Dynamics cheat sheet

magnitude of ⃗r
period T
mean satellite speed n
r = a ( 1 − e 2)
1 + e cos θ = h2 1
µ 1 + e cos θ
r cos θ = a (cos E − e)
r = a (1 − e cos E)
√
T = 2 h πab = 2π a 3
µ
n = 2π
T = √
µ
a 3
eccentric anomaly E tan θ 2 = √
1+e
1−e tan E 2
area sweep rate
dA
dt = h 2
equation of motion ¨⃗r +
µ
r 3 ⃗r = 0
(eq 4.2-14 Bate book)
spherical coordinates relation cos(i) = sin(A z ) cos(φ) where i is the inclination
and A z is the azimuth and φ is latitude
2
Notice in the above, that the period T of satellite depends only on a (for same µ)
In the above, µ = GM where M is the mass of the body at the focus of the ellipse and G is the gravitational
constant. h is the specific mass angular momentum (moment of linear momentum) of the satellite. Hence the
units of h2
µ
is length.
To draw the locus of the satellite (the small body moving around the ellipse, all what we need is the
eccentricity e and a, the major axes length. Then by changing the angle θ the path of the satellite is drawn. I
have a demo on this here
See http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html for earth facts
This table below is from my class EMA 550 handouts (astrodynamics, spring 2014)
2 Image is from my class notes, page 7-9, EMA 550, Univ. Of Wisconsin, Madison by professor S. Sandrik
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