Intuition The one core ideaAn orbit is a single fixed ellipse floating in space, and there are two complete ways to name it: six numbers that say "here I am, moving this fast" or six numbers that say "this is the ellipse's size, shape, tilt, and where I sit on it." Converting between them is just re-describing that one ellipse — so before we can convert, we must earn every symbol that either language uses. (The formal names for those two schemes — state vectors and orbital elements — are defined in sections 3 and 8 once their pieces exist.)
This page defines every symbol the parent note Converting between orbital elements and state vectors (r, v) leans on, in an order where each idea rests on the one before it. If a symbol below feels obvious, read it anyway — the conversion recipes break the instant one meaning is fuzzy.
Everything happens around one heavy object (Earth, the Sun). We treat it as a fixed point at the origin, and a tiny body (a satellite) moves around it.
Definition The central body and the orbiting body
The central body sits at the origin — the "pin" the orbit is anchored to.
The orbiting body is the moving dot whose position and velocity we track.
We assume the orbiting body is so light it does not tug the central body — this is the setup of the two-body problem .
Before "position vector" means anything, "vector" must.
a vector is an arrow : it has a length (how long) and a direction (which way it points). We write it with a little arrow, a
We will soon lay down three perpendicular measuring sticks (section 2). Once those exist, a vector can also be written as three numbers ( a x , a y , a z )
Definition Magnitude of a vector — written∣ a ∣ magnitude is just the length of the arrow, a single positive number. Once we have the three components (next section) it is computed as
∣ a ∣ = a x 2 + a y 2 + a z 2 The picture: stretch the arrow out straight and measure it with a ruler. Why the topic needs it: r = ∣ r ∣ v = ∣ v ∣
Common mistake The plain-letter naming convention (and one clash to watch)By convention we often drop the arrow and bars and write the magnitude as the plain letter : r = ∣ r ∣ v = ∣ v ∣ h = ∣ h ∣ n = ∣ n ∣ e = ∣ e ∣ r r One clash to keep straight: the plain letter a not used for the length of some vector a a semi-major axis (section 9). Whenever you meet a generic example arrow ; its length we always write in full as ∣ a ∣ a
Definition Reference frame and unit vectorsx ^ , y ^ , z ^ reference frame is three perpendicular measuring sticks meeting at the origin. The little hats mean unit vectors — arrows of length exactly 1 pointing along each stick. z ^
a x , a y , a z a three numbers : a x x ^ a y y ^ a z z ^ components . This is exactly why the magnitude formula ∣ a ∣ = a x 2 + a y 2 + a z 2
Why the topic needs it: the tilt i from the plane spanned by x ^ , y ^ z ^ n = z ^ × h ECI frame; a second, orbit-hugging frame (PQW) appears later — see Reference frames — perifocal (PQW) vs ECI .
Definition Position vectorr position vector is the arrow from the origin to the satellite right now . Its length r
Definition Velocity vectorv velocity vector is the arrow showing which way the satellite is moving and how fast — it is tangent to the path. Its length v
Together, ( r , v ) state vectors — this is the first of the two naming schemes the intro promised.
Intuition Why "state" = 6 numbersr v six numbers completely pin down the motion: give me where you are and how you move, and Newton's law tells me your entire future orbit. This is the "sensor language" the parent note describes.
We need to talk about "the angle between two arrows" before the dot product, because the dot product's meaning is built on it.
θ angle θ how much you turn to go from one direction to another — from 0 9 0 ∘ 18 0 ∘ 36 0 ∘
Definition Cosine and sine on a right triangleOn a right triangle with an angle θ cos θ = hypotenuse adjacent sin θ = hypotenuse opposite angle into a ratio of side lengths — the bridge between "how much turn" and "which numbers." Two facts we will lean on: cos 0 = 1 cos 9 0 ∘ = 0 cos 18 0 ∘ = − 1
cos − 1 cos − 1 ( x ) "which angle has this cosine?" — it undoes cosine. Crucial limit: it only ever answers between 0 18 0 ∘ sign check to each angle (spelled out in section 10): cosine alone cannot tell ν = 6 0 ∘ ν = 30 0 ∘
The conversions live and die on two ways to combine two arrows. Both use the angle θ
a ⋅ b dot product multiplies two arrows into a single number :
a ⋅ b = a x b x + a y b y + a z b z = ∣ a ∣ ∣ b ∣ cos θ θ Why this tool? It answers "how much do these two arrows point the same way?" Because cos θ positive for angles under 9 0 ∘ zero at 9 0 ∘ negative past 9 0 ∘ r ⋅ v sign tells us whether the satellite is moving away from the central body (positive) or toward it (negative) — the key that unlocks the true-anomaly quadrant.
a × b cross product multiplies two arrows into a new arrow perpendicular to both :
a × b = ( a y b z − a z b y , a z b x − a x b z , a x b y − a y b x ) area of the parallelogram the two arrows make. Why this tool? It answers "what direction is perpendicular to this plane?" Point right-hand fingers from a b a × b
Why the topic needs it: h = r × v manufactures the axis the orbit spins about . That is how one operation fixes both the tilt i Ω
Definition Specific angular momentumh h = r × v perpendicular to the entire orbit plane . It is called specific because it is per unit mass (no mass in the formula).
h h twice the rate at which the arrow r . A satellite that sweeps area at a constant rate has constant h h Angular momentum conservation in orbits .
Why the topic needs it: h conserved (constant), so it is a rock-solid signpost. Its direction fixes the orbit plane; its magnitude sets the parameter p = h 2 / μ
μ = GM G M μ how strongly this central body pulls . For Earth μ = 398600 km 3 / s 2 Why bundle them? We can measure μ G M
Definition Specific orbital energyε ε = 2 v 2 − r μ kinetic ("energy of motion"), the second is gravitational potential ("energy of position"). Their sum per unit mass is ε constant along the whole orbit .
Why the topic needs it: energy fixes the size through a = − μ / ( 2 ε ) ε vis-viva relation.
The sign of ε
Now we meet the geometry itself. Follow figure s04 as you read — it labels every quantity below on one drawing.
Definition Semi-major axisa longest diameter — the "size" number, written with the plain letter a 2 a a
e how squished the ellipse is. e = 0 e → 1 0 ≤ e < 1
Definition Periapsis, apoapsis, andp
Periapsis is the point of closest approach to the central body; distance a ( 1 − e ) Apoapsis is the farthest point; distance a ( 1 + e ) The semi-latus rectum p = a ( 1 − e 2 ) = h 2 / μ a e
ν ν angle at the central body, measured from periapsis to the body's current position . At periapsis ν = 0 ν = 18 0 ∘
Two more arrows must be built before the orientation angles have anything to grab onto.
n ascending node is where the orbit crosses the reference plane going upward . The node vector n = z ^ × h points from the origin toward that crossing . Its length is n = ∣ n ∣
i tilt of the orbit plane away from the reference plane, measured between z ^ h cos i = h z / h i 0 18 0 ∘ cos − 1 no sign check needed .
Definition Right ascension of the ascending nodeΩ swivel angle of the node line around z ^ x ^ cos Ω = n x / n Quadrant check: if n y < 0 Ω = 36 0 ∘ − Ω y n
Definition Argument of periapsisω from the ascending node (n e . cos ω = ( n ⋅ e ) / ( n e ) Quadrant check — do it right: the correct companion test is the sign of ( n × e ) ⋅ h negative , then ω = 36 0 ∘ − ω Why this and not "e z < 0 We need to know whether periapsis lies on the upward-swung side of the node within the orbit plane , and h z ^ e z for prograde orbits with the reference plane as equator is "if e z < 0 ω = 36 0 ∘ − ω
Definition True-anomaly quadrant checkFor ν cos ν = ( e ⋅ r ) / ( e r ) Quadrant check: if r ⋅ v < 0 ν = 36 0 ∘ − ν r ⋅ v toward periapsis, so it is on the "return" half.
Definition Orbital elementsThe set ( a , e , i , Ω , ω , ν ) orbital elements — the second naming scheme the intro promised. Only ν
The definitions in section 10 quietly assume the arrows n e undefined — this is not a bug, it is geometry running out of landmarks.
Common mistake Circular orbit:e = 0 no periapsis — every point is equally close. So e = 0 ω ν Fix: replace them with the argument of latitude u
Common mistake Equatorial orbit:i = 0 ∘ i = 18 0 ∘ never crosses that plane, so there is no ascending node. Then h z ^ n = z ^ × h = 0 Ω ω Fix: use the true longitude of periapsis ϖ x ^
Common mistake Both at once:e = 0 i = 0 neither a node nor a periapsis, so Ω ω ν a e i true longitude ℓ x ^ The systematic cure for all these gaps is a re-parameterisation with no singularities: equinoctial elements .
position r and velocity v
sign checks for quadrants
Convert elements and state vectors
Definition A rotation matrixA rotation matrix is a grid of numbers that, when multiplied onto a vector, spins that arrow by a fixed angle without changing its length. R 3 ( θ ) z ^ R 1 ( θ ) x ^ c = cos θ s = sin θ R 3 ( θ ) = c s 0 − s c 0 0 0 1 , R 1 ( θ ) = 1 0 0 0 c s 0 − s c
Why the topic needs it: Direction B first builds r , v R 3 ( − Ω ) R 1 ( − i ) R 3 ( − ω ) R 3 z x y R 1 x y z time you additionally need Kepler's equation and time-of-flight .
Cover each answer and test yourself — if any is shaky, re-read its section before the parent note.
What is the difference between r r r is the arrow (position, 3 numbers);
r = ∣ r ∣ is just its length (distance), one positive number.
Why is the plain letter a not a vector magnitude on this page? Because
a is reserved for the semi-major axis; a generic arrow's length is always written
∣ a ∣ , never bare
a .
What is the dot product formula in terms of θ cos θ a ⋅ b = ∣ a ∣∣ b ∣ cos θ ;
cos θ measures how much the two arrows point the same way (1 aligned, 0 perpendicular, −1 opposite).
What does the sign of r ⋅ v Positive → moving away from the central body (ν < 18 0 ∘ toward it (ν > 18 0 ∘
What does the cross product r × v A new arrow (
h ) perpendicular to the orbit plane; its length is twice the areal sweep rate.
Why is h It is constant and perpendicular to the plane, so it fixes both inclination i Ω
What does the eccentricity vector e It points toward periapsis and its length is exactly the eccentricity,
∣ e ∣ = e ; its formula is
e = μ 1 [( v 2 − μ / r ) r − ( r ⋅ v ) v ] .
What is the correct quadrant test for ω The sign of
( n × e ) ⋅ h ; if negative,
ω = 36 0 ∘ − ω (the
e z < 0 shortcut only works for equatorial-referenced prograde orbits).
What does specific energy ε The orbit size: a = − μ / ( 2 ε )
What is μ G M μ = GM G M
Define a e p a e p = a ( 1 − e 2 ) r = p / ( 1 + e cos ν )
Why does the orbit equation give smallest r ν = 0 Because cos 0 = 1 1 + e r = p / ( 1 + e ) = a ( 1 − e )
Which companion sign fixes each angle's quadrant? n y for
Ω ,
( n × e ) ⋅ h for
ω ,
r ⋅ v for
ν ;
i needs none.
When do ω Ω ω , ν e = 0 u Ω i = 0
Write R 3 ( θ ) R 1 ( θ ) R 3 [ c , − s , 0 ; s , c , 0 ; 0 , 0 , 1 ] z R 1 [ 1 , 0 , 0 ; 0 , c , − s ; 0 , s , c ] x
