How to find the relationship between the eccentricity and the flight path angle?
- Saturday Oct 31,2009 01:57 AM
- By diddy
- In Others
A satellite is injected into an orbit at radius r from the center of attraction. The velocity is exactly the desired speed to go into circular orbit but the flight path angle (gamma) does not equal zero. The flight path angle is the angle between the velocity vector and the plane normal to the position vector. How do a develop the relationship between the eccentricity and sin(gamma).
The answer is in terms of cos(gamma). How do you determine it for sin(gamma)?
Center Of Attraction, Circular Orbit, Eccentricity, Flight Path, Gamma, Position Vector, Radius, Relationship, Satellite, Sin, Velocity Vector
One Comment
State vector.
x = r
y = 0
z = 0
Vx = V sin ?
Vy = V cos ?
Vz = 0
Earth’s gravitational parameter.
GM = 3.986004418e14 m³ sec?²
semimajor axis
a = 1 / { 2/r ? V²/(GM) }
angular momentum per unit mass
hx = y Vz ? z Vy = 0
hy = z Vx ? x Vz = 0
hz = x Vy ? y Vx = rV cos ?
h² = (hx)² + (hy)² + (hz)² = r²V² cos²?
eccentricity
e = sqrt[ 1 ? h² / (GMa) ]
e = sqrt{ 1 ? cos²? [ 2rV²/(GM) ? r²V?/(GM)² ] }
Q = rV²/(GM)
e = sqrt{ 1 ? cos²? [ 2Q ? Q² ] }
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