ultra long orbital tethers behave highly non-keplerian and … · ultra long orbital tethers behave...

Ultra Long Orbital Tethers Behave Highly Non-Keplerian and Unstable

RADU D. RUGESCU1, DANIELE MORTARI2

1Fulbright Senior Researcher, Texas A&M Univ., USA, [email protected]. Professor, Texas A&M University, www.aero.tamu.edu, [email protected],

701A HR Bright Building, 3141 TAMU, College Station, Texas, USA,

Abstract: - Large twin tethers are investigated as possible competitive-cost tools for non-gasdynamic descent,landing, takeoff and return from target celestial bodies and as passive tools for debris retrieval from orbit. Theparticular behavior of orbiting bodies connected with long cables is a recent preoccupation in astrodynamicsand proves being full of unexpected results. The investigation here presented is focused on the non-Keplerianbehavior of such large tether systems, considered in a first approximation as rigid or very stiff and massless.The investigation starts with the feasibility of non-gasdynamic orbital deployment of twin tethers without anyinvolvement of expensive rocket propulsion means. The free tether release systems are associated to ahorizontal impulsive separation (HIS) and eventual friction-free deployment to the desired length. Thishorizontal deployment seems to supply the most productive means of continuous separation and departure ofmasses in orbit. The relative motion during separation is studied and the observation is made that a considerablekinetic moment of the system preserves during all eventual phases of the flight. After the friction-freedeployment the extending cable is instantly immobilized at the so-called connection moment. From here afterthe tether length remains constant. The evolution of the deployed tether is followed in order to record thespecific behavior when the length of the tether is extremely great. The motion of the two connected masses andof the mass center proves completely non-Keplerian, beginning with the libration around local vertical due tothe considerable residual kinetic moment at connection. A practical application of the quasi-vertical libration isin orbital passive debris collector, when a sandwich composite large panel is orbited for long periods of timefor collecting small mass, high velocity Earth orbit debris. The most promising and controversial application ofsuch long tethers resides in the anchoring technique to achieve the skeleton of a future space elevator. Thestability of motion is an important aspect which is approached my numerical simulations.

Key-words: - Astrodynamics, Space tethers, Tether dynamics, Large space structures, Tether instability.

1 IntroductionTether systems were first proposed as replacingmeans of the gas-dynamical propulsion systems fordescent, landing and return from far celestial bodies,orbital transfer maneuvers, orbital launch from sub-orbital flight trajectories, entry into the atmosphereby space scoop devices and finally the anchoring ofthe presumed space elevator from orbit. Generallythey are considered as potential substitutes of thecostly and mass expensive conventional rocketpropulsion systems. The phase of a free, un-powered deployment and the subsequent orbitalevolving of tethered masses are under consideration.A series of published results are present in the openliterature, as listed in the reference chapter,including those from the five InternationalConferences on Space Elevators (2001-2005), fromthe five International Conferences on Tethers inSpace and recent presentations to the IAC dedicatedsessions.

A lot of work was devoted to the study of thedynamics of large tethers, early researches showingthat great challenges are related to the stability ofsuch large systems [19], [21], [25], [26].

The relative reluctance towards tether systemscan only be overcome when significant advantagesfrom the use of space tethers, either for Earth orbitalmissions or for distant planetary missions, areproved. Regarding the orbital descent and ascent forexample, the question is if tethers offer a betteralternative to rocket propulsion and this could provemainly important when deployment and anchoringto the Earth surface of a space elevator isconsidered.

The analysis here presented is based on theassumption that the technological problems relatedto ultra-long space cables are solved and now theproblems of their exploitation in the spaceenvironment must be analyzed, prior to decide onthe efficiency and specific design solutions.

WSEAS TRANSACTIONS on MATHEMATICS Radu D. Rugescu, Daniele Mortari

ISSN: 1109-276987

Issue 3, Volume 7, March 2008

The deployment phase is decisive for thedynamics of tethers. The technique here adopted iscalled the “free tether”, as far as friction-like orother kinematical restrictions on the deployingstring are considered as negligible. The deploymentstarts by implementing a given, small impulsebetween the main orbital object and the tetheredmass, or between equal tethered masses. Therelative impulse is achieved by small size spring orpyrotechnical means. Consequently he masses areinserted into different orbits that end in a continuousdeparture between each-other, up to a convenientdistance that will be eventually set constant atconnection [2].

It is known that a continuous departure of theobjects only occurs in the condition of a horizontalinitial impulse, while a vertical separation inducesrepeated rejoining at each completed orbit. Theeventual motion in free space must be observed byspecific equipment that must continuously followthe line of sight between the departing masses, atever increasing distances of the order of tens ofkilometers and more. The attitude control duringdeployment is compulsory as the only means toguarantee a frictionless liberation of the requiredcable length during the departure of the two masses.In other words, the cable releasing system mustpredict the tether deployment under normalconditions and avoid any brake or friction lossduring this essential phase of free deployment.

The next problem of connection is essentially amater of mechanical collision, when the radialrelative velocity is bluntly suppression at theconnection moment. Some shock-typediscontinuities in the motion of the tethered systemintervene, absorbed in part by the elasticity of thesystem but tending to end in global tether systeminstability. A residual rotational momentum alwayspersists and sends the system into the known“dumbbell libration” (Lorenzini [3]). The problemof minimizing the collision stresses in the cable isalso a problem of optimal tether design and thenumerical simulations play an important role. It isthe scope of this research to demonstrate thepossibilities of the proposed free deployment bynumerical simulations.

Design of the attitude control system duringdeployment is essentially based on the prediction ofthe relative motion of the end-masses during thisphase and the simulation must begin with thisaspect. The amplitude of motion and the relatedoverall characteristics of the optical/microwavetransducers for line of sight control will result. Thereal design must also consider the large errors thatmight occur and must be properly balanced.

2 Prediction in the free deploymentThe following assumptions are consistent:- the gravity field is spherical, free of any non-symmetrical components;- astronomical perturbations of the Moon and Sunare negligible;- the orbiting masses are material points associatedto their center of mass;- the mass and weight of the tether string arenegligible;- the tether is flexible but inextensible;- no time relaxation occurs.

Consequently, the elliptical motion of the splittingmasses remains to be considered here, after theinitial separation by the horizontal impulse (HIS).Prior to HIS, a circular satellite orbit is assumed forthe non-deployed tether, with the connected objectsmoving at the constant altitude 0h with the circularabsolute velocity

00 rKw ⊕= . (1)Hereafter ⊕K denotes the gravity constant of theEarth with K=fM in general.

At the HIS moment ( 0=t ) the velocity impulse0w∆ is administered in the direction of flight and,

denoting by Mm=µ the ratio of masses of theaccelerated and decelerated objects respectively, thetwo bodies enter separate orbits with the insertionvelocities

000 11 wwv ∆+

+=µ

, 000 1wwV ∆

+−=

µµ . (2)

The corresponding orbits (polar coordinates) of theaccelerated and lower-orbiting masses respectively,behaving with eccentricities ε and E , are describedby:

)(cos1 0θθε −+=

pr , )(cos1 0Θ−Θ+

=E

PR . (3)

In the given circumstances the insertion point for theupper mass is located in its apoapsis and for thelower one in its periapsis. Assuming the commonspace-time origin at these specific points ( 0θ and

00 =Θ , namely Pr = AR =r0) and using thekinematics of motion [7] and the vis-viva equation,the two orbits are described by

θε cos1+=

pr ,

−= ⊕ ar

Kv 122 , (4)

Θ−=

cos1 EPR ,

−= ⊕ AR

KV 122 , (5)


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cossin1 cos

p θ = θ+ ε θ

r , sincos

Kep

⊕ − θ = + θ

v . (6)

The velocity of the upper object m develops lowerand the speed of the lower object M performshigher, with the orbital parameters

120

0 −=⊕K

vrε ,

ε−=

10ra , ( )ε+= 10rp , (7)

⊕+=

KV

rE2

001 ,

Er

A+

=1

0 , ( )ErP −= 10 . (8)

On each orbit the speed v has a particular value anddirection in space, depending on the position r onthe orbit and given by [7]:

0 11 cos

1sin

K ee

p p⊕

− + ϕ = ⋅ ⋅

ϕ

v r . (9)

The relative distance d gradually increases, standingthe mechanism of the free tether deployment. Notethat the right anomaly is a cyclic coordinate. TheKepler local time is:

)sin(2

νενπ

−= mTt ,

εε

−+

=11

0vaTm , (10)

)sin(2

NENT

t M +=π

, EE

VATM −

+=

11

0. (11)

The eccentric anomaly are ν and N for the upper mand lower M masses, while the correspondingperiodic times are mT and MT . Time is measuredfrom the common HIS origin, from the periapsis ofmass m and apoapsis of M. These periodic times are

00

22 12vr

aTmεπ −

= , 00

22 12vr

EATM−

=π . (12)

The computational sequence for this twin-constellation flight is thus

},,,{,,)( 22112121 yxyxcommontData →→→→ θθνν .

The Gauss substitution [2] transfers the eccentricanomalies into the true anomalies by the non-lineartransforms:

211

2θ

εεν tgtg

+−

= , 21

12

Θ+−

= tgEENtg . (13)

These are the space references for eccentric and trueanomalies and must be based on the eigen-periapsisof each orbit (3). They play the role of the primaryreferential in the present simulations, where themethod of the running origin is finally used forvisualization of the relative motion duringdeployment and further.

3 Free deployment phaseAlthough the constellation flight problem is direct,special precautions are mandatory. The correctpositioning of time origin on each orbit is one. TheKepler equations (2) require a Newton-Raphsoniterating solution for the eccentric anomaly at acommon time value, although some other efficientnumerical techniques are under development.Because the relative motion of the constellationbodies manifests extreme amplitudes, while theEarth-body distance remains always high, thenumerical code can only be run in double precision.In the case of Fig. 1 the common duration ofdeployment is 2100 seconds and the common heightangle is 34.9º, no matter how great the separationimpulse 0w∆ is.

Fig.1: Free tether deployment for 2100 s.

The angular amplitude of the motion does notdepend on the initial relative velocity and aoutstanding similitude ot the family of trajectoriesresults. At such a local scale the motion of the twinmasses in respect to the starting pole O preservesalmost symmetrical, where the components of thespeeds are given by

r r= + θv ρ τ&& , R R= + ΘV R T& & ,( ( )cos sin ) /d r r R R rR rR rR d= + − + δ+ δ δ& && && & (14)

These are the entering values for the connectionprocess of the orbital system, as the mainparameters in the conservation lows which follow.

23

45

∆w0=1


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2rmK⊕

2RMK⊕

F

m

M

r

Rψ

Ψ

θ-Θ

ℓ

4 Tether connectionAt the connection point (CP), meaning at the verymoment when the planned tether length is attainedand the tether free deployment is instantly blocked,during that short time interval the conservation lawsof momentum and moment of momentum show thatthese parameters keep constant as far as internalforces are only present. Negligible small variationsin the external forces occur and consequently onemay describe the status of the momentum as:

ttec ∆≈=− ∫ FFHH d , 0→∆ t , ec HH = ,(15)

ttec ∆≈+=− ∫ Fd)^^( MFRfrKK ,

0→∆ t , ec KK = . (16)

While the kinetic moment, moment of momentumand the position of the masses are insignificantlyaltering, the speed components manifest drasticchanges, similar to the variations encounteredduring collisions. The following relations (see alsoFig.2) account for the conservation laws:

M

M

m

m

sin cos ,1

sin cos ,1

sin cos ,1

sin cos ,1

e lec

le ec

e lec

le ec

K Hrml

H Krml

K HRMl

H KRMl

= ψ + ψ+µ

θ = ψ − ψ+ µ

= Ψ − Ψ+µ

Θ = − Ψ − Ψ+µ

&

&

&

&

(17)

The symbols used and the geometry of the tetheredorbiters are shown in Fig. 2 and summarized on thenext page.

Fig.2: Tether geometry in a central field.

The set (17) of kinematical conditions assure thetransfer from the preliminary free motion to thedefinite motion after stop of deployment. Whilewriting the above formulae the notations were used

m

M

( cos sin ) cos sin ,

sin cos ,

sin cos .

le e e e e

ee e

ee e

H r r R RK R RMlK r rml

=µ ψ+ θ ψ − Ψ− Θ Ψ

= Ψ− Θ Ψ

= ψ− θ ψ

& & &&

& &

&&

(18)

The tether motion is eventually subjected to animportant gravity gradient, visible due to the largescale of the tethered structure in respect to the radiusof the central body. The behavior is highly nonlinearand this ends in a markedly non-Keplerian motionand even instability, as numerically proved below.No closed, analytical solutions for this motion areavailable [2], [4], [6], [20], [24], although a series ofanalytical analyses of the character of the motionwere already performed [19], [21], [25], [26].Numerical computations of the actual tether orbitalevolution were much less emphasized and are herepresented in detail, as performed by proprietarysimulation codes [18].

5 Numerical simulationThe description of tether motion presents intrinsicdifficulties18, due to the inconvenient geometry,almost parallel directions of the position vectors andlarge variations in the angular reciprocal positionsof the masses involved. The related computationalinstabilities were specifically solved. The 2-Dproblem is governed by 3 degrees of freedom,namely 3 dependent variables.

5.1 The differential equationsConsequently, the three second order differentialequations have the following form (9) of ODE-s:

−⋅µ+

−Θ−θ+=

⋅µ+

Θ−θµ+−=Θ

⋅µ+

Θ−θ−−=θ

ω+

Ψ+

ψ≡

⊕

⊕

.1

)cos(

,)1(

)sin(2

,)1(

)sin(2

,coscos

22

222

r

KQrRurr

QR

rRVZ

Qr

Rruw

Rr

KQ

&&

&&

&&

l

(19)

The geometric relations follow the data in Fig.2:


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( ) ( )( ) ( )

2 2

2

cos cos( ) ,cos cos( ) ,sin sin( ), (10)sin sin( ),

2 sin cos

2 sin cos .

R rr RR

r

R R R R

r r r r

ψ = θ−Θ − Ψ = θ−Θ − ψ = − θ−Θ Ψ = θ−Θω = Θ+ Θ Ψ+ − Θ Ψ+

+ θ+ θ ψ+ − θ ψ

l

l

l

l

&& & & && &l

&& & && &&

In the Kane form, ready for numerical integration,the equations of motion read

Θθ=≡=Θθ=

≡=Θ

Θθ=≡=θ

Θ

θ

).,,,,(,)(

),,,,,,(

,)(

),,,,,,(,)(

uRrfwwwfr

ZVRrfV

VVf

wuRrfuuuf

w

r

V

u

&

&

&

&

&

&

(11)

6 Numerical resultsThe Earth constants used all along are

⊕K =3.987624457·1014 m3/s2 for the gravity and=⊕r 6371221 m for the Earth radius. A regular and

apparently periodical gravity coupling appears forLunar tethers with lengths comparable to the Lunarorbit (Fig. 3).

Fig. 3: Lunar gravity coupling.

Similar resonance manifests at least for satellitetethers of Europa and Rhea. Values of KE=3.20419804·1012 m3/s2 and rE =1561000 m are forEuropa and KR =1.542020305·1011 m3/s2 and rR=764000 m for Rhea10,11. Decagonal geometry isobserved as given in Fig. 4 for an orbit at 1000 kmabove Europa. For carefully implemented Speeds atHIS at different given altitudes of the orbit theperiodicity of motion develops remarkable. A line ofsymmetry also appears, always inclined as referredto the HIS point on the circular initial orbit (therunning origin in the draft).

Fig. 4: Europan tether decagonal resonance.

A double-pentagonal configuration of resonanceappears for a specific motion around Rhea (Fig. 5).The length of the tether system for thisconfiguration represents 54% from the Saturniansatellite radius. To be more visible, this motion isdrawn in figure 5 for the mass center of the tether(yellow line) only. The initial circular orbit is drawnin blue and the entire relative motion is againregarded to the running origin of the deployment onthe circular initial orbit. Due to the curvature of theorbit, the momentum conservation of the system atconnection produces a decrease in the kinetic energyand the eventual orbit of the structure alwaysremains lower than the starting circular orbit. Theinitial conditions of the tether at connection asure alength of the cable of 416 km, while the Rheanradius is of 764 km. The cable length remains wellbelow the radius of the central body.


ISSN: 1109-276991


The drop in kinetic energy is more visible for largertethers and, as discussed by several authors, finallyends in a total instability of the motion, when thetether length becomes extremely great.

Fig. 5: Rhean double-pentagonal resonance.

Up to that point however, the motion of the systemperforms always below the circular orbit, as visiblein all numerical simulations here presented. Theproblem of flying tether systems with length greaterthen the orbital radius is not physically possible, asit results in immediate crash of the lower mass onthe planet surface.

7 Orbital instabilityPrevious analytical observations [19], [26] regardingthe instability of motion for very long tethers,nearing the orbital radius, show that the tethermotion is rather unstable. The condition is deducedthat the unstable tether should be longer than itsorbital radius (Troger [19]), but this condition isalready practically impossible. They are confirmedby direct numerical computations. At the same timethe observation must be made that structures withsizes comparable to the radius of the initial circularorbit are in fact impossible. Due to the residualnegative amount of energy after connection, a tethersystem with the length equal to the altitude of theinitial orbit will crash before even completing itsfirst revolution around the body. As structures largerthan the radius of the orbit are nonsense, theproblem of the stability must be worked out forstructure sizes smaller than the altitude of thecircular orbit above the planet surface (Fig. 6).

Fig. 6: Orbit instability for a large Lunar tether.

The initial circular orbit selected for thisapplications hovers at 4500 km above the Lunarsurface, where a relative HIS velocity of142.4584311 m/s is implemented between the twotethered masses. A deployment for 194 minutesproduces a tether length of 2,397,804 m, muchlarger than the Lunar radius (1,737,400 m). Duringthe first seven revolutions, the trajectories seem toperform stable and regular, each turn around thecentral body goes very near to the previous one, asseen in the draft 5. A small lag is recorded howeverand this accumulates to produce a rapid change intoinstability after revolution 8, ending in a crashduring the 18th revolution. Changes as small as one

sm /µ suffice to change the character of the motion,occurrence that also shows, after Liapunov’s theory,the presence of an unstable region.A relevant situation regarding the effect of verysmall changes in the initial conditions is also givenin figure 7. For the same orbital altitude the HISvelocity is increased to 142.4584311 m/s and thisproduces a positive lag of the orbits, ending in a latecrash after 27 orbits and 448 hours of evolution.

Fig. 7: Late instability for very large tethers.


ISSN: 1109-276992


The effect of the majestic gravity gradient is visibleas far as the libration of the tether during theevolution on the lower part of the trajectory, verynear to the Lunar surface, is completely suppressed.Here the effect of centrifugal forces within thetethered structure is overwhelming. The lower massis fastly sweeping the Lunar surface with double ofthe circular velocity (1372.7 m/s) at an altitude of afew kilometers. It is once more obvious that thecondition of a tether longer than the orbital altitude,to say nothing of more than the orbital radius, isimpossible to consider. This criterion of Troger [19]or Beletsky [20] remains in fact purely theoretical.

8 Orbital stability after ScheeresA lot of research is devoted to the so-calledgeneralized two-bodies problem (GTBP), where thebodies engaged into reciprocal gravitationalinfluence are no more replaceable by two masspoints or particles. They manifest a visible size andirregular mass distribution in comparison to thedistance between the two, coupled bodies. Thisinduces important gravity gradient effects when thebodies are rigid or even tidal effects when they areelastic or fluid, with sensible energy dissipation andloss of stability. For the rigid body assumption themechanical interaction ends into non-Keplerianeffects and even high instability. Scheeres hadshown that the condition of stability is that thewhole mechanical energy be negative,

0),,( <ωvhE (9)

and this condition should be proved by the presentsimulation also. The twin tether is in fact theextreme case of the non-spherical mass distribution,presenting the most elongated ellipsoid of inertiaamong all possible mass distributions achievable inpractice. The non-Keplerian behavior and all othergravity gradient effects outlined in the theory ofGTBP are thus maximal in this study case.

9 ConclusionsThe ultra-large scale twin space tethers prove beingthe extreme cases of non-Keplerian celestial bodies,where the non-linear effects of the gravitationalgradient are maximal. These effects are highlysurpassing all possible situations for commonplanetary bodies like the small, very non-sphericalasteroid systems, largely approached by recentstudies. Consequently the tether systems play therole of a benchmark research tool in non-linearastrodynamics.

A lot of works had shown through theoreticalinvestigations that for such systems the orbitalmotion could get unstable. The accurate numericalsimulation of both equal and non-equal, twin-masstether systems in LEO and around other bodiesshows possible unpowered deployment scenarioswith consequent stable librating motion of thesystem, when the connection between the two end-masses is settled in advance of the smooth point ofconnection and the tether length is smaller than theradius of the central body.

For time intervals below one sideral day little ifany extinguish process was found on the oscillatorymotion that performs regular and at least quasi-periodic. The entire study was performed underconvenient, simplifying assumptions. The stabilityof the tethered masses was also accounted bycalculating the variation of the force in the string. Astable configuration means a positive value of thetension. The librating motion proves almost alwaysstable, while the tumbling rotation ends always ininstability. This is a real challenge for tetherlanding, as repeated approach of the lunar soil ishighly desirable for a safe landing.

Combinations of HIS velocities and altitudes arealways found which provide conveniently stablemotion, but this greatly depends on the initialconditions of flight and the type of deployment. Ahigh precision of the maneuvers is required to assurethe success of the landing. Tether evolution in caseof ultra-high length of the cable is usually unstable,but offers support for a number of orbital evolutionsin quasi-stable configuration. This is an importantobservation regarding landing projects by tethers.

College Station, on August 24, 2006.

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ultra long orbital tethers behave highly non-keplerian and … · ultra long orbital tethers behave...

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