a note of lambert´s theorem - r.devaney, and r. blanchard

8/16/2019 A note of Lambert´s Theorem - R.Devaney, and R. Blanchard

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X-

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


2/9

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


4/9

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)

3


5/9

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


6/9

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.

5


7/9

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.

6


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Figure 1-E vs.

T

for e l l i p t i c case

7


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i

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i

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a note of lambert´s theorem - r.devaney, and r. blanchard

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