Electrodynamic Tether

The deorbiting principle of an electrodynamic tether relies on the Lorentz force that is generated as the tether moves through the Earth magnetic field. The operation relies on a current that flows through the tether. The current loop is closed by allowing the tether to exchange charge with the ionosphere plasma at both ends of the tether.

An electrodynamic tether used for deorbiting operates as a generator – by taking energy out of the orbit and converting it into heat. The reverse is also possible, allowing orbit raising manoeuvres, although this application will not be considered here.

In a deorbiting application, the tether will be deployed and stabilized so that it is aligned with the local vertical direction. A tip-mass at the end of the tether will aid in stabilization through a gravity gradient effect. Most of the tether will be uninsulated so that it can collect electrons from the ionosphere plasma. The electrons will travel along the length of the tether towards the tip-mass where an electron emitter will expel them back into the ionosphere plasma, closing the current loop.

The current that flows up the tether, that is moving through the Earth magnetic field, will result in a Lorentz force that opposes the orbital motion of the satellite, causing it to spiral towards the Earth similar to aerodynamic drag.

Theory – Lorentz force

The Lorentz force generated by an electrodynamic tether moving through the Earth magnetic field is dependent on the current flowing through the tether. The force is found by integrating along the length of the tether (Pardini, et al., 2009):

𝐅𝑡𝑒𝑡ℎ𝑒𝑟= ∫ 𝐼

𝐿 0

𝑑𝐥 × 𝐁 10-8

In the equation, L is the length of the tether, I is the current, dl is a differential element of the tether and

B is the local magnetic field.

The current flows through the tether as a result of the induced voltage

Φ = ∫ (𝐯 × 𝐁) ∙ 𝑑𝐥 𝐿

0

10-9

Where v is the velocity of the tether relative to the magnetic field. Ideally the tether would be aligned with the nadir direction so as to maximize the Lorentz force. This is usually achieved by having a tip-mass at the end of the tether and relying on gravity gradient stabilization. In reality the tether would have a

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bowed shape due to the Lorentz force interaction (Xianren, et al., 2010). The tether is also likely to exhibit dynamic motion and oscillate over time.

For simplicity it is assumed that the tether will be straight, and oriented at a fixed angle, α, with respect to the local vertical. It is also assumed that the orbit is circular.

The tether length vector can then be expressed in the orbit reference frame as

𝐋𝒐= [ −𝐿 sin 𝛼 0 𝐿 cos 𝛼 ] 10-10

For the circular orbit the velocity vector is 𝐯 = [𝑣 0 0]𝑇 _{and the local magnetic field vector in the orbit} reference frame is 𝐁_𝑂= [𝐵𝑂𝑥 𝐵𝑂𝑦 𝐵𝑂𝑧]

𝑇

. The induced voltage is then

Φ = 𝑣𝐿 cos 𝛼 𝐵𝑂𝑦 10-11

The current flowing through the tether is

𝐼 =Φ 𝑅

10-12

Where R is the tether resistance. The latter will also be a function of the tether length, the resistivity of the tether material, ρ, and the cross sectional area of the tether, AT.

𝑅 =𝜌𝐿 𝐴𝑇

10-13

The resulting tether force vector is

𝐅𝑡𝑒𝑡ℎ𝑒𝑟=

𝑣𝐴𝑇

𝜌 cos 𝛼 𝐵𝑂𝑦(𝐋 × 𝐁)

10-14

The component of the tether force along the velocity direction is called the electrodynamic drag. It is this force that will act to decelerate the satellite and cause it to deorbit. For the above configuration the electrodynamic drag force is (Hoyt & Forward, 2000)

𝐹𝑡𝑒𝑡ℎ𝑒𝑟 𝑑𝑟𝑎𝑔= − 𝑣𝐿𝐴𝑇 𝜌 cos 2_{𝛼 𝐵} 𝑂𝑦 2 10-15

From the above equation it can be seen that the drag force will be optimal if the tether is aligned with the local vertical (α = 0). It should also be evident why a tether will not work well in a polar orbit. For a polar orbit the component of the magnetic field along the orbit normal (𝐵𝑂_𝑦) will be very small.

This can be seen also in Figure 10-1 where the time to reach atmospheric re-entry is plotted for a 700 kg satellite from initial 780 km circular orbit with attached electrodynamic tether. The plot shows the deorbit time for a 0°, 40° and 80° inclined orbit. For the 80° orbit the deorbit time is considerably longer.

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Figure 10-1 Time-to-deorbit for a 700kg satellite as a function of tether length, under various orbit inclinations

The tether satellite will continue to experience aerodynamic drag as well as solar radiation pressure disturbing forces. In this case the surface area will be that of the host satellite itself and the tether. A 5 km long 1 mm diameter tether will have a drag surface area of 5 m2_.

Mass Requirement

The tether drag force (and as a result the time to deorbit) can be changed by scaling the tether length. The tether material and cross section also plays a role.

The mass of the tether system includes the mass of the tether itself, and the mass of the deployment mechanics, housing, electron emitter, current controller and tip mass. The latter items have all been lumped together as 𝑚𝑠𝑢𝑏−𝑠𝑦𝑠𝑡𝑒𝑚 and the total mass for the tether is then

𝑚𝑡𝑒𝑡ℎ𝑒𝑟= 𝐿𝐴𝑇𝑑 + 𝑚𝑠𝑢𝑏−𝑠𝑦𝑠𝑡𝑒𝑚 10-16

The density of the tether material is d. Aluminium is usually chosen for the tether material due to its low density and high electrical conductivity (inverse of resistivity). To select values for the mass parameters in the above equation, existing designs are again consulted. A commercially available tether system, the Terminator Tether, served as reference for the mass properties. The mass properties are listed below (Hoyt & Forward, 2000).

Table 10-2 Mass properties for an electrodynamic tether

Parameter Value

𝐴𝑇 2 mm2

𝑑 2700 kg/m3

𝑚𝑠𝑢𝑏−𝑠𝑦𝑠𝑡𝑒𝑚 16.45 kg

The mass properties from Table 10-2 are again applicable to large satellites (>1000 kg) and it is likely that smaller versions can be designed for smaller satellites. The same commercial venture responsible for the Terminator Tether also has a passive solution for nanosatellites that does not include active management of the current in the tether using an electron emitter. It augments the “passive” electrodynamic force with increased aerodynamic drag by increasing the surface area of the tether.

10 100 1000 10000 50 100 200 400 800 1600 Tim e to d eo rb it (d ay s) Tether length (m)

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Collision energy and fragmentation risk

Tethers are susceptible to small particle impacts. Due to their long lengths it is inevitable that they will be impacted by small particles, capable of severing the tether. Tether manufacturers have devised robust designs that make use of multiple strands to improve the survivability. An example is the Hoytether (Hoyt & Forward, 2000) that combines three aluminium strands.

In spite of efforts to increase robustness, tethers remain highly vulnerable to small particle impacts. Even multiple tether strands may fail due to multiple particle impacts, and knot points still contribute to a single point of failure.

It also remains to be shown if a collision between a tether and larger satellite will lead to fragmentation. In the same manner as before the energy-to-mass ratio can be calculated. For a large object impacting with the tether, the energy-to-impactor-mass-ratio is below the threshold for fragmentation. A 100 kg satellite will collide with about 6 g of tether material with resulting collision energy-to-impactor-mass ratio of 3.1 J/g at 10 km/s relative velocity. The 100 kg object is less likely to fragment, but higher energy collisions, or collisions where the geometry is unfavourable might still lead to fragmentation of the 100 kg object. Regardless of fragmentation of the larger object, the tether will be destroyed.

The debris-generating cross section area for a tether deorbiting satellite can be calculated in a similar approach as for the sail in Figure 6-15. We will assume that it is only the host satellite bus area and tip- mass that contributes to the debris generating area, but the total cross-section area will be quite large due to the long tether.

Operation, Integration and sub-system requirements

A true electrodynamic tether will actively control the flow of current through the tether, and as such is not a purely passive solution. Keeping the tether in a stable attitude will also require active control, with inputs from attitude sensors. There is thus a need for a functioning power system and attitude and tether dynamics sensing. (Attitude control can make use of controlled tether current).

It is possible to deorbit using a tether without the need for operations support. It may not be necessary to communicate with the host satellite to change settings or check telemetry. The tether concept presented by (Hoyt & Forward, 2000) does however include an RF transceiver for telemetry monitoring and allowing the descent rate to be controlled.

The operational and active sub-system constraints imply that tether deorbiting will not likely make use of the full 25 year allowed time. The sub-system reliability and operating cost will influence this, but in keeping with the philosophy for active solar sailing deorbiting presented in Chapter 7, tether deorbiting should also target a 1 or 2 year deorbit duration. This is also in line with existing tether analysis results where deorbit times shorter than a year is achieved with tethers in the order of a few of kilometres (Hoyt & Forward, 2000).

The electrodynamic force from a tether is also a low thrust force making a controlled re-entry impossible. The same constraints as drag augmentation apply when considering the re-entry breakup and fragmentation. The re-entry analysis should also analyse the survivability of the tether.

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The previous three sections introduced the alternative strategies that can be applied to LEO satellite deorbiting. In the following sections these strategies will be evaluated on the hand of a number of case studies.