the quasi-elliptic motion of the moon

7/29/2019 The Quasi-Elliptic Motion of the Moon

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CHINESE JOURNAL OF PHYSICS VOL. 50, NO. 5 October 2012

The Quasi-Elliptic Motion of the Moon

Maurizio M. DEliseo

Osservatorio S. Elmo-Via A. Caccavello 22, 80129 Napoli, Italy

(Received January 12, 2012)

Among the many perturbing effects exerted by the Sun on what would otherwise be theKeplers ellipse of the Moon around the Earth, it is possible to set apart a particular classof terms of periodic and secular nature related to the size of the eccentricity and to thelocation of the perigee. These matters can be jointly treated by studying the behavior of asingle mathematical entity, the eccentricity vector, a complex number that in polar form hasthe eccentricity as magnitude and the angle marking the position of perigee as argument.The comparison between the theoretical behavior of this vector and the observations hashistorically represented an important test of the inverse-square law of gravitation.

PACS numbers: 96.20.-n, 45.10.Hj, 96.12.De

I. INTRODUCTION

The solar perturbation makes the Moons motion relative to the Earth not amenableto a simple geometric and kinematic description, but not so much that the basic model ofan elliptic orbit need be abandoned altogether. It turns out that, as a first approximation,

one may keep Keplers picture of the motion with the changes that are reflected in twostatements.

First, the apsidal line (joining apogee and perigee) of the Keplerian ellipse rotatesslowly in the same direction as the Moon itself making one complete revolution in aboutnine years, and to this motion is superimposed a semiannual oscillation.

Second, the ellipse becomes pulsating, that is its shape is periodically more or lessflattened because the eccentricity oscillates around a mean value, with the same period anda phase shift of /2 with respect to that of the apsidal line.

Both occurrences are expected from mathematical analysis and confirmed by obser-vation (historically, this path has been followed in reverse order). Qualitatively, the directrotation of the apsidal line can be inferred from a careful reasoning on the imbalance of the

effects of the solar perturbation in some specific geometric configurations of the Sun andMoon in the course of time, but we omit the details for brevity [1]. With regard to thevariation of the eccentricity, one can say that when the apsidal line is parallel with the di-rection toward the Sun (that is when the Moon is full at perigee and new at apogee, or viceversa) the attraction of the Sun tends to elongate the ellipse of the relative lunar motionand increase its eccentricity. If, on the other hand, the apsidal line is perpendicular to the

Electronic address: [email protected]

http://PSROC.phys.ntu.edu.tw/cjp 720 c 2012 THE PHYSICAL SOCIETYOF THE REPUBLIC OF CHINA


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VOL. 50 MAURIZIO M. DELISEO 721

line joining the positions of the full and new Moons, the solar attraction will tend again to

widen the ellipse and diminish its eccentricity. In particular, for a terrestrial observer, thevariation of the eccentricity is reflected in the periodic perturbation known as the evection,a gain and a loss in the angular position of the Moon compared with that provided by apure Kepler motion, with a period of about 32 days.

All these effects are entangled with each other in the behavior of the eccentricityvector e, a complex number whose polar form has the eccentricity as its magnitude and theangle marking the position of perigee (and therefore of the apsidal line) as its argument. Ifthere were no attraction of the Sun on the Moon this vector would be constant, implying theimmutability of the ellipse, but actually the vector e is a dynamical variable that changeswith time. A careful inspection of the function e(t) allows us to quickly isolate the expressionof the position vector of the Moon that incorporates all the mentioned occurrences and make

the orbit in this approximation as quasi-elliptic, meaning by this that part of the actuallunar motion described in terms of an elliptic orbit of which some parameters slowly changewith time and are mathematically represented at the lowest order of their variability. In thepast, the study of this matter has been instrumental in reconciling observation and theoryin support of the inverse-square law of gravitation [2].

II. THE ECCENTRICITY VECTOR

We define first the eccentricity vector for an Earth-Moon system thought of as beingfree from any outside influence. We fix a geocentric coordinate system. The position of

the Moon in the orbital plane that we mathematize as the complex plane, is given bythe vector r. Then r2 = |r|2 = rr, r being the complex conjugate of r, while the realand the imaginary parts of a complex quantity, such as r are Re(r) = (r + r)/2 andIm (r) = i(r r)/2, respectively. In polar coordinates, r() = r()exp(i), where thepolar angle is the true longitude measured from a suitably chosen oriented axis. Thefunction r(t) = r[(t)] will be known as soon as we find the time dependence of , and itstime derivative is

r =

r

r

+ i

r. (1)

Let E be the mass of the Earth, M that of the Moon, G the gravitational constant. These

two bodies for all practical purposes can be supposed to be punctiform. At a given time, letthe Earth be located at the origin of the coordinates, and let r be the geocentric vector ofthe Moon. The force per unit mass by which the Moon is attracted toward the Earth andthe force per unit mass by which the Earth is attracted toward the Moon are, respectively,

GErr3

,GMr

r3

. (2)

In order that the Earth stays at the origin even at later times, it should be put to rest withrespect to the Moon. This can be formally achieved by submitting the Moon at any time


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722 THE QUASI-ELLIPTIC MOTION . . . VOL. 50

3

+

F

GEr

r3

GMr

r3

+

F

G(E+M)rr3

to a force per unit mass equal and opposite to that by which it attracts the Earth. On thisassumption one can write the equation of the geocentric motion of the Moon as

r =

r

r3

,

G(E+ M). (3)

We multiply Eq. (3) by r, and take the imaginary part [3]. Since

Im (rr) =d

dtIm (rr),

r3

Im (rr) = 0, (4)

we get, upon integration, the angular momentum integral of the two-body motion:

Im (rr) = . (5)

With the help of Eq. (1), we immediately deduce first the polar expression of :

= Im ( rr) = r2 = 2 dAr

dt = const., (6)

so that is twice the constant areal velocity, Ar being the area swept out by r, andfurthermore the formal operation

d

dt=

d

d=

r2

d

d, (7)

from which it follows, in particular, that

d

dt

r

r

=

r2

d

dei =

iei

r2

= ir

r3

. (8)

Using this identity, Eq. (3) may be written in the form

0 = ddt

i

r +

r

r

, (9)

which, upon integration, becomes the Hermann-Laplace integral [4]:

e = i

r rr

, (10)


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where e

e exp(i) is a complex constant, the eccentricity vector. Now we multiply both

sides of Eq. (10) by r and take the real part. We find, with simple steps,

Re (er) =

Re (irr) Re (rr)

r

, (11)

Re (er) =

Im (rr) r, (12)

r[1 + e cos( )] = 2

. (13)

Solving for r and multiplying by exp(i), we get the polar expression of the lunar vector:

r = rei =2

ei

[1 + e cos( )] . (14)

Comparing this equation with the representation of an ellipse in polar coordinates, wededuce that the eccentricity vector e = e exp(i) has the eccentricity as magnitude andthe angle marking the position of perigee (where r = rmin) as argument; besides, we obtainan expression of in terms of two orbital elements:

2

= a(1 e2) =

a(1 e2). (15)

By denoting with T the orbital period and with n the real number such that T = 2/n, weobtain, from Eq. (6),

T0

dt =

2/n,2Ar,

2a2

1 e2,(16)

by the well-known formula for the area of the ellipse, from which it follows that

= na2

1 e2, (17)

that confirms and clarifies the meaning of relationship (15). Notice that = na2 to ordere, and further, by Eqs. (15), (17), that n =

/a3, so that n is Keplers mean motion of

the Moon. Since e is already present in e, actually contains and controls also anotherconstant of motion, the semi-major axis a.

We can find very simply the function r(t), the solution of Eq. (3), to order e [5]. FromEqs. (14), (15), (17), we have

r = rei a[1 Re (eei)] ei

= aei 12

ae 12

aeei2, (18)


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=

r2 na2

r2 n + 2en cos( ). (19)Integrating the last expression we get

nt + + 2esin(nt + ) (20)= nt + i

eei(nt+) eei(nt+)

, (21)

where the integration constant , called the mean longitude at epoch, is the element thatspecifies the position along the orbit at the initial time, and is therefore an arbitrary constant(the fourth, after a, e, ) of the elliptic motion. The sine term in the right-hand side ofEq. (20), named the equation of center, represents the gain or loss in longitude due to theeccentricity compared with the uniform circular motion described by the secular term nt.By the way, from this observed inequality one deduces the eccentricity of the lunar orbit,which is e 0.0549.

From Eq. (21) it is easily seen, by developing in series the complex exponentials in ,that in Eq. (18) one can assume

ei e + ei(nt+) + eei2(nt+), (22)

ei2 ei2(nt+), (23)so that we find

r = aeil 32

ae +1

2aeei2l, (24)

where we have introduced the mean longitude of the Moon l

nt + . The lunar vector

r is thus expressed to order e as the composition of a circular motion and of two termscontaining the eccentricity vector, one constant and the other periodic of period one-halfcompared to that of the circular motion and with a phase-shift determined by the positionof perigee. The check of this solution is immediate, because we have

r = an2(

eil + 2eei2l)

,

rr3

= n2a3r

r3

n2 [1 + 3e cos(l )] r

= n2 [1 + 32 eeil +

3

2 eeil] r

an2(

eil + 2eei2l)

.

The knowledge of the unperturbed position vector r(t) of Eq. (24) is all that one needsto find the main perturbations of the eccentricity vector due to the Sun. In particular, itallows the approximate symbolic computation of the secular motion of the lunar perigeefor that part (by far the greatest) dependent only on m = n/n 0.0748013, the ratio ofthe Suns sidereal mean motion to the Moons sidereal mean motion (n = 0.9856090/day,n = 13.1763631/day, respectively).


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III. THE TIME VARIATION OF THE ECCENTRICITY VECTOR

Assume, in full generality, that the elliptic motion of the Moon around the Earth isperturbed by any cause whatsoever, so that the equation of motion (3) takes the form

r = R + P rr3

+ P, (25)

where, for brevity, R stands for the inverse-square centripetal acceleration and P is aperturbing acceleration. In this situation the eccentricity vector is no longer a constant ofmotion. In order to find the time variation of e, we start from Eq. (10) written in the form

e =

i

Im (rr)r

r

r

, (26)

and consider its time derivative

e = i

[Im(rr)r + Im (rr)r] ddt

r

r

. (27)

The substitution, in Eq. (27), of r = R + P will produce the following expression

e = i

[Im(rr)R + Im (Pr)r + Im (rr)P] ddt

r

r

. (28)

Equation (28) must be solved perturbatively, because the functions r, r, r, r, that are

present in its right-hand side refer to the actual, and still unknown, orbit of the Moon. So,in order to write the first approximation of Eq. (28), sufficient for our purposes, we feedthe right-hand side of the equation with the values derived from the two-body solution.That we can stop at the first approximation is beneficial, because we must not attemptto find the equations of perturbation related to the (former) constants a and , which arenecessary for the computation of the next values of the quartet r, r, r, r, in order to carryout the second approximation to achieve a greater precision. With this understanding,using Eq. (8) to cancel the term containing R, we obtain the perturbation equation for e:

e =d

dt(eei) = (e/e + i) e

= i [Im(Pr) r + P] , (29)

from which we immediately deduce the corresponding equations for e and :

e = Re(e e/e) , (30)

= Im (e/e) . (31)


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IV. THE SOLAR PERTURBING ACCELERATION

Now we deal with the expression of the perturbing acceleration of the Sun in theactual motion of the Moon around the Earth. In order to simplify the calculations as muchas possible, we suppress all the nonessential details and emphasize the core of the problem.We assume that the lunar motion takes place in the ecliptic (the plane in which the Earthrevolves around the Sun) and that the relative motion of the Sun around the Earth iscircular and uniform, so neglecting the inclination of the Moons orbit to the ecliptic andthe eccentricity of the orbit of the Earth. This is theoretically possible because within ourapproximations the motion of the Moon projected on the ecliptic is decoupled from themotion perpendicular to it. Let S be the mass of the Sun and r its geocentric vector. Asthe Sun attracts both the Moon and the Earth, it is clear that in the geocentric system

one must take into account the fact that on the Moon, other than the Earths attraction,will act only the difference between the Suns accelerative attraction on the Moon and theSuns attraction on the Earth:

GS(r r)| r r|3

GSr

r3

, (32)

so that the geocentric equation of motion of the Moon is

r = rr3

+

r r

|r r|3 r

r3

r

r3

+ P, (33)

where

= GS, the standard gravitational parameter of the Sun, while in the right side arenow present the inverse-square and the perturbing accelerations. Ultimately, P representsthe tidal acceleration of the Moon caused by the Sun. Note that we have implicitly setS+ E+ M = S, that is we have assumed that the masses of Earth and Moon are negligiblewith respect to that of the Sun. The Sun is about 330,000 times more massive than theEarth, while its distance is about 400 times the distance of the Moon. It follows that themaximum tidal force is about 70 times less than the inverse-square force.

Since the motion of the Sun relative to the Earth is assumed to be circular anduniform, we can write r = r exp[i(nt + )] where r = const. The constant n =

/r3

is the sidereal mean motion of the Sun, while the phase is its longitude at t = 0. If weleave out the second and higher powers of r/r 1/400 and denote with l nt + thesolar mean longitude, the perturbing acceleration P becomes

P = n2(r r)(

1 + 3r

r

cos(l ) + . . .)

n2r

n2r + 3 n2r rr

cos(l )

= n2r + 32

n2eil

r

(ei(l

) + ei(l

)

)

=1

2n2r +

3

2n2rei2l

, (34)


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and Eq. (33) takes the form

r = rr3

+1

2n2r +

3

2n2rei2l

. (35)

It is worth noting that on the right-hand side the ratio r/r has disappeared, and with itany reference to the finite distance of the Sun (the solar parallax), and we are left with onlytwo solar terms, the first of which is a repulsive force between the Earth and the Moon thatdepends linearly on the Earth-Moon distance and therefore is of central type, representingan average solar perturbation.

V. THE MAIN PERTURBATIONS OF THE ECCENTRICITY VECTOR

It is useful to write Eq. (35) in the form

r = R + P = rr3

+1

2m2n2r Pr

+3

2m2n2rei2l

Pr

, (36)

where we have indicated that the solar perturbation P is the sum two parts, Pr, Pr, of whichthe first is a subtractive (i.e., directed away from the central body) radial acceleration. Iteasy to verify that P is a perturbation of the two-body motion. As r is proportional to a,the lunar vector r in the first term on the right of Eq. (36) has the prefactor /a3 = n2,

and therefore the ratio between P and the inverse-square acceleration has m2

0.005595as a factor, thus justifying a perturbative approach to the problem. To go on, we explicitlyput in the right side of Eq. (29),

r = a

eil 3

2e +

1

2eei2l

, (37)

r = an(

ieil + ieei2l)

, (38)

and = na2, since we work to first-order in e, so that

Im (Pr)r = I m (Prr)r

=3

2m2n3a3Im (r 2ei2l

)r, (39)

P = m2n3a3

1

2r +

3

2rei2l

, (40)

where in Eq. (39) we have taken advantage of the fact that Im (Prr) = 0. Since the products

in the right members of Eqs. (39), (40), have the same symbolic coefficients, from now on,


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to facilitate the calculations, we shall write the expressions of r, r, , P to be multiplied

together in a normalized form, without the presence of a,n,m, which will appear once andfor all together as prefactor in the perturbation equation. This way Eq. (29) will assumethe form (with = n2a3)

e = m2n3a3i

n2a3[Im(Prr

)r + P]

= m2ni [Im(Prr)r + P] . (41)

Once one knows the right-hand side of Eq. (41), which is the simplest differential equationof the normal type, the solution of the equation in the first approximation is given by theindefinite integral

e(t) = m2ni

[Im(Prr)r + P] dt, (42)

where, in the integrand, e, e are treated as constants. Let us look first at the possibleform of the expressions constituting the function e(t). The term by term integration of theperiodic terms resulting from the products indicated in Eq. (42) leads evidently to complexexponentials of the form

Cm2ni

ei(hl

+kl)dt =Cm2n

hn + knei(hl

+kl), (43)

where h, k are integers not both zero, and where the constant C is real or complex. Inaddition, non-periodic terms will also be produced, which are evidently those that causesecular effects. Since we want to confine our study only to some specific aspects of the lunarmotion, the greatest part of the terms (some tens) that constitute the complete expressionof e are not of our concern here, so they must be discarded by an opportune selective actionto significantly reduce their number. A first criterion is that they contain the first powerof e, e. This is required by the structure of the perturbation equations (30), (31), and isconsistent with the first-order approximation of our work. Among the constant terms, onlyone has this feature. With regard to the choice of the periodic terms, it is worth notingthat letting k = 0 in Eq. (43) we obtain a term modulated by the position of the Sun alonefor which

Cm2nieihldt = Ch

m2 nn

eihl = Ch

meihl , (44)

and so the integration lowers in the coefficient the power from m2 to m, enhancing itsgreatness. So we choose precisely this term that is by far the largest among those periodic.In this way we get the differential equation governing the time behavior of the eccentricityvector:

e(t) =3

4m2nie +

15

4m2nieei2l

. (45)


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This equation can be solved by an iterative perturbation procedure. As said before, we

begin considering e = eexp(i) (and so e) in the right side as constant, with the value ofe opportunely fixed and that of as referred at the instant of time assumed as initial. Thisis emphasized by the particular notation we have used, by distinguishing between e(t) ande. Thus integrating Eq. (45) we obtain

e(t) =3

4m2niet +

15

8meei2l

+ e, (46)

e(t) = 34

m2niet +15

8me ei2l

+ e. (47)

The non-secular part of these expressions shows that the eccentricity vector is subject to aperiodic variation with a frequency of n times per interval of time. Inserting this solutionback in the right-hand side of Eq. (45) and taking into account the terms up to order m3,we obtain

e(t) =3

4m2nie +

15

4m2ni

15

8meei2l

ei2l

+15

4m2nieei2l

, (48)

which modifies the previous evaluation of e(t), and so Eq. (45) must be replaced by

e(t) =

3

4+

225

32m

m2nie +

15

4m2nieei2l

. (49)

It is worth noting that the coefficient of the periodic term is unchanged. Now we can useEqs. (30), (31), to obtain the functions e(t) and (t). We have

e(t) =

Re[ee(t)/e] dt

=15

4m2ne

Re

(iei(2l

2)

)dt

=15

8me cos(2l 2) + e, (50)

where the tilde over in the result reminds us of having to take into account the lineargrowth of the argument of perigee with time, according to the outcome of the integrationof performed below. We find that the lunar eccentricity oscillates periodically within thelimits

e

1 158

m e1 1

7

, (51)

about its mean value, which can be consistently interpreted as that empirically deducedfrom the equation of center. Further we have

(t) =

Im [ e(t)/ e ] dt =

3

4m2 +

225

32m3

nt +

15

4m2n

Im

(iei(2l

2)

)dt

=

3

4m2 +

225

32m3

nt +

15

8m sin(2l 2) + . (52)


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VI. THE QUASI-ELLIPTIC ORBIT

The dynamical variables e(t) and (t) we have worked display a constant rotation ofperigee of the Moon (i.e., the orientation of the ellipse) in the direction of an increasing, as well as its semiannual libration, with an accompanying pulsation in the size of theMoons orbital eccentricity, to which we alluded at the beginning.

The m3 term in (t) is that found for the first time after a very exacting work bydAlembert [6]. Together, the two secular terms represent approximately five-sixths of theobserved motion, and the perigee makes a full turn in the forward direction in

T =2

n 2

[(3/4)m2 + (225/32)m3]n 3, 827 (53)

days or about 10.47 years, 1.62 years off from the observed period of 8.85 years, not badconsidered the approximations made. Notice that the constant term of e(t) in Eq. (45)originates only from the radial part of the perturbing acceleration, so the result (3 /4)m2ntprovided by the first lunar theorists (Newton, Clairaut, dAlembert, and Euler) shows thatthe combination of the inverse-square centripetal acceleration and the subtractive radialcomponent of the perturbative acceleration yields a forward motion only half that observed.This is because the ratio between the second and the first term of the secular part of (t)is, for the Moon,

75

8m 7

10, (54)

and this means that for the Moon the m3 term gives a contribution comparable to that ofthe m2 term, something totally unexpected, with the result that the rotation period of thelunar perigee is nearly halved, making it close to that observed. This explains the inabilityof the first lunar theorists, people who grew up in the spirit of infinitesimals, to predictsuch an occurrence before undertaking a more complete and detailed calculation. Therelationship in Eq. (54) is small only when m is very small, at least an order of greatnesslesser than it is for the Moon. For example, it is nearly negligible in the theory of theJupiters four Galileian satellites, where 4 104 < m < 4 103.

The secular term, whose nature we have clarified, is easily managed by considering itimplicitly attached to the element , so that it can be formally discarded when consideringthe correction to the position vector of the Moon due to the variation ofe, by agreeing that

now e = e exp(i), with + nt. Then

r = aeil 32

ae(t) +1

2ae(t)ei2l, (55)

with e(t), e(t) given by the two last elements of Eqs. (46), (47) only. This way the quasi-elliptic position vector rq of the Moon that jointly describes the rotating ellipse, and theoscillations of perigee and eccentricity is found to be

rq = aeil 3

2ae +

1

2aeei2l 45

16ame ei2l

+15

16ame ei(2l2l

). (56)


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It should be noted that, taken by itself, rq is not an approximate solution of Eq. (35),

as indeed can be inferred from the procedure we have followed, but it is rather only asignificant part of the complete first-order solution of the equation.

The knowledge of rq rq exp(iq) allows the derivation of the correspondent polarcoordinates rq, q of the Moon. The radial coordinate rq is given by

rq

a=

rqr

q

a2 1 e cos(l ) 15

8mecos(l 2l + ), (57)

We have added to the elliptic term a periodic variation of the length of the radius vectorwith an amplitude of about 2,960 km. For the polar angle q we get

q = i lnrqrq = i lneilrqe

il

rq

= i

il + ln(

rq

aeil

) ln rq

a

l + 2e sin(l ) + 154

mesin(l 2l + ). (58)

For both rq, q, the periodic m-part has a period of about 31.8 days. We have thus verifiedthat in the angular motion of the Moon around the Earth the two largest deviations fromthe uniformity, depending on the first power of the eccentricity, are in the order the equationof center and the evection (the first has an amplitude of about 6.2815, the second of about1.2759). As regards the second approximation to the secular motion of lunar perigee in the

literal form given by dAlembert, it has been interesting to see how, avoiding the detailedcalculations the early lunar theorists had to run to obtain it for the first time, the conceptof eccentricity vector provides an easy way to find a term that has historically representedan important support to Newtons law of gravitation.

References

[1] F. R. Moulton, An Introduction to Celestial Mechanics, 2nd Ed., Art. 197, p. 352, (DoverPublications, 1984).

[2] M. C. Gutzwiller, Rev. Mod. Phys. 70, 589 (1998).[3] M. M. DEliseo, Am. J. Phys. 75, 352 (2007).

[4] M. Nauemberg, Archive for History of Exact Sciences, Springer, Vol. 64, Number 3, 281 (2010).[5] M. M. DEliseo, J. Math. Phys. 50, Issue 2, 022901 (2009).[6] J. dAlembert, Recherches sur Differents Points du Systeme du Monde Vol. 1, Art 66, 70 (1754).

the quasi-elliptic motion of the moon

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