The Automated Radiation Measurements for Aerospace Safety (ARMAS) program has successfully deployed a fleet of six instruments measuring the ambient radiation environment at commercial aircraft altitudes. ARMAS transmits real‐time data to the ground and provides quality, tissue‐relevant ambient dose equivalent rates with 5 min latency for dose rates on 213 flights up to 17.3 km (56,700 ft). We show five cases from different aircraft; the source particles are dominated by galactic cosmic rays but include particle fluxes for minor radiation periods and geomagnetically disturbed conditions. The measurements from 2013 to 2016 do not cover a period of time to quantify galactic cosmic rays' dependence on solar cycle variation and their effect on aviation radiation. However, we report on small radiation “clouds” in specific magnetic latitude regions and note that active geomagnetic, variable space weather conditions may sufficiently modify the magnetospheric magnetic field that can enhance the radiation environment, particularly at high altitudes and middle to high latitudes. When there is no significant space weather, high‐latitude flights produce a dose rate analogous to a chest X‐ray every 12.5 h, every 25 h for midlatitudes, and every 100 h for equatorial latitudes at typical commercial flight altitudes of 37,000 ft (~11 km). The dose rate doubles every 2 km altitude increase, suggesting a radiation event management strategy for pilots or air traffic control; i.e., where event‐driven radiation regions can be identified, they can be treated like volcanic ash clouds to achieve radiation safety goals with slightly lower flight altitudes or more equatorial flight paths.

The Nowcast of Atmospheric Ionizing Radiation for Aviation Safety climatological model and the Automated Radiation Measurements for Aerospace Safety (ARMAS) statistical database are presented as polynomial fit equations. Using equations based on altitude, L shell, and geomagnetic conditions an effective dose rate for any location from a galactic cosmic ray (GCR) environment can be calculated. A subset of the ARMAS database is represented by a second polynomial fit equation for the GCR plus probable relativistic energetic particle (REP; Van Allen belt REP) effective dose rates within a narrow band of L shells with altitudinal and geomagnetic dependency. Solar energetic particle events are not considered in this study since our databases do not contain these events. This work supports a suggestion that there may be a REP contribution having an effect at aviation altitudes. The ARMAS database is rich in Western Hemisphere observations for L shells between 1.5 and 5; there have been many cases of enhanced radiation events possibly related to effects from radiation belt particles. Our work identifies that the combined effects of an enhanced radiation environment in this L shell range are typically 15% higher than the GCR background. We also identify applications for the equations representing the Nowcast of Atmospheric Ionizing Radiation for Aviation Safety and ARMAS databases. They include (i) effective dose rate climatology in comparison with measured weather variability and (ii) climatological and statistical weather nowcasting and forecasting. These databases may especially help predict the radiation environment for regional air traffic management, for airport overflight operations, and for air carrier route operations of individual aircraft.

SummaryThe necessary conditions of Clebsch and Weierstrass and of the multiplier rule in the calculus of variations, which arise from the study of the first variation of a function, are summarised. A further necessary condition associated with the second variation is stated. The latter condition is applied to two problems: (i) the determination of the thrust-time programme which maximises the altitude of a sounding rocket, (ii) the determination of the thrust direction programme for a rocket with a known propellant expenditure programme which yields a maximum range. In both problems it is found that the additional necessary condition is satisfied.

The problem of transferring a space vehicle between two points in a given gravitational field such that the minimum amount of fuel is used has been called the fundamental navigational problem of astronautics. In such a problem it may be required to find the optimum thrust magnitude and thrust direction which yields a minimum fuel trajectory. Furthermore, certain end conditions may be specified which the optimal trajectory must satisfy. In a large number of published papers the velocity of the vehicle is supposed known both at the beginning of the transfer and at the end whereas the time taken to complete the manoeuvre may or may not be given. Also, other performance criteria have been chosen besides minimum fuel. For example, minimum time of transit or maximum orbital altitude at perigee. Papers mentioned in this review deal mainly with flight in two dimensions apart from those sections on general theory. Furthermore, all space vehicles considered are assumed to have a fixed exhaust velocity unless otherwise stated.

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