Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft

This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

Search space pruning and global optimisation of Multiple Gravity Assist spacecraft trajectories

We introduce and describe the Multiple Gravity Assist problem, a global optimisation problem that is of great interest in the design of spacecraft and their trajectories. We discuss its formalization and we show, in one particular problem instance, the performance of selected state of the art heuristic global optimisation algorithms. A deterministic search space pruning algorithm is then developed and its polynomial time and space complexity derived. The algorithm is shown to achieve search space reductions of greater than six orders of magnitude, thus reducing significantly the complexity of the subsequent optimisation.

On the importance of interplanetary magnetic field (By)on polar cap patch formation

A number of poleward moving events were observed between 1130 and 1300 UT on 11 February 2004, during periods of southward interplanetary magnetic field (IMF), while the steerable antenna of the European Incoherent Scatter (EISCAT) Svalbard radar (ESR)and the Tromsø VHF radar pointed nearly northward at low elevation. In this interval, simultaneous SuperDARN CUTLASS Finland radar measurements showed poleward moving radar aurora forms (PMRAFs) which appeared very similar to the density enhancements observed by the ESR northward pointing antenna. These events appeared quasiperiodically with a period of about 10 min. Comparing the observations from the above three radars, it is inferred that there is an almost one‐to‐one correspondence between the poleward moving plasma concentration enhancements (PMPCEs) observed by the ESR and the VHF radar and the PMRAFs measured by the CUTLASS Finland radar. These observations are consistent with the interpretation that the polar cap patch material was generated by photoionization at subauroral latitudes and that the plasma was structured by bursts of magnetopause reconnection giving access to the polar cap. There is clear evidence that plasma structuring into patches was dependent on the variability in IMF |By|. The duration of these events implies that the average evolution time of the newly opened flux tubes from the subauroral region to the polar cap was about 33 min.

Implications of non-cylindrical flux ropes for magnetic cloud reconstruction techniques and the interpretation of double flux rope events

Magnetic clouds (MCs) are a subset of interplanetary coronal mass ejections (ICMEs) which exhibit signatures consistent with a magnetic flux rope structure. Techniques for reconstructing flux rope orientation from single-point in situ observations typically assume the flux rope is locally cylindrical, e.g., minimum variance analysis (MVA) and force-free flux rope (FFFR) fitting. In this study, we outline a non-cylindrical magnetic flux rope model, in which the flux rope radius and axial curvature can both vary along the length of the axis. This model is not necessarily intended to represent the global structure of MCs, but it can be used to quantify the error in MC reconstruction resulting from the cylindrical approximation. When the local flux rope axis is approximately perpendicular to the heliocentric radial direction, which is also the effective spacecraft trajectory through a magnetic cloud, the error in using cylindrical reconstruction methods is relatively small (≈ 10∘). However, as the local axis orientation becomes increasingly aligned with the radial direction, the spacecraft trajectory may pass close to the axis at two separate locations. This results in a magnetic field time series which deviates significantly from encounters with a force-free flux rope, and consequently the error in the axis orientation derived from cylindrical reconstructions can be as much as 90∘. Such two-axis encounters can result in an apparent ‘double flux rope’ signature in the magnetic field time series, sometimes observed in spacecraft data. Analysing each axis encounter independently produces reasonably accurate axis orientations with MVA, but larger errors with FFFR fitting.

Effects of Thomson-scattering geometry on white-light imaging of an interplanetary shock: synthetic observations from forward magnetohydrodynamic modelling

Stereoscopic white-light imaging of a large portion of the inner heliosphere has been used to track interplanetary coronal mass ejections. At large elongations from the Sun, the white-light brightness depends on both the local electron density and the efficiency of the Thomson-scattering process. To quantify the effects of the Thomson-scattering geometry, we study an interplanetary shock using forward magnetohydrodynamic simulation and synthetic white-light imaging. Identifiable as an inclined streak of enhanced brightness in a time–elongation map, the travelling shock can be readily imaged by an observer located within a wide range of longitudes in the ecliptic. Different parts of the shock front contribute to the imaged brightness pattern viewed by observers at different longitudes. Moreover, even for an observer located at a fixed longitude, a different part of the shock front will contribute to the imaged brightness at any given time. The observed brightness within each imaging pixel results from a weighted integral along its corresponding ray-path. It is possible to infer the longitudinal location of the shock from the brightness pattern in an optical sky map, based on the east–west asymmetry in its brightness and degree of polarisation. Therefore, measurement of the interplanetary polarised brightness could significantly reduce the ambiguity in performing three-dimensional reconstruction of local electron density from white-light imaging.

Reconstruction of geomagnetic activity and near-Earth interplanetary conditions over the past 167 yr – Part 1: A new geomagnetic data composite

We present a new composite of geomagnetic activity which is designed to be as homogeneous in its construction as possible. This is done by only combining data that, by virtue of the locations of the source observatories used, have similar responses to solar wind and IMF (interplanetary magnetic field) variations. This will enable us (in Part 2, Lockwood et al., 2013a) to use the new index to reconstruct the interplanetary magnetic field, B, back to 1846 with a full analysis of errors. Allowance is made for the effects of secular change in the geomagnetic field. The composite uses interdiurnal variation data from Helsinki for 1845–1890 (inclusive) and 1893–1896 and from Eskdalemuir from 1911 to the present. The gaps are filled using data from the Potsdam (1891–1892 and 1897–1907) and the nearby Seddin observatories (1908–1910) and intercalibration achieved using the Potsdam–Seddin sequence. The new index is termed IDV(1d) because it employs many of the principles of the IDV index derived by Svalgaard and Cliver (2010), inspired by the u index of Bartels (1932); however, we revert to using one-day (1d) means, as employed by Bartels, because the use of near-midnight values in IDV introduces contamination by the substorm current wedge auroral electrojet, giving noise and a dependence on solar wind speed that varies with latitude. The composite is compared with independent, early data from European-sector stations, Greenwich, St Petersburg, Parc St Maur, and Ekaterinburg, as well as the composite u index, compiled from 2–6 stations by Bartels, and the IDV index of Svalgaard and Cliver. Agreement is found to be extremely good in all cases, except two. Firstly, the Greenwich data are shown to have gradually degraded in quality until new instrumentation was installed in 1915. Secondly, we infer that the Bartels u index is increasingly unreliable before about 1886 and overestimates the solar cycle amplitude between 1872 and 1883 and this is amplified in the proxy data used before 1872. This is therefore also true of the IDV index which makes direct use of the u index values.

Reconstruction of geomagnetic activity and near-Earth interplanetary conditions over the past 167 yr – Part 2: A new reconstruction of the interplanetary magnetic field

We present a new reconstruction of the interplanetary magnetic field (IMF, B) for 1846–2012 with a full analysis of errors, based on the homogeneously constructed IDV(1d)composite of geomagnetic activity presented in Part 1 (Lockwood et al., 2013a). Analysis of the dependence of the commonly used geomagnetic indices on solar wind parameters is presented which helps explain why annual means of interdiurnal range data, such as the new composite, depend only on the IMF with only a very weak influence of the solar wind flow speed. The best results are obtained using a polynomial (rather than a linear) fit of the form B = χ · (IDV(1d) − β)α with best-fit coefficients χ = 3.469, β = 1.393 nT, and α = 0.420. The results are contrasted with the reconstruction of the IMF since 1835 by Svalgaard and Cliver (2010).

Implications of the recent low solar minimum for the solar wind during the Maunder minimum

The behavior of the Sun and near-Earth space during grand solar minima is not understood; however, the recent long and low minimum of the decadal-scale solar cycle gives some important clues, with implications for understanding the solar dynamo and predicting space weather conditions. The speed of the near-Earth solar wind and the strength of the interplanetary magnetic field (IMF) embedded within it can be reliably reconstructed for before the advent of spacecraft monitoring using observations of geomagnetic activity that extend back to the mid-19th century. We show that during the solar cycle minima around 1879 and 1901 the average solar wind speed was exceptionally low, implying the Earth remained within the streamer belt of slow solar wind flow for extended periods. This is consistent with a broader streamer belt, which was also a feature of the recent low minimum (2009), and yields a prediction that the low near-Earth IMF during the Maunder minimum (1640-1700), as derived from models and deduced from cosmogenic isotopes, was accompanied by a persistent and relatively constant solar wind of speed roughly half the average for the modern era.

The heliospheric magnetic field

The heliospheric magnetic field (HMF) is the extension of the coronal magnetic field carried out into the solar system by the solar wind. It is the means by which the Sun interacts with planetary magnetospheres and channels charged particles propagating through the heliosphere. As the HMF remains rooted at the solar photosphere as the Sun rotates, the large-scale HMF traces out an Archimedean spiral. This pattern is distorted by the interaction of fast and slow solar wind streams, as well as the interplanetary manifestations of transient solar eruptions called coronal mass ejections. On the smaller scale, the HMF exhibits an array of waves, discontinuities, and turbulence, which give hints to the solar wind formation process. This review aims to summarise observations and theory of the small- and large-scale structure of the HMF. Solar-cycle and cycle-to-cycle evolution of the HMF is discussed in terms of recent spacecraft observations and pre-spaceage proxies for the HMF in geomagnetic and galactic cosmic ray records.

Using coordinated observations in polarized white light and Faraday rotation to probe the spatial position and magnetic field of an interplanetary sheath

Coronal mass ejections (CMEs) can be continuously tracked through a large portion of the inner heliosphere by direct imaging in visible and radio wavebands. White light (WL) signatures of solar wind transients, such as CMEs, result from Thomson scattering of sunlight by free electrons and therefore depend on both viewing geometry and electron density. The Faraday rotation (FR) of radio waves from extragalactic pulsars and quasars, which arises due to the presence of such solar wind features, depends on the line-of-sight magnetic field component B ∥ and the electron density. To understand coordinated WL and FR observations of CMEs, we perform forward magnetohydrodynamic modeling of an Earth-directed shock and synthesize the signatures that would be remotely sensed at a number of widely distributed vantage points in the inner heliosphere. Removal of the background solar wind contribution reveals the shock-associated enhancements in WL and FR. While the efficiency of Thomson scattering depends on scattering angle, WL radiance I decreases with heliocentric distance r roughly according to the expression Ir –3. The sheath region downstream of the Earth-directed shock is well viewed from the L4 and L5 Lagrangian points, demonstrating the benefits of these points in terms of space weather forecasting. The spatial position of the main scattering site r sheath and the mass of plasma at that position M sheath can be inferred from the polarization of the shock-associated enhancement in WL radiance. From the FR measurements, the local B ∥sheath at r sheath can then be estimated. Simultaneous observations in polarized WL and FR can not only be used to detect CMEs, but also to diagnose their plasma and magnetic field properties.

Reconstruction of geomagnetic activity and near-Earth interplanetary conditions over the past 167 yr - Part 4: Near-Earth solar wind speed, IMF, and open solar flux

In the concluding paper of this tetralogy, we here use the different geomagnetic activity indices to reconstruct the near-Earth interplanetary magnetic field (IMF) and solar wind flow speed, as well as the open solar flux (OSF) from 1845 to the present day. The differences in how the various indices vary with near-Earth interplanetary parameters, which are here exploited to separate the effects of the IMF and solar wind speed, are shown to be statistically significant at the 93% level or above. Reconstructions are made using four combinations of different indices, compiled using different data and different algorithms, and the results are almost identical for all parameters. The correction to the aa index required is discussed by comparison with the Ap index from a more extensive network of mid-latitude stations. Data from the Helsinki magnetometer station is used to extend the aa index back to 1845 and the results confirmed by comparison with the nearby St Petersburg observatory. The optimum variations, using all available long-term geomagnetic indices, of the near-Earth IMF and solar wind speed, and of the open solar flux, are presented; all with ±2sigma� uncertainties computed using the Monte Carlo technique outlined in the earlier papers. The open solar flux variation derived is shown to be very similar indeed to that obtained using the method of Lockwood et al. (1999).

Reconstruction of geomagnetic activity and near-Earth interplanetary conditions over the past 167 yr - Part 3: improved representation of solar cycle 11

Svalgaard (2014) has recently pointed out that the calibration of the Helsinki magnetic observatory’s H component variometer was probably in error in published data for the years 1866–1874.5 and that this makes the interdiurnal variation index based on daily means, IDV(1d), (Lockwood et al., 2013a), and the interplanetary magnetic field strength derived from it (Lockwood et al., 2013b), too low around the peak of solar cycle 11. We use data from the modern Nurmijarvi station, relatively close to the site of the original Helsinki Observatory, to confirm a 30% underestimation in this interval and hence our results are fully consistent with the correction derived by Svalgaard. We show that the best method for recalibration uses the Helsinki Ak(H) and aa indices and is accurate to ±10 %. This makes it preferable to recalibration using either the sunspot number or the diurnal range of geomagnetic activity which we find to be accurate to ±20 %. In the case of Helsinki data during cycle 11, the two recalibration methods produce very similar corrections which are here confirmed using newly digitised data from the nearby St Petersburg observatory and also using declination data from Helsinki. However, we show that the IDV index is, compared to later years, too similar to sunspot number before 1872, revealing independence of the two data series has been lost; either because the geomagnetic data used to compile IDV has been corrected using sunspot numbers, or vice versa, or both. We present corrected data sequences for both the IDV(1d) index and the reconstructed IMF (interplanetary magnetic field).We also analyse the relationship between the derived near-Earth IMF and the sunspot number and point out the relevance of the prior history of solar activity, in addition to the contemporaneous value, to estimating any “floor” value of the near-Earth interplanetary field.

Comparison of interplanetary signatures of streamers and pseudostreamers

If the source of the slow solar wind is a web comprising pseudostreamer belts connected to the streamer belt, then one expects the properties of interplanetary pseudostreamer flows to be similar to those of streamer flows. That expectation is tested with data from the slow wind preceding stream interfaces in stream interaction regions at 1 AU, where the interfaces separate what was originally slow and fast wind. Pseudostreamer cases were separated from streamer cases with the aid of the streamer identification tool developed by Owens et al. (2013), and superposed epoch analysis was performed to compare the patterns of a number of plasma and composition parameters. The results reveal that pseudostreamer flows have all of the slow-wind characteristics of streamer flows except that they are slightly less pronounced than streamer characteristics when compared to fast wind. The results are consistent with the concept that the solar wind displays a continuum of dynamic states rather than only slow and fast states.

Comparative Cluster/Double Star observations of the high and low latitude dayside magnetopause

The launch of the Double Star mission has provided the opportunity to monitor events at distinct locations on the dayside magnetopause, in coordination with the quartet of Cluster spacecraft. We present results of two such coordinated studies. In the first, 6 April 2004, both Cluster and the Double Star TC-1 spacecraft were on outbound transits through the dawn-side magnetosphere. Cluster observed northward moving FTEs with +/- polarity, whereas TC-1 saw -/+ polarity FTEs. The strength, motion and occurrence of the FTE signatures changes somewhat according to changes in IMF clock angle. These observations are consistent with ongoing reconnection on the dayside magnetopause, resulting in a series of flux transfer events (FTEs) seen both at Cluster and TC-1. The observed polarity and motion of each FTE signature advocates the existence of an active reconnection region consistently located between the positions of Cluster and TC-1, lying north and south of the reconnection line, respectively. This scenario is supported by the application of a model, designed to track flux tube motion, to conditions appropriate for the prevailing interplanetary conditions. The results from the model confirm the observational evidence that the low-latitude FTE dynamics is sensitive to changes in convected upstream conditions. In particular, changing the interplanetary magnetic field (IMF) clock angle in the model predicts that TC-1 should miss the resulting FTEs more often than Cluster, as is observed. For the second conjunction, on the 4 Jan 2005, the Cluster and TC-1 spacecraft all exited the dusk-side magnetosphere almost simultaneously, with TC-1 lying almost equatorial and Cluster at northern latitudes at about 4 RE from TC-1. The spacecraft traverse the magnetopause during a strong reversal in the IMF from northward to southward and a number of magnetosheath FTE signatures are subsequently observed. One coordinated FTE, studied in detail by Pu et al, [this issue], carries and inflowing energetic electron population and shows a motion and orientation which is similar at all spacecraft and consistent with the predictions of the model for the flux tube dynamics, given a near sub-solar reconnection line. This event can be interpreted either as the passage of two parallel flux tubes arising from adjacent x-line positions, or as a crossing of a single flux tube at different positions.