[1,2]Ashok Mondal

[1]Department of Applied Mathematics, University of Calcutta, 92 A.P.C Road,
Kolkata, 700009, West Bengal, India

2]Department of Applied Science and Humanities, Haldia Institute of
Technology, Hatiberia, ICARE Complex, Haldia, 721657, West Bengal, India

# A study on star formation in rotating and magnetized filamentary molecular
clouds

<EMAIL_ADDRESS>*  [

###### Abstract

A 2D dynamic model is utilized to investigate star formation in rotating
filamentary molecular clouds (FMCs) amidst magnetic fields. The study reveals
that the emergence of field stars is possible under both weak and strong
magnetic fields due to the presence of low-density structures. The presence of
a strong rotation in a strongly magnetized FMC Plays a crucial role in forming
a combination of two binary pairs and two field stars. An intermediate-density
structure in the presence of a moderate magnetic field can form a combination
of binary stars and field stars, whereas, in very high dense filamentary
molecular clouds, a low magnetic field may help to form binary stars or
stellar associations. In a rapidly rotating and weakly magnetized highly dense
cloud binary stars are formed, whereas, strong rotation and moderate magnetic
field help to form stellar associations.

###### keywords:

Filamentary molecular cloud, double well potential, interstellar magnetic
field, star formation.

##  1 Introduction

The interstellar medium (ISM) displays a predominantly filamentary structure
across various scales, as evidenced by studies such as Schneider and Elmegreen
[40], Hartmann [17], [32] and Flagey et al [12]. These self-gravitating
filaments are generally found in dense molecular clouds that serve as the
birthplaces of stars. Star-forming molecular clouds are threaded by magnetic
fields, which certainly influence cloud morphology and evolution (Krumholz and
Federrath 26). In Galactic molecular clouds, magnetic fields are frozen-in
fields from the interstellar medium of the Milky Way, which arose from
galactic-scale dynamo amplification of seed fields generated by Biermann
batteries in early Active Galactic Nuclei and/or in Population III stars.
Currently, the study of magnetic fields in molecular clouds is an important
and challenging topic in itself to know the role of magnetic fields in the
formation and evolution of interstellar clouds and in star formation. Despite
significant progress in recent years, the basic physics of star formation is
not clearly explained. One of the most fundamental questions about star
formation that has remained unanswered is what drives the star formation
process. Depending on the role of magnetic fields there are two major classes
of star formation theory: the strong-field models and the weak-field models.
In strong magnetic field models, magnetic fields with ambipolar diffusion
control the formation and evolution of the molecular clouds from which stars
form. Ambipolar diffusion drives to form dense cores and their gravitational
collapse to form protostars. In this theory, clouds are formed with
subcritical masses, and hence, the magnetic pressure is sufficiently strong to
counteract gravity and prevent gravitational collapse. Neutral gases and dust
contract gravitationally through the magnetic field and ions because the
magnetic field is only frozen into the ionized gas and dust, increasing mass
in the cloud cores. The magnetic field strength increases more slowly than
does mass. In weak magnetic field models, turbulent flows control the
formation of clouds and dense cores, with cores either dissipating back into
the general interstellar medium or collapsing and forming stars if they are
self-gravitating when formed. In this theory, clouds are formed with
supercritical masses, hence in general, clouds dissipate as they do not become
gravitationally bound; those that are self-gravitating form stars in
essentially a free-fall time. This study delves into the evolution of a
collection of nonlinear dynamical equations, derived from an extended form of
the double-well potential. I explore diverse physical scenarios to unravel the
mechanisms giving birth to field, binary, and multiple star systems in a
rotating and magnetized molecular clouds.
In Section 2, the mathematical model is expounded upon. Following this,
Section 3 outlines and deliberates upon the initial parameter values. Section
4 encompasses the presentation and discourse of results, while the concluding
remarks can be found in Section 5. Additional mathematical derivations are
provided in Appendix 6.

##  2 Mathematical Model

In this work, I have studied the dynamical evolution of filamentary molecular
clouds (hereafter FMCs) in the presence of the magnetic field. Mondal et al
[30] studied the evolution of rotating FMCs. For this purpose, they considered
double-well potential (hereafter DWP), which has similar density structures to
FMCs. They introduced the following generalized DWP (Eq.1) in a two-
dimensional system and compared the similarity between DWP and FMCs.

$V(X,Y,Z^{2}_{x},Z^{2}_{y})=-Z^{2}_{x}X^{2}-Z^{2}_{y}Y^{2}+\lambda(a_{xx}X^{4}+2a_{xy}X^{2}Y^{2}+a_{yy}Y^{4}).$
(1)

Details of these parameters $Z_{x}^{2}$, $Z_{y}^{2}$, $a_{xx}$, $a_{xy}$,
$a_{yy}$, and $\lambda$, and their significance can be found in Mondal et al
[30]. In this work, I further study the evolution in presence of a constant
magnetic field first, and then the evolution of rotating and magnetized FMCs.

###  2.1 Presence of a constant magnetic field

The role of the magnetic field on the formation and evolution of FMCs have
been studied by many researchers through theoretical and observational
investigations ( Khesali et al 24; Kainulainen et al 23; [1]; Tokuda et al
42). This field firmly impacts the shape of interstellar gas through the
formation of filaments and reduction of the numbers of clumps, cores, and
stars (Inutsuka [21]; Hennebelle and Inutsuka [19]). Self- gravitating FMCs
consistently attempt to get steady against fragmentation while the magnetic
field influences interstellar gas motion for fragmentation. Numerous FMCs are
associated with magnetic fields having energy comparable to the gravitational
binding energy(Myers and Goodman 34; Carlberg and Pudritz 9; Li and Berkert
27; Klassen et al 25). In our model, I have considered the presence of a
constant magnetic field in the filamentary molecular cloud. The direction of
the magnetic field in the spiral galaxy is along the direction of spiral arms
(Sofue et al 41, Beck 5). So it has no component along z-direction, i.e,
$\vec{B}=(B_{x},B_{y},0)$, and the corresponding magnetic force components
along x and y direction are $-\frac{B^{2}x}{x^{2}+y^{2}}$ and
$-\frac{B^{2}y}{x^{2}+y^{2}}$ respectively. So in the presence of a constant
magnetic field strength B along the spiral arm, the scaled equations (See
Appendix 6) of motion for general DWP are,

$\ddot{x}=2Z^{2}_{x}x-\lambda(4a_{xx}x^{3}+4a_{xy}xy^{2})-\frac{1}{\rho}\frac{B^{2}x}{x^{2}+y^{2}},$
(2)
$\ddot{y}=2Z^{2}_{y}y-\lambda(4a_{yy}y^{3}+4a_{xy}x^{2}y)-\frac{1}{\rho}\frac{B^{2}y}{x^{2}+y^{2}},$
(3)

where $\rho$ is the density of the molecular clouds. Here, I have made
dimensionless (See Appendix 6) X, Y, and T by $X=x\times 10^{17}$ cm,
$Y=y\times 10^{17}$ cm, and $T=t\times 10^{13}$ s for observational
compatibility.
For stability analysis, I transformed the aforementioned second-order
differential equations into a system of first-order differential equations.
Then equations 2 and 3 reduce to the following system of equations.
$\dot{x}=X^{\prime}$
$\dot{X}^{\prime}=2Z^{2}_{x}x-\lambda(4a_{xx}x^{3}+4a_{xy}xy^{2})-\frac{1}{\rho}\frac{B^{2}x}{x^{2}+y^{2}}$
$\dot{y}=Y^{\prime}$
$\dot{Y}^{\prime}=2Z^{2}_{y}y-\lambda(4a_{yy}y^{3}+4a_{xy}x^{2}y)-\frac{1}{\rho}\frac{B^{2}y}{x^{2}+y^{2}}$.
The number of solutions or fixed points are 16 of which 8 of them can be
written as
(i) $\\{x=0,y=\pm\frac{\sqrt{\rho Z^{2}_{y}\pm\sqrt{(\rho
Z^{2}_{y})^{2}-4\lambda\rho B^{2}a_{yy}}}}{2\sqrt{\rho\lambda a_{yy}}}\\},$
(ii) $\\{x=\pm\frac{\sqrt{\rho Z^{2}_{x}\pm\sqrt{(\rho
Z^{2}_{x})^{2}-4\lambda\rho B^{2}a_{xx}}}}{2\sqrt{\rho\lambda
a_{xx}}},y=0\\},$
The mathematical expression for the remaining stationary points are very large
to be quoted. I have studied the stability of this system of stationary points
in Table LABEL:TAble_potential_magnetic by considering suitable arbitrary
values of the parameters $\lambda$, $a_{xx}$, $a_{xy}$, $a_{yy}$, $Z^{2}_{x}$
and $Z^{2}_{y}$, for which the eigenvalues are real negative or their real
parts are close to zeros as found through numerical computations. The
eigenvalues of the corresponding Jacobian matrix can be expressed as,
$\pm\frac{\sqrt{(P_{2}+R_{2})\pm\sqrt{(P_{2}-R_{2})^{2}+4Q^{2}_{2}}}}{\sqrt{2}}$
where, $P_{2}=2Z^{2}_{x}-12\lambda a_{xx}x^{2}-4\lambda
a_{xy}y^{2}+\frac{B^{2}(x^{2}-y^{2})}{\rho(x^{2}+y^{2})^{2}}$,
$Q_{2}=-8\lambda a_{xy}xy+\frac{2B^{2}xy}{\rho(x^{2}+y^{2})^{2}}$ and
$R_{2}=2Z^{2}_{y}-12\lambda a_{yy}y^{2}-4\lambda
a_{xy}x^{2}-\frac{B^{2}(x^{2}-y^{2})}{\rho(x^{2}+y^{2})^{2}}$. One sufficient
condition for which a stationary point becomes a stable center is
$P_{2}R_{2}>Q^{2}_{2}$. In stable center scenario, particles near a stable
point will move around that stable point, and this situation might attract
other points to come closer, which helps later to collapse towards the stable
point. I have not studied chaos as it has no relevance in the present study
for star formation.

Figure 1: Phase space for the gravitational potential in the presence of a
constant magnetic field $B=10\mu G$ without any rotation for the values of the
parameters $\lambda=0.25\times
10^{-5},a_{xx}=1,a_{xy}=1,a_{yy}=2,Z^{2}_{x}=0.05,Z^{2}_{y}=-0.05.$. The
stable center stationary points are marked by green dots,The blue curves are
particular solutions starting at the positions indicated by red stars.

###  2.2 Presence of both rotation and magnetic field

In this section I discuss the fragmentation and stability of FMCs rotating
with a constant angular velocity in the presence of a constant magnetic field
throughout the cloud. The scaled equations(See Appendix 6) of motion for the
general DWP are

$\ddot{x}=2Z^{2}_{x}x-\lambda(4a_{xx}x^{3}+4a_{xy}xy^{2})-\frac{1}{\rho}\frac{B^{2}x}{x^{2}+y^{2}}+\Omega^{2}x+2\Omega\dot{y}$
(4)
$\ddot{y}=2Z^{2}_{y}y-\lambda(4a_{yy}y^{3}+4a_{xy}x^{2}y)-\frac{1}{\rho}\frac{B^{2}y}{x^{2}+y^{2}}+\Omega^{2}y-2\Omega\dot{x}.$
(5)

I have converted this system of second order differential equations to a
system of first order differential equations and proceeded as in Section 2.1
for stability analysis. The number of solutions or fixed points are 16 out of
which 8 of them can be written as,
(i)
$\\{x=0,y=\pm\frac{\sqrt{\rho(2Z^{2}_{y}+\Omega^{2})\pm\sqrt{(\rho^{2}(2Z^{2}_{y}+\Omega^{2})^{2}-16\lambda\rho
B^{2}a_{yy}}}}{2\sqrt{2\rho\lambda a_{yy}}}\\},$
(ii)
$\\{x=\pm\frac{\sqrt{\rho(2Z^{2}_{x}+\Omega^{2})\pm\sqrt{(\rho^{2}(2Z^{2}_{x}+\Omega^{2})^{2}-16\lambda\rho
B^{2}a_{xx}}}}{2\sqrt{2\rho\lambda a_{xx}}},y=0\\},$
The mathematical expression for remaining stationary points are large to be
quoted. I have studied the stability for the system of stationary points in
Table LABEL:TAble_potential_magnetic_rotation by considering suitable
arbitrary values of the parameters $\lambda$, $a_{xx}$, $a_{xy}$, $a_{yy}$,
$Z^{2}_{x}$ and $Z^{2}_{y}$ for which the eigenvalues have real values or real
parts are close to zero. The eigenvalues of the corresponding Jacobian matrix
can be expressed as
$\pm\frac{\sqrt{(P_{4}+R_{4}-4\Omega^{2})\pm\sqrt{(P_{4}+R_{4}-4\Omega^{2})^{2}-4(P_{4}R_{4}-Q^{2}_{4}))}}}{\sqrt{2}}$,
where, $P_{4}=2Z^{2}_{x}-12\lambda a_{xx}x^{2}-4\lambda
a_{xy}y^{2}+\frac{B^{2}(x^{2}-y^{2})}{\rho(x^{2}+y^{2})^{2}}+\Omega^{2}$,
$Q_{4}=-8\lambda a_{xy}xy+\frac{2B^{2}xy}{\rho(x^{2}+y^{2})^{2}}$ and
$R_{4}=2Z^{2}_{y}-12\lambda a_{yy}y^{2}-4\lambda
a_{xy}x^{2}-\frac{B^{2}(x^{2}-y^{2})}{\rho(x^{2}+y^{2})^{2}}+\Omega^{2}$.
One sufficient condition for which a stationary point becomes a stable center
is, $P_{4}R_{4}>Q^{2}_{4}.$

##  3 Initial values of the parameters

I have studied the stability of different types of molecular clouds (MCs) for
a wide range of the parameters. Khesali et al [24] showed that the magnetic
field in the MCs is of the order $\mu$G. The range of mean magnetic field
strength in giant molecular cloud (GMC) is from $1.4\,\mu G$ to $14\,\mu G$
[37]. In the present work, I have considered the range of B from $1.4\,\mu G$
to $14\,\mu G$. Several authors (Herbst and Klemperer [20]; Goldsmith and
Langer [15]; Bally et al [2, 3]; Caselli et al 10) found the initial number
density of MCs as $\sim$ $10^{4}$ cm-3 to $10^{5}$ cm-3, and the peak density
as $\sim$ $10^{5}$ cm-3 [33]. Although some authors (Gammie and Ostriker 13;
Ostriker et al 36)) showed the mean cloud density $n_{H_{2}}$ is typically
$\sim$ 25 cm-3 to 100 cm-3 in cloud complexes, whereas in clumps it is
$10^{3}$ cm-3. Throughout the current work, I have considered the initial
number density in MCs as $10^{4}$ cm-3 i.e, $\rho=1.67\times 10^{-20}$ g cm-3.
Brainel et al [7] shown the angular velocity of most of the MCs is less than
$0.03$ km s-1 pc-1 [7], although it may be as high as $1.5$ km s-1 pc-1 in few
cases [14]. The angular velocity of different types of cores in a MC is higher
than the average angular velocity of that MC. The observed value of $\Omega$
of cloud cores is in the range from 0.3 km s-1 pc-1 to 3 km s-1 pc-1 (Goodman
et al 16; Barranco and Goodman 4; Burkert and Bodenheimer 8). In this work, I
have considered the full range of rotational velocity to understand full
effect of rotation i.e. $\Omega$ varies from 0.01 km s-1 pc-1 to 3 km s-1
pc-1.

##  4 Results and discussion

In this study, I have developed a synthetic model to investigate the potential
of DWP to facilitate the formation of field stars, binary pairs, or stellar
associations within FMCs. Mondal et al [30] showed that the density patterns
associated with a DWP can exhibit resemblances to those observed in FMCs ,
which are the remnants of strong shock waves passing through them in the
presence of a helical magnetic field ( Pudritz and Kevlahan 38; Fiege and
Pudritz 11; McKee et al 28; Heiles et al 18; Schleuning 39; Mondal and
Chattopahdyay 31). I have explored the existence of stable stationary points
for various values of the parameters in the DWP along with various values of
the strength of the magnetic field ($B$) and rotational velocity ($\Omega$).
The results have been shown in Tables LABEL:TAble_potential_magnetic \-
LABEL:TAble_potential_magnetic_rotation and Figs. 1 \- 2. Figs. 1 and 2 show
that the particles very close to a stable center point come around again and
again. I get ellipsis in the phase plane, and consequently, particles of the
clouds will be exaggerated around the stable points. Thus, mass around a
stable center increases, and subsequently, when it crosses its jeans mass,
collapse starts, and the star formation process begins. Drawing from Jiménez-
Esteban et al [22] analysis of 3741 commoving binary and multiple stellar
systems in the Gaia DR2 catalogue, Mondal et al [30] have considered the
maximum distance between two binary stars as 2.45pc. Table
LABEL:TAble_potential_magnetic compiles the results in the presence of
magnetic field. The following facts are observed from Table
LABEL:TAble_potential_magnetic, which is obtained for magnetized self-
gravitating FMC.

When $\lambda$ is very small ($\sim 10^{-5}$ ) and DWP is asymmetric (here,
$a_{xx}=a_{xy}=1,a_{yy}=2,Z_{x}^{2}=0.05$ and $Z_{y}^{2}=-0.05$) or when
$\lambda$ is higher ($\sim 1$ ) and the remaining parameters
($a_{xx}=a_{xy}=10^{-7},a_{yy}=2\times 10^{-7},Z_{x}^{2}=10^{-2}$ and
$Z_{y}^{2}=1.96\times 10^{-2}$) are small, then field star formation is very
likely. These values of the parameters indicate a small depth of the DWP,
i.e., the density distribution of the DWP is lighter, which corresponds to
scattered FMC (Mondal et al 30). In these self-gravitating FMCs, a higher
magnetic field only reduces the distance between two stable points. Thus, when
B increases the distance between stable points decreases in lower dense FMC.

When the density of an FMC is moderate ( $\lambda\sim 10^{-3}$ and
$a_{xx}=a_{xy}=1,a_{yy}=2,Z_{x}^{2}=1$ and $Z_{y}^{2}=1.96$), then in the
presence of a higher magnetic field ( $B\sim 10\mu G$ ) binary pairs form,
whereas in the presence of a very low magnetic field ($B\sim 1.4\mu G$ )
stellar associations are formed. Since $\lambda$ is smaller and the remaining
parameters are higher compared to previous values, their combined effect
indicates a moderate depth of the DWP i.e., moderately dense FMC.

In a highly dense FMC ($\lambda\sim 10^{-2}$ and
$a_{xx}=a_{xy}=1,a_{yy}=2,Z_{x}^{2}=1$ and $Z_{y}^{2}=1.96$ ) stars may or may
not form depending on the strength of the magnetic field. In presence of a
lower magnetic field binary pairs or stellar associations are formed, whereas
in the presence of a strong magnetic field no stars will form i.e. there are
no stable centers. In high-dense FMCs, the strict magnetic flux freezing can
not be assumed to hold always, especially when the ambipolar diffusion time
scale becomes much shorter. Then, the neutrals in a magnetized cloud drift
past the ions and the magnetic field lines, and consequently the radius of the
FMC decreases. As we have considered the presence of a constant magnetic
field, the magnetic critical mass decreases, so, FMCs get a favour to collapse
and fragment into smaller parts. These fragments may undergo further
subsequent fragmentation due to their low magnetic critical mass, and hence
very low-mass fragments may be formed. The fragmentation of molecular cloud
can give rise to gaseous fragments with masses 0.001 $M_{\odot}$ (Mondal and
Chattopadhyay 29, Whitworth and Stamatellos 43, Boss 6), but The lowest
critical mass for a star to support nuclear fusion is 0.08 $M_{\odot}$
(Nakamura et al 35). Thus, highly magnetized and highly dense FMCs sometimes
may remain starless.

Table 1: Number of stationary points and stable centers for various types of self-gravitating magnetized filamentary MC. |  |  |  |  |  |  | No. of | No. of
---|---|---|---|---|---|---|---|---
$\lambda$ | $a_{xx}$ | $a_{xy}$ | $a_{yy}$ | $Z^{2}_{x}$ | $Z^{2}_{y}$ | $B$ | stationary | stable centers
|  |  |  |  |  | ($\mu$G) | points | (X, Y):distance (pc)
|  |  |  |  |  |  |  | (pc, pc)
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 3.135,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 14 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 3.012,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 5 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 3.215,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.5 | -0.005 | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 32.404,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.6 | 0.5 | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 11.223,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.5 | 0.6 | 10 | 12 | 4
|  |  |  |  |  |  |  | $(\pm 9.162,\pm 4.583)$
$0.25\mathrm{e}{-4}$ | 1 | 1 | 2 | $0.05\mathrm{e}{+1}$ | $-0.05\mathrm{e}{+1}$ | 6 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 3.237,0)$
$0.25\mathrm{e}{-4}$ | 1 | 1 | 2 | $0.05\mathrm{e}{+1}$ | $-0.05\mathrm{e}{+1}$ | 1.4 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 10.247,0)$
$1$ | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 10 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 7.057)$
$1$ | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 5 | 12 | 4
|  |  |  |  |  |  |  | $(\pm 1.141,\pm 7.099)$
$1$ | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 14 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 6.935)$
$2$ | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 10 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 4.900)$
$1$ | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-1}$ | $1.96\mathrm{e}{-1}$ | 10 | 12 | 4
|  |  |  |  |  |  |  | $(\pm 4.548,\pm 22.450)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 9.912,0)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 5 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 10.167,0)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 14 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 9.524,0)$
$0.25\mathrm{e}{-1}$ | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 32.301,0)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | 0.5 | -0.5 | 10 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 102.455,0)$
$1\mathrm{e}{-3}$ | 1 | 1 | 2 | 1 | 1.96 | 10 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 0.7057)$
$1\mathrm{e}{-3}$ | 1 | 1 | 2 | 1 | 1.96 | 1.4 | 12 | 4
|  |  |  |  |  |  |  | $(\pm 0.1428,\pm 0.7099)$
$1\mathrm{e}{-3}$ | 100 | 100 | 200 | 1 | 1.96 | 10 | 0 | 0
$1\mathrm{e}{-3}$ | 100 | 100 | 200 | 1 | 1.96 | 1.4 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 0.0693)$
$0.25\mathrm{e}{-4}$ | 1 | 1 | 2 | 0.05 | -0.05 | 10 | 0 | 0
$0.25\mathrm{e}{-4}$ | 1 | 1 | 2 | 0.05 | -0.05 | 6 | 4 | 2
|  |  |  |  |  |  |  | $(\pm 0.8484,0)$
$0.25\mathrm{e}{-5}$ | 1000 | 1000 | 2000 | 0.05 | -0.05 | 10 | 0 | 0
$0.25\mathrm{e}{-5}$ | 1000 | 1000 | 2000 | 0.05 | -0.05 | 1.4 | 0 | 0
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 10 | 0 | 0
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 14 | 0 | 0
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 1.4 | 12 | 4
|  |  |  |  |  |  |  | $(\pm 0.0382,\pm 0.2242)$
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 5 | 8 | 2
|  |  |  |  |  |  |  | $(0,\pm 0.2168)$

Table LABEL:TAble_potential_magnetic_rotation compiles the results of the
analysis in the presence of both rotation and magnetic field. The following
facts have been observed.

Table 2: Number of stationary points and stable centers for various types of self-gravitating, magnetized, and rotating filamentary MC. |  |  |  |  |  |  |  | No. of | No. of
---|---|---|---|---|---|---|---|---|---
$\lambda$ | $a_{xx}$ | $a_{xy}$ | $a_{yy}$ | $Z^{2}_{x}$ | $Z^{2}_{y}$ | $\Omega$ | $B$ | stationary | stable centers
|  |  |  |  |  |  (km s-1 pc${}^{-1})$ |  ($\mu$G) | points | (X, Y):distance (pc)
|  |  |  |  |  |  |  |  | (pc, pc)
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 3 | 10 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.2730)$, $(\pm 10.474,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 3 | 14 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.3824)$, $(\pm 10.471,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 3 | 5 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.1361)$, $(\pm 10.476,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 0.01 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(\pm 3.216,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 1.5 | 10 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.3422)$, $(\pm 5.939,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 3 | 10 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.2751)$, $(\pm 3.304,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.05 | -0.05 | 3 | 6 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.1641)$, $(\pm 3.310,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.5 | -0.005 | 3 | 10 | 8 | 2
|  |  |  |  |  |  |  |  | $(\pm 33.901,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.6 | 0.5 | 3 | 10 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.1798)$, $(\pm 15.008,0)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 0.5 | 0.6 | 3 | 10 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 13.537,\pm 4.583)$
1 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 10 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 49.837,\pm 7.099)$,$(\pm 0.2552,0)$
1 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 5 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 49.837,\pm 7.099)$,$(\pm 0.1276,0)$
1 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 14 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 49.837,\pm 7.099)$,$(\pm 0.2552,0)$
1 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 0.01 | 8 | 2
|  |  |  |  |  |  |  |  | $(0\pm 7.057)$
2 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 10 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 35.240,\pm 5.020)$,$(\pm 0.2552,0)$
1 | $1\mathrm{e}{-7}$ | $1\mathrm{e}{-7}$ | $2\mathrm{e}{-7}$ | $1\mathrm{e}{-1}$ | $1.96\mathrm{e}{-1}$ | 3 | 10 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 50.027,\pm 22.450)$,$(\pm 0.2552,0)$
1 | $1\mathrm{e}{-6}$ | $1\mathrm{e}{-6}$ | $2\mathrm{e}{-6}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 10 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 15.758,\pm 2.245)$,$(\pm 0.2553,0)$
1 | $1\mathrm{e}{-6}$ | $1\mathrm{e}{-6}$ | $2\mathrm{e}{-6}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 5 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 15.760,\pm 2.245)$,$(\pm 0.1276,0)$
1 | $1\mathrm{e}{-4}$ | $1\mathrm{e}{-4}$ | $2\mathrm{e}{-4}$ | $1\mathrm{e}{-2}$ | $1.96\mathrm{e}{-2}$ | 3 | 5 | 12 | 6
|  |  |  |  |  |  |  |  |  $(\pm 1.571,\pm 0.2245)$,$(\pm 0.1280,0)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 10 | 6 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.2592)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 5 | 6 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.1296)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 0.01 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(\pm 9.918,0)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 1.5 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.5270)$
$0.25\mathrm{e}{-1}$ | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 10 | 6 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.2592)$
$0.25\mathrm{e}{+1}$ | $1\mathrm{e}{-7}$ | 0 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 10 | 6 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.2592)$
0.25 | $1\mathrm{e}{-7}$ | 0 | 0 | 0.5 | -0.5 | 3 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(\pm 142.901,0)$
0.25 | $1\mathrm{e}{-5}$ | 1 | 0 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 10 | 6 | 2
|  |  |  |  |  |  |  |  | $(\pm 10.011,0)$
0.25 | $1\mathrm{e}{-5}$ | 0 | 1 | $0.5\mathrm{e}{-2}$ | $-0.5\mathrm{e}{-2}$ | 3 | 10 | 8 | 4
|  |  |  |  |  |  |  |  | $(\pm 10.011,\pm 0.0313)$
$1\mathrm{e}{-3}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 10 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.4972,\pm 0.7099)$
$1\mathrm{e}{-3}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 1.4 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.5184,\pm 0.7099)$
$1\mathrm{e}{-3}$ | 10 | 10 | 20 | 1 | 1.96 | 3 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.2141)$
$1\mathrm{e}{-3}$ | 100 | 100 | 200 | 1 | 1.96 | 3 | 10 | 0 | 0
$1\mathrm{e}{-3}$ | 100 | 100 | 200 | 1 | 1.96 | 3 | 1.4 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.0474,\pm 0.0710)$
$1\mathrm{e}{-3}$ | 100 | 100 | 200 | 1 | 1.96 | 3.09 | 1.4 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.0491,\pm 0.0710)$
$0.25\mathrm{e}{-5}$ | 1 | 1 | 2 | 1 | 1 | 3 | 10 | 8 | 0
$0.25\mathrm{e}{-4}$ | 1 | 1 | 2 | $0.05\mathrm{e}{+1}$ | $-0.05\mathrm{e}{+1}$ | 3.09 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(\pm 4.583,0)$
$0.25\mathrm{e}{-5}$ | 1000 | 1000 | 2000 | 0.05 | -0.05 | 3 | 10 | 0 | 0
$0.25\mathrm{e}{-5}$ | 1000 | 1000 | 2000 | 0.05 | -0.05 | 3 | 1.4 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.0388)$,$(\pm 0.3295,0)$
$0.25\mathrm{e}{-5}$ | 1000 | 1000 | 2000 | 0.05 | -0.05 | 3 | 0.01 | 8 | 4
|  |  |  |  |  |  |  |  |  $(0,\pm 0.0003)$,$(\pm 0.3313,0)$
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 10 | 4 | 2
|  |  |  |  |  |  |  |  | $(0,\pm 0.2142)$
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 14 | 0 | 0
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 1.4 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.1627,\pm 0.2242)$
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 3 | 5 | 12 | 4
|  |  |  |  |  |  |  |  | $(\pm 0.1452,\pm 0.2242)$
$1\mathrm{e}{-2}$ | 1 | 1 | 2 | 1 | 1.96 | 0.01 | 10 | 0 | 0

When $\lambda$ = 0.25 $\times 10^{-5}$, $a_{xx}=a_{xy}=1$, $a_{yy}=2$,
$Z^{2}_{x}=0.05$ and $Z^{2}_{y}=-0.05$ then in presence of both magnetic field
and rotation ( Table LABEL:TAble_potential_magnetic_rotation), the number of
stationary points as well as the number of stable centers increase. As B
increases the distance between the stable centers on the Y-axis increases and
the distance between the stable centers on the X-axis slightly decreases. On
the other hand, the effect of rotation is opposite to that magnetic field.
When $\Omega$ increases the stable centers on both axes come closer to each
other. The depth of the DWP is small (Mondal et al 30) for this set of
parameter values. Here, a combination of two binary pairs and two field stars
form at higher values of both B and $\Omega$. But, the distance between the
field stars is higher compared to the previous case i.e., in the presence of
only magnetic field. In the presence of very low rotation, there are no
significant differences from the case of only magnetic fields. Again when B is
high ($\sim 10$ $\mu G$) and $\Omega$ is small ($\sim 0.01$ km s-1 pc-1 ) then
the formation of field stars is more likely than binary pairs or multiple
systems. When $\lambda=1$, $a_{xx}=a_{xy}=10^{-7}$, $a_{yy}=2\times 10^{-7}$,
$Z^{2}_{x}=10^{-2}$ and $Z^{2}_{y}=1.966\times 10^{-2}$, then in the presence
of a strong magnetic field and strong rotation formation of a combination of
four field stars and two binary pairs is possible. These types of scenarios
may be seen in the filaments having less or moderate gravitational attraction.

In a moderately dense FMC ($\lambda=10^{-3}$, $a_{xx}=a_{xy}=1$, $a_{yy}=2$,
$Z^{2}_{x}=1$ and $Z^{2}_{y}=1.966$), the presence of strong rotation and a
strong magnetic field increase the distances between the stable centers and
four field stars are formed. On the other hand, the presence of strong
rotation with low magnetic field forces to form a combination of two binary
pairs and two field stars. Binary star formation is more frequent than field
stars in the presence of both a strong magnetic field and strong rotation in a
moderately high dense ($\lambda=10^{-3}$, $a_{xx}=a_{xy}=10$, $a_{yy}=20$,
$Z^{2}_{x}=1$ and $Z^{2}_{y}=1.966$) FMC.

When $\lambda=10^{-2}$, $a_{xx}=a_{xy}=1$, $a_{yy}=2$, $Z^{2}_{x}=1$ and
$Z^{2}_{y}=1.96$, then the depth of the potential is higher compared to the
depth for the previous set of parameters. In this case, the effect of the
magnetic field is more pronounced than rotation. Here, in presence of strong
rotation with a low magnetic field binary pairs are formed. For strong
rotation and lower (B $\sim 1.4$ $\mu G$) or moderate (B$\sim 5$ $\mu G$)
values of B stars are formed in stellar associations, whereas, when B is high
enough ($\sim 10$ $\mu G$ or more) there is no stable center irrespective of
the strength of rotation, that means when the depth of the DWP is greater i.e
for dense filaments if the magnetic field is high then the dynamical system of
molecular clouds would diverse rather than converge around some stable
centers.

Figure 2: Phase space for a self-gravitating and magnetized rotating cloud
with $\Omega=3\times 10^{-13}s^{-1}$, and $B=10\mu G$ as in Fig. 1 for the
same values of the parameters. For these values one binary pairs and two
isolated stars are formed.

##  5 Conclusions

This study employs a synthetic model that leverages the double well potential
concept to forecast various star formation scenarios within different
categories of rotating FMCs under the influence of a magnetic field.
(i) When the depth of the potential is low, i.e., for lesser-dense FMC most of
the stars are formed as field stars in the presence of weak as well as strong
magnetic fields. The effect of the magnetic field on the number of stars is
less. On the other hand, rotation acts opposite to that magnetic field. When
clouds rotate fast the distance between the stable centers decreases. The
presence of a strong rotation in a strongly magnetized FMC Plays a crucial
role in forming a combination of two binary pairs and two field stars. (ii)
When the FMCs have an intermediate density distribution, then in the presence
of a strong magnetic field binary stars form, whereas, in the presence of a
low magnetic field stellar association may form. But, from strongly magnetized
and strongly rotating moderately dense clouds field stars are formed. In a
weakly magnetized cloud, the presence of strong rotation influences to form a
combination of two binary pairs and two field stars. (iii) For moderately high
or very high dense FMC, a low magnetic field may help to form binary stars or
stellar associations. But, a strong magnetic field strongly acts against star
formation, and no star forms. In this case, the effect of magnetic field is
more pronounced than rotation. In a rapidly rotating and weakly magnetized
cloud binary stars are formed, whereas, strong rotation and moderate magnetic
field help to form stellar associations.

##  6 Appendix

###  6.1 Scaling

$\ddot{X}=2Z^{2}_{x}X-4\lambda(a_{xx}X^{3}+a_{xy}XY^{2})-\frac{1}{\rho}\frac{B^{2}X}{X^{2}+Y^{2}}+\Omega^{2}X+2\Omega\dot{Y}.$
Let us put,$X=x\times 10^{17}\,cm$, $Y=y\times 10^{17}\,cm$, and $T=t\times
10^{13}\,s$. Then the above equation reduces to
$\frac{10^{17}}{10^{26}}\ddot{x}=2Z^{2}_{x}(x\times
10^{17})-4\lambda(a_{xx}x^{3}+a_{xy}xy^{2})\times
10^{51}-\frac{1}{\rho}\frac{B^{2}x}{x^{2}+y^{2}}\frac{1}{10^{17}}+\Omega^{2}x\times
10^{17}+2\Omega\dot{y}\times\frac{10^{17}}{10^{13}}.$ Or,
$\ddot{x}=2(Z^{2}_{x}\times 10^{26})x-4(\lambda\times
10^{60})(a_{xx}x^{3}+4a_{xy}xy^{2})-\frac{1}{\rho}\frac{B^{2}x}{x^{2}+y^{2}}10^{-8}+\Omega^{2}x\times
10^{26}+2\Omega\dot{y}\times 10^{13}.$ Or,
$\ddot{x}=2Z^{{}^{\prime}2}_{x}x-4\lambda^{\prime}(a_{xx}x^{3}+a_{xy}xy^{2})-(\frac{B^{2}\times
10^{-8}}{\rho})\frac{x}{x^{2}+y^{2}}+(\Omega\times
10^{13})^{2}x+2(\Omega\times 10^{13})\dot{y}.$ Where
$Z^{{}^{\prime}2}_{x}=Z^{2}_{x}\times 10^{26}$ and
$\lambda^{\prime}=\lambda\times 10^{60}$. Now let a, b are two constants such
that their ranges are from 1.4 to 14, and from 0.01 to 3 multiplied by 0.3241
respectively.So, I can write $B=a\times 10^{-6}\,G$, and $\Omega=b\times
10^{-13}$ $s^{-1}$. Also $\rho=1.67\times 10^{-20}$ $g\,cm^{-3}$. Therefore,
$\frac{B^{2}}{\rho}\simeq 10^{8}$, $\Omega^{2}\simeq 10^{-26}$, and
$\Omega=10^{-13}$. Thus the equation of motion along the x axis in reduce form
is $\ddot{x}=2Z^{\prime
2}_{x}x-4\lambda^{\prime}(a_{xx}x^{3}+a_{xy}xy^{2})-\frac{1}{\rho^{\prime}}\frac{B^{\prime
2}x}{x^{2}+y^{2}}+\Omega^{\prime 2}x+2\Omega^{\prime}\dot{y}$, Where B’=a,
$\rho^{\prime}=1.67$, $\Omega^{\prime}=b$. Since $Z^{\prime 2}_{x}$ and
$\lambda^{\prime}$ are arbitrary constants, the form of this reduce equation
is same with original equation before scaling. For the equation of motion
along y axis, after scaling I get the same type of equation as original
equation with B’=a, $\rho^{\prime}=1.67$, $\Omega^{\prime}=b$.

=========================================================

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# Innovative approaches to high school physics competitions: Harnessing the
power of AI and open science

D Borovský1 J Hanč1 and M Hančová2 1Institute of Physics, Faculty of Science,
Pavol Jozef Šafárik University in Košice, Slovakia  1Institute of Mathematics,
Faculty of Science, Pavol Jozef Šafárik University in Košice, Slovakia
<EMAIL_ADDRESS>
###### Abstract

High school physics competitions serve as a platform for talented students to
showcase their skills, engage in challenging problems, and foster a passion
for science. This paper explores innovative approaches to enhance these
competitions by harnessing the power of open science and artificial
intelligence (AI) tools. Particularly we delve into the capabilities of state-
of-the-art AI chatbots, i.e. ChatGPT, Bard, Claude, related to problem solving
in physics. Together with open science tools like SageMath and Jupyter AI,
they have the potential to serve as intelligent, powerful co-pilots, tutors,
and assistants in understanding and applying physics, as well as knowledge
from connected STEM fields. Furthermore, these innovative approaches can
revolutionize high school physics competitions, providing students and their
tutors with powerful resources to excel in their scientific pursuits.

##  1 Introduction

High school physics competitions and preparation for them are an essential
part of informal physics education at the secondary level. Special places in
these high school competitions are held by the Physics Olympiad, the PhO
(https://www.ipho-new.org), and the Young Physicist Tournament, the YPT
(https://www.iypt.org), both of which have long-standing traditions with
international significance, ranking them among the most prestigious. In these
competitions, talented students can not only refine and deepen their
theoretical and experimental knowledge and skills but also compare with their
peers.

In the broader context, participation in the PhO or/and YPT not only provides
an opportunity for competition but also has a strong impact on shaping the
students’ future careers. The successful participation often guides talented
students not only for university studies in natural science and technical
fields but equips them with a high level of critical thinking and scientific
literacy essential also in the study of medicine, economics, or law. In other
words, such competitions appear also important from the perspective of current
trends in STEM education and the needs of our society, where STEM disciplines
are the primary drivers of the economy and where our jobs, health, education
$\&$ environment, and even roles as citizens become strongly dependent on
STEM, digital, and data literacy.

With the ubiquitous rise and implementation of artificial intelligence (AI) in
recent years, particularly exemplified this year (2023) by phenomenal AI
chatbots like ChatGPT (https://chat.openai.com), we have decided to explore
the potential of such technologies in preparing students for competitions like
PhO and YPT. In this article, we present our ongoing research and latest
findings dealing with this task.

##  2 Our Research and Methodology

Our research concerning generative AI in physics education (which is a part of
the Ph.D. study of DB) is connected to the following research questions:

  * •

How can state-of-art AI Chatbots, like ChatGPT, be integrated into physics
education to support physics teaching and learning?

  * •

What are the perceptions of students and teachers concerning the effectiveness
of the generative AI tools in enhancing understanding and engagement in
physics?

  * •

How does the use of generative AI impact students’ performance in physics, as
quantified by empirical data?

As for research design and methodology, to find answers we have decided to
apply an exploratory sequential mixed-methods design [1] with a focus on
qualitative perspectives supported by quantitative data (design QUAL
$\Rightarrow$ quan). Currently, in the qualitative phase, we are undertaking a
literature review to understand the theoretical framework and potential of
generative AI like ChatGPT in physics education. This qualitative stage also
encompasses a pilot case study, where we are exploring the capabilities and
performance of generative AI in diverse physics tasks, as well as its
integration into physics education. One of our prime examples of such
integration is the field of informal preparation for competitions like the PhO
and YPT. The forthcoming planned quantitative phase will provide data to
measure the impact of AI within the existing conditions of physics education.

##  3 Technology and Models behind the State-of-Art AI Chatbots

To uncover the capabilities and truly grasp the performance of generative AI
tools in the context of current published works related to education, we
cannot solely rely on direct interaction with ChatGPT through an experimental
approach, as illustrated in references [2, 3, 4, 5]. It’s also crucial to
delve into a more detailed theoretical analysis and understanding of what
ChatGPT actually is, and how the model behind ChatGPT works from technology
and mathematics perspective.

At the heart of today’s phenomenal chatbots, like ChatGPT [6], lies a unique
and revolutionary artificial neural network architecture known as the
transformer. Introduced by researchers from Google in 2017 [7], the
transformer model has since been the backbone of many breakthroughs in AI. We
do not explain in detail how transformers work (explanations at different
levels can be found in [8, 9, 10]). For our purposes, it is sufficient to look
at a transformer in analogy with a human brain.

As a type of neural network, transformer as a neural network is similar to the
neurons network in our human brain. The size of transformers is given by a
number of parameters and if a number of parameters is more than $10^{9}$ we
call a transformer for natural language processing a Large Language Model
(LLM) [11]. Every transformer has to be taught on a dataset after creation and
initialization, similarly as a newborn has to learn about the world. This
process is called pre-training, in analogy with the human brain, it is like
early learning when a baby learns basic concepts, sounds, visuals, languages,
etc.

After pre-training, the model is fine-tuned (together with implementing
ethical norms) on a smaller, specific dataset for particular tasks like
translation, question-answering, or summarization. ChatGPT was fine-tuned with
the aim of improving my performance for user interactions. Fine-tuning
corresponds to schooling or formal education in human life which can be
diverse and in the final stage is tailored towards specific content and skills
of the chosen profession (career). A pre-trained and fine-tuned transformer
designed for natural language processing is called a Generative Pre-trained
Transformer or GPT (for more accurate details on what type of transformer GPT
exactly is, see [8]).

ChatGPT with model GPT-3.5 was trained on a gigantic and diverse dataset
equivalent to reading 45 million books in 26 languages. The language
capabilities of ChatGPT are therefore phenomenal compared to its predecessors.
It can interactively communicate with you indistinguishably from a human,
remembering the content of the conversation within a single session. It can
discuss and draw conclusions on virtually any topic, including history,
science, art, or technology. In interactions with a chatbot like ChatGPT, any
text or question we pose, expecting a response, is called a prompt (see Fig.
1).

Figure 1: The problem from the Physics Olympiad (2022, regional round,
category C) as prompts in AI chatbots: a simple prompt (on the left) and with
an improving advanced prompt using priming (on the right).

It’s crucial to recognize that re-inputting the same prompt doesn’t guarantee
an identical response from the chatbot. This variability isn’t a glitch, but
it is intentional. In an LLM model, it is determined by a technical parameter
called temperature. By default, many chatbots set the temperature parameter at
0.7 for their large language models. When the temperature is set to zero, the
chatbot’s output remains the same, reflecting the highest probability response
from the neural network structure. With any non-zero value, the chatbot has
the possibility to randomly select a different answer with a probability close
to the maximum. This means that the higher the temperature, the further from
the optimal answer the chatbot can deviate. It also means that a higher
temperature will result in less boring, more creative and unpredictable text.
It is analogous to the concept of temperature in physics, where a higher
temperature means the more chaotic and unpredictable motion of particles and
vice versa.

From a technological perspective, the release of ChatGPT in November 2022
wasn’t so groundbreaking event, as numerous LLM transformers, including GPT-3,
were already available (see Tab.1, Fig. 2 in [11]). However, the release of
ChatGPT became one of the milestones igniting accelerated research in the LLM
field. Another important milestone was the release of LLM with a free, open
license from Meta called LLaMA [12].

##  4 Current Capabilities of Generative AI Tools

###  4.1 Problem Solving with the State-of-Art AI Chatbots

To determine the current capabilities of State-of-the-Art AI Chatbots in
solving physics tasks and problems from PhO and YPT, we conducted a simple
pilot case study. We have chosen currently the most powerful chatbots using
the strongest LLMs (according to chatbot arena at https://chat.lmsys.org):

  * •

the aforementioned ChatGPT from OpenAI [6], supported by Microsoft (with the
free GPT-3.5 model and the paid GPT-4 model and plugins available at
https://chat.openai.com)

  * •

Bard from Google using PaLM2 [13] (https://bard.google.com; via a Google
account),

  * •

Claude from the Google-backed Anthropic [14] (https://claude.ai; via the Opera
with VPN),

  * •

Vicuna, the open-source chatbot [15] from Berkeley University based on the
LLaMA model from Meta (accessible in the chatbot arena
https://chat.lmsys.org).

####  4.1.1 PhO Problem.

As the first problem in the study, we chose a task from the 63rd edition of
the Slovak Physics Olympiad (regional round, category C – 2nd year of high
school) from the academic year 2021/2022 (authors: Bystrický & Konrád). The
task is shown in Figure 1 as a simple prompt. In Figure 2, we see the best
solutions (chosen from 10 times repeatedly produced solutions to the same
prompt) of part a) provided by OpenAI GPT-3.5 and Google Bard. Both chatbots
were characterized by great speed (they generated the solution almost
instantly). Additionally, Bard provided a preview of three different responses
in the sense of the temperature parameter described earlier, from which we can
choose and continue the conversation accordingly.

We see that both chatbots provided annotated structured solutions, with the
correct application of the physical law. Such annotations are what we expect
in solutions to PhO tasks. However, both chatbots made errors. Google Bard did
not convert quantities into SI units, an oversight sometimes also seen among
younger PhO solvers. In both cases, the chatbots failed in mathematics.

Figure 2:  Comparing solutions for part a) of the PhO problem from Fig. 1
provided by free versions of AI chatbots from OpenAI (GPT-3.5) and Google
(Bard).

The correct values are $n=0.403~{}\mathrm{mol}$, $m=11.3~{}\mathrm{g}$ which
means that GPT-3.5 value was 2 orders of magnitude higher. We’ll just mention
here that the Vicuna-33B chatbot solved the problem as competently as GPT-3.5
did but was significantly slower and also with numerical errors
($n=0.219~{}\mathrm{mol}$, $m=6.0698~{}\mathrm{g}$). None of the chatbots were
able to correct numerical errors, even after multiple prompts.

Strictly speaking, from the perspective of the rules of the PhO, none of the
chatbots demonstrated the required general solution. This behavior is
understandable since we didn’t prompt the chatbots to do so. In such cases, we
can make further prompts iteratively to get the desired general solution.
However, a challenge explaining general solutions is not a trivial matter.
Based on our multiple interactions with chatbots and also the literature
review, it seems that a very effective prompt strategy is called priming.
Generally, the right priming improves the performance of language models by
providing them with context or initial information before generating text.
This is typically done by supplying the model with a more advanced prompt or a
set of input tokens related to the task or context in question. There are
several priming techniques (see [16, 11]), and an example is the advanced
prompt in Fig. 1 (on the right).

In Fig. 3, we can see the completely correct annotated general solution from
Claude and ChatGPT, along with the correct numerical results. We want to
highlight that, in the case of the Claude chatbot, it’s possible to upload
instructions in pdf form which simplifies and clarifies communication.
Initially, Claude provided a numerically incorrect result
($m=0.70~{}\mathrm{g}$), which he corrected only after our challenge.

Figure 3:  The state-of-art AI chatbots solutions using priming and plugins:
Claude v2 (on the left; with instructions as a pdf attachment) and GPT-4
solution (on the right; with plugin Wolfram).

In the case of ChatGPT, we see the solution in the paid GPT-4 model with the
Wolfram plugin. This means that every numerical result is secretly calculated
not by GPT-4, but by Wolfram Alpha, a computational knowledge engine developed
by Wolfram Research (https://www.wolframalpha.com). Thanks to this plugin,
ChatGPT almost certainly provides correct numerical results. In rare cases,
the Wolfram plugin fails, gets stuck in a loop, and provides no answer. In
such situations, it is necessary to solve the problem from scratch.

####  4.1.2 YPT Problem.

From the perspective of physical complexity, task a) was of a basic knowledge
level related to the ideal gas theme, which even an average high school
student should manage. In our second instance, we tackled a much more
challenging problem from the YPT competition (problem 12 from the 34th IYPT).
The problem’s statement was as follows:

> A Wilberforce pendulum consists of a mass hanging from a vertically oriented
> helical spring. The mass can both move up and down on the spring and rotate
> about its vertical axis. Investigate the behavior of such a pendulum and how
> it depends on relevant parameters.

We were able to solve the problem, concerning its dynamics, successfully with
the Chatbot ChatGPT using its two paid models (applying an advanced modelling
approach – Lagrange analytical method; here it is recommended to install a
browser extension [17] for rendering LaTeX equations):

  * •

GPT-4 model:
https://sharegpt.com/c/MZPHMS2

  * •

GPT-4 code interpreter:
https://chat.openai.com/share/0d059dc6-1632-4e9f-af46-4c5926b91304

###  4.2 Generative AI in Open Science Jupyter Notebooks

Problems faced by AI in using mathematics, whether related to the accuracy of
numerical results or algebraic manipulations, can be addressed by integrating
AI with open science tools designed for scientific computing and data
processing. Moreover, this integration also facilitates a more straightforward
and efficient interaction with AI during scientific computing.

One of such open science tools is a free Python-based mathematics software
called SageMath (https://www.sagemath.org). In our previous works [18, 19] we
introduced and described the capabilities of SageMath in research and STEM
education. We also demonstrated that SageMath is ideally suited for solving
PhO and YPT problems. Technologically, from a visual standpoint, SageMath is a
Jupyter Notebook – an interactive web page that runs on any modern web browser
and can be easily edited as a Word document. It allows users to insert text,
images, and videos, as well as computational code representing variables,
equations, or commands for plotting graphs. Moreover, any Python library or
extension designed for Jupyter can be installed and utilized in SageMath. In
this article, we show two possibilities for integrating AI into Jupyter that
we identified through our research in relevant resources.

####  4.2.1 AI Assistant Jupyternaut.

In March 2023, also a team from Project Jupyter (https://jupyter.org) began
intensively working on AI implementation through a sub-project named Jupyter
AI [20]. The culmination of this effort is the `jupyter_ai` extension, which
integrates the Jupyter notebook with AI, allowing users to choose from various
large language models (several dozens), including the OpenAI GPT-3.5, which
was chosen by us. Upon its installation, an AI chatbot named Jupyternaut
appears.

This chatbot acts as a general-purpose AI virtual assistant – serving mainly
for coding purposes and, as for physics (or natural sciences), becomes a
helpful guide for computations, explaining and applying physics concepts, and
laws. Remarkably, with the aid of GPT-3.5, it can do so in 26 languages,
including Slovak. Figure 4 (on the left) illustrates the AI’s interaction
within SageMath, showing a conversation during solving our PhO problem. Using
the `/learn` command, we requested Jupyternaut to study all materials within
the Documents folder. In our case, it was a text with general instructions for
solving physics problems (Fig. 1, on the right), complemented by guidelines
stating that explanations should be rendered in SageMath as text cells, while
calculations and symbolic manipulations should be done as computation cells.
Then, with the `/generate` command, we asked Jupyternaut to create a Jupyter
notebook as SageMath, tasked with solving the given physics problem. In a
brief moment, the virtual assistant produced a notebook, which after our minor
adjustments is displayed in Fig. 4 (on the right).

Figure 4:  A solution generated by Jupyternaut (with GPT-3.5) with
corresponding conversation (on the left) and the resulting SageMath Jupyter
notebook (on the right).

Now, we can see the general solution, with numerical results, accurate and
precise up to 15 decimal places. By highlighting any section of the notebook
on the right, Jupyternaut can provide a detailed explanation (tailored to our
prompts) or fix errors.

Employing an AI assistant directly within a digital scientific environment as
SageMath, compared to a web-based chatbot interface, is substantially more
straightforward. The need for copying content back and forth is eliminated.
Concurrently, the synergy with SageMath gives us better interactivity at
intermediate steps, easy modifiability, reproducibility, and direct
visualization capabilities.

####  4.2.2 ChatGPT – Jupyter – AI Assistant.

A second, very easy solution for bringing AI into the SageMath Jupyter
notebook is through the use of a browser extension, named ChatGPT – Jupyter –
AI Assistant [21], which is compatible with browsers like Chrome, Edge, or
Opera. This extension using only GPT-3.5 can be installed with a single click,
appearing as a toolbar with various control buttons within the Jupyterlab
environment. Similar to Jupyternaut, it can comment on the highlighted
computational code, format it, explain it, fix its errors, and also generate
new code based on the prompt.

##  5 Conclusion

In this paper, we explored the role and capabilities of generative artificial
intelligence through state-of-art chatbots like ChatGPT to enhance high school
physics competitions. The results indicate the undeniable benefits of using
chatbots in the context of the Physics Olympiad (PhO) and Young Physicist
Tournament (YPT). These AI tools, when combined with open science platforms
like SageMath, can remove problems with the mathematics that even the best
chatbots struggle with alone. They could also guide students through intricate
calculations, offer explanations for physical phenomena, and suggest various
approaches to tackle a given problem, while ensuring every step is transparent
and reproducible.

However, over-reliance on chatbots may inadvertently reduce a student’s
critical thinking and problem-solving skills. Since general-purpose language
models like ChatGPT are also pre-trained on a large body of Internet data they
produce misconceptions and AI hallucinations. To address these concerns, such
AI tools must be adapted in the context of active learning to be facilitators
and not authorities. An example of such adaptation is experimentally tested
Khanmigo, a fine-tuned version of ChatGPT [22]. While Khanmigo has been
created to suit educational needs in math teaching, we believe that thanks to
rapid progress fine-tuned open-source language models will serve specifically
to the demands of physics education and competitions.

In a world where STEM and digital literacy are becoming increasingly crucial,
merging cutting-edge AI tools with traditional methods of education might just
be the perfect recipe for nurturing the next more powerful generation of
scientists and researchers.

##  6 References

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## Acknowledgments

This work is supported by the Slovak Research and Development Agency under the
Contract no. APVV-22-0515, APVV-21-0216 and APVV-21-0369.