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Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin

:&2-;-)(*!"<-$2-"(=*%>*/-?"=)(*/%/="#* o Gyrating particle constitutes an electric current loop with a dipole moment: 1/2mv2 µ = " B o The dipole moment is conserved, i.e., is invariant. Called the first adiabatic invariant. ! o µ = constant even if B varies spatially or temporally. If B varies, then vperp varies to keep µ = constant => v|| also changes. o Gives rise to magnetic mirroring. Seen in planetary magnetospheres, magnetic bottles, coronal loops, etc.

Bz o Right is geometry of mirror Br from Chen, Page 30. @-?"=)(*/2$$%$2"?*

o Consider B-field pointed primarily in z-direction and whose magnitude varies in z- direction. If field is axisymmetric, B! = 0 and d/d! = 0.

o This has cylindrical symmetry, so write B = Brrˆ + Bzzˆ

o How does this configuration give rise to a force that can trap a charged particle? !

o Can obtain Br from " #B = 0 . In cylindrical polar coordinates:

1 " "Bz (rBr )+ = 0 r "r "z ! " "Bz => (rBr ) = #r "r "z

o If " B z / " z is given at r = 0 and does not vary much with r, then

r $Bz 1 2&$ Bz ) ! rBr = " #0 r dr % " r ( + $z 2 ' $z *r =0 ! 1 &$ Bz ) Br = " r( + (5.1) 2 ' $z *r =0

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o Now have Br in terms of BZ, which we can use to find Lorentz force on particle.

o The components of Lorentz force are: Fr = q(v" Bz # vzB" ) (1)

F" = q(#vrBz + vzBr ) (2) (3)

Fz = q(vrB" # v" Br ) (4)

o As B! = 0, two terms vanish and terms (1) and (2) give rise to Larmor gyration. Term (3) vanishes on the axis and! causes a drift in radial direction. Term (4) is therefore the one of interest.

o Substituting from Eqn. 5.1: Fz = "qv# Br qv r $B = # z 2 $z

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Averaging over one gyro-orbit, and putting and r = r o v " = !v# L

qv"rL #Bz Fz = ! 2 #z o This is called the mirror force, where! -/+ arises because particles of opposite charge orbit the field in opposite directions.

! 2 1 v" $Bz o Above is normally written: Fz = ! q 2 #c $z 1 mv2 $B = % " z 2 B $z

2 #Bz 1/2mv# or Fz = "µ where µ " is the magnetic moment. #z ! B dB o In 3D, this can be generalised to: F = "µ = "µ# B || ds || ! ! where F|| is the mirror force parallel to B and ds is a line element along B.

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A2$.#*-&2-;-)(*2"<-$2-"#* o As particle moves into regions of stronger or weaker B, Larmor radius changes, but µ remains invariant. o To prove this, consider component of equation of motion along B:

dv dB m || = "µ dt ds o Multiplying by v||: dv dB mv || = "µv || dt || ds o Then, d " 1 % ds dB ! $ mv2 ' = (µ dt # 2 || & dt ds ! dB = (µ dt (5.2) o The particle’s energy must be conserved, so d # 1 2 1 2 & % mv + mv ( = 0 dt $ 2 || 2 " ' 1/2mv2 o Using µ " # ! B d " 1 % $ mv2 +µB' = 0 dt # 2 || & !

! ! A2$.#*-&2-;-)(*2"<-$2-"#* o Using Eqn 5.2, dB d "µ + (µB) = 0 dt dt dB dB dµ "µ + µ + B = 0 dt dt dt dµ => B = 0 dt ! o As B is not equal to 0, this implies that dµ = 0 dt o That is, µ = constant in time! (invariant). o µ is known as the first adiabatic! invariant of the particle orbit. o As a particle moves from a weak-field to a strong-field region, it sees B increasing and therefore vperp must increase in order to keep µ constant. Since total energy must remain constant, v|| must decrease.

o If B is high enough, v|| eventually -> 0 and particle is reflected back to weak-field.

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o Consider B0 and B1 in the weak- and strong-field regions. Associated speeds are v0 and v1.

B0 B1

mv2 mv2 o The conservation of µ implies that µ = 0 = 1 2B0 2B1 o So, as B increases, the perpendicular component of the particle velocity increases: particle moves more and more perpendicular to B. ! o However, since we have E=0, the total particle energy cannot increase. Thus as v⊥ increases, v|| must decrease. The particle slows down in its motion along the field.

o If field convergence is strong enough, at some point the particle may have v|| = 0. B%".=C'="(=*%>*2"<-$2-"(=*%>*1D*@-?"=)(*/2$$%$2"?*

o Say that at B1 we have v1,|| = 0. From conservation of energy: 2 2 2 v1 = v1,"= v0 mv2 mv2 o Using µ = 0 = 1 we can write 2B0 2B1 v 2 !2 B v v v⊥ 0 = 0," = 0," ! 2 2 ! B1 v1," v0 B v|| o But sin(!) = v⊥/ v0 where ! is the pitch angle.

B o !Therefore 0 = sin2(") B1 o Particles with smaller ! will mirror in regions of higher B. If ! is too small, B >> B and particle does not mirror. 1 ! 0

B%".=C'="(=*%>*2"<-$2-"(=*%>*1D*@-?"=)(*/2$$%$2"?* o Mirror ratio is defined Bm Rm = B0

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o So, the smallest ! of a confined particle is

B 1 sin2(") = 0 = Bm Rm o This defines region in velocity space in the shape of a cone, called the loss cone.

! E0=*,%..*(%"=* o Particles are confined in a mirror with ratio Rm if they have ! > !m. o Otherwise they escape the mirror. That portion of velocity space occupied by escaping particles is called the loss-cone. o The opening angle of the loss cone is not dependent on mass or charge. Electrons and protons are lost equally, if the plasma is collisionless. o Coulomb collisions scatter charged particles, changing their pitch angle. Thus a particle which originally lay outside the loss cone can be scattered into it. o Electrons are more readily Coulomb-scattered than ions; thus electrons will be scattered out of the mirror trap more quickly.

F%'"(=*/%)%"*;=#G=="*#G%*/2$$%$.* o Early magnetic confinement machines involved designs using magnetic mirror. The original idea was based on the fact that an electric current generates a magnetic field, and that the currents flowing in the plasma will "pinch" the plasma, containing it within its own magnetic field. F%'"(=*/%)%"*;=#G=="*#G%*/2$$%$.* o A particle outside the loss-cone in a field structure which is convergent at both ends (such as the Earth’s magnetic field) will be reflected by both mirrors and bounce between them.

o Superposed on this bounce will be drift motions. For example, particle orbits in the Earth’s magnetic field are a combination of gyromotion, bounce motion, and grad-B drift.

H%..*(%"=*&2.#$2;')%".*-"&*2".#-;2,2)=.* o Consider an initially Maxwellian distribution of particles inside a magnetic o mirror. In velocity space, the distribution is spherical. o As particles exit via the loss cone, the distribution becomes anisotropic

o The distribution will try to relax back to an isotropic one (higher entropy); one of the ways it does this is by radiating energy in the form of plasma waves. These waves are somewhat more complex than cold plasma wave - they involve the distribution functions of particles. They are generally called kinetic plasma waves, driven by a kinetic instability.