# Magnetic structures and turbulence in SN 1006 revealed with imaging X-ray
polarimetry

Ping Zhou School of Astronomy and Space Science, Nanjing University, Nanjing
210023, PR China  Dmitry Prokhorov Anton Pannekoek Institute for Astronomy &
GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands  Riccardo Ferrazzoli INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Yi-Jung Yang
Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong
Laboratory for Space Research, The University of Hong Kong, Hong Kong  Patrick
Slane Center for Astrophysics — Harvard & Smithsonian, 60 Garden St,
Cambridge, MA 02138, USA  Jacco Vink Anton Pannekoek Institute for Astronomy &
GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands  Stefano Silvestri Istituto Nazionale di Fisica Nucleare, Sezione
di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy  Niccolò Bucciantini INAF
Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze,
Italy  Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze,
Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino
(FI), Italy  Estela Reynoso  Instituto de Astronomía y Física del Espacio
(IAFE), Av. Int. Guiraldes 2620, Pabellón IAFE, Ciudad Universitaria, 1428
Ciudad Autónoma de Buenos Aires, Argentina  David Moffett  Department of
Physics, Furman University, Greenville, SC 29613, USA  Paolo Soffitta INAF
Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere
100, 00133 Roma, Italy  Doug Swartz Science and Technology Institute,
Universities Space Research Association, Huntsville, AL 35805, USA  Philip
Kaaret NASA Marshall Space Flight Center, Huntsville, AL 35812, USA  Luca
Baldini Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B.
Pontecorvo 3, 56127 Pisa, Italy  Dipartimento di Fisica, Università di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy  Enrico Costa INAF Istituto di
Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  C.-Y. Ng Department of Physics, The University of Hong Kong,
Pokfulam, Hong Kong  Dawoon E. Kim INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Dipartimento di
Fisica, Università degli Studi di Roma ”La Sapienza”, Piazzale Aldo Moro 5,
00185 Roma, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Roma ”Tor
Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy  Victor
Doroshenko Institut für Astronomie und Astrophysik, Universität Tübingen, Sand
1, 72076 Tübingen, Germany  Steven R. Ehlert NASA Marshall Space Flight
Center, Huntsville, AL 35812, USA  Jeremy Heyl University of British Columbia,
Vancouver, BC V6T 1Z4, Canada  Frédéric Marin Université de Strasbourg, CNRS,
Observatoire Astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France
Tsunefumi Mizuno Hiroshima Astrophysical Science Center, Hiroshima University,
1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan  Melissa Pesce-
Rollins Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B.
Pontecorvo 3, 56127 Pisa, Italy  Carmelo Sgrò Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Toru
Tamagawa RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan  Martin C. Weisskopf NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Fei Xie Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China  INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Iván Agudo Instituto de
Astrofísica de Andalucía—CSIC, Glorieta de la Astronomía s/n, 18008 Granada,
Spain  Lucio A. Antonelli INAF Osservatorio Astronomico di Roma, Via Frascati
33, 00078 Monte Porzio Catone (RM), Italy  Space Science Data Center, Agenzia
Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy  Matteo Bachetti
INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy  Wayne H. Baumgartner NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Ronaldo Bellazzini Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Stefano
Bianchi Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre,
Via della Vasca Navale 84, 00146 Roma, Italy  Stephen D. Bongiorno NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Raffaella Bonino
Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1,
10125 Torino, Italy  Dipartimento di Fisica, Università degli Studi di Torino,
Via Pietro Giuria 1, 10125 Torino, Italy  Alessandro Brez Istituto Nazionale
di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
Fiamma Capitanio INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy  Simone Castellano Istituto
Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127
Pisa, Italy  Elisabetta Cavazzuti ASI - Agenzia Spaziale Italiana, Via del
Politecnico snc, 00133 Roma, Italy  Chien-Ting Chen Science and Technology
Institute, Universities Space Research Association, Huntsville, AL 35805, USA
Stefano Ciprini Istituto Nazionale di Fisica Nucleare, Sezione di Roma ”Tor
Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy  Space Science
Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma,
Italy  Alessandra De Rosa INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Ettore Del Monte
INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del
Cavaliere 100, 00133 Roma, Italy  Laura Di Gesu ASI - Agenzia Spaziale
Italiana, Via del Politecnico snc, 00133 Roma, Italy  Niccolò Di Lalla
Department of Physics and Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California 94305, USA  Alessandro Di
Marco INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del
Cavaliere 100, 00133 Roma, Italy  Immacolata Donnarumma ASI - Agenzia Spaziale
Italiana, Via del Politecnico snc, 00133 Roma, Italy  Michal Dovčiak
Astronomical Institute of the Czech Academy of Sciences, Boční II 1401/1,
14100 Praha 4, Czech Republic  Teruaki Enoto RIKEN Cluster for Pioneering
Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  Yuri Evangelista INAF
Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere
100, 00133 Roma, Italy  Sergio Fabiani INAF Istituto di Astrofisica e
Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
Javier A. Garcia California Institute of Technology, Pasadena, CA 91125, USA
Shuichi Gunji Yamagata University,1-4-12 Kojirakawa-machi, Yamagata-shi
990-8560, Japan  Kiyoshi Hayashida  Osaka University, 1-1 Yamadaoka, Suita,
Osaka 565-0871, Japan  Wataru Iwakiri International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  Svetlana G. Jorstad
Institute for Astrophysical Research, Boston University, 725 Commonwealth
Avenue, Boston, MA 02215, USA  Department of Astrophysics, St. Petersburg
State University, Universitetsky pr. 28, Petrodvoretz, 198504 St. Petersburg,
Russia  Fabian Kislat Department of Physics and Astronomy and Space Science
Center, University of New Hampshire, Durham, NH 03824, USA  Vladimir Karas
Astronomical Institute of the Czech Academy of Sciences, Boční II 1401/1,
14100 Praha 4, Czech Republic  Takao Kitaguchi  RIKEN Cluster for Pioneering
Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  Jeffery J.
Kolodziejczak NASA Marshall Space Flight Center, Huntsville, AL 35812, USA
Henric Krawczynski Physics Department and McDonnell Center for the Space
Sciences, Washington University in St. Louis, St. Louis, MO 63130, USA  Fabio
La Monaca INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso
del Cavaliere 100, 00133 Roma, Italy  Luca Latronico Istituto Nazionale di
Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
Ioannis Liodakis Finnish Centre for Astronomy with ESO, 20014 University of
Turku, Finland  Simone Maldera Istituto Nazionale di Fisica Nucleare, Sezione
di Torino, Via Pietro Giuria 1, 10125 Torino, Italy  Alberto Manfreda Istituto
Nazionale di Fisica Nucleare, Sezione di Napoli, Strada Comunale Cinthia,
80126 Napoli, Italy  Andrea Marinucci ASI - Agenzia Spaziale Italiana, Via del
Politecnico snc, 00133 Roma, Italy  Alan P. Marscher Institute for
Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA
02215, USA  Herman L. Marshall MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA  Giorgio Matt Dipartimento di Matematica e Fisica,
Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
Ikuyuki Mitsuishi  Graduate School of Science, Division of Particle and
Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi
464-8602, Japan  Fabio Muleri INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Michela Negro
University of Maryland, Baltimore County, Baltimore, MD 21250, USA  NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771,
USA  Stephen L. O’Dell NASA Marshall Space Flight Center, Huntsville, AL
35812, USA  Nicola Omodei Department of Physics and Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford, California
94305, USA  Chiara Oppedisano Istituto Nazionale di Fisica Nucleare, Sezione
di Torino, Via Pietro Giuria 1, 10125 Torino, Italy  Alessandro Papitto INAF
Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone
(RM), Italy  George G. Pavlov Department of Astronomy and Astrophysics,
Pennsylvania State University, University Park, PA 16802, USA  Abel L. Peirson
Department of Physics and Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California 94305, USA  Matteo Perri
Space Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy  INAF Osservatorio Astronomico di Roma, Via Frascati 33,
00078 Monte Porzio Catone (RM), Italy  Pierre-Olivier Petrucci Université
Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Maura Pilia INAF
Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius
(CA), Italy  Andrea Possenti INAF Osservatorio Astronomico di Cagliari, Via
della Scienza 5, 09047 Selargius (CA), Italy  Juri Poutanen Department of
Physics and Astronomy, 20014 University of Turku, Finland  Simonetta Puccetti
Space Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy  Brian D. Ramsey NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  John Rankin INAF Istituto di Astrofisica e
Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
Ajay Ratheesh INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy  Oliver Roberts Science and
Technology Institute, Universities Space Research Association, Huntsville, AL
35805, USA  Roger W. Romani Department of Physics and Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford, California
94305, USA  Gloria Spandre Istituto Nazionale di Fisica Nucleare, Sezione di
Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Fabrizio Tavecchio INAF
Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy
Roberto Taverna Dipartimento di Fisica e Astronomia, Università degli Studi di
Padova, Via Marzolo 8, 35131 Padova, Italy  Yuzuru Tawara  Graduate School of
Science, Division of Particle and Astrophysical Science, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan  Allyn F. Tennant NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Nicholas E. Thomas
NASA Marshall Space Flight Center, Huntsville, AL 35812, USA  Francesco
Tombesi Dipartimento di Fisica, Università degli Studi di Roma ”Tor Vergata”,
Via della Ricerca Scientifica 1, 00133 Roma, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Roma ”Tor Vergata”, Via della Ricerca Scientifica
1, 00133 Roma, Italy  Department of Astronomy, University of Maryland, College
Park, Maryland 20742, USA  Alessio Trois INAF Osservatorio Astronomico di
Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy  Sergey S.
Tsygankov Department of Physics and Astronomy, 20014 University of Turku,
Finland  Roberto Turolla Dipartimento di Fisica e Astronomia, Università degli
Studi di Padova, Via Marzolo 8, 35131 Padova, Italy  Mullard Space Science
Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5
6NT, UK  Kinwah Wu Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK  Silvia Zane Mullard
Space Science Laboratory, University College London, Holmbury St Mary,
Dorking, Surrey RH5 6NT, UK

###### Abstract

Young supernova remnants (SNRs) strongly modify surrounding magnetic fields,
which in turn play an essential role in accelerating cosmic rays (CRs). X-ray
polarization measurements probe magnetic field morphology and turbulence at
the immediate acceleration site. We report the X-ray polarization distribution
in the northeastern shell of SN1006 from a 1 Ms observation with the Imaging
X-ray Polarimetry Explorer (IXPE). We found an average polarization degree of
$22.4\pm 3.5\%$ and an average polarization angle of $-45.4\pm 4.5^{\circ}$
(measured on the plane of the sky from north to east). The X-ray polarization
angle distribution reveals that the magnetic fields immediately behind the
shock in the northeastern shell of SN 1006 are nearly parallel to the shock
normal or radially distributed, similar to that in the radio observations, and
consistent with the quasi-parallel CR acceleration scenario. The X-ray
emission is marginally more polarized than that in the radio band. The X-ray
polarization degree of SN 1006 is much larger than that in Cas A and Tycho,
together with the relatively tenuous and smooth ambient medium of the remnant,
favoring that CR-induced instabilities set the turbulence in SN 1006 and CR
acceleration is environment-dependent.

Supernova remnants (1667); Polarimetry (1278); X-ray astronomy (1810); Shocks
(2086); Cosmic rays (329)

††software:  ixpeobssim (vers. 30.0) (Baldini et al., 2022), DS9 (Joye &
Mandel, 2003; Smithsonian Astrophysical Observatory, 2000), SAS (SAS
development team, 2014), Xspec (vers. 12.9.0u, Arnaud, 1996).

##  1 Introduction

Supernova remnants (SNRs) are regarded as factories of relativistic particles.
Baade & Zwicky (1934) first proposed supernovae as a potential origin of
cosmic-ray (CRs) in the 1930s. The first observational evidence for their
hypothesis was the detection of radio emission from SNRs. The radio emission
is found to be non-thermal, generated by the relativistic electrons gyrating
in the magnetic fields (e.g., van der Laan, 1962). Nevertheless, it lacked
direct evidence for SNR accelerating CRs up to energies of $100$ TeV energy
until the discovery of non-thermal X-ray emission from bilateral shells of SN
1006 (Koyama et al., 1995). In the high-energy regime, H.E.S.S and Fermi-LAT
observations show that SN 1006 has a shell-like morphology in the TeV and GeV
band (Acero et al., 2010; Xing et al., 2016; Condon et al., 2017), further
confirming the particle acceleration in the shock front. X-ray synchrotron
emission has also been detected in a few other young SNRs that drive fast
shocks with speeds of a few thousand km s-1. As synchrotron emission is
produced by relativistic electrons gyrating in the magnetic fields, it keeps
crucial information on particle acceleration, radiative processes, and
magnetic fields.

Two key questions in the study of CRs are how magnetic fields influence CR
acceleration and how SNRs and their CRs modify magnetic fields. The diffusive
shock acceleration (DSA) theory has predicted that magnetic fields can be
strongly amplified through either the streaming instability (Bell, 2004) or
turbulent dynamo progress via density fluctuations (e.g., Giacalone & Jokipii,
2007). This prediction is supported by the presence of thin X-ray synchrotron
filaments and rapid variability of nonthermal X-ray spots in young SNRs, where
the magnetic fields are found to be orders of magnitude stronger than the
typical value in the Galaxy (Vink & Laming, 2003; Parizot et al., 2006;
Uchiyama et al., 2007). The X-ray observations before the launch of IXPE had
not directly revealed how the amplified magnetic fields are orientated and
ordered. Magnetic field amplification and turbulence are suggested to occur
close to the shock fronts, which are best traced by the X-ray synchrotron
emission. X-ray polarization measurements thus directly probe the magnetic
field direction and turbulence, helping in understanding the mechanisms of
magnetic field amplification. The X-ray polarization angle (PA; measured on
the plane of the sky from north to east; International Astronomical Union
standard) orients perpendicularly to the magnetic fields, while the
polarization degree (PD) reflects the turbulence level, with more ordered
magnetic fields probed with a high PD.

IXPE (Weisskopf et al., 2022) opened the imaging X-ray polarization window to
probe magnetic field distribution and turbulence in high-energy sources,
including SNRs. The very recent X-ray polarization measurements of the young
SNRs Cas A (Vink et al., 2022b) and Tycho (Ferrazzoli et al., 2023) showed
radially distributed magnetic fields. Cas A has a relatively low polarization
degree PD$\sim 5\%$ at the outer rim, while Tycho is more polarized with
PD$\sim 12\%$. The difference in PD suggests a different turbulence level or
scale between these two SNRs, but both are smaller than the theoretical
maximum value of PD${}_{\max}=\Gamma/(\Gamma+2/3)\sim 80\%$, with $\Gamma$ the
photon index.

SN 1006 (G327.6$+14.6$) was the third SNR observed with IXPE. It is a Galactic
SNR with a diameter of $\sim 0.5^{\circ}$ at a distance of $2.18\pm 0.08$ kpc
(Winkler et al., 2003). The IXPE observation covered the SNR’s northeastern
shell, which is composed of two rims (see circles in Figure 1). This remnant
has some unique properties for studying magnetic turbulence and CR
acceleration. Firstly, SN 1006 is among the best sources for probing spatially
resolved magnetic properties since its nonthermal rims with typical widths of
$\sim 20$–30′′ (Bamba et al., 2004; Ressler et al., 2014) can be roughly
resolved by IXPE with an angular resolution of $\sim 30^{\prime\prime}$
(Weisskopf et al., 2022). The bright nonthermal rims are produced by TeV
electrons in magnetic fields amplified to tens of the Galactic level (Parizot
et al., 2006), and are not contaminated by thermal emission in 2–8 keV.
Secondly, SN 1006 evolves in a tenuous ambient medium high above the Galactic
plane ($\sim 560$ pc), in contrast to Cas A and Tycho, which are both located
in the Galactic plane. The density fluctuation of the northeastern and eastern
shells appears small, explaining the faint and fairly uniform surface
brightness of the Hα emission (Winkler et al., 2003; Giuffrida et al., 2022).
SN 1006 is thus a test bed to study magnetic turbulence driven predominately
by CRs, because the turbulence from small-scale density fluctuations is not
considered an essential factor in this remnant. Thirdly, the bilateral
morphology SN 1006 in both X-ray (Koyama et al., 1995) and radio (Reynolds &
Gilmore, 1986) bands has been attributed to the magnetic field orientation
(Fulbright & Reynolds, 1990; Petruk et al., 2009; West et al., 2016). SN 1006
has been the focus target of observations for a long-standing question on
whether magnetic fields “quasi-parallel” or “quasi-perpendicular” to shocks
are more efficient in accelerating particles (Bamba et al., 2004; Reynoso et
al., 2013; Winkler et al., 2014). In particular, Rothenflug et al. (2004)
suggested that the bright X-ray synchrotron is associated with the polar caps,
suggesting that there the magnetic field runs parallel to the shock normal. A
direct way to answer the question is to measure the magnetic field
distribution and turbulence in SN 1006.

##  2 Observations and data reduction

###  2.1 IXPE

IXPE is an imaging X-ray polarimetry explorer working in the 2–8 keV band
(Weisskopf et al., 2022). It contains three X-ray telescopes. A detector unit
(DU) is placed at the focus of each telescope, and each contains a gas-pixel
detector (Costa et al., 2001; Bellazzini et al., 2006) that probes the
polarization of X-ray photons. IXPE provides an angular resolution of
$24^{\prime\prime}$–$30^{\prime\prime}$ (half-power diameter) and a total
effective area of $\sim 76~{}\,{\rm cm}^{2}$ at 2.3 keV, and a field-of-view
of $12\farcm{9}$ in diameter (Weisskopf et al., 2022), which allow for imaging
polarimetry of extended sources such as SNRs.

IXPE observed the northeast shell of SN 1006 (see Figure 1) in 2022 August
5–19 with the ID 01006801 and 2023 March 3–10 with ID 02001701, respectively.
The livetime of the three detectors is 628–641 ks for the observation in 2022
and 339 ks in 2023. A small fraction of observations suffered from the
contamination of solar flares, which caused high photon flux in the light
curves of the data. To assess that, we binned the 2–8 keV photons from the
source region (northeastern shell) every 120 s and obtained the light curves.
To remove the time with solar flare contamination, we used a Gaussian function
to fit the count rates distribution and filtered out the time intervals
showing count rates higher than $3\sigma$ above the mean level. Consequently,
3–5 ks were removed in 2022 and 2-3 ks in 2023 for DU1–3.

Figure 1:  Field-of-view of the two IXPE observations overlaid on the XMM-
Newton 2–8 keV image of SN 1006 (Li et al., 2016). The plus sign denotes the
SNR center (${15}^{{\rm h}}{02}^{{\rm m}}{51}\fs{7}$,
${-41}^{\circ}{56}^{\prime}{33}^{\prime\prime}$, J2000) located by Reynolds &
Gilmore (1986) and Winkler & Long (1997).

The software ixpeobssim (vers. 30.0) (Baldini et al., 2022) was applied to
reduce and analyze the IXPE data. We used version 12 of the ixpeobssim
instrument response functions of the HEASARC CALDB. The algorithm PMAPCUBE in
xpbin under the ixpeobssim software is applied to produce images of Stokes I,
Q, and U, polarization degree, and polarization angle. A few regions of
interest are selected for in-depth analysis. Further polarization analysis for
binned (e.g., within given regions) data products used the PCUBE algorithm in
xpbin. We extracted spectra of Stokes I, Q, and U with PHA1, PHAQ1, and PHAU1
algorithms in xpbin, respectively. In the spectropolarimetric analysis, we
used weighted analysis and the weight response files (Di Marco et al., 2022).
The Stokes I spectra were grouped to reach at least 50 counts per bin. A
constant 0.4 keV energy binning is applied to the Stokes Q and U spectra. The
spectra were analyzed using $Xspec$ package (vers. 12.12.1) in HEASoft
software.

In the polarimetric analysis, we did not correct the vignetting effect. The
vignetting does not influence the instrumental background. The emission
crossing the mirror module assemblies is affected, but the Stokes parameters
I, Q, and U are subject to the vignetting in the same way, and thus the source
PD and PA are not much influenced: PD$=\sqrt{Q^{2}+U^{2}}/I$, $Q/I=\cos(2{\rm
PA})$, and $U/I=\sin(2{\rm PA})$. Nevertheless, for large regions covering on-
axis and off-axis positions, vignetting puts more weight on the on-axis
position, where the effective area is larger.

###  2.2 Other data

We retrieved the XMM-Newton EPIC data of SN 1006 taken in 2009 (obs. ID:
0555630301; PI: A. Decourchelle), with a total exposure of 125 ks. After
filtering out the intervals with heavy proton flares, the net exposure is
87/89/60 ks for the MOS1/MOS2/pn camera. The data reduction was conducted
using the Science Analysis System software (SAS; vers. 19.1, SAS development
team, 2014), and spectra were analyzed using Xspec.

For comparison purposes, we also used the 1.4 GHz radio polarization data of
SN 1006 (Reynoso et al., 2013) taken with the Australia Telescope Compact
Array (ATCA). We used the Stokes I, Q, and U radio images to calculate the
degree and angle of polarization (PDr and PAr) in a few regions of interest,
which will be elaborated in Section 4.4. The polarized radio emission suffered
from Faraday rotation in the ionized foreground medium, which causes a
rotation of the polarization angle with the square of the wavelength
$\lambda^{2}$: PA${}_{r}^{\rm obs}$=PA${}_{r}+$RM $(\lambda/1~{}{\rm m})^{2}$.
Following Reynoso et al. (2013), a uniform rotation measure RM=$12~{}{\rm
rad~{}m}^{-2}$ is applied in this work to obtain the original polarization
angle PAr, although there could be local variations of RM in the SNR.

###  2.3 Position correction of the IXPE data

There was an offset of the IXPE pointing in the two epochs of observations for
SN 1006. A misalignment was also found in the Cas A observations with the
order of $2\farcm{5}$ (Vink et al., 2022b) and also in one of Tycho SNR
observations (Ferrazzoli et al., 2023). The focal pointing ($x$, $y$) has an
offset due to the boom bending and the difficulty of the bending correction
for extended sources (Weisskopf et al., 2022). We did not consider the
rotation angle correction, as it is small compared to the angular resolution
of IXPE. The roll angle of IXPE is determined to an accuracy of $0.7^{\circ}$.
Across the field of view with a diameter of $12\farcm{9}$, the maximum shift
in the position of an X-ray source due to roll angle error is $4\farcs{7}$.

To correct for the positions of the two-epoch IXPE observations, we simulated
a reference image using the XMM-Newton X-ray image of SN 1006 taken in 2009
(obs. ID: 0555630301; PI: A. Decourchelle). We are aware that the SNR shell
has a proper motion of $\sim 0\farcs{48}$ yr-1 (Katsuda et al., 2009), which
can cause a northeastern motion of $\sim 6^{\prime\prime}$ for the shell in a
baseline of 13 yr. Nevertheless, given the angular resolution of
$30^{\prime\prime}$ for IXPE, such a small offset does not significantly
influence our polarization studies. From the XMM-Newton 2–8 keV image and
spectrum, we simulated a 10 Ms IXPE observation with $ixpeobssim$ and obtained
the reference image with a pixel size of $2\farcs{46}$. Then we compared the
reference image, $img1$, and the observed image, $img2$, (also binned with a
pixel size of $2\farcs{46}$) with an offset $(dx,dy)$ added. The best-fit
offset $(dx,dy)$ is obtained when the deviation between the two images
$\sum_{i=1}^{n}[img2(i)-img1(i)]^{2}$ reaches the minimum, where $i$ and $n$
are the $i$th pixel and pixel number, respectively. With this algorithm, we
estimate the offset ($-14\farcs{8},0^{\prime\prime}$) for the observation in
2022 and ($49\farcs{3},24\farcs{6}$) for the observation in 2023, but the
offset correction could have an uncertainty. Based on a visual check, the
final offset is much smaller than the angular resolution of IXPE.

###  2.4 Background analysis

The background analysis is crucial for faint or extended sources observed with
IXPE (Xie et al., 2021; Ferrazzoli et al., 2023; Di Marco et al., 2023). SN
1006 is the faintest among the three SNRs observed so far by IXPE, and thus
the background contribution needs to be considered. The background photons
have two contributions: the diffuse X-ray background near SN 1006 and the
particle-induced instrumental background, which is nearly uniform across the
detector. Since SN 1006 is located at the high Galactic latitude
($b=14\fdg{6}$), the Galactic diffuse background is small. The 2–8 keV
background level of previous IXPE observations was found in the range
8.7–$12.1\times 10^{-4}$ counts s-1 arcmin-2 (Di Marco et al., 2023). We
conducted an energy-dependent background rejection by selecting the events
with the event-track region of interest following the strategy by Di Marco et
al. (2023). This removes up to 40% instrumental background events.

The residual background is still non-negligible, so we selected a source-free
region outside SN 1006 to estimate the background (see the dashed regions in
Figure 3). Emission in this region includes the local (sky) background of SN
1006 and residual instrumental background. We obtained the background level of
5.1–$5.9\times 10^{-4}$ counts s-1 arcmin-2 in 2–8 keV in DU1–3, while at the
SNR shell, we obtained 1.9–$2.0\times 10^{-3}$ counts s-1 arcmin-2. Therefore,
the background contributes $\sim 30$% of the photons from the shell region.

Figure 2:  IXPE images of the northeastern shell of SN 1006 in the 2–4 keV
band. Upper left: Stokes I image with a pixel size of $10^{\prime\prime}$.
Upper right: polarized signal ($\sqrt{Q^{2}+U^{2}}$; right) image binned with
a pixel $1\farcm{5}$. Lower left: polarization degree (PD) distribution for
pixels with Stokes I$>158$ counts (to mask out pixels with low X-ray
brightness), overlaid with magnetic vectors and their $1~{}\sigma$ errors. The
blue and red vectors correspond to pixels with a pre-trial significance of
2–$3~{}\sigma$ and $>3~{}\sigma$, respectively. The length of the magnetic
vectors scale with the polarized degree. Lower-right: the post-trial
confidence level for the detection of the polarized signal. The white contours
show the Stokes I levels.

##  3 Results

###  3.1 Polarization images

IXPE allows a spatially resolved polarization study of the double-rim
structure in the northeast shell of SN 1006. Figure 2 (left) shows the Stokes
I count morphology in the 2–4 keV band with a pixel size of
$10^{\prime\prime}$. We selected the 2–4 keV energy band to minimize the
contamination from the strong instrumental background above 4 keV (see
Appendix and Figure 5). The distribution of polarized X-ray emission in the
northeast of SN1006 with a bin size of $1\farcm{5}$ is also shown in Figure 2
(right). The polarized emission is made using the Stokes Q and U maps
($\sqrt{Q^{2}+U^{2}}$) and generally follows the distribution of Stokes I.

The PD distribution in the shell is shown in the lower-left panel of Figure 2
(pixel size of $1\farcm{5}$). The magnetic vectors and the $1~{}\sigma$
uncertainty range are overlaid on pixels with a pre-trial confidence level
${\rm CL_{pre}}>95\%$ ($2~{}\sigma$). In these pixels, the PDs range from 14%
to 25%, and the uncertainty is $\approx 6\%$, except for the southern pixel
with a larger error of $9\%$. The PA values for most of these pixels are
around $-45^{\circ}$ with an error of 8–$11^{\circ}$, except for the southern
pixel with PA$=-3^{\circ}$. The overall magnetic orientation tends to be
parallel to the shock normal, consistent with the radio polarization
measurement (Reynoso et al., 2013).

The quoted significance level per pixel is pre-trial significance, not
corrected for the number of pixels. Each independent pixel contributes to a
trial, and as the number of trials (pixels) increases, the probability of
finding a “significant” random fluctuation also increases. In the shell with
$N=23$ independent pixels, the probability of finding no random deviation
above a significance threshold (here taking ${\rm CL_{pre}}$) is ${\rm
CL_{post}=CL_{pre}}^{N}$. Assuming a uniform polarization degree error across
the shell, we obtained the post-trial confidence level distribution as shown
in the lower-right panel of Figure 2. The most significant pixel in the shell
has ${\rm CL_{post}}=97\%$ and two more pixels show ${\rm
CL_{post}}=87\%$–88%. To increase the significance, we need to enlarge the
pixel size or select larger regions. Moreover, the PD values are not corrected
for the background, which introduces unpolarized photons that may reduce the
PD values.

Figure 3:  Stokes I image with regions and the polar plots of these regions in 2–4 keV band. In the first panel, the regions delineated with solid and dashed lines correspond to the source and background regions, respectively. Each polar plot shows the polarization degree (%) and angles in the radial and position angle coordinates, respectively, with the background subtracted. The best-fit PD and PA values are denoted by black dots. The 50%, 90%, 99%, and 99.9% confidence levels (based upon $\chi^{2}$ on 2 degrees of freedom) are coded in red, orange, green, and blue, respectively.  Table 1: Polarization results of 5 regions by IXPE and radio observations.  | IXPE (polarimetric) | IXPE (spectropolarimetric) | radio
---|---|---|---
Region | PD | PA | $\sigma$ | PD | PA | $\Gamma$ | PDr | PAr
| (%) | (∘) |  | (%) | (∘) |  | (%) | (∘)
Shell | $22.4\pm 3.5$ | $-45.4\pm 4.5$ | 6.3 | $20.2\pm 3.3$ | $-49.4\pm 4.7$ | $2.77\pm 0.04$ | $14.5\pm 0.2$ | $-36.3\pm 0.4$
A | $20.7\pm 4.5$ | $-49.5\pm 6.2$ | 4.6 | $21.3\pm 4.7$ | $-56.1\pm 6.2$ | $2.77\pm 0.05$ | $12.5\pm 0.5$ | $-58.4\pm 1.0$
B | $22.8\pm 4.9$ | $-43.7\pm 6.2$ | 4.6 | $18.5\pm 4.8$ | $-44.8\pm 7.5$ | $2.65\pm 0.06$ | $19.1\pm 0.6$ | $-28.2\pm 0.9$
C | $21.9\pm 6.4$ | $-40.0\pm 8.3$ | 3.4 | $27.5\pm 6.6$ | $-45.3\pm 6.5$ | $2.82\pm 0.07$ | $18.2\pm 0.4$ | $-28.0\pm 0.7$
D | $<38.8$ | undetermined | 2.5 | $<31.8$ | undetermined | $3.11\pm 0.09$ | $13.9\pm 0.4$ | $-39.8\pm 0.8$

###  3.2 Polarization properties of regions

To accurately measure the polarization parameters for SN 1006, we need to
consider the contribution of background photons in the source regions. We
calculated the polarization parameters of five regions by taking the
background from a nearby source-free region (see the dashed regions in Figure
3). The regions include “Shell” for the whole northeastern shell of SN1006,
regions “A” and ”B” for the outer filaments, and regions “C” and “D” for the
inner filament (see Figure 3). This region division allows studying
polarization differences between filaments and investigating polarization
angle (PA) variation with the position angle of the structures relative to the
SNR center.

The polarization degrees and angles with their 50%, 90%, 99%, and 99.9%
confidence level contours of the five regions are shown as polar plots in
Figure 3, and the values are tabulated in Table 1. The polarization signal
from the whole northeastern shell is detected at a significance of $7\sigma$,
while the four regions A–D are detected at the 2.5 – 4.6$\sigma$ level. The
average polarization of the northeastern shell is $22.4\%\pm 3.5\%$, and the
polarization angle is $-45\fdg{4}\pm 4\fdg{5}$. Three smaller regions show a
polarization degree of 21% – $23\%$, and polarization angles between
$-50^{\circ}$ and $-40^{\circ}$, consistent with those of the entire shell.
The outer filaments (A and B) and the inner filament (C and D) reveal
identical PD, implying a similar turbulence level of the magnetic fields. The
PA in the southern regions (B and C) and northern regions (A and D) cannot be
distinguished considering the error range.

Different methods exist to divide the northeastern shell and select regions.
We also selected regions based on X-ray brightness distribution, as elaborated
in Appendix. We found that the PD is near uniform (PD$\sim 20\%$–26%) and does
not show a correlation with the X-ray brightness.

Previous radio polarization measurements of SN 1006 found that the magnetic
fields are predominantly radially distributed but also contain components
nearly parallel to the Galactic plane (Reynoso et al., 2013). To test whether
the IXPE measurements support a radial magnetic geometry, we reconstruct
events files based on the circular polarization geometry model using the
$xpstokesalign$ command in $ixpeobssim$ and check whether this approach
improves the detection of polarization. The initial Stokes parameters $q_{k}$
and $u_{k}$ of each event are rotated with respect to the SNR center at R.A
(J2000)$={15}^{{\rm h}}{02}^{{\rm m}}{51}\fs{7}$, Decl.
(J2000)=${-41}^{\circ}{56}^{\prime}{33}^{\prime\prime}$ (Reynolds & Gilmore,
1986; Winkler & Long, 1997) so that the zero for the photoelectron direction
is aligned with a circularly symmetric polarization model at the position of
the event. If the PA in SN 1006 northeast has a circular symmetry rather than
a uniform distribution, we would find significantly larger PD due to less
depolarization. Using the aligned events files, region “Shell” has PD${}_{\rm
align}=21.9\pm 3.5\%$ at a significance of $6.2\sigma$, similar to that
obtained from the uniform PA model (PD=$22.4\pm 3.5\%$, see Table
1).111Similar results are obtained by taking the SNR center (${15}^{{\rm
h}}{02}^{{\rm m}}{54}\fs{9}$, ${-41}^{\circ}{56}^{\prime}{8}\farcs{0}$, J2000)
measured by Katsuda et al. (2009): PD${}_{\rm align}=21.8\pm 3.5\%$,
PA${}_{\rm align}=1.5\pm 4.6^{\circ}$. This means that we cannot distinguish
the radial and uniform magnetic orientation models based on current IXPE
observations, which only covered the northeastern X-ray bright limb of SN
1006. The obtained PA${}_{\rm align}=0\fdg{9}\pm 4\fdg{6}$ strongly favors the
radial magnetic field (PA${}_{\rm align}=0^{\circ}$) over a tangential
magnetic morphology (PA${}_{\rm align}=90^{\circ}$).

###  3.3 Spectropolarimetric analysis of regions

A different approach to calculate the polarization of the five regions is to
use the I, Q, and U spectra. We extracted source spectra from regions “Shell”
and “A–D” and background spectra from “bkg 2022” for the 2022 observation and
from “bkg 2023” for the 2023 observations (see regions in Figure 3). For each
region, we jointly fit the background-subtracted I, Q, and U spectra with a
model combining a foreground absorption, a power-law spectrum, and a
polarization degree and angle constant over the 2–4 keV energy band
($tbabs*powerlaw*constpol$ in XSpec). Since the foreground absorption can
hardly be constrained with the $>2$ keV photons, we needed to calculate
$N_{\rm H}$ by fitting the XMM-Newton spectra using the $tbabs*powerlaw$
model. The XMM-Newton spectral fit gives the foreground gas column density
$N_{\rm H}=1.93(\pm 0.03)\times 10^{21}~{}\,{\rm cm}^{-2}$ for the shell
region. Then we fixed $N_{\rm H}$ in the IXPE spectropolarimetric analysis.

Figure 4: Left: Stokes I, Q, U spectra of the region “Shell” (data points) and
the best-fit model (solid lines). Right: The residuals in terms of
(data$-$model)/error. The 6 Stokes I spectra from two observations are coded
using different colors. The 6 Stokes Q and U spectra are grouped,
respectively, for better visualization.

Figure 5 shows the I, Q, and U spectra of region “Shell”, the best-fit model,
and the residuals. The spectral fits produced good fit statistics with
$\chi^{2}/dof=335.63/349=0.96$ for the “Shell ” region, and
$\chi^{2}/dof=0.98$–1.18 for regions A–D. The best-fit PD, PA, and photon
index $\Gamma$ in each region are shown in Table 1. The polarization
parameters obtained from the spectropolarimetric analysis agree with the
polarimetric results (using PCUBE algorithm in $xpbin$). Moreover, in the
“Shell” region, the photon index $\Gamma=2.77\pm 0.04$ of the IXPE spectra is
consistent with that given by the XMM-Newton spectral fit $\Gamma=2.74\pm
0.01$. The consistency is also found for smaller-scale regions.

##  4 Discussion

###  4.1 Magnetic orientation

IXPE observations reveal the magnetic field aligned with the shock normal in
the northeastern shell of SN 1006 (see Section 3.2). The measured magnetic
position angle of $44\fdg{6}\pm 4\fdg{5}$ is compatible with the position
angle $24^{\circ}$ – $58^{\circ}$ of the region “Shell” with respect to the
SNR center (position angle $=90^{\circ}$ for the east of SN 1006 and
$=0^{\circ}$ for the north). This is also consistent with a radial magnetic
field configuration in the observed region.

The origin of radial magnetic orientation in young SNRs is still not fully
understood, but a few explanations have been proposed (see detailed
discussions in Vink et al., 2022b). One explanation invokes MHD instabilities
introduced by density fluctuation. Young SNRs in an inhomogeneous ambient
medium drive Richtmyer-Meshkov instabilities (RMI) which lead to amplifying
the radial component of the magnetic fields (Giacalone & Jokipii, 2007; Inoue
et al., 2013). This scenario requires density fluctuations in the ambient
medium, which could be valid for remnants in a cloudy medium. However, SN 1006
evolves in a fairly uniform, tenuous medium 560 pc above the Galactic plane
(Acero et al., 2007; Giuffrida et al., 2022). It lacks observational evidence
of a large density fluctuation for driving strong magnetic turbulence.

An alternative explanation of the radial magnetic fields in the northeast
shell of SN 1006 is related to efficient particle acceleration (West et al.,
2017). If the CRs are more efficiently accelerated when the magnetic fields
are quasi-parallel to the shock normal, the CR electron density will be
amplified in the radial magnetic fields and suppressed in the tangential
fields. In this case, even if the magnetic orientation is totally random, the
X-ray synchrotron emission will tend to probe the enhanced CR electrons that
align the radial magnetic field, resulting in an apparent radial magnetic
distribution. However, as simulated by West et al. (2017), purely random
magnetic fields would predict a spherically symmetric rim, which does not
explain the limb-brightened morphology of SN 1006 (see Figure 1). Moreover, a
problem with this explanation for the radial magnetic field orientation in the
X-ray regime is that the 10–100 TeV electrons emitting X-rays have diffused
away from the region from which they originated. How much of an effect that
has depends on the length scales of the regions with radial magnetic-field
orientations compared to the diffusion length scale (see Vink et al., 2022b).

It is likely that the pre-existing magnetic fields of SN 1006 consist of a
northeast-southwest orientated component, which it kept, to some extent, after
shock propagation (Reynoso et al., 2013). Our IXPE observations did not cover
the whole SNR to test this hypothesis, but this was supported by the radio
polarization measurement of the southeastern and northwestern portions of the
shells in SN 1006 (Reynoso et al., 2013). This northeast-southwest magnetic
distribution is favored by 3D MHD simulations, which found that the quasi-
parallel CR acceleration model best reproduces the radio morphology (Bocchino
et al., 2011) and polarization images of SN 1006 (Schneiter et al., 2015).

The X-ray polarization angle distribution of SN 1006 is consistent with
efficient quasi-parallel CR acceleration, which enhances the CR electron
density and thus the synchrotron emission in the northeastern and southwestern
limbs of SN 1006. Moreover, turbulent magnetic fields can be generated in the
quasi-parallel case and vanish in the quasi-perpendicular case (Caprioli &
Spitkovsky, 2014). This can explain the PD$<$PDmax in the northeast shell of
SN 1006.

###  4.2 Magnetic turbulence

The X-ray polarization degree of the northeastern shell of SN 1006
(PD$=22.4\%\pm 3.5\%$) is smaller than the theoretical maximum value of
PD${}_{\max}=80\%$ for a photon index of $\Gamma=2.74$, supporting that the
magnetic fields are turbulent.

The turbulent magnetic fields in the upstream can be generated by streaming
instabilities driven by the CR precursors (Bell, 2004). The length scale of
non-resonant Bell instability for the most rapid growth mode is given as
(Bell, 2004):

$l_{\rm Bell}\sim 2\times 10^{17}~{}\,{\rm cm}~{}\left(\frac{V_{\rm
s}}{5\times 10^{3}~{}\,{\rm km}\,{\rm s}^{-1}}\right)^{-3}\\\
\left(\frac{n_{0}}{0.05~{}\,{\rm cm}^{-3}}\right)^{-1}\left(\frac{E_{\rm
max}}{100~{}{\rm TeV}}\right)\left(\frac{B_{0}}{3~{}\mu{\rm G}}\right)$ (1)

where we take the shock velocity $V_{\rm s}\sim 5\times 10^{3}~{}\,{\rm
km}\,{\rm s}^{-1}$ (Katsuda et al., 2009), ambient density $n_{0}\sim
0.05~{}\,{\rm cm}^{-3}$ (Acero et al., 2007), and a maximum energy $E_{\rm
max}\sim 100$ TeV (Acero et al., 2010).

Assuming the magnetic turbulence in the postshock has similar properties to
that in the preshock, we can compare the maximum length scale of the Bell
instability with the spatial resolution of IXPE $l_{\rm IXPE}$. One expects to
observe some highly polarized structures when $l_{\rm Bell}>{\rm min}(l_{\rm
X},l_{\rm IXPE})$, where $l_{\rm X}$ is the synchrotron rim width, and $l_{\rm
IXPE}$ is the structure size observed with IXPE. The FWHM rim width in the 2–7
keV band is $5^{\prime\prime}$–$48^{\prime\prime}$ (Ressler et al., 2014),
with an averaged value of $25^{\prime\prime}$ ($8\times 10^{17}~{}\,{\rm cm}$)
for the outer rim and $15^{\prime\prime}$ ($5\times 10^{17}~{}\,{\rm cm}$) for
the inner rim. The angular resolution of IXPE is $\sim 30^{\prime\prime}$,
corresponding to $10^{18}~{}\,{\rm cm}$ for SN 1006, but the regions we
selected are a few arcminutes in size. Therefore, the magnetic turbulence
scale is smaller than the typical rim width and cannot be resolved with IXPE,
which could explain the relatively low PD. We caution that the Bell
instability is predicted for the preshock region, while magnetic turbulence
properties in the postshock are unclear. Moreover, it is unclear whether the
Bell instability can result in radially distributed magnetic fields observed
with IXPE. In a more realistic case, the PD distribution depends not only on
the telescope resolution or rim width, but also on the initial turbulence
models and the nonlinear physics of turbulent cascades (Bykov et al., 2020).

The Bell mechanism is not the only process to drive turbulent magnetic fields.
Shock interaction with the inhomogeneous medium can lead to magnetic
amplification and turbulence in both preshock and postshock regions via
turbulent dynamo (Giacalone & Jokipii, 2007; Beresnyak et al., 2009; Drury &
Downes, 2012; Inoue et al., 2013; Xu & Lazarian, 2017). The characteristic
scale of the turbulent modification of the magnetic field $l_{\rm RMI}$
depends on the density fluctuation scale $l_{\Delta\rho}$ as (Inoue et al.,
2013): $l_{\rm RMI}\sim l_{\Delta\rho}/(r_{c}A)$, where $r_{c}$ is the shock
compression ratio, $\Delta\rho$ is the density fluctuation, and
$A\simeq(\Delta\rho/\rho)(1+\Delta\rho/\rho)$. This scale could be large in SN
1006, compared to $l_{\rm Bell}$, since the remnant evolves in a fairly
uniform interstellar medium in the northeast and the south (Giuffrida et al.,
2022). The northeastern shell of SN 1006 has a very low density of 0.05–0.085
cm-3 (Acero et al., 2007; Katsuda et al., 2009), although there is evidence
for denser gas in the northwest with enhanced Hα and FUV emission (e.g.
Winkler et al., 2003; Korreck et al., 2004). X-ray proper motion measurements
found that the northeast shell across different azimuthal angles expands at
nearly constant velocities (Katsuda et al., 2009), which requires a
homogeneous ambient medium beyond the shell.

Due to the small density fluctuation, it is likely that the turbulent dynamo
is less efficient in SN 1006 than in young SNRs in the clumpy medium. We
suggest that CR-induced instabilities could set the magnetic turbulence in SN
1006.

###  4.3 Comparison between SNRs observed with IXPE

Table 2: X-ray polarization degree, shock velocity and ambient density $n_{0}$ of Cas A, Tycho, and SN 1006 northeast  | PD (rim) | PD (SNR) | PD (peak) | $V_{s}$ | $n_{0}$ | ref.
---|---|---|---|---|---|---
| (%) | (%) | (%) |  ($\,{\rm km}\,{\rm s}^{-1}$) |  (cm-3) |
Cas A | $4.5\pm 1.0$ | $2.5\pm 0.5$ | $\sim 15$ | $\sim 5800$ | $0.9\pm 0.3$ | [1][2][3]
Tycho | $12\pm 2$ | $9\pm 2$ | $23\pm 4$ | $\sim 4600$ |  $\sim 0.1$–0.2 | [4][5][6]
SN 1006 NE | $22.4\pm 3.5$ | $\cdots$ | $31\pm 8$ | $\sim 5000$ |  $\sim 0.05$–0.085 | [7][8][9][10]

To date, we have the X-ray polarization measurements for three young SNRs, Cas
A, Tycho, and SN 1006. IXPE has already revealed some interesting differences
and similarities among them.

A remarkable similarity is the presence of predominantly radial magnetic
fields at the immediate postshock. In contrast, old SNRs show tangential
magnetic field orientation (Gao et al., 2011), which is believed to originate
from compressing the tangential components of ambient magnetic fields. This
reinforces that the magnetic fields in young SNRs probed by X-ray polarimetry
are not generated by compressing but need other mechanisms to work. As
discussed in Section 4.1, these mechanisms invoke MHD instabilities and
efficient CR acceleration that are crucial for understanding CR acceleration
mechanisms in SNRs. But these three SNRs do not necessarily share the origin
of their radial magnetic morphology since one mechanism may dominate over
others in specific cases. Compared to Cas A and Tycho (Vink et al., 2022b;
Ferrazzoli et al., 2023), the radial magnetic morphology of SN 1006 northeast
likely favor the efficient quasi-parallel CR acceleration in the pre-existing
field over the MHD instabilities from density fluctuations.

The three young SNRs show significant differences in the degree of
polarization as listed in Table 2. At the forward shock, the X-ray PD in SN
1006 is around twice of that in Tycho ($12\%\pm 2\%$, Ferrazzoli et al., 2023)
and 5 times of that in Cas A ($4.5\%\pm 1.0\%$; Vink et al., 2022b, see Table
2). This comparison hints that the three young SNRs do not have the same level
of turbulent magnetic fields in the immediate postshock region.

Bandiera & Petruk (2016) has provided analytic formulae for calculating the
polarization parameters of SNRs in turbulent magnetic fields. Under a
simplifying assumption that the magnetic field is a combination of an ordered
component $\bar{B}$ and a random component with isotropic Gaussian
distribution (with a kernel of $\sigma_{B}$), the PD is a function of the
turbulence level $\sigma_{B}/\bar{B}$ and the photon index $\Gamma$ (see their
Eq. 31). For the “Shell” region of SN 1006 with a photon index $\Gamma=2.74$,
we obtain the $\sigma_{B}/\bar{B}=1.2^{+0.2}_{-0.1}$. We also derived
$\sigma_{B}/\bar{B}\approx 1.8$ in Tycho rim with $\Gamma=2.82$ and
$\sigma_{B}/\bar{B}\approx 3.3$ for Cas A in its outer rim with $\Gamma=3.0$.
The comparison shows that the ratio between random and ordered magnetic fields
is largest in Cas A and smallest in SN 1006. Note that the
$\sigma_{B}/\bar{B}$ values obtained above are based on a simplified
assumption of the magnetic-field decomposition. They have a different meaning
in physics from the magnetic amplification ratio of $10$–$100$ (between the
actual field strength in the downstream and ambient field strength) derived
from the profiles of the nonthermal X-ray filaments (Vink & Laming, 2003).

What causes the different turbulent levels among the young SNRs needs more
investigation, but a possible cause is the different environments. As
discussed in Section 4.2, SN 1006 is a peculiar case with a tenuous and smooth
ambient medium, so the magnetic turbulence is likely purely generated by CR-
induced instability. Cas A is a young core-collapse SNR evolving in its highly
clumpy progenitor winds (Vink et al., 1996; Chevalier & Oishi, 2003; Koo et
al., 2023). The diffuse part of the circumstellar medium has an average
density of $0.9\pm 0.3$ cm-3, while the dense winds contain many knotty
structures a few orders of magnitude denser than the diffuse gas (Peimbert &
van den Bergh, 1971; Hwang & Laming, 2009; Lee et al., 2014; Koo et al.,
2023). Due to the large density fluctuation, the length scale of turbulence-
modified magnetic fields $l_{\rm RMI}$ should be small, and the turbulent
dynamo can play an important role in Cas A. Tycho has a Type Ia origin, and
the average ambient density is 0.1–0.2$~{}\,{\rm cm}^{-3}$ (e.g., Williams et
al., 2013). Nevertheless, the surrounding medium of Tycho is not uniform. The
radio and X-ray proper motion observations of Tycho have revealed that the SNR
is experiencing strong deceleration in the northeast and east (Reynoso et al.,
1997; Williams et al., 2016; Tanaka et al., 2021). The deceleration is
suggested to result from a very recent interaction of the SNR with a
surrounding molecular bubble swept up by the progenitor winds (Lee et al.,
2004; Zhou et al., 2016). Therefore, the density fluctuation can cause some
magnetic turbulence in Tycho.

Given the aforementioned environmental properties of Cas A, Tycho, and SN
1006, the MHD turbulent dynamo can work at a decreasing level for these
remnants. This might explain the decreasing magnetic turbulence and thus
increasing PD in the three SNRs, although we do not exclude the possibilities
that other mechanisms might also play a part and CR-induced instability alone
might cause different PDs. Our IXPE observations suggest that magnetic
turbulence, and ultimately CR acceleration in SNRs, is environment-dependent
(Xu & Lazarian, 2022).

###  4.4 Comparison between the X-ray and radio polarization

The IXPE observation of SN 1006 reveals a possible difference with the radio
polarization (Reynoso et al., 2013).

As shown in Table 1, the average angle of polarization in the radio band is
PA${}_{r}=-36\fdg{3}\pm 0\fdg{4}$, while the X-ray value is PA$=-45\fdg{4}\pm
4\fdg{5}$. We caution that a uniform RM=$12~{}\rm{rad~{}m}^{-2}$ is used to
correct the Faraday rotation and the error of RM is not included. Due to the
large uncertainty of RM value (Reynoso et al., 2013) and its influence PAr, we
cannot conclude that there is a large discrepancy in PA between radio and
X-ray bands.

X-ray PD ($22.4\%\pm 3.5\%$) in the SNR shell is larger than the radio value
($14.5\%\pm 0.2\%$) at a $\sim 2\sigma$ significance level. Although the
significance is not sufficiently large, a discrepancy in PD between X-ray and
radio measurements is not unexpected. Firstly, the intrinsic PD of the X-rays
can be larger than that of the radio emission. The maximum polarization
PD${}_{\rm max}=\Gamma/(\Gamma+2/3)$ is 80% in the X-ray band with
$\Gamma=2.74$ and 70.5% in the radio band with $\Gamma=1.6$ (Reynoso et al.,
2013). Secondly, the X-ray emission is radiated from a smaller layer than the
radio emission, and thus may have less depolarization. The synchrotron
emissivity profile reflects the strength of the magnetic fields and particle
distribution. The X-ray synchrotron emission originates from electrons freshly
accelerated to TeV energy, which lose their energy $E$ quickly with a
characteristic timescale of $\tau_{\rm loss}=640/(B^{2}E)$ s, with the
electron energy, E, in units of erg and the magnetic field strength, B, in
units of Gauss (e.g., Reynolds, 1998). Radiative energy losses as particles
advect downstream can shrink the rim width in the higher energy band $l_{\rm
loss}\approx v_{d}\tau_{\rm loss}$, with $v_{d}$ the fluid velocity downstream
of the shock. The rim width might also be regulated by the diffusion transport
of particles or/and magnetic damping, but Ressler et al. (2014) compared with
the observations and supported that the filament widths in SN 1006 decrease
with increasing photon frequency $\nu$ (width$\propto\nu^{m_{E}}$,
$m_{E}\sim-0.3$ to $-0.8$). With a larger emitting volume than the X-rays, the
radio emission also tends to be depolarized as the polarization angles vary
along the line of sight. Thirdly, if the RM is nonuniform but we incorrectly
used a uniform RM, the calculated PAr value may vary across the shell.
Consequently, combining the radio polarization data from a large region might
cause a depolarization.

##  5 Summary

In this article, we report the first X-ray polarization imaging observations
of the northeastern shell of SN 1006. The main results are summarized as
follows:

  1. 1.

We show the spatial distribution of PD and PA of the shell. The PA in
different pixels shows a tangential distribution along the shell, suggesting
that the overall magnetic orientation parallels the shock normal or has a
radial distribution.

  2. 2.

The overall degree of polarization in the northeastern shell is $22.4\%\pm
3.5\%$ and PA$=-45.5^{\circ}\pm 4.5^{\circ}$ (background removed). We checked
the variation of polarization properties by separating the shell into a few
regions, but did not found significant differences of PD and PA across the
shell.

  3. 3.

The magnetic field orientation in SN 1006 supports the efficient quasi-
parallel CR acceleration.

  4. 4.

We compared X-ray polarization measurements of Cas A, Tycho, and SN 1006. The
remarkable similarity is the radially distributed magnetic fields. But they
also show an increasing PD, likely reflecting a decreased turbulent level of
the magnetic fields. We compared their environmental density and interpreted
that the different density and density fluctuations could cause the PD
discrepancy between the three SNRs. The IXPE observations thus support that
magnetic turbulence is environment-dependent.

  5. 5.

The X-ray PD in the shell is larger than the radio PD ($14.5\%\pm 0.2\%$) at a
$\sim 2\sigma$ level. The possible explanation includes a larger intrinsic PD
in X-rays, a larger radio emission emitting area that causes a depolarization,
or a depolarization due to applying incorrect RM distribution.

The Imaging X-ray Polarimetry Explorer (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), and by the Istituto Nazionale di Astrofisica (INAF) and the 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 Goddard Space Flight Center (GSFC). P.Z.
acknowledges the support from NSFC grant No. 11590781 and Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (NWO) Veni Fellowship, grant no.
639.041.647. C.-Y. Ng and Y. J. Yang are supported by a GRF grant of the Hong
Kong Government under HKU 17305419. N.B. was supported by the INAF MiniGrant
“PWNnumpol - Numerical Studies of Pulsar Wind Nebulae in The Light of IXPE”.

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Figure 5 shows the IXPE DU1 spectrum of the region “Shell”, with the
background spectrum subtracted. The solid line shows the modeled background
spectrum. Above $\sim 4$ keV, the background flux becomes comparable to or
brighter than the source flux. Therefore, we select only the 2–4 keV energy
band to examine the polarization properties.

Figure 5:  Background-subtracted Stokes I spectrum from DU1 (data points) of the region “Shell” and the background level (solid line).  Figure 6:  Stokes I image overlaid with regions E–H with decreasing brightness.  Table 3: X-ray polarization results of 4 regions with different X-ray brightness  Region | PD | PA | $\sigma$
---|---|---|---
| (%) | (∘) |
E | $20.5\pm 5.0$ | $-47.1\pm 6.9$ | 4.1
F | $22.5\pm 5.4$ | $-36.2\pm 6.9$ | 4.2
G | $21.7\pm 5.6$ | $-45.6\pm 7.4$ | 3.9
H | $26.0\pm 7.7$ | $-39.0\pm 8.4$ | 3.4

To investigate whether the polarization parameters vary with the X-ray surface
brightness, we calculated the background-subtracted PD and PA in four new
regions, E, F, G, and H, with decreasing X-ray brightness (see Figure 6). The
obtained results are shown in Table 3. The PD and PA values of the four
regions are consistent with each other, suggesting that PD is near uniform
across the shell and independent of the X-ray surface brightness.