Visual walkthrough — Rendezvous and proximity operations — Clohessy-Wiltshire equations

3.2.40 · D2 · Physics › Orbital Mechanics & Astrodynamics › Rendezvous and proximity operations — Clohessy-Wiltshire equ

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Step 1 — Do ships, ek origin (Frame KAYA hai?)

KAYA. Socho Earth ek bada dot hai. Do spacecraft ek hi path ke aas-paas circle kar rahe hain: target (jis cheez se hum dock karte hain) aur chaser (hum). Dono ko planet ke ird-gird loop karte dekhne ke bajaye, hum ek camera target pe chipka dete hain aur use target ke saath spin karne dete hain. Is "chase-cam" mein target kabhi move nahi karta — woh origin pe baitha rehta hai — aur hum sirf chaser ko nearby hilte dekhte hain.

KYUN. Target lagaataar curve (accelerate) kar raha hai Earth ke around. Ek still frame mein woh screen se bahar zoom ho jaata. Ek frame jo usi rate pe turn karta hai use pin rakhta hai, toh sirf choti si relative separation bachti hai — jis cheez ki ek docking pilot ko hamesha zarurat hoti hai.

PICTURE. Teeno axes hain:

• $x$ = radial, seedha upar Earth se door,
• $y$ = along-track, woh direction jisme ships fly kar rahi hain,
• $z$ = cross-track, page ke bahar (orbital plane ka normal).

Yeh LVLH / Hill frame hai. Yeh ek rotating reference frame hai, isliye fake forces appear honge — yeh ek moving camera ki keemat hai, aur hum ise Step 4 mein pay karte hain.

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Step 2 — $n$ kahan se aata hai (Woh spin rate KYUN?)

KAYA. Frame bilkul usi rate pe spin karta hai jis rate pe target orbit karta hai. Ek circular orbit ke liye woh rate fixed hoti hai. Hum ise $n$ kehte hain aur ise abhi pin karte hain, kisi equation mein use karne se pehle.

KYUN. Ek circle par, gravity hi ek maatra inward pull hai, aur ise exactly woh centripetal force supply karni chahiye jo path ko circle mein bend karne ke liye chahiye. In dono ko balance karne se $n$ milta hai. Yeh Kepler's third law ka hi disguise mein roop hai.

PICTURE. Gravity target ko $\mu /R_0^2$ ke strength se andar kheenchti hai. Circular motion ko $n^2{R}_0$ ka inward pull chahiye. Inhe equal karo:

$$\underset{\text{gravity pull}}{\underbrace{\frac{\mu }{R_0^2}}}=\underset{\text{needed to circle}}{\underbrace{n^2{R}_0}}⟹n=\sqrt{\frac{\mu }{R_0^3}}.$$

Yahan $\mu$ planet ka gravitational parameter hai (mass $\times$ Newton's $G$) — ek akela number jo "is planet ki gravity kitni strong hai" ko package karta hai.

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Step 3 — Gravity, lekin sirf change (Taylor-expand KYUN?)

KAYA. Chaser par real gravity hai $\vec{g}=-\mu \vec{r}/r^3$, jahan $\vec{r}$ Earth se chaser ki taraf point karta hai. Woh $1/r^3$ nasty aur nonlinear hai. Hum ise ek straight-line approximation se replace karte hain jo chhote $(x,y,z)$ ke liye kaafi hai.

KYUN yeh tool — Taylor expansion? Humein sirf yeh jaanna hai ki gravity kitni alag hai target aur chaser ke beech jab woh ek kilometres-wide orbit par metres apart hain. Jab ek change tiny hota hai, toh honest kaam hai linearization: value plus uski pehli slope rakhna, baaki curved cheez phenkna. Yeh exactly yeh sawaal answer karta hai "jab main target se thoda hatoon toh gravity kitni tilt hoti hai?" — bas itna, kuch nahi.

PICTURE. Chaser $\vec{r}=(R_0+x)\hat{x}+y\hat{y}+z\hat{z}$ par baitha hai. Earth se uski doori hai

$$r=\sqrt{(R_0+x{)}^2+y^2+z^2}.$$

Chhote offsets ke liye, $r^{-3}\approx R_0^{-3}\left(1-\frac{3x}{R_0}\right)$ — $x$ (radial) term dominate karta hai kyunki upar/neeche jaane se tumhari altitude badlti hai, jabki sideways moves Earth se tumhari doori barely change karti hain. First order tak multiply karne par:

$$g_x\approx \underset{\text{baseline pull}}{\underbrace{-\frac{\mu }{R_0^2}}}+\underset{\text{tidal stretch}}{\underbrace{\frac{2\mu }{R_0^3}x}},g_y\approx \underset{\text{sideways squeeze}}{\underbrace{-\frac{\mu }{R_0^3}y}},g_z\approx \underset{\text{sideways squeeze}}{\underbrace{-\frac{\mu }{R_0^3}z}}.$$

Arrows padho: $x$ ke along gravity tumhe stretch karti hai (positive coefficient — upar jao, gravity weak hoti hai, tum aur upar drift karte ho); $y$ aur $z$ ke along yeh tumhe orbital line ki taraf pinch karti hai.

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Step 4 — Do fake forces (Ek moving camera unhe KYUN invent karta hai)

KAYA. Kyunki hamara camera spin karta hai, Newton's law mein do extra apparent forces aa jaati hain: Coriolis force aur centrifugal force. Hum inhe chaser par compute karte hain.

KYUN yeh tool — cross product $\times$? Rotation sideways act karta hai: camera spin karo aur ek moving object apni velocity ke right angle mein curve karta hua lagta hai. Cross product $\vec{a}\times \vec{b}$ woh machine hai jo dono ke perpendicular ek vector produce karta hai — exactly woh sideways kick jo rotation deliver karta hai. Frame $z$ ke around spin karta hai, toh $\vec{\omega }=n\hat{z}$.

PICTURE — Coriolis. Yeh sirf un cheezon par act karta hai jo frame mein move kar rahi hain:

$$-2\vec{\omega }\times \dot{\vec{r}}=-2n\hat{z}\times (\dot{x},\dot{y},\dot{z})=(\underset{\text{along-track speed kicks radial}}{\underbrace{+2n\dot{y}}},\underset{\text{radial speed kicks along-track}}{\underbrace{-2n\dot{x}}},0).$$

Note karo: forward move karna ($\dot{y}$) tumhe upar dhakelta hai ($x$); upar move karna ($\dot{x}$) tumhe backward dhakelta hai ($-y$). Yeh right-angle swap hi poori wajah hai ki "target ki taraf burn karo" ulta pad jaata hai.

PICTURE — Centrifugal. Yeh position par act karta hai, tumhe spin plane mein bahar phenk kar:

$$-\vec{\omega }\times (\vec{\omega }\times \vec{r})=n^2(x,y,0).$$

Yeh dono fictitious forces poori tarah Rotating Reference Frames — Coriolis and Centrifugal mein explain ki gayi hain.

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Step 5 — The great cancellation (Target KYUN hover karta hai)

KAYA. Teeno contributions jodo — real gravity (Step 3), Coriolis (Step 4), centrifugal (Step 4) — component by component. Step 2 use karke har $\mu /R_0^3$ ko $n^2$ se swap karo.

KYUN. Target ko origin par still baithna chahiye. Yeh tabhi ho sakta hai jab uس par leftover constant force zero ho. Dekho constants kaise takraate hain.

PICTURE — $x$ (radial) equation. $x$ ke along act karne wala har piece line up karo:

$$\ddot{x}=\underset{\text{gravity baseline}}{\underbrace{-n^2{R}_0}}\underset{\text{tidal}}{\underbrace{+2n^{2}x}}\underset{\text{Coriolis}}{\underbrace{+2n\dot{y}}}\underset{\text{centrifugal}}{\underbrace{+n^{2}x}}\underset{\text{centrifugal on }{R}_0}{\underbrace{+n^2{R}_0}}.$$

Do bade constants $-n^2{R}_0$ aur $+n^2{R}_0$ annihilate ho jaate hain — target, $x=0$ par baitha, kuch bhi feel nahi karta. Yeh hai hover. Baaki collect karne par:

$$\boxed{\ddot{x}-2n\dot{y}-3n^{2}x=0}(2n^{2}x+n^{2}x=3n^{2}x).$$

PICTURE — $y$ (along-track) equation. Gravity $-n^{2}y$ deti hai, centrifugal $+n^{2}y$ deta hai — woh bhi cancel ho jaate hain — aur Coriolis $-2n\dot{x}$ chhodta hai:

$$\boxed{\ddot{y}+2n\dot{x}=0}.$$

PICTURE — $z$ (cross-track) equation. Koi centrifugal nahi (woh $x,y$ plane mein rehta hai), koi Coriolis coupling nahi, bas pinch $-n^{2}z$:

$$\boxed{\ddot{z}+n^{2}z=0}.$$

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Step 6 — Solution ka drift padhna (Coasting KYUN fail karta hai)

KAYA. Boxed pair ko solve karne par (ek baar $y$-equation integrate karo, use $x$ mein daalo, driven oscillator solve karo) neeche diye closed forms milte hain. Hum unhe yahan sirf padhte hain — algebra parent note mein hai.

$$x(t)=(4-3\cos nt)x_0+\frac{\sin nt}{n}{\dot{x}}_0+\frac{2}{n}(1-\cos nt){\dot{y}}_0,$$ $$y(t)=\underset{\text{grows forever}}{\underbrace{6(\sin nt-nt)x_0}}+y_0-\frac{2}{n}(1-\cos nt){\dot{x}}_0+\underset{\text{grows forever}}{\underbrace{\frac{1}{n}(4\sin nt-3nt){\dot{y}}_0}}.$$

KYUN yeh matter karta hai. Do terms mein ek bare $nt$ hai jo kabhi rukta nahi — secular drift. Ek chhota radial offset $x_0$ ya forward speed ${\dot{y}}_0$ tumhe along-track bina ruke march kara deta hai.

PICTURE. $x$–$y$ plane mein chaser ka path plot karo: galat initial velocity ke saath woh spiral away karta hai (drift); magic choice ${\dot{y}}_0=-2n{x}_0$ se $nt$ terms vanish ho jaate hain aur path ek sahi 2:1 ellipse mein close ho jaata hai — woh famous "football."

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Step 7 — Degenerate & edge cases (Taki kuch surprise na kare)

KAYA. Har special starting condition, draw ki gayi, taki tumhe koi unseen scenario na mile.

• $x_0=0,{\dot{y}}_0=0$ (pure radial kick ${\dot{x}}_0=v_r$): dono drift terms vanish → bounded ellipse $x=\frac{v_r}{n}\sin nt,y=\frac{2v_r}{n}(\cos nt-1)$.
• $x_0=0$, sab velocities zero, $y_0\ne 0$ (V-bar loiter): har trig term mar jaata hai, drift drive karne ke liye koi $x_0$ nahi → chaser bas $y=y_0$ par baitha rehta hai. Isi liye docking se pehle along-track axis par park karte hain.
• $x_0<0$ (lower orbit mein drop karo): $-6nt{x}_0$ term ab positive hai → tum forward creep karte ho, catch up karte ho. Ek poore period mein ($nT=2\pi$) oscillatory part zero par wapas aa jaata hai, ek net $\Delta y=-12\pi x_0$ chhodta hai.
• $z$ akele: pure SHM, sab se independent, $z(t)=z_0\cos nt+\frac{{\dot{z}}_0}{n}\sin nt$ — yeh kabhi drift nahi kar sakta, sirf oscillate karta hai.

KYUN. Har case sirf "kaunse terms bachte hain jab yeh initials zero hain" hai. Survivors padhna har baar ODE dobara solve karne se fast hai.

PICTURE. Chaar chhoti trajectories side by side: football, parked dot, forward-creeping spiral, aur flat cross-track sine.

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Ek-picture summary

Yeh single frame poori story stack karta hai: Earth aur spinning frame (Step 1–2), teeno forces chaser par milte hue dono constants cancel hote hue (Step 3–5), aur resulting football-or-drift path (Step 6–7). Ise left se right trace karo aur tumne CW equations ek blank page se dobara derive kar li hain.

State-marching machinery jo $x(t),y(t),z(t)$ ko ek single matrix mein package karti hai woh State Transition Matrix mein hai; eccentric-orbit generalisation Tschauner–Hempel Equations hai; aur underlying orbital dynamics Two-Body Problem aur Orbital Maneuvers — Hohmann Transfer se aate hain.

Recall Feynman retelling — plain words mein wapas bolo

Humne ek camera target par bithaaya aur ise spin kiya taki target centre par freeze ho jaaye. Kyunki camera spin karta hai, do make-believe forces appear hoti hain: Coriolis (jab tum move karte ho tumhe sideways kick karti hai) aur centrifugal (tumhe bahar phenk ti hai). Humne real gravity likhi lekin sirf uski tiny difference dono ships ke beech rakhi. Gravity plus do fake forces jodne par, do giant constant pushes cancel ho gaye — isi liye target bas hover karta rehta hai. Jo bachta hai woh teen neat equations hain: upar/neeche aur aage/peechhe wali Coriolis se lock hain, aur sideways wali ek plain spring hai. Unhe solve karne par, ek hidden term time ke saath grow karta hai: ek chhota altitude offset tumhe endlessly along-track le jaata hai. Yeh trap bhi hai ("sirf coast mat karo") aur tool bhi ("catch up karne ke liye neeche drop karo"). Us term ko ${\dot{y}}_0=-2n{x}_0$ se kill karo aur tumhara path 2:1 football mein close ho jaata hai.

Recall Quick self-test

Target origin par kyun hover karta hai? ::: Constant gravity term $-n^2{R}_0$ aur centrifugal term $+n^2{R}_0$ exactly cancel ho jaate hain. Kaun sa tool $1/r^3$ gravity ko linear terms mein convert karta hai, aur kyun? ::: Taylor expansion / linearization — kyunki humein sirf gravity ka chhota change ek tiny separation par chahiye. Ek forward ($+y$) burn tumhe peechhe kyun kar deta hai? ::: Coriolis radial/along-track velocities ko ek dusre mein turn karta hai; ek forward push orbit raise karta hai, tumhe slow karta hai. Drift kill karne ke liye initial velocity ka kaun sa choice hai? ::: ${\dot{y}}_0=-2n{x}_0$. Ek orbit mein radial offset $x_0$ se net along-track shift? ::: $\Delta y=-12\pi x_0$.