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Self-Similar Scaling of Solitons and Compactons in Relativistic
JetsKeith Andrew, Michael Carini, Brett [email protected] [email protected] [email protected]
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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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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The field components F are representative of vector potential components orelectric field or magnetic field components.
2jFL
Luminosity ~ Field Amplitude SquaredWidth constrained by jet diameter, velocity constrained by dispersion
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MHD Equations
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
0
1
2
Nonlinearity 0
10
20
30
Kerr Radius
-50
0
50
Amplitude
-2
-1
0
1
2
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
