Continuity in Morality and Law

According to an influential and intuitively appealing argument, morality is usually continuous, namely, a gradual change in one morally significant factor triggers a gradual change in another; the law should usually track morality; therefore, the law should often be continuous. This argument is illustrated by cases such as the following example: since the moral difference between a defensive action that is reasonable and one that is just short of being reasonable is small, the law should not impose a severe punishment when the action is almost reasonable and no punishment at all when the action is reasonable. In this Article, I consider two doubts regarding this argument. First, the premise that morality is continuous in such cases is incompatible with the common view that the moral status of actions is not continuous since there is an important difference between actions that are permissible and actions that are wrong— even if this difference is due to a difference that is very small, such as the one between an action whose consequences are the best and an action whose consequences are just slightly less good. This view extends also to the overall moral status of agents given the common assumption that it depends on the moral status of their actions. This is an important challenge that the Continuity Argument should confront. However, I argue that the best account of morality is more scalar than the common view in these respects. Therefore, I conclude that the first premise of the Continuity Argument is correct in this regard, although it is based on a minority view. The second doubt concerns the scope of the second premise: since there are reasons both in favor and against legal continuity, and the applicability and force of these reasons depend not only on various moral propositions but also on contingent non-moral facts, we often lack the evidence to determine the degree to which the law, at a certain place and time, should be continuous, and specifically that it should often be continuous

On The Synchrotron Self-Compton Emission from Relativistic Shocks and Its Implications for Gamma-Ray Burst Afterglows

We consider the effects of inverse Compton scattering of synchrotron photons from relativistic electrons in GRB afterglows. We compute the spectrum of the inverse Compton emission and find that it can dominate the total cooling rate of the afterglow for several months or even years after the initial explosion. We demonstrate that the presence of strong inverse Compton cooling can be deduced from the effect it has on the time-evolution of the cooling break in the synchrotron spectral component, and therefore on the optical and X-ray afterglow lightcurves. We then show how the physical interpretation of the observed characteristics of the synchrotron spectrum must be modified to take into consideration this extra source of cooling, and give a revised prescription for computing physical parameters characterizing the expanding shock wave from the observed quantities. We find that for a given set of observables (synchrotron break frequencies and fluxes) there is either no consistent physical interpretation or two of them. Finally we discuss the prospects of directly detecting the inverse Compton emission with Chandra. We argue that such a detection is possible for GRBs exploding in a reasonably dense (n>1 cm^-3) medium.Comment: 21 pages, ApJ submitte

GeV Emission from Prompt and Afterglow Phases of Gamma-Ray Bursts

We investigate the GeV emission from gamma-ray bursts (GRBs), using the results from the Energetic Gamma Ray Experimental Telescope (EGRET), and in view of the Gamma-ray Large Area Space Telescope (GLAST). Assuming that the conventional prompt and afterglow photons originate from synchrotron radiation, we compare an accompanying inverse-Compton component with EGRET measurements and upper limits on GeV fluence, taking Klein-Nishina feedback into account. We find that EGRET constraints are consistent with the theoretical framework of the synchrotron self-Compton model for both prompt and afterglow phases, and discuss constraints on microphysical parameters in both phases. Based on the inverse-Compton model and using EGRET results, we predict that GLAST would detect GRBs with GeV photons at a rate >~20 yr^{-1} from each of the prompt and afterglow phases. This rate applies to the high-energy tail of the prompt synchrotron emission and to the inverse-Compton component of the afterglow. Theory predicts that in a large fraction of the cases where synchrotron GeV prompt emission would be detected by GLAST, inverse-Compton photons should be detected as well at high energies >~10 GeV. Therefore GLAST will enable a more precise test of the high-energy emission mechanism. Finally, we show that the contribution of GRBs to the flux of the extragalactic gamma-ray background measured with EGRET is at least 0.01% and likely around 0.1%.Comment: 11 pages, 7 figures; accepted by Ap

Jets in GRBs: Tests and Predictions for the Structured Jet Model

The two leading interpretations of achromatic breaks that are observed in the light curves of GRBs afterglow are (i) the manifestation of the edge of a jet, which has a roughly uniform energy profile and a sharp edge and (ii) a line of sight effect in jets with a variable energy profile. The first scenario requires the inner engine to produce a jet with a different opening angle each explosion, while the latter requires a standard engine. The physical structure of the jet is a crucial factor in understanding GRB progenitors, and therefore discriminating the two jet scenarios is particularly relevant. In the structured jet case, specific predictions can be made for the distribution of observed break angles $\theta_{\rm break}$, while that distribution is arbitrary in the first scenario. We derive the theoretical distribution for the structured jet model. Specifically, we predict the most common angle to be about 0.12 rad, in rough agreement with the sample. If this agreement would hold as the sample size increases, it would strengthen the case for the standard jet hypothesis. We show that a prediction of this model is that the average viewing angle is an increasing function of the survey sensitivity, and in particular that a mission like {\em Swift} will find the typical viewing angle to be about 0.3 rad. The local event rate predicted by this model is $R_{\rm GRB}(z=0)\sim 0.5$ Gpc$^{-3}$ yr$^{-1}$.Comment: 14 pages, 3 figures; accepted to Ap

Discrete Self-Similarity in Type-II Strong Explosions

We present new solutions to the strong explosion problem in a non-power law density profile. The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-\omega}$, where $\omega>3$ (Type-II solutions). The perturbations we consider are spherically symmetric and log-periodic with respect to the radius. While the unperturbed solutions are continuously self-similar, the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations, i.e. the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. Finally we show that this method can be generalized to treat any small, spherically symmetric density perturbation by employing Fourier decomposition

The prompt energy release of gamma-ray bursts using a cosmological k-correction

The fluences of gamma-ray bursts (GRBs) are measured with a variety of instruments in different detector energy ranges. A detailed comparison of the implied energy releases of the GRB sample requires, then, an accurate accounting of this diversity in fluence measurements which properly corrects for the redshifting of GRB spectra. Here, we develop a methodology to ``k-correct'' the implied prompt energy release of a GRB to a fixed co-moving bandpass. This allows us to homogenize the prompt energy release of 17 cosmological GRBs (using published redshifts, fluences, and spectra) to two common co-moving bandpasses: 20-2000 keV and 0.1 keV-10 MeV (``bolometric''). While the overall distribution of GRB energy releases does not change significantly by using a k-correction, we show that uncorrected energy estimates systematically undercounts the bolometric energy by ~5% to 600%, depending on the particular GRB. We find that the median bolometric isotropic-equivalent prompt energy release is 2.2 x 10^{53} erg with an r.m.s. scatter of 0.80 dex. The typical estimated uncertainty on a given k-corrected energy measurement is ~20%.Comment: Accepted to the Astronomical Journal. 21 pages (LaTeX) and 4 figure

Formation of Kuiper Belt Binaries

The discovery that a substantial fraction of Kuiper Belt objects (KBOs) exists in binaries with wide separations and roughly equal masses, has motivated a variety of new theories explaining their formation. Goldreich et al. (2002) proposed two formation scenarios: In the first, a transient binary is formed, which becomes bound with the aid of dynamical friction from the sea of small bodies (L^2s mechanism); in the second, a binary is formed by three body gravitational deflection (L^3 mechanism). Here, we accurately calculate the L^2s and L^3 formation rates for sub-Hill velocities. While the L^2s formation rate is close to previous order of magnitude estimates, the L^3 formation rate is about a factor of 4 smaller. For sub-Hill KBO velocities (v << v_H) the ratio of the L^3 to the L^2s formation rate is 0.05 (v/v_H) independent of the small bodies' velocity dispersion, their surface density or their mutual collisions. For Super-Hill velocities (v >> v_H) the L^3 mechanism dominates over the L^2s mechanism. Binary formation via the L^3 mechanism competes with binary destruction by passing bodies. Given sufficient time, a statistical equilibrium abundance of binaries forms. We show that the frequency of long-lived transient binaries drops exponentially with the system's lifetime and that such transient binaries are not important for binary formation via the L^3 mechanism, contrary to Lee et al. (2007). For the L^2s mechanism we find that the typical time, transient binaries must last, to form Kuiper Belt binaries (KBBs) for a given strength of dynamical friction, D, increases only logarithmically with D. Longevity of transient binaries only becomes important for very weak dynamical friction (i.e. D \lesssim 0.002) and is most likely not crucial for KBB formation.Comment: 20 pages, 3 figures, Accepted for publication in ApJ, correction of minor typo

Eccentricity Evolution for Planets in Gaseous Disks

We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances which are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances which overlap the planet's orbit. If the planet clears a gap in the disk, the rate of eccentricity damping by co-orbital Lindblad resonances is reduced. Density gradients associated with the gap activate eccentricity damping by corotation resonances at a rate which initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque drives a mass flux which reduces the density gradient near the resonance. Sufficient partial saturation of corotation resonances can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of a few percent is required to overcome viscous diffusion which acts to unsaturate corotation resonances by reestablishing the large scale density gradient. Thus eccentricity growth is a finite amplitude instability. Formally, interactions at the apsidal resonance, which is a special kind of co-orbital Lindblad resonance, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, apsidal waves have such long wavelengths that they do not propagate in protoplanetary disks. This reduces eccentricity damping by the apsidal resonance to a modest level.Comment: Submitted to Ap

Sustainable Transitional Justice Policies in Africa: A Searchlight on Pan African Youth Organizations

The teeming literature on transitional justice policies and peacebuilding suggests that regional organizations are veritable platforms for advancing transitional justice policies for attaining a relatively peaceful and justice-compliant society. The recognition of institutionalized transitional justice processes and the growing emphasis on the role of regional organizations has led to the undervaluation of non-traditional regional organizations. This paper examines the utility of such non-traditional regional bodies such as the Pan African Youth Union and Afrika Youth Movement in promoting effective transitional justice policies that will culminate in the creation of a culture of peace amongst African youth. The study analyzes and compares the history, structure, practices, and projects of these organizations to demonstrate a movement away from traditional liberal institutionalism in peacebuilding processes.Keywords: Transitional justice, policies, Pan African and Youth Organization

Composite self-similar solutions for relativistic shocks: the transition to cold fluid temperatures

The flow resulting from a strong ultrarelativistic shock moving through a stellar envelope with a polytrope-like density profile has been studied analytically and numerically at early times while the fluid temperature is relativistic--that is, just before and just after the shock breaks out of the star. Such a flow should expand and accelerate as its internal energy is converted to bulk kinetic energy; at late enough times, the assumption of relativistic temperatures becomes invalid. Here we present a new self-similar solution for the post-breakout flow when the accelerating fluid has bulk kinetic Lorentz factors much larger than unity but is cooling through $p/n$ of order unity to subrelativistic temperatures. This solution gives a relation between a fluid element's terminal Lorentz factor and that element's Lorentz factor just after it is shocked. Our numerical integrations agree well with the solution. While our solution assumes a planar flow, we show that corrections due to spherical geometry are important only for extremely fast ejecta originating in a region very close to the stellar surface. This region grows if the shock becomes relativistic deeper in the star.Comment: 25 pages, 8 figures; submitted to Physics of Fluid