# Small-angle scattering interferometry with neutron orbital angular momentum
states

Dusan Sarenac<EMAIL_ADDRESS>Department of Physics, University at
Buffalo, State University of New York, Buffalo, New York 14260, USA  Institute
for Quantum Computing, University of Waterloo, Waterloo, ON, Canada, N2L3G1
Melissa E. Henderson  Institute for Quantum Computing, University of Waterloo,
Waterloo, ON, Canada, N2L3G1  Department of Physics and Astronomy, University
of Waterloo, Waterloo, ON, Canada, N2L3G1  Huseyin Ekinci  Institute for
Quantum Computing, University of Waterloo, Waterloo, ON, Canada, N2L3G1
Department of Physics and Astronomy, University of Waterloo, Waterloo, ON,
Canada, N2L3G1  Charles W. Clark  Joint Quantum Institute, National Institute
of Standards and Technology and University of Maryland, College Park, Maryland
20742, USA  David G. Cory  Institute for Quantum Computing, University of
Waterloo, Waterloo, ON, Canada, N2L3G1  Department of Chemistry, University of
Waterloo, Waterloo, ON, Canada, N2L3G1  Lisa DeBeer-Schmitt  Neutron
Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Michael G. Huber  National Institute of Standards and Technology,
Gaithersburg, Maryland 20899, USA  Owen Lailey  Institute for Quantum
Computing, University of Waterloo, Waterloo, ON, Canada, N2L3G1  Department of
Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada, N2L3G1
Jonathan S. White  Laboratory for Neutron Scattering and Imaging, Paul
Scherrer Institut, CH-5232 Villigen PSI, Switzerland  Kirill Zhernenkov
Jülich Centre for Neutron Science at Heinz Maier-Leibnitz Zentrum,
Forschungszentrum Jülich GmbH, 85748 Garching, Germany  Dmitry A. Pushin
<EMAIL_ADDRESS>Institute for Quantum Computing, University of
Waterloo, Waterloo, ON, Canada, N2L3G1  Department of Physics and Astronomy,
University of Waterloo, Waterloo, ON, Canada, N2L3G1

###### Abstract

Access to the neutron orbital degree of freedom has been enabled by the recent
actualization of methods to prepare and characterize neutron helical waves
carrying orbital angular momentum (OAM) at small-angle neutron scattering
(SANS) facilities. This provides new avenues of exploration in fundamental
science experiments as well as in material characterization applications.
However, it remains a challenge to recover phase profiles from SANS
measurements. We introduce and demonstrate a novel neutron interferometry
technique for extracting phase information that is typically lost in SANS
measurements. An array of reference beams, with complementary structured phase
profiles, are put into a coherent superposition with the array of object
beams, thereby manifesting the phase information in the far-field intensity
profile. We demonstrate this by resolving petal-structure signatures of
helical wave interference for the first time: an implementation of the long-
sought recovery of phase information from small-angle scattering measurements.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Small-angle neutron scattering (SANS) is a versatile technique that is used to
probe the nanoscale structure and dynamics of materials such as polymers,
biomolecules, magnetic nanoparticles, porous media, and metal–organic
frameworks Schmitt  _et al._ (2020); Perera  _et al._ (2018); Lai  _et al._
(2019); Heller (2023); Draper  _et al._ (2020); Heller (2021); Allen (2023).
Applications of SANS include studying self-assembly, phase diagrams,
interactions of soft matter systems, investigating the magnetic properties and
functionalization of nanomaterials, and combining SANS with neutron imaging to
access multiple length scales from micrometers to nanometers Fuhrman  _et al._
(2015); Qian  _et al._ (2018); Dewhurst  _et al._ (2015); Nagaosa and Tokura
(2013); Mühlbauer  _et al._ (2019). SANS is complementary to other scattering
methods such as small-angle X-ray scattering (SAXS), and can benefit from the
use of contrast variation and isotope labelling Schmatz  _et al._ (1974);
Jeffries  _et al._ (2021); Hollamby (2013).

In a typical SANS measurement neutrons propagate for several meters after
passing through the sample and its environment before they are detected by a
position–sensitive neutron detector. Phase information is sacrificed in
detection since the far-field neutron intensity profile is proportional to the
square modulus of the Fourier transform of the outgoing wave function. Recent
work, inspired by computed tomography methods, examined phase-recovery by
employing multi-angle SANS measurements Heacock  _et al._ (2020); Henderson
_et al._ (2023).

The need for phase retrieval in SANS is heightened by the developments in
production and detection of structured neutron waves, and their use as probes
of material properties Clark  _et al._ (2015); Sarenac  _et al._ (2016, 2019);
Larocque  _et al._ (2018); Geerits  _et al._ (2023); Sarenac  _et al._ (2018);
Nsofini  _et al._ (2016); Jach and Vinson (2022); Sherwin (2022); Afanasev
_et al._ (2021, 2019); Sarenac  _et al._ (2022); Le Thien  _et al._ (2023).
These techniques rely on tailoring the transverse phase profile of the
wavefront to induce non-trivial propagation characteristics such as orbital
angular momentum (OAM), non-diffraction, and self-acceleration. Rubinsztein-
Dunlop  _et al._ (2016); Bliokh  _et al._ (2023); Ivanov (2022); Chen  _et
al._ (2021); Ni  _et al._ (2021). For example, imprinting an azimuthal phase
gradient upon the wave function prepares helical waves that carry OAM and
display a doughnut–like intensity profile in the far-field. Allen  _et al._
(1992); Bazhenov  _et al._ (1990); Berry (2004).

To advance SANS techniques for disambiguating the phase profile of the wave
function, we adopt concepts from neutron interferometry Rauch and Werner
(2015). We create a coherent superposition of structured object and reference
beams, thereby yielding observable interference amplitudes to reveal phase
information in the far field. We demonstrate this method by observing the
petal-structure interference between two helical beams that have been prepared
using arrays of fork-dislocation phase-gratings, and whose transverse
amplitude profile takes on a doughnut-like form with propagation.

Figure 1: a) Pictorial depiction of the implemented SANS interferometry with
orbital angular momentum (OAM) states. Ref. Sarenac  _et al._ (2022) showed
that a single array of fork-dislocation phase-gratings prepares helical waves
in the diffraction orders that carry well-defined OAM values and manifest
doughnut-like intensity profiles. Here, to observe the transverse phase
profile of these helical waves we create a coherent superposition between
$\ell$ and $-\ell$ OAM states. Note that we have neglected the higher order
interference as the direct beam is orders of magnitude brighter than the
diffraction orders. b) The general idea behind the SANS interferometry
technique to observe the phase profile of a structured neutron beam with non-
trivial propagation characteristics. We consider an incoming neutron with wave
vector $k_{0}$ and two arrays with periodicity $a$ and separation distance
$d$. The array of reference beams need to spatially overlap the array of
object beams in order to interfere at the camera. Therefore, the amplitude of
the reference beams needs to diffract similar to the object beams. One
straightforward approach to achieve this is by ensuring that the phase profile
of the reference beams $\phi_{R}(x,y)$ and the object beams $\phi_{O}(x,y)$
possess a matching carrier wave vector $k_{x}$, complementary structured phase
profiles $f(x,y)$, and enforcing a translational shift $\Delta x$ that ensures
overlap (See Eq. 1).

##  II SANS interferometry with structured waves

A conventional Mach-Zehnder (MZ) perfect-crystal neutron interferometer (NI)
prepares a coherent superposition of two paths at its output, which provides
the phase information through a direct measurement of interference Rauch and
Werner (2015). A distinguishing factor from the standard optical MZ
interferometer setups is that the neutron transverse coherence length is
orders of magnitude smaller than the beam. Therefore, the action of the MZ NI
is applied to each neutron and the observed intensity at the output is their
incoherent sum.

Figure 2: The observed petal-structure signatures of neutron helical wave
interference in the first diffraction orders as the pitch and yaw of the
double-sided arrays of fork-dislocation phase-gratings was varied. a) We find
agreement for a simulation that considers a transverse coherence length of
$\sigma_{{}_{\mkern-6.0mu\perp}}$$=3~{}\mu$m and translational offsets between
the two arrays of $x_{i}=0$, $y_{i}=800$ nm. b) The SANS image after
incrementing the yaw by $\approx 0.6$ mrad corresponding to an updated offset
of $x_{i}=0$ nm, $y_{i}=961$ nm. c) The profile following an additional
increment in pitch by $\approx 0.5$ mrad corresponding to $x_{i}=-135$ nm,
$y_{i}=961$ nm. In this proof of principle experiment, the poor visibility is
mainly attributed to the wavelength spread, as depicted on Fig. 3.

A challenge arises to integrate interferometry methods and structured neutron
waves as their transverse amplitude profiles significantly change with
propagation. In order to maximize the interference signal the object beams and
the reference beams of the NI need to spatially overlap at the detector plane.
Ref. Sarenac  _et al._ (2022) introduced methods to prepare well-defined OAM
states and showed how the amplitude of the neutron wave packets deviates away
from their central phase singularity with increasing OAM value. As a result,
the radius of the doughnut-like profile of the OAM=3 (OAM=7) states at the
neutron camera was measured to be $\approx 3$ cm ($\approx 5$ cm) which was
significantly bigger than the measured half width half maximum, $\approx 1$
cm, of the OAM=0 state.

One way to achieve an appreciable overlap between the reference beams and the
object beams is to utilize reference beams that possess complementary spatial
phase profiles resulting in similar time-dependent amplitude profiles. In the
case of the OAM preparation methods of Ref. Sarenac  _et al._ (2022) this
entails creating a coherent superposition between $\ell$ and $-\ell$ OAM
states in each diffraction order, as shown in Fig. 1a. The intensity at the
camera would take on the form of the well-known petal structure indicative of
the OAM Fickler  _et al._ (2013), while the OAM dependent phase shift between
the two beams would cause the petal profile to rotate. The general concept is
shown in Fig. 1b where we consider two complementary arrays of fork-
dislocation phase-gratings with periodicity $a$ and separation distance $d$.
The interference at the detector is achieved by ensuring that the phase
profile of the reference beams $\phi_{R}(x,y)$ and the object beams
$\phi_{O}(x,y)$ possess matching carrier wave vector $k_{x}$ and complementary
structured phase profiles $f(x,y)$. It follows that there are two
interferometric conditions. First, the presence of a translational offset
$(\Delta x)$ between the object and reference elements that ensures the
reference and object beams are collinear:

$\displaystyle\Delta x\approx
d\cdot\tan(k_{x}/k_{0})=d\cdot\tan(\lambda/p)\approx d\lambda/p,$ (1)

where $\lambda~{}(k_{0})$ is the neutron wavelength (wave vector), and
$p~{}(k_{x})$ is the periodicity (wave vector) of the object and reference
elements. If $\Delta x$ is greater than the array period $a$, then the
effective translation becomes $\Delta x$ mod $a$. Note that the highest
contrast occurs for the case where the translational offset is a multiple of
the array periodicity, $\Delta x=n\cdot a$ for some integer $n$. The second
condition is that the transverse coherence length needs to be greater than
$\Delta x$, as the interference is between the first diffraction orders of the
incoming beam.

Figure 3: The effect of array structure, distance between the reference and
object elements, phase shift between the two interfering OAM states, and
wavelength distribution on the petal-structure intensity profile produced in
the first diffraction order by the double-sided arrays of fork-dislocation
phase gratings. a) Considering an ideal coherent and monochromatic neutron
beam incident on object and reference elements with negligible distance
between them, one would be able to resolve the interference pattern that is
both indicative of the OAM interference and the two-dimensional array
structure. b) Removing the higher order frequencies corresponding to the two-
dimensional array structure results in the OAM petal-structure being
emphasized. c) Increasing the distance between the reference and object
elements to the experimental value of $280~{}\mu$m introduces a phase shift
and propagation diffraction that manifests a winding structure in the
interference pattern. d) Adding to the object beams an additional OAM
dependent $\pi$ phase shift effectively rotates the petal pattern. e)
Increasing the wavelength distribution to the experimental
$\Delta\lambda/\lambda\approx 0.13$ smears the interference pattern.

##  III Materials and Methods

To aid in alignment we employed double-sided nanofabrication methods. An array
of fork-dislocation phase-gratings was fabricated on both sides of a silicon
wafer where on one side there was a $q=3$ topology and the other side
possessed $q=-3$, which results in them being 180$\degree$ relative to each
other. The arrays covered a 0.5 cm by 0.5 cm area and consisted of $6,250,000$
individual 1 $\mu$m by 1 $\mu$m fork-dislocation phase-gratings, where each
one possessed a period of $120$ nm, height $500$ nm, and was separated by 1
$\mu$m on each side from the other fork-dislocation phase-gratings. The fab
procedure for a single array can be found in the Supplementary Material of
Ref. Sarenac  _et al._ (2022). After fabricating the front side of the wafer,
it was spun with a thick PMMA to protect the fabricated gratings for the
future steps. Using the back-side alignment feature at a maskless aligner (MLA
150, Heidelberg Instrument NIS ), cross-shape marks were placed on the other
face of the wafer via the e-beam evaporation and lift-off techniques (Cr/Au
20/100 nm). The back-side alignment at MLA has an alignment accuracy of better
than 1 $\mu$m. Based on these alignment marks, the second array of gratings
was fabricated on the other face of the wafer.

A preliminary study was done at the SANS-I beamline at the Paul Scherrer
Institute Kohlbrecher and Wagner (2000), as a result of which an improved
setup was devised and implemented at the GP-SANS beamline at the High Flux
Isotope Reactor at Oak Ridge National Laboratory Wignall  _et al._ (2012). The
experimental parameters were the same as those in Ref. Sarenac  _et al._
(2022) which described the observation of neutron helical waves. The double-
sided wafer was placed inside a rotation mount which was placed 17.8 m away
from a 20 mm diameter source aperture. A 4 mm diameter sample aperture was
placed right in front of the sample. The distance from the sample to the
camera was 19 m, and the camera size spans an area of $\approx 1$ m2 with each
pixel being $\approx 5.5$ mm by $4.3$ mm in size. The wavelength distribution
was triangular with $\Delta\lambda/\lambda\approx 0.13$ and a central
wavelength of $12$ Å, and the standard deviation of the resolution
distribution was estimated to be $\sigma_{Q}=0.00016$ Å-1. The neutron
wavelength is selected by a turbine-like velocity selector which eliminates
the fractional $\lambda$ contributions that are present when a monochromator
crystal is used.

An empty beam scan without a sample and a background scan for a plain silicon
wafer of equivalent size and thickness were collected. These measurements were
used to take into account the factors that contribute to losses in intensity
and increased background scattering noise. The final SANS data was passed
though a low-pass filter to remove the Poissonian noise that varies from pixel
to pixel.

##  IV Implementation and Results

The silicon wafer sample with the double-sided arrays of fork-dislocation
phase-gratings was mounted in a holder that was then placed on a manual tip-
tilt stage with access to pitch ($\approx 2.9$ mrad/rev) and yaw ($\approx
6.9$ mrad/rev) degrees of rotation. The experiment consisted of measuring the
SANS intensity profile for 90 min as the pitch and yaw were systematically
varied by $1/6$ rev and $1/12$ rev, respectively. Given the neutron wavelength
of $12$ Å, the phase grating period $p=120$ nm, and the thickness of the wafer
$d=280~{}\mu$m, we can determine the optimal translational offset (see Eq. 1)
to be $\Delta x\approx 2.8~{}\mu$m; which from the fact that the period of the
array is $2~{}\mu$m reduces to $\Delta x\approx 0.8~{}\mu$m or a wafer
rotation angle of $\approx 3$ mrad.

To determine the behaviour of the interference pattern at the camera we
analyze the propagation of the neutron wave function through the object and
reference elements given the experimental parameters. Let us consider the wave
function of a neutron wave packet after passing through the first array:

$\displaystyle\psi_{i}(x,y)=\frac{1}{\sigma_{{}_{\mkern-6.0mu\perp}}\sqrt{\pi}}e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}_{{}_{\mkern-6.0mu\perp}}}}e^{iNb_{c}\lambda
hF_{3}[x,y]},$ (2)

where $\sigma_{x}=\sigma_{y}=$ $\sigma_{{}_{\mkern-6.0mu\perp}}$ is the
neutron transverse coherence length, $Nb_{c}$ is the scattering length density
of the sample material, $\lambda$ is the neutron wavelength, $h$ is the height
of the grating structures, and $F_{q}(x,y)$ is the profile of an array of
fork-dislocation phase-gratings whose individual profiles are given by:

$\displaystyle\frac{1}{2}\text{sgn}\left(\text{sin}\left[\frac{2\pi}{p}x+q\tan^{-1}(y/x)\right]\right)+\frac{1}{2},$
(3)

where $p$ is the grating period and $q$ is its topological charge. Note that
for $F_{q}(x,y)$ in our simulations we considered the SEM profiles found in
the Supplementary Material of Ref. Sarenac  _et al._ (2022). The wavefunction
is then propagated a distance of $280~{}\mu$m as per the wafer thickness, at
which point the phase profile corresponding to the second array is added:
$Nb_{c}\lambda hF_{-3}[x-x_{i},y-y_{i}]$, where $\\{x_{i},y_{i}\\}$ is the
translational offset between the two arrays that results from the pitch and
yaw increments. The intensity $I(x^{\prime},y^{\prime})$ at the camera is well
approximated by:

$\displaystyle I\left(x^{\prime}\equiv k_{x}\lambda z,y^{\prime}\equiv
k_{y}\lambda z\right)=|FT\\{\psi_{f}(x,y)\\}|^{2}.$ (4)

where $FT\\{\\}$ is the Fourier Transform, $z$ is the distance from the wafer
to the camera, and $\psi_{f}(x,y)$ is the wave function after the second
array. To account for the experimental parameters that set the resolution of
$\sigma_{Q}=0.00016$ Å-1, we add a low-pass filter with the corresponding
cutoff frequency to the obtained intensity profile. This removes the
diffraction signatures from the $2~{}\mu$m periodicity of the arrays.

Fig. 2a shows the observed petal-structure signatures of neutron helical wave
interference in the first diffraction orders. Using
$\sigma_{{}_{\mkern-6.0mu\perp}}$$=3~{}\mu$m we find good agreement for Fig.
2a for $x_{i}=0$, $y_{i}=800$ nm. Fig. 2b shows the SANS image after
incrementing the yaw by $\approx 0.6$ mrad and Fig. 2c shows the profile
following an additional increment in pitch by $\approx 0.5$ mrad. The
simulated profiles are then determined by the addition of the corresponding
translational offsets of $x_{i}=0$ nm, $y_{i}=961$ nm for Fig. 2b, and
$x_{i}=-135$ nm, $y_{i}=961$ nm for Fig. 2c.

Fig. 3 shows how the petal-structure intensity profile is affected by array
periodicity, distance between the reference and object elements, phase shift
between the two interfering OAM states, and wavelength distribution. Note that
in Fig. 3 we consider the described experimental parameters but with the
mentioned optimal case scenario of array period equaling the transitional
offset $a=\Delta x=2.8~{}\mu$m.

Lastly, we can note that although our model considers the full diffraction
spectra, the main contributing factors are the two complementary OAM states as
shown in Fig. 1a. Taking note that each wavelength results in a displaced
intensity profile at the camera as per the diffraction angle of
$\approx\lambda/p$, it can be shown that for the considered parameters the
superposition of OAM $=-3~{}\&~{}3$ simulates almost identical petal-structure
intensity profiles as shown in Fig. 2 & 3.

##  V Conclusion and Discussion

We have introduced and demonstrated a novel neutron interferometry technique
for extracting phase information that is typically lost in SANS measurements.
By preparing reference beams with complementary spatial phase profiles to that
of the object beams we observed the petal-structure signatures of neutron
helical wave interference for the first time. With the growing number of
neutron-imaging and -scattering user facilities worldwide Garoby  _et al._
(2017); Alarcon  _et al._ (2023), the demonstrated techniques are set to make
a high impact in the next-generation of neutron scattering studies.
Furthermore, it is also possible that this approach could be extended to
small–angle x–ray scattering Woods  _et al._ (2021). Lastly, these methods can
be extended to implementations with helical waves of atoms and molecules Luski
_et al._ (2021), which also rely on arrays of fork-dislocation gratings.

The employed neutron optics element, which was composed of an array of fork-
dislocation phase-gratings on both sides of a silicon wafer, adds an
additional capability to the toolbox of structured neutron waves for the
studies of helical features. Analogous to a $\pi/2$ spin flipper that prepares
a coherent superposition of two orthogonal spin states, this element prepares
a coherent superposition of two OAM states. Future experiments employing this
element will study helical phase shifts from materials.

There are several ways to improve the observed visibility of the presented
helical wave interference. In this proof of principle experiment, the poor
visibility is mainly attributed to the width of the wavelength distribution.
Furthermore mechanical drifts and ambient conditions, which typically are not
a concern for SANS measurements of grating structures, can induce dephasing
that degrades the visibility of the interference pattern. We can also note
that as nanofabrication methods advance it is expected that a much higher
signal in the diffraction orders will become obtainable. For example, if
amplitude-gratings or phase-grating with heights of $5~{}\mu$m were feasible
then the intensity in the diffraction orders would be around a hundred times
higher compared to this experiment that employed phase-grating heights of 500
nm. The corresponding measuring time of $<1$ min would enable a wide range of
possibilities.

## Acknowledgements

This work was supported by the Canadian Excellence Research Chairs (CERC)
program, the Natural Sciences and Engineering Research Council of Canada
(NSERC), the Canada First Research Excellence Fund (CFREF), and the US
Department of Energy, Office of Nuclear Physics, under Interagency Agreement
89243019SSC000025. This work was also supported by the DOE Office of Science,
Office of Basic Energy Sciences, in the program “Quantum Horizons: QIS
Research and Innovation for Nuclear Science” through grant DE-SC0023695. A
portion of this research used resources at the High Flux Isotope Reactor, a
DOE Office of Science User Facility operated by the Oak Ridge National
Laboratory. This work is based partly on experiments performed at the Swiss
spallation neutron source SINQ, Paul Scherrer Institute, Villigen,
Switzerland.

## References

  * Schmitt  _et al._ (2020) Julien Schmitt, Jan Joost Zeeuw, Jeroen Plomp, Wim G Bouwman, Adam L Washington, Robert M Dalgliesh, Chris P Duif, Michel A Thijs, Fankang Li, Roger Pynn, _et al._ , “Mesoporous silica formation mechanisms probed using combined spin–echo modulated small-angle neutron scattering (semsans) and small-angle neutron scattering (sans),” ACS applied materials & interfaces 12, 28461–28473 (2020).
  * Perera  _et al._ (2018) Suchithranga MDC Perera, Udeep Chawla, Utsab R Shrestha, Debsindhu Bhowmik, Andrey V Struts, Shuo Qian, Xiang-Qiang Chu, and Michael F Brown, “Small-angle neutron scattering reveals energy landscape for rhodopsin photoactivation,” The Journal of Physical Chemistry Letters 9, 7064–7071 (2018).
  * Lai  _et al._ (2019) Samson Y Lai, Kenneth D Knudsen, Benjamin T Sejersted, Asbjørn Ulvestad, Jan Petter Mæhlen, and Alexey Y Koposov, “Silicon nanoparticle ensembles for lithium-ion batteries elucidated by small-angle neutron scattering,” ACS Applied Energy Materials 2, 3220–3227 (2019).
  * Heller (2023) William T Heller, “Small-angle neutron scattering study of a phosphatidylcholine–phosphatidylethanolamine mixture,” ACS omega 8, 33755–33762 (2023).
  * Draper  _et al._ (2020) Emily R Draper, Bart Dietrich, Kate McAulay, Christopher Brasnett, Haleh Abdizadeh, Ilias Patmanidis, Siewert J Marrink, Hao Su, Honggang Cui, Ralf Schweins, _et al._ , “Using small-angle scattering and contrast matching to understand molecular packing in low molecular weight gels,” Matter 2, 764–778 (2020).
  * Heller (2021) William T Heller, “A small-angle neutron scattering study of the physical mechanism that drives the action of a viral fusion peptide,” Chemistry and Physics of Lipids 234, 105022 (2021).
  * Allen (2023) Andrew J. Allen, “Selected advances in small-angle scattering and applications they serve in manufacturing, energy and climate change,” Journal of Applied Crystallography 56, 787–800 (2023).
  * Fuhrman  _et al._ (2015) WT Fuhrman, J Leiner, P Nikolić, L DeBeer-Schmitt, J-M Mignot, SM Koohpayeh, _et al._ , “Interaction driven subgap spin exciton in the kondo insulator smb 6,” Physical review letters 114, 036401 (2015).
  * Qian  _et al._ (2018) Fengjiao Qian, Lars J Bannenberg, Heribert Wilhelm, Grégory Chaboussant, Lisa M Debeer-Schmitt, Marcus P Schmidt, Aisha Aqeel, Thomas TM Palstra, Ekkes Brück, Anton JE Lefering, _et al._ , “New magnetic phase of the chiral skyrmion material cu2oseo3,” Science Advances 4, eaat7323 (2018).
  * Dewhurst  _et al._ (2015) CD Dewhurst, M Mochizuki, K Yanai, _et al._ , “Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor gav4s8,” Nature materials 14, 1116–1122 (2015).
  * Nagaosa and Tokura (2013) Naoto Nagaosa and Yoshinori Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature nanotechnology 8, 899–911 (2013).
  * Mühlbauer  _et al._ (2019) Sebastian Mühlbauer, Dirk Honecker, Élio A Périgo, Frank Bergner, Sabrina Disch, André Heinemann, Sergey Erokhin, Dmitry Berkov, Chris Leighton, Morten Ring Eskildsen, _et al._ , “Magnetic small-angle neutron scattering,” Reviews of Modern Physics 91, 015004 (2019).
  * Schmatz  _et al._ (1974) W Schmatz, T Springer, J t Schelten, and K Ibel, “Neutron small-angle scattering: experimental techniques and applications,” Journal of Applied Crystallography 7, 96–116 (1974).
  * Jeffries  _et al._ (2021) Cy M Jeffries, Jan Ilavsky, Anne Martel, Stephan Hinrichs, Andreas Meyer, Jan Skov Pedersen, Anna V Sokolova, and Dmitri I Svergun, “Small-angle x-ray and neutron scattering,” Nature Reviews Methods Primers 1, 70 (2021).
  * Hollamby (2013) Martin J Hollamby, “Practical applications of small-angle neutron scattering,” Physical chemistry chemical physics 15, 10566–10579 (2013).
  * Heacock  _et al._ (2020) B Heacock, D Sarenac, DG Cory, MG Huber, JPW MacLean, H Miao, H Wen, and DA Pushin, “Neutron sub-micrometre tomography from scattering data,” IUCrJ 7, 893–900 (2020).
  * Henderson  _et al._ (2023) ME Henderson, B Heacock, M Bleuel, DG Cory, C Heikes, MG Huber, J Krzywon, O Nahman-Levesqué, GM Luke, M Pula, _et al._ , “Three-dimensional neutron far-field tomography of a bulk skyrmion lattice,” Nature Physics 19, 1617–1623 (2023).
  * Clark  _et al._ (2015) Charles W Clark, Roman Barankov, Michael G Huber, Muhammad Arif, David G Cory, and Dmitry A Pushin, “Controlling neutron orbital angular momentum,” Nature 525, 504–506 (2015).
  * Sarenac  _et al._ (2016) Dusan Sarenac, Michael G Huber, Benjamin Heacock, Muhammad Arif, Charles W Clark, David G Cory, Chandra B Shahi, and Dmitry A Pushin, “Holography with a neutron interferometer,” Optics express 24, 22528–22535 (2016).
  * Sarenac  _et al._ (2019) Dusan Sarenac, Connor Kapahi, Wangchun Chen, Charles W Clark, David G Cory, Michael G Huber, Ivar Taminiau, Kirill Zhernenkov, and Dmitry A Pushin, “Generation and detection of spin-orbit coupled neutron beams,” Proceedings of the National Academy of Sciences 116, 20328–20332 (2019).
  * Larocque  _et al._ (2018) Hugo Larocque, Ido Kaminer, Vincenzo Grillo, Robert W Boyd, and Ebrahim Karimi, “Twisting neutrons may reveal their internal structure,” Nature Physics 14, 1 (2018).
  * Geerits  _et al._ (2023) Niels Geerits, Hartmut Lemmel, Anna-Sophie Berger, and Stephan Sponar, “Phase vortex lattices in neutron interferometry,” Communications Physics 6, 209 (2023).
  * Sarenac  _et al._ (2018) D Sarenac, J Nsofini, I Hincks, M Arif, Charles W Clark, DG Cory, MG Huber, and DA Pushin, “Methods for preparation and detection of neutron spin-orbit states,” New Journal of Physics 20, 103012 (2018).
  * Nsofini  _et al._ (2016) Joachim Nsofini, Dusan Sarenac, Christopher J Wood, David G Cory, Muhammad Arif, Charles W Clark, Michael G Huber, and Dmitry A Pushin, “Spin-orbit states of neutron wave packets,” Physical Review A 94, 013605 (2016).
  * Jach and Vinson (2022) Terrence Jach and John Vinson, “Method for the definitive detection of orbital angular momentum states in neutrons by spin-polarized he 3,” Physical Review C 105, L061601 (2022).
  * Sherwin (2022) Joseph A Sherwin, “Scattering of slow twisted neutrons by ortho-and parahydrogen,” Physics Letters A , 128102 (2022).
  * Afanasev  _et al._ (2021) AV Afanasev, DV Karlovets, and VG Serbo, “Elastic scattering of twisted neutrons by nuclei,” Physical Review C 103, 054612 (2021).
  * Afanasev  _et al._ (2019) Andrei V Afanasev, DV Karlovets, and VG Serbo, “Schwinger scattering of twisted neutrons by nuclei,” Physical Review C 100, 051601 (2019).
  * Sarenac  _et al._ (2022) Dusan Sarenac, Melissa E Henderson, Huseyin Ekinci, Charles W Clark, David G Cory, Lisa DeBeer-Schmitt, Michael G Huber, Connor Kapahi, and Dmitry A Pushin, “Experimental realization of neutron helical waves,” Science Advances 8, eadd2002 (2022).
  * Le Thien  _et al._ (2023) Quan Le Thien, Sam McKay, Roger Pynn, and Gerardo Ortiz, “Spin-textured neutron beams with orbital angular momentum,” Physical review B 107, 134403 (2023).
  * Rubinsztein-Dunlop  _et al._ (2016) Halina Rubinsztein-Dunlop, Andrew Forbes, MV Berry, MR Dennis, David L Andrews, Masud Mansuripur, Cornelia Denz, Christina Alpmann, Peter Banzer, Thomas Bauer, _et al._ , “Roadmap on structured light,” Journal of Optics 19, 013001 (2016).
  * Bliokh  _et al._ (2023) Konstantin Y Bliokh, Ebrahim Karimi, Miles J Padgett, Miguel A Alonso, Mark R Dennis, Angela Dudley, Andrew Forbes, Sina Zahedpour, Scott W Hancock, Howard M Milchberg, _et al._ , “Roadmap on structured waves,” Journal of Optics 25, 103001 (2023).
  * Ivanov (2022) Igor P Ivanov, “Promises and challenges of high-energy vortex states collisions,” Progress in Particle and Nuclear Physics , 103987 (2022).
  * Chen  _et al._ (2021) Jian Chen, Chenhao Wan, and Qiwen Zhan, “Engineering photonic angular momentum with structured light: a review,” Advanced Photonics 3, 064001 (2021).
  * Ni  _et al._ (2021) Jincheng Ni, Can Huang, Lei-Ming Zhou, Min Gu, Qinghai Song, Yuri Kivshar, and Cheng-Wei Qiu, “Multidimensional phase singularities in nanophotonics,” Science 374, eabj0039 (2021).
  * Allen  _et al._ (1992) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
  * Bazhenov  _et al._ (1990) V. I. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with wave front screw dislocations,” Pisma v Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 52, 1037–1039 (1990).
  * Berry (2004) MV Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” Journal of Optics A: Pure and Applied Optics 6, 259 (2004).
  * Rauch and Werner (2015) Helmut Rauch and Samuel A Werner,  _Neutron Interferometry: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement_ , Vol. 12 (Oxford University Press; 2 edition, 2015).
  * Fickler  _et al._ (2013) Robert Fickler, Mario Krenn, Radek Lapkiewicz, Sven Ramelow, and Anton Zeilinger, “Real-time imaging of quantum entanglement,” Scientific reports 3, 1914 (2013).
  * (41) Certain trade names and company products are mentioned in the text or identified in an illustration in order to adequately specify the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the products are necessarily the best available for the purpose.
  * Kohlbrecher and Wagner (2000) Joachim Kohlbrecher and Werner Wagner, “The new sans instrument at the swiss spallation source sinq,” Journal of applied crystallography 33, 804–806 (2000).
  * Wignall  _et al._ (2012) George D Wignall, Kenneth C Littrell, William T Heller, Yuri B Melnichenko, Kathy M Bailey, Gary W Lynn, Dean A Myles, Volker S Urban, Michelle V Buchanan, Douglas L Selby, _et al._ , “The 40 m general purpose small-angle neutron scattering instrument at oak ridge national laboratory,” Journal of Applied Crystallography 45, 990–998 (2012).
  * Garoby  _et al._ (2017) Roland Garoby _et al._ , “Progress on the ess project construction,” Proceedings of IPAC (2017).
  * Alarcon  _et al._ (2023) R Alarcon, A Aleksandrova, S Baeßler, DH Beck, T Bhattacharya, M Blatnik, TJ Bowles, JD Bowman, J Brewington, LJ Broussard, _et al._ , “Fundamental neutron physics: a white paper on progress and prospects in the us,” arXiv preprint arXiv:2308.09059 (2023).
  * Woods  _et al._ (2021) Justin S. Woods, Xiaoqian M. Chen, Rajesh V. Chopdekar, Barry Farmer, Claudio Mazzoli, Roland Koch, Anton S. Tremsin, Wen Hu, Andreas Scholl, Steve Kevan, Stuart Wilkins, Wai-Kwong Kwok, Lance E. De Long, Sujoy Roy, and J. Todd Hastings, “Switchable x-ray orbital angular momentum from an artificial spin ice,” Phys. Rev. Lett. 126, 117201 (2021).
  * Luski  _et al._ (2021) Alon Luski, Yair Segev, Rea David, Ora Bitton, Hila Nadler, A Ronny Barnea, Alexey Gorlach, Ori Cheshnovsky, Ido Kaminer, and Edvardas Narevicius, “Vortex beams of atoms and molecules,” Science 373, 1105–1109 (2021).


# Disconnecting Open Solar Magnetic Flux

C.E. DeForest∗, T.A. Howard∗, and D.J. McComas∗† ∗Southwest Research
Institute, 1050 Walnut Street Suite 300, Boulder CO 80302  †University of
Texas at San Antonio, San Antonio TX 78249

Disconnection of open magnetic flux by reconnection is required to balance the
injection of open flux by CMEs and other eruptive events. Making use of recent
advances in heliospheric background subtraction, we have imaged many abrupt
disconnection events. These events produce dense plasma clouds whose
distinctive shape can now be traced from the corona across the inner solar
system via heliospheric imaging. The morphology of each initial event is
characteristic of magnetic reconnection across a current sheet, and the newly-
disconnected flux takes the form of a “U”-shaped loop that moves outward,
accreting coronal and solar wind material.

We analyzed one such event on 2008 December 18 as it formed and accelerated at
20 m s-2 to 320 km s-1, expanding self-similarly until it exited our field of
view 1.2 AU from the Sun. From acceleration and photometric mass estimates we
derive the coronal magnetic field strength to be $8\,\mu T$, 6$R_{\odot}$
above the photosphere, and the entrained flux to be $1.6\times 10^{11}Wb$
($1.6\times 10^{19}Mx$). We model the feature’s propagation by balancing
inferred magnetic tension force against accretion drag. This model is
consistent with the feature’s behavior and accepted solar wind parameters.

By counting events over a 36 day window, we estimate a global event rate of
$1\,d^{-1}$ and a global solar minimum unsigned flux disconnection rate of
$6\times 10^{13}Wb\,y^{-1}$ ($6\times 10^{21}Mx\,y^{-1}$) by this mechanism.
That rate corresponds to $\sim-0.2\,nT\,y^{-1}$ change in the radial
heliospheric field at 1 AU, indicating that the mechanism is important to the
heliospheric flux balance.

##  1 Introduction

The solar dynamo generates magnetic flux inside the Sun, whch is transported
outward and emerges through the Sun’s surface into the corona. Magnetic loops
build up “closed” magnetic flux (connected to the Sun at both ends) in the
corona. Some of these closed loops subsequently “open” into interplanetary
space – that is, they are connected to the Sun at only one end with the other
extending to great distances in the heliosphere or beyond. Owing to the very
high electrical conductivity, open magnetic flux is frozen into the solar wind
and carried out with it. The magnetized solar wind expands continuously
outward from the Sun in all directions, filling and inflating our heliosphere
and protecting the inner solar system from the vast majority of galactic
cosmic rays. The balance between the opening and closing of magnetic flux from
the Sun is thus critical and fundamental both to the solar wind and to the
radiation environment of our solar system.

Magnetic flux opens when coronal mass ejections (CMEs) erupt through the
corona, carrying previously closed magnetic loops beyond the critical point
where the solar wind exceeds the Alfvén speed (typically <$20\,R_{\odot}$) and
can no longer return to the Sun. CMEs were first studied in OSO-7 and Skylab
observations of the corona (e.g.Tousey 1973; Gosling et al. 1974; Hundhausen
1993), and since then continued work has provided an increasingly detailed
picture of these transient magnetic structures both during their formation and
ejection, and as they continue to evolve and interact with the solar wind.

Long lasting, radial "legs" are often observed along the flanks of a CME and
persisting behind it. These legs are generally interpreted as evidence for at
least some continued magnetic connection of CMEs back to the Sun and hence the
opening of new magnetic flux with CME ejections. That picture is further
supported by observatation, _in situ_ , of beamed suprathermal halo electrons
streaming in both directions along the local interplanetary magnetic field
(IMF) during the passage of an interplanetary CME (ICME) cloud (e.g. Gosling
1990, 1993 & references therein), which are commonly interpreted as signatures
of direct connection of the ICME magnetic field to the solar corona in both
directions, and hence of newly opening magnetic flux. However, less is known
about the equally necessary process of disconnection that must be present to
remove newly opened flux and prevent the IMF from growing without limit.

Because of the continual opening of magnetic flux through CMEs, McComas and
coworkers in the early 1990s pursued a series of studies to determine how
magnetic flux could be closed back off and avoid a so-called magnetic flux
“catastrophe” of ever increasing magnetic field strength in the interplanetary
magnetic field (McComas 1995& references therein). The amount of open magnetic
flux in interplanetary space can be approximated with the "total flux
integral" which removes the effects of variations in the solar wind speed in
determining the amount of magnetic flux crossing 1 AU (McComas et al. 1992a).
Using this integral, McComas et al. (1992a; 1995) showed that if all
counterstreaming electron events represent simply connected opening magnetic
loops, then for solar maximum CME rates, the amount of flux crossing 1 AU
would double over only ~9 months. For flux rope CMEs, significantly more
magnetic flux may be observed in the loops crossing 1 AU than what remains
attached to the Sun along the CMEs’ legs; however, it must be stressed that if
CMEs retain any solar attachment whatsoever, the flux catastrophe will
ultimately occur in the absence of some other process to close off previously
open fields.

Of course a magnetic flux catastrophe is not observed in the solar wind and,
in fact, the overall magnitude of the IMF and amount of open flux seems to
vary over the solar cycle. For cycle 21, the average magnitude varied by ~50%
(Slavin et al. 1986) while the total flux integral varied by ~60% (McComas et
al. 1992a; 1992b), with maxima shortly after solar maximum and minima shortly
after solar minimum McComas 1994. Since these studies, the solar wind has gone
through a prolonged (multi-cycle) reduction in both solar wind power (the
dynamic pressure of the solar wind that ultimately inflates the heliosphere)
(McComas et al. 2008) and magnetic field magnitude (Smith & Balogh 2008). The
lack of a flux catastrophe, solar cycle variation and now long-term reduction
in the open magnetic flux from the Sun all show that there must be some
process for closing off previously open field regions and returning magnetic
flux to the Sun.

Magnetic reconnection plays an important role in regulating the topology of
solar magnetic flux, however, once the top of a loop passes the critical
point, its magnetic flux remains open until some other process occurs to close
it off below the critical point. That is, reconnection above the critical
point can only rearrange the topology of open magnetic flux in the heliosphere
– only reconnection between two oppositely directed (inward and outward field)
regions of open flux close to the Sun can close off previously open magnetic
flux. The most obvious method of reducing the amount of magnetic flux open to
interplanetary space is via reconnection between oppositely directed,
previously open field lines (McComas et al. 1989), which creates closed field
loops that can return to the Sun and the release of disconnected U-shaped
field structures into interplanetary space. An example of such a coronal
disconnection event was shown by McComas et al. 1991 using SMM coronagraph
images from 1 June 1989\. An even older example of a likely coronal
disconnection event can be found as far back as the 16 April 1893 solar
eclipse (e.g., Cliver 1989), where sketches (data in 1893) made in time
ordered sequence from Chile, Brazil, and Senegal, indicate the outward motion
of a large U-shaped structure (McComas 1994).

For the opening and closing of the solar magnetic flux to maintain some sort
of equilibrium, there must be some type of feedback between these two
processes. McComas et al. (1989; 1991) suggested that this feedback occurs
through transverse magnetic pressure in the corona, where the expansion of
newly opened field regions must enhance transverse pressure and compress
already open flux elsewhere around the Sun. When enough pressure builds up,
reconnection between oppositely direct open flux would reduce the pressure and
amount of open flux. The sequence of images from the 27 June 1988 coronal
disconnection event, in fact showed just such a compression, indicated by the
deflection of the streamers in the corona, just prior to and appearing to
precipitate the coronal disconnection event. Another line of supporting
evidence was provided by numerical simulations (Linker et al. 1992), which
indicated that increased magnetic pressure could lead to reconnection across a
helmet streamer and the release of disconnected flux. Schwadron et al. (2010)
recently reexamined the flux balance issue in light of the anomalously long
solar minimum between cycles 23 and 24 and modeled the level of magnetic flux
in the inner heliosphere as a balance of that flux injected by CMEs, lost
through disconnection, and closed flux lost through interchange reconnection
near the Sun.

McComas et al. (1992c)conducted a statistical study of three months of SMM
coronagraph observations (Hundhausen (1993)) to assess the frequency of
coronal disconnection events. These authors found that while the initial
survey (St. Cyr & Burkepile 1990) found no obvious disconnections, six of the
53 transient events during this interval (11%) showed some evidence of
disconnection in more than one frame and 13 (23%) showed a single frame with
an outward "U" or "V" structure. Given the imaging and analysis technology of
the day, McComas et al. (1992c) concluded that magnetic disconnection events
on previously open field lines may be far more common than previously
appreciated. With today’s imaging and exceptional analysis capabilities, the
question of coronal disconnection events should finally be resolvable.

For this study we used image sequences, collected by the SECCHI (Howard et al.
2008) instrument suite on board NASA’s _STEREO-A_ spacecraft, of Thomson-
scattered sunlight from free electrons in the interplanetary plasma. The
observations span from the deep solar corona to beyond 1 AU at elongation
angles of up to $70^{\circ}$ from the solar disk; this continuous observation
is enabled by recently developed background subtraction techniques (DeForest
et al. 2011) operating on the _STEREO_ data. The signature U-shaped loops of
disconnected plasma are far clearer in the processed heliospheric data far
from the Sun than in the coronagraph data close to the Sun, and we detect 12
characteristic departing “V” or “U” events in 36 days – far more than the
expected number based on scaling the results of McComas et al. (1992c). For
this initial report, we focus on quantitative analysis of a single event. In
Sections 2.1-2.7, we describe the observations and calculate the geometry, the
mass evolution, and (by assuming the U-loop is accelerated by the tension
force) the coronal magnetic field sand entrained flux in the disconnecting
structure. As a plausibility check, we explore the tension force scenario and
its consequences for the long-term evolution of the feature, and find that the
scenario is consistent with accepted values for the solar wind density and
speed. In Section 3, we discuss broader consequences of the observation,
including estimating the disconnection rate based on the number of similar
events in our data set, and discuss implications for the global magnetic flux
balance.

##  2 Observations

The SECCHI suite on STEREO was intended to be used as a single integrated
imaging instrument (e.g. Howard et al. 2008). It consists of an EUV imager
(EUVI) observing the disk of the Sun, and four visible light imagers (COR-1,
COR-2, HI-1, and HI-2) with progressively wider overlapping fields of view, to
cover the entire range of angles between the solar disk and the Earth. The
visible light imagers view sunlight that has been Thomson scattered off of
free electrons in the corona and interplanetary space; the theory of Thomson
scattering observations has been recently reviewed by Howard & Tappin (2009a).
We set out to view coronal and heliospheric events in the weeks around 2008
December, using newly developed background subtraction techniques to observe
solar wind features in the HI-1 and HI-2 fields of view (DeForest et al.
(2011); Howard & DeForest (2011)). In the initial 36 day data set we prepared,
we observed 12 disconnection events identified by a clear “V” or “U” shaped
bright structure propagating outward in the heliosphere. We chose a
particuarly clearly presented one, which was easily traceable to its origin in
the low corona on 2008-Dec-12 at 04:00, for further detailed study.

###  2.1 Image Preparation

Data preparation followed standard and published techniques. For COR-1 and
COR-2, we downloaded Level 1 (photometrically calibrated) data, and further
processed them by fixed background subtraction: we acquired images for an
11-day period, and found the 10 percentile value of each pixel across the
entire 11-day dataset. This image was median filtered over a 5x5 pixel window
to generate a background image that included the F corona, any instrumental
stray light, and the smooth, steady portion of the K corona. Subtracting this
background from each image yielded familiar coronal images of excess feature
brightness compared to the smooth, steady background. We performed one
additional step: motion filtration to suppress stationary image components.
This step matches the motion filtration step used for HI-1 and HI-2 (below),
and suppresses the stationary streamer belt while not greatly affecting the
moving features under study.

The heliospheric imagers required further processing to remove the starfield,
which is quite bright compared to the faint Thomson scattering signal far from
the Sun in the image plane. We processed the STEREO-A HI-2 data as described
by DeForest et al. (2011). The HI-1 data used a similar process adapted to the
higher background gradients in that field of view and described by Howard &
DeForest (2011).

All the imagers yielded calibrated brightness data in physical units of the
mean solar surface brightness ( $B_{\odot}=2.3\times
10^{7}W\,m^{-2}\,SR^{-1}$). Because of the wide field of view, as a subsequent
processing step we distorted the images into azimuthal coordinates, in which
one coordinate is azimuth (solar position angle) in the image plane and the
other is either elongation angle $\epsilon$ (“radius” on the celestial sphere)
or its logarithm. The latter projection, if scaled properly, is conformal: it
preserves the shape of features that are small compared to their distance from
the Sun. To equalize brightness, we applied radial filters to the images for
presentation, with either a $\epsilon^{3.5}$ scaling (for coronal images) or a
$\epsilon^{3}$ scaling (for heliospheric images).

Figure 1 shows snapshots of the disconnected plasma and associated cusp, as
observed by four separate instruments over the course of four days as it
propagated outward. Shortly after 2008 Dec 18 04:00, the streamer belt at
$160^{\circ}$ ecliptic azimuth ($20^{\circ}$ CCW of the Sun-Earth line)
pinched and separated, forming a “U” loop that retracted outward, with a
trailing cusp, over the course of the following three days. The feature
remained visible in Thomson scattered light because of plasma scooped up
during the early acceleration period in the lower corona: this plasma remained
denser than the surrounding medium, yielding a bright feature throughout the
data set. The disconnected plasma completely missed the ecliptic plane and was
therefore not observed _in situ_ by any of the near-Earth or STEREO probes.

Figure 1: The disconnection event of 2008 Dec 18 in context \- 8 still images
showing formation and evolution of the U-loop: left to right, top to bottom.

###  2.2 Observing Geometry and 3-D structure

The observing geometry for the 2008 Dec 18 event is shown in Figure 2, from an
overhead (northward out-of-ecliptic) point of view co-rotating with the
STEREO-A orbit. The event departure angle was measured both using direct
triangulation between the coronagraphs in STEREO-A and STEREO-B. We used the
triangulation method described by Howard & Tappin (2008). Although the
disconnection event is small compared to most CMEs, subtending just a few
degrees in latitude, it is still large enough to cast doubt on the simple
triangulation results, so we also used “TH model” semi empirical transient
event reconstruction tool (Tappin & Howard 2009) to extract the departure
angle. TH was developed to reconstruct CME leading edge (“sheath”) overall
envelope and propagation speed, but is also applicable to smaller transient
events such as this one. Details, applications, and limitations of the TH
model are further described by Tappin & Howard 2009 and by Howard & Tappin
(2009b; 2010). Departure longitude was measured to be
$-10^{\circ}\text{\textpm}5^{\circ}$ in heliographic coordinates, with an
estimated event width of under $5^{\circ}.$

We took the disconnected feature’s trajectory to have constant radial motion
(the “Fixed-$\Phi$ aproximation”) in the solar inertial frame – this leads to
the slightly curved aspect to the trajectory in the co-rotating heliographic
ecliptic frame, which maintains the prime meridian at the Earth-Sun line.
Figure 2 shows an out-of-ecliptic projected view of the observing geometry,
including construction angles and distances used in Section 2.4 for trajectory
calculations.

Figure 2: Observing geometry in the ecliptic plane on 2008 Dec 18 - 2008 Dec
22

###  2.3 Feature Evolution

To analyze the feature’s evolution across a two-order-of-magnitude shift in
scale over its observed lifetime, we transformed the processed _STEREO-A_
source images into local heliographic radial coordinates – i.e. zero azimuth
is due solar West from the viewpoint of _STEREO-A_ , with azimuthal coordinate
increasing clockwise around the image plane; this follows early work by
DeForest et al. (2001) in imaging polar plumes. Distances from Sun center are
recorded as elongation angle $\epsilon$ from the center of the Sun, as a
reminder of the angular nature of the wide-field observations. To avoid
aliasing in the resampling process, we resampled the images using the
optimized resampling package described by DeForest (2004). Figures 3 and 4
show the liftoff and propagation of the feature across 65 degrees of
elongation from its origin in the solar streamer belt. Both figures have a
radial gain filter applied to equalize the feature’s brightness, which varies
by over seven orders of magnitude: from $1.5\times 10^{-9}B_{\odot}$ in the
low streamer belt at 2008 Dec 18 04:30 to $5.7\times 10^{-17}B_{\odot}$ six
days later, at $\epsilon=65^{\circ}$.

The bright feature takes the classic wishbone shape of reconnecting field
lines emerging from a current sheet (e.g. Chapter 4 of Priest et al. 2000).
The aspect ratio of the wishbone may be estimated by dividing the vertical
height from cusp to the top of the visible horns, by the width between the
horns. This aspect ratio varies from ~10:1 when the horns are first clearly
resolved near 2008 Dec 18 08:00, to approximately 2:1 some four hours later
and 1:1 by 2008 Dec 19 04:00 – one full day after the first pinch is observed
in the streamer belt. After 2008 Dec 19, the feature expands approximately
self-similarly as it propagates, subtending approximately $16^{\circ}$ of
azimuth and not changing its aspect ratio throughout the rest of its
trajectory.

Note that aspect ratio is _not_ preserved by the linear azimuthal mapping used
in Figure 3, which was selected to show the early acceleration clearly; aspect
ratio is preserved by the (conformal) logarithmic mapping used in Figure 4,
which shows nearly self-similar expansion in the image plane despite
perspective effects that come into play above about $\epsilon=30^{\circ}$

The scaling of brightness is reassuring because, in a uniformly propagating
wind with no acceleration, density must decrease as $r^{-2}$ and feature
column density must thus decrease as $r^{-1}$, while illumination decreases as
$r^{-2}$, so feature brightness is expected to decrease as $r^{-3}$. The fact
that brightness levels do not change much across Figure 4, which is scaled by
$\epsilon^{3}$, suggests that the disconnected flux and material entrained in
it are indeed propagating approximately uniformly. The fact that they _do_
change slightly, with brighter images to the right, indicates that the feature
is gaining intrinsic brightnsss by accumulating material as it propagates.

The horizontal positions and error bars in Figures 3 and 4 are the results of
manual feature location of the cusp, with a point-and-click interface. The
white error bars are based on the sharpness of the feature. In the excess
brightness plot, the feature is easy to see but blurs near the top of Figure 3
due to the higher levels of both photon noise and motion blur as the feature
accelerates to the top of the coronagraph field of view. The running
difference plot highlights fine scale feature and helps identify the cusp
location near the top of the COR-2 field of view.

Figure 3: Formation and early acceleration of the 2008 Dec 18 disconnection
event through the STEREO-A COR-1 and COR-2 fields of view The trailing edge of
the event is marked, with error bars based on feature identification. TOP:
direct excess-brightness images show feature formation and overall structure.
BOTTOM: running-difference images show detail. These stack plots include a
small image of the feature at each sampled time to show evolution. Intensities
are scaled with $\epsilon^{3.5}$ to equalize brightness vs. height. The
individual images have been resampled into linear azimuthal (radial)
coordinates, and the horizontal range is $160^{\circ}-174^{\circ}$ of azimuth.
Note that this projection does not preserve aspect ratio: despite appearances,
the event widens as it rises. Figure 4: Propagation and evolution of the 2008
Dec 18 disconnection event through the STEREO-A HI-1 and HI-2 fields of view.
The trailing edge of the event is marked, with error bars based on feature
identification. These stack plots include a small image of the feature at each
sampled time to show evolution. The individual images have been resampled into
logarithmic azimuthal (radial) coordinates, and the horizontal range is
$162^{\circ}-178\text{\textdegree}$of azimuth. This projection is conformal,
so the shape of the feature is preserved in each image. Intensities are scaled
with $\epsilon^{3}$ to equalize brightness vs. height. Note self-similar
expansion: the angular width and shape of the feature are preserved.

###  2.4 Acceleration profile

Converting angular observed coordinates to examine the inertial behavior of
the plasma requires triangulation using the Law of Sines. Using the “fixed
$\Phi"$ approximation (assuming the feature’s cusp is small and that it
propagates in a radial line from the Sun), the feature’s radius from the Sun
is easily calculated:

$r_{ev}=r_{A}\frac{{sin\left(\epsilon^{\prime}\right)}}{sin\left(\epsilon\right)},$
(1)

where the variables take the meanings in Figure 2: $\epsilon$ is the solar
elongation of the feature as seen from _STEREO-A_ , $\epsilon^{\prime}$ is the
solar elongation of _STEREO-A_ as seen from the feature), and the _STEREO-A_
solar distance $r_{A}$ is found by spacecraft tracking and is supplied by the
mission. Although no camera was present at the event itself,
$\epsilon^{\prime}$ is calculated by noting that
$\epsilon^{\prime}=180^{\circ}-\epsilon-\left(L-L_{ev}\right)$, as the
feature, _STEREO-A,_ and the Sun form a triangle. Figure 5 shows the results
of the tracking from Figures 3 and 4, propagated through Equation 1.

As expected, the event rapidly accelerates during the early phase, reaching a
peak acceleration of $20\,m\,s^{-2}$ as the aspect ratio changes in the
initial hours. The acceleration peaks 4-5 hours after the initial pinch in the
streamer belt, or 2-3 hours after the first observation of a well-formed cusp.
The feature reaches its final speed of $\sim 320\pm 15\,km\,s^{-1}$ within
just 8 hours of the initial pinch at 04:00 and within 6 hours of the first
observation of the well formed cusp at 06:00, and undergoes no further
significant acceleration nor deceleration during its obsered passage to beyond
1 AU over the next five days.

Figure 5: Inferred position, speed, and acceleration of the disconnected
plasma from the 2008-Dec-18 event, during onset (LEFT) and over the full
observation period (RIGHT). Error bars are derived by propagating _a priori_
location error and geometric error in the longitude of the event. The shaded
region indicates the full time range of the left-side plots.

###  2.5 Mass profile

We extracted photometric densities using the feature brightness in suitable
frames. The feature brightness is determined from the density via the Thomson
scattering equation (see, e.g., Howard & Tappin 2009a for a clear exposition).
Compact features can be treated as nearly point sources, and the line-of-sight
integral for the optically thin medium reduces to:

$B=B_{\odot}\Omega_{\odot}(r)\sigma_{e}\left(1+cos^{2}\chi\right)\rho\mu_{av}^{-1}d$
(2)

where $B$ is the measured feature brightness (in units of emissivity:
$Wm^{-2}SR^{-1}$), $B_{\odot}$ is (still) the solar surface brightness;
$\Omega_{\odot}(r)$ is the solid angle subtended by the Sun at the point of
scatter, well approximated by $\pi r_{\odot}^{2}r_{ev}^{-2}$ everywhere above
about 4 $r_{\odot}$; $\sigma_{e}$ is the differential Thomson scattering cross
section, given by half of the square of the classical electron radius
$r_{e}^{2}/2=4.0\times 10^{-30}m^{2}$; $\chi$ is the scattering angle (equal
to $\epsilon^{\prime}$ in Figure 2); $\rho$ is the mass density; $\mu_{av}$ is
the average mass per electron in the coronal plasma; and $d$ is the depth of
the feature.

$mu_{av}$ may be calculated from the spectroscopically measured 5% He/H number
ratio in the corona (Laming & Feldman 2000) and the assumption that the helium
is fully ionized (yielding two electrons per ion). This yields
$mu_{av}=1.1m_{p}=1.84\times 10^{-27}kg$.

Solving for the line-of-sight integrated mass surface density $\rho d$ gives:

$\rho
d=\mu_{av}\frac{B}{B_{\odot}}\Omega_{\odot}^{-1}(r)\sigma_{e}^{-1}\left(1+cos^{2}\epsilon^{\prime}\right)^{-1}$
(3)

and therefore

$m_{ev}=\left(\rho
d\right)wh=\mu_{av}\frac{B}{B_{\odot}}\Omega_{\odot}^{-1}(r)\sigma_{e}^{-1}\left(1+cos^{2}\epsilon^{\prime}\right)^{-1}\Omega_{ev}^{\left(1+cos^{2}\epsilon^{\prime}\right)}S^{2}$
(4)

where $w$ and $h$ are the dimensions shown in Figure 7; $\Omega_{ev}$ is the
solid angle subtended by the feature in the images; and $S$ is the calculated
spacecraft-feature distance, calculated by the law of sines as for $r_{ev}.$

To extract the mass profile from the data, we generated an image sequence
containing the feature, and marked the locus of the feature visually using a
pixel paint program. Using the generated masks, we summed masked pixels in the
feature for each photometric image, thereby integrating the feature brightness
over the solid angle represented by the corresponding pixels, to obtain an
intensity and an average brightness within the feature. To account for errors
in visual masking, we assigned error bars based on one-pixel dilation and one-
pixel contraction of the masked locus. We omitted frames with excessive noise,
encroachment of an image boundary, or a star or cosmic ray in or near the
feature. The results of the calculation are given in Figure 6, which shows
steady accretion of material through most of the journey through the
heliosphere.

Because our photometric analysis is based on subtraction of a calculated
background derived from the data set itself, we measure only excess feature
brightness (not absolute brightness) from Thomson scattering; thus our
brightness measurements and mass estimates are biased low, because we cannot
measure the absolute density of the background. The initial derived mass of
20-25 Tg translates to an electron number density of $2\times 10^{7}\,cm^{-3}$
in the lower corona, which is comparable to the density in bright coronal
features – so the total mass may be up to a factor of order two higher.

The final “feature excess” mass is $8\pm 2\times 10^{10}kg$, and the final
subtended solid angle is 0.028 SR, for a presented cross-section of $4.3\pm
0.1\times 10^{20}m^{2}$. Taking the depth to be the square root of the
observed cross section yields an estimated volume at 1 AU of $8.9\pm 0.3\times
10^{31}m^{3}$, for a total estimated excess electron density of $5\pm
1.4\,cm^{-3}$ at 1 AU, which is in good agreement with slow solar wind
densities ($3-10\,cm^{-3}$ when scaled to 1 AU) that were observed by
_Ulysses_ in situ in the same heliographic latitude range (e.g. McComas et al.
2000). Approximately 2/3 of this excess density appears to have been
accumulated enroute from the surroundign solar wind; this is further described
in Section 2.7.

Figure 6: Photometrically determined excess mass profile of the retracting
disconnected feature of 2008 Dec 18. Error bars are based on identification of
the feature boundary in the images. The trendline is extracted from regression
of the HI-1 and HI-2 data. The mass shown is excess mass in the feature
compared to the background solar wind (see text).

###  2.6 Entrained magnetic flux

Figure 7: Cartoon of the initial acceleration process of a disconnection
event. Tension force along newly released field lines is balanced by mass
entrained on the field lines. By measuring the acceleration and mass we infer
the amount of magnetic flux that was disconnected.

From the mass of the feature, and its acceleration, it is possible to extract
the entrained magnetic field by measuring the rate of change of momentum and
inferring a magnetic tension force via $f=ma$. The system is sketched in
Figure 7. The magnetic tension force is conserved along the open field lines,
so we can calculate it at any convenient cut plane including the one shown.
Tension force is frequently referred to as a “curvature force” and calculated
locally; here we integrate around the “U”, and notice that the integrated
force is just the unbalanced tension on the field lines contained in the “U”
shape. It is therefore given by

$f=m_{ev}a_{ev}=f_{B}=\frac{B^{2}A}{2\mu_{0}}=\frac{{\Phi^{2}}}{2\mu_{0}A}$
(5)

where, here, $B$ is the magnetic field strength (not brightness, as before).
Solving for $\Phi$,

$\Phi=\sqrt{2\mu_{0}dwm_{ev}a_{ev}}$ (6)

taking $m_{ev}$ to be $25Tg$ ($2.5\times 10^{10}kg$) during the peak of the
acceleration, and taking $w=d=0.2R_{\odot}$ (based on the measured width of
the feature’s fork during maximum acceleration, at $6R_{s}$ from the surface
($7R_{s}$ from Sun center) gives $\Phi=1.6\times 10^{11}Wb$ ($1.6\times
10^{19}Mx$ ), corresponding to an average field strength of 8$\mu T$ ( 0.08
Gauss) at that altitude, or an equivalent $r^{2}$-scaled field of 400$\mu T$
(4 Gauss) at the surface; this is comparable to accepted values of the open
flux density at the solar surface at solar minimum. Because of the way $m$ was
calculated (section 2.5 above) this figure is probably low by a factor of
order $\sqrt{2}$.

###  2.7 Accretion and force balance

As the disconnectioned structure travels outward, it accretes new material.
This effect is dramatic: as seen in Figure 6, the mass increases by a factor
of 3 from the corona to 1 AU. We conjecture that the material is accreted by
“snowplow” effects from the plasma ahead of the disconnected cusp as it
propagates. For the observed mass growth in the feature, new material must be
compressed to become visible in our Thomson scattering images, and the most
plausible way for it to be compressed is via ram effects. This scenario also
neatly explains the constant speed of the feature, by balancing the continued
tension force from the cusp with accretion momentum transfer. Here we explore
the concept of force balance between accretion and the tension force, to
identify whether some other model is required in addition to this simple one.

Extending Newton’s law to include momentum transfer by accretion, and
neglecting all but the tension force,

$\frac{\Phi^{2}}{2\mu_{0}A_{\Phi}}=m_{ev}a_{ev}+\frac{dm_{ev}}{dt}\Delta v,$
(7)

where the LHS is just the tension force from Equation 5, with the modification
that the cross section of the exiting field lines is written $A_{\Phi}$;
$a_{ev}=0$ after the initial acceleration; and the second term represents
momentum transfer into accreted material, with $\Delta v$ being the difference
between the feature speed and surrounding wind speed. The feature is thus in
equilibrium between accretion drag and continued acceleration by the tension
force. This accretion drag is important to the observed increase in feature
mass, because ram pressure against the surrounding wind material is what
compresses incoming material and renders it visible in the data against the
subtracted background.

Applying conservation of mass, we can relate $\Delta v$ and the average
density of the background solar wind through which the feature is propagating:

$\rho_{sw}=\frac{dm_{ev}/dt}{A_{ev}\Delta v}.$ (8)

where $A_{ev}$ is the geometrical area presented by the feature to the slow
wind ahead of it. Solving Equations 7 and 8 to eliminate $\Delta v$ gives

$\rho_{sw}=\left(\frac{dm_{ev}}{dt}\right)^{2}\left(\frac{2\mu_{0}}{\Phi^{2}}\right)\left(\frac{A_{\Phi}}{A_{ev}}\right),$
(9)

which gives the background solar wind density in terms of the accumulation
rate of mass in the observed feature, assuming constant outflow for both the
wind and the feature, and acceleration by the tension force. Given the
conservation of mass and the approximately constant speed of the solar wind,
$\rho_{sw}$ falls as approximately $r^{-2}$. Further, we observe nearly self-
similar expansion throughout most of the heliospheric range, so
$A_{\Phi}/A_{ev}$ is constant in that part of the trajectory – hence
$dm_{ev}/dt$ must also fall as $r^{-1}$ during the approximately constant
speed portion of the feature’s lifetime. Using this functional form, we can
extract an analytic expresson for the feature mass versus radius. We introduce
the $r^{-1}$ dependence by switching from the linear regression used in Figure
6, to a semi-log regression that assumes $dm_{ev}/d(log_{e}(r)$ to be
constant. Figure 8 shows such a regression, with the result that
$dm_{ev}/dr=34\pm 3\times 10^{9}\left(R_{\odot}/r\right)kgR_{\odot}^{-1}$.
Including the measured outflow speed of $315\pm 15\,km\,s^{-1}$, we find that
$dm_{ev}/dt=\left(1AU/r\right)\left(7.1\pm 1\times 10^{4}kg\,s^{-1}\right)$.

Including all of these values into Equation 9, together with the average
particle mass from Section 2.5, yields a background wind numeric density of
$n_{sw}\left(1\,AU\right)$ of $30\pm 6\,cm^{-3}\left(A_{\Phi}/A_{ev}\right)$.
From the morphology of the feature in Figure 4, we conservatively estimate
$A_{\Phi}/A_{ev}<0.25$, i.e. the forward cross section of the “horns” of the
vee appears to be well under 1/4 of the cross section of the vee itself. This
value yields a derived background solar wind density of $n_{sw}<8\,cm^{-3}$ at
1 AU to maintain the force balance in Equation 7. That figure is again in line
with the wind measurements from _Ulysses_ at $15^{\circ}$ heliographic
latitude (McComas et al. 2000), adding to the plausibility of the accretion
force balance picture. The corresponding mass density limit is
$\rho_{sw}<8\times 10^{-19}\,kg\,m^{-3}$ at 1 AU

As a sanity check, we can use this $\rho_{sw}$ limit and Equation 8 to find
that $\Delta v$ must then be a few tens of $km\,s^{-1},$ i.e. the background
wind speed must be close to $300\,km\,s^{-1}$.

We conclude that the picture of force balance between snowplow accretion and
the tension force is at least broadly consistent with the observed feature,
though further study of more events (preferably with corresponding _in situ_
measurements of the feature itself) is necessary.

Figure 8: Semilog regression fit of $m_{ev}(r)$

##  3 Discussion

Using data from STEREO/SECCHI, we have identified and measured the
characteristics of a single flux disconnection event and associated cusp
feature, similar to that discovered by McComas et al. (1992), from initial
detection in the lower corona to distances beyond 1 AU. The cusp feature is
formed in the classic X-point geometry and rapidly accelerates under the
tension force to approximately $320\,km\,s^{-1}$, which it reaches in under 4
hours at an altitude of approximately 10$R_{\odot}$. Thereafter the feature
continues to accumulate mass but maintains approximately constant speed until
it is lost to sight 1.2 AU from the Sun.

Based on photometry, we are able to estimate the onset mass of the event as
$25\,Tg$ and the entrained flux as $160\,GWb,$ corresponding to a coronal
field strength of $0.08\,G$ and an $r^{2}$-normalized surface open field of
$4\,G$ over the projected surface footprint of the feature. These estimates
are likely low by a factor of order $\sqrt{2}$, because they make use of
feature excess brightness rather than absolute Thomson-scattered brightness in
the coronagraph images; using polarized-brightness imagery could improve the
measurement by separating the non-transient component of the Thomson
scattering signal from the unwanted F coronal background.

Because our measurements are all based on morphology and photometry, we have
performed several consistency checks to build confidence in the calculated
parameters of the feature as it propagates. In particular, a model of simple
force balance between the tension force and mass accretion is consistent with
both the inferred magnetic field and accepted values for background slow solar
wind density and speed.

Simple accretion models such as we developed here demonstrate clearly why
ejected features such as U-loops or CMEs seem frequently to propagate at near
constant speed: under continuous weak driving, an equilibrium forms rapidly
between the driving force and momentum transfer by mass accretion. The
equilibrium outflow speed is the sum of a large, fixed (or at least driver-
independent) speed – that of the surrounding wind – with a smaller offset
speed that drives mass accretion. Thus the feature speed is quite insensitive
to the driver. In our case, doubling the tension force would only increase the
outflow rate by $\sim 10\%$.

The event under study is well presented, but is not unusual at all; such
events are easy to identify in heliospheric image sequences, because of their
distinctive “U” and cusp shape; they are readily traced back to the corona.
This technique represents a new, very effective way of finding these
disconnection events, which are small and hard to identify __ in the
coronagraph sequences alone, but are strongly and easily visible in the
processed heliospheric images.

In an initial reduced data set of 36 days near the deepest part of the recent
extended solar minimum (2008 Dec – 2009 Jan), we identified 12 such events;
all of them were identified by tracking “V” or “U” shapes back from the
heliospheric images to the corona. Assuming the present feature to be typical,
and considering that the single viewpoint affords clear coverage of about 1/4
of the circumference of the Sun, we estimate the global disconnection feature
rate at that time to be over $1\,event\,d^{-1}$, and the flux disconnection
rate to thus be at least of order $60\,TWb\,y^{-1}$. Expanded to a 1 AU
sphere, this amounts to a rate of change of the open field of order
$0.2\,nT\,y^{-1}$, which is a significant fraction of the observed cycle-
dependent rate of change of the open heliospheric field (e.g. Schwadron,
Connick, & Smith 2010). These figures are based on a single calculated flux
and an event rate obtained by initial visual inspection of a single 36-day
data set, and hence are merely rough estimates – but they indicate that flux
disconnections of this type are important to the global balance of open flux.
Further study, in the form of a systematic survey, is needed to determine
whether they are the primary mechanism of flux disconnection from the Sun.

The authors thank the STEREO instrument teams for making their data available.
Our image processing made heavy use of the freeware Perl Data Language
(http://pdl.perl.org). The work was enhanced by enlightening conversations
with J. Burkepile, C. Eyles, and N. Schwadron, to whom we are indebted. This
work was supported by NASA’s SHP-GI program, under grant NNG05GK14G.

##  4 References

## References

  * Cliver (1989) Cliver, E. W. 1989, Sol. Phys., 122, 319
  * DeForest (2004) DeForest, C. E. 2004, Sol. Phys., 219, 3
  * DeForest et al. (2011) DeForest, C. E., Howard, T. A., & Tappin, S. J. 2011, ApJ, 738, 103
  * DeForest et al. (2001) DeForest, C. E., Plunkett, S. P., & Andrews, M. D. 2001, ApJ, 546, 569
  * Gosling (1990) Gosling, J. T. 1990, Washington DC American Geophysical Union Geophysical Monograph Series, 58, 343
  * Gosling (1993) —. 1993, J. Geophys. Res., 981, 18937
  * Gosling et al. (1974) Gosling, J. T., Hildner, E., MacQueen, R. M., Munro, R. H., Poland, A. I., & Ross, C. L. 1974, J. Geophys. Res., 79, 4581
  * Howard et al. (2008) Howard, R. A., et al. 2008, Space Sci. Rev., 136, 67
  * Howard & DeForest (2011) Howard, T. A., & DeForest, C. E. 2011, ApJ, (submitted)
  * Howard & Tappin (2008) Howard, T. A., & Tappin, S. J. 2008, Sol. Phys., 252, 373
  * Howard & Tappin (2009a) —. 2009a, Space Sci. Rev., 147, 31
  * Howard & Tappin (2009b) —. 2009b, Space Sci. Rev., 147, 89
  * Howard & Tappin (2010) —. 2010, Space Weather, 80, S07004
  * Hundhausen (1993) Hundhausen, A. J. 1993, J. Geophys. Res., 981, 13177
  * Linker et al. (1992) Linker, J. A., van Hoven, G., & McComas, D. J. 1992, J. Geophys. Res., 971, 13733
  * McComas (1994) McComas, D. J. 1994, NASA STI/Recon Technical Report N, 951, 10770
  * McComas (1995) —. 1995, Tongues, bottles, and disconnected loops: The opening and closing of the interplanetary magnetic field (95RG00124), Tech. rep.
  * McComas et al. (2008) McComas, D. J., Ebert, R. W., Elliott, H. A., Goldstein, B. E., Gosling, J. T., Schwadron, N. A., & Skoug, R. M. 2008, Geophys. Res. Lett., 351, L18103
  * McComas et al. (1992a) McComas, D. J., Gosling, J. T., & Phillips, J. L. 1992a, J. Geophys. Res., 97, 171
  * McComas et al. (1992b) McComas, D. J., Gosling, J. T., & Phillips, J. L. 1992b, in Solar Wind Seven Colloquium, ed. E. Marsch & R. Schwenn, 643–646
  * McComas et al. (1989) McComas, D. J., Gosling, J. T., Phillips, J. L., Bame, S. J., Luhmann, J. G., & Smith, E. J. 1989, J. Geophys. Res., 94, 6907
  * McComas et al. (1991) McComas, D. J., Phillips, J. L., Hundhausen, A. J., & Burkepile, J. T. 1991, Geophys. Res. Lett., 18, 73
  * McComas et al. (1992c) McComas, D. J., Phillips, J. L., Hundhausen, A. J., & Burkepile, J. T. 1992c, in Solar Wind Seven Colloquium, ed. E. Marsch & R. Schwenn, 225–228
  * McComas et al. (2000) McComas, D. J., et al. 2000, J. Geophys. Res., 105, 10419
  * Priest et al. (2000) Priest, E., Forbes, T., & Murdin, P. 2000, Magnetohydrodynamics, ed. Murdin, P.
  * Schwadron et al. (2010) Schwadron, N. A., Connick, D. E., & Smith, C. 2010, ApJ, 722, L132
  * Slavin et al. (1986) Slavin, J. A., Jungman, G., & Smith, E. J. 1986, Geophys. Res. Lett., 13, 513
  * Smith & Balogh (2008) Smith, E. J., & Balogh, A. 2008, Geophys. Res. Lett., 352, L22103
  * St. Cyr & Burkepile (1990) St. Cyr, O. C., & Burkepile, J. T. 1990, NASA STI/Recon Technical Report N, 911, 17996
  * Tappin & Howard (2009) Tappin, S. J., & Howard, T. A. 2009, Space Sci. Rev., 147, 55
  * Tousey (1973) Tousey, R. 1973, in Space Research XIII, Vol. 2, p. 713 - 730, ed. M. J. Rycroft & S. K. Runcorn, 713–730