Kepler's second law — equal areas in equal times, from angular momentum conservation

3.2.6 · Physics › Orbital Mechanics & Astrodynamics

LAW kya hai?

Toh $\frac{dA}{dt}=\text{const}$. Kepler ka yeh vague astronomical observation (1609) pure mechanics se kyun follow karta hai? Kyunki yeh secretly angular momentum ke baare mein ek statement hai.

KYU: iske peeche physics

KAISE: scratch se derive karo

Step 1 — Ek patli triangular slice ka area. Ek choti si time $dt$ mein radius vector $\vec{r}$ move karke $\vec{r}+d\vec{r}$ ho jaata hai. Woh tiny area swept triangle hai jo $\vec{r}$ aur $d\vec{r}$ se bana hai:

$dA=\frac{1}{2}|\vec{r}\times d\vec{r}|$

Yeh step kyun? Do edge vectors $\vec{a},\vec{b}$ waale triangle ka area $\frac{1}{2}|\vec{a}\times \vec{b}|$ hota hai. Yahan edges $\vec{r}$ aur displacement $d\vec{r}$ hain.

Step 2 — $dt$ se divide karo areal velocity pane ke liye.

$\frac{dA}{dt}=\frac{1}{2}\left|\vec{r}\times \frac{d\vec{r}}{dt}\right|=\frac{1}{2}|\vec{r}\times \vec{v}|$

Yeh step kyun? $d\vec{r}/dt=\vec{v}$, velocity hai. Cross product linear hota hai, toh hum $1/dt$ ko andar pull kar sakte hain.

Step 3 — Angular momentum laao. Angular momentum hai $\vec{L}=\vec{r}\times m\vec{v}$, toh $\vec{r}\times \vec{v}=\vec{L}/m$. Isliye

$\boxed{\frac{dA}{dt}=\frac{L}{2m}}$

Yeh step kyun? Yeh ek geometric quantity (area rate) ko ek dynamical conserved quantity ke terms mein rewrite karta hai. Magic tab ho jaata hai jab hum show kar dein ki $L$ constant hai.

Step 4 — Dikhao ki $L$ constant hai (torque = 0).

$\frac{d\vec{L}}{dt}=\frac{d}{dt}(\vec{r}\times m\vec{v})=\underset{=0}{\underbrace{\vec{v}\times m\vec{v}}}+\vec{r}\times m\vec{a}=\vec{r}\times \vec{F}$

Gravity ke liye $\vec{F}=-\frac{GMm}{r^2}\hat{r}$, jo $\vec{r}$ ke parallel hai, toh $\vec{r}\times \vec{F}=0$.

$\Rightarrow \frac{d\vec{L}}{dt}=0\Rightarrow L=\text{const}\Rightarrow \frac{dA}{dt}=\frac{L}{2m}=\text{const.}■$

KYU equal areas → paas fast, door slow

Kyunki $\frac{1}{2}{r}^2\dot{\theta }$ fixed hai, jab $r$ chota hota hai (perihelion) toh $\dot{\theta }$ bada hona chahiye (fast angular sweep); jab $r$ bada hota hai (aphelion) toh $\dot{\theta }$ chota hota hai. Isse famous perihelion–aphelion speed relation milti hai:

$m{v}_p{r}_p=m{v}_a{r}_a\Rightarrow v_p{r}_p=v_a{r}_a$

(valid hai kyunki peri/aphelion par $\vec{v}\perp \vec{r}$, toh $L=mvr$ directly milta hai).

Worked examples

Common mistakes

Active recall

Connections

• Angular Momentum Conservation — is law ke peeche ka engine
• Central Forces and Torque — kyun $\tau =0$
• Kepler's First Law (Ellipse) — specifically inverse-square chahiye
• Kepler's Third Law (T² ∝ a³) — $\pi ab=(L/2m)T$ use karta hai
• Vis-viva Equation — kisi bhi radius par speed
• Cross Product and Area — $\frac{1}{2}|\vec{a}\times \vec{b}|$ ki geometry

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