra obtained from archival high-dispersion ($R\approx 33,000$) Very Large Telescope/UVES data suggest that the emission consists of two velocity components spatially separated by ~ 0."3, or a projected distance of ~ 60 AU. The stronger redshifted component is centered at ~ +21 km/s with a line width of ~ 140 km/s, and the weaker blueshifted component at ~ -90 km/s with a line width of ~ 190 km/s. The two components trace velocity centroids of the known microjets and show large line widths that extend across the systemic velocity, suggesting their potential origins in wide-angle winds that may eventually collimate into jets. Optical line ratios indicate that the microjets are hot ($T\lesssim1.6\times10^4$ K) and ionized ($n_e\gtrsim5.7\times10^4$ cm$^{-3}$). The blueshifted component has ~ 13% higher temperature and ~ 46% higher electron density than the redshifted counterpart, forming a system of asymmetric pair of jets. The detection of the [Ne III]{\lambda}3869 line with the distinct velocity profile suggests that the emission originates in flows that may have been strongly ionized by deeply embedded hard X-ray sources, most likely generated by magnetic processes. The discovery of [Ne III]{\lambda}3869 emission along with other optical forbidden lines from Sz 102 support the picture of wide-angle winds surrounding magnetic loops in the close vicinity of the young star. Future high sensitivity X-ray imaging and high angular-resolution optical spectroscopy may help confirm the picture proposed.Comment: 33 pages, 9 figures, 2 tables; accepted for publication in the ApJ (minor typo and reference list fixed

A Unified Model for Bipolar Outflows from Young Stars: Apparent Magnetic Jet Acceleration

We explore a new, efficient mechanism that can power toroidally magnetized jets up to two to three times their original terminal velocity after they enter a self-similar phase of magnetic acceleration. Underneath the elongated outflow lobe formed by a magnetized bubble, a wide-angle free wind, through the interplay with its ambient toroid, is compressed and accelerated around its axial jet. The extremely magnetic bubble can inflate over its original size, depending on the initial Alfv\'en Mach number $M_A$ of the launched flow. The shape-independent slope $\partial{}v_r/\partial{}r=2/3t$ is a salient feature of the self-similarity in the acceleration phase. Peculiar kinematic signatures are observable in the position--velocity (PV) diagrams and can combine with other morphological signatures as probes for the density-collimated jets arising in toroidally dominated magnetized winds. The apparent second acceleration is powered by the decrease of the toroidal magnetic field but operates far beyond the scales of the primary magnetocentrifugal launch region and the free asymptotic terminal state. Rich implications may connect the jets arising from the youngest protostellar outflows such as HH 211 and HH 212 and similar systems with parsec-scale jets across the mass and evolutionary spectra.Comment: 21 pages, 8 figures, 1 table, to appear in Astrophysical Journal Letters (2023

Method for Thermo-optic Analysis in a Star Sensor

An autonomous star sensor is a highly accurate attitude-measuring instrument used in spacecraft, and its performance is restricted by ambient temperature of the outer space. This paper puts forward an effective scheme to the thermooptic analysis using finite element analysis (FEA) and ray tracing in star sensor. Specific difficulties: (a) how to evaluate thermo-optic effect in star sensor, and (b) how to make FEA results useful in optical design mode have been resolved using the scheme. Based on this scheme, the errors of star sensor, which are caused by thermo-optic effects, can be investigated in any complicated temperature condition, and the required temperature scope for the thermal design can be achieved. For example, the errors of the star sensor were 0.0863" and 2.2933", when the temperature differences of the experimental optical system were 10 °C and 5 °C in axial and lateral, respectively.Defence Science Journal, 2010, 60(3), pp.276-281, DOI:http://dx.doi.org/10.14429/dsj.60.35