A system of parameters for describing orbits of space vehicles

Planet. Space Sci. 1964. Vol. 12, pp. 255 to 258. Pergamon Press Ltd.

Prinred in Northern

Ireland

A SYSTEM OF PARAMETERS FOR DESCRIBING ORBITS OF SPACE VEHICLES G. V. SAMOILOVICH Translated by H. S. H. Massey from Zskusstvennye Sputniki Zemli No. 16, p. 136 (1963).

Osculating elements are a convenient system of parameters for describing the evolution of an orbit acted upon by disturbing forces. However, since the components of the Laplace vectorf, and fi (in an inertial geocentric system of Cartesian co-ordinates) have the same order of magnitude with respect to the eccentricity e (at e + 0), the expression for the angular distance of the perigee at the point e = 0 becomes an uncertainty of the type O/O; and in the system of differential equations in osculating elements there appears a feature which makes difficult the study of circular and nearly circular orbits. The componentsf, and fiapproach 0 (as e + 0) simultaneously and circular orbits are described by three instead of five parameters; therefore by including the constantsf, and fi instead of e and w in the system of orbit elements we can obtain universal parameters for describing orbits with any eccentricity. To write the equations in osculating elements in the simplest form it is best, however, to use the quantities q and k which are connected withf, and fi by the relationships: fi = 4 cos (2 - k sin Q cos

i,

:=qsin*+kcosQcosi 3 P of the ascending loop of the orbit, i is the inclination constant.

where R is the longitude of the orbit and ~1is the gravitational Therefore q and k, when expressed in terms of the parameters e and w by means of the

formulae, q =

e

cos w,

k = e sin LL),

are components of the Laplace vector in the system of co-ordinates (6,

FIG.

1.

ARRANGEMENT

OF THE CO-ORDINATE 255

SYSTEMS

xyz,

Et/< AND

THE

7,

5) (Fig. 1). The

VECTOR

f.

G. V. SAMOILOVICH

256

system of parameters p, q, k, Q i can be used to describe orbits with any eccentricity (p is the focal parameter of an orbit). After expressing the radius in the new parameters P

r=

1 + k sin u + q cos u ’ the dependence of q and k on the co-ordinates and velocities of a satellite takes the form:

(1) I

1(rlq2.

p =

P

Here V,.and V, are the radial and transversal velocities and u is the argument of the satellite’s latitude. From these relationships we can obtain (without using the well-known differential equations with respect to e and co) differential equations (in osculating motion) with respect to q and k. For this purpose we use the relationships x = r(cos 24cos Q - sin 24sin R cos i), y = r(cos 24sin Q + sin u cos Q cos i), z = r sin u sin i

to remove cos u and sin u from (1). Differentiating the expressions obtained according to the basic rule(l) and once again eliminating x, y, z we obtain after reduction of like terms the differential equations in terms of q and k which, together with the usual equations in terms ofp, Q, i are a complete system for describing the orbits: sin u cos u + k sin u)-’ K sin i

+(l+q

$=(I +qcos u + dp - = 2p(l + dt

2

k sin u)-’ cos u w,

q cos u + k sin u)-‘F,

(2)

= (4 sin2u+tksin2u)coti~+~(q+cosu)$r dt 2p

+(ksinu-+qsin2u)cosi$+#sinu+~cosu,

2 = (k sin2 u cos ie

+

dt

$4 sin 2u) cot i Ft - (q sin2 u + &k sin 2~)

’

Zp(k+sinu)%-

s”cosu+

Fsinu.

FOR DESCRIBING

A SYSTEM OF PARAMETERS

ORBITS OF SPACE VEHICLES

257

Here we are using the notations:

where S, T and W are the radial, transversal and binormal components respectively of the acceleration due to the disturbance. After elimination of the derivatives dp/dt, dQ/dt, dildt the equations for dqldt and dkldt take the form: & -=

dt

dk

dt=-

ksinucoti

q + GOSu

w+

[ 1 + q cos u + k sin u

1 + q cos u + k sin u

q sin u cot i w+ 1 + q cos u + k sin u

+ cosu

k + sin u [ 1 + q cos u + k sin r4

1 1

P-:+ S”sinu,

+ sinu

F--

fcosu.

These equations are valid over the whole range of eccentricities. Their singularity (at e 2 1) belongs to the non-singular point of the osculating orbit. We can change from the disturbances 6q, 6k to the disturbances de, 6w (which are easier to visualize geometrically) for orbits with an initial eccentricity of e, # 0 by using the following equations: = ‘, (cos w 6k - sin u) 6q), de = ‘, (k 6k + q 6q).

Here the values of e, cu, q, k are taken at the time of the preceding osculation. then 6e = e = 1/4” + k2 and

If e, = 0,

h=w=-N+GM, 2

where N=

(1 - sgnq) + (1 - sgnqsgnk)U M= sgnqsgnk.

+ sgnq),

When numerical integration of the differential equations is used to determine the disturbances, the unavoidable inaccuracy of the calculation gives the value of 0 as the arithmetic mean : CT,=-

1

2(

arcsin-

k

e

+ arccos4- . e1

Let us explain the physical meaning of the Laplace vector f whose components are q and k. For this purpose we write the Laplace vector in the form f=[VC]-pfl.

(3)

Here V isthe vector of the orbital velocity, C is the vector of the moment of the amount of R

motion, r0 = - is the point of the focal vector-radius, ,urO= R2A,,, where &, is the centrifIRI ugal acceleration acting on satellite moving instantaneously in a circular orbit. The first item on the right-hand side of (3) can be re-expressed as follows: [VC] = [VR%.@‘] = Rp cos ypco*p]

= Rp cos yA.

G. V. SAMOILOVICH

258

Here zi and o* are the angular velocities of the satellite’s motion relative to the focus and the instantaneous centre of curvature respectively, p is the instantaneous radius of curvature, y is the angle between the orbital velocity vector and the transverse direction, A is the centrifugal acceleration acting on the satellite in question and P is the position of the positive normal to the plane of the orbit. Using the expressions obtained and eliminating p by means of the equation p = p/cos3 y (p is the focal parameter) we obtain

or (V,, is the velocity of a satellite moving instantaneously

in a circular orbit)

,=.,($A-.,,).

(4)

The Laplace vector is thus expressed in terms of dynamic quantities. From (4) we can derive the equation for the eccentricty (< a = Rp) :

e~=~-&tCOsa+!!x At the perigee we obtain

haA,

Vo4A,2

*

,=V”A-1 V,” A,,

*

In conclusion I should like to avail myself of the opportunity simskii for his valuable discussion of the whole subject.

to thank D. Ye. Okhot-

REFERENCE 1. G. N. DUBCWIN, Znrroduction to Celestial Mechanics Moscow (1938).

(in Russian). United Scientific and Technical Press,