The obliquity of Pluto

ICARUS

55,

231-235 (1983)

The Obliquity ANTHONY Jet Propulsion

Laboratory,

of Pluto

R. DOBROVOLSKIS

AND

183-501, California

of Technology,

Institute

ALAN W. HARRIS Pasadena,

California

91109

Received January 28, 1983; revised March 22, 1983 Pluto’s obliquity (the angle between -126” over a period of about 3 million stable, leading to only modest changes slightly retrograde ever since its current

its spin axis and orbit normal) varies between -102 and years. These oscillations are nearly sinusoidal and quite in the insolation regime. Thus, Pluto’s rotation has been orbit and rotation rate were established.

INTRODUCTION

Pluto is a fascinating object from a dynamical standpoint. It has the greatest orbital eccentricity and inclination of any major planet in the solar system. Because of the former, Pluto periodically comes closer to the Sun than Neptune (as at present), but because of the latter, the orbits are still well separated. Furthermore, Cohen and Hubbard (1965) have numerically simulated the motions of the outer planets over 120,000 years and discovered that Pluto’s mean motion is commensurate with that of Neptune, so that the two planets come into conjunction only when Pluto is near aphelion. Subsequently, Dirk Brouwer (1966) suggested that the precession of Pluto’s orbit could cause it to intersect Neptune’s orbit on occasion, which might disrupt this resonance. In order to test its stability, Williams and Benson (1971) undertook to integrate Pluto’s orbital variations over 4,500,OOO years. They found that the longitude 0 of Pluto’s node regresses more or less normally with a circulation period of 3.7 millions years. On the other hand, its argument of perihelion o librates near its present value so that the orbits of Neptune and Pluto never intersect. With respect to the invariable plane of the solar system, Williams and Benson’s (1971) results are well represented as a = 39.45 AU,

(1)

e = 0.244 + 0.022 cos +!I + 0.005 cos 3$/J, (2) w = 90?0 + 24’10sin I,!J,

(3)

i = 15”91 - l”O4 cos $I/,

(4)

R = 111’?428- 97”209 t - l”5 sin IJ,

(5)

where J, = 72?8 + 91?0 t and t is time in millions of years A.D. The longitude of the node, R, is measured from the vernal equinox along the ecliptic to its intersection with the invariable plane, and then along the invariable plane to the ascending node of the orbit on the invariable plane. More recently Christy and Harrington (1978) have detected a satellite orbiting Pluto, which has come to be known as Charon (name not yet approved by the IAU). Its orbital period (6.387 days) is equal to the period of Pluto’s lightcurve, but Charon is not bright enough to account for the total light variation; furthermore, Charon’s orbital plane seems to lie in the equator of Pluto (Christy and Harrington, 1978; Harrington and Christy, 1980, 1981). These facts strongly suggest that both bodies are tidally locked into synchronous rotation with their mutual circular orbit (Christy and Harrington, 1978; Harrington and Christy, 1980, 1981; see also Farinella et al., 1978). The estimated dimensions of the system are R, = 2000 km for the radius of Pluto and Rc = 1000 km for the radius of Charon (Bonneau and Foy, 1980), and R =

231 0019-1035/83 $3.00 Copyright B 1983 by Academic press, Inc. All rights of reproduction in any form reserved.

232

DOBROVOLSKIS AND HARRIS l4R M+m

I

R

I

FIG.

I

1. Schematic of the Plutonian system.

18?0 relative to the node of its orbit on the invariable plane (Harrington, private communication, 1980). The time evolution of the obliquity under both axial and orbital precession is given by the simultaneous first-order differential equations dd

20,000 km for the distance between their centers (Harrington and Christy, 1981). This system is drawn roughly to scale in Fig. 1. In a recent review paper, Harris and Ward (1982) estimated that the axial precession period of Pluto is comparable to the nodal period of its orbit. A similar coincidence in the case of Mars causes the Martian obliquity to oscillate between -21 and -38”, and may drive dramatic climate changes (Ward, 1973, 1974, 1979). By analogy, Harris and Ward (1982) suggested that the history of Pluto’s obliquity is quite complex. Motivated by this claim, we have investigated the past history of the spin axis by numerical integration using the analytic approximations (l)-(5) above for the orbit of Pluto. We find that the variation of the spin axis is very regular and of modest amplitude, contrary to the above expectation. The reason is that since Pluto’s spin is retrograde, the motion of its spin axis is prograde with respect to the orbit normal, while the orbit node is moving in a retrograde direction with respect to the invariable plane. Hence the two motions, which are indeed of comparable magnitude, are in opposite directions, so that the near coincidence of frequencies does not lead to a high-amplitude beat phenomenon. PRECESSIONAL

THEORY

A planet’s obliquity is the angle 8 between its rotational pole and its orbit normal, both taken in the right-hand sense. To specify its axial orientation, the azimuth + is defined so that the longitude of the ascending node of the planet’s equator on its orbital plane is $ + 90”. Pluto’s present pole position is described by 8 = 118?5,+ =

z=

-sin ices +($I

d+

da -ff cos 8 - cos i dt

dt=

+ sin 4 ($1,

(

(6)

i

ds1 + sin i sin + cot 19 dt ( 1

+ cos 4 cot 8 -& di ( 1

(7)

(Ward, 1973, 1974), where i is the orbit’s inclination to the invariable plane and fl is the longitude of its ascending node on that plane. This system can be expressed also in Cartesian coordinates as dX x=aYZ+

Ycosi

(

da dt

1

(8) dY dt=

-C&Z -Xcosi(z) dZ

-=Xsini(g) dt

+Z($)

(9)

- Y(z),

(10)

where we define X = sin 8 cos $, Y = sin 8 sin 4, and Z = cos 8 (Ward, 1974). Note that system (8)-( 10) contains only two independent equations because of the constraint Xz + y2 + 22 = 1. The quantity (Yappearing in Eq. (7), (8), and (9) above is given by a

=

; d

~(1

_

e2)-312,

(11)

where (+ is the planet’s rotation rate, n is its orbital mean motion, e its orbital eccentricity, and His a dimensionless measure of the planet’s oblateness called the precessional

THE OBLIQUITY

233

OF PLUTO

constant. This definition applies to a planet without satellites, or to one like Mars whose satellites have a negligible effect on its precession. The situation is more complicated when the satellite’s orbit contains more angular momentum than the planet’s spin; this is the case in the Earth-Moon system, and probably for the Pluto-Charon system as well. However, since the latter are presumably locked into synchronous rotation with their mutual, circular orbit, the entire system can be treated as a single rigid body (see Fig. 1). Now the precessional constant H in (11) can be written H = (C - B/2 - A/2)lC,

(12)

where A d B I C are the principal moments of inertia of the system. Along the line joining the centers of Pluto and Charon, the moment of inertia is A = Zp + Z,,

(13)

where Zp and Zc are the moments of inertia of Pluto and Charon, respectively, approximated as spheres. Along the other principal axes, however, we have B=C=Zp+Zc+ m2MR2 + (M + m)2 =

FIG. 2. Possible precession trajectories of Pluto. The circle represents a projection of the celestial sphere along Pluto’s ascending node on the invariable plane. The arrows labeled 0, I, P, and R, respectively, indicate Pluto’s orbit normal, the invariable pole, Pluto’s prograde Cassini state, and its retrograde Cassini state. Pluto’s current right-hand rotational pole is marked X; over a precessional cycle Pluto’s axis is confined to the corresponding solid curve. The dashed continuations of the trajectories are explained in the text.

mMZR2 (M + m>2 ZP +

k

+

g!$

CASSINI STATES

(14)

where M and m are the respective masses of Pluto and Charon and R is the separation between their centers (see Fig. 1). Finally, substituting (13) and (14) into (12) gives H = (C - B/2 - Al2)lC = (Cl2 - Al2)lC, = 4 - f(AIC) = 4 - 0.02 = 0.48,

(15)

where we have estimated AIC as a few percent. The exact dimensions of the system are poorly known, but this uncertainty is not important since the value of H is determined mostly by the large aspect ratio of the system. It is this fact that enables us to predict accurately the long-term behavior of Pluto’s obliquity when so little else is known about the planet.

It is clear from Eqs. (6) and (7) that if the planet’s orbital plane is fixed, its obliquity is

constant also $

= $

= $ = 0 , while

its azimuth undergoes a uniform circulation d+ -(Y cos 6 . Since (Y = 1.71 X 10m6 1 ( dt= year-’ and 8 = 118’?5 at present, this “zeroth-order” formula suggests that Pluto’s free precessional period is about 9 million years. As noted previously, however, Pluto’s orbit precesses also on a similar time scale. If only uniform circulation of the node is considered

da di dQ - = - = 0, - = constant i dt dt dt

1.7 X 10m6year-’ tions may be

=

the precessional equafrom a Hamiltonian

234

DOBROVOLSKIS sd

I

138 -10

”

I

1

”

I

,

”

-5

I

,

,

,

”

”

TIME FIG. 3. History

,

,

a cos20 -

[cos i cos 8 + sin i sin 0 sin 41

,

,

,

“1

”

0

8

"1

+5

+10

(MY)

of Pluto’s

(Colombo, 1966; Peale, 1969, Ward, 1975a,b). For our purposes it suffices to say that the axial motion may be associated with an energy E given by E = -;

AND HARRIS

obliquity.

minimum of E over the sphere, respectively. These represent stable pole positions (Cassini states) located at obliquities of 40 and 171”. Note that only two Cassini states are possible since the mutual axis of the parabolae (indicated by the horizontal line at Z = - lla(dWdt) cos i = 1.16) just misses the unit sphere.

(16) = -)aP

-

($)

[Z cos i + Y sin i]

(Ward, 1975a). Since E(8, 4) is a constant of the motion, the obliquity must be a periodic function of time (and +)! This solution is illustrated in Fig. 2. This is a view of the celestial sphere along the ascending node of Pluto’s orbit on the invariable plane, so that the vertical coordinate is Z = cos 0 and the horizontal coordinate is Y = sin 0 sin 4. The orbit normal and invariable pole are indicated by arrows labeled 0 and I, respectively. The parabolae represent contours of constant E; since Yz + Zz 5 1, these contours are dashed except within the unit circle. The resulting stripes (and their mirror images on the reverse side) represent precessional trajectories over the celestial sphere. Pluto’s present pole position is marked by a cross. Since it must stay always on the same contour, Pluto’s obliquity varies periodically between about 102 and 126”. The arrows labeled P and R in Fig. 2 indicate the maximum and

NUMERICAL

CALCULATIONS DISCUSSION

AND

In order to study the history of Pluto’s obliquity more accurately, we undertook to integrate the precessional equations including the orbital variations given by Eqs. (I)(5). We both used the same predictor-corrector algorithm (JPL library subroutine SVDQ), but one of us (A.R.D.) integrated the spherical form, Eqs. (7) and (8), while the other (A.W.H.) solved the Cartesian system (8)-(10). As an accuracy check, A.W.H. computed x2 + YL + 22 at each timestep and found it within 10m5of unity even after integrating backwards and forwards for 100 million years. We both confirmed independently that Pluto’s equinox advances almost uniformly while its obliquity varies between -101 and -127, with a period of about 3 million years (see Fig. 3). Even for a wide range of initial conditions, the cycle is quite stable, indicating that the approximation of the preceding section is rather accurate.

THE OBLIQUITY

The inevitable conclusion is that Pluto’s rotation has always been somewhat retrograde-or at least for as long as the representation (l)-(5) of its orbit has been valid. Unfortunately, it is not possible to draw definite conclusions about Pluto’s dynamics before the establishment of its current orbit and rotation rate. ACKNOWLEDGMENTS The authors wish to thank Bill Ward for suggesting the Cassini state treatment and for other discussions. A.R.D. was supported by an NRC Resident Research Associateship. This research was supported at the Jet Propulsion Laboratory, Caltech, by the Planetary Program, NASA, under Contract NAS7-100.

OF PLUTO

235

COHEN, C. J., AND E. C. HUBBARD(1964). Libration of the close approaches of Pluto to Neptune. Astron. /. 70, 10-13.

COLOMBO, G. (1966). Cassini’s Second and Third Laws. Astron. J. 71, 891-896. HARRINGTON,R. S.,AND J. W. CHRISTY(1980). The satellite of Pluto. II. Astron. J. 85, 168-170. HARRINGTON,R. S.,AND J. W. CHRISTY(1981). The satellite of Pluto. III. Astron. J. 86, 442-443. HARRIS, A. W., AND W. R. WARD (1982). Dynamical constraints of the formation and evolution of planetary bodies. Annu. Rev. Earth Planet. Sci. 10,61108. PEALE, S. J. (1967). Generalized Cassini’s Laws. Astron. J. 74, 483-489.

WARD, W. R. (1973). Large-scale variations in the obliquity of Mars. Science 181, 260-262. WARD, W. R. (1974). Climatic variations on Mars. I. Astronomical theory of insolation. J. Geophys. Res. 79, 3375-3386.

REFERENCES BONNEAU,D., AND R. FOY (1980). Interferometrie au 3.60 m CFH. I. Resolution du systeme PlutonCharon. Astron. Astrophys. 92, Ll-L4. BROUWER,D. (1966). The orbit of Pluto over a long interval of time. The Theory of Orbits in the Solar System and in Stellar Systems; (G. Contopoulos, Ed.), pp. 227-229. IAU Symposium No. 25. CHRISTY,J. W., AND R. S. HARRINGTON(1978). The satellite of Pluto. Astron. J. 83, 1005-1008.

WARD, W. R. (1975a). Tidal friction and generalized Cassini’s laws in the solar system. Astron. J. 80,6470.

WARD, W. R. (1975b). Past orientation of the lunar spin axis. Science 189, 377-379. WARD, W. R. (1979). Present obliquity oscillations of Mars: Fourth-order accuracy in orbital e and I. J. Geophys. Res. 84, 237-241. WILLIAMS, J. G., AND G. S. BENSON (1971). Resonances in the Neptune-Pluto system. Astron. J. 76, 167-177.