#  Tuned electron-nucleus resonance as a tool of producing the 229mTh isomer

F. F. Karpeshin

###### Abstract

The possibility of refining the energy of the 8.36-eV 229mTh nuclear isomer —
the most likely candidate for the role of a nuclear frequency standard — by
means of resonant optical pumping is discussed. Attention is focused on
considering the problem of broadening the resonance in order to reduce
scanning time. The two-photon method proposed exploits the radical broadening
of the isomer line due to mixing with the electron transition. This is not
burdened with the cross-section reduction, in contrast with internal-
conversion-based resonance broadening or intended extra-broadening of the
spectral line of a pumping laser. In the case under consideration, according
to the calculations, it turns out to be two orders of magnitude more
efficient. It is applicable to both ionized and neutral thorium atoms.
Realization of the method supposes excitation of the both nucleus and electron
shell in the final state.

D. I. Mendeleyev Institute for Metrology, Saint-Petersburg, Russia

e-mail<EMAIL_ADDRESS>

##  1 Introduction

Much attention is being paid to the problem of creating nuclear optical clocks
and, accordingly, the next generation frequency standard. Record samples of
atomic clocks demonstrate an error within several units of $10^{-18}$ [1],
while in order to solve challenging fundamental and applied problems it is
necessary to further reduce the errors by another order of magnitude. The
development of heavy-ion clocks [2] has good prospects. A further reduction in
the error would allow to resolve the long-standing question about the possible
drift of the fundamental constants [3]. The most pressing task of modern
physics is the search for dark matter and energy. Here the fundamental idea is
to detect wave oscillations of particles of ultralight matter in its
interaction with ordinary matter [4]. The strengths of nuclear clocks are
indicated in many reviews and original works (for example, [5] and references
therein). Some of them seem self-evident, such as protection from external
fields. And their use to search for the drift of fundamental constants has
irreplaceable features, since the contribution from the nuclear component,
compared to the Coulomb component, to the transition frequency is much
stronger than in optical ones. Some projects are based on the joint use of
atomic and nuclear clocks, using the specified features of the latters [2, 6].

The number one candidate for the creation of nuclear clocks is the unique
nuclide of 229Th, whose excited state $3/2^{+}[631]$ lies at a height of only
$\omega_{n}$ = 8.355740(3) eV above the ground state $5/2^{+}[633]$ [7]. For
estimates, I set its lifetime in neutral atoms to be within 10 $\mu$s [8, 9,
10, 11]. The proper lifetime of the nucleus is much longer due to internal
conversion (IC), which greatly enhanced the decay rate of the isomer: ICC
(internal conversion coefficient) $\alpha(M1)=0.987\times 10^{9}$. Thus,
accounting for the IC leads to an increase in the natural width of the
isomeric line from $\Gamma_{n}=0.667\times 10^{-19}$ eV ($10^{-5}$ Hz) to
$\Gamma_{a}=0.7\times 10^{-10}$ eV ( 10 kHz). An indication has been obtained
that in a crystalline environment it can be several times shorter than [7,
12], this not changing the qualitative conclusions, though.

The recent breakthrough [7] was achieved using a highly doped CaF2 crystal and
a four-wave mixing laser. As a result, the thorium isomer transition was
optically excited resonantly for the first time. Theory of direct nuclear
excitation in strong electromagnetic fields was advanced in Refs. [13, 14, 15,
16, 17] and cited therein. In this concern, I remind that another effective
way to search for nuclear optical resonance is offered by exploiting the
resonant properties of the electron shell. The interaction of the nucleus with
the electron shell is carried out through internal conversion (IC), which in
the subthreshold region turns into discrete, or resonance conversion (RC). The
RC concept was formulated yet in relation to deexcitation of fission fragments
in muonic atoms [18, 19]. RC was confirmed experimentally in Ref. [20]. A
method for laser-induced radiative decay of the 76-eV isomer 235U by means of
RC was proposed in Ref. [21]. The method is based on two-photon resonance
absorption by the atomic electrons: one photon — virtual — is emitted by the
nucleus, the other comes from the external radiation of a tunable laser, which
is used in order to tune the resonance. In this way, the question of scanning
the laser frequency range at the isomer energy was first posed. Common
features of RC with the electronic bridges (EB), introduced by Krutov [22],
were noted. It is worth noting that EB was observed experimentally in [23] in
the decay of the 30.7-keV isomeric level 93Nb. RC is actually the cross-
invariant continuation of EB into the subthreshold field. In the form of bound
internal conversion (BIC) it was discovered in the 35-keV transition in highly
charged 125Te ions [24].

Based on Ref. [25], application of inverse EB was proposed in Ref. [26].
Morita suggested that creation of holes in inner shells can induce a non-
radiative electron transition accompanied with excitation of the nucleus
(known as NEET). Actually guided by the misunderstood paper by I. S. Batkin
[27], Tkalya suggested that lifting up a valence electron to an excited level
may induce excitation of the nucleus. For the nucleus to be effectively
excited, however, the intermediate electron state should have exactly the same
energy as the nuclear transition energy. In fact, Batkin never told that such
a resonance can be realized in nature indeed, considering energy transfer
through a virtual electron level.

In view of the improbability of such a coincidence, as well as the uncertainty
of the isomer energy, a “non-resonance” method was proposed in Ref. [28].
According to the method, resonating electronic levels are populated in a
spontaneous transition from a higher shell. Moreover, it was shown that in
fact, there is a number of the levels which form resonances with the isomer
transition. As a result, all these resonance intermediate states are populated
automatically, contrary to the method of Ref. [26]. An experiment aimed at
testing this mechanism was carried out in Ref. [29]. The problem, however, was
that the most probable isomer energy at that time was considered to be
3.5$\pm$1 eV. The value of 7.6$\pm$1 eV [30], close to the present one,
appeared much later.

In Ref. [31] the authors realized that existence of a resonant pair of levels
is hardly possible, supposing that the resonance condition may be not so
important if the intermediate electron state is in the continuum. Numerical
estimations were performed in Ref. [32]. Furthermore, when it became clear
that the energy of the isomer lies in the region of 7 – 8 eV, this induced
appearance of projects based on the absorption of two or three laser photons
([33, 34, 35, 36, 37, 38, 39] and Refs. cited therein). At the same time,
single-photon projects are also considered, like Ref. [40], which became the
prototype of the state-of-the-art project [11].

Whatever project we take, however, we will definitely be faced with the issue
of optimizing the scanning step by spectral broadening the line of the pumping
beam. Thus, in Ref. [38] it was noted that the effective width of the isomer
line is 40 nuclear widths, which, however, is still too small to make it as a
scanning step. In the [11] project, the line broadening due to the IC reaches
9 orders of magnitude, but exactly the same amount is lost in the excitation
cross section. In the latest paper Ref. [7] the laser had an inherent
bandwidth of about 1 GHz, but it was deliberately modulated to an effective
bandwidth of 10-20 GHz to have a broader lineshape available for better
coverage of the search region. As a consequence, the resonance excitation
cross section of the isomer again decreases by the same factor. Different
physics lies in the idea of excitation of the nucleus through electron-nuclear
resonance. For example, the natural width of the $7p$ level is eleven orders
of magnitude greater than the nuclear one, whilst its excitation cross section
does not suffer at all, but on the contrary, gains many orders of magnitude in
comparison to the nuclear cross section. The combination of these factors
creates attractive prospects for scanning. In order to study them in more
detail, I conduct a comparative analysis of the above-mentioned methods, in
order to outline ways to further improve the efficiency of research. My
narration comes as follows. In the next section I remind the state-of-the-art
project in more details in order to highlight the principled points by the
comparison. Section 3 outlines the physical principles of the present method.
Neutral atoms are considered. Values needed for assessment of effectiveness of
the method are presented in section 4. Required scan time is assessed in
section 5. The main results obtained are discussed in section 6. Prospects for
further research are also outlined there.

##  2  Application of internal conversion in the frequency comb method

Start with the state-of-the-art project Ref. [11]. Two decisive factors
underlie the project. First, it is frequency comb consisting of $10^{5}$
teeth. Broadening of the isomer line by a factor of $10^{9}$ due to
deexcitation via IC comprises the second factor, contracting the necessary
scanning time.

Project [11] is based on making use of the seventh harmonic of the reference
beam from an ytterbium-doped fiber laser with a wavelength of 1070 nm . Pulse
trains from that setup are repeated with a frequency of 77 MHz. As a result,
the Fourier spectrum of the transformed beam takes the form of a frequency
comb around the target line, consisting of $1.2\times 10^{5}$ equidistant
teeth. The radiation power in each of the teeth is 10 nW in the center, the
half-width is 490 Hz ($2\times 10^{-12}$ eV), the distance between the teeth
is 77 MHz = $3\times 10^{-8}$ eV. This project was calculated at a time when
the isomer energy was considered to be $\omega_{n}$ = 8.338 eV with an
uncertainty of 0.024 eV, according to [12]. Thus, the entire comb covers all
the uncertainty interval of the isomer energy.

The project was calculated at a time when the isomer energy was considered to
be $\omega_{n}$ = 8.338 eV with an uncertainty of 0.024 eV, according to [12].
Thus, the entire comb was designed to cover the entire uncertainty interval of
the isomer energy, and to find the resonance it was necessary to scan only the
interval between the teeth.

Furthermore, each tooth of the comb carries a photon flux of $J=1.06\times
10^{13}\text{ c}^{-1}\text{cm}^{-2}$. This flux is focused onto a target,
which is a thin circle with an area of $7.07\times 10^{-4}\text{ cm}^{2}$.
$1.6\times 10^{13}$ atoms of 229Th are deposited onto its surface by
sputtering or otherwise. If the isomer energy coincides with a photon one, the
nucleus transfers to the isomeric state. The latter decays back through IC.
Detection of the conversion electrons shows that the resonance is established.

The cross-sections of photon absorption are related to the radiative widths of
the reverse transitions by the detailed balancing principle. For the scanning
purposes, it is most effective to use a light beam with a spectral width which
is within the resonance width. Then the photoexcitation cross section for a
level with energy $\omega$ (we use the relativistic system of units
$\hbar=c=m_{e}$ = 1) can be estimated by means of the following interrelation
[43, 45]:

$\sigma_{\gamma}=\Gamma_{\gamma}\frac{2I_{\text{is}}+1}{2I_{0}+1}\left(\frac{\pi}{\omega}\right)^{2}S_{\omega}\,,$
(1)

where $S_{\omega}$ is the spectral density of the beam, $\Gamma_{\gamma}$ —
the radiative width of the reverse transition. Designating $S_{\omega}\approx
1/\Gamma$, $\Gamma$ being the total width of the isomeric state, and
supposing, where this is possible, $\Gamma_{\gamma}\approx\Gamma$, one arrives
at the universal expression as follows:

$\sigma_{\gamma}=\frac{2I_{\text{is}}+1}{2I_{0}+1}\left(\frac{\pi}{\omega}\right)^{2}\,,$
(2)

Eqs. (1) – (2) are equally valid for both nuclear and atomic systems. Such a
universality is extremely important when considering the phenomena of
electron-nuclear resonance. In particular, Eq. (2) describes the cross-section
for resonance scattering on a bare nucleus.

In the case of a neutral thorium atom, however, $S_{\omega}=1/\Gamma_{c}$,
where $\Gamma_{c}$ is the conversion width of the isomeric state. Substituting
it in formula (1), one arrives at the expression for the excitation cross-
section of the isomer in neutral atoms as follows:

$\displaystyle\sigma_{\gamma}=\frac{\Gamma_{\gamma}}{\Gamma_{c}}\
\frac{2I_{\text{is}}+1}{2I_{0}+1}\left(\frac{\pi}{\omega}\right)^{2}=\alpha^{-1}(M1)\frac{2I_{\text{is}}+1}{2I_{0}+1}\left(\frac{\pi}{\omega}\right)^{2}=3.73\times
10^{-20}\text{ cm}^{2}\,.$ (3)

Now one can estimate the rate of isomer formation
$p=\sigma_{\gamma}J=3.95\times 10^{-7}\text{ c}^{-1}$. It is significantly
lower than the rate of their spontaneous deexcitation $\gamma=10^{5}\text{
c}^{-1}$. As a result,
$N^{\text{(imp)}}=\sigma_{\gamma}JN_{0}\tau_{\text{imp}}=633$ isomeric nuclei
are formed per pulse, 90% of which, however, decay by the beginning of the
measurement, about 60 of them surviving. Furthermore, if the laser frequency
is changed each second with a step equal to the width of the target line, that
is, $10^{-10}$ eV, then scanning of the supposed uncertainty interval of 0.024
eV will require 5000 steps, which takes 5000 seconds. To determine which tooth
resonates, one needs to change the interval between the teeth. Therefore, to
clarify the energy of the isomer, several more scanning cycles will be
required.

The above project teaches a lesson how the acceleration of the isomeric
transition, arising due to IC, can be used in order to reduce the scan time.
Strictly speaking, this can be called a kinematic gain as no resonance is
exploited. And this way is only suitable for neutral atoms. Already in singly-
charged ions, the ionization energy becomes greater than the isomer energy,
turning off the IC channel. At the same time, most clock projects are aimed at
usage of 229Th ions. Let us see how the Warsaw effect in combination with
resonance can be exploited both in neutral and ionized atoms.

##  3 Physical premises of application of the resonantly enhanced Warsaw
effect

Quite recently, another way of measuring the energy of an isomer was proposed
in Ref. [41], based on the Warsaw effect of mixing nuclear states with
different spins through interaction with an orbital electron. It descends from
the fact that the excitation of the isomer occurs through the sudden removal
of one of the $s$-shell electrons, according to the Feinberg—Migdal theory
[42]. Therefore, in the case of the photoelectric effect, the ejected
electrons have different energies depending on whether the nucleus remains in
the ground or isomeric state. By measuring this difference, one obtains the
energy of the isomer. Let us see how electron transfer to a discrete $p$-level
can be used to accurately determine the isomer energy. Some directions for the
development of these ideas were discussed at the KVNO-23 conference and in its
Proceedings [44].

First, I recall the results of Ref. [46], where acceleration of the isomer
decay in hydrogen-like ions was demonstrated, occurring due to mixing of the
ground and isomeric levels of the nucleus in atomic states with a total
angular momentum $F$ = 2. In the ground state of such a hydrogen-like system
the spins of the electron and the nucleus can be parallel or antiparallel. The
corresponding total moment will be equal to $F$ = 3 or 2. The spins of the
electron and the isomeric nucleus are added in a similar way, forming states
with $F$ = 2 or 1. Thus, the state of an atom with $F$ = 2 can be of two
types: 1) the nucleus is in the ground state, the spins are antiparallel, and
2) the nucleus is in the isomeric state, the spins are parallel. And since
they interact through RC, then, according to the principles of quantum
mechanics, these states form a superposition, which will be the real wave
function. Denoting the amplitudes of mixing of atomic states of the two types
as $\alpha$ and $\beta$, the wave function of the atom in the ground or
isomeric states can be written as follows:

$\displaystyle\Psi_{g}=\alpha|g\rangle+\beta|is\rangle$
$\displaystyle\Psi_{\text{is}}=\alpha|is\rangle-\beta|g\rangle\,,$ (4)

where $\alpha\approx$ 1 and $\beta$ can be determined through the interaction
matrix element and the energy difference between two states in the first order
of perturbation theory. And since the interaction is nothing more than a
conversion amplitude, the probability of mixing in Eq. (4) can be expressed in
terms of radiative nuclear width times ICC as the product of the radiative
width of the nuclear transition and discrete conversion coefficient
$\alpha_{d}(M1;1s-1s)$ [46]):

$\beta^{2}=\frac{2I_{\text{is}}+1}{2I_{0}+1}\frac{\alpha_{d}(1s-1s;\omega_{n})\Gamma_{n}^{(\gamma)}/2\pi}{\omega_{n}^{2}}\,.$
(5)

Here $\omega_{n}$ is the isomer energy. $\Gamma_{n}^{(\gamma)}$ — the
radiative partial nuclear width, $\Gamma_{n}^{(\gamma)}\equiv\Gamma_{n}$ in
the present case. $\alpha_{d}(1s-1s;\omega_{n})$ — dimensional IC coefficient
(ICC) for the $M1$ nuclear transition with energy $\omega_{n}$, and what is
more, in this case a diagonal transition between $1s$-states [18, 19, 24]
takes place. It can be concluded that if such an atom is in a state with total
momentum $F=2$, then in its ground state the nucleus can nevertheless be in
the isomeric state with the probability $|\beta|^{2}$, and vice versa, if the
atom is in the isomeric state, however, the nucleus can be in the ground one
with the same mix probability $|\beta|^{2}$. Now imagine that one suddenly
removes a $7s$ electron from a neutral thorium atom. Then the interaction
leading to the formation of an isomer is switched on at the same moment.
According to the Feinberg—Migdal theory, the nucleus will remain in the
isomeric state with the probability of $\beta^{2}$.

Figure 1: Feynman graph of the resonance optical pumping the isomer. The
thicker line specifies the nuclear transition from the ground to the isomeric
state.

The Feynman graph of this process is shown in Fig. 1 with the only difference
that the electron is transferred to the $7p$ level. Concerning the excitation
of the isomer, such a transfer has essentially the same effect, since in the
$7p$ state the electron’s inelastic interaction with the nucleus can be
neglected. In more detail, the nucleus spontaneously transfers into the
isomeric state as a result of virtual exchange with a photon with a valence
$7s$ electron. The electron falls to $\omega_{n}$ energy level below the mass
shell. The energy balance is restored after it absorbs a photon from the laser
field and settles in the $7p$ state of the atom. The photon energy $\omega$ is
determined from the energy conservation:

$\omega=\omega_{n}+\epsilon_{7p}\,,$ (6)

where $\epsilon_{7p}$ is the energy of the $7p$ level. At this frequency, the
amplitude of the process passes through a resonance with a width
$\Gamma_{a}=\Gamma_{7p}$, determined by the width of the $7p$ level.

To calculate the cross section of the process in Fig. 1, one can apply formula
(1) for the resonance scattering of photons by the $7s$-electron with its
transfer to the $7p$-state, provided that the nucleus simultaneously passes
into the isomeric state. Since these two processes are independent, one
arrives at the following expression for the cross section:

$\sigma_{\gamma}=\beta^{2}\frac{2j_{7p}+1}{2j_{7s}+1}\frac{\Gamma_{\gamma}^{(a)}(7p-7s;\omega)}{\Gamma_{7p}}\left(\frac{\pi}{\omega}\right)^{2}\,,$
(7)

with $\Gamma_{\gamma}^{(a)}(7p-7s;\omega)$ being the radiative width of the
$7p-7s$ transition provided its energy $\omega$ is out of the mass shell, and
$\Gamma_{7p}$ — the total width of the $7p$ level. In view of that the
radiative width is proportional to the energy in cube, one can put down
$\Gamma_{\gamma}^{(a)}(7p-7s;\omega)=f\Gamma_{\gamma}^{(a)}(7p-7s;\omega_{a})=f\Gamma_{7p}$,
$f=(\omega/\omega_{a})^{3}$, $\omega_{a}$ being the energy of the $7p-7s$
transition in a sole atom. By means of these expressions, the cross-section
acquires the following form:

$\sigma_{\gamma}=\frac{(2j_{7p}+1)(2I_{is}+1)}{(2j_{7s}+1)(2I_{g}+1)}\left(\frac{\pi}{\omega}\right)^{2}fWQ\,,$
(8)

where $Q\leq 1$ is the ratio of the partial radiative width to the total
width. This value characterizes the quality factor of the atomic resonance. In
our case, $Q$ = 1 — the maximum value. $W\equiv\beta^{2}$ is the value of the
Warsaw effect (5), in this case — on the $7s$ electrons instead of $1s$. The
cross section (8) can be transparently represented as follows:

$\sigma_{\gamma}=\frac{2j_{7p}+1}{2j_{7s}+1}\frac{2I_{0}+1}{2I_{is}+1}fWQ\tilde{\sigma}_{\gamma}^{(b.n.)}\,,$
(9)

where $\tilde{\sigma}_{\gamma}^{(b.n.)}$ is approximately equal to the cross
section on a bare nucleus, differing from the latter in that it is calculated
for the photon energy $\omega$ instead of $\omega_{n}$. It follows from Eq.
(9) that it is reasonable to choose wide levels with good quality factor, for
which $Q\approx 1$, as the final state. The same formula can be obtained in
another way, by substituting into Eq. (9) the radiative width of the process
inverse to Fig. 1, which was calculated in Ref. [43].

Note that the full resonance width is in the denominator of the formula (8),
as well as in Eq. (1). It does not differ much from the conversion width of
the isomer. At the same time, tremendous amplification comes from the vertex
corresponding to the electric dipole absorption of a laser photon by the
atomic electron. It is by 12 orders of magnitude as large as the magneto-
dipole nuclear vertex corresponding to the direct absorption of the photon by
the nucleus.

Finally, for the purposes of qualitative analysis, I introduce the effective
cross-section parameter $V$, multiplying the cross-section (9) in my case, and
(3) — in the case of the project [11] by the width of the corresponding
resonance, which can be identified with the scanning step, and dividing by the
isomer’s own width $\Gamma_{n}$. In the latter case, I arrive at the following
transparent expression:

$V\equiv\sigma_{\gamma}^{(b.n.)}\,,$ (10)

In the case of shake-up mechanism in Fig. 1, the width is $\Gamma_{7p}$,
consequently,

$V=fWQ\frac{\Gamma_{7p}}{\Gamma_{n}}\tilde{\sigma}_{\gamma}^{(b.n.)}\,.$ (11)

In the case of a scheme similar to Ref. [7], based on intended spectral
broadening of the laser beam, Eq. (10) holds for the effective cross section.

##  4  Calculation results

For the purpose of a qualitative assessment of the scanning time using the
mechanism in Fig. 1, one can use the Dirac—Fock method. Within this method,
required parameters were calculated using the computer code RAINE [47]. They
are published e.g. in Ref. [43] and elsewhere. Those calculations had been
performed using the isomer energy of 3.5 eV. Extrapolation to modern energy is
performed directly, using the energy dependence of the radiative widths
$\sim\omega^{3}$ and ICC as $\sim\omega^{-3}$. The obtained values were
published in a number of works, for example, in [43]. Extrapolation to modern
energy is performed directly using the energy dependence of radiation widths
$\sim\omega^{3}$ and conversion coefficients $\sim\omega^{-3}$. As a result,
ICC for the $M1$ transition with energy $\omega_{n}$ = 8.36 eV in Th I turned
out to be equal to $\alpha(M1)=0.987\times 10^{9}$ [9]. With the lifetime in
neutral atoms of 10 $\mu$s, the above-mentioned proper width of the isomer
$\Gamma_{n}=0.667\times 10^{-19}$ eV follows.

Furthermore, within the framework of the method, the resonance cross-section
consists of the two components corresponding to the final levels either
$7p_{1/2}$ or $7p_{3/2}$, the intensity of the second component being twice as
high as the first one. Energy of the final state $\epsilon_{7p_{3/2}}$ = 1.64
eV [43], therefore $\omega$ = 10.75 eV. In turn, $\epsilon_{7p_{1/2}}$ = 1.64
eV, respectively $\omega$ = 9.98 eV for this level. I also get
$\alpha_{d}(7s-7s;\omega_{n})=1.23\times 10^{10}$ eV. Radiative width of an
atomic transition $\Gamma_{\gamma}^{(a)}(7p_{3/2}-7s;\omega)=3.1\times
10^{-6}$ eV, which is thirteen orders of magnitude greater than the natural
width of the $\Gamma_{n}$ isomeric line.
$\Gamma_{\gamma}^{(a)}(7p_{1/2}-7s;\omega)=2.4\times 10^{-6}$ eV.

##  5  Scan time estimate

For evaluation purposes, I consider a thought experiment involving scanning
with a monochromatic light beam with a power of 1 mW. Such is the total power
of all the teeth of the frequency comb in the project [11]. Put $\omega$ =
10.75 eV — resonance energy corresponding to the excitation of the $7p_{3/2}$
level in the final state. Let the target area be $10^{-4}\text{ cm}^{2}$. Now
the photon flux $J=5.47\times 10^{18}\text{ cm}^{-2}\text{ c}^{-1}$, which is
five orders of magnitude greater than in the frequency comb project.

By means of Eq. (8) I find the cross section $\sigma_{\gamma}=6.24\times
10^{-21}\text{ cm}^{2}$ and estimate the rate of excitation of the isomer in
the laser radiation field: $p=\sigma_{\gamma}J=0.034\text{ c}^{-1}\ll\gamma$.
Thus, for the same $N_{0}$, in a time of 10-5 s, $N_{is}\approx
nN_{0}=5.4\times 10^{6}$ isomeric atoms are formed, which provides $10^{5}$
times greater statistics as compared to the IC project. Therefore, this
interval, that is, 10-5 s, looks to be sufficient as the duration of the
scanning pulse. If one puts the same time for checking if the condition of
resonance is fulfilled, then each step will take $\tau_{\text{sc}}=2\times
10^{-5}$ s. Furthermore, since the width of the $7p_{3/2}$ level is
$\Gamma_{7p_{3/2}}=3.13\times 10^{-8}$ eV, then scanning an interval of 1 eV
will need $N_{\text{ sc}}=1\text{ eV}/\Gamma_{7p_{3/2}}=3.3\times 10^{7}$
steps, that is $N_{\text{sc}}\tau_{\text{sc}}$ = 11 minutes. And for scanning
the uncertainty interval of 0.024 eV half minute is enough.

Figure 2: Resonance landscape resulting from the fragmentation of the $7p$
levels according to [49].

Taking into account interelectronic interactions, the final $7p$ states are
fragmented into several strong levels [48], between which the resonance cross
section found above is distributed. One can find the spectrum of the final
states and the corresponding resonance cross sections using NIST data on the
relative intensity of the Th I absorption lines from the ground state.
According to [49], there is a strong component in the spectrum, corresponding
to the transition from the ground state to a level with an energy of
2.1520074477 eV. Accordingly, this line forms the main resonance in scanning,
achieved at a laser photon energy of $\omega$ = 10.490 eV. In Fig. 2, the
resonance cross sections are presented for the strongest lines, appearing at
energies $\omega$ = 11.476, 11.096, 10.067 eV and others. They can be used as
reference points, the detection of which will help one to reliably identify
these components in an experiment and, accordingly, determine the energy of
the isomer according to (6) with an accuracy corresponding to laser
measurement methods. The cross-section scale is given under the assumption
that the strongest line retains 1/3 of the total resonance strength of the
$7p_{1/2}$\- and $7p_{3/2}$ levels, as calculated within the framework of the
Dirac-Fock method.

##  6  Discussion

To summarize, one can conclude that when scanning neutral atoms, one can take
advantage of the broadening of the isomeric line due to IC by 9 orders of
magnitude, up to $\sim 10^{-10}$ eV. At the same time, this leads to a
decrease in the photoexcitation cross section by the same factor. A frequency
comb can help to reduce the scan time due to the large number of $\sim 10^{5}$
teeth. A similar effect can be achieved with deliberately broadened spectral
line of the scanning laser [7].

The resonance mechanism of the line broadening, presented above, exploits big
natural width of the atomic resonances involved. According to this approach,
the atom remains in an excited state with a good quality factor, as well as
the nucleus does. This increases the line width to atomic values of $\sim
10^{-8}$ eV. The isomer is excited due to the Feinberg—Migdal shaking
mechanism. This involves the main smallness in the cross section of the
process related with small probability of excitation of the isomer (Warsaw
effect) $W=4\times 10^{-12}$ on the valence $7s$ electrons. This value
coincides with that calculated in Ref. [41]. The $W$ values increase on the
inner shells, up to 10-4 on the $1s$-shell [43, 41, 50]. Another beautiful
consequence of the Warsaw effect for the gyromagnetic ratio of the 229Th
nucleus in the ground state was drawn attention to in Ref. [51]. The above
estimates demonstrate robustness of the scheme. Thus, when comparing formulas
(10) with (11), the section $\sigma_{\gamma}^{(b.n.)}$ can be compared with
the section $\tilde{\sigma}_{\gamma}^{(b.n.)}$ as quantities of the same
order. However, the additional factor
$fWQ\frac{\Gamma_{7p}}{\Gamma_{\gamma}^{(n)}}$= 92 speaks eloquently in favor
of the two-photon resonance method. A similar conclusion was also obtained in
Ref. [43].

What is said above fully applies to any other excitation schemes based on the
absorption of two, three or more photons: excited electronic state, left after
the nucleus is transferred to the isomeric state, radically contracts scanning
time needed. For example, this moment was not taken into account in Ref. [33].
It was shown already in Ref. [52] that the cross-section radically increases
if the electron shell is left in an excited state. Multiplied by
inconsistencies with multipolities of the comprising transitions, this
shortcoming gave rise to the thorium puzzle (e.g. [53] and Refs. therein). The
above arguments sum up its solution.

The author would like to express his gratitude to P. G. Thirolf, L. F.
Vitushkin and L. von der Wense for fruitful discussions.

## References

  * 1. E Peik, T Schumm, M S Safronova, A Palffy J. Weitenberg and P G Thirolf, Quantum Sci. Technol. 6 (2021) 034002.
  * 2. Steven A. King et al.. Nature, 611, 43 (2022).
  * 3. Flambaum V 2006 Phys. Rev. Lett. 97 0 92502.
  * 4. D. Antypas et al., arXiv:2203.14915 [hep-ex].
  * 5. E. Peik, M. Okhapkin. Nuclear clocks based on resonant excitation of $\gamma$-transitions, C. R. Physique 16, 516 (2015).
  * 6. V V Flambaum and V A Dzuba. Search for variation of the fundamental constants in atomic, molecular, and nuclear spectra, Can. J. Phys. 87, 25 (2009).
  * 7. J. Tiedau, M. V. Okhapkin, K. Zhang et al., Laser excitation of the Th-229 nucleus. Phys. Rev. Lett., in print.
  * 8. L. Von der Wense, B. Seiferle, M. Laatiaoui et al. Direct detection of the 229Th nuclear clock transition, Nature, 47, 533 (2016).
  * 9. F. F. Karpeshin and M. B. Trzhaskovskaya. Impact of the electron environment on the lifetime of the 229mTh low-lying isomer. Phys. Rev. C76, 054313 (2007).
  * 10. N. Minkov, A. Palffy, Phys. Rev. Lett. 122, 162502 (2019).
  * 11. L. von der Wense and Z. Chuankun. Concepts for direct frequency-comb spectroscopy of 229mTh and an internal-conversion-based solid-state nuclear clock. Eur. Phys. J. Ser. D 74, 146 (2020).
  * 12. S. Kraemer, J. Moens, M. Athanasakis-Kaklamanakis et al. Observation of the radiative decay of the 229Th nuclear clock isomer. Nature 617, 706 (2023).
  * 13. Adriana Palffy, Jorg Evers, and Christoph H. Keitel, Phys. Rev. C 77, 044602 (2008).
  * 14. Adriana P?lffy, Oliver Buss, Axel Hoefer, and Hans A. Weidenm?ller, Phys. Rev. C 92, 044619 (2015).
  * 15. A. Ya. Dzyublik, G. Gosselin, V. Meot and P. Morel, EPL, 102, 62001 (2013).
  * 16. A. Ya. Dzyublik, JETP Lett., 92, 130 (2010).
  * 17. Lars von der Wense et al., EPJ A 56, 176 (2020).
  * 18. D. F. Zaretsky et al. Mesoatomic X-ray radiation of muons on fragments of instantaneous fission. Yad. Fiz. 29, 306 (1979) [Sov. J. Nucl. Phys. 29, 151 (1979)].
  * 19. F.F.Karpeshin, Resonance Internal Conversion as the way of accelerating nuclear processes. Particles and Nuclei, 37, 522 (2006).
  * 20. C. Rösel et al. Experimental evidence for muonic X-rays from fission fragments. Z. Phys. A345, 425 (1993).
  * 21. B.A.Zon, F.F.Karpeshin, Acceleration of the decay of 235mU by laser-induced resonant internal conversion. Zh. Eksp. i Teor. Fiz., 97, 401 (1990) [Sov. Phys. — JETP (USA), 70, 224, 1990.]
  * 22. V. A. Krutov. Ann. Phys. (Leipzig), 21, 291 (1968); Pis’ma v ZhETF, 52, 1176 (1990) [JETP Lett. 52, 584 (1990)].
  * 23. D. Kekez et al.. Phys. Rev. Lett. 55, 1366 (1985).
  * 24. F. F. Karpeshin, M.R.Harston, F.Attallah, J.F.Chemin, J.N.Scheurer, I.M.Band, M. B. Trzhaskovskaya, Subthreshold internal conversion to bound states in highly-ionized 125Te ions, Phys. Rev. C53 (1996) 1640.
  * 25. M. Morita, Prog. Theor. Phys. 49, 1574 (1973).
  * 26. E.V.Tkalya. JETP Lett. 55, 216 (1992); Nucl. Phys. A 539, 209 (1992).
  * 27. I. S. Batkin. Compton excitation of the nuclear levels. Yad. Fiz. 29, 903 (1979) [Sov. J. Nucl. Phys. 29, 464 (1979)].
  * 28. F.F.Karpeshin, I.M.Band, M.B.Trzhaskovskaya and B.A.Zon, Study of 229Th through laser-induced resonance internal conversion. Phys. Lett. 282B, 267 (1992).
  * 29. T. T. Inamura and H. Haba, Search for a “3.5-eV isomer” in 229Th in a hollow-cathode electric discharge. Phys. Rev. C 79, 034313 (2009).
  * 30. B. R. Beck et al., Phys. Rev. Lett., 98, 142501 (2007).
  * 31. P. V. Borisyuk, N. N. Kolachevsky, A. V. Taichenachev, E. V. Tkalya, I. Yu. Tolstikhina, and V. I. Yudin. Excitation of the low-energy 229mTh isomer in the electron bridge pro-cess via the continuum. Phys. Rev. C 100, 044306 (2019).
  * 32. A.Ya. Dzyublik. Quasiclassical theory of 229mTh excitation by laser pulses via electron bridges. Phys. Rev. C 106, 064608 (2022).
  * 33. S. G. Porsev et al. Excitation of the Isomeric 229mTh Nuclear State via an Electronic Bridge Process in 229Th+. Phys. Rev. Lett. 105, 182501 (2010).
  * 34. F. F. Karpeshin, M. B. Trzhaskovskaya, Bound internal conversion versus nu-clear excitation by electron transition: Revision of the theory of optical pumping of the 229mTh isomer. Phys. Rev. C 95, 034310 (2017).
  * 35. R.A. Müller, A.V. Volotka, S. Fritzsche, A. Surzhykov, Theoretical analysis of the electron bridge process in 229Th3+. NIM B 408, 84 (2017).
  * 36. Pavlo V. Bilous et al., Phys. Rev. Lett. 124, 192502 (2020).
  * 37. Haowei Xu, Hao Tang, Guoqing Wang, Changhao Li, Boning Li, Paola Cappellaro and Ju Li, Phys. Rev. A 108, L021502 (2023).
  * 38. Lin Li et al. Scheme for the excitation of thorium-229 nuclei based on electronic bridge excitation, Nucl. Sci. Techn. (2023) 34:24; https://doi.org/10.1007/s41365-023-01169-4.
  * 39. Neng-Qiang Cai, Guo-Qiang Zhang, Chang-Bo Fu and Yu-Gang Ma. Populating 229mTh via two-photon electronic bridge mechanism, Nucl. Sci. Techn. (2021) 32:59; https://doi.org/10.1007/s41365-021-00900-3.
  * 40. L. von der Wense, B. Seiferle, S. Stellmer, J. Weitenberg, G. Kazakov, A. Palffy, and P. G. Thirolf. Phys. Rev. Lett. 119, 132503 (2017).
  * 41. F.F. Karpeshin, Measurement of the Energy of the 8.3-eV 229Th Isomer Using the Photoelectric Effect. JETP Letters, 118, 548 (2023).
  * 42. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Pergamon, New York, 1977).
  * 43. F. F. Karpeshin, I.M. Band, and M.B. Trzhaskovskaya, 3.5-eV isomer of ${}^{229m}Th$: how it can be produced. Nucl. Phys. A654, 579 (1999).
  * 44. F.F. Karpeshin and L.F. Vitushkin. On the problems of creating a nuclear-optical frequency standard based on 229Th, https://doi.org/10.48550/arXiv.2307.08711; F.F.Karpeshin and M.B.Trzhaskovskaya, Zh. Exp. Teor. Fiz. [JETP] 165, 145 (2024).
  * 45. A.I.Akhiezer, V.B.Berestetskii. Quantum Electrodynamics. John Wiley & Sons, New-York-London-Sydney, 1965.
  * 46. F.F.Karpeshin, S.Wycech, I.M.Band, M. B. Trzhaskovskaya, M. Pfützner and J.Żylicz, Rates of transitions between the hyperfine-splitting components of the ground-state and the 3.5-eV isomer in 229Th89+. Phys. Rev. C57, 3085 (1998).
  * 47. I. M. Band and M. B. Trzhaskovskaya, Internal Conversion Coefficients for Low-Energy Nuclear Transitions, At. Data Nucl. Data Tables 55, 43 (1993).
  * 48. F.F.Karpeshin and M.B.Trzhaskovskaya, Impact of the electron environment on the lifetime of the 229mTh low-lying isomer. Phys. Rev. C76, 054313 (2008).
  * 49. A. Kramida and Yu. Ralchenko, J. Reader and NIST ASD Team (2022), NIST Atomic Spectra Database (ver. 5.10), https://physics.nist.gov/asd. National Institute of Standards and Technology, Gaithersburg, MD; DOI: https://doi.org/10.18434/T4W30F.
  * 50. M. G. Kozlov, A. V. Oleynichenko, D. Budker, D. A. Glazov, Y. V. Lomachuk, V. M. Shabaev, A. V. Titov, I. I. Tupitsyn, and A. V. Volotka, Excitation of the 229Th nucleus by the hole in the inner electronic shells. Arxive: 2308.05173.
  * 51. Shabaev V.M., Glazov D. A., Ryzhkov A.M. et al.. Ground-state $g$ factor of highly charged 229Th ions: an access to the $M1$ transition probability between the isomeric and ground nuclear states. Phys. Rev. Lett. 128, 043001 (2022).
  * 52. F.F.Karpeshin and M.B.Trzhaskovskaya, Yad. Fiz. 78, 765 (2015) [Phys. At. Nucl. 78, 715 (2015)].
  * 53. F.F.Karpeshin and M.B.Trzhaskovskaya, A proposed solution for the lifetime puzzle of the 229mTh+ isomer, Nucl. Phys. A1010, 122173 (2021).
  * 54. F.F. Karpeshin and L.F. Vitushkin. On the problems of creating a nuclear-optical frequency standard based on 229Th, https://doi.org/10.48550/arXiv.2307.08711


# A Model for the Sources of the Slow Solar Wind

S. K. Antiochos  NASA Goddard Space Flight Center, Greenbelt, MD, 20771
<EMAIL_ADDRESS>Z. Mikić, V. S. Titov, R. Lionello, J. A. Linker
Predictive Science, Inc., San Diego, CA 92121

###### Abstract

Models for the origin of the slow solar wind must account for two seemingly-
contradictory observations: The slow wind has the composition of the closed-
field corona, implying that it originates from the continuous opening and
closing of flux at the boundary between open and closed field. On the other
hand, the slow wind also has large angular width, up to $\sim 60^{\circ}$,
suggesting that its source extends far from the open-closed boundary. We
propose a model that can explain both observations. The key idea is that the
source of the slow wind at the Sun is a network of narrow (possibly singular)
open-field corridors that map to a web of separatrices and quasi-separatrix
layers in the heliosphere. We compute analytically the topology of an open-
field corridor and show that it produces a quasi-separatrix layer in the
heliosphere that extends to angles far from the heliospheric current sheet. We
then use an MHD code and MDI/SOHO observations of the photospheric magnetic
field to calculate numerically, with high spatial resolution, the quasi-steady
solar wind and magnetic field for a time period preceding the August 1, 2008
total solar eclipse. Our numerical results imply that, at least for this time
period, a web of separatrices (which we term an S-web) forms with sufficient
density and extent in the heliosphere to account for the observed properties
of the slow wind. We discuss the implications of our S-web model for the
structure and dynamics of the corona and heliosphere, and propose further
tests of the model.

Sun: magnetic field — Sun: corona — Sun: solar wind

##  1 Introduction

Decades of in situ measurements of the heliosphere have firmly established
that the Sun’s wind consists of two distinct types: “fast” and “slow”. In
terms of its origins at the Sun, the best understood is the fast wind, which
typically exhibits speeds in excess of 600 km/s at 1 AU and beyond (e.g.,
McComas et al., 2008). The fast wind is measured to be approximately steady,
except for some Alfvénic turbulence (e.g., Bame et al., 1977; Bruno & Carbone,
2005). This wind is known to originate from coronal holes, regions that appear
dark in XUV and X-ray images, due to a plasma density that is substantially
lower ($<50\%$) than in surrounding coronal regions (Zirker, 1977). As implied
by eclipse and coronagraph images, the magnetic field in coronal holes is
open—appearing mainly radial and stretching out without end—whereas the field
in the surrounding regions is closed, looping back down to the photosphere.
Hence, the fast wind corresponds to the steady wind predicted by Parker in his
classic work (Parker, 1958, 1963).

The slow wind, however, is much less understood. In particular, its origin at
the Sun has long been one of the major unsolved problems in solar/heliospheric
physics. This wind has a number of observed features that distinguish it
physically from the fast wind. First, its speeds are typically slower,
$<500\,{\rm km/s}$. More important, the slow wind appears to be inherently
non-steady when compared to the fast wind (e.g., Bame et al., 1977; Schwenn,
1990; Gosling, 1997; McComas et al., 2000). It exhibits strong and continuous
variability in both plasma (for example, speed and composition) and magnetic
field properties; variability that cannot be described as simply Alfvénic
disturbances superimposed on a steady background (Zurbuchen & von Steiger,
2006; Bruno & Carbone, 2005). Finally, its location in the heliosphere is
distinct; it is generally found surrounding the heliospheric current sheet
(HCS) (e.g., Burlaga et al., 2002). A key point is that the HCS is always
embedded inside slow wind, never fast. From the presently available spacecraft
observations, it is not possible to rule out the possibility that slow wind
also occurs in locations unconnected to the HCS, in other words, that there
are pockets of slow wind with no embedded HCS and surrounded completely by
fast wind. However, the present data are certainly consistent with the picture
that, at least, during solar minimum when the corona-wind mapping can be
determined with some accuracy, all slow wind originates from a band that
encompasses the HCS, so that the mapping of the slow wind down to the Sun
appears to connect to or near the helmet streamer belt (e.g., Gosling, 1997;
Zhao et al., 2009).

Another key feature of the slow wind is its latitudinal extent, which
typically ranges from $40^{\circ}$–$60^{\circ}$ near solar minimum, a time
when it is easiest to distinguish the sources of fast and slow wind. Within
this broad region of slow wind the actual HCS, across which the magnetic field
changes direction, is very narrow. As for any current sheet, one can identify
in the heliospheric data a scale over which the field becomes small and the
plasma beta, defined as the ratio of the gas pressure $P_{g}$ to the magnetic
pressure $B^{2}/8\pi$, becomes large. This region is termed the plasma sheet
and is usually of the order of a few degrees in angular width (e.g.,
Winterhalter et al., 1994; Bavassano et al., 1997; Wang et al., 2000; Crooker
et al., 2004). It is important to note that the HCS is often not symmetrically
located within the broad band of slow wind, but is often found nearer to one
edge of the slow wind region (Burlaga et al., 2002). It is also important to
note that the field almost never vanishes at the HCS, as would be expected for
a true steady-state. This observation implies that, at least, the wind near
the HCS must be continuously dynamic.

The final and most critical feature of the slow wind that distinguishes it
from the fast is the plasma composition (Geiss et al., 1995; von Steiger et
al., 1995). It is well-known that in the closed field corona, the ratio of the
abundances of elements with low first ionization potential (FIP), such as Mg
and Fe, to those with high FIP, such as C and Ne, is a factor 4 or so higher
than in the photosphere (e.g., Meyer, 1985; Feldman & Widing, 2003). This so-
called FIP effect is not seen in the fast wind, which has abundances similar
to those of the photosphere; but, it is present in the slow wind, which has
abundances similar to that of the closed corona (Gosling, 1997; Zurbuchen &
von Steiger, 2006; Zurbuchen, 2007).

Along with the difference in elemental abundances, the slow and fast wind also
exhibit clear differences in their ion charge state abundances, for example,
the ratio of ${\rm O}^{7}/{\rm O}^{6}$. This ratio can be used to determine
the “freeze-in” temperature of the ion charge states at the source of the
wind. Close to the Sun where the time scales for ionization and recombination
are much shorter than the plasma’s expansion time-scales, the ion charge
states are approximately in ionization equilibrium with the local electron
temperature. As the solar wind plasma expands outward, however, the electron
density drops rapidly and the recombination time scales become so large that
the ionic charge states stop changing, freezing-in the electron temperature at
this point. The freeze-in radius varies for the different ions, but is
typically 1 - 3 R⊙. The data show that the slow wind has a higher freeze-in
temperature ($\geq 1.5\times 10^{6}\,{\rm K}$) than the fast wind ($\leq
1.2\times 10^{6}\,{\rm K}$) (von Steiger et al., 1997, 2001; Zurbuchen et al.,
1999, 2002). Note, however, that this freeze-in temperature corresponds only
to the electron temperature in the low corona. The proton and ion temperatures
measured in situ and in coronal holes by UVCS, for example, (e.g., Kohl et
al., 2006) show the opposite trend in that the ion temperatures are
substantially higher in the fast wind than in the slow (Marsch, 2006). The
origin of these differences in the ion temperatures between the two winds is
still not clear, but in any case, both the ion and freeze-in temperatures
suggest that the sources of the two winds near the Sun are physically
different.

The elemental abundances track very well the ionic abundances, indicating that
there is a consistent compositional distinction between the two winds.
Furthermore, the two winds have markedly different temporal variability in
elemental and ionic composition. The fast exhibits an approximately constant
composition; whereas the slow exhibits large and continuous variability, so
that its elemental composition varies from coronal to near photospheric. The
composition results suggest that the fast wind has a unique origin, presumably
in coronal holes, but that the slow wind originates from a mixture of sources.

In fact, Zurbuchen and coworkers have argued that the compositional
differences, rather than the speed, are what truly distinguish the two winds,
because it is possible to find solar wind whose composition and constancy
match that of the “fast wind,” but that has relatively slow speed, $<500\,{\rm
km/s}$ (Zhao et al., 2009). Note also that, as determined by the composition
measurements (Zurbuchen et al., 1999), the boundary between the slow and fast
wind in the heliosphere is sharp, of order a few degrees in angular extent,
much smaller than the angular width of the slow wind region, but comparable to
that of the plasma sheet. An important point is that the observed sharpness of
the composition transition is not merely a dynamical effect, because it does
not depend on whether the stream-stream transition is fast to slow or slow to
fast (Geiss et al., 1995; Zurbuchen, 2007). We conclude, therefore, that the
fast and slow winds are far more appropriately described as the steady and
unsteady winds, and that the boundary layer between the two winds is much
narrower than the width of either wind.

Since the differences in plasma composition of the two winds must be due to
differences in their origins at the Sun, the composition data place severe
constraints on the possible sources of the slow wind. In particular, the data
imply that the slow wind originates in the dynamic opening of closed magnetic
flux, which releases closed-corona plasma into the wind. Such a process would
also naturally explain the difference in variability between the fast and slow
wind.

It should be emphasized, however, that this constraint on the slow wind’s
origin is not universally accepted. Several authors have argued that the slow
wind originates from open-field coronal holes, just like the fast wind, but
from the edges of the holes, where the field expands super-radially as it
extends from the photosphere out to the heliosphere (e.g., Kovalenko, 1981;
Wang & Sheeley, 1991; Cranmer & van Ballegooijen, 2005; Cranmer et al., 2007;
Wang et al., 2009). The hypothesis is that a large expansion factor can both
slow down the wind by affecting the location of wave energy deposition in
coronal flux tubes, and change the plasma composition by the FIP mechanism
proposed by Laming (2004). Note that in the expansion factor model, as in all
steady state wind solutions, the properties of the wind in a given flux tube
are determined uniquely, in most cases, by the flux tube geometry and the
forms of the heating and momentum deposition (Cranmer et al., 2007). Of course
the detailed forms of the heating and momentum deposition will depend on the
flux tube geometry, and may depend on other factors, as well, but the
dependence on these other factors cannot be dominant; otherwise the calculated
wind speed would not be well correlated with expansion factor. In other words,
two flux tubes on the Sun with identical geometry should have similar
heating/momentum deposition and end up with the same wind properties.
Therefore, the steady-state models inherently predict a tight correlation
between speed and composition (e.g., Cranmer et al., 2007).

The problem, however, is that observations indicate that wind speed is not
tightly correlated with composition. The wind from small equatorial coronal
holes with a large expansion factor is indeed slow, with speeds $<500\,{\rm
km/s}$, in good agreement with the predictions of the expansion factor models.
But this wind has photospheric FIP ratios, so it is still considered to be
“fast wind” (Zhao et al., 2009). Furthermore, this not-so-fast wind has the
temporal quasi-steadiness of the fast wind, rather than the quasi-chaotic time
variation of the slow wind.

We conclude, therefore, that the most likely source for the true slow wind,
that with FIP-enhanced coronal composition, is the closed-field corona. In
this case, the process that releases the coronal plasma to the wind must be
either the opening of closed flux or interchange reconnection between open and
closed magnetic field lines. This latter process is the underlying mechanism
invoked by Fisk and co-workers (Fisk et al., 1998; Fisk, 2003; Fisk & Zhao,
2009) in their model for the heliospheric field. These authors argue that open
flux can diffuse freely throughout the solar surface, even deep inside the
helmet streamer region. If so, then the interchange reconnection between open
and closed magnetic field lines would naturally account for both the
composition and geometrical properties of the slow wind. The difficulty with
this model is that it has not been demonstrated that such open flux diffusion
can actually occur. In fact, detailed MHD simulations indicate that it is
difficult to bring open fields into closed-field regions without having them
close down (Edmondson et al., 2010; Linker et al., 2010). The simulation
results are in agreement with Antiochos et al. (2007), who argued that, for
the low-beta corona, basic MHD force balance forbids the presence of open flux
deep inside the closed helmet streamer region.

Within the context of MHD models, the most likely location for the release of
closed-field plasma is from the tops of helmet streamers (the Y-point at the
bottom of the HCS), where the balance between gas pressure and magnetic
pressure is most sensitive to perturbations. A number of authors have argued
that streamer tops are unstable and should undergo continual opening and
closing as a result of thermal instability (Suess et al., 1996; Endeve et al.,
2004; Rappazzo et al., 2005). Even if streamer tops are stable, it seems
inevitable that the constant emergence and disappearance of photospheric flux
and the constant motions of the photospheric would force them to be
continuously evolving. Furthermore, coronagraph observations often show the
ejection of “blobs” from the tops of streamers and into the HCS (Sheeley et
al., 1997).

Although this streamer top model seems promising in that it naturally explains
both the composition and variability, it has difficulty in accounting for the
large angular widths of the slow wind. One would expect the instabilities to
be confined to the high-plasma beta region about the current sheet. In fact,
the plasma emanating from the streamer tops, the so-called stalks, is observed
to be only $\sim 3^{\circ}$–$6^{\circ}$ wide, which agrees well with the
plasma sheet width in the heliosphere (Bavassano et al., 1997; Wang et al.,
2000). Even if the plasma sheet width were to be widened by the Kelvin-
Helmholtz instability (e.g., Einaudi et al., 1999), there would not be enough
mass flux from the narrow region at the streamer tops to account for the slow
wind. The streamer-top models can account for a thin band of slow wind around
the HCS, but it seems unlikely that this is the origin of the bulk of the slow
wind, which can extend as far as $30^{\circ}$ in latitude from the HCS.

In order to be compatible with the in situ data, we require some process that
releases closed-field plasma onto open field lines that, in the heliosphere,
can be far from the HCS. This requirement seems impossible to satisfy, because
the plasma release must occur at the boundary between the open and closed
field in the corona, which maps directly to the HCS. We describe below,
however, a magnetic topology that resolves this slow wind paradox: the flux
associated with an open-field corridor can be simultaneously near to and far
from the open-closed boundary!

##  2 The Topology of an Open-Field Corridor

Figure 1 illustrates the magnetic connectivity from the photosphere to the
heliosphere that results from an open-field corridor. The dark yellow inner
sphere in the figure represents the photosphere, while the light yellow, semi-
transparent one represents an arbitrary radial surface in the open-field
heliosphere, say at $5R_{\odot}$ The green line on the photosphere marks the
boundary between open (gray) and closed (yellow) field regions, which is
mapped by the magnetic field (red lines) to the HCS (thick green line) at the
$5R_{\odot}$ surface. The green line at the HCS is also the polarity inversion
line at this surface. Note that the four points, a, b, c, and d, which are
meant to represent the end-points of the corridor at the Sun, map sequentially
to the corresponding points a′, b′, c′, and d′ along the HCS.

The open field pattern at the photosphere of Fig. 1 consists of a large polar
coronal hole and, as is often seen, a smaller low-latitude hole. In recent
work, we argued that if the two holes are in the same photospheric polarity
region, then by our uniqueness conjecture the holes must be connected by an
open field corridor, as illustrated above (Antiochos et al., 2007). It is
evident from the figure that the flux in the corridor maps on the heliospheric
surface to a thin arc (light gray band), bounded at both ends by the HCS. The
flux between the arc and the HCS maps to the low-latitude extension while the
flux outside the arc maps to the main part of the polar coronal hole. The
corridor and its associated arc are the footprints of two quasi-separatrix
layers (QSLs, e.g., Priest & Démoulin, 1995; Démoulin et al., 1996) that
combine into a hyperbolic flux tube, as has been described in detail by Titov
et al. (2002, 2008) for the case of closed magnetic configurations. In
contrast, the HCS is a true separatrix.

The key point for understanding the origin of the slow wind is that, just like
the HCS, the QSL arc in the heliosphere can also be a source region for slow
wind. If the open-field corridor at the Sun is sufficiently narrow, then the
continual evolution of the photosphere, driven by the ever-present
supergranular flow and flux emergence/submergence in particular, will
continually change the exact location of this corridor. But, by the uniqueness
conjecture (Antiochos et al., 2007), the corridor is a topologically robust
feature, similar to a null-point, and must be present on the photosphere as
long as the low-latitude coronal hole extension is present. Its location and
shape, however, will vary in response to local photospheric changes. These
variations require field line opening/closing and interchange reconnection,
thereby releasing closed-field plasma all along the QSL arc in the
heliosphere. Therefore, if the QSL arc extends to high latitudes, this will
naturally produce slow wind with an extent far from the HCS.

To determine whether the QSL resulting from an open field corridor does,
indeed, reach high heliospheric latitudes, we have calculated an example of a
field such as that of Fig. 1 using the source surface model (Altschuler &
Newkirk, 1969; Schatten et al., 1969; Hoeksema, 1991). The field is most
easily determined from the image-dipole formula derived by Antiochos et al.
(2007). For a dipole with moment ${\bf d}$ located at a point ${\bf r}_{d}$
inside the Sun, and a source surface at radius $R_{S}$, the magnetic field
${\bf B}$ is determined from the potential $\Phi$ via ${\bf B}=-\nabla\Phi$,
where $\Phi$ is given by:

$\Phi=\frac{{\bf d}\cdot({\bf r}-{\bf r}_{d})}{|{\bf r}-{\bf
r}_{d}|^{3}}-\frac{R_{S}r_{d}^{3}{\bf d}\cdot(R_{S}^{2}{\bf r}-r^{2}{\bf
r}_{d})}{|r_{d}^{2}{\bf r}-R_{S}^{2}{\bf r}_{d}|^{3}}.$ (1)

This field satisfies the source-surface boundary condition that
$B_{\theta}=B_{\phi}=0$ at $r=R_{S}$, since $\Phi=0$ there. The advantage of
this formulation is that most active regions can be approximated by a
collection of dipoles, and one can build up a field of arbitrary complexity by
simply adding a series of dipoles of the form of Eq. (1). Each dipole is
specified in terms of its position in spherical coordinates ${\bf
r}_{d}=r_{d}{\bf\hat{r}}(\theta_{d},\phi_{d})$, where $r_{d}$, $\theta_{d}$,
and $\phi_{d}$ specify the location of the dipole, and the spherical
components of its dipole moment, ${\bf d}=(d_{r},d_{\theta},d_{\phi})$.

Figure 2 shows the field computed from Eq. (1) for the case of two dipoles: a
sun-centered global dipole with a dipole moment of unit magnitude directed
along the north polar axis, and an equatorial “active region” dipole at ${\bf
r}_{d}=0.8R_{\odot}{\bf\hat{r}}(90^{\circ},0^{\circ})$ with a northward-
pointing dipole moment ${\bf d}=(0,-0.2,0)$. The source surface radius is
chosen as $R_{S}=4R_{\odot}$, though the exact value is not critical for our
argument. Note that for convenience in viewing the magnetic field, we have
selected the dipole parameters so that the system has symmetry across both the
equatorial $(\theta=90^{\circ})$ and meridional $(\phi=0)$ planes. Also, for
ease of viewing, we show in the Fig. 2 only the front hemisphere defined by
the angular region $(15^{\circ}\leq\theta\leq 90^{\circ})$ and
$(-90^{\circ}\leq\phi\leq 90^{\circ})$.

The solar surface, the photosphere, corresponds to the gray grid in Fig. 2.
The colored contours on this surface correspond to contours of radial flux,
indicating the presence of the active region dipole at the equator. We
selected the parameters for the active region dipole so that its structure
would be easily resolved. It is evident from Fig. 2 that the region is large
compared to real active regions, which are generally only a few degrees in
angular extent. On the other hand, the maximum field strength at the dipole
center is only $\sim 20$ times that of the polar region, which is much less
than the corresponding ratio for solar active regions, so the flux ratio
between the active region and global background field is approximately
correct. This ratio is the important parameter to obtain a coronal hole
extension.

The thick black line along the equator is the $B_{r}=0$ contour, i.e., the
polarity inversion line. The thick black line above the solar surface is the
polarity inversion line at the source surface, i.e., the bottom of the HCS.
Red field lines are traced at equal intervals along the HCS down to the solar
surface. These define the boundary between open and closed field lines. As
expected, the effect of the equatorial dipole is to pull the open-closed
boundary down to lower latitudes; in other words, to create a low-latitude
extension of the coronal hole, which can be seen as the gray shaded region in
the Figure. Far from the dipole, the coronal hole boundary is at a latitude of
$\sim 54^{\circ}$, whereas at the meridional symmetry plane the boundary drops
down to $\sim 26^{\circ}$.

For the large spatial scale of our active region dipole, the extension of the
coronal hole down to low latitudes is gradual rather than in the form of a
distinct “elephant trunk”, but the basic effect is clearly present. There is
no open-field corridor in Fig. 2, but let us now add another dipole to the
system, displaced $20^{\circ}$ in both latitude and longitude from the
equatorial one and a factor of five times weaker. This dipole is located at
${\bf r}_{d}=0.8R_{\odot}{\bf\hat{r}}(70^{\circ},20^{\circ})$ with a primarily
southward-pointing dipole moment ${\bf d}=(0,0.05,0)$. In order to maintain
the equatorial and meridional symmetry, as mentioned earlier, we actually add
4 dipoles symmetrically located about the equatorial and meridional planes.

The resulting field is shown in Figure 3. The effect of the additional dipoles
is to add high-latitude polarity inversion lines to the system. These
“squeeze” the open-flux extension of Fig. 2 to form a narrow corridor and a
low-latitude coronal hole. As in Fig. 2, red field lines are traced from
equidistant footpoints along the HCS down to the solar surface. The red
footpoints at the photosphere appear to traverse the boundary of the low-
latitude hole and then jump abruptly to the polar hole boundary, which implies
that the mapping defined by the field develops extreme gradients in the region
connecting the two holes. To clarify this point, we have traced two sets of
field lines, colored in blue, from footpoints that are closely located at the
HCS. The corresponding solar footpoints are much more widely spaced, running
along the corridor. The resulting structure, Fig. 3, looks very similar to the
mapping drawn in Fig. 1, in that the closely spaced pairs of points a′,b′ and
c′,d′ at the HCS map to far-separated points a,b and c,d at the solar surface.
Note also that although the footpoints of the two sets of blue lines approach
each other very closely at the photosphere, they are far separated at the HCS,
by a distance of order $R_{\odot}$. This result indicates that even though the
low-latitude coronal hole has small area, it contains a substantial magnetic
flux. As is evident from the colored contours in Fig. 3, the photospheric
field strength in the low-latitude hole is large due to the presence of the
active region dipole.

The analytic model underlying Fig. 3 has similar topology to the case shown
schematically in Fig. 1. The low-latitude coronal hole extension in Fig. 3 is
connected to the main polar hole by a corridor that becomes very narrow.
Furthermore, this type of topology is not difficult to obtain. It is often
observed in quasi-steady MHD solutions for observed photospheric fields, as
will be shown below. A similar corridor was found for Carrington rotation 1922
(Antiochos et al., 2007).

The question now is whether the open flux in the corridor connects to large
latitudes in the heliosphere. To answer this question, we trace field lines
from a set of photospheric footpoints lying on a latitudinal line segment
spanning the narrowest width of the corridor, which is only of order
$5{,}000\,{\rm km}$ at the photosphere. Fig. 4b shows the footpoints and the
field lines (green) near the photosphere and Fig. 4a shows where they map to
on the source surface. We note that the corridor maps to high latitudes. In
fact, for this analytic case, the corridor mapping defines a QSL arc that
reaches latitudes $>45^{\circ}$, greater than that of the observed slow wind.

This result, that the corridor maps to heliospheric latitudes far above the
HCS, is robust in that it is not sensitive to the exact position of the
secondary dipole. The position and geometry of the corridor, on the other
hand, is very sensitive to the photospheric flux distribution. For example,
its width would change or even become singular (Titov et al., 2011), and its
location would change substantially if the secondary dipoles were moved in
longitude. Based on flux conservation arguments, and the fact that the
heliospheric magnetic field is almost uniform in latitude, we can argue that
the angular extent of the QSL arc, however, would be expected to depend
primarily on the ratio of the flux in the low-latitude coronal hole extension
to that in the polar hole. For example, in the extreme case that the fluxes
were equal, the corridor mapping would be expected to reach the heliospheric
pole ($90^{\circ}$ from the HCS!), irrespective of the geometry of the
corridor or of the coronal holes.

##  3 The S-Web Model

If the width of the corridor at the photosphere is small compared to the scale
of typical motions there, such as the supergranular flow, we expect that the
whole corridor will continuously disrupt and reform at the photosphere and,
consequently, closed-field plasma will be released by reconnection all along
the QSL arc in the heliosphere. Therefore, the topology of Fig. 2 may be able
to resolve the slow wind paradox. The overriding question, however, is whether
there are enough such corridors and corresponding QSL arcs in the heliosphere
to account for the slow wind that is observed. The flux distribution of Fig. 2
produces only one such arc, which would certainly not be sufficient to
reproduce the observed slow wind. There are two issues that must be addressed,
the number of arcs (their density and extent on the Sun and heliosphere), and
the amount of mass and energy that each arc can be expected to release. In
this paper we concentrate on the first issue and only briefly discuss the
second in Section 4 below, because addressing this issue requires fully
dynamic calculations.

In order to address the issue of the number of QSL arcs, we calculated the
quasi-steady model for an observed photospheric flux distribution. Figure 5a
shows the photospheric radial field as derived from MDI observations on SOHO
(Scherrer et al., 1995) for a time period preceding the August 1, 2008 total
solar eclipse. This calculation was used to predict the structure of the
corona prior to the eclipse, using magnetic field data measured during the
period June 25–July 21, 2008\. The prediction compares very favorably with
images of the corona taken during the eclipse in Mongolia (Rušin et al.,
2010). Note that the high resolution of the calculation captures the details
of many small-scale bipoles in the photospheric magnetic field (Harvey, 1985).
This has been incorporated into the idea of the “magnetic carpet” (Schrijver
et al., 1997). We also show the polarity inversion line $B_{r}=0$ slightly
above the photosphere, at $r=1.05R_{\odot}$ to delineate the magnetic polarity
of the large-scale structures. (The polarity inversion line in the photosphere
itself shows an enormous complexity that overshadows its usefulness to discern
the large-scale magnetic polarity.)

The quasi-steady model was calculated by using the 3D MHD code MAS. The MAS
code and its implementation are described in detail by Mikić & Linker (1994),
Mikić et al. (1999), Linker et al. (1999), and Lionello et al. (2009). MAS
solves the time-dependent MHD equations, including a realistic energy equation
with optically thin radiation and thermal conduction parallel to the magnetic
field. Given the magnetic field at the photosphere and an assumption for the
coronal heating source, the MHD equations are advanced until the magnetic
field settles down close to steady state. MHD models are generally considered
to be the most sophisticated implementation of Parker’s solar wind theory
because they incorporate all the essential physics, including the balance
between gas pressure and Lorentz force. An important assumption is the form of
coronal heating, which is prescribed empirically at the present time since the
coronal heating process is still unknown. The parameters of the empirical
heating model are constrained by observations of coronal emission in EUV and
X-rays (e.g., Lionello et al., 2009), as well as by solar wind measurements.
Details on the assumed form for the heating and on the thermodynamics used in
the MAS code can be found in Mikić et al. (2007) and Lionello et al. (2009).

In order to capture as much of the photospheric magnetic structure as
possible, we ran the MAS code with unprecedented resolution. Our calculation
used more than 16 million mesh cells and was run on over 4000 processors of
NSF’s Ranger supercomputer at the Texas Advanced Computing Center, making it
possible to include much of the small-scale structure of the photospheric
field in both the quiet sun and in coronal holes, as shown in Fig. 5a. These
calculations are unique in the degree to which they capture the small-scale
structure of the measured magnetic field.

Figure 5b shows the distribution of open and closed magnetic field regions at
the solar surface as determined by the model. It is evident that there are
many low-latitude coronal hole extensions, similar to that in Fig. 3, but with
much more structure. Several of these extensions appear to be disconnected
from the main polar holes, but this is partly due to the limited resolution of
the figure. A few of these coronal hole extensions are indeed connected by
very thin corridors in the photosphere, though many are only linked to the
polar coronal holes in a singular manner, as described in detail by Titov et
al. (2011), and as discussed further below.

The open field pattern in Fig. 5b is clearly complex, but the important issue
is the degree of complexity of the mapping into the heliosphere and, in
particular, the structure of the separatrices and QSLs there. We determined
the open field mapping in great detail by tracing tens of millions of magnetic
field lines. The topology of this mapping, as evidenced by structures such as
separatrices and QSLs, is most easily seen by analyzing the squashing factor
$Q$ (Titov et al., 2002; Titov, 2007). $Q$ is a measure of the distortion in
the magnetic field mapping, and is directly related to the gradients in the
connectivity. QSLs are regions of very large $Q$; we generally define them as
any region with $Q>10^{3}$. True separatrices such as the HCS have infinite
$Q$, because the mapping is singular there, but when computed numerically they
appear as surfaces with very large (unresolved) values of $Q$. The gray arc at
$r=5R_{\odot}$ in Fig. 1 is a QSL in the open field, and consequently would be
a region of high $Q$. The green HCS would also be a region of high (infinite)
$Q$. As will be seen below, a high-resolution analysis of the $Q$ properties
of our MHD simulation is extremely informative.

Figure 6a shows $Q$ in a meridional plane at a central Carrington longitude of
$23.33^{\circ}$ at the time of the eclipse at 10:21UT, while Figure 6b shows
magnetic field lines traced from the vicinity of the solar limbs at the same
time. We see that $Q$ outlines the boundary between open and closed field,
which is a true separatrix surface, but it is apparent that there is much more
detailed structure in both the closed and open field regions. The complex
structure of $Q$ in the closed-field region is expected; it simply reflects
the fact that the photospheric field consists of many small bipoles; but,
there is also substantial structure in the open field near the open-closed
boundary. Note the presence of a “pseudostreamer” on the NE limb, a feature
that has been discussed by Wang et al. (2007). The relationship of
pseudostreamers to open hole corridors and the S-web is discussed in detail in
Titov et al. (2011)

Figure 7a shows $Q$ in the spherical surface at $r=10R_{\odot}$ using a
logarithmic scale. This is the structure that is expected to map into the
inner heliosphere (appropriately wrapped into a spiral magnetic field by solar
rotation), since the magnetic field has reached its asymptotic structure by
this radius. The thick black line is the heliospheric current sheet (at which
$B_{r}$ reverses sign). Figure 7b shows the magnitude of $B_{r}$ at the same
radial surface $r=10R_{\odot}$. Note that the choice of $10R_{\odot}$ is not
crucial. Any surface in the heliosphere (where the field is all open) yields
similar results.

It is important to emphasize that the apparent structure in $Q$ expresses only
the connectivity of the open field, not its actual magnitude. In spite of the
enormous magnetic complexity at the solar surface, the radial field
distribution in the heliosphere is completely unremarkable, Fig. 7b. There is
a single polarity inversion line denoting a single HCS, as is generally
observed near solar minimum, and this HCS runs more or less equatorial. The
radial field is essentially uniform away from the HCS, as would be expected
from simple pressure balance. (Careful examination of the plot of $B_{r}$
shows that there is a faint semblance of the structure that can be seen in
$Q$, but it is only a small perturbation.)

On the other hand, the $Q$ map at this surface is remarkable, indeed, Fig. 7a.
We see that surrounding the HCS is a broad web of separatrices and QSLs of
enormous complexity. There are at least four striking features of this S-web.
First, it has an angular extent in latitude of approximately $40^{\circ}$,
sufficient to account for the observed extent of the slow wind. Note also that
the angular extent does vary with longitude, but only by a factor of two or
so. Second, the HCS is not necessarily in the center of the S-web, but is
sometimes near its edge. This can explain the frequent observation that the
HCS is usually not centrally located within slow wind streams (e.g., Burlaga
et al., 2002). Third, the boundary between the S-web layer and the featureless
polar hole region is sharp; it is narrow compared to the width of the S-web.
This can explain the observation that the transition from slow to fast wind as
measured by the composition data is narrow compared to the slow wind region
itself (Zurbuchen et al., 1999).

In order to explore the details of how coronal hole extensions connect to the
polar holes, we calculated coronal hole areas at different heights in the
corona. Figure 8 shows the location of a region near longitude $75^{\circ}$
and latitude $15^{\circ}$N in which we explored the connection between the
low-latitude coronal hole extensions (of negative polarity, shown in blue) in
detail. It is evident that the coronal hole extensions in this region appear
disconnected from the north polar hole in the photosphere, but connect with it
low in the corona (at heights approximately between $0.01R_{\odot}$ and
$0.02R_{\odot}$ above the photosphere). Figure 9 shows explicitly how these
coronal holes connect in the low corona. The three-dimensional shape of the
coronal hole boundary is shown as a green semi-transparent surface in the low
corona in the region detailed in Figure 8. This is the boundary between open
and closed field regions. The regions marked by A, B, and C show examples in
which the extensions of coronal holes are not connected in the photosphere, at
least by any measurable open-field corridor, but appear to connect above the
photosphere in the low corona. These regions are also indicated in Figure 8
for ease of cross-reference. Despite the fact that these coronal holes are
“disconnected” in the photosphere, they always remain topologically linked in
a singular manner with the polar coronal hole, as discussed by Titov et al.
(2011).

Finally, note that the connections of the high-$Q$ lines between the
neighborhood of the HCS and the photosphere and low corona that were
postulated by the uniqueness conjecture (Antiochos et al., 2007) are largely
present, even though the insight from these new high-resolution MHD
simulations has led us to generalize the uniqueness conjecture. We have found
that, in general, coronal hole extensions are sometimes connected to the polar
holes in the photosphere via narrow corridors, as originally postulated
(Antiochos et al., 2007), but in other instances they are disconnected in the
photosphere, but remain topologically linked to the polar holes (Titov et al.,
2011). In either case, these connections are responsible for the formation of
the S-web. It should be emphasized that in order to capture the intricate
structure of these connections, very high resolution models are required that
can incorporate some of the complexity of the photospheric magnetic carpet
fields. Given sufficient resolution, the S-web should appear as a generic
feature of all quasi-steady models, including the PFSS. In fact, the PFSS
models should be more effective than the MHD for studying the complex topology
of the S-web, because they allow for much higher spatial resolution than is
possible with an MHD code. On the other hand, for quantitative comparison with
observations, the MHD models should be more effective, because they include
the gas thermal and kinetic pressure forces and Lorentz forces that we know
are present in the real corona.

##  4 Discussion

The major conclusion from our results is that the underlying premise of the
streamer top model is valid. The slow wind is expected to originate from the
release of closed-field plasma due to the dynamic rearrangement of the open-
closed field boundary. The key new addition of our S-web model to this picture
is that the inherent complexity of the photospheric field leads to a network
of narrowly connected and disconnected coronal holes that nevertheless always
remain linked. This produces a separatrix web in the heliosphere that extends
the release of slow wind to regions that significantly depart from the HCS.
Hence, our model accounts for both the observed composition and the broad
extent of the slow wind.

One immediate prediction from the model is that the angular width of the slow
wind is determined primarily by the complexity of the flux distribution in the
photosphere. This complexity produces a very convoluted polarity inversion
line in the low corona and an intricate coronal hole pattern (Figure 5). Our
ability to identify the S-web and its manifestations rests on high-resolution
calculations that are beginning to capture the multitude of small dipoles in
the photospheric magnetic field. If the solar field were a pure dipole,
producing an inversion line that runs straight along the equator, then only
the polar coronal holes would be present and there would be no separatrix web
in the heliosphere. For this “basal” (though idealized) slow wind case, if we
assume that the dynamic broadening of the open-closed boundary at the Sun is
of order the scale of a supergranule, $\sim 30{,}000\,{\rm km}$, the angular
extent of the wind would be only of order $3^{\circ}$–$5^{\circ}$, and would
be centered about the HCS. Of course, the solar field is never a simple
dipole.

At the present time we do not know if the complexity seen in Figures 5–7 is
typical, or whether it is particular to this late declining phase of Cycle 23.
It should be noted that the present minimum appears to be somewhat different
than the previous few minima. In particular, the polar field strength is
significantly weaker (e.g., Luhmann et al., 2009).

The S-web model predicts that for time periods during which extensions of
coronal holes away from the main polar holes are less prevalent than in Cycle
23, the angular extent of the slow wind region would be smaller. In fact,
there is clear evidence from radio scintillation data (Tokumaru et al., 2010)
and recent Ulysses solar wind measurements that the Cycle 23 minimum has a
substantially broader and more structured slow wind region than that of the
previous cycle. Indeed, during the previous minimum (circa 1996), equatorial
coronal hole extensions were less common than during the recent solar minimum.
Further high-resolution numerical calculations will be needed to address this
result.

Another prediction of the model is that the slow wind region is actually a
mixture of winds. It is evident from Fig. 7 that the separatrix web is not
space-filling. There are regions within the broad S-web band where the wind
emanates from the low-latitude coronal hole extensions. These regions are
likely to have large expansion factor, so that the wind will be slow compared
to the fast wind from the polar regions, but its composition will be different
than that of closed-field plasma. Our model, therefore, naturally explains the
observed variability of the slow wind composition.

A key aspect of the S-web model that has yet to be calculated is the dynamic
release of closed-field plasma. Although our quasi-steady calculations allow
us to investigate the topology of the field, and to identify the structure of
the separatrix web in the heliosphere, they do not actually produce a slow
wind with closed-field composition. For this we need fully dynamic simulations
that include the driving due to photospheric motions (e.g., resulting from
differential rotation) and flux emergence. Such simulations are now being
performed in 3D (e.g., Edmondson et al., 2009, 2010; Linker et al., 2010) for
simplified photospheric flux distributions and driving flows. These
simulations do verify the basic idea of the S-web model that open-field
corridors will form and evolve in response to photospheric motions (Edmondson
et al., 2009). Higher resolution simulations will be needed, however, to test
the model in detail. On the other hand, it seems unlikely that dynamic
calculations with the degree of structure present in Fig. 7 will be feasible
in the near future. It is likely that a definitive treatment of the slow wind
will require the development of a statistical theory of the dynamics of the
S-web model.

This work has been supported by the NASA TR&T, SR&T, and HTP Programs. The
work has benefited greatly from the authors’ participation in the NASA TR&T
focused science team on the solar-heliospheric magnetic field. SKA thanks J.
Karpen for invaluable scientific discussions and help with the graphics.

## References

  * Altschuler & Newkirk (1969) Altschuler, M. D. & Newkirk, G. 1969, Sol. Phys., 131
  * Antiochos et al. (2007) Antiochos, S. K., DeVore, C. R., Karpen, J. T., & Mikić, Z. 2007, ApJ, 671, 936
  * Bame et al. (1977) Bame, S. J., Asbridge, J. R., Feldman, W. C., & Gosling, J. T. 1977, J. Geophys. Res., 82, 148
  * Bavassano et al. (1997) Bavassano, B., Woo, R., & Bruno, R. 1997, Geophys. Res. Lett., 24, 1655
  * Bruno & Carbone (2005) Bruno, R. & Carbone, V. 2005, Living Reviews in Solar Physics, 2, no. 4, http://solarphysics.livingreviews.org/Articles/lrsp-2005-4/
  * Burlaga et al. (2002) Burlaga, L. F., Ness, N. F., Wang, Y.-M., & Sheeley, N. R. 2002, J. Geophys. Res., 107(A11), 1410, doi:10.1029/2001JA009217
  * Cranmer & van Ballegooijen (2005) Cranmer, S. R. & van Ballegooijen, A. A. 2005, ApJS, 156, 265
  * Cranmer et al. (2007) Cranmer, S. R., van Ballegooijen, A. A., & Edgar, R. J. 2007, ApJS, 171, 520
  * Crooker et al. (2004) Crooker, N. U., Huang, C.-L., Lamassa, S. M., Larson, D. E., Kahler, S. W., & Spence, H. E. 2004, J. Geophys. Res., 109, A03107, doi:10.1029/2003JA010170
  * Démoulin et al. (1996) Démoulin, P., Henoux, J. C., Priest, E. R., & Mandrini, C. H. 1996, A&A, 308, 643
  * Edmondson et al. (2009) Edmondson, J. K., Lynch, B. J., Antiochos, S. K., De Vore, C. R., & Zurbuchen, T. H. 2009, ApJ, 707, 1427
  * Edmondson et al. (2010) Edmondson, J. K., Antiochos, S. K., De Vore, C. R., & Zurbuchen, T. H. 2010, ApJ, submitted
  * Einaudi et al. (1999) Einaudi, G., Boncinelli, P., Dahlburg, R. B., & Karpen, J. T. 1999, J. Geophys. Res., 104, 521
  * Endeve et al. (2004) Endeve, E., Holzer, T. E., &Leer, E. 2004, ApJ, 603, 307
  * Feldman & Widing (2003) Feldman, U. & Widing, K. G. 2003, Space Sci. Rev., 107, 665
  * Fisk et al. (1998) Fisk, L. A., Schwadron, N. A., & Zurbuchen, T. H. 1998, Space Sci. Rev., 86, 51
  * Fisk (2003) Fisk, L. A. 2003, J. Geophys. Res., 108, 1157
  * Fisk & Zhao (2009) Fisk, L. A. & Zhao, L. 2009, in Universal Heliospheric Processes, Proc. IAU Symp. 257, 109
  * Geiss et al. (1995) Geiss, J., Gloeckler, G, & von Steiger, R. 1995, Space Sci. Rev., 72, 49
  * Gosling (1997) Gosling, J. T. 1997, in AIP Conf. Proc. 385, Robotic Exploration Close to the Sun: Scientific Basis, ed. S. R. Habbal (Woodbury: AIP), 17
  * Harvey (1985) Harvey, K. L. 1985, Aust. J. Phys., 38, 875
  * Hoeksema (1991) Hoeksema, J. T. 1991, Adv. Space Res., 11, 15
  * Kohl et al. (2006) Kohl, J. L., Noci, G., Cranmer, S. R., & Raymond, J. C. 2006, ARA&A, 13, 31
  * Kovalenko (1981) Kovalenko, V. A. 1981, Sol. Phys., 73, 383
  * Laming (2004) Laming, J. M. 2004, ApJ, 614, 1063
  * Luhmann et al. (2009) Luhmann, J. G. et al. 2009, Sol. Phys., 256, 285
  * Linker et al. (1999) Linker, J. A., Mikić, Z., Biesecker, D. A., Forsyth, R. J., Gibson, S. E., Lazarus, A. J., Lecinski, A., Riley, P., Szabo, A., & Thompson, B. J. 1999, J. Geophys. Res., 104, 9809
  * Linker et al. (2010) Linker, J. A., Lionello, R., Mikić, Z., Titov, V. S., & Antiochos, S. K. 2010, ApJ, submitted
  * Lionello et al. (2009) Lionello, R., Linker, J. A., & Mikić, Z. 2009, ApJ, 690, 902
  * Marsch (2006) Marsch, E. 2006, Living Reviews in Solar Physics, 3, no. 1, http://solarphysics.livingreviews.org/Articles/lrsp-2006-1/
  * McComas et al. (2000) McComas, D. J., Barraclough, B. L., Funsten, H. O., Gosling, J. T., Santiago-Munoz, E., Skoug, R. M., Goldstein, B. E., Neugebauer, M., Riley, P., & Balogh, A. 2000, J. Geophys. Res., 105, 10419
  * McComas et al. (2008) McComas, D. J., et al. 2008, Geophys. Res. Lett., 35, L18103
  * Meyer (1985) Meyer, J.-P. 1985, ApJS, 57, 173
  * Mikić & Linker (1994) Mikić, Z., & Linker, J. A. 1994, ApJ, 430, 898
  * Mikić et al. (1999) Mikić, Z., Linker, J. A., Schnack, D. D., Lionello, R., & Tarditi, A. 1999, Phys. Plasmas, 6, 2217
  * Mikić et al. (2007) Mikić, Z., Linker, J. A., Lionello, R., Riley, P., & Titov, V. 2007, in Solar and Stellar Physics Through Eclipses, ASP Conference Series, Vol. 370, eds. O. Demircan, S. O. Selam, & B. Albayrak, (San Francisco: Astronomical Society of the Pacific), p.299
  * Parker (1958) Parker, E. N. 1958, ApJ, 128, 664
  * Parker (1963) Parker, E. N. 1963, Interplanetary Dynamic Processes, (New York: Interscience Publishers)
  * Priest & Démoulin (1995) Priest, E. R., & Démoulin, P. 1995, J. Geophys. Res., 100, 23443
  * Rappazzo et al. (2005) Rappazzo, A. F.,Velli, M., Einaudi, G., & Dahlburg, R. B. 2005, ApJ, 633, 474
  * Rušin et al. (2010) Rušin, V., Druckmüller, M., Aniol, P., Minarovjech, M., Saniga, M., Mikić, Z., Linker, J. A., Lionello, R., Riley, P., & Titov, V. S. 2010, A&A, 513, A45
  * Schatten et al. (1969) Schatten, K., Wilcox, J. W., & Ness, N. F. 1969, Sol. Phys., 9, 442
  * Scherrer et al. (1995) Scherrer, P. H. et al. 1995, Sol. Phys., 162, 129
  * Schrijver et al. (1997) Schrijver, C. J. et al. 1997, Nature, 48, 424
  * Schwenn (1990) Schwenn, R., 1990, in Physics of the Inner Heliosphere I, eds. R Schwenn & E. Marsch, (Berlin:Springer-Verlag), p.99
  * Sheeley et al. (1997) Sheeley, N. R., Jr. et al. 1997, ApJ, 484, 472
  * von Steiger et al. (1995) von Steiger, R., Schweingruber, R. F., Wimmer, R., Geiss, J., & Gloeckler, G. 1995, Adv. Space Res.y, 15(7), 3
  * von Steiger et al. (1997) von Steiger, R., Geiss, J., & Gloeckler, G. 1997, in Cosmic Winds and the Heliosphere, eds. J. R. Jokipii, C. P. Sonett, & M. S. Giampapa (Tucson: Arizona U Press), p. 581\.
  * von Steiger et al. (2000) von Steiger, R., et al., 2000, J. Geophys. Res., 105, 27217.
  * von Steiger et al. (2001) von Steiger, R., Zurbuchen, T. H., Geiss, J., Gloeckler, G., Fisk, L. A., & Schwadron, N. A. 2001, Space Sci. Rev., 97, 123
  * Suess et al. (1996) Suess, S.T., Wang, A-H. & Wu, S. T. 1996, J. Geophys. Res.101, 19957
  * Titov et al. (1999) Titov, V. S., Démoulin, P., & Hornig, G. 1999, in Magnetic Fields and Solar Processes, ESA SP-448, 715T
  * Titov et al. (2002) Titov, V. S., Hornig, G., & Démoulin, P. 2002, J. Geophys. Res., 107, 1164
  * Titov (2007) Titov, V. S. 2007, ApJ, 660, 863
  * Titov et al. (2008) Titov, V. S., Mikić, Z., Linker, J. A., & Lionello, R. 2008, ApJ, 675, 1614
  * Titov et al. (2011) Titov, V. S., Mikić, Z., Linker, J. A., Lionello, R., & Antiochos, S. K. 2011, ApJ, in press
  * Tokumaru et al. (2010) Tokumaru, M., Kojima, M., & Fujiki, K. 2010, J. Geophys. Res., 115(A4), 4102, doi:10.1029/2009JA014628
  * Wang & Sheeley (1991) Wang, Y.-M., & Sheeley, N. R. 1991, ApJ, 372, L45
  * Wang et al. (2000) Wang, Y.-M., Sheeley, N. R., Socker, D. G., Howard, R. A., & Rich, N. B. 2000, J. Geophys. Res., 105(A11), 25133
  * Wang et al. (2007) Wang, Y.-M., Sheeley, N. R., Jr., & Rich, N. B. 2007, ApJ, 658, 1340
  * Wang et al. (2009) Wang, Y.-M., Ko, Y.-K., & Grappin, R. 2009, ApJ, 691, 760
  * Winterhalter et al. (1994) Winterhalter, D., Smith, E. J., Burton, M. E., Murphy, N., & McComas, D. J. 1994, J. Geophys. Res., 99(A4), 6667
  * Zhao et al. (2009) Zhao, L., Zurbuchen, T. H., & Fisk, L. A. 2009, Geophys. Res. Lett., 36, CiteID L14104, doi:10.1029/2009GL039181
  * Zirker (1977) Zirker, J. B. 1977, Coronal Holes and High Speed Wind Streams, (Boulder: Colorado Assoc. University Press)
  * Zurbuchen et al. (1999) Zurbuchen, T. H., Hefti, S., Fisk, L. A., Gloeckler, G., & von Steiger, R. 1999, Space Sci. Rev., 87, 353
  * Zurbuchen et al. (2002) Zurbuchen, T. H., Fisk, L. A., Gloeckler, G., & von Steiger, R. 2002, Geophys. Res. Lett., 29, 66, doi: 10.1029/2001GL013946
  * Zurbuchen & von Steiger (2006) Zurbuchen, T. H., & von Steiger, R. 2006, in SOHO 17: 10 years of SOHO and Beyond, ed. H. Lacoste & B. Fleck, (Noordwijk, ESA), SP-617, p. 7.1
  * Zurbuchen (2007) Zurbuchen, T. H. 2007, ARA&A, 45, 297

Figure 1:  Magnetic field topology of an open field region consisting of a
large polar coronal hole and a smaller low-latitude hole connected by an open-
field corridor. The inner surface is the photosphere, with the dark gray and
bright yellow regions corresponding to open and closed field respectively. The
outer transparent surface is a radial surface in the heliosphere. The dark
green line is the polarity inversion line and the light gray arc indicates
where the open-field corridor maps to on this outer surface.

Figure 2:  (Top) Open-closed magnetic field topology for a photospheric flux
distribution due to a global dipole and an equatorial dipole. The gray shaded
region indicates the polar coronal hole (the open flux region). The contours
on the inner surface indicate radial field magnitude at the photosphere. The
black lines correspond to the polarity inversion line at the photosphere and
source surface. The red lines are magnetic field lines. (Bottom) Close-up near
the solar surface of the magnetic field above.

Figure 3:  As in Figure 2, but for a flux distribution that includes
additional high-latitude dipoles. Two additional polarity inversion lines can
be seen at the photosphere. The blue field lines outline an open-field
corridor. Note that the system is symmetric about the meridional plane
$\phi=0$.

Figure 4:  (Top) Open field lines (green) traced from photospheric footpoints
along a line segment spanning the narrowest part of the corridor. The lines
clearly extend to high latitude above the HCS. (Bottom) Close-up near the
solar surface showing the photospheric footpoints of the corridor field lines.
Figure 5:  (a) Distribution of the radial component of the magnetic field in
the photosphere that was used in the MHD simulation to predict the structure
of the corona for the August 1, 2008 eclipse, as deduced from MDI
measurements. (b) The open and closed field regions in the photosphere as
determined from the MHD solution. The polarity inversion line ($B_{r}=0$) at a
height $r=1.05R_{\odot}$ is superimposed on these images to aid in identifying
the polarity of the large-scale magnetic flux.  Figure 6:  (a) Plot of the
squashing factor $Q$ on a logarithmic scale in a meridional plane at the time
of the eclipse on August 1, 2008 at 10:21UT. In this view, solar north is
vertically up and the $B_{0}$ angle is zero. [At the time of the eclipse
$B_{0}=5.8^{\circ}$, so this view is slightly different than what would have
been observed.] The Sun’s surface is colored by the value of $B_{r}$ with the
same scaling as that in Fig. 5. (b) Magnetic field lines traced from the
vicinity of the limbs at the same time, showing the structure of the open and
closed field regions.  Figure 7:  (a) Plot of the squashing factor $Q$ in the
spherical surface $r=10R_{\odot}$ on a logarithmic scale versus longitude and
latitude. (b) Plot of $B_{r}$ in the same spherical surface. The HCS (i.e.,
the location of $B_{r}=0$) is superimposed on these images as a thick black
line. The complex structure in $Q$ in the vicinity of the HCS is produced by
the S-web.  Figure 8:  The variation of coronal hole shape with height above
the photosphere. The top panel shows coronal holes at the photosphere, as in
Fig. 5b. The black square shows a $100^{\circ}\times 100^{\circ}$ region
centered at longitude $75^{\circ}$ and latitude $15^{\circ}$N that was used to
compute the variation of coronal hole shape with radius in the lower panels.
Note that the extended coronal holes connect to the polar holes low in the
corona. The regions denoted by A, B, and C are cross-referenced with the
corresponding regions in Fig. 9.  Figure 9:  The three-dimensional shape of
the coronal hole boundary (semi-transparent surface) in the region detailed in
Figure 8, showing that some of the coronal hole extensions (blue areas on the
surface of the sphere) connect with the north polar hole low in the corona.
The top panel shows a view in which the surface is artificially stretched in
radius by a factor of 3$\times$ to show details near the photosphere. The
bottom left panel shows the same view without the radial stretching. The
bottom right panel shows the region detailed in the context of the whole Sun.
The regions denoted by A, B, and C are cross-referenced with the corresponding
regions in Fig. 8.