3.2.36 · D1Third-body perturbations
Earth ke around orbit karne wala satellite Moon aur Sun se bhi pull hota hai — lekin jo actually uski orbit ko disturb karta hai woh unka raw pull nahi hai, balki yeh hai ki woh satellite ko kitna zyada (ya kam) pull karte hain, Earth ko pull karne ki tulna mein. Kyunki hum satellite ko ek aisi Earth se dekhte hain jo khud Moon ki taraf gir rahi hai, sirf woh tiny difference bachta hai jo Earth-se-satellite ke gap mein hota hai — aur wahi difference orbit ko dheere dheere bigaadta hai.
Yeh page parent note ke har ek symbol aur idea ko build karta hai, ek samajhdar 12-saal ke bachche se shuru karte hue jisne kabhi koi arrow hat-wala nahi dekha. Upar se neeche padhein; har block agla earn karta hai.
Definition Vector aur notationr vector ek arrow hai: iske paas ek length hai (kitni door) aur ek direction (kis taraf). Hum ise upar ek chhoti arrow ke saath likhte hain, jaise r length ki parwah karte hain, toh arrow hata ke r
Pehla figure dekho. Neela arrow ek point se shuru hokar doosre point par khatam hota hai. Woh poora arrow hi r r
Intuition Topic ko arrows ki zaroorat kyun haiMoon sirf "thoda" nahi kheenchta — woh ek khaas direction mein kheenchta hai (Moon ki taraf), aur kitna kheenchta hai yeh depend karta hai kahan satellite baitha hai. Sirf arrows hi dono baatein ek saath carry kar sakte hain. Ek plain number humein kabhi nahi bata sakta ki pull sideways tilts hoti hai jab satellite side mein hota hai.
Position ek arrow
r kisi chosen origin se kisi cheez ke location tak
Bare symbol r r ki
length , ek akela number
Har arrow ko kahin se shuru hona padta hai. Woh starting point origin kehlata hai.
r s r E r 3 inertial origin — jo accelerate nahi kar raha). Usme se teen arrows kheencho:
r s satellite ki taraf,r E Earth ki taraf,r 3 third body ki taraf (Moon ya Sun). Chhota "3" ka matlab sirf "kahani ka teesra mass" hai.
Definition Relative position — arrows ghataanar = r s − r E Earth se satellite ki taraf jaata hai. Ek position arrow ko doosre se ghataane par milta hai "A, B ke nazariye se kahan hai." Isi tarah d = r 3 − r E Earth se third body ki taraf jaata hai.
Doosra figure dekho: teen lambe arrows door origin se nikalte hain, aur chhota green arrow r r s r E
Intuition "Relative" kyun matter karta hai yahanHum satellite ko us door-wale origin se track nahi karte — hum ise Earth se track karte hain, kyunki Earth humara home base hai. Yahi poori wajah hai ki perturbation ek difference ke roop mein nikalta hai: hamara home base move kar raha hai, toh humein hamesha poochna padta hai "Earth ke relative." Yeh Two-body problem wali soch hai: do moving bodies ko ek relative arrow r
r satellite ki position
Earth ke relative ,
r s − r E d third body ki position Earth ke relative,
r 3 − r E Hum r E kyunki hum Earth se observe karte hain, inertial origin se nahi
Ab pull khud. Hum famous law ko bite-sized symbols mein todenge.
G M μ = GM
G gravitational constant , prakriti ka ek fixed number (6.674 × 1 0 − 11 strength set karta hai.M mass (kilograms).Inका product μ = GM standard gravitational parameter kehlata hai. Hum inhe bundle karte hain kyunki orbit maths mein G M hamesha saath chalte hain — toh hum dono ko ek naam dete hain. Earth ke liye, μ = G M E μ 3 = G m 3
r ^ Hat ka matlab hai "length exactly 1, sirf direction." Toh r ^ = r / r r direction rakhta hai aur size chod deta hai. Iska use karo jab tum kahin point karna chahte ho bina yeh bataye ki kitni door.
Teesra figure dekho yeh dekhne ke liye ki r 3 r r 2 r ^ r r r r
Common mistake "Hat aur arrow same hain."Kyun sahi lagta hai: dono same direction mein point karte hain. Kyun galat hai: r r r ^ Fix: r ^ = r / r sirf-direction version hai.
Intuition Humein "force" nahi "acceleration" kyun chahiyeKisi bhi spot par har object ko kisi given mass se same acceleration milti hai, chahe uska apna weight kuch bhi ho (bhaari aur halka saath girte hain). Kyunki hum compare karte hain ki Moon satellite ko kaise accelerate karta hai vs Earth ko, accelerations mein kaam karne se satellite ka apna mass poori tarah nikal jaata hai — yahi aage aane wale cancellation ka raaz hai.
μ bundle GM
r ^ unit vector —
r ki direction, length exactly 1
− μ r / r 3 gravity inward kheenchti hai, mass ki taraf
r / r 3 r 3 yeh
r ^ / r 2 ko rewrite kiya hai; ek
r r ko
r ^ mein convert karta hai
r ¨ r ˙ Do dots, r ¨ acceleration , velocity kitni tezi se change hoti hai. Yeh time ke saath do baar liya gaya derivative hai.
r ¨ acceleration ke baare mein ek statement hai: force yeh set karta hai ki velocity kaise change hoti hai. Toh poori derivation ek aisi sentence hai "r ¨ =
r ˙ velocity — position har second kaise change hoti hai
r ¨ acceleration — velocity har second kaise change hoti hai
Tidal formula d ^ ⋅ r r
a ⋅ b dot product ek akela number hai: unki lengths ko multiply karo aur phir yeh dekho ki woh kitne aligned hain. Khaas taur par, d ^ ⋅ r d ^ r shadow ki length jo d ^ r
Chautha figure dekho: green arrow r d ^ d ^ ⋅ r r d ^ r
Intuition Topic ko dot product kyun chahiyeTidal effect Moon ki line ke along stretch karta hai aur sideways squeeze karta hai . Ise mathematically likhne ke liye r d ^ 3 ( d ^ ⋅ r ) d ^ d ^
d ^ ⋅ r r ka
d ^ direction par shadow (projection) (uska "along-the-Moon" part)
Perpendicular arrows ka dot product zero
Tidal formula ise kyun use karta hai stretch direction ko squeeze direction se alag karne ke liye
Ab har piece haath mein hai, toh hum central object ko naam de sakte hain.
s = r 3 − r s s satellite se third body ki taraf jaata hai — Moon satellite ko kaise "dekhta" hai. Parent note ki key quantity ek pull nahi balki ek difference hai: Moon ka satellite par pull minus uska Earth par pull. Kyunki Earth Moon ki taraf satellite ke saath freely fall kar rahi hai, pull ka shared part cancel ho jaata hai, aur sirf gap ke across variation bachti hai. Woh bacha hua tidal (differential) acceleration kehlata hai.
Intuition "Difference," pictured kyunImagine karo do log ek hi lift mein gir rahe hain. Koi bhi apna girna feel nahi karta — tabhi unke beech drift hoti hai jab ek doosre se zyada khicha jaata hai. Satellite aur Earth dono gir rahe hain; Moon unhe bahut slightly alag tarike se kheenchta hai, aur wahi tiny drift perturbation hai. Yahi woh free-fall logic hai jo astronauts ko float karata hai.
s satellite-se-third-body arrow,
r 3 − r s Raw pull μ 3 / s 2 perturbation kyun nahi hai Earth lagbhag utna hi kheencha jaata hai; common part cancel ho jaata hai, sirf difference bachta hai
Bachne wala difference kehlata hai tidal / differential acceleration
≪ r / d r ≪ d r d bahut chhota hai." Yahan r d r / d tiny number hai. Jab bhi koi quantity tiny ho, hum uska square throw away kar sakte hain (tiny² aur bhi tinier hota hai) — isliye parent "first order in r / d
Intuition Approximate kyun karte hainDo-pulls-ka-difference wala exact formula sahi hai par clumsy hai. Kyunki r / d tidal form milta hai d 3 μ 3 ( 3 ( d ^ ⋅ r ) d ^ − r )
r ≪ d Earth–satellite distance, Earth–third-body distance se bahut chhoti hai
( r / d ) 2 yeh rakhe gaye first-order term ke comparison mein negligibly small hai
Vector arrow r-hat and length r
Relative position r equals r_s minus r_E
G and mass give mu equals GM
Newton pull minus mu r over r cubed
Double dot r means acceleration
Split along versus perpendicular
Tidal approximation with r much less than d
Khud test karo — tum parent note ke liye ready ho tabhi jab har reveal obvious lage.
Main bata sakta hoon arrow aur hat ka matlab r ek poora arrow hai (direction + length
r );
r ^ = r / r sirf direction hai, length 1
Main r r s r E r = r s − r E , Earth se satellite tak ka arrow
Mujhe pata hai μ μ = GM μ 3 = G m 3
Main Newton ka pull padh sakta hoon a = − μ r / r 3 = − μ r ^ / r 2 , inward point karta hai,
1/ r 2 ki tarah kamzor hota hai
Mujhe pata hai r ¨ acceleration, position ka second time-derivative
Main d ^ ⋅ r r ka third body ki direction mein shadow (projection)
Main samajhta hoon perturbation difference kyun hai Earth bhi free-fall karta hai; common pull cancel hota hai, sirf tidal gradient bachta hai
Mujhe pata hai r / d r ≪ d r / d
