Stellar dynamics

Stellar dynamics
Days on Diffraction 2023
Authors: O. M. Lecian, B. B. Tirozzi
Speaker: B. B. Tirozzi
June 5 - 9, 2023.
Authors: O. M. Lecian, B. B. Tirozzi Speaker: B. B. Tirozzi Stellar dynamics Days on Diffraction 2023

Abstract
Analyses of stellar dynamics of virialised systems are pursued. The
phenomena of undamped oscillations of virialised systems are studied for
systems evolving upon the non-Gaussianities to the present time. The
time-dependent harmonic-oscillator potentials apt for the analyses are
scrutinised. The asymptotic expansion of the Lewis invariant is
established for the proper potentials. The method of the asymptotic
expansion of small time parameters is set. According to the method, new
conservation laws defining new constants of motions are found. The
mathematical implementation is presented. The physical application
allows one to infer that the constants of motion are functions of the
virialised mass and of the virialised radius; and that the effects leading to
behaviours different from the undamped oscillations must be due to
modifications of the CBE (collisionless Boltzmann equation) hypotheses.
The scheme is demonstrated to be apt not only at the time of
non-Gaussianities, but also at any age of the Universe at which a stellar
system is hypothesised to be formed. The phenomena of the described
oscillations are analysed and framed within observational evidences.

Summary
• Oscillations of stellar systems after the time of virialisation:
•Tools:
- the virial theorem,
- Hamiltonian analysis, and
- description of oscillations.
• Methodologies:
1: new small time parameter asymptotic expansion:
new infinite series of conservation laws of the constants of motion;.
2 new recursive direct integrations:
new approximations of the direct results of the constants of
motions
• Prospective applications:
new WKB paradigms, and
new expressions of the velocities.

Introduction
The presence of undamped oscillations in stellar systems is studied within
the framework of the Hamiltonian formalism.
The Jeans theorem provides with a characterization of the phase space in
terms of the integrals of motion in the case of the collisionless Boltzmann
equations.
The stellar dynamics is qualified at the time of the virialisation
corresponding to the epoch of non-Gaussianities.
The development of new techniques for the calculation of the new
constants of motions provides with new expressions of the Hamiltonian
velocities.
The new techniques allow for a comparison in the precision obtained with
respect to the calculation of the Hamiltonian velocities.
The new analytical expressions allow for a juxtaposition with the cases of
unknown stress-energy tensors.

Evolution of the universe
- t = 0 cosmological singularity;
- t < 10−43s Planck epoch;
- semiclassicalization;
D. Polarski, A. A Starobinsky, Semiclassicality and decoherence of cosmological
perturbations, Class. Quantum Grav. 13, 377 (1996). - General Relativity in the
strong-field regime;
- t = 10−33s to t = 10−32s inflation;
D. Brizuela, T. Pawl’owski, Quantum fluctuations and semiclassicality in an
inflaton-driven evolution, JCAP 10, 080 (2022).
- primordial non-Gaussianities;
X. Chen, Primordial Non-Gaussianities from Inflation Models, Advance in Astronomy
2010, Article ID 638979 (2010).
J. Kristiano, Jun’ichi Yokoyama, Why Must Primordial Non-Gaussianity Be Very
Small? Phys. Rev. Lett. 128, 061301 (2022).
- asymptotic Minkowski.
In the present work, the big-bang scheme is considered, with an FLRW
(Fridman-Lemai’tre-Robertson-Walker) metric, inflation scenario and the
thermal history.

Structures formation
- t < 1s to t = 380.000 years: between inflation and the release of
CMB;
- t = 380.000 years to t = a few hundred million years: CMB and
formation of the first stars and galaxies;
- t = a few hundred million years to t = now: after the formation
of the first stars and galaxies.

The virial theorem
R̄ gravitational virialised radius :
average measure of the linear dimension of the system
(i.e., also, the radius of the sphere in a spherical system),
virialised mass M:
the mass of the virialised system:
⇒ gravitational potential energy W
W = −1
2
GM2
R̄
;
inertia-moment tensor Iij
d2
dt2 Iij = 2Tij + Wij , with
kinetic-energy tensor Tij , and
potential-energy tensor Wij ;
total energy of the system E, remaining constant:
d2
I
dt2 = 2T + W = 2E − W;
⇒ now possible to analyse the evolution of the dynamics of the studied
spherical system.
S. Chandrasekhar, D. D. Elbert, Some elementary applications of the virial theorem to stellar dynamics, MNRAS
155, 435 (1972);
S. Chandrasekhar, E. P. Lee, A Tensor Virial-Equation for Stellar Dynamics, MNRAS 139, 39135 (1968).

The Jeans theorem
Asymmetric drift: difference between the azhimutal velocity and the
circular velocity at a given radius, averaged over a volume element at a
fixed position in space for a steady state.
i) Any steady-state solution of the collisionless Boltzmann
equation depends on the 6-dimensional phase-space coordinates
only through the 3 isolating integrals of motion descending from
the galactic potential; and
ii) any function of the the integrals of motion yields a steady-state
solution of the collisionless Boltzmann equation.
MIT Lectures Series, Chapter 1 Galaxies: dynamics, potential theory, and equilibria, e
print https : //ocw.mit.edu/courses/8 − 902 − astrophysics − ii − fall −
2004/15bf 4ed2d8b1247fc667162259f 251d0 lec11.pdf .
Isolating integral of motion: for particular values, it isolates
hyper-surfaces in phase space.
G. Contopoulos, Astrophysical Journal 138, 1297 (1963).

Stellar Hamiltonian dynamics
- Poisson equation of self-gravitating stellar systems
∇2
Φ = 4πGρ̃(⃗
r, t) ≡ 4πG
R
f (⃗
r, ⃗
v, t)d3
v;
- acceleration from the Poisson equation as due to gravitational
interaction within the stellar system;
- inward gravitational acceleration a(r) from circular velocity v(r) at
radius r as a(r) = −v(r)
r ;
- the distribution function must be one of negative energy;
Chandrasekhar, S.; Elbert, D. D., Some elementary applications of the virial theorem
to stellar dynamics, MNRAS, 155, 435 (1972);
A Tensor Virial-Equation for Stellar Dynamics , S. Chandrasekhar, Edward P. Lee,
MNRAS 139, 1968, 13539.
- collisionless Boltzmann distribution function;
- isolating integrals, the Jeans theorem and the distribution functions:
gravitating systems are parameterised after a distribution function, a
mass density, related after a gravitational potential which obey the
simplified Vlasov equations
A. M. Fridman, V. L. Polyachenko, Physics of Gravitating Systems I- Equilibrium and
Stability, Springer-Verlag New York Inc., Harrisonburg, Virginia, USA (1984).

Galactic oscillations
- oscillations in ancient star reveal galactic merger:
- kinematically distinct structures within the Milky Way identify univoquely stellar
population accreted after acquisition of multiple smaller satellite galaxies;
- after NASA Transiting Exoplanet Survey Satellite;
- naked-eye visible star;
- age of star dated after presence in the star of elements heavier than carbon produced
by nuclear reactions involving helium and compared with Galactic-disk stars;
- asteroseismic observations, spectroscopic, astrometric, and kinematic ones.
-calculation of oscillation: measurement of the positions and velocities ,
comparison with orbital integrations for the populations with low and
high [Mg/Fe]: resulting distributions of the eccentricity and maximum
vertical excursion from the Galactic mid-plane evaluated.
W. J. Chaplin et al., Age dating of an early Milky Way merger via asteroseismology of the naked-eye star ν Indi,
Nat. Astron. 4, 382 (2020).

Solar-System oscillations
Vertical oscillations: oscillations of the system in the direction
orthogonal to the galactic plane, due to the resultant of the vector
sum of the gravitational accelerations from the regions of the
Galaxy.
- vertical oscillation of the solar system with respect to the
Galactic plane can be estimated from the mass distribution;
M. R. Rampino, Galactic cycle, In: Encyclopedia of Planetary Science. Encyclopedia
of Earth Science. Springer, Dordrecht, 1997.
- within the Solar System:
long-term cyclic motions during its path along the Miky Way, and
vertical oscillation of the solar system with respect to the
Galactic plane.
R. H. Miller, B. F. Smith, Galactic oscillations, AIP Conference Proceedings 278, 130
(1992);
H. Ro et al., Transverse Oscillations of the M87 Jet Revealed by KaVA Observations,
Galaxies 11, 33 (2023).

Observational evidence of oscillataions and constants of
motions
Constants of motion calculated within symplectic transformations
to be compared with
M. Hoeft, J. P. Mücket, S. Gottlöber, Velocity Dispersion Profiles in Dark Matter
Halos, The Astrophysical Journal 602, 162 (2004);
K. Dolag, S. Borgani, S. Schindler, A. Diaferio, A.M. Bykov, Simulation techniques for
cosmological simulations, e-Print: 0801.1023 [astro-ph];
P. M. S. Namboodiri, Oscillations in galaxies, Celestial Mechanics and Dynamical
Astronomy 76, 69 (2000);
Z. Xu, Evolution of energy, momentum, and spin parameter in dark matter ow and
integral constants of motion, e-print [arXiv:2202.04054 [astro-ph.CO]];
Z. Xu, Inverse mass cascade in dark matter flow and effects on halo deformation,
energy, size, and density profiles, e-Print: 2109.12244 [astro-ph.CO].
with specific initial conditions of structures formation.
Vogelsberger et al. Cosmological simulations of galaxy formation, Nat. Rev. Phys. 2,
42 (2020).

Potentials
Mathematical outline
Regularity assumptions about the time-dependent-oscillators
potentials
G. Fiore, The time-dependent harmonic oscillator revisited, e-print math-ph
arXiv:2205.01781.
Generalized time-dependent oscillator Hamiltonian spelled out as
H = f (t) p2
2m + 1
2g(t)w2
0 x2
- for a point particle of mass m, i.e. for a
system in which both the kinetic term and the potential term are
generalised as time-dependent by means of the functions f (t) and
g(t), respectively; furthermore,
- the initial values for the two functions f (t) and g(t) are
established.
L. F. Landovitz, A. M. Levine, W. M. Schreiber, Time-dependent harmonic oscillators,
Phys. Rev. A 20, 1162 (1979).

Generalised harmonic potentials:
Astrophysical implementations
- (pulsed) Plummer potential
V (r, t) = − m(t)
√
1+r2
;
H. C.Plummer, On the problem of distribution in globular star clusters, Mon. Not. R.
Astron. Soc. 71, 460 (1911).
- (pulsed) Dehnen potentials
V (r, t) = −m(t)
2−γ
h
1 − r2−γ
(1+r)2−γ
i
,
with γ = 0, γ = 1/2 and γ = 1;
W. Dehnen, A Family of Potential-Density Pairs for Spherical Galaxies and Bulges,
Monthly Notices of the Royal Astronomical Society 265, 250, (1993).
- Terzic’-Kandrup potentials
spherically-symmetric potentials which lead to a periodic dynamics
(which can be cast in the Mathieu Equation)
V (r, t) = V (r)(1 + m0sinωt),
B. Terzic’, H. E. Kandrup, Orbital structure in oscillating galactic potentials, Mon.
Not. R. Astron. Soc. 347, 957 (2004).

- Eddington-Jeans potentials
V (r, t) = V (r)M0Mn
(t)
J. H. Jeans, Mon. Not. Roy. Astron. Soc. 85, 912 (1924);
E. N. Polyachova, Uchen. Zap. Saratovsk. Gosud. Chernyshevsk. Univ., 424, 104
(1989).
- Jaffe potential and Hernquist potential:
can be considered as particular cases of the Dehnen potential
W. Jaffe, A simple model for the distribution of light in spherical galaxies, MNRAS
202, 995 (1983); L. Hernquist , An Analytical Model for Spherical Galaxies and
Bulges, ApJ 356, 359 (1990).

Cosmological model with a generic Terzic’-Kandrup pulsed potential:
the Ermakov-Lewis-Leach invariants
Constants of motion descending from time-dependent potentials.
- one-dimensional generalised time-dependent harmonic oscillator
H = ẍ + ω2(t)x
- generic Terzic’-Kandrup pulsed potential
V (r, t) ≡ V (r)(1 + m0sinωt)
- auxilliary function ρ(t) obeying
ρ̈(t) + ω2(t)ρ(t) − 1
ρ3(t)
= 0
defines the
- Ermakov-Lewis-Leach invariant of motion IELL
IELL = 1
2

ρ−2x2 + (ρp − ρ̇x)2

V.P. Ermakov, Univ. Izv. (Kiev) 20, 1 (1880);
H. R. Lewis, Class of Exact Invariants for Classical and Quantum Time Dependent
Harmonic Oscillators, J. Math. Phys. 9, 1976 (1968);
P. G. L. Leach, On a generalization of the Lewis invariant for the time-dependent
harmonic oscillator, Siam J. Apll. Math. 34, 496 (1978).

New solution techniques
- new asymptotic small-time- parameter expansion
ρ = ρ0 + ϵρ1 + ϵ2ρ2 + ϵ3ρ3 + ϵ4ρ4..,
with ϵ parameter to be addressed in the following.
⇒ New infinite series of conservation laws
d
dt ρ̇2
2n = (−1)−1+n/22ρ̇2
2nf2n(ρ2n, ρ2n−2, ..., ρ2, ρ0)
for the auxiliary variable ρ defining a
new invariant of motion:
according to the new following two methodologies.

New methodology 1: asymptotic small-time-parameters
expansion
case: third-order approximation
ρ0 ̸= 0, Ω2
ρ0 −
1
ρ3
0
= 0,
Ω2
ρ1 + 3
ρ1
ρ4
0
= 0,
¨
ρ0 + Ω2
ρ2 + 3
ρ0ρ2 − 2ρ2
1
ρ5
0
= 0,
¨
ρ1 + Ω2
ρ3 −
12ρ1ρ2 − 10ρ3
1 − 3ρ3ρ2
0
ρ6
0
= 0
at the considered orders in ϵ formally solved as
Ω ≡ ±
1
ρ2
0
,
ρ1 = 0,
d
dt
˙
ρ0
2
= 2 ˙
ρ0

−Ω2
ρ2 − 3
ρ2
ρ4
0

,
ρ3 = 0.

new evaluation of the constants of motion at the time of
non-Gaussianities during a small time integration interval
∆t = tf − ti
during which
the virialised radius and the virialised mass are constant
ρ0 = A0 + A1(t − ti ) + A2(t − ti )2 + ...
ρ1 = 0,
ρ2 = B0 + B1(t − ti ) + B2(t − ti )2 + ...
ρ3 = 0

The following expansion is found
2A2 + 4B0V (r)(1 + m0sinωti ) + (4B0V (r)m0ωcos(ωti ) +
4B1V (r)(1 + m0sin(ωti )))(t − ti ) + (−2B0V (r)m0sin(ωti )ω2 +
4B1V (r)m0cos(ωti )ω + 4B2V (r)(1 + m0sin(ωti )))(t − ti)2 +
O((t − ti )3) = 0
newly solved as
A0 ≡
1
(V (r)(1 + m0sinωti ))1/4
,
A1 ≡ −
1
4
m0cosωti
(V (r)(1 + m0sinωti )1/4(1 + m0sinωti )
,
A2 ≡
1
32
m0ω2
(m0cos2
ωti + 4sinωti + 4m0)
(V (r)(1 + m0sinωti ))1/4 (−m2
0cos2ωti + 2m0sinωti + m2
0 + 1)
B0 ≡ −
1
64
m0ω2
(m0cos2
(V (r)(1 + m0sinωti ))5/4 (−m2
0 + 1)
,
B1 ≡ −
1
64
m2
0ω3
V (r)cosωti (m0cos2
(V (r)(1 + m0sinωti ))9/4 (−m2
0 + 1)
,
B2 ≡ −2ωı̀2m0V (r)sinωti −
1
64
m0ω2
(m0cos2
(V (r)(1 + m0sinωti ))5/4 (−m2
0 + 1)
+
+ 4m0ωcosωti
1
64
m2
0ω3
V (r)cosωti (m0cos2
(V (r)(1 + m0sinωti ))9/4 (−m2
0 + 1)
.

New methodology 2
ρ0 = 1
(1+m0sin(ωt))1/4
new assumption:
ρ2(t) ≡ ρ̃2(t) changing slowly during the integration time interval
˜
ρ2 = (1+m0sin(ωt))1/4
ω2
√
1−m2
0
h
2ωtarctan

m0+tan
ωti
2
ω
√
1−m2
0

− 2ωti arctan

m0+tan
ωti
2
ω
√
1−m2
0

+
−2ω(t − ti )arctan

m0+tan
ωti
2
ω
√
1−m2
0

− iωtLog
1−ieiωt
m0+
√
1−m2
0
1+
√
1−m2
0
+
+iωtLog
−1−ieiωt
m0+
√
1−m2
0
−1+
√
1−m2
0
− iωti Log
1−ieiωti m0+
√
1−m2
0
1+
√
1−m2
0
+
iωti Log
−1−ieiωti m0+
√
1−m2
0
−1+
√
1−m2
0
+ PolyLog

2, − ieiωt
m0
−1+
√
1−m2
0

−
PolyLog

2, − ieiωti m0
−1+
√
1−m2
0

− PolyLog

2, ieiωt
m0
1+
√
1−m2
0

+
PolyLog

2, ieiωti m0
1+
√
1−m2
0
i
.

Next issues
• verification of the small time asymptotic expansion parameter ϵ;
• the Ermakov-Lewis (EL) invariant: a new class of conservation
laws of the constants of motions ruled after the generic
Terzic’-Kandrup pulsed potential:
H = 1
2η

ẍ + Ω2
(t)x

- auxiliary function σ(t) obeying
η2
σ̈(t) + ω2
(t)σ(t) − 1
σ3(t) = 0
defines the
- invariant of motion
IEL = 1
2

σ−2
x2
+ (σp − ησ̇x)2

new asymptotic small time asymptotic expansion of the
auxililary variable σ according to the infiinitesimal parameter ϵ:
generalised constants of motion
- depending also on the virialised radius, and
- bringing new light on the virialisation of the mass.
Harmonic Oscillators, J. Math. Phys. 9, 1976 (1968).

• the Ermakov-Lewis adiabatic invariant: a new class of
adiabatic invariants ruled after the generic Terzic’-Kandrup
pulsed potential:
the series decomposition of IEL with the auxiliary variable σ
decomposed as positive powers of η brings
a new class of adiabatic invariants.
Harmonic Oscillators, J. Math. Phys. 9, 1976 (1968).
• new generalisation to three-dimensional models
including
- the anisotropic problem, and
the singular quadratic perturbation problem.
N. J. Guenther, G. P. L. Leach, Generalized Invariants for the time-dependent
harmonic oscillator, J. Mathematical Phys. 18, 572 (1977).

Perspectives
• WKB methods:
- linear WKB method,
linear oscillator: the distribution of the energy is always the arcsine
distribution;
M. Robnik, V.G. Romanovski, Let’s Face Chaos through Nonlinear Dynamics,
Proceedings of the 7th International Summer School and Conference Let’s Face Chaos
through Nonlinear Dynamics, 2008, AIP Conf. Proc. 1076, M. Robnik, V.G.
Romanovski Ed.s, AIP, Melville, New York (2008).
- non-linear WKB method for time-dependent power law potentials,
in which the isolating integral Energy is not constant:
for homogeneous power law potentials
which includes also the quartic oscillator;
M. Robnik, Recent results on time-dependent Hamiltonian oscillators, Eur. Phys. J.
Spec. Top. 225, 1087 (2016);
• characterization of the variable ẋ from a Hamiltonian point of
view.
L. Lucie-Smith, H. V. Peiris, A. Pontzen, Explaining dark matter halo density profiles
with neural networks, (2023) e-print 2305.03077 [astro-ph.CO].

Thank You for Your attention.

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