МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ
DOI: https://doi.org/10.15407/itm2025.04.067 The goal of this article is to theoretically substantiate the applicability of the classical formula of the single cylindrical probe theory to determining the electron density based on measurements of the currents of a floating probe system in ionospheric...
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Technical Mechanics| _version_ | 1870649989771821056 |
|---|---|
| author | LAZUCHENKOV, D. N. LAZUCHENKOV, N. M. |
| author_facet | LAZUCHENKOV, D. N. LAZUCHENKOV, N. M. |
| author_institution_txt_mv | [
{
"author": "D. N. LAZUCHENKOV",
"institution": "Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, 15 Leshko-Popel St., Dnipro 49005, Ukraine; e-mail: lazuch.dn@gmail.com"
},
{
"author": "N. M. LAZUCHENKOV",
"institution": "Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, 15 Leshko-Popel St., Dnipro 49005, Ukraine"
}
] |
| author_sort | LAZUCHENKOV, D. N. |
| baseUrl_str | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/oai |
| collection | OJS |
| datestamp_date | 2026-07-13T20:26:18Z |
| description | DOI: https://doi.org/10.15407/itm2025.04.067
The goal of this article is to theoretically substantiate the applicability of the classical formula of the single cylindrical probe theory to determining the electron density based on measurements of the currents of a floating probe system in ionospheric conditions. Probe measurements in the ionosphere are modeled using a cylindrical probe and the body of a very small satellite placed transversely in the incident supersonic flow of a collisionless plasma. The ionospheric plasma is considered to be Maxwellian and consists of electrons and singly charged atomic ions of oxygen and hydrogen. A mathematical model of current collection by the floating probe system “probe – plasma – satellite body” was developed based on classical relationships for the electron and ion currents to a thin cylinder placed transversally in the incident flow. To model the collection of the hydrogen ion current by the satellite body, the results of numerical calculations of the ion current to the cylinder using the two-dimensional Vlasov-Poisson model were approximated. The probe bias potential was determined such that, under ionospheric conditions, the electron region of the floating probe system's current-voltage characteristic is closest to that of a single probe. The determination of the electron density using the classical calculation formula of the single-probe theory was simulated for probe current measurements in the low voltage portion of the electron region of the current-voltage characteristic of the floating probe system. The limiting methodological error in determining the electron density under ionospheric conditions within the framework of the probe system model considered was estimated. The effect of probe current measurement errors on the determination of the electron density using the calculation formula of the single-probe theory was studied. The obtained results may be used in the preparation and interpretation of ionospheric plasma diagnostics experiments using ultra-small satellites.
REFERENCES
1. Lebreton J. P., Stverak S., Travnicek P. et al. The ISL Langmuir probe experiment processing onboard DEMETER: Scientific objectives, description and first results. Planetary and Space Science. 2006. V. 54. No. 5. Pp. 472 - 486. https://doi.org/10.1016/j.pss.2005.10.017
2. Liu D., Zeren Z., Shen X. et al. Typical ionospheric disturbances revealed by the plasma analyzer package onboard the China Seismo-Electromagnetic Satellite. Advances in Space Research. 2021. V. 68. Pp. 3796 - 3805. https://doi.org/10.1016/j.asr.2021.08.009
3. Davis B. Studying the ionosphere with Langmuir probe with an application to seismic monitoring. Final report. ASEN 5168: Remote Sensing, 2012. 11 pp.
4. Oyama K. DC Langmuir probe for measurement of space plasma : A brief review. Journal Astronomy and Space Science. 2015. V. 32. No. 3. Pp. 167-180. https://doi.org/10.5140/JASS.2015.32.3.167
5. Boyd R. Langmuir probes on spacecraft. Plasma Diagnostics. W. Lochte-Holtgreven (Ed.). New York : AIP Press, 1995. Pp. 732-776.
6. Chung, P. M., Talbot L., Touryan K. J. Electric Probes in Stationary and Flowing Plasmas. Springer-Verlag, 1975. 150 pp. https://doi.org/10.1007/978-3-642-65886-0
7. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of probe measurements in a supersonic flow of a four-component collisionless plasma. Teh. Meh. 2020. No. 4. Pp. 97 - 108.https://doi.org/10.15407/itm2020.04.097
8. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical simulation of ionospheric plasma diagnostics by electric current measurements using an insulated probe system. Teh. Meh. 2024. No. 2. Pp. 112 - 123. https://doi.org/10.15407/itm2024.02.112
9. Mott-Smith H., Langmuir I. The theory of collectors in gaseous discharges. Phys. Rev. 1926. V. 28. No. 5. Pp. 727-763. https://doi.org/10.1103/PhysRev.28.727
10. Lazuchenkov D. N., Lazuchenkov N. M. Calculation of the ion current to a conducting cylinder in a supersonic flow of a collisionless plasma. Teh. Meh. 2022. No. 3. Pp. 80 - 88.https://doi.org/10.15407/itm2022.03.091
11. Hoegy W. R., Wharton L. E., Current to a moving cylindrical electrostatic probe. Journal of Applied Physics. 1973. V. 44. No. 12. Pp. 5365-5371. https://doi.org/10.1063/1.1662157
12. Laframboise J. G. Theory of Spherical and Cylindrical Langmuir Probes in a Collisionless Maxwellian Plasma at Rest. Report, No. 100. Univ. of Torto, Institute of Aerospace Studies. 1966. 210 pp. https://doi.org/10.21236/AD0634596
13. Godard R., Laframboise J. Total current to cylindrical collectors in collisionless plasma flow. Planetary Space Science. 1983. V. 31, No. 3. Рp. 275-283. https://doi.org/10.1016/0032-0633(83)90077-6
14. Choiniere E. Theory and experimental evaluation of a consistent steady-state kinetic model for two-dimensional conductive structures in ionospheric plasmas with application to bare electrodynamic tethers in space: Ph.D. dissertation. University of Michigan, 2004. 288 pp.
15. International Reference Ionosphere-2012 (IRI-2012). https://ccmc.gsfc.nasa.gov/modelweb/models/iri2012_vitmo.php. |
| first_indexed | 2025-12-17T12:05:41Z |
| format | Article |
| fulltext |
67
UDC 533.9 https://doi.org/10.15407/itm2025.04.067
D. N. LAZUCHENKOV, N. M. LAZUCHENKOV
MATHEMATICAL MODELING OF ELECTRON DENSITY DETERMINATION
USING A STATIONARY CYLINDRICAL LANGMUIR PROBE UNDER
IONOSPHERIC CONDITIONS
The Institute of Technical Mechanics of National Academy of Sciences of Ukraine and State Space Agency of
Ukraine, 15 Leshko-Popelya St., Dnipro 49005, Ukraine; e-mail: lazuch.dn@gmail.com
Метою статті є теоретичне обґрунтування застосування класичної формули теорії одиночного
циліндричного зонда для визначення концентрації електронів за окремими вимірюваннями струмів
плаваючої зондової системи в умовах іоносфери. Моделювання зондових вимірювань в іоносфері
виконано на прикладі циліндричних зонда і корпусу надмалого супутника, що розташовані поперечно в
надзвуковому беззіштовхувальному потоці плазми. Іоносферна плазма вважається максвелівською,
складається з електронів та однозарядних атомарних іонів кисню та водню. Розроблено математичну
модель збирання електричних струмів плаваючою зондовою системою "зонд – плазма – корпус
супутника". Модель побудована на основі класичних співвідношень для електронного та іонного струмів
на тонкий циліндр, що поперечно обтікається. Для моделювання збирання корпусом супутника іонів
водню отримано апроксимацію результатів числових розрахунків іонного струму на циліндр за
двовимірною моделлю Власова–Пуассона. Знайдено значення потенціалів зсуву зонда, за яких в умовах
іоносфери електронна область вольтамперної характеристики плаваючої зондової системи найбільш
близька до вольтамперної характеристики одиночного зонда. Виконано моделювання визначення
концентрації електронів за класичною розрахунковою формулою теорії одиночного зонда при
вимірюваннях зондових струмів на низьковольтній ділянці електронної області вольтамперної
характеристики зондової системи. Отримано граничні оцінки методичної похибки визначення
концентрації електронів в умовах іоносфери в рамках моделі розглянутої зондової системи. Досліджено
вплив похибок виміру зондового струму на визначення концентрації електронів за розрахунковою
формулою теорії одиночного зонда. Отримані результати можуть бути використані при підготовці та
інтерпретації експериментів із діагностики іоносферної плазми з використанням надмалих супутників.
Ключові слова: іоносферна плазма, іони атомарного кисню та водню, надмалі космічні апарати,
плаваюча зондова система, одиночний циліндричний зонд, математична модель збирання струму,
достовірність визначення концентрації електронів.
The goal of this article is to theoretically substantiate the applicability of the classical formula of the single
cylindrical probe theory to determining the electron density based on measurements of the currents of a floating
probe system in ionospheric conditions. Probe measurements in the ionosphere are modeled using a cylindrical
probe and the body of a very small satellite placed transversely in the incident supersonic flow of a collisionless
plasma. The ionospheric plasma is considered to be Maxwellian and consists of electrons and singly charged
atomic ions of oxygen and hydrogen. A mathematical model of current collection by the floating probe system
“probe – plasma – satellite body” was developed based on classical relationships for the electron and ion currents
to a thin cylinder placed transversally in the incident flow. To model the collection of the hydrogen ion current by
the satellite body, the results of numerical calculations of the ion current to the cylinder using the two-dimensional
Vlasov-Poisson model were approximated. The probe bias potential was determined such that, under ionospheric
conditions, the electron region of the floating probe system's current-voltage characteristic is closest to that of a
single probe. The determination of the electron density using the classical calculation formula of the single-probe
theory was simulated for probe current measurements in the low voltage portion of the electron region of the
current-voltage characteristic of the floating probe system. The limiting methodological error in determining the
electron density under ionospheric conditions within the framework of the probe system model considered was
estimated. The effect of probe current measurement errors on the determination of the electron density using the
calculation formula of the single-probe theory was studied. The obtained results may be used in the preparation
and interpretation of ionospheric plasma diagnostics experiments using ultra-small satellites.
Key words: ionospheric plasma, atomic oxygen and hydrogen ions, ultra-small spacecraft, floating probe
system, single cylindrical probe, mathematical model of current collection, reliability of electron density
determination.
Introduction. Stationary cylindrical Langmuir probes are traditionally used for
ionospheric plasma diagnostics on spacecraft [1 – 4] due to the simplicity of the
measuring equipment design, well-developed theory, and acceptable accuracy of
measurements of local plasma parameters (electron density and temperature) in the
vicinity of the spacecraft.
Exposed part of the spacecraft body is typically used as the reference electrode
for the probe. The probe measurement system onboard the spacecraft is floating – the
reference electrode (spacecraft) potential acquires such value that the total current on
© D. N. Lazuchenkov, N. M. Lazuchenkov, 2025
Техн. механіка. – 2025. – № 4.
https://doi.org/10.15407/itm2025.04.0
68
the spacecraft equals zero [5]. It is known [5, 6] that to utilize the classical
relationships of a single cylindrical probe in the electron saturation regime (at high
bias potentials of the probe relative to the spacecraft), the area of the reference
electrode must be 3-4 orders of magnitude greater than the area of the probe. In the
ionosphere, researchers often conduct measurements in the low voltage side of the
electron part of the current-voltage characteristic (CVC), at bias potentials of about
10 V, where the requirement for the ratio of the reference and measuring electrodes
areas is less strong [1 – 4].
Currently, the miniaturization of electronics and power supplies offers
promising prospects for the use of stationary cylindrical Langmuir probes for
organizing global ionospheric monitoring using a large number of inexpensive nano
satellites. However, trying to increase the signal level (probe current) conflicts with
the limitations of the theory of a single stationary Langmuir probe [5]. Therefore,
quantitative estimating the applicability of the single-probe theory onboard nano
satellites is relevant.
In [7 – 8], ionospheric plasma diagnostics using a floating (isolated from the
spacecraft body) probe system is theoretically substantiated, and a mathematical
model for current collection is developed for an arbitrary ratio of the reference and
measuring electrodes areas. Calculation formulas for determining the plasma's
charged particle density are obtained for the electron saturation region, i.e., for
sufficiently high probe bias potentials. This article provides a theoretical
substantiation for the applicability of the relations of the single cylindrical probe
theory to determining the electron density in the low voltage side of the electron
region of the CVC of a floating probe system. The influence of the electrodes areas
ratio and electric current measurement errors on the reliability of electron density
determination is estimated.
Problem formulation. We model probe measurements onboard a very small
spacecraft under the following assumptions. The spacecraft body is a fairly long
circular cylinder with a base radius of cpr with conductive side surface and end
surfaces insulated from the plasma. The probe is a long circular cylinder with a
significantly smaller base radius pr , p cpr r= . From here on, the subscript "cp"
refers to the spacecraft (counter probe), and the subscript "p" refers to the probe.
Ionospheric plasma is a weakly ionized gas mixture whose charged particles are
electrons and singly charged atomic ions of oxygen O+
and hydrogen H + . For the
scale of the probe measurement problem, the interaction of charged and neutral
components of the gas mixture can be neglected. The unperturbed plasma is
considered to be Maxwellian, quasi-neutral, and nonisothermal. The ion temperatures
are identical, iH O
T T T+ += = , and the degree of nonisothermality is characterized by
the parameter e iT T= , where eT is the electron temperature. The ion composition
of the plasma is characterized by the parameter ( )n eH H O H
n n n n n+ + + + = + ,
where
H
n + ,
O
n + are the densities of ions H + and O+
, respectively, en is the
electron density.
During probe measurements, the spacecraft body is used as a reference
electrode. We consider that the axes of symmetry of the spacecraft body and probe
are perpendicular to the orbital velocity V ; the electrostatic and gas dynamic
interactions between the probe and the spacecraft in the plasma are weak; the side
69
surfaces completely absorb the charge of incident particles (electrons are absorbed,
ions are neutralized), and there is no emission current from the surfaces; the flow
around the probe and the spacecraft is free-molecular; the effect of the magnetic field
on the probe current is negligible; the presence of different species of ions in the
plasma does not significantly alter the self-consistent electric field in the vicinity of
the cylinder.
The main geometric parameter of the considered floating probe system is the
current-collecting surface areas ratio s cp pS S S= , where сpS , is the area of the
reference electrode, pS is the area of the probe ( p cpS S ). The theory of a single
cylindrical probe corresponds to the case sS → . The purpose of this article is to
evaluate the applicability of the relationships of the theory of a single cylindrical
probe for determining the electron density en by measuring probe currents in the low
voltage side of the electron region of the CVC of a floating probe system.
Mathematical model of current collection. The theory of supersonic free-
molecular transverse plasma flow around a long conducting cylinder is familiar and
studied quite well. The main parameters determining the flow regime are: the ion
velocity ratio i iS V u= , the ratio of the cylinder base radius cr to the Debye length
c dr = , and the dimensionless potential of the surface relative to the undisturbed
plasma potential c c eeU kT = . Here, V is the flow velocity, 2i i iu kT m= is the
thermal velocity of ions of mass im , cU is the dimensional potential of the surface,
k is the Boltzmann constant, and e is the unit charge.
During orbital motion in the ionosphere, the velocity ratio is 4iS for oxygen
ions and 2iS for hydrogen ions.
In the ionosphere, the Debye length for electrons is d 5 mm. Consequently,
the characteristic size of a cylindrical probe of pr 0.5 mm corresponds to
p p dr = 0.1 << 1, and the characteristic size of a spacecraft (nano satellite) of
cpr 5 cm corresponds to cp cp dr = 10.
Thus, ion flow around the probe occurs at 4iS , <<1 for O+
, and 2iS ,
<<1 for H + ; ion flow around the reference electrode (spacecraft body) occurs at
4iS , 10 for O
+
, and 2iS , 10 for H + .
There is a classical Langmuir asymptotic relation c− for the ion current in
supersonic flow around a thin cylinder ( 1 ) with ion-attracting surface potential
c [6, 9]. The applicability of the Langmuir asymptotic relation for the flow
parameters of 1iS , 1 and 3iS , 10 for the body potential of
50 0c− is substantiated based on the results of numerical modeling [10].
A mathematical model of current collection by a cylindrical electrode in a
supersonic flow of collisionless plasma with two-species ions is developed in [7 – 8].
The model is developed on the basis of asymptotic formulas for electron and ion
currents on a thin cylinder [9, 11] taking into account the results of works [12 – 13,
14]. In dimensionless quantities, the total current on the cylinder with the potential
70
relative to the undisturbed plasma potential, is estimated by the relations [8] (the
electron current on the cylinder is positive):
( ) ( ) ( ) ( ) ( )2 24 1c e n nH O
I I I I+ + = − − − , 4iS , (1)
( )
( )
2 4 , 0
exp , 0
eI
+
=
, (2)
( )
( )2 2
2 2
2 exp ,
2 1 2 ,
i i
O
i i
S S
I
S S
+
−+
=
+ −
, (3)
( )
( )2 2
2 2
2 exp 16 , 16
2 1 2 16 , 16
i i
H
i i
S S
I
S S
+
−+
=
+ −
, (4)
where cI , eI are the total and electron currents on the cylinder, respectively, which
are normalized by the thermal electron current;
O
I + ,
H
I + are ion currents on the
cylinder, respectively, which are normalized by the thermal currents of ions of
corresponding species; eeU kT= is the dimensionless electric potential (U is the
dimensional potential); 2 e O
m m + = is the mass ratio of charged particles;
i O
S V u += is the velocity ratio for O +
ions . The thermal current of particles of
species is ,0 cI j S = , where 2j en u = is the density of the thermal
current, 2u kT m = is the thermal velocity, T and m are the temperature
and mass of the particles, cS is the area of the current-collecting surface of the
cylinder. From here on, the index = i refers the value to the ions, the index =O +
to atomic oxygen ions, the index = H + to atomic hydrogen ions, and the index
= e to electrons.
Relations (1)–(4) fairly well approximate the current collection by the cylinder
at 1 = , 1iS and, within the framework of the adopted assumptions, can be used
to model the currents on the stationary cylindrical probe. These same relations are
applicable to currents on cylinder at 10 , 3iS . Therefore, to model the electron
and O +
ions currents to the reference electrode (spacecraft body) at potentials
50 − , relations (2) and (3) are used.
To model the collection of H + ions current by the spacecraft body, we use the
results of numerical calculations of ion currents to the cylinder using the two-
dimensional Vlasov-Poisson model [10, 14] at 1 10 , 1 3iS and potentials
50 0− . Based on the results of the numerical calculations, the ion current to the
cylinder is approximated as follows:
71
( )
( )
( )
2
1
2 2
2
exp log 1 2 16 , 0
2
1 2 16 , 0
i
H
i
S
I
S
+
− + +
=
+ −
, (5)
where
( )( )
( )( )
( )( )
( ) ( )
2 2 2 2
2
22 2 2 2 2
96 1 7 1 49
0.163 0.38 0.315
33.271 33 11 6 1
i
i
S
S
+ − + − + = + +
+ + + + + +
.
Fig. 1 shows the results of calculating the dimensionless ion current iI as a
function of the dimensionless potential of the cylinder at velocity ratios of iS =1
(Fig. 1, a)), iS =2 (Fig. 1, b)), iS =3 (Fig. 1, c)) for different . Curves
(approximations) and markers (calculations, see Fig. 1, c) for markers designation)
correspond to = 1 (1), =1 (2, 6), =3 (3, 7), =5 (4, 8), =10 (5, 9, 10).
Markers 6, 8, 9 represent calculations [14]; 7, 10 – calculations [10].
a) b) c)
Fig. 1
As one can see at Fig. 1, relation (5) approximates the results of numerical
calculations of ion current on the cylinder with satisfactory accuracy.
Thus, in a supersonic plasma flow with two ions species, the total current on the
probe and the spacecraft body is determined by (1) through the electron current (2),
the oxygen ion current (3), and the hydrogen ion current (4) for the probe and (5) for
the spacecraft body.
Direct problem of probe measurements. A mathematical model for collecting
currents by a floating probe system with cylindrical electrodes is developed in [7, 8].
In the ionosphere, the floating probe system always has such equilibrium potential, at
which the total current of charged particles through all collecting surfaces of the
electrodes equals zero [5]. The potential of the spacecraft body cpU relative to the
undisturbed plasma potential is almost always negative. The probe potential relative
to the undisturbed plasma is p iz cpU U U= + , where izU is the probe potential
relative to the reference electrode (bias potential). Probe measurement is the
1
3
5
7
-50 -25 0
2
4
6
8
-50 -25 0
3
5
7
9
-50 -25 0
iI
□ 6
△ 7
○ 8
◇ 9
+ 10
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
iI
iI
72
recording the current pI in the "probe – plasma – reference electrode" circuit while
modulating the bias potential izU .
In dimensionless variables, calculating the CVC of the probe ( )p izI , taking
into account relations (1) – (5), reduces to a system of nonlinear equations [8]
( ) ( )p iz c iz cpI I = + , (6)
( ) ( ) 0s c cp c iz cpS I I + + = , (7)
where cp is the equilibrium potential of the reference electrode relative to the
undisturbed plasma potential, which corresponds to the bias potential iz .
Since approximation (1)–(5) of the current cI on the cylinder in a supersonic
plasma flow is a continuous piecewise analytic function of the potential and
parameters n , 2 , , iS , the solution to the nonlinear equation (7) of the current
balance of the floating probe system relative to the equilibrium potential cp exists
and is unique for all values of the bias potential iz and parameter sS . The solution
to equation (7) can be found by an iterative method [7].
Thus, relations (6), (7), (1)–(5) determine the electrical and gas-dynamic
interaction (the dimensionless VAC ( )p izI ) in the floating "probe–plasma–
reference electrode" system through the dimensionless parameters n , 2 , , iS ,
sS and the bias potential iz .
The dimensionless parameters , iS , sS , iz are determined through the
parameters of the undisturbed plasma, probe, and reference electrode: eT , iT , V ,
pS , cpS , izU . The dimensional CVC of a floating probe system writes
( ) ( )p iz e p p izI U j S I= , where ( )2e e e ej e k m n T= is the density of the
thermal electron current to the probe.
In [7, 8], the influence of electrodes areas ratio sS and plasma ion composition
n on the CVC of a probe system insulated from the spacecraft body is studied at
sufficiently high bias potentials iz . It is shown that in the electron saturation regime
the influence of n and sS on the collected probe current is the greatest.
Fig. 2 illustrates the influence of the bias potential iz and parameters n , sS
on the probe current pI of a floating probe system in the low voltage side of the
electron region of the CVC. Calculations are performed for 5iS = ,
53.4 10− = and
1.3 = . Fig. 2, a) is the dependence of pI on iz for n =0.1 (solid curves), n =0.3
(dashed curves) for sS = 100 (1), 150 (2), 200 (3), 250 (4), 2000 (5). Fig. 2, b) is the
dependence of pI on n for iz =40, sS = 2000 (1), 300 (2), 250 (3), 200 (4), 150
(5), 100 (6) and for iz =15, sS = 2000 (7), 300 (8), 250 (9), 200 (10), 150 (11), 100
(12).
It is clearly seen that the influence of n and sS on the probe current increases
as the bias potential iz increases. Increasing the electrode areas ratio sS makes the
73
CVC ( )p izI closer to the CVC of a single probe (for which sS → ). The CVC of
the floating probe system approaches to the CVC of a single probe in the low voltage
side of the electron region, at iz <40.
Let’s model numerically the probe measurements for such base values of
parameters 112 10en = m-3, 32.8 10eT = K, 1.3 = , 37.5 10V = km/s that
correspond to flow conditions in the ionosphere at an altitude of about 700 km [15].
A bias potential of iz 40 corresponds to a dimensional potential of izU 10 V
and a probe current density of about 0.018 A/m2.
a) b)
Fig. 2
Inverse problem of probe measurements. We model ionospheric
measurements by a floating probe system using the mathematical model of current
collection (6), (7), (1) – (5). Let’s consider the problem of determining the electron
density (parameter en of the mathematical model) by the results of probe current
measurements ( )p izI U in the low voltage side of the electron region of the CVC of
the floating probe system. We use the calculation formula of the theory of a single
stationary cylindrical probe [6] to determine en :
( )2
,0
z 2
p ize
e
iz
dI Um
n
eS e dU
= . (8)
Due to the linearity of ( )2
p izI U in the electron region of the CVC, (8) writes as
( ) ( )2 2
,0
z 2
p iz p ize
e
I U dU I Um
n
eS e dU
+ −
= . (9)
Here, ( )p izI U is the probe current, corresponding to the bias potential izU
within the framework of the single probe theory, 0dU is the bias potential
increment.
0
2
4
6
8
10
12
0 20 40 60 80 100
3
4
5
6
7
0 0,2 0,4 0,6 0,8 1
pI
iz n
1 2 3 4 5
1 2 3 4 5 6
7 8 9 10 11 12
pI
74
Methodological error of the procedure for applying calculation formula (9) to
determine the density en within the framework of the mathematical model of current
collection is estimated by calculating the value en by (9) using the currents found by
solving (6), (7), (1) – (5) and comparing the result with en . In this case, the relative
error ( )
en e e en n n = − corresponds to the specific measurement conditions (bias
potentials izU during probe current ( )p izI U measurements ) and the method of
processing the results (the method of numerical differentiation of ( )2
p izI U ).
The dependence of the relative methodological error n on sS is presented at
Fig. 3. Curves at Fig. 3, a) represent various n = 0 (curve 1), 0.1 (2), 0.3 (3), 0.5 (4),
0.7 (5), 0.9 (6); Fig. 3, b) – various iV = 7000 (curve 1), 7500 (2), 8000 (3); Fig. 3, c)
– various eT = 1500 (curve 1), 2000 (2), 2500 (3), 3000 (4), 3500 (5). The values of
the parameters n , eT , V cover the ranges of their variation during the ionospheric
measurements. The curves at plot 3, d) are the maximal values of the dependence
en on sS for the considered ranges of n , V , eT variations at various bias
potentials izU = 5V (curve 1), 10V (2), 15V (3), 20V (4).
a) b)
c) d)
Fig. 3
-0,1
-0,08
-0,06
-0,04
-0,02
0
100 200 300 400 500
-0,1
-0,08
-0,06
-0,04
-0,02
0
100 200 300 400 500
-0,1
-0,08
-0,06
-0,04
-0,02
0
100 200 300 400 500
-0,1
-0,08
-0,06
-0,04
-0,02
0
100 200 300 400 500
n
sS
n
sS
n
sS
n
sS
1
2
3
4
5
6
1
2
3
1
2
3
4
5
1
2
3
4
75
Analysis of the results presented in Fig. 3 shows that with an increase in the
electrodes areas ratio sS and a decrease in the bias potential izU in the low voltage
side of the electron region of the CVC, the methodological error n decreases for all
considered n , eT , V . The effect of variation of parameters n , V on the error in
determining the density is monotonic for sS >100, a variation in the electron
temperature eT , as it can be seen at Plot 3, c), leads to an intersection of the curves at
sS 220.
Variation in the flow velocity V does not significantly affect n . The influence
of n on n is the strongest, and the maximum methodological error is observed at
n = 0 (Plot 3, a)). The results presented at Plot 3, d) are the upper estimate of the
methodological error n for the procedure for determining the electron density in
ionospheric conditions using the calculation formula (9) within the framework of the
current collection model (6), (7), (1) – (5).
From the results of modeling it follows that at a bias potential of izU 10 V, the
required electrodes areas ratio is sS > 350 to ensure a methodological error of n
1%, sS 300 for n 2%, sS 250 for n 5%. The methodical error in
determining the electron density n < 4% at izU 5 V and sS 200.
Probe measurements error. The structure of formula (9) is similar to
calculation formulas for determining the electron density in a dissociated diatomic
gas flow and plasma with single-species ions [6, 8]. For the electron density en%
calculated from measured (obtained experimentally) probe currents ( )p izI U% % , the
following estimate satisfies:
( ) ( )1 2e e e n iz In n n U dU− + % . (10)
Here, n is the maximum relative error of the determination en using (9), I is
the maximum relative error of the probe current measurement, ( )p p p II I I− % .
We notice that the relative error n doesn’t depend explicitly on the geometric
parameter sS . However, the total error, including the methodological error of
formula (9), as shown in Fig 3, depends on the electrodes areas ratio sS . For the
given n (accuracy of determining the electron density) we use (10) and the results
presented at Plot 3, d) to select sS , izU , dU to estimate I (the required
measurement accuracy).
It follows from (10) that increasing the potential increment dU leads to a
decrease in n , which is typical for a numerical differentiation problem. As the bias
potential izU decreases, the error n decreases monotonically. However, in this case
probe measurements occur in the transition region of the CVC, where high-precision
measurement is difficult to conduct due to strong plasma noise.
Conclusions. The applicability in ionospheric conditions of classical relations
of the single cylindrical probe theory for determining the electron density from
individual current measurements in the low voltage side of the electron region of the
CVC of a floating probe system is theoretically substantiated. A mathematical model
76
is developed that determines the electrical and gas-dynamic interaction (CVC) in the
“floating probe–plasma–spacecraft body” system through the parameters of the
plasma, the probe system, and the probe bias potential. Calculations confirm that in
the low voltage side of the electron region of the CVC, a decrease in the probe bias
potential and an increase in the ratio of the spacecraft body and probe areas leads to a
convergence of the CVC of the floating probe system and the CVC of a single probe.
A modeling of the determination of the electron density using the classical
calculation formula of the single probe theory by measuring probe currents in the low
voltage side of the electron region of the CVC of a floating probe system is
performed. Upper estimates of the methodological error in determining the electron
density under ionospheric conditions are obtained using the floating probe system
model. It is shown that, for bias potentials not greater than 10 V and a spacecraft-to-
probe areas ratio greater than 300, the methodological error does not exceed 2 %.
Presented numerical and analytical estimates of the electron density error enable
the selection of the floating probe system's geometric parameters, probe bias
potentials, and the assessment of the required measurement accuracy when planning
and conducting ionospheric plasma diagnostic experiments.
The obtained results can be used in the preparation and interpretation of
ionospheric plasma diagnostic experiments onboard ultra-small spacecraft.
1. Lebreton J. P., Stverak S., Travnicek P. at al. The ISL Langmuir probe experiment processing onboard
DEMETER: Scientific objectives, description and first results. Planetary and Space Science. 2006. V. 54,.
No. 5. Pp. 472–486. https://doi.org/10.1016/j.pss.2005.10.017
2. Liu D., Zeren Z., Shen X. at al. Typical ionospheric disturbances revealed by the plasma analyzer package
onboard the China Seismo-Electromagnetic Satellite. Advances in Space Research. 2021. V. 68. Pp. 3796–
3805. https://doi.org/10.1016/j.asr.2021.08.009
3. Davis B. Studying the ionosphere with Langmuir probe with an application to seismic monitoring. Final report.
ASEN 5168: Remote Sensing, 2012. 11 pp.
4. Oyama K. DC Langmuir probe for measurement of space plasma : A brief review. Journal Astronomy and
Space Science. 2015. V. 32, No. 3. Pp. 167–180. https://doi.org/10.5140/JASS.2015.32.3.167
5. Boyd R. Langmuir Probes on Spacecraft. In: Plasma Diagnostics. W. Lochte-Holtgreven (Ed.). New York: AIP Press,
1995. Pp. 732–776.
6. Chung, P. M., Talbot L., Touryan K. J. Electric Probes in Stationary and Flowing Plasmas. Springer-Verlag, 1975.
150 p. https://doi.org/10.1007/978-3-642-65886-0
7. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of probe measurements in a supersonic flow of
a four-component collisionless plasma. Teh. Meh. 2020. No. 4. Pp. 97–108.
https://doi.org/10.15407/itm2020.04.097.
8. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of determination the ionospheric plasma
charged particles density by electric current measurements using an insulated probe system. Teh. Meh. 2024.
No. 2. Pp. 112–123. https://doi.org/10.15407/itm2024.02.112
9. Mott-Smith H., Langmuir I. The theory of collectors in gaseous discharges. Phys. Rev. 1926. V. 28, No. 5.
Pp. 727–763. https://doi.org/10.1103/PhysRev.28.727
10. Lazuchenkov D. N., Lazuchenkov N. M. Calculation of the ion current to a conducting cylinder in a supersonic
flow of a collisionless plasma. Teh. Meh. 2022. No. 3. Pp. 80–88. https://doi.org/10.15407/itm2022.03.080.
11. Hoegy W. R., Wharton L. E., Current to a moving cylindrical electrostatic probe. Journal of Applied Physics.
1973. V. 44, No. 12. Pp. 5365–5371. https://doi.org/10.1063/1.1662157
12. Laframboise J. G. Theory of Spherical and Cylindrical Langmuir Probes in a Collisionless Maxwellian Plasma
at Rest. Report, No. 100. Univ. of Toronto, Institute of Aerospace Studies. 1966. 210 pp.
https://doi.org/10.21236/AD0634596
13. Godard R., Laframboise J. Total current to cylindrical collectors in collision less plasma flow. Planetary Space
Science. 1983. V. 31, No. 3. Рp. 275–283. https://doi.org/10.1016/0032-0633(83)90077-6
14. Choiniere E. Theory and experimental evaluation of a consistent steady-state kinetic model for two-
dimensional conductive structures in ionospheric plasmas with application to bare electrodynamic tethers in
space : Ph.D. dissertation. University of Michigan, 2004. 288 pp.
15. International Reference Ionosphere–2012 (IRI-2012).
https://ccmc.gsfc.nasa.gov/modelweb/models/iri2012_vitmo.php.
Received on October 27, 2025,
in final form on December 2, 2025
https://www.researchgate.net/profile/Stepan-Stverak
https://www.researchgate.net/scientific-contributions/P-Travnicek-2004835648
https://www.researchgate.net/journal/Planetary-and-Space-Science-0032-0633
https://doi.org/10.1007/978-3-642-65886-0
https://doi.org/10.1103/PhysRev.28.727
https://doi.org/10.1063/1.1662157
https://doi.org/10.21236/AD0634596
https://doi.org/10.1016/0032-0633(83)90077-6
https://ccmc.gsfc.nasa.gov/modelweb/models/iri2012_vitmo.php
|
| id | oai:ojs2.journal-itm.dp.ua:article-155 |
| institution | Technical Mechanics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-14T01:00:50Z |
| publishDate | 2025 |
| publisher | текст 3 |
| record_format | ojs |
| resource_txt_mv | journal-itmdpua/d8/0e53f2f2835482bd756dbaa240fe50d8.pdf |
| spelling | oai:ojs2.journal-itm.dp.ua:article-1552026-07-13T20:26:18Z MATHEMATICAL MODELING OF ELECTRON DENSITY DETERMINATION USING A STATIONARY CYLINDRICAL LANGMUIR PROBE UNDER IONOSPHERIC CONDITIONS МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ LAZUCHENKOV, D. N. LAZUCHENKOV, N. M. ionospheric plasma, atomic oxygen and hydrogen ions, ultra-small spacecraft, floating probe system, single cylindrical probe, mathematical model of current collection, reliability of electron density determination. іоносферна плазма, іони атомарного кисню та водню, надмалі космічні апарати, плаваюча зондова система, одиночний циліндричний зонд, математична модель збирання струму, достовірність визначення концентрації електронів. DOI: https://doi.org/10.15407/itm2025.04.067 The goal of this article is to theoretically substantiate the applicability of the classical formula of the single cylindrical probe theory to determining the electron density based on measurements of the currents of a floating probe system in ionospheric conditions. Probe measurements in the ionosphere are modeled using a cylindrical probe and the body of a very small satellite placed transversely in the incident supersonic flow of a collisionless plasma. The ionospheric plasma is considered to be Maxwellian and consists of electrons and singly charged atomic ions of oxygen and hydrogen. A mathematical model of current collection by the floating probe system “probe – plasma – satellite body” was developed based on classical relationships for the electron and ion currents to a thin cylinder placed transversally in the incident flow. To model the collection of the hydrogen ion current by the satellite body, the results of numerical calculations of the ion current to the cylinder using the two-dimensional Vlasov-Poisson model were approximated. The probe bias potential was determined such that, under ionospheric conditions, the electron region of the floating probe system's current-voltage characteristic is closest to that of a single probe. The determination of the electron density using the classical calculation formula of the single-probe theory was simulated for probe current measurements in the low voltage portion of the electron region of the current-voltage characteristic of the floating probe system. The limiting methodological error in determining the electron density under ionospheric conditions within the framework of the probe system model considered was estimated. The effect of probe current measurement errors on the determination of the electron density using the calculation formula of the single-probe theory was studied. The obtained results may be used in the preparation and interpretation of ionospheric plasma diagnostics experiments using ultra-small satellites. REFERENCES 1. Lebreton J. P., Stverak S., Travnicek P. et al. The ISL Langmuir probe experiment processing onboard DEMETER: Scientific objectives, description and first results. Planetary and Space Science. 2006. V. 54. No. 5. Pp. 472 - 486. https://doi.org/10.1016/j.pss.2005.10.017 2. Liu D., Zeren Z., Shen X. et al. Typical ionospheric disturbances revealed by the plasma analyzer package onboard the China Seismo-Electromagnetic Satellite. Advances in Space Research. 2021. V. 68. Pp. 3796 - 3805. https://doi.org/10.1016/j.asr.2021.08.009 3. Davis B. Studying the ionosphere with Langmuir probe with an application to seismic monitoring. Final report. ASEN 5168: Remote Sensing, 2012. 11 pp. 4. Oyama K. DC Langmuir probe for measurement of space plasma : A brief review. Journal Astronomy and Space Science. 2015. V. 32. No. 3. Pp. 167-180. https://doi.org/10.5140/JASS.2015.32.3.167 5. Boyd R. Langmuir probes on spacecraft. Plasma Diagnostics. W. Lochte-Holtgreven (Ed.). New York : AIP Press, 1995. Pp. 732-776. 6. Chung, P. M., Talbot L., Touryan K. J. Electric Probes in Stationary and Flowing Plasmas. Springer-Verlag, 1975. 150 pp. https://doi.org/10.1007/978-3-642-65886-0 7. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of probe measurements in a supersonic flow of a four-component collisionless plasma. Teh. Meh. 2020. No. 4. Pp. 97 - 108.https://doi.org/10.15407/itm2020.04.097 8. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical simulation of ionospheric plasma diagnostics by electric current measurements using an insulated probe system. Teh. Meh. 2024. No. 2. Pp. 112 - 123. https://doi.org/10.15407/itm2024.02.112 9. Mott-Smith H., Langmuir I. The theory of collectors in gaseous discharges. Phys. Rev. 1926. V. 28. No. 5. Pp. 727-763. https://doi.org/10.1103/PhysRev.28.727 10. Lazuchenkov D. N., Lazuchenkov N. M. Calculation of the ion current to a conducting cylinder in a supersonic flow of a collisionless plasma. Teh. Meh. 2022. No. 3. Pp. 80 - 88.https://doi.org/10.15407/itm2022.03.091 11. Hoegy W. R., Wharton L. E., Current to a moving cylindrical electrostatic probe. Journal of Applied Physics. 1973. V. 44. No. 12. Pp. 5365-5371. https://doi.org/10.1063/1.1662157 12. Laframboise J. G. Theory of Spherical and Cylindrical Langmuir Probes in a Collisionless Maxwellian Plasma at Rest. Report, No. 100. Univ. of Torto, Institute of Aerospace Studies. 1966. 210 pp. https://doi.org/10.21236/AD0634596 13. Godard R., Laframboise J. Total current to cylindrical collectors in collisionless plasma flow. Planetary Space Science. 1983. V. 31, No. 3. Рp. 275-283. https://doi.org/10.1016/0032-0633(83)90077-6 14. Choiniere E. Theory and experimental evaluation of a consistent steady-state kinetic model for two-dimensional conductive structures in ionospheric plasmas with application to bare electrodynamic tethers in space: Ph.D. dissertation. University of Michigan, 2004. 288 pp. 15. International Reference Ionosphere-2012 (IRI-2012). https://ccmc.gsfc.nasa.gov/modelweb/models/iri2012_vitmo.php. DOI: https://doi.org/10.15407/itm2025.04.067 Метою статті є теоретичне обґрунтування застосування класичної формули теорії одиночного циліндричного зонда для визначення концентрації електронів за окремими вимірюваннями струмів плаваючої зондової системи в умовах іоносфери. Моделювання зондових вимірювань в іоносфері виконано на прикладі циліндричних зонда і корпусу надмалого супутника, що розташовані поперечно в надзвуковому беззіштовхувальному потоці плазми. Іоносферна плазма вважається максвелівською, складається з електронів та однозарядних атомарних іонів кисню та водню. Розроблено математичну модель збирання електричних струмів плаваючою зондовою системою "зонд – плазма – корпус супутника". Модель побудована на основі класичних співвідношень для електронного та іонного струмів на тонкий циліндр, що поперечно обтікається. Для моделювання збирання корпусом супутника іонів водню отримано апроксимацію результатів числових розрахунків іонного струму на циліндр за двовимірною моделлю Власова–Пуассона. Знайдено значення потенціалів зсуву зонда, за яких в умовах іоносфери електронна область вольтамперної характеристики плаваючої зондової системи найбільш близька до вольтамперної характеристики одиночного зонда. Виконано моделювання визначення концентрації електронів за класичною розрахунковою формулою теорії одиночного зонда при вимірюваннях зондових струмів на низьковольтній ділянці електронної області вольтамперної характеристики зондової системи. Отримано граничні оцінки методичної похибки визначення концентрації електронів в умовах іоносфери в рамках моделі розглянутої зондової системи. Досліджено вплив похибок виміру зондового струму на визначення концентрації електронів за розрахунковою формулою теорії одиночного зонда. Отримані результати можуть бути використані при підготовці та інтерпретації експериментів із діагностики іоносферної плазми з використанням надмалих супутників. ПОСИЛАННЯ 1. Lebreton J. P., Stverak S., Travnicek P. at al. The ISL Langmuir probe experiment processing onboard DEMETER: Scientific objectives, description and first results. Planetary and Space Science. 2006. V. 54,. No. 5. Pp. 472–486. https://doi.org/10.1016/j.pss.2005.10.017 2. Liu D., Zeren Z., Shen X. at al. Typical ionospheric disturbances revealed by the plasma analyzer package onboard the China Seismo-Electromagnetic Satellite. Advances in Space Research. 2021. V. 68. Pp. 3796–3805. https://doi.org/10.1016/j.asr.2021.08.009 3. Davis B. Studying the ionosphere with Langmuir probe with an application to seismic monitoring. Final report. ASEN 5168: Remote Sensing, 2012. 11 pp. 4. Oyama K. DC Langmuir probe for measurement of space plasma : A brief review. Journal Astronomy and Space Science. 2015. V. 32, No. 3. Pp. 167–180. https://doi.org/10.5140/JASS.2015.32.3.167 5. Boyd R. Langmuir Probes on Spacecraft. In: Plasma Diagnostics. W. Lochte-Holtgreven (Ed.). New York: AIP Press, 1995. Pp. 732–776. 6. Chung, P. M., Talbot L., Touryan K. J. Electric Probes in Stationary and Flowing Plasmas. Springer-Verlag, 1975. 150 p. https://doi.org/10.1007/978-3-642-65886-0 7. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of probe measurements in a supersonic flow of a four-component collisionless plasma. Teh. Meh. 2020. No. 4. Pp. 97–108. https://doi.org/10.15407/itm2020.04.097 8. Lazuchenkov D. N., Lazuchenkov N. M. Mathematical modeling of determination the ionospheric plasma charged particles density by electric current measurements using an insulated probe system. Teh. Meh. 2024. No. 2. Pp. 112–123. https://doi.org/10.15407/itm2024.02.112 9. Mott-Smith H., Langmuir I. The theory of collectors in gaseous discharges. Phys. Rev. 1926. V. 28, No. 5. Pp. 727–763. https://doi.org/10.1103/PhysRev.28.727 10. Lazuchenkov D. N., Lazuchenkov N. M. Calculation of the ion current to a conducting cylinder in a supersonic flow of a collisionless plasma. Teh. Meh. 2022. No. 3. Pp. 80–88. https://doi.org/10.15407/itm2022.03.080 11. Hoegy W. R., Wharton L. E., Current to a moving cylindrical electrostatic probe. Journal of Applied Physics. 1973. V. 44, No. 12. Pp. 5365–5371. https://doi.org/10.1063/1.1662157 12. Laframboise J. G. Theory of Spherical and Cylindrical Langmuir Probes in a Collisionless Maxwellian Plasma at Rest. Report, No. 100. Univ. of Toronto, Institute of Aerospace Studies. 1966. 210 pp. https://doi.org/10.21236/AD0634596 13. Godard R., Laframboise J. Total current to cylindrical collectors in collision less plasma flow. Planetary Space Science. 1983. V. 31, No. 3. Рp. 275–283. https://doi.org/10.1016/0032-0633(83)90077-6 14. Choiniere E. Theory and experimental evaluation of a consistent steady-state kinetic model for two-dimensional conductive structures in ionospheric plasmas with application to bare electrodynamic tethers in space : Ph.D. dissertation. University of Michigan, 2004. 288 pp. 15. International Reference Ionosphere–2012 (IRI-2012). https://ccmc.gsfc.nasa.gov/modelweb/models/iri2012_vitmo.php. текст 3 2025-12-11 Article Article application/pdf https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/155 Technical Mechanics; No. 4 (2025): Technical Mechanics; 67-76 Институт технической механики Национальной академии наук Украины и Государственного космического агентства Украины; № 4 (2025): Technical Mechanics; 67-76 ТЕХНІЧНА МЕХАНІКА; № 4 (2025): ТЕХНІЧНА МЕХАНІКА; 67-76 en https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/155/65 Copyright (c) 2025 Technical Mechanics |
| spellingShingle | іоносферна плазма іони атомарного кисню та водню надмалі космічні апарати плаваюча зондова система одиночний циліндричний зонд математична модель збирання струму достовірність визначення концентрації електронів. LAZUCHENKOV, D. N. LAZUCHENKOV, N. M. МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title | МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title_alt | MATHEMATICAL MODELING OF ELECTRON DENSITY DETERMINATION USING A STATIONARY CYLINDRICAL LANGMUIR PROBE UNDER IONOSPHERIC CONDITIONS |
| title_full | МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title_fullStr | МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title_full_unstemmed | МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title_short | МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВИЗНАЧЕННЯ ЕЛЕКТРОННОЇ ГУСТИНИ ЗА ДОПОМОГОЮ СТАЦІОНАРНОГО ЦИЛІНДРИЧНОГО ЗОНДА ЛЕНГМЮРА В ІОНОСФЕРНИХ УМОВАХ |
| title_sort | математичне моделювання визначення електронної густини за допомогою стаціонарного циліндричного зонда ленгмюра в іоносферних умовах |
| topic | іоносферна плазма іони атомарного кисню та водню надмалі космічні апарати плаваюча зондова система одиночний циліндричний зонд математична модель збирання струму достовірність визначення концентрації електронів. |
| topic_facet | ionospheric plasma atomic oxygen and hydrogen ions ultra-small spacecraft floating probe system single cylindrical probe mathematical model of current collection reliability of electron density determination. іоносферна плазма іони атомарного кисню та водню надмалі космічні апарати плаваюча зондова система одиночний циліндричний зонд математична модель збирання струму достовірність визначення концентрації електронів. |
| url | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/155 |
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