Self-Similar Scaling of Solitons and Compactons in Relativistic Jets Keith Andrew, Michael Carini,...

Self-Similar Scaling of Solitons and Compactons in Relativistic

JetsKeith Andrew, Michael Carini, Brett BolenKeith.Andrew@wku.edu Mike.Carini@wku.edu Brett.Bolen@wku.edu

Department of Physics and AstronomyWestern Kentucky University

Bowling Green KY Soliton- long lived nondissipative wave form where nonlinear amplitude growth is balanced by dissipative losses, need not be topological in originCompacton-long lived wave form with well defined functional relationship between amplitude, width and speed of propagation, no exponential envelope (3,7)

Existence Requirements

• Convection

• Dispersion

• Diffusion

• Nonlinear

://rst.gsfc.nasa.gov/Sect20/h_accretion_disk_02.jpg&imgrefurl=http://rst.gsfc.nasa.gov/

Quasar 3C120

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Large Amplitude Nonlinear Fields-Solitons Nonlinear Schrödinger Equation

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Multiscale Wavelet Similarity Analysis For Nonlinear PDEs (1,5,6) Waves characterized by:1. Amplitude A2. Width w3. Velocity v-limited by dispersion relationship

Localized soliton and compacton solutions expanded with GaussianFamily Wavelets

For a given scale, j, the similarity transformationMaps the NPDE->single scale algebraic constraintof the form F(A,w,v)=0 for localized soliton like solutions.

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Luminosity ~ Field Amplitude SquaredWidth constrained by jet diameter, velocity constrained by dispersion

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U-internal energyP-pressure-ρ-densityΦ-external gravitational potentialV-velocity vector fieldB-magnetic field

Buckingham’s Pi TheoremOnly dimensionless quantities needed (1)

G-Newton’s constant of gravitationB- magnetic fieldl- characteristic length of the jet-ρ-density of surrounding medium -c-speed of light-v-jet’s ejection velocityM- core massdM/dt- mass accretion rateL-jet luminosity

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1. Espinosa, M. H., Mendoza, S. Hydrodynamical scaling laws for astrophysical Jets, arXiv.astro ph/0503336 v1, (Mar 2005)

2. Tevecchio, F., Jets at all scales, arXiv.astro-ph/0212254v1, (Dec. 2002)3. Marklund, M., Tskhakaya, D.D, Shukla, P.K., Quantum Electrodynamical shocks and solitons

in astrophysical plasmas, arXiv.astro-ph/0510485 v1 Jan. 2002)4. Schwinger, J., On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951)5. P G Kevrekidis, V V Konotop, A R Bishop and S Takeno 2002 J. Phys. A: Math. Gen. 35

L641-L6526. Ludu, A, O’Connell, R.F., Draayer, J.P., Nonlinear Equations and Wavelets, Mulit-Scale

Analysis, arXiv.math-ph/0201043 v1(Oct 2005)7. Tatsumo, T., Berezhiani, V. I., Mahajn, S. M., Vortex Solitons-Mass, Energy and Angular

momentum bunching in relativistic electron-positron plasmas, arXiv.astro-ph/0008212 v1, (Aug 2000)

HST Image of Quasar Jets

Wavelet Scaling Rules

Abstract

The jet forming inner region of an object containing a massive Kerr black hole will contain a hot turbulent lepton plasma that can be modeled by a system of relativistic MHD-NPDE. The nonlinearities in these equations give rise to long lived localized soliton solutions and soliton like solutions known as compactons that exist at all length scales. These objects could give rise to structure formation at all locations along the jet that appear as shock bows, vortices or knots that would cause luminosity variations along the jet axis. Here we study the scaling behavior of these solutions in jet environments by using the dimensionless scaling rules from the Buckingham Pi Theorem with the self-similar scaling of nonlinear wavelets in a system of relativistic NPDE to estimate the resulting fractional change in jet luminosity.

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ConclusionsFractional change in luminosity

From BU website:http://www.bu.edu/blazars/research.html

From Dr. Marsher BU websiteFrame from a conceptual animation of 3C 120 created by COSMOVISION                         

Luminosity Nonlinear

-2

-1

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1

2

Nonlinearity 0

10

20

30

Kerr Radius

-50

0

50

Amplitude

-2

-1

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1

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Nonlinearity

Luminosity Nonlinear

-2

-1

0

1

2

Nonlinearity 0

10

20

30

Kerr Radius

-50

0

50

Amplitude

-2

-1

0

1

2

Nonlinearity

Odd power η=3

Even power η=4