Predicting atmospheric delays for rapid ambiguity resolution in precise point positioning

Integer ambiguity resolution in precise point positioning (PPP) can shorten the initialization and re-initialization time, and ambiguity-fixed PPP solutions are also more reliable and accurate than ambiguity-float PPP solutions. However, signal interruptions are unavoidable in practical applications, particularly while operating in urban areas. Such signal interruptions can cause discontinuity of carrier phase arc, which introduces new integer ambiguities. Usually it will take approximately 15 min of continuous tracking to a reasonable number of satellites to fix new integer ambiguities. In many applications, it is impractical for a PPP user to wait for such a long time for the re-initialization. In this paper, a method for rapid ambiguity fixing in PPP is developed to avoid such a long re-initialization time. Firstly, the atmospheric delays were estimated epoch by epoch from ambiguity-fixed PPP solutions before the data gap or cycle slip occurs. A random walk procedure is then applied to predict the atmospheric delays accurately over a short time span. The predicted atmospheric delays then can be used to correct the observations which suffer from signal interruptions. Finally, the new ambiguities can be fixed with a distinct WL-LX-L3 (here LX denotes either of L1, L2) cascade ambiguity resolution strategy. Comprehensive experiments have demonstrated that the proposed method and strategy can fix zero-difference integer ambiguities successfully with only a single-epoch observation immediately after a short data gap. This technique works even when all satellites are interrupted at the same time. The duration of data gap bridged by this technique could be possibly extended if a more precise atmospheric delay prediction is found or on-the-fly (OTF) technology is applied. Based on the proposed method, real-time PPP with integer ambiguity fixing becomes more feasible in practice.     

Chlorella vulgaris culture as a regulator of CO2 in a bioregenerative life support system

It is the primary task for a bioregenerative life support system (BLSS) to maintain the stable concentrations of CO₂ and O₂. However, these concentrations could fluctuate based on various factors, such as the imbalance between respiration/assimilation quotients of the heterotrophic and autotrophic components. They can even be out of balance through catastrophic failure of higher plants in the emergency conditions. In this study, the feasibility of using unicellular Chlorella vulgaris of typically rapid growth as both “compensatory system” and “regulator” to control the balance of CO₂ and O₂ was analyzed in a closed ecosystem. For this purpose, a small closed ecosystem called integrative experimental system (IES) was established in our laboratory where we have been conducting multi-biological life support system experiments (MLSSE). The IES consists of a closed integrative cultivating system (CICS) and a plate photo-bioreactor. Four volunteers participated in the study for gas exchange by periodical breathing through a tube connected with the CICS. The plate photo-bioreactor was used to cultivate C. vulgaris. Results showed that the culture of C. vulgaris could be used in a situation of catastrophic failure of higher plant under the emergencies. And the productivity could recover itself to the original state in 3 to 5 days to protect the system till the higher plant was renewed. Besides, C. vulgaris could grow well and the productivity could be affected by the light intensity which could help to keep the balance of CO₂ and O₂ in the IES efficiently. Thus, C. vulgaris could be included in the design of a BLSS as a “compensatory system” in the emergency contingency and a “regulator” during the normal maintenance.     

Using only observation station data for PPP ambiguity resolution by UPD estimation

This article proposes a new method for uncalibrated phase delay (UPD) estimation to improve the accuracy of precise point positioning (PPP), which uses only observation station data. This means that the station used to generate the UPDs is the same station to which they are applied. First, dual-frequency observation equations based on a raw PPP model are developed. Then, the UPDs are calculated from integer linear combinations of float ambiguities. Third, with the UPD corrections, the least-squares ambiguity decorrelation adjustment (LAMBDA) method is utilized to obtain the integer ambiguities. Since only observation station data are used for UPD estimation, the partial ambiguity resolution (PAR) method is adopted to increase the possibility of finding a subset of integer ambiguities. The UPD estimation and ambiguity resolution are performed in each epoch. To obtain the correct integer ambiguity, the ratio test and success rate (bootstrapping) are used to evaluate the estimated integer ambiguity. Finally, by treating the integer ambiguities as constants, fixed solutions can be obtained. Quality control is also applied throughout the entire data processing procedure to obtain high quality float and fixed solutions. Data from 22 stations of the International Global Navigation Satellite System (GNSS) Service (IGS) in East Asia on day of year (DOY) 206, 2017, are used to verify the feasibility of this method. The experimental results show that compared with the float solution, the proposed method can significantly improve the accuracy in the east, north and up directions by 24%, 21% and 18% for static PPP and 36%, 18% and 34% for dynamic PPP, respectively. However, the accuracy of the proposed method is still lower than that of the fixed solutions obtained by the PRIDE-PPPAR software, in which the fractional cycle bias is computed based on reference network data. These findings sufficiently show that the proposed method can offer better solution accuracy than the float solution. However, the quality of the UPDs estimated only from observation station data is not as good as that of the estimates obtained based on reference network data.     

Testing a new multivariate GNSS carrier phase attitude determination method for remote sensing platforms

GNSS (Global Navigation Satellite Systems)-based attitude determination is an important field of study, since it is a valuable technique for the orientation estimation of remote sensing platforms. To achieve highly accurate angular estimates, the precise GNSS carrier phase observables must be employed. However, in order to take full advantage of the high precision, the unknown integer ambiguities of the carrier phase observables need to be resolved. This contribution presents a GNSS carrier phase-based attitude determination method that determines the integer ambiguities and attitude in an integral manner, thereby fully exploiting the known body geometry of the multi-antennae configuration. It is shown that this integral approach aids the ambiguity resolution process tremendously and strongly improves the capacity of fixing the correct set of integer ambiguities. In this contribution, the challenging scenario of single-epoch, single-frequency attitude determination is addressed. This guarantees a total independence from carrier phase slips and losses of lock, and it also does not require any a priori motion model for the platform. The method presented is a multivariate constrained version of the popular LAMBDA method and it is tested on data collected during an airborne remote sensing campaign.     

Selection of candidate salad vegetables for controlled ecological life support system

Higher plants, as one of the essential biological components of CELSS, can supply food, oxygen and water for human crews during future long-duration space missions and Lunar/Mars habitats. In order to select suitable leaf vegetable varieties for our CELSS Experimental Facility (CEF), five varieties of lettuce (“Nenlvnaiyou”, “Dasusheng”, “Naichoutai”, “Dongfangkaixuan” and “Siji”), two of spinach (“Daye” and “Quanneng”), one of rape (“Jingyou No. 1”) and one of common sowthistle were grown and compared on the basis of edible biomass, and nutrient content. In addition, two series of experiments were conducted to study single leaf photosynthetic rates and transpiration rates at 30 days after planting, one which used various concentrations of CO₂ (500, 1000, 1500 and 2000 μmol mol⁻¹) and another which used various light intensities (100, 300, 500 and 700 μmol m⁻² s⁻¹). Results showed that lettuce cvs. “Nenlvnaiyou”, “Siji” and “Dasusheng” produced higher yields of edible biomass; common sowthisle would be a good source of β-carotene for the diet. Based on the collective findings, we selected three varieties of lettuce (“Nenlvnaiyou”, “Dasusheng” and “Siji”) and one of common sowthistle as the candidate crops for further research in our CEF. In addition, elevated CO₂ concentration increased the rates of photosynthesis and transpiration, and elevated light intensity increased the rate of photosynthesis for these varieties. These results can be useful for determining optimal conditions for controlling CO₂ and water fluxes between the crops and the overall CELSS.     

The thermospheric composition different responses to geomagnetic storm in the winter and summer hemisphere measured by “SZ” Atmospheric Composition Detectors

Three “SZ” Atmospheric Composition Detectors (ACDs) on board spacecraft “SZ-2”, “SZ-3” and “SZ-4” were launched on 10th January 2001, 26th March 2002 and 31st December 2002 separately. A large quantity of thermospheric composition data at the orbital altitude ranging from 330 to 362 km were collected from the in-situ measurement of ACDs. The spacecrafts’ lifetime was just in the second peak period of the 23rd solar cycle which includes two peaks and the solar activity value F_10.7 was from 89 to 228. During this period, several intense geomagnetic disturbances happened.     

两步法快速解算编队卫星GPS模糊度

为克服卫星编队飞行实时相对定位中双频模糊度解算速度慢的缺点,结合扩展Kalman滤波(EKF,Extended Kalman Filter),首先采用少数个历元(如10个)相位平滑伪距相对定位结果与 L₆的平均值对滤波初始化,再根据两步法解算双频模糊度,即先解算并正确固定宽巷模糊度,获得较准确的基线分量估值,然后采用选权拟合方法,将基线分量作为约束条件解算并固定双频模糊度.仿真算例计算结果表明,当宽巷模糊度正确固定后,编队卫星间相对定位误差在5cm以内,两步法可以在较短时间(约3min)内固定双频模糊度,为精确解算编队卫星的相对状态提供保障.     

Old tail lobes effect on the solar-wind – Magnetosphere energy transport for the 27 August 2001 substorm

The magnetic flux of tail lobes Ψ is divided in two parts of comparable values Ψ₁ and Ψ₀₂, with the first that appears during substorm and the second, observed before substorm start. The first was named “new magnetic flux”, the second – “old magnetic flux”. The first, Ψ₁, is known to play a definitive role in the energy transport from the solar wind into the magnetosphere-ionosphere-atmosphere system, but the role of Ψ₀₂ in this transport is not well known. From the 27 August 2001 substorm data we study the involvement in the above transport process of the old flux Ψ₀₂. This involvement is observed in the polar cap (PC) area, which existed prior to the substorm and is called respectively “the old PC”. In this study, as distinct from earlier works, we use the balance equation of the energy stored in magnetosphere and energy consumed. Activation of the old PC magnetic flux Ψ₀₂ was found to increase the energy input by ∼85% in the event under consideration.     

Formation control using μ-synthesis for Inner-Formation Gravity Measurement Satellite System

Inner-Formation Gravity Measurement Satellite System (IFGMSS) is used to map the gravity field of Earth. The IFGMSS consists of two satellites in which one is called “inner satellite” and the other one is named as “outer satellite”. To measure the pure Earth gravity, the inner satellite is located in the cavity of the outer satellite. Because of the shield effect of the cavity, the inner satellite is affected only by the gravitational force, so it can sense Earth gravity precisely. To avoid the collision between the inner satellite and the outer satellite, it is best to perform a real-time control on the outer satellite. In orbit, the mass of the outer satellite decreases with the consumption of its propellant. The orbit angular rate of the inner satellite varies with time due to various disturbing forces. These two parameters’ uncertainties make the C–W function be not so accurate to describe the formation behavior of these two satellites. Furthermore, the thrusters also have some uncertainties due to the unmodelled dynamics. To cancel the effects caused by the above uncertainties, we have studied the robust control method based on the μ-synthesis. This μ-synthesis eliminates the conservativeness and improves the control efficiency comparing with the H_∞ method. Finally, to test the control method, we simulate an IFGMSS mission in which the satellite runs in a sun synchronous circular orbit with an altitude of 300 km. The simulation results show the effectiveness of the robust control method. The performances of the closed-loop system with the μ-controller are tested by the μ-analysis. It has found that the nominal performance, the robust stability and the robust performance are all achieved. The transient simulation results further prove the control response is fast and the accuracy of the relative position meets the demand of the gravity measurement.     

Accuracy assessment of single and double difference models for the single epoch GPS compass

The single epoch GPS compass is an important field of study, since it is a valuable technique for the orientation estimation of vehicles and it can guarantee a total independence from carrier phase slips in practical applications. To achieve highly accurate angular estimates, the unknown integer ambiguities of the carrier phase observables need to be resolved. Past researches focus on the ambiguity resolution for single epoch; however, accuracy is another significant problem for many challenging applications. In this contribution, the accuracy is evaluated for the non-common clock scheme of the receivers and the common clock scheme of the receivers, respectively. We focus on three scenarios for either scheme: single difference model vs. double difference model, single frequency model vs. multiple frequency model and optimal linear combinations vs. traditional triple-frequency least squares. We deduce the short baseline precision for a number of different available models and analyze the difference in accuracy for those models. Compared with the single or double difference model of the non-common clock scheme, the single difference model of the common clock scheme can greatly reduce the vertical component error of baseline vector, which results in higher elevation accuracy. The least squares estimator can also reduce the error of fixed baseline vector with the aid of the multi-frequency observation, thereby improving the attitude accuracy. In essence, the “accuracy improvement” is attributed to the difference in accuracy for different models, not a real improvement for any specific model. If all noise levels of GPS triple frequency carrier phase are assumed the same in unit of cycles, it can be proved that the optimal linear combination approach is equivalent to the traditional triple-frequency least squares, no matter which scheme is utilized. Both simulations and actual experiments have been performed to verify the correctness of theoretical analysis.     

Seismic deformation of the Mw 8.0 Wenchuan earthquake from high-rate GPS observations

High-rate GPS positioning has been recognized as a powerful tool in estimating epoch-wise station displacement which is particularly useful for seismology. In this study, station displacements during the 12 May 2008 M_w 8.0 Wenchuan earthquake are derived from the 1-Hz GPS data collected at a set of stations in China. The impacts of integer ambiguity resolution and station environment-dependent effects are investigated in order to yield more accurate results. The position accuracy of horizontal components of better than 1 cm suggests that GPS can sense the rapid position oscillation of about 2 cm in amplitude. Temporal and spatial analysis is applied to the surface displacement at station XANY and the characteristics of the movements due to Rayleigh and Love waves are detected and discussed. The comparison of GPS-derived displacement with relevant synthetic data computed based on a recently published rapture model shows a reasonable agreement in waveform. The various differences in amplitude need further investigation and also imply that rapture inversion might be improved if GPS-derived displacement is assimilated.     

RTK model and positioning performance analysis using Galileo four-frequency observations

This paper proposes a real-time kinematic (RTK) model that uses one common reference satellite for the Galileo system with four frequency observations. In the proposed model, the double-differenced (DD) pseudorange and carrier phase biases among the different frequencies are estimated as unknown parameters to recover the integer features of the DD ambiguities among the different frequencies for ambiguity resolution and precise positioning. Analysis results show that the E5a, E5b, and E5 frequencies have virtually the same performance in terms of the positioning accuracy, observation residuals, and ratio values of ambiguity resolution. However, the E1 frequency performs worse than the E5a, E5b, and E5 frequencies. The RTK results for the combination of multiple frequencies are much better than those for a single-frequency observation, the coordinates’ standard deviation is improved about 20–30%, and the ambiguity fix time is improved about 10%.     

Stellar β-decay rates of iron isotopes and its implications in astrophysics

β-Decay and positron decay are believed to play a consequential role during the late phases of stellar evolution of a massive star culminating in a supernova explosion. The β-decay contributes in maintaining a respectable lepton-to-baryon ratio, Y_e, of the core prior to collapse which results in a larger shock energy to produce the explosion. The positron decay acts in the opposite direction and tends to decrease the ratio. The structure of the presupernova star is altered both by the changes in Y_e and the entropy of the core material. Recently the microscopic calculation of weak interaction mediated rates on key isotopes of iron was introduced using the proton–neutron quasiparticle random phase approximation (pn-QRPA) theory with improved model parameters. Here I discuss in detail the improved calculation of β^±-decay rates for iron isotopes (^54,55,56Fe) in stellar environment. The pn-QRPA theory allows a microscopic “state-by-state” calculation of stellar rates as explained later in text. Excited state Gamow–Teller distributions are much different from ground state and a microscopic calculation of decay rates from these excited states greatly increases the reliability of the total decay rate calculation specially during the late stages of stellar evolution. The reported decay rates are also compared with earlier calculations. The positron decay rates are in reasonable agreement with the large-scale shell model calculation. The main finding of this work includes that the stellar β-decay rates of ^54,55,56Fe are around 3–5 orders of magnitude smaller than previously assumed and hence irrelevant for the determination of the evolution of Y_e during the presupernova phase of massive stars. The current work discourages the inclusion of ^55,56Fe in the list of key stellar β-decay nuclei as suggested by former simulation results.     

Integer phase ambiguity estimation in next-generation geodetic Very Long Baseline Interferometry

Next-generation Very Long Baseline Interferometry (VLBI) system designs are aiming at 1 mm global position accuracy. In order to achieve this, it is not only necessary to deploy improved VLBI systems, but also to develop analysis strategies that take full advantage of the observations taken. Since the new systems are expected to incorporate four independent radio frequency bands, it should be feasible to resolve phase ambiguities directly from post-correlation data, providing roughly an order of magnitude improvement in precision of the delay observable. As the unknown ambiguities are of integer nature, it is discussed here how they the can be resolved analytically using algorithms which have been developed for Global Navigation Satellite System (GNSS) applications. Furthermore, it will be shown that ionosphere contribution and source structure effects, so-called core-shifts, can be solved simultaneously with the delay, which is the main geodetic observable for follow-on analysis. In order to verify the proposed algorithm, simulated observations were created using parameters from actual design studies. It is shown that, even in the case of low signal-to-noise ratio observations, reliable phase ambiguity resolution can be achieved and it is discussed how the integer ambiguity recovery depends on the number of observations and signal-to-noise ratio.     

Towards PPP-RTK: Ambiguity resolution in real-time precise point positioning

Integer ambiguity resolution at a single station can be achieved by introducing predetermined uncalibrated phase delays (UPDs) into the float ambiguity estimates of precise point positioning (PPP). This integer resolution technique has the potential of leading to a PPP-RTK (real-time kinematic) model where PPP provides rapid convergence to a reliable centimeter-level positioning accuracy based on an RTK reference network. Nonetheless, implementing this model is technically subject to how rapidly we can fix wide-lane ambiguities, stabilize narrow-lane UPD estimates, and achieve the first ambiguity-fixed solution. To investigate these issues, we used 7 days of 1-Hz sampling GPS data at 91 stations across Europe. We find that at least 10 min of observations are required for most receiver types to reliably fix about 90% of wide-lane ambiguities corresponding to high elevations, and over 20 min to fix about 90% of those corresponding to low elevations. Moreover, several tens of minutes are usually required for a regional network before a narrow-lane UPD estimate stabilizes to an accuracy of far better than 0.1 cycles. Finally, for hourly data, ambiguity resolution can significantly improve the accuracy of epoch-wise position estimates from 13.7, 7.1 and 11.4 cm to 0.8, 0.9 and 2.5 cm for the East, North and Up components, respectively, but a few tens of minutes is required to achieve the first ambiguity-fixed solution. Therefore, from the timeliness aspect, our PPP-RTK model currently cannot satisfy the critical requirement of instantaneous precise positioning where ambiguity-fixed solutions have to be achieved within at most a few seconds. However, this model can still be potentially applied to some near-real-time remote sensing applications, such as the GPS meteorology.     

Using S-band dual polarized radar for convective/stratiform rain indexing and the correspondence with AMSR-E GSFC profiling algorithm

The separation of rain types in convective and stratiform regimes has long been a goal in microwave remote sensing of precipitation research. In this essence, a dual polarized radar based indexing scheme that provides information on convective and stratiform (C/S) rain regimes has been presented in correspondence with advanced microwave scanning radiometer – earth observing system (AMSR-E) GSFC profiling algorithm estimate of convective rain percentage. The dual polarized radar based C/S indexing scheme first retrieves the normalized gamma drop size distribution parameters, median volume drop diameter (D₀) and concentration parameter (N_w), from dual polarized radar measurements Z_H and Z_DR, representing reflectivity and differential reflectivity respectively, by means of the genetic programming approach. Next, the C/S rain index is calculated based on the formulation of an empirical relation in N_w–D₀ domain. The scheme has been inspected and applied on measurements from the S-band Chilbolton dual polarized radar. A considerable number of “coincident” cases from the radar and the AMSR-E observations are investigated. It has been revealed that the dual polarized radar based C/S rain indexing is in a similar pattern with the AMSR-E GSFC profiling algorithm estimate of convective rain percentage. Generally, as C/S rain index value increases, which signifies a stratiform to convective trend, the AMSR-E convective rain percentage also increases.     

Using high-energy proton fluence to improve risk prediction for consequences of solar particle events

The potential for exposure to large solar particle events (SPEs) with high energy levels is a major concern during interplanetary transfer and extra-vehicular activities (EVAs) on the lunar and Mars surface. Previously, we have used data from the last 5 solar cycles to estimate percentiles of dose to a typical blood-forming organ (BFO) for a hypothetical astronaut in a nominally shielded spacecraft during a 120-d lunar mission. As part of this process, we made use of complete energy spectra for 34 large historical SPEs to calculate what the BFO mGy-Eq dose would have been in the above lunar scenario for each SPE. From these calculated doses, we then developed a prediction model for BFO dose based solely on an assumed value of integrated fluence above 30 MeV (Φ₃₀) for an otherwise unspecified future SPE. In this study, we reasoned that since BFO dose is determined more by protons with higher energies than by those with lower energies, more accurate BFO dose prediction models could be developed using integrated fluence above 60 (Φ₆₀) and above 100 MeV (Φ₁₀₀) as predictors instead of Φ₃₀. However to calculate the unconditional probability of a BFO dose exceeding a pre-specified limit (“BFO dose risk”), one must also take into account the distribution of the predictor (Φ_30,Φ₆₀, or Φ₁₀₀), as estimated from historical SPEs. But Φ₆₀ and Φ₁₀₀ have more variability, and less available historical information on which to estimate their distributions over many SPE occurrences, than does Φ₃₀. Therefore, when estimating BFO dose risk there is a tradeoff between increased BFO dose prediction at a given energy threshold and decreased accuracy of models for describing the distribution of that threshold over future SPEs as the threshold increases. Even when taking the second of these two factors into account, we still arrived at the conclusion that overall prediction improves as the energy level threshold increases from 30 to 60 to 100 MeV. These results can be applied to the development of approaches to improve radiation protection of astronauts and the optimization of mission planning for future space missions.     

Real-time clock comparison and monitoring with multi-GNSS precise point positioning: GPS,GLONASS and Galileo

The precise point positioning (PPP) technique is widely used in time and frequency applications. Because of the real-time service (RTS) project of the International GNSS Service, we can use the PPP technique for real-time clock comparison and monitoring. As a participant in the RTS, the Centre National d’Etudes Spatiales (CNES) implements the PPPWIZARD (Precise Point Positioning with Integer and Zero-difference Ambiguity Resolution Demonstrator) project to validate carrier phase ambiguity resolution. Unlike the Integer-PPP (IPPP) of the CNES, fixing ambiguities in the post-processing mode, the PPPWIZARD operates in the real-time mode, which is also called real-time IPPP (RT-IPPP). This paper focuses on applying the RT-IPPP for real-time clock comparison and monitoring. We review the principle of real-time clock comparison and monitoring, and introduce the methodology of the RT-IPPP technique. The observations of GPS, GLONASS and Galileo were processed for the experiments. Five processing modes were provided in the experiment to analyze the benefits of ambiguity resolution and multi-GNSS. In the clock comparison experiment, the average reduction ratios of standard deviations with respect to the G PPP mode range from 9.7% to 35.0%. In the clock monitoring experiment, G PPP mode can detect clock jumps whose magnitudes are larger than 0.9 ns. The RT-IPPP technique with GRE PPP AR (G) mode allows for the detection of any clock jumps larger than 0.6 ns. For frequency monitoring, G PPP mode allows detection of frequency changes larger than 1.1 × 10⁻¹⁴. When the RT-IPPP technique is applied, monitoring with GRE PPP AR (G) mode can detect frequency changes larger than 6.1 × 10⁻¹⁵.     

An evaluation of the IRI-2007 storm time model at low latitude stations

This paper discusses the ability of the International Reference Ionosphere IRI-2007 storm time model to predict foF2 ionospheric parameter during geomagnetic storm periods. Experimental data (based on availability) from two low latitude stations: Vanimo (geographic coordinates, 2.7 °S, 141.3 °E, magnetic coordinates, 12.3 °S, 212.50 °E) and Darwin (geographic coordinates, 12.45 °S, 130.95 °E, magnetic coordinates, 22.9 °S, 202.7 °E) during nine storms that occurred in 2000 (Rz₁₂ = 119), 2001(Rz₁₂ = 111) and 2003 (Rz₁₂ = 64) are compared with those obtained by the IRI-2007 storm model. The results obtained show that the percentage deviation between the experimental and IRI predicted foF2 values during these storm periods is as high as 100% during the main and recovery phases. Based on the values of “relative deviation module mean” (RDMM) obtained (i.e. between 0.08 and 0.60), it is observed that there is a reasonable to poor agreement between measured foF2 values and the IRI-storm model prediction values during main and recovery phases of the storms under investigation. As a result, in addition to other studies that have been carried out from different sectors, more studies are required to be carried out. This will enable IRI community to improve on the present performance of the model. In general the IRI-storm model predictions follow normal trend of the foF2 measured values but does not reproduce well the measured values.     

Magnetar spectra and twisted magnetospheres

The huge potential drop between the footpoints of the closed field lines in the twisted magnetospheres of magnetars may accelerate electrons up to very high energies, γ ? 10⁶. On the other hand, the comparison between the observed spectra of magnetars and spectra obtained by accurate theoretical models seems to favor of a picture in which the magnetosphere is filled by “slow” electrons (v ? 0.8c), rather than by ultra-relativistic particles.