Third-body perturbations

3.2.36 · Physics › Orbital Mechanics & Astrodynamics

WHAT is a third-body perturbation?

"Difference" KYUN? Newton's law inertial frame mein absolute accelerations deta hai. Lekin hum satellite ko Earth ke relative track karte hain. Earth Moon ki taraf free-fall kar rahi hai, bilkul jaise satellite karti hai. Ek free-falling frame mein, acceleration ka common part cancel ho jaata hai — sirf gradient (Moon ke field ki variation jo Earth–satellite gap mein hai) bachti hai. Yahi reason hai ki astronauts orbit mein weightless feel karte hain: woh aur unka capsule ek saath girate hain.

HOW to derive it from first principles

Maano (saare vectors ek inertial origin se):

• ${\vec{r}}_s$ = satellite position, ${\vec{r}}_E$ = Earth position, ${\vec{r}}_3$ = third body position.
• $\mu =G{M}_E$, aur third body ki mass $m_3$ hai jiske liye ${\mu }_3=G{m}_3$.

Step 1 — Absolute accelerations (Newton).

Satellite Earth + third body dono ko feel karta hai: ${\ddot{\vec{r}}}_s=-\mu \frac{{\vec{r}}_s-{\vec{r}}_E}{|{\vec{r}}_s-{\vec{r}}_E{|}^3}+{\mu }_3\frac{{\vec{r}}_3-{\vec{r}}_s}{|{\vec{r}}_3-{\vec{r}}_s{|}^3}$

Yeh step kyun? Har mass $-GM\hat{r}/r^2$ produce karta hai jo uski taraf point karta hai; satellite par dono pulls likho.

Earth third body ko feel karti hai (hume Earth ki apni motion ki chinta hai): ${\ddot{\vec{r}}}_E={\mu }_3\frac{{\vec{r}}_3-{\vec{r}}_E}{|{\vec{r}}_3-{\vec{r}}_E{|}^3}$

Kyun? Moon poori Earth ko kheenchta hai, hamare observation frame ko accelerate karta hai. Hume yeh subtract karna hoga.

Step 2 — Earth-centered relative coordinates mein jaao.

Define karo $\vec{r}={\vec{r}}_s-{\vec{r}}_E$ (satellite w.r.t. Earth) aur $\vec{d}={\vec{r}}_3-{\vec{r}}_E$ (third body w.r.t. Earth). Subtract karo: $\ddot{\vec{r}}={\ddot{\vec{r}}}_s-{\ddot{\vec{r}}}_E=-\mu \frac{\vec{r}}{r^3}+\underset{\text{perturbation }{\vec{a}}_{3b}}{\underbrace{{\mu }_3\left(\frac{{\vec{r}}_3-{\vec{r}}_s}{|{\vec{r}}_3-{\vec{r}}_s{|}^3}-\frac{{\vec{r}}_3-{\vec{r}}_E}{|{\vec{r}}_3-{\vec{r}}_E{|}^3}\right)}}$

Yeh step kyun? Pehla term familiar Keplerian pull hai. Bracket third-body perturbing acceleration hai — literally satellite par pull minus Earth par pull.

Step 3 — Clean form. $\vec{s}={\vec{r}}_3-{\vec{r}}_s=\vec{d}-\vec{r}$ ke saath:

Tidal approximation (useful 80/20 core)

Almost hamesha $r\ll d$ hota hai (satellite–Earth distance ≪ Earth–Moon/Sun distance). $r/d$ mein first order tak expand karo.

Step 1 — $|\vec{d}-\vec{r}{|}^{-3}$ expand karo. Likho $|\vec{d}-\vec{r}{|}^2=d^2-2\vec{d}\cdot \vec{r}+r^2$. Maano $\hat{d}=\vec{d}/d$: $|\vec{d}-\vec{r}{|}^{-3}=d^{-3}{\left(1-2\frac{\vec{d}\cdot \vec{r}}{d^2}+\frac{r^2}{d^2}\right)}^{-3/2}\approx d^{-3}\left(1+3\frac{\vec{d}\cdot \vec{r}}{d^2}\right)$

Kyun? Binomial $(1+x{)}^{-3/2}\approx 1-\frac{3}{2}x$; $r/d$ mein linear terms rakhho.

Step 2 — Substitute karo aur $O((r/d{)}^2)$ drop karo: ${\vec{a}}_{3b}\approx {\mu }_3\left[\frac{(\vec{d}-\vec{r})}{d^3}(1+3\frac{\vec{d}\cdot \vec{r}}{d^2})-\frac{\vec{d}}{d^3}\right]\approx \frac{{\mu }_3}{d^3}(3(\hat{d}\cdot \vec{r})\hat{d}-\vec{r})$

Key scaling insight: disturbance $\propto {\mu }_3/d^3$ hai, ${\mu }_3/d^2$ nahi. Isliye Sun (huge ${\mu }_3$ lekin dur) aur Moon (small ${\mu }_3$ lekin paas) compete karte hain. Numbers daalo: Earth-orbit satellites par Moon ka tidal effect Sun ke effect se do guna zyada hai — halanki Sun kaafi zyada massive hai.

Worked examples

Common mistakes (steel-manned)

Flashcards

Connections

• Two-body problem — woh unperturbed baseline jisse hum subtract karte hain.
• Tidal forces — same $3(\hat{d}\cdot \vec{r})\hat{d}-\vec{r}$ gradient physics.
• J2 perturbation and oblateness — doosra main perturbation, low altitude par dominant.
• Gauss and Lagrange planetary equations — kaise ${\vec{a}}_{3b}$ $i,\Omega ,e$ mein slow drifts mein turn hota hai.
• GEO stationkeeping — jahan luni-solar N–S drift regular burns force karta hai.
• Restricted three-body problem — full treatment jab $r/d$ small nahi hota.

Concept Map