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Tracing magnetic field-lines of the Earth's magnetosphere using beams of relativistic electrons will open up new insights into space weather and magnetospheric physics. Analytic models and a single-particle-motion code were used to explore the dynamics of an electron beam emitted from an orbiting satellite and propagating until impact with the Earth. The impact location of the beam on the upper atmosphere is strongly influenced by magnetospheric conditions, shifting up to several-degrees in latitude between different phases of a simulated storm. The beam density cross-section evolves due to cyclotron motion of the beam centroid and oscillations of the beam envelope. The impact density profile is ring shaped, with major radius $\sim 22$ meters, given by the final cyclotron radius of the beam centroid, and ring thickness $\sim 2$ meters given by the final beam envelope. Motion of the satellite may also act to spread the beam, however it will remain sufficiently focused for detection by ground-based optical and radio detectors. An array of such ground stations will be able to detect shifts in impact location of the beam, and thereby infer information regarding magnetospheric conditions.

The Sun's connection with the Earth's magnetic field and atmosphere is carried out through the exchange of electromagnetic and mass flux and is regulated by a complex interconnection of processes. During space weather events, solar flares, or fast streams of solar atmosphere strongly disturb the Earth's environment. Often the electric currents that connect the different parts of the Sun-Earth system become unstable and explosively release the stored electromagnetic energy in one of the more dramatic expressions of space weather—the geomagnetic storm and substorm. Some aspects of the magnetosphere-ionosphere connection that generates auroral arcs during space weather events are well-known. However, several fundamental problems remain unsolved because of the lack of unambiguous identification of the magnetic field connection between the magnetosphere and the ionosphere. The correct mapping between different regions of the magnetosphere and their foot-points in the ionosphere, coupled with appropriate distributed measurements of plasma and fields in focused regions of the magnetosphere, is necessary to establish unambiguously that a given magnetospheric process is the generator of an observed arc. We present a new paradigm that should enable the resolution of the mapping ambiguities. The paradigm calls for the application of energetic electron beams as magnetic field tracers. The three most important problems for which the correct magnetic field mapping would provide closure to are the substorm growth phase arcs, the expansion phase onset arcs and the system of arcs that emerge from the magnetosphere-ionosphere connection during the development of the early substorm expansion phase phenomenon known as substorm current wedge (SCW). In this communication we describe how beam tracers, in combination with distributed measurements in the magnetosphere, can be used to disentangle the mechanisms that generate these critical substorm phenomena. Since the application of beams as tracers require demonstration that the beams can be injected into the loss cone, that the spacecraft potentials induced by the beam emission are manageable, and that sufficient electron flux reaches the atmosphere to be detectable by optical or radio means after the beam has propagated thousands of kilometers under competing effects of beam spread and constriction as well as effects of beam-induced instabilities, in this communication we review how these challenges are currently being addressed and discuss the next steps toward the realization of active experiments in space using relativistic electron beams.

Using electron beam accelerators attached to satellites in Earth orbit, it may be possible to measure arc length and curvature of field-lines in the inner magnetosphere if the accelerator is designed with the capability to vary the beam energy. In combination with additional information, these measurements would be very useful in modeling the magnetic field of the inner magnetosphere. For this purpose, a three step data assimilation modeling approach is discussed. The first step in the procedure would be to use prior information to obtain an initial forecast of the inner magnetosphere. Then, a family of curves would be defined that satisfies the observed geometric attributes measured by the experiments, and the prior forecast would then be used to optimize the curve with respect to the allowed degrees of freedom. Finally, this approximation of the field-line would be used to improve the initial forecast of the inner magnetosphere, resulting in a description of the system that is optimally consistent with both the prior information and the measured curvature and arc length. This article details the method by which a family of possible approximations of the field-line may be defined via a numerical procedure, which is central to the three step approach. This method serves effectively as a pre-conditioner for parameter estimation problems using field-line curvature and arc length measurements in combination with other measurements.

Magnetosphere-ionosphere coupling is a particularly important process that regulates and controls magnetospheric dynamics such as storms and substorms. However, in order to understand magnetosphere-ionosphere coupling it is necessary to understand how regions of the magnetosphere are connected to the ionosphere. It has been proposed that this connection may be established by firing electron beams from satellites that can reach an ionospheric footpoint creating detectable emissions. This type of experiment would greatly aid in identifying the relationship between convection processes in the magnetotail and the ionosphere and how the plasma sheet current layer evolves during the growth phase preceding substorms. For practical purposes, the use of relativistic electron beams with kinetic energy on the order of 1 MeV would be ideal for detectability. However, Porazik et al. \cite{Porazik2014} has shown that, for relativistic particles, higher order terms of the magnetic moment are necessary for consideration of the ionospheric accessibility of the beams. These higher order terms are related to gradients in the magnetic field and curvature and are typically unimportant unless the beam is injected along the magnetic field direction, such that the zero order magnetic moment is small. In this article, we address two important consequences related to these higher order terms. First, we investigate the consequences for satellites positioned in regions subject to magnetotail stretching and demonstrate systematically how curvature affects accessibility. We find that curvature can reduce accessibility for beams injected from the current sheet, but can increase accessibility for beams injected just above the current sheet. Second, we investigate how detectability of ionospheric precipitation of variable energy field-aligned electron beams could be used as a constraint on field-line curvature, which would be valuable for field-line reconstruction and/or stability analysis.