a note of lambert´s theorem - r.devaney, and r. blanchard
TRANSCRIPT
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43-
66- 35
A
NOTE ON LAMBERT S THEOREM
bY
E.
R. Lancaster , C.
Blanchard'
and R.
A.
Devaneyf
February
1966
GODDARD
SPACE FLIGHT CENTER
Greenbelt, Maryland
* Aerospace Engineer
f Mathematician
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A
NOTE ON LAMBERT'S THEOREM
This note combines all
the
various cases of Lambert s Theorem into
a
single form which
is
particularly comenient for numerical work. This
is
made
possible by appro pria te choice of par ame ter and independent variable.
Suppose
a
particle in a gravitational central
force
field has distances r I
and
r , at
t imes t and
t ,
fro m the cente r of
attraction.
Let c be the distance
and 8 the cen tra l angle between the positions of t he pa rti cle at the two times.
Define
s
-
r l
+
r, + c ) / 2
K = 1
-
c / s
q
= + K H
The sign of q is taken care of by the angle 8
if
we make use of
c 2 =
r: + r 2 r 1 r 2 cos e
to derive
We fur the r define
G =
universal gravitational constant
M = ma ss of attracting body
=
GM
a =
semimajor
axis
of transfer orbit
E
= -s/2a
for elliptic transfer
= s / 2 a for hyperbolic transfer
T = (8p /S ) ( t , -
tl)/s
1
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m =
number of complete cir cu its during tr an sf er time.
Note
that 1 5 q 5
1,
0
< E <
fo r hyperbolic tra nsf er, -1
5 E
0 f ,I'
elliptic transfer, and
E
= 0 for parabolic transfer.
A l s o 0
5
H 5
7 i f 0
5 q
1.
1
and
71
<
0
5 2 n i f
-1
f
q < 0.
Lam bert's Theorem' fo r elliptic tra ns fe r gives
For
hyperbolic transfer,
T =
- y -
6 ( s i n h y s inh 6
E
=
s i n h 2
(y /2)
If
E
is chosen2 as the independent variable,
a
is ambiguous. We avoid any
ambiguity by choosing as the independent variable
x
=
cos ( a / 2 ) , l ( x < 1,
-
cosh (y /2 ) , x
1.
For both elliptic and hyperbolic transfer
E x 2
-
1.
2
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For the elliptic case let
y =
sin
2/2)
=
(-E)'
z
=
COS
pi21
=
1
.+
KE)
f = sin ( ) a - P ) = y(z
-
qx )
g =
cos
( ) a ,D l =
x z
-
qE
O l a P z 2 7 ~
ince
O c f z 1
h
= ( A) (sin a
sin p = y ( X 92
A
=
arctan f / g ) ,
0
5
A
5
I t
then follows for the elliptic case that
T
=
2(m77 + A
- h)/y3
For the hyperbolic case l et
y =
sinh
y / 2 )
=
EH
z = cosh
6/2)
= 1 +
KE)
f
= sinh
( A)
( y
- 6 = y ( z - qx)
g
=
cosh
34) ( y
-
6 )
=
x z - qE
h
=
( A)
(sinh
y
- sinh
6 ) =
y(x qz)
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Thus fo r the hyperbolic ca se
T =
2
[h
l n f
+ g ) l / y 3 4)
..
When
m
0 ,
equations
(l),
(2),
3)
and
(4)
break down fo r
x
=
1
and suffer
rom a cri tical l os s of signif icant digits in the neighborhood of x
remedy
t h i s
1) is written
1 . To
Replacing arcs in U and 1 -
u)
by series3,
@ ( u ) 4/3
+
C
a n u n ,
lul < 1
m
n = 1
a n
1
2n
...
A simila r procedure produces the same ser ie s fo r the hyperbolic case. In
fact
(5)
holds fo r the elliptic rn
=
0 )
,
parabolic, and hyperbolic cases provided
o < x < 2 .
It
is
now
apparent that, given
q
and
x
the following st eps produce
T
for
all cases:
1 . K
q2
2 .
E
=
x 2 - 1
4.
If p is near
0
compute T from
(5).
7 . f
=
y z
q x )
4
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9. I f
E < 0
A
=
arctan
f / g ) , d - m n
- A,
5 A 7-r
I f
E > 0 ,
d
= l n f
+ g)
The following formula for the derivative holds fo r
all cases
except for
x
=
Owith K = 1 andfor x
= 1 .
dT/dx = 4
4qKx/z - 3xT)/E
If x
is
near 1 , the series representation should be differentiated.
If
q =
1
we
have
a
left-hand derivative of 8 and
a
right-hand derivative
of 0 at
x
= 0 .
If q = -1
we have a left-hand derivative
of
0 and
a
right-hand derivative of -8
a t x
=
0. (See Figure
1.)
In the derivation of Lambert's Theorem for the elliptic case a and are
defined
in
such a
way
that
(5)
E, - E,
-
,8
i-
2m77
where
E
nd
E re
the values of the eccentric anomaly at times t and t .
Thus
from equation 1)
E, - E,
=
- E I 3 l 2T i-
s i n a
-
s i n p
=
y 3 T
i-
2 y x - q z ) .
We now obtain a formula fo r the sc al ar product
S, r , v , -
r l v l
sin
J1
\
of the position
and
velocity vectors at time
t v
nd
J1
being
the
speed
and
flight path angle.
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Kepler's equation can be written4
( p / a 3 ) t 2 - t l )
=
E, E , -I-
S
[l -
cos(E2
E,) l / (pa )
- 1
.-
r l / a )
s i n ( E 2
E l ) .
Substituting a = - s / 2E , t - t
=
s3 2 T / ( 8 p ) and making use of (5) and
6) we have, after some algebra,
A simi lar procedure produces the sa me formula fo r S, in the hyperbolic
case. It also holds fo r the parabolic case.
Figures
1
and
2
show
T
as
a
function of
x
fo r el liptic and hyperbolic
tran sfe r, the parabolic case occurring for x
pare these curves with those in Reference
2
showing T as
a
double-valued
function
of
E with infinite slope at E
=
-1.
1.
We suggest the rea der com-
No solutions of Lambert 's equation ex is t in the shaded regions of figu res 1
T - 0 as x -,a
nd
2.
x =
1
m >
0 )
and
x
= -1 are vertical asymptotes.
REFERENCES
1.
Plummer,
H.
C.
,
An Introductory T re at is e on Dynamical Astronomy,''
Cambridge University Press, 1918, al so Dover Publications, Inc. N.
Y.
,
1960,
paperback rep rin t, chapter 5.
2.
Breakwell, John V. Gillespie, Rollin W. and Ross, Stanley, Researches
in Interp laneta ry Transfer, ARS J. 31, 201-208 (Feb. 1961).
3. Gedeon, G. S. ILambertian Mechanics, Proceedings of the XII International
Astronautical Congress (Springer-Verlag, Wien, and Academic Pr e s s , Inc.
New York, 1963).
4.
Battin, Richard
H.
, Astronautical Guidance, McGraw-Hill Book Co nc. ,
1964, page 46.
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T
X
Figure 1-E vs.
T
for e l l i p t i c case
7
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