In accordance with the contract, two problem were studied: the upper-bound limit of current collection taking into account the current's magnetic field and preparation toward numerical computation of current collection. Also investigated was a proposed scheme of measuring the location of the end body

Krivorutsky, E. N.

English

NASA Technical Reports Server

ALK3 _"45 '97 09:18 FROM E583-2_-544-5244 TO 989065?5 P_,_3E. _1
yz_ 76 _'_
Current Collection in a Maenetic Field
Final Report of E. N. Kriyorutsky, Spring 1997
In accordance with the contract, two problems were studied:
1. The upper-bound limit of ctment collection taking into account the cun'ent's
magnetic field
2. Preparation toward numerical computation of current collection
Also investigated was a proposed scheme of measuring the location of the end body.
Upper-Bound Limie of Cuz'ront Colloetion
The upper.boundlimitofcurrentcone_tionm aunifon_magneticfieldisgivenby thewell-
known formulaofParke_and Murphy. Ifthemagneticfield is notessential,thenthe
currentis10-20timeslessthanthis.The questionunderconsiderationisthis:what isthe
roleofthetethercurrentmagneticfield?Does itchangetheupper-boundlimitofcturent
collection7Also,istherea dependenceon theshapeof the current collectingwi_?
Cy11ndrlual wire
Magnetic field of the current
Itissupposedbelow thatthefieldsand plasmaareuniformalongthewire.Inthe
coordinatesystemgiveninFigureIwiththecurrent7 alongthez axis,theEar,h's
magneticfield/}oisinthey-zplaneandthecurrentmagneticfieldB, isinthex-yplane.
The complete magnetic field/_ is then
.a[_,sm_+i,(Vp+co,_)+_,cot_] (1)
a ffiS_im_ p ffir/r. re --O. 2[/a
https://ntrs.nasa.gov/search.jsp?R=19980214886 2020-06-18T00:33:51+00:00Z
Hcre, I is the current in amperes, Bo is in Gauss, and r and ro are in cm. From this, the
equation of the field lines in the x-y plane is _en
lap + pcosqp = g = const (2)
The separatrix at g'_l divides the closed lines around the wire and the uaclosed lines
farther out.
Close to the wire, 8 cbecomes greater than Bo. This is true for r<3 cm for 1=5 A,
and also at r<6 cm for 1=I0 A. On the surface of the wire, 8/B_=60 for/ffi-10 A and
r_,v=O. 1 cm. The magnetic force is the _m¢ order as the electric force for ,"<15 cm if l-- 10
A and the w_ potential is .¢w ~l(P V.
magnetic field potcntisl can be w_cn
,4 " -a_i_(r, q)) = _,8 (3)
Particle motion and the upper-bound limit of current
collection
The Lagrangian of tbc particl© motion in this magnetic field is
where ¢ is the clcctrostatic potenti,,1. L does not depend on z, so o_L/o3_-- con.st, or
m£+-CA,= P, =,coast
C
Taking into account that
E=_(/z + r2¢ =+ z2) +e¢ -- const
(4)
(5)
(e)
Ab_ 25 "97 09:19 FROM E583-L:_-544-5244 TO 98906575 PAGE.O03
I
and neglecting the thermal motion, it can be found that a pmicle can reach the wire ff
e 2_ • e..._.A 2__me
_,_- <__A,. < mc + _"
wlm'e the subscript w indicaws quantities relat_ to the w_. It follows from (7) that on
the region of the wire _th fixed An, particles are coming from the region y--+_, where
_[,._ - x givesaregionofcunentcollec_on
mC _-2t
For thecylindrical_ allpointson thewirehavenearlythesame magne_ potential
,4.u_onst, and the collected current is then
j = 2enoVr _-m _'.
oJmsinor
Th_ result is exactly the same as P_er-Murphy. It should be noted that the limit t_ -+ 0
can be found only if _ as a function of u is known.
O)
(s)
(9)
JC/_-e of arbi_raz7 shape
The inequality (7) is valid for an arbitrary shape ofthe wire, but in this case different points
of the wire surface have a different magnetic potential, A_V For fixed A w, the region of
current collection is still given by (8). For different points on the wire, A wis different and
the ends of the region Ax=x2-x I will aLso be differenL The region of current collection will
/be _-Ax where
,5. _.'-'-A, - - ---_,-- (lo)
Bosin_ Bosina
Here, a is the characteristic width of the wire and_/a-B_.
For 1-5-10 A, a--0.5-1.0 cm and fCzlx-,0.1-0.5. Therefore, th© upper-bound of the
curt'cat collection wealdy depends on the shape of the wire and can be estimated by the
Parker-Murphy formula.
Kinoti¢ Xoaols for a Numerical CalauZation of
the Current Colle=t£on
For a kinetic description of the current colIecdon, a particle simulation code as weft
as the kinetic calculation can be u6ed. A particle simu/ation code [such as that of Dr.
Singh] can be adopted for current collection by mcIuding the magnetic field of the current.
However,inclusionof particlescattering/nthecodewill bedifficult.Thetwo
charactefist/cscales,theDebyeradiusandthemeanfreepath,are0.2-1.0cmand400-600
cm, respectively, are too much different.
In a model based on the kinetic equatio.q, the last problem can be solved in two
ways. The f_t,h due to greater time of calculation gives an opportunity to include the
smallest length scale. The second method uses different models for the different lengths.
The region of cur_nt collection can be divided into a region where scattering|s essential,a
region where it is not important but where the drift equation is still correct, and a region
from out near the separatrix/n to the wire. An approach for ¢alculatiom in the second
region on an analytical and partly computational level has been developed (together with
Drs. O. V. Khazanov and M. W. Liemohn).
Boheme got Measur£ng the End Body Pos£t£on
Suppose a sowce of li$ht (laser) located on the end body spins with some period T
at an angle o_from the tether (Figure 2). In position A, the time between two signals
registered at O (mother body) is the same. If the laser is in point B, however, the time will
be different. It is supposed that the laser is oriented in the same way to the gravity field in
points A and B. Dr. Stone proposed a hanging laser to achieve this. For a_-lO ',
/=OA=106 cm, A=AB=I0 cm, and T---60 s, the difference in the time is At--10 _ s. If the
aoun:e of light gives a circle or an arc with a radius close to the maximum displacement AB
at the mother body (horizontal plane containing O), then the position of the end body in the
Satel_
0
A B
_dB_y
(LaseO
plane perpendicular to the plane of the figure can be found.
Two light receivers separated by about 10 cm on the mother
body are needed. The 10 cm end body displacement can be
registered, cornicing the time when the signal is registered
by the different receivers. The difference is around 1(_s s.
Such times are easily measured. The oscillation of the end
body arou_ its axis also can be easily registered using a
radial dark lane on the light source. As the, body rotates, the
dark line in the laser point also rotates and can be measured.
The main question is the power of the laser needed
during the daytime. It can be lowered using fdters and
different receiving schemes. During the night, the sigtleJ will
be well registered.
"9 "_ 09:20 Fl:tOt'l E383-L_S-544-5244 TO 989065",_ #AGE._3
8_tmmlk_¥
1. It is found that the upper-hotrod limit for current collection in the case of strong
magnetic field from the current is close to that given by the Parker-Murphy formula.
This conclusion is consistent with the results obtained in laboratory experiments. This
limit weakly depends on the shape of the wire. The adiabatic limit in this case will be
easily surpassed due to strong magnetic field gradients near the separatfix. The
calculations can be done using the kinetic equation in the drift approximation.
AIlalytic_ i_tmlt$ are o_aJ_ed for the region wher_ the Earth's magtletic f'_]d is
domiua_ (i_m 2 above).
2. The cmt_t collection can be calculami (neglecting r,cattcrLug) using a particle
simu]a6on Code. Dr. Singh has agrcexl to collaborate, allowing the me of his particle
code. The code can be _l_ted for the case whea the current rnRgn_.ic field is strong.
(The results from item 1 above are valid only for the upp_-bound limit, but net for the
real current collection.) The needed time for these modifications b 3-4 months.
3. The analytical description and essendal part of the program is Ixepared for the
calculation ofthecurrentintheregionwhere theadiabaticdescriptioncanbe used.
Thiswas completedwiththecollaborationf_. Khazanov andLiemohn.
4. A scheme of _ the end body position is also proposed. The initial estimations
are shown in item 3 above. The scheme was discussed in the laboratory (with Dr.
Smile) and it was concluded that it can be proposed for engineering analysis.
The all above mentioned items can be presented in more details if it will be necessary.


Current Collection in a Magnetic Field

It is found that the upper-bound limit for current collection in the case of strong magnetic field from the current is close to that given by the Parker-Murphy formula. This conclusion is consistent with the results obtained in laboratory experiments. This limit weakly depends on the shape of the wire. The adiabatic limit in this case will be easily surpassed due to strong magnetic field gradients near the separatrix. The calculations can be done using the kinetic equation in the drift approximation. Analytical results are obtained for the region where the Earth's magnetic field is dominant. The current collection can be calculated (neglecting scattering) using a particle simulation code. Dr. Singh has agreed to collaborate, allowing the use of his particle code. The code can be adapted for the case when the current magnetic field is strong. The needed dm for these modifications is 3-4 months. The analytical description and essential part of the program is prepared for the calculation of the current in the region where the adiabatic description can be used. This was completed with the collaboration of Drs. Khazanov and Liemohn. A scheme of measuring the end body position is also proposed. The scheme was discussed in the laboratory (with Dr. Stone) and it was concluded that it can be proposed for engineering analysis

F Iq_DM E583-2_F3-544-5244 TO 98_'75 PA_. _ I
,_ 7-" 2064.72
Current Collection in a Magnetic Field
Final Report of E. N. Kriyorutsky, Spring 1997
/
//. -'7o -_'P_-"
o_ lT-
l..Z g 77 7
In accordance with the contract, two problems were studied:
1. The upper-hotrod limit of cunent collection taking into account the current's
magnetic field
2. Preparation toward numerical computation of current collection
Also investigated was a proposed scheme of measuring the location of the end body.
Upper-Bound Limit: of Cut'rent Collection
The upper.bound limit of current collection in a uniform magnetic field is given by the well-
known formula of Parke_ and Murphy. If the magnetic field is not essential, then the
cm'rent is 10-20 times less than this. The question under co-sideration is this: what is the
role of the tether current magnetic field.'? Does it change the upper-bound limit of current
coUection7 Also, is there a dependence on the shapeof the current collecting wire?
Cylindrical wire
Magnetic field of the current
h is supposedbelow thatthefieldsand plasmaareuniformalongthewire.Inthe
coordinatesystemgiveninFigureIwiththecurrent7 alongthez axis,theEarth's
magnetic field Be is in the y-z plane and the current magnetic field B: isin the x-y plane.
The complete magnetic field B is then
_= a[_,s=, + _,(_p + coil)+ _,cot=] (l)
a = aosm= p = r/to ro = 0.2Ua
https://ntrs.nasa.gov/search.jsp?R=19980004569 2020-06-16T00:34:03+00:00Z
_U5 # ' £70:J : 19 FF::_'Otl E_83-205-544-5244 TO _5-P5 #AGE.
Il
Here, I is the current in amperes, B o is in Gauss, and r and r o are in cm. From this, the
equation of the field lines in the x-y plane is then
lnp + pcoscp = g = const (2)
The separatrix at g_l divides the closed lines around the wire end the unclosed fines
fmher out.
Close to the wire, B cbecomes greater than Bo- This is true for r<3 cm for 1=5 A,
and also at r<6 cm for I=10 A. On the surface of the wire, B/B#60 for/= I0 A and
r._.=O.lcm. The magneticforceisthegram orderastheelectricforceforr<15cm ifI--I0
A and the wire potential is .¢w ~10 3 V.
Themagneticfieldpotentialcan bewritten
._--a,.4.(,,_)' _,8 (3)
Particle motion and the upper-bound limit: of current
collection
The Lagrengian of thc particle motion in this raagnc_ field is
L-_(,:,,': +:)+_(_,+,_,,,,)-,,,
where ¢ is the electrostatic potential. L does not depend on z, so o_L/O_.= corot, or
mi ÷ e At = p, = const
C
Taking into account that
_o_(:,_,,,,+:) +_. oo,,,,
(4)
(5)
(6)
ALI(] _5 '_7 _; 19 FR(3?I E_83-_05-544-5244 TO 9890e5-75 P/:_iGF.{}03
i
and neglectingthethermalmotion,itcanbe foundthatapardclecanreachthewireif
e 2_< • < e + /'--_ (7)
where the subscript w indicates quantiti©s related to the wire. It follows from (7) that on
the region of the wire with fixed Azw, particl©s are coming from the region y--_, where
:_1,-..-x givesaregionfcurrentcollection
mc
For thecylindricalwireallpointson Thewirehavenearly the same magnetic potential,
h,_.cconst,and thecollectedcurrentisthen
j = 2_oV r "_I-mCP"
oJasina (9)
This result is exactly the same as Parker-Murphy. It should be noted that the limit a --* 0
canbe found onlyifO_,asafunctionofu isknown.
Wire of az-/_it:z'a_ a_apa
The inequality (7) is valid for an arbitrary shape of the wire,butinthis case different points
of the whe surface have a different magn_ potential. A,, For fix_! Aw, the region of
current collection is still given by (8). For different points on the wire. Awls different and
theends oftheregionAx=x2-xIwillalsobe _ The regionofcurrentcollectionwill
,be8+Ax where
8-Ar'_ - At'" -_ (lo)
Bosincc Bosina
Here,a isthecharacteristicwidth ofthewireanddA/a-B c.
For/~5-I0 A, a-,O.5-1.0cm and _/zlx--O.l-0.5,Therefore,theupper-boundof the
collection weakly depends on the shape of the w_'e and can be estimated by the
Parker-Murphy formula.
Kinetio Models for a NUmer£cal Calculatlon o£
the Current Collection
For a_ne_c d¢_ption ofthecurrentcollecuon,apm_cle B_u_on codeasw_
theki.ed¢calculationcan beused.A particle_n_a_on code [such_ dmt ofI_.
Singh] can be adoptedforcunentcoIIec_onby mcludingthemagnet/cfieldofthecurrent.
AL)O 25 '_7 09:20 FROM E583-L:_5-544-5244 TO 98906575 PAGE._
However, inclusion of particle scattering in the code will be difficult. The two
characteristic scales, the De,bye radius and the mean free path, are 0.2-1.0 cm and 400-600
cm, respectively, are too much different.
In a model based on the kinetic equation, the last problem can be solved in two
ways, The f'ust,b due to g_eatcr time of calculation gives an opportunity to include the
smallest leagth scale. The second method uses different models for the different |engths.
The region of current coUPon can be divided into a region where scauering is essential, a
region where it is not important but where the drift equation is still correct, and a region
from out aear the separatfix in to the wire. An approach for calculations in the second
region on an analytk:al and partly computational level b_ been developed (together with
Drs. O. V. Khazanov and M. W. Liemoha).
8ohm for N6asur£ng the End Body Pom£t£on
Suppose a source of light 0aser) located on the end body spins with some period T
at an angle _ from the tether (Figure 2). In position A, the time between two signals
registered at O (mother body) is the same. If the laser is in point B, however, the time will
be different. It is suplx_sed that _ laser is oriented in tl_ same way to the gravity field in
points A and B. Dr. Stone proposed a hanging laser 1:oachieve this. For w-lO',
/---OA=10 e cm, _AB=IO cm, and T'-.-60s, the difference in the time is At~10 4 s. If the
source of light fives a circle or an arc with a radius close to the maximum displer, ement AlE}
at the mother body (horizontal plane containing O), then the position of the end body in the
Satellite
0
A B
End Body
(Laser)
plane perpendicular to the plane of the figure can be found.
Two light receivers separated by about 10 cm on the mother
body are needed. The lO cm end body displacement can be
registered, com_g the time when the signal is registered
by the different receivers. The difference is around 10.5 s.
Such times are easily measured. The oscillation of the end
body around its axis also can be easily registered using a
radial dark lane on the light source. As the body rotates, the
dark line in the las_ point also rotates and can be measured.
The main question is the power of the laser needed
during the daytime. It can be lowered using filters and
different receiving schemes. During the night, the s/gald w_
be wetl registered.
Summary
I. It is found that the upl_'-bound limit forcunent collection in the case of strong
magnetic field from the current is close to that given by the Parker-Murphy formula.
This conclusion is consistent with the results obtsmed in laboratory experiments. This
limit weakly depends on the shape ofthe wire. The adiabatic limR in this case will be
emily sm'passed due to strong magnetic field gradients near the separatrix. The
calcul_ons can be done using the kinetic equation in the drift approximation.
Al_ytice/results are obtained for the region where the Earth's magnetic £'teld is
(item 2 above).
2. The current collection can be calculated (neglecting scatterins) using a particle
simulation code. Dr. Singh has agreed to collaborate, allowing the use of his perticle
code. The code can be edapted for the case when the current magnetic field h _trong.
(The results from item 1 above are valid only for the upper-bound limit, but not for the
real cterent collection.) The needed time for these modifications is 3-4 months.
3. The analytical description and essential part of the program is prepm_l for the
calculation of the current in the region where the adiabatic description can be used.
This was completed with thecollaboration ofDrs.Khazanov and Liemohn.
4. A scheme of measuring the end body position is also proposed. The initial estimations
are shown in item 3 above. The scheme was discussed in the laboratory (with Dr.
Stone) and it was concluded that it can be proposed for engineering analysis.
The aU abovementioneditemscan bepresentedinmore detailsifitwillbe necessary.


Current Collection in a Magnetic Field

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