Abstract A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution functionf0(r,p) at cylindrical radiusrfrom the jet axis (assumed to lie along thez-axis) is given by convolving the particle momentum spectrum with the Green’s function , which describes the monoenergetic spectrum solution in which asr→ ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for . The flow velocityu=u(r)ezis along the axis of the jet (thez-axis).uis independent ofz, andu(r) is a monotonic decreasing function ofr. The scattering time in the shear flow region 0 <r<r2, and , wheres> 0 in the regionr>r2is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r,p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution that particles observed at (r,p) originated fromr→ ∞ with momentum . The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.
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Half-time range description for the free space wave operator and the spherical means transform
Abstract The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphereS. The datagrepresent the time-dependent values of the pressure on the observation surfaceS. Finding initial pressureffrom the known values ofgconsitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means offwith centers onS. Here we consider a problem of range description of the wave operator mappingfintog. Such a problem was considered before, with datagknown on time interval at least (assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range . However, such data are redundant. We present necessary and sufficient conditions for functiongto be in the range of the wave operator, forggiven on a half-time interval . This also implies range conditions on spherical means measured for the radii in the range .
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- Award ID(s):
- 2405348
- PAR ID:
- 10571961
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 41
- Issue:
- 3
- ISSN:
- 0266-5611
- Format(s):
- Medium: X Size: Article No. 035005
- Size(s):
- Article No. 035005
- Sponsoring Org:
- National Science Foundation
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