Functional equations and the Cauchy mean value theorem

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy mean value theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.

Local Interstellar Medium: Six Years of Direct Sampling by IBEX

The Interstellar Boundary Explorer (IBEX) has been directly observing neutral atoms from the local interstellar medium for the last six years (2009–2014). This paper ties together the 14 studies in this Astrophysical Journal Supplement Series Special Issue, which collectively describe the IBEX interstellar neutral results from this epoch and provide a number of other relevant theoretical and observational results. Interstellar neutrals interact with each other and with the ionized portion of the interstellar population in the “pristine” interstellar medium ahead of the heliosphere. Then, in the heliosphereʼs close vicinity, the interstellar medium begins to interact with escaping heliospheric neutrals. In this study, we compare the results from two major analysis approaches led by IBEX groups in New Hampshire and Warsaw. We also directly address the question of the distance upstream to the pristine interstellar medium and adjust both sets of results to a common distance of ~1000 AU. The two analysis approaches are quite different, but yield fully consistent measurements of the interstellar He flow properties, further validating our findings. While detailed error bars are given for both approaches, we recommend that for most purposes, the community use “working values” of ~25.4 km s⁻¹, ~75°7 ecliptic inflow longitude, ~−5°1 ecliptic inflow latitude, and ~7500 K temperature at ~1000 AU upstream. Finally, we briefly address future opportunities for even better interstellar neutral observations to be provided by the Interstellar Mapping and Acceleration Probe mission, which was recommended as the next major Heliophysics mission by the NRCʼs 2013 Decadal Survey.

Selective sampling during catastrophic disruption: Mapping the location of reaccumulated fragments in the original parent body

In this paper, we simulate numerically the catastrophic disruption of a large asteroid as a result of a collision with a smaller projectile and the subsequent reaccumulation of fragments as a result of their mutual gravitational attractions. We then investigate the original location within the parent body of the small pieces that eventually reaccumulate to form the largest offspring of the disruption as a function of the internal structure of the parent body. We consider four cases that may represent the internal structure of such a body (whose diameter is fixed at 250 km) in various early stages of the Solar System evolution: fully molten, half molten (i.e., a 26 km-deep outer layer of melt containing half of the mass), solid except a thin molten layer (8 km thick) centered at 10 km depth, and fully solid. The solid material has properties of basalt. We then focus on the three largest offspring that have enough reaccumulated pieces to consider. Our results indicate that the particles that eventually reaccumulate to form the largest reaccumulated bodies retain a memory of their original locations in the parent body. Most particles in each reaccumulated body are clustered from the same original region, even if their reaccumulations take place far away. The extent of the original region varies considerably depending on the internal structure of the parent. It seems to shrink with the solidity of the body. The fraction of particles coming from a given depth is computed for the four cases, which can give constraints on the internal structure of parent bodies of some meteorites. As one example, we consider the ureilites, which in some petrogenetic models are inferred to have formed at particular depths within their parent body. (C) 2014 Elsevier Ltd. All rights reserved.