# Reflection and Relay Dual-Functional RIS Assisted MU-MISO Systems ††thanks:
This work is supported in part by the National Natural Science Foundation of
China (Grant No. 61971088, 62071083, U1808206, and U1908214), the Natural
Science Foundation of Liaoning Province (Grant No. 2020-MS-108), in part by
the Fundamental Research Funds for the Central Universities (Grant No.
DUT20GJ214, DUT21GJ208 and DUT20RC(3)029), in part by Dalian Science and
Technology Innovation Project (Grant No. 2020JJ25CY001), and in part by the
Open Research Fund of National Mobile Communications Research Laboratory,
Southeast University (Grant No. 2021D08).

Yanan Ma∗, Rang Liu∗, Ming Li∗†, Yang Liu∗, Qingqing Wu‡ and Qian Liu∗

∗Dalian University of Technology, Dalian, Liaoning 116024, China
E-mail: {mayanan<EMAIL_ADDRESS>
{mli, yangliu_613<EMAIL_ADDRESS>†Southeast University, Nanjing,
Jiangsu 210096, China  ‡University of Macau, Macau, 999078, China
E-mail<EMAIL_ADDRESS>
###### Abstract

Reconfigurable intelligent surface (RIS) is a promising solution to adaptively
manipulate wireless propagation with low-cost passive devices. However, the
traditional passive RIS can offer sufficient signal strength only when
receivers are very close to it. Moreover, the users at the back side of it
cannot be well served due to its reflective property. In this paper we
introduce a novel reflection and relay dual-functional RIS architecture, which
can simultaneously realize passive reflection and active relay
functionalities. The problem of joint transmit beamforming and dual-functional
RIS design is investigated to maximize the achievable sum-rate of a multiuser
multiple-input single-output (MU-MISO) system. Based on fractional programming
(FP) theory and majorization-minimization (MM) technique, we propose an
efficient iterative transmit beamforming and RIS design algorithm. Simulation
results demonstrate the superiority of the introduced dual-functional RIS
architecture and the effectiveness of the proposed algorithm.

###### Index Terms:

Reconfigurable intelligent surfaces, beamforming, relay, fractional
programming (FP), majorization-minimization (MM).

##  I Introduction

Reconfigurable intelligent surface (RIS) has been proposed as an inherent
component of future wireless communication systems owing to its superior
ability of shaping the wireless environment [1]-[10]. However, due to “double
fading” effect (i.e., the equivalent path loss of the transmitter-RIS-receiver
link is the product of the path losses of transmitter-RIS link and RIS-
receiver link, which is usually an order-of-magnitude larger than that of the
direct link), RIS should be deployed close to either the base station (BS) or
users to effectively improve communication quality-of-service (QoS) [1], [2].
Besides, the research on the comparison of RIS and relay illustrates that RIS
can be easily outperformed by conventional full-duplex (FD) amplify-and-
forward (AF) relay unless very large RIS is employed [3], [4]. However, very
large number of reflecting elements will lead to high training overhead for
channel estimation and high complexity of beamforming designs [5]. Meanwhile,
the control of a large RIS will also become a difficult task.

Moreover, the users at the back of RIS cannot be well served due to its
reflective property. The service coverage of RIS is limited to only one side
of the surface. To overcome this limitation, simultaneously transmitting and
reflecting RIS (STAR-RIS) was proposed [6], [7], which can reflect and
transmit the incident signals to serve the users in front of RIS or behind it.
Thus, the STAR-RIS can provide service for users at both sides of it and
improve coverage in a reflective and transmissive way [8]. However, the
performance improvement of passive STAR-RIS is still limited due to severe
signal attenuation.

In order to overcome the double fading and extend the coverage, the authors in
[9] proposed a novel relay-aided RIS architecture consisting of two RIS
surfaces connected via an FD relay, which can achieve the same performance
with conventional RIS while only requiring much fewer reconfigurable elements.
However, they only consider single-input single-output (SISO) system and
derive the theoretical upper bound of achievable rate. The designs of BS
beamforming and RIS reflection have not been well investigated. Another
limitation of this work is that the first RIS is adjusted to reflect the
signal towards the relay only and cannot simultaneously serve the users around
it, which does not take the full use of RIS. In [10], [11], the authors
proposed a semi-active RIS-aided architecture, which can not only passively
reflect the signal, but also actively amplify it at the same time. However,
the practical implementation of this hybrid-RIS remains an open problem, while
it needs expensive radio frequency (RF) chains. Active RIS [2], [12] has
recently been proposed to overcome the practical implementation problems of
passive RIS. However, the hardware implementation is still an issue needs to
be addressed.

In this paper we propose a novel reflection and relay dual-functional RIS
architecture to overcome the double fading effect, enhance the coverage and
reduce hardware complexity. The dual-functional RIS unites two RISs back-to-
back with a full-duplex amplifier and can simultaneously realize passive
reflection and active relay functionalities. Particularly, one RIS passively
adjusts its phase shifters to serve the users close to it and at the same side
as this RIS. Meanwhile, the signal is amplified and transmitted by the other
RIS to serve the users far away from BS/RIS and at the back side of it. Based
on the proposed dual-functional RIS architecture, a sum-rate maximization
problem is considered for an MU-MISO system. A joint transmit beamforming and
RIS reflection design algorithm is developed based on fractional programming
(FP) theory and majorization-minimization (MM) technique. The superiority of
the dual-functional RIS architecture and the effectiveness of the proposed
design algorithm are validated by simulation results.

##  II System Model

We consider dual-functional RIS assisted wireless communication system as
illustrated in Fig. 1. The BS equipped with $N$ antennas communicates to $K$
single-antenna users with the aid of a dual-functional RIS. Specifically, the
proposed dual-functional RIS is composed of two conventional passive RISs
connected via a full-duplex amplifier. Each conventional RIS comprises a large
number of passive elements, each being able to reflect the incident signal
with a controllable phase-shift. As shown in Fig. 1, user-1, …,
user-${(K-1)}$, are located around RIS and served by the RIS facing them
(which is labelled as RIS-1 in Fig. 1). Meanwhile, the $K$-th user cannot
receive sufficient strong signals from the BS or RIS-1 due to the fact that it
is located in the opposite of it and the direct link to BS is blocked.
Therefore, we propose the dual-functional RIS scheme to provide satisfactory
QoS to all users, no matter where they are located. Specifically, the signal
from BS is first reflected by RIS-1 to horn antenna-1, then amplified and
emitted to the second (right) side of the RIS (labeled as RIS-2), which
transmits the signal towards the $K$-th user. We assume that there exists no
direct link from RIS-1 to the $K$-th user and vice versa.

Figure 1: The proposed reflection and relay dual-functional RIS architecture
in an MU-MISO system.

$g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})=\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)-\sum_{k=1}^{K}\gamma_{k}\\\
+\sum_{k=1}^{K-1}(2\sqrt{1+\gamma_{k}}\Re\left\\{\tau_{k}^{\ast}\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}\right\\}-\left|\tau_{k}\right|^{2}\sum_{i=1}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}-\left|\tau_{k}\right|^{2}{\sigma_{k}^{2}})\\\
$

$+2\sqrt{\beta\left(1+\gamma_{K}\right)}\Re\left\\{\tau_{K}^{\ast}\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{K}\right\\}-\left|\tau_{K}\right|^{2}\beta\sum_{i=1}^{K}\left|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}-\left|\tau_{K}\right|^{2}\xi_{K}^{2}$
(10)

Each side of the dual-functional RIS has $M$ passive reconfigurable elements,
whose phase shifters can be adjusted by a controller through a dedicated
control link. We define the reflection coefficient vector
$\bm{\phi}_{1}\triangleq\left[e^{j\theta_{1,1}},\ldots,e^{j\theta_{1,M}}\right]^{T}$
and reflection matrix
$\bm{\Phi}_{1}\triangleq\operatorname{diag}\left(\bm{\phi}_{1}\right)$, with
$\theta_{1,m}$ denoting the phase-shift of the $m$-th element of RIS-1.
Similarly, the notations $\bm{\phi}_{2}$ and $\bm{\Phi}_{2}$ are defined in
the same way to represent the phase-shift of RIS-2. We consider a challenging
scenario where no direct link exists between the transmitter and the users due
to, for example, the blockage of buildings. Since the $k$-th user,
$k=1,\cdots,K-1$, is solely served by reflective RIS-1, the received signal at
these users can be modeled as

$y_{k}=\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\sum_{i=1}^{K}\mathbf{w}_{i}s_{i}+n_{k},~{}~{}~{}k=1,\ldots,K-1,$
(1)

where $s_{i}\in\mathbb{C},i=1,\ldots,K$, is the transmitted symbol,
$\mathbb{E}\\{\left|s_{i}\right|^{2}\\}=1$, $\mathbf{w}_{i}\in\mathbb{C}^{N}$
is the beamformer at the BS for the $i$-th user,
$\mathbf{G}\in\mathbb{C}^{M\times N}$ and $\mathbf{h}_{k}\in\mathbb{C}^{M}$
denote the channels from BS to RIS-1, from RIS-1 to user-$k$, respectively,
and $n_{k}\sim\mathcal{CN}\left(0,\sigma^{2}_{k}\right)$ denotes the complex
additive white Gaussian noise with variance $\sigma^{2}_{k}$ at user-$k$. We
assume that the channel state information (CSI) of all channels involved is
perfectly known. The practical CSI acquisition techniques can be found in
[13]. Thus, the signal-to-interference-plus-noise ratio (SINR) of the $k$-th
user can be expressed as

$\displaystyle\mathrm{SINR}_{k}=\frac{|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}|^{2}}{\sum_{i\neq
k}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+{\sigma_{k}^{2}}},k=1,\ldots,K-1.$
(2)

Similarly, the received signal at the input of amplifier via the horn
antenna-1 can be written as

$y_{\mathrm{r}}=\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\sum_{i=1}^{K}\mathbf{w}_{i}s_{i}+n_{0},$
(3)

where $n_{0}\sim\mathcal{CN}\left(0,\sigma_{0}^{2}\right)$ denotes the thermal
noise with variance $\sigma_{0}^{2}$,
$\mathbf{g}_{\mathrm{t}}\in\mathbb{C}^{M}$ denotes the channel from RIS-1 to
horn antenna-1. Then, the signal $y_{\mathrm{r}}$ is amplified and transmitted
by RIS-2 towards the $K$-th user, whose received signal can be written as

$\displaystyle y_{K}=$
$\displaystyle\sqrt{\beta}\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\left(\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\sum_{i=1}^{K}\mathbf{w}_{i}s_{i}+n_{0}\right)+n_{K}$
(4) $\displaystyle=$
$\displaystyle\sqrt{\beta}\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\sum_{i=1}^{K}\mathbf{w}_{i}s_{i}+\sqrt{\beta}\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}n_{0}+n_{K},$

where $\mathbf{g}_{\mathrm{r}}\in\mathbb{C}^{M}$ denotes the channel from horn
antenna-2 to RIS-2 and $\beta$ denotes the amplification gain which is assumed
to be a constant. Considering that the RISs and the horn antennas are deployed
very closely, we use near-field propagation model to describe channels
$\mathbf{g}_{\mathrm{t}}$ and $\mathbf{g}_{\mathrm{r}}$ [14], [15], whose
details are omitted due to space limitations. Thus, the SINR of the $K$-th
user can be expressed as

$\displaystyle\mathrm{SINR}_{K}=\frac{\beta|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{K}|^{2}}{\beta\sum_{i\neq
K}^{K}\left|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+\xi_{K}^{2}},$
(5)

where
$\xi_{K}^{2}\triangleq\beta\sigma_{0}^{2}|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}|^{2}+\sigma_{K}^{2}$
denotes the power of the received noise at the $K$-th user.

This paper aims at maximizing the sum-rate of the MU-MISO downlink system by
jointly designing the BS beamforming $\mathbf{w}_{k},\forall k$, and RIS
reflection matrices $\bm{\Phi}_{1},\bm{\Phi}_{2}$ of the dual-functional RIS,
subject to the following constraints: i) Transmit power at the BS is less than
or equal to power budget $P_{\mathrm{T}}$, i.e.,
$\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq P_{\mathrm{T}}$; ii)
Constant magnitude of reflection coefficients, i.e.
$|\bm{\phi}_{1}(m)|=1,|\bm{\phi}_{2}(m)|=1,\forall m$. Therefore, the
corresponding optimization problem can be formulated as

$\displaystyle\mathrm{(P1)}\max_{\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k}}$
$\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\operatorname{SINR}_{k}\right)$
(6a) $\displaystyle\mathrm{s.t.}~{}~{}~{}$
$\displaystyle\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{T}},$ (6b)
$\displaystyle|\bm{\phi}_{1}(m)|=1,~{}|\bm{\phi}_{2}(m)|=1,~{}\forall m.$ (6c)

Obviously, the optimization problem (6) is non-convex and difficult to solve
due to the unit-modulus constraints and the coupling between variables. To
tackle the difficulties in (6), in the following we propose an iterative BS
beamforming and RIS reflection design algorithm based on fractional
programming (FP) theory.

##  III Joint Beamforming and RIS Design

In this section, we first equivalently transform the original problem into a
more tractable form based on the theory of FP. With the transformed objective
function, BS beamforming and RIS reflection matrices are designed iteratively.

###  III-A FP-Based Transformation of Objective Function

The objective in (6) is a typical function of multiple fractional parameters,
which is usually difficult to solve. Motivated by [16], [17] which use FP to
solve this multiple-ratio problem, we attempt to equivalently transform the
original problem into a more tractable form by extracting the ratio terms
SINRk, $k=1,\ldots,K,$ from the logarithmic function. Based on Lagrangian dual
transform [16], the optimization problem (6) is equivalent to

$\displaystyle\mathrm{(P2)}\max_{\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma}}$
$\displaystyle
f_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma})$ (7)
$\displaystyle\mathrm{s.t.}{}{}{}{}{}$
$\displaystyle\eqref{eq:problem12},\eqref{eq:problem15},$

where the objective function in (7) is

$\displaystyle
f_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma}$
$\displaystyle)=\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)-\sum_{k=1}^{K}\gamma_{k}$
(8) $\displaystyle+$
$\displaystyle\sum_{k=1}^{K-1}\frac{\left(1+\gamma_{k}\right)|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}|^{2}}{\sum_{i=1}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+{\sigma_{k}^{2}}}$
$\displaystyle+$
$\displaystyle\frac{\beta\left(1+\gamma_{K}\right)|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{K}|^{2}}{\beta\sum_{i=1}^{K}\left|\mathbf{h}_{K}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+\xi_{K}^{2}},$

and $\bm{\gamma}\triangleq[\gamma_{1},\ldots,\gamma_{K}]^{T}$ is an auxiliary
variable vector. Unfortunately, optimization problem (7) is still intractable
due to the complicated form of the sum of $K$ fractional terms. Next, we apply
quadratic transform [17] on the fractional term to transform it into a
solvable formula by introducing another auxiliary variable vector
$\bm{\tau}\triangleq[\tau_{1},\ldots,\tau_{K}]^{T}$. Then the optimization
problem (7) can be transformed into

$\displaystyle\mathrm{(P3)}\max_{\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau}}$
$\displaystyle
g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})$
(9)
$\displaystyle\mathrm{s.t.}~{}~{}~{}~{}~{}\eqref{eq:problem12},\eqref{eq:problem15},$

where the objective function
$g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})$
is formulated as (10) presented at the top of this page.

To efficiently solve the above problem, we adopt the block coordinate ascent
(BCA) methodology to alternatively update each block of variables while keep
others being fixed. The sub-problems of updating each block will be specified
in details in the following.

###  III-B Update Auxiliary Variable Vectors

1) Update $\bm{\gamma}$

When $\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k}$ and $\bm{\tau}$ are held
fixed, the objective function $g_{\mathrm{R}}$ in (9) is a concave
differentiable function with respect to the variables $\gamma_{k}$, $\forall
k$. By solving $\frac{\partial g_{\mathrm{R}}}{\partial\gamma_{k}}=0$, the
optimal ${\gamma}_{k}^{\star}$ can be found in a closed-form as

$\gamma_{k}^{\star}=\left\\{\begin{aligned}
&\frac{|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}|^{2}}{\sum_{i\neq
k}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+{\sigma_{k}^{2}}},k=1,\ldots,K-1,\\\
&\frac{\beta|\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}|^{2}}{\beta\sum_{i\neq
k}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+\xi_{K}^{2}},k=K.\end{aligned}\right.$
(11)

2) Update $\bm{\tau}$

Similarly, given $\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k}$ and
$\bm{\gamma}$, $g_{\mathrm{R}}$ in (9) is a concave differentiable function
with respect to $\tau_{k}$, $\forall k$. The optimal variable
$\tau_{k}^{\star}$ can be obtained by setting $\frac{\partial
g_{\mathrm{R}}}{\partial\tau_{k}}=0$ and has the following optimal value

$\tau_{k}^{\star}=\left\\{\begin{aligned}
&\frac{\sqrt{1+\gamma_{k}}\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}}{\sum_{i=1}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+{\sigma_{k}^{2}}},k=1,\ldots,K-1,\\\
&\frac{\sqrt{\beta\left(1+\gamma_{k}\right)}\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}}{\beta\sum_{i=1}^{K}\left|\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{i}\right|^{2}+\xi_{K}^{2}},k=K.\end{aligned}\right.$
(12)

###  III-C Update BS Beamforming

With other variables being fixed, the update of beamforming $\mathbf{w}_{k}$
is transformed into solving the following problem

$\displaystyle\mathrm{(P4)}~{}\max_{\mathbf{w}_{k}}~{}$ $\displaystyle
g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})$
(13a) $\displaystyle\mathrm{s.t.}~{}~{}$
$\displaystyle\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{T}},$ (13b)

which is a convex problem. Fortunately, the optimal solution of problem (13)
can be obtained in an analytic form via checking its optimality conditions. By
introducing a multiplier $\mu$ for the power constraint in (13b), we can form
a Lagrangian function as

$\displaystyle\mathcal{L}=g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})+\mu\left(P_{\mathrm{T}}-\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\right).$
(14)

Then, the optimal solution of BS beamforming $\mathbf{w}_{k}$ can be
determined by setting the partial derivative of $\mathcal{L}$ with respect to
$\mathbf{w}_{k}$ and $\mu$ to zero, i.e.

$\left\\{\begin{aligned}
&\frac{\partial\mathcal{L}}{\partial\mathbf{w}_{k}}=0,~{}~{}\forall k,\\\
&\frac{\partial\mathcal{L}}{\partial\mu}=0,\\\ \end{aligned}\right.$ (15)

which yields the optimal BS beamforming as

$\mathbf{w}_{k}^{\star}=\left(\mathbf{A}+\mu^{\star}\mathbf{I}\right)^{-1}\sqrt{1+\gamma_{k}}\widetilde{\mathbf{h}}_{k},~{}~{}\forall
k,$ (16)

with

$\widetilde{\mathbf{h}}_{k}^{H}\triangleq\left\\{\begin{aligned}
&\tau_{k}^{\ast}\mathbf{h}_{k}^{H}\bm{\Phi}_{1}\mathbf{G},~{}~{}k=1,\ldots,K-1,\\\
&\sqrt{\beta}\tau_{k}^{\ast}\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G},k=K,\end{aligned}\right.$
(17)
$\displaystyle\mathbf{A}\triangleq\sum_{k=1}^{K}\widetilde{\mathbf{h}}_{k}\widetilde{\mathbf{h}}_{k}^{H}.$
(18)

The optimal BS beamforming $\mathbf{w}_{k}$ can be easily obtained after
having the optimal multiplier $\mu^{\star}$ by a bisection search.

###  III-D Update RIS Reflection

Given $\bm{\gamma}$, $\bm{\tau}$ and BS beamforming $\mathbf{w}_{k},\forall
k$, the sub-problem with respect to the RIS reflection matrices
$\bm{\Phi}_{1}$ and $\bm{\Phi}_{2}$ can be presented as:

$\displaystyle\mathrm{(P5)}~{}~{}\max_{\bm{\Phi}_{1},\bm{\Phi}_{2}}~{}~{}$
$\displaystyle
g_{\mathrm{R}}(\bm{\Phi}_{1},\bm{\Phi}_{2},\mathbf{w}_{k},\bm{\gamma},\bm{\tau})$
(19a) $\displaystyle\mathrm{s.t.}~{}~{}~{}~{}$
$\displaystyle|\bm{\phi}_{1}(m)|=1,~{}~{}|\bm{\phi}_{2}(m)|=1,~{}~{}\forall
m.$ (19b)

Considering that $\bm{\Phi}_{1}$ and $\bm{\Phi}_{2}$ are coupled and jointly
yield a nonconvex objective, we turn to separately optimize either of them
with the other being fixed. When $\bm{\Phi}_{2}$ is given, by defining

$\mathbf{r}_{k,i}^{H}\triangleq\left\\{\begin{aligned}
&\tau_{k}^{\ast}\mathbf{h}_{k}^{H}\mathrm{diag}\left(\mathbf{G}\mathbf{w}_{i}\right),~{}k=1,\ldots,K-1,~{}\forall
i,\\\
&\sqrt{\beta}\tau_{k}^{\ast}\mathbf{h}_{k}^{H}\bm{\Phi}_{2}\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\mathrm{diag}\left(\mathbf{G}\mathbf{w}_{i}\right),k=K,\forall
i,\end{aligned}\right.$ (20) $\displaystyle\widetilde{\mathbf{R}}$
$\displaystyle\triangleq\sum_{k=1}^{K}\sum_{i=1}^{K}\mathbf{r}_{k,i}\mathbf{r}_{k,i}^{H},$
(21) $\displaystyle\mathbf{d}$
$\displaystyle\triangleq\sum_{k=1}^{K}\sqrt{\left(1+\gamma_{k}\right)}\mathbf{r}_{k,k},$
(22)

and removing constants, the optimization of $\bm{\phi}_{1}$ is equivalent to
solving the following problem

$\displaystyle\mathrm{(P6)}\min_{\bm{\phi}_{1}}~{}$
$\displaystyle-\sum_{k=1}^{K}\left(2\sqrt{1+\gamma_{k}}\Re\left\\{\mathbf{r}_{k,k}^{H}\bm{\phi}_{1}\right\\}-\sum_{i=1}^{K}\left|\mathbf{r}_{k,i}^{H}\bm{\phi}_{1}\right|^{2}\right)$
$\displaystyle=\bm{\phi}_{1}^{H}\widetilde{\mathbf{R}}\bm{\phi}_{1}-2\Re\left\\{\bm{\phi}_{1}^{H}\mathbf{d}\right\\}$
(23a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle|\bm{\phi}_{1}(m)|=1,~{}m=1,\cdots,M.$ (23b)

Problem (23) is still difficult to solve due to the constant modulus
constraint. To effectively solve this problem, we adopt the majorization-
minimization (MM) algorithm to perform the updating. The essential idea of MM
framework is to sequentially surrogate the original objective function with a
tightly upper-bounded approximation, which is designed to render the
transformed problem more tractable [18]. Given the update value
$\bm{\phi}_{1}^{t}$ in the previous iteration, we propose to majorize the
objective function of problem (23) via executing its second-order Taylor
expansion at the point of $\bm{\phi}_{1}^{t}$ followed by replacing the
associated quadratic term with an upper bound. Specifically, the quadratic
term in (23) is majorized by

$\displaystyle\bm{\phi}_{1}^{H}\widetilde{\mathbf{R}}\bm{\phi}_{1}\leq$
$\displaystyle\bm{\phi}_{1}^{H}\bm{\Lambda}\bm{\phi}_{1}-2\Re\left\\{\bm{\phi}_{1}^{H}(\bm{\Lambda}-\widetilde{\mathbf{R}})\bm{\phi}_{1}^{t}\right\\}$
(24)
$\displaystyle+\left(\bm{\phi}_{1}^{t}\right)^{\mathrm{H}}(\bm{\Lambda}-\widetilde{\mathbf{R}})\bm{\phi}_{1}^{t},$

where $\bm{\Lambda}\triangleq\lambda_{\max}\mathbf{I}_{M}$ and
$\lambda_{\max}$ is the maximum eigenvalue of $\widetilde{\mathbf{R}}$. The
subproblem to be solved at the $(t+1)$-th iteration is given by

$\displaystyle\min_{\bm{\phi}_{1}}~{}~{}$
$\displaystyle\bm{\phi}_{1}^{H}\bm{\Lambda}\bm{\phi}_{1}-2\Re\left\\{\bm{\phi}_{1}^{H}(\bm{\Lambda}-\widetilde{\mathbf{R}})\bm{\phi}_{1}^{t}\right\\}$
$\displaystyle+\left(\bm{\phi}_{1}^{t}\right)^{\mathrm{H}}(\bm{\Lambda}-\widetilde{\mathbf{R}})\bm{\phi}_{1}^{t}-2\Re\left\\{\bm{\phi}_{1}^{H}\mathbf{d}\right\\}$
(25a) $\displaystyle\mathrm{s.t.}~{}~{}$
$\displaystyle|\bm{\phi}_{1}(m)|=1,~{}m=1,\cdots,M.$ (25b)

Since $\bm{\phi}_{1}^{H}\bm{\phi}_{1}=M$, we have
$\bm{\phi}_{1}^{H}\bm{\Lambda}\bm{\phi}_{1}=M\lambda_{\max}$, which is a
constant. By removing the constants, problem (25) can be rewritten as follows

$\displaystyle\max_{\bm{\phi}_{1}}~{}~{}$ $\displaystyle
2\Re\left\\{\bm{\phi}_{1}^{H}\mathbf{p}^{t}\right\\}$ (26a)
$\displaystyle\mathrm{s.t.}~{}~{}$
$\displaystyle|\bm{\phi}_{1}(m)|=1,~{}m=1,\cdots,M,$ (26b)

where
$\mathbf{p}^{t}\triangleq(\lambda_{\max}\mathbf{I}_{M}-\widetilde{\mathbf{R}})\bm{\phi}_{1}^{t}+\mathbf{d}$.
Thus, the optimal solution of problem (26) is given by

$\bm{\phi}_{1}^{t+1}=e^{j\angle\mathbf{p}^{t}}.$ (27)

After obtaining $\bm{\phi}_{1}$, we can solve $\bm{\phi}_{2}$ by repeating the
same procedure. By defining

$\displaystyle\mathbf{v}_{K,k}^{H}$
$\displaystyle\triangleq\sqrt{\beta}\tau_{K}^{\ast}\mathbf{h}_{K}^{H}\mathrm{diag}\left(\mathbf{g}_{\mathrm{r}}\mathbf{g}_{\mathrm{t}}^{H}\bm{\Phi}_{1}\mathbf{G}\mathbf{w}_{k}\right),\forall
k,$ (28) $\displaystyle\widetilde{\mathbf{V}}$
$\displaystyle\triangleq\sum_{k=1}^{K}\mathbf{v}_{K,k}\mathbf{v}_{K,k}^{H},$
(29) $\displaystyle\mathbf{b}$
$\displaystyle\triangleq\sqrt{\left(1+\gamma_{K}\right)}\mathbf{v}_{K,K},$
(30)

and removing the other constant terms unrelated to $\bm{\phi}_{2}$, problem
(19) can be rearranged as

$\displaystyle\mathrm{(P7)}\min_{\bm{\phi}_{2}}$
$\displaystyle-2\sqrt{\left(1+\gamma_{K}\right)}\Re\left\\{\mathbf{v}_{K,K}^{H}\bm{\phi}_{2}\right\\}+\sum_{i=1}^{K}\left|\mathbf{v}_{K,i}^{H}\bm{\phi}_{2}\right|^{2}$
$\displaystyle=\bm{\phi}_{2}^{H}\widetilde{\mathbf{V}}\bm{\phi}_{2}-2\Re\left\\{\bm{\phi}_{2}^{H}\mathbf{b}\right\\}$
(31a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle|\bm{\phi}_{2}(m)|=1,~{}m=1,\cdots,M.$ (31b)

Following the identical argument as before, the optimal solution is given by

$\bm{\phi}_{2}^{t+1}=e^{j\angle\mathbf{q}^{t}},$ (32)

where
$\mathbf{q}^{t}\triangleq(\zeta_{\max}\mathbf{I}_{M}-\widetilde{\mathbf{V}})\bm{\phi}_{2}^{t}+\mathbf{b}$
and $\zeta_{\max}$ is the maximum eigenvalue of $\widetilde{\mathbf{V}}$.

If low-resolution phase shifters are employed to realize the RIS, the
corresponding discrete reflection coefficient can be obtained using a simple
quantization operation

$\displaystyle\widetilde{\bm{\phi}}_{1(2)}^{t+1}=\exp\left\\{j\left\langle\frac{\arg\left(\mathbf{p}^{t}(\mathbf{q}^{t})\right)}{\Delta}\right\rangle\times\Delta\right\\},$
(33)

where $\langle\cdot\rangle$ denotes the rounding operation, and
$\Delta\triangleq 2\pi/2^{b}$ is the angle resolution controlled by $b$ bits.

Now, the complete procedure of finding the optimal auxiliary vectors
$\bm{\gamma}^{\star}$ and $\bm{\tau}^{\star}$, the BS beamforming
$\mathbf{w}_{k}^{\star}$, and RIS reflection matrices
$\bm{\Phi}_{1}^{\star},\bm{\Phi}_{2}^{\star}$ is straightforward. With
appropriate initialization, we iteratively update $\bm{\gamma}$, $\bm{\tau}$,
$\mathbf{w}_{k}$ and $\bm{\Phi}_{1},\bm{\Phi}_{2}$ until convergence. For
clarity, the proposed FP-based BS beamforming and RIS reflection design
algorithm is summarized in Algorithm 1.

Algorithm 1 Joint Beamforming and RIS Reflection Design

0: $\mathbf{h}_{k},\forall k$, $\mathbf{G}$, $\mathbf{g}_{\mathrm{r}}$,
$\mathbf{g}_{\mathrm{t}}$, $b$, $\sigma^{2}$, $\sigma_{0}^{2}$, $\beta$,
$P_{\mathrm{T}}$, $T_{max}$.

0: $\mathbf{w}_{k}^{\star},\forall k$, $\bm{\Phi}_{1}^{\star}$,
$\bm{\Phi}_{2}^{\star}$.

1: Initialize $\mathbf{w}_{k},\forall k$, $\bm{\Phi}_{1}$, $\bm{\Phi}_{2}$.

2: Repeat

3: Update $\bm{\gamma}^{\star}$ by (11).

4: Update $\bm{\tau}^{\star}$ by (12).

5: Update each beamforming vector $\mathbf{w}_{k}^{\star},\forall k$, by
solving (16).

6: Set $t=1$.

7: while $t<T_{max}$ & no convergence of $\bm{\Phi}_{1}$ do

8: Update $\bm{\phi}_{1}^{t+1}$ by (27)/(33).

9: $t:=t+1$.

10: end while

11: Set $t=1$.

12: while $t<T_{max}$ & no convergence of $\bm{\Phi}_{2}$ do

13: Update $\bm{\phi}_{2}^{t+1}$ by (32)/(33).

14: $t:=t+1$.

15: end while

16: Until convergence

17: Return $\mathbf{w}_{k}^{\star},\forall k$, $\bm{\Phi}_{1}^{\star}$,
$\bm{\Phi}_{2}^{\star}$.

###  III-E Complexity Analysis

In this subsection, we provide a brief computational complexity analysis for
the proposed joint BS beamforming and RIS reflection design for the sum-rate
maximization problem, i.e., Algorithm 1.

The overall computational complexity of proposed algorithm is mainly caused by
the update of variables. In each iteration, obtaining the optimal solution of
$\bm{\gamma}$ and $\bm{\tau}$ requires approximately $\mathcal{O}(K^{2}M^{2})$
and $\mathcal{O}(K(K+1)M^{2})$ operations, respectively; updating the BS
transmit beamforming $\mathbf{w}_{k}$ requires about $\mathcal{O}(M^{2}N^{3})$
operations, which is mainly caused by the matrix inversion operation.
Computing the RIS phase-shift matrices $\bm{\Phi}_{1}$ and $\bm{\Phi}_{2}$ has
a complexity of approximately $\mathcal{O}(I_{1}M^{2})$ and
$\mathcal{O}(I_{2}M^{2})$ with $I_{1}$ and $I_{2}$ being the number of
iterations. Therefore, the total computational complexity of Algorithm 1 can
be approximated by
$\mathcal{O}\left(I_{R}\left(M^{2}N^{3}+I_{1}M^{2}+I_{2}M^{2}+K^{2}M^{2}\right)\right)$,
wherein $I_{R}$ represents the required number of iterations for the algorithm
convergence.

##  IV Simulation Results

In this section, we provide simulation results to demonstrate the advancement
of the proposed dual-functional RIS architecture and illustrate the
effectiveness of our proposed algorithm. The noise power at all receivers is
$\sigma_{k}^{2}=-80$dBm and the thermal noise power introduced by the
amplifier is $\sigma_{0}^{2}=-70$dBm [1]. The path-loss is modeled as
$\text{PL}(d)=C_{0}\left(d_{0}/d\right)^{\kappa}$, where $C_{0}=-30$dB,
$d_{0}=1$m, $d$ is the link distance, and $\kappa$ is the path-loss exponent.
The BS-RIS channel is assumed to follow a small-scale Rician fading model with
a Rician factor of 3dB and a path-loss exponent of 2.5, while the other
channels only have non-line-of-sight (NLoS) components and a path-loss
exponent of 3. The channel $\mathbf{g}_{\mathrm{t}}$ from RIS-1 to horn
antenna and the channel $\mathbf{g}_{\mathrm{r}}$ from horn antenna to RIS-2
adopt near-field model [14], [15]. The BS is equipped with $N=6$ antennas and
serves $K=4$ users. The RIS is 50m away from the BS, while the three users
($k=1,2,3$) are 2m away from the dual-functional RIS and the fourth user
($k=4$) is 20m away from the RIS.

Figure 2: Sum-rate versus the number of iterations ($P_{\mathrm{T}}$ = 40dBm,
$M$ = 64, $\beta$ = 30dB). Figure 3: Sum-rate versus the transmit power
$P_{\mathrm{T}}$ for different amplifier gain $\beta$ ($M$ = 64, $b=\infty$.
The BS power consumption of “STAR-RIS” is the same with the total power
consumption of dual-functional RIS assisted system with $\beta$ = 40dB).

We start with presenting the convergence of the proposed joint BS beamforming
and RIS reflection design by plotting the sum-rate versus the number of
iterations in Fig. 2. Simulation result illustrates that the proposed
algorithm can converge fast and within 15 iterations.

Fig. 3 shows the sum-rate with the proposed dual-functional RIS architecture
under different realistic values for the amplification gain $\beta$. For fair
comparison, we also include the full-duplex relay equipped with $4$ antennas
and STAR-RIS proposed in [6] as the benchmark which consumes the same power as
the proposed dual-functional RIS aided system. It is noted that the dual-
functional RIS with a reasonable amplification gain can result in better
performance compared with full-duplex relay since the large-scale RIS has
significant beamforming gain. When the amplification gain $\beta$ is 40dB,
only 30dBm transmit power is required to achieve the same performance, for
example, sum-rate is around 37 bps/Hz, while 50dBm transmit power is need when
STAR-RIS is deployed.

Fig. 4 illustrates the sum-rate versus the number of RIS elements $M$. The
same conclusion can be drawn that the proposed dual-functional RIS
architecture can substantially enhance the communication quality compared with
the conventional passive STAR-RIS scheme. As we can see, only 36 elements
adopting 1-bit resolution phase shifters are needed for the proposed dual-
functional RIS to achieve the same performance as passive STAR-RIS with 256
elements, which means the dual-functional RIS architecture can improve
wireless communication performance without increasing the number of elements.

##  V Conclusions

In this paper, we introduced a novel reflection and relay dual-functional RIS
architecture to overcome the double fading effect and enhance the coverage of
downlink MU-MISO systems. Aiming at maximizing the achievable sum-rate, an
iterative algorithm based on FP and MM was proposed to design the transmit
beamforming and RIS reflection. Simulation results demonstrated the
superiority of the introduced dual-functional RIS architecture and the
effectiveness of the proposed algorithm.

Figure 4: Sum-rate versus the number of RIS elements $M$ ($P_{\mathrm{T}}$ =
40dBm, $\beta$ = 30dB).

## References

  * [1] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” IEEE Commun. Mag., vol. 58, no. 1, pp. 106-112, Jan. 2020.
  * [2] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. V. Poor, “Active RIS vs. passive RIS: Which will prevail in 6G?” Mar. 2021. [Online]. Available: https://arxiv.org/abs/2103.15154
  * [3] E. Björnson, Ö. Özdogan, and E. G. Larsson, “Intelligent reflecting surface versus decode-and-forward: How large surfaces are needed to beat relaying?” IEEE Wireless Commun. Lett., vol. 9, no. 2, pp. 244-248, Oct. 2020.
  * [4] K. Ntontin, M. D. Renzo, and F. Lazarakis, “On the rate and energy efficiency comparison of reconfigurable intelligent surfaces with relays,” in Proc. IEEE Conf. Signal Process. Advances Wireless Commun. (SPAWC Workshop), Atlanta, USA, May 2020, pp. 1-5.
  * [5] C. You, B. Zheng, and R. Zhang, “Channel estimation and passive beamforming for intelligent reflecting surface: Discrete phase shift and progressive refinement,” J. Sel. Areas Commun., vol. 38, no. 11, pp. 2604-2620, Nov. 2020.
  * [6] J. Xu, Y. Liu, X. Mu, and O. A. Dobre, “STAR-RISs: Simultaneous transmistting and reflecting reconfigurable intelligent surfaces,” IEEE Commun. Lett., vol. 25, no. 9, pp. 3134-3138, Sep. 2021.
  * [7] H. Niu, Z. Chu, F. Zhou and Z. Zhu, “Simultaneous transmission and reflection reconfigurable intelligent surface assisted secrecy MISO networks,” IEEE Commun. Lett., to appear.
  * [8] W. Cai, R. Liu, Y . Liu, M. Li, and Q. Liu, “Joint beamforming designs for intelligent omni surface assisted wireless communication systems,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Madrid, Spain, Dec. 2021.
  * [9] X. Ying, U. Demirhan, and A. Alkhateeb, “Relay aided intelligent reconfigurable surfaces: Achieving the potential without so many antennas,” Jun. 2020. [Online]. Avaliable: https://arxiv.org/abs/2006.06644
  * [10] N. T. Nguyen, Q.-D. Vu, K. Lee, and M. Juntti, “Hybrid relay-reflecting intelligent surface-assisted wireless communication,” Mar. 2021. [Online]. Available: https://arxiv.org/abs/2103.03900
  * [11] N. T. Nguyen, Q.-D. Vu, K. Lee and M. Juntti, “Spectral efficiency optimization for hybrid relay-reflecting intelligent surface,” in IEEE Int. Conf. Commun. Workshops (ICC Workshops), Montreal, Canada, Jun. 2021.
  * [12] R. Long, Y.-C. Liang, Y. Pei and E. G. Larsson, “Active reconfigurable intelligent surface-aided wireless communications,” IEEE Trans. Wireless Commun., vol. 20, no. 8, pp. 4962-4975, Aug. 2021.
  * [13] A. L. Swindlehurst, G. Zhou, R. Liu, C. Pan, and M. Li, “Channel estimation with reconfigurable intelligent surfaces - A general framework,” Oct. 2021. [Online]. Available: https://arxiv.org/abs/2110.00553
  * [14] X. Cheng and Y. He, “Channel modeling and analysis of ULA massive MIMO systems,” in Proc. Int. Conf. Advanced Commun. Tech. (ICACT), Gangwon-do, South Korea, Feb. 2018, pp. 411-416.
  * [15] E. Björnson and L. Sanguinetti, “Power scaling laws and near-field behaviors of massive MIMO and intelligent reflecting surfaces,” IEEE Open J. Commun. Society, vol. 1, pp.1306-1324, Sep. 2020.
  * [16] K. Shen and W. Yu, “Fractional programming for communication systems-Part I: Power control and beamforming,” IEEE Trans. Signal Process., vol. 66, no. 10, pp. 2616-2630, May 2018.
  * [17] K. Shen and W. Yu, “Fractional programming for communication systems-Part II: Uplink scheduling via matching,” IEEE Trans. Signal Process., vol. 66, no. 10, pp. 2631-2644, May 2018.
  * [18] Y. Liu and J. Li, “Linear precoding to optimize throughput, power consumption and energy efficiency in MIMO wireless sensor networks,” IEEE Trans. Commun., vol. 66, no. 5, pp. 2122-2136, May 2018.


∎

11institutetext: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, MS 83, Cambridge, MA 02138, USA
Tel.: +161-74-967981
11email<EMAIL_ADDRESS>
# The Origins and Physical Properties of the Complex of Local Interstellar
Clouds

Jonathan D. Slavin

(Received: date / Accepted: date)

###### Abstract

The Complex of Local Interstellar Clouds (CLIC) is a relatively tight grouping
of low density, warm, partially ionized clouds within about 15 pc of the Solar
System. The Local Interstellar Cloud (LIC) is the cloud observed on most lines
of sight and may be the cloud that immediately surrounds our Solar System, the
properties of which set the outer boundary conditions of the heliosphere.
Using absorption line data toward nearby stars, _in situ_ observations of
inflowing interstellar gas from spacecraft in the Solar System, and
theoretical modeling of the interstellar radiation field and radiative
transfer, we can deduce many characteristics of the LIC. We find that the LIC
is partially ionized with modest electron density, $n_{e}\approx 0.07$ cm-3.
The combination of its temperature and ionization favor
photoionization/thermal equilibrium over a non-equilibrium cooling cloud
picture. The abundances in the LIC suggest moderate dust destruction for
silicate dust but complete destruction of carbonaceous grains. An origin for
the LIC as a density enhancement in the ambient medium that has been overrun
by a shock seems likely, while its velocity away from the Sco-Cen association
points to a possible connection to that region and the Loop I bubble.

###### Keywords:

Interstellar medium: Physical properties Interstellar medium: Solar
neighborhood Interstellar medium: Atomic processes

††journal: SSRv

##  1 Introduction

The interstellar medium (ISM) that surrounds the Solar System impinges on the
outflowing Solar wind creating the heliosphere. The particular characteristics
of that circumheliospheric interstellar medium (CHISM), such as its density,
ionization, temperature and magnetic field, determine in detail the
interaction of the gas with the solar wind. Thus understanding the nature of
the heliosphere requires knowledge of the state of the CHISM while at the same
time interpretation of some of the _in situ_ observations relevant to
discerning the nature of the CHISM demands an understanding of the
heliosphere.

It has long been assumed, because it appears on so many lines of sight, that
the velocity component (or cloud) identified as the Local Interstellar Cloud
(LIC) surrounds the heliosphere. Recently Redfield and Linsky (2008) have
analyzed many lines of sight toward nearby stars and identified 15 distinct
velocity vectors, which are attributed to coherent clouds in the local ISM.
The LIC component, found in by far the most lines of sight, was found to have
an associated temperature of $T=7500\pm 1300$ K. Redfield and Linsky (2008)
claim that the heliosphere is located in a transition zone between the LIC and
G cloud based on the fact that the temperature and velocity of the gas are
intermediate between those found for the LIC and G cloud components. We
believe that this is somewhat speculative as yet since the temperature of the
inflowing gas is within the error bars on the LIC while falling outside of the
error bars for the G cloud value. The inflowing gas velocity does appear to be
somewhat discrepant ($2-3$ km s-1) with their LIC vector, but that may be due
to some weak disturbance in the cloud. Much of what we discuss does not depend
specifically on whether the heliosphere is within the LIC as long as their
properties are not very different, which appears to be the case. The primary
assumption that we make related to this is that the portion of the line of
sight between the Solar System and the star $\epsilon$ CMa that traverses the
LIC can be treated as passing through a cloud that is in thermal and
photoionization equilibrium.

##  2 The Physical Characteristics of the Complex of Local Interstellar Clouds

###  2.1 Primary Observational Evidence

The complex of local interstellar clouds, of which the Local Interstellar
Cloud (LIC) that surrounds the Solar System is one member, are nearby, low
density and partially ionized patches of gas that are surrounded by much lower
density and (probably) hot ISM, the Local Bubble. We have the most information
about the LIC since most sight lines to nearby stars pass through it, but the
LIC is not extremely different from other clouds within the CLIC (see Redfield
and Linsky, 2008, also Redfield, this volume).

Among the information on the LIC that we need to understand its origins are:

  * •

ionization – is the LIC gas in photoionization equilibrium or not?,

  * •

abundances – important for evaluating ionization corrections, radiative
cooling rate and as evidence for dust destruction or enrichment,

  * •

temperature/density/pressure – where does the LIC fit into typical ISM phase
picture? Is the LIC in thermal equilibrium?,

  * •

velocity – can provide hints as to the dynamics, history and past environment
of the LIC,

  * •

magnetic field – both the strength and orientation affect the size and shape
of the heliosphere. The field also provides pressure support for the cloud and
can play an important role in determining the nature of the cloud boundary.

Some of this information is relatively directly measureable, e.g. the
temperature of the cloud at the Solar System via He0 temperature observations
(Witte, 2004), but most require some modeling to infer their values.

Evidence on the nature of the LIC comes from a wide variety of sources. There
is a large database of ion absorption lines toward nearby stars wherein
particular velocity components have been identified as being due to the LIC
(see Redfield and Linsky, 2008, and references therein). This has been
accomplished by finding a consistent velocity vector for the cloud given the
observed projections for many lines of sight.

Among the important ions for which column densities have been observed for the
LIC are: Mg ii, Mg i, C ii∗, S ii, and Fe ii. The importance of these ions
lies in the constraints they place on the physical state of the cloud. As we
discuss below, with Mg ii and Mg i we can determine the electron density. With
the addition of C ii∗, Si ii and Fe ii we gain information on the elemental
abundances of dust components. These along with S ii provide necessary
information on the total cooling within the cloud, which is primarily due to
forbidden line emission. In order to understand the ionization and thermal
properties of the cloud it is necessary to have data on these column densities
and data on more ions will generally help to constrain the models better.

The most direct evidence on the state of the CHISM comes from observations of
neutral He. Because of its small cross section for charge transfer reactions,
He0 is believed to sail through the bowshock, heliopause and termination shock
essentially unaffected. As a result the measurements of He0 density and
temperature, $n(\mathrm{He}^{0})=0.015\pm 0.003$ cm-3 and
$T(\mathrm{He}^{0})=6300\pm 340$ K (Witte, 2004), are the best constraints we
have on the CHISM. Other _in situ_ data including backscattered H Lyman
$\alpha$, anomalous cosmic rays and pickup ions provide further constraints,
but each comes with a more model-dependent interpretation. Direct observations
of interstellar dust flowing into the Solar System (Baguhl et al., 1995;
Landgraf et al., 2000) is very interesting for what it tells us about the
chemical composition of the LIC and clues about its history, but does not
directly inform us about the gas phase abundances that govern its thermal
balance. Dust can be an important heat source in the ISM via photoelectron
ejection, but we find the dust heating is small relative to photoionization
heating for the conditions of the LIC.

An additional set of input data needed to understand the physical conditions
in the LIC is the ionizing radiation field. Both the current ionization and
the sources of photoionization are needed if we are to make sense of the
present state of the cloud. Portions of the field have been directly observed,
namely the far UV and extreme UV from stellar sources. There have also been
direct observations of diffuse soft X-rays, though the source of those remain
controversial (see Koutroumpa, this volume). As of now it appears that most of
the softest X-rays (Wisconsin B and C band, $E=70-280$ eV) do come from hot
gas within the Local Bubble and these are the most important X-rays for the
ionization of the cloud. Unfortunately the potentially even more important
diffuse EUV background has yet to be observed and instrumental limitations
make it unlikely that such an observation can be made for some time to come.

Another important observational fact is that the LIC, and indeed the CLIC,
exist in a large, extremely low density cavity. The location of these clouds
within the Local Bubble is clearly important to understanding their origins
and evolution.

Figure 1: Constraints on the electron density and temperature in the LIC
derived from Mg ii, Mg i, C ii∗, C ii and _in situ_ He0 temperature
observations.

###  2.2 Electron Density and Temperature of the LIC

As noted above, the observations of Mg ii and Mg i place constraints on the
electron density of the LIC. Balance of FUV ionization, radiative and
dielectronic recombination and charge transfer gives us

$n_{e}=\frac{\Gamma(\mathrm{Mg}^{0})}{\alpha(\mathrm{Mg}^{+})+C^{CT}(n(\mathrm{H}^{+})/n_{e})}\frac{N(\mathrm{Mg\,I})}{N(\mathrm{Mg\,II})}$
(1)

where $\Gamma($Mg${}^{0})$ is photoionization rate, $\alpha(\mathrm{Mg}^{+})$
is (total) recombination rate and $C^{CT}$ is charge transfer rate. We derive
the photoionization rate using the observed and modeled FUV background from
Gondhalekar et al. (1980). If we have $N(\mathrm{C\,II})$ and
$N(\mathrm{C\,II}^{*})$ we can also get $T$:

$\frac{N(\mathrm{C\,II}^{*})}{N(\mathrm{C\,II})}=\frac{\gamma_{12}(T)\,n_{e}}{A_{21}+\gamma_{21}(T)\,n_{e}}$
(2)

Unfortunately, the most easily observable C ii line at 1334.5Å is nearly
always saturated, making the column density difficult to derive with any
certainty.

For the $\epsilon$ CMa line of sight, $n_{e}$ and $T$ have been derived by Gry
and Jenkins (2001) by making assumptions for the abundance of C to S and using
the well observed S ii line results to derive a upper limit for $N($C ii). If
we use the constraints on $T$ from _in situ_ observations and up-to-date
values for the Mg+ recombination coefficients, we find that we need an
abundance ratio of C/S $>20$ to find a viable solution for $n_{e}$ and $T$.
The combination of the upper limit on $N($C ii), the observed limits on Mg
ii/Mg i and the observed limits on $T$ lead to tight limits,
$n_{e}=0.05-0.104$ cm-3, as illustrated in Figure 1. We discuss these limits
and the implications for the C abundance in more detail in Slavin and Frisch
(2006). When we carry out more detailed modeling including models for the
interstellar radiation field, thermal balance and radiative transfer (Slavin
and Frisch, 2008), we derive even tighter limits on $n_{e}$, $n_{e}=0.07\pm
0.01$ cm-3.

###  2.3 Radiative Transfer Models

To go from ion column densities to abundances requires ionization corrections,
especially for $N($H). In general elements with first ionization potential
$E_{0}>13.6$ eV will require an ionization correction to derive the total
element column density and these include He, N, O, Ne, and Ar. Oxygen is a
special case, however, because its ionization is tightly coupled to H
ionization by charge transfer. Nitrogen is also similarly coupled to H
ionization but much less tightly. These corrections are particularly important
because the H i column is not very well determined in many cases (e.g.
$\epsilon$ CMa). Data from the Extreme Ultraviolet Explorer (EUVE) can give us
the total H i column, but does not give us the fraction of the total
attributable to each velocity component if there are multiple velocity
components along the line of sight.

In order to derive the ionization corrections one must have a model for the
ionization of the cloud and in general this demands a model for the ionizing
radiation field, as well as the radiative transfer within the cloud. To model
the field we use the sum of emission from stellar sources (FUV and EUV) and
diffuse sources including the hot gas that we assume fills the Local Bubble
and an evaporative boundary between the warm cloud and the hot gas.

The nature of the boundary between the warm cloud gas and the surrounding hot
gas is still quite uncertain as there have been no definitive observations
establishing the existence of an evaporative boundary. It is known that the
magnetic field will suppress thermal conduction across field lines, yet a
completely magnetically isolated cloud would seem unlikely. In our models we
assume a magnetic suppression of conductivity of a factor of 2, appropriate to
the case in which the field is at a 45∘ angle with the cloud surface. This
part of the radiation field is clearly quite uncertain and we intend to
explore possible variants in future work.

We find that the details of the field do not seem to strongly influence our
results for the ionization of the cloud. We show in Figure 2 the H ionizing
radiation field at the position of the Sun in several models that are
consistent with the observations. The requirements we impose on the models to
match $n($He${}^{0})$, $T($He${}^{0})$ and the ion column densities act to fix
the H ionization, $X(\mathrm{H})$ and $n_{e}$.

Figure 2: Model ionizing radiation fields for several different models from
Slavin and Frisch (2008), all of which match the observational constraints.

We find successful models for range of values for $N(\mathrm{H\,I})$ towards
$\epsilon$CMa and $T$(hot gas). Our results for the Solar location are:

  * •

$n(\mathrm{H}^{0})=0.19-0.20$ cm-3,

  * •

$n_{e}=0.07\pm 0.01$ cm-3,

  * •

$n(\mathrm{H})=0.23-0.27$ cm-3,

  * •

$B=2.1,2.5\mu$G for two best models (the magnetic field affects emission
intensity from cloud boundary).

##  3 Dust and Elemental Abundances in the LIC

Our approach to modeling the abundances in the LIC is to force our models to
match the column densities observed by adjusting the elemental abundances.
Thus the abundances are an output of the modeling rather than in input. Figure
3 illustrates the abundance results for C, N, O, S, Si, Mg and Fe relative to
a particular assumed solar abundance of the elements (Asplund et al., 2005).
The x-axis is the condensation temperature for the element, a quantity often
presumed to correlate with the amount of depletion from the gas phase. In
Table 1 we list the derived abundances in our models that are consistent with
observations.

Figure 3: Abundances relative to Asplund et al. (2005) solar abundances vs. condensation temperature. Results from models of the ionization for the $\epsilon$ CMa line of sight including radiative transfer (Slavin and Frisch, 2008). The symbol shape indicates the assumed H i column density for the model, while the color indicates the assumed temperature for the hot gas of the Local Bubble. The abundances are fixed so as to match the observed ion column densities. Table 1: Elemental Gas Phase Abundances (ppm) | Element
---|---
Model No. | C | N | O | Mg | Si | S | Fe
14 | 589 | 40.7 | 295 | 5.89 | 7.24 | 14.1 | 2.24
25 | 631 | 66.1 | 437 | 7.76 | 10.0 | 19.5 | 3.09
26 | 661 | 46.8 | 331 | 6.61 | 8.13 | 15.8 | 2.51
27 | 759 | 64.6 | 437 | 8.71 | 10.7 | 20.9 | 3.31
28 | 708 | 45.7 | 331 | 7.08 | 8.32 | 16.6 | 2.57
29 | 813 | 64.6 | 437 | 9.33 | 11.0 | 21.9 | 3.39
30 | 741 | 46.8 | 331 | 7.41 | 8.51 | 17.0 | 2.63
42 | 724 | 39.8 | 295 | 6.76 | 7.76 | 15.1 | 2.34

The derived abundances indicate modest depletion of the constituents of
silicate dust, Si, Fe, Mg, and O, implying that at least some destruction of
this type of dust. Depending on ones assumptions about the initial depletion
of Si (Savage and Sembach, 1996, quote values of 70-95%), the total Solar
abundance of Si (recent determinations range from about 30 to 43 ppm) and its
current gas phase abundance (we find $\sim 7.2-11.5$ ppm) the fraction of the
silicate grain mass destroyed (i.e. returned to the gas phase) ranges from 0
to 35%. The high C abundance on the other hand seems to indicate that all the
carbonaceous dust has been destroyed. Radiative shocks destroy dust via
various processes: sputtering, vaporization, shattering. Detailed calculations
by Jones et al. (1996) find that silicate dust should be _more_ destroyed than
carbonaceous dust by shocks, as illustrated in Figure 4. Therefore either some
other process is strongly influencing the gas phase abundances in the LIC or
the models of shock processing of grains need revision.

Figure 4: Percentage of initial total grain mass lost because of dust
destruction in the shock vs. shock speed. To get the Si gas phase abundance
determined for the LIC we need $\lesssim$35% destruction of silicate dust
which is consistent with $v_{\mathrm{shock}}\lesssim 150$ km s-1. Such shocks
should destroy less than 15% of the cabonaceous dust, however, in
contradiction with the derived large gas phase abundance of C.

##  4 LIC Ionization and Thermal Balance

Observations with the Extreme Ultraviolet Explorer (EUVE) toward nearby stars
found unexpected results for the ratio of H i to He i column density (Dupuis
et al., 1995). Instead of the expected ratio of $\leq 10$ that one would get
if the cosmic He abundance is 0.1 and H is more ionized than He, it was found
that $N(\mathrm{H\,I})/N(\mathrm{He\,I})\sim 14$, indicating that He is more
ionized than H. This unusual ionization of the local ISM has long been
considered puzzling and has led to the suggestion that the LIC is out of
ionization equilibrium, being overionized for its temperature because of an
earlier ionizing event (e.g. a shock) (see, e.g., Lyu and Bruhweiler, 1996).
The long timescale for recombination, particularly of H, it was reasoned,
makes it likely that the LIC is out of ionization equilibrium. However, the
cooling rate of the gas also has to be considered in such a model. Doing this
one finds that in fact the cooling time for the gas in any likely scenario is
considerably less than the recombination time. As a result, if the LIC were
cooling from a hotter and more ionized state, it should still be quite highly
ionized by the time that it has cooled to the observed temperature of
$T\approx 6300$ K. In Figure 5 we illustrate this by showing the temperature
and ionization evolution behind a 100 km s-1 shock. Even greater disparity
between the cooling time and recombination time is found for a simple isobaric
cooling model. Therefore the fact that the LIC is in fact mostly neutral,
$X(\mathrm{H}^{+})\sim 0.2$, implies that the cloud has had time to recombine
while being maintained at a warm temperature. This requires a heat source to
balance the cooling. While alternative sources have been proposed, such as
turbulent dissipation (Minter and Spangler, 1997), the most likely heat source
appears to be photoionization heating. Such heating is also accompanied by
ionization, suggesting that the cloud is at least close to thermal and
photoionization equilibrium.

Figure 5: Time evolution of temperature and H ion fraction behind a 100 km s-1
shock (J. Raymond, private communication). The dotted lines show the time,
temperature and H ion fraction when the gas cools to the LIC temperature,
while the dashed lines show the same for the LIC H ion fraction. Isobaric
cooling shows a similar lag of H recombination. The gas cools too fast to
allow the relatively low degree of ionization in the LIC at temperatures of
$\sim 6300$ K without some substantial heating source.

Another argument in favor of the cloud being in ionization equilibrium was
first suggested by Jenkins et al. (2000) based on Ar i and O i data. Since O
ionization is tied to H ionization by charge exchange, if we assume an O
abundance we can then compare the ionization of Ar and H. In recombining gas,
it is found that Ar and H have roughly equal ionization fractions because H+
and Ar+ have similar recombination coefficients. However, for gas in
photoionization equilibrium, Ar i is deficient relative to H i (or Oi) because
the photoionization cross section for Ar0 is 5-30 times larger than that for
H0. Observations for the LIC (Jenkins et al., 2000) find
$X(\mathrm{Ar}^{0})/X(\mathrm{H}^{0})\sim 0.4$ toward nearby white dwarfs.
Detailed NEI calculations for cooling gas show that
$X(\mathrm{Ar}^{0})/X(\mathrm{H}^{0})$ remains $\sim 1$ until gas nears
equilibrium _and_ is photoionized.

Despite this evidence that the LIC is currently close to ionization and
thermal equilibrium, there are reasons to believe that it has not always been
so. The LIC is clearly many times denser than its surrounding gas in the Local
Bubble as can be deduced from the low absorption by neutral gas within the
bubble and lack of observable optical emission from possible warm ionized gas
that could conceivably fill the cavity. That leaves only highly ionized and
very low density gas as the primary volume filling gas in the bubble.
Therefore it appears highly likely that the gas that presently makes up the
LIC and other nearby clouds was at one time substantially overdense compared
with the surrounding medium before becoming incorporated into the Local
Bubble. The most likely scenario is that cold neutral medium gas, with $n\sim
100$ cm-3, $T\sim 100$ K, that was embedded in warm gas was hit by a shock.
However, it is important to note that any shock _no matter what speed_ hitting
such a dense cloud will go radiative in the cloud. Thus one needs to find a
means to heat the cloud to warm neutral medium temperatures. An origin in a
fragmented shell implies a similar radiative shock and heating requirements.
The means to heat the shocked warm clouds seems to require their expansion to
lower density as the pressure of the bubble drops at the same time as ionizing
flux from the hot gas and possibly from the cloud boundary regions provides
heating.

For diffuse ISM conditions, calculated heating and cooling rates typically
lead to the possibility of thermal balance with two stable thermal phases
within a limited range of thermal pressures, with a cold neutral phase and a
warm neutral or (perhaps partially) ionized phase. Figure 6 shows a density
vs. pressure plot or phase diagram showing two different phase equilibrium
curves. The one for “Low Ionization” comes from the work of Wolfire et al.
(2003) and assumes low ionizing flux whereas the “LIC ionization” one is
calculated using one of our model ionizing radiation fields for the LIC. We
note that the thermal pressure will generally not dominate the total dynamical
pressure because other pressure forms including magnetic, cosmic ray and
turbulent, are typically estimated to be of the same order of magnitude as the
thermal pressure. This does not affect the phase curves, however, since it is
the components of the thermal pressure (i.e. density and temperature) that
directly affect the heating-cooling balance. In the diffuse ISM cosmic ray
heating is small compared to dust and photoionization heating. It may be that
turbulent dissipation, particularly in concert with MHD turbulence, provides
significant heating, however the rate for that remains quite uncertain and is
neglected in the Figure.

The arrows on the plot indicate how gas parcels will evolve under the
influence of shocks, adiabatic cooling (cooling via expansion) and evaporation
via thermal conduction. The shock arrow indicates a relatively small increase
in pressure, which would require only a mach 2.5 (relative to the cold gas)
shock. A shock that could heat typical warm (ionized or neutral) medium gas to
about $10^{6}$ K would need to be much faster, $v_{s}\approx 270$ km s-1, or
mach 27 in the warm medium. The pressure would thus be increased to
$P/k_{B}\sim 2\times 10^{6}$ cm-3 K. In order for the local clouds to become
warm would require the pressure to drop by more than two orders of magnitude
after the shock passed over them. This could be achieved after sufficient
expansion, e.g. a factor of $\sim 4$ in radius assuming adiabatic expansion of
a spherical bubble. This requires that the clouds were close enough to the
center of the superbubble that the shock had not gone radiative yet and that
the clouds (or at least a fraction of them) could survive long enough to
persist until our current state in which the surrounding bubble has a
relatively low pressure.

Figure 6: Phase diagram for the diffuse interstellar medium for either low
ionizing flux conditions (dashed curve, using rates from Wolfire et al., 2003)
or the moderate ionizing flux as modeled for the LIC (solid curve using our
modeled radiation field). Points on the curve are conditions of thermal
equilibrium, below the curve heating exceeds cooling and above the curve
cooling exceeds heating. The diagonal lines are curves of constant
temperature. A shock will tend to move a gas parcel up and to the right in the
diagram, while adiabatic cooling moves points down and to the left as
illustrated by the arrows. Evaporation via thermal conduction moves points to
left. Note that the pressure in the plot is only thermal pressure and thus
neglects the dynamically important magnetic, cosmic ray and turbulent
pressures in the ISM.

##  5 The Origin the Complex of Local Interstellar Clouds

The above discussion lays out some of the challenges facing any model for the
origin of the complex of local interstellar clouds. In summary we would like a
theory to explain these facts:

  * •

The density, temperature and ionization of the clouds are in sharp contrast to
the surrounding Local Bubble gas (though we don’t know all the properties of
that gas),

  * •

the CLIC has a significant velocity relative to the LSR and direction roughly
away from Galactic center,

  * •

the ionization of the LIC is unusual with He apparently more ionized than H,

  * •

the abundances in the gas seem to imply that carbonaceous dust has been
destroyed, and yet interstellar dust observed in the Solar System implies a
relatively low gas-to-dust ratio.

A number of theories have been put forward to explain the CLIC. The clouds
have been variously proposed to be: 1) pieces of the Sco-Cen bubble from an
earlier epoch of star formation (Frisch, 1981), 2) a fragment from Sco-Cen/LB
interaction (Breitschwerdt et al., 2000), and 3) a flux tube/filament that has
broken away from the bubble wall (Cox and Helenius, 2003). We would add to
this list, 4) a dense cloud in the ambient medium overrun by an expanding
bubble shock, a model that we have discussed briefly above but that has yet to
be fully explored.

Each of these models has its problems. The velocity and relative positions of
the CLIC and Loop I bubble strongly suggest a connection between them but
detailed modeling of how these clouds could have come from that bubble is
lacking. Breitschwerdt et al. (2000) propose that the Local Bubble and the
Loop I bubble are interacting and that the CLIC is associated with the wall
that separates the bubbles. In their model the clouds are created by
instabilities generated in the interaction region. The LIC is currently about
70 pc from that neutral wall and moving about 20 km s-1 away from the Sco-Cen
association that is believed to be responsible for creating the Loop I bubble.
It is unclear how cloudlets like the CLIC could have been traveling for 3.5
million years away from this interaction zone and yet the wall is apparently
intact between the two bubbles. Frisch (1981) suggests that the clouds as well
as the Local Bubble are associated with a previous epoch of star formation of
Sco-Cen. This requires that somehow a cold neutral wall was reformed within
the bubble between these epochs of star formation. The mechanism for doing
that is left unexplained. The flux tube theory of the origins of the CLIC by
Cox and Helenius (2003) requires that a flux tube sprang from the wall of the
Local Bubble pulling warm gas along with it into the bubble interior. The
magnetohydrodynamics of this explanation seem questionable however, in
particular that one flux tube can spring from the bubble wall while the rest
of the bubble is not collapsing. Finally, our idea that the clouds originated
as cold clouds in a warm intercloud medium seems reasonable but does not
explain why the velocity of the CLIC is directed away from the Sco-Cen
association and towards the center of the Local Bubble rather than away from
it. We must appeal to a random velocity of the gas prior to being overrun by
the expanding Local Bubble to explain this.

##  6 Summary

The wide range of data that we have on the LIC has lead to a fairly complete
picture of the cloud. We find that it is:

  * •

partially ionized, $X(\mathrm{H}^{+})\sim 0.2-0.3$, $X(\mathrm{He}^{+})\sim
0.3-0.4$, $n_{e}\approx 0.07$ cm-3

  * •

has experienced mixed dust destruction – moderate for silicate dust, complete
for carbonaceous,

  * •

at or close to ionization equilibrium

An origin as a cloud embedded in a lower density medium that was shocked seems
likely, and some association with the Loop I bubble and Sco-Cen OB association
remains a possibility. Many mysteries remain about its abundances and origins
within the local ISM.

###### Acknowledgements.

I would like to thank the organizers of the “Outer Heliosphere to the Local
Bubble” conference for inviting me to give this talk and Priscilla Frisch, my
collaborator in much of the work I presented. This research was supported by
NASA Solar and Heliospheric Physics Program grants NNG05GD36G and NNG06GE33G
to the University of Chicago.

## References

  * Asplund et al. (2005) M. Asplund, N. Grevesse, A. J. Sauval, In ASP Conf. Ser. 336: Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, pages 25–37 (2005)
  * Baguhl et al. (1995) M. Baguhl, E. Grün, D. P. Hamilton, G. Linkert, R. Riemann, P. Staubach, Space Science Reviews 72, 471 (1995)
  * Breitschwerdt et al. (2000) D. Breitschwerdt, M. J. Freyberg, R. Egger, Astron. Astrophys. 361, 303–320 (2000)
  * Cox and Helenius (2003) D. P. Cox, L. Helenius, Astrophys. J. 583, 205–228 (2003)
  * Dupuis et al. (1995) J. Dupuis, S. Vennes, S. Bowyer, A. K. Pradhan, P. Thejll, Astrophys. J. 455, 574 (1995)
  * Frisch (1981) P. C. Frisch, Nature 293, 377–379 (1981)
  * Gondhalekar et al. (1980) P. M. Gondhalekar, A. P. Phillips, R. Wilson, Astron. Astrophys. 85, 272 (1980)
  * Gry and Jenkins (2001) C. Gry, E. B. Jenkins, Astron. Astrophys. 367, 617–628 (2001)
  * Jenkins et al. (2000) E. B. Jenkins, W. R. Oegerle, C. Gry, J. Vallerga, K. R. Sembach, R. L. Shelton, R. Ferlet, A. Vidal-Madjar, D. G. York, J. L. Linsky, K. C. Roth, A. K. Dupree, J. Edelstein, Astrophys. J. Letters 538, L81–L85 (2000)
  * Jones et al. (1996) A. P. Jones, A. G. G. M. Tielens, D. J. Hollenbach, Astrophys. J. 469, 740 (1996)
  * Landgraf et al. (2000) M. Landgraf, W. J. Baggaley, E. Grün, H. Krüger, G. Linkert, J. Geophys. Res. 105, 10,343–10,352 (2000)
  * Lyu and Bruhweiler (1996) C.-H. Lyu, F. C. Bruhweiler, Astrophys. J. 459, 216 (1996)
  * Minter and Spangler (1997) A. H. Minter, S. R. Spangler, Astrophys. J. 485, 182 (1997)
  * Redfield and Linsky (2008) S. Redfield, J. L. Linsky, Astrophys. J. 673, 283–314 (2008)
  * Savage and Sembach (1996) B. D. Savage, K. R. Sembach, Astrophys. J. 470, 893 (1996)
  * Slavin and Frisch (2006) J. D. Slavin, P. C. Frisch, Astrophys. J. Letters 651, L37–L40 (2006)
  * Slavin and Frisch (2008) J. D. Slavin, P. C. Frisch, Astron. Astrophys. submitted (2008)
  * Witte (2004) M. Witte, Astron. Astrophys. 426, 835–844 (2004)
  * Wolfire et al. (2003) M. G. Wolfire, C. F. McKee, D. Hollenbach, A. G. G. M. Tielens, Astrophys. J. 587, 278–311 (2003)