Dynamics cheat sheet

18.6 showing that energy is constant
Showing that energy E = v2
2 − µ r
is constant.
Most of such relations starts from the same place. The equation of motion of satellite under the assumption
that its mass is much smaller than the mass of the large body (say earth) it is rotating around. Hence we can
use ν = GM and the equation of motion reduces to
¨⃗r + µ r 3⃗r = 0
In the above equation, the vector ⃗r is the relative vector from the center of the earth to the center of the satellite.
The reason the center of earth is used as the origin of the inertial frame of reference is due to the assumption
that M ≫ m where M is the mass of earth (or the body at the focal of the ellipse) and m is the mass of the
satellite. Hence the median center of mass between the earth and the satellite is taken to be the center of earth.
This is an approximation, but a very good approximation.
The first step is to dot product the above equation with ˙⃗r giving
˙⃗r · µ
r 3 ⃗r = µ ṙ
r 2
˙⃗r · ¨⃗r + ˙⃗r · µ
r3⃗r = 0 (1)
( )
ṙ2
2
and we also see that
And there is the main trick. We look ahead and see that ˙⃗r · ¨⃗r = ṙ¨r but ṙ¨r = d dt
but µ ṙ = d ( −µ
)
r 2 dt r Hence equation 1 above can be written as
Hence
(
d v
2
dt 2 − r )
= 0
µ
E = v2
2 − r µ
Where E is a constant, which is the total energy of the satellite.
18.7 Earth satellite Transfer orbits
18.7.1 Hohmann transfer
This diagram shows the Hohmann transfer
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