NASA Technical Memorandum 101992 

Satellite Relocation by Tether Deployment 

Geoffrey A. Landis and Frank J. Hrach 

Lewis Research Center 

Cleveland, Ohio 

April 1989 

(NASA-TH-101992) SATELLITE BELOCATION BY 19 8 9- 268 77 

TETHER DEPLOYMENT (NASA. Lewis Research 

Center) 16 p CSCL 22B 

Unclas 

G3/16 0224579 
Satellite Relocation by Tether Deployment 

Geoffrey A. Landis* and Frank J. Hrach 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, OH 44135 

Abstract 

Several new uses of satellite tethers are discussed, including: (1) using tether 

extension to reposition a satellite in orbit without fuel expenditure by extending a 

mass on the end of a tether; (2) using a tether for energy storage to power the satellite 

during eclipse; and (3) using a tether for eccentricity pumping to correct perturbations 

in the orbit and as a means of adding energy to the orbit for boosting and orbital 

transfer. 

1. Orbital Repositioning of a Satellite 

In trod u c ti on 

One of the limitations of the working lifetime of a satellite is the expenditure 

of fuel required for changing and maintaining the orbit. In geosynchronous orbit, 

the orbit is subject to perturbations, primarily due to the gravitational effects of the 

sun and the moon. Fuel is also required for relocating a satellite, for example, if a 

synchronous satellite positioned over Indonesia is desired to be relocated over South 

America. 

A tether is a long, flexible cable which connects one part of a satellite with another. 

Tethers have recently been well covered in the aerospace literature [ref. 1-51. In the 

equilibrium configuration, as shown in figure 1, the tether is oriented radially outward, 

with a tension on the tether due to the gravitational gradient (or “tidal”) force. 

*National Research Council-NASA Research Associate. 
Mass of the cable is an important figure. A figure of merit for material strength is 

the critical length L,, the length of untapered cable that could be suspended in a 1-g 

gravitational field. One proposed cable material is Kevlar, for one common type of 

which the critical length is 250 km [6]. The effective acceleration due to the gravity 

gradient a distance x from the center of mass (CM) is, to first order: 

where go is the gravity at the Earth’s surface, r, is the orbital radius and re is the 

radius of the earth. At geosynchronous Earth orbit (GEO), 3 rq/r: = 1.6.10-~ km-1. 

The minimum mass m, of untapered cable required to support an end mass m, is 

thus to first order (for x in km): 

Thus, in GEO a 1000 km long tether can easily be made much less massive than 

the satellite it supports. 

Most analyses of tether orbits assume that the center of mass of a tethered 

satellite system remains in the original orbit [1,2]; %.e., that the angular velocity of 

the tethered satellite does not change as the tether is extended or retracted. We note 

that this is true only to the first order approximation in tether length. Briefly, the 

mass that extends outward experiences an increase in centrifugal force that increases 

linearly with distance, but the mass that extends inward experiences an increase in 

gravity that increases faster than linearly. Thus, the center of mass of the orbit is 

pulled inward, and to conserve angular momentum, the angular velocity of the orbit 

increases. 

2 
Mathematical Analysis 

In the following discussion we assume a tether of negligible mass in circular orbit. 

The extension of the analysis to tethers of non-negligible mass is straightforward. 

Consider a satellite of mass mi consisting of two pieces of mass ml=rn2=m/2 

connected by a tether. The initial orbit is assumed to be circular, with an angular 

velocity w,, and an initial orbital radius (measured from the Earth’s center) ro. With 

the tether at initial length zero, the orbit has initial angular momentum 

(3) 2 Li = miworo. 

Force balance (outward centrifugal force equals inward gravitational force) 

requires w,2r:=GM, where GM is the gravitational constant times the mass of the 

Earth. Now assume that the tether is extended to length 2 in each direction from 

the CM, as shown in figure 1. The total length is 22. Note that energy is not 

conserved, since in deploying a tether work is done by the effective tidal force. Angular 

momentum is still conserved, 

(4) 2 2 L = mlwrl + m2wr2, 

where rem is the orbital radius of the CM, and rl=rcm - and r2=rem + 2. The 

system must also have a force balance, which implies (for a massless tether) that the 

inward tension on the low end of the tether equals the outward tension on the high 

end of the tether, 

w2(q + r2) = GM . (rr2 + rc2). (5) 

If we expand this to second order in 2, then set equation (3) equal to equation 

(4) to solve equations (5) for w and rcm as a function of tether extension 2, we find 

3 
22 

r,, = r, - 5 - 

PO 

and the orbital period P is 

22 

PO 

P = P,[1 - 9(-) 1. (7) 

The result is that the orbital period decreases as the tether length is extended. 

For example, a GEO satellite consisting of two equal masses on a 1000 km long tether 

will have a period faster than that of an untethered satellite by 0.44" per day. 

Inclusion of higher order terms results in an increase in the effect. 

If the two masses are allowed to differ, the orbital period change is proportional 

to mlm2/(ml + m2), which is maximum when the two masses are equal. 

Sat ellit e Relo cat ion 

We propose to use this effect to relocate a satellite without expenditure of fuel. 

Consider a satellite in orbit which consists of two masses connected by a tether with a 

motorized spool which allows it to be pulled out or in. Preferably the spool contains 

a motor/generator, such that the energy used pulling the tether in can be regained 

(except for friction losses) when the tether is let out. It is also important that the 

spool have a controller that can be programmed to damp lateral oscillations in the 

tether during the reeling process. 

The spool motor can be powered by the satellite solar array, which does not 

expend of fuel. 

Pulling the tether in will result in a longer orbital period for the satellite, and 

thus drift the satellite backward (west) in longitude; while letting the tether out will 

result in a shorter orbital period, and thus drift the satellite forward (east). This is 

shown schematically in figure 2. 

4 
Maximum relocation capability is achieved when the masses on both ends of 

the tether are equal. The countermass could be the spent booster used to boost 

the satellite up from low Earth orbit or the apogee kick motor. Alternatively, the 

countermass could be systems which do not require to be located in close proximity 

to the lower part of the satellite, or could be an entirely separate satellite system. 

Finally, if the tether is sufficiently long, no counterweight would be needed, with the 

mass of the tether itself providing this function. In this case a single tether could 

extend either inward or outward. 

In the normal operating position, the tether would be extended to a length l/fi 

of full length (about 0.707). If the tether is 1000 km long, this allows the satellite 

to be moved forward as fast as 0.22"/day by fully extending the tether, or to lag 

backward up to 0.22"/day by fully retracting it. Thus, relocation by a few degrees in 

longitude could be accomplished in a week or so, and the worst possible case, 180" 

relocation, in two years. 

The preceding analysis has assumed equilibrium conditions, i. e., that the orbit 

remains circular during the extension and deployment of the tether. This assumption 

is true only if the tether is deployed or retracted over a time greater than an orbital 

period. Faster deployment will result in dynamic changes to the orbital eccentricity. 

This is discussed in more detail in section 3. 

2. Energy Storage 

Note that the tether need not be an otherwise inactive part of the satellite. Since 

energy is gained in deploying the tether out and used in pulling the tether in, it 

could be used for energy storage. An optimal design for a geosynchronous satellite 

might well use this energy storage in place of batteries to power the satellite electrical 

systems during the eclipse period (which lasts about 1.1 hours per day for a period of 

several days around the equinox), when no solar energy is available to power the solar 

cells. The small orbital change produced by this could be easily corrected by reeling 

5 
the tether in to compensate during the sunlit portion of the orbit. By integrating 

equation 1, the total energy available is: 

E = 1.5 mgoL2rZ/r: 

For E the energy in watt-hrs, L the half-length in km, and m the satellite mass 

in kg, in GEO the specific energy is 0.55 W-hrs per kilogram of satellite mass. From 

equation 2, the maximum energy that can be stored per kilogram tether mass is 

- 0.5 goL, E 

mt 

-- (9) 

independent of orbital radius. For a kevlar cable, this is about 350 W-hr/kg. 

Even including a large safety margin, this compares favorably to the -25-35 W-hrs/kg 

produced by current technology Ni-H batteries [7]. 

The tether may have additional uses on the satellite as well. In addition to 

correction of small perturbations in the orbital period (e.g., due to lunar and solar 

perturbations) by adjusting the tether length, modulation of the length in phase 

with the orbital period could be used to adjust changes in the orbital eccentricity as 

discussed in the following section. The eccentricity of an orbit can be changed without 

changing the angular momentum, by adding orbital energy by modulating the length 

of a tethered satellite. This process can be used for orbit-to-orbit propulsion. Another 

possible use for the tether is as an element of the initial boost of the satellite into 

position, by the well-researched process of tether boost [1,2], as shown in figure 3. 

3. Eccentricity Pumping 

Introduction 

It has previously been shown that the orbital eccentricity of a tethered spacecraft 

can be changed by modulating the tether length [8,9]. In this paper the use of 

6 
this effect as a means for orbital propulsion that does not require expenditure of 

reaction mass is discussed. In the process energy is added to the orbit (from a power 

source on board the spacecraft), while the orbital angular momentum is constant. 

Particular applications discussed are injection of a spacecraft into an escape orbit 

from an initially circular orbit, and use of the process for transfer orbits, e.g., LEO 

to GEO. 

The method is similar to oblateness pumping [2], but does not require deviation 

from spherical symmetry and can be done from an equatorial orbit. 

The reverse process, circularizing an eccentric orbit by removal of energy by a 

viscoelastic tether, has been discussed in detail by Columbo et al. [lo]. 

We also note that, contrary to expectations, the eccentricity pumping process 

is most effective when the orbit is nearly circular. Although the amount of energy 

available per orbit decreases as the orbit becomes nearly circular, the sensitivity of the 

eccentricity to small changes in energy increases as l/c, and this factor dominates over 

the decrease in energy. For a perfectly circular orbit, higher order terms contribute 

as well. 

The method is straightforward. A mass is deployed away from the spacecraft on 

the end of a tether. The stable configuration is with the tether oriented radially from 

the central body. The tether is mounted on a reel with a motor which can pull it in 

or let it out. The method of orbit pumping consists of pulling the tether in at perigee 

(more generally, periapsis) and letting the tether out at apogee. Since gravitational 

gradient (tidal) forces are to first order proportional to the inverse radial distance 

cubed, more mechanical work is done against the tidal force in pulling the tether 

in than is returned when the tether is let back out. Thus, energy is added to the 

orbit. Since the orbital angular momentum is unchanged, the eccentricity c of the 

orbit increases. 

Eccentricity pumping can only be done if the initial orbit is high enough that the 

minimum perigee does not impact the primary. For pumping from an initial circular 

7 
orbit at distance a to escape, recognizing that at perigee of the escape parabola v 

= Jz times orbital velocity, this implies that a, 2 2 re. In general, the minimum 

perigee will also desirably be high enough not to intersect the atmosphere; re(ejjectiue) 

= re+ -200 km. This corresponds to an initial orbital radius of -13,150 km, or a 

minimum initial orbital altitude of 6,575 km. 

Figure 4 shows the pumping method in schematic. 

Mathematical Analysis 

We assume here that the tether itself is of negligible mass. The case of a massive 

tether can be straightforwardly calculated by integration over the mass distribution. 

We also assume that the masses on each end of the tether are equal, the case which 

for fixed tether length maximizes the effect. Again, extrapolation to unequal masses 

is straightforward. 

The gravity gradient for an orbit a distance z from the center of mass is (from 

equation (1): 

where re is the radius of the Earth, and ro the distance of the orbit from the center 

of the planet. Now assume equal masses m/2 extended on the ends of a tether of full 

length d (i. e., half-length z). The mechanical energy stored in the tether is thus 

Now assume the orbit is elliptical. The orbital energy is 

where a is the semimajor axis of the orbit. From conservation of angular momentum, 

the semimajor axis of the ellipse must be related to the eccentricity c by 

a, 

1 - €2 

a=- 

8 
where a, is the semimajor axis of a circular orbit of the same angular momentum and 

E is the eccentricity. Thus, 

The apogee and perigee altitudes are: 

and 

a0 rperigee = a( 1 - E) = - 

(1+E) 

Thus, the amount of energy required to retract the tether at perigee minus the amount 

recovered in extending the tether at perigee is: 

3 2 

- AE = - 3 mg,d2L [(I +E) 3 - (1 - E)"] = - mg,d2 46+ c2) orbit 8 4 8 4 

In the real case, the tether length d will not reeled in all the way to zero length. An 

effective value of d can be used, de;, = d,,$, - dZii,. 

By solving (5) for E and differentiating, the sensitivity of E to changes in energy is 

The orbital period is 

r=- reao 2n [ - 

(1-c) ao2 1: 

or, expressed in terms of the orbital period of the initial circular orbit, 

= ( 2 8 0 w / k g ) a 0 9 / 2 L 2 ~(2.45 + 0.41r2)(1 - c2) 

By dividing equation (8) by equation (11) we calculate the average power 

expended, 

-- dE 3 r," - -mg, - d2 ~(6 + t2)(1 - E~) dt 8 a:ro 

The function of E has a maximum of 2.45 at ~0.6. In practical units, this is an 

average specific power of 

9 
where a, is the initial semimajor axis in multiples of the earth radius re and L is 

the tether length in thousands of kilometers. Note that the function of e has been 

normalized to a maximum value of 1. Since for most applications d will be << 1000 

km and a, will be >-1.5 re, this specific power is well within achievable levels. The 

peak power levels required will be higher, and depend on how fast the tether is reeled 

in. 

Then the rate of change of eccentricity is 

(6 + f2)(1 - e’)# dc de dE 3 d2 

dt dE dt 870 at 

_- --= -- (23) 

Figure 5 shows the rate plotted against eccentricity. As the eccentricity approaches 

1 (ie., the orbit approaches a parabolic escape orbit), the rate drops to zero. 

To find the time required to reach a given eccentricity, this expression is inverted 

and integrated. The integral can be done exactly, 

8 a: 1 7€ € t(€) = -7,- - 

3 d27&2 

Figure 6 shows the time to reach a given eccentricity plotted. Escape is approached 

asymptotically. It is also of interest to look at the time required to reach a given 

apogee altitude; this is shown in figure 7. 

As an example, consider the case where eccentricity pumping is used to move 

from LEO to geosynchronous transfer orbit. a~~o=6.63 re, so aLEO-GEO = 3.82 re, 

and cLEo-GEO = 0.738. The minimum perigee requirement leads to a minimum initial 

orbital radius of a, = 1.74 re, i.e., initial orbital altitude 4700 km and initial orbital 

period 7, N 200 min. From figure 6, the time needed is 0.43 ~~(2)~. 

For a 500 km tether length, (%)2 -500, and the orbital pumping process takes 

725 hours, or about 31 days. 

A efficient technique for the apogee kick would be to continue eccentricity 

pumping until the apogee is well past GEO, perform the apogee kick, then use 

eccentricity pumping in reverse to circularize the orbit. The amount of velocity change 

10 
AV required to give the orbit sufficient angular momentum to attain circular orbit at 

GEO is inversely proportional to the distance. Thus, in theory, the AV needed for 

apogee kick could be made arbitrarily low by pumping the apogee to a high enough 

initial value, although this would take a long time. 

The decrease in eccentricity rate with increasing eccentricity is due to several 

factors, one of them being that the energy transfer per orbit is proportional to the 

square of the tether length over the orbital semimajor axis. As the eccentricity, 

and thus the semimajor axis, increases, the rate decreases. Since the tether is 

retracted at perigee, the effect can be eliminated by increasing the tether length as the 

apogee distance increases. (As tether length increases, this will require initiating the 

retraction slightly before perigee.) Note that since the total stress due to tidal force 

goes as d2/r3, the maximum stress at apogee decreases despite the increased tether 

length. The requirement for a minimum initial altitude comes from the necessity that 

the minimum perigee of the orbit not intercept the atmosphere. This requirement 

can be alleviated if the pumping maneuver is combined with an incremental AV at 

each apogee to increase the angular momentum just sufficiently to keep the perigee 

from decreasing. 

4. Conclusions 

A new method has been described for a satellite in orbit to be repositioned in 

orbital longitude by use of a tether connecting the active satellite with an inert mass, 

such as the expended booster. In addition to allowing repositioning, the system also 

allows correction of small changes in orbital period and eccentricity. 

The tether can be used for energy storage, for example, to power a satellite during 

eclipse. 

The tether can be used for changing the orbital eccentricity of the satellite without 

changing the angular momentum. This process can be used to correct eccentricity 

11 
perturbations. Since the process adds energy to the orbit, it can also be used as an 

orbital boost method, or for orbit to orbit transfer. 

5. References 

1. I. Berkey and P.A. Penzo, “Tether Propulsion,” NASA/AIAA/PSN 

International Conference on Tethers in Space, 17-19 Sept. 1986, Arlington VA; 

reprinted in Aerospace America July 1986. 

2. W.A. Baracat and C.L. Butner, Tethers in Space Handbook, printed by NASA 

Office of Space Flight, N88-14123, August 1986. 

3. D.A. Arnold, “The Behavior of Long Tethers in Space,” J. Astronautical Sci., 

VO~. $5, NO. 1, 3-18 (1987). 

4. W.A. Baracat, ed., Applications of Tethers an Space, NASA Conference 

Publication 2422, 1986. 

5. P.M. Bainum et al. (ed), Advances in Astronautical Science Volume 62, 

Tethers in Space, AAS, San Diego (1987). 

6. E. Levy, Jr., “U.S. Air Force Design Nickel-Hydrogen Cells: Flight Status and 

Recent Improvements,” paper 869348, Proc. 21 st Intersociety Energy Conversion 

Engineering Conference, 1537-1540 (1986). 

7. A.S. Brown, “Spreading Spectrum of Reinforcing Fibers, Aerospace America, 

Jan. 1989, p. 16. 

8. W.A. Baracat and C.L. Butner, Op. cit., p. 3-53. 

9. G. von Tiesenhausen, “The Roles of Tethers on Space Station,” NASA TM 

86519, Oct. 1985. 

10. G. Columbo, M. Grossi, D. Arnold, and M. Martinez-Sanchez, Orbital 

Transfer and Release of Tethered Payloads, Final Report 1 Sept. 1979-28 February 

1983, Report Number N83-25751, Mar. 1983. 
4 

T~~~~~~ 

TETHER -’ 

ORBIT OF EARTH CENTER 

Figure 1. - Tether orbit and definitions. 

ETRACTED. 

BACKWARD 

TO NORMAL 

SATELLITE ORBIT 

\ 2. NORMAL TETHER POSITION 

1. TETHER FULLY EXTENDED. 

SATELLITE DRIFTS FORWARD IN 

ORBIT RELATIVE TO NORMAL 

POSITION 

Figure 2. - Satellite relocation by extension and retraction of a tether. 

APOGEE: EXTEND 4. 

PERIGEE: RETRACT 

TETHER 

TETHER 

Figure 4.. 

perigee 

Eccenrriuty pumping (schematic). Tether is retracted at 

of orbit. extended at apogee. 

13 
2.5 r 

T 2.0 

t? 

b 1.5 

NX 

v) 

t 

2 1.0 

P 

W 0.5 

2 L 

XI 

8 

v) 

t 

Z 

W 

I2 

r 

1 .o 

0.8 

0.6 

0.4 

0.2 

0 0.2 0.4 0.6 0.8 1 .o 

ECCENTRICITY 

Figure 5. - Rate of eccentricity change vs. eccentridty. 

0 0.2 0.4 0.6 0.8 1 .o 

ECCENTRICITY 

Figure 6. -Time required to amin a given eccentricity. 

- 

0 

NN 

e 

a01 x 

0.5 

0.4 

0.3 

0.2 

0.1 

V 

1 2 3 4 5 

APOGEE ALTITUDE (IN UNITS OF ad 

Figure 7. - Time required to reach a given apogee altitude. 

14 
1. Report No. 

NASA TM-101992 

7. Author@) 

Geoffrey A. Landis and Frank J. Hrach 

2. Government Accession No. 

~~~ 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

3. Recipient's Catalog No. 

5. Report Date 

April 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-4696 

10. Work Unit No. 

506-41-11 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

5. Supplementary Notes 

Geoffrey A. Landis, National Research Council-NASA Research Associate; Frank J. Hrach, NASA Lewis 

Research Center. 

6. Abstract 

Several new uses of satellite tethers are discussed, including: (1) using tether extension to reposition a satellite in 

orbit without fuel expenditure by extending a mass on the end of a tether; (2) using a tether for energy storage to 

power the satellite during eclipse; and (3) using a tether for eccentricity pumping to correct perturbations in the 

orbit and as a means of adding energy to the orbit for boosting and orbital transfer. 

7. Key Words (Suggested by Author@)) 

Tethers 

Space propulsion 

18. Distribution Statement 

Unclassifled - Unlimited 

Subject Category 16 

3. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No of pages 22. Price' 

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