The Two-body Equation of Motion
Newton’s Laws gives us:
GM Earth
accel satellite  
2
R
• The solution is an orbit described by a conic section (circle,
ellipse, parabola, or hyperbola) that is fixed in space
• The satellite will trade kinetic energy for potential energy
(speed for altitude) as it moves around in orbit
• We need six initial conditions to solve this equation
– The six Classical Orbital Elements (a, e, i, W, w, n) are often used to
visualize the location and motion of the satellite
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 0
Classical Orbital Elements
Orbit Size:
Semi-major Axis
a
Orbit Shape:
Eccentricity
e
Orbit Tilt:
Inclination
i
Orbit Twist:
Right Ascension of
the Ascending Node
Ω
Orbit Rotation:
Argument of Perigee
ω
Satellite Location:
True Anomaly
n
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 1
Size: Semi-Major Axis (a)
• How big is an orbit? We measure the length of the longest
side of the ellipse and, by convention, divide it in half
US: Fig. 5-2
• Orbit size depends on how fast we “throw” our satellite into
orbit
– The faster we throw it, the more energy its orbit has and the bigger
its orbit is
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 2
Shape: Eccentricity (e)
e = .8
e = .5
e = .7
US, Fig. 5.3
e 
e = 0 (circular)
c
a
Circle
e = 0.0
Ellipse
e = 0.0 to 1.0
Parabola
e = 1.0
Hyperbola
e > 1.0
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 3
Tilt: Inclination (i)
Inclination
K
h
i
Angular momentum
vector
Equatorial Plane
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 4
Tilt: Inclination (i)
Inclination, i
Orbit Type
0o or 1800
Equatorial
90o
Polar
0o  i  900
Direct or prograde
(satellite moves in same
direction as Earth’s rotation)
90o  i  1800
Indirect or retrograde
(satellite moves in opposite
direction of Earth’s rotation)
US: Table 5-2
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 5
Twist: Right Ascension of the Ascending Node (W)
We measure how an orbit is twisted by locating its ascending node
relative to the vernal equinox direction (in the equatorial plane)
Vernal Equinox Direction
(Originally pointed to the constellation Aries, the Ram)
Equatorial Plane
Ascending Node
W
Right Ascension of the Ascending Node
(Also called the Longitude of the Ascending Node)
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 6
Rotation: Argument of Perigee (w)
We locate perigee relative to the ascending node
(in the orbit plane)
Argument of Perigee
w
Equatorial Plane
Ascending Node
Perigee
(Point Closest to the Earth)
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 7
Satellite Location: True Anomaly (n)
Finally, we locate the satellite relative to perigee,
(in the orbit plane)
Equatorial Plane
n
True Anomaly
Perigee
(Point Closest to the Earth)
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 8
Classical Orbital Elements
w
2a
K
i
h
e = .8
n
W
Vernal Equinox
Direction
Ascending Node
Perigee
SP200, Block III, 1 Dec 05, Orbits and Trajectories
UNCLASSIFIED
Slide 9