Foundations — Orbit equation r = p - (1 + e·cos θ) — derivation from equations of motion

3.2.3 · D1 · Physics › Orbital Mechanics & Astrodynamics › Orbit equation r = p - (1 + e·cos θ) — derivation from equat

Yeh page har symbol ko build karta hai jis par parent derivation tika hua hai, bilkul zero se shuru karke. Agar koi symbol parent mein aata hai, toh woh yahaan pehle define hoga — ek picture aur ek reason ke saath ki woh exist kyun karta hai.

0 — Woh picture jis par hum baar baar point karte rahenge

Kisi bhi symbol se pehle, scene dekho. Ek bada mass (use Sun kaho) fixed baitha hai. Ek chhota body (use planet kaho) uske around move kar raha hai. Hum Sun par khade hain aur describe karte hain ki planet kahaan hai.

Neeche jo kuch bhi hai woh is ek drawing ke kisi na kisi hisse par chipka hua label hai.

1 — Position distance aur angle se: $r$ aur $\theta$

Yeh topic $x,y$ ki jagah ise kyun use karta hai? Gravity arrow ke saath point karti hai — Sun ki taraf. Jo coordinate system apni pehli direction "arrow ke saath" rakhta hai woh physics ko cleanly nikalta hai. $x,y$ mein force dono axes par phail jaati; $(r,\theta )$ mein woh poori tarah $r$ direction mein rehti hai. Dekho Central Force Motion.

2 — Do direction-arrows: $\hat{r}$ aur $\hat{\theta }$

Yeh topic inhe kyun use karta hai? Kyunki yeh humein kisi bhi motion ko "Sun ki taraf/Sun se door" (woh $\hat{r}$ wala hissa) aur "chakkar lagana" (woh $\hat{\theta }$ wala hissa) mein todne dete hain. Gravity pure $\hat{r}$ hai; yahi poora reason hai ki sideways swing untouched rehti hai — jo §6 mein angular momentum ban jaati hai.

3 — Change ki rates: dot, $\dot{}$ aur $\ddot{}$

Picture: $\dot{r}$ arrow ki length ka speedometer hai; $\dot{\theta }$ arrow ke sweeping ka speedometer hai.

Yeh topic ise kyun use karta hai? Motion hi change hai. "Gravity acceleration cause karti hai" likhne ke liye humein acceleration ka symbol chahiye, aur woh $\ddot{\vec{r}}$ hai. Dot sirf shorthand hai taaki equations chhoti rahein.

4 — Vectors aur arrow-multiply: $\vec{r}$, dot, aur cross product

Do arrows ko do alag tareekon se "multiply" kiya ja sakta hai, aur topic unme se ek use karta hai:

Do facts jo hum baar baar use karenge:

• Agar do arrows parallel hain (same ya opposite direction), toh parallelogram bilkul flat ho jaata hai — zero area — toh $\vec{a}\times \vec{a}=0$ aur $\vec{a}\times (-k\vec{a})=0$.
• Result ki length naapti hai ki arrows ke beech kitna sideways swing packed hai.

Yeh topic ordinary multiplication ki jagah cross product kyun use karta hai? Topic ko prove karna hai ki "sideways swing" kabhi nahi badlta. Sirf cross product sideways-ness naapti hai (us perpendicular area ke zariye). Ordinary number multiplication direction dekh hi nahi sakti. Yeh tool bilkul isi sawal ke liye bana hai. Yeh Specific Angular Momentum h ko power karta hai.

5 — Gravity kahaan se aati hai: $G$, $M$, $m$, aur $\mu$

$\mu$ mein bundle kyun karein? Kyunki $G$ aur $M$ hamesha $GM$ ki tarah glued hoke aate hain. Pair ko ek naam $\mu$ dene se baad ke har formula chhota ho jaata hai aur yaad dilata hai ki planet ka khud ka mass cancel ho gaya hai — same jagah har planet ko same acceleration feel hoti hai. Dekho Kepler's Laws.

6 — Show ka star: specific angular momentum $h$

Topic ko $h$ kyun chahiye? Do reasons, dono crucial:

1. Kyunki gravity $\vec{r}$ ke parallel hai (dono $\hat{r}$ ke along), cross product $\vec{r}\times \ddot{\vec{r}}=0$ hai (parallel arrows, §4), toh $\vec{h}$ kabhi nahi badlta. Ek constant sona hai: hum messy time variable ko angle se replace kar sakte hain.
2. $h=r^2\dot{\theta }$ woh bridge ban jaata hai jo derivation mein "rate per second" ko "change per angle" mein convert karta hai.

7 — Polar form mein radial acceleration: $\ddot{r}-r{\dot{\theta }}^2$

Yeh woh ek line hai jo scary lagti hai lekin §2 ke turning arrows ki pure geometry hai.

• $\ddot{r}$ = genuine "distance accelerate ho rahi hai" wala hissa.
• $-r{\dot{\theta }}^2$ = centripetal term: chahe $r$ steady bhi rahe, ek circle mein swinging khud hi ek inward acceleration hai (yeh wohi $v^2/r$ hai jo merry-go-round par feel hoti hai, kyunki $v_{\theta }=r\dot{\theta }$).

Extra term kyun? Kyunki $\hat{r}$ rotate karta hai (§2 mistake box). Jab tum spinning frame of arrows mein acceleration likhte ho, toh rotation ek $-r{\dot{\theta }}^2$ radial direction mein leak karta hai. Yeh naya physics nahi hai — yeh turning coordinates ka bookkeeping hai. Right side sirf §5 wali gravity hai.

8 — Final curve ke do dials: $p$ aur $e$

Yeh payoffs hain — woh numbers jo poori derivation chase kar rahi hai.

Do alag dials kyun? Kyunki ek conic section ko bilkul do pieces of information chahiye: kitna bada aur kitna stretched. $p$ aur $e$ precisely woh carry karte hain, aur dono physics se automatically drop out hote hain ($h$, $\mu$, aur ek integration constant se). Gehri meaning Eccentricity and Orbital Energy, Conic Sections, aur Vis-viva Equation mein hai.

Prerequisite map

Ise upar se neeche padho: coordinate aur calculus symbols acceleration line build karte hain; cross product $h$ build karta hai; gravity inward pull build karti hai; teeno streams orbit equation mein merge hote hain, $p$ aur $e$ output dials ke roop mein.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi answer fuzzy hai, derivation se pehle woh section dobara padho.