It has been recently suggested that the magnetic field

created by the current in a bare tether could sensibly

reduce its electron collection capability in the magnetised ionosphere, a region of closed magnetic surfaces  disconnecting the cylinder from infinity. In this paper, the ohmic voltage drop along the tether is taken into account in considering self-field effects. Separate analyses are carried out for the thrust and power generation and drag modes of operation, which are affected in different ways. In the power generation and drag modes, bias decreases as current increases along the tether, starting at the anodic, positively-biased end (upper end in the usual, eastward-flying spacecraft); in the thrust mode of operation, bias increases as current increases along the tether, starting at the lower end. When the ohmic voltage drop is considered, self-field effects are shown to be weak, in all cases, for tape tethers, and for circular cross-section tethers just conductive in a thin outer layer. Self-field effects might become important, in the drag case only, for tethers with fully conductive cross sections that are unrealistically heavy

Estes, Robert D.

Sanmartín Losada, Juan Ramón

It has been recently suggested that the magnetic field created by the current in a bare tether could sensibly reduce its electron collection capability in the magnetised ionosphere, a region of closed magnetic surfaces disconnecting the cylinder from infinity. In this paper, the ohmic voltage drop along the tether is taken into account in considering self-field effects. Separate analyses are carried out for the thrust and power generation and drag modes of operation, which are affected in different ways. In the power generation and drag modes, bias decreases as current increases along the tether, starting at the anodic, positively-biased end (upper end in the usual, eastward-flying spacecraft); in the thrust mode of operation, bias increases as current increases along the tether, starting at the lower end. When the ohmic voltage drop is considered, self-field effects are shown to be weak, in all cases, for tape tethers, and for circular cross-section tethers just conductive in a thin outer layer. Self-field effects might become important, in the drag case only, for tethers with fully conductive cross sections that are unrealistically heavy

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Magnetic self-field effects on bare-tether current collection

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1MAGNETIC SELF-FIELD EFFECTS
ON BARE-TETHER CURRENT COLLECTION
J. R. Sanmartín
E. T. S. I. Aeronáuticos, Universidad Politécnica de Madrid
Pza. Cardenal Cisneros 3, 28040 Madrid, Spain
Phone 34 91 336 6302,  fax  34 91 336 6303,  email  jrs@faia.upm.es
R. D. Estes
Smithsonian Astrophysical Observatory
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, US
Abstract
It has been recently suggested that the magnetic field
created by the current in a bare tether could sensibly
reduce its electron collection capability in the
magnetised ionosphere, a region of closed magnetic
surfaces disconnecting the cylinder from infinity. In this
paper, the ohmic voltage drop along the tether is taken
into account in considering self-field effects. Separate
analyses are carried out for the thrust and power
generation and drag modes of operation, which are
affected in different ways. In the power generation and
drag modes, bias decreases as current increases along
the tether, starting at the anodic, positively-biased end
(upper end in the usual, eastward-flying spacecraft); in
the thrust mode of operation, bias increases as current
increases along the tether, starting at the lower end.
When the ohmic voltage drop is considered, self-field
effects are shown to be weak, in all cases, for tape
tethers, and for circular cross-section tethers just
conductive in a thin outer layer. Self-field effects might
become important, in the drag case only, for tethers with
fully conductive cross sections that are unrealistically
heavy.
Introduction
      Electrodynamic bare tethers work as cylindrical
Langmuir probes. Tether bias varies along its length, but
typical values of the length-to-thickness ratio are of
order 106;  bare tethers thus collect current per unit
length, at each point, as if uniformly polarised at the
local bias. An analysis of tether collection requires a
detailed theory for the electron-attracting branch of 2D
probes with thickness comparable to the electron Debye
length  λDe  and bias highly positive.
       The electron current  I  to a positively polarised
cylinder of circular cross section, has an upper bound,
IOML  (orbital-motion-limited current), which, at high
bias, is proportional to the square root of tether bias  ΦP,
in addition to being proportional to its length  L,  cross-
section perimeter  p,  and plasma density  N∞,
    emPeNe
p
LpOMLI /2)( Φ×∞
××≈
π
.
This bound is reached for radius less than some
maximum value,  R  =  p / 2 π  ≤  Rmax,  which has been
recently determined by way of an asymptotic analysis
[J. R. Sanmartín and R. D. Estes,  Physics of Plasmas 6,
395 (1999)]. Figure 1 shows general results for
Rmax/λDe.  For  typical values  Ti  ≈  Te  and  eΦP  ≈  102-
104 k Te  one has  Rmax  ≈  λDe.
       For  R  >  Rmax   the ratio  I  / IOML (p)  is a function
of  R /Rmax,  eΦP / k Te,  and  Ti /Te.  This is shown in
Fig.2 [R. D. Estes and J. R. Sanmartín, Physics of
Plasmas 7,.4320 (2000)]. The current  I  is thus, in
general, a function of both radius and perimeter.
            The above results are valid for an arbitrary
convex cross section if the radius  R  is replaced by
some ‘equivalent’ radius  Req.  As examples, one finds
Req = p / 8  for a thin tape and  Req ≈ p / 6.78  for a
square cross section. Results are again valid for non-
convex cross sections when both radius and perimeter
are replaced by equivalent radius  Req  and perimeter
peq;  for a cross section made of two adjoining circles of
radius  c,  one finds  Req = πc / 2 ≈  peq / 6.55.  For cross
sections made of disjoint parts, say two non-adjoining
circles, one can still determine equivalent radius  Req
and ‘effective’ perimeter  peff,  to be used in the
formulae [J. R. Sanmartín and R. D. Estes,  Physics of
Plasmas, to appear].
Self-Field Effects
         All the above results apply to unmagnetised
plasmas at rest, with unperturbed electron distribution
function isotropic, and no trapped-electron population.
Tethers, however, move through a plasma at relative
speed  Usat  in the presence of the geomagnetic field  B0.
Effects related to both  B0  and  Usat  (and to trapped
electrons) were discussed by Sanmartín and Estes,
(1999), and Estes and Sanmartín (2000).  There are also
preliminary results from laboratory tests and PIC
2calculations that tentatively indicate that any such
effects would not decrease the current collected.
           Recently, however, it has been suggested that the
magnetic field  Bs  created by the tether current itself
could be so large as to substantially affect the total
current to the tether [G. V. Khazanov et al.,  J. Geophys.
Res. 105, 15835 (2000)]. No such effects would
influence electron collection by tether-end devices.
      Let us first recall the (Parker-Murphy) upper bound
that the geomagnetic fielf sets up on the current, in the
absence of self-field; at high bias one has
          
OML
I
R
el
PM
I ××≈
π
2
where  le ≡ vth / Ωe  is a thermal gyroradius,  vth ≡
√kTe/me  and  Ωe  being the electron thermal velocity
and gyrofrequency. Clearly, if  le >> R,  this upper
bound is well above  IOML;  the geomagnetic field should
then have no effect on the current. Actually, the Parker-
Murphy bound ignores any space-charge effect;  it has
been suggested [Sanmartín and Estes, (1999)  that an
additional condition  (le >> λDe)  is required for no  B0-
effects.
        Khazanov et al. (2000) modified the Parker-
Murphy analysis to take into account the self-field  Bs
and found a reduction in the PM bound. Further, they
gave a rough criterion to determine when the actual
current is strongly reduced (say, supressed). The
argument involves the new topology of magnetic field
lines: there is now a separatrix, with field lines being
open outside.the separatrix and closed inside. The
characteristic dimension of the separatrix is the radial
distance  r*  that results from equating fields  B0  and  Bs
                      
rc
I
s
B
2
0
2 επ
= ;                         (1)
with  B0  near perpendicular to the tether one has
                      
0
2
0
2
*
Bc
I
r
επ
= .
      The simple criterion for full self-field effects,
roughly requiring the plasma sheath to lie inside the
separatrix, is the condition
rsh (Φp)  +  lef  <  a r* (I).             (2)
Values  0.3  and  0.5  were suggested for the coefficient
a;  we shall take  a = 0.5, which is conservative for our
argument that condition (2) does not apply (where it
could lead to significant effects on the current). Also,
we shall drop, from the condition, the term  lef   which
represents certain gyroradius for the full field; this is
again conservative for our argument. Finally,  rsh  is a
'sheath radius', which Khazanov et al  (2000)  wrote as
ekTPesh
r /
0
Φ×≅ ρ
with  ρ0 = λDe × a function of  R / λDe.  Exact
calculations in Sanmartín and Estes (1999) and Estes
and Sanmartín (2000)  show  ρ0 / λDe  to be a function of
R / λDe,  eΦp / kTe,  and  Ti / Te,,  which is comparable to
Khazanov's expression for most values of interest.
Condition (2) for strong self-field effects is now
                             rsh (Φp)  <  a r* ( I ) .                (3)
Note that this condition can be applied to, say, tape
tethers, Eq.(1)  holding at distance  r*  >>  Req.
Ohmic effects
      Use of condition  (3)  requires knowledge of the
relation  between current  I   and bias  Φp. For current
high enough the ohmic voltage drop can not be ignored.
Universal relations can be obtained in dimensionless
form for all three modes of tether operation by
introducing dimesionless current and bias,
                                I  ≡  σc Ac Em  ×  i
                               Φp  ≡ Em L*  ×  ϕ
where the motional induced field is typically  Em ∼  102
V / km  and where we introduced a characteristic length
for ohmic effects  L*,  defined by writing
       
c
A
m
E
c
LmeEeN
p
L σ
π
4
3*2* ≡×
∞
×× ,
with  Ac  the electrically conducting area of the cross
section and  σc  the conductivity. Condition  (3)  now
reads
                             √ϕ  <  C i                           (4)
substantial self-field effects setting up when the ratio
√ϕ /i,  which varies along the tether, drops below a
constant  C,
3   ( )
×
⊥
≡







 3/1
6
5
0
3/13
6/1
2
c
th
v
sat
U
B
Ba
C
π
                 
3/1
23
0
3
23/2
0 2






Ω
×
De
c
e
c pA
λρπε
σ
.
with  B⊥  the component of the geomagnetic field
perpendicular to the orbit. Taking  B⊥/B0 = 0.8,  Usat =
7.5 km/s,  Te = 0.12 eV,  B0 = 0.3 gauss,  σc = 3 × 107 /
Ωm,  one finds
   
0
3/2
04.2
ρλ
eqR
De
h
C ×≈ 


 3/2
02.1 



×≈
De
h
λ
for a tape of thickness  h.  In the final expression we
took  Ti  ∼  Te ,  Req  ∼ Rmax ∼ λDe,  and  eΦP /kTe  large,
leading to  ρ0 ∼2 λDe.  Similarly we find
            
0
3/2
54..2
ρλ
δ R
De
C ×≈ 



  
3/2
27.1 



×≈
Deλ
δ
for a circular tether conductive in an outer layer of
thickness  δ.  Also,
               8.0
0
3/2
60.1 ≈×≈ 



ρλ
R
De
R
C
for a fully conductive circular tether. Clearly,  C  will be
small for the first two types of tether.
      For the power generation and deboost cases, with
the self-field ignored, one has [J. R. Sanmartín, M.
Martínez-Sánchez and E. Ahedo, J. Propul. Power 9,
353 (1993)]
       
3/13/1
)(2  −+−= imaxiimaxiϕ .       (5)
Power generation, if efficient, will require the current to
be a small fraction of the short circuit value,  σc Ac Em.
With  imax  small, condition (4)  leads to a reduced
maximum current,  maxi
~ ,  given by
              C
max
i
max
imaxi
i
=
−
→ ~
3/1)~(3/12ϕ
                  132
2
1
1
~
≈−≈ C
max
i
maxi
maxi ,
self-field effects being clearly weak for all three types
of tethers.
      For deboost, maximum current can reach the value
imax  =  1,  if the tether is long enough,  L  ≥  4 L*   [E.
Ahedo and J. R. Sanmartín, to be published].  We would
then have
                       
( )
C
maxi
maxi
i
=
−
→ ~
3/2~1ϕ
If  C  is small,  we find  maxi
~   ≈ 1 - C3/2 ≈ 1,  that is,
effects are again weak. However, if  C  is not small, the
case of fully conductive, circular cross sections with  R
comparable to  λDe,  maxi
~   will be well less than  1.
Such tethers would be unrealistically heavy.  With  the
minimum  λDe  about  3 mm (at  N∞  1012  m-3), taking an
aluminum  tether,  Em  ∼  135 V/km,  N∞  1012  m-3,  and
R   > 3 mm,  L  >  4 L*,  one finds a tether mass
exceeding 1600 kg.
     In the reboost case, with the self-field ignored, one
has [J. R. Sanmartín, R. D. Estes, and E. Lorenzini, in
Space Technology and Applications International
Forum 2001, ed. M. S. El-Genk, AIP, New York, pp.
479-487 (2001)]
                   √ϕ  =  (2i  +  i2)1/3,
an expression that does not involve  imax.  Self-field
effects then require a maximum current given by
                           C
max
i
max
i
i
=
+
→ 3/2~
3/1)~2(ϕ
.
For  C  small,  self-field effects require  imax  to exceed
maxi
~  ∼ 1 / C3  >> 1.  For C  ∼ 0.8  self-field effects
require  imax  to exceed  maxi
~  ∼ 3. The design parameter
imax  never reaches such high values, however.
4Fig. 1
Fig. 2
0 4 8 12
0.4
0.6
0.8
1.0
R / Rmax
(300,1)
(300,0.3)
(300,3)
(3000,1)
(3000,0.3)
(3000,3)
(eΦp/kTe, Te/Ti)
101 102 103
10-2
10-1
100
101
R 
   
  /
λ D
e
m
a
x
eΦ  / kTp
T  / Tei
30.
1.0
0.2
e


Magnetic self-field effects on bare-tether current collection

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