3.2.16 · D1True anomaly from eccentric anomaly
Ek planet ek dabī huī circle (ek ellipse ) par move karta hai, jisme Sun ek special point par off-center baitha hai jise focus kehte hain. Easily compute ho sakne wale "helper angle" ko — jo beech se measure hota hai — real angle mein convert karne ke liye, jo actually Sun se dikhta hai, bas yeh jaanna zaroori hai ki circle kitni dabī gayi aur Sun kitna off-center hai — yahi parent topic ka poora content hai.
Yeh page har symbol ko build karta hai jo parent note use karta hai, bilkul zero se, ek aise order mein jahan har piece pehle wale par rests karta hai. (Woh do angles jinhe intro mein naam diya gaya — "helper angle" aur "real angle" — unke formal symbols §4 mein aate hain; hum jaanbujhkar unhe letters se tab tak naam nahi dete.) Agar koi word ya letter parent mein kabhi bhi dikhe aur tum 100% sure nahi ho ki uski picture kya hai, toh woh yahan define hai.
Ek ellipse ek circle hai jo ek direction mein dabī (squash) ho gayi hai. Ek perfect circle lo aur use uske horizontal axis ki taraf press karo: jo milega woh ek oval hai. Wahi oval woh path (woh orbit ) hai jis par planet travel karta hai.
Figure dekho. Sabse lambe taraf (left–right, red) ko major axis kehte hain. Chhoti taraf (up–down, blue) ko minor axis kehte hain. Inme se har ek ke aadhe ka ek naam hai:
Definition Semi-major axisa b
==a sabse lambi width ka aadha. Red arrow ko picture karo, exact middle se far right edge tak.
==b sabse chhoti width ka aadha. Blue arrow ko picture karo, middle se seedha upar top edge tak.
"Semi" ka matlab sirf "aadha" hai. Yeh do numbers ellipse ki size aur uski squashedness poori tarah fix kar dete hain. Dekho Orbit geometry — semi-major axis and eccentricity .
Topic ko yeh kyun chahiye: parent ka pehla formula body ki height ko b sin E b / a yeh batata hai ki circle kitni flatten hui, aur b
Kisi bhi angle, cos sin origin kahan hai, x Parent page par har formula silently neeche diye convention ko use karta hai — yahan hum use loud karte hain.
Definition Perifocal convention (is topic mein har jagah use hoti hai)
==x center se perihelion ki taraf point karta hai (orbit ka near end, §4 mein define hai).
==y x 90° counter-clockwise point karta hai (har figure mein seedha "upar").
Angles ko positive x counter-clockwise (CCW) measure kiya jaata hai. CCW positive direction hai; clockwise mein ghoomna negative angles deta hai.
Do origins use hoti hain: center O (§1) aur focus F (§2b). Ek subscript batata hai kaun sa — coordinates ( x O , y O ) ( x F , y F )
Yeh perifocal frame hai. Yeh picture ek baar fix karo aur poore topic ka har cos sin
Topic ko yeh kyun chahiye: ek akela "cos E E + x ν
O F
==Center O ( x O , y O )
==Focus F center se perihelion ki taraf shifted . Sun (heavy mass) yahan baitha hai, center par nahi. Yeh ( x F , y F )
Center aur focus ke beech ki jagah ka apna naam hai:
c c O F + x
Topic ko yeh kyun chahiye: jo angle tum actually observe karte ho woh focus par measure hota hai, lekin easy helper angle center par measure hota hai. Poora conversion sach mein bas yahi hai: "inhe do viewing points ke beech ke shift c
e e 0 1 kitni squash hai:
e = a c = semi-major axis focus offset .
e = 0 perfect circle .e 1 lamba patla cigar .
e c = e a a e = 0.6 a = 10 c = 6 c = a e
Teeno key lengths ek Pythagorean-flavoured relation se bndhi hain:
Topic ko yeh kyun chahiye: term 1 − e 2 e
Astronomy mein anomaly ka word bas yeh mean karta hai: "ek angle jo batata hai ki body orbit par kahan hai." Inme se teen hain, aur poora topic unme se do ke beech convert karne ke baare mein hai. Key trap yeh hai: ek angle meaningless hai jab tak tum na kaho ki kis point se aur kis starting direction se tum use measure kar rahe ho — exactly isliye §2 pehle aaya.
Definition Perihelion — starting linePerihelion = orbit par woh point jo focus (Sun) ke sabse kareeb hai. Yeh major axis ke near end par, + x
Definition Teen anomalies
==True anomaly ν real angle, focus F , CCW. Yeh wahi hai jo tum actually Sun se bahar dekhne par dekhoge. Yeh woh answer hai jo tum chahte ho. (Yeh intro ka "real angle" hai.)
==Eccentric anomaly E helper angle center O measure kiya gaya, CCW, ellipse ke around draw ki gayi ek imaginary circle use karke. Compute karna easy hai, lekin yeh woh nahi jo tum dekhte ho. (Yeh intro ka "helper angle" hai.)
==Mean anomaly M fake angle jo clock ke saath perfectly evenly badhta hai. Orbit par iska koi direct geometric picture nahi hai — yeh time ka stand-in hai. Dekho Mean anomaly and time .
Common mistake "Focus par measure kiya gaya" ko "center par measure kiya gaya" samajhnaYeh ek jaisa kyun lagta hai: dono perihelion direction se angles hain. Yeh kyun nahi hai: tum alag points par khade ho. Center se tum angle E alag angle ν
Yeh woh picture hai jo E define karti hai — ise yaad karo, kyunki parent ka pehla step poori tarah iske andar rehta hai.
Definition Auxiliary circle aurE a O auxiliary circle hai. Ab ellipse par point P seedha upar jao (vertically) jab tak circle par point P ′ P ′ center O aur CCW, ==eccentric anomaly E
Kyunki P ′ a ( x O , y O ) = ( a cos E , a sin E )
Intuition Circle ke through kyun jaate hain?Angles on a circle dead simple hote hain: angle E ( cos E , sin E ) b / a ( x O , y O ) = ( a cos E , b sin E ) E
True anomaly ν focus par measure hota hai, isliye hume planet ki position ko F F ( c , 0 ) = ( a e , 0 ) origin ko right mein c :
Definition Focus-relative coordinatesx F , y F x F = x O − c = a cos E − a e = a ( cos E − e ) , y F = y O = a 1 − e 2 sin E .
==x F right mein hai (positive x
==y F upar hai (positive y
Yeh wahi symbols hain jo parent ka shift-to-focus step produce karta hai. Neeche wala figure F
r r focus F P r = x F 2 + y F 2 . r = a ( 1 − e cos E ) Orbital radius equation .
Topic ko yeh kyun chahiye: true anomaly padhne ke liye tum cos ν = x F / r sin ν = y F / r x F y F r
Formulas teen trig ideas use karte hain. Yahan har ek hai, tumhare dimag mein ek picture se anchored.
cos sin 1 θ positive x ghoomkar pahuncha, woh yahan baitha hai:
cos θ horizontal position (kitna right/left),sin θ vertical position (kitna up/down).
Isliye ( a cos E , a sin E ) a a
tan tan θ = c o s θ s i n θ = horizontal vertical = adjacent opposite . tan θ tan
Definition Half-angle identity — workhorsetan 2 2 θ = 1 + c o s θ 1 − c o s θ , aur yeh bhi tan 2 θ = 1 + c o s θ s i n θ . Topic is par kyun lean karta hai: raw conversion cos ν sin ν tan ( ν /2 ) = 1 − e 1 + e tan ( E /2 ) Half-angle trigonometric identities mein stocked hai.
Common mistake Branch aur sign: square root lena aurν tan 2 ( θ /2 ) ± kaun sa sign? Ek poore orbit ke liye E ν 0 360° E /2 ν /2 [ 0° , 180° ) tan positive root ke saath liya jaata hai: tan ( ν /2 ) = + 1 − e 1 + e tan ( E /2 ) tan ( θ /2 ) monotonic hai, har E ν ν ν /2 = atan2 ( 1 + e sin 2 E , 1 − e cos 2 E ) atan2 ( sin ν , cos ν )
arccos cos ν Yeh tempt kyun karta hai: tumhe cos ν Yeh kyun break hota hai: arccos [ 0° , 180° ] sin ν < 0 tan atan2) poore orbit par monotonic hai — koi ambiguity nahi. Yeh sabse important akela reason hai kyun topic half-angle formula prefer karta hai.
Intuition Yeh map kaise padheinHar box ek foundation hai jo upar build hui hai. Arrows ka matlab "feeds into" hai. Unhe top-to-bottom follow karo: shape facts (a b c e r ν
Eccentricity e equals c over a
Squash factor root of 1 minus e squared
Auxiliary circle point a cosE and a sinE
Ellipse point squashed height
Coordinate convention x to perihelion CCW
Shift origin to focus xF yF
Radius r and angle nu at focus
Clean tan nu over 2 formula
Test karo khud ko — reveal karne se pehle jawab zor se bolo.
Semi-major axis a Ellipse ki sabse lambi width ka aadha — center se long axis ke far end tak.
b a e b = a 1 − e 2 (vertical squash factor times
a ).
Is topic mein use hone wale coordinate convention ko state karo. Origin at center ya focus; + x + x
Eccentricity e e = c / a
Mass (Sun) actually kahan baitha hai — center par ya focus par? Focus F + x c = a e
Perihelion kahan hai? Orbit ka woh point jo focus ke sabse kareeb hai, + x
True anomaly ν Perihelion direction se, CCW, focus par measure kiya gaya.
Eccentric anomaly E Perihelion direction se, CCW, center par measure kiya gaya, auxiliary circle use karke.
Focus-relative coordinates x F , y F x F = a ( cos E − e ) ,
y F = a 1 − e 2 sin E .
Auxiliary-circle point se ellipse point kaise milta hai? ( a cos E , a sin E ) b / a ( a cos E , b sin E )
tan 2 ( θ /2 ) tan 2 ( θ /2 ) = 1 + cos θ 1 − cos θ
E ν Positive root, kyunki E /2 ν /2 [ 0° , 180° ) tan ( θ /2 ) ν atan2 use karo.
arccos ( cos ν ) Yeh sirf [ 0 , 180° ]
r Focus-to-planet distance
r = x F 2 + y F 2 ;
r = a ( 1 − e cos E ) .
