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16.346 Astrodynamics
Fall 2008
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Lecture 16
Non-Singular ``Gauss-Like'' Method for the BVP
Derivation of the Time Equation
Start with the Lagrange time equation and the equation for the mean point radius
3
1√
2
2 µ(t2 − t1 ) = a (ψ − sin ψ cos φ)
r0 =
r0p =
FS =
a(1 − cos φ) = r0p (1 + tan2 12 ψ )
√
1 1
r1 r2 cos 12
θ
2 2 (r1 + r2 ) +
√
r1 r2 cos 12 θ
(1)
(2)
(3)
(4)
Eliminate cos φ and compare with the elementary form of Kepler’s equation
1 µ
r0
sin ψ
−
t
)
=
ψ
−
sin
ψ
+
(t
2
1
E ⇐⇒ ψ
2 a3
a
Identify
q ⇐⇒ r0
q
µ
(t − τ ) = E − sin E + sin E
3
a
a
t − τ ⇐⇒ 12 (t2 − t1 )
since t2 − t1 = (t2 − τ ) + (τ − t1 ) = 2(t2 − τ ).
Fig. 7.4 from An Introduction to the
Mathematics and Methods of
Astrodynamics. Courtesy of AIAA.
Used with permission.
From the classical relation between the true and eccentric anomalies:
tan2
1
2
f=
1+e
tan2
1−e
1
2
2a − q
tan2
q
E=
where we have defined
F S
cos f = 0 =
F0 P2
16.346 Astrodynamics
x = tan2
1
2
1
2
E
E
and
r1 r2 cos 21 θ
1
2 (r1 + r2 )
then
2 tan2 12 E
q
=
a
tan2 12 f + tan2
=⇒
= tan2
√
=
1
2
f .
1
2
E
Since
1 − cos f
F P − F0 S
= 0 2
1 + cos f
F0 P2 + F0 S
Lecture 16
=
2x
+x
Relation to Gauss’ Classical Method
The time equation
2 tan 12 E
q
q
1 µ
q3
−
t
)
×
=
E
−
sin
E
+
sin
E
=
E
−
sin
E
+
×
(t
1
a3
a
a 1 + tan2 12 E
2 q3 2
= sin E
becomes
4 tan3 21 E
1 µ
8
3 1
−
t
)
×
×
tan
E
=
E
−
sin
E
+
(t
1
2
2 q3 2
( + x)(1 + x)
( + x)3
Then, since
q = r0 = r0p (1 + tan2
1
2
E ) = r0p (1 + x)
we have an expression for the transfer time as a function only of E ≡ ψ :
4 tan3 12 E
4 tan3 12 E
µ
3 = E − sin E +
3 (t2 − t1 ) ×
8r0p
( + x)(1 + x)
[( + x)(1 + x)] 2
E − sin E
m
m3
=m
+
1
3
3
( + x)(1 + x)
[( + x)(1 + x)]
4 tan 2 E
Following Gauss, we can define y as
y2 =
m
( + x)(1 + x)
so that
where
µ(t2 − t1 )2
m=
3
8r0p
y3 − y2 = m
E − sin E
4 tan3 12 E
√
r1 + r2 − 2 r1 r2 cos
=
√
r1 + r2 + 2 r1 r2 cos
and
1
2
1
2
θ
θ
Note: The new y does not have the same geometric significance as Gauss’ y .
Parameter and Semimajor Axis
1
2x
2xy 2
=
=
mr0p
a
r0p (1 + x)( + x)
√
sin φ
c
c(1 + x)2
y 2 (1 + x)2
cy 2 (1 + x)2
p
=
=
=
=
=
pm
8ax
4mr0p
sin ψ
m
2a sin2 ψ
r0p (1 + x)
2x
q
=
=
a
+x
a
=⇒
where we have used the equation
c = 2a sin ψ sin φ
from Lecture 9, Page 1
16.346 Astrodynamics
Lecture 16
Universal Form
The equations are universal when x is extended to include the other conics:
⎧
⎨
tan2 14 (E2 − E1 )
x=
0
⎩
− tanh2 14 (H2 − H1 )
ellipse
parabola
hyperbola
Comparing the Structure of Gauss’ Method and the New Method
Gauss’ Method
D≡
√
r1 r2 cos 12 θ
√
r1 + r2 − 2 r1 r2 cos 12 θ
=
4D
2
µ(t2 − t1 )
m=
8D3
2 1
x ≡ sin 2 ψ
m
y2 =
+x
2ψ − sin 2ψ
y3 − y2 = m
sin3 ψ
p
cy 2
=
pm
4mD
2y 2 x
1
=
(1 − x)
a
mD
16.346 Astrodynamics
New Method
√
D ≡ 14 (r1 + r2 + 2 r1 r2 cos 12 θ)
√
r1 + r2 − 2 r1 r2 cos 12 θ
=
4D
µ(t2 − t1 )2
m=
8D3
x ≡ tan2 12 ψ
m
y2 =
( + x)(1 + x)
ψ − sin ψ
y3 − y2 = m
4 tan3 12 ψ
cy 2
p
=
(1 + x)2
pm
4mD
2y 2 x
1
=
a
mD
Lecture 16