Simulating the charging of isolated free-falling masses from TeV to eV energies: detailed comparison with LISA Pathfinder results

In studying the interaction of the primary cosmic ray particles with the spacecraft, the simulations used in [6] are revisited, implementing a number of updates. Firstly the geometry of the LPF spacecraft has been updated. While a full reproduction of the as-flown spacecraft was deemed unnecessarily complex, a number of changes were made to the implementation of the payload elements closest to the TM to reflect design changes that occurred late in the development of the mission. A number of dimensions of electrodes within the housing have been adjusted, the material of the GRS electrodes has been changed from sapphire to molybdenum and the tungsten gravitational balance masses surrounding the GRS were adjusted to match the size and location implemented in the final design of the instrument. As with previous simulations, the total mass simulated in the spacecraft is approximately 80% of the total in-orbit mass of LPF.

Secondly, simulations have been executed using Geant4 version 10.3, which has had significant development of the software toolkit since the publication of the previous work [6] which used Geant4 version 6.0. Of key importance to the test-mass charging process are the hadronic models that deal with interactions of primary cosmic rays and their secondaries with energies above 100 MeV/nucleon, and electromagnetic processes that govern the production of low-energy particles, close to the TM.

The physics models used by the Geant4 simulations and the energies over which they are applied are defined by the simulation physics list. A comprehensive overview of the physics models in Geant4 can be found in the physics reference manual for release 10.3 used in this work ¹¹1http://cern.ch/geant4-userdoc/UsersGuides/ForApplicationDeveloper/ BackupVersions/V10.3/fo/BookForAppliDev.pdf. Highlighted below are changes relevant to the high and low energy regimes relative to the previous implementation in Ref. [6].

The baseline simulations use the FTFP_Bert reference physics list for hadronic interactions which is most commonly used for high-energy and space applications [21]. The QGSP_BiC physisc list was also used to investigate the effect of these models on the net charging rate.

Low-energy charged particle interactions including the photoelectric effect, Compton scattering, Rayleigh scattering, gamma conversion, bremsstrahlung and ionization are governed by electromagnetic (EM) physics models. These processes are well understood and validated but careful choices still need to be made in striking the balance between computational efficiency and accuracy. Any charged particle can contribute to charging irrespective of its energy and so the priority is accuracy of the simulation over computational cost, especially in the production of low-energy secondary particles. All particles created in the simulation are tracked to zero energy, but the minimum energy for secondary particle production (especially delta electrons from ionisation) depends on the EM physics model and on the incident particle type. The earlier work [6] used a customised physics list with a 250 eV minimum production cut for electrons. For the Livermore EM physics list used in this work, the recommended minimum for secondary ionisation produced by primary electrons is also around 250 eV, although the coverage of some models can be extended by extrapolation to as low as 10 eV, (see for example Ref. [22]). This work limits the minimum energy of electrons to 100 eV to avoid compromising the accuracy of the simulation. For ionisation produced by primary protons, the minimum production cut in the Livermore EM physics is at 790 eV, the mean ionisation energy for gold. The uncertainties resulting from these production cuts is discussed in Sec. IV. Atomic deexcitation processes were set to default (with fluorescence enabled, but Auger emission, Auger cascades and PIXE not enabled) for the full simulation model. However, their full contributions in terms of additional electron production were checked by a simpler simulation. In the case of 1 MeV electron primaries for example there is a small increase at the few percent level. For 300 MeV protons there was no detectable increase seen in a simulation with 10⁶ primaries. The Livermore EM physics list makes use of the G4UrbanMSCoption4 physics model for multiple scattering physics which provides the most accurate representation of scattering at boundaries.

Several low-energy physical processes below 100 eV are not modelled by Geant4 but have the potential to significantly change the results; these were identified in the original work [6]. In particular, low-energy electrons can be ejected when a higher energy charged particle crosses a surface boundary. These kinetic electrons can be ejected by either electrons (EIEE) or ions (IIEE). The KE will come from the surface layers of the TM and EH and these are all gold plated with sufficient thickness that contains the process to that layer.

KE yield. The KE yield is energy dependent and Ref. [6] used several literature measurements to estimate the full curves for gold from electrons, protons and alpha particles at normal incidence, to make a preliminary estimate of contribution to TM charging. This work parameterizes the energy and angular dependence of the KE yield to calculate the KE emission probability particle by particle.

A number of sources in the literature employ semi-empirical models to parameterize the KE yield as a function of angle and energy due to electrons [23, 24, 25] and ions [26] incident on gold.

Preliminary calculations using several of these models with an angular dependence close to $\cos^{-1}\theta$ produced an average yield with a very large number of KEs in strong disagreement with TM charging noise results [27] and with the new measurements of the dependence of charging rate on $V_{\textsc{tm}}$ presented in Ref. [5]. Very high yield at grazing incidence can be avoided by decreasing the negative exponent of the cosine dependence. It was found that the model of Furman and Pivi [24] provided a reasonable match to the experimental results. They describe the KE yield, $Y$, due to an electron with energy $E$ incident at an angle relative the surface normal $\theta$ as

where $E_{r}=E/E_{\mathrm{max}}(\theta)$, with

and

$Y_{0}$ and $E_{0}$ are the values of yield and peak energy, respectively, for particles normally incident on the surface. A number of values for $Y_{0}$ in EIEE, can be found in the literature ranging from 1.28 to 2.5 [28, 29, 30] and thus this was treated as a somewhat free parameter to be constrained by the experimental data. Indeed, the actual yield realized for real surfaces is known to depend on its condition and contamination state [31, 32]. $E_{0}$ defines the peak of the yield curve measured between 500 and 800 eV [28, 29, 30] for electrons. The parameter $s$ describes the spectral index of the yield curve for $E_{r}>E_{\mathrm{max}}$. Literature values range from 1.5 to 1.7 [24, 28, 29]. The effect on the overall yield of varying both $E_{0}$ and $s$ is degenerate with $Y_{0}$ and for this work we choose to fix values of $s=1.64$ and $E_{0}=700$ eV for EIEE as these parameters are relatively well constrained by the literature. A quantitative discussion of the impact of these parameters on the agreement between our simulations and measured data is given in Section IV. The right-hand panel in Fig. 1 illustrates how the yield curves for incident primary electrons and protons vary with angle of incidence. For IIEE due to protons, which is the subdominant process in our simulation we fix $Y_{0}=2.18$, $E_{0}=180$ keV and $s$=1.54 [33]. Note that $E_{0}$ scales approximately with the energy of maximum stopping power for the corresponding particle. IIEE due to alpha particles is small and the angular dependence of the yield is not considered. In principle the emission properties for EIEE and IIEE may vary from surface to surface within the GRS, however, in order not to introduce too many parameters to the model, the emission properties of all surfaces are assumed identical.

KE spectrum. In the context of scanning electron microscopy, there is extensive literature on the properties of EIEE, with Ref. [34] providing a comprehensive review. KE are emitted with a cosine angular distribution and there is no experimental evidence for the KE yield being different for a primary particle entering or exiting a surface. In addition, for primary electrons with energy greater than 100 eV, the resulting EIEE energy distribution is expected to be independent of the primary energy [35]. However, the shape of the energy distribution is dependent on an effective work function for kinetic emission, $\phi_{\textsc{ke}}$, and Ref. [32] gives its form as:

where $A$ is a normalization factor, and $E_{k}$ is the kinetic electron energy. Both $A$ and $\phi_{\textsc{ke}}$ are dependent on any surface contamination [32]. This form also seems to fit the predictions [19] using a specific new model in Geant4 which deals with electron production from gold down to very low energies [22], albeit with a relatively high value of $\phi_{\textsc{ke}}$. For LPF the behaviour of the gold surfaces on TMs and surrounding electrodes/enclosures was examined in the context of the photoelectric discharge system. An average photoelectric work function of 4.2 eV was found [36] which is lower than that of clean pristine gold, but is comparable to subsequent representative laboratory-based measurements [37, 38]. Taking Eq. 4 and this work function gives the KE energy distribution shown in Fig. 2. As a consequence of the energy scale of this distribution, it is expected that the transfer of secondary electrons between TM and electrode housing will be influenced by Volt-scale potentials. Actuation voltages used to control the TM position and the TM potential itself can reach this level. Therefore, depending on the number of secondary electrons emitted, an electrostatic dependence on the net TM charging rate is expected. Given the parallel-plate geometry of the GRS, the fraction of secondary electrons affected by the TM potential will depend on the distribution of energies associated with the perpendicular component of the emitted electron velocities and this has been derived from Eq. 4 using a cosine-weighted angular emission to give the dashed curve in the insert in Fig. 2.

The Geant4 simulation tool for calculating test-mass charging produced by cosmic rays is largely unchanged from previous work; hence only the principle of the simulation is described as further details are available in  [6, 7]

Within the simulation, cosmic ray particles of a given species are generated individually with an energy drawn from a distribution defined in the simulation setup. Each particle is generated on the surface of a generator sphere surrounding the spacecraft model with a random direction drawn from a cosine distribution about the surface normal. The result is an isotropic flux of particles within the simulation sphere. When simulating a differential flux of particles $J(E)$, the simulation time can be calculated from $T=N/A\int{J(E)}dEd\Omega$, where $A=\pi r^{2}$ is the cross-sectional area of the generator sphere, and $N$ is the number of particles simulated. This also allows us to allocate a mean time to the $N^{\mathrm{th}}$ simulated event.

The primary particle flux spectra used in the simulations have been updated to be as representative as possible of the flux experienced by LPF in orbit, and specifically for the times of the relevant experiments described in [5]. Although protons and ⁴He are the dominant species, making up more than 96% of cosmic ray particles in the energy regime relevant for charging, the simulations have also been carried out with ²H, ³He, ¹²C, and electrons, to include the next most abundant cosmic ray elements, and ⁵⁶Fe nuclei to examine any effects associated with heavier elements.

Proton flux data for the date of the analyzed experiment were taken from daily measurements of the cosmic ray flux by the Alpha Magnetic Spectrometer (AMS) on board the International Space Station [39]. Helium fluxes were also derived from AMS flux data, albeit with lower temporal resolution [40]; these data are available until April 2017. For comparison with LPF measurements from June 2017 the latest available data have been used. For ³He AMS measurements [41] (of the ³He/⁴He ratio) are used. PAMELA data are used for the galactic cosmic ray electron flux at solar minimum [42], and for ²H nuclei [43]. AMS data are used for ¹²C [44] and ⁵⁶Fe [45].

Since contemporaneous data for heavier ions are not available, the uncertainty associated with these flux estimates is larger than for the dominant species. However, the abundance is low enough so that the contribution to the uncertainty in the total charge rate is small, as discussed later. In the case that measurement data do not cover the relevant energy range required as input for the simulation, an extrapolation was carried out so that all primary particles extend over the range 100 MeV/nucleon to 1 TeV/nucleon for hadrons. For electrons, a low-energy cutoff of 10 MeV was chosen, just below the energy required to penetrate the 5.7 g/cm² of material shielding the TMs. For proton and helium data from AMS, the lowest energy flux data is at around 500 MeV/nucleon. To extrapolate to lower energies, an analytical description of the cosmic ray spectrum is adopted using the force-field approximation so that the flux $J$ as a function of energy, $E$, and the solar modulation parameter $\Phi$ [46, 47] can be written

Here $E_{0}=mc^{2}$, with $m$ being the mass of the particle, and $J_{\textsc{LIS}}$ is the local interstellar cosmic ray spectrum. For best agreement with the data from AMS the parameterization of [48] is chosen for $J_{\textsc{LIS}}$:

where $P(E)=\sqrt{E(E+2E_{0})}$. Values of $\Phi=445$ MV for protons and $\Phi=360$ MV for alphas give best agreement with the data. Figure 3 shows the energy distributions of primary particles used in the Geant4 simulations.

The total charge entering and leaving the TM in each event is tallied and recorded. In an extension from the previous work, the species of every particle that crosses a TM or opposing electrode housing surface is also recorded together with its location, direction, energy and event number. These particles serve as the primaries for a separate calculation of the KE emission process.

For each particle traversing a relevant boundary during a given event generated by the Geant4 simulation, the energy and angle-dependent KE yield curve (Fig. 1) is used to determine the Poisson probability of KE emission. Each electron is assigned a location based on the position where the primary particle traversed the surface and a time based on the simulation event number. Each kinetic electron is assigned an energy sampled from the distribution shown in Fig. 2, assumed independent of primary particle energy and species.

An electron-tracking code developed for analysing the behaviour of the UV discharge system [36] is used to determine the influence of electrostatic fields on the low-energy component of the TM charging rate. As in that work, the sensor is segmented into 24 parallel plate regions between the TM and housing corresponding to the equipotential surfaces on the electrode housing (18 electrodes plus 6 faces). An additional region of the housing is defined to account for corner regions without a parallel TM surface opposite and the grounded, exposed gaps around the electrodes.

To model the effect of electric fields within the sensor, fully representative voltages are applied to each electrode region. This includes all time-varying actuation and injection voltages and the instantaneous TM potential contributed by charge accumulation and capacitively induced bias.

The ability of the emitted electron to traverse the gap between TM and housing is determined based on the instantaneous potential difference at the emission location. If the electron can traverse the gap between the TM and its surroundings, the change in the TM charge is recorded. This calculation is carried out for all KEs emitted by a primary event originating in the Geant4 simulation, with the electrode voltages evaluated at a time determined by the event number.

In this simplified treatment, edge field effects are ignored. In reality, since electrons are emitted from the surface with some angular distribution, it is possible for them to escape the parallel field region and miss the TM even if they are energetic enough to traverse the local potential barrier. To account for this effect, an additional factor ($<1$) is applied to the probability of emission. This factor was determined for each surface in the sensor assuming a cosine angular distribution of emission and uniform spatial distribution. The fraction of KEs emitted from each region and directed toward or away from the TM was calculated by a simple Monte Carlo model.

The final result of the simulation is a timeline of charging events produced by primary cosmic ray interactions and secondary kinetic electron emission. Summing the two timelines, event by event, produces the final net charge timeline on each TM from which the net charging rate and charge-rate fluctuations are determined which can be compared with measurements made during the LPF mission.

In order to understand the features of the sensor design which influence the overall charging rate and its dependence on potential, a simple model is presented to supplement the Monte Carlo simulation method in relation to the behaviour of the low-energy secondary electron population.

The assumptions used in the model are:

• (i)

  The TM charging rate caused by the high energy cosmic ray processes is constant and unaffected by any applied potentials. This overall rate is denoted by $H$.

• (ii)

  The low-energy secondary electron emission rate per unit area, $R_{s}$, is the same from all surfaces, both TM and EH.

• (iii)

  When $V_{\textsc{tm}}=0$ the kinetic electrons will travel in straight lines away from their point of origin. This means that all electrons leaving the TM will end up on the EH, whereas only a fraction, $\alpha$ of those leaving the EH will end up on the TM due to the gaps between the EH and the TM.

• (iv)

  When $V_{\textsc{tm}}<0$ the flow of kinetic electrons away from the TM will be unaffected as they still all leave the TM. However, those flowing from the EH will be repelled away from the TM and some fraction $\beta(V_{\textsc{tm}})$ will no longer reach it.

• (v)

  When $V_{\textsc{tm}}>0$ the flow of kinetic electrons away from the TM will be reduced as some fraction $\beta(V_{\textsc{tm}})$ will no longer leave the TM. It is assumed that this fraction, $\beta$, has the same functional form as that for $V_{\textsc{tm}}<0$. In addition the same fraction of electrons from the EH which would otherwise miss the TM (see point (iii) above) are now attracted to the TM.

• (vi)

  In the regime of interest for the data acquired by LPF, $\beta(V_{\textsc{tm}})$ is determined by the perpendicular component of the energy distribution shown in Fig. 2. Its form depends on $\phi_{\textsc{ke}}$ and is best determined by numerical simulation.

Hence, there are four free parameters in the model, $H$, $R_{s}$, $\alpha$, and $\phi_{\textsc{ke}}$. The model predictions are then

and

where $A_{h}$ and $A_{t}$ are the surface areas of the electrode housing and TM.

In routine operations of LPF there was a continuous sinusoidal TM potential of 0.6 V applied at 100 kHz. Hence, in deriving results, from Eqs. 7 and 8 a sine-weighted average must be performed around $V_{\textsc{tm}}$.

The variation in charging rate for TM 1 and TM 2 was measured on two occasions by LPF [5]. These are referred to by the dates on which the measurements were conducted. The first, 2017-01-30, was a narrow-band data set with $V_{\textsc{tm}}$ between $\pm$0.3 V. The second, 2017-06-22, was over a wider band with $V_{\textsc{tm}}$ between $\pm$1 V, together with two extra individual data points taken under different conditions.

Figure 4 shows the evaluation of the charging rate at each step of TM potential on TM 1 and TM 2 during the wide-band measurements of 2017-06-22 reproduced from Ref. [5]. The environmental charging rates of both TMs show similar, approximately linear dependencies on $V_{\textsc{tm}}$ with a slope of $\sim-30$ e s^-1 V^-1 and a charge rate at $V_{\textsc{tm}}$=0 V of $28.3\pm 0.8$ e s^-1 on TM 1 and $30.9\pm 0.8$ e s^-1 on TM 2. The steep negative slope immediately confirms that low-energy electrons are playing a significant role and the symmetry either side of $V_{\textsc{tm}}$=0 V implies that these are originating from both the TMs and from the surrounding surfaces. The data show that for $V_{\textsc{tm}}\approx+1$ V the negative low-energy electron charging balances the positive charging from the high-energy cosmic rays and the overall charging rate goes to zero. The additional data point for TM 1 near 0 V shows the result of the charge rate measurement made with a charged TM biased to $V_{\textsc{tm}}=0$ V with dc voltages of 4.8 V applied to the $y$ and $z$ electrodes. This measurement allows a comparison of the model predictions with measured data in a higher energy regime compared to probing the charge-rate measurement dependence on test mass potential alone. Under these conditions with the overall TM 1 potential at $-0.03$ V the TM charging rate was 19.3 e s^-1. At the same time, the rate measured on TM 2 with no applied voltages and $V_{\textsc{tm}}=+0.03$ V was 30.2 e s^-1.

Using the combination of charging models described in Sec. II the environmental charging rate in the electrostatic conditions described above can be estimated.

In Table 1 the predicted charging rates and noise contributions from the different cosmic ray species are given, as well as the KE contributions at $V_{\textsc{tm}}=0$ V for nominal values of the parameters for the KE models. The results for TM 1 are presented; for TM 2 the results are consistent within the model uncertainties. The results are based on 1000 s of simulation for all cosmic ray species. The total charging rate predicted by the model is higher than that measured during the 2017-06-22 experiment by 9.2$\pm$0.2 e s^-1; this discrepancy is discussed further in the following sections. At $V_{\textsc{tm}}=0$ V, the contribution of the total charge rate from kinetic electrons is small (12% of the total) and positive. The overall charging rate from the high-energy component of the model is lower than that reported in Ref. [7, 6] for solar minimum conditions by around 10 e s^-1 though the relative contributions to charging rate of protons, ⁴He and ³He are comparable. As was found in the previous work, heavier elements make a larger contribution to charging relative to their flux levels. Nonetheless, the overall contribution from each of the elements heavier than ⁴He including ⁵⁶Fe is less than 1 e s^-1.

From Ref. [6] the near balance of the rate of negative and positive charging events combined with the occasional high multiplicity event within the high-energy processes resulted in a predicted effective stochastic noise rate of $\lambda_{\mathrm{eff}}\sim$200–300 e s^-1 in the absence of KE, with an increase to $\sim$400 e s^-1 if the then estimated KE contribution was included. Measurements of the stochastic noise rate made by LPF showed that the effective noise rate is even higher at $\lambda_{\mathrm{eff}}\sim$1100–1400 e s^-1333Note these figures need to be adjusted for incoming cosmic-ray flux (and spectrum) and to compare with Table 1 for example they need to be increased by 20% despite the mean charging rate being close to the predictions [4]. Table 1 shows the latest estimate for the KE noise contribution during the measurement of 2017-06-22. Nominally, there are equal numbers of KE electrons per second per unit area being emitted from the EH and TM surfaces. However, there is 18% more surface area on the EH than on the TM which might imply an overall reduction in the positive charging rate. The results in Table 1 show that the combined effect of some of the electron trajectories from the EH not hitting the TM, and some ac bias voltages being present even when $V_{\textsc{tm}}=0$ conspire to produce an additional small positive charging rate. However, many more electrons are exchanged between TM and EH and these will contribute to the noise. In addition, some of these will be associated with high-multiplicity events. As shown by Fig. 1 the most efficient production of KE is from sub-keV electrons, sub-MeV protons and $\sim$MeV alpha particles and not from primary cosmic ray protons or helium nuclei. Hence they will come from heavily degraded secondary particles of which there may be many per cosmic ray primary. Figure 5 illustrates the multiplicity of charging events from 1000 s of simulated data. As was demonstrated in previous work a broad range of charging event multiplicities is observed in the HE contribution to charging. Also shown is the multiplicity of KE charging events which shows net charge transfer as large as $\pm$22 e due to KE in single events. The noise contribution from the high-energy component of the charging model is 375 e s^-1, slightly higher than previous work. The additional contribution from KE is 202 e s^-1, two times higher than the level calculated by the simple analytical model which does not take into account high multiplicity events.

As seen in Table 1 the total noise predicted by the Geant4 simulations, coupled to the low-energy KE model, and following the approach of [6], is only 40% of the value actually measured (once incoming cosmic ray flux has been suitably adjusted from the in-situ radiation monitor (RM) count rate). In principle, it would be easy to enhance the low-energy electron flux to explain the full noise budget. However, as will be seen in the next section, this is constrained by requiring a fit to the observed dependence of charging rate on TM potential.

There will be additional charge rate noise caused by any variations in the GCR flux, but these are expected to be small. Inherent variability in the incoming cosmic ray fluxes during the LPF mission were monitored by its radiation monitor (RM) [50, 51, 52]. Fluctuations seen on short timescales ($f>10\,\mu$Hz) were dominated by RM counting statistics rather than real fluctuations [50]. Between 1 and 10 $\mu$Hz the RM singles count rate revealed an amplitude spectral density with $1/f$ shape. Using the measured correlation coefficient between RM single count rate and TM charging rate [5] the corresponding charge-rate noise amplitude spectral density is $1.4\times\frac{0.1\,\mathrm{mHz}}{f}$ e s${}^{-1}\,\rm{Hz}^{-1/2}$. This is a small contribution (1% of the noise power) to the measured value of 10 e s${}^{-1}\,\rm{Hz}^{-1/2}$ between 0.1 and 1 mHz [4].

Monte Carlo Simulations: The observed slope in Fig. 4 depends on the steeply falling part of the dashed curve in the insert of Fig. 2 (applicable to the perpendicular emission component of velocity), and it can be seen that an applied bias of $\pm 1$ V can actually affect a relatively large fraction of the total KE population, with perpendicular energies below the bias potential. The value of the coefficient $Y_{0}$ in Eq. 1 determines the overall scale factor for the KE yield and larger values result in larger predicted slopes. Figure 6 shows the prediction of the model compared to the measured data using a set of parameters that give the best agreement with the TM 1 data points. The shaded error region around the model prediction indicates the statistical uncertainties from the Monte Carlo simulation. In order to achieve the level of agreement shown, a constant offset has been applied to all simulation data. The uncorrected results are shown as a dashed curve. Based on results from the toy model of the KE charging process (see next subsection), an equal shift in the charging rate for all values of TM potential can be achieved through a change in the parameter $H$, equivalent to the charging rate in the absence of kinetic electrons, rather than a change in the KE emission model. A detailed discussion of the justification of the applied offset is provided in the following section. The best fit parameters for the electron yield model for both TM 1 and TM 2 data are $Y_{0}=1.15$, $E_{0}=700$ eV and $s=1.64$. The value for the peak yield is at the lower end of yield estimates but still consistent with the literature. With these parameters, the best fit to the TM 1 data is achieved with an offset of $-9.3$ e s^-1 and $-8.5$ e s^-1 for TM 2. With the inclusion of the offset, the simulation model shows good agreement with the measured data ($\chi^{2}_{\mathrm{red}}=4.3$ and 5.4), including the additional measurement with applied bias in which the environmental charging rate is suppressed by around 10 e s^-1 compared to the measurement at the same TM potential with no applied bias on TM 2.

Simple Model Results: Equations 7 and 8, suitably averaged over the 100 kHz bias, predict the variation in charging rate expected as a function of TM potential. Allowing four free parameters in the simple analytical model does not result in a unique set of values for the best fit to the experimental data, and it is necessary to fix those values which are already reasonably well justified. This includes the effective KE work function which is taken to be similar to the photoelectric work function, with $\phi_{\textsc{ke}}=4.2$ eV [36, 37]. Once this value has been set there is still too much degeneracy and the value of $\alpha$ is determined by geometry and by the KE angular distribution function, and should have a value just below 1. A preliminary scan of the residual $\chi^{2}$ for a range of values of $\alpha$ between 0.70 and 1.0, revealed a minimum at a value of 0.86, similar to the value from the Monte Carlo simulation, and this was then fixed.

With the two remaining free parameters, $H$ and $R_{s}$, the simple model gives best-fit curves as shown in the upper panels of Fig. 7 for the two measurement campaigns. The best fit values of the two free parameters and their uncertainties from the covariance matrices of the fit are shown in Table 2. The lower panels in Fig. 7 show how the charging rate curves change as the parameters are varied by $\pm 5\sigma$. From the plots it can be seen that $H$ moves the curve vertically as expected, whereas $R_{s}$ causes a rotation about a centre on the curve between $V_{\textsc{tm}}=-0.25$ V and $V_{\textsc{tm}}=+0.25$ V depending on the value of $\alpha$. Altering the input cosmic ray flux will simultaneously change $H$ and $R_{s}$ causing a rotation about the point where the charging rate goes to zero.

The last row for each range in Table 2 gives the value of $R_{s}\times(A_{h}+A_{t})$ which is the total number of low-energy kinetic emission electrons implicated in the population, $N_{\textsc{ke}}$. However, it should be noted that these values are sensitive to $\alpha$, $\phi_{\textsc{ke}}$, the assumed form of $\beta(V_{\textsc{tm}})$ and to whether any other voltages are present, so care should be taken not to over-interpret the results from this simple model but more to use it to consolidate the understanding of the results from the full simulation. Comparing $N_{\textsc{ke}}$ with the overall KE charging rate and KE noise in Table 1 it can be seen that at $V_{\textsc{tm}}=0$ only a few percent of the electrons contribute to the charging rate and that there is a clear multiplicity enhancement in the KE noise contribution. Although the narrow range measurements require a statistically higher kinetic electron flux, implying a higher incoming cosmic ray flux, this is probably an artefact from having a shorter lever arm to fit over and if a fit is performed on the corresponding central points of the wide range data the values of $R_{s}$ become statistically consistent.

In Table 2 results for a high value of $\phi_{\textsc{ke}}$ (15 eV) are given. This is motivated both by an observation in [32] that such a high value has been noted for Au after cleaning by sputtering, and by the recent work of [19] which shows a Geant4 simulation of the yield from Au which has a broad energy distribution which can be approximated to a function of the type shown in Equation 4 with $\phi_{\textsc{ke}}\approx 15$ eV. A harder KE spectrum means fewer of the electrons will be affected by the applied bias, $V_{\textsc{tm}}$, and hence more are needed to obtain the observed dependence on $V_{\textsc{tm}}$. Adopting $\phi_{\textsc{ke}}=15$ eV increases the overall number of KE electrons by 60%, simultaneously increasing the associated KE charging noise contribution, shown in Table 1, by the same proportion, thus partially closing the gap between model and observation in this respect. However, even this increase in $\phi_{\textsc{ke}}$ is poorly motivated as the surfaces in LPF were large area and likely to be contaminated by exposure to air, which is in itself responsible for a lowered photoelectric work function [37], and a lowered value of $\phi_{\textsc{ke}}$ compared to a very clean surface [32].