Electrostatic interaction between Interball-2 and the ambient plasma. 1. Determination of the spacecraft potential from current calculations

The Interball-2 spacecraft travels at altitudes extending up to 20 000 km, and becomes positively charged due to the low-plasma densities encountered and the photoemission on its sunlit surface. Therefore, a knowledge of the spacecraft potential 8s is required for correcting accurately thermal ion measurements on Interball-2. The determination of8s is based on the balance of currents between escaping photoelectrons and incoming plasma electrons. A three-dimensional model of the potential structure surrounding Interball-2, including a realistic geometry and neglecting the space-charge densities, is used to find, through particle simulations, current-voltage relations of impacting plasma electronsIe(8s) and escaping photoelectrons Iph(8s). The inferred relations are compared to analytic relationships in order to quantify the effects of the spacecraft geometry, the ambient magnetic fieldB0 and the electron temperature Te. We found that the complex geometry has a weak effect on the inferred currents, while the presence of B0 tends to decrease their values. Providing that the photoemission saturation current densityJph0 is known, a relation between 8s and the plasma density Ne can be derived by using the current balance. Since Jph0 is critical to this process, simultaneous measurements of Ne from Z-mode observations in the plasmapause, and data on the potential difference 8s − 8p between the spacecraft and an electric probe (p) are used in order to reverse the process. A value Jph0 ∼= 32μA m−2 is estimated, close to laboratory tests, but less than typical measurements in space. Using this value, Ne and 8s can be derived systematically from electric field measurements without any additional calculation. These values are needed Correspondence to: M. Bouhram (mehdi.bouhram@cetp.ipsl.fr) for correcting the distributions of low-energy ions measured by the Hyperboloid experiment on Interball-2. The effects of the potential structure on ion trajectories reaching Hyperboloid are discussed quantitatively in a companion paper.


Introduction
The charging of a conducting spacecraft in sunlight and its influence on low-energy plasma measurements are longstanding problems in magnetospheric physics, particularly when the spacecraft body potential s is comparable to the measured plasma energies.Previous studies based on electric field measurements showed that typical values of s range from a few volts positive in the inner magnetosphere up to 50 V in the tail lobes (Pedersen, 1995).Primarily, ions with energies lower than s are repelled by the spacecraft, while ions with higher energies may reach the instruments, but at shifting energies.Consequently, only a fraction of the ion population is measured.Furthermore, the potential can enhance or decrease the number of particles collected through the influence of the electric fields in the spacecraft sheath on the particle trajectories.Concerning the Interball-2 spacecraft, electric-field measurements in the polar regions show that s ranges from 0 up to 12 V (Torkar et al., 1999).The Hyperboloid experiment (Dubouloz et al., 1998), on board Interball-2, is devoted to measure three-dimensional distributions of low-energy ions (< 80 eV).Since the energy of the measured ions is comparable to the typical values of s , The spacecraft spins around the z-axis, which is sunward directed.Therefore, the sunlit parts of the spacecraft are the surfaces perpendicular to the z-axis, i.e. the four solar panels and the top cylindrical surface.a knowledge of s and the three-dimensional potential distribution around the spacecraft body are essential for correcting ion distributions recorded by Hyperboloid.These concerns are especially relevant to ion measurements when the spacecraft enters in polar regions, where ion mean energies may be very low.
The value of s is determined by the balance of currents associated with ambient electrons and ions impacting on the spacecraft body surface and photoelectrons emitted from it, which can be expressed as (Garrett, 1981): where I ph , I e , I se and I i denote the currents of photoelectrons flowing to the plasma, incident plasma electrons, secondary electrons due to incident electrons, and plasma ions, respectively.For positive potentials, I i is smaller than I e due to the ion to electron mass ratio, and can be totally neglected.
I se has to be taken into account when the spacecraft enters in eclipse (Grard, 1973).Due to the fact that Interball-2 was always in sunlight beyond the plasmasphere, I se is negligible in our case.Hence, Eq. ( 1) is reduced to: The incoming plasma electron current I e mainly depends on s and the characteristics of the electron distribution (i.e.electron density N e and temperature T e ).The photoelectron current-voltage relation I ph ( s ) is also determined by the photoelectron energy distribution and the photoemission saturation current density or the photoemission production rate J ph0 = I ph ( s = 0)/A s , where A s denotes the sunlit area of the spacecraft body.
If we consider a spherical body immersed into an unmagnetized collisionless maxwellian plasma, I e (N e , s ) can be determined analytically (Mott-Smith and Langmuir, 1926), while an expression of I ph ( s ) can be inferred on the basis of laboratory measurements (Grard, 1973) and in-flight investigations (see Pedersen, 1995;Nakagawa et al., 2000;Scudder et al., 2000).Using this current equilibrium as described by Eq. ( 2), a density-potential relation N e ( s ) can be inferred.In this way, measurements of s from electric field double-probe experiments on various satellites (see Pedersen, 1995;Escoubet et al., 1997;Torkar et al., 1999) have been used as a diagnostic method to derive the plasma density N e .These previous works assumed a simple geometry for the spacecraft body and neglected the ambient magnetic field.
In our case study, the geometric structure of Interball-2 is very complex, as described in Fig. 1.Furthermore, at Interball-2 altitudes from about 8000 up to 20 000 km, the ambient magnetic field B 0 ranges from 1000 to 5000 nT, corresponding to an electron gyroradius from 0.7 to 3.4 m for an energy about 1 eV.Since these values are comparable to the spacecraft body size, the ambient magnetic field should affect the current-voltage characteristics.Therefore, the method using Eq. ( 2) cannot be applicable to the Interball-2 case without taking into consideration the spacecraft geometry and the ambient magnetic field B 0 .
Recently, a Laplace solution of the three-dimensional potential structure around the Interball-2 spacecraft was carried out by Zinin et al. (1995).The model neglects the space charge effects, but includes a realistic geometry of the spacecraft body and is especially designed for computing particle trajectories through a three-dimensional potential field.In the present paper, we use this potential structure in the presence of an ambient magnetic field B 0 in order to find, through particle simulations, current-voltage relationships of incoming plasma electrons I e ( s ) and escaping photoelectrons I ph ( s ).Providing that the photoemission production rate J ph0 is known, a relation between the ambient density N e and s is inferred from the equilibrium of currents by considering different values of B 0 .Since J ph0 is critical to this process, in-flight measurements of N e and s are needed to reverse the process and estimate a value of J ph0 on the Interball-2 sunlit surface.The main objective of this work is to infer a systematic table of spacecraft potentials and plasma densities from electric-field double probe measurements.These values are then used in correcting the ion distributions measured by the Hyperboloid instrument on board Interball-2 through particle trajectories and for estimating the unmeasured ion densities at low energy.The ion trajectory problem is discussed in a companion paper (Hamelin et al., this issue).
The outline of the paper is as follows.A brief description of the simulation technique is given in Sect. 2. Currentvoltage characteristics of I e ( s ) and I ph ( s ) and comparisons with analytic theories are described in Sect.3. The dependence of the currents on the different parameters (e.g.magnetic field, photoemission, electron temperature) is stud- ied.In Sect.4, a procedure for determining the photoemission rate J ph0 is described.The conclusions of the paper and some discussions of the results are given in Sect. 5.

A three-dimensional model of the potential distribution
The Interball-2 satellite is modelled according to the drawing in Fig. 1.The Hyperboloid instrument was included in the model in order to study the electric-field perturbations on thermal ion trajectories reaching the instrument.It is also important to note that a bias potential bias = −6 (from August 1996 to April 1997) or −8 V (from April 1997 to September 1998) was applied between the instrument and the spacecraft body in order to clamp the instrument potential to the plasma potential.The effect of potential distribution on ion trajectories reaching the Hyperboloid instrument is discussed by Hamelin et al. (this issue).
The Laplace equation solved to calculate the 3D potential distribution is linear, so that the potential distribution corresponding to any set of spacecraft and Hyperboloid potentials can be deduced from the two basic cases: -spacecraft potential = 1 and Hyperboloid potential = 0, giving a normalized solution u S (r); -spacecraft potential = 0 and Hyperboloid potential = 1, giving a normalized solution u H (r).
The potential solution (r) from the Laplace equation can be then expressed as: where s is the floating spacecraft potential in V.
Since the shape of the Interball-2 satellite is very extended, an accurate description of such a geometric structure requires the use of several grids.First a coarse grid (sizes: ±40 × ±40 × ±30 m, grid spacing: 0.5 m) is defined in the whole of the simulation domain .Second a finer grid (sizes: ±13×±13×±13 m, grid spacing: 0.25 m) overhangs the first grid and defines a subdomain on .The internal bound of corresponds to the spacecraft body surface.Finally a third grid (sizes: ±5 × ±5 × ±1.5 m, grid spacing: 0.025 m), finer than the second grid, defines a subdomain ω on .The spacecraft body is centered inside each domain.The method used to solve the Laplace equation is based on an especially designed multi-grid algorithm.Details of the method are described in Zinin et al. (1995Zinin et al. ( , 1998)).
Figure 2 shows an example of equipotential contours of the 3D potential structure for a given value of s .The spatial extent of the potential structure from the center of the body is about 15 m in the x − y plane and 10 m along the z-direction.We can see in the x − y plane wings of positive potentials extended along diagonal directions.These wings are due to the electric antennas located below the solar panels at z = −0.6 m (see Fig. 1).The corresponding 3D electricfield model was used in calculating particle trajectories.

Current calculations
This section describes the method of calculating the electron and photoelectron current-voltage characteristics.Since the potential distribution is not calculated self-consistently, electrons and photoelectrons can be computed separately.
The ambient electrons are simulated by using a reservoir which blankets the simulation system , and contains a drifting maxwellian electron population with a density N e , and a temperature T e .The thickness of the wall of the particle reservoirs is chosen to be sufficiently large in order to describe correctly the velocity distribution.A fixed number of electrons is kept inside the reservoir in order to conserve the electron density N e inside .Two values of T e Electrons Φ s =4 V, B 0 =0 are considered: 1 and 10 eV, corresponding to thermal populations, while the suprathermal electron populations are not computed in our model.Therefore, this model can be applied only when the satellite travels in regions where the thermal plasma is dominant.This is the case in most of the regions, but not always, especially above aurora, where the electron thermal density can be lower than the density of energetic particles.
The photoelectrons are uniformly emitted from all the sunlit parts of the spacecraft body, and are distributed in velocity according to a maxwellian distribution with a temperature T ph = 1.5 eV (Grard, 1973) and a saturation current density J ph0 .In some works based on in-flight measurements, such as Escoubet et al. (1997) or Nakagawa et al. (2000), an additional term is found in the photoelectron current for potentials greater than about 10 V. Since s is less than 12 V in our case (Torkar et al., 1999), this term is not needed in the calculations.
Each particle (electron or photoelectron) is characterized by a negative charge q e , and a mass m e .The particle trajectories are performed by solving the equation of motion of computer particles (electrons and photoelectrons) given by: where B 0 is the ambient magnetic field and E is the electric field due to the spacecraft charging.The particle motions were advanced in each time step t using a leapfrog integration technique.The electric field E(r) was obtained from the 3D potential solution (r) from the Laplace equation.E(r) was interpolated with a scheme which provides a field accuracy of about 10 −4 (Hamelin et al., this issue).

Numerical results
Particle trajectories were computed by using a Laplace solution for the 3D potential field, as described in Sect. 2. The main plasma parameters are summarized in Table 1.Several values of the floating spacecraft potential s have been considered, ranging from 0 to 10 V.The bias potential bias applied between Hyperboloid and the spacecraft body is −8 V.However, the Hyperboloid area is insignificant compared to the spacecraft body area for disturbing the electron and photoelectron current-voltage relations.We performed calculations with and without ambient magnetic field B 0 , in order to separate geometric and magnetic effects on the currents.The magnitude of B 0 ranges from 1 to 5 µT, corresponding to altitudes about 20 000 and 8000 km, respectively.When the satellite travels from the dayside to the nightside auroral zone, the angle α between B 0 and the solar panels (x − y plane) ranges from −30 • to 30 • .In our simulations, B 0 is contained in the x − z plane and different values of α are studied, as listed in Table 1.Using analytic calculations, Escoubet et al. (1997) pointed out that in a tenuous plasma (N e < 1 cm −3 ), the electron temperature T e may act as a sensitive parameter in determining the relation between N e and s .Since N e may be lower than 1 cm −3 when Interball-2 enters in polar regions, it is necessary to study the effect of T e on the incoming electron current I e .

Incoming plasma electron current-voltage relation
The simulation for ambient electrons starts at t = 0 with an empty volume.When t > 0, the electrons are simulated with the reservoir blanketing the volume, and progressively filling the box.Figure 3 shows the time history of electron collection by the spacecraft and the total electron density, as seen from a particular run.It took here about 70 µs for the collected current and the ambient density to reach a quasisteady state, which corresponds roughly to the average time for an electron from the reservoir to reach the spacecraft body across the simulation system.Electron current-voltage relationships were established by repeating simulations, for various values of B 0 and T e .Figure 4 shows the resulting curves for α = 0.

Geometrical effects
The collected current without magnetic field (circles) can be compared to the current collected by an electrostatic probe with sizes smaller than the electron Debye length and given by (Garrett, 1981): where I e0 denotes the electron thermal current given by: This current corresponds for a maxwellian distribution to the electron current collected by a body at the plasma potential ( s = 0).A T is the total spacecraft body area about 32 m 2 for Interball-2, N e is the plasma density in cm −3 , and T e is the electron temperature in eV.The curve for B 0 = 0 is very close to the curve corresponding to Eq. ( 5).This points out that the electron collected current is not sensitive to complex geometrical effects.

Magnetic field and electron temperature effects
For T e = 1 eV, the collected currents for B 0 = 1 µT (squares) and B 0 = 5 µT (triangles) are smaller than the collected current in an unmagnetized medium.This effect has been already identified in previous theoretical and numerical works of current collection by a probe in a magnetized plasma (see Laframboise and Sonmor, 1993;Singh et al., 1994).The electrons collected by the body come from a bunch of field lines forming a cylindrical volume aligned with the magnetic shadow of the body, the transverse size depending mainly on the electron gyroradius for moderate potentials.For T e = 10 eV, the electron gyroradius becomes greater than the spacecraft dimensions, and therefore the collected currents for B 0 = 1 µT (squares) and B 0 = 5 µT (triangles) are found closer to the collected current in an unmagnetized medium.This means that for T e > 10 eV and altitudes above 19 000 km, electrons can be considered as unmagnetized.
Figure 5 shows the plasma electron current for B 0 = 1 µT and different values of α.For any value of s , we found a difference on the currents of about less than 10%.This points out that for the angular range considered, there is a minor effect in the orientation of B 0 .

Escaping photoelectron current-voltage relation
The simulation for emitted photoelectrons starts at t = 0 by distributing uniformly a maxwellian population on the spacecraft sunlit surfaces.When t > 0, the photoelectron motion is followed by solving Eq. (4) for each computer particle.For a particular run, I ph ( s ) is determined by the fraction of photoelectrons which reached the ambient plasma by leaving the simulation domain .Figure 6 shows the resulting photoelectron current-voltage characteristics for α = 0 • , and different values of B 0 .

Geometrical effects
The ejected current for B 0 = 0 (circles) is compared to the currents ejected from a small spherical electrostatic sample or a point source (dashed curve) and from a planar surface (dash-dot), given by (Grard, 1973): Planar surface: where J ph0 is the photoelectron production rate, and A S = 12 m 2 is the total sunlit area.The ejected current is maximum in the point source case, when the spacecraft body size is lower than the photoelectron Debye length, as previously reported by Grard (1973).In the point source case, the equipotential surfaces are spherical, and therefore the photoelectrons are always emitted parallel to the electric field lines, and are reflected at a distance depending on their energy and not on the direction along which they have been emitted.The situation is somewhat different in the planar probe case: the equipotential surfaces are then planar, and therefore the distance at which a photoelectron is reflected also depends on the orientation of the emitted velocity vector.Therefore, all the photoelectrons of energies just higher than s can escape into the plasma in the point-source case, against only photoelectrons emitted close enough to the perpendicular direction in the planar surface case.This results for a given spacecraft body potential in the velocity phase space, in a lower number of ejected particles for a planar surface.The case of a spacecraft (e.g.Interball-2) is obviously intermediate between these two extreme cases.

Magnetic field effects
The ejected current-voltage characteristics for B 0 = 1 µT (squares) and B 0 = 5 µT (triangles) are smaller than the ejected currents in an unmagnetized medium.By taking, for example, s = 4 V and B 0 = 0, about 18% of the photoelectrons leave the simulation box.This fraction decreases down to 9% and 4% for B 0 = 1 µT and B 0 = 5 µT, respectively.This is due to the gyration motion of photoelectrons curving some trajectories back to the spacecraft body, since the photoelectron gyroradius is less than the spacecraft sizes.For this reason, when B 0 increases, a significant part of the photoelectron distribution returns back to the spacecraft body surface.This effect acts as an additional process on the photoelectrons with energies below s and returning to the spacecraft.
Figure 7 shows the photoelectron current for B 0 = 1 µT and different angles between B 0 and the solar panel plane x −y.For α = ± 30 • , the net photoemission current I ph ( s ) is found to be about 10% higher than in the case of α = 0 • .This points out that in our case, the orientation of B 0 plays a minor role in calculating I ph ( s ).

Density-potential relation
In space, the spacecraft potential s floats with respect to the ambient plasma conditions, as described by the balance of currents in Eq. (2).By using the current-voltage characteristics derived above, Eq. ( 2) provides a relationship between s and the various ambient parameters, providing that the full emitted photoelectron current density or the photoemission production rate on the spacecraft sunlit surface is known.Figure 8 shows the resulting N e ( s ) relations for different values of B 0 .The cases where B 0 is not in the solar panel plane x − y are not displayed in Fig. 8, but these cases are discussed in the next section.We took a photoemission production rate J ph0 about 50 µA m −2 , which corresponds to the typical value inferred for previous magnetospheric missions (Pedersen, 1995;Escoubet et al., 1997).The unmagnetized curves are compared to the analytical solution for an unmagnetized point source with the same total area, as obtained by setting Eq. ( 5) equal to Eq. ( 7) and solving for N e : It is found that the N e ( s ) curve for B 0 = 0 is close to the analytical solution for a point source.This result is comprehensible, as discussed earlier in Sect.3.2, because the effect of the complex geometrical surface is found negligible in the current calculations.For an electron temperature T e = 1 eV, the effect of B 0 is weak on N e ( s ).This is due to the fact that both collected electron and ejected photoelectron currents are reduced under the effect of B 0 , but by the same factor, because T ph is comparable to T e in this case.
In contrast, for T e = 10 eV, ambient electrons are found as unmagnetized, while the photoelectron population remains magnetized.Therefore, for high electron temperatures, the influence of B 0 is more significant in the resulting N e ( s ) curves in our altitude range of interest (8000-20 000 km).

Yes
Step 0 Step i>0 Laboratory measurements of photoemission properties of materials have been published by Grard (1973), who used the solar spectrum, together with these laboratory measurements, in order to determine photoelectron characteristics.The photoemission production rate J ph0 is about 30 µA m −2 for indium oxide, which is the coating material of Interball-2, and 13 µA m −2 for vitreous carbon, which is used for the electric field probes.Actually, inferred values in space are higher than from laboratory tests (Pedersen, 1995), probably because gas contamination during the pre-launch can produce a surface layer of higher photoemission rate when exposed to solar radiations over a longer period.Conversely, when the perigee altitude is lower than 1000 km, as for Interball-2, the value of J ph0 can be significantly reduced, presumably due to impacts of atmospheric oxygen on the spacecraft body surface (Pedersen, 1995).All these unlinked effects suggest how difficult it is to determine the variations of J ph0 on the spacecraft body surface along its orbit.Previous missions (Pedersen, 1995) showed that J ph0 ranges from 10 µA m −2 for low-altitude orbits (e.g.Viking, CRRES) to 80 µA m −2 for high-altitude orbits (e.g.ISEE, GEOS).The aim of this section is to determine a value of J ph0 for the Interball-2 case.For doing so, current-voltage characteristics from the simulations showed earlier, and in-flight measurements are used.

Method
When the Interball-2 satellite enters the plasmapause at altitudes about 15 000 km, the electron gyrofrequency f ce is about 25-50 kHz, and becomes lower than the electron plasma frequency f pe .Under these conditions, cold plasma theory predicts the existence of four separately identifiable plasma wave modes at frequencies near f ce and f pe (Stix, 1962).These modes are the free-space right-hand extraordinary (R-X) mode, the free-space left-hand ordinary (L-O) mode, the Z-mode, and the whistler mode.The lowfrequency cutoff of the R-X and L-O free-space modes are at f pe , and the respectively.The Z-mode is limited by the upper hybrid resonance, f U H = f 2 ce + f 2 pe 1/2 , and the L = 0 cutoff, whistler mode propagates at frequencies below f ce .In these conditions, the values of f pe and f R are very close to f U H .However, only the Z-mode has a upper-frequency boundary above f ce .
An example of plasmapause crossing by Interball-2 is given in Fig. 9. Panel (a) shows the electric component power spectrum, as measured by the POLRAD experiment (Hanasz et al., 1998) below 100 kHz, with a frequency resolution of 4 kHz.Figures 9b and 9c show spacecraft potential measurements by electric field double-probes, and the low-energy ion fluxes measured by the Hyperboloid experiment, respectively.The electron gyrofrequency deduced from magnetic field measurements is represented by a dashed curve in the wave spectrum.From about 13:50 UT, Interball-2 progressively enters the plasmasphere, as evidenced by a cold and dense proton population on Hyperboloid data.From about the same time, a intense emission is observed at frequencies above f ce , and is tracked by crosses in the wave spectrum.Unfortunately, the magnetic wave-field components were unavailable during this time period.However, the narrow banded nature of the emission, and the fact that it is obviously of non-gyroharmonic nature, leads us to assume that this emission corresponds to the upper-hybrid resonance of the Z-mode.The plasma density can be then inferred from the formula defining f U H (in kHz): Simultaneously, data on the spacecraft potential (panel b) are gathered by the IESP experiment (Perraut et al., 1998), which comprises double probes in order to measure the electric field as well as the potential between the spacecraft body (s) and the probe (p): A bias current I bias = 110 nA was sent to the probes in order to clamp p to near the plasma potential.I bias is added to the plasma electron current in order to compensate for the photoemission current on the probe's surface.The value of p is adjusted to maintain the current balance in the probe's surface: The probe has a radius r s = 4 cm, smaller than the photoelectron Debye length (∼ 1 m) and the photoelectron gyroradius (4 m).Under these conditions, the photoelectron current rejected from the probe is the same as for an unmagnetized point source: For the same reasons, the electron current can be expressed as: Using the current equilibrium on the spacecraft's surface, as inferred from numerical simulations, we obtain an additional relation between J ph0 , N e and s (Sect.3): Equations ( 10), ( 11), ( 12) and (15) form a system of 4 equations in 5 unknowns: N e , T e , s , p , and J ph0 .Setting one of the unknowns allows for the system to be solved.When Interball-2 travels in the plasmapause, the angle between B 0 and the solar panel plane x − y is about +30 • and the magnitude of the field is about 1 µT.Furthermore, the value of T e in the plasmasphere is about 1 eV, as confirmed by measurements from the KM7 experiment on Interball-2 (Afonin et al., 2000).Therefore, we used for Eq. ( 15) the numerical relation according to these conditions, i.e.T e = 1 eV and B 0 = 1 µT with an angle of +30 • .The system of equations is solved using an iterative procedure according to the flow chart of Fig. 10.Initially, we start setting the probe potential to the plasma potential : p (0) = 0.At the first step, the spacecraft potential s (1) is obtained from IESP measurements (Eq.10) s (1) + sp .The value s (1) is then used with the measurement of N e to find the photoemission rate J ph0 (1) from Eq. ( 15).Then, we determine the probe potential p (1) from Eq. ( 12).At the next step, the latter value of the probe potential p ( 1) is added to sp to fix the spacecraft potential s (2) .We then iterate this process until all the unknowns (i.e s (i) , p (i) , J ph0 (i) ) converge, attaining an minimum accuracy ε.
We assumed here that the photoemission rate is nearly the same for the spacecraft body and the probe.This hypothesis was justified for previous magnetospheric missions (Pedersen, 1995), showing that the potential difference between the spacecraft body and a floating probe (i.e.no bias current applied) was found about a fraction of a volt in a wide range of plasma conditions.The calculation method also imposes the condition that J ph0 has to keep roughly the same value during the period of the measurements, which is clearly the case in the absence of the eclipse and due to the narrow altitude range considered.

Results
Figure 11 shows the values for sp and N e related to conjugate measurements by IESP and Z-mode observations during the period between July 1997 and October 1997.After solving the system of equations for all the measurements, we found an average value of J ph0 = 32 ± 5 µA m −2 .We have represented in the graph the N e ( sp ) relation for T e = 1 eV and J ph0 = 32 µA m −2 .It is interesting to compare our estimated value of J ph0 with the values inferred from laboratory measurements and previous studies for other satellites.This value is in the range [10, 80 µA m −2 ] inferred from electric field double-probe measurements on board previous missions (Pedersen, 1995;Escoubet et al., 1997;Nakagawa et al., 2000).The value J ph0 = 32 µA m −2 is very close to the value inferred from laboratory measurements (Grard, 1973) which is about 30 µA m −2 for indium oxide surfaces.If we compare J ph0 to the values inferred from satellites coated in indium oxide, such as Geotail (Nakagawa et al., 2000), our value differs approximately by a factor of 2.5 (32 µA m −2 against 80 µA m −2 ).Pedersen (1995) points out that J ph0 values are generally higher in space.However, the same au- thor shows that these values can be reduced for satellites having a low-altitude perigee (< 1000 km).Another possible effect on Interball-2 is that frequent gas releases (once in 12 h) used to damp the nutation of the satellite can keep J ph0 low (Galperin, private communication).

Spacecraft potential and density tables
A value of the photoemission production rate J ph0 has been deduced from in-flight measurements and simulation results, as described in the last section.Using this value, the plasma density N e can be deduced from s by using the relations inferred numerically from Laplace simulations.Along the Interball-2 orbit, the IESP experiment provides the potential difference sp between the spacecraft body (s) and the electric probes (p).Therefore, it is possible to determine systematically the values of the plasma density N e and the floating spacecraft body potential s , with respect to the plasma.In this way, Eqs. ( 11), ( 12) and ( 15) can be computed numerically for J ph0 = 32 µA m −2 , and values of sp , ranging from 0 to 10 V, with the following unknowns N e , s , p and T e .Two values of T e are considered: T e = 1 eV and T e = 10 eV, and the ambient magnetic field B 0 is taken to be about 1 µT, with α ranging from −30 • to 30 • .This magnitude of B 0 corresponds to altitudes ranging from 15 000 to 20 000 km along the Interball-2 orbit.
Figure 12 shows the resulting curves of s and N e versus sp .For T e = 1 eV, the s ( sp ) and N e ( sp ) depend weakly on the angle α between B 0 and the solar panel plane x − y.The floating spacecraft body potential versus sp is not sensitive to the electron temperature for measurements above 2 V.An asymptotic linear shape is found above 2 V.This is due to the fact that when N e decreases very low, the electron current collected by the probe (see Eq. 14) becomes negligible in Eq. ( 12).Therefore, the value p insures the equilibrium between the bias current and the photoelectron current, giving a constant value of about 2 V, and s can be asymptotically expressed as: However, the plasma density remains more sensitive to the electron temperature T e when sp is less than 4 V.This result was previously reported in Sect.3.3, and is due to the fact that the electron population becomes unmagnetized when T e is high (above 10 eV), modifying significantly the current equilibrium.While T e is undetermined, N e can be estimated only with a limited accuracy.Therefore, the electron temperature has to be taken into account when the satellite enters into regions where suprathermal electrons are observed, such as the auroral zones.
During the working periods of the IESP experiment, these diagnostic curves will be put as input parameters for determining the floating spacecraft potential with respect to the plasma.An example is given in Hamelin et al. (this issue), where the knowledge of sp and therefore s is used to perform both energy and angular corrections on ion distributions measured by the Hyperboloid instrument.

Summary
A method for determining the floating potential s of the Interball-2 spacecraft as a function of the different plasma pa-rameters has been developed on the basis of the current balance between photoelectrons rejected from the spacecraft's sunlit surface and incoming plasma electrons.In contrast to previous works based on this method (see Pedersen, 1995;Escoubet et al., 1997), the spacecraft model is not approximated to a simple geometry and consequently, analytic formulas are not useful.In this way, current-voltage relations of escaping photoelectrons I ph ( s ) and incoming plasma electrons I e ( s ) are inferred numerically from particle trajectories in a realistic three-dimensional model of the potential distribution surrounding Interball-2 in the infinite Debye length limit (i.e.Laplace solution).By comparing the simulation results with analytic relationships, we point out that the inferred current-voltage relations are weakly modified by the complex geometrical effects.Furthermore, we have studied the dependence of the currents on the different parameters, such as the electron temperature T e , the magnitude and the direction of the ambient geomagnetic field B 0 .For the orbital conditions considered, the magnitude of B 0 has a more larger effect on the current-voltage relations than its orientation in the spacecraft frame.Actually, the main effect, when B 0 is included, is to reduce both I ph ( s ) and I e ( s ), because of particle gyroradii comparable to the spacecraft dimensions.When B 0 is fixed, the current equilibrium between I ph ( s ) and I e ( s ) provides a relation between s and the plasma parameters (electron density N e and temperature T e ).Meanwhile, in order to obtain realistic values of I ph , we need to know the photoemission rate or the saturation current density J ph0 = I ph ( s = 0)/A s , where A s denotes the spacecraft's sunlit area.In this way, we have developed a reversed method, using in-flight measurements of N e and the potential difference sp = s − p between the spacecraft and an electric probe.The method consists of solving, by setting T e = 1 eV, a system of four equations with N e , s , p and J ph0 as unknowns.The inferred photoemission rate J ph0 is about 32 µA m −2 , comparable to laboratory predictions (Grard, 1973).
Once J ph0 is determined on Interball-2 and providing that B 0 and T e are known, values of the plasma density N e and the spacecraft potential s can be found systematically from inflight measurements of sp , without any analytic calculation.Measurements of B 0 are currently available from DC magnetometers.The situation is somewhat different for T e .Since measurements of T e are not systematic, we have considered two extreme values T e = 1 to 10 eV in order to have an idea of the uncertainty when determining N e and s .Such values of T e correspond to plasma conditions where cold electron populations are dominant, such as in the polar regions or in the plasmapause.It is found that s can be inferred precisely from sp , while the accuracy in determining N e is still limited.
All these results are fundamental for correcting thermal ion measurements on Interball-2.In particular on Interball-2, the knowledge of s is essential for studying the spacecraft charging effects on the ion distributions recorded by Hyperboloid, such as the distortion effects on the ion trajectories reaching the vicinity of the instrument (see Hamelin et al., this issue).Systematic estimations of s and N e , along with angular corrections providing that measurements of sp are available, will be used in correcting ion distributions.Furthermore, the knowledge of N e provides an estimation of the density of low-energy ions repelled by the potential structure and missed by the instrument.Ultimately, a Hyperboloid database will take into account these corrections for the twoyear working period of the instrument.

Fig. 1 .
Fig. 1.Geometric structure of the Interball-2 satellite.The main body is a cylinder (radius 0.75 m, height 1.60 m) along the z-axis.Four flat solar panels looking like petals, are located in the x − y plane with extended booms at the end of each panel.Four other antennas (length 10 m) are located in a plane parallel to the x − y plane.A monopole (length 10 m) is inclined at 5 • to the z-axis.The spacecraft spins around the z-axis, which is sunward directed.Therefore, the sunlit parts of the spacecraft are the surfaces perpendicular to the z-axis, i.e. the four solar panels and the top cylindrical surface.

Fig. 2 .
Fig. 2. Equipotential contours in V of the three-dimensional potential distribution near Interball-2 for s = 4 V, and bias = −8 V in different planes: (a) in the solar panels plane x − y, and (b) in a meridian plane x − z through the booms.

Fig. 3 .
Fig. 3. Temporal evolution of collected current (thick curve, I e ) and electron density (thin curve, N e ) in macroparticle unit, for s = 4 V, T e = 1 eV and without magnetic field.The electron current is obtained by counting the number of macroparticles impacting the spacecraft body surface during a time step.

Fig. 4 .
Fig.4.Electron current-voltage characteristics for electron temperatures of T e = 1 eV (solid) and T e = 10 eV (dashed), for B 0 = 0 (circles), B 0 = 1 µT (squares) and B 0 = 5 µT (triangles).The two other curves correspond to the characteristic for a small Langmuir probe.All the characteristics are normalized with respect to the electron current for T e = 1 eV, B 0 = 0 and s = 0.

Fig. 6 .
Fig.6.Photoelectron current-voltage characteristics for B 0 = 0 (circles), B 0 = 1 µT (squares) and B 0 = 5 µT (triangles).The dashed and dash-dot curves display the currents ejected from a point source and from a planar surface, respectively.All the curves are normalized with respect to the photoelectron saturation current.

Fig. 9 .
Fig. 9. Interball-2 data obtained on 13 July 1997 between 13:30 and 14:00 UT, around the plasmapause.From top to bottom: (a) wave spectrogram in dB from the POLRAD electric (Y) antenna, (b) spacecraft potential measurements in V with respect to the electric probes from the IESP experiment, (c) H + ion fluxes in (m 2 eV s sr) −1 versus time and energy from Hyperboloid.

Fig. 10 .
Fig. 10.Flow chart of the procedure used for determining the photoemission rate.

Fig. 11 .
Fig. 11.Results of diagnostic measurements.The diamonds show the measurements of sp and N e inferred from the IESP and POL-RAD experiments, respectively.The curve corresponds to a fitted N e ( sp ) relation for T e = 1 eV, B 0 = 1 µT, α = 30 • , and by assuming a value of the photoemission rate J ph0 = 32 µA m −2 .

Table 1 .
List and values of numerical parameters used in the simulations