Visual walkthrough — Space environment — LEO radiation (SAA, Van Allen), atomic oxygen, MMOD debris

3.6.34 · D2 · Physics › Spacecraft Structures & Systems Engineering › Space environment — LEO radiation (SAA, Van Allen), atomic o

Yeh visual companion hai the parent topic ka.

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Step 1 — Ek moving charge, aur woh ek arrow jo ise push karta hai

KYA HAI. Ek single tiny charged speck imagine karo — ek proton — jo space mein fly kar raha hai. Pehle hum ek coordinate frame fix karte hain taaki left/right, up/down aur in/out of the page sab ka precise meaning ho: $\hat{x}$ right point kare, $\hat{y}$ up point kare, aur $\hat{z}$ page se bahar tumhari taraf point kare (yeh standard right-handed frame hai). Ab velocity ko ek arrow $\vec{v}=v\hat{x}$ (right pointing) draw karo aur magnetic field $\vec{B}=B\hat{y}$ (up pointing) switch on karo.

WHY cross product aur ordinary multiplication kyun nahi. Ek magnetic field simply ek charge ko forward ya backward push nahi karta. Experiment se pata chalta hai ki force sideways hoti hai — motion aur field dono ke perpendicular. Ek hi mathematical tool hai jo do arrows khaata hai aur ek teesra arrow dono ke right angle mein ugalta hai — woh hai cross product $\times$. Isliye hi Lorentz law iska use karta hai:

$\vec{F}=q(\vec{v}\times \vec{B})$

• $q$ — charge (ek number; proton ke liye positive). Yeh poori force ko scale karta hai.
• $\vec{v}$ — velocity arrow: particle kitna tez aur kis direction mein move kar raha hai.
• $\vec{B}$ — magnetic field arrow: field kitni strong hai aur kis direction mein point kar rahi hai.
• $\vec{v}\times \vec{B}$ — ek naya arrow dono ke perpendicular, jiska length $vB\sin \theta$ hai ($\theta$ = $\vec{v}$ aur $\vec{B}$ ke beech ka angle). "Sideways" ka precisely yahi matlab hai.

Right-hand rule se direction nikalna. Apne right hand ki fingers $\vec{v}$ ke along (right, $+\hat{x}$) point karo aur $\vec{B}$ (up, $+\hat{y}$) ki taraf curl karo; tumhara thumb page se bahar ($+\hat{z}$) point karega. Symbols mein $\hat{x}\times \hat{y}=+\hat{z}$, toh $\vec{v}\times \vec{B}=vB\hat{z}$. Hamare positive proton ($q>0$) ke liye force isliye page se bahar, tumhari taraf point karti hai.

PICTURE. Figure mein teal arrow ($\vec{v}$) right point karta hai, plum arrow ($\vec{B}$) up point karta hai, aur orange symbol — ek circle ke andar dot — force $\vec{F}$ ko mark karta hai jo seedha page se bahar tumhari taraf pop kar raha hai, un do arrows ke span kiye hue plane ke perpendicular.

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Step 2 — Velocity ko decompose karna taaki "woh part jo bend hota hai" ke baare mein baat kar sakein

KYA HAI. Koi bhi clean force balance likhne se pehle, hume velocity arrow ke do pieces ko naam dena hoga — kyunki Step 1 ka cross product already hint kar chuka hai ki field motion ka sirf ek hissa hi pakadta hai. Velocity arrow $\vec{v}$ ko do perpendicular pieces mein toddo, field line ke against measure karte hue:

• $v_{\perp }$ — $\vec{v}$ ka component jo $\vec{B}$ ke across (perpendicular to) hai.
• $v_{\parallel }$ — $\vec{v}$ ka component jo $\vec{B}$ ke along (parallel to) hai.

In do right-angle pieces ke Pythagoras se, $v^2=v_{\perp }^2+v_{\parallel }^2$.

Abhi kyun split karein — cross product iska demand kyun karta hai. $\vec{v}\times \vec{B}$ ki length $vB\sin \theta$ hai, aur $v\sin \theta$ exactly $v_{\perp }$ hai, yaani across-the-field part. Along-the-field part $\sin 0°=0$ contribute karta hai: ise koi force nahi lagti aur yeh simply coast karta hai. Toh magnetic force sirf $v_{\perp }$ par act karti hai. Koi bhi honest equation likhne ke liye, $v_{\perp }$ pehle ek defined symbol hona chahiye — isliye hum circle se pehle, yahan decompose karte hain.

PICTURE. Teal arrow $\vec{v}$ ek orange leg $v_{\perp }$ (field line ke across) aur ek teal leg $v_{\parallel }$ (field line ke along) mein split hota hai; inke beech ka aur field line ke saath right angle mark hai, taaki tum $v^2=v_{\perp }^2+v_{\parallel }^2$ ko ek right triangle ke roop mein dekh sako.

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Step 3 — Ek force jo hamesha sideways hoti hai woh ek circle banati hai

KYA HAI. Sirf $v_{\perp }$ par focus karo (Step 2 mein define kiya tha). Iska force hamesha iske saath 90° par hai. Toh yeh force ise kabhi speed up ya slow down nahi karti — yeh sirf direction bend karti hai. Direction ko same amount bend karo, baar baar, aur across-motion ek circle mein close ho jaata hai.

Circle kyun hota hai aur koi runaway curve kyun nahi. Ek ball imagine karo jo overhead string par ghoom rahi ho. String inward (tumhare haath ki taraf) pull karti hai ball ki motion ke 90° par; ball circle mein jaati hai. Magnetic force exactly us string ka role play karti hai. Isliye hum centripetal force ka idea borrow kar sakte hain — circular motion hold karne ke liye zaroori inward pull:

$F_{\text{centripetal}}=\frac{m{v}_{\perp }^2}{r}$

• $m$ — particle ki mass (heavier → bend karna mushkil).
• $v_{\perp }$ — across-the-field speed jo Step 2 mein define hui (woh hissa jo actually bend hota hai).
• $r$ — us circle ka radius jisme yeh settle hota hai.
• Fraction kehta hai: tez ya bhaari particle → bada circle; tight turn ke liye zyada force chahiye.

PICTURE. Orange force arrow dekho jo hamesha center ki taraf aim kar raha hai jabki teal velocity arrow rim par ride karta hai — dono poore circle mein 90° par locked rehte hain.

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Step 4 — Dono forces ko equal set karna gyroradius $r_L$ earn karta hai

KYA HAI. Ek hi inward force ke do descriptions agree karne chahiye. Magnetic force centripetal force supply karti hai. Unhe equal set karo. Kyunki force magnitude kabhi negative nahi hoti, hum charge ki magnitude use karte hain, $|q|$ likhte hain ($q$ ka size, plus/minus sign ignore karke).

Equal kyun set karein. "Supplies" key word hai: magnetic push wahi cheez hai jo path curve kar rahi hai, toh iska size us size ke equal hona chahiye jo circle demand karta hai. Equal cause, equal effect.

$\underset{\text{magnetic push}}{\underbrace{|q|v_{\perp }B}}=\underset{\text{circle demands}}{\underbrace{\frac{m{v}_{\perp }^2}{r}}}$

Ab $r$ ke liye solve karo. Dono sides se ek $v_{\perp }$ cancel karo:

$|q|B=\frac{m{v}_{\perp }}{r}⟹r_L=\frac{m{v}_{\perp }}{|q|B}$

• $r_L$ — Larmor radius (gyroradius): woh chhota circle kitna bada hai jo particle ek field line ke around trace karta hai.
• $|q|$ — charge ki magnitude. $|q|$ use karne se guarantee hoti hai ki $r_L$ positive aaye, jaisa ek radius hona chahiye; $q$ ka sign sirf decide karta hai ki particle kis direction mein circle karta hai, size kitna bada hai yeh nahi.
• Bada $m$ ya $v_{\perp }$ → bada circle (turn karna mushkil).
• Bada $|q|$ ya $B$ → chhota, tighter circle (stronger grip).

Yeh exactly woh formula hai jo parent note ne state kiya tha ($|q|$ explicit ke saath) — ab yeh humara hai.

PICTURE. Plum circle apne radius $r_L$ ke saath labelled hai; orange arrow magnetic push $|q|v_{\perp }B$ ko andar point karte hue dikhata hai, teal arrow circle ki demand $m{v}_{\perp }^2/r$ dikhata hai, aur neeche ki equation inhe equal padhti hai.

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Step 5 — $v_{\parallel }$ wapas add karo: spiral janam leta hai

KYA HAI. Steps 3–4 ne sirf $v_{\perp }$ use kiya. Ab coasting piece $v_{\parallel }$ (Step 2 se) wapas daalo. Ise koi force nahi lagti, toh yeh simply steadily field line ke along glide karta hai jabki across-part circling karta rehta hai.

Yeh helix kyun deta hai. Circle ($v_{\perp }$ se) + line ke along steady glide ($v_{\parallel }$ se) = ek helix. Pitch angle $\alpha$ use karte hue ($\vec{v}$ ka field line se tilt), do pieces hain:

$v_{\perp }=v\sin \alpha v_{\parallel }=v\cos \alpha$

• $\alpha$ — pitch angle: velocity arrow aur field line ke beech ka angle.
• $v_{\perp }=v\sin \alpha$ — circling part. $\alpha =90°$ par yeh poori speed hai.
• $v_{\parallel }=v\cos \alpha$ — glide-along part. $\alpha =0°$ par yeh poori speed hai.

Woh corkscrew belt ki teeno motions mein se pehli hai.

PICTURE. Plum spiral dashed field line ke around wind karta hai; orange arrow $v_{\perp }=v\sin \alpha$ circling drive karta hai aur teal arrow $v_{\parallel }=v\cos \alpha$ $\vec{B}$ ke along steady climb drive karta hai.

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Step 6 — Particle poles ke beech bounce kyun karta hai (magnetic mirror)

KYA HAI. Earth ka field uniform nahi hai — yeh ek dipole hai, toh field lines magnetic poles ke paas crowd (zyada strong hoti hain) aur equator ke paas spread out (zyada weak hoti hain). Ek particle jo pole ki taraf spiral karta hai woh ek stronger $B$ mein jaata hai.

WHY $\mu$ constant rehta hai — visual argument. Gyrating particle ka magnetic moment define karo:

$\mu =\frac{m{v}_{\perp }^2}{2B}$

• $m{v}_{\perp }^2/2$ — kinetic energy jo circling motion mein tied up hai.
• $B$ — local field strength jahan particle currently hai.
• $\mu$ — yeh ratio; physically yeh us tiny current loop ka magnetic moment hai jo gyration banata hai.

Yeh kyun $\mu$ barely changes. Ek gyrating charge ek tiny current loop hai area $A=\pi r_L^2$ ke saath. Us loop se guzrne wala magnetic flux $\Phi =BA=B\pi r_L^2$ hai. $r_L=m{v}_{\perp }/(|q|B)$ substitute karo:

$\Phi =B\pi {\left(\frac{m{v}_{\perp }}{|q|B}\right)}^2=\frac{\pi m^2{v}_{\perp }^2}{|q{|}^{2}B}=\frac{2\pi m}{|q{|}^2}\mu .$

Toh $\Phi$ aur $\mu$ same quantity hain fixed constant $2\pi m/|q{|}^2$ tak. Slowly-varying fields ka ek basic result (Faraday's law ek loop par apply kiya jise field ek orbit se tez change nahi kar sakta) yeh hai ki gyro-loop se guzrne wala flux conserved hota hai — loop simply shrink karta hai taaki same number of field lines usme se guzrein. Kyunki $\Phi$ fixed hai, $\mu$ fixed hai. Yeh sirf tab hold karta hai jab field slowly change kare: space mein ek loop-width $r_L$ pe barely, time mein ek gyration mein barely — jo Earth ka dipole comfortably satisfy karta hai. Iska matlab yahi hai "adiabatic invariant."

Quantitative reflection. Ab do conservation laws saath act karte hain:

$\mu =\frac{m{v}_{\perp }^2}{2B}=\text{const},v^2=v_{\perp }^2+v_{\parallel }^2=\text{const}$

Doosra isliye hold karta hai kyunki magnetic force koi work nahi karta (Step 3), toh total speed $v$ kabhi nahi badlati. Pehle se, $v_{\perp }^2=\frac{2\mu }{m}B$ — toh jaise particle stronger $B$ mein climb karta hai, $v_{\perp }^2$ exactly $B$ ke proportion mein grow karta hai. Doosre mein substitute karo:

$v_{\parallel }^2=v^2-\frac{2\mu }{m}B.$

• $v_{\parallel }$ shrink hota hai jaise $B$ rise karta hai, aur exactly wahan zero hit karta hai jahan $\frac{2\mu }{m}B=v^2$.

Equator se anchor karo, jahan particle ka pitch angle ${\alpha }_{\text{eq}}$ aur field $B_{\text{eq}}$ hai. Wahan $v_{\perp }^2=v^2{\sin }^2{\alpha }_{\text{eq}}$, toh $\frac{2\mu }{m}=\frac{v^2{\sin }^2{\alpha }_{\text{eq}}}{B_{\text{eq}}}$. Turning point ($v_{\parallel }=0$) ko $\frac{2\mu }{m}{B}_{\text{mirror}}=v^2$ chahiye, jo clean mirror condition deta hai:

$\boxed{\frac{B_{\text{mirror}}}{B_{\text{eq}}}=\frac{1}{{\sin }^2{\alpha }_{\text{eq}}}}$

• $B_{\text{mirror}}$ — reflection point par field strength.
• ${\alpha }_{\text{eq}}$ — equator par wapas pitch angle.
• Read karo: chhota equatorial pitch angle wale particle ko turn around karne ke liye bada field ratio chahiye — ise pole ki taraf deep climb karna padta hai.

Particle pole-to-pole bounce karta hai jaise do mirrors ke beech ek bead — magnetic mirror. Yeh doosri motion hai.

PICTURE. Converging plum field lines ek funnel banati hain; spiral tighten hota hai (uska $r_L$ shrink hota hai, flux $\Phi$ constant rakhte hue — shrinking loop scale pe drawn hai) jaise yeh climb karta hai, $v_{\parallel }$ marked point $B_{\text{mirror}}=B_{\text{eq}}/{\sin }^2{\alpha }_{\text{eq}}$ par zero tak dwindle karta hai, aur path turn around karta hai.

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Step 7 — Edge cases: extreme pitch angles par kya hota hai

KYA HAI. Hume har particle cover karna hai, sirf tidy ones nahi. Pitch angle $\alpha$ sabka fate decide karta hai.

Yeh cases kyun matter karte hain. Yeh "trapped" ki boundaries hain. Inhe miss karo aur tum galti se claim karte ki sab particles safely bottled hain.

• $\alpha =90°$ (pure perpendicular). Poori speed $v_{\perp }$ hai; $v_{\parallel }=0$. Mirror condition se, $B_{\text{mirror}}/B_{\text{eq}}=1/{\sin }^{2}90°=1$ — yeh right at the equator mirror karta hai aur line ke along barely drift karta hai: most deeply trapped case.
• $\alpha =0°$ (pure parallel). Poori speed $v_{\parallel }$ hai; $v_{\perp }=0$, toh $r_L=0$ — bilkul koi circle nahi, aur $\mu =0$. Mirror condition infinite field ratio demand karti hai, jo kabhi nahi hota; particle seedha field line ke neeche stream karta hai, pole ke paas atmosphere mein dive karta hai, aur lost ho jaata hai. Yeh loss cone ka core hai: koi bhi particle jiska pitch angle bahut chhota ho woh atmosphere ke neeche mirror karta hai aur escape ho jaata hai.
• $B\to 0$ (field vanishes, e.g. dipole se dur). Tab $r_L=\frac{m{v}_{\perp }}{|q|B}\to \infty$: "circle" ek straight line ban jaata hai. Koi field nahi, koi trapping nahi. Yeh ek reason hai ki belts khatam hoti hain — jahan field bahut weak hai, particles free fly karte hain.
• $q=0$ (ek neutral particle). Force $=0$. Yeh field ko bilkul ignore karta hai aur straight line mein travel karta hai. Sirf charged particles trap hote hain — isliye belts protons aur electrons se bani hain, neutrons ya dust se nahi.

PICTURE. Char mini-panels: mota equatorial circle ($90°$), loss cone mein straight dive ($0°$), $B\to 0$ par straightening path, aur ek neutral particle ka indifferent straight shot.

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Step 8 — Teesri motion: gradient aur curvature drift ek full ring banate hain

KYA HAI. Ab chhote gyro-circle ke center ko watch karo, particle ko nahi. Dipole field ki do features har loop mein us center ko thoda sideways push karti hain, aur woh tiny steps add up hote hain jab tak center poori Earth ke around nahi ghoom jaata.

Center step kyun karta hai — gradient drift, concretely. Ek gyro-circle follow karo. Kyunki $\vec{B}$ us loop ke Earth-facing side par stronger hai aur far side par weaker, local radius $r_L=\frac{m{v}_{\perp }}{|q|B}$ strong-field arc par chhota aur weak-field arc par bada hota hai. Particle isliye strong side par ek tight half-loop aur weak side par ek wide half-loop sweep karta hai. Ek tight arc aur ek wide arc se stitched circle close nahi ho sakta — ek full gyration ke baad particle wahan se thoda side mein land karta hai jahan se shuru hua tha. Woh leftover displacement drift step hai. Geometry carry karne se gradient-drift velocity milti hai:

${\vec{v}}_{\nabla B}=\frac{m{v}_{\perp }^2}{2|q|B^2}\frac{\vec{B}\times \nabla B}{B}$

• $\nabla B$ — woh arrow jo direction dikhata hai jis taraf $B$ badh raha hai (yahan Earth ki taraf).
• $\vec{B}\times \nabla B$ — cross product jo step ko sideways bhejta hai, dono field aur jis direction mein yeh strong hota hai, dono ke perpendicular.
• $\propto 1/|q|$ with $q$ ka sign cross product ke andar carry hota hai: protons ek direction (westward) drift karte hain, electrons opposite direction (eastward). Charge reverse karo aur poori drift direction reverse ho jaati hai — do species ki woh opposite motion literally ek electric current hai, belt ka ring current.
• Bada $v_{\perp }$ (mota loop) → bada side-step, kyunki tight/wide arcs zyada differ karte hain.

Doosra drift kyun — curvature drift. Dipole mein field lines bhi curved hain, straight nahi. Ek particle jo curved line ke along glide kar raha hai (via $v_{\parallel }$) curve se outward ek centrifugal push feel karta hai; woh push, $\vec{B}$ se cross hoke, same sense mein ek doosra sideways step produce karta hai jaise gradient drift:

${\vec{v}}_{\text{curv}}=\frac{m{v}_{\parallel }^2}{|q|B^2}\frac{{\vec{R}}_c\times \vec{B}}{R_c^2}$

• ${\vec{R}}_c$ — field line ka radius-of-curvature arrow (woh kitna sharply bend karta hai).
• $v_{\parallel }$ — glide speed (Step 5); ek sharp bend ke along fast glider zyada drift karta hai.
• Same $1/|q|$ sign rule → protons aur electrons phir opposite drift karte hain, gradient drift mein add karte hain rather than fight karte hain.

Saath mein, gradient + curvature drift har particle ke guiding centre ko poore planet ke around circle karte hain.

Ek number taaki accumulation feel ho. Ek belt proton ka $r_L\sim 100\text{km}$ hai aur uska drift speed sirf uski gyration speed ka tiny fraction hai, toh har single gyration center ko sideways ek gyroradius se bahut kam nudge karta hai. Lekin ek pass mein millions of gyrations hoti hain, aur steps sab ek hi direction point karte hain, toh woh coherently add hote hain: ek typical inner-belt proton poori Earth ke around ek loop minutes to hours mein complete karta hai. Per turn slow, aggregate mein unstoppable — exactly waise jaise ek ghadi ki second hand crawl karti hai phir bhi dial lap karti hai.

PICTURE. Earth ke upar top-down view: Earth-facing arc par har chhota loop tight draw hai, far arc par wide; mismatch har successive loop ke centre ko ek small step sideways nudge karta hai (steps short arrows se mark hain), aur ~dozen loops stack karna belt ki full ring trace karta hai — inset do consecutive loops zoom karta hai ek single sideways step dikhane ke liye unke centres ke beech.

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Ek-picture summary

Teen motions, ek particle: field line ke around ek fast tight gyration (Steps 3–5), do magnetic poles ke beech ek slow bounce $B_{\text{mirror}}/B_{\text{eq}}=1/{\sin }^2{\alpha }_{\text{eq}}$ se set hota hua (Step 6), aur uneven, curved field se poori Earth ke around ek bahut slow drift (Step 8). Inhe layer karo aur tumhe Van Allen belts ka doughnut milta hai. Single formula $r_L=\frac{m{v}_{\perp }}{|q|B}$ in sab ka seed hai: yeh circle size set karta hai, yeh mirror explain karta hai (bada $B$ → chhota $r_L$, zyada $v_{\perp }$, $v_{\parallel }$ zero tak), aur yeh drift explain karta hai (uneven $B$ → uneven arc → sideways step).

Recall Feynman retelling — plain words mein wapas batao

Pehle axes fix karo: right $\hat{x}$ hai, up $\hat{y}$ hai, page se bahar $\hat{z}$ hai. Ek proton upar pointing field mein right zip karta hai; right-hand rule kehta hai $\vec{v}\times \vec{B}$ page se bahar point karta hai, toh ek positive charge tumhari taraf shove hota hai (ek electron, page ke andar). Kyunki woh cross product sirf velocity ka woh hissa pakadta hai jo field ke across hai, pehle $\vec{v}$ ko $v_{\perp }$ (across, bend hota hai) aur $v_{\parallel }$ (along, coast karta hai) mein split karo. Hamesha-sideways shove $v_{\perp }$ ko ek circle mein curl karta hai, aur "magnetic shove = circle's demand" match karne se iska size milta hai, $r_L=m{v}_{\perp }/(|q|B)$ — hum $|q|$ use karte hain taaki radius positive rahe. Coasting $v_{\parallel }$ wapas add karo aur circle + glide = ek corkscrew. Pole ki taraf head karo aur field crowd up hoti hai; gyro-loop se guzrne wala flux pinned hai, toh jaise loop shrink hota hai $v_{\perp }$ climb karta hai, aur kyunki total speed fixed hai forward glide exactly wahan drain ho jaata hai jahan $B_{\text{mirror}}=B_{\text{eq}}/{\sin }^2{\alpha }_{\text{eq}}$ — particle bounce karta hai. Meanwhile field har chhote loop ke Earth's side par stronger hai, toh loop ek side par tight aur doosri par wide hai aur close nahi ho sakta: uska centre har turn sideways ek baal step karta hai (gradient drift), aur curved field lines ek second sideways nudge add karti hain (curvature drift) same direction mein — protons west chaltein hain, electrons east. Millions of coherent chhote steps har particle ko minutes to hours mein planet ke around le jaate hain. Gyrate, bounce, drift — use months par spread karo aur tumne ek glowing doughnut paint kar diya. Escape clauses: line ke seedha neeche point karo (loss cone) aur tum hawa mein dive karte ho aur die karte ho; koi charge nahi carry karo, ya koi field nahi milo, aur trap tumhe simply jaane de deta hai.

Recall Quick self-check

Magnetic force velocity ke hamesha perpendicular kyun hoti hai? ::: Kyunki yeh cross product $q\vec{v}\times \vec{B}$ se aati hai, jo by definition dono $\vec{v}$ aur $\vec{B}$ ke 90° par point karta hai. $\vec{v}$ right aur $\vec{B}$ up ke saath, proton par force kis direction mein hogi? ::: Page se bahar ($+\hat{z}$), right-hand rule $\hat{x}\times \hat{y}=\hat{z}$ se; electron ke liye yeh page ke andar flip ho jaati hai. Gyroradius formula $|q|$ kyun use karta hai? ::: Ek radius positive hona chahiye; $|q|$ charge ka sign drop karta hai, jo sirf circling ki direction decide karta hai, size nahi. Bounce force karne wali conserved quantity kya hai, aur yeh conserved kyun hai? ::: Magnetic moment $\mu =m{v}_{\perp }^2/(2B)$; yeh fixed hai kyunki gyro-loop se guzrne wala magnetic flux conserved hota hai jab field ek orbit ke dauran slowly change kare. Particle kahan mirror karta hai? ::: Jahan $B_{\text{mirror}}=B_{\text{eq}}/{\sin }^2{\alpha }_{\text{eq}}$ hai, yaani jahan $v_{\parallel }$ zero reach karta hai. Belts ring-shaped kyun hain, aur har species kis direction drift karta hai? ::: Gradient aur curvature drift har loop ke centre ko har turn sideways step karte hain, particle ko Earth ke around ek torus mein walk karte hain; protons west drift karte hain, electrons east ($q$ ke sign ki wajah se opposite).

See also: Van Allen Probes mission · Magnetic field modeling · Single-event effects · Spacecraft materials selection