Electromagnetic Models of Gamma Ray Bursts: A Tutorial

1
Electromagnetic Models of Gamma Ray BurstsA
Tutorial

• Roger Blandford
• KIPAC
• Stanford
• With thanks to Jonathan Granot

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Outline

• Some Reasons to Consider EM Models
• Variations on a Theme
• General Principles
• Formal Approaches
• Sketch EM Model for Long GRBs

Physics not phenomenology
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Some reasons to consider EM models

• Fireball model presents some difficulties
• Hard to create large entropy outflows
• Like the early universe S(GRB)106k S(COS)
  1010k
• Hard to make high Mach number flows
• M G300
• Real flows shock and dissipate long before
  reaching this level of organization
• Pairs may annihilate and radiation may decouple
  from the jet before it accelerates the baryons to
  ultrarelativistic speed
• Within 10 stellar radii
• If magnetic field is invoked to create the jet
  power, how do you get rid of it so efficiently?
• Must become less important than a small admixture
  of protons

4
More reasons to consider EM models

• Other collimated, relativistic, bipolar outflows
  are not radiation-dominated electro/hydromagnetic
  models are most commonly invoked now.
• Pulsar wind nebulae
• AGN jets
• Galactic superluminal sources
• Magnetic fields provide active collimation
• Hoop stress
• Some AGN jets (eg Pictor A) are extremely
  collimated

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Pulsar Wind Nebulae

• Spinning, magnetized neutron stars release energy
  electromagnetically
• Polar jets are common and an appealing
  interpretation is that you see X-ray synchrotron
  emission from rapidly cooling electrons where the
  electrical currents flow

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Pictor A
Magnetic Pinch?
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Electromagnetic Models

• Basic hypothesis is that the energy is released
  electromagnetically by a central spinning object
  and then transported to the main emission site
  where it is dissipated in the form of electrons
  and positrons which radiated Synchrotron and
  inverse Compton radiation
• Applies to all essentially ultrarelativistic
  outflows

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Variations on a theme

• GRB Sources
• Spinning black holes
• Up to 29 of the mass of a hole is extractable
• Relativistic accretion disks/tori
• More energy may come from the disk than the hole
• Millisecond magnetars
• Can release their spin energy in minutes as
  required

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More Variations

• Energy transport
• AC transmission
• e.g. chaotic electromagnetic fields with length
  scale 100-1000 km, characteristic of
  the source variation
• EB as relativistic
• Dynamically like radiation-dominated outflow
• Scalar pressure
• No active collimation
• Natural particle acceleration mechanisms

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More Variations

• Energy transport
• Global DC transmission
• Large scale order in magnetic field
• Large scale current circuits
• Toroidal magnetic field dominates parallel field
  far from the source
• If flux is conserved, parallel field (Area)-1
• If current conserved toroidal field (Area)-1/2
• E B still and energy carried by Poynting flux
  B2c
• Center of momentum frame moves relativistically
• Need equipartition particle pressure along axis
  to oppose hoop stress of toroidal field in
  comoving frame.

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More Variations

• Energy Transport
• Local DC transmission
• Episodic ejection of magnetically-confined jet
  segments
• No large scale current circuits
• Relativistic motion
• Changing polarity of parallel field reflects
  changing polarity of disk field
• Disk may eject loops of toroidal field or be
  launched and collimated by vertical field

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General Principles

• Generation
• Collimation
• Propagation
• Dissipation

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Generation

• Disks advect magnetic field and generate it
  through the magnetorotational instability.
• Most electromagnetic source models are some sort
  of unipolar induction mechanism the details vary
• Generally, a rotating magnetic field generates an
  EMF W B, which drives a current
• The relevant impedance is generally that of free
  space under relativistic, electromagnetic
  conditions - 100W.
• The power is roughly EMF x Current x Coefficients
• A current description (plus boundary conditions)
  is equivalent to an electromagnetic field
  description. Both descriptions can convey insight

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Generation

• Unipolar Induction
• Rules of thumb
• F B R2 V W F
• I V / Z0 P V I
• PWN AGN GRB
• B 1012 G 104 G 1016 G
• n 10 Hz 10-5 Hz 103 Hz
• R 106 cm 1015 cm 106 cm
• V 1016.5 V 1020.5 V 1022.5 V
• I 1014.5 A 1018.5 A 1020.5 A
• P 1031 W 1039 W 1043 W (W 107
  erg/s)

?
M
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Simulations are transforming our understanding

• MHD
• 3D
• GR
• Plot of magnetic energy density

Villiers et al
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Consequences of large EMFs

• Particle energy density / EM energy density
• Can be as small as rL/L mec2/eV
• In practice, it wont be!
• Vacuum is an excellent conductor thanks to QED
• Electric field ? rapid breakdown
• accelerate electron, scatter photon, create pair
  and repeat
• This ensures that B2 - E2 gt 0
• It is hard to produce entropy under these
  conditions
• Not a criticism of neutrino models!

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Dipolar vs Quadrupolar Current Flow

• It has generally been assumed that disk field has
  odd parity and the currents are even parity
• If large disks trap only a tiny fraction of the
  radial field present at their outer radii, then
  the opposite may be closer to the truth.

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Asymmetric Outflows/Jets
B
I
X
Even Field Odd Current
Odd Field Even Current
Hybrid Mixed Parity
Can you measure the toroidal field pattern?
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Collimation

• Rotating stars (or gas clouds) can provide
  hydrodynamic collimation of outflow
• Twin exhaust mechanism
• Magnetic collimation is much more powerful and
  will operate in spherically symmetric
  surroundings
• Principles illustrated by cylindrically symmetric
  jet

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Simple Model of Cylindrical Jet
I
I
Pext
r
Pjet
R

• Current I flows along jet walls radius r
• Return current flows along cylinder radius R
• Magnify confining pressure PjetPext(R/r)2
• Equivalently, cavity adjusts to R(2pPext)-1/2I
• Pjet is mixture of particles and tangled field

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Propagation

• As can be seen from the preceding table, putative
  electromagnetic sources generate 1022.5 V EMFs
  (ample for the most energetic cosmic rays!).
• Most fireball models implicitly assume that the
  associated current 1020.5 A shorts out and
  dissipates close to the source and creates heat.
• Electromagnetic jet models propose that the
  current flows out into the emission region and we
  observe the dissipation - like a luminous light
  filament.

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Magnetic field
Nonthermal emission is ohmic dissipation of
current flow?
1018 not 1017 A
DC not AC
Electromagnetic models of extragalactic radio
sources and pulsar wind nebulae
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Propagation

• Jets terminate, sharing their momentum with a
  shocked external medium
• In the case of GRBs the jet only lasts for
  minutes and becomes a spherical cap while the
  external shock remains relativistic
• AGN and PWN jets evolve differently but the
  underlying physical processes should be similar

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Dissipation

• Relativistic particles by shock Fermi
  acceleration.
• This is demonstrably true in the solar system and
  probably the case in the SNR, though the details
  are controversial
• This is problematic in the case of GRBs
• If shocks are relativistic and especially pair
  dominated simulations do not exhibit acceleration
  Spitkovsky?
• Strong fields ? weak shocks
• Other dissipation mechanisms worth considering

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Cylindrical Jet (again)

• Jets are likely to have a relativistic velocity
  gradient G(r) and there has to be internal
  pressure to balance magnetic hoop stress
• Force balance
• Electromagnetic and fluid jet powers
• Rough equipartition of energy

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Dissipation

• Relativistic electromagnetic jets are likely
  pair-dominated
• They are also likely to be locally unstable,
  though velocity gradients may convey global
  stability
• The best candidate acceleration mechanism seems
  to be to develop a turbulence spectrum of EM
  modes cascading down to short wavelengths where
  they are absorbed by stochastic particle
  acceleration
• Needs simulation!

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Other Particle Acceleration Mechanisms

• Internal shocks are ineffectual
• Shear flow in jets
• Full potential difference is available for
  particles accelerated via polarization drift
  along E
• UHECR??
• Fast/intermediate wave spectrum
• Nonlinear wave acceleration (Blandford 1973)
• Charge starvation (Thompson Blaes 1997)
• Force-free allows EgtB - catastrophic breakdown

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Let there be Light

• Faraday
• Maxwell
• Definition
• Initial Condition

? Maxwell Tensor, Poynting Flux
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Electromagnetic Velocity

• L-C Circuit
• Near Solenoid, E lt B
• U (E x B)/B2 lt 1
• Near Capacitor, E gt B
• U (E x B)/E2 lt 1
• Resistive wire ? E ? E x B into wire where the
  energy dissipated
• Astrophysical Sources
• V 10 15 - 1022 Volts
• QED effects ? E lt B
• Cosmic sources have inductance.
• Velocity of frame in which E 0

U
U
I
ExB
U
Q
-Q
U
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Force-Free Limit of Relativistic MHD

• Ignore inertia of matter s UM/UPgtgtG2, 1
• Electromagnetic stress acts on electromagnetic
  energy density
• Fast and intermediate wave characteristics
• Simpler than RMHD

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Electromagnetic GRB Model

• Gravitational binding energy ? EM energy flux
• Organized, anisotropic, axisymmetric current
  flow/Poynting flux
• VEME/B c
• Electromagnetic acceleration ? G 100, M 1
• Pairs combine, gs escape, E,B dominate
• Poynting flux catches shocked CSM 1016cm
• Current dissipation-gt pairs -gtGRB
• Relativistic internal motions -gt variability
• Sweep up CSM at 1017cm
• Field incorporated from magnetic piston, electron
  shock acceleration
• Anisotropic afterglow

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Sketch EM Model of Long Bursts

• I Energy Release
• II Bubble Inflation
• III Shell Expansion
• IV Blast Wave

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I Energy Release

• Long bursts
• Spinning Black Hole Torus
• Millisecond Magnetar
• LEM R2B2c
• B 1014 G, n? 3 kHz, E 1052 erg, ts 100 s
  106 tdyn
• V 1022.5 V , I 1020.5 A
• Stationary, axisymmetric DC current flow
• Short bursts admit more possibilities
• e.g. coalescing neutron stars

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II Bubble Inflation

• Collapsar/hypernova within stripped star, R
  1011 cm
• Surface return current, surface stress
  (I/Rsinq)2
• Anisotropric expansion in absence of rotation
• Dissipation inevitable if V lt c/ln(qmax/qmin)
  0.1c otherwise not
• cf PWN
• Rationale for fireball model?
• Compute evolution given envelope dynamics
  tbreakout 10 s
• Biconical expansion outside star dictated by CSM
• Shell forms when r gt cts 3 x 1012 cm
  ultrarelativistic expansion
• Thermal precursor measure of dissipation?

Toroidal magnetic field self-collimating
G 104
Pairs combine, gs escape
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Underlying geometry ? scalings

• B 2I/cr, r R sin?
• L? R2B2c ? (sin?)-2
• ? E? ? (sin?)-2 ?-2
• ? E? ?2E? const (as observations imply Frail
  et al. 2001)

I
B
B
I
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III Shell Expansion
Shocked Circumstellar Medium

• rGRB G2cts (Lts2/rc)1/4 1016 cm
• V ExB/B2 G 100
• Piston thickness cts 3x1012 cm
• Instability ? variable g-ray emission
• Facilitates escape of hardest g-rays

t
?
?C
ts
q
r
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Simple derivation of radial scaling

• The EM outflow travels essencially at the speed
  of light (i.e. ?EM (1-U2)-1/2 ? ?CD)
• Energy emitted at time te catches up with the CD
  at ti R(te)/?1-??ti te/1-?(ti) te ?2(ti)
• ? E(ti) Lti/?2(ti) A(cti)3-kc2?2(ti)
  r????R?k
• ? ?(ti) (L/Ac5-kti2-k)1/4 for te lt ts (ti lt
  Rad/c)
• Rad (Lts2/Ac)1/(4-k) RGRB (forward shock
  becomes adiabatic)
• ?(Rad) (L/Ac5-kts2-k)1/2(4-k)

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IV Blast Wave

• rGRB lt r lt rNR (Lts/rc2)1/3 1018 cm G
  100-2
• Achromatic break when G q-1
• Magnetic field mixed in from CD?
• Particles accelerated at shock?
• Constant energy per decade in ?
• Standard qualitative interpretation of
• afterglow spectra
• More variation than in shock models
• ? is important parameter
• Axial currents ? short bursts?
• Becomes more spherical at r gt rNR

q
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Possible Tests

• Early afterglow evolution
• Shape -VLBI
• Thermal precursors no reverse shock
• Polarization -constant PA?
• GLAST not AMANDA UHECR
• Orientation statistics, orphan afterglows, XRF
• Fluctuation statistics

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Summary

• EM GRB models different from fluid models
• Circumvent hydrodynamic middleman
• Closer to models of other relativistic outflows
• Many interesting physical processes, poorly
  understood
• Simulation methods developing rapidly
• Proceed with care when drawing conclusions from
  observations!