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Astrodynamics Quick Reference
Updated 6 Sep 2006 : Keric Hill
Two-body Equations
Orbital Period P =
2 π
n
P = 2π
a^3
μ
Mean Motion n =
μ/a^3
2 π
P
Specific Mechanical Energy ξ =
V 2
μ
r
ξ = −
μ
2 a
Semiparameter p =
h^2
μ
p = a(1 − e^2 ) p =
b^2
a
Angular Momentum ¯h = ¯r × V¯ h =
μp h = rV cos φf pa h = raVa h = rpVp h = r^2 V˙
Radial Rate r˙ =
r V e˙ sin ν
1 + e cos ν
Velocity Rate V˙ =
na 2
r^2
1 − e^2 V˙ =
μ(1 + e cos ν)^2
(1 − e^2 )^3 /^2
a^3
V^ ˙ = n
(1 + e cos ν) 2
(1 − e^2 )^3 /^2
Parameter Values
AU = 149 , 597 , 870 km
c = 299 , 792 , 458 km/s
G = 6. 67259 × 10 − 11 (N m 2 )/kg 2
R = 696 , 000. 000 km
R⊕ = 6 , 378. 1363 km
R$ = 1 , 738 km
R ♂ = 3397. 2 km
μ = 1. 32712428 × 10 11 km 3 /s 2
μ⊕ = 3. 986004415 × 10 5 km 3 /s 2
μ$ = 4902. 799 km 3 /s 2
μ♂ = 4. 305 × 10 4 km 3 /s 2
a⊕ = 149 , 598 , 023 km
a$ = 384 , 400 km
a ♂ = 227 , 939 , 186 km
Elliptical Orbit Equations
Eccentricity e =
c
a
e =
ra − rp
ra + rp
e =
ra
a
rp
a
e =
2 ξh^2
μ^2
e¯ =
(V 2 − μ/r) ¯r −
¯r · V¯
V¯
μ
e =
r 2 − r 1
r 1 cos ν 1 − r 2 cos ν 2
Flight path angle tan φf pa =
e sin ν
1 + e cos ν
sin φf pa =
e sin E √ 1 − e^2 cos^2 E
cos φf pa =
1 − e^2
1 − e^2 cos^2 E
Radius r =
a(1 − e^2 )
1 + e cos ν
r =
rp(1 + e)
1 + e cos ν
r =
p
1 + e cos ν
Apoapsis radius ra = a(1 + e) =
p
1 − e
ra = 2a − rp ra = rp
1 + e
1 − e
Periapsis radius rp = a(1 − e) =
p
1 + e
rp = 2a − ra rp = ra
1 − e
1 + e
Semimajor axis a =
ra + rp
2
a =
μr
2 μ − V 2 r
a =
rp
1 − e
ra
1 + e
a = −
μ
2 ξ
a =
μ
P
2 π
) 2 )^1 /^3
a =
μ
n^2
Time > periapsis t − tp =
E − e sin E
n
t − tp =
M
n
t − tp = (E − e sin E)
a 3
μ
Mean anomaly M = E − e sin E M = n(t − tp)
Ecc. anomaly cos E =
e + cos ν
1 + e cos ν
sin E =
sin ν
1 − e^2
1 + e cos ν
E = tan−^1
sin E
cos E
True anomaly cos ν =
p
re
e
cos ν =
rp(1 + e)
re
e
cos ν =
a(1 − e^2 )
re
e
cos ν =
cos E − e
1 − e cos E
sin ν =
sin E
1 − e^2
1 − e cos E
Velocity V =
2 μ
r
μ
a
V =
ξ +
μ
r
V =
μ
r
1 − e^2
1 + e cos ν
Vescape =
2 μ
r
rpVp = raVa Vcirc =
μ
r
