Astrodynamics 101 - DerAstrodynamics.com

Newton / Classical Formulation
Equations of motion
2
d r
= r a
2 3 d t r
Given : r(t 0), v(t
0), t 0 , t
Find : r(t ) , v(t
)
Kepler method:
Assume zero perturbation:
,
a 0
d =
resulting in spherical
gravity effect.
Kepler analytic solution:
r fI =
v
fI gI
gI
r0
v
where f , g, f , and g are
analytic and functions of
r r(t ) and v v(t
)
d
0
0 0 0 0
UNCLASSIFIED
Comparison of Analytic Solutions
Vinti / Hamilton-Jacobi Formulation
Equations of motion
dpk H(q, p, t ) dqk
H(q,
p, t )
,
d t qk d t pk
where q's and p's are respectively
coordinates and momenta, and k 1,
2, 3
Given : r(t 0), v(t
0), t 0 , t
Find : r(t ) , v(t
)
Vinti method:
Assume non-zero perturbation:
,
a 0
d
resulting in spheroidal gravity effect so that
zonal geopotentials J , J , J are included.
2 3 4
Vinti analytic solution:
See Vinti AIAA book, V177, 1998,
"Orbital and Celestial Mechanics",
Chapter 8.
Also see GTDS, Chapter 5.12 on
attributes of "Vinti Theory", singularity free
and applicability to satellites and missiles.