Non-linear diffusive shock acceleration with free escape boundary

We present here a semi-analytical solution of the problem of particle acceleration at non-linear shock waves with a free escape boundary at some location upstream. This solution, besides allowing us to determine the spectrum of particles accelerated at the shock front, including the shape of the cutoff at some maximum momentum, also allows us to determine the spectrum of particles escaping the system from upstream. This latter aspect of the problem is crucial for establishing a connection between the accelerated particles in astrophysical sources, such as supernova remnants, and the cosmic rays observed at the Earth. An excellent approximate solution, which leads to a computationally fast calculation of the structure of shocks with an arbitrary level of cosmic ray modification, is also obtained.Comment: 11 pages, 2 figures, Accepted for publication in APh

The role of forecasts in monetary policy

Forecasts of future economic developments play an important role for the monetary policy decisions of central banks. For example, forecasts of goal variables can help central banks achieve their goals and make them more accountable to the public. There are two primary explanations for the benefits of forecasts. The first is that monetary policy affects goal variables such as inflation and output only with substantial lags. Policy actions should, therefore, be based on forecasts of goal variables at horizons consistent with policy lags and be taken when these forecasts are inconsistent with policy goals. Under such an approach, the quality of a central bank's forecasts and the effectiveness of its actions to bring forecasts into alignment with targets provide a basis for judging the performance of policymakers and for holding them accountable.> The second, and less intuitive, explanation is that by focusing on a forecast of only one variable -- inflation -- a central bank can potentially achieve multiple goals. This approach can be successful even if there are tradeoffs among the various goal variables. For example, the approach can combine a commitment to long-run price stability with concern for the effects of monetary policy on output.> Amato and Laubach argue that the lagged effects of monetary policy make the use of forecasts necessary. They also argue that delegating a single goal—such as inflation stabilization—to the central bank facilitates accountability, but at the risk of not achieving other goals. They then examine how the Eurosystem and the Bank of England, both of which have been assigned a single goal, address the existence of tradeoffs among goals. Finally, the authors provide evidence that a monetary policy aimed primarily at stabilizing inflation forecasts—as practiced by the Bank of England, for example—can, in fact, achieve multiple goals.Forecasting ; Monetary policy ; Banks and banking, Central

The value of interest rate smoothing : how the private sector helps the Federal Reserve

Most central banks conduct monetary policy by setting targets for overnight interest rates. During the 1990s, central banks have tended to move these interest rates in small steps without reversing direction quickly, a practice called interest rate smoothing. For example, the majority of Federal Reserve policy moves in the last decade and a half have come in a sequence of 25 basis point moves, in striking contrast to the early 1980s, when short-term interest rates fluctuated widely. In light of this historical contrast, it is natural to ask whether interest rate smoothing is a desirable way to conduct monetary policy.> Amato and Laubach argue that interest rate smoothing is beneficial because the private sector is forward-looking. The private sector bases its decisions on expectations of the future. Thus, a monetary policy move today will be more effective if it is expected to persist over time. By smoothing interest rates, the size of changes in interest rates required to reduce fluctuations in the economy can be smaller than would otherwise be necessary.Monetary policy ; Interest rates ; Monetary policy - United States

On the escape of particles from cosmic ray modified shocks

Stationary solutions to the problem of particle acceleration at shock waves in the non-linear regime, when the dynamical reaction of the accelerated particles on the shock cannot be neglected, are known to show a prominent energy flux escaping from the shock towards upstream infinity. On physical grounds, the escape of particles from the upstream region of a shock has to be expected in all those situations in which the maximum momentum of accelerated particles, $p_{max}$, decreases with time, as is the case for the Sedov-Taylor phase of expansion of a shell Supernova Remnant, when both the shock velocity and the cosmic ray induced magnetization decrease. In this situation, at each time $t$, particles with momenta larger than $p_{max}(t)$ leave the system from upstream, carrying away a large fraction of the energy if the shock is strongly modified by the presence of cosmic rays. This phenomenon is of crucial importance for explaining the cosmic ray spectrum detected at Earth. In this paper we discuss how this escape flux appears in the different approaches to non-linear diffusive shock acceleration, and especially in the quasi-stationary semi-analytical kinetic ones. We apply our calculations to the Sedov-Taylor phase of a typical supernova remnant, including in a self-consistent way particle acceleration, magnetic field amplification and the dynamical reaction on the shock structure of both particles and fields. Within this framework we calculate the temporal evolution of the maximum energy reached by the accelerated particles and of the escape flux towards upstream infinity. The latter quantity is directly related to the cosmic ray spectrum detected at Earth.Comment: Version accepted for publication in MNRA

Dynamical effects of self-generated magnetic fields in cosmic ray modified shocks

Recent observations of greatly amplified magnetic fields ($\delta B/B\sim 100$) around supernova shocks are consistent with the predictions of the non-linear theory of particle acceleration (NLT), if the field is generated upstream of the shock by cosmic ray induced streaming instability. The high acceleration efficiencies and large shock modifications predicted by NLT need however to be mitigated to confront observations, and this is usually assumed to be accomplished by some form of turbulent heating. We show here that magnetic fields with the strength inferred from observations have an important dynamical role on the shock, and imply a shock modification substantially reduced with respect to the naive unmagnetized case. The effect appears as soon as the pressure in the turbulent magnetic field becomes comparable with the pressure of the thermal gas. The relative importance of this unavoidable effect and of the poorly known turbulent heating is assessed. More specifically we conclude that even in the cases in which turbulent heating may be of some importance, the dynamical reaction of the field cannot be neglected, as instead is usually done in most current calculations.Comment: 4 pages, 1 figure, accepted for publication in ApJ Letter

The contribution of supernova remnants to the galactic cosmic ray spectrum

The supernova paradigm for the origin of galactic cosmic rays has been deeply affected by the development of the non-linear theory of particle acceleration at shock waves. Here we discuss the implications of applying such theory to the calculation of the spectrum of cosmic rays at Earth as accelerated in supernova remnants and propagating in the Galaxy. The spectrum is calculated taking into account the dynamical reaction of the accelerated particles on the shock, the generation of magnetic turbulence which enhances the scattering near the shock, and the dynamical reaction of the amplified field on the plasma. Most important, the spectrum of cosmic rays at Earth is calculated taking into account the flux of particles escaping from upstream during the Sedov-Taylor phase and the adiabatically decompressed particles confined in the expanding shell and escaping at later times. We show how the spectrum obtained in this way is well described by a power law in momentum with spectral index close to -4, despite the concave shape of the instantaneous spectra of accelerated particles. On the other hand we also show how the shape of the spectrum is sensible to details of the acceleration process and environment which are and will probably remain very poorly known.Comment: 19 pages, 8 figures, published version (references updated

Frequency domain analysis of the mean and osculating trajectories of LAGEOS-1

Several crucial developments in orbital mechanics are obtained through the method of averaging, which is equivalent to integrating only the dominant harmonics of the Fourier series of the dynamics. Frequency analysis of the trajectory expressed in orbital elements facilitates an in-depth understanding of Earth satellite dynamics, and facilitates the interpretation of numerical results from special perturbations within the theoretical framework of celestial mechanics. In this work, we examine recent Two-Line Element series for LAGEOS-1 in the frequency domain, highlighting the presence of short-periodic terms that are likely introduced in the generation of TLEs from observations. We also propose employing well-established methods from signal processing, such as spectrograms, to examine the evolution of dissipative systems such as satellites in LEO