Foundations — Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

3.2.4 · D1 · Physics › Orbital Mechanics & Astrodynamics › Orbit shape from eccentricity — circle (e=0), ellipse (0 - e

Yeh page har woh symbol build karta hai jis par parent note rely karta hai, ek smart 12-saal-ke-bacche ke toolkit se shuru karke aur kuch nahi. Upar se neeche padho — har block sirf wahi cheezein use karta hai jo uske upar define ho chuki hain.

1. Distance, angle, aur do masses

Saath mein $(r,\theta )$ polar coordinates hain — ek "kitni door, kis taraf" address. Isliye poora topic $r(\theta )$ ke roop mein likha gaya hai: mujhe direction do, main distance bata dunga.

2. Focus — woh special point jiske around orbit draw hoti hai

Yeh matter karta hai kyunki — ek baar jab hum baad mein orbit equation build kar lenge — distance $r$ hamesha focus se measure hogi, na ki geometric centre se. Sun focus par hai, ellipse ke ek taraf — kabhi uske middle mein nahi.

3. Cone ko slice karna — sab chaar shapes kahan se aati hain

4. Geometry se do cheezein milti hain: $a$, $c$, aur $p$

Energy aane se pehle, ek ellipse ki shape ko teen lengths chahiye.

Ab hamare paas $p$ purely geometry se hai. Baad mein, jab angular momentum aur gravity ki strength define ho jaayegi (Sections 6 aur 8), wohi $p$ dobara $p=h^2/\mu$ ke roop mein appear hoga — same width ki physics value. $p$ ka yeh double life exactly wahi hai jo orbit ki shape ko body ki motion se jodta hai, lekin hum woh formula honestly tab tak nahi likh sakte jab tak woh symbols exist na karein.

5. Trig tool: cosine, aur kyun woh (sine nahi) appear karta hai

Ab jab hamare paas $p$, $e$, aur $\cos \theta$ hain, hum finally woh orbit equation padh sakte hain jo parent state karta hai: $r(\theta )=\frac{p}{1+e\cos \theta }$ Isme har symbol ab earn ho chuka hai. ($\theta =90^{\circ }$ par, $\cos 90^{\circ }=0$, isliye $r=p$ — exactly isliye $p$ "width at the focus" hai jaise humne upar define kiya.)

Yeh single fact parent ke poore "read off the denominator" argument ko drive karta hai: denominator shrink hota hai jaise $\theta$ $180^{\circ }$ ki taraf badhta hai ($\cos$ $-1$ ki taraf fall karta hai), aur $r$ badhta hai.

6. Rates of change: dot, aur specific angular momentum $h$

Connection dekho Angular Momentum in Orbits: $h$ woh width $p=h^2/\mu$ fix karta hai (isme $\mu$ Section 8 mein define hai).

7. Energy: $U$, $\epsilon$, $E$, aur woh sign jo sab decide karta hai

Figure s04 exactly yahi draw karta hai. Curved chalk line potential well $U=-GMm/r$ hai: centre ke paas steep aur deep, door "rim" (energy $=0$) ki taraf flat hoti jaati hai. Body ki total energy ek horizontal level line hai — kyunki energy conserved hai, body move hone par flat rehti hai. Figure aise padho: blue level rim ke neeche baith'ti hai ($\epsilon <0$) isliye body kabhi right edge tak nahi pahunch sakti → woh trapped hai → ellipse. Pink dashed level exactly rim par hai ($\epsilon =0$) → yeh barely infinity tak pahunchti hai → parabola. Yellow level rim ke upar hai ($\epsilon >0$) → yeh height bachao ke saath infinity tak pahunchti hai → hyperbola.

Dekho Specific Orbital Energy aur Escape Velocity ($\epsilon =0$ boundary), aur Vis-viva Equation (jo $\epsilon$ ko kisi bhi $r$ par speed mein convert karta hai).

8. Gravity ke constants: $G$ aur $\mu$

$G$ aur $M$ ko bundle kyun karein? Kyunki yeh orbit maths mein hamesha saath appear hote hain, isliye ek symbol $\mu$ carry karna cleaner hai. Yeh $p=h^2/\mu$ mein appear karta hai (ab isme har symbol — $p$, $h$, $\mu$ — define hai) aur $e=\sqrt{1+2\epsilon h^2/{\mu }^2}$ mein.

9. Master relation ko plain words mein samajhna

$\epsilon$ ka sign seedha padho:

• $\epsilon <0\Rightarrow$ square-root ke andar 1 se kam → $e<1\Rightarrow$ ellipse/circle.
• $\epsilon =0\Rightarrow e=\sqrt{1}=1\Rightarrow$ parabola.
• $\epsilon >0\Rightarrow e>1\Rightarrow$ hyperbola.

Woh single formula poora parent topic compressed hai. Baaki sab decoration hai.

Prerequisite map

Neeche diya diagram is page ka dependency order dikhata hai: geometry symbols (left) orbit equation build karte hain, motion symbols (right) eccentricity–energy relation build karte hain, aur dono streams us four-shape classification mein merge hoti hain jo parent topic hai. Arrow ko "pehle chahiye" padho. Yeh reading order visible ho gaya — koi bhi downstream symbol koi aisa symbol use nahi karta jo uske upstream na ho.

Saari streams parent topic ko feed karti hain.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle har ek ka jawab do.