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Variation of the orbital elements for parabolic trajectories due to a small impulse using Gauss equations Osman M. Kamela,∗ , M.K. Ammarb a Department of Astronomy and Space Science, Faculty of Science, Cairo University, Giza, Egypt b Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt

Received 17 March 2003; accepted 23 March 2006

Abstract Firstly, we derive Gauss’ perturbation equation for parabolic motion using Murray–Dermott and Kovalevsky procedures. Secondly, we easily deduce the variations of the orbital elements for the parabolic trajectories due to a small impulse at any point along the path and at the vertex of the parabola. © 2006 Elsevier Ltd. All rights reserved. Keywords: Gauss’ perturbation equations; Parabolic motion; Rocket Dynamics; Impulsive motion; Perturbations

1. Introduction There are different forms for the perturbative equations for the derivatives with respect to physical time of the elliptic, hyperbolic and parabolic orbital elements. The famous Lagrangian form of the derivatives of elliptic elements is widely known as the variation of arbitrary constants method. It is a mathematical rigorous derivation written in a typically elegant Lagrangian manner [1–3]. Le Verrier’s work on the theory of motion of the principal planets is an outstanding application [4]. Gauss’ form for the classical perturbation theory is also fundamental and well known [1,2]. In the Gaussian form of the equations the three mutually perpendicular components of the perturbative acceleration is used. A direct derivation is also available for the Gaussian form [1,5].

Moreover Burns [6] displayed a straight forward derivation of the equations using elementary dynamics. It is somewhat amazing that Lagrange’s equations for elliptic and hyperbolic orbits could be deduced easily from Gauss’ equations [5], since we are capable of ﬁnding the expressions in terms of the partial derivatives with respect to the elements. The Lagrangian and Gaussian equations for hyperbolic motion are obtained by Kamel et al. [7]. In this article we derive the derivatives of the elements for parabolic trajectories by two different procedures. In a subsequent step we ﬁnd easily the change of the parabolic parameters due to the application of a small arbitrary impulse along the parabolic path. 2. Fundamental formulae for parabolic orbits The equation of a parabolic orbit can be written as [8]

∗ Corresponding author. 31, Eskandaz Akbar Street, Heliopolis,

Cairo 11341, Egypt. E-mail address: [email protected] (O.M. Kamel). 0094-5765/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2006.03.008

r=

p f p = sec2 , 1 + cos f 2 2

(1)

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O.M. Kamel, M.K. Ammar / Acta Astronautica 59 (2006) 1111 – 1116

where p and f are the semi-latus rectum and true anomaly, respectively. The integral of area is r 2 f˙ = h,

(2)

where p = h2 /. The velocity V of a body in a parabolic orbit is given by V2 =

2 . r

(3)

Substituting Eq. (1) into (2) and integrating, we obtain 1/2 f 1 f 4 3 (4) (t − ) = tan + tan3 , P 2 3 2 where is the time of perihelion passage. Deﬁne the quantities n¯ and M by the expressions n¯ 2 p 3 = ,

M = n(t ¯ − ).

f 1 f ¯ − ). + tan3 = 2M = 2n(t 2 3 2

x = r[cos cos( + f ) − sin sin( + f ) cos I ], y = r[sin cos( + f ) + cos sin( + f ) cos I ], z = r[sin( + f ) sin I ].

(5)

Eq. (4), will take the form tan

in a parabolic orbit about the primary O. The point M is any point on the orbit and its longitude in the orbital plane, r the radius vector. The components of the disturbing function F in the reference plane are Fx , Fy , Fz . In the r, , Z coordinates the components of the disturbing function F are R, S, W such that R is the radial component directed outwards along the radius vector, S the azimuthal component in the orbital plane at right angles to R, and W is the orthogonal component to the orbital plane. The coordinates of the body referred to the reference plane are given by

(6)

(7)

We shall use the following expressions [9]: The constant of area is given by ˙ zx˙ − x z˙ , x y˙ − y x), ˙ h = (hx , hy , hz ) = (y z˙ − zy, where

3. Gauss planetary equations

hx = ±h sin I sin ,

In Fig. 1 consider the rectangular axes Oxyz have its origin in the reference plane and Ox, Oy lie in the reference plane such that Ox is taken in a ﬁxed direction (vernal equinox), Oy is perpendicular to Ox, and Oz is normal to the reference plane with the orbiting body

hy = ∓h sin I cos , hz = h cos I ,

(8)

where the upper sign in Eqs. (8) is taken for hz > 0, and the lower sign for hz < 0. Using (8) we obtain sin = ±

hz , h sin I

cos = ∓

hy , h sin I

cos I =

hz . h (9)

Let R, S, and W be, respectively, the magnitudes of the radial, azimuthal, and the orthogonal components of ˆ zˆ are the standard unit the disturbing forces, and rˆ , , vectors, then the disturbing forces may be written as F = R rˆ + S ˆ + W zˆ .

(10)

Since the rate of change of angular momentum is equal to the moment of the applied forces, then we have dh = r ∧ F = (0, −rW , rS). dt

(11)

This implies that dh = rS, dt Fig. 1.

(12)

since −rW ˆ changes the direction of h but does not affect its magnitude.

O.M. Kamel, M.K. Ammar / Acta Astronautica 59 (2006) 1111 – 1116

3.1. Gauss’ equation of p˙

Using relation (8)

Performing time differentiation of the relation √ (13) h = p and using Eq. (12) we obtain dh 1 = p˙ = rS. dt 2 p

2 rS. np ¯

(14)

Performing time differentiation of the third relation of (9), we obtain

Using Eqs. (8), this relation can be reduced to (15)

We can easily deduce that [9] h˙ x = r{S. sin I sin + W [sin( + f ) cos + cos( + f ) sin cos I ]}, h˙ y = r{−S. sin I cos + W [sin( + f ) sin − cos( + f ) cos cos I ]}, h˙ z = r{S. cos I − W cos( + f ) sin I }.

(18)

˙ 3.4. Gauss’ equation of M Put relation (5) in the form √ M = 3/2 (t − ). p Differentiating with respect to time we obtain √ 3 ˙ n(t ¯ − )p. ˙ M = 3/2 − p 2p

hz h˙ − h˙ z h I˙ = . h2 sin I 1 ˙ h − h˙ z / hz ). (h/ tan I

˙ ˙ ˙ = hy sin + hx cos = r sin( + f ) W . h sin I h sin I ˙ = 1 r sin( + f ) W . np ¯ 2 sin I

3.2. Gauss’ equation of I˙

I˙ =

˙ ˙ ˙ = hy (hx / h sin I ) − hx (hy / h sin I ) , h sin I

Hence

Making use of (5), we can write p˙ =

1113

(19)

Substituting (6) and (14) into (19) we obtain after some reductions ˙ = n¯ − 1 tan f 2 + cos f r.S. M np ¯ 2 2 1 + cos f r 1 f ˙ 1+ r.S. (20) M = n¯ − tan np ¯ 2 2 p 3.5. Gauss’ equation of ˙ The integral of area in the orbital plane is given by

(16)

d = h, dt

Substituting (12) and (16) into (15) we obtain after making use of (5) and (13)

r2

r cos( + f ) W. I˙ = np ¯ 2

where is the angle made by the radius vector with the ﬁxed direction (vernal aquinox). Thus we may put [5]

(17)

˙ 3.3. Gauss’ equation of

(21)

d = df + d + d cos I . Substituting into (21) we obtain

From Eq. (9) we obtain hx tan = − . hy Differentiation with respect to time, and using the second relation of (8) we obtain ˙ ˙ ˙ = h x hy − h y hx . h2 sin2 I

˙ =

h ˙ cos I . − f˙ − r2

(22)

To obtain an expression of f˙, we use Eq. (6), which after differentiation and using Eqs. (1), (5), (13), and (20) can be written in the form r h 1 p2 ˙ 1+ sin f.S, (23) = 2− f˙ = 2 M r r np ¯ p

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˙ I˙, 4. Alternative procedure for the derivation of p, ˙ and

where we have used the relation f p tan = sin f . r 2

(24) 4.1. Fundamental perturbative formulas

Substituting (23) into (22) we obtain r 1 ˙ cos I . 1+ sin f.S − ˙ = np ¯ p

We consider the system of equations (25)

x¨ = −xr −3 + Fx , z¨ = −zr −3 + Fz .

y¨ = −yr −3 + Fy ,

We now change our orbital elements p, l, , , M to another elements, p, I, , , ¯ such that

The ﬁrst integral

¯ = + ,

f (x, y, z; x, ˙ y, ˙ z˙ ) = (c1 , c2 , . . . , cj ) = const.

M = nt ¯ + − . ¯

(26)

Hence, using (18) and (25) we can write ˙ ˙¯ = 1 1 + r sin f.S + 2 sin2 I . np ¯ p 2

(27)

Also, we have ˙ − n¯ + ˙¯ ˙ = M 1 r = 1+ sin f.S np ¯ p 1 I ˙ r r f − 1+ tan .S + 2 sin2 . np ¯ p p 2 2

where x˙ = dx/dt. By simple algebraic calculus we reach the fundamental relation

Collecting the results, we can obtain Gauss’ equations for the variations of the elements in a parabolic orbit in the form

j dcj jf jf jf = Fx + Fy + Fz jcj dt jx˙ jy˙ jx˙ j − → −−→ = F .Gradvel f ,

1 r sin( + f ) d = .W , dt np ¯ 2 sin I I ˙ 1 2 + cos f d = sin f.S + 2 sin2 , dt np ¯ 1 + cos f 2 r2 1− 2 p

(34)

In the coordinances r, , z the components of the veloc− → − → − → ˙ z˙ ; ity along the axes R , S , W are respectively r˙ , r , −−→ whence the components of Gradvel f are

1 dI = .r cos( + f ).W , dt np ¯ 2

(33)

where the gradient is taken in the velocity space, and df = 0. dt

2 dp = r.S, dt np ¯

2 + cos f 1 + cos f I ˙ + 2 sin2 . 2

(32)

j

I ˙ r r2 1 1+ 1 − 2 sin f.S + 2 sin2 . (28) ˙ = np ¯ p p 2

(31)

could be obtained for the two-body problem, i.e. the same Eq. (30) with Fx = Fy = Fz = 0. The function may depend on any one of the orbital elements c1 , c2 , . . . , cj . Replacing c1 , c2 , . . . , cj by the osculating elements of the perturbed motion, we can write the derivative of Eq. (31) in the form jf dx j dcj jf d2 x , = + jcj dt jx dt jx˙ dt 2 x,y,z

Again using relation (24), we can write

d 1 = dt np ¯

(30)

jf , j˙r

1 jf , ˙ r j

jf . j˙z

Therefore Eq. (33) could be written in the form j dcj jf 1 jf jf = R+ W S+ ˙ jcj dt j˙r r j j˙z j

sin f.S (29)

(35)

which is the fundamental formula that enables us to derive Gauss’ equations.

O.M. Kamel, M.K. Ammar / Acta Astronautica 59 (2006) 1111 – 1116

˙ − h cos I cos I˙ + h sin I sin = rW (sin u sin − cos u cos cos I ),

4.2. Gauss’ equation of p˙ ˙ = √p. We have h = r 2 √ ˙ Put f = p and c = r 2 . Application of Eq. (35) gives

h sin I I˙ = rW cos u sin I ,

2 r.S. np ¯

r cos( + f ) W, I˙ = h (36)

˙ 4.3. Gauss’ equations of I˙, Use of Eq. (8), the integrals of area projected on the three axes x, y, z are respectively √ p sin I sin = y z˙ − zy, ˙ √ − p sin I cos = zx˙ − x z˙ , √ p cos I = x y˙ − y x. ˙

(37)

Successive application of Eq. (33) gives ˙ rS sin I sin + h cos I sin I˙ + h sin I cos = yF z − zF y ,

(38)

We note that the right hand side of (37) are the projections on the three axes. Oxyz of the moment of forces − → − → r ∧ F . When projected along the r., Z axes it is equal to (0, −rW , rS). The projections of this vector along the Oxyz axes may be written as h˙ x = rS sin I sin + rW (sin u cos + cos u sin cos I ), h˙ y = − rS sin I cos + rW (sin u sin − cos u cos cos I ), h˙ z = rS cos − rW cos u sin I . Substitution of (38) into (37) gives ˙ h cos I sin I˙ + h sin I cos = rW (sin u cos + cos u sin cos I ),

˙ = r sin( + f ) W , h sin I √ where h = p = np ¯ 2.

(41)

5. Changes in parabolic orbital elements due to a small impulse We consider the effects on the parabolic orbital elements when applying a small impulse I at an arbitrary angle to the orbit. Since the radius vector does not change during the operation, all changes in the elements depend upon the velocity vector’s change in magnitude and direction caused by the application of the impulse. The impulse’s change V in the velocity vector can be split into a component at right angles to the orbital plane VW , and two mutually perpendicular components lying in the orbital plane, along and at right angles to the radius vector VR and VS , respectively. Thus V = VW + VR + VS .

˙ − rS sin I cos − h cos I cos I˙ + h sin I sin = zF x − xF z , rS cos − h sin I I˙ = xF y − yF x .

(40)

˙ we obtain where u = + f . Solving for I˙, ,

d √ p = r.S, dt p˙ =

1115

Eqs. (29) may be rewritten to give the change C in any orbital element C of a parabolic orbit due to a small impulse I . Writing VR = Rt,

From our derivation of the variation equations, we deduce that the radial impulse VR does not affect the parabolic elements, thus the equations become p =

2 r.VS , np ¯

I =

1 .r cos( + f ).VW , np ¯ 2

1 r sin( + f ) .VW , np ¯ 2 sin I 1 2 + cos f I = sin f.VS + 2 sin2 , np ¯ 1 + cos f 2 1 2 + cos f r2 = 1 − 2 sin f.VS np ¯ 1 + cos f p I (42) + 2 sin2 . 2 =

(39)

VS = St, VW = W t.

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If the impulse is applied at the vertex of the parabola i.e. when r =p/2; f =0, we have the following reduced equations: 1 p = .VS , n¯ 1 I = cos .VW , 2np ¯ =

1 sin .VW , 2np ¯ sin I

=

1 I tan sin .VW , 2np ¯ 2

=

1 I tan sin .VW . 2np ¯ 2

6. Conclusion The equations of Gauss for the perturbed parabolic motion are derived by two procedures, for the ﬁrst time. The two procedures led to the same results for the variations of the parabolic orbital elements p, I, . For the elements , ; J . Kovalevsky procedure failed to reach the desired results. Murray–Dermott treatment could attain a solution for the problem for all the parameters of the parabola. We considered the variations of the parabolic orbital elements p, I, , , when applying a small impulse at an arbitrary angle to the orbit. All changes in the elements depend upon the velocity vector alternation in both magnitude and direction caused by the application of the impulse I , since the radius vector does not change during the operation. The impulse’s change V in the velocity vector could be analyzed into three components, the ﬁrst at right angles to the orbital plane VW , the other two are mutually perpendicular components lying in the orbital plane, along and at right angles to the radius vector (VR and VS ). Thus V = VW + VR + VS . Appendix Notations f h I M n¯ p r

true anomaly integral of area orbital inclination ¯ − ) deﬁned by M = n(t deﬁned by n¯ 2 p 3 = semi-latus rectum radius vector

R S t V W x, y, z

radial component of the disturbing force referred to the moving coordinates transverse component of the disturbing force referred to the moving coordinates physical time velocity of the body in the orbital plane orthogonal component of the disturbing force referred to the moving coordinates rectangular coordinates of the body in the reference plane mean longitude at the epoch constant of gravitation time of pericenter passage longitude of the body referred to a ﬁxed direction argument of the perihelion measured from the ascending node argument of the perihelion measured from the vernal equinox longitude of the node

References [1] D. Brouwer, G.M. Clemence, Methods of Celestial Mechanics, Academic Press, New York, 1965. [2] W.M. Smart, Celestial Mechanics, Longmans, New York, 1960. [3] D. Boccaletti, G. Pucacco, Theory of Orbits Part 2, Springer, Berlin, 1999. [4] U.J.J. Le Verrier, Annales de l’Observatoire de Paris, 1855. [5] J. Kovalevsky, J.J. Levallois, G’eod’esie G’ene’rale, Tome IV, Editions Eyrolles, 1971. [6] J.A. Burns, Elementary derivation of the perturbation equations of celestial mechanics, American Journal of Physics 44 (10) (1976) 944–949. [7] O.M. Kamel, A.S. Soliman, M.K. Ammar, The change in the hyperbolic orbital elements due to application of a small impulse, Mechanics and Mechanical Engineering, 2002, submitted for publication. [8] A.E. Roy, Orbital Motion, Adam Hilger, Bristol, Boston, MA, 1982. [9] C.D. Murray, S.F. Dermott, Solar System Dynamics, Cambridge University Press, Cambridge, 1999.

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