Earth’s Dipole Magnetic Field

Ling-Hsiao Lyu Institute of Space Science, NCU Dipole Magnetic Field −M • Earth’s dipole magnetic field: B = E (e 2cosθ + e sinθ) r 3 r θ 2 • The dipole magnetic field line equation: r = req sin θ

• We define the L value to be L = req / RE B • The dipole magnetic field: − 0 B = 3 (er 2cosθ + eθ sinθ) −5 (r / RE ) where B0 ≈ 3 × 10 T = 0.3 Gauss • The dipole magnetic field line equation of a given L 2 value is r = LRE sin θ 2 2 2 3/2 3 • The L value of a given r (x + y + z ) / RE L = 2 = 2 2 2 satellite location ( x , y , z ) is RE sin θ (x + y ) / RE

• The ionospheric foot point ( r i ono = R E ) latitude π 1 = L sin2 θ ⇒ θ = sin−1(1 / L ) ⇒ λ = − sin−1(1 / L ) iono iono iono 2 2 Dipole Magnetic Field Line

• How to determine the dipole magnetic field line equation: −M B = E (e 2cosθ + e sinθ) r 3 r θ dr rdθ dr rdθ r sin2 θ = ⇒ = ⇒ d lnr = d ln(sin2 θ) ⇒ = Br Bθ 2cosθ sinθ req 1 2 r(θ) = req sin θ

其中，積分過程中，我們取 r eq = r ( θ = 90 ° ) ，也就是這根磁場線通過磁赤道 面(magnetic equatorial plane)時與地心的距離。當我們說一根磁場線 通過磁赤道面時的位置，距離地心三萬八千三百公里時，你聽起來不會有

任何感覺。可是如果我改說，r eq 約為6個地球半徑，你就很有感覺了。所

以太空物理學家，常將 r e q 改寫為 r e q = L R E 。並用 L-shell 來描述所有 req = LRE 的磁偶極場線所組成的一圈環狀結構。 3 Earth’s dipole magnetic field 發現者 William Gilbert (1544‐1603) the royal physician to Queen Elizabeth I

http://www.plasma-universe.com/Terrella Earth’s dipole magnetic field 強度的測量者 Gaussian distribution Normal distribution

Carl Friedrich Gauss (1777–1855), painted by Christian Albrecht Jensen

B0 ≈ 0.3 Gauss In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism (including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of Kirchhoff's circuit laws in electricity. 當電荷不累積時，才成立。 NRL Plasma Formulary http://wwwppd.nrl.navy.mil/nrlformulary/ The Unit of the Magnetic Field

• SI Units B = ∇ × A ⇒ O(B) = O(A / L) O(eA) = O(mv) ⇒ O(B) = O(M / QT ) • Gaussian Units

B = ∇ × A ⇒ O(B) = O(A / L) eA O( ) = O(mv) & O(e) = O( ML3 / T ) c M (L / T )2 M ⇒ O(B) = O(A / L) = O[ ] = O( ) L ML3 / T LT Gyro Motion

• Equations of Motion of a charge particle in a

uniform magnetic field B = ez B dx = v dt dv m = q v × B dt • Home Work 1.1: For a given set of initial conditions:

At t = 0 , x = y = z = 0 & v x = v 0 & vy = vz = 0 Solve the above equations of motion, find the

solutions of x(t), y(t),vx (t),vy (t) 8 Gyro Frequency & Gyro (Larmor) Radius eB • Electron gyro frequency Ωce = me eB Joseph Larmor • Ion gyro frequency Ωci = (1857-1942) mi

• Gyro radius of an electron v⊥ v⊥ me rLe = = with perpendicular velocity v⊥ Ωce eB

• Gyro radius of an ion v⊥ v⊥ mi rLi = = with perpendicular velocity v⊥ Ωci eB

• Gyro radius of a thermal vthe/thi kBTe/i me/i re/i = = electron/ion Ωce/ci eB 9 Joseph Larmor (1857‐1942) 誤判 • Larmor opposed Albert Einstein's theory of relativity • Larmor rejected both the curvature of space and the special theory of relativity, to the extent that he claimed that an absolute time was essential to astronomy (Larmor 1924, 1927). • Larmor held that matter consisted of particles moving in the aether. 貢獻 • Larmor believed the source of electric charge was a "particle" (which as early as 1894 he was referring to as the electron). • Larmor held that the flow of charged particles constitutes the current of conduction (but was not part of the atom). • Larmor calculated the rate of energy radiation from an accelerating electron. • Larmor explained the splitting of the spectral lines in a magnetic field by the oscillation of electrons. • In 1919, Larmor proposed sunspots are self-regenerative dynamo action on the Sun's surface. Three Cyclic Motions -- Action principle & three adiabatic motions

11 Action Principle • Action variable ( J =  ∫ p dq ) is adiabatic invariant under slow change of parameters (Goldstein, 1980). • Action of a quasi-periodic motion is conserved if parameters, which affect the periodic motion, are nearly steady and nearly uniform. • Three periodic motions may be found when a charge particle moving in a dipole magnetic field: 先教 (1) gyro motion around the magnetic field 先教 (2) bounce motion in a magnetic mirror machine (dipole field) 以後教 (3) periodic drift motion around a magnetic mirror machine 12 The First Adiabatic Invariant

• For gyro motion around the magnetic field

v⊥ p = mv⊥ dq = rgyrodθ = dθ Ωc v m J = pdq = mv ⊥ dθ = 2πmv2 1 ∫ ∫ ⊥ Ω ⊥ eB c • If within one gyro period and two gyro radii, the changes on the magnetic field and the particle speed are negligible, the is conserved. J1

13 Magnetic Moment

• By definition magnetic momentum of a single charged particle is

dQ 2 e v⊥ 2 µ = IA = πrgyro = π( ) dt 2π / Ωc Ωc

where Ωc = eB / m

it yields 1 2 mv⊥ 2 W⊥ 2 m 4πm µ = = and J = 2πmv = µ B B 1 ⊥ eB e

If the J 1 is conserved, the µ is conserved.

14 the first-order approximation of grad-B drift • A quick way, but incorrect in physics, to get the first-order approximation of grad-B drift: Consider a force due to “gradient of perpendicular kinetic energy”

W mv2 F = −∇W = −∇(µB) = −µ∇B = − ⊥ ∇B = − ⊥ ∇B ⊥ B 2B (−∇W ) × B mv2 (−∇B) × B mv2 (−∇ B) × B v = ⊥ = ⊥ = ⊥ ⊥ drift qB2 2B qB2 2B qB2

15 The Second Adiabatic Invariant

• For bounce motion in a magnetic mirror machine

p = mv dq = ds J = mv ds  2 ∫   • If within one bounce period the changes on the magnetic field configuration that can affect the

bounce motion are negligible, the J 2 is conserved. • What is bounce motion? 1 1 2 1 2 2 W mv mv constant µ = mv⊥ / B = constant = ⊥ +  = 2 2 2 v ⊥ increases with increasing B

v decreases with increasing v ⊥

Bounce motion occurs at strong B, such that v = 0 16 Bounce Motion & Loss Cone Distribution • Show that the magnetic field increases with decreasing the cross section of the magnetic flux tube • Show that the mirror point of a given magnetic field

line depends on the pitch angle α e q of the particle at the equator, where the pitch angle is the angle between the particle’s velocity and the B-field line. 2 1 2 1 2 1 2 veq 1 W = mv⊥ eq + mveq = mv⊥ eq (1+ 2 ) = µBeq ( 2 ) 2 2 2 v⊥ eq sin α eq

1 2 2 W = mv⊥ mirror = µBmirror ⇒ Beq / Bmirror = sin α eq ≤ 1 2 Particles will be lost in the ionosphere if Bmirror > Bionosphere How to determine Beq & Bionosphere along a given magnetic field line? 17 How to Determine Beq & Bionosphere & Loss-cone Angle of a Given Magnetic Field Line? −B B = 0 (e 2cosθ + e sinθ) • The dipole magnetic field: 3 r θ (r / RE ) 3 • Magnetic field at the equator of a given L: Beq = B0 / L −1 • The ionospheric foot point co-latitude θiono = sin (1 / L )

−B0 L − 1 1 Biono = 3 (er 2cosθiono + eθ sinθiono ) = −B0 (er 2 + eθ ) (riono / RE ) L L

3 L = 1 Biono = B0 ⇒ Biono = B0 4 − check results: L L = 3 Biono = 3B0 < 2B0

2 Beq Beq 1 L sin α eq = ≥ = 3 Bmirror Biono L 4L − 3 1 1 ⇒ sinα = eq−loss− cone 4 L L(4L − 3) 18 Discussion

• 其實上一頁有些計算並不重要。在此舉這些例子，只是想要加 深各位同學對 Earth’s dipole magnetic field, magnetic field line, gyro motion, bounce motion, 等等的印象。 • 未來要介紹被太陽風吹得變形的外磁層。不過為了說明整個變 形的過程，我們需要先介紹一些簡單的Charge particle drift motion and the diamagnetic current in the plasma. 希望各位 的電漿物理導論，先學了一些，我們就可以教得快一點。 • 地球 dipole magnetic field 會隨時間改變 ，甚至發生 geomagnetic reversal • Dynamo Theory：解釋產生磁場以及磁場反轉的過程 • 大西洋中洋脊，記錄了地球磁場反轉的情形，也因此造成南大 西洋地表與電離層磁場過弱現象 (South Atlantic Anomaly)。因 此常在劇烈磁暴後，出現 trapped energetic particles. • 紅字部份，可上google 查詢更多資訊，包括圖片與影片。 北極附近的磁南極（S）過去200年的動向

位在西伯利亞（東半球）的可能年代約為 西元0、800、 1600、 2300年（前後200年）（我猜的！）

位在加拿大（西半球）的可能年代約為 西元400、1200、 1900年（前後200年）（我猜的！） 北極附近的磁南極（S）過去200年的 南極附近的磁北極（N）過去200年 動向：由低緯移動到高緯 的動向：由高緯移動到低緯 北極附近的磁南極（S）未來50年的動向：快速跨越地理北極

地理北極與磁南極的夾角為 19.1º (100年前） 11.9º (25年前） < 7.3º （現在）

North Magnetic Pole (2001) 81.3°N 110.8°W (2004 est) 82.3°N 113.4°W (2005 est) 82.7°N 114.4°W South Magnetic Pole (1998) 64.6°S 138.5°E. (2004 est) 63.5°S 138.0°E (2008 est) 65°S 138°E 地球磁場反轉 （大西洋海底的 證據） 地球磁場反轉 （大西洋海底的 證據）