On the rotation of comet Borrelly’s nucleus

V.V.Sidorenko(Keldysh Institute of Applied Mathematics,

Moscow, RUSSIA)D.J. Scheeres

(University of Colorado, Boulder CO,USA),S.M. Byram

(University of Michigan, Ann Arbor MI, USA)

Abstract

We consider the secular effect of outgassing torques on the rotation of a cometnucleus. An averaging method is applied to obtain evolutionary equations whichallow us to study the long-term variations in the nucleus spin state. Since the spinaxis direction of 19P/Borrelly’s nucleus is close to the line of apsides direction,a simplified version of these equations can be written to analytically study themost important qualitative effects. In particular, a correlation between the driftof the rotation axis direction and the possible spin up/spin down of the nucleusis revealed.

1 Introduction

The comet 19P/Borrelly was discovered by Alphonse Borrelly in December 1904 andis a Jupiter family short-period comet with a period of 6.76 years. In September 2001NASA’s space probe “Deep Space 1” (DS1) returned detailed images of the comet’snucleus. In particular, the observations made during the 19P/Borrelly flyby by theMiniature Integrated Camera and Spectrometer (MICAS) imaging system onboardDS1 revealed a highly elongated nucleus which consists morphologically of two unequalcomponents with a total length of 8 km (Figure 1). The intermediate zone is coveredby a complicated network of cracks and there exists a hypothesis that 19P/Borrelly’snucleus is actually two pieces in loose contact [2].

From the analysis of the DS1 images, Hubble Space Telescope (HST) and Earth-based observations it was found that the nucleus of 19P/Borrelly most likely rotatesabout the main inertia axis, and its angular momentum vector is directed approxi-mately along the line of apsides [10, 18, 30, 33]. The proposed nucleus rotation periodis 26± 1 h [22].

The aim of our note is to investigate the stability of 19P/Borrelly’s nucleus ro-tation taking into account the most important physical factors. Using an averagingmethod, we construct evolutionary equations allowing us to analyze secular effects in19P/Borrelly’s rotational motion over long time intervals. Also we discuss the parame-ters that control the qualitative properties of the nucleus motion. Previously, averaging

1

Figure 1: 19P/Borrelly nucleus: a schematic reconstruction based on DS1 images

the equations of motion was used to study the long-term evolution of comet nuclei spinstates due to outgassing in [16, 23, 24, 25].

2 Physical model

2.1 Preliminary discussion

Physical factors that influence a nucleus rotation are reviewed in [14, 29, 31]. Theycan be divided into two groups: constantly present and sporadically present. The mainfactors which are constantly present are the reactive torque, Mr , due to anisotropicsublimation of nucleus matter [41] and energy dissipation at non-stationary nucleusdeformations caused by the difference of the rotational state from principal axis rota-tion. In [19, 20, 38], the influence of the solar gravity torque, Mg , was also consideredas an essential factor for the nucleus dynamics. This is not the case for 19P/Borrelly.According to our estimations the change in its angular momentum orientation due tothe Mg torque would be smaller than 1 over a time interval on the order of hundredsof comet apparitions.

Examples of the sporadic factors can be found in [31]. In particular, they includepossible impacts of various small celestial bodies (as long as they do not result indestruction of the nucleus) and redistribution of nucleus matter at a certain stage of itsevolution. In our analysis, we will take into account only the main constantly presentfactors mentioned above. Observations of the nucleus made by the imaging systemonboard DS1 and those made by the Planetary Camera (WFPC2) on the HST [18] donot imply existence of other factors that make essential contributions to the dynamics.

We assume that the variation of the angular momentum, L , and the kinetic energy,T , of the nucleus’ rotation can be described with the following equations:

dL

dt= Mr, (1)

dT

dt= (ω ,Mr)−Rint , (2)

where ω is the angular velocity and Rint is the internal dissipation rate.

2

2.2 Evaluation of reactive torque due to sublimation

To calculate the torque due to anisotropic ice sublimation we use the formula

Mr = −N∑j=1

βV∗Qj(Rj × nj) , (3)

where N is the number of faces of the polyhedron approximating the nucleus shape inany reasonable way, β is the momentum transfer efficiency, V∗ is the effective velocityof the ejected matter, Qj is the rate of ice sublimation from j -th face, Rj is theradius vector of the face’s center in the body-fixed coordinate system, and nj is theouter normal to the face.

The ice sublimation rate depends in a complex way on the heliocentric distance, r ,and on local insolation conditions [6]. Following [16, 27, 28], to calculate Qj we usethe empirical expression

Qj = sjg(r)f(δj)Q∗ (4)

Here Q∗ is the sublimation rate from a plane surface of area equal to the total surfacearea of the nucleus oriented perpendicularly to the Sun line of sight at perihelion( q = 1.358 AU), r is the heliocentric distance, sj is the relative intensity (ratio of themaximum possible mass ejection rate from the j -th face at this heliocentric distanceto Q∗ ), δj is the angle between the outer normal to the face, nj , and the unit vectorpointing to the Sun, es .

The function g(r) describes the dependence of the mass ejection rate on the helio-centric distance and is given by the expression [21]

g(r) = g0

(r

r0

)−c1 [1 +

(r

r0

)c2]−c3, (5)

wherec1 = 2.15, c2 = 5.093, c3 = 4.6142, r0 = 2.808, g0 = 0.23476.

The specialists who are familiar with Marsden’s formula (5) will recognize that thevalue of the coefficient g0 is different from what was proposed in [21]. It is explainedby the reasons of convenience: we normalize g(r) in such a way that g(q) = 1 .

The function f(δj) defines the dependence of the mass ejection rate on the anglebetween the direction to the Sun and the normal to the j -th face. We assume theusual hypothesis that the mass ejection on non-illuminated faces is zero, leading to afunctional form

f(δj) =

cos δj = (es,nj), δj ≤ π

2

0 , δj >π2

(6)

Attempts to apply more realistic models for reactive torque calculation were under-taken, for example, in [11, 38]. Of course, the empirical relations (4) and (5) provide avery rough description of the processes on the nucleus surface. Nevertheless, we hopethat the qualitative analysis we make will provide specialists with some landmarksand guidelines which can be specified in further investigations based on more complexphysical models.

3

2.3 Internal dissipation

Attempts to evaluate the influence of internal dissipation on a celestial body’s tumblingmotion have been made in [3, 8, 9, 36]. When the secular effects are discussed, it issufficient to calculate the averaged power of the dissipation process, ⟨Rint⟩ , instead ofconsidering the much more difficult problem of describing the instantaneous attitude’smotion of a non-rigid body.

Below we will be interested mainly in the case where the maximum value of theangle between the nucleus main inertia axis and the angular momentum vector, L , issmall (in other words, when the nucleus is close to principal axis rotation). Let thismaximum angle be denoted as Θ . It follows from the results obtained in [9] that theaverage rate of the energy dissipation in this case is

⟨Rint⟩ ∼L5Θ2

(µQ)m3R9∗. (7)

Here L = |L| , µ is the shear modulus of the nucleus matter, Q is a factor character-izing energy dissipation in the nucleus, R∗ is the nucleus’ typical size, and m is itsmass. Note that the averaging is done over the Euler-Poinsot motion. The formula (7)is written without non-dimensional scaling factor, since its value is a matter of ongoingdiscussions. For us it is more important to demonstrate an opportunity to include theeffects due to the internal dissipation into the evolution equations and to understand,at least on the qualitative level, their interplay with effects due to the outgassing.

3 Evolutionary equations describing secular evolu-

tion of the attitude motion

3.1 Basic equations

In our analysis of the nucleus’ attitude dynamics, we use three right-hand orthogonalcoordinate systems with their origins at the center of mass, O .

OXY Z : the “perihelion” system, with the OZ -axis parallel to the line of apsidesand directed to the Sun at perihelion, the OX -axis normal to the plane of the or-bit, and the OY -axis parallel to the tangent to the orbit at perihelion and directedcorresponding to the orbit motion.

Oxyz : the frame connected with the angular momentum vector of the nucleus, L ,with the Oz axis directed along L and the Oy axis in the plane OXY . We definethe orientation of the coordinate system Oxyz with respect to the “perihelion” systemOXY Z through the angles ρ and σ . A turn through the angle σ about the OZ axisfollowed by a turn through the angle ρ about the Oy axis puts the trihedron Oxyzinto its current position from an initial position coinciding with the trihedron OXY Z(Figure 2).

Oξηζ : the body-fixed system, the axes Oξ, Oη, Oζ being the principal inertiaaxes. We assume that the inertia moments with respect to the axes Oξ, Oη, Oζ ,denoted by A,B,C respectively, satisfy

A < B < C (8)

The orientation of the system Oξηζ with respect to the system Oxyz is defined throughthe Euler angles φ, ϑ, ψ (Figure 3).

4

Figure 2: Angles used to define the orientation of the coordinate system Oxyz withrespect to the “perihelion” system OXY Z

Figure 3: Angles used to define the orientation of the body-fixed coordinate systemOξηζ with respect to the system Oxyz

5

It is easy to find the projections of the angular velocity vector, ω , on the axesOξ, Oη, Oζ :

ωξ =(L

A

)sinϑ sinψ, ωη =

(L

B

)sinϑ cosψ, ωζ =

(L

C

)cosϑ. (9)

Inserting (9) into the formula for the kinetic energy

T =1

2(Aω2

ξ +Bω2η + Cω2

ζ ), (10)

provides us with the expression for T as the function of L, ϑ, ψ :

T =L2

2

[sin2 ϑ

(sin2 ψ

A+

cos2 ψ

B

)+

cos2 ϑ

C

]. (11)

Let’s continue with the derivation of the evolutionary equations. Taking into ac-count (1), the following equations describing the variation of magnitude and directionof angular momentum vector, L , in the “perihelion” coordinate system are obtained:

dρ

dτ=M r

x

L,

dσ

dτ=

M ry

L sin ρ,

dL

dτ=M r

z . (12)

The values M rx ,M

ry ,M

rz in equations (12) are the projections of the reactive torque

onto the corresponding axes of the Oxyz coordinate system .The parameter Θ , the maximum angle between the Oζ -axis and L in unperturbed

motion, can be used as an “osculating” variable describing qualitative properties ofthe motion close to rotation around the main axis of inertia. The variation of Θ isdescribed by the equation resulting from (2)

dΘ

dt=

(dT

dΘ

)−1

[(eL × ω , eL ×Mr)−Rint] . (13)

Here eL denotes the unit vector directed along the angular momentum L . The kineticenergy of the nucleus rotation, T , is given by the formula

T =L2

2C

[1 +

(C

B− 1

)sin2Θ

]. (14)

Since maximum values of ϑ are achieved at ψ = 0 mod π , to obtain the relation (14)one should substitute ϑ = Θ, ψ = 0,mod π into (11).

In summary, the system of equations which will be the departure point for subse-quent consideration is:

dΘ

dt= hΘ(ρ, σ, L, ν, φ, ψ, ϑ), (15)

dρ

dt= hρ(ρ, σ, L, ν, φ, ψ, ϑ),

dσ

dt= hσ(ρ, σ, L, ν, φ, ψ, ϑ),

dL

dt= hL(ρ, σ, L, ν, φ, ψ, ϑ),

6

where ν is the true anomaly considered as a known periodic function of time withperiod T = 2π/Ω and Ω is the orbital mean motion of the comet.

The system (15) is still nonclosed since the right hand side of the equations dependon the Euler angles φ, ψ, ϑ . To obtain a closed system of equations, we take theexpressions for φ, ψ, ϑ as functions of time in the case of Euler-Poinsot motion (theseexpressions depend on Θ and L as parameters) and apply the averaging procedure. Asecond averaging over orbital motion gives us the system of the autonomous differentialequations which will be used to study the secular effect in nucleus spin state evolution.

3.2 Averaging procedure

Preliminarily, we would like to recall certain properties of the motion of a rigid body’sattitude in the absence of any perturbation (the so called Euler-Poinsot case). In theabsence of any external torques the angular momentum vector, L , does not vary, sothe variables ρ, σ, L preserve their initial values. The behavior of the Euler anglesφ, ϑ, ψ is governed by the equations

dϑ

dt= L sinϑ sinψ cosψ

(1

A− 1

B

), (16)

dφ

dt= L

(sin2 ψ

A+

cos2 ψ

B

),

dψ

dt= L cosϑ

(1

C− sin2 ψ

A− cos2 ψ

B

).

Here L should be considered as a parameter. The derivation of the equations (16) aswell as the explicit formulas for the variation of the Euler angles in torque-free motioncan be found, for example, in [42] (in another designations). Since these formulas arerather complicated (written in terms of the higher transcendental functions), it is im-portant to mention an opportunity to classify the unperturbed motions geometrically.To start we present in Figure 4 the inertia ellipsoid

Aξ2 +Bη2 + Cζ2 = 1 (17)

with several polhodes (recall that a polhode is a curve consisting of the points wherethe inertia ellipsoid is intersected by an instantaneous angular velocity vector, ω , forrigid body motion). If the polhode encircles the Oξ -axis, the corresponding motion iscalled complex long axis mode (complex LAM). If the polhode encircles the Oζ -axis,the motion is called complex short axis mode (complex SAM). Rotations about theOξ -axis or Oζ -axis are called simple LAM and simple SAM respectively. Althoughthese terms are not introduced in classical textbooks on rigid body dynamics, theyare widely used in publications on the rotational motion of celestial bodies ( [15, 17],etc). To determine the mode for given values of the kinetic energy, T , and angularmomentum, L , the quantity

w =2BT

L2(18)

can be used. If w ∈ (1, B/A) then we have complex LAM, while in the case w ∈(B/C, 1) we have complex SAM. In complex SAM, the quantity w and the maximum

7

Figure 4: The inertia ellipsoid and polhodes

value of the angle between the main inertia axis Oζ and the angular momentum vectorare related in the following way:

w(Θ) =B

C+(1− B

C

)sin2Θ. (19)

As mentioned, observations suggest that 19P/Borrelly’s nucleus is in a simple SAMor close to it. Nevertheless, we start by averaging over the Euler-Poinsot motion ina more general case for a complex SAM where the averaged equations for close to asimple SAM motion will be obtained as a limit. In unperturbed complex SAM, theintegration of equations (16) can be reduced to the integration of a single differentialequation

dψ

dt= −L

B

√(C −B)(B − wA)

ACU(ψ,Θ). (20)

Here

U(ψ,Θ) =

√AC

(C −B)(B − wA)× (21)

√(B

A− w

)−(B

A− 1

)cos2 ψ

√(B

A− B

C

)−(B

A− 1

)cos2 ψ

∣∣∣∣∣∣w=w(Θ)

,

To derive the equation (21) we used an integral of the unperturbed motion (11) toexpress ϑ as the function ψ .

Let φ(t), ϑ(t), ψ(t) correspond to complex SAM motion. Then the functions φ(t)and ψ(t) can be written as

φ(t) = ωφt+ φ1(t), ψ(t) = ωψt+ ψ1(t) (22)

8

where the quantities ωφ and ωψ are called the frequencies of the Euler-Poinsot motionand are given by the formulas:

ωφ =L

C

[1−

(1− C

A

)Π(λ, k)

K(k)

], (23)

ωψ = − π

2K(k)

(L

B

)√(C −B)(B − wA)

AC.

Here K(k) and Π(λ, k) denote complete elliptic integrals of the first and third kindrespectively and their parameters are

λ =C(B − A)

A(C −B), k =

√√√√(B − A)(B − wC)

(B − C)(B − wA), (24)

In general, the frequencies ωφ and ωψ are incommensurable. The functions φ1(t) andψ1(t) in (22) are (Tψ/2) -periodic functions of time, where Tψ = 2π/|ωψ| . The varia-tion of the angle ϑ in complex SAM is also described by a (Tψ/2) -periodic functionof time where the following relation is valid:

ϑ = arccos

√√√√√(BA− w

)−(BA− 1

)cos2 ψ(

BA− B

C

)−(BA− 1

)cos2 ψ

. (25)

An example illustrating the behavior of the variables in unperturbed complex SAMcan be seen in Figure 5. In this case the maximum value of the angle between thenucleus’ main inertia axis and the angular momentum vector is equal to 3 so themotion is close to simple SAM. Choosing the initial angular velocity close to the angularvelocity of 19P/Borrelly’s nucleus, ω∗ ≈ 6.712 · 10−5s−1 , we obtain the Euler-Poinsotmotion with the frequencies ωφ ≈ 8.751 · 10−5s−1 (Tφ = 2π/ωφ ≈ 0.831 days) andωφ ≈ −2.034 · 10−5s−1 (Tψ ≈ 3.576 days). Values of the inertia moments, A,B,C ,were assumed to be the same as in the example of the perturbed nucleus motion givenin Section 3.4.

In addition to averaging over the Euler-Poinsot motion, we average the right handside of equations (15) over the orbital motion. This approach (based on the jointaveraging over the Euler-Poinsot motion and over the orbital motion) was first appliedto study the weakly perturbed attitude motion of a rigid body in a Keplerian orbitin [4]. Formally it means the calculation of the integrals

1

TTφTψ

∫ T

0dt′∫ Tφ

0dt′′

∫ Tψ

0hΘ,ρ,σ,L(ρ, σ, L, ν(t

′), ωφt′′+φ1(t

′′′), ωψt′′′+ψ1(t

′′′), θ(t′′′))dt′′′

(26)Taking into account that in unperturbed motion

dν

dt=

Ω

(1− e2)3/2(1 + e cos ν)2 (27)

where e is the comet’s eccentricity and using the relations (20) and (25), the integrals(26) can be rewritten in the form that is more convenient for calculations:

1

TTφTψ

∫ T

0dt′∫ Tφ

0dt′′

∫ Tψ

0hΘ,ρ,σ,L(ρ, σ, L, ν(t

′), ωφt′′+ (28)

9

Figure 5: The behavior of the angles φ, ϑ, ψ in unperturbed complex SAM

10

Table 1: Orientation of 19P/Borrelly’s angular momentum vector

Authors RA(deg) Dec(deg) ρ(deg) σ(deg)

Chesley [5] 208± 2.5 −4± 2.5 42.2 113.4

Farnham, Cochran [10] 214± 4 −5± 4 36.4 111.0

Samarasinha, Mueller [30] 221± 11 −7± 5 29.2 108.8

Schleicher et al [33] 214± 1 −5.7± 1 36.1 112.0

IAU/IAG working group [34] 218.5± 3 −12.5± 3 29.6 121.0

Note: The comet orbital parameters were taken from the JPLwebsite devoted to Solar System dynamics (http://ssd.jpl.nasa.gov)

φ1(t′′′), ωψt

′′′ + ψ1(t′′′), θ(t′′′))dt′′′ =

(1− e2)3/2

16π2K(k)

∫ 2π

0

dν

(1 + e cos ν)2

∫ 2π

0dφ∫ 2π

0

hΘ,ρ,σ,L(ρ, σ, L, ν, φ, ψ, ϑ(ψ))dψ

U(ψ,Θ)

To calculate these integrals without any additional assumptions about the attitude’smotion properties is possible only numerically. More details on the application of nu-merical averaging in theoretical studies of spin state evolution under non-gravitationaltorques can be found in [32].

3.3 Approximate equations

As it was mentioned in Section 1, the spin state of 19P/Borrelly’s nucleus can becharacterized as simple SAM or close to it. Table 1 summarizes the information onthe spin axis orientation in the standard equatorial coordinate system. This tablealso contains the corresponding orientation of the angular momentum vector, L , inthe “perihelion” system OXY Z under the assumption of principal axis rotation. Agood agreement between the results provided by different authors is evident. Theobservations allow us to conclude that the orientation of the angular momentum vectorof the nucleus is relatively close to the direction of the apsides line (only in [5] theorientation was predicted with ρ slightly greater 40 ).

Realistically only a small fraction of 19P/Borrelly’s nucleus surface is active (theestimation ranges from 4% [33] to 8% [18]). The DS1 images indicate that the mostactive area is located in the central part of the nucleus (waist) near the rotation axisand is constantly illuminated by the Sun at the perihelion passage. This simplifies theaveraging procedure since we can avoid the necessity to check the insolation condition(6).

Finally we obtain the following approximate evolutionary equations valid at smallvalues of the angles ρ and Θ :

dΘ

dt= − 1

L

[L4

κ+

3β

2(R∗V∗Q∗)ΦD1

]Θ, (29)

dρ

dt=β(R∗V∗Q∗)

2L[Φ1D1 − Φ2(D1 sin σ +D2 cos σ) sin σ] ρ,

11

dσ

dt=β(R∗V∗Q∗)

2L[Φ1D2 + Φ2(D1 cosσ −D2 sin σ) sin σ] ,

dL

dt= β(R∗V∗Q∗)Φ1D1,

where

κ ∼ (µQ)m2R7∗, Φ1 =

(1− e2)3/2

π

∫ π/2

0

cos νg(r(ν))dν

(1 + e cos ν)2≈ 5.219 · 10−2,

Φ2 =(1− e2)3/2

πg(r(π

2

))≈ 1.822 · 10−2

D1 =N∑j=1

sj(nj, eζ)(eζ ,dj), D2 =N∑j=1

sj(eζ ,nj × dj),

dj = nj ×(Rj

R∗

), j = 1, . . . , N.

The designation eζ is used for the unit vector directed along the axis Oζ of thebody-fixed coordinate system.

Parameters D1 and D2 describe the dependence of the resulting reactive torqueon the geometry of the active zones (i.e. areas of active nucleus matter sublimation)and Φ1 is a certain integral characteristic specifying the comet’s activity.

3.4 Numerical verification

To test the validity of the discussed approximation, we compared the results of thenumerical integration of non-averaged motion equations, averaged motion equationsand evolutionary equations (29) with the same initial conditions.

The nucleus geometrical parameters were taken from [25]. These parameters cor-respond to a 19P/Borrelly-like nucleus consisting of two ellipsoidal components withmajor semi-axes 1.6, 1.8, 3.0 km and 0.96, 1.08, 1.8 km respectively, with a distancebetween the centers of these components of 3.7 km (Figure 1). The nucleus is as-sumed to be homogeneous with a mean density of 0.3 g cm−3 (in agreement with re-cent estimation of 19P/Borrelly’s nucleus density as 0.1− 0.3 g cm−3 [7]). After somestraightforward calculations we obtain

A = 1.3287 · 1019kgm3, B = 5.1327 · 1020kgm3, C = 5.2886 · 1020kgm3, (30)

andm = 1.2718 · 1013kg. (31)

The nucleus waist can be interpreted as the origin of the bright jet seen in DS1images, although the highest resolution images reveal a complex structure of severalinteracting jets on the base of the main jet. For definiteness we assume the existence ofthree jets corresponding to the active zones defined in the body-fixed reference frameOξηζ by radius-vectors R1,R2,R3 (Table 2). Under the assumption of the equalintensity of the matter ejected from these zones ( si = 1/3, i = 1, 2, 3 ), the geometricparameters D1 and D2 have the following values:

D1 = 1.71 · 10−3, D2 = 1.597 · 10−2 (32)

In [18] the H2O production rate was estimated as 2.05× 1028 molecules sec −1 ata heliocentric distance 1.41 AU. Since this distance is close to 19P/Borrelly’s perihe-lion distance, q , it corresponds to the case when all the active areas are illuminated

12

Table 2: Position and orientation of the active zones

Zone 1R1 R∗(−0.09958, 0.10890, 0.36281)TOξηζn1 (−0.17789, 0.23609, 0.95531)TOξηζ

Zone 2R2 R∗(−0.18462, 0.01054, 0.35605)TOξηζn2 (−0.24927, 0.03398, 0.96784)TOξηζ

Zone 3R3 R∗(−0.09540,−0.15139, 0.35139)TOξηζn3 (−0.17362,−0.30169, 0.93747)TOξηζ

Note: The accepted value of the typical linear size R∗ = 4 km

simultaneously and the total mass loss due to sublimation does not differ substantiallyfrom hypothetical value Q∗ . So we can estimate Q∗ as 600 kg sec −1 . The effectivevelocity V∗ and the momentum transfer efficiency β were assumed to be 250m sec−1

and 0.833 respectively (nearly the same value of the momentum transfer efficiency wasused in papers [11]).

The influence of the nucleus’ non-rigidity on its attitude dynamics in our tests wasnot taken into account. Although it can easily be justified using an argument of theshort duration of the considered time interval in comparison with the characteristictime estimations of the secular effects due to internal dissipation, actually it was donedue to another reason. Expression (7) allows us to imitate the dynamical consequenceof the non-rigidity dealing with the equations of motion of a rigid body but it only worksfor the averaged equations. It is a much more difficult problem of how to introducea correction factor into non-averaged rigid body equations to imitate the non-rigidity.Until now a conceptually clear and convincing solution of this problem has not beengiven. So instead of dealing with some empiric formulas (e.g., [40] Sec. 2.3.2), welimited the integration interval.

Figure 6 shows the typical results of integrating the equations of motion (bothaveraged and non-averaged) where the following initial conditions were assumed

Θ = 3, ρ = 30, σ = 110, L = Cω∗. (33)

One can see nice agreement between the results obtained. Similar results were obtainedin the integrations of several other initial spin axis orientations satisfying the conditionof the active zones permanent illumination at the perihelion passage. This allows usto conclude that the approximate equations (29) describe the secular evolution of thenucleus in a rather accurate way and can be used to study its properties.

The predicted evolution of the Borrelly’s nucleus spin state is relatively slow incomparison with what was proposed for some other comets (e.g., [11, 12, 13, 38]). Itis not so surprising if we take into account that active zones on Borrelly’s nucleus areplaced in the most non-effective way for creation a reactive torque. Also the analysis inthe mentioned papers was based on more refined thermophysical models, in which theice sublimation rate (and, respectively, a reactive torque) drops with increase of theheliocentric distance not so fast as it is follows from Marsden’s formula (5). We supposethat the main qualitative properties of the rotation evolution should be robust withrespect to the model used. Nevertheless, we consider as the direction of the future work

13

Figure 6: Example of the long-term evolution of the rotation state. Wiggly linesrepresent the solution of non-averaged motion equations. Solid straight lines correspondto the solution of numerically averaged motion equations. Dashed lines represent thesolution of the approximate evolutionary equations

14

the application of the modern thermophysical models to compute the reactive torquesand to increase the reliability of our conclusions regarding the attitude dynamics ofBorrelly’s nucleus.

4 Some qualitative aspects of 19P/Borrelly’s nu-

cleus attitude dynamics

It follows from (29) that the long-term evolution of the nucleus attitude motion dependson unknown values of parameters D1, D2 and κ . Currently, we can not estimate D1

and D2 , nor do we know their sign. Nevertheless, if future observations show that19P/Borrelly’s nucleus spins up (dL

dt> 0) , it would imply that D1 > 0 (respectively,

spin down of the nucleus would imply D1 < 0 ). If it is possible to measure the nucleusprecession at least on the short arc, then the second and third equations in (29) allowsus to estimate D1 and D2 and to predict the further evolution of the orientation ofL .

On the other hand, it is reasonable to assume that the factor before Θ (i.e., theexpression in the square brackets) in the first equation of (29) has positive value forBorrelly’s nucleus. We have mentioned about a region of cracks seen in the DS1images [2]. This suggests small rigidity of the nucleus, at least locally. Small rigiditytogether with possible existence of inner cracks produced by the same processes thathave broken the nucleus’ surface, make the inner dissipation Rint more intensive.Therefore, we believe that even under destabilizing reactive torques (in the case D1 <0 ), the inner dissipation could provide stability of the rotation about the main inertiaaxis.

Currently 19P/Borrelly is not the only short-period comet with well determinedattitude dynamics. Information obtained during spacecraft flybys in combination withground based observations allows to assume that the nuclei of the comets 9P/Tempel 1and 81P/Wild 2 are in the principal axis rotation state or close to it (more definitely inthe case 9P/Tempel 1 [39] and more presumably in the case 81P/Wild 2 [35]). This isdrastically different from the non-principal axis rotation of comet 1P/Halley (accordingto [1], the long axis of 1P/Halley’s nucleus is inclined to the total angular momentumvector L by ≈ 66 and rotates around L with a period of 3.69 days where the spincomponent about the long axis has a period of 7.1 days). An important open questionis the reason for such a difference. This difference becomes paradoxical, if we takeinto account that 19P/Borrelly, P/Wild 2 and P/Tempel 1 are Jupiter family cometsand more often come close to the Sun (their nuclei are more often exposed to thenon-gravitational torques) than the comet 1P/Halley 1. Nevertheless, the properties offour objects only can not provide a reliable basis for any conclusion about the typicalrotation states of the comet nuclei in general.

5 Concluding remarks

Our paper can be considered as an additional example of how the application of theaveraging procedure reveals the secular effects in the rotational motion of the celestialbodies. Although we used a very simple model of matter sublimation, the averaging ofthe equations of motion is an effective tool to increase effectiveness of the dynamicalstudies based on more realistic and respectively more complex models of the involved

15

physical processes. Also it is worth while to mention that the developed technique canbe used in studies of another phenomena which attracts attention of specialists, the socalled YORP effect in asteroids rotational motion [26].

Acknowledgements. The authors would like to express their thanks to M.Efroimsky,S.Feraz-Mello, P.J. Gutierrez, Yu.V.Skorov for useful discussions and advice during theaccomplishment of this work. The comments provided to us by anonymous refereeswere very important for improving the quality of the manuscript. V.V.S. acknowledgessupport from the Russian Foundation for Basic Research via Grant NSh-1123.2008.1.

References

[1] Belton, M.J.S., W.H.Julian, A.J.Anderson, and B.E.A.Mueller: 1991, ‘The spinstate and homogeneity of comet Halley’s nucleus’, Icarus, 93, 183–193.

[2] Britt, D.T., and 9 co-authors: 2002, ‘The geology of comet 19P/Borrelly’, LPI,33, 1686.

[3] Burns, J.A. and V.Safronov: 1973, ‘Asteroid nutation angles’, MNRAS, 165, 403–411.

[4] Chernousko, F.L.: 1963, ‘On the motion of a satellite about the mass centreunder the action of gravitational moments’, Prikl. Mat. Mekh., 27, 474-483 (inRussian)=Appl. Math. Mech., 27, 708–722 (English translation)

[5] Chesley, S. R.: 2002, ‘Modeling Comet Nongravitational Accelerations with Dis-crete Rotating Jets’, Bull. Am. Astron. Soc., 34, 869.

[6] Crifo, J.F., and A.V.Rodionov: 1999, ‘Modelling the circumnuclear coma ofcomets: objectives, methods and recent results’, Planetary and Space Science,47, 797–826.

[7] Davidsson, B.J.R., and P.J.Gutierrez: 2004, ‘Estimating the nucleus density ofComet 19P/Borrelly’, Icarus, 168, 392–408.

[8] Efroimsky, M. and A.Lazarian: 2000, ‘Inelastic dissipation in wobbling asteroidsand comets’, Mon. Not. R. Astron. Soc., 311, 269–278.

[9] Efroimsky, M.: 2000, ‘Precession of a freely rotating rigid body. Inelastic relaxationin the vicinity of poles’, J. of Math. Physics, 41, 1854-1888.

[10] Farnham,T.L., and A.L.Cochran: 2002, ’A McDonald Observatory Study ofComet 19P/Borrelly: Placing the Deep Space 1 Observations into a Broader Con-text’, Icarus, 160, 398-418.

[11] Gutierrez, P.J., and B.J.R.Davidsson: 2007, ‘Non-gravitational force modeling ofComet 81P/Wild 2’, Icarus, 191, 651–664.

[12] Gutierrez, P.J., L.Jorda, J.L.Ortiz, and R.Rodrigo: 2003, ’Long-term simulationof the rotational state of small irregular cometary nucleui’, Astron. Astrophys.,406, 1123–1133.

16

[13] Gutierrez, P.J., L.Jorda, N.H.Samarasinha, and P.Lamy: 2005, ’Outgassing-induced effects in the rotational state of comet 67P/Churyumov-Gerasimenkoduring the Rosetta mission’, Planetary and Space Science, 53, 1135-1145.

[14] Jewitt, D.: 1997, ‘Cometary rotation: an overview’, Earth, Moon, and Planets,79, 35–53.

[15] Julian, W.H.: 1987, ’Free precession of the comet Halley nucleus’, Nature, 326,57–58.

[16] Julian, W.H.: 1990, ‘The comet Halley nucleus: random jets’, Icarus, 88, 355–371.

[17] Kinoshita, H.: 1992, ‘Analytical expansions of torque-free motions for short andlong axis modes’, Celes. Mech. Dyn. Astronomy, 53, 365–375.

[18] Lamy, P.L., I.Toth, and H.A.Weaver: 1998, ‘Hubble Space Telescope observationsof the nucleus and inner coma of comet 19P/1904 Y2 (Borrelly)’, Astron. Astro-phys., 337, 945–954.

[19] Oberti, P., E. Bois, and C. Froeschle: 1991, ‘Rotational behavior of comet nu-clei under gravitational perturbations’,Asteroids, Comets, Meteors, LPI, Houston,439-442.

[20] Bois, E., P. Oberti, and C. Froeschle: 1992, ‘Rotational behavior of cometary-typebodies’, in: S.Ferraz-Mello (Ed.), Chaos, Resonance and Collective DynamicalPhenomena in the Solar System, Kluwer, Dordrecht, 291–296.

[21] Marsden, B.G., Z.Sekanina, and D.K.Yeomans: 1973, ‘Comets and nongravita-tional forces.V’, Astron. J., 78, 211–225.

[22] Mueller, B.E.A. and N.H.Samarasinha : 2002, ‘Visible Lightcurve Observations ofComet 19p/Borrelly’, Earth, Moon, and Planets, 90, 463-471.

[23] Mysen, E.: 2007, ’The dynamics of globally active triaxial comets, with applica-tions to asteroid rotation’, Mon. Not. R. Astron. Soc., 372, 1345–1350.

[24] Neishtadt, A.I., D.J.Scheeres, V.V.Sidorenko, and A.A.Vasiliev: 2002, ‘Evolutionof comet nucleus rotation’, Icarus, 157, 205–218.

[25] Neishtadt, A.I., D.J.Scheeres, V.V.Sidorenko, P.Stooke, and A.A.Vasiliev: 2002,‘The influence of reactive torques on comet nucleus rotation’, Cel. Mech. Dyn.Astronomy, 90, 307–330.

[26] Rubincam, D.P.: 2000, ‘Radiative spin-up and spin-down of small Asteroids’,Icarus, 148, 2–11.

[27] Peale, S.J. and J.J.Lissauer: 1989, ‘Rotation of Halley’s comet’ Icarus, 79, 396–430.

[28] Samarasinha, N.H. and M.J.S.Belton: 1995, ‘Long-term evolution of rotationalstates and nongravitational effects for Halley-like cometary nuclei’, Icarus, 116,340–358.

17

[29] Samarasinha, N.H: 2007, Rotation and activity of comets, Advances in Space Re-search, 39, 421–427.

[30] Samarasinha, N.H. and B.E.A.Mueller: 2002, ‘Spin axis direction of comet19P/Borrelly based on observations from 2000 and 2001’, Earth, Moon, and Plan-ets, 90, 473–482.

[31] Samarasinha, N.H., B.E.A.Mueller, M.J.S.Belton, and L.Jorda: 2004, ’Rotation ofcometary nuclei’, in: M.C. Festou, H.U. Keller, and H.A. Weaver (Eds.), CometsII, University of Arizona Press, Tucson, 281–299.

[32] Sazonov, V.V.: 1994, ’Motion of an asteroid about its center of mass due to torquefrom light pressure’, Sol. Syst. Res., 28, 152-162.

[33] Schleicher, D.G., L.M.Woodney, and R.L.Mills: 2003, ‘Comet 19P/Borrelly atmultiple apparations: seasonal variations in gas production and dust morphology’,Icarus, 162, 415–442.

[34] Seidelmann, P.K., and 14 co-authors: 2007, ’Report of the IAU/IAG WorkingGroup on cartographic coordinates and rotational elements: 2006’,Cel. Mech. Dyn.Astronomy, 98, 155–180.

[35] Sekanina, Z., D.E.Brownlee, T.E.Economou, A.J.Tuzzolino, and S.F.Green: 2004,’Modeling the nucleus and jets of comet 81P/Wild 2 based on the stardust en-counter data’, Science, 304, 1769–1774.

[36] Sharma, I., J.A.Burns, and C.-Y.Hui: 2005, ’Nutational damping times in solidsof revolution’, Mon. Not. R. Astron. Soc., 359, 79–92.

[37] Soderblom, L.A., and 20 co-authors: 2002, ‘Encounter with comet 19P/Borrelly:results from the Deep Space 1 Miniature Integrated Camera and Spectrometer’,LPI, 33, 1256.

[38] Szego, K., J.F.Crifo, L.Foldy, J.S.V.Lagerros, and A.V.Rodionov: 2001, ‘Dynam-ical effects of comet P/Halley gas production’, Astron. Astrophys., 370, L35–L38.

[39] Thomas, P.C. and 13 co-authors: 2007, ’The shape, topography, and geology ofTempel 1 from Deep Impact observations’, Icarus, 187 , 4–15.

[40] Vokrouhlicky, D., S.Breiter, D.Nesvorny, and W.F.Bottke: 2007, ’GeneralizedYORP evolution: onset of tumbling and new asymptotic states’, Icarus, 191,636–650.

[41] Whipple, F.L.: 1951, ‘A comet model.I. Acceleration of comet Encke’, Astrophys.J., 113, 464–474.

[42] Whittaker, E.T.: 1917, A Treatise on the Analytical Dynamics of Particles andRigid Bodies, 2nd edition, University Press, Cambridge, UK.

18