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The defining trait of magnetars, the most strongly magnetized neutron stars (NSs), is their transient activity in the X/$\gamma$-bands. In particular, many of them undergo phases of enhanced emission, the so-called outbursts, during which the luminosity rises by a factor $\sim$10$-$1000 in a few hours to then decay over months/years. Outbursts often exhibit a thermal spectrum, associated with the appearance of hotter regions on the surface of the star, which subsequently change in shape and cool down. Here we simulate the unfolding of a sudden, localized heat injection in the external crust of a NS with a 3D magneto-thermal evolution code, finding that this can reproduce the main features of magnetar outbursts. A full 3D treatment allows us to study for the first time the inherently asymmetric hot-spots which appear on the surface of the star as the result of the injection and to follow the evolution of their temperature and shape. We investigate the effects produced by different physical conditions in the heated region, highlighting in particular how the geometry of the magnetic field plays a key role in determining the properties of the event., Comment: 12 pages, 10 figures, accepted for publication in ApJ

Full list of authors: Di Gesu, Laura; Donnarumma, Immacolata; Tavecchio, Fabrizio; Agudo, Ivan; Barnounin, Thibault; Cibrario, Nicolo; Di Lalla, Niccolo; Di Marco, Alessandro; Escudero, Juan; Errando, Manel; Jorstad, Svetlana G.; Kim, Dawoon E.; Kouch, Pouya M.; Liodakis, Ioannis; Lindfors, Elina; Madejski, Grzegorz; Marshall, Herman L.; Marscher, Alan P.; Middei, Riccardo; Muleri, Fabio; Myserlis, Ioannis; Negro, Michela; Omodei, Nicola; Pacciani, Luigi; Paggi, Alessandro; Perri, Matteo; Puccetti, Simonetta; Antonelli, Lucio A.; Bachetti, Matteo; Baldini, Luca; Baumgartner, Wayne H.; Bellazzini, Ronaldo; Bianchi, Stefano; Bongiorno, Stephen D.; Bonino, Raffaella; Brez, Alessandro; Bucciantini, Niccolo; Capitanio, Fiamma; Castellano, Simone; Cavazzuti, Elisabetta; Ciprini, Stefano; Costa, Enrico; De Rosa, Alessandra; Del Monte, Ettore; Doroshenko, Victor; Dovciak, Michal; Ehlert, Steven R.; Enoto, Teruaki; Evangelista, Yuri; Fabiani, Sergio; Ferrazzoli, Riccardo; Garcia, Javier A.; Gunji, Shuichi; Hayashida, Kiyoshi; Heyl, Jeremy; Iwakiri, Wataru; Karas, Vladimir; Kitaguchi, Takao; Kolodziejczak, Jeffery J.; Krawczynski, Henric; La Monaca, Fabio; Latronico, Luca; Maldera, Simone; Manfreda, Alberto; Marin, Frederic; Marinucci, Andrea; Massaro, Francesco; Matt, Giorgio; Mitsuishi, Ikuyuki; Mizuno, Tsunefumi; Ng, C-Y; O'Dell, Stephen L.; Oppedisano, Chiara; Papitto, Alessandro; Pavlov, George G.; Peirson, Abel L.; Pesce-Rollins, Melissa; Petrucci, Pierre-Olivier; Pilia, Maura; Possenti, Andrea; Poutanen, Juri; Ramsey, Brian D.; Rankin, John; Ratheesh, Ajay; Romani, Roger W.; Sgro, Carmelo; Slane, Patrick; Soffitta, Paolo; Spandre, Gloria; Tamagawa, Toru; Taverna, Roberto; Tawara, Yuzuru; Tennant, Allyn F.; Thomas, Nicolas E.; Tombesi, Francesco; Trois, Alessio; Tsygankov, Sergey; Turolla, Roberto; Vink, Jacco; Weisskopf, Martin C.; Wu, Kinwah; Xie, Fei; Zane, Silvia.--This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited., Particle acceleration mechanisms in supermassive black hole jets, such as shock acceleration, magnetic reconnection, and turbulence, are expected to have observable signatures in the multiwavelength polarization properties of blazars. The recent launch of the Imaging X-Ray Polarimetry Explorer (IXPE) enables us, for the first time, to use polarization in the X-ray band (2–8 keV) to probe the properties of the jet synchrotron emission in high-synchrotron-peaked BL Lac objects (HSPs). We report the discovery of X-ray linear polarization (degree Πx = 15% ± 2% and electric vector position angle ψx = 35° ± 4°) from the jet of the HSP Mrk 421 in an average X-ray flux state. At the same time, the degree of polarization at optical, infrared, and millimeter wavelengths was found to be lower by at least a factor of 3. During the IXPE pointing, the X-ray flux of the source increased by a factor of 2.2, while the polarization behavior was consistent with no variability. The higher level of Πx compared to longer wavelengths, and the absence of significant polarization variability, suggest a shock is the most likely X-ray emission site in the jet of Mrk 421 during the observation. The multiwavelength polarization properties are consistent with an energy-stratified electron population, where the particles emitting at longer wavelengths are located farther from the acceleration site, where they experience a more disordered magnetic field. © 2022. The Author(s). Published by the American Astronomical Society., IXPE is a joint US and Italian mission. The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The Italian contribution is supported by the Italian Space Agency (Agenzia Spaziale Italiana, ASI), through contract ASI-OHBI-2017-12-I.0, agreements ASI-INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by Istituto Nazionale di Astrofisica (INAF) and Istituto Nazionale di Fisica Nucleare (INFN) in Italy. This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-energy Astrophysics Science Archive Research Center (HEASARC), at NASA's Goddard Space Flight Center (GSFC). The Instituto de Astrofisica de Andalucia (IAA)-CSIC group acknowledges financial support from the Spanish "Ministerio de Ciencia e Innovación" (MCINN) through the "Center of Excellence Severo Ochoa" award for IAA-CSIC (SEV-2017-0709) and through grants AYA2016-80889-P and PID2019-107847RB-C44. The Polarimetric Monitoring of AGN at Millimeter Wavelengths (POLAMI) observations were carried out at the IRAM 30 m Telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). Some of the data reported here are based on observations made with the NOT, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku, and the University of Oslo, representing Denmark, Finland, and Norway; the University of Iceland; and Stockholm University at Observatorio del Roque de los Muchachos, La Palma, Spain, of Instituto de Astrofisica de Canarias. E.L. was supported by Academy of Finland projects 317636 and 320045. The data presented here were obtained (in part) with the Alhambra Faint Object Spectrograph and Camera (ALFOSC), which is provided by IAA under a joint agreement with the University of Copenhagen and NOT. Part of the French contribution is supported by the Scientific Research National Center (CNRS) and the French space agency (CNES). The research at Boston University was supported in part by National Science Foundation (NSF) grant AST-2108622, NASA Fermi Guest Investigator grant 80NSSC21K1917, and NASA Swift Guest Investigator grant 80NSSC22K0537. This research was conducted in part using the Mimir instrument, jointly developed at Boston University and Lowell Observatory and supported by NASA, NSF, and the W. M. Keck Foundation.

Pulsating Ultra Luminous X-ray sources (PULXs) are thought to be X-ray bright, accreting, magnetized neutron stars, and could be the first and only evidence for the existence of magnetars in binary systems. Their apparent soft (< 20 keV) X-ray luminosity can exceed the Eddington luminosity for a neutron star (NS) by a few orders of magnitude. Although several scenarios have been proposed to explain the different components observed in the X-ray spectra and the characteristics of the X-ray lightcurve of these system, detailed quantitative calculations are still missing. In particular, the observed soft X-ray lightcurves are almost sinuosidal and show an increase in the pulsed fraction (from 8% up to even 30%) with increasing energy. Here, we present how emission originating from an optically thick envelope, expected to be formed during super-Eddington accretion, can result in pulsed fractions similar to observations.

The observational properties of isolated NSs are shaped by their magnetic field and surface temperature. They evolve in a strongly coupled fashion, and modelling them is key in understanding the emission properties of NSs. Much effort was put in tackling this problem in the past but only recently a suitable 3D numerical framework was developed. We present a set of 3D simulations addressing both the long-term evolution (≈ 104–106 yrs) and short-lived outbursts (≲ 1 yr). Not only a 3D approach allows one to test complex field geometries, but it is absolutely key to model magnetar outbursts, which observations associate to the appearance of small, inherently asymmetric hot regions. Even though the mechanism that triggers these phenomena is not completely understood, following the evolution of a localised heat injection in the crust serves as a model to study the unfolding of the event.

Anomalous X-ray pulsars (AXPs) and Soft gamma repeaters (SGRs) form together a single class of astrophysical sources, commonly associated to magnetars. New-generation X-ray polarimeters will play a key role in assessing the nature of these sources by directly probing the star magnetic field. In the highly magnetized environment radiation is expected to be strongly polarized and such a measure will be easily within reach of IXPE and eXTP. Polarization measurements will eventually confirm the presence of ultra-strong magnetic fields, probing the magnetar scenario. In this work we will discuss theoretical expectations for the polarization signature of AXPs and SGRs and present numerical simulations for the detector response of the polarimeters currently under construction. We will also show how these sources can be used to test vacuum birefringence, a QED effect predicted by Heisemberg and Euler in the Thirties and not experimentally verified as yet.

X-ray emission from the surface of isolated neutron stars (NSs) has been now observed in a variety of sources. The ubiquitous presence of pulsations clearly indicates that thermal photons either come from a limited area, possibly heated by some external mechanism, or from the entire (cooling) surface but with an inhomogeneous temperature distribution. In a NS the thermal map is shaped by the magnetic field topology, since heat flows in the crust mostly along the magnetic field lines. Self-consistent surface thermal maps can hence be produced by simulating the coupled magnetic and thermal evolution of the star. We compute the evolution of the neutron star crust in three dimensions for different initial configurations of the magnetic field and use the ensuing thermal surface maps to derive the spectrum and the pulse profile as seen by an observer at infinity, accounting for general-relativistic effects. In particular, we compare cases with a high degree of symmetry with inherently 3D ones, obtained by adding a quadrupole to the initial dipolar field. Axially symmetric fields result in rather small pulsed fractions ($\lesssim 5\%$), while more complex configurations produce higher pulsed fractions, up to $\sim25\%$. We find that the spectral properties of our axisymmetric model are close to those of the bright isolated NS RX~J1856.5-3754 at an evolutionary time comparable with the inferred dynamical age of the source., 14 pages, 11 figures, accepted for publication in ApJ

The radio-quiet pulsar PSR J2021+4026 is mostly known because it is the only rotation-powered pulsar that shows variability in its {\gamma}-ray emission. Using XMM-Newton archival data, we first confirmed that its flux is steady in the X-ray band, and then we showed that both the spectral and timing X-ray properties, i.e. the narrow pulse profile, the high pulsed fraction of 80-90% and its dependence on the energy, can be better reproduced using a magnetized atmosphere model instead of a simply blackbody. With a maximum likelihood analysis in the energy-phase space, we inferred that the pulsar has, in correspondence of one magnetic pole, a hot spot of temperature T~1 MK and colatitude extension {\theta}~20{\deg}. For the pulsar distance of 1.5 kpc, this corresponds to a cap of R~5-6 km, greater than the standard dimension of the dipolar polar caps. The large pulsed fraction further argues against emission from the entire star surface, as it would be expected in the case of secular cooling. An unpulsed (, Comment: Accepted for publication in A&A, 10 pages, 3 figures, 3 tables

Photon-axion mixing can create observable signatures in thermal spectra of isolated, cooling neutron stars. Their shape depends on the polarization properties of the radiation, which, in turn, are determined by the structure of stellar outermost layers. Here we investigate the effect of mixing on the spectrum and polarimetric observables, polarization fraction and polarization angle, using realistic models of surface emission. We focus on RX J1856.5-3754, the only source among the X-ray dim isolated neutron stars for which polarimetric measurements in the optical band were performed. Our results show that in the case of a condensed surface in both fixed and free-ion limits, the mixing can significantly limit the geometric configurations which reproduce the observed linear polarization fraction of 16.43%. In the case of an atmosphere, the mixing does not create any noticeable signatures. Complementing our approach with the data from upcoming soft X-ray polarimetry missions will allow to obtain constraints on $g_{\gamma a} \sim 10^{-11}$ GeV$^{-1}$ and $m_a \lesssim 10^{-6}$ eV, improving the present experimental and astrophysical limits., Comment: 12 pages, 8 figures, accepted for publication in ApJ

The "Magnificent Seven" (M7) are a group of radio-quiet Isolated Neutron Stars (INSs) discovered in the soft X-rays through their purely thermal surface emission. Owing to the large inferred magnetic fields ($B\approx 10^{13}$ G), radiation from these sources is expected to be substantially polarised, independently on the mechanism actually responsible for the thermal emission. A large observed polarisation degree is, however, expected only if quantum-electrodynamics (QED) polarisation effects are present in the magnetised vacuum around the star. The detection of a strongly linearly polarised signal would therefore provide the first observational evidence of QED effects in the strong-field regime. While polarisation measurements in the soft X-rays are not feasible yet, optical polarisation measurements are within reach also for quite faint targets, like the M7 which have optical counterparts with magnitudes $\approx 26$--$28$. Here, we report on the measurement of optical linear polarisation for the prototype, and brightest member, of the class, RX\, J1856.5$-$3754 ($V\sim 25.5$), the first ever for one of the M7, obtained with the Very Large Telescope. We measured a polarisation degree $\mathrm{P.D.} =16.43\% \pm5.26\%$ and a polarisation position angle $\mathrm{P.A.}=145\fdg39\pm9\fdg44$, computed east of the North Celestial Meridian. The $\mathrm{P.D.}$ that we derive is large enough to support the presence of vacuum birefringence, as predicted by QED., 9 pages, 7 figures, accepted for publication on MNRAS

In this paper we present the science potential of the enhanced X-ray Timing and Polarimetry (eXTP) mission for studies of strongly magnetized objects. We will focus on the physics and astrophysics of strongly magnetized objects, namely magnetars, accreting X-ray pulsars, and rotation powered pulsars. We also discuss the science potential of eXTP for QED studies. Developed by an international Consortium led by the Institute of High Energy Physics of the Chinese Academy of Sciences, the eXTP mission is expected to be launched in the mid 2020s., Accepted for publication on Sci. China Phys. Mech. Astron. (2019)

We present the results of an XMM-Newton observation of the slowly rotating ($P = 3.4$ s), highly magnetized ($B \approx 3 \times 10^{13}$ G) radio pulsar PSR J0726-2612. A previous X-ray observation with the Chandra satellite showed that some of the properties of PSR J0726-2612 are similar to those of the X-ray Dim Isolated Neutron Stars (XDINSs), a small class of nearby slow pulsars characterized by purely thermal X-ray spectra and undetected in the radio band. We confirm the thermal nature of the X-ray emission of PSR J0726-2612, which can be fit by the sum of two blackbodies with temperatures $kT_1 = 0.074_{-0.011}^{+0.006}$ keV and $kT_2 = 0.14_{-0.02}^{+0.04}$ keV and emitting radii $R_1 = 10.4_{-2.8}^{+10.8}$ km and $R_2 = 0.5_{-0.3}^{+0.9}$ km, respectively (assuming a distance of 1 kpc). A broad absorption line modeled with a Gaussian profile centred at $0.39_{-0.03}^{+0.02}$ keV is required in the fit. The pulse profile of PSR J0726-2612 is characterized by two peaks with similar intensity separated by two unequal minima, a shape and pulsed fraction that cannot be reproduced without invoking magnetic beaming of the X-ray emission. The presence of a single radio pulse suggests that in PSR J0726-2612 the angles that the dipole axis and the line of sight make with the rotation axis, $\xi$ and $\chi$ respectively, are similar. This geometry differs from that of the two radio-silent XDINSs with a double peaked pulse profile similar to that of PSR J0726-2612, for which $\xi \sim 90^\circ$ and $\chi \sim 45^\circ$ have been recently estimated. These results strengthen the similarity between PSR J0726-2612 and the XDINSs and support the possibility that the lack of radio emission from the latter might simply be due to an unfavourable viewing geometry., Comment: Accepted for publication in A&A, 10 pages, 10 figures

PSR B0943+10 is a mode-switching radio pulsar characterized by two emission modes with different radio and X-ray properties. Previous studies, based on simple combinations of blackbody and power law models, showed that its X-ray flux can be decomposed in a pulsed thermal plus an unpulsed non-thermal components. However, if PSR B0943+10 is a nearly aligned rotator seen pole-on, as suggested by the radio data, it is difficult to reproduce the high observed pulsed fraction unless magnetic beaming is included. In this work we reanalyze all the available X-ray observations of PSR B0943+10 with simultaneous radio coverage, modeling its thermal emission with polar caps covered by a magnetized hydrogen atmosphere or with a condensed iron surface. The condensed surface model provides good fits to the spectra of both pulsar modes, but, similarly to the blackbody, it can not reproduce the observed pulse profiles, unless an additional power law with an ad hoc modulation is added. Instead, the pulse profiles and phase-resolved spectra are well described using the hydrogen atmosphere model to describe the polar cap emission, plus an unpulsed power law. For the X-ray brighter state (Q-mode) we obtain a best fit with a temperature kT~0.09 keV, an emitting radius R~260 m, a magnetic field consistent with the value of the dipole field of 4x10^12 G inferred from the timing parameters, and a small angle between the magnetic and spin axis, $\xi$=5. The corresponding parameters for the X-ray fainter state (B-mode) are kT~0.08 keV and R~170 m., Comment: 16 pages, 10 figures, accepted for publication in ApJ

Bursts and flares are among the distinctive observational manifestations of magnetars, isolated neutron stars endowed with an ultra-strong magnetic field ($B \approx 10^{14}$--$10^{15}$ G). It is believed that these events arise in a hot electron-positron plasma which remains trapped within the closed magnetic field lines. We developed a simple radiative transfer model to simulate magnetar flare emission in the case of a steady trapped fireball. After dividing the fireball surface in a number of plane-parallel slabs, the local spectral and polarization properties are obtained integrating the radiative transfer equations for the two normal modes. We assume that magnetic Thomson scattering is the dominant source of opacity, and neglect contributions from second-order radiative processes, although double-Compton scattering is accounted for in establishing local thermal equilibrium in the fireball atmospheric layers. The observed spectral and polarization properties as measured by a distant observer are obtained summing the contributions from the patches which are visible for a given viewing geometry by means of a ray-tracing code. The spectra we obtained in the 1-100 keV energy range are thermal and can be described in terms of the superposition of two blackbodies. The blackbody temperature and the emitting area ratio are in broad agreement with the available observations. The predicted linear polarization degree is in general greater than 80% over the entire energy range and should be easily detectable by new-generation X-ray polarimeters, like IXPE, XIPE and eXTP., 19 pages, 14 figures, 1 table, accepted for publication in MNRAS

The evolution of magnetic field in isolated neutron stars is one of the most important ingredients in the attempt to build a unified description of these objects. A prediction of field evolution models is the existence of an equilibrium configuration, in which the Hall cascade vanishes. Recent calculations have explored the field structure in this stage, called the Hall attractor. We use X-ray data of near-by, cooling neutron stars to probe this prediction, as these sources are surmised to be close to or at Hall attractor phase. We show that the source RX J1856.5-3754 might be closer to the attractor than other sources of its class. Our modelling indicates that the properties of surface thermal emission, assuming that the star is in the Hall attractor, are in contradiction with the spectral data of RX J1856.5-3754., Comment: 9 pages, accepted to MNRAS

Recent optical polarimetry observations of an X-ray dim isolated neutron star, RX J1856.5-3754, showed a first evidence for QED vacuum birefringence induced by a strong magnetic field. This important result can be confirmed by performing systematically polarimetry observations in the X-ray band for other strongly magnetized neutron stars, such as transient or persistent magnetars. We computed the phase averaged polarization fraction (PF) and polarization angle (PA) expected in the thermal emission from transient magnetars in the soft X-ray energy band. We found that the detection of a PF higher than 60% is a strong evidence for vacuum birefringence. We also found that a steady change in the PA measured from transient magnetars during their outburst decay (up to 23 degrees for a magnetospheric untwisting of 0.5 rad) is a strong signature of vacuum birefringence. This latter detection would also provide an independent check of the magnetospheric untwisting model for these sources. Simulations show that these measurements are achievable by future polarimetric missions such as XIPE and IXPE with 20-380 ks of observational time, and with eXTP with 3-60 ks.

The observed polarization properties of thermal radiation from isolated, cooling neutron stars depend on both the emission processes at the surface and the effects of the magnetized vacuum which surrounds the star. Here we investigate the polarized thermal emission from X-ray Dim Isolated Neutron Stars, taking RX J1856.5-3754 as a representative case. The physical conditions of the star outermost layers in these sources is still debated, and so we consider emission from a magnetized atmosphere and a condensed surface, accounting for the effects of vacuum polarization as the radiation propagates in the star magnetosphere. We have found that, for a significant range of viewing geometries, measurement of the phase-averaged polarization fraction and phase-averaged polarization angle at both optical and X-ray wavelengths allow us to determine whether this neutron star has an atmosphere or a condensed surface. Our results may therefore be relevant in view of future developments of soft X-ray polarimeters., Comment: 12 pages, 12 figures, accepted for publication in MNRAS

Recently, numerical calculations of the magnetic field evolution in neutron stars demonstrated the possible existence of a Hall attractor, a stage at which the evolution of the field driven by the Hall cascade ends. The existence of such a stage in neutron star magnetic evolution is very important, and can be potentially probed by observations. Here we discuss three types of objects which could have reached this stage. First, we briefly describe the evolution of normal radio pulsars with ages about a few hundred thousand years. Then we analyse in more detail observations of RX J1856.5-3754, one of the Magnificent Seven, focusing on the surface temperature distribution and comparing model predictions with the temperature map inferred from X-ray observations. Finally, we discuss the necessity of the Hall attractor stage to explain the hypothetical existence of accreting magnetars. We conclude that at the moment there is no direct confirmation of the Hall attractor stage in known sources. However, more detailed observations in the near future can demonstrate existence (or absence) of this stage of the crustal magnetic field evolution., Comment: 5 pages, Proc. of PNS-2017, to be published in Open Access Journal of Physics: Conference Series (JPCS)

New-generation X-ray polarimeters currently under development promise to open a new window in the study of high-energy astrophysical sources. Among them, neutron stars appear particularly suited for polarization measurements. Radiation from the (cooling) surface of a neutron star is expected to exhibit a large intrinsic polarization degree due to the star strong magnetic field ($\approx 10^{12}-10^{15}$ G), which influences the plasma opacity in the outermost stellar layers. The polarization fraction and polarization angle as measured by an instrument, however, do not necessary coincide with the intrinsic ones derived from models of surface emission. This is due to the effects of quantum electrodynamics in the highly magnetized vacuum around the star (the vacuum polarization) coupled with the rotation of the Stokes parameters in the plane perpendicular to the line of sight induced by the non-uniform magnetic field. Here we revisit the problem and present an efficient method for computing the observed polarization fraction and polarization angle in the case of radiation coming from the entire surface of a neutron star, accounting for both vacuum polarization and geometrical effects due to the extended emitting region. Our approach is fairly general and is illustrated in the case of blackbody emission from a neutron star with either a dipolar or a (globally) twisted magnetic field., 13 pages, 9 figures, accepted for publication in MNRAS

The study of magnetars is of particular relevance since these objects are the only laboratories where the physics in ultra-strong magnetic fields can be directly tested. Until now, spectroscopic and timing measurements at X-ray energies in soft gamma-repeaters (SGRs) and anomalous X-ray pulsar (AXPs) have been the main source of information about the physical properties of a magnetar and of its magnetosphere. Spectral fitting in the ~ 0.5-10 keV range allowed to validate the "twisted magnetosphere" model, probing the structure of the external field and estimating the density and velocity of the magnetospheric currents. Spectroscopy alone, however, may fail in disambiguating the two key parameters governing magnetospheric scattering (the charge velocity and the twist angle) and is quite insensitive to the source geometry. X-ray polarimetry, on the other hand, can provide a quantum leap in the field by adding two extra observables, the linear polarization degree and the polarization angle. Using the bright AXP 1RXS J170849.0-400910 as a template, we show that phase-resolved polarimetric measurements can unambiguously determine the model parameters, even with a small X-ray polarimetry mission carrying modern photoelectric detectors and existing X-ray optics. We also show that polarimetric measurements can pinpoint vacuum polarization effects and thus provide an indirect evidence for ultra-strong magnetic fields., Comment: 12 pages, 8 figures, accepted for publication in MNRAS