# Monte Carlo simulations in anomalous radiative transfer: a tutorial

Tiziano Binzoni  Department of Basic Neurosciences, University of Geneva,
Geneva, Switzerland  Department of Radiology and Medical Informatics,
University Hospital, Geneva, Switzerland  Corresponding author:
<EMAIL_ADDRESS>Fabrizio Martelli  Dipartimento di Fisica e
Astronomia dell’Università degli Studi di Firenze, Sesto Fiorentino, Firenze,
Italy

###### Abstract

Anomalous radiative transfer (ART) theory is a generalization of classical
radiative transfer theory. The present tutorial wants to show how Monte Carlo
(MC) codes describing photons transport in anomalous media can be implemented.
It is shown that the heart of the method consists in suitably describing, in a
“non-classical” manner, photons steps starting from fixed light sources or
from boundaries separating regions of the medium with different optical
properties. To give a better intuition of the importance of these particular
photon step lengths, it is also shown numerically that the described approach
is essential to preserve the invariance property for light propagation. An
interesting byproduct of the MC method for ART, is that it allow us to
simplify the structure of “classical” MC codes, utilized e.g. in biomedical
optics.

##  1 Introduction

Monte Carlo (MC) simulations represent the tool of choice when describing
“classical” photons migration in propagating media [1]. This is specially true
when exact solutions (except for statistical noise) of complex media are
needed. In fact, when dealing with the simultaneous presence of different
media types, complex geometries and varying optical parameters, it becomes
extremely difficult to use alternative approaches such us analytical solutions
or finite differences based methods. Moreover, the continuous increase of the
computational power, even for simple desktops, renders in general more and
more attractive and easy to use the MC methods. These are some of the reasons
why data generated by MC simulations have become nowadays a “gold standard”,
allowing to test alternative methods.

In recent years, a generalization of the classical theory allowing also to
describe anomalous radiative transfer (ART), has been introduced [2, 3]. ART
theory permits to modelize photons migration in anomalous media, i.e., media
where the classical Beer–Lambert–Bouguer’s law is not valid.

The release of the Beer–Lambert–Bouguer’s law hardly complexify the definition
of the analytical models describing ART and their related solutions. This
increase in complexity may be generated, e.g., by the need of the introduction
of models based on fractional (integro)-differential equations, where the
solutions are often difficult to obtain and manipulate. Moreover, when the
investigated optical medium is not infinite, and thus necessitates the
introduction of boundary conditions, the definition of the analytical models
and the obtainment of their solutions may represent a real challenge.

This is why in communities such as the biomedical optics, where the main aim
is not specifically to understand the mathematical theory but above all to
obtain solutions related to complex practical problems in photon migration,
more efficient and general purpose tools would be particularly appreciated. It
is precisely in the context of ART that the MC methods show their extreme
power, i.e., thanks to their relative simplicity in describing and solving the
considered problems.

In general, the MC methods describe how photons propagate step-by-step in the
media, through scattering, absorption, reflection and refraction events. The
main difference between the classical and the ART MC-approaches is given by
the specific choice of the function describing the probability $p(s)ds$ for a
photon to reach a distance $s\in[s-ds,s+ds]$ in the medium, without
interactions; where $p(s)$ is a probability density function (pdf).

As it is well known, in the classical case

$p(s)=\frac{1}{\ell}e^{-s/\ell},$ (1)

where $\ell$ is the mean free path for the investigated medium (in biomedical
optics $\ell$ is often reported as $1/\ell=\mu_{t}$; where $\mu_{t}$ is the
extinction coefficient). Historically, Eq. (1) has been determined by
considering the fact that a classical medium must satisfy
Beer–Lambert–Bouguer’s law.

In ART, it is not mandatory for $p(s)$ to be an exponential function and a
panoply of choices, depending on the specific optical properties of the
medium, are possible.

At this point, the fundamental remark is that to simulate ART models, by means
of the MC method, it is not sufficient to substitute in the MC code the
classical $p(s)$ [Eq. (1)] with the relative anomalous one. In fact, if we
proceed in such a simple way the fundamental reciprocity law (RL) at the basis
of optics will not be correctly reproduced by the simulations. As a short
reminder, in the present context the RL tells us that if we exchange light
sources and detectors (by inverting the direction of the detected photons; see
e.g. red line in Fig. 1), we must always measure (detect) the same photons
flux. For a more indepth approach to the RL see e.g. Ref. [4].

Thus, based on the findings derived from ART theory, the aim of the present
tutorial is to describe the suitable way to treat this problem. This will be
done by directly giving the necessary practical “recipes” allowing to build MC
codes for ART. We will see that the core of the solution is given by the
particular choice of the pdf allowing to generate the length $s$ of the first
step of photons that are starting from a medium boundary (defined by adjacent
regions with different optical parameters) or from a fixed light source. The
remaining steps, and the photons interactions with absorbers/scatterers, being
treated as in a classical MC code.

In this tutorial, it is considered that the reader already has a basic
knowledge on MC simulations applied to classical problems in photons
migration, typical e.g. for biomedical optics. Very simplified numerical
examples will be given, allowing to intuitively understand the influence of a
correct/incorrect approach.

To simplify the explanations, even if ART theory clearly include the classical
case, in the following sections we will sometimes adopt the convention that
the expression ART does not include the classical case. The context easily
clarifies this use.

##  2 Basic MC tools in ART

###  2.1 Basic probabilty density functions for photons steps

ART theory is a generalization of the classical radiative transfer theory. We
will recall here how in ART photons steps are generated and, in particular,
how the interactions with the boundaries and light sources are treated [5, 6].

In ART, photons steps, $s_{c}$, occurring inside the medium are generated by
giving a pdf

$p_{c}(s_{c};a,b,\dots),$ (2)

defining the probability $p_{c}(s_{c};a,b,\dots)ds_{c}$ to reach a distance
$s_{c}\in[s_{c}-\frac{ds_{c}}{2},s_{c}+\frac{ds_{c}}{2}]$ in the investigated
anomalous medium. Thus, the generalization of the classical radiative tranfer
theory consists in the fact that $p_{c}(s_{c};a,b,\dots)$ must not necessarily
have an exponential behavior [Eq. (1)]. In other words, ART can treat photons
propagation in media that are not memory-less. The parameters $a,b,\dots$
represent the constants defining a specific pdf, depending on the optical
characteristics of the medium. The index $c$ is to remember that the positions
of the sites where the scattering or absorbing events occur in the medium are
statistically correlated. In fact, these positions can statistically be
described by the use of a probability distribution function
$v_{k}(s_{c};a,b,\dots)$ which, in principle, can directly be derived from
$p_{c}(s_{c};a,b,\dots)$ (see Sec. 2.2.2). The specific choice of the model
for $p_{c}(s_{c};a,b,\dots)$ comes from the actual optical properties of the
physical system (medium) we want to describe. Thus, $p_{c}(s_{c};a,b,\dots)$
can in principle be assessed theoretically or experimentally.

Until this point, the ART approach appears to be similar to the classical,
i.e., once the suitable $p_{c}(s_{c};a,b,\dots)$ is defined, we can generate
the photons step lengths, and decide if the photons are scattered, absorbed,
reflected or refracted at the different interaction sites. However, the
fundamental difference clearly appears when a photon hits a boundary. In this
case, if the boundary is encountered when traveling along the step $s_{c}$,
then in ART the photon simply stops at the boundary. From this reached
position a new step is immediately generated, without the need to terminate
the previous path by applying the well known procedure usually utilized in the
classical case (see e.g. Ref. [7]).

When we generate a new step $s_{u}$ starting from a fixed boundary or a fixed
light source position (i.e. these positions do obviously not result from the
probability law $v_{k}(s_{c};a,b,\dots)$), the pdf can no more be $p_{c}(.)$.
In this case, a different pdf must be considered, i.e. [5],

$p_{u}(s_{u};a,b,\dots)=\frac{1-\int_{0}^{s_{u}}p_{c}(s_{c};a,b,\dots)ds_{c}}{\int_{0}^{+\infty}s_{c}p_{c}(s_{c};a,b,\dots)ds_{c}},$
(3)

where the index $u$ is to remember that the photon starts from an uncorrelated
origin (i.e. the position is fixed). Thus, $p_{u}(s_{u};a,b,\dots)ds_{c}$
represents the probability to reach a distance
$s_{u}\in[s_{u}-\frac{ds_{u}}{2},s_{u}+\frac{ds_{u}}{2}]$, if starting from a
fixed position (boundary or light source). All remaining steps are generated
with the law $p_{c}(s_{c};a,b,\dots)$.

As we mentioned in Sec. 1, Eq. (3) derives from the fact that, similarly to
the classical case, ART must also satisfy the fundamental RL. In other words,
if we do not take into account Eq. (3), e.g. by simply putting
$p_{u}(.)=p_{s}(.)$, the RL is no more satisfied (except for the classical
case, see below) and photons propagation in the medium will be described in a
wrong manner. Note that, Eq. (3) permits the RL to be satisfied for any choice
of optical parameters (scattering coefficients, phase functions, refractive
indexes, etc.) or geometries.

The random steps, $s_{c}$ and $s_{u}$, based on laws $p_{c}(s_{c};a,b,\dots)$
and $p_{u}(s_{u};a,b,\dots)$, can numerically be generated as usual by
analytically solving

$\xi=\int_{0}^{s_{i}}p_{i}(s_{i}^{\prime};a,b,\dots)ds_{i}^{\prime};\quad
i\in\\{c,u\\},$ (4)

as a function of $s_{i}$, where $\xi\in(0,1)$ is a uniformly distributed
random variable. If an analytical solution of Eq. (4) does not exist, a
numerical solution can be found, and a look-up table relating $\xi$ to $s_{i}$
can be derived.

This is all we need to know to generate an ART MC simulation, the remaining
part of the MC code being the same as in the classical case.

###  2.2 Extracting general medium properties from $p_{c}(s_{c};a,b,\dots)$

Once $p_{c}(s_{c};a,b,\dots)$, the geometry, the optical parameters and the
phase functions are given, it becomes possible to perform MC simulations in
the ART domain. However, before to run a specific simulation, it is maybe
interesting to know that from $p_{c}(s_{c};a,b,\dots)$ we can derive some very
general properties of the investigated anomalous medium. Strictly speaking,
the properties presented here hold for an infinite medium with pdf
$p_{c}(s_{c};a,b,\dots)$, but are of fundamental importance in understanding
the physics described by $p_{c}(s_{c};a,b,\dots)$. Thus, for the sake of
completeness, in the following paragraphs we will report four functions,
directly derived from $p_{c}(s_{c};a,b,\dots)$, describing some of these
properties. The reported equations are derived from well known results in
renewal theory and the interested reader can refer e.g. to Refs. [8, 9]. The
equations hold for a photon propagation through any homogeneous part of the
medium.

1) The first interesting function is the survival probability
$P_{c}(s_{c};a,b,\dots)$, i.e., in the language of optics, the probability
that a photon survive after a path of length $s_{c}$. This probability is
expressed as

$P_{c}(s_{c};a,b,\dots)=\int_{s_{c}}^{+\infty}p_{c}(s_{c}^{\prime};a,b,\dots)ds_{c}^{\prime}.$
(5)

In the classical case, $P_{c}(s_{c};a,b,\dots)$ corresponds to the
Beer–Lambert–Bouguer’s law (within a multiplicative factor).

2) Now, let be

$\tilde{p}_{c}(w_{c};a,b,\dots)=\mathcal{L}\\{p_{c}(s_{c};a,b,\dots);w_{c}\\},$
(6)

where $\mathcal{L}\\{.;.\\}$ represents the Laplace transform and $w_{c}$ the
obtained variable in the transformed domain. Then, we can express the average
number of photon interactions along a path of length $s_{c}$ as

$\langle
m(s_{c};a,b,\dots)\rangle=\mathcal{L}^{-1}\left\\{\frac{\tilde{p}_{c}(w_{c};a,b,\dots)}{w_{c}[1-\tilde{p}_{c}(w_{c};a,b,\dots)]};s_{c}\right\\};$
(7)

3) The probability that $k$ photon interactions occur along a path of length
$s_{c}$ is

$v_{k}(s_{c};a,b,\dots)=\mathcal{L}^{-1}\left\\{\frac{1-\tilde{p}_{c}(w_{c};a,b,\dots)}{w_{c}}\right.\\\
\left.\times[\tilde{p}_{c}(w_{c};a,b,\dots)]^{k};s_{c}\right\\};$ (8)

4) The probability that the sum of the first $k$ photon steps does not exceed
$s_{c}$ is

$F_{k}(s_{c};a,b,\dots)=\mathcal{L}^{-1}\left\\{\frac{[\tilde{p}_{c}(w_{c};a,b,\dots)]^{k}}{w_{c}};s_{c}\right\\}.$
(9)

The above equations allow us to better describe the physical system under
study, determined by a given choice of $p_{c}(s_{c};a,b,\dots)$.

##  3 ART MC simulations: explanatory examples

The scope of the following examples is to show the importance of Eq. (3) (the
core of the ART MC simulations). This will be done by demonstrating, with easy
to understand numerical examples, that if we neglect Eq. (3), a fundamental
physical property of the system cannot be reproduced, i.e., the invariance
property (IP) will not be satisfied (see below). Note that, the IP represents
a very powerful test allowing us to check the reliability of any MC code [10].

###  3.1 Two general analytical models

We will define here two possible models for $p_{c}(s_{c};a,b,\dots)$, i.e.,
the power law and the constant step models.

####  3.1.1 Power law

The first model we will consider is called a “power law” and is expressed as

$p_{c}(s_{c};\ell,a)=\frac{a(a+1)\ell(a\ell)^{a}}{(a\ell+s_{c})^{a+2}};\quad
a>0,$ (10)

where $\int_{0}^{\infty}p_{c}(s_{c};\ell,a)ds_{c}=1$ and
$\int_{0}^{\infty}s_{c}p_{c}(s_{c};\ell,a)ds_{c}=\ell$. The asymptotic limit
($s_{c}\rightarrow+\infty$) of Eq. (10) is

$p_{c}(s_{c};\ell,a)\sim\frac{a(a+1)\ell(a\ell)^{a}}{s_{c}^{a+2}};\quad
s_{c}\gg 1.$ (11)

The specific $s_{c}$ dependence of Eq. (11) is particularly interesting
because it implies that it may exist an asymptotic diffusion limit of this
anomalous model [11] (related to a “classical” or a “fractional” diffusion
equation, depending on the choice of $a$; see below). Thus, this simple model
[Eq. (10)] represents a kind unified summary for the classical and anomalous
approach.

Then, the pdf $p_{u}(s_{u},\ell,a)$ can be derived by applying Eq. (3), to
obtain

$p_{u}(s_{u},\ell,a)=\frac{1}{\ell}\left(\frac{a\ell}{a\ell+s_{u}}\right)^{a+1}.$
(12)

Finally, the algorithms allowing to generate random $s_{c}$ and $s_{u}$ values
in the MC codes can be obtained by introducing Eqs. (10) and (12) into Eq.
(4), and by solving as a function of $s_{c}$ and $s_{u}$, respectively. This
gives

$s_{c}=a\ell\left[(1-\xi)^{-\frac{1}{a+1}}-1\right],$ (13)

and

$s_{u}=a\ell\left[(1-\xi)^{-\frac{1}{a}}-1\right].$ (14)

####  3.1.2 Constant step

The second model we will consider in this tutorial is called “constant step”,
because it describes photons that “jump” always with steps of the same length
$\ell$ , i.e.,

$p_{c}(s_{c};\ell)=\delta(s_{c}-\ell),$ (15)

where $\int_{0}^{\infty}p_{c}(s_{c};\ell)ds_{c}=1$ and
$\int_{0}^{\infty}s_{c}p_{c}(s_{c};\ell)ds_{c}=\ell$.

Equation (3) is then written as

$p_{u}(s_{u};\ell)=\frac{1-\Theta(s_{u}-\ell)}{\ell},$ (16)

where $\Theta(.)$ is the Heaviside function.

The generating functions for $s_{c}$ and $s_{u}$ are then expressed as [Eq.
(4)]

$s_{c}=\ell;\quad\forall\xi,$ (17) $s_{u}=\ell\xi.$ (18)

###  3.2 Numerical MC examples: IP test

Let’s now define four different examples of MC simulations based on Eqs. (10)
and (15). In the following sub-sections we will give the geometry of the
problem, all the necessary optical quantities, together with the light sources
and detectors positions. The general properties (geometry independent) of the
four chosen tutorial examples are treated, and a short summary concerning the
IP is also reported.

####  3.2.1 Monte Carlo simulations

Geometry:  The common geometrical model chosen in this tutorial consists of
two spheres centered at the axis origin, with radious $r_{in}<r_{out}$ and
matched refractive indexes at all the boundaries (Fig. 1).

Figure 1: Schematic of the geometrical model utilized for the Monte Carlo
simulations, representing two 3D spheres centered at the axis origin.

The mean free paths inside the spheres are $l_{in}$ and $l_{out}$. The
absorption coefficients and the anisotropy parameters of the spheres are set
to zero (mandatory conditions to test the IP in ART; see Sec. 3.3.2.6).

Forward measurements:  An isotropic and uniform radiance illumination
impinging on the surface of the external sphere (continuous uniform
distribution of Lambertian sources) is utilized as a light source. Note that,
in Fig. 1 only one representative photon is reported (black path; forward
propagation). Photons are detected on the whole surface of the external
sphere. The pathlength of each single photon, $s$, is computed and the mean
pathlength $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ is assessed. Each
simulation consists of 100 independent simulations (each of them has $10^{7}$
photons trajectories) in order to evaluate the standard error of the mean.
Isotropic scattering is used. Random photons steps for the three examples of
power law are generated using Eqs. (13) and (14), for three representative $a$
values, i.e., $a\in\\{1,1/2,+\infty\\}$ with
$\ell\in\\{\ell_{in},\ell_{out}\\}$ (the choice of $\ell_{in}$ or $\ell_{out}$
depends on where the photon is situated along the path). For the constant step
example, photons steps are obtained using Eqs. (17) and (18) with
$\ell\in\\{\ell_{in},\ell_{out}\\}$.

Adjoint measurements:  Adjoint measurements are assessed by inverting the
direction of each photon reaching the sphere surface (detector). Then, these
photons, with their new inverted directions (red-dashed path in Fig. 1), are
utilized as light source to generate the adjoint measurements, with the same
conditions utilized for the forward measurements. Even in this case, photons
are detected on the whole surface of the external sphere. At the end of the
simulation, the mean pathlength, $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$, is then computed.

The importance of $p_{u}(s_{u},\ell,a)$:  Forward and adjoint MC simulations
are repeated two times: 1) By using the rules for the MC ART presented in Sec.
2.2.1, giving $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ and $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ and; 2) By setting $p_{u}(.)=p_{c}(.)$,
as it is usually done in classical MC, resulting in this case in two mean
paths that we will note $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$ and $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$.

####  3.2.2 Power law: $a=1$

In this case random steps $s_{c}$ and $s_{u}$ are generated as [Eqs. (13) and
(14)]

$s_{c}=\ell\left[(1-\xi)^{-\frac{1}{2}}-1\right],$ (19)
$s_{u}=\frac{\ell\xi}{1-\xi}.$ (20)

To better understand the physical properties of this model, we will compare it
with the classical one (see below), through the functions presented in Sec.
2.2.2. In this case, Eq. (5) is expressed as

$P_{c}(s_{c};\ell,a=1)=\left(\frac{\ell}{\ell+s_{c}}\right)^{2}.$ (21)

To obtain the remaining Eqs. (7), (8) and (9) let first compute Eq. (6), i.e.,

$\tilde{p}_{c}(w_{c};\ell,a=1)=2(w_{c}\ell)^{2}\Gamma(-2,w_{c}\ell)e^{w_{c}\ell}.$
(22)

where $\Gamma(.,.)$ is the incomplete gamma function. Then, insert Eq. (22) in
Eqs. (7), (8) and (9). Being the obtained expressions too complex to be
derived analytically, numerical solutions can be assessed by applying the
Talbolt’s inverse Laplace transform [12] (see Fig. 2).

Note that, the asymptotic limit [Eq. (11)] of this power law with $a=1$ can be
described by a classical diffusion equation [11].

####  3.2.3 Power law: $a=1/2$

The random steps $s_{c}$ and $s_{u}$ for $a=1/2$ are generated as [Eqs. (13)
and (14)]

$s_{c}=\frac{\ell}{2}\left[(1-\xi)^{-\frac{2}{3}}-1\right],$ (23)
$s_{u}=\frac{\ell}{2}\left[(1-\xi)^{-2}-1\right].$ (24)

This is an interesting case, because the asymptotic limit of this model is
related to a fractional diffusion equation with derivative of order $3/2$
[11]. In this case [Eq. (5)],

$P_{c}(s_{c};\ell,a=1/2)=\left(\frac{\ell}{\ell+2s_{c}}\right)^{\frac{3}{2}}$
(25)

and Eq. (6) can be expressed as

$\tilde{p}_{c}(w_{c};\ell,a=1/2)=\frac{\sqrt{2}}{2}(\ell
w_{c})^{\frac{3}{2}}\Gamma(\frac{1}{2},\frac{\ell w_{c}}{2})e^{\frac{\ell
w_{c}}{2}}-\ell w_{c}+1$ (26)

Then, Eqs. (7), (8) and (9) can be obtained numerically with the same
procedure presented in Sec. 3.3.2.2 (see Fig. 2).

####  3.2.4 Power law: $a=+\infty$

The random step $s_{c}$ and $s_{u}$ are generated as

$s_{c}=s_{u}=-\ell\ln(1-\xi).$ (27)

In this case, $s_{c}=s_{u}=s$, because the correlated and uncorrelated pdfs
are the same, i.e.,

$p_{c}(s;\ell,a)=p_{u}(s;\ell,a)=\frac{1}{\ell}e^{-s/\ell}.$ (28)

We immediately recognize here the pdf of the classical model, related to the
underlying Beer–Lambert–Bouguer’s law. In fact [Eq. (5)],

$P_{c}(s_{c};\ell,a=+\infty)=e^{-s_{c}/\ell}$ (29)

It is in this sense, i.e., when $a=+\infty$, that the classical model is
naturally included in the tutorial ART power model of Sec. 3.3.1.

In this special case, by deriving before Eq. (6), i.e.,

$\tilde{p}_{c}(w_{c};\ell,a=+\infty)=\frac{1}{\ell w_{c}+1},$ (30)

Eqs. (7), (8) and (9) can be completely obtained analytically, i.e.,

$\langle m(s_{c};\ell,a=+\infty)\rangle=s_{c}/\ell$ (31)
$v_{k}(s_{c};\ell,a=+\infty)=\frac{(s_{c}/\ell)^{k}}{k!}e^{-s_{c}/\ell}$ (32)
$F_{k}(s_{c};\ell,a=+\infty)=\sum_{n=k}^{+\infty}\frac{(s_{c}/\ell)^{n}}{n!}e^{-s_{c}/\ell}$
(33)

####  3.2.5 Constant step

By following the same procedure utilized for the power law, we obtain for the
constant step model

$P_{c}(s_{c};\ell)=1-\Theta(s_{c}-\ell),$ (34)
$\tilde{p}_{c}(w_{c};\ell)=e^{-w_{c}\ell},$ (35) $\langle
m(s_{c};\ell)\rangle=\left\lfloor\frac{s_{c}}{\ell}\right\rfloor,$ (36)

where $\left\lfloor.\right\rfloor$ is the floor function and,

$v_{k}(s_{c};\ell)=\Theta(s_{c}-k\ell)-\Theta(s_{c}-(k+1)\ell),$ (37)
$F_{k}(s_{c};\ell)=\Theta(s_{c}-k\ell).$ (38)

####  3.2.6 Invariance property (IP)

As it was mentioned in Sec. 2.2.1, the introduction of Eq. (3) allows us to
suitably taking into account the RL in the MC ART simulations. Neglecting this
rule, the RL may not be satified. In this context, the examples chosen for
this tutorial represent a special instructive case. In fact, it is easy to see
that, due to the absence of absorption and the particular geometry of the
light source and detector, the RL will always be satisfied, independently of
the remaining optical parameters or other functions utilized for the MC
simulation. The reason of this special behavior is due to the fact that all
the lunched photons always reach the detector (i.e., same photon flux through
the detector in forward or adjoint direction). This means that testing the RL
is not always a good test to check a MC code, and that a more universal method
is needed. This is why the IP is introduced in the following paragraph and is
then exploited to show the importance of the right choice of $p_{u}(.)$ [Eq.
(3)].

The IP [13, 14] is a very powerful property that in the case of anomalous
transport states the following: let be a uniform and isotropic radiance
illumination applied at the external boundary, $S$, of a finite, scattering
and non-absorbing medium of any shape, volume V, matched refractive indexes
and isotropic phase function. Then, if we apply the rules reported in Sec.
2.2.1 (to suitably describe ART), the mean pathlength, $\langle
s\rangle_{\textrm{theory}}$, spent inside V by photons outgoing from $S$, is
an invariant quantity independent of the scattering strength. Note that the
volume $V$ can also be the union of any number of sub-volumes with different
scattering coefficients but matched refractive indexes (external medium also
matched). The quantity $\langle s\rangle_{\textrm{theory}}$ is expressed as

$\langle s\rangle_{\textrm{theory}}=4\frac{V}{S}.$ (39)

This property holds for anomalous and classical photon propagation and can be
utilized to test the validity of the MC codes. It is worth to note that in the
case of classical photon transport (Beer–Lambert–Bouguer’s law), the
hypothesis of an isotropic phase function is not necessary.[13, 14, 15] A
generalization of Eq. (39) for unmatched refractive indexes also exists [16,
15], but this goes far from the aim of the present tutorial. However, in this
context we need a general formulation valid for ART, implying the use of an
isotropic phase function.

##  4 Results

###  4.1 General comparisons of the classical and the ART models

Before to proceed with the MC simulations, let us compare (semi)-analytically
the power law and the constant step models with the classical one [Eq. (28)].

In Fig. 2

Figure 2: Probabilistic functions (see Sec. 2.2.2) characterizing the models
chosen as a tutorial cases (see Secs. 3.3.2.2, 3.3.2.3, 3.3.2.4) and 3.3.2.5).

the power law [Eqs. (21) and (25) ], for $a=1$ and $a=1/2$, and the constant
step [Eq. (34)] models, are compared with the special case $a=+\infty$ [Eq.
(29)], related to the classical Beer–Lambert–Bouguer’s law. For simplicity we
have chosen the case $\ell=1$, but the general considerations remain valid for
other $\ell$ values.

In Fig. 2a the classical model for $P_{c}(.)$ appears as a straight line
(logarithmic scale). It is known from renewal theory that this model is
representative of a Markovian or memoryless process. This means that a future
photon step length only depends on the present state and not on the past
history of the photon propagation. The remaining models appearing in Fig. 2a,
that deviate from this behavior, are not memoryless. This explains in part why
it may be more complex to express these models through an
(integro)-differential equation formalism. In general, $P_{c}(.)$ may be
assessed experimentally by the measurement of the so called ballistic photons,
while the remaining functions in Fig. 2 may only be indirectly derived by
means of a mathematical procedure. Figure 2a clearly shows that the power low
curves for ART go slower to zero compared to the classical case. This means
that there is a higher probability for a photon to suddenly undergo very long
steps. This may be e.g. the case of photons that pass through a non-
homogeneous distribution of clouds.

Figure 2b represents the average number of scattering events (or steps),
$\langle m_{c}(.)\rangle$, along a path of length $s_{c}$. The figure
highlights the fact that even if in general the average step is always $\ell$
for the power law [Eq. (10)] and the constant step [Eq. (15)] models, for the
specific ART condition it is not possible to simply divide the path length
$s_{c}$ by $\ell$ to find the $\langle m_{c}(.)\rangle$ value, as it can be
done in the classical case [Eq. (31)] (black color; straight line). In fact,
Fig. 2b shows that ART curves are not linear and thus a specific calculation
must be performed [Eq. (7)]. This particular behavior is generated by the
“memory” of the ART models. Function $\langle m_{c}(.)\rangle$ is extremely
important because renewal theory tells us that it uniquely determines the
process generating the photons’ steps.

Figure 2c reports the probability $v_{k}(.,.)$ that k scattering events occur
on a path of length $s_{c}=5$ mm. As expected, in the classical case
$v_{k}(.,.)$ is a Poisson distribution [Eq. (32)], due to the exponentially
distributed photons steps. The function $v_{k}(.,.)$ can be utilized to
reproduce the statistical spatial distribution of the scattering sites inside
the medium, and gives us an idea on how the medium is structured.

Figure 2d reports the probability $F_{k}(.)$ that the sum of the first $k$
photon steps does not exceed $s_{c}=5$ mm. It can be seen that the classical
case has smaller values than the ART power law (apart for the cases $k=1,2$).
The exceptions for $k=1,2$ can be intuitively explained by looking e.g. Fig.
2a from which we can deduce e.g. that the probability to have a long step
larger than 5 mm is higher for the classical case.

Thus, Fig. 2 gives us the main features of a given ART model compared to the
classical one. The above (semi)-analytical approach may be sometimes useful
for the interpretation of the MC simulations.

###  4.2 MC results

The results of the MC simulations, for the three cases and the geometrical
model appearing in Fig. 1, are summarized in Table 1.

Table 1: Explanatory tests of the IP for the ART theory. Parameters of the simulations: $r_{out}=5$ mm, $r_{in}=2$ mm, $\ell_{out}=1$ mm, $\ell_{in}=3$ mm, $\langle s\rangle_{\textrm{theory}}=6.\bar{6}$ mm (for the remaining parameters see text). PL: power law; CS: constant step; BLB: Beer–Lambert–Bouguer’s. Values in parenthesis are the standard error of the mean. pdf model | $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ | $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$ | $\langle s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ | $\langle s\rangle^{\textrm{Adj}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$
---|---|---|---|---
PL $(a=1)$ | $6.6666\;(0.0004)$ | $5.4777\;(0.0002)$ | $6.6668\;(0.0002)$ | $5.4242\;(0.0002)$
PL $(a=1/2)$ | $6.6667\;(0.0002)$ | $4.9731\;(0.0002)$ | $6.6667\;(0.0002)$ | $4.9030\ (0.0002)$
CS | $6.6666\;(0.0004)$ | $9.3866\;(0.0004)$ | $6.6668\;(0.0004)$ | $9.6856\ (0.0004)$
BLB $(a=+\infty)$ | $6.6668\;(0.0003)$ | $6.6664\;(0.0002)$ | $6.6666\;(0.0003)$ | $6.6668\;(0.0003)$

The theoretical mean pathlength that we must reproduce with the MC simulations
is $\langle s\rangle_{\textrm{theory}}=\frac{4}{3}r_{out}$ mm [Eq. (39)].

From Table 1 it clearly appears that, $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ perfectly reproduces the theoretical
$\langle s\rangle_{\textrm{theory}}$ IP-value (within the statistical error)
for all cases, i.e., the power law, the constant step and the classical model.
Moreover, $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}=\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ (within the statistical error).

However, if the condition for the RL is not satisfied, i.e., if we neglect the
special behavior for uncorrelated scattering events and set $p_{u}(.)$ equal
to $p_{s}(.)$, then the MC simulations generates results in disagreement with
the laws of optics, i.e., $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}$, $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle_{\textrm{theory}}$ and $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle_{\textrm{theory}}$, when $a\in\\{1,1/2\\}$. The only exception being
the classical case ($a=+\infty$), where all results remain valid (last line of
Table 1). This is obviously due to the fact that in this case
$p_{u}(.)=p_{s}(.)$ [Eq. (28)].

Considering the geometry of the problem, the inequality $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}$, for the
ART, immediately tells us that the light reaching the detector in the forward
direction do not represent an isotropic and uniformly distributed radiance. In
other words, one of the conditions for the IP is not satisfied for the adjoint
case. As expected, in the classical case this problem does not subsist, i.e.,
$\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-
reciprocity}}$}}=\langle s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-
reciprocity}}$}}$.

In Table 1, the data are exact up to the third decimal but, obviously, if we
increase the number of photons trajectories per simulation, we can also
increase the precision of the results at any desired level.

##  5 Discussion and conclusions

In Sec. 3 and 4 we have seen, through some tutorial example, the importance to
distinguish between correlated and uncorrelated scattering events, through the
pdfs $p_{c}(.)$ and $p_{u}(.)$. In fact, this approach not only satisfies (by
construction of the theory) the RL for photon propagation, but it is also
mandatory if one want to satisfy the universal IP.

In Table 1 we have also seen that there is an exception for the case
$a=+\infty$, corresponding to the classical case where photons propagate under
the Beer–Lambert–Bouguer’s law. In fact, in this special case (the only one)
$p_{u}(.)=p_{s}(.)$, and as a consequence the results remain exact in all
cases (Table 1; last line).

In practice, the ART theory is a generalization of the classical one, the
latter appearing as a particular case. Thus, the ART MC method can be applied
to solve classical problems in photon transport, e.g., in biomedical optics.
This may greatly simplify the writing of MC codes (the photon step just stops
at the boundary, before the next step), because it does not necessitate to
treat the photons reaching the boundaries with more complex algorithms as it
is classically done [7].

But then, do we have two different algorithms to treat the photons
interactions with the boundaries in classical MC ? Which is the right one?
Actually, the classical and the ART based MC methods give the same results.
Intuitively, it is probably hard to see that the two approaches are
equivalent. For this reason, another simplified example might maybe help the
reader to have a more concrete idea. Let be a medium with only one internal
boundary and mean free paths $\ell_{1}$ and $\ell_{2}$. Let generate a photon
step, $s=s_{1}+s_{2}$, as in Fig. 3.

Figure 3: Schematic of a photon that crosses a boundary separating two regions
with different mean free path.

It is well known that in classical MC the random step $s$ is obtained as

$s=\begin{cases}-\ell_{1}\ln\xi;&\mbox{if }s\leq s_{1}\\\
\frac{\ell_{2}}{\ell_{1}}(-\ell_{1}\ln\xi-s_{1})+s_{1};&\mbox{if }s\mbox{
computed above}>s_{1},\end{cases}$ (40)

where $\xi\in(0,1)$ is a uniform distributed random variable.

In the same way, as it was explained in this tutorial, in ART the MC random
step $s$ is generated as

$s=\begin{cases}-\ell_{1}\ln\xi_{1};&\mbox{if }s\leq s_{1}\\\
-\ell_{2}\ln\xi_{2}+s_{1};&\mbox{if }s\mbox{ computed
above}>s_{1},\end{cases}$ (41)

where $\xi_{1},\xi_{2}\in(0,1)$ are independent uniform distributed random
variables. It easy to see (e.g. numerically) that Eqs. (40) and (41) generate
random $s$ with the same pdf. This equivalence is valid in general for any
choice of complex geometries, number of boundaries, optical properties, etc.
This is the reason why, in the case of classical simulations where Eq. (1)
holds, ART and the classical MC approaches are equivalent.

At this point, without entering to much in too technical details, it is maybe
worth making a general consideration concerning the RL and the IP. In fact,
historically, the pdf $p_{u}(.)$ has been derived independently for the RL and
for the IP theory. Each theory having its own independent (but compatible)
physical assumptions allows us to assess $p_{u}(.)$. In this didactical
contribution, we have chosen a point of view that is not historical but that
should better highlight the relationship existing between RL and IP. In fact,
the RL is a more “general” law than the IP. This, because RL holds e.g. also
for any light source, detector configuration, optical properties values, or
phase function choice. Thus, the pdf for a step starting from a fixed position
must be $p_{u}(.)$, if we want RL to be satisfied.

On the other hand, historically, the pdf $p_{u}(.)$ necessary to satisfy the
IP has also been derived independently [14] (the obtained $p_{u}(.)$ equals
the one satisfying the RL), but by using different hypothesis (actually, a
subset of conditions of the RL). We realize here that this derivation for the
IP is not strictly necessary, because $p_{u}(.)$ is already given by the RL.
In practice, it is possible to demonstrate the IP by considering $p_{u}(.)$
already known. We invite the interested reader to revisit the original papers
on IP (e.g. Ref. [14]) with this point of view. This might help to have a more
coherent vision on the strong relationship existing between RL and IP.

The few examples presented in this tutorial had the aim to show how to
generate MC simulations for any $p_{c}(.)$ describing a desired medium.
However, we must be aware of the fact that to describe the exact physics of
the investigated medium, the first moment of $p_{c}(.)$ must be finite [Eq.
(3)]. From a practical point of view this constraint on $p_{c}(.)$ is not a
real problem, because when simulating media representing actual physical
systems, it is unlikely that photons goes to infinity without any interaction.
This keeps the average step length finite.

In conclusion, it may be worth to note that an MC code compatible with the ART
theory may also be of a more general interest, than a simple technicality, for
tissue optics. In fact, it has been shown that some biological tissues, such
as bone and lung, have a particular fractal-like structure [17, 18, 19]. As
reported in Ref. [20] by Davis and Mineev-Weinstein, (multi-)fractal
structures may produce anomalous photon propagation. This observation may open
a new domain of exploration in biomedical optics, and adapted MC codes may
represent one of the important tools to address this interesting topic.

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# The creation of a massive UCD by tidal threshing from NGC 936

Sanjaya Paudel,1,2 Pierre-Alain Duc,3 Sungsoon Lim,1 Mélina Poulain,4 Francine
R. Marleau,5 Oliver Müller,6 Rubén Sánchez-Janssen,7 Rebecca Habas,3,8 Patrick
R. Durrell,9 Nick Heesters,6Daya Nidhi Chhatkuli,10 Suk-Jin Yoon,1,2
1Department of Astronomy, Yonsei University, Seoul, 03722, Republic of Korea
2Center for Galaxy Evolution Research, Yonsei University, Seoul, 03722,
Republic of Korea
3Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France
4Space Physics and Astronomy Research Unit, University of Oulu, P.O. Box 3000,
FI-90014, Oulu, Finland
5Institut für Astro- und Teilchenphysik, Universität Innsbruck,
Technikerstraße 25/8, Innsbruck, A-6020, Austria
6Institute of Physics, Laboratory of Astrophysics, École Polytechnique
Fédérale de Lausanne (EPFL), 1290 Sauverny, Switzerland
7UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh,
EH9 3HJ, UK
8INAF Osservatorio Astronomico di Teramo, via Maggini, I-64100, Teramo, Italy
9Youngstown State University, One University Plaza, Youngstown, OH 44555 USA
10Central Department of Physics, Tribhuvan University, Kirtipur 44618,
Kathmandu, Nepal E-mail<EMAIL_ADDRESS>
(Accepted 2023 August 30. Received 2023 August 28; in original form 2023
August 03.)

###### Abstract

We study a compact nucleus embedded in an early-type dwarf galaxy, MATLAS-167,
which is in the process of disruption by the tidal force of the neighboring
giant S0 galaxy, NGC 936, in a group environment. Using the imaging data of
the MATLAS survey, we analyze the stellar tidal tail of MATLAS-167 and its
central compact nucleus, designated as NGC 936_UCD. We find that NGC 936_UCD
has a luminosity of Mg = $-$11.43$\pm$0.01 mag and a size of 66.5$\pm$17 pc,
sharing the global properties of Ultra Compact Dwarf galaxies (UCDs) but
significantly larger and brighter compared to the typical UCD populations
observed in the Virgo cluster. By integrating the total luminosity of both the
tidal stream and MATLAS-167, we estimate that the disrupted dwarf progenitor
possesses a luminosity of Mg = $-$15.92$\pm$0.06 mag, a typical bright dE
luminosity. With the help of the optical spectrum observed by the SDSS survey,
we derive the simple stellar population properties of NGC 936_UCD: a light-
weighted age of 5.6$\pm$0.7 Gyr and metallicity of [Z/H] = $-$0.83$\pm$0.3
dex. Our findings suggest that tidal threshing is a possible formation
mechanism of bright UCD populations in close proximity to giant galaxies.

###### keywords:

galaxies: dwarf — galaxies: evolution — galaxies: groups: general — galaxies:
interactions — galaxies: nuclei

††pubyear: 2022††pagerange: The creation of a massive UCD by tidal threshing
from NGC 936–References

##  1 INTRODUCTION

Ultra-compact dwarf galaxies (UCDs) bridge the gap between galaxies and star
clusters in terms of mass, size, and luminosity, making it difficult to
clearly distinguish between the two classes of stellar systems (Hilker et al.,
1999; Drinkwater et al., 2000; Phillipps et al., 2001; Evstigneeva et al.,
2008; Norris et al., 2014). The question at the heart of this discussion is
whether UCDs are the largest star clusters or the smallest galaxies (Mieske et
al., 2002; Kissler-Patig et al., 2006). UCDs are larger, brighter, and more
massive than the typical globular clusters (GCs) with typical half-light radii
of 10 $\lesssim$ Rh $\lesssim$ 100 pc, and luminosities Li $\gtrsim$ 105 L☉
(Haşegan et al., 2005; Mieske et al., 2008; Misgeld & Hilker, 2011; Norris et
al., 2014; Voggel et al., 2016). Their stellar population is old ($\gtrsim$5
Gyr), with a wide range of metal content, mostly sub-solar (Firth et al.,
2009; Paudel et al., 2010; Chilingarian et al., 2011; Janz et al., 2016; Zhang
et al., 2018; Forbes et al., 2020; Fahrion et al., 2019). The central velocity
dispersions ($\sigma_{v}$) of UCDs are similar to dwarf galaxies, with a
typical value of 20 $\lesssim$ $\sigma_{v}$ $\lesssim$ 50 $km\,s^{-1}$. Their
dynamical mass estimates show that they have mass-to-light ratios, which are,
on average, about twice as large as those of GCs (Hilker et al., 2007;
Baumgardt & Mieske, 2008; Frank et al., 2011; Mieske et al., 2013; Janz et
al., 2015). Recent high spatial resolution spectroscopic observations show
that a fraction of UCDs also hosts a central intermediate-mass black hole
(Seth et al., 2014; Ahn et al., 2017, 2018; Afanasiev et al., 2018; Voggel et
al., 2019).

Figure 1: Comparison between the SDSS and the MATLAS image. The left panel
shows a tri-color image of NGC 936 from SDSS, created by combining $g$-, $r$-,
and $i$-band images. The right panel shows a deep $g$-band image from the
MATLAS, which clearly reveals the filament and low surface brightness plumes
around NGC 936. Both images have a field of view of
5.5$\arcmin$$\times$5.5$\arcmin$. The position of the disrupted dwarf,
MATLAS-167, is highlighted by a green circle in both images. While only the
star cluster is visible in the SDSS image, the underlying low surface
brightness host is revealed in the MATLAS image.

Since the discovery of UCDs, there has been a significant amount of research
focused on understanding their origins. It has become clear that UCDs are not
a uniform population and can be formed through a variety of different
processes (Hilker, 2011). Two main formation pathways are frequently discussed
in the literature (e.g., Fellhauer & Kroupa, 2002). The first involves tidal
disruption, with UCDs proposed as the remnant nuclei of tidally disrupted
galaxies (Drinkwater et al., 2003; Gregg et al., 2003; Goerdt et al., 2008;
Pfeffer & Baumgardt, 2013; Pfeffer et al., 2014). In this scenario, a
nucleated dwarf galaxy in a cluster or group environment may undergo complete
tidal disruption, leaving behind a naked dense stellar core (known as a
nuclear star cluster). The remnant dense nuclear star cluster is
gravitationally strong enough to retain its stars against tidal disruption
(Bekki et al., 2003). Evidence in support of the tidal disruption origin of
UCDs includes the presence of features such as tidal tails, extended haloes,
SMBH, and asymmetries around these objects (Voggel et al., 2016; Wittmann et
al., 2016; Schweizer et al., 2018; Evstigneeva et al., 2008; Liu et al.,
2020). Other nucleated dwarf galaxies undergoing disruption have been
discovered. They include the Sagittarius dwarf galaxy around the Milky Way, a
so-called dog-leg tidal stream around NGC 1407 (Galianni et al., 2010;
Amorisco et al., 2015) and extremely diffuse nucleated dwarf galaxies at the
Virgo cluster (Mihos et al., 2015).

The second scenario suggests that UCDs are the high-mass end of the GC mass
function (Kroupa, 1998; Fellhauer & Kroupa, 2002; Mieske et al., 2002; Brüns
et al., 2011) and bright UCDs might have formed through the merger of GCs
(Kissler-Patig et al., 2006). It is also argued that UCDs can be primordial
objects formed in an intense burst of star formation (Murray, 2009).

There is a wide range of properties among known UCDs, and they share
characteristics with both GCs and the nuclei of dwarf galaxies. This suggests
multiple formation processes contribute to their creation (Francis et al.,
2012). However, it is likely that stripped nuclei account for at least some
percentage of the UCD population due to various similarities to compact galaxy
nuclei (Drinkwater et al., 2003; Paudel et al., 2010). These include
overlapping luminosity distributions and similar size$-$luminosity
relationships (Evstigneeva et al., 2008), internal velocity dispersions
(Drinkwater et al., 2003), positions on the color$-$magnitude diagram, and
stellar population properties (Côté et al., 2006; Evstigneeva et al., 2008;
Paudel et al., 2010; Brodie et al., 2011; Chilingarian et al., 2011; Spengler
et al., 2017; Zhang et al., 2018).

In this work, we identify a star cluster located at the end of a tidal stream
that is likely to have originated from the disruption of an early-type dwarf
galaxy (dE), MATLAS-167. The star cluster is bright, $M_{g}$ =
$-$11.43$\pm$0.01 mag, and compact, likely a surviving nucleus of MATLAS-167
disrupted by the tidal force of the nearby giant galaxy NGC 936 located at
23.0 Mpc111The distance is measured using the surface brightness fluctuation
method by Tonry et al. (2001). away from us. We propose that the nuclear star
cluster is in the process of forming a UCD through tidal stripping.

##  2 Data and Analysis

The aim of the Mass Assembly of early Type gaLAxies with their fine Structures
(MATLAS) project is to conduct a comprehensive imaging survey of local
elliptical galaxies that were selected from the ATLAS3D legacy survey
(Cappellari et al., 2011; Duc et al., 2015). Its primary objective is
identifying and documenting low surface brightness features such as stellar
streams, filaments, and shells surrounding giant early-type and dwarf galaxies
(Bílek et al., 2020; Habas et al., 2020; Marleau et al., 2021). This project
has a magnitude limit of 29 mag arcsec2 for extended low surface brightness
objects. Through a thorough visual examination of all the galaxies in the
survey, we have identified a system of ongoing disruption of a dwarf galaxy
around the giant S0 galaxy, NGC 936. The disrupting dwarf galaxy is
MATLAS-167, and it is cataloged as a dE galaxy in the dwarf galaxy catalog,
which has a prominent bright nucleus at the center (Poulain et al., 2021).

In Figure 1, we compare the SDSS color image and the MATLAS $g$-band image. As
expected, the SDSS image does not reveal any stream, and only a compact source
is visible (see the green circle). On the other hand, the deeper MATLAS
$g$-band image displays a spectacular view of the tidal stream around NGC 936.
The compact star cluster is embedded in a stellar stream, which we have marked
by a green circle. We consider it a putative UCD (hereafter NGC 936_UCD). It
is located at the end of the stream, which forms an almost semi-circular
trajectory around NGC 936. The focus of this study is the nature of the
interaction between NGC 936 and MATLAS-167 and the evolution of NGC 936_UCD.

Figure 2: On-sky position of member galaxies in the NGC 936 group. Green
symbols represent giant galaxies, while dEs and star-forming dwarf galaxies
are represented by red and blue symbols, respectively. Black dots indicate
nucleated dEs. Additionally, the diagram includes two large symbols to
indicate the position of NGC 936 itself and its disrupted satellite,
MATLAS-167.

NGC 936 is a barred S0 galaxy classified as S0Bb in the RC3 catalog (de
Vaucouleurs et al., 1991). It is the most dominant galaxy in the group, which
includes three other massive galaxies. It has a face-on orientation with an
inclination of $<$10 degrees as shown in Figure 1, and it has a prominent
central bar. The MATLAS search for dwarf galaxies identified 27 dwarf galaxies
around NGC 936, and their distribution around NGC 936 is shown in Figure 2.
Only 7 out of 27 dwarf galaxies are star-forming dwarf galaxies (Habas et al.,
2020; Poulain et al., 2021). Among 20 dEs, the nucleated fraction is nearly
50%.

Table 1: Physical properties of NGC 936_UCD

Properties | Values | Unit | Note
---|---|---|---
R.A. | 02:27:32.88 | h:m:s | 1
Decl. |  $-$01:13:49.31 | d:m:s | 2
$M_{g}$ |  $-$11.43$\pm$0.01 | mag | 3
$z$ | 0.0039 |  | 4
$g-r$ | 0.60$\pm$0.01 | mag | 5
Re | 66.5$\pm$17 | pc | 6
$M_{g}$ |  $-$15.92$\pm$0.06 | mag | 7
$\Delta$d | 23 | kpc | 8
$\Delta$vr | 260 | $km\,s^{-1}$ | 9

1) R.A. of NGC 936_UCD
2) Decl. of NGC 936_UCD
3) The absolute $g$-band magnitude of NGC 936_UCD
4) Redshift measured from the SDSS spectrum of NGC 936_UCD
5) $g-r$ color of NGC 936_UCD
6) Effective radius of NGC 936_UCD
7) The $g$-band integrated magnitude of putative dwarf galaxies along the
streams
8) Sky-projected separation between NGC 936_UCD and NGC 936
9) Relative line-of-sight velocity between NGC 936_UCD and NGC 936

###  2.1 Imaging and Photometry

####  2.1.1 The nucleus

In this work, we used Megacam CHFT images obtained by the MATLAS survey (Duc
et al., 2015; Bílek et al., 2020). The MATLAS survey consisted of $g$, $r$,
and $i$-band images, where the $g$-band is the deepest and $i$-band is the
best in image quality. We, therefore, used the $g$-band images for the
photometric measurement and surface photometry of the host galaxy. The
$i$-band, particularly, was used for size measurement of compact nucleus,
which can provide a better spatial resolution than others. All $g$, $r$, and
$i$-band images are observed in 0.19$\arcsec$ pixel-1 spatial resolution, and
the $i$-band has a median PSF of 0.89$\arcsec$ which corresponds to 99 pc at
the distance of NGC 936 (23 Mpc).

Figure 3: The $g$-band surface brightness profile of MATLAS-167 along its
major axis. The best-fit Sérsic function is shown by the red line. We also
show a 45$\arcsec$$\times$45$\arcsec$ $g$-band image of MATLAS-167 and the
residual after subtracting the best-fit model image in the inset. The vertical
dash line represents the size of NGC 936_UCD.

To accurately measure the flux of the nucleus, ensuring that the surrounding
galaxy light does not contaminate it, we employed a method that involves
subtracting the host galaxy light. To accomplish this, we utilized the IRAF
$ellipse$ task, which outputs an azimuthally averaged value along an elliptic
path with a function of galactocentric radii. Figure 3 depicts the $g$-band
light profile of MATLAS-167 along the major axis, with the black dots
representing the observed data points and the red line representing the best-
fitted Sérsic function. To avoid any interference from the central nucleus, we
excluded the inner (r $\leq$ 4$\arcsec$) data points during the fit.

The best-fitted parameters derived from the best-fitted Sérsic function are an
effective radius (Re) of 12.23$\arcsec$ and a Sérsic index (n) of 1.4. To
construct a two-dimensional representation of the observed galaxy, we
incorporated the one-dimensional best-fitted flux into the output of the IRAF
ellipse fit and employed the $bmodel$ task. Considering this best-fit Sérsic
model represents the bound component of MATLAS-167, it has a luminosity of
$M_{g}$ = $-$12.76 mag.

Subsequently, we performed aperture photometry of the compact nucleus in the
resulting model-subtracted residual images. For the measurement of the total
flux and its magnitude, we used an aperture that is roughly twice the size of
the Full Width at Half Maximum (FWHM). To determine the FWHM, we utilized
multiple bright, unsaturated stars in the field as references. To eliminate
background contributions, we selected an annulus with inner and outer radii of
twice and thrice the FWHM, respectively. Total brightness is $M_{g}$ =
$-$11.43$\pm$0.01 mag, and $g-r$ color is 0.6$\pm$0.01 mag.

Figure 4: The $g$-band surface brightness map of the field around NGC 936. The
unrelated foreground and background objects are masked out manually. The green
box in the left panel represents the zoom-in area shown in the left panel,
which is prepared after subtracting the model of NGC 936\. The green polygon
in the left panel delineates the aperture used to carry out the photometry.
Figure 5: The SDSS fiber spectrum (black), together with its best-fit SSP
model spectrum (red). The residuals are shown in the lower panel. The fit is
generally consistent within 5 percent of the observed flux (the horizontal
lines).

The size of NGC 936_UCD was determined by analyzing the galaxy-subtracted
$i$-band image, where it was partially resolved. To perform the measurement,
we utilized the publicly available software $ishape$, and explored both MOFFAT
and KING profiles (Larsen, 1999). The software convolves a model light profile
with a provided PSF and fits it to the source. The analysis resulted in a size
estimate of 0.56$\arcsec$ for the KING15 profile and 0.64$\arcsec$ for the
MOFFAT15 profile, both exhibiting a similar uncertainty of 0.16$\arcsec$. When
translated into physical units, these values correspond to sizes of 62 pc and
71 pc for the KING15 and MOFFAT15 profiles, respectively. The discrepancies in
residuals between the two models were not statistically significant.
Consequently, we opted to adopt the average of these two measurements,
yielding a final size estimate of 66.5$\pm$17 pc. This is relatively large for
a typical NSC of stellar mass $<$107 M☉ and NSC of the stellar mass of
$\approx$107 M☉ or $M_{g}$$\approx$$-$12 mag typically have effective radius
of $\approx$50 pc (Böker et al., 2004; Georgiev et al., 2016)

####  2.1.2 Surface photometry of the tidal stream

A ring filter was utilized to remove foreground stars and compact background
galaxies from the images, and any residual artifacts were manually subtracted
using the IRAF task $imedit$. In Figure 4, we show the $g$-band surface
brightness map of the field around NGC 936 after cleaning and masking
unrelated foreground and background objects. The background gradient of halo
light from the nearby giant galaxy NGC 936 is subtracted. First, a constant
sky-background level is subtracted across the entire image. The constant sky
background level is derived using 10 independent sky regions of size
10$\times$10 pixel boxes from which we sampled the sky background and
calculated an overall median. Subsequently, we masked MATLAS-167 and its tidal
tail region and ran $ellipse$ task to model NGC 936, and then subtracted this
model of NGC NGC 936 from the image.

To measure the total flux of filamentary structure, we conducted aperture
photometry using a polygonal aperture (see the green polygon in Figure 4).
Since the surface brightness of the faint filaments was too low for automatic
detection, an aperture is defined visually. We excluded pixels below the S/N
threshold from the measurement, and the resulting values are presented in
Table 1. The total brightness in $g$-band we measured was Mg =
$-$15.92$\pm$0.04 mag. However, we want to emphasize that this estimate may
not account for additional starlight below our detection threshold or behind
NGC 936, and it may also include contamination from faint point sources.
Therefore, caution should be exercised when interpreting these measurements as
the accreted galaxy luminosity. We followed a similar procedure in the
$r$-band image, measuring the flux within the identical polygonal aperture,
and found that the color of the full stream is $g-r$ = 0.72$\pm$0.06 mag.

###  2.2 Spectroscopy

The SDSS targeted NGC 936_UCD for spectroscopic observation, which we
retrieved from the SDSS archive server, and it proved to be of sufficient
quality and high signal-to-noise ratio to perform a detailed stellar
population study. The SDSS spectrum is observed with a fiber of radius
1.5$\arcsec$, which is nearly three times NGC 936_UCD size. However, the light
contribution of NGC 936_UCD in the fiber is dominant, i.e., $>$90%.

To extract the maximum amount of information from the spectrum, we employed a
full-spectrum fitting method, which exploits the extensive wavelength coverage
of SDSS optical spectroscopy. This fitting method involves modeling the
spectrum using a combination of simple stellar populations (SSPs) defined by
their age and metallicity. We utilized the publicly available code UlySS by
Koleva et al. (2008) for this purpose. We used an SSP model provided by
Vazdekis et al. (2010), based on MILES stellar library (Sánchez-Blázquez et
al., 2006). This model considers the effects of different stellar evolutionary
phases, such as the main sequence, red giant branch, and asymptotic giant
branch. We fitted the observed spectrum of wavelength range 4100 to 7000 Å
after smoothing the SDSS spectrum by a three-pixel Gaussian kernel. The
quality of the model comparison with the SDSS spectrum is shown in Figure 5,
where the observed spectrum typically matches within 5 percent of the modeled
flux. The analysis yielded a light-weighted SSP age of 5.6$\pm$0.8 Gyr and
[Z/H] of $-$0.83$\pm$0.3 dex.

##  3 Discussion

Figure 6: Relation between the luminosity of dEs and their nuclei. NGC 936_UCD
is shown in red and the comparison sample of the Virgo cluster dEs are shown
in blue, which we obtained from Sánchez-Janssen et al. (2019). The arrow in
NGC 936_UCD represents our measurement of MATLAS-167 flux is a lower limit.

the In the right panel, we show the magnitude distribution of the Virgo
cluster dE nuclei and UCDs. The Virgo cluster UCDs magnitudes are obtained
from Liu et al. (2020).

###  3.1 Comparison of UCDs and dE Nuclei Properties

UCDs and dE nuclei are compact and dense stellar systems of high mass. They
often contain predominantly old stellar populations, indicating their
formation in the early stages of galaxy evolution. In this section, we make a
comparative analysis between UCDs and dEs nuclei and find the position of NGC
936_UCD. For this purpose, we use the Virgo cluster UCDs and dE nuclei as
reference samples.

The relationship between the luminosity of dEs and their nuclei is depicted in
Figure 6. The Virgo cluster dE sample is obtained from Sánchez-Janssen et al.
(2019), shown in blue. NGC 936_UCD is represented by a red dot. As
anticipated, a well-established correlation emerges between the luminosity of
dEs and that of their nuclei, placing NGC 936_UCD among the brightest objects
situated in the upper-right corner. It is important to note, however, that the
estimated luminosity of the NGC 936_UCD host galaxy represents a lower limit,
implying that its actual position on the plot may have been even further to
the right.

Figure 7 illustrates the relationship between the derived SSP properties and
local projected density. In this analysis, we utilized UCD and dE nuclei
samples from the Virgo cluster, as studied by Paudel et al. (2010). The local
density was determined by calculating the circular projected area enclosing
the 10th neighbor. The results indicate a weak correlation between the local
projected density and the ages of the nuclei. More importantly, an age break
is observed at approximately $\sim$ 4 (100 kpc)2. Almost all UCDs are located
in the high-density region as defined above, and their age distribution
overlaps with that of dE nuclei situated in high-density environments. A
similar trend is identified in the metallicity distribution of dE nuclei,
where those in high-density environments exhibit lower metallicity compared to
nuclei of dE located in low-density environments. Notably, the SSP properties
of NGC 936_UCD, being situated in a relatively dense region, $>$4 (100 kpc)2,
resemble those of Virgo UCDs or dE nuclei located in dense regions.

Figure 7: Comparison of age and metallicity of the Virgo cluster UCDs (black)
and dEs nuclei (blue) with respect to the local projected density. The data
are from Paudel et al. (2010). NGC 936_UCD is shown in red. Figure 8: Relation
between the distance of UCDs from their nearest bright galaxy and their
luminosities. The blue symbol represents the median distance in the magnitude
bin, accompanied by an error bar that indicates the normalized standard
deviation. NGC 936_UCD is shown in red.

###  3.2  Tidal Interaction and Formation of UCDs

Observations have shown that UCDs have a size$-$luminosity distribution and
internal velocity dispersion similar to compact nuclei (Drinkwater et al.,
2003; Evstigneeva et al., 2008; Pfeffer & Baumgardt, 2013). The high dynamical
mass-to-light ratios of UCDs suggest that they may contain a significant
amount of dark matter, which may have been inherited from the parent dwarf
galaxies during the tidal disruption (Baumgardt & Mieske, 2008; Mieske et al.,
2008). Several UCDs display indications of asymmetrical or tidal features,
while others reveal the presence of stellar envelopes or the status of
transitional objects from dwarf galaxies to UCDs (Wittmann et al., 2016).

State-of-the-art high-resolution imaging and spectroscopic observations of
these compact objects have allowed us to search for the presence of
supermassive black holes (SMBH). Particularly, recent observations have
revealed that all the top three most massive UCDs of the Virgo cluster possess
SMBH (Ahn et al., 2017, 2018; Seth et al., 2014) and these SMBHs account for a
substantial portion of their overall mass. These trends provide compelling
evidence of their tidal stripping origin (Voggel et al., 2019).

The tidal threshing scenario has been proposed to account for the origin of
intra-cluster GCs, which is quite faint compared to the UCDs (West et al.,
1995). Our analysis suggests that massive UCDs are likely to form through
tidal stripping. Based on Figure 6, it is evident that the disrupted nucleated
dE, MATLAS-167, stands out as one of the brighter dEs, and its nucleus
luminosity is comparable to the brighter UCDs observed in the Virgo cluster.
In fact, considering the combined luminosity of MATLAS-167 and its tidal
stream, it surpasses the luminosity of all other dEs identified by the MATLAS
dwarf galaxy survey around NGC 936 (Habas et al., 2020).

The close proximity of NGC 936_UCD to a giant galaxy raises questions about
whether the special environment plays a role in the formation and evolution of
bright UCDs. To shed light on this issue, we show a relation between UCD
brightness and distance to the nearest bright galaxy (Mr $<$$-$19 mag) of the
Virgo cluster UCD sample studied by Liu et al. (2020) in Figure 8. The figure
reveals that bright UCDs tend to be closer to bright galaxies than faint UCDs,
indicating a potential link between bright UCD formation and proximity to a
bright galaxy. We find that almost all UCDs of $M_{g}$ < $-$12 mag are located
within 20 kpc sky-projected distance from their nearest giant neighbor galaxy.
NGC 936_UCD, located at a sky-projected distance of 19 kpc away from a giant
galaxy, NGC 936, is consistent with the observed trend in the Virgo cluster.
To quantify the observed trend, we sub-sample the UCD sample into faint (Mg
$>$ $-$11 mag) and bright categories and compute the two-point correlation
coefficient between these subsets and massive galaxies. We find a significant
disparity in the correlation coefficients. Specifically, the correlation
coefficient between bright galaxies and bright UCDs is almost double (2.05)
compared to that of bright galaxies and faint UCDs (0.96).

Indeed, the destruction of a bright dwarf necessitates a strong tidal force,
which can typically be attained in the vicinity of a massive galaxy or within
a densely populated cluster core. Consequently, the substantial tidal force
exerted by giant galaxies appears to be advantageous in destroying bright
dwarf galaxies, thereby resulting in exposed luminous nuclei commonly referred
to as UCDs. This line of reasoning strongly supports the hypothesis that the
tidal stripping mechanism is not only accountable for the formation of low-
mass intra-cluster GC but also massive UCDs. Remarkably, these objects
represent the two extremes of the mass function of compact stellar systems.

## Acknowledgements

S.-J.Y. and S.P. acknowledge support from the Basic Science Research Program
(2022R1A6A1A03053472) through the National Research Foundation (NRF) of Korea.
S.P. and S.-J.Y., respectively, acknowledge support from the Mid-career
Researcher Program (No. RS-2023-00208957) and the Mid-career Researcher
Program (No. 2019R1A2C3006242) through the NRF of Korea. O.M. is grateful to
the Swiss National Science Foundation for financial support under the grant
number PZ00P2_202104. Mélina Poulain is supported by the Academy of Finland
grant n:o 347089.

## DATA AVAILABILITY

Most of the data underlying this article are publicly available. The derived
data generated in this research will also be shared on reasonable request to
the corresponding author.

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Polarization parameters of the quasi–elastic $(p,2p)$ reaction with nuclei at
1 GeV
O.V. Miklukho, G.M. Amalsky, V.A. Andreev, S.V. Evstiukhin, O.Ya. Fedorov,
G.E. Gavrilov, A.A. Izotov, A.Yu. Kisselev, L.M. Kochenda, M.P. Levchenko,
V.A. Murzin, D.V. Novinsky, A.N. Prokofiev, A.V. Shvedchikov, S.I. Trush, A.A.
Zhdanov

B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina, RUSSIA

New experimental data on the polarization and spin-correlation parameters in
the (p,2p) reaction with nuclei at 1 GeV are presented. The experiment was
aimed to study a modification of the proton–proton scattering matrix.

Comments: 11 pages, 5 figures. Presented at XV International Workshop on
High–Energy Spin Physics, Dubna, October 2013. Submitted to the Workshop
proceedings.

Category: Nuclear Experiment (nucl–ex)

##  1 Introduction

There were some speculations on modifications of nucleon and meson masses and
sizes, and of meson–nucleon coupling constants, and, as a consequence, of
nucleon–nucleon scattering matrix in nuclear medium [1–3]. These speculations
were motivated by a variety of theoretical points of view, including the
renormalization effects due to strong relativistic nuclear fields,
deconfinement of quarks, and partial chiral symmetry restoration.

This work is a part of the experimental program in the framework of which the
medium–induced modifications of the nucleon–nucleon scattering amplitudes are
studied at the PNPI synchrocyclotron with the 1 GeV proton beam [4–8]. The
intermediate–energy quasi–free $(p,2p)$ reaction is a good experimental tool
to study such effects, since in the first approximation, this reaction can be
considered as a proton–proton scattering in the nuclear matter. Usage of
S–shell protons (with zero orbital momentum) is preferred because
interpretation of obtained data in this case is essentially simplified since
the effective polarization is not involved [9]. The polarization observables
in the reaction are compared with those in the elastic $pp$ scattering. In our
exclusive experiment, a two–arm magnetic spectrometer is used, the shell
structure of the nuclei being evidently distinguished. To measure polarization
characteristics of the reaction, each arm of the spectrometer was equipped
with a multi-wire–proportional chamber polarimeter.

In the early PNPI–RCNP experiment [5], the polarizations $P_{1}$ and $P_{2}$
of both secondary protons from the $(p,2p)$ reactions at 1 GeV with the
1$S$–shell protons of the nuclei 6Li, 12C and with the 2$S$–shell protons of
the 40Ca nucleus has been measured at nuclear proton momenta close to zero.
The polarization observed in the experiment, as well as the analyzing power
$A_{y}$ in the RCNP experiment at the 392 MeV polarized proton beam [10, 11],
drastically differed from those calculated in the framework of
non–relativistic Plane Wave Impulse Approximation (PWIA) and of spin–dependent
Distorted Wave Impulse Approximation (DWIA) [12], based on free space
proton–proton interaction. This difference was found to have a negative value
and to increase monotonously with the effective mean nuclear density
$\bar{\rho}$ [10]. The latter is determined by the absorption of initial and
secondary protons in nucleus matter. The observed inessential difference
between the non–relativistic PWIA and DWIA calculations pointed out only to a
small depolarization of the secondary protons because of proton–nucleon re-
scattering inside a nucleus. All these facts strongly indicated a modification
of the proton–proton scattering amplitudes due to the modification of the main
properties of hadrons in the nuclear matter.

Later, the results of the experiment with a 4He target broke the
above–mentioned dependence of the difference between the experimental
polarization values and those calculated in the framework of the PWIA on the
effective mean nuclear density [6]. The difference for the 4He nucleus proved
to be smaller than that for the 12C nucleus. This evidently contradicts the
elastic proton–nucleus scattering experiment. According to the experiment, the
4He nucleus has the largest mean nuclear density. The important feature of the
experiment with the 4He nucleus was a possibility to see the medium effect
without any contribution from multi–step processes (for instance, from the
$(p,2pN)$ reactions). These processes could take place when there were
nucleons of outer shells as in other nuclei. Therefore, they could not cause
the systematic difference between the polarizations $P_{1}$ and $P_{2}$
clearly obtained for the first time in the experiment [6].

Here we present the polarization data for the reaction with the nuclei 4He,
6Li, 12C (1$S$–shell), and 40Ca (2$S$–shell) obtained with a much better
statistical accuracy in our last experiments. New data on the polarization in
the reaction with the 1$S$–shell protons of the 28Si nucleus are presented.
The 1$S$–state of the 28Si nucleus has a larger value of the mean proton
binding energy $E_{s}$ (50 MeV) than that of the 12C nucleus (35 MeV). We also
present the polarization measured in the reaction with the $P$–shell and
$D$–shell protons of the 12C and 28Si nuclei, respectively.

In recent experiments, the research program was extended to measure the spin
correlation parameters $C_{ij}$ in the $(p,2p)$ reaction with the 4He and 12C
nuclei. Measurements of the parameters in the reaction with nuclei were for
the first time performed. The main attention was concentrated on the spin
correlation parameter $C_{nn}$ since its value is the same in the
center–of–mass and laboratory systems. Besides, this parameter is not
distorted by the magnetic fields of the two–arm spectrometer because of the
proton anomalous magnetic moment [13]. Since the polarization and the spin
correlation parameter $C_{nn}$ are expressed differently through the
scattering matrix elements [3], the measurement of both these polarization
observables can provide a more comprehensive information about a modification
of the hadron properties in the nuclear medium.

##  2 Experimental method

The general layout of the experimental setup is shown in Fig. 1 [14]. The
experiment is performed at non–symmetric scattering angles of the final state
protons in the coplanar quasi–free scattering geometry with a complete
reconstruction of the reaction kinematics. The measured secondary proton
momenta $K_{1}$, $K_{2}$ (kinetic energies $T_{1}$, $T_{2}$) and the
scattering angles $\Theta_{1}$, $\Theta_{2}$ are used together with the proton
beam energy $T_{0}$ to calculate nuclear proton separation energy $\Delta E$ =
$T_{0}$-$T_{1}$-$T_{2}$ and the residual nucleus momentum $K_{r}$ for each
$(p,2p)$ event. In the impulse approximation, the $K_{r}$ is equal to the
momentum $K$ of the nuclear proton before the interaction (${\bf K_{r}}$ =
-${\bf K}$).

External proton beam of the PNPI synchrocyclotron was focused onto the target
TS of a two–arm spectrometer consisting of the magnetic spectrometers MAP and
NES. The beam intensity was monitored by the scintillation telescope M1, M2,
M3 and was at the level of about 5$\cdot$1010 protons/(s$\cdot$cm2).

Solid nuclear targets TS made of CH2 (for the setup calibration), 6Li, 12C,
28Si, and 40Ca, as well as a cryogenic target made of liquid helium 4He (or
liquid hydrogen for calibration) were used in the experiment [6, 14].

The spectrometers were used for registration of the secondary protons from the
$(p,2p)$ reaction in coincidence and for measurement of their momenta and
outgoing angles. The polarization of these protons $P_{1}$ and $P_{2}$,

Figure 1: The experimental setup. TS is the target of the two–arm
spectrometer; Q1$\div$Q4 are magnetic quadrupoles; D1, D2 are dipole magnets;
C1, C2 are collimators; S1$\div$S4 and M1$\div$M3 are scintillation counters;
PC1$\div$PC4, PC1’, PC4’ (PC5$\div$PC8, PC5’, PC8’) and A1 (A2) are the
proportional chambers and the carbon analyzer of the high–momentum
(low–momentum) polarimeter, respectively.

and the spin correlation parameters $C_{ij}$ were measured by the polarimeters
located in the region of focal planes of the spectrometers MAP and NES (Fig.
1). The first index of the $C_{ij}$, $i$ (where $i$ is $n$ or $s^{,}$), and
the second index $j$ (where $j$ is $n$ or $s^{,,}$) correspond to the forward
scattered proton analyzed by the MAP polarimeter and the recoil proton
analyzed by the NES polarimeter, respectively. The unit vector ${\bf n}$ is
perpendicular to the scattering plane of the reaction. Unit vectors
${\bf{s^{,}}}$ and ${\bf s^{,,}}$ are perpendicular to the vector ${\bf n}$
and to the coordinate axes z, and z,, (Fig. 1) of the polarimeters.

The overall energy resolution (on $\Delta E$) of the spectrometer estimated
from the elastic proton–proton scattering with the 22 mm thick cylindrical CH2
target was about 5 MeV (FWHM). The spectra which was analysed is presented in
Fig.2 [14].

Figure 2: Proton separation energy spectra for elastic $pp$–scattering (left
panel) and for the $(p,2p)$ reaction with 12C nuclei (right panel). In the 12C
spectrum the accidental background contribution is subtracted.

The track information from the proportional chambers of both polarimeters was
used in the off-line analysis to find the azimuthal $\phi_{1}$, $\phi_{2}$ and
polar $\theta_{1}$, $\theta_{2}$ angles of the proton scattering from the
analyzers A1, A2 for each $(p,2p)$ event.

The polarization parameters were estimated by folding the theoretical
functional shape of the azimuthal angular distribution into experimental one
[8], using the CERNLIB MINUIT package and a $\chi^{2}$ likelihood estimator.
This method permits to realize the control over $\chi^{2}$ in the case the
experimentally measured azimuthal distribution is distorted due to the
instrumental problems.

The time difference (TOF) between the signals from the scintillation counters
S2 and S4 was measured. It served to control the accidental coincidence
background. The events from four neighboring proton beam bunches were
recorded. Three of them contained the background events only and were used in
the off-line analysis to estimate the background polarization parameters and
the background contribution at the main bunch containing the correlated events
[14].

The recoil proton spectrometer NES was installed at a fixed angle
$\Theta_{2}\simeq$ 53.2∘. At a given value of the $S$–shell mean binding
energy of the nucleus under investigation, the angular and momentum settings
of the MAP spectrometer and the momentum setting of the NES spectrometer were
chosen to get a kinematics of the $(p,2p)$ reaction close to that of the
elastic proton–proton scattering. In this kinematics, the momentum $K$ of the
nuclear $S$–proton before the interaction is close to zero. At this condition,
the counting rate of the $S$-shell proton knockout reaction should be maximal.

##  3 Results and discussion

In Fig. 3, the polarizations $P_{1}$, $P_{2}$ in the $(p,2p)$ reaction with
the $S$–shell protons of the nuclei 4He, 6Li, 12C, 28Si, 40Ca are plotted
versus the $S$–shell proton binding energy $E_{s}$ [14]. For all nuclei
(excluding 4He), the effective mean nuclear density $\bar{\rho}$, normalized
on the saturation nuclear density $\rho_{0}\approx$ 0.18 fm-3, is given. The
actual calculation of the effective mean nuclear density $\bar{\rho}$, which
is determined by absorption of the incident and both outgoing protons, was
carried out following a procedure [10] using the computer code THREEDEE [12].
The potential model of a nucleus employed by the code is not correct for the
4He nucleus. The calculated value of the $\bar{\rho}$ in this case is strongly
unreliable [6]. The 4He data should be excluded in comparison with theoretical
models which differ from the PWIA.

The points ($\circ$) and ($\bullet$) in the figure correspond to the
polarizations $P_{1}$

Figure 3: Polarizations $P_{1}$ and $P_{2}$ of the protons scattered at the
angles $\Theta_{1}$ ($\circ$) and $\Theta_{2}$ ($\bullet$) in the $(p,2p)$
reaction with the $S$–shell protons of nuclei at 1 GeV. The points at $E_{s}$
= 0 correspond to the elastic proton-proton scattering. The curves correspond
the theoretical calculations described in the text.

and $P_{2}$ of the forward scattered protons at the angle $\Theta_{1}$ =
21${}^{\circ}\div$25∘ (with energy $T_{1}$ = 745$\div$735 MeV) and of the
recoil protons scattered at the angle $\Theta_{2}\simeq$ 53.2∘ (with energy
$T_{2}$ = 205$\div$255 MeV). The points at the $E_{s}$=0 are the polarizations
$P_{1}$ and $P_{2}$ in the elastic proton-proton scattering at the angles
$\Theta_{1}$ = 26.0∘ and $\Theta_{2}$ = 53.2∘ ($\Theta_{cm}$ = 62.25∘).

In Fig. 3, the experimental data are compared with the results of the non-
relativistic PWIA, DWIA calculations (the dashed and solid curves,
respectively) [14] and the DWIA* calculation with the relativistic effect, the
distortion of the nucleon Dirac spinor in nuclear medium, taken into account
(the dotted, $M^{*}_{N}$, curve) [2,14]. For the calculations, the computer
code THREEDEE was used [12] using an on–shell factorized approximation and the
final energy prescription. A global optical potential, parametrized in the
relativistic framework and converted to the Shrödinger–equivalent form, was
used to calculate the distorted wave functions of incident and outgoing
protons in the case of DWIA and DWIA*. A conventional well–depth method was
used to construct the bound–state wave function. The DWIA* calculations were
carried out in the Shrödinger–equivalent form [5]. In this approach, a
modified $NN$ interaction in medium is assumed due to the effective nucleon
mass (smaller than the free mass), which affects the Dirac spinors used in the
calculations of the $NN$ scattering matrix. A linear dependence of the
effective mass of nucleons on the nuclear density was assumed in the
calculations.

The results of the polarization studies:

1\. The difference of the final proton polarizations $P_{1}$ and $P_{2}$ found
in the PWIA, DWIA and DWIA* is quite small (less than 0.005) for all nuclei
under investigation.

2\. The difference between the PWIA and DWIA results is small. This indicates
that the distortion in the conventional non-relativistic framework does not
play any essential role in the polarization for the kinematic conditions under
consideration (the transferred momenta $q$ = 3.2$\div$3.7 fm-1).

3\. Predictions of the DWIA* with relativistic corrections (distortion of the
proton Dirac spinor in nuclear medium) are close to experimental data for the
forward scattered proton polarization $P_{1}$.

4\. A significant difference is observed between the measured polarization of
the scattered proton $P_{1}$ and that of the recoil proton $P_{2}$.

Note that the difference between the measured polarizations $P_{1}$ and
$P_{2}$ was also observed in the reaction with the $D$–shell protons of the
28Si nucleus and was not seen in the reaction with the $P$–shell protons of
the 12C (Fig. 4).

Figure 4: Polarization in the $(p,2p)$ reaction with the external shell
protons of the 12C and 28Si.

The experimental data on the spin correlation parameters $C_{ij}$ in the
reactions with the 4He and 12C are given in Fig. 5. The dashed and dotted
curves in the figure correspond to the PWIA calculations of the $C_{nn}$ and
$C_{s^{,}{s^{,,}}}$ parameters using the current Arndt’s group phase-shift
analysis (SP07). The mixed $C_{s^{,}{s^{,,}}}$ parameter was found by taking
into account its distortion in the magnetic field of the spectrometers. The
points at the $E_{s}$ = 0 correspond to the elastic proton-proton scattering.

As seen in Fig. 5, the $C_{nn}$ data (as well as the $C_{s^{,}{s^{,,}}}$ data)
are described in the framework of the PWIA. The question arises, there is no
the nuclear medium modification of the $C_{nn}$ parameter as it is for the
polarization of the final protons (Fig. 3)? Whether this fact is connected to
the strong

Figure 5: Spin correlation parameters $C_{ij}$ in the $(p,2p)$ reaction at 1
GeV with the $S$–shell protons of the 4He and 12C nuclei at the secondary
proton scattering angles $\Theta_{2}$ = 53.22∘, $\Theta_{1}$ = 24.21∘ and
$\Theta_{2}$ = 53.22∘, $\Theta_{1}$ = 22.71∘, respectively. The points at
$E_{s}$ = 0 correspond to the elastic proton-proton scattering ($\Theta_{1}$ =
26.0∘, $\Theta_{cm}$ = 62.25∘). The curves are the results of calculations
described in the text.

polarization dropping for the recoil proton? It is possible that some spin-
flip mechanism compensates the nuclear medium effect in the $C_{nn}$.

Due to the parity conservation in the elastic proton-proton scattering, the
spin correlation parameters $C_{ns^{\prime\prime}}$ and $C_{s^{\prime}n}$
should be equal to 0. This is confirmed by the experimental data at the
$E_{s}$ = 0 in the Fig. 5. For the $(p,2p)$ reaction, we see some deviation of
the parameters from zero. It may be related to the spin-flip mechanism
mentioned above. Note that test calculations of all spin correlation
parameters for the accidental coincidence background give zero values as
should be expected.

To find an explanation of the observed effects, let us assume that there is a
spin-flip interaction of the recoil (nuclear) proton with the residual
nucleus, which is not taken into account by the theoretical approaches. This
additional interaction mechanism, governed by the Pauli exclusion principle in
a nucleus, reverses the proton spin direction and, as a consequence, changes
the signs of the polarization and the spin correlation parameter $C_{nn}$.

The relative contribution ($\alpha$) of the spin–flip mechanism in the
interaction with a residual nucleus, which is mainly determined by the proton-
nucleon re-scattering at small angles, can be found from experiment via the
relative polarization dropping ($g_{p}$) for the recoil proton. First define
the averaged polarization of the recoil proton:

$<{P}_{2}>~{}=~{}\frac{P_{2}+\alpha(-P_{2})}{1+\alpha}~{}=~{}\frac{P_{1}+\alpha(-P_{1})}{1+\alpha}~{}=~{}\frac{(1-\alpha)P_{1}}{1+\alpha}.$
(1)

In the equation we used the fact that all employed theories give equal values
of the polarizations $P_{1}$ and $P_{2}$. The averaged value of the $C_{nn}$
can also be calculated using the equation:

$<{C}_{nn}>~{}=~{}\frac{C_{nn}+\alpha(-C_{nn})}{1+\alpha}~{}=~{}\frac{(1-\alpha)C_{nn}}{1+\alpha}.$
(2)

The relative polarization dropping $g_{p}$ is determined as:

$g_{p}=\frac{P_{1}-<{P}_{2}>}{P_{1}}=g_{C_{nn}}=\frac{C_{nn}-<{C}_{nn}>}{C_{nn}}=\frac{2\alpha}{1+\alpha}.$
(3)

It can be seen that the proposed spin–flip interaction couples in simple form
the relative dropping of the polarization and the $C_{nn}$ parameter $g_{p}$ =
$g_{C_{nn}}$. From experimental data we find $g_{p}$(4He) = 0.153$\pm$0.018,
$g_{p}$(12C) = 0.325$\pm$0.031 and make corrections to the PWIA calculations
using the formula $C_{nn}$-cor = (1-$g_{p}$)$C_{nn}$ (the solid curve, PWIA-C,
in Fig. 5). One can see from the figure that the the experimental $C_{nn}$
points lie above the curve. So it can be expected that the nuclear medium
modification enhances the $C_{nn}$ parameter, while the polarization is
reduced.

From the experimental $g_{p}$ data, the probability of the spin–flip
interaction can be defined for the corresponding residual nuclei: $\alpha$(3H)
= 0.083$\pm$0.010, $\alpha$(11B) = 0.194$\pm$0.022.

What could be the nature of the considered spin–flip interaction? It was first
time proposed by D.I. Blokhintsev that there are the fluctuations of nuclear
density in nuclei, or the dense nucleon associations [15]. The reflection of
the recoil proton off the objects is similar to the spin–flip interaction
considered above. As a result, a proton belonging to a correlation, with
opposite spin direction (due to the Pauli principle) is detected.

Nucleon correlations are intensively studied in the JLAB using electron beam.
The CLAS collaboration gives the probability for a given nucleon to belong to
a two-nucleon correlation in nucleus with A nucleons $a_{2N}$(3He) =
0.080$\pm$0.016, $a_{2N}$(12C) = 0.193$\pm$0.041 [16].

We can see that there is a coincidence between the PNPI $\alpha$ and the JLAB
$a_{2N}$ for the corresponding residual nuclei. The model of the spin–flip
interaction for explanation of the PNPI polarization data is currently being
developed. Preliminary results suggest that the ratio of the $\alpha$ and
$a_{2N}$ is very close to unity.

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