Conditions for jet breakout in neutron stars’ mergers

ABSTRACT We consider conditions for jet breakout through ejecta following mergers of neutron stars and provide simple relations for the breakout conditions. We demonstrate that: (i) break-out requires that the isotropic-equivalent jet energy Ej exceeds the ejecta energy Eej by Ej ≥ Eej/βej, where βej = Vej/c, Vej is the maximum velocity of the ejecta. If the central engine terminates before the breakout, the shock approaches the edge of the ejecta slowly ∝ 1/t; late breakout occurs only if at the termination moment the head of the jet was relatively close to the edge. (ii) If there is a substantial delay between the ejecta’s and the jet’s launching, the requirement on the jet power increases. (iii) The forward shock driven by the jet is mildly strong, with Mach number M ≈ 5/4 (increasing with time delay td); (iii) the delay time td between the ejecta and the jet’s launching is important for $t_\mathrm{ d} \gt t_0= ({3 }/{16}) {c M_{\mathrm{ ej}} V_{\mathrm{ ej}}}/{L_\mathrm{ j}} = 1.01 {\rm \mathrm{ s}} M_{\mathrm{ ej}, -2} L_{\mathrm{ j}, 51} ^{-1} \left({\beta _{\mathrm{ ej}}} /{0.3} \right)$, where Mej is ejecta mass, Lj is the jet luminosity (isotropic equivalent). For small delays, t0 is also an estimate of the break-out time.

Clinical Utility of Two- and Three-Dimensional Visuoconstructional Tasks in Mild Cognitive Impairment and Early Alzheimer's Disease.

Objective The aim of this study was to investigate whether different types of visuoconstructional abilities are useful to distinguish individuals with mild cognitive impairment (MCI) and Alzheimer’s disease (AD) from healthy controls (HCs). Method We selected 20 patients with MCI and 14 with AD diagnosis based on standard criteria. The neuropsychological performance of MCI and AD groups were compared with that of a group of 11 HCs using a standard neuropsychological battery and visuoconstructional tasks that differed difficulty and type of implicated skills (graphomotor vs. non-graphomotor): two-dimensional (Clock Drawing Test, CDT; Block Design, BD; and Visual Puzzles, VP) and three-dimensional Block Construction (TBC). Results AD group scored significantly lower than HCs in BD, VP and TBC tasks, but no significant differences were found between HCs and MCI. CDT (copy condition) scores did not differ significantly among the groups. The receiver operating characteristic analysis showed that BD [sensitivity (se) = .85, specificity (sp) = .90, Youden index (J) = .76], VP (se = .78 and sp = .72, J = .51) and TBC (se = .71, sp = 100, J = .71) were accurate tasks to discriminate between AD and HCs. Moreover, BD tasks (se = .85, sp = .70, J = .55) and TBC (se = .71, sp = .80, J = .51) showed fair accuracy to differentiate between MCI and AD groups. Conclusions These findings indicate that non-graphomotor visuoconstructional tasks are already impaired in the early stages of AD, but are preserved in MCI individuals when compared with HCs.