Converting between orbital elements and state vectors (r, v)

The six orbital elements (WHAT each means)

Foundations we DERIVE everything from

Two conserved quantities do all the heavy lifting.

WHY these matter: $\vec h$ is perpendicular to the orbit plane, so it fixes $i$ and $\Omega$ . Energy $\varepsilon$ fixes the size $a$ . The eccentricity vector (below) points to periapsis, fixing $e$ and $\omega$ .

Formula Eccentricity vector (derived, not memorised)

Start from $\ddot{\vec r} = -\dfrac{\mu}{r^3}\vec r$ . Cross with $\vec h$ (constant): $\ddot{\vec r}\times\vec h = -\frac{\mu}{r^3}\,\vec r\times(\vec r\times\vec v)$ Using $\vec r\times(\vec r\times\vec v)=\vec r(\vec r\cdot\vec v)-\vec v r^2$ and $\dfrac{d}{dt}\!\left(\dfrac{\vec r}{r}\right)=\dfrac{\vec v r^2-\vec r(\vec r\cdot\vec v)}{r^3}$ , the left side is $\dfrac{d}{dt}(\dot{\vec r}\times\vec h)$ and the right is $\mu\dfrac{d}{dt}(\vec r/r)$ . Integrating: $\vec e = \frac{\vec v\times\vec h}{\mu} - \frac{\vec r}{r}$ Its magnitude is the scalar $e$ , and it points toward periapsis.

Direction A: State vectors → Orbital elements

Formula Step-by-step recipe (each step WHY'd)

Given $\vec r,\vec v,\mu$ . Let $r=|\vec r|$ , $v=|\vec v|$ .

$\vec h=\vec r\times\vec v$ , $h=|\vec h|$ . — fixes the plane.
$\vec n=\hat z\times\vec h$ , $n=|\vec n|$ . — node vector, points to ascending node.
$\displaystyle \vec e=\frac{1}{\mu}\!\left[(v^2-\tfrac{\mu}{r})\vec r-(\vec r\cdot\vec v)\vec v\right]$ , $e=|\vec e|$ . — shape + periapsis direction.
$\varepsilon=\dfrac{v^2}{2}-\dfrac{\mu}{r}\ \Rightarrow\ a=-\dfrac{\mu}{2\varepsilon}$ . — energy sets size.
$\cos i=\dfrac{h_z}{h}$ . — tilt of $\vec h$ from $\hat z$ .
$\cos\Omega=\dfrac{n_x}{n}$ (quadrant: if $n_y<0$ , $\Omega=2\pi-\Omega$ ).
$\cos\omega=\dfrac{\vec n\cdot\vec e}{n\,e}$ (if $e_z<0$ , $\omega=2\pi-\omega$ ).
$\cos\nu=\dfrac{\vec e\cdot\vec r}{e\,r}$ (if $\vec r\cdot\vec v<0$ , $\nu=2\pi-\nu$ ).

WHY the quadrant checks? $\cos^{-1}$ only returns $[0,\pi]$ . The sign of a companion quantity (e.g. $\vec r\cdot\vec v$ : positive means moving away from periapsis, so $\nu<\pi$ ) restores the full $[0,2\pi)$ .

Direction B: Orbital elements → State vectors

Figure — Converting between orbital elements and state vectors (r, v)

Worked example 1 — State → Elements

Given (units: km, km/s, $\mu=398600\ \text{km}^3/\text{s}^2$ ): $\vec r=(7000,0,0)$ , $\vec v=(0,7.5,1.0)$ .

$\vec h=\vec r\times\vec v=(0\cdot1-0\cdot7.5,\ 0\cdot0-7000\cdot1,\ 7000\cdot7.5-0)=(0,-7000,52500)$ . Why: cross product fixes the plane; $h=52964.6$ .
$r=7000$ , $v=\sqrt{7.5^2+1^2}=7.566$ . $\varepsilon=\frac{7.566^2}{2}-\frac{398600}{7000}=-28.31$ . Why: energy → $a=-\mu/2\varepsilon=7040$ km.
$\cos i=h_z/h=52500/52964.6=0.9912\Rightarrow i=7.60^\circ$ . Why: small $v_z$ gives a slight tilt.
$\vec e=\frac1\mu[(v^2-\mu/r)\vec r-(\vec r\cdot\vec v)\vec v]$ ; here $\vec r\cdot\vec v=0$ , so $\vec e=\frac{v^2-\mu/r}{\mu}\vec r=(0.00579)(7000,0,0)$ ... giving $e\approx0.0057$ . Why: $\vec r\cdot\vec v=0$ means we're at periapsis/apoapsis, so $\nu=0$ or $180^\circ$ ; since orbit is nearly circular here it's tiny $e$ .

Worked example 2 — Elements → State

Given $a=8000$ km, $e=0.1$ , $i=30^\circ$ , $\Omega=40^\circ$ , $\omega=60^\circ$ , $\nu=0^\circ$ (at periapsis).

$p=a(1-e^2)=8000(0.99)=7920$ km.
$r=p/(1+e\cos0)=7920/1.1=7200$ km. Why: periapsis distance $=a(1-e)=7200$ . ✓
$\vec r_{pqw}=7200(\cos0,\sin0,0)=(7200,0,0)$ .
$\vec v_{pqw}=\sqrt{\mu/p}(-\sin0,\ e+\cos0,0)=7.094(0,1.1,0)=(0,7.803,0)$ . Why: at periapsis velocity is purely tangential — no radial component, matching $-\sin\nu=0$ .
Apply $\tilde R$ with those angles to get $\vec r,\vec v$ in inertial frame.

Common mistake Steel-manning the classic errors

1. Forgetting quadrant checks. Feels right because $\cos^{-1}$ "gives the angle." Fix: always check the companion sign ( $n_y$ for $\Omega$ , $e_z$ for $\omega$ , $\vec r\cdot\vec v$ for $\nu$ ).

2. Using $R_3(+\omega)$ instead of $R_3(-\omega)$ . Feels right because you "rotate by $\omega$ ." Fix: we transform from perifocal to inertial, which is the inverse of the frame rotation → negative angles.

3. Circular/equatorial singularities. When $e=0$ , $\omega$ undefined; when $i=0$ , $\Omega$ undefined (node vector $\to 0$ ). Feels like a bug. Fix: use alternate elements (argument of latitude $u=\omega+\nu$ , or true longitude).

4. Wrong $a$ sign for hyperbolas. Forgetting $\varepsilon>0\Rightarrow a<0$ . Fix: $a=-\mu/2\varepsilon$ handles all conics automatically.

Recall Feynman: explain to a 12-year-old

Imagine a race car on an oval track. One way to describe it: "it's here on the map going that fast" (state vector). Another way: "the track is this big, this squished, tilted this much, and the car is this far past the start line" (orbital elements). Both tell you everything about the car. To translate, you use two facts that never change: how fast the car sweeps area (angular momentum) and how much total energy it has. From those you rebuild the whole track and the car's spot — then just spin your map to line it up with north.

Flashcards

What two conserved vectors/scalars drive the state→element conversion?

Specific angular momentum

\vec h=\vec r\times\vec v

(fixes plane,

i

\Omega

) and specific energy

\varepsilon=v^2/2-\mu/r

(fixes

a

Formula for semi-major axis from energy?

a=-\mu/(2\varepsilon)

where

\varepsilon=v^2/2-\mu/r

What does the eccentricity vector point toward?

Toward periapsis; its magnitude equals

e

\vec e=\frac{\vec v\times\vec h}{\mu}-\frac{\vec r}{r}

Why do we need quadrant checks after using

\cos^{-1}

\cos^{-1}

only returns

[0,\pi]

; a companion sign (e.g.

\vec r\cdot\vec v

for

\nu

) recovers the full

[0,2\pi)

Sign check for true anomaly

\nu

\vec r\cdot\vec v<0

(moving toward periapsis), then

\nu=2\pi-\cos^{-1}(\cdots)

Position vector in perifocal frame?

\vec r_{pqw}=r(\cos\nu,\sin\nu,0)

with

r=p/(1+e\cos\nu)

Velocity vector in perifocal frame?

\vec v_{pqw}=\sqrt{\mu/p}\,(-\sin\nu,\ e+\cos\nu,\ 0)

Rotation sequence from perifocal to inertial frame?

R_3(-\Omega)R_1(-i)R_3(-\omega)

(swivel Ω, tilt i, spin ω), applied to both

\vec r

and

\vec v

What is

p

(semi-latus rectum) in terms of

a,e

and of

h

p=a(1-e^2)=h^2/\mu

Which elements become undefined for a circular orbit (

e=0

\omega

(no periapsis); use argument of latitude

u=\omega+\nu

instead.

Connections

Two-body problem and the vis-viva equation
Angular momentum conservation in orbits
Kepler's equation and time-of-flight
Reference frames — perifocal (PQW) vs ECI
Orbital singularities and equinoctial elements
Hohmann transfers (needs state↔element conversions)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek orbit ko do tareeke se describe kar sakte ho. Pehla: state vectors $(\vec r,\vec v)$ — matlab satellite abhi kahan hai aur kitni tezi se move kar raha hai. Doosra: orbital elements $(a,e,i,\Omega,\omega,\nu)$ — jo orbit ki shape, tilt, aur position batate hain. Dono same ellipse ki alag-alag bhaasha hain. Sensor tumhe $\vec r,\vec v$ deta hai, par insaan ke samajhne ke liye elements zyada useful hain (circular hai kya? kitna tilted hai? perigee kahan hai?).

Conversion ka jaadu do conserved cheezon par tika hai: angular momentum $\vec h=\vec r\times\vec v$ (jo orbit plane ke perpendicular hota hai, isse $i$ aur $\Omega$ milta hai) aur energy $\varepsilon=v^2/2-\mu/r$ (jisse size $a=-\mu/2\varepsilon$ milta hai). Ek aur cheez — eccentricity vector — seedha periapsis ki taraf point karta hai, isse $e$ aur $\omega$ mil jaate hain.

Ulta direction (elements se state) mein pehle perifocal frame mein $\vec r,\vec v$ nikaalo — ye easy hai kyunki periapsis $\hat p$ axis par hota hai. Phir teen rotations $R_3(-\Omega)R_1(-i)R_3(-\omega)$ lagakar inertial frame mein le aao. Bas map ko north ke saath align karna, samjho.

Sabse important galti: $\cos^{-1}$ sirf $0$ se $180^\circ$ deta hai, isliye quadrant check zaroori hai — jaise $\vec r\cdot\vec v<0$ ho to $\nu$ ko $2\pi$ se subtract karo. Aur circular ya equatorial orbits mein $\omega$ / $\Omega$ undefined ho jaate hain, tab alternate elements use karo. Ye topic har mission planning, Hohmann transfer, aur tracking ka base hai.

Converting between orbital elements and state vectors (r, v)

The six orbital elements (WHAT each means)

Foundations we DERIVE everything from

Direction A: State vectors → Orbital elements

Direction B: Orbital elements → State vectors

Worked example 1 — State → Elements

Worked example 2 — Elements → State

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Orbital Mechanics & Astrodynamics

Connections