Approximate analytical solution for low thrust propulsion in space

< ~ H

van

KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE

16

APPROXIMATE ANALYTICAL SOLUTION

FOR

LOW THRUST PROPULSION IN SPACE

by

Lo MOULIN

RHODE-SAINT-GENESE, BELGIUM

...

~~"o~

~~

-TECHNICAL NOTE 16

APPROXIMATE ANALYTICAL SOLUTION FOR LOW THRUST PROPULSION IN SPACE

by

Lo MOULIN

The research reported in this document has been sponsored by the Air Force Office of Scienti~ic Re~earch, OAR, through the European Office, Aerospacè Research, United States Air Force. under Grant N° 63-390

Approximate analytical solutions are derived for the

problem of low t hrust propulsion, in the case of constant thrust, set at a constant angle to the velocity vector, for

any type of initialorbit (elliptic, parabolic or hyperbolic) • Simple expressions are obtained, giving energy, angular

momentum and excentricity in terms of the excentric anomaly . The solutions allow for calculation of the fuel consumption o Their validity is restricted to t he field of orbit correction .

c

e

effective exhaust velocity

". 5e_

V's

orbit excentricity

E(e .. ",,)

incomplete elliptic integral of the second kind F (e,4') incomplete elliptic integral of the first kind

~o acceleration of gravity at sea level

h

non dimensional angular momentum

J

3[

2.

]Y2

H

(e,~) ;:

e

2

cc.!.h

x -1 eh:. o

J

E-2. .2.

]_1

12.

I

(e,~) =

L

e

c.o.sh x. -1 d:.c. o

I

specific impulse

M

mass of the vehicle

hl =

R

s

T

radius vector

radius of the planet

distance along the trajectory thrust

U

non dimensional specific energy

V

velocity

circular velocity at sea level

Ij

-

Vs

mean elliptic anomaly

angle between velocity vector and local horizon angle between thrust and velocity vectors

parabolic parameter mean hyperbolic anomaly

'5

hyperbolic excentric anomaly

f

mass flow of propelIants

....

p

= R <J ;

s

R

1: :

. I

-Mo3o

cp

elliptic 'Y=~-cp SUl.;f3cripts initial acceleration excentric anomaly o i ni t i al 0 rb::. t ~ perturbations

The problem of low thrust propulsion ~n space has received much attenti on in t he past o

Tsien (refol) first treated the problem by power

series expansion, in the case of radial or circumferential

thrust~ for circular orbits o A similar solution for tangential thrust has been obtained by Benney (refo2)o A parametric

study, in the case of constant acceleration, has been made by

Perkins (ref.3) , who developed a general differential equation

to be numerically integratedo Numerical studies of the problem

have been made by Moeckel (refo

4)

for constant tangential

thrust o The problem of take-off from circular orbits with low

tangential thrust has been analyzed by Shapiro (refo5) using

the Krylov-Bogoliuboff methodo More recently, Tsien's solution

has been extended by Ting and Brofman (refo6) to the case

where the thrust vector is set at a constant angle to the radius vectoro

The purpose of the present analysis is to derive an

approximat e analytic~l solution to the problem, which would be

valid for any type of ini tialorbit and which, in addition.

would allow for the calculation of fuel consumption, whi ch is

not the case of other available solutions o

To develop the analysis, it has been assumed that the vehicle was only subjected to one inverse square law

gravita-tional fieldt neglecting thus other perturbations o

Further-moree the vehicle is considered as powered by constant thrust, set at a constant angle to the velocity vector o The latter

which would in any case require application of variational

calculus and lead to numerical solutionso It is considered that a first analytical approximatiqn would be a valuable step

towards trajectory optimizationo

The solutions are not expected to be valid for wide variations of velocity as encountered in the problem of take-off from circular velocity up to escape velocity. Their field of application lS merely confined to such problems as low thrust transfer between close orbits of any type, or orbit correction.

I . Basic equations

Denoting by ~ the angle between the velocity vector

and the :ocal horizon at one particular point of the trajee -tory (Figol ) , ~ being defined positive below the local horizon;

and by Ó the angle between velocity and thrust vector

T,

t

being defined positive below the vel ocity vector, the two basic equations of motion are

2. d Y M ( R

):1.

2--MV . .:...JL =- ~ - co~y

....

MV

(.,(po _

Ts'''Ö

ct5 0 r 0 r

(1 .2)

where M denotes the mass of the vehicle, ~o the acceleration

of gravity at the surface of the planet of radius

R,

r the

distance from the vehicle to the center of the planet . The

quantity S represents the distance travelled along the trajec-tory.

There is one additional differential equation for

the radius vector :

defined as the product of the mass flow

r

of the exhaust gases. by the effective exhaust velo city Ce.

The exhaust velocity, which is considered as constant for given propelIants or propulsion system, is related to the specific impulse I by the equation

The case of constant thrust will only be considered here, which then corresponds to a constant value of~ o Under those circumstances, one additional differential equation is obtained for the variation of mass, which simply states that

JI'

i s a con st an t :

20 Non dimensional system

The analysis of the problem should preferably be worked out ~n terms of non dimensional quantitieso A very

convenient system can be defined by using the radius of the planet as unit length, the value of the circular velo city at the surface of the planet as unit velocity, and the mass Mo of

the vehiele at the point where propulsion ~s initiated as unit mass o

The fol1owing non dimensional quantities ean aeeordin-gly be defined :

(1 08 )

(1,10)

Substituting these definitions in Eqs o (lol ) , (1 02)\1

(1 03) and (1.6) respective1y, one obtains the fo11owing system of non dimensiona1 differential equations

3. Definition of orbit parameters

It will be found more convenient 1n the subsequent

analysis to chanee variables~ and USe as independent variables

energy, angular momentum and excentricity of the orbit which

wou1d be constant, within the frame of assumptions,in the case

of no propulsion.

The definitions of non dimensiona1 enerey U , angu1ar momentum

hand excentricity e are the fo110wing

(1.16)

2.

hl.

e

.::.-1+2.U

Differentiating these equations with respect to ~ ~

an dus in g E q s . (1 . 11) t 0 (1. 1 3), y iel d s

Two of the above equations, plus Eq.(1 .14) may be

regarJ.ed as the basic system of differential equations to

represent the actual problem o It should be noted that according

to the assumptions made, the term"t: is a constant .

4

.

Method of solution

Exact analytical i ntegration of the system of diffe

-rential equations is impossible , One consequently has to

resolve to an approximate solution; the nature of the problem

suggests immediately a perturbation method, since the parameters U ,

h

t

e

and

m

would be constants in absence of propulsion.

Thrust can be regarded as a perturbation function, with the further restrietion that ~ , which represents the initial

value of the accel~ration delivered by the system of propulsion, should be small; such a solution can consequently be expected to be valid for advanced electrical propulsion.

Application of the suggested method makes it impossible to derive a nalyti-cal solutions which would be valid for all possible cases . It is necessary to distinguish between the type of orbit (elliptic. parabolic or hyperbolic) which the vehicle follows prior to

starting propulsion.

2 0 ELLIPTIC ORBITSo

The solution of t he problem is grent1y eased i f the independent variabIe is modifiedo Instead of 0- , the best

quantity to use appears to be the excentric anonaly o

10 Excentri c anomalyo

In dimensional form, the excentric anona1y ~ can be

deflned by the relationshir

where

a

is the semi major .azi s of t he el1i se . The ab ve

equation can be easily t ransformed into

:z.

e

(;..OS cf> ::.

p

UT' - 1 _(201-2)

The analysis is further simplified by the i ntroduction

of the complement ~ of t he excentric anomaly :

The particular values of ~ are 900 _{at perigee and 270}0

at apogee.

The variables

p

,

w-

and

r

can then be easily expressed

ln terms of ~ and t he parameters of the orbi t , by the following

re lat ion shi ps

r

:0

UI" 2. ::. 2. U (..::e_s_""'_ .

....Jtl--+_-t_)

eSV'/\*-1

(2 01-8)

It should be noted that t o each value of ~ correspond

two values of

r ,

one positive, the other nesative, depending upon the direction of rotation on the ellipse o The convention

will be made that in all double signs, like i n Eqo(2 01-7),

the upper one corresponds to such a direction of mation that

the vehicle moves towards perigee, while the lower one corresponds to a motion towards apogee .

USl.ng Eqs. (1.11) to (1 .13). one obtains the folloHing diffe-rential equation for the complement of the excentric anomaly ~

with

(2 01-10)

2. ?-1e an anomaly.

A useful parameter to consider is the mean anomalYt which is proportional to the t ime. The complement ~ of the mean anomaly is defined as

(2.2-1)

30 System of differential e3uations .

Substituting Eqs. (2 .1-5) to (2.1-8) l.n the system of equations (1.18) to (1 020) and Eqo(1.14). one finally

obtains the basic system of differential equations for elliptic orbits :

dm

- = cf 0--C [

]-Yz [" -

eSm

f

J''''

- - -2.U C -t + e ~w. 'f _{(2 .3-1)}

cl..,

-ec.osu.. - =+ I dG"

where f(u,h'I,e,-r)is given by Eq. (2 ,l-lO) .

(2.3-2)

(2,. 3-4)

One of Eqs (2 .3-2), (2.3-3) or (2 .3-4) l S redundant,

but will be kept in the integration to allow for a

veririca-tion.

4. Introduction of the perturbation.

Each independent variable ~ can be perturbed 1n terms of ~ by writing

~ :: ~o + 1: ~1 + .--

(2

.

4-1)

where ~o denotes the value of the variable on the initial

orbit.

differen-tial eQuations, and neglecting terms 1n ,; » one obtains the fundamental solution, which yields h10 , U_{o ,}

e

_o and ho constantSt withMo_-t ·

EQo{203-5} then becomes

[

Zo . l.

]yz.

1-e,,;~ .... 0 {2 .4-2}

which ean be immediately integrated to yield the measure of elliptie arc

{2 04-3}

denotes the elliptic integral of second kind. of modulus

e

o and argument ~o

5. System of first order equations .

The first order solution lS obtained by only retaining terms of order ~ in the system of differential eQuations.

Doing so, and eliminating ~with the aid of Eq. {2.4-2}, one obtains the following system of differential eQuations, with

i'o as the independent vari abl e :

dU ..

_ : : . ct'i'o

(2 0

5- 4 )

6

0 Integration of the system.

Written in this form, the system of equations can be

direct1y integrated . Returning to the definition (2 04-1) of

the perturbations, one obtains t he fo11owing fina1 resu1t :

(2 .6-1) (2.6-2) (2 .6-3) (2.6-4) with (2 .6-5)

(206-6)

(2.6-7)

As verifications, one may note that Eqs.(2 .6-1) and

(2 .2-1) show that

1lYl,,=

and that consequent1y, the variation of mass lS proportiona1

to time, as assumed origina11yo

Combination of Eqs.(2.6-2} and (2.4-3) give s

which lS ln accordance with Eq.(1 01-8).

Fina11y, introuucing the definitions of the

perturba-tions ln Eq. (101-7) and retaining terms of order ~ on1y yie1ds

(2 .6-10)

(2.6-3) and (2.6-4) 0 Constants appear~ng ~n the solution must be evaluated in terms of initial conditions for the powered fli ght 0

6

.

Perturbation of the excentric anomaly o

For complet ion, the solution must allow for the

calculation of the value of the independent variable ~ on the

new orbit .The solut ion can be obtained but takes a form which

is rather involved, although it contains only simple

3. CIRCULAR ORBlTS.

Circular orbits can be treated as a particu1ar case of elliptic orbits, with zero excentricity. Therefore,

formu1as for circu1ar orbits can be immediately derived from the resu1ts of the previous section, by putting

e

o equal to

zero.

Since excentric anoma1y is not privileged on a Clrcu-lar orbit, its value at the point where propulsion lS started can be taken equa1 to zero .

Furthermore, the el1iptic integrals degenerate in this particular case, and one has

Consequently, Eqs. (2 .6-1), (2 ,6-2), (2.6-3) and(2 .6-4) become respectively iYY) :: ~

-

ï:

'+'0

_{( 3 .2)} (-2.U_{o )} 3/~ C. U ::

U

_o

-

"'C

c.o.s

~

'+'0

2. Uo ( 3 .3) 2.

e :.

0 _{(3 .4)}

One can also verify that substituting Eqs o(3 03) and (3 05) ~n

Eqo(2 0

6-7)

yields Eqo(3.4).

The fact that the excentricity rema~ns equal to zero means that one departs from circular orbits by following a

succession of cireular orbitso The solution ean consequently

not be applied to such problem as take-off from circular orbit with low thrust propulsion, s~nee it would break-down before escape velocity ~s reachedo However, the approximation is

4.

HYPERBOLIC ORBlTS.

The treatment for hyperbolic orbits is identical to the one for elliptic orbits, if one again introduces the excentric anomaly as independent variabIe.

1 . Excentric anomaly.

In the hyperbolic case, the excentric anomaly

1

can be defined by the non dimensional relationship

(4.1-1)

which ~s analogous to Eq.(2 .1-4).

The variables

f,

lAT and

r

can similarly be expressed

~n terms of

J

by the fol1owing relationships, using the same convention of sign for

r

:

e..eosh ~ -1 2U

w-

2

_=-

_{2U eeosh] + ....}

eeos.h

'l-1

(4.1-2) (4 .1-4)

(4 .1-5)

Comparison with Eqs . (2.1-5) to (2 .1-8) obtained in the case of elliptic orbits suggests immediately a possible transformation between elliptic and hyperbolic expressions :

(4 .1- 6)

which 1S equivalent to

(4 .1- 8)

The differential equation for the hyperbolic

excentric anomaly can then be directly obtained by substitu-tin g E q s • ( 4 • 1-7) an d (4. 1 - 8) i n E q s • (2 • 1-9) an d ( 2 • 1-10) 0 One obtains

(4

0

1-9)

+ ((.2_1) ~hZs ~[

V,.,...

(o\ ..

e.c..osh!)

.± ~2. _I) v.t

,sw.h

1

~

\,

~

2.U'l"n

(4.1-10)

Furthermore, a mean hyperbolic anoma1y

r

can a1so be

introduced, introducine the tr&nsformation (4.1-6) in the

definition of the mean e11iptic anoma1y. One obtains the f0110wing expres sion

(4.1-11 )

which is proportiona1 to the time.

2. System of first order equations.

Transformation (4.1-6) can be direct1y app1ied to

the system of differential equations obtained in the e11iptic

case. Ey ana10gy with e11iptic integrals, two hyperbolic

integrals wi11 be defined :

(1+.2-1)

(4.2-2)

These inteerals can be eva1uated l.n terms of el1iptic

If one notes that Eqo(1 0 4-2) beeomes

and eonsequent1y,

(f"=

(4 02-4)

the system of first order differentia1 equations, (20 5-1) to

(2 0 5-4) ean be direet1y transformed into

dm,

clIo

=

_(h.2-5)

(4 .2-6)

3. Integration of the systemo

Integration of the system ~s straight forward and

yields the final solution

(4 .3-1) (4.3-2) (4.3-3) (4.3-4) with (4.3-5)

(2 .6-9) and (2.6-10) can also be performed on the above system. Constants must be evaluated in terms of initial conditions for

the propulsion.

4.

Perturbation of the hyperbolic cxcentric anomaly.

The problem of the perturbation of the hyperbolic

excentric anomaly is similar to the elliptical case. Develop-ments are given in appendix 2.

5.

PARABOLIC ORBlTS

The characteristics of parabo1ic orbits are excentri-city equa1 to unity and energy equa1 to zero. In such a case, the concept of excentric anoma1y loses its significance. As easi1y verified, the solutions obtained previous1y for e11ip-tic and hyperbo1ic orbits become undetermined in the 1imiting case of parabo1ic orbits. Consequent1y, special attent ion

must be given to this particu1ar prob1em.

1. Introduction of the perturbations.

To deve10p the ana1ysis, one shou1d return to the origina1 system of differentia1 equations (1.18). (1 .19) s

( ] • 2 0 ). (1. 1 4) an d (1. 13) •

In addition, one shou1d note that the fo11owing expression can be derived for the inc1ination ang1e

0 :

Furthermore, at any point of a parabo1ic trajeetory, one has

express~on (1.4-1) and ~n particu1ar

r

'='

Po

+- T

f ..

of.

-.-Eq. (l ol-3) can now be written as

Squaring the above expression, introducing the

definitions of the perturbations and taking advantage of the

fa c t th at th een e r gy U_o 0 f th e par ab ol i c 0 rb i t i s e q u a l t 0

zero, one obtains the fo11owing re1ationship :

Retaining on1y terms independent of ~ , one obtains

the fo11owing fundamental equation on1y va1id for parabo1ic

)

orbits :

2,0

(dPo\Z,=

2,0

_hZ.

10 ~a-

J

10 0

(5.1-6)

Keeping terms of order ~ on1y, one obtains the fo11owing

which ean be integrated. provided the quantities

U,

and

h,

have a1ready been ea1cu1ated.

Eq.(5.1-6) suggests the fo11owing transformation

(5.1-8)

which is justified sin ce cos~

"\

1.S equa1 to 1 at per1.gee. and 1argerthan 1 at any other point . Substituting in Eq.(5.1-6)

one obt ain s

and integrating. the measure of the parabo1ie arc beeomes

(5.1-10)

The above transformation wi11 be used to integrate the system of equations.

2. Differential system of first order.

Before applying the transformation, one can easily

verify that

Eq.(5.2-1) shows that the transformation which has been

intro-duced is actually that of Gudermann.

Performing thc transfornation and retaining terms

of order 1: only, Eqs .(l.lG), (l o]-9) ~ (1. 20) .and (lell+)

become respectively, af ter eliminati~n of ~ with the aid of

Eq.(5.1-9)

dU

)2.

,2.

("'"

_ _ I ::

+

t) ~I'l

i

co.so

cl""

0 (5.2-4) (5 .2-6)

3

.

Solution of the system.

Integration of the above system lS immediate, and

yields the following answer

Ver ifications similar to those of the previous cases

can be applied to t he above system. Constants are evaluated

in terms of initial conditions.

4

.

Perturbation of the radius vector.

Perturbation

P1

of the radius vector can be obtained

6.

CONCLUDITIC RE1ARKS

Approximate analytical solutions have been obt ained

for the problem of low-thrust propulsion in space, in the case

of two-body confie;uration, and constant thrust set at a constant

angle to the velocity vector.

It can be observed that difficuities encountercd in

obtai nine; such solutions are stronely dependent uran the rroper

choice of thc variables. It turns out thut the best parameters

to use are energy, ane;ular momcntum and excentricity of orbi ts,

while excentric anomaly rlays an i mportant role as the

indepen-dent variabie. Furthermorc , it is necessary to dist ine;uish

between thc particular types of init i a l orbits (ell iptic,

parabalie or hyperbolic) •

The existence of a simple imacinary transformntion .

has been established, which allows one to apply the results

of the elliptic case to the hyperbolic one, and vice- versa.

Bath solutions break down at the l imit represented by a pa

ra-bolic orbit, which can be analysed independently using

Gude rmann 's t ran sfo rmat ion.

Comparisons have been carried out ln all cases between

the actual approximate solution and numeri cal integration

of the original system of differential equations. Agreement lS

good all over a portion of trajeetory which correponds to

moderate fuel consumpt ion ; solution holds over a l arger range

Actual solutions fail to provide a simple means to calculate the exact point on the final orbit where propulsion

terminates . They can nevertheless be applied to a wide class of problems where the latter point does not have to be spec~­

APPENDIX 1 EVALUATION OF HYPERBOLIC INTEGRALS

Hyperbolie integrals have been introdueed by Eqs .

(4.2-1) and (4 .2- 2) respeetively as

(Al-l)

(Al-2)

They ean be rather easily evaluated by first putting

(Al-3)

an d int rodu ei ng t he t ran s format i on

(Al-4)

with standard notation for Jaeobian elliptie funetions .

SUbstituting Eqs . (Al-3) and (Al-4) in Eqs . (Al-l) and (Al-2).

and integratine, one obtains respeetively

and _(Al-6)

where _(Al-7)

Af ter transformation and rearrangement, the

expressions for the two hyperbolic integrals can be written as

(Al-8)

I-t

r ...

,~.)

= -

e.

t (,. :.,.)

F

['';':'('':'~IJ, ~]

.

f

[&.,;.

o'G':".),

~

1

(Al-9)

APPENDIX 2 PERTURBATION OF THE INDEPENDENT VARIABLE

I . El1iptic and hyperbolic orbits.

For el1iptic orbits, the differential equation of the independent variabIe ~ is given by Eqs.(2.1-9) and (2.1-10). By putting

(A2-1)

Eq.(2.1-9) can be written as

(A2-2)

Introducing the definition of the perturbations 1n the above equation, and retaining only terms independent of

~ • one obtains first the fundamental solution

(A2-3 )

which is identical to Eq. (1.4-2) obtained previously.

Retaining only terms of order r one obtains the first order

~

differential equation

The above differentia1 equation ean be integrated by eonsidering first the ho~ogeneous part whieh yields

(A2-5)

or, eliminating 0- with the aid of Eq. (A2-3)

C

~ I~). ";)'1-' 0

+ ..

(A2-6 ) Integrating, one has

(A2-7 )

or, ln the present case

(A2-8 )

Substituting Eq.(A2-7) ln Eqo(A2-5) yie1ds the differentia1 equation for the funetion C l.~) as

(A2-9 )

where partia1 derivations are easi1y ea1eu1ated from Eq.(A2-1); funetions U, ,

e ,

2. and

t(

\}o te..., 'fo) are respeetive1y given by

Eqs,(2.6-3) t (2,6-5) and (2.1-10).

Eq.(A2-9) ean be deve10ped by substitution of the

simple quadratures. HOvlever, the final result is rather involvei and not practical for any analytical discussion.

A similar re sult can be obtained for hyperbolic orbits with the equation

with

and where

f ..

(U,e.

,m,'n

1S glven by Eq. (1+ ol-la) e

One obtains the following results

(A2-l 0)

(A2-ll)

(A2-12 )

where

cCSo)

must be obtained by integration of the differential equation

(A2-13 )

(4 .3-5) •

2. Parabolic orbits.

Instead of investigating the differential eQuation of the perturbation of the independent variabIe .~ , one can as weIl consider the perturbation of the radius vector

p,

which is directly given by EQ. (5.1-7). Af ter elimination of ~ with the aid of Eq. (5 .1-9). and rearrangement, Eq. (5.1-7) c an be w rit ten as

(A2-14 )

Integration 1S straight forward, S1nce the homogeneous part yields

wit h th e f 0110 win g di f fe r e nt i al e q u at ion f 0 r C (

't))

(A2.J.6 )

where UI and

h.

are given by Eqs . (5.3-2) and (5.3-5) respecti vely.

REFERENCES

1. TSIEN, H.S., "Take-off from sate11ite orbit".Jet Propulsion 23.

233-236, August,1953.

2. BENNEY, D.J., "Escape from a circu1ar orbit using tangentia1

thrust". Jet Propulsion 167-169, March,1958 .

3. PERKINS, F.M. , "F1ight mechanics of low-thrust spacecraft" .

JAS.26,291-297, May, 1959.

4.

MOECKEL, W.E. , "Trajectories with constant tangentia1 thrust

in central gravitationa1 fie1ds." NASA TR R-53.

1960.

5. SHAPIRO,

G.,

"Orbits with low tangentia1 thrust" . AFOSR 1479

1961.

6. TING, L., BROFHAN, S. , "On take-off from circu1ar orbit by

---FigoI - Notations horizon

v

T Surface of the lanet

von Karman Institute for F1uid Dynamics,

May

1964.

APPROXIMATE ANALYTICAL SOLUTION FOR LOW THRUST PROPULSION IN SPACE. By L. Mou1in.

Approximate analytica1 solutions are derived

for the problem of low thrust propulsion, in the case of constant thrust, set at a constant

ang1e to the velocity vector, for any type of

initialorbit (elliptic, parabolic or

hyper-bolie). Simple expressions are obtained, givi~ energy, angular momentum and excentricity in

·uo1~oa~~oo ~1q~o

JO Pla1J aq~ o~ pa~o1~~sa~ S1 Ä~1Pl1~A ~laq~

·uo1~dwnsuoo lanJ aq~ JO uOl~~lnol~o ~OJ MOll~

suol~nlos' aq~ ·Äl~WOU~ 01~~uaoxa aq~ JO sw~a~

terms of the excentric anomaly. The solutions

allow for calculation of the fue1 eonsumption. Their validity is restrieted to the field of

von Karman Institute for Fluid Dynamics,

May

1964

0

APPROXIMATE ANALYTICAL SOLUTION FOR LOW THRUST PROPULSION IN SPACE o By Lo Moulin.

Approximate analytical solutions are derived for the problem of low thrust propulsion, in the case of constant thrust, set at a constant angle to the velooity veotor, for any type of

initial orbit(elliptic, parabolio or

hyper-bolie). Simple expressions are obtained,giving energy, angular momentum and exoentricity in

Their validity is restricted to the field of orbit correction.

Ul ·~~lolJ~UaOXa pu~ wn~uawow J~tn~u~ '~~Jaua

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