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!"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789* !"#$%&"'()'*+,-".#'*/&&0&/-,' Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin :&2-;-)(*!"<-$2-"(=*%>*/-?"=)(*/%/="#* o Gyrating particle constitutes an electric current loop with a dipole moment: 1/2mv"2 B o The dipole moment is conserved, i.e., is invariant. Called the first adiabatic invariant. µ= ! 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$$%$2"?* o Consider B-field pointed primarily in z-direction and whose magnitude varies in zdirection. If field is axisymmetric, B! = 0 and d/d! = 0. o This has cylindrical symmetry, so write B = Br rˆ + Bz zˆ o How does this configuration give rise to a force that can trap a charged particle? ! o Can obtain Br from " # B = 0. In cylindrical polar coordinates: "B 1" (rBr ) + z = 0 r "r "z ! "B " => (rBr ) = #r z "r "z o If "Bz /"z is given at r = 0 and does not vary much with r, then ! ! $Bz 1 &$B ) dr % " r 2 ( z + $z 2 ' $z *r=0 1 &$B ) Br = " r( z + 2 ' $z *r=0 rBr = " #0r r (5.1) ! @-?"=)(*/2$$%$2"?* 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 # vz B" ) (1) F" = q(#vr Bz + vz Br ) (2) (3) Fz = q(vr B" # 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 $Bz 2 $z @-?"=)(*/2$$%$2"?* o Averaging over one gyro-orbit, and putting v" = !v# and r = rL qv"rL #Bz 2 #z o This is called the mirror force, where ! -/+ arises because particles of opposite charge orbit the field in opposite directions. Fz = ! ! o Above is normally written: 1 v2 $Bz Fz = ! q " 2 # c $z =% or Fz = "µ #Bz where #z ! µ" 1/2mv#2 B o In 3D, this can be generalised to: ! 1 mv"2 $Bz 2 B $z F|| = "µ is the magnetic moment. dB = "µ# ||B ds ! where F|| is the mirror force parallel to B and ds is a line element along B. ! A2$.#*-&2-;-)(*2"<-$2-"#* o As particle moves into regions of stronger or weaker B, Larmor radius changes, but µ remains invariant. o To prove this, consider component of equation of motion along B: dv|| dB = "µ dt ds Multiplying by v||: dv|| dB mv|| = "µv|| dt ds Then, d "1 2 % ds dB ! $ mv|| ' = (µ & dt # 2 dt ds ! dB = (µ (5.2) dt d #1 2 1 2 & The particle’s energy must be conserved, so % mv|| + mv" ( = 0 ' dt $ 2 2 1/2mv#2 ! Using µ " % d "1 2 B $ mv|| + µB ' = 0 & dt # 2 ! m o o o o ! ! A2$.#*-&2-;-)(*2"<-$2-"#* o Using Eqn 5.2, dB d + (µB) = 0 dt dt dB dB dµ "µ +µ +B =0 dt dt dt "µ => B ! to 0, this implies that o As B is not equal dµ =0 dt dµ =0 dt ! (invariant). o That is, µ = constant in time o ! invariant of the particle orbit. µ is known as the first adiabatic o As a particle moves from a weak-field to a strong-field region, it sees B increasing and therefore vperp must increase in order to keep µ constant. Since total energy must remain constant, v|| must decrease. o If B is high enough, v|| eventually -> 0 and particle is reflected back to weak-field. B%".=C'="(=*%>*2"<-$2-"(=*%>*1D*@-?"=)(*/2$$%$2"?* o Consider B0 and B1 in the weak- and strong-field regions. Associated speeds are v0 and v1. B0 o The conservation of µ implies that B1 µ= mv02 mv12 = 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$$%$2"?* o Say that at B1 we have v1,|| = 0. From conservation of energy: o Using µ = ! mv02 mv12 = we can write 2B0 2B1 B0 = B1 v0,2" = v1,2" v12 = v1,2"= v02 v ! v0,2" v02 v⊥ ! o But sin(!) = v⊥/ v0 where ! is the pitch angle. ! o Therefore v|| B B0 = sin2 (" ) B1 o Particles with smaller ! will mirror in regions of higher B. If ! is too small, B1>> B0 and particle does not mirror. ! B%".=C'="(=*%>*2"<-$2-"(=*%>*1D*@-?"=)(*/2$$%$2"?* o Mirror ratio is defined Rm = Bm B0 ! o So, the smallest ! of a confined particle is sin2 (" ) = B0 1 = 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.