Session : Electrodynamic Tethers

Transcription

1 Session : Eectrodynaic Tethers Eectrodynaic tethers are ong, thin conductive wires depoyed in space that can be used to generate power by reoving kinetic energy fro their orbita otion, or to produce thrust when adding energy fro an on-board source. In either case, the frictiona or thrust force, is produced eectrodynaicay, through the interaction between oving charges and agnetic fieds. Fro here, we notice that the appication of eectrodynaic tethers is restricted to ceestia objects that have a non-zero agnetic fied and ionospheric pasa (e.g., Earth, Jupiter, Saturn, etc, not the oon or Mercury, etc). The fundaenta aspect of tethers is that, as ooked fro a reference frae oving at soe veocity Ev with respect to a fixed frae (for exape, an object in orbit oving with respect to the earth), an (EFM) induced eectric fied wi be generated, E ' = E + Ev B E (1) Where E and E ' are the eectric fieds in the stationary and oving fraes, respectivey. There is agnetic fied B E. The orbita situation is depicted in the figure, where there is no eectric fied in the stationary frae, E = 0. Fro here, the induced EMF (E ) wi be just the product of the agnitude of the veocity vector and the horizonta coponent of the agnetic fied, E = vb H. Assuing an eevation β of the agnetic fied with respect to the horizonta, and a agnetic co-atitude of θ, we can write this fied as, E = vb 0 sin θ cos β (2) where B 0 is the oca agnitude of the agnetic fied. Since in genera the orbit wi be incined by an ange i, both the eevation and co-atitude wi change and in consequence the agnitude of the E fied wi change over tie. MN α N α MN 300 E ( V k) i S t ( hrs) As an exape of the agnitude of E, consider a tether in a 300 k circuar orbit. At this atitude, the orbita veocity is about 7700 /s with a baseine fied of B Tesa. We then get E vb V/k. If the tether is aigned in the direction of the fied E ', then a potentia of V = 4 kv wi be generated fro end to end of a 20 k ong tether. In principe, one coud ake use of this (non-constant) votage source to generate power. 1

2 Tethers as Power Generators In this case, we have a spacecraft depoying an insuated tether upwards as shown in the figure. We assue the tether oves with a veocity Ev under a perpendicuar agnetic fied B E. The tether has ength e and a corresponding resistance T. anode V S T I φ pasa E φ tether E L cathode V L ΔV C As the tether oves, we have a fied E = vb that wi generate an open circuit votage V oc = vbe. If there is a oad resistance L in series, the a (positive) current wi fow, as ionospheric eectrons are coected on an exposed anode at the top of the tether, V oc ΔV I = (3) T + L where ΔV is the (reativey sa) tota potentia drop due to pasa sheaths at the anode and cathode. To copete the circuit, a cathode reeases the eectrons at the botto of the tether. Given this current fow, eectrica power wi be generated at the oad, 2 V oc ΔV P = I 2 L = L (4) T + L We observe that the generated power is both zero when the oad resistance is either zero or. This eans there is an optiu resistance that wi yied axiu power. To find this, we differentiate Eq. (4) and set it to zero, therefore, (V oc ΔV ) 2 P ax = which occurs when L = T (5) 4 T 2

3 The efficiency of this power-generating tether can be defined as, ( P I 2 L L ΔV η g = = = 1 V ) (6) IV oc IV oc T + L V oc We can easiy verify that for axiu power generation, the efficiency is η g 0.5. Ideay, we woud ike L» T for axiu efficiency, but then we get ower power. We can aso verify that the input power IV oc is identica to the rate of change of the eectrodynaic drag work (force F E, force density f E ), Ẇ = F E Ev = f E Ev dv = Ej B E Ev dv = IBev = IE e = IV oc (7) Tethers as Thrusters Now, we have a spacecraft that depoys the tether downwards. In this case, the anode at the botto coects eectrons fro the ionosphere. The E fied wi point aso upwards, however, we now have an on-board power suppy that forces the net (positive) current in the opposite direction, as shown. cathode ΔV C anode V S V S T I φ pasa T I φ tether E L anode cathode V Instead of reoving energy fro the orbit, this configuration wi add energy, as the force density f E = Ej B E points in the direction of the veocity vector Ev, With the current give by, EF = f E Ev dv = IBe v 3 (8)

4 We coud aso write the agnitude of this force (thrust) as, Therefore, the rate of work added to the orbit is, V s ΔV E e I = (9) T IBve IE e W F = = = (10) v v v The efficiency of the tether as a thruster is, Ẇ = F v (11) Ẇ IE e V oc η t = = = < 1 (12) IV s IV s V s It is then possibe to use a tether both as a generator and a propeant-ess thruster. In fact, since the tether can be used as a generator, an interesting question woud be: what is the tether efficiency copared against a fue ce? Fue ces ake use of soe consuabe (ike propeant) to produce eectric power. As an exercise, cacuate the efficiency of a tether producing power when its drag is copensated by a thruster consuing propeant. This is a tether working as a fue ce. Tether Dynaics As we have seen, it is iportant that the tether is oriented aong the radia vector in its orbit for it to be used as an eectrodynaic generator or thruster. F c θ T c θ = 1 F T 1 = 2 We need to anayze the tether dynaics to verify that such aignent can be achieved without the use of copicated active contro. 4

5 eferring to the scheatic, we have a ass-ess tether of ength = 1 + with two point asses at its ends, such that =. Let us anayze the dynaics of 2, assuing that the center of ass (c) is orbiting at a radius. There wi be a centrifuga force acting on 2 given by, which is baanced by a gravitationa force, 2 v 2 2 v 2 = ( ) (13) + cos θ 1 + cos θ 2 µ 2 µ = (14) ( + 2 cos θ) 2 ( ) cos θ In addition, the orbita veocity is, ( ) v = Ω 1 + cos θ (15) and since µ/ = Ω, then µ = 3 Ω 2. difference between these two forces, F net The net (upwards) force acting on 2 is the ( ) = 2 Ω cos θ 2 Ω 2 1 ( ) 2 (16) 1 + cos θ Even for a ong wire, we have, therefore we can approxiate, the net force on 2 is then, 1 ( ) cos θ cos θ F net = 3 Ω cos θ = 3 Ω 2 cos θ (17) The projection of this force on the direction of the wire is the tension, T = F cos θ (18) The tension on a 20 k tether for a ass of 100 kg woud be 7.4 N, which is not a strong force enabing the use of ight and thin wires. The projection of the net force in the orthogona direction generates a torque, 1 2 τ = F sin θ and for sa θ, τ = 3 Ω 2 θ (19) 5

6 It is the cear that the gravity gradient force gives a torque that goes in the opposite direction to the tether defection fro the vertica and therefore the it tends to stabiize the tether in the direction favorabe for its use as eectrodynaic actuator. Given its oent of inertia J, the tether rotationa dynaics are described by, dω J = τ where J = 1 e e 2 = e (20) dt Cobining Eq. (19) and Eq. (20), we get, which eans the tether wi osciate at a frequency, θ + 3 Ω 2 θ = 0 (21) f ω = 3 Ω (22) For exape, at 500 k atitude, we have Ω 10 3 rad/s, which corresponds to an orbita period of 94 inutes. The corresponding osciation tie for the tether woud then be 54 inutes, which eans we have a azy osciation. The dynaic anaysis in the off-pane direction is soewhat ore copicated, but yieds an osciation frequency in the sae order, 2Ω. 6

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