# Group properties and invariant solutions of a sixth-order thin film equation
in viscous fluid

Ding-jiang Huang1,2,3<EMAIL_ADDRESS>Qin-min Yang1 Shuigeng Zhou2,3
1Department of Mathematics, East China University of Science and Technology,
Shanghai 200237, China  2School of Computer Science, Fudan University,
Shanghai 200433, China  3Shanghai Key Lab of Intelligent Information
Processing, Fudan University, Shanghai 200433, China

###### Abstract

Using group theoretical methods, we analyze the generalization of a one-
dimensional sixth-order thin film equation which arises in considering the
motion of a thin film of viscous fluid driven by an overlying elastic plate.
The most general Lie group classification of point symmetries, its Lie
algebra, and the equivalence group are obtained. Similar reductions are
performed and invariant solutions are constructed. It is found that some
similarity solutions are of great physical interest such as sink and source
solutions, travelling-wave solutions, waiting-time solutions, and blow-up
solutions.

PACS numbers

47.15.G-,02.20.Sv,02.30.Jr

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

In the past several decades there is an increasing interest in physics and
mathematics literatures in higher-order nonlinear diffusion equations because
they are models of various interesting phenomenon in fluid physics and have
surprising mathematic structure and properties. Probably, one of the most
famous example is the fourth-order thin film equation in the form

$u_{t}=(u^{\alpha}u_{xxx})_{x},\quad\quad\alpha>0$ (1)

which was first introduced by Greenspan in 1978 Greenspan (1978). This
equation describes the surface-tension-dominated motion of thin viscous films
for the film height $u(t,x)$ and spreading droplets in the lubrication
approximation Greenspan (1978). In particular, for $\alpha=3$ it describes a
classical thin film of Newtonian fluid, as reviewed in Oron  _et al._ (1997),
$\alpha=1$ occurs in the dynamics of a Hele-Shaw cell Constantin  _et al._
(1993) and $\alpha=2$ arises in a study of wetting films with a free contact
line between film and substrate Bertozzi (1998). There also exist many
interesting generalizations of the famous equation (1) (see King (2001); Yarin
_et al._ (1993) and reference therein).

Apart from the fourth-order equations, another interesting higher-order
diffusion model is the sixth-order nonlinear thin film equation in the form

$u_{t}=(u^{m}u_{xxxxx})_{x},$ (2)

which appear in flow modeling. The case $m=3$, for instance, was first
introduced by King in King (1986) as a model of the oxidation of silicon in
semiconductor devices King (1989) or for a moving boundary given by a beam of
negligible mass on a surface of a thin film Smith  _et al._ (1996a). Here
$u\geq 0$ will be treated as the thickness of a fluid film beneath an elastic
plate and $p=u_{xxxx}$ as the pressure within the film King (1986). The other
derivatives of $u$ can in the usual way be assigned different physical
meaning, for instance $\Gamma=-u_{xx}$ is the bending moment on the overlying
plate and $\Sigma=u_{xxx}$ is the shearing force Landau  _et al._ (1986), here
all such expressions are dimensionless. An equation of this type can be used
to model the motion of a thin film of viscous fluid overlain by an elastic
plate King (1989); see also Hobart et al. Hobart  _et al._ (2000) and Huang et
al. Huang and Suo (2002) for possible applications of such modelling
approaches to the wrinkling upon annealing of SiGe films bonded to Si
substrates. Other plausible applications of Eq. (2), and suitable
generalizations thereof, include a simple model for the influence of a crust
on a solidifying melt or for a microfluidic pump (see Koch et al. Koch  _et
al._ (1997), for instance).

Eqs. (1) and (2) are also the second and the third member of a hierarchy
arising from the generalized Reynolds equation

$u_{t}=(u^{m}p_{x})_{x},$ (3)

under different driving forces respectively. For gravity driven flows, we have
$p=u$, giving the very widely studied porous-medium equation (see, for
example, Aronson Aronson (1986)). For surface-tension driven flows we have
$p=-u_{xx}$, leading to the fourth-order thin film equation (1). For elastic
plate driven flows, we have $p=u_{xxxx}$, which give the sixth-order thin film
equation (2) King (1989).

Up to now, the mathematic structure and properties of the fourth-order thin
film equation (1) have been widely investigated, including (non-)uniqueness,
wetting behaviour and contact line motion, in particular optimal propagation
rates, waiting time or dead core phenomena and self-similar solutions(see
Hulshof Hulshof (2001), for instance). Recent years, there are also many
researches devoted to symmetry group structure and exact solutions of the
fourth-order thin film equations (1) and their generalizationsSmyth and Hill
(1988); Bernis and McLeod (1991); Choudhury (1995); Bernis  _et al._ (2000);
Gandarias and Bruzon (2000); Bruzon  _et al._ (2003); Qu (2006); Gandarias and
Ibragimov (2008); Gandarias and Medina (2001); Cherniha  _et al._ (2010), or
searching for special invariant finite vector spaces of solutions Galaktionov
and Svirshchevskii (2007).

However, the sixth-order thin film equation (2) has been much less extensively
investigated. It was only a few researches that were devoted to qualitative
mathematic properties such as the existence of weak solutions, initial
boundary value problems (see Bernis Friedman Bernis and Friedman (1990), King
et al Flitton and King (2004), Smith et al. Smith  _et al._ (1996b), Evans et
al. Evans  _et al._ (2000), Barrett et al. Barrett  _et al._ (2004) for
existing studies, the first two being analytical and the others primarily
numerical), while the symmetry group properties and corresponding algebraic
structure as well as explicit exact solutions of Eq. (2) still remain open.
Therefore, the aim of the present work is to find such group properties,
algebraic structure and exact solutions. To do this, we investigate
alternatively a more general sixth-order nonlinear diffusion equation in the
form

$u_{t}=(f(u)u_{xxxxx})_{x},$ (4)

than the original equation (2), where $f(u)$ is an arbitrary smooth function
depending on the geometry of the problem and $f_{u}\neq 0$( i.e., (4) is a
nonlinear equation).

We use the method of Lie groups, one of the powerful tools available to solve
nonlinear PDEs, and which was discovered and applied firstly by S. Lie in the
nineteenth century, but only in the last decades has it become a common tool
for both mathematicians and physicists (see for examples Bluman and Kumei
(1989); Ovsiannikov (1982); Olver (1986); Bluman  _et al._ (2010); Alfinito
_et al._ (1995); Carbonaro (1997); Pulov  _et al._ (1998); Senthilvelan  _et
al._ (2002); Struckmeier and Riedel (2002); Stewart and Momoniat (2004)). The
method consists of looking for the infinitesimal generators of a group of
point transformations which leave the equation under study invariant. An
important point of the Lie theory is that the conditions for an equation to
admit a group of transformations are represented by a set of linear equations,
the so-called “determining equations”, which are usually completely solvable.
Having once found the groups of transformations, one can obtain a number of
interesting results, which include the possibility to reduce a partial
differential equation with two independent variables to an ordinary
differential equation with one independent variable, etc.. Solving these
reduced equations, one can obtain some particular solutions for the original
equations. These particular solutions are usually called “similarity
solutions” or “invariant solutions” Bluman and Kumei (1989); Ovsiannikov
(1982); Olver (1986); Bluman  _et al._ (2010). When the equation contains
“arbitrary elements”(a variable coefficient deriving from the particular
equation of state chosen to characterize the physical mechanism. “Arbitrary
elements” are functions or variable parameters whose form is not strictly
fixed and can be assigned freely on the grounds of physical hypotheses about
the nature of the medium under consideration.), the theory gives rise to the
problem of group classification of differential equations which is the core
stone of modern group analysis Ovsiannikov (1982); Bluman  _et al._ (2010). In
particular, in the past several years, a numbers of novel techniques, such as
algebraic methods based on subgroup analysis of the equivalence group Basarab-
Horwath  _et al._ (2001); Zhdanov and Lahno (1999); Gazeau and Winternitza
(1992); Popovych  _et al._ (2010), compatibility and direct integration
Ovsiannikov (1982); Ibragimov (1994) (also referred as the Lie-Ovsiannikov
method) as well as their generalizations (eg. method of furcate split Nikitin
and Popovych (2001), additional and conditional equivalence transformations
Popovych and Ivanova (2004); Huang and Ivanova (2007), extended and
generalized equivalence transformation group, gauging of arbitrary elements by
equivalence transformations Ivanova  _et al._ (2010); Huang and Zhou (2011))
have been proposed to solve group classification problem for numerous
nonlinear partial differential equations. Although a great deal of
classification was solved by these methods, almost all of them are limited to
the equations whose order are lower than four (see Huang and Zhou (2011) for
details).

In this paper we extend these new techniques, specific compatibility and
direct integration as well as equivalence transformation techniques, to sixth-
order nonlinear diffusion equations. We first carry out group classification
of Eq. (4) under the usual equivalence group. The Lie group of point
symmetries of Eq. (2), as a special case of Eq. (4), and its Lie algebra are
also obtained. Then similar reductions of the classification models are
performed and invariant solutions are also constructed. It is found that some
similarity solutions are solutions with physical interest: sink and source
solutions, travelling-wave solutions, waiting-time solutions and blow-up
solutions.

The rest of this paper is organized as follows: In Sec. II we derive the
equivalence group and perform the group classification related to Eq. (4). In
Sec. III, similar reductions of classification models are carried out. Sec. IV
contains examples of some specific exact solutions, including sink and source
solutions, travelling-wave solutions, waiting-time solutions and blow-up
solutions, while in Sec. V some concluding remarks are reported.

##  II SYMMETRY CLASSIFICATION

Background and procedures of the modern Lie group theory are well described in
literature Bluman and Kumei (1989); Ovsiannikov (1982); Olver (1986); Bluman
_et al._ (2010); Popovych and Ivanova (2004); Huang and Ivanova (2007).
Without going into the details of the theory, we present only the results
below.

Let

${\bf
Q}=\tau(t,x,u)\partial_{t}+\xi(t,x,u)\partial_{x}+\phi(t,x,u)\partial_{u}$

be a vector field or infinitesimal operator on the space of independent and
dependent variables $t,~{}x,~{}u$. A local group of transformations $G$ is a
symmetry group of Eq. (4) if and only if

$\rm{pr}^{(6)}{\bf Q}|(\Delta)=0,$ (5)

whenever $\Delta=u_{t}-\big{[}f(u)u_{xxxxx}\big{]}_{x}=0$ for every generator
of $G$, where $\rm{pr}^{(6)}{\bf Q}$ is the sixth-order prolongation of ${\bf
Q}$.

Expanding Eq. (5) we get

$\displaystyle\phi^{t}=$ $\displaystyle\phi
f^{\prime\prime}(u)u_{x}u_{xxxxx}+f^{\prime}(u)u_{xxxxx}\phi^{x}+f^{\prime}(u)u_{x}\phi^{xxxxx}$
(6) $\displaystyle+\phi f^{\prime}(u)u_{xxxxxx}+f(u)\phi^{xxxxxx}$

which must be satisfied whenever Eq. (4) is satisfied. Substituting the
formulae of $\phi^{t}$, $\phi^{x}$, $\phi^{xxxxx}$ and $\phi^{xxxxxx}$ into
Eq. (6) we get an equation of $t,x,u$ and the derivatives of
$\tau,\xi,\phi,u$. Replacing $u_{t}$ by the right hand side of Eq. (4)
whenever it occurs, and equating the coefficients of the various independent
monomials to zero, we obtain the determining equations

$\begin{cases}\tau_{x}=\tau_{u}=\xi_{u}=\phi_{uu}=0\\\
3(2\phi_{xu}-5\xi_{xx})f(u)+\phi_{x}f^{\prime}(u)=0\\\
(\phi_{xu}-2\xi_{xx})f^{\prime}(u)=0\\\ 3\phi_{xxu}-4\xi_{xxx}=0\\\
(\phi_{xxu}-\xi_{xxx})f^{\prime}(u)=0\\\ (\tau_{t}-6\xi_{x})f(u)+\phi
f^{\prime}(u)=0\\\ 4\phi_{xxxu}-3\xi_{xxxx}=0\\\
\phi_{xxxxxx}f(u)-\phi_{t}=0\\\
\phi_{xxxxx}f^{\prime}(u)+\xi_{t}+6\phi_{xxxxxu}f(u)-\xi_{xxxxxx}f(u)=0\\\
5\phi_{xxxxu}-2\xi_{xxxxx}=0\\\ (5\phi_{xxxxu}-\xi_{xxxxx})f^{\prime}(u)=0\\\
(2\phi_{xxxu}-\xi_{xxxx})f^{\prime}(u)=0\end{cases}$

The first three equations imply that $\phi_{xu}=\xi_{xx}=0$, which together
with the sixth, the eighth and the ninth equations imply that
$\phi_{t}=\phi_{x}=\xi_{t}=0$, so the determining equations reduce to

$\begin{cases}\tau_{x}=\tau_{u}=0\\\ \xi_{t}=\xi_{u}=\xi_{xx}=0\\\
\phi_{t}=\phi_{x}=\phi_{uu}=0\\\ (\tau_{t}-6\xi_{x})f(u)+\phi
f^{\prime}(u)=0,\end{cases}$ (7)

which is equivalent to

$\begin{cases}\xi=ax+b\\\ \tau=ct+d\\\ \phi=pu+q\\\
(c-6a)f(u)+(pu+q)f^{\prime}(u)=0\end{cases}$ (8)

where $a$, $b$, $c$, $d$, $p$, and $q$ are arbitrary constants.

In order to make the classification as simple as possible, we next look for
equivalence transformations of class (4), and then solve system (8) under
these transformations. An equivalence transformation is a nondegenerate change
of the variables $t$, $x$ and $u$ taking any equation of the form (4) into an
equation of the same form, generally speaking, with different $f(u)$. The set
of all equivalence transformations forms the equivalence group $G^{\sim}$. To
find the connected component of the unity of $G^{\sim}$, we have to
investigate Lie symmetries of the system that consists of Eq. (4) and some
additional conditions, i.e.

$\begin{cases}u_{t}=f_{u}u_{x}u_{xxxxx}+fu_{xxxxxx},\\\ f_{t}=0,\\\
f_{x}=0.\end{cases}$ (9)

That is to say we must seek for an operator of the Lie algebra $A^{\sim}$ of
$G^{\sim}$ in the form

$\displaystyle{\bf X}=$
$\displaystyle\tau(t,x,u)\partial_{t}+\xi(t,x,u)\partial_{x}$ (10)
$\displaystyle+\phi(t,x,u)\partial_{u}+\psi(t,x,u,f)\partial_{f}.$

Here $u$ and $f$ are considered as different variables: $u$ is on the space
$(t,x)$ and $f$ is on the extended space $(t,x,u)$. The coordinates $\tau$,
$\xi$, $\phi$ of the operator (10) are sought as functions of $t$, $x$, $u$
while the coordinates $\psi$ are sought as functions of $t$, $x$, $u$ and $f$.

Applying $\rm{pr}^{(6)}{\bf X}$ to Eq. (9) we get the infinitesimal criterion

$\begin{cases}\phi^{t}=u_{x}u_{xxxxx}\psi^{u}+f_{u}u_{xxxxx}\phi^{x}\\\
\qquad+f_{u}u_{x}\phi^{xxxxx}+u_{xxxxxx}\psi+f\phi^{xxxxxx},\\\
\psi^{t}=0,\quad\psi^{x}=0\end{cases}$ (11)

which must be satisfied whenever Eq. (9) is satisfied. Substituting the
formulae of $\phi^{t}$, $\phi^{x}$, $\phi^{xxxxx}$, $\phi^{xxxxxx}$,
$\psi^{t}$, $\psi^{x}$, and $\psi^{u}$ into Eq. (11) we get equations of $t$,
$x$, $u$, $f$, and the partial derivatives of $\tau$, $\xi$, $\phi$, $u$, $f$,
and $\psi$. Replacing $u_{t}$, $f_{t}$ and $f_{x}$ by the right hand side of
Eq. (9) whenever they occur, and equating the coefficients of various
independent monomials to zero, we obtain

$\begin{cases}\xi_{xx}=0,\quad\xi_{t}=0,\quad\xi_{u}=0\\\
\tau_{x}=0,\quad\tau_{u}=0\\\ \phi_{x}=0,\quad\phi_{t}=0,\quad\phi_{uu}=0\\\
\psi_{x}=0,\quad\psi_{t}=0,\quad\psi_{u}=0\\\
\psi=(6\xi_{x}-\tau_{t})f\end{cases}$

which can be reduced to

$\begin{cases}\tau=c_{4}t+c_{1}\\\ \xi=c_{5}x+c_{2}\\\ \phi=c_{6}u+c_{3}\\\
\psi=(6c_{5}-c_{4})f\end{cases}$ (12)

where $c_{1},c_{2},\ldots,c_{6}$ are arbitrary constants.

Thus the Lie algebra of $G^{\sim}$ for class (4) is

$A^{\sim}=\langle\partial_{t},\partial_{x},\partial_{u},t\partial_{t}-f\partial_{f},x\partial_{x}+6f\partial_{f},u\partial_{u}\rangle.$

Continuous equivalence transformations of class (4) are generated by the
operators from $A^{\sim}$. In fact, $G^{\sim}$ contains the following
continuous transformations:

$\displaystyle\tilde{t}=t{\varepsilon_{4}}+\varepsilon_{1},\qquad\tilde{x}=x{\varepsilon_{5}}+\varepsilon_{2},$
$\displaystyle\tilde{u}=u{\varepsilon_{6}}+\varepsilon_{3},\qquad\tilde{f}=f\varepsilon_{4}^{-1}\varepsilon_{5}^{6},$

where $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{6}$ are arbitrary
constants.

Solve the last equation of system (8) under the above equivalence group
$G^{\sim}$, we can obtain three inequivalent equations of class (4) with
respect to the transformations from $G^{\sim}$:

Case 1: $f(u)$ is an arbitrary nonconstant function, the symmetry algebra of
class (4) is a three-dimensional Lie algebra which is generated by the
operators

$Q_{1}=\partial_{t},\quad Q_{2}=\partial_{x},\quad
Q_{3}=t\partial_{t}+\frac{1}{6}x\partial_{x};$ (13)

Case 2: $f=e^{\lambda u}\mod G^{\sim}(\lambda\neq 0)$, the symmetry algebra of
class (4) is a four-dimensional Lie algebra which is generated by the
operators

$\displaystyle Q_{1}=\frac{1}{\lambda}\partial_{u}-t\partial_{t},$
$\displaystyle Q_{2}=\partial_{t},$ (14) $\displaystyle
Q_{3}=-x\partial_{x}-\frac{6}{\lambda}\partial_{u},$ $\displaystyle
Q_{4}=\partial_{x};$

Case 3: $f=u^{m}\mod G^{\sim}(m\neq 0)$, the symmetry algebra of class (4) is
a four-dimensional Lie algebra which is generated by the operators

$\displaystyle Q_{1}=\frac{1}{m}u\partial_{u}-t\partial_{t},\quad$
$\displaystyle Q_{2}=\partial_{t},$ (15) $\displaystyle
Q_{3}=-x\partial_{x}-\frac{6}{m}u\partial_{u},\quad$ $\displaystyle
Q_{4}=\partial_{x}.$

From the above results, it is easy to see that equation (2) is exactly
corresponding to case 3, thus possess a four-dimensional symmetry algebra.

##  III SIMILARITY REDUCTION

In order to obtain all the inequivalent reductions, we look for the one-
dimensional optimal systems (see Ovsiannikov (1982)). These systems,
similarity variables and reduced equations are listed below. In the following
tables III.1, III.2, III.3, each row shows the infinitesimal generators
$Q_{i}$ of each optimal system, as well as its similarity variable, similarity
solution and reduced equation. $\alpha$ is an arbitrary constant, while
$\lambda$ is a non-vanishing arbitrary constant. Note that in the case
$f(u)=u^{m}$ which corresponds to Eq. (2), we only consider $m\neq 0$,
otherwise the equation is linear.

###  III.1 $f(u)$ is an arbitrary nonconstant function

In this case, the symmetry operators are Eq. (13). These operators satisfy the
commutation relations

$[Q_{1},\ Q_{3}]=Q_{1},\ \ [Q_{2},\ Q_{3}]=\frac{1}{6}Q_{2}$

and thus the corresponding symmetry algebra is a realization of the algebra
$A_{3,5}^{a}(0<|a|<1)$ Patera and Winternitz (1977). According to the results
of Patera and Winternitz Patera and Winternitz (1977), an optimal system of
one-dimensional subalgebras is those spanned by

$Q_{1},\quad Q_{2},\quad Q_{3},\quad Q_{1}+\alpha Q_{2}.$

Therefore, the corresponding similarity variables and reduced ODEs can be
easily calculated. Such results are listed in Table III.1.

Table 1. Reduced ODEs for arbitrary nonconstant $f(u)$
(let $E=\big{[}f(v)v_{yyyyy}\big{]}_{y}$).  $i$ Subalgebra Ansatz $u=$ $y$
Reduced ODEi 1 $\langle Q_{1}\rangle$ $v(y)$ $x$ $E=0$ 2 $\langle
Q_{2}\rangle$ $v(y)$ $t$ $v_{y}=0$ 3 $\langle Q_{3}\rangle$ $v(y)$
$xt^{{-}1/6}$ $E=-yv_{y}/6$ 4 $\langle Q_{1}+\alpha Q_{2}\rangle$ $v(y)$
$x-\alpha t$ $E=-\alpha v_{y}$

###  III.2 $f(u)=e^{\lambda u}$ ($\lambda\neq 0$)

In this case, the symmetry operators are given by Eq. (14), which satisfy the
commutation relations

$[Q_{1},\ Q_{2}]=Q_{2},\ \ [Q_{3},\ Q_{4}]=Q_{4}$ (16)

and thus the corresponding symmetry algebra is a realization of the algebra
$2A_{2}$. According to the results of Patera and Winternitz Patera and
Winternitz (1977) again, an optimal system of one-dimensional subalgebras is
those generated by

$Q_{2},Q_{3},Q_{4},Q_{1}+\alpha Q_{3},Q_{1}+\alpha Q_{4},Q_{2}+\alpha
Q_{4},Q_{2}+\alpha Q_{3}.$

The corresponding similarity variables and reduced ODEs are listed in Table
III.2.

Table 2. Reduced ODEs for $f(u)=e^{\lambda u}$
(let $\lambda\neq 0$, $E=\big{(}e^{\lambda v}v_{yyyyy}\big{)}_{y}$).  $i$
Subalgebra Ansatz $u=$ $y$ Reduced ODEi 5 $\langle Q_{2}\rangle$ $v(y)$ $x$
$E=0$ 6 $\langle Q_{3}\rangle$ $v(y)+\frac{6}{\lambda}\ln x$ $t$
$144e^{\lambda v}-\lambda v_{y}=0$ 7 $\langle Q_{4}\rangle$ $v(y)$ $t$
$v_{y}=0$ 8 $\langle Q_{1}+\alpha Q_{3}\rangle$
$v(y)+\frac{6\alpha-1}{\lambda}\ln t$ $xt^{-\alpha}$
$E=\frac{6\alpha-1}{\lambda}-\alpha yv_{y}$ 9 $\langle Q_{1}+\alpha
Q_{4}\rangle$ $v(y)-\frac{1}{\lambda}\ln t$ $x+\alpha\ln t$ $E=\alpha
v_{y}-\frac{1}{\lambda}$ 10 $\langle Q_{2}+\alpha Q_{4}\rangle$ $v(y)$
$x-\alpha t$ $E=-\alpha v_{y}$ 11 $\langle Q_{2}+\alpha Q_{3}\rangle$
$v(y)-\frac{6\alpha t}{\lambda}$ $xe^{\alpha t}$ $E=\alpha
yv_{y}-\frac{6\alpha}{\lambda}$

###  III.3 $f(u)=u^{m}$ ($m\neq 0$)

In this case, the symmetry operators are Eq. (15). These operators share the
same commutation relations Eq. (16). Hence an optimal system of one-
dimensional subalgebras is the same as the case $f(u)=e^{\lambda u}$. The
corresponding similarity variables and reduced ODEs are listed in Table III.3.

Table 3. Reduced ODEs for $f(u)=u^{m}$
(let $m\neq 0$, $E=\big{(}v^{m}v_{yyyyy}\big{)}_{y}$).  $i$ Subalgebra Ansatz
$u=$ $y$ Reduced ODEi 12 $\langle Q_{2}\rangle$ $v(y)$ $x$ $E=0$ 13 $\langle
Q_{3}\rangle$ $v(y)x^{\frac{6}{m}}$ $t$
$v^{m+1}\prod\limits_{k=-4}^{1}(\frac{6}{m}{+}k)=v_{y}$ 14 $\langle
Q_{4}\rangle$ $v(y)$ $t$ $v_{y}=0$ 15 $\langle Q_{1}+\alpha Q_{3}\rangle$
$v(y)t^{\frac{6\alpha-1}{m}}$ $xt^{-\alpha}$ $E=\frac{6\alpha-1}{m}v-\alpha
yv_{y}$ 16 $\langle Q_{1}+\alpha Q_{4}\rangle$ $v(y)t^{{-}\frac{1}{m}}$
$x+\alpha\ln t$ $E=\alpha v_{y}-\frac{1}{m}v$ 17 $\langle Q_{2}+\alpha
Q_{4}\rangle$ $v(y)$ $x-\alpha t$ $E=-\alpha v_{y}$ 18 $\langle Q_{2}+\alpha
Q_{3}\rangle$ $v(y)e^{-\frac{6\alpha t}{m}}$ $xe^{\alpha t}$ $E=\alpha
yv_{y}-\frac{6\alpha}{m}v$

##  IV INVARIANT SOLUTIONS

Using the above reduced ODEs, we can construct some invariant solutions for
the original equations (4). It is easy to see that some of the similarity
variables in the tables III.1, III.2 and III.3 have a clear physical
interpretation. Besides, for some higher order reduced ODEs, they can be
further reduced by using new symmetries. Below, we discuss some facts related
with some types of similarity solutions with physical interest and obtain some
particular solutions. Different types of solutions are separately analyzed.

###  IV.1 Source and Sink Solutions

There are two ODEs, i.e., ODE${}_{\ref{PowerFunction4}}$ and
ODE${}_{\ref{PowerFunction7}}$, among our reduced equations are related to
this type of solutions. In fact, if we choose $\alpha=\frac{1}{m+6}$ in
ODE${}_{\ref{PowerFunction4}}$, then the similarity solution has the form

$u(t,x)=\frac{1}{t^{\frac{1}{m+6}}}v(\frac{x}{t^{\frac{1}{m+6}}}).$

Thus, if $m>-6$ it is clear that $u(t,x)\to\delta(x)$ as $t\to 0$ and the
similarity solution is a source solution; if $m<-6$ it is clear that
$u(t,x)\to\delta(x)$ as $t\to+\infty$ and the similarity solution is a sink
solution. Furthermore, we can also observe that, for the above choice of
$\alpha=\frac{1}{m+6}$, ODE${}_{\ref{PowerFunction4}}$ can be integrated once
to obtain

$v^{m}v_{yyyyy}+\frac{1}{m+6}yv=k,$

where $k$ is an arbitrary constant. Thus, we can obtain a class of source
solutions and sink solutions for the general thin-film equations (4) with
$f(u)=u^{m}$ (i.e., equation (2)) by solving the above fifth-order ODE. If we
further choose $k$ as zero, we have

$v_{yyyyy}=-\frac{1}{m+6}yv^{1-m}.$

This equation admits the symmetry group corresponding to the infinitesimal
generator ${\bf v}=y\partial_{y}+\frac{6}{m}v\partial_{v}$. Taking into
account that the invariants of its first prolongation and setting

$x_{1}=vy^{-\frac{6}{m}},\quad
u_{1}=y^{\frac{6}{m}}(yv^{\prime}-\frac{6}{m}v)^{-1},$

this equation becomes a fourth-order ODE:

$\displaystyle-m^{5}u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}}+5m^{4}(3mu_{1x_{1}}+2mu_{1}^{2}$
$\displaystyle-6u_{1}^{2})u_{1}^{2}u_{1x_{1}x_{1}x_{1}}+10m^{5}u_{1}^{2}u_{1x_{1}x_{1}}^{2}$
$\displaystyle-5m^{3}[21m^{2}u_{1x_{1}}^{2}+20m(m-3)u_{1}^{2}u_{1x_{1}}$
$\displaystyle+(7m^{2}-48m+72)u_{1}^{4}]u_{1}u_{1x_{1}x_{1}}+105m^{5}u_{1x_{1}}^{4}$
$\displaystyle+150m^{4}(m-3)u_{1}^{2}u_{1x_{1}}^{3}+15m^{3}(7m^{2}-48m$
$\displaystyle+72)u_{1}^{4}u_{1x_{1}}^{2}+10m^{2}(m-3)(5m^{2}-48m$
$\displaystyle+72)u_{1}^{6}u_{1x_{1}}+[m^{5}x_{1}^{-m+1}/(m+6)$
$\displaystyle+72(m-2)(m-3)(m-6)(2m-3)x_{1}]u_{1}^{9}$
$\displaystyle+12m(2m^{4}-50m^{3}+315m^{2}$
$\displaystyle-720m+540)u_{1}^{8}=0.$

Thus, source and sink solutions can be also obtained by solving the above
fourth-order ODE.

If we choose $m=-6$ in ODE${}_{\ref{PowerFunction7}}$, then the similarity
solution has the form

$u(t,x)=\frac{1}{e^{-\alpha t}}v(\frac{x}{e^{-\alpha t}}),$

thus, if $\alpha>0$ it is clear that $u(t,x)\to\delta(x)$ as $t\to+\infty$ and
the similarity solution is a sink solution; if $\alpha<0$ it is clear that
$u(t,x)\to\delta(x)$ as $t\to-\infty$ and the similarity solution is a source
solution. As in the previous case, for the choice of $m=-6$,
ODE${}_{\ref{PowerFunction7}}$ can be integrated once to obtain a fifth-order
ODE

$v^{-6}v_{yyyyy}-\alpha yv=k,$

where $k$ is an arbitrary constant. Consequently, source and sink solutions
can be computed by solving a fifth-order ODE.

###  IV.2 Travelling-wave Solutions

This types of solution corresponds to the reductions III.1, III.2 and III.3.
In fact, in these three reductions the similarity variables are given by
$y=x-\alpha t$, $u=v$, so that $u(t,x)=v(x-\alpha t)$, thus the corresponding
solutions are travlling-wave solutions. Due to the physical interest of this
type of solutions, in what follows we study further symmetries of the
associated ODEs and then construct some such kinds of solutions. First of all,
we integrate these three equations once trivially and obtain

$\displaystyle\mbox{ODE}^{\prime}_{\ref{ArbitraryFunction4}}:f(v)v_{yyyyy}+\alpha
v=k,$ $\displaystyle\mbox{ODE}^{\prime}_{\ref{ExponentFunction6}}:e^{\lambda
v}v_{yyyyy}+\alpha v=k_{1},$
$\displaystyle\mbox{ODE}^{\prime}_{\ref{PowerFunction6}}:v^{m}v_{yyyyy}+\alpha
v=k_{2},$

where $k,k_{1},k_{2}$ are arbitrary constants. Because
ODE${}_{\ref{ExponentFunction6}}$ and ODE${}_{\ref{PowerFunction6}}$ are given
by ODE${}_{\ref{ArbitraryFunction4}}$ for $f(u)=e^{\lambda u}$ and
$f(u)=u^{m}$ respectively, so we will focus on the
ODE${}^{\prime}_{\ref{ArbitraryFunction4}}$ below. This equation is invariant
under the group of translations in the $y$-direction, with infinitesimal
generator $\frac{\partial}{\partial y}$. Set

$x_{1}=v,\quad u_{1}=v^{-1}_{y},$

then the equation becomes a fourth-order ODE:

$\displaystyle
f(x_{1})(105u^{4}_{1x_{1}}-105u_{1}u^{2}_{1x_{1}}u_{1x_{1}x_{1}}+10u_{1}^{2}u^{2}_{1x_{1}x_{1}}+$
(17) $\displaystyle
15u_{1}^{2}u_{1x_{1}}u_{1x_{1}x_{1}x_{1}}-u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}})+\alpha
x_{1}u_{1}^{9}=ku_{1}^{9}.$

We further suppose that

${\bf v}=\xi(x_{1},u_{1})\partial_{x_{1}}+\phi(x_{1},u_{1})\partial_{u_{1}}$

is an infinitesimal generator of the last equation, then the coefficients
$\xi(x_{1},u_{1})$ and $\phi(x_{1},u_{1})$ are satisfied with

$\begin{cases}\xi=ax_{1}+b,\\\ \phi=cu_{1},\\\ [5(a+c)\alpha
x_{1}+b\alpha-(4a+5c)k]f(x_{1})\\\ +[-a\alpha
x_{1}^{2}+(ak-b\alpha)x_{1}+kb]f^{\prime}(x_{1})=0.\end{cases}$ (18)

If $f(u)=e^{\lambda u}$, from the above system we can obtain $\xi=\phi=0$,
which means that ODE${}^{\prime}_{\ref{ExponentFunction6}}$ has no nontrivial
symmetry. Thus, it can not be reduced again. Consequently, the travelling wave
solutions for Eq. (4) with $f(u)=e^{\lambda u}$ can be computed by solving a
fourth-order ODE:

$\displaystyle e^{\lambda
x_{1}}(105u^{4}_{1x_{1}}-105u_{1}u^{2}_{1x_{1}}u_{1x_{1}x_{1}}+10u_{1}^{2}u^{2}_{1x_{1}x_{1}}+$
$\displaystyle
15u_{1}^{2}u_{1x_{1}}u_{1x_{1}x_{1}x_{1}}-u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}})+\alpha
x_{1}u_{1}^{9}=ku_{1}^{9}.$

If $f(u)=u^{m}$, then we have three cases from system (18)

$\begin{array}[]{ll}(i)\quad\ k\neq 0,\quad\xi=0,\quad\phi=0;\\\ (ii)\quad
k=0,\quad m=1,\quad\xi=ax_{1}+b,\quad\phi=-\frac{4}{5}au_{1};\\\ (iii)\quad
k=0,\quad m\neq 1,\quad\xi=ax_{1},\quad\phi=\frac{m-5}{5}au_{1}.\end{array}$

Due to the triviality, the first case is excluded from the consideration. From
the second case, we can get a rational travelling wave solution for Eq. (4)
with $f(u)=u$ in the form

$u(t,x)=-\frac{1}{120}\alpha(x-\alpha t)^{5}+\sum_{i=0}^{4}c_{i}(x-\alpha
t)^{i}.$

For the third case, we can set

$x_{2}=u_{1}x_{1}^{\frac{5-m}{5}},\quad
u_{2}=x_{1}^{\frac{m-5}{5}}(x_{1}u_{1}^{\prime}+\frac{5-m}{5}u_{1})^{-1},$

then Eq. (17) can be reduced to:

$\displaystyle
625x_{2}^{3}u_{2}^{2}u_{2x_{2}x_{2}x_{2}}-125[50x_{2}u_{2x_{2}}+(11m$
$\displaystyle-25)x_{2}u_{2}^{2}+75u_{2}]x_{2}^{2}u_{2}u_{2x_{2}x_{2}}+9375x_{2}^{3}u_{2x_{2}}^{3}$
$\displaystyle+125[3x_{2}u_{2}(11m-25)+275]x_{2}^{2}u_{2}u_{2x_{2}}^{2}$
$\displaystyle+25[125(5m-12)x_{2}u_{2}+(46m^{2}-225m$
$\displaystyle+250)x_{2}^{2}u_{2}^{2}+2625]x_{2}u_{2}^{2}u_{2x_{2}}+(24m^{4}+875m^{2}$
$\displaystyle-250m^{3}-1250m+625\alpha x_{2}^{5}+625)x_{2}^{4}u_{2}^{7}$
$\displaystyle+10(48m^{3}-375m^{2}+875m-625)x_{2}^{3}u_{2}^{6}$
$\displaystyle+125(38m^{2}-195m+225)x_{2}^{2}u_{2}^{5}+13125(2m$
$\displaystyle-5)x_{2}u_{2}^{4}+65625u_{2}^{3}=0.$

Consequently, the travelling wave solutions for Eq. (4) with $f(u)=u^{m}(m\neq
1)$ can be computed by solving a third-order ODE.

Finally, we consider a special situation when $f(u)=ue^{-u}$ and $k=0$, in
which system (18) infers that ${\bf
v}=-5\partial_{x_{1}}+u_{1}\partial_{u_{1}}$. Let

$x_{2}=u_{1}e^{\frac{x_{1}}{5}},\quad
u_{2}=-(5u_{1x_{1}}+u)^{-1}e^{-\frac{x_{1}}{5}},$

then Eq. (17) is reduced to:

$\displaystyle
x_{2}^{3}u_{2}^{2}u_{2x_{2}x_{2}x_{2}}-x_{2}^{2}u_{2}(10x_{2}u_{2x_{2}}+11u_{2}^{2}x_{2}$
$\displaystyle+15u_{2})u_{2x_{2}x_{2}}+15x_{2}^{3}u_{2x_{2}}^{3}+11x_{2}^{2}u_{2}(5$
$\displaystyle+3x_{2}u_{2})u_{2x_{2}}^{2}+x_{2}u_{2}^{2}(46x_{2}^{2}u_{2}^{2}+125x_{2}u_{2}$
$\displaystyle+105)u_{2x_{2}}+x_{2}^{4}(625\alpha
x_{2}^{5}+24)u_{2}^{7}+96x_{2}^{3}u_{2}^{6}$
$\displaystyle+190x_{2}^{2}u_{2}^{5}+210x_{2}u_{2}^{4}+105u_{2}^{3}=0$

Therefore, the travelling wave solutions for equation (4) with $f(u)=ue^{-u}$
can be computed by solving a third-order ODE too.

###  IV.3 Waiting-time Solutions

ODE${}_{\ref{PowerFunction2}}$ is a first-order equation that can be easily
solved, in this way we obtain a family of waiting-time solutions for the
sixth-order thin film equation (4) corresponding to $f(u)=u^{m}$ (if $m\neq
3/2,2,3$ or $6$). These solutions are given by

$u(t,x)=\left\\{\begin{aligned}
x^{\frac{6}{m}}\big{[}m(t_{0}-t)\prod\limits_{k=-4}^{1}(\frac{6}{m}+k)\big{]}^{-\frac{1}{m}},\quad
x\geq 0,\\\ 0,\quad\quad\quad\quad\quad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ x<0.\end{aligned}\right.$

where $t_{0}$ being an arbitrary constant.

###  IV.4 Blow-up Solutions

ODE${}_{\ref{ExponentFunction2}}$ is also a first-order equation. Solving it
we get for the sixth-order thin film equation (4) with $f(u)=e^{\lambda u}$
the corresponding similarity solution

$u(t,x)=\frac{1}{\lambda}\ln\frac{(x-x_{0})^{6}}{144(t_{0}-t)}$

where $t_{0}$ is an arbitrary constant. This solution describes a localized
blow-up at $x=x_{0}$. Note that the solution is only valid if
$\frac{(x-x_{0})^{6}}{144(t_{0}-t)}\leq 1$, then, it ceases before $t=t_{0}$.

##  V CONCLUDING REMARKS

We have carried out a detailed group-theoretical analysis for the generalized
one-dimensional sixth-order thin film equation (4) which arises in considering
the motion of a thin film of viscous fluid driven by an overlying elastic
plate. A complete Lie point symmetry group classification for the class (4)
have been performed under the continuous equivalence transformation group.
Based on these, a complete list of symmetry reductions of the classification
cases have been derived by making use of the optimal system of one-dimensional
subalgebras of the corresponding Lie symmetry algebras. Furthermore, invariant
solutions of the Eq. (4) with different functional form of $f$ have been
constructed by solving the reduced ODEs. In particular, by focusing our
attention in those aspects with physical interest, we have found:

  1. 1.

The thin film equation (4), for the case $f(u)=u^{m}$ (which corresponds to
equation (2)), $m>-6$ admits source solutions and $m<-6$ admits sink
solutions. These solutions are related to the solutions of a fourth-order ODE.
If $m=-6$, Eq. (4) admits source and sink solutions. In this case these
families of solutions are related to a fifth-order ODE.

  2. 2.

The thin film equation (4) has travelling-wave solutions. In the case
$f(u)=u^{m}$, for $m=1$ the equation admits a rational travelling-wave
solutions, for $m\neq 1$ the problem of finding these solutions can be
transformed into the problem of solving third-order ODEs. In the case
$f(u)=e^{\lambda u}$, the travelling wave solutions can be computed by solving
a fourth-order ODE. While for the case $f(u)=ue^{-u}$, the travelling wave
solutions of equation (4) can be computed by solving a third-order ODE.

  3. 3.

Waiting-time solutions in the case $f(u)=u^{m}$, and blow-up solutions in the
case $f(u)=e^{\lambda u}$ are obtained in the context of symmetry reductions.
However, it should be noted that these two types of solutions can also be
obtained by means of variable separation. In the first case one takes
$u(t,x)=T(t)X(x)$ and in the second case $u(t,x)=T(t)+X(x)$.

These results may lead to further applications in physics and engineering such
as tests in numerical solutions of Eq.(4) and as trial functions for
application of variational approach in the analysis of different perturbed
versions of Eq.(4). Other topics including nonclassical symmetry, non-Lie
exact solutions and physical applications of class (4) will be studied in
subsequent publication.

###### Acknowledgements.

This work was partially supported by the National Key Basic Research Project
of China under Grant No. 2010CB126600, the National Natural Science Foundation
of China under Grant No. 60873070, Shanghai Leading Academic Discipline
Project No. B114, the Postdoctoral Science Foundation of China under Grant
Nos. 20090450067, 201104247,, Shanghai Postdoctoral Science Foundation under
Grant No. 09R21410600 and the Fundamental Research Funds for the Central
Universities under Grant No. WM0911004.

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# Interstellar Polarization Survey II: General Interstellar Medium

M.J.F. Versteeg Department of Astrophysics/IMAPP, Radboud University, PO Box
9010, 6500 GL Nijmegen, The Netherlands  A.M. Magalhães Depto. de Astronomia,
IAG, Universidade de São Paulo, Brazil  M. Haverkorn Department of
Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The
Netherlands  Y. Angarita Department of Astrophysics/IMAPP, Radboud University,
PO Box 9010, 6500 GL Nijmegen, The Netherlands  C.V. Rodrigues Divisão de
Astrofísica, Instituto Nacional de Pesquisas Espaciais (INPE/MCTI), Av. dos
Astronautas, 1758, São José dos Campos, SP, Brazil  R. Santos-Lima Depto. de
Astronomia, IAG, Universidade de São Paulo, Brazil  Koji S. Kawabata Hiroshima
Astrophysical Science Center, Hiroshima University, Kagamiyama, Higashi-
Hiroshima, Hiroshima, 739-8526, Japan

###### Abstract

of a new catalog of optical observations of linearly polarized starlight in
the diffuse interstellar medium (ISM), the Interstellar Polarization Survey,
General ISM (IPS-GI). We used data from the IPS-GI, focusing on 38 fields
sampling lines of sight in the diffuse medium. The fields are about 0.3∘ by
0.3∘ in size and each of them contains $\sim 1000$ stars on average. The IPS-
GI catalog has polarimetric measurements of stars, over of which have
${P}/\sigma_{P}>5$. We added distances and other parameters from auxiliary
catalogs to over of these stars. We analyzed parameter distributions and
correlations between parameters of a high-quality subsample of stars (i.e.
$\sim 275$ stars per field). As expected, the degree of polarization tends to
increase with the extinction, producing higher values of polarization at
greater distances or at lower absolute Galactic latitudes. Furthermore, we
large-scale ordered Galactic magnetic field.

Optical astronomy (1776), Starlight polarization (1571), Interstellar medium
(847), Polarimetry (1278), Interstellar magnetic fields (845)

##  1 Introduction

It has been recognized that the magnetic fields in galaxies significantly
affect the interstellar medium (ISM). Magnetic fields play a fundamental role
in how stars form . The magnetic field controls the origin and the confinement
of cosmic rays in galaxies, high energy particles that play important roles
themselves in the galactic environment . In addition, magnetic fields are an
important component of the energy balance of the Galactic ecosystem .

Light emitted by stars can become polarized as it travels through the ISM
(Hiltner, 1949a, b; Hall, 1949). This is caused by interstellar anisotropic
extinction. Non-spherical dust grains embedded in the ISM align their short
axes with the Galactic Magnetic Field (GMF), polarizing optical starlight
parallel to the magnetic field, see for example Andersson et al. (2015) for a
recent review. By studying the resulting polarized starlight, we can learn
about the dust properties and distribution, as well as the magnetic field
structure between the star and observer. More specifically, optical starlight
polarization traces the line-of-sight average of the plane-of-sky component of
the magnetic field, weighted by the local dust density. Although the observed
polarization is the integrated polarization along the whole line of sight to
the star, by observing many stars and taking the distance to each of the stars
into account, the GMF can be studied in 3 dimensions.

Optical starlight polarization has been used to study magnetic fields of
distinct objects and dense regions of the sky, see for example Crutcher
(2012); Medhi et al. (2008); Pereyra & Magalhães (2007). The largest currently
available optical polarization catalog, containing 9286 stars, is an
agglomeration of many smaller catalogs (Heiles, 2000). The program111
(Magalhães et al., 2005, Magalhães et al. in prep.) is an effort to increase
the number and density of these observations.

Section 2 outlines the survey and describes the acquisition and processing of
the data. We cross-correlate our data with auxiliary catalogs in Section 3.
Section 4 describes the verification of the data, including a comparison to
the Heiles (2000) catalog. Section 5 explores various parameter distributions
(spatial, photometric and polarimetric) of the stars. Section 6 discusses
correlations between stellar parameters. Finally, we discuss our findings in
Section 7, which are summarized in Section 8.

##  2 Observations

###  2.1 Data acquisition

The optical polarization data were obtained as a part of the Interstellar
Polarization Survey (IPS) program (see Magalhães et al., 2005, Magalhães et
al. in prep.), based on observations made at the Observatório do Pico dos
Dias/LNA (Brazil). The polarimeter, a modified CCD camera, consists of a
Savart prism and a rotating half-wave waveplate (see Magalhães et al. (1996)
for more details). A remarkable feature of the polarimeter is the simultaneous
imaging of both the ordinary and the extraordinary beams, which allows for
photon-noise limited observations even under nonphotometric conditions, as
well as cancellation of any sky polarization. The polarimeter was mounted onto
at Observatorio Pico dos Dias (OPD), operated by Laboratório Nacional de
Astrofísica (LNA) in Brazil. IPS covered different regions of the sky aiming
at various distinct science goals such as open clusters or dense clouds.

For this paper, we looked at the 38 “General ISM” fields of view (IPS-GI)
observed in V band, for which the observations were carried out between April
2000 and August 2003. The Galactic coordinates and number of objects (after
applying quality filters, see Section 4.2) of these fields can be found in
Table 1. The fields are approximately 0.3 x 0.3 degrees in size and mostly
located close to the Galactic plane ($|b|<10\arcdeg$). ach field contains, on
average, about 1000 stars. The fields are primarily centered on stars in the
Heiles (2000) catalog. Each field was selected carefully to avoid dense
structures and clouds, thus focusing on the diffuse ISM Some fields were
observed more than once using different exposure times, typically ranging from
10 to 30 seconds per waveplate position. The different exposure times ensure a
good covering of a wide magnitude range, achieving high signal-to-noise ratio
measurements of $P/\sigma_{P}>5$ for some faint stars of about $V_{mag}=18$,
although due to the different exposure times we cannot claim completeness.

###  2.2 Data processing

The data was reduced using the SOLVEPOL pipeline, developed specifically for
imaging polarimetry data of this type (Ramírez et al., 2017). SOLVEPOL uses
the astrometry.net software (Lang et al., 2010) to calibrate the astrometry
and the Guide Star Catalog version 2.3 (Lasker et al., 2008) for the magnitude
calibration. Images were corrected for bias and flat-field whenever possible
and if no bias frames and flat-fields were available, the raw images were used
in the processing. This did not lead to significantly lower quality results.
SOLVEPOL produces tables with polarimetric (degree of linear polarization P,
polarization angle $\theta$) and photometric (V-band magnitudes) measurements
and their associated uncertainties for each detected star. SOLVEPOL uses the
following equations to calculate the degree of polarization and the
polarization angle (see Ramírez et al. (2017) for more details):

$P=\sqrt{Q^{2}+U^{2}},$ (1)

$\theta=\frac{1}{2}\arctan\frac{U}{Q},$ (2)

in which Q and U are the normalized Stokes’ parameters and P refers to the
degree of linear polarization as a fraction of the total intensity. We follow
the conventions from Ramírez et al. (2017) and refer to that paper for more
details. Degrees of polarization will be presented as a percentage throughout
.

The output tables were used for further processing. Entries from stars that
were observed more than once were merged. The final value is an average of all
observations, weighted with the inverse of the squared uncertainties in the
Stokes parameters. Observations of standard stars of known polarization were
used to determine the correction of the polarization angle to the equatorial
coordinate system. The weights were set equal to the inverse of the squares of
the uncertainties in Q and U. The Stokes parameters were then used to find the
polarization, following the equations above. This led to an average
instrumental polarization of 0.07%. This is smaller than the observational
uncertainties, as the average observational error in the degree of
polarization across the entire data set is approximately , and thus we did not
correct for instrumental polarization.

In this paper, the averages of the degree of linear polarization P and
polarization angle $\theta$ are calculated following the conventions from
Pereyra & Magalhães (2007). The polarimetric averages are weighted by the
inverse of the squared uncertainty in the Stokes parameters Q and U.

###  2.3 Debiased Degree of Polarization

Measurements of the degree of linear polarization are biased at low
${P}/\sigma_{P}$ (Clarke & Stewart, 1986). The polarization values in this
paper have not been debiased. We instead filtered the data to only include
sources with a ${P}/\sigma_{P}>5$, thus avoiding the need to debias data.
However, for verification, we compared our biased degrees of polarization to
values that have been debiased using the estimator from Plaszczynski et al.
(2014):

$P_{d}=P-\sigma_{P}^{2}\frac{1-e^{\frac{-P^{2}}{\sigma_{P}^{2}}}}{2P},$ (3)

where $P_{d}$ is the debiased polarization. Correcting the polarization
degrees using this estimator did not lead to significant differences in the
values. Differences between the biased and debiased polarization values are of
the order $10^{-3}\%$, well below the typical uncertainties. This is
consistent with the findings of Plaszczynski et al. (2014), who find that
their estimator becomes indistinguishable from other estimators beyond
${P}/\sigma_{P}>3$. We therefore conclude the ${P}/\sigma_{P}$ filter to be an
effective limiting criterion in avoiding statistical difficulty. Applied to
the full, unfiltered dataset, the difference between the biased and debiased
values is larger (). This shows the need for debiasing if the data were used
without stringent ${P}/\sigma_{P}$ filters. We note that the applied
${P}/\sigma_{P}$ filter may inadvertently remove lower polarization stars.

##  3 Auxiliary catalogs

###  3.1 Distances

Our analysis required each star to have an estimate of its distance and
interstellar extinction in addition to its interstellar polarization.
Therefore, we cross-correlated our initial polarimetric sample with other
catalogs as described below. Each observed field was assigned a unique
identifier and each star in the catalog was matched to a GAIA using Topcat
(Taylor, 2005). Matches are based on position (a 3” margin, corresponding to
$1.5\sigma_{position}$ for IPS) and magnitude (2 mag margin). This margin in
magnitude is to account for differences in GAIA’s G-band magnitude and IPS’
V-band magnitude. Visual inspection of the positions of the stars in each
catalog showed that there is little chance of mismatches– most stars are close
to only one other potential match. Even if more than one potential match were
nearby, inclusion of the magnitude filter finds the best match. For of the IPS
sources, a GAIA counterpart was found within the given margins. In addition,
we added data from the Anders et al. (2022) StarHorse-based catalog, which
includes not only distances, but also V-band extinctions and many other
stellar parameters. Anders et al. (2022) parameters were matched to each star
using the unique GAIA EDR3 id.

###  3.2 V-band extinction

In addition to the parameters from IPS-GI, , we also used the V-band
extinction, $A_{\rm{V50}}$, to derive values for the reddening E(B-V) and
hydrogen column density NH using the following equations:

$E(B-V)=\frac{A_{V50}}{3.1},$ (4)

$N_{\mathrm{H}}=5.8\times 10^{21}cm^{-2}E(B-V),$ (5)

##  4 Data verification

###  4.1 Comparison to Heiles’ catalog

Although large databases of starlight polarization in the diffuse medium are
rare, we compared our findings to other available measurements to verify their
accuracy. To this end, we compared the IPS-GI with those from the Heiles
(2000) catalog. We used Topcat to match sources common to both catalogs. We
found matches within 1.5 arcsec for stars. The comparisons between the
polarization degrees and polarization angles are shown in Fig. 1.

Figure 1: Comparisons between the polarimetric measurements of Heiles (2000)
and IPS-GI. Red circles indicate variable stars. Left: degree of linear
polarization. Right: angle of linear polarization. Dashed lines show the x=y
diagonal. Error bars indicate the measurement uncertainties. The measurements
of both data sets show strong agreement.

Figure 1 shows good agreement between Heiles’ catalog and the IPS-GI
measurements. stars are O- and B-type, and are classified variable stars in
the _General Catalogue of Variable Stars_ (GCSV, Version 5.1,). The variable
stars are marked red in Figure 1. Many OB stars are known to exhibit
polarimetric variability, see for example Bjorkman (1994), which could partly
explain the difference in measurements between the IPS and Heiles (2000). We
note an apparent systemic deviation in seven stars, for which we found a
polarization higher than the Heiles measurement. One outlier stands out in the
polarization angle comparison. The polarimetric measurements for this star
from Heiles’ catalog are taken from Hiltner (1956). This star, identified as
BD-14 4922, has also been observed polarimetrically as part of the Cikota et
al. (2016) analysis of archival data of polarized standard stars. This study,
which also includes V-band observations, finds a polarization angle for BD-14
4922 of $\theta=50\pm 0.07\arcdeg$. This is in agreement with our own
polarimetry of this star, where we found a polarization angle of . We
therefore excluded this star from further comparison of the polarization
angles with Heiles (2000). Furthermore, we also excluded all known variable
stars from further comparison, see marked stars in Fig. 1.

We calculated a modified reduced $\chi^{2}$ statistic using the following
equation:

$\chi^{2}_{red}=\frac{1}{N}\sum\frac{X_{dif,i}^{2}}{\sigma_{i}^{2}},$ (6)

where $N$ is the number of observations ( in the case of polarization degree,
for the polarization angle), $X_{dif}$ is the absolute difference between the
observations, either the degree of polarization or the polarization angle, and
$\sigma_{i}$ is the square root of the sum of squared error in both
measurements. This lead to a $\chi^{2}_{red,P}=12.2$ for the degree of
polarization P, and a $\chi^{2}_{red,\theta}=4.9$ for the polarization angle
$\theta$. To visualize the differences between the measurements of the two
catalogs, we created Bland-Altman plots for both parameters. These are
presented in Figure 2. Following the process outlined in Bland & Altman
(1986), we defined a mean value, as well as limits of agreement out to
2$\sigma$ for both parameters. These plots show that for the degree of
polarization P, within the 2$\sigma$ boundaries. For the polarization angle
$\theta$, one star falls outside this limit.

Figure 2: Bland-Altman plots for the differences in measurements between
Heiles and IPS-GI. Variable stars were removed from the comparison. Left:
degree of linear polarization. Right: angle of linear polarization with
BD-144922 removed. Solid lines indicate the mean difference between the two
data sets. Dashed and dotted lines represent the 1$\sigma$ and 2$\sigma$
limits of agreement, respectively. Error bars indicate the errors of both
measurements, added in quadrature.

###  4.2 Applying quality filters

  1. 1.

$p/\sigma_{p}>5$, which ensures only high-quality polarimetric observations
are taken into consideration,

  2. 2.

$dp<0.8\%$, which removes sources with spurious polarimetric errors,

  3. 3.

$fidelity>0.5$, which removes spurious results from GAIA fitting routines, see
Rybizki et al. (2022),

  4. 4.

$sh_{outflag}=0$, which removes spurious results from Anders et al. (2022),

  5. 5.

$C^{*}/\sigma_{C}<5$, which exclude sources with spurious color excess, see
also Riello et al. (2021, eq. 18 therein.)

  6. 6.

Only high-certainty matches, which removes results for which the cross-
matching between the IPS and GAIA EDR3 catalogs was uncertain.

##  5 Statistical properties of the data

###  5.1 Low polarization sources

The filtered subsample shows a lack of low polarization sources. This appears
to be caused by the ${P}/\sigma_{P}>5$ filter. This introduced a bias towards
stars with higher polarization, inadvertently filtering out low polarization
sources. However, although stringent, this filter ensures that all analysis is
applied only to the highest quality observations. In addition, it removes the
need for debiasing as explained in section 2.3.

###  5.2 High polarization sources

###  5.3 Spatial distributions

Assuming the starlight to be initially unpolarized, all measured polarization
is a direct result of starlight passing through the ISM. As the polarization
is a function of the location where it is produced, which can be defined by
Galactic longitude _l_ , Galactic latitude _b_ and distance _d_ , it is
important to study the spatial distribution of the polarization.

The IPS-GI observations are focused on certain lines of sight in the diffuse
ISM, which leads to a very inhomogeneous source distribution across the sky,
see Fig 4. Figure 3 shows an example of a typical ordered field (C47, cf Table
1) located at $l=320.5\arcdeg$, $b=-1.2\arcdeg$. After applying quality
filters (see Section 4.2), this field contains stars. Each star’s polarization
vector was plotted– its orientation is equal to the Galactic polarization
angle and its length depends on the degree of polarization. The uniformity of
the polarization vectors is typical for this data set and is indicative of a
dominant large-scale magnetic field even over the full range of distances, see
Figs. 7, middle and right.

Figure 3: Example field C47 containing stars, located at l=320.5°, b=-1.2°.
The vectors represent the polarization angle and degree. Red example vector
shows a polarization degree of $5\%$ and $\theta_{gal}=90\arcdeg$. See Fig. 7
for more details. Figure 4: Positions of lines of sight in IPS-GI projected on
the Milky Way. Each marking represents an observed field of stars. Longitude
increases towards the right. Labels indicate the two high Galactic latitude
fields and example field C47. Background image adapted from ESA/Gaia/DPAC.

Figure 4 shows the locations of all observed fields within the Milky Way. In
addition, the distributions for Galactic longitude, Galactic latitude and
distance in Fig. 5 show that the observed fields lie within a longitude range
of $220\arcdeg<l<50\arcdeg$ and that most fields are within 15° latitude from
the Galactic plane. Two sparsely populated fields are positioned further above
and below the Galactic plane, see annotations in Fig. 4. We also indicated the
location of example field C47 (see Fig. 3). Finally, most stars are located
nearby (around $d\sim 2\textrm{ kpc}$), with .

Figure 5: Distributions of spatial parameters for all filtered IPS-GI data.
Left: Galactic longitude _l_ in degrees. Middle: Galactic latitude _b_ in
degrees. .

###  5.4 Photometric and polarimetric distributions

Next, we looked at the distributions of the V-band magnitude and the
polarimetric parameters P and $\theta$. Figure 6, left, shows the distribution
of the magnitudes for the filtered survey data: IPS-GI covers a range of
magnitudes roughly from 8 to 19. The majority of stars have V <15\. We again
note that this survey is not complete in magnitude due to the different
integration times of different fields.

Figures 6, middle and right, show the distributions of the degree of
polarization P and the angle of polarization $\theta$, respectively. The
degree of polarization peaks around 3%, with a notable absence of very low
polarization sources (see Section 5.1). Furthermore, some stars show very high
(P >10%) polarization (see Section 5.2). The distribution of polarization
angles peaks around 90°, which is expected for a line-of-sight-averaged
magnetic field that is parallel to the Galactic plane. This is in agreement
with observations based on radio polarimetry (see for example Haverkorn
(2015)) as well as observations of external spiral galaxies (Beck, 2015). The
distribution also shows a second peak around $\theta=40\arcdeg$ and a somewhat
flattened structure around $\theta=140\arcdeg$. These features will be
discussed in more detail in Section 5.5.

Figure 6: Distributions of photometric and polarimetric parameters for all
filtered IPS-GI data. Left: V-band magnitude. Middle: degree of polarization P
as percentage. Right: angle of polarization $\theta$ in degrees in the
Galactic coordinate system. $\theta_{Gal}=90\arcdeg$ is oriented along the
Galactic plane.

Returning to our example field C47, we took a closer look at the distributions
of the degree of polarization and the polarization angle. Histograms showing
the distributions of those parameters are presented in Figure 7. These figures
quantify the structure seen in Figure 3. The degrees of polarization vary over
a relatively wide range, but the polarization angle is highly peaked around an
average $\theta_{Gal}=96\arcdeg$.

Figure 7: Distributions of polarimetric parameters and distances for example
field C47. Left: degree of polarization _P_ as percentage. Middle: angle of
polarization __in degrees in the Galactic coordinate system.

###  5.5 Averages per field

Table 1 describes the main properties of the IPS-GI filtered subsample. The
weighted averages of the polarimetric parameters P and were calculated
following conventions from Pereyra & Magalhães (2007). From this table, we
found that some fields have an average orientation that deviates greatly from
$\theta_{Gal}\sim 90\arcdeg$, a value expected for an average magnetic field
direction in the plane of the Milky Way. Some of these fields account for
features mentioned above, e.g. the histogram of polarization angles in Figure
6, right. For example, field C4 has an average Galactic polarization angle of
$\theta_{Gal}=45\arcdeg$. With 455 stars, this field contributes significantly
to the shape of Figure 6, right. Furthermore, fields C7, C50 and C57 have
average polarization angles of $\theta_{Gal}=155\arcdeg$,
$\theta_{Gal}=132\arcdeg$ and $\theta_{Gal}=142\arcdeg$, respectively. These
fields explain the apparent flattening of the histogram in Figure 6, right.

Table 1 has also been summarized in Figure 8, which shows the average
polarization vectors per field on a sky plot. The background shows the
Galactic magnetic field as detected by Planck. The optical polarization
vectors are mostly in agreement with an ordered large-scale magnetic field
parallel to the Galactic plane as can be seen in the background. Any turbulent
components along the line of sight at different distances are cancelled out or
diminished. This can only be concluded based on starlight polarization data in
conjunction with distance measurements, as Planck and similar sub-mm surveys
show the result of an integration along an entire line of sight.

Figure 8: Sky plot with vectors of weighted average degree of polarization
(length) and weighted average polarization angle (orientation) per field.
Background image copyright ESA/Planck collaboration, credit Marc-Antoine
Miville-Deschênes. Longitude increases toward the right. Figure 9:

Table 1: Mean values per IPS-GI field

Field ID aaThe letter C refers to the word _Campo_ , which is Portuguese for field. | l | b | P | dP | $\theta_{Gal}$ | d$\theta$ | $\sigma\theta_{Gal}$ | V | $A_{V}$ | E(B-V) | Dist | N stars
---|---|---|---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (%) | (%) | (deg) | (deg) | (deg) | (mag) | (mag) | (mag) | (kpc) |
C0 | 327.57 | -0.83 | 2.44 | 0.15 | 92.08 | 1.79 | 12.81 | 15.22 | 1.79 | 0.58 | 1.49 | 397
C1 | 330.42 | 4.59 | 1.62 | 0.13 | 76.03 | 2.23 | 12.17 | 15.61 | 1.75 | 0.56 | 2.39 | 545
C2 | 359.34 | 13.47 | 3.14 | 0.16 | 107.62 | 1.49 | 9.41 | 16.51 | 1.11 | 0.36 | 3.52 | 594
C3 | 339.49 | -0.42 | 1.72 | 0.11 | 77.36 | 1.88 | 13.53 | 14.98 | 1.94 | 0.63 | 1.59 | 448
C4 | 18.63 | -4.46 | 2.46 | 0.18 | 45.67 | 2.12 | 8.64 | 15.14 | 1.54 | 0.5 | 2.91 | 455
C5 | 25.06 | -0.74 | 1.68 | 0.12 | 78.9 | 1.96 | 14.55 | 15.5 | 2.23 | 0.72 | 1.97 | 421
C6 | 298.61 | 0.64 | 1.66 | 0.14 | 79.44 | 2.45 | 15.32 | 14.25 | 1.43 | 0.46 | 1.85 | 210
C7 | 41.68 | 3.39 | 1.18 | 0.13 | 155.16 | 3.15 | 15.52 | 15.74 | 2.3 | 0.74 | 2.62 | 326
C11 | 307.53 | 1.33 | 1.5 | 0.1 | 76.0 | 1.85 | 14.32 | 14.58 | 1.32 | 0.43 | 1.87 | 372
C12 | 331.04 | -4.7 | 2.6 | 0.13 | 103.04 | 1.41 | 8.29 | 15.28 | 1.33 | 0.43 | 3.0 | 881
C13 | 333.24 | 3.75 | 3.81 | 0.25 | 69.99 | 1.91 | 6.11 | 15.84 | 2.53 | 0.82 | 1.73 | 287
C14 | 312.87 | 5.48 | 2.23 | 0.12 | 104.62 | 1.53 | 11.61 | 15.79 | 1.24 | 0.4 | 3.04 | 653
C15 | 304.71 | -0.17 | 2.16 | 0.11 | 77.46 | 1.46 | 13.67 | 15.8 | 2.48 | 0.8 | 1.91 | 461
C16 | 301.97 | -8.77 | 3.13 | 0.14 | 104.99 | 1.3 | 7.43 | 15.44 | 1.03 | 0.33 | 2.4 | 561
C30 | 222.54 | -1.63 | 0.33 | 0.09 | 84.32 | 7.9 | 35.57 | 13.96 | 1.44 | 0.47 | 1.7 | 37
C34 | 34.81 | -45.02 | 0.27 | 0.16 | 11.12 | 16.78 | 35.26 | 15.38 | 0.17 | 0.05 | 1.15 | 9
C35 | 271.45 | -1.07 | 1.49 | 0.11 | 115.2 | 2.02 | 8.75 | 15.5 | 1.97 | 0.64 | 1.39 | 144
C36 | 343.29 | 11.97 | 1.06 | 0.11 | 104.59 | 2.99 | 16.89 | 15.21 | 0.96 | 0.31 | 2.59 | 201
C37 | 331.16 | 7.62 | 1.55 | 0.11 | 86.19 | 1.99 | 11.46 | 15.52 | 0.85 | 0.27 | 2.84 | 539
C38 | 334.0 | 55.86 | 0.58 | 0.08 | 98.33 | 3.82 | 23.13 | 14.86 | 0.21 | 0.07 | 1.44 | 11
C39 | 303.07 | 1.63 | 2.12 | 0.15 | 83.2 | 1.98 | 11.59 | 15.59 | 1.94 | 0.62 | 1.68 | 309
C40 | 302.16 | -1.21 | 2.03 | 0.17 | 64.45 | 2.42 | 16.38 | 15.69 | 2.28 | 0.74 | 1.99 | 264
C41 | 257.34 | -0.48 | 0.8 | 0.12 | 17.56 | 4.13 | 20.24 | 15.01 | 1.87 | 0.6 | 2.5 | 116
C42 | 245.49 | -0.11 | 0.35 | 0.08 | 76.74 | 6.63 | 30.2 | 14.55 | 0.8 | 0.26 | 2.61 | 88
C43 | 273.17 | -0.82 | 1.58 | 0.17 | 110.77 | 3.1 | 20.49 | 14.05 | 1.43 | 0.46 | 1.71 | 69
C44 | 305.47 | 1.93 | 0.99 | 0.18 | 140.57 | 5.19 |  | 13.93 | 1.08 | 0.35 | 0.5 | 1
C45 | 21.81 | 0.71 | 1.53 | 0.15 | 25.27 | 2.84 | 31.16 | 15.81 | 3.15 | 1.02 | 1.33 | 97
C46 | 44.21 | 2.66 | 1.38 | 0.21 | 79.04 | 4.47 | 20.5 | 13.92 | 2.15 | 0.69 | 1.62 | 36
C47 | 320.49 | -1.23 | 2.82 | 0.13 | 95.19 | 1.29 | 7.43 | 14.63 | 1.98 | 0.64 | 1.76 | 240
C50 | 20.27 | 1.04 | 2.05 | 0.15 | 132.38 | 2.07 | 14.6 | 15.55 | 2.47 | 0.8 | 1.55 | 255
C52 | 318.76 | 2.78 | 1.25 | 0.1 | 97.68 | 2.27 | 10.11 | 13.63 | 1.3 | 0.42 | 1.75 | 177
C53 | 359.17 | 4.97 | 3.56 | 0.32 | 32.58 | 2.59 | 6.4 | 13.42 | 2.3 | 0.74 | 0.91 | 11
C54 | 305.17 | 1.31 | 1.53 | 0.11 | 75.46 | 1.97 | 15.25 | 14.37 | 1.33 | 0.43 | 1.34 | 137
C55 | 15.15 | 1.68 | 1.08 | 0.12 | 126.81 | 3.26 | 19.57 | 15.22 | 2.3 | 0.74 | 1.89 | 278
C56 | 14.97 | -0.96 | 0.55 | 0.11 | 126.17 | 5.84 | 36.34 | 14.43 | 1.41 | 0.46 | 1.29 | 55
C57 | 40.59 | 4.14 | 1.75 | 0.13 | 142.0 | 2.1 | 10.57 | 16.4 | 2.81 | 0.91 | 2.18 | 343
C58 | 351.32 | 0.6 | 1.26 | 0.11 | 54.96 | 2.62 | 22.59 | 15.74 | 2.48 | 0.8 | 1.3 | 193
C61 | 0.48 | 2.19 | 4.08 | 0.12 | 49.25 | 0.84 | 6.7 | 15.82 | 2.4 | 0.77 | 1.44 | 295

Note. — Main properties of the filtered IPS-GI subsample, mean values per
field. The columns are: 1) Field number; 2) Galactic Longitude; 3) Galactic
Latitude; 4) Degree of Polarization (weighted); 5) Error on Degree of
Polarization (weighted); 6) Polarization angle in Galactic coordinates
(weighted); 9) V band magnitude; 10) V band extinction taken from Anders et
al. (2022); 11) Reddening (Using $E(B-V)=A_{V}/3.1$); 12) Distance taken from
Anders et al. (2022); 13) Number of stars in field.

##  6 Correlations between stellar parameters

###  6.1 Polarimetry and spatial parameters

Due to the nature of the polarization, we expect to see correlations between
certain parameters. For example, a decrease of the polarization for higher
Galactic latitudes due to the decrease in integrated dust content along those
lines of sight. To investigate this as well as other correlations, the data
was divided into nearby ($d<4$ kpc) and distant ($d>4$ kpc), as well as high
($|b|>15\arcdeg$) and low ($|b|<15\arcdeg$) latitude. The weighted average
presented in Table 2. The values were calculated following the method of
Pereyra & Magalhães (2007). This first coarse look appeared to reveal two
expected correlations. Firstly, the polarization increases as a function of
distance. We expected to see this, because the polarization of the starlight
is caused by dichroic extinction in the ISM. As the starlight travels through
more ISM dust, the light becomes more polarized if the magnetic field
projection does not change considerably along the line of sight. In the low
latitude subsample of stars, the average polarization increases by percentage
point when comparing nearby and distant stars.

Table 2: Polarimetric averages at different latitudes and distances.

Lat. | Dist. | N Stars (%) | P | $\theta_{Gal}$
---|---|---|---|---
|  |  | (%) | (deg)
| Total | 20 (100) | $0.44\pm 0.10$ | $98.04\pm 6.22$
High | Nearby | 19 (95) | $0.44\pm 0.09$ | $97.46\pm 6.42$
| Distant | 1 (5) | $4.87\pm 0.47$ | $129.18\pm 2.76$
| Total | 10496 (100) | $1.39\pm 0.13$ | $89.94\pm 1.73$
Low | Nearby | 9272 (88) | $1.37\pm 0.12$ | $89.19\pm 2.53$
| Distant | 1224 (12) | $2.07\pm 0.21$ | $101.62\pm 2.91$

Note. — Weighted average degree of polarization and angle of polarization and
their associated errors for several bins. High and low latitude stars are
located at $|b|>15\arcdeg$ and $|b|<15\arcdeg$, respectively. Distant and
nearby stars have $d>4$ kpc and $d<4$ kpc, respectively.

Secondly, high latitude stars show less polarization compared to low latitude
stars. The starlight from low latitude stars passes through more dust than
their high latitude counterparts, thus leading to an increase in average
polarization. Comparing low and high latitude nearby stars, the degree of
polarization increases by percentage point. However, it remains important to
acknowledge the low number of stars in the high latitude subsample.

We next present the distributions of Galactic latitude and longitude in
conjunction with polarimetric parameters P and $\theta$ on a field-by-field
basis, revealing a rich structure, as can be seen in Figures 10 and 11.
Firstly, looking at latitude confirmed the findings from Table 2. As can be
seen in Figure 10, left, the highest weighted average degrees of polarization
are found in fields located near the center of the Galactic plane at b = 0°.
As for the polarization angle in Figure 10, right, we note that the fields
around the b = 0°show a wide range of polarization angles centered around
$\theta_{Gal}\approx 90\arcdeg$, whereas the two bins

Figure 10: Average polarimetric measurements per field . Marker sizes scale
with the number of stars in that field. Left: weighted average degrees of
polarization _P_ in percentage per field. Error bars represent the weighted
error. Right: average polarization angles $\theta$ in degrees per field. Gray
error bars denote the standard deviation of the distribution within each
field. The black error bars represent the average measurement error within
each field.

We also investigated average values by longitude. Figure 11, left shows the
weighted average degrees of polarization per field. The polarization appears
to be highest in the direction of the Galactic center and its immediate
surroundings. This is caused by the higher densities of dust associated with
that region (see for example Rezaei Kh. et al., 2017; Misiriotis et al.,
2006). Plotting the average polarization angle per field as in Figure 11,
right, reveals an average polarization angle $\theta_{Gal}\sim 90\arcdeg$
throughout the longitude range, with significant outliers above and below that
value, see also the vectors in Fig. 8. As described in Section 5.5 and Table
1, these fields may be indicative of variations of the magnetic field across
the Galactic plane or intervening structures such as spiral arms throughout
the line of sight.

Figure 11: Average polarimetric measurements per field . Marker sizes scale
with the number of stars in that field. Left: weighted average degrees of
polarization _P_ in percentage per field. Error bars represent the weighted
error. Right: average polarization angles $\theta$ in degrees per field. Gray
error bars denote the standard deviation of the distribution within each
field. The black error bars represent the average measurement error within
each field.

Figure 12: Left: V-band extinction $A_{V}$ versus distance for low-latitude
($|b|<5\arcdeg$) and high-latitude ($|b|>5\arcdeg$) stars. Right: degree of
polarization versus distance for low-latitude ($|b|<5\arcdeg$) and high-
latitude ($|b|>5\arcdeg$) stars. Distances taken from Anders et al. (2022)

. Firstly, we binned the data in bins of 400 pc. Figure 13 shows the relevant
distributions. Figure 13, right, shows a stable average position angle
$\theta_{Gal}\sim 90\arcdeg$ out to $d\sim 6\mathrm{kpc}$. This is indicative
of a dominant ordered magnetic field. . Another possibility is a lack of dust
beyond a certain distance. As light passes through magnetic field-aligned
dust, we expect the degree of polarization to increase. As we go beyond the
dusty thick Galactic disc, we expect the dust content to decrease
dramatically, indicated by a flattening of the increase of polarization.
However, the precise profile will depend on the line of sight.

Figure 13: Average polarimetric measurements per 400 pc distance bin. Marker
sizes scale with the number of stars in that bin. Right: weighted average
polarization angles $\theta$ in degrees per distance bin.

To further investigate the correlation between distance and degree of
polarization, we applied a polynomial fit to the data presented in Figure 13,
left. The overall trend appears to agree well with a third-degree polynomial.
A fit to the data finds the following coefficients

$P=0.087+1.526d-0.396d^{2}+0.033d^{3}$ (7)

where P is the degree of polarization in percent and d is the distance in
kiloparsec. The bottom of Figure 13, left show the residuals for each data
point. The third degree polynomial trend is in agreement with findings from
Fosalba et al. (2002) based on Heiles (2000) data, see also the dotted line in
Fig. 13, left, although the coefficients found therein differ slightly from
those in Eq. 7. (Fosalba et al. (2002) find
$P=0.13+1.81d-0.47d^{2}+0.036d^{3}$, see Eq. 1 therein). We note that there is
no physical basis for applying a third-degree polynomial and physical
information cannot be derived from the coefficients of the fit. Furthermore,
we emphasize that the distribution across the sky of stars and the depth of
the survey differ greatly between Fosalba et al. (2002) and the IPS-GI.
Nonetheless, it is remarkable that we independently found an overall trend
that is very similar to Fosalba et al. (2002).

###  6.2 Polarimetry and photometric parameters

Next, we investigated the correlations between degree of polarization and
various photometric parameters. Figure 14, left, shows the degree of
polarization as a function of the V-band magnitude for the entire sample of
stars. Although the density of stars increases greatly towards higher
magnitude, peaking at V $\sim$ 16, two trends are apparent. Firstly, we note
the increase of the error in the degree of polarization towards higher
magnitude. Fainter stars appear to have a larger error in the degree of
polarization, which is expected. Despite employing longer observing times to
include higher magnitude stars in the sample, a higher error cannot fully be
eliminated. Secondly, the degree of polarization itself appears to increase
with higher magnitude. A similar trend is visible in the Heiles (2000) data,
although it must be noted that the highest magnitude therein is V $\sim$ 13\.
It is important to note that in this case, the lack of lower polarization,
higher magnitude sources is partially caused by the filtering, especially the
${P}/\sigma_{P}$ filter that excludes low polarization stars. As the error in
polarization is relatively higher for those sources, they are inadvertently
filtered out. Finally, stars observed under less than ideal weather conditions
will tend to have larger errors due to, for example, a lower photon count.
They would contribute to the spread in accuracy beyond what could be expected
from the (unattenuated) magnitudes.

Figure 14, right shows the average polarization per V-band extinction (AV,
taken from ) bin. The data was binned to extinction bins of 0.21 mag wide. As
the extinction is a reliable tracer of the dust contents along a line of
sight, the apparent increase of polarization as a function of that extinction
is expected. A detailed analysis of the polarization behavior with extinction
will be presented in a forthcoming paper (Angarita et al., in prep).

Figure 14: Left: degrees of polarization _P_ in percentage versus V-band
magnitude. The error bars represent the measurement error. Right: weighted
average degrees of polarization _P_ in degrees per V-band extinction bin.
Extinctions taken from Anders et al. (2022). The error bars represent the
average measurement error within each bin. Marker sizes scale with the number
of stars in that bin.

##  7 Discussion

We presented a first look at the IPS-GI data, which contains data of linearly
polarized stars in 38 fields distributed over the Southern Sky, mainly close
to the Galactic plane that sample the diffuse ISM. analyzed a high-quality
subsample of 10516 stars. The degree and position angle of the linear
interstellar polarization depend on the quantity and alignment properties of
the dust along the line of sight. The polarization angle denotes the line-of-
sight averaged orientation of the magnetic field component in the plane of the
sky. As the polarization is not a scalar, the effect of more than one dust
cloud or other intervening structure is not necessarily additive in the
polarization degree.

The observed distribution of degree of polarization centers around 2% (Fig 6,
middle) and the average polarization degree is higher than in the Heiles
(2000) catalog (Fosalba et al., 2002), which is likely because the IPS-GI
fields are more concentrated towards than the Heiles compilation. In addition,
the IPS-GI contains more distant stars, which may show a more strongly
polarized signal.

The observed distribution of polarization angle is surprisingly uniform over
each 0.3x0.3deg field, with a clear preferred angle for most of the fields (as
for example in Fig 3). This is a clear indication that the turbulent component
of the magnetic field, thought to have maximum correlation scales up to a few
$\sim 100$ pc (Beck et al., 2016) is averaged out along the (kpc) lines of
sight towards the IPS-GI stars .

Furthermore, in most fields the field-averaged polarization angle has the same
orientation as the magnetic field orientation averaged over the whole line of
sight through the Milky Way as inferred by Planck polarized dust data (). As
the IPS-GI stars only probe part of this line of sight, the similarity of
angle orientation between IPS-GI and Planck points to a remarkable uniformity
of the large-scale magnetic field direction along these lines of sight.
However, there are exceptions to this uniformity in polarization angles, as
will be discussed in a forthcoming paper.

Figure 15: Left: polarization angle dispersion $\sigma\theta_{Gal}$ in degrees
versus Galactic longitude. Right: polarization angle dispersion
$\sigma\theta_{Gal}$ in degrees versus Galactic latitude. Marker sizes scale
with the number of stars in that bin.

analysis is based on the assumption that the starlight we observed is
intrinsically unpolarized. While this may be true for most stars, we cannot
eliminate all intrinsically polarized sources through the applied filters.
When studying individual stars in this catalog, it is therefore important to
keep this in mind. However, considering the uniformity of parameters and their
agreement with the assumption that our stars are intrinsically unpolarized, we
do not expect intrinsically polarized stars to significantly influence the
overall statistics of the catalog. We do note that stars with deviating
polarization angle (see for example in Figure 7, right) are candidates for
further a more detailed analysis of their polarimetric properties.

In addition, it is of the utmost importance to keep in mind the highly
fragmented spatial distribution of sources in the plane of the sky. While it
is tempting to use the data presented in this paper to draw generalizing
conclusions about the ISM and GMF, this must be done with caution. Due to
their small size, 0.3x0.3$\arcdeg$, the fields may not be representative of
the general structure and conditions of the surrounding ISM. Any outliers or
stand-out features presented in the above sections may be falsely interpreted
as characteristics of the whole ISM or the GMF, while it may be one (dominant)
field that allows the average values to deviate from a general trend.
Therefore, it remains important to compare the data presented here to other
starlight polarization sources such as Heiles (2000) as well as complementary
data at other wavelengths (e.g. Planck dust polarization (Abergel et al.,
2014) as in or GPIPS infrared data (Clemens et al., 2020)), but this is beyond
the scope of this paper.

##  8 Conclusions

We presented a first look at the Interstellar Polarization Survey - General
ISM (IPS-GI) catalog. This new catalog contains polarimetric measurements . We
applied quality filters, presenting an analysis limited to 10516 of the
highest quality observations. We presented various distributions of
polarimetric and photometric parameters, as well as correlations between those
parameters. As expected, the measured degree of polarization correlates with
sky position, as well as distance and V-band extinction as expected. We found
evidence of a dominant ordered magnetic field, oriented parallel to the
Galactic plane, indicated by a stable Galactic polarization angle $\theta\sim
90\arcdeg$ across different Galactic latitudes, longitudes and distances . The
measured degree of polarization varies across the sky, although we can
associate higher dust contents with a more strongly polarized signal, as
expected based on the nature of the polarization. The catalog should enable
further research into the structure of the ISM, including a new, more detailed
view of the GMF.

MJFV acknowledges Wouter Veltkamp for his contributions to Figures 3 and 8.
MJFV, MH and YAA acknowledge funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 772663). AMM’s work and Optical/NIR Polarimetry at IAG has
been supported over the years by several grants from São Paulo state funding
agency FAPESP, especially 01/12589-1 and 10/19694-4. AMM has also been
partially supported by Brazilian agency CNPq (grant 310506/2015-8). AMM
graduate students have been provided grants over the years from Brazilian
agency CAPES. CVR acknowledges Conselho Nacional de Desenvolvimento Científico
e Tecnológico (CNPq), Proc. 303444/2018-5. This work has made use of data from
the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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