Rigidly circling charged tori
Astronomical Institute
Czech Academy of Sciences
Prague
Institute of Physics
Silesian University in Opava
Czech Republic
CZ.1.07/2.3.00/20.0071: Synergy
Jiří Kovář, Petr Slaný, Claudio Cremaschini , Zdeněk Stuchlík
Vladimír Karas, Audrey Trova
Rigidly rotating torus - beginning of the story
Phys. Rev. D 90, 044029 – Published 13 August 2014
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rigidly rotating fluid torus - W = const
charged torus
MHD approach – general relativistic
test particles approximation (very low densities)
kinetic theory (low densities)
MHD (high densities)
Basic equations
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basic GRMHD equations
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the model
- test non-conductive charged fluid
- axial symmetry and azimuthal motion
- self field is neglected
Pressure equations – general case
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pressure equations
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integrability condition
integrable
non-integrable
integrable
The Wald background and basic assumptions
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Geometry
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Electric and magnetic test fields
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Angular momentum - velocity distribution
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Energy density
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Equation of state
Auxiliary functions and integrability condition solution
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auxiliary function
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specific charge distribution function
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integrability condition implies
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generic functions for charge distribution
Tori existence
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Conditions for the existence
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Equatorial tori existence
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Polar cap existence
The solution
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Equatorial torus
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Polar cap
Characteristic quantities profiles
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pressure
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mass density
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specific charge
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temperature
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magnetic self-field
fixing G=5/3
B,Q,k0, w - center
h-function (B, Q, k0, h0, w)
- space characteristic (volume, edges)
- density (=> pressure, charge density), specific charge, temperature
- k parameter - density (=> pressure, charge density)
Tori construction and astrophysical setup – center setting
Center setting - B, w, Q,k0
polar cap
equatorial torus
h-function profiles
polar cap
equatorial torus
Specific charge a density profiles
polar cap
equatorial torus
Summary
1. We have considered GR toy model of charged thick dielectric torus rotating in
gravitational and electromagnetic fields.
2. Within the model, we have prove the existence of the equatorial and also of
polar tori rigidly rotating in homogeneous magnetic field close to a charged
black hole.
3. Along with the pure theoretical results, we have tried and still trying to fit the
physical parameters to the astrophysical scales.
4. We have considered many simplifying assumptions (test fields, completely
dielectric matter, neglecting the gravitational and electromagnetic self fields,
neglecting the electrostatic pressure). These are, however, within the assumed
model and chosen parameters very reasonable, simplifying the investigation
and not substantially changing the main results.