Download Tue_10.00-Cadez

The role of electromagnetism in
tidal disruption events
Andrej Čadež
University in Ljubljana
Why should global electomagnetic fields be
important in large scale phenomena?
• Freefall of pressureless fluid into a black hole.
• Question: How can the tidal energy be “almost instantaneously”
transformed into X rays?
• An out of a sleeve proposal?
Gravity directly couples to
electromagnetism through
dynamics.
Pulsar slow down as an example of coupling
dynamics to electromagnetism --- the case of Crab
• It is often assumed that pulsars slow down as the result of
electromagnetic radiation that they emit as rotating magnets and that
their plerionic nebulae are illuminated by synchrotron radiation
emitted in the hot corona of the neutron star.
• However, detailed observations of the famous Crab nebula and its
pulsar present some striking displays which, I believe, point to
organized structures on a large scale and thus hint on the presence of
the direct coupling of dynamics to electromagnetism.
3D Structure of Crab nebula (by spectroscopy)
• Filaments, formed
by line radiation
from H, N and S (Hred, N-green, Sblue) form a regular
cage spread over an
almost spherical
oval with a parsec
across. The
gravitational field
alone can not form
such a structure
X-rays are emitted in a regular disk-jet pattern,
completely contained within the line radiation cage
Pulsar slow-down
• If energy loss is magnetic
dipole radiation then:
•
•
Rotational phase residuals with respect to the braking law of the episode
and braking index during the episode. The total number of turns during
the ~30 years is 2 × 1010 .
𝑑𝐸𝑟𝑜𝑡
𝜔4
∝−
𝑟𝑝 4 𝐵𝑠 2 so:
𝑑𝑡
4𝜋𝜇0
𝑑𝜔
= 𝜔 = −𝐾𝜔3
𝑑𝑡
𝜔𝜔
Braking index: 𝑛 = 2 = 3
𝜔
• But: Crab pulsar slow down
closely follows a braking
index law with the braking
index staying constant for a
few years and then abruptly
changes to a different value
between 2.2 and 2.6
• Braking index less than 3 can
be understood, if the nebula
has a resistive component
Pulsar EM field interacting with nebula
• Magnetic dipole radiation into dispersive medium described by conductivity and
permittivity: 𝑗 = 𝜎𝐸 and 𝐷 = 𝜀𝐸
• The dispersion relation between 𝑘 and 𝜔 is
𝑘2
𝜔 2
𝑐
=𝜀
+ 𝑖𝜇0 𝜎𝜔
• Poynting flux from pulsar: magnetic moment 𝑝𝑚 , angular velocity 𝜔 if dispersive
shell is at radius 𝑟𝑠 is:
• 𝑖𝑓 𝑘𝑟𝑝 ≪ 1
3
2
𝜔2
𝑐3
• 𝜀 > 0: 𝐿1 =
𝜇0 2
𝜔
6𝜋
• 𝜀 < 0: 𝐿1 =
𝜇0 2 𝜇0 𝜎
𝜔
𝑝𝑚 2
6𝜋
𝑟𝑠
𝜀
+
𝜇0 𝜎
𝑟𝑠
𝑝𝑚
2
𝜔𝜔
𝜔2
⟹
𝑛=
⟹
𝑛=1
=3−
2
𝜔 2 𝑟𝑠
3/2
1+𝜀
𝑐 𝜇0 𝜎𝑐
(can not explain Crab braking)
What is needed to explain average Crab
braking index 𝑛 ≈ 2.5?
𝑛 =3−
1. 𝜎 =
𝑛𝑒 𝑒 2
; 𝜈𝑐
𝑚𝑒 𝜈𝑐
2
2
3
𝜔
𝑟𝑠
1+𝜀 2
𝑐 𝜇0 𝜎𝑐
= 2 .5 requires 𝜎 =
1 3/2 𝜔 2 𝑟𝑠
𝜀
3
𝑐
𝜇0 𝑐
… electron collision frequency ⟹ 𝜈𝑐 =
2. 𝜈𝑐 = 𝜈𝑒𝑒 + 𝜈𝑒𝑖 + 𝜈𝑒𝛾 =
𝑒2
𝑛𝑒
3
3
4𝜋𝜀0
3 𝑚𝑒 𝑘 𝑇
4 𝜋
2
𝑛𝑒
3
𝜀2
= 3.5 × 10−12
𝐴
𝑉𝑚
𝜀 3/2
𝑟𝑠
𝑟𝑝
× 8.1 × 109 𝑐𝑚3 𝑠 −1 very high!
2
𝑛𝑒 + 2𝑍 𝑛𝑖 𝐿𝑜𝑔 Λ +
𝜎𝑇 𝐵2
𝛾
𝑚𝑒 𝑐 𝜇0 𝑒
Assume 𝑟𝑠 = 𝑟𝑝 , 𝜀~1, magnetic field on pulsar surface 𝐵0 = 108 T.
1. If 𝜈𝑐 =
𝑊𝐵 =
𝐵2
2
2
𝜈𝑒𝛾 then 𝐿𝑛𝑒𝑏 = 𝑚𝑒 𝑐 𝛾𝑒 𝜈𝑐 𝑛𝑒 𝑑𝑉 ≈ 2 𝛾𝑒 𝑛𝑒 𝜎𝑇 𝑐
𝑑𝑉 ; magnetic energy
2𝜇0
𝐵2
𝐵0
𝑟
8𝜋 3 𝑩0 2
3
𝑑𝑉; if 𝐵 is dipole 𝐵 = 𝑟𝑝
− 3𝑟 ∙ 𝐵0 5 , then 𝑊𝐵 = 𝑟𝑝
~7 × 1034 𝐽
3
2𝜇0
𝑟
𝑟
3
𝜇0
𝑛𝑒
𝛾𝑒 2
𝑒𝑟𝑔
37
38 [𝑒𝑟𝑔 ]
𝐿𝑛𝑒𝑏 ~3 × 10 [
,
𝐿
~2
×
10
𝑠]
𝑠
𝑠𝑙𝑜𝑤𝑑𝑜𝑤𝑛
1000𝑐𝑚−3
1000
and
Can the nebular plasma satisfy the calculated
conditions?
•
i.e.
1.
2.
3.
4.
5.
𝛾𝑒 2
~3, 𝜀 > 0
1000
𝑛𝑒 ~0.3𝑐𝑚−3 , 𝛾𝑒 ~1 × 105 , 𝜀~1
𝑛𝑒
1000𝑐𝑚−3
𝑛𝑒 is consistently lower than electron density in filaments
𝛾𝑒 ~1 × 105 is not inconsistent with energy spectrum
𝜀~1 ??? Neutral plasma 𝑘 2 𝑐 2 = 𝜔2 − 𝜔𝑝 2 , so for 𝜔 < 𝜔𝑝 : 𝜀 = 0.
𝑛𝑒 𝑒 2
Plasma frequency 𝜔𝑝 = 𝜀 𝑚 ∗ is higher than the frequency of pulsar
0 𝑒
EM wave, so wave can not propagate and heat plasma, only locally
non-neutral plasma allows propagation.
To keep the population of high energy electrons tied to the nebula,
long range electric and magnetic fields are required. Plasma can not
be locally neutral!
1
𝜀
Size of heating region 𝑙ℎ ~ ℐ𝑚[𝑘] = 𝜇 𝜎𝑐 ~7.6 × 108 𝑚: about solar
0
size!!!
Kohri, Ohira, Ioka: MNRAS2012
The role of the nebula and assumed structure
• Reprocessing the power emitted by the central source
• Providing electromagnetic field to prevent electron depletion in the central source
Structure:
• no line radiation detected inside the “cage” of filaments, but gamma emission present
• high energy electrons must be present at very low density
• Collision frequencies :
•
•
1
𝑛𝑖
𝑐 3
2
6
= 75 × 10 𝑍𝑖
𝐿𝑜𝑔
𝜈𝑒𝑖
13 𝑐𝑚−3
𝑣𝑒
1
𝑛𝑖
𝑐 3
3
= 41 × 10
𝐿𝑜𝑔 Λ
𝜈𝑒𝑒
13 𝑐𝑚−3
𝑣𝑒
Λ year
year
• Conclude: mean free path of both ions and electrons is longer than the size of the “cage”,
electrons and ions effectively do not exchange energy.
• Assume that the “cage” represents a cold (~10.000K) neutral plasma confinement
Boltzmann and generation of electric field
• Consider the ensemble of electrons and ions inside the “cage” as a two component ideal gas of
particles acted on only by the electric field that they mutually create and is constrained to move
inside the “cage”. The central engine is very small and heats electrons to very high temperature,
while ions remain relatively cold because of their small 𝑒 𝑚 ratio. The distributions of electrons
and ions in the phase space are described by their respective distribution functions 𝑓𝑒 𝑟, 𝑝 and
𝑓𝑖 𝑟, 𝑝 . In stationary state the distribution functions are constants of motion, so their Poisson
brackets with the system Hamiltonian vanishes: 𝑓𝑒 , ℋ = 0, 𝑓𝑖 , ℋ = 0.
𝑁
𝑁
𝑒
𝑖
• System Hamiltonian: ℋ = 𝑖=1
𝐻𝑒 𝑟𝑖 , 𝑝𝑖 + 𝑖=1
𝐻𝑖 𝑟𝑖 , 𝑝𝑖 , here 𝐻𝑒 and 𝐻𝑖 are single particle
Hamiltonians for electrons and ions (protons) respectively:
• 𝐻𝑒(𝑖) = ±𝑒 𝑐𝐴0 − 𝑐𝛽 𝑖 𝑝𝑖 ± 𝑒𝐴𝑖 + 𝛼𝑔 𝑐 𝛾 𝑖𝑗 𝑝𝑖 ± 𝑒𝐴𝑖 𝑝𝑗 ± 𝑒𝐴𝑗 + 𝑚𝑒(𝑖) 2 𝑐 2
Electromagnetic 4-potential: 𝐴 = 𝐴0 , 𝐴1 , 𝐴2 , 𝐴3
𝑔00 𝑔𝑜𝑗
−𝛼𝑔 2 + 𝛽 𝑘 𝛽𝑘 𝛽𝑗
Metric tensor: 𝑔
𝑔𝑖𝑗 =
𝑖0
𝛽𝑖
𝛾𝑖𝑗
Neglecting gravity (𝛼𝑔 =1, 𝛽 = 0,0,0 , 𝛾𝑖𝑗 = 𝛿𝑖𝑗 ) and magnetic field (𝐴 = 0,0,0 ) as well as taking
the nonrelativistic limit (even if not justified it does not but essentially change the result in this case)
one can write the distribution functions as:
𝑝2
2𝑚𝑒 𝑖
𝑓𝑒(𝑖) = 𝐸𝑥𝑝 −𝛼𝑒 𝑖 − 𝛽𝑒 𝑖
∓ 𝑒Φ , where α and β are the Lagrange multipliers eventually
determining the numbers of electrons and ions and their energy. The charge density is:
𝜚𝑒 = 𝑒
𝑓𝑖 −
𝑑3 𝑝
𝑓𝑒 ℎ3
; Coulomb equation ΔΦ = 𝜌 𝜀0 .
A similar development as in Debye-Hückel theory leads to:
Electric field in rarefied plasma of electrons
and ions of different temperature
• Field equation: Δ𝛿 = 𝜅 𝑒 −𝛿 − 𝑒
𝜏+ 𝜏− 𝛿
,
𝛼𝑓 𝜆𝑐
ℕ ),
2𝜋𝜏+ 𝑅 0
𝑚𝑒 𝑐 2
,
𝛽±
𝑒(Φ−Φ )
0
• where: 𝜅 … (electrostatic energy/thermal energy) (κ =
𝜏± =
𝛿=
, 𝑅 … radius of
𝑚𝑒 𝑐 2 𝜏+
the “cage”, ℕ0 … total number of particles to within a numerical factor, 𝛼𝑓 … fine structure constant, 𝜆𝑐 …
𝑟
Compton wavelength, Δ dimensionless Laplacian with respect to coordinates 𝑟, 𝜗, 𝜑 ⟶ 𝜉 = , 𝜗, 𝜑 .
𝑅
• Domain: 𝜉𝑚𝑖𝑛 < 𝜉 < 1 … 𝜉𝑚𝑖𝑛 𝑅 is the radius at which the central heating source takes over and 𝛿
reaches a constant value 𝛿2 .
𝑑𝛿
• Boundary conditions: total charge inside “cage”=0 (lim = 0), electric potential of “cage” with respect to ∞
𝜉→1 𝑑𝜉
𝑒Φ0
is 0 (lim 𝛿 = −
= 𝛿1 ).
2
𝜉→1
𝑚𝑒 𝑐 𝜏+
• Parameters determining solution: number of particles, total energy, ratio 𝜏+ 𝜏− ,
value 𝛿1 (related to potential difference between “cage” and central engine 𝛿2 ) must be determined
a posteriori as the value at which the system reaches highest entropy for given volume energy and number of
particles
15
Equation of state for 10 particles in R=1m
sphere: 𝜏− = 𝜏+
a) Entropy (𝑆 𝑘𝑁𝑎𝑙𝑙) as function of energy
(ℰ 𝑚𝑒 𝑐 2𝑁𝑎𝑙𝑙) and the electric parameter 𝛿0 . Entropy
has local maxima for a fixed energy with respect to
variation of 𝛿0 (marked by black), but, the
maximum at 𝛿0 = 0 is higher than any other local
maximum. The preferred thermodynamic state of
equal electron-ion temperature plasma is locally
neutral plasma with equation of state described by
the Sackur-Tetrode equation
𝑉 4𝜋𝑚𝑒 ℇ𝑒
𝑆 = 𝑘𝑁𝑒 𝐿𝑜𝑔
𝑁𝑒 3ℎ2 𝑁𝑒
3/2
5
𝑉 4𝜋𝑚𝑝 ℇ𝑝
+ + 𝑘𝑁𝑝 𝐿𝑜𝑔
2
𝑁𝑝 3ℎ2 𝑁𝑝
𝑉 4𝜋 𝑚𝑝 𝑚𝑒 ℇ
𝑆 = 𝑘𝑁 𝐿𝑜𝑔
𝑁
3ℎ2
𝑁
3
2
+
5
2
3/2
+
5
2
15
Equation of state for 10 particles in R=1m
sphere: 𝜏− = 2𝜏+
States colored with colors of the bottom code all have higher
entropy than that of the electrically neutral state at the same
energy , many local entropy maxima generate states that are
more probable than the state of local charge neutrality.
Properties of non-neutral states
• Charge density as a function of radius (for
spherically symmetric boundary conditions!)
The higher temperature component yields
some entropy to allow the low temperature
component to occupy more phase space,
which makes the entropy of the complete
system to increase (A process similar to
crystallization?).
The deep
electrostatic well
allows electrons
in the central
source to have
very high kinetic
energy, even if
the pressure at
𝜉 = 1 is very low.
Boltzmann and generation of magnetic field
• The angular momentum lost by rotating pulsar is first absorbed by the
nebula, so on must expect that in stationary state the nebula must
have some constant angular momentum, which may generate
magnetic field. The Boltzmann distribution functions must contain an
additional vector Lagrange multiplier 𝛾 related to the total angular
momentum. To make further calculations manageable we again take
the nonrelativistic limit and write:
• 𝑓𝑒(𝑖) = 𝐸𝑥𝑝 −𝛼𝑒
𝑖
− 𝛽𝑒
𝑝−𝑒 𝐴 𝑝2
𝑖
2𝑚𝑒 𝑖
∓ 𝑒Φ − 𝛾 ∙ 𝑟 × 𝑝
Field equations
• 𝜌𝑒(𝑖) = 𝒆
• 𝑗𝑥 𝒆(𝒊) = 𝒆
• 𝑗𝑦
𝒆(𝒊)
=𝒆
𝑑3 𝑝
𝑓𝑒(𝑖) 3
ℎ
= ∓𝒆
𝑝𝑥 −𝒆𝐴𝑥 𝑑 3 𝑝
𝑓 3
𝑚
ℎ
𝑝𝑦 −𝒆𝐴𝑦
𝑚
𝑑3 𝑝
𝑓 3
ℎ
𝟑
2𝜋𝑚𝑒(𝑖) 𝟐
𝛽𝑒(𝑖) ℎ2
= ∓𝒆
= ±𝒆
𝑚𝑒(𝑖) 𝛾2
∓𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽
𝑥 2 +𝑦 2
𝑒(𝑖)
𝑒 −𝛼𝑒(𝑖) ±𝛽𝑒(𝑖) 𝒆Φ𝑒 𝑒
𝟑
2𝜋𝑚𝑒(𝑖) 𝟐
𝛽𝑒(𝑖) ℎ2
𝟑
2𝜋𝑚𝑒(𝑖) 𝟐
𝛽𝑒(𝑖) ℎ2
𝑚𝑒(𝑖) 𝛾2
𝒆𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽
𝑥 2 +𝑦 2
𝑒(𝑖)
𝑒 −𝛼𝑒(𝑖) ±𝛽𝒆Φ𝑒 𝑒
• 𝐴=𝑊
𝑥 2 + 𝑦 2 , 𝑧 {𝑦, −𝑥, 0} (since j is toroidal)
• Maxwell:
• ∆Φ𝑒 =
𝜌𝑒 +𝜚𝑖
𝜀0
• 𝛻 × 𝛻 × 𝐴 = 𝜇𝑜 𝑗𝑒 + 𝑗𝑖
𝑚𝑒(𝑖) 𝛾2
𝒆𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽
𝑥 2 +𝑦 2
𝑒(𝑖)
𝑒 −𝛼𝑒(𝑖) ±𝛽𝒆Φ𝑒 𝑒
• 𝑗𝑧 𝒆(𝒊) =0
𝛾
𝛾
𝑦
𝛽𝑒(𝑖)
𝑥
𝛽𝑒(𝑖)
Simplified field equations and solutions
• Boundary conditions all positive charge at center, total charge in spherical
enclosure zero, enclosure “superconductive”
• ∆Φ𝑒 =
• ∆𝑊 𝑊 = −
𝒆 2𝜋𝑚
𝜀0 𝛽ℎ2
𝜇0 𝒆 2𝜋𝑚
𝛽
𝛽ℎ2
𝟑
𝟐
𝟑
𝟐
𝑒 −𝛼+𝛽𝒆Φ𝑒 𝑒
𝑒 −𝛼+𝛽𝒆Φ𝑒 𝑒
𝑚𝛾
𝛾𝒆 𝑊+2𝒆𝛽 𝑟 2 𝑆𝑖𝑛2 𝜃
+
𝑞 3
𝛿
𝜀0
𝑟
Δ =
𝜕2
𝜕𝑟
2 +
2 𝜕
𝑟 𝜕𝑟
+
1
𝜕2
𝑟2
𝜕𝜃2
+ 𝐶𝑜𝑡 𝜃
𝜕
𝜕𝜃
𝑚𝛾
𝛾𝒆 𝑊+2𝒆𝛽 𝑟 2 𝑆𝑖𝑛2 𝜃
𝑞
𝛾 𝜏
60 0.133
0.05
0.1950.25
0.4
20.25
0.25
60
0.1
𝛾
Δ𝑊 =
𝜕2
𝜕𝑟
2 +
4 𝜕
𝑟 𝜕𝑟
+
1
𝜕2
𝑟2
𝜕𝜃2
+ 3 𝐶𝑜𝑡 𝜃
𝜕
𝜕𝜃
Magnetic field
behavior
Increasing angular momentum
makes magnetic stress stronger
at equator
𝑞 𝛾
𝜏
60 0.05 0.25
𝑞 𝛾
𝜏
60 0.1 0.25
Increasing temperature with high angular
momentum increases the pressure of the jet
𝑞 𝛾
𝜏
60 0.4 0.25
𝑞 𝛾 𝜏
60 0.4 0.5
Conclusions
• The preliminary results suggest that very rarified collision-less plasma
segregates into regions with alternating density of electrons and ions
• Such plasma can exist in cavities behind shock fronts in interstellar
space.
• The potential difference between the center of the cavity and its
surface grows with disparity between ion and electron temperature
and allows electrons at the center to have large kinetic energy.
• If angular momentum of the heating source is fed to the plasma, it
naturally induces magnetic field and forms jets (and probably
synchrotron disk as well)