Magnetar powered GRBs: Explaining the extended emission and X-ray plateau of short GRB light curves

Extended emission (EE) is a high-energy, early time rebrightening sometimes seen in the light curves of short gamma-ray bursts (GRBs). We present the first contiguous fits to the EE tail and the later X-ray plateau, unified within a single model. Our central engine is a magnetar surrounded by a fall-back accretion disc, formed by either the merger of two compact objects or the accretion-induced collapse of a white dwarf. During the EE phase, material is accelerated to super-Keplarian velocities and ejected from the system by the rapidly rotating ($P \approx 1 - 10$ ms) and very strong ($10^{15}$ G) magnetic field in a process known as magnetic propellering. The X-ray plateau is modelled as magnetic dipole spin-down emission. We first explore the range of GRB phenomena that the propeller could potentially reproduce, using a series of template light curves to devise a classification scheme based on phenomology. We then obtain fits to the light curves of 9 GRBs with EE, simultaneously fitting both the propeller and the magnetic dipole spin-down and finding typical disc masses of a few $10^{-3}$ $M_{\odot}$ to a few $10^{-2}$ $M_{\odot}$. This is done for ballistic, viscous disc and exponential accretion rates. We find that the conversion efficiency from kinetic energy to EM emission for propellered material needs to be $\gtrsim 10\%$ and that the best fitting results come from an exponential accretion profile.Comment: 13 pages, 5 figures, accepted to MNRA

Broadband modelling of short gamma-ray bursts with energy injection from magnetar spin-down and its implications for radio detectability

The magnetar model has been proposed to explain the apparent energy injection in the X-ray light curves of short gamma-ray bursts (SGRBs), but its implications across the full broadband spectrum are not well explored. We investigate the broadband modelling of four SGRBs with evidence for energy injection in their X-ray light curves, applying a physically motivated model in which a newly formed magnetar injects energy into a forward shock as it loses angular momentum along open field lines. By performing an order of magnitude search for the underlying physical parameters in the blast wave, we constrain the characteristic break frequencies of the synchrotron spectrum against their manifestations in the available multi-wavelength observations for each burst. The application of the magnetar energy injection profile restricts the successful matches to a limited family of models that are self-consistent within the magnetic dipole spin-down framework.We produce synthetic light curves that describe how the radio signatures of these SGRBs ought to have looked given the restrictions imposed by the available data, and discuss the detectability of these signatures with present-day and near-future radio telescopes. Our results show that both the Atacama Large Millimetre Array and the upgraded Very Large Array are now sensitive enough to detect the radio signature within two weeks of trigger in most SGRBs, assuming our sample is representative of the population as a whole. We also find that the upcoming Square Kilometre Array will be sensitive to depths greater than those of our lower limit predictions.Comment: 15 pages, 4 figures, 6 tables, accepted for publication in MNRA

Constraining properties of GRB magnetar central engines using the observed plateau luminosity and duration correlation

An intrinsic correlation has been identified between the luminosity and duration of plateaus in the X-ray afterglows of Gamma-Ray Bursts (GRBs; Dainotti et al. 2008), suggesting a central engine origin. The magnetar central engine model predicts an observable plateau phase, with plateau durations and luminosities being determined by the magnetic fields and spin periods of the newly formed magnetar. This paper analytically shows that the magnetar central engine model can explain, within the 1$\sigma$ uncertainties, the correlation between plateau luminosity and duration. The observed scatter in the correlation most likely originates in the spread of initial spin periods of the newly formed magnetar and provides an estimate of the maximum spin period of ~35 ms (assuming a constant mass, efficiency and beaming across the GRB sample). Additionally, by combining the observed data and simulations, we show that the magnetar emission is most likely narrowly beamed and has $\lesssim$20% efficiency in conversion of rotational energy from the magnetar into the observed plateau luminosity. The beaming angles and efficiencies obtained by this method are fully consistent with both predicted and observed values. We find that Short GRBs and Short GRBs with Extended Emission lie on the same correlation but are statistically inconsistent with being drawn from the same distribution as Long GRBs, this is consistent with them having a wider beaming angle than Long GRBs.Comment: MNRAS Accepte

Metastability of life

The physical idea of the natural origin of diseases and deaths has been presented. The fundamental microscopical reason is the destruction of any metastable state by thermal activation of a nucleus of a nonreversable change. On the basis of this idea the quantitative theory of age dependence of death probability has been constructed. The obtained simple Death Laws are very accurately fulfilled almost for all known diseases.Comment: 3 pages, 4 figure

Evidence for the Gompertz Curve in the Income Distribution of Brazil 1978-2005

This work presents an empirical study of the evolution of the personal income distribution in Brazil. Yearly samples available from 1978 to 2005 were studied and evidence was found that the complementary cumulative distribution of personal income for 99% of the economically less favorable population is well represented by a Gompertz curve of the form $G(x)=\exp [\exp (A-Bx)]$, where $x$ is the normalized individual income. The complementary cumulative distribution of the remaining 1% richest part of the population is well represented by a Pareto power law distribution $P(x)= \beta x^{-\alpha}$. This result means that similarly to other countries, Brazil's income distribution is characterized by a well defined two class system. The parameters $A$, $B$, $\alpha$, $\beta$ were determined by a mixture of boundary conditions, normalization and fitting methods for every year in the time span of this study. Since the Gompertz curve is characteristic of growth models, its presence here suggests that these patterns in income distribution could be a consequence of the growth dynamics of the underlying economic system. In addition, we found out that the percentage share of both the Gompertzian and Paretian components relative to the total income shows an approximate cycling pattern with periods of about 4 years and whose maximum and minimum peaks in each component alternate at about every 2 years. This finding suggests that the growth dynamics of Brazil's economic system might possibly follow a Goodwin-type class model dynamics based on the application of the Lotka-Volterra equation to economic growth and cycle.Comment: 22 pages, 15 figures, 4 tables. LaTeX. Accepted for publication in "The European Physical Journal B

Exact Solution of an Evolutionary Model without Ageing

We introduce an age-structured asexual population model containing all the relevant features of evolutionary ageing theories. Beneficial as well as deleterious mutations, heredity and arbitrary fecundity are present and managed by natural selection. An exact solution without ageing is found. We show that fertility is associated with generalized forms of the Fibonacci sequence, while mutations and natural selection are merged into an integral equation which is solved by Fourier series. Average survival probabilities and Malthusian growth exponents are calculated indicating that the system may exhibit mutational meltdown. The relevance of the model in the context of fissile reproduction groups as many protozoa and coelenterates is discussed.Comment: LaTeX file, 15 pages, 2 ps figures, to appear in Phys. Rev.

The Heumann-Hotzel model for aging revisited

Since its proposition in 1995, the Heumann-Hotzel model has remained as an obscure model of biological aging. The main arguments used against it were its apparent inability to describe populations with many age intervals and its failure to prevent a population extinction when only deleterious mutations are present. We find that with a simple and minor change in the model these difficulties can be surmounted. Our numerical simulations show a plethora of interesting features: the catastrophic senescence, the Gompertz law and that postponing the reproduction increases the survival probability, as has already been experimentally confirmed for the Drosophila fly.Comment: 11 pages, 5 figures, to be published in Phys. Rev.

Interplay between distribution of live cells and growth dynamics of solid tumours

Experiments show that simple diffusion of nutrients and waste molecules is not sufficient to explain the typical multilayered structure of solid tumours, where an outer rim of proliferating cells surrounds a layer of quiescent but viable cells and a central necrotic region. These experiments challenge models of tumour growth based exclusively on diffusion. Here we propose a model of tumour growth that incorporates the volume dynamics and the distribution of cells within the viable cell rim. The model is suggested by in silico experiments and is validated using in vitro data. The results correlate with in vivo data as well, and the model can be used to support experimental and clinical oncology

When the optimal is not the best: parameter estimation in complex biological models

Background: The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values. Results: We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit. Conclusions: The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system, and point to the need of a theory that addresses this problem more generally