The exact dimensions are not important as the purpose of ... · Newton's Second Law of Motion...

Attitude dynamics during landing on an active comet

AJ. Oria, T.S. Bowling

Department of Aerospace Science, Cranfield University, Cranfield,

Bedfordshire MK43 OAL, UK

Abstract

On May 2012 the European Space Agency Rosetta spacecraft will rendezvouswith Comet P/Wirtanen and begin an in-situ investigation of the cometarynucleus. One, or possibly two, probes will be deployed to analyze the surfacematerial. The surface science packages (SSP's) are small and will carry nomeans for active attitude or trajectory control during the descent. It is crucialfor the probe to land in an upright position to ensure a proper data relay backto Earth via the orbiting spacecraft. During deployment, stability will beaffected by an incoming stream of gas and dust particles caused by thesublimation of surface ices, but this may actually be used to stabilize the vehicleand ensure an upright orientation upon impact on the surface.

Introduction

The surface lander is a multi-sensor instrument unit designed for in-situ analysisom the cometary surface for a period of several tens of hours. There are twobasic versions under study: a passive soft lander option and an active softlander option. The probe's primary structure consists of a cylinder supportingthe spacecraft's separation interface and some sensors, such as the cameras ofthe in-situ imaging system, and a circular plate supporting all other items ofequipment. This equipment comprises four lithium/sulphur-dioxide primarybatteries, a control unit common to all the equipment items and science sensors,a power distribution unit, and a VHF-band communications package for datatransmission. The single communications link is the science data transmission at0.2 kbit/s from surface to spacecraft; e.g. Schwehm & Hechler". In its closedconfiguration, the probe is shaped roughly like a prolate ellipsoid (Fig. 1). In

Transactions on the Built Environment vol 19, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

624 Structures in Space

this analysis it is assumed that the basic dimensions are: 80 cm wide and 40 cmlong, with a total mass (including structure and instruments) of about 30 kg.The exact dimensions are not important as the purpose of this study is toanalyze the effects of outgassing in general, and not for one particular design.

80cm

Centerofpressum^-— ^ Centerofmass

Z "X

Figure 1: Basic configuration of the Champollion passive soft lander

The probe opens up like a petal after it has landed, very much like the SovietLuna class vehicles that landed on the Moon in the 1960's. The center of massof the vehicle is assumed to be located 10 cm below the center of pressure and,because of this offset, translational forces due to cometary outgassing createmoments about the center of mass which affect the attitude of the probe duringthe descent. There are no retro-rockets to slow the vehicle down as it reachesthe surface, so the probe is covered with impact-damping material to absorbshocks upon impact.

Theory

Free Molecular Flow

Aerodynamic forces caused by the stream of gas and dust particles radiatingfrom the nucleus become more severe as the probe moves closer to the surface.Once the surface probe is released there are no means to adjust its trajectory orcontrol the attitude. Aerodynamic forces and moments are obtained byintegrating the pressure and shear stresses acting on the exposed surfaces. Allfluid flowfields are governed by the continuity equation (conservation of mass),Newton's Second Law of Motion (conservation of momentum), and the FirstLaw of Thermodynamics (conservation of energy). Individual gas moleculesare so far apart in cometary atmospheres that the flow loses its continuumnature and becomes rarefied flow or free molecular flow. The flowfield isdescribed by the Boltzmann equation, which describes the mechanics ofindividual molecules. Gases at a microscopic level consist of a large numberof particles which are in constant random motion and they collide with eachother, and any solid boundary present; e.g. AHearn and Festou*, Keller*, Regan& Anandakrishnan". Properties like temperature and velocity are obtained by


Structures in Space 625

taking the average of all the particles within a volume over an interval of time.The distance a particle has to travel until it becomes indistinguishable from allothers is defined as the relaxation distance. If the relaxation distance is smallcompared to the characteristic dimension of the fluid, macroscopic propertiesare considered to vary continuously and the flow is described as a continuum.However, in free molecular flow the particles are much farther apart and therelaxation distance will be much larger. One measure of the relaxation distanceis the mean free path, e.g. A'Hearn and Festou*,

where a is the effective collision diameter (diameter of the gas molecules) andn is the number density (number of molecules per unit volume). The mean freepath is defined in a reference frame that moves along with the flow. TheKnudsen number is defined as

where A is the mean free path and L is the characteristic flowfield length. Ahigh value for Kn means that the gas molecules are far apart and the Boltzmannequation should be used. In general, we can say that Kn » 1 for continuumflow, and Kn « 1 for free molecular flow. Continuum flow is dominated byintermolecular collisions, whereas in free molecular regime the flow isdominated by molecule-surface interactions. For a Halley-sized comet at 3.0AU, a typical number density of l.Oxl 0" n/m* (molecules per unit volume) anda molecular cross section area of 1.0x10"** m* (cross section of water moleculesat - 160°K), yields a molecular mean free path of, approximately 707.1 m.Since the scale of the surface probe is approximately 1.0 m, the Knudsennumber is quite high and the flow can be modelled as a free molecular regime.However, it should be noted that at a distance of 1.0 AU the number densityincreases to 1.0x10" n/m* which given a mean free path of only 0.7071 m.This means that the flow has shifted into the transition regime as intermolecularcollisions become more important.

Aerodynamic Forces and Moments

When modelling free molecular flow we assume that the mean free path ismuch larger than the characteristic dimension of the body. The flowfield in thevicinity of the body is dominated by particle-surface collisions and the numberof collisions between incident and reflected particles is negligible, so theincident stream of molecules is undisturbed. The main problem with thisanalysis is that the molecule-surface interaction is not fully understood. Despiteextensive studies in this area the most useful models are the specular anddiffuse reflection postulated by Maxwell in 1879. If the reflection is specularthe collision of the particles on the surface is elastic and the inbound velocity isequal to the outbound velocity (with opposite sign). This model predicts that



there is only normal momentum transfer and no tangential energy transfer. Onthe other hand, the diffuse model assumes that the velocity of the reflectedparticles would have the same velocity distribution as if emanating from animaginary gas behind the reflecting surface. The normal momentumaccommodation coefficient (<%,) and the tangential momentum accomodationcoefficient (a,) are defined as,

where p and r are the normal and tangential momentum flux of the incidentparticle (/), reflected particle (r), and at the surface (w). They define therelative transfer of energy from the gas onto the probe. For a diffuse reflectionOn = Ot = 1, and for a specular reflection a» = o> = 0. Assuming a Maxwellianvelocity distribution the momentum and energy flux at the surface of the bodycan be determined. The pressure and shear stress coefficients (Cp and G)acting upon an elemental area inclined to the freestream at an angle 0 are, e.g.Regan & Anandakrishnan**,

2.T

(1 \ n (T \(2 - crV - + s' sm\0)\ + —- — -Jxs • sin(0)

C,

where q is the dynamic pressure. The error function is defined as,

(5)

(6)

(7)

Parameter s is defined as the free stream molecular speed ration, which issomewhat analogous to the Mach number. We define a parameter called /?which is used to obtain the free stream molecular speed ratio,

J. = K./L where /L=#UL (8,9)

where V is the free stream velocity, T is the temperature of the gas, and R isthe molecular gas constant. Since water molecules are not diatomic (i.e. O:,N%), and water vapor is far from being a perfect gas it is somewhat difficult tocalculate the gas constant. The problem now is to compute the molecular gasconstant R for the stream of sublimated cometary material. For water vapor atvery low pressures, the specific heat at constant pressure (Cp) can beapproximated as, e.g. Van Wylen & Sonntag ,

c, = 143.05 -183.540°" + 82.7510°* - 3.69890 (10)

where


100.0The specific heat at constant volume is defined as


(11)

(12)

where y is the ratio of specific heats (y = 1.33 for water vapor). Therefore, thegas constant is defined as,

5 = c,-fv (13)

where Cy and c> are obtained from eqns. 10, 1 1 & 12. Finally, we substitute thegas constant into eqn. 14. The mean molecular mass is assumed to be 18.02 aswater is the most abundant species.

R = -R (14)18.02 ^ *

From eqns. 6, 7 and 8 it is apparent that the skin friction depends only onthe incident stream and reflection characteristics, where as the pressure dependsonly on the wall pressure . For simple shapes, eqns. 6 and 7 can be integratedover the surface of the body. However, this is easy to say and difficult to carryout in practical cases. Where closed-form integration is not possible it isnecessary to divide the surface of the body into small panels. The force andmoment coefficient vectors ((/, and Cm) on each individual panel are computedand then all added; e.g. Regan & Anandakrishnan" Hence,

(15)1=1 * * ' *'

and

(16)

where p is the total number of panels, r is the position vector of the centroid ofeach panel, n the unit vector normal to each panel, and K is the freestreamvelocity unit vector. Therefore, the total force and moment on the body is,

FpVlSCf and M = pV2SLC. (17,18)

where p is the freestream mass density, S is the reference area, and L is thecharacteristic length of the body. These equations assume that a particle thathas been reflected onto one panel will not impinge onto another panel. Thusthe model is, strictly speaking, only valid for convex bodies. These equationsallow us to compute the aerodynamic forces produced by the stream ofcometary material on different configurations for the surface probe. This isimportant because, even if we select a descent trajectory which does notintersect any cometary jet, there is still a residual outgassing from the surfacewhich amounts to, approximately, 10 % of all the material that is ejected intospace by the comet nucleus; e.g. Keller*, Rickman'*.



Results

Descent Trajectory

Once the landing site is identified Rosetta will manoeuvre into position torelease the Champollion surface science package from a distance of at least 3km above the comet surface. The problem of navigating at such close range toa comet nucleus has been discussed by several authors; e.g. Champetier et al/,Guillaud & Lafontaine*, Lafontaine*, Oria & Bowling**, Prado et al.*\ In thisanalysis it is assumed that Rosetta is inserted in a circular orbit. The probe isfired at a velocity of 1 m/s because escape velocity at the surface is only 0.5 m/sand we want to avoid the probe bouncing right off the surface and be ejectedback into space. At such low speeds the momentum dissipates upon impact,assuming that the surface material is mainly ash and a porous mixture of ice anddust. The probe will settle quickly on the surface.

Trajectory

Error Ellipsoid

Figure 2: Geometry of the descent trajectory for the SSP

One option is to fire the probe aiming directly to the nucleus (y = -90°),which is also the origin of the inertial set of axes. However, the probe stillmoves downrange as the circular velocity of the orbiter is directly transferredonto the lander. This velocity component will depend on the distance at whichthe orbiter is located, as well as the mass of the comet nucleus. The trajectory



frame (X\Y\Z*) is defined so that the Z axis is always pointing "up", the Y?axis the downrange, and the X^ axis is the crossrange directed towards theright of the velocity vector. The descent trajectory is confined to the Z*Vplane as out of plane forces are small. The average radius of the nucleus is 945m, which corresponds to a sphere with identical volume to a 3.0 x 1.51 x 1.49km triaxial ellipsoid with a mean density of 500 kg/m*. These parameters try torepresent what an average comet nucleus may turn out to look like, but itshould be understood that this choice is somewhat heuristic; e.g. Etienne*,Huebner*, McDonnell et al.*°, Mohlmann & Kuhrt", Rickman". The gasproduction rate has been scaled to this size, and assumes an active surface areaof 1 km ; McDonnell et al.'°, Oria & Bowling". The levels of outgassing arevery hard to predict and may fluctuate by up to 50 %; e.g. Manson™. This willnot radically change the trajectory of the probe, but is could move the landingspot a few tens of meters back and forth. This results in a positionaluncertainty, or error ellipsoid, which grows in time (Fig. 2).

Simulationsof the descent are carried out for a heliocentric distance of 3.0AU. The surface temperature is only 160°K and the total gas production rate is8.0x10" mol/sec; e.g. McDonnell et al*°, Rickman". The gas stream velocityat the surface is approximately 300 m/s, Keller*. The descent trajectorydepicted in Fig.3 is chosen so as to avoid any cometary jet. The spacecraft isonly affected by residual outgassing, which accounts to only 10 % of the total.Results show that residual outgassing is not strong enough to perturb thedescent trajectory. The gravitational pull is so weak that the trajectory isessentially a straight line. Plans are that Rosetta will encounter CometP/Wirtanen at a distance of 4.4 AU and critical near nucleus operations will notcommence until it moves within a heliocentric distance of 3.25 AU; Schwehmand Hechler". There could, however, be some surprises and the activity couldbe much higher than expected, but even an increase of one order of magnitudein the levels of gas production affect the descent trajectory.

z'(m)

Y"(m)

Figure 3: Descent trajectory. The shaded area represents the 945 m radiusequivalent sphere, which is elliptical because the axes are not to the same scale.



Attitude Dynamics

The surface probe does not have any mechanism for active attitude control andtakes almost one hour to reach the comet surface and, during this time,aerodynamic torques perturb the attitude. Two different sets of pitch and yawangles are defined, one with respect to the flight path line, and the other withrespect to the local surface normal (Fig. 4). The pitch angle measures attitudechanges contained in the Z*V plane, and the yaw angle those perpendicular tothis plane. Angles 6p and XJ/F define the pitch and yaw with respect to the flightpath line, whereas GL and IJ/L are defined with respect to the local surfacenormal (Z* in Fig. 2).

Trajectory Plane

Local Normal

Flight Path Line

Figure 4: Reference directions used for the pitch and yaw angles.

Figure 5 depicts the attitude dynamics when the probe is not spun (o>i = 0).The flight path angle y changes from -90° to -45° during the descent becausethe velocity vector is rotated by the incoming flow. The average pitchoscillation frequencies is 4.5x10"* Hz. Note that there is an overall change of35° in pitch with respect to the flight path line. The reason for such a largechange in 0p is that the local normal vector is defined as line than joins theorigin of the coordinate axes X**,Y**,Z** and the center of mass of the probe.Because comet nuclei are so small, the direction of the stream of gas and dustparticles (i.e. the local normal) relative to the spin axis changes as the vehiclemoves down. The probe is fired almost straight down at the surface, so the



aerodynamic torques are initially very small. However, as the probe gets closerto the surface the direction of the flow relative to the spin vector changes andthe aerodynamic torques get stronger. The steady state pitch angle is laggingapproximately one degree behind the local surface normal. This result is veryimportant as it shows that, even though the translational forces are very weak(Fig. 3), aerodynamic torques are strong enough to stabilize the probe and keepit aligned within one degree of the local surface normal. Because the probe isnot spin stabilized pitch and yaw motion remain uncoupled and the effects ofoutgassing on the yaw angle is negligible.

r(nui) 7 y(rad)

xHT* -' xio"

T(iec) T(sec)Figure 5: Spacecraft attitude during the descent (coi = 0)

The next step is to analyze the effects of spin stabilization on attitude. Thespacecraft is symmetric about its spin axis, so roll motion is uncoupled frompitch and yaw motions. The spin rate Q)\ remains unchanged throughout thesimulation because energy dissipation effects in a free molecular flow regimeare negligible. However, if the probe is spun then pitch and yaw motioncouple. Figure 6 depicts the attitude time history attitude when it is spinning ata rate of 30 rpm. Spin stabilization introduces a stiffness term that tries tomaintain the spacecraft aligned with the flight path line. Towards the end ofthe simulation aerodynamic forces become stronger and the spin axis isperturbed from its original orientation by 0.0002° in pitch and 0.2° in yaw.Even though the aerodynamic torque is perpendicular to the Z*V plane in Fig4, gyroscopic precession moves the spin axis out of the plane. In other words,a torque about the pitch axis results in a perturbation along the yaw axisbecause the motion is coupled.

The attitude of the descent probe changes very little for a 30 rpm spin rate.However, this may be a problem because it results in a 35° angle of incidence



upon impact on the surface. The reason for such a large angle is that theeffects of curvature are very significant due to the small size of the cometnucleus. During the 53 minutes it takes for the probe to reach the surface itmoves more than 500 m downrange. This is quite a lot for a comet nucleuswhose dimensions are just 3.0 x 1.51 x 1.49 km. The probe may tip over overif the ground slope too large, or it may even skip off the surface if not enoughtranslational energy is dissipated upon landing. A possible solution to thisproblem is to force the orbiter into a cometosynchronous orbit at the time thesurface lander is deployed. There are considerable risks involved with thismanoeuvre which could lead to Rosetta crashing onto the surface shouldsomething go wrong. This would get rid of the downrange velocity componentand reduce the effects of curvature. However, there are unavoidable pokingerrors which must still be taken into account. An angle misalignment of only 3°would result in a 12.6° angle of incidence upon landing.

Y (rad) ,

' 0 800 1600 2400 3200 4000T(sec)

2400 1200 4000T(scc)

Figure 6: Spacecraft attitude during the descent (coi = 30 rpm).

Figures 7 and 8 depict the attitude time history for 15 and 3 rpm spin rates.The total change in yaw angle is 0.4° for 15 rpm, and 2.0° for 3 rpm. Theseresults are consistent as it shows that the perturbation caused by outgassing isdirectly proportional to the spin rate. The perturbation for 15 rpm is exactlydouble that for 30 rpm, and for 3 rpm it is ten times larger. These results aresummarized in Table 1, and shows that a spin rate of just 3 rpm is sufficient tocontrol aerodynamic torques caused by residual outgassing.



COi (rpm)30153

12

905

FXX0.

n10"10-'9

VF(°)0.20.42.0

Table 1: Comparison of results for spin stabilization

y (rad) y(rad)7

xKT* '

YL(');

800 1600 Z4@0 3200 4000T(sec)

800 1600 2100 3200 4000T(sec)

Figure 7: Spacecraft attitude during the descent (0)1 = 15 rpm)

VL C) ~

800 I BOO Z400 3700 4000T(scc) 1600 2400 3200 4000

T(sec)

Figure 8: Spacecraft attitude during the descent (coi = 3 rpm)



Euler Equations & Stability Analysis

The next step is to provide some means to independently verify the accuracy ofthe results obtained from the computer simulations of Euler's equations (eqns.19a,b,c). The accuracy of the numerical integration results can be verified byanalyzing the eigenfrequencies for a simple model. These results are thencompared with the frequencies obtained in the simulations. First we determinethe natural frequencies of the probe with environmental torques present. TheEuler dynamic equations for the surface probe are

A/, =1,6), +(/3 -/2^2^3 (roll) (19a)

A/2 = /jtyj + (/i - h}®\<»3 (pitch) (19b)

M, =I,d), +(/2 -/,)#,#2 (yaw) (19c)

where M» are the environmental torques, 0% are the body rates, I; are the massmoments of inertia, and the indices 1,2,3 denote principal axes. The reason forusing body axes is to obtain a diagonal inertia tensor. Principal axes (1) is theaxis of maximum inertia and coincides with spin vector a)j. The probe issymmetric so the x,y,z body axes coincide with the principal axes and h = h.The outflow of cometary gas molecules and dust particles is, essentially, a freemolecular flow and momentum is only transferred normal to the surface of thebody. There are no torques about the longitudinal axis (M\ = 0) and there is noviscous dissipation of rotational energy. Hence, Euler's equations can berewritten as,

/>, =0 (roll) (20a)

/2 6) 2 + (/j - 12 }<o, a) 2 = A/2 (pitch) (20b)

/>3 + (/2 -/,)#,#3 = &4 (yaw) (20c)

The perturbed angular velocities due to rotations ((>, 6, v|/ are

<Q, = '<I> + N (21a)

(o^=e-N\i/ (21b)

co, = ij/+Nd (21c)

where N is the spin rate. Substituting eqns. 21a,b,c into eqns. 20a,b,c andlinearizing for small perturbations gives

7,JUO (22a)

/2&- (2/2 - /, )AV + (/, - /2 )N*0 = A/, (22b)

/2y> + (2A-/iW + (/,-/2)Ar = M, (22c)

The external aerodynamic torques can be approximated asA/2=-Afsin(0)cos(y) (23a)

A/3 = Afsin( ) cos(0) (23b)

where M is the maximum torque, which occurs when the incoming flow isnormal to the longitudinal axis of the spacecraft. This result is substituted into



eqns. 23a,b,c. After linearizing and some manipulations we obtain the matrixform of the differential equations of motion,

"e

0

0

0

and the transformed Laplacian equation is

"7,5* +(7, -7,)#:+M/7, (7, -27,)A& . -,-, , , - .

(7,-27,)A& 7,5*+(7,-7,)tf* "" ' - -'-"'"' " ' * '

The determinant of eqn. 25 is a quartic polynomial that gives two complexconjugate roots, which are the natural frequencies of a system that is perturbedfrom an initial state. For example, if a spinning satellite is hit by a large objectthe spin vector will be suddenly shifted away from the total angular momentumvector. As a result, the rotation vector will begin to wobble around the angularmomentum bector with two periodic motions, a nutation frequency and anotherinvolving cross coupling motion.

For the lander analysis things are somewhat different. The basic analysis isthe same, but the initial conditions are different. At the beginning of thedescent manoeuvre the the lander is not disturbed in any manner. The probe ispointing straight down and the stream of cometary gas is coming from thefront. However, as it moves down to the surface the direction of the flowrelative to the longitudinal axis of the probe is changing. Hence, the forcingfunction is time varying. This response moment will try to realign the probe soas to be parallel to the incoming flow. As a result the probe will oscillate.Next we will calculate the frequencies of this oscillation and compare them withthe numerical integration results.

To carry out this analysis a simplified model for the surface lander has to beused. The surface penetrator is approximates as a cylinder and the soft landeras a sphere. The maximum aerodynamic torque is given by

M = |fJ.lfFJ C, (26)

where r,c is the distance from the center of mass to the center of pressure, p isthe mass density of gas and dust particles, V is the velocity of the gas stream,Sref is the projected area, and C<, is the drag coefficient. The drag coefficient inNewtonian flow is 1.0 for a sphere, and 1.3 for a circular cylinder; e.g. Griffin& French*.

The frequencies are computed at four distances above the surface: 0, 1000,2000, 3000, and 4000 m, and the nucleus is modelled as a circular sphere. Thereason for this is that, because of the small size of a comet nucleus, the particledensity changes by almost one order of magnitude over a distance of only 3000m. For the case where <DI = 0 rpm two identical pairs of roots are obtained



from eqn. 25 (Table 2). The frequencies obtained from the computersimulation is 4.5x10"* Hz, which is calculated by counting the number of peaksin Fig. 5 and dividing by the total simulation time. These values represent an"average" oscillation frequency throughout the descent manoeuvre. The reasonfor not repeating this comparison for the spin stabilized case is that the theduration of the manoeuvre is too short to infer any oscillatory motion from thegraphs.

Altitude (m)0

100020003000

Eigenfrequency5.39x10"*2.63x10"*1.73x10"*1.29x10"*

(Hz)

Table 2: Theoretical eigenfrequencies for coi = 0.

The results are in agreement., which proves that the numerical simulation isvalid. It is important to find ways to validate results obtained using computerprograms, since numerical methods are not infallible. There are manynumerical integration algorithms around. However, they only apply for aspecific set of conditions and the programmer must be very careful in usingthem. To mention a few, in fixed step methods like Runge-Kutta numericaldamping may be introduced into the system if the step size is too small. Othermethods, like Gear's Stiff, can only be used for cases where there is initially avery high frequency that then gradually damps out, as in explosions. Choosingthe wrong method for a particular situation may result in wrong answers.

Conclusion

Results have shown that residual cometary outgassing is strong enough toaffect the stability of the surface science package. However, it may also beused to stabilize the vehicle and ensure that it impacts the surface in an uprightposition. In order to achieve this it is necessary to design the Champollionlander so that the center of mass is at least 10 cm below the center of pressure.This configuration results in a angle of incidence upon landing of just 2°. Spinstabilization should only be used if the Rosetta orbiter can be forced intocometosynchronous orbit to deploy the surface probes.



Bibliography

1. AHearn M.F., Festou, MC, Physics and Chemistry of Comets - TheNeutral Coma, (Huebner, W.F., Ed.), Springer Verlag, 1990, pp 68-112.

2. Champetier, C, Serrano, J., and Lafontaine, J., Autonomous and AdvancedGNC Techniques for the Interplanetary Rosetta Mission, Proceedings of the1st ESA International Conference (SP-323), ESTEC, Noordwijk, TheNetherlands, 4-7 June, 1990.

3. Etienne, C, Comet Nucleus Sample Return: Mission Definition Study, ESAContract 6714/86/NL/SK, January 1988.

4. Griffin, M.D., French, J.R., Space Vehicle Design, AIAA Education Series,Washington DC, 1991.

5. Guillaud, V, Lafontaine, J., Navigation in the Vicinity of a CometaryNucleus, Advances in the Astronautical Sciences, AAS 89-207, 1989.

6. Huebner, W.F., Physics and Chemistry of Comets - Introduction (Huebner,W.F., Ed.), Springer Verlag, 1990, pp 1-12.

7. Keller, H.U., Physics and Chemistry of Comets - 7%e Nucleus (Huebner,W.F., Ed.), Springer Verlag, 1990, pp 13-68.

8. Lafontaine, J., Autonomous Spacecraft Navigation and Control for CometLanding, Journal of Guidance, Control, and Dynamics, Vol. 15, No.3,AIAA, 1992.

9. Manson, J.W. (Ed.), Comet Halley: Investigations, Results, Interpretations,Vols. I & H, John Wiley & Sons, Chichester, UK, 1994.

lO.McDonnell, J.A., Beard, R, Green, S.F., Schwehm, G.H.., Cometary ComaPaniculate Modelling for the ROSETTA Mission Aphelion Rendezvous,Ann. Geophysicae 10, 150-156, EGS - Springer Verlag, 1992.

ll.Mohlmann, D, Kuhrt, E, Comet Nucleus Models, Advances in SpaceResearch, Vol. 9, No.3, pp 17-23, 1989.

12.Oria, A. J, Bowling, T.S., Orbit Perturbations in the Vicinity of ActiveComets, Planetary and Space Science, Vol.43, No. 12, November 1995.

13.Prado, J Y, Rodriguez-Canabal, J., and Cotin, V., Vesta Trajectories andNavigation, Advances in the Astronautical Sciences, AAS 89-210, 1989.

H.Regan, F J. and Anandakrishnan, S.M., Dynamics of Atmospheric Re-Entry,AIAA Education Series,Washington DC, USA, 1993.

15.Rickman, H., The Nucleus of Comet Halley: Surface Structure, MeanDensity, Gas and Dust Production, Advances in Space Research, Vol. 9,No.3, pp 59-71, 1989.

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17. Van Wylen, G.J., Sonntag, RE, Fundamentals of ClassicalThermodynamics, John Wyley & Sons, New York, USA, 1985.


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