Foundations — Kepler's third law — T² ∝ a³ — derivation

3.2.7 · D1 · Physics › Orbital Mechanics & Astrodynamics › Kepler's third law — T² ∝ a³ — derivation

Pehle aapko parent page par di gayi derivation padhni hai — Kepler's Third Law — lekin usse padhne se pehle aapko har ek squiggle apna banana hoga, bina kisi excuse ke. Yeh page har ek cheez ko zero se build karta hai, ek aisi order mein jahan har symbol sirf pehle wale symbols pe tikaa ho.

0 · Ek symbol ko kaise padhein

Ek symbol ek picture ka shorthand hota hai. Agar aap picture nahi bana sakte, toh abhi woh symbol aapka nahi hua. Isliye neeche har entry ke teen hisse hain: plain words → picture → topic ko yeh kyun chahiye. Inhe order mein padhein; har ek agli cheez ke liye ek eent hai.

1 · $r$ — centre se doori

Figure 1 dekho. Sun beech mein baitha hai. Planet ring par baitha hai. $r$ label wala white arrow woh gap hai, metres mein measure kiya gaya.

Topic ko yeh kyun chahiye. Orbital mechanics mein sab kuch doori ke saath kamzor ya slow hota jaata hai. Gravity $r$ ke saath fade karti hai, speed $r$ ke saath drop hoti hai, lap-time $r$ ke saath badhta hai. Agar "kitna door hai" ke liye koi naam hi nahi hoga, toh aap yeh sab kuch keh hi nahi sakte. $r$ pehli eent hai.

2 · $\pi$ aur circumference $2\pi r$ — ek lap ki length

Figure 2 dekho. Circle ko ek seedhi strip mein unroll karo: uski length exactly $2\pi r$ hai. Radius double karo, aur strip bhi double ho jaati hai — lap-length $r$ ke saath saath badhti hai.

Topic ko yeh kyun chahiye. Orbit ka ek lap ek circumference hi hoti hai. Jab parent note likhta hai "planet ek period mein $2\pi r$ cover karta hai," yahi picture hai. Yahi door ke planets ke liye "longer track" penalty hai: bada $r$ matlab literally lamba rasta.

3 · $v$ — orbital speed

Agar ek lap $2\pi r$ metres lamba hai aur use complete hone mein $T$ time lagta hai, toh

$v=\frac{2\pi r}{T}\text{(lap length divided by lap time)}.$

Topic ko yeh kyun chahiye. Speed orbit ke size aur usmein lagne wale time ke beech ka bridge hai. Gravity speed fix karegi (agla section), aur speed period fix karegi. $v$ ke bina hum orbit ki shape ko clock reading se connect hi nahi kar sakte.

4 · $T$ — period, aur "squared" ka matlab

$T^2$ kyun, sirf $T$ kyun nahi? Parent law $T^2\propto a^3$ ke roop mein bataya gaya hai, $T\propto \text{kuch}$ nahi. Yeh isliye hai kyunki honest relationship, $T\propto r^{3/2}$, mein ek awkward square root ($r^{1/2}=\sqrt{r}$) hai. Dono sides ko square karne se root clear ho jaata hai: $(r^{3/2}{)}^2=r^3$. Toh $T^2$ deliberately choose kiya gaya hai — yeh ek ugly half-power ko ek clean whole-power cube mein badal deta hai. Yahi ek wajah hai ki "2" aur "3" saath mein dikhte hain.

5 · $M$ aur $m$ — do masses (inhe kabhi mix mat karo)

Figure 3 dekho: centre mein bada amber dot ($M$), ring par chhota cyan dot ($m$). Mass bas "kitना stuff hai" matlab hai, kilograms mein measure kiya jaata hai.

Topic ko yeh kyun chahiye. Gravity dono masses par depend karti hai. Lekin Kepler's Third Law ka punchline yeh hai ki chhota mass $m$ cancel out ho jaata hai aur answer se gaayab ho jaata hai — sirf central $M$ bachta hai. Jab tak aap dono masses ko strictly alag nahi rakhte, tab tak aap us magic trick ki appreciation nahi kar sakte.

6 · $G$ — gravity ki strength

Topic ko yeh kyun chahiye. Gravity ka pull $\frac{GMm}{r^2}$ likha jaata hai (dekho Newton's Law of Universal Gravitation). $G$ woh dial hai jo "masses aur distance" ko newtons mein actual force mein convert karta hai. Yeh poori derivation mein chalta rehta hai aur final constant $\frac{4{\pi }^2}{GM}$ mein land karta hai.

7 · Force, centripetal pull, aur $\frac{m{v}^2}{r}$

Figure 4 dekho. Pull ke bina planet dashed tangent line ke along ud jaata. Gravity (amber arrow) woh leash hai jo usse har instant ring par waapas bend karti hai.

Yeh tool $\frac{m{v}^2}{r}$ kyun aur koi nahi? Hume ek formula chahiye jo kahe "is speed ko is size ke circle par hold karne ke liye mujhe kitna inward pull karna hoga?" Yahi exactly centripetal force result hai Centripetal Force & Circular Motion se. Notice karo yeh speed-squared ke saath badhta hai (fast cheezein turning resist karti hain) aur radius ke saath ghatta hai (gentle wide circles mein kam pull chahiye). Poori derivation ek sentence hai: gravity ke pull ko is required pull ke barabar set karo.

8 · $\propto$ — "proportional to" arrow

Topic ko yeh kyun chahiye. Kepler ne $T^2\propto a^3$ purely telescope data se discover kiya tha, isse kaafi pehle ki kisi ko pata chala ki multiplier $4{\pi }^2/GM$ hai. $\propto$ sign hume pehle pattern batane deta hai aur exact constant baad mein fill karne deta hai. Jab aap ek hi planet ke do moons compare karte hain, constant cancel ho jaata hai aur sirf $\propto$ pattern bachta hai.

9 · $a$ — semi-major axis (ellipse ka honest radius)

Kyunki $r$ elliptical orbit par badalta rehta hai (section 1), hume law mein plug karne ke liye ek steady number chahiye. Woh number $a$ hai, closest approach aur farthest point ka average:

$a=\frac{r_{\min }+r_{\max }}{2}.$

Topic ko yeh kyun chahiye. Parent law ka final form hai $T^2=\frac{4{\pi }^2}{GM}{a}^3$. Real orbits ellipses hoti hain, aur $a$ hai — koi single distance nahi — jo year ko control karta hai. Circle derivation $r$ use karti hai; honest ellipse version simply $r\to a$ swap karta hai. Circle par jo kuch aap seekhte hain woh sab transfer ho jaata hai.

Prerequisite map

Ise upar se neeche padhein: distance aur masses do forces ko feed karte hain; un forces ko equal set karne se, period aur average size $a$ ke saath milkar, law nikalta hai.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi jawaab aapko surprise kare, toh parent derivation kholne se pehle uska section upar dobara padhein.