Fundamental Tether Concepts

By definition, in the absence of external torque, the angular momentum of an orbiting body remains constant.

Equation 1: Conservation of Angular Momentum (Note: Click on Image for Symbol Definitions)

Within the realm of space tether applications, the sum of the angular momentum for the linked bodies and the tether remains constant. For most applications the narrow tether represents a negligible amount of mass when compared to the system mass, and can therefore be removed from the analysis. In this sense, the tether only provides a means to exchange energy or angular momentum between the two bodies. With angular velocity equated by the linkage, one body can increase its momentum relative to the central body only at the expense of the other body, which loses momentum. This trade, termed momentum exchange, provides the basis for many interesting applications.

Equation 2: Change in Angular Momentum

A primary use of momentum exchange involves tether release or capture, whereby the momentum exchange is translated directly into an orbital change. The scenario of a tether release orbital boost provides an illustrative example. In this problem, two masses, m1 and m2, start out joined together in a circular orbit. The tether then deploys moving the masses apart from each other by the tether length, L. A gravity gradient stabilized position (discussed later) establishes a vertical orientation, with one mass on top and one on the bottom, with the length oriented along the radial direction. Severing the tether completes the momentum exchange, releasing the two bodies into their new orbits.

Figure 1: Angular Momentum Transfer by Tether Release (Capture)

A few simplifying assumptions allow for rough calculations. With an initial circular orbit and both masses initially joined together, the initial velocity of the masses equal the circular orbital velocity. As the tether deploys, it is assumed that the radius of the orbit greatly exceeds the deployed length of the tether. From this assumption, one can also conclude that the velocity of each mass remains roughly constant, at the circular orbital velocity. The tether contrains the sum of the change in radial distance for both masses to equal the tether length, L. The radial distance change is determined by the ratio of the two masses. Using this knowledge produces a simplified equation for the change in angular momentum for each body.

Equation 3: Change in Angular Momentum as a Function of the Mass Ratio

Solving for specific angular momentum allows for the calculation of orbital change through equation 5.

Equation 4: Specific Angular Momentum

Equation 5: Dependence of a, e on Specific Angular Momentum

The sign of delta h_specific depends upon the position of the body, with positive corresponding to the upper body, which gets boosted. The boosted (upper) body starts at its perigee position, while the de-boosted (bottom) body starts at apoapsis.

Equations 6-7: Apoapsis & Perigee Radius

Combining equation six though seven allows for the simultaneous solving of the eccentric and semi-major axis for the two new elliptical orbits. For L=10 km, m1=.25*m2, and an initial 6678 km circular orbit, the table below provides the new orbital parameters.

Body:	Mass 1	Mass 2
Semi-Major Axis	6694.05 km	6674 km
Eccentricity	0.0023974	0.000599

Tether release provides the most benefits when excess momentum from one body can be transferred to another. For example, as cited by Cosmo [3] the shuttle/ external fuel tank combination provides an excellent opportunity. Using a tethered release of the tank instead of a pure jettison can save or salvage some of the excess momentum of the waste tank and provide the shuttle with an extra boost. Tether capture works in the same manor, only in the reverse order. These examples assumed that both masses started or finished joined together. However, a swinging capture and release combination enhances the magnitude of the momentum exchange. This type of problem employs a tether, with one mass, rotating as it progresses though its orbit. For a boosting situation, the second mass rendezvouses with the free end of the tether at the bottom side of the tether. The mass then rides the rotation of the tether for one half of a revolution, and then releases on the topside. The additional velocity imparted by the rotating motion provides greater momentum exchange. This type of swing might be suitable for interplanetary journeys, with the swing providing the necessary hyperbolic velocity.

The Earth's magnetic field has profound effects for space tether analysis. To a rough approximation, the magnetic field of the Earth resembles a magnetic dipole, with the magnetic pole offset from the rotational axis by 11.5°.

Figure 2: Orientation of Earth's Magnetic Field

A conductive tether material acts as a long wire moving though a magnetic field. This induces an electromotive force and corresponding current to move though the wire, with the surrounding plasma completing the circuit. The electromotive force, or voltage potential, depends directly on the field strength, the orbital velocity, and the tether length.

Equation 8: Generated EMF

Equation 9: Induced Current-Ohm's Law

Thus, orbital velocities of approximately 7 km/sec give rise to extremely large voltage potentials. For example, the TSS-1R mission produced a potential of 3500 V with a 19.5 km tether. [4] The induced current flows up the tether in the positive radial direction, to the upper mass as dictated by the right hand rule. Interaction with the surrounding plasma field usually occurs at both ends of the tether (serving as the cathode and anode) where electrons are collected or disseminated by field emission devices. Some of this potential can be harnessed to provide power for the spacecraft. Of course, this power generation comes at a cost. The current flow dissipates power according to the multiplication of voltage and current, which reduces the total energy of the spacecraft. To compute the effect on the orbital elements, the electrodynamic force due to the tether current is computed according to the Lorenz force acting on the wire.

Equation 10: Lorenz Force

For convenience, an inertial frame is set up by orienting the vertical z-direction with magnetic pole, instead of the rotational axis. This requires defining a new inclination, lambda, rotated by the angle between the magnetic pole and the rotational axis. This angle changes as the Earth rotates, varying between +11.5° and -11.5° once per day.

Figure 3: Variation in Lambda Over 12 Hours

In this system the electrodynamic force, and the component of this force in the direction of motion, termed the electrodynamic drag force, depend on the angle lambda, and the angle of the tether with respect to the radial direction, alpha.

Equation 11: Electrodynamic Force

Equation 12: Electrodynamic Drag Force

The angle alpha depends on an equilibrium between the drag forces and the gravity gradient forces, as discussed in a subsequent section. Forward develops an equation for the secular rate of change of the semi-major axis on a circular orbit due to the electrodynamic drag force. [4]

Equation 13: Time Rate of Change in the Semi-Major Axis

The term <(cos l)^2> averages out the variation in the modified inclination due to Earth's rotation. The drag has negligible secular effect on the inclination.

Electrodynamic drag provides a passive method to rapidly reduce the orbit of a spacecraft, often by a few kilometers per day. Removal of space debris provides an important application for this technique. The proliferation of dysfunctional satellites, spent boosters, and other miscellaneous man-made debris poses an increasing threat to future near Earth activities. Furthermore, decay rates due to atmospheric drag and other factors will not force much of the debris down for several thousand years. Using inexpensive space tethers to remove some items may help to partially alleviate the problem. The relatively light tether could deploy after the completion of the object's mission and force the structure into the atmosphere.

By contrast, space tethers can also provide passive propulsion. By reversing the flow of current with an opposing potential provided by the spacecraft, the direction of the electrodynamic force switches, aligning it in the direction of motion. This accelerates the spacecraft and boosts the orbit. Thus, spacecraft can use excess power, when other instrument are not in use for instance, to maintain the orbit instead of having to carry extra fuel on board.

Many of the aforementioned concepts require determining the equilibrium position of the tether/two body system. Examining the forces acting on the system provides the means to resolve the orientation. For the one body problem, the forces felt by the body at the center of mass equate to zero as the gravitational force offsets the centrifugal force.

Equation 14: Gravitational Force

Equation 15: Centrifugal Force

This remains true at the center of mass of the tether system, but the masses at either end experience a net force. As discussed previously the tether constrains the angular velocity of the end bodies, thereby forcing the upper mass to speed up relative to the untethered angular velocity and the lower mass to slow down. This alteration of the angular velocity creates an imbalance between the gravitational and centrifugal forces acting on the end masses.

Equation 16: Force Imbalance

For example, a vertical tether system with a 20 km tether and a lower, or ballast, mass of 10 kg, experiences a net force of .807 N in the negative radial direction. Here it is assumed that the lower mass is much smaller than the upper main mass, leading L_cm to approach L.

Equation 17: Example Force Imbalance

An approximation of this force, put forth by Cosmo [3], produces a similar result.

Equation 18: Force Imbalance Approximation

The equation defines the gradient field strength, gamma, as the system angular velocity squared. Examining the force equation shows that the upper and lower masses experience equal and opposite forces, with the upper mass forced away from the central body, the lower mass forced towards the central body, and the force magnitude equal to the tension in the connecting tether. In the absence of other disturbing forces, such as drag, the system will move to a stable vertical equilibrium with the tether length oriented along the radial direction of the system. Which mass ends up on top depends on the deployment. One interesting application of the gradient field is the creation of controlled gravity laboratories. Placing a third body, the laboratory, on the tether between the two end masses, allows for the artificial gravity in the laboratory to vary. It varies from nearly zero (for micro-gravity experiments) when the laboratory resides at the center of mass, to a maximum value when the laboratory moves to the end of the tether.

For conducting tethers, the primary disturbing force is the electrodynamic drag, which tends to push the tether away from the vertical direction. The new equilibrium arises by balancing the torque acting on the system, again with the simplifying assumption that one mass dominates the other.

Equation 19: Torque Balance

Figure 4: Forces Acting on the Tether System

Solving for unknown angle alpha, and using the drag equations presented earlier allows for a mission designer to calculate the effect of the electrodynamic drag on the orbit.

Finally, the tether material proves critical to the successful implementation of these concepts. Ultimately, the application sets the desired material properties. This section outlines some of the important material considerations. As discussed previously, the angular velocity constraint introduces a tensile force, and stress in the tether.

Equation 20: Tether Stress

Additionally, the material density defines the mass of the tether.

Equation 21: Tether Mass

Solving for the unknown design variable, the area of the tether, in equation 15 allows the minimum required mass to prevent failure to be expressed as a function of the yield stress and density. Thus, a desirable tether material possesses high specific strength, minimizing the mass while carrying the tensile load safely.

Equation 22: Yield Area

Equation 23: Minimum Mass

Previous sections developed equilibrium equations for the simple case of balancing gradient and drag torques. However, the system experiences other disturbing forces that introduce vibrations into the tethered system. For example, if one mass, the space shuttle, executes a thrust maneuver along the transverse direction with respect to the tether length, a transverse vibration will develop in the cable. Describing the motion of the cable requires introducing the wave equation for a continuous medium, with w representing displacement in the transverse direction, and x representing position along the tether length.

Equation 24: Wave Equation

Therefore, the motion depends upon the speed of sound of the material, c, a function of the Young's Modulus, E, and the density, rho.

Equation 25: Speed of Sound

A designer needs to select materials that can limit the magnitude of such vibrations to reasonable levels.

Many applications of tethered systems operate in low earth orbits, where the interaction with the atmosphere and the magnetic field has profound affects. Several material properties control these effects. First, the resistivity controls the overall resistance of the tether, and in turn the resulting electrodynamic drag.

Equation 26: Tether Resistance

Maximizing or minimizing the resistivity depends on whether drag is desirable to the mission. Interaction with the atmosphere requires examining the ability of the material to withstand heat generated by friction, and exposure to elemental oxygen. Another environmental hazard involves impact with micrometeorites that can sever the tether. Cosmo [3] provides a rough approximation for estimating the number of failures per year due to impacts for tethers with a diameter, D, greater than 1mm.

Equation 27: Fatel Micrometeorites Impact Rate

Thus a 10 km long, 2mm diameter tether suffers 1.65 failures per year, or has a predicted average lifetime of 221 days. Reducing this risk is often accomplished by using a mesh cylindrical tether, similar to a cylinder formed with chicken wire, instead of a single rod. Under this scheme if one link fails, due to impact, the rest of the mesh can still carry the load safely.

Proposed tether missions have focused on three main tether materials; metals, carbon and silicon carbides, and polymers and organic fibers. Metals promote plasma interaction for power generation or passive thrusting, while silicon carbides and polymers present higher specific strengths necessary for higher stress applications like interplanetary swings. Many polymers also have a distinct weakness in resistance to atomic oxygen, therefore requiring some covering over the tether. Tethers often employ a combination of materials to garner the advantages of each. The TSS-1R tether was a copper wire, for conduction, insulated with Teflon, and then covered with braided wraps of Kevlar, to provide strength, and Nomex, to provide chemical resistance. [9] Missions