# Influence of dark matter on gravitational stability of isothermal gas clouds

I.Yu. Kalashnikov,1 V.M. Chechetkin,1,2,3
1Keldysh Institute of Applied Mathematics, 4 Miusskaya sq., Moscow, 125047,
Russia
2National Research Center ”Kurchatov Institute”, 1 Akademika Kurchatova sq.,
Moscow, 123182, Russia
3National Research Nuclear University MEPhI, 31 Kashirskoe sh., Moscow,
115409, Russia E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

To date, the presence of dark matter (DM) can be judged only by its
gravitational interaction on the visible matter. It is therefore important to
find the consequences of this interaction, which can then help to determine
both the DM properties and parameters and the dynamics and evolution of
visible matter. The gravitational influence of dark matter on the stability of
interstellar medium (ISM), the progenitor of stars and star clusters, was
considered. An isothermal self-gravity gas was taken as a suitable model
describing ISM, particles interacting only gravitationally were considered as
DM. The results obtained by analytical methods show that even a small amount
of fast DM particles significantly increases the stable radius of the gas
cloud and the corresponding mass while a higher relative density of DM
destabilizes the gas. It was shown that with typical parameters of ISM and DM,
its presence increases the maximum stable mass of isothermal cloud by a factor
of four and the radius by five.

###### keywords:

dark matter – ISM: clouds – ISM: evolution – galaxies: star formation –
hydrodynamics – instabilities.

††pubyear: 2022††pagerange: Influence of dark matter on gravitational
stability of isothermal gas clouds–Influence of dark matter on gravitational
stability of isothermal gas clouds

##  1 Introduction

Since obtaining of the first observational evidences to this day, the nature
of dark matter (DM) remains one of the most mysterious questions of the modern
astrophysics. Intensive research in this area (Bertone & Hooper (2018)) has
led to currently the most common view that DM consists of neutral stable
particles with a small cross-section of an interaction with particles of the
visible baryonic matter (BM). There is usually no reason to introduce a DM
particle interaction other than gravitational one. Thus, when talking about
observational manifestations of DM, one is usually referring to its
gravitational influence on BM, such as the non-Keplerian rotation curves of
galaxies (Rubin & Ford (1970); Bosma (1978); Rubin et al. (1980)), the
dynamics of galaxy clusters (Zwicky (1937); Allen et al. (2011)) and
gravitational lensing (Clowe et al. (2004)). It is worth mentioning the so far
unsuccessful attempts both to detect dark matter directly (Undagoitia & Rauch
(2015)) and to construct alternative theories of gravity (Milgrom (1983);
Bekenstein (2004)). Since so far the presence of DM is known only through the
gravitational interaction, it is important to find other consequences of this
interaction, which may further help in determining the DM properties and
parameters and also to find out how the behaviour of BM changes in the
presence of DM with parameters determined by other methods. In this paper, we
considered a gravitational effect of DM presence on the stability of
interstellar gas, the progenitor of stars and star clusters.

The consideration of self-gravitating gas stability problems was started by
Jeans (1902), who studied homogeneous state of gas at rest. Later on this
theory was extended to the case of an expanding universe (Bonnor (1957)),
taking into account the effects of general relativity (Lifshitz (1946)). The
unquenchable interest in this problem is caused by open questions about the
formation of the large-scale structure of the universe (Shandarin et al.
(1983); Longair (2008); Demiański et al. (2011)). Investigation of kinetic
theory of Jeans instability were initiated by Lynden-Bell (1962) and Sweet
(1962) and it continues to be developed (Trigger et al. (2004); Rozina et al.
(2017); Yang et al. (2020); Kremer (2021)). Besides, studies were carried out
taking into account the presence of turbulence (Chandrasekhar (1951);
Bonazzola et al. (1987)), magnetic fields (Parker (1966); Kumar & Srivastava
(1990)) and dust (Pandey et al. (1994); Rozina et al. (2017)).

In his consideration Jeans did not take into account thermodynamic state of
the self-gravitating gas, holding its density as a constant. However, such a
stationary configuration cannot exist (see Section 2.2). Easy-to-analyze and
at the same time sufficiently accurate models of a self-gravitating gas have
been known for a long time. They have barotropic equation of state
$p=K\rho^{\gamma}$, where $p$, $\rho$ are pressure and density, $K$, $\gamma$
are free parameters. The case $\gamma>1$ corresponds to the polytropic models
widely used in the study of stellar evolution (Chandrasekhar (1939)). If
$\gamma=1$ and $K=c_{s}^{2}$, where $c_{s}$ is speed of sound, the equation
describes an isothermal gas, which, when self-gravity is taken into account,
gives a description of a cloud of gas, commonly used as a model for ISM –
interstellar medium (Draine (2011)). The results of the stability
investigation of this configuration, done by Ebert (1955); Bonnor (1956),
showed only a slight difference from Jeans theory. Subsequently, these studies
were continued mainly in connection with the general stability properties of
self-gravity systems and the development of the corresponding methods for
their investigation (Antonov (1962); Lynden-Bell et al. (1968); Katz (1978);
Padmanabhan (1989); Chavanis (2002a, b)).

In the early days of the search for DM particle candidates, neutrinos were
considered (Lee & Weinberg (1977); Zeldovich et al. (1980)). However, further
studies of the formation of large-scale structures of the universe have shown
the failure of this assumption (White et al. (1983)). After that, researchers’
attention shifted to other possible candidates, the most discussed of which
are hypothetical weakly interacting massive particles – WIMPs (Steigman &
Turner (1985); Cerdeno (2009)). In this work the effects of the presence of
such particles on the stability of isothermal self-gravity gas as an
appropriate model describing ISM are investigated. In doing so, our
consideration is not limited to WIMPs, but can also be applied to other
candidates whose behavior can be considered statistically in a limited area
without self-interaction other than gravitational one.

##  2 preview

###  2.1 Isothermal gas spheres

To begin with let us shortly consider the theory of isothermal gas spheres and
their stability. The problem of finding the equilibrium configuration of such
a sphere is reduced to finding the gravitational potential $\phi$ with a
density $\rho$ known from the Boltzmann distribution:

$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}r^{2}\frac{d\phi}{dr}=4\pi G\rho,$
(1) $\displaystyle\rho=\rho_{c}\exp\left(-\frac{m_{b}\phi}{kT}\right),$ (2)
$\displaystyle\phi|_{r=0}=\frac{d\phi}{dr}\bigg{|}_{r=0}=0,$ (3)

where the first boundary condition means that the potential is counted from
zero at the center, the second one corresponds to zero gravity at the center
of the sphere; $\rho_{c}$ is density in the center, $m_{b}$ is a mean mass of
BM particles, $T$ is a constant temperature of the gas, $G$, $k$ –
gravitational and Boltzmann constants respectively. Introducing the notations
$\psi=m_{b}\phi/kT$, $\zeta=r/r_{0}=r\sqrt{4\pi G\rho_{c}m_{b}/kT}$ we come to
the following problem:

$\displaystyle\frac{d^{2}\psi}{d\zeta^{2}}+\frac{2}{\zeta}\frac{d\psi}{d\zeta}=e^{-\psi},$
(4) $\displaystyle\psi|_{\zeta=0}=\frac{d\psi}{d\zeta}\bigg{|}_{\zeta=0}=0.$
(5)

The mass $M$ enclosed in a sphere of radius $r$ is the integral of $4\pi
r^{2}\rho$, which may be rewritten using (4) in the form:

$\displaystyle
M(r)=M(r_{0}\zeta)=4\pi\rho_{c}r_{0}^{3}\zeta^{2}\frac{d\psi}{d\zeta}.$ (6)

Next, we may express the central density in terms of the whole mass
$M_{\text{tot}}$ of the considering sphere with radius $R=r_{0}\varkappa$:

$\displaystyle\rho_{c}=\frac{\mu_{s}^{2}}{4\pi
M_{\text{tot}}^{2}}\left(\frac{kT}{m_{b}G}\right)^{3},$ (7)

where $\mu_{s}=\varkappa^{2}\psi^{\prime}(\varkappa)$ is the dimensionless
sphere mass, $\varkappa$ – dimensionless radius of the sphere, the prime
denotes $\zeta$ derivative.

The gas pressure on the surface of the sphere should be balanced by the
external pressure $P_{s}$. Taking into account the ideal equation of state and
(2)(7) relations we have:

$\displaystyle P_{s}=\frac{1}{4\pi
G^{3}M_{\text{tot}}^{2}}\left(\frac{kT}{m_{b}}\right)^{4}\cdot\varkappa^{4}\psi^{\prime
2}(\varkappa)e^{-\psi(\varkappa)}\stackrel{{\scriptstyle\text{def}}}{{=}}\Pi_{s}\cdot\pi_{s},$
(8)

where the dimensionless pressure $\pi_{s}=\varkappa^{4}\psi^{\prime
2}(\varkappa)e^{-\psi(\varkappa)}$ was introduced. Thus, setting $m_{b}$,
$M_{\text{tot}}$, $T$ and the external pressure $P_{s}$ we may find out the
radius of such gas sphere with solving
$P_{s}/\Pi_{s}(M_{\text{tot}},T,m_{b})=\pi_{s}(\varkappa)$ equation relatively
$\varkappa$, thereby to reveal the remaining characteristics using above
relations.

Figure 1: Dependence of the dimensionless external pressure $\pi_{s}$ on the
dimensionless radius $\varkappa$ of an isothermal sphere. Stable
configurations correspond to the solid line, unstable – dashed. The potential
$\psi$ is represented via dotted line.

The dependence $\pi_{s}(\varkappa)$ is shown on Figure 1, according to which
the dimensionless pressure does not increase monotonically with increasing of
sphere radius. As it was shown by Ebert (1955); Bonnor (1956), the global
maximum $\varkappa^{\text{max}}=6.45$, $\pi_{s}^{\text{max}}=17.65$ separates
regions of stable and unstable configurations. If the external pressure is
higher than this critical value $P_{s}>\Pi_{s}\pi_{s}^{\text{max}}$ then the
elasticity of the gas is insufficient to withstand it and compression must
occur. The critical mass corresponding to this pressure is called the Bonnor -
Ebert mass, it may be gotten from the definition (8) of $\Pi_{s}$:

$\displaystyle
M_{\text{tot}}^{\text{max}}=\frac{\alpha}{\sqrt{P_{s}G^{3}}}\left(\frac{kT}{m_{b}}\right)^{2},$
(9)

where $\alpha=\mu_{s}e^{-\psi(\varkappa)/2}/\sqrt{4\pi}=1.18$. The radius of
such a sphere may be expressed as:

$\displaystyle R^{\text{max}}=\frac{\beta}{\sqrt{P_{s}G}}\frac{kT}{m_{b}},$
(10)

with $\beta=\varkappa e^{-\psi(\varkappa)/2}/\sqrt{4\pi}=0.49$, which has
nonmonotonic dependence on $\varkappa$. Further consideration is devoted to
the question of the influence of DM on the Bonnor - Ebert mass and the
corresponding radius.

###  2.2 Jeans swindle

The gravitational stability of a homogeneous medium at rest was considered by
Jeans (1902). The initial prerequisites for such a consideration were not
entirely correct, since within the framework of Newtonian theory, there is no
equilibrium stationary distribution of matter. The fact is that from the Euler
equation at zero velocities and pressure gradients it follows that the
gravitational potential must be constant, while according to the Poisson
equation this is possible only in the absence of matter. Nevertheless,
consideration of the perturbations propagation against the background of a
constant gravitational potential (i.e. zero unperturbed gravitational force)
leads to the correct results. The assumption that the unperturbed
gravitational field is inessential for the propagation of perturbations is
called the Jeans swindle.

A correct stability analysis of a uniformly distributed gas against the
background of an expanding universe was made by Bonnor (1957). Before this, a
similar problem within the framework of the general relativity was considered
by Lifshitz (1946). These investigations showed an almost exact match with
critical mass of a gas obtained by Jeans (1902). Future consideration
(Kiessling (2003); Falco et al. (2013)) showed that Jeans swindle is not a
trick, but a completely physically grounded assumption, which at one time
could not appear in a self-consistent form due to the absence of the general
theory of relativity. In this work, we proceeded from Newton’s theory of
gravity and the uniform distribution of DM, assuming the validity of Jeans
swindle because of the reasons described above.

##  3 Problem formulation

###  3.1 Governing equations

We continued to describe the motion of a fluid hydrodynamically, while the
motion of DM particles was described using the collisionless kinetic equation.
These components interact with each other through the gravitational potential
$\phi$. Defining by $\mathbf{v}$ the velocity of dark matter particles with
mass $m_{d}$ and through $\mathbf{u}$ the velocity of the gas with density
$\rho$ and pressure $p$ we have the following equations:

$\displaystyle\frac{\partial f}{\partial t}+\mathbf{v}\frac{\partial
f}{\partial\mathbf{r}}-\frac{\partial\phi}{\partial\mathbf{r}}\frac{\partial
f}{\partial\mathbf{v}}=0,$ (11) $\displaystyle\frac{\partial\rho}{\partial
t}+\nabla{\rho\mathbf{u}}=0,$ (12)
$\displaystyle\rho\frac{\partial\mathbf{u}}{\partial
t}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}=-\nabla p-\rho\nabla\phi,$ (13)
$\displaystyle\Delta\phi=4\pi G\left(\rho+m_{d}\int d\mathbf{v}f\right),$ (14)

with ideal equation of state $p={kT}\rho/{m_{b}}$.

Let the initial configuration be stationary and consider small perturbations:
$\rho=\rho_{0}(\mathbf{r})+\rho_{1}(\mathbf{r},t)$,
$p=p_{0}(\mathbf{r})+p_{1}(\mathbf{r},t)$,
$\mathbf{u}=0+\mathbf{u}(\mathbf{r},t)$,
$f=f_{0}(\mathbf{v},\mathbf{r})+f_{1}(\mathbf{v},\mathbf{r},t)$,
$\phi=\phi_{0}(\mathbf{r})+\phi_{1}(\mathbf{r},t)$. Then the initial
configuration is described with the following equations:

$\displaystyle\nabla p_{0}=-\rho_{0}\nabla\phi_{0},$ (15)
$\displaystyle\mathbf{v}\frac{\partial
f_{0}}{\partial\mathbf{r}}-\nabla{\phi_{0}}\frac{\partial
f_{0}}{\partial\mathbf{v}}=0,$ (16) $\displaystyle\Delta\phi_{0}=4\pi
G\left(\rho_{0}+m_{d}\int d\mathbf{v}f_{0}\right),$ (17) $\displaystyle
p_{0}=c_{s}^{2}\rho_{0},$ (18)

where $c_{s}^{2}={kT}/{m_{b}}$ is the sound speed.

Equation (16) has the formal solution
$f_{0}(\mathbf{r},\mathbf{v})=F_{0}(\phi_{0}(\mathbf{r})+\mathbf{v}^{2}/2)$
with an arbitrary $F_{0}$. Supposing that kinetic energy of DM particles is
much more than gravitational energy (see Section 3.2 for details) we have
homogeneous spatial distribution: $f_{0}=n_{0}F(\mathbf{v})$, where
$n_{0}=const$ is the DM particles concentration. We limited ourselves to
considering such a case without assuming any coordinate dependence of the DM
density. Therefore the Jeans swindle should work for the DM component of the
potential (see Section 2.2), i.e. propagation of perturbations occurs in the
presence of only BM initial potential. For the spherical symmetric case the BM
unperturbed distribution is described in Section 2.1. Since the initial
configuration is stationary, we may set the perturbations to be proportional
to $e^{-i\omega t}$, thus we were not interested in the initial conditions of
the perturbations. Then the equations describing evolution of small
perturbations are following:

$\displaystyle-i\omega f_{1}+\mathbf{v}\frac{\partial
f_{1}}{\partial\mathbf{r}}-\nabla{\phi_{0}}\frac{\partial
f_{1}}{\partial\mathbf{v}}=n_{0}\frac{\partial
F}{\partial\mathbf{v}}\nabla\phi_{1},$ (19)
$\displaystyle-i\omega\rho_{1}+\nabla\rho_{0}\mathbf{u}=0,$ (20)
$\displaystyle-i\omega\rho_{0}\mathbf{u}=-c_{s}^{2}\nabla\rho_{1}-\rho_{0}\nabla\phi_{1}-\rho_{1}\nabla\phi_{0}$
(21) $\displaystyle\Delta\phi_{1}=4\pi G\left(\rho_{1}+m_{d}\int
d\mathbf{v}f_{1}\right).$ (22)

The positive sign of the imaginary part of $\omega$ corresponds to an unstable
initial configuration. Thus, solving these equations with the corresponding
boundary conditions discussed below, and finding $\omega$ as an eigenvalue,
one can make conclusion about the stability of an isothermal gas cloud in the
presence of DM for given $m_{b}$, $T$, $P_{s}$ and $m_{d}$, $n_{d}$,
$F(\mathbf{v})$.

###  3.2 Longwave approximation

Since the assumption has been made that the kinetic energy of the DM particles
is much greater than the energy of the gravitational field, the third term of
(19) can be neglected in comparison with the second one. Considering small
increments for the corresponding terms one can conclude that this means that
the work done by the gravitational field on a particle is small compared to
the kinetic energy that the particle originally had. For interstellar gas
clouds with their small gravitational gradients this approximation holds well
(see also Section 4.2.3).

Since a spatially endless DM distribution is considered, the localized gas
distribution can be represented in zero approximation as

$\displaystyle\rho_{1}^{(0)}=M_{1}\delta(\mathbf{r}),\,\,\,M_{1}=const,$ (23)

and then (19) and (22) may be solved to continue with (20)(21)(22) with
$f_{1}$ obtained in this way. Using Fourier transform it is easy to get the
solution of (19)(22)(23):

$\displaystyle f_{1}=-4\pi
Gn_{0}M_{1}\int\frac{d\mathbf{k}}{(2\pi)^{3}}\frac{1}{\mathbf{k}\mathbf{v}-\omega}\frac{e^{i\mathbf{k}\mathbf{r}}}{k^{2}+\chi(\mathbf{k},\omega)}\mathbf{k}\frac{\partial
F}{\partial\mathbf{v}},$ (24)

where

$\displaystyle\chi(\mathbf{k},\omega)=4\pi
Gm_{d}n_{0}\int\frac{d\mathbf{v}}{\mathbf{k}\mathbf{v}-\omega}\mathbf{k}\frac{\partial
F}{\partial\mathbf{v}}.$ (25)

Defining the presenting in (22) integral as $\eta$ we have:

$\displaystyle\eta(\mathbf{r})=m_{d}\int
d\mathbf{v}f_{1}=\int\frac{d\mathbf{k}}{(2\pi)^{3}}\frac{\chi(\mathbf{k},\omega)e^{i\mathbf{k}\mathbf{r}}}{k^{2}-\chi(\mathbf{k},\omega)},$
(26)

then the (22) reads:

$\displaystyle\Delta\phi_{1}=4\pi
G\left(\rho_{1}+M_{1}\eta(\mathbf{r})\right).$ (27)

Considering (20)(21)(27) we may see that they must have the same temporal
behavior (the same increment $\omega$). Thus, the solution of these equations
with appropriate boundary conditions will give the seeking increment $\omega$.

The initial assumption about localized BM density distribution implies that
the propagating in DM medium perturbations have wave length which is larger
than spatial scale of a gas cloud, i.e. $k$ is small. Therefore (25) may be
simplified:

$\displaystyle\chi(\mathbf{k},\omega)=-\frac{k^{2}}{\omega^{2}\tau_{d}^{2}}\left(1+\frac{2}{\omega}\overline{(\mathbf{k}\cdot\mathbf{v})}+\frac{3}{\omega^{2}}\overline{(\mathbf{k}\cdot\mathbf{v})^{2}}\right)+\mathcal{O}(k^{3}),$
(28)

where the overline means the averaged over a distribution values and
$\tau_{d}=(4\pi Gm_{d}n_{0})^{-1/2}$. We neglected the second term supposing
the absence of a directed DM movement. Then, using (26) definition,
$\eta(\mathbf{r})$ was gotten:

$\displaystyle\eta(\mathbf{r})=-\frac{1}{1+\omega^{2}\tau_{d}^{2}}\left(\delta(\mathbf{r})-\frac{3\tau_{d}^{2}}{1+\omega^{2}\tau_{d}^{2}}\overline{v_{\alpha}v_{\beta}}\frac{\partial^{2}}{\partial
r_{\alpha}r_{\beta}}\delta(\mathbf{r})\right),$ (29)

where the presence of the delta-functions is a consequence of the assumption
about the zero-order BM density spatial distribution (23).

###  3.3 Spherical symmetric case

Figure 2: Representation of a spherical symmetric baryonic matter (BM)
configuration in the presence of endless dark matter (DM).

We restricted ourselves to consider only spherical symmetric case of initial
gas configuration and perturbation propagation (see Figure 2). Therefore,
following by Chavanis (2002a), we introduced mass variables for $\rho_{1}$ and
$\eta$ as follows:

$\displaystyle\rho_{1}=\frac{1}{4\pi r^{2}}\frac{dq}{dr},$ (30)
$\displaystyle\eta=\frac{1}{4\pi r^{2}}\frac{d\Theta}{dr}.$ (31)

Then the perturbed Poisson equation (27) takes the just Newton’s law form:

$\displaystyle\frac{d\phi_{1}}{dr}=G\frac{q+M_{1}\Theta}{r^{2}},$ (32)

which is a representation of the Gauss theorem. From continuity equation (20)
we got:

$\displaystyle u=\frac{i\omega}{4\pi r^{2}\rho_{0}}q.$ (33)

Then, after substituting (32)(33) to (21) and some simple transformations the
equation, describing the radial mass perturbation behavior, was gotten:

$\left(\frac{\omega^{2}}{4\pi\rho_{0}}+{G}\right)\frac{q}{r^{2}}+\frac{d}{dr}\frac{c_{s}^{2}}{4\pi
r^{2}\rho_{0}}\frac{dq}{dr}=-G\frac{\Theta(r)}{r^{2}}M_{1}.$ (34)

This is the second order heterogeneous equation on eigenvalues $\omega$.
Therefore the three boundary conditions are required – two for determination
arbitrary constants of homogeneous solutions and third for finding possible
values of $\omega$. The Lagrangian derivative of perturbed pressure should
equals to zero at the sphere bounder, which in our terms has the form:

$\displaystyle\left(\frac{dq}{dr}-\frac{q}{\rho_{0}}\frac{d\rho_{0}}{dr}\right)_{r=R}=0.$
(35)

Also, by definition (30) we have:

$\displaystyle q(0)=0,$ (36)

which corresponds to zero mass including in zero radius sphere. Third
heterogeneous boundary condition was gotten considering $M_{1}$ as full
perturbed mass including in the sphere, so $M_{1}=4\pi\int\rho_{1}r^{2}dr$,
which with (30) leads to:

$\displaystyle q(R)=M_{1}.$ (37)

For spherically symmetric case the shape of $\Theta(r)$ is following:

$\displaystyle\Theta(r)=-\frac{1}{1+\omega^{2}\tau_{d}^{2}}\left(1+\frac{3\tau_{d}^{2}\overline{v^{2}}}{1+\omega^{2}\tau_{d}^{2}}\left(\frac{2\delta(r)}{r}-\delta^{\prime}(r)\right)\right),$
(38)

where the prime denotes $r$ derivative.

Thus, the formulated problem (34)–(38) completely determines the behavior of
the perturbations and the stability of isothermal sphere in presents of DM
with predefined BM density and DM velocity distributions. This problem may be
solved analytically for an arbitrary $\Theta(r)$ in case of constant density.

##  4 Stability investigation

###  4.1 Constant density

Despite the fact that a stationary, spatially limited configuration
configuration with $\rho_{0},T=const$ cannot be realized (see also Section
2.2), we considered it because there is an exact solution for perturbations
evolution in this case. It helped to construct a problem for isotherm
distribution (see Section 4.2.1) and reveal some quality characteristics which
should be expected from consideration of realizable equilibrium
configurations.

By setting $\rho_{0}=const$ we made $q(r)=r^{3/2}g(r)$ replacement which
transforms (34)–(37) problem to the form:

$\displaystyle
r^{2}g^{\prime\prime}+rg^{\prime}+\left(\nu^{2}r^{2}-\frac{9}{4}\right)g=-l^{-2}r^{1/2}\Theta(r)M_{1},$
(39) $\displaystyle g(0)=0,$ (40) $\displaystyle
Rg^{\prime}(R)+\frac{3}{2}g(R)=0,$ (41) $\displaystyle g(R)=R^{-3/2}M_{1},$
(42)

where $l^{-2}=4\pi G\rho_{0}c_{s}^{-2}=\tau_{b}^{-2}c_{s}^{-2}$ and
$\nu^{2}=c_{s}^{-2}\omega^{2}+l^{-2}$. The general solution of homogeneous
part is the combination of Bessel functions of $3/2$ and $-3/2$ orders
therefore it is possible to set up solution of heterogeneous equation via
Wronskian. After some transformations the general solution may be written as:

$\displaystyle g(r)=$ $\displaystyle C_{1}J_{3/2}(\nu r)+C_{2}J_{-3/2}(\nu
r)+$
$\displaystyle+\frac{M_{1}}{l^{2}\nu^{3}}r^{-3/2}\int_{0}^{r}dz\,\,z^{-2}\Theta(z)\Big{\\{}\nu(r-z)\cos[\nu(r-z)]\Big{.}$
$\displaystyle\Big{.}-(1+\nu^{2}rz)\sin[\nu(r-z)]\Big{\\}}.$ (43)

With the boundary conditions (40)(41)(42) we came to the equation for the
increment:

$\displaystyle\frac{R}{l^{2}}\sqrt{\frac{\pi}{2}}\int_{0}^{R}\frac{J_{3/2}(\nu
z)\Theta(z)}{\sqrt{\nu z}}dz=\sin\nu R.$ (44)

In the absent of DM (i.e. $\zeta=0$) the usual Jeans radius may be gotten.
Substituting there (38) and dividing it on $\nu R\neq 0$ we got the equation:

$\displaystyle\frac{\sin\left(\varkappa\sqrt{1+\lambda^{2}}\right)}{\varkappa\sqrt{1+\lambda^{2}}}=-\sigma\frac{\sigma+\lambda^{2}+\eta(1+\lambda^{2})}{\lambda^{2}(\sigma+\lambda^{2})(1+\sigma+\lambda^{2})},$
(45)

where $\lambda^{2}=\tau_{b}^{2}\omega^{2}$ – dimensionless increment,
$\varkappa=R/c_{s}\tau_{b}$,
$\sigma=\tau_{b}^{2}/\tau_{d}^{2}=m_{d}n_{0}/\rho_{b}$ – ratio of the
densities of DM and BM, $\eta=3\overline{v^{2}}/c_{s}^{2}$ – the ratio of the
mean square speed of DM particles to the sound speed in gas.

Figure 3: Dependence of the maximum stable dimensionless radius $\varkappa$ of
an homogeneous sphere on the relative concentration of DM particles
$\sigma=m_{d}n_{0}/\rho_{0}$ at various relative mean square velocities
$\eta=3\overline{v^{2}}/c_{s}^{2}$: $\,\,3\cdot 10^{6}$ (solid line), $3\cdot
10^{4}$ (dashed), $10$ (dotted), $1$ (dot-dashed) and $10^{-2}$ (double-dot-
dashed).

Figure 4: Dependence of the maximum stable dimensionless radius $\varkappa$
(top left panel), dimension radius ($\beta$ from equation (10), left bottom),
mass ($\alpha$ from equation (9), top right) and ratio of average-to-central
densities $\vartheta=\overline{\rho}/\rho_{c}$ (right bottom) of an isothermal
sphere on the relative density of DM particles $\sigma=m_{d}n_{0}/\rho_{c}$
with various relative mean square velocities
$\eta=3\overline{v^{2}}/c_{s}^{2}$: $\,\,3\cdot 10^{6}$ (solid line), $3\cdot
10^{4}$ (dashed), $10$ (dotted), $1$ (dot-dashed) and $10^{-2}$ (double-dot-
dashed).

First of all it should be noted that, as follows from (45), when DM particles
are at rest (i.e. $\eta=0$), any arbitrarily small portion of DM leads to the
contraction of the gas sphere. This is because the resting gravitating DM
particles cannot stay in equilibrium and begin to collapse, pulling along the
gas, which interacts with them via the gravitational potential. However, this
case does not apply to our consideration due to the assumption made in
Sections 3.1, 3.2 about the relatively large kinetic energy of the DM
particles. To check how the mean square velocity affects the stability,
several values of $\eta$ were chosen, for each of which the dependence of the
critical radius $\varkappa$ on the relative density of DM particles $\sigma$
was calculated.

The results of the calculations are presented in Figure 3, from which it can
be seen that the small addition of slow DM particles leads to a increase in
the radius $\varkappa$, and, accordingly, the mass, of a homogeneous gas
cloud. Herewith, an increase of the DM particle velocity $\eta$ leads to a
significant increase of the stable radius. Moreover, as expected, an increase
in the relative density $\sigma$ of the DM leads to a decrease of $\varkappa$
at any values of the particle velocity. For large values of $\eta$ at low DM
densities, extremely large values of the radius/mass of a homogeneous cloud
were obtained. The physically meaningful case of an isothermal sphere need not
have such large stable radius values, but we expected it to behave similarly.
The whole case with constant density can be considered only as a toy model.
Having dealt with the characteristic features of this case, we proceeded to
study of the stability of isothermal spheres.

###  4.2 Isothermal sphere stability

####  4.2.1 Problem determination

For the convenience of further numerical calculations, we transferred the
values associated with the delta-function from the right side of (34)(38) to
the boundary condition (36). To do this, we solved the problem on the interval
$[0,\epsilon]$, where $0<\epsilon\rightarrow 0$, therefore the density can be
considered being unchanged in this interval. The corresponding general
solution was already obtained in the Section 4.1. To satisfy the boundary
condition (36) in the solution we had to nullify $C_{2}$ in (43), so it turns
out that the solution is expressed only in terms of the integral, which can be
easily calculated for $\epsilon\rightarrow 0$. Thus, we got an expression for
$q(\epsilon)$ and may solve the problem on the interval $[\epsilon,R]$.
Defining $\mu=q/M_{1}$ and dimensionless values according to Sections 2.1, 4.1
the following problem was gotten:

$\displaystyle(\lambda^{2}e^{\psi}+1)\mu+\zeta^{2}\frac{d}{d\zeta}\frac{e^{\psi}}{\zeta^{2}}\frac{d\mu}{d\zeta}=\frac{\sigma}{\sigma+\lambda^{2}},$
(46)
$\displaystyle\mu(\epsilon)=-\frac{\sigma}{(\sigma+\lambda^{2})^{2}}\eta,\,\,\,\,\,0<\epsilon\rightarrow
0,$ (47) $\displaystyle\mu(\varkappa)=1,$ (48)
$\displaystyle\frac{d\mu}{d\zeta}\Big{|}_{\varkappa}=-\frac{d\psi}{d\zeta}\Big{|}_{\varkappa},$
(49)

which may be examined as an eigenvalue problem for the set of parameters
$\varkappa$, $\sigma$, $\eta$. To test the obtained problem (46)–(49) we
calculated $\lambda$ for some arbitrary parameters with $\psi=0$. The
resulting $\lambda$ were the same as in the case of $\rho=const$ for any small
enough $\epsilon$ like $10^{-5}$ or $10^{-10}$. Therefore, believing that the
problem was set correctly, we proceeded to the stability analysis with
$\psi(\zeta)$ obtained from the equations (4)(5).

####  4.2.2 Analysis of results

The results of the calculations are presented in Figure 4 in terms of
$\sigma=m_{d}n_{0}/\rho_{c}$, although instead of $\rho_{c}$ another gas
characteristic calculated from the relations in Section 2.1 may be used. It
can be seen that, as in the case of constant density, the presence of slow
($\eta=10^{-2}$) DM particles only leads to a decrease in both the
dimensionless stable radius and mass of a gas cloud. Corresponding dimensional
radius $\beta$ from equation (10) increases slightly up to $\sigma\approx
0.03$ and then decreases thereafter. It can be seen that for faster particles
the dimensionless radius considerably increases up to $\sigma\approx 0.03$,
after which it sharply decreases. At the same time, with increasing $\sigma$,
the values of the dimensional radius decreases to $\beta=0.37$, after which,
at DM dominant state $\sigma\approx 10$, they return to values corresponding
to the absence of DM ($\beta=0.49$), and then rapidly decrease. The mass of
the cloud $\alpha$ from (9) behaves similarly, dropping to $\alpha=0.64$ at
$\sigma\lesssim 10$, i.e. decreasing by $2.5$ times compared to a gas cloud
without DM. Moreover, for very fast particles ($\eta=3\cdot 10^{6}$) this
occurs at very low DM densities ($\sigma\sim 10^{-7}$).

The above means that presence of DM particles makes it possible to realize
configurations of an isothermal self-gravitating gas sphere lying in
inaccessible without DM areas. Indeed, as it can be seen from the Figure 1,
the critical equilibrium point shifts to the right beyond the global maximum
of $\pi_{s}$ as $\sigma$ increases. The greater the relative speed $\eta$, the
further along the curve the equilibrium point can move before it starts to
move backwards at the value $\sigma\approx 0.03$. When the DM amount
sufficiently exceeds the amount of BM ($\sigma\gtrsim 10$), only small, low-
mass stationary configurations are possible, corresponding to the near-zero
positions in the Figure 1. As far as it can be seen from our numerical
calculations, the point $\sigma\approx 0.03$ is not the same for all $\eta$
considered, but differs in the third digit. Its presence was quite expected
(see Section 4.1), but the calculations above were needed to determine its
value.

Since at collapse the DM particles are carried along by the collapsing gas,
the collapse of DM and BM must start simultaneously (Zeldovich et al. (1980)).
That is why an increase in velocity of DM particles prevents the beginning of
collapse, while an increase in their concentration promotes it. At the regime
with dominating DM the BM contribution is insignificant and stability of such
configuration is described by Jeans theory applied to the homogeneous DM we
consider.

Since the dimension radius $\beta$ from (10) has non-monotonic dependence on
dimensionless one $\varkappa$ and the behavior of $\beta$ and the mass
$\alpha$ are similar, it is difficult to judge about the structure of a stable
configuration of an isothermal sphere. To clarify this, we calculated
$\vartheta$ – the ratio of average density to density in the center for the
critical configurations using (7)(9)(10) relations:

$\displaystyle\vartheta=\frac{\overline{\rho}}{\rho_{c}}=3\frac{\psi^{\prime}(\varkappa)}{\varkappa},$
(50)

which is represented at Figure 4. It can be seen that the presence of
relatively small amounts of DM leads to the formation of a gas cloud with a
high concentration of matter in the center and low one at the periphery.
Moreover, $\vartheta$ can achieve values $\sim 10^{-3}$, i.e. almost all gas
is concentrated around the center of a sphere. And, as in the case of the
quantities discussed earlier, a small number of very fast DM particles are
sufficient to produce this effect. Then, as $\sigma$ increases so does
$\vartheta$, tending to values equal to to $1$, which means that in the case
of DM dominance the density of an isothermal sphere is homogeneous, although,
as it was shown above, such configurations must be spatially small.

Without aiming at a detailed analysis of ISM conditions, we estimated effects
of DM presence on an isothermal gas sphere. We took the typical for solar
neighborhood parameters: $\rho_{c}=5\cdot 10^{-22}\text{ g cm}^{-3}$,
$T=50\text{ K}$, $m_{b}=2\cdot 10^{-24}\text{ g}$ with corresponding
$c_{s}=8.3\cdot 10^{4}\text{ cm}^{2}\text{ s}^{-2}$ and usual ones for DM
consisting of WIMPs: $m_{d}n_{0}=5\cdot 10^{-25}\text{ g cm}^{-3}$,
$(\overline{v^{2}})^{1/2}=6\cdot 10^{7}\text{ cm}^{2}\text{ s}^{-2}$. The
obtained results are presented in Table 1, which shows that the presence of DM
increases the maximum stable mass of such a cloud by a factor of four, the
radius by five, and decreases the average density by a factor of almost
thirty.

| $\varkappa$ | Mass, $M_{\odot}$ | Radius, ly | Mean density,$\text{ g cm}^{-3}$
---|---|---|---|---
Without DM | 6.45 | $1.2\cdot 10^{3}$ | 19.6 | $8.8\cdot 10^{-23}$
With DM | 31.36 | $4.8\cdot 10^{3}$ | 95 | $3.2\cdot 10^{-24}$
Table 1: Values of an critical isothermal gas cloud at typical interstellar
conditions parameters without and with DM having $\sigma=10^{-3}$ and
$\eta=3\cdot 10^{6}$.

####  4.2.3 Applicability limits

To check the limits of applicability we calculated the work done by the
gravitational field of a gas cloud on a DM particle to move it from its
boundary to the center, which turned out to be
$\psi(\varkappa)m_{d}c_{s}^{2}$. According to Section 3.2 this work must be
much less that the kinetic energy $m_{d}\overline{v^{2}}/2$, therefore we got
the following criterion:

$\displaystyle\eta\gg\psi(\varkappa).$ (51)

$\psi(\varkappa)$ is represented at Figure 1, from which it can be seen that
for $\eta=10^{-2},1$ this condition is only fulfilled at large values of
$\sigma$ when the radius of the sphere $\varkappa$ is exceptionally small. But
for large values of the relative velocity of DM particles $\eta$ this
condition is well satisfied, which means that both the spatial DM density
distribution and the third term in (19) might be neglected.

##  5 Conclusions

The stability of self-gravitating isothermal gas spheres in the presence of
homogeneously distributed DM was analyzed. The behavior of the gas was
described hydrodynamically, while the behavior of the DM particles was
described using the kinetic equation. In describing the evolution of small
perturbations of the distribution function, the gravitational effect of the
gas cloud on these perturbations was neglected (the third term in (19)),
which, according to Section 4.2.3, is a good approximation for relatively fast
particles. In addition, in Section 3.2 we choose the BM spatial distribution
in zero approximation as delta-function since it is a finite configuration in
the background of infinite distributed DM. Then we obtained the problem
(34)–(38) of finding first approximation of the perturbations and the
instability increment for the spherically symmetric case. Further, having
obtained an exact solution for the case of constant density, we formulated a
numerically convenient problem (46)–(49) for isothermal spheres. The
calculation results presented in Figure 4 show that increasing DM density up
to $0.03\rho_{c}$, where $\rho_{c}$ is a central density, allows the
inaccessible stable configurations located to the right of the global maximum
in Figure 1 to be realized. Moreover, the greater the root mean square
velocity of DM particles, the more distant from the global maximum
configuration can be realized. A further increase in the DM density leads to a
shift of a critical stable configuration in the opposite direction. At low DM
densities, the mass of the gas in the cloud and its radius can increase
several times, reducing its average density. Calculations performed for
typical for solar neighbourhood parameters of ISM and DM consisting of WIMPs
showed (see Table 1) that DM presence increases the maximum stable mass of an
isothermal cloud by a factor of four, the radius by five, and decreases the
average density by a factor of thirty. It should be noted that parameters of
both DM and ISM can differ widely both for different parts of a galaxy and for
a considered cosmological epoch.

Thus, it was shown that the presence of even a small number of fast DM
particles leads to a significant change in the parameters and structure of
isothermal gas clouds, making the realization of unstable configurations
possible. We have considered the simplest model of isothermal clouds, which
does not take into account many effects important for ISM (see Section 1). In
order to get some accurate observational implications that can tell us
something about the nature of DM, it seems necessary to take those effects
into account. However, as follows from our simplified consideration, such
effects should be quite significant.

When hydrogen in the central part of a star is depleted, the nuclear energy
release here ceases and passes to the upper layers surrounding the burnt-out
core. As a result, an isothermal gas sphere appears in the center of the star,
which is under pressure from the overlying non-isothermal layers. Such stars
that left the main sequence are called red giants. If the star is massive
enough ($M\gtrsim 2.5M_{\odot}$) then its core consists of non-degenerate gas
and may be described as in Section 2.1. Our work has shown that the presence
of even a small amount of fast DM particles interacting with the gas only
gravitationally leads to a significant increase in the maximum possible mass
and size of the isothermal sphere, which should be reflected in the structure
of the red giant and its subsequent fate. Although in this case our
consideration may not be entirely appropriate because the influence of the
star’s gravitational field on the initial DM spatial distribution and the
development of the instabilities may be quite significant. However, the DM
presence must have a significant influence on the entire evolution of stars,
from the protostellar clouds to the compact objects formation. Several recent
works (Raen et al. (2021); Ellis (2021); Sagun et al. (2021),Lopes, José &
Lopes, Ilídio (2021)) have focused on the effects of DM particle captured by
stars and theirs possible observational manifestations. In addition, it would
be useful to consider the gravitational stability of stars in the approach
presented in this paper to find out the influence of DM on their fates.

## Acknowledgements

IK thanks G. Kolomiytsev for productive discussions. The authors are grateful
to A. Doroshkevich for useful final remarks.

Make science, not war

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# Generalized optical theorem for Rayleigh scattering approximation

Irving Rondón111corresponding author<EMAIL_ADDRESS>and Jooyoung Lee
School of Computational Sciences,
Korea Institute for Advanced Study,
85 Hoegi-ro, Seoul 0245,
Republic of Korea

###### Abstract

A general expression for the optical theorem for probe sources given in terms
of propagation invariant beams is derived. This expression is obtained using
the far field approximation for Rayleigh regime. In order to illustrate this
results is revisited the classical and standard scattering elastic problem of
a dielectric sphere for which the incident field can be any propagation
invariant beam.

##  1 Introduction

In electromagnetic theory the Optical Theorem (OT) has a very long an
interesting history (See R. Newton [1] and references therein). This physical
concept of the OT is a useful result in scattering theory, relating the
extinction cross section of a structure to the scattering amplitude in the
forward direction [2, 3]. In applied sciences to understand this mechanics and
how the absorption and the scattering affect wave propagation throughout a
medium can help to obtain a meaningful features of the object such as the
characterization and its physical properties [4].
Recently, some interesting reviews have been published [5, 6] concerning the
development of generalized Lorenz-Mie theories, the Extended Optical theorem
(EOT), using structured beam shape methods are presented, and the description
of several electromagnetic effects by experiments, some applications using
T-matrix methods for structured beam illumination.
The optical theorem has generated important and new applications, such as
calculating the complex scattering amplitude on spherical and cylindrical
objects, application in acoustic backscattering [7], to understand correctly
the physical effects of propagation in radiation force and torque [8], without
input nonphysical effect or based on numerical calculations (See [9] and
references).
A generalized optical theorem for acoustic waves has been published by Marston
and Zhang [10, 11], recently extended for arbitrary beams [12]. For
electromagnetic fields [13, 14, 15, 16] quantum mechanics [17], in time
domain, transmission lines, propagation in acoustic and electromagnetic waves
in anisotropic medium [18, 19, 20], seismologic waves [21], the manipulation
of the scattering pattern using non-Hermitian particles [22].
In the following section, a general derivation EOT applicable to any
“nondiffracting beams”using the Huygens principle in the far field
approximation for Rayleigh regime. A generalized scattering amplitude function
is presented. These results are illustrated using a Bessel beam.

##  2 Theoretical background analysis

Let us consider a scattering particle of arbitrary form and size, with volume
$V$ a boundary surface $\partial\Omega$ and, a complex permittivity
$\varepsilon\left(\vec{r}\right)=\varepsilon_{0}\left[\varepsilon_{r}^{\prime}(\vec{r})+i\varepsilon_{r}^{\prime\prime}(\vec{r})\right],$
where $\varepsilon_{0}$ is the vacuum permittivity. For simplicity the medium
surrounding the scattering particle is lossless and its permittivity is
$\varepsilon$ and not magnetic. Let us consider the incident arbitrary beams
as $\vec{E}_{i}(\vec{r}),\,\vec{H}_{i}(\vec{r})$ that strikes the scattering
particle, $\vec{E}_{s}(\vec{r}),\,\vec{H}_{s}(\vec{r})$ the scattered
electromagnetic field and $\vec{E}(\vec{r}),\,\vec{H}(\vec{r})$ as the total
fields [3, 2, 23, 24].

$\vec{E}=\vec{E}_{i}+\vec{H}_{s},$ (1) $\vec{H}=\vec{H}_{i}+\vec{H}_{s}.$ (2)

The time-averaged power absorbed by the scattering object is given by

$P_{a}=-\frac{1}{2}\int_{\partial\Omega}\Biggl{[}\vec{n}\cdot\left[\left(\vec{E}_{i}+\vec{E}_{s}\right)\times\left(\vec{H}_{i}^{*}+\vec{H}_{s}^{*}\right)\right]\Biggr{]}dS,$
(3)

this expression can be written in term of the Poynting vector as

$P_{a}=-\frac{1}{2}\int_{\partial\Omega}\vec{n}\cdot\left[\vec{S}_{i}+\vec{S}_{s}+\vec{S}^{\prime}\right]dS,$
(4)

Where the first term of Eq. (4) is the incident Poynting vector

$\vec{S}_{i}=\frac{1}{2}\mathbf{Re}\left[\vec{E_{i}}\times\vec{H}^{*}_{i}\right],$
(5)

followed by the scattered vector

$\vec{S}_{s}=\frac{1}{2}\mathbf{Re}\left[\vec{E}_{s}\times\vec{H}_{s}^{*}\right],$
(6)

and the cross term interaction vector, which depend in term of the initial and
scatter vectors

$\vec{S}^{\prime}=\frac{1}{2}\mathbf{Re}\left[\vec{E}_{i}\times\vec{H}_{s}^{*}+\vec{E}_{s}\times\vec{H}_{i}^{*}\right].$
(7)

Using these equations and the right boundary condition which depend of the
specific geometry. It is possible to obtain analytical and numerical solution
for general scattering problem (See [23, 24] where the authors has reported a
pedagogical tutorials).
In this letter, we follow and adapt the approach presented in ( See. Refs [25,
26]) for an arbitrary structured beams, where in general the electromagnetic
fields can be expressed as $\vec{E}(r,t)=\mathbf{Re}[{E(r)e^{-i\omega t}}]$
and $\vec{H}(r,t)=\mathbf{Re}[{H(r)e^{-i\omega t}}]$, where $\omega$ is the
harmonic frequency of the wave (single frecuency).
Let us define the following incident fields as

$\vec{E}_{i}=\hat{e}_{i}\varphi(x,y)e^{ik_{z}z}e^{-i\omega t},$ (8)

$\vec{H}_{i}=\frac{1}{i\omega\mu_{0}}\nabla\times\vec{E}_{i},$ (9)

where $\varphi(x,y)$ physically represent a structured beam (scalar function)
which satisfies the Maxwell’s equations and the transversal Helmholtz equation
$\nabla_{t}^{2}\varphi+k_{t}^{2}\varphi=0$, in Cartesian, circular, parabolic
cylindrical, and elliptical coordinates [30, 27, 28, 29] and $k_{t}$ is the
transversal wave vector. For simplicity the factor $e^{-i\omega t}$ is
omitted.
In the following is considered a polarized electric field, with a complex
amplitude $E_{0}$.

$\vec{E}_{i}=\hat{e}_{x}E_{0}\varphi(x,y)e^{ik_{z}z},\\\ $ (10)

$\vec{H}_{i}\approx\hat{e}_{y}\frac{E_{0}}{\omega\mu_{0}}k_{z}\varphi(x,y)e^{ik_{z}z},$
(11)

The Poynting vector for these incident fields lies the plane XY [3].
In this approximation has been assumed that any impinging electric field can
be written as invariant structured beam using the plane wave spectrum
representation [30, 27, 28, 29].

$\varphi(x,y)=\int_{0}^{2\pi}A(\phi)e^{ik_{t}(x\cos\phi+y\sin\phi)}d\phi,$
(12)

where $A(\phi)$ is an angular function. Note that for
$A(\phi)=\delta(\phi-\phi_{0})$, the plane wave case is recovered, $k_{t}$ is
the transversal wave vector. This equation has the following interpretation. A
structured or non-diffracting beam is given by a superposition of multiple
plane waves having transversal wave vectors $k_{t}$ on a circle. Only for
particular functions $A(\phi)$ on the Whittaker’s integral can be expressed
analytically. For example higher order Bessel beams are defined by
$A(\phi)=e^{im\phi}$, where $m$ is the azimuthal order of the beam. Mathieu
beams are defined by $A(\phi)=C(m,q,\phi)+iS(m,q,\phi)$ are the Mathieu cosine
and sine functions. Weber beam are defined by
$A(a,\phi)=\frac{1}{2(\pi|\sin\phi|^{1/2})}e^{ia\ln|\tan\phi/2|}$ in turn is
divided into even and odd case [27].

After substituting Eqs. (10) and (11) into Eq. (4) is written as

$P_{a}+P_{s}=-\frac{1}{2}\int_{\partial\Omega}\mathbf{Re}\left[\vec{E}_{i}\times\vec{H}_{s}^{*}+\vec{E}_{s}\times\vec{H}_{i}^{*}\right]\cdot\vec{n}dS,$
(13)

Figure 1: Differential scattering cross section (23) normalized and the
polarization function (24) vs scattering angle for a plane wave and Bessel
beam. In the Bessel beam case the sphere is on the beam axis and the beam
order is $m=0$

.

It is possible to recast Eq. (13) after some vector algebra, as follow

$P_{a}+P_{s}=\frac{\mathbf{Re}}{2}\int_{\partial\Omega}\Biggl{[}\hat{e}_{x}\cdot\left(\vec{n}\times\vec{H}_{s}\right)E_{0}^{*}\varphi^{*}(x,y)e^{-ik_{z}z}+\hat{e}_{y}\cdot\left(\vec{n}\times\vec{E}_{s}\right)\varphi^{*}(x,y)\frac{E_{0}^{*}e^{-ik_{z}z}}{\eta}\Biggr{]}dS$
(14)

where $\eta=\sqrt{\mu/\varepsilon}$ is denoted for the impedance. This is a
general expression that represent from the physical point of view the
interaction between the incident and scattered fields in term of the Green
function, also related with the Huygens principle[2, 25, 26]. Therefore, the
scattered electric field in the far field can be expressed as

$\vec{E}_{s}=\int_{\partial\Omega}i\omega\mu
G(\vec{r},\vec{r^{\prime}})\cdot\left[\hat{n}\times\vec{H^{\prime}}_{s}\right]+\nabla\times
G(\vec{r},\vec{r^{\prime}})\cdot\left[\hat{n}\times\vec{E^{\prime}}_{s}\right]dS^{\prime}.$
(15)

Alternatively, this field in the scattered vector in the far approximation can
also be expressed as

$\vec{E}_{s}(\vec{r})=\frac{e^{ikr}}{r}F(\hat{k}_{i},\hat{k}_{s})\cdot\hat{e}_{i},$
(16)

where the function $F(\hat{k}_{i},\hat{k}_{s})$ is called the scattering
amplitude function, $\hat{e}_{i}$ is an unitary vector. A similar expression
is related with the pressure in acoustic [10, 11, 12] .
This function can provide itself very interesting physics in scattering
phenomena. Indeed, in Refs. [31, 32] the authors proposed a method how to
obtain analytical expressions for the scattering amplitude function in order
to explore an acoustic Bessel beam, later extended for Mathieu [33], and Weber
[34] waves. For instance , in order to obtain an OT expression for arbitrary
beam, we use Eq. (16) in the forward direction i.e $\hat{k}_{s}=\hat{k}_{i}$
[38], taking the product with
$\hat{e}_{i}\varphi^{*}(x^{\prime},y^{\prime})E_{0}^{*}$ , and making the
integration over the scattered object in the far field approximation for the
electromagnetic case [35, 36, 37, 15, 16]. Also a clear derivation for the ETO
theorem for acoustic waves using Jones lemma can be reviewed [10, 11, 12]

$\hat{e}_{i}\varphi^{*}(x^{\prime},y^{\prime})\cdot\vec{E}_{s}=\int_{\partial\Omega}\varphi^{*}(x^{\prime},y^{\prime})e^{-i\hat{k}_{s}\cdot\vec{r}^{\prime}}\left[\hat{e}_{i}\cdot(\hat{n}\times\vec{H^{\prime}}_{s})+(\hat{k}_{i}\times\hat{e}_{i})\cdot(\hat{n}\times\vec{E^{\prime}}_{s})\right]dS^{\prime},$
(17)

Therefore, without loss generality applying the same physical condition as
[26, 25, 35, 36, 38] using Eq. (16) and (17), the optical theorem can be
written as

$\sigma_{\text{ext}}=\frac{4\pi}{k}\mathbf{Im}\left[\hat{e}_{i}\cdot\varphi^{*}(x^{\prime},y^{\prime})F(\hat{k}_{i},\hat{k}_{i})\cdot\hat{e}_{i}\right].$
(18)

This expression recover the OT for the plane wave case,

$\sigma_{\text{ext}}=\frac{4\pi}{k}\mathbf{Im}\left[\hat{e}_{i}\cdot
F(\hat{k}_{i},\hat{k}_{i})\cdot\hat{e}_{i}\right]$ (19)

##  3 Generalized scattering amplitude function

In this section, we show an application of the scattering amplitude function.
Let us consider an electric field

$\vec{E}=\frac{3}{2+\varepsilon_{r}}\vec{E}_{i}.$ (20)

this equation describe the electric field $\vec{E}$ inside a dielectric sphere
immersed in term of the incident electric field $\vec{E}_{i}$ [2, 3, 26]

In order to avoid redundancy, we put forward the scattering amplitude function
the Rayleigh regime reported [38], but here written in term of Whittaker
integral as

$F(\hat{i},\hat{o})=k^{2}a^{3}\left(\frac{n^{2}-1}{n^{2}+2}\right)\left[\hat{e}_{i}-\left(\hat{o}\cdot\hat{e}_{i}\right)\hat{o}\right]e^{ik_{z}z}\int_{0}^{2\pi}A(\phi)e^{ik_{t}(x\cos\phi+y\sin\phi)}d\phi,$
(21)

where $n$ is the complex refraction index, $k$ is the wave vector and $a$ is
the sphere radius.
The square of this equation an integration over the solid angle $d\Omega$
gives the differential scattering cross section as

$\frac{d\sigma_{\text{s}}}{d\Omega}=|F(\hat{i},\hat{o})|^{2}=k^{4}a^{6}\left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}I_{\text{beam}},$
(22)

where
$I_{\text{beam}}=\left[1-\left(\hat{o}\cdot\hat{e}_{i}\right)^{2}\right]|\varphi(x,y)|^{2},$
is the intensity expressed in term of a scalar potential, it is related with
the incident angle and observation point, if $\hat{e}_{i}$ is perpendicular to
the scattering plane, and $\hat{o}\cdot\hat{e}_{1}=\sin\theta$, if
$\hat{e}_{i}$ lies in the plane, $|\varphi(x,y)|^{2}$ is related with the
intensity. Using this expression several probe fields can be used to measure
the scattering cross section. In the following, it is denoted
$\sigma_{\text{d}}\equiv d\sigma_{\text{s}}/d\Omega$.

Figure 2: Normalized differential scattering cross section and the
polarization scattered function for a Bessel beam with $m=0$, using the
equations (23) and (24) with $\beta=35^{\circ}$.

###  3.1 The Rayleigh scattering approximation using a Bessel beam

In this section, we consider an incident electric field expressed using Eq.
(20) taking the angular spectrum as $A(\phi)=e^{im\phi}$, which represent a
linear polarized Bessel beam of $m$ order is expressed as
$\vec{E}_{i}(\vec{r})=\hat{e}_{i}J_{m}(k_{t}r)e^{ik_{z}z}$. Using this result
it is possible to obtain the polarization function of the scattered radiation
as

$\sigma_{\text{d}}(\theta)=\frac{k^{4}a^{6}}{2}\left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}J_{m}^{2}(k_{t}r)(1+\cos^{2}\theta).$
(23)

In this problem,we have used the standard scattering geometry [41, 42, 40] to
relate the scattering angle
$\theta=\arccos\left(\hat{e}_{i}\cdot\hat{o}\right)$ to the unit vectors
$\hat{i}$ and $\hat{o}$ in a particular point P, where the scattered radiation
is observed. In the other hand, if the incident field is unpolarized, the
differential scattering function is the average over parallel and
perpendicular incidents fields, then
$\sigma_{\text{d}}(\theta)=\frac{1}{2}\left[\sigma^{\perp}_{\text{d}}(\theta)+\sigma^{\parallel}_{\text{d}}(\theta)\right]$,
and $\hat{o}\cdot\hat{e}_{i}=0$ if $\hat{e}_{i}$ is perpendicular to the
scattering plane and $\hat{o}\cdot\hat{e}_{1}=\sin\theta$ if $\hat{e}_{i}$
lies in the plane. Using these information we can calculate the polarization
scattered function as

$\Pi(\theta)=\frac{\sigma^{\perp}_{\text{d}}(\theta)-\sigma^{\parallel}_{\text{d}}(\theta)}{\sigma^{\perp}_{\text{d}}(\theta)+\sigma^{\parallel}_{\text{d}}(\theta)}=\frac{\sin^{2}\theta}{1+\cos^{2}\theta}J_{m}^{2}\left(k_{t}r\right).$
(24)

Figure 3:  Normalized differential scattering cross section and the
polarization scattered function for a Bessel beam with $m=1$ and with
$\beta=60^{\circ}$.

It is important to note, that integrating the equation Eq. (23) over $d\Omega$
is straightforward to obtain $\sigma_{\text{s}}$ [38], the scattering
transversal section, while the absorbed transversal section
$\sigma_{\text{a}}$ have to be obtained with the Poynting vector [2, 3, 25,
26], since in other way, this would result in $\sigma_{\text{ext}}=0$. Note
that taking $r\rightarrow 0$ into (23) and (24) is recovered the plane wave
case [2].

Finally, in order to validate our results, we show in Fig. (1) some numerical
results, in which we compare the scattered radiation as function of the
dispersion angle for both the differential cross section and the polarization
scattered function, considering a plane wave and a Bessel beam with $m=0$. For
simplicity, we make the analysis in one dimension and vary the angle from $0$
to $\pi$. For the Bessel beam case, $k_{t}=k\sin\beta$ is the transversal wave
vector, and $\beta$ is the value of the half-cone angle.
In addition, this model reproduce the expected behavior for a plane wave [2].
For a plane wave case, the differential cross section present a minimum at
$\pi/2$ and the polarization scattered function a maximum at $\pi/2$ See
Figure 3, these angles the scattered radiation is linear polarized, this
effect was proposed by Rayleigh to explain the blue sky [2, 42, 41]. At these
same values for a zero Bessel beam, the differential cross section shows a
reduction respect to the plane wave case, with a maximum around $\pi/3$, while
the scattered polarization function shows a clear decreasing of the scattered
radiation.
In Figure (2), we show the three dimensional behavior of the differential
cross section $\left(\sigma_{s}\right)_{\text{BB}}$ and the polarization
scatter function $\Pi(\theta)_{\text{BB}}$ with $m=0$ and $\beta=35^{\circ}$
for a Bessel beam. In addition, to illustrate how the scattering cross section
and the polarization scattered function change with the variation of the
incident field, we show in Figure (3) the behavior of these quantities for
$m=1$ and $\beta=60^{\circ}$. Bi-dimensional scattered pattern were reported
using only a zero order Bessel EM scattering in a dielectric sphere [38] where
the author compare the incident Bessel beam as a function of the impinging
angle $\beta$.

##  4 Conclusions

A general optical theorem for any arbitrary beam was derived, using the
amplitude scattering function and the Huygens principle in the far field
approximation. the presented ordinary form of the optical theorem renders the
particular case for free space derived in previous works [9, 10, 11, 12]. A
general representation for the extinction in Rayleigh scattering regime was
studied and the effect for a linearly polarized Bessel beam of $m$ order as a
function of the incident impinging angle $\beta$. From the physical point of
view this method Eq. (22) can be extended using another spectral beam wave
representation [39], it allows to study waves such as X waves, Airy, Frozen
waves, among others. It would be interesting to explore as was stated [12] the
physics of differential cross section of scattering
$d\sigma_{\text{s}}/d\Omega$ and for the extinction
$d\sigma_{\text{ext}}/d\Omega$, the geometrical features and analogies between
acoustic and electromagnetism. Several application such as the Rayleigh
scattering of nanoparticles and optical forces [43] using focused femtosecond
laser pulses are in development. Applications measuring the optical extinction
has been recently experimentally explored using a radial polarized cylindrical
beams [44].

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