Single-point in situ measurements of thermal ions during the KiNET-X ionospheric sounding rocket mission

Moses, M. L.; Lynch, K.; Delamere, P. A.; Lessard, M.; Pfaff, R.; Larsen, M.; Hampton, D. L.; Conde, M.; Barnes, N. P.; Damiano, P. A.; Otto, A.; Moser-Gauthier, C.

doi:10.1063/5.0253729

The KINetic-scale Energy and momentum Transport eXperiment (KiNET-X) sounding rocket mission investigated energy and momentum transport in the ionosphere caused by the known input of two barium releases. Here we investigate (1) the coupling of the injected barium ion (⁠ $B a^{+}$ ⁠) cloud to the ambient ionospheric plasma and (2) associated wave-particle interactions. This study uses the main payload platform's measurements of ambient ionospheric thermal ions (⁠ $O^{+}$ ⁠) and injected $B a^{+}$ ⁠. The observations show ambient $O^{+}$ heating following each release, coinciding with observed lower-hybrid waves and ion cyclotron oscillations. Comparisons to theoretical expectations reveal that ion cyclotron oscillations were the primary heating mechanism for both releases. The observed $B a^{+}$ density (⁠ $n_{B a^{+}}$ ⁠) showed a differently shaped $n_{B a^{+}}$ profile for the two releases, and, for event 2, a precursor spike in $n_{B a^{+}}$ ⁠. We developed an idealized model of each release cloud's $B a^{+}$ particles' trajectories under the Lorentz force to estimate $n_{B a^{+}}$ in the main payload's region. Comparison of our modeled $n_{B a^{+}}$ to that observed shows release 1's $B a^{+}$ was almost immediately trapped in gyro-motion about the geomagnetic field. However, release 2's model results indicated additional processes may have played a role in the observed $B a^{+}$ profile. Thus, we perform a sensitivity study of the impact of skidding and delayed ionization on the modeled output. As release 2 yielded about three times the barium of release 1, release 2's $B a^{+}$ mass growth rate was much larger than the rate of ambient Alfvénic plasma coupling to the injected $B a^{+}$ plasma, resulting in some features that can be attributed to skidding.

I. INTRODUCTION

The KINetic-scale Energy and momentum Transport eXperiment (KiNET-X) sounding rocket mission proposed a study of the momentum and energy transport in the Earth's magnetized ionosphere caused by the known input of two neutral barium releases into the twilight mid-latitude conditions. The scientific objectives of KiNET-X, discussed in (Delamere , 2024), that this paper focuses on are

Observe and investigate the coupling of the injected barium ion cloud to the ambient ionospheric plasma (Delamere , 2024).
Observe the mechanisms by which electromagnetic energy is converted into plasma kinetic and thermal energy.

Objective 1 was motivated by a previous experimental observation during the Combined Release and Radiation Effects Satellite (CRRES) campaign. CRRES's optical measurements showed that the injected plasma clouds moved perpendicular to the magnetic field for a time following each release (Delamere , 2000). This phenomenon, called “skidding,” indicated that the injected ions were not fully coupled to the background ionosphere (i.e., magnetic field) for some time (Delamere , 2000; 2024). Simulations to investigate the skidding observed during CRRES indicated that the formation of parallel electric fields (i.e., fields along the local geomagnetic field direction) were an important prerequisite condition for skidding to occur (Delamere , 2000). The skidding motion is hypothesized to be the result of an increased period of time required for the transfer of momentum from the neutral motion to ion gyro-motion. Barium pickup ions produce currents that close along the ambient magnetic field and through the Alfvén wave front via polarization currents. In the rest frame of the ambient plasma, the JxB force accelerates the ambient plasma and decelerates the barium ion cloud. Momentum is therefore transferred from the ion cloud to the ambient medium until the injected cloud is stopped (Delamere , 2002). Objective 2 aims to better understand the wave-particle interaction(s) that transfer energy from the induced electromagnetic waves to the ambient ionospheric plasma. These scientific objectives are of fundamental interest to the investigation of how ionizing neutral particles couple to ambient plasma environments such as in Jupiter-Io coupling (and also perhaps Jupiter-Europa) where the ionized particles from Io's ejected volcanic clouds interact with Jupiter's magnetic field, creating the plasma torus in the Jovian ionosphere (Achilleos , 2015; Delamere , 2000; 2015).

KiNET-X's science objectives require measurements of the ambient ionospheric thermal ions as well as the $B a^{+}$ thermal ions created from the releases. Thermal ions' temperatures and densities can be determined from in situ measurements made by Petite-Ion-Probes (PIPs) (Fraunberger , 2020) during the mission. The PIPs are small retarding potential analyzers that measure the anode current from the ion flux into the sensor at a range of screen bias retarding voltages. From these anode current (I) vs screen bias voltage (V) profiles, called IV curves, the temperatures and densities of the measured ions can be determined via a forward-modeling process.

The KiNET-X main payload platform carried eight PIPs. Also, two small instrument platforms (PIP-Bobs) (Roberts , 2017a; 2017b), each carrying two PIPs, were ejected from the main assembly at different points along the flight upleg in order to provide multi-point thermal ion measurements at about 0.5 and 1 km from the main platform during the two releases on the downleg. In this publication, only the main payload PIPs' data are examined, leaving the PIP-Bobs' measurements and multi-point PIP observations to a future publication. However, many of the improvements made to the PIPs' forward-modeling for this mission are applicable to PIPs both on the main payload platform and on the PIP-Bobs. Significantly, the forward-modeling process is adapted here for measurements made in a multi-species plasma, as will be discussed in Sec. III. By imposing different constraints on the PIP fitting model based on other in situ and remote sensing data from the mission, we extract the time-dependent injected barium ion (⁠ $B a^{+}$ ⁠) density (⁠ $n_{B a^{+}}$ ⁠) and ambient oxygen ion (⁠ $O^{+}$ ⁠) temperature (⁠ $T_{O^{+}}$ ⁠) profiles from the PIPs' measurements.

The PIPs' $n_{B a^{+}}$ data from the two releases confirm that barium ion clouds were formed following each neutral barium release. The $n_{B a^{+}}$ profile for the second release is greater in magnitude and duration, and has a distinctly different shape, than the $n_{B a^{+}}$ profile from the first release. Additionally, there are brief sharp enhancements in the second release's $n_{B a^{+}}$ profile during the $B a^{+}$ cloud's growth period that are not present in the first release's $n_{B a^{+}}$ profile. In order to understand the underlying physics behind these differences, we developed a simple, idealized, kinetic model of the motion of barium ions, created by ionization of a spherically expanding neutral cloud, in a geomagnetic field. Comparison of this simple model's results to PIPs' measurements for the first release shows that this model is a good representation of the first release. However, the shape of the density profile following release 2 has features that can be attributed to non-idealized processes, including skidding and/or delayed ionization. Furthermore, reproducing the onset signature of the second release requires that some portion of the barium population travels to the observation point at a higher expansion velocity. Thus, the PIPs provide observations of non-idealized and idealized barium ions from the second and first releases, respectively. Finally, the PIPs' $T_{O^{+}}$ profiles both show increases in the ambient $O^{+}$ population's temperature following each release. Comparison with the wave data from the onboard electric field probes indicates that these $O^{+}$ temperature enhancements are consistent with cyclotron heating.

Section II describes the design of the KiNET-X mission as well as the instruments of importance to this study. Then, in Sec. III, the PIPs' forward-modeling procedure is reviewed and the improvements made to it discussed. The results are presented, and then in Sec. IV interpretations are drawn through comparison to the idealized kinetics model. Section V concludes the paper. The appendixes provide examples of permutations of the different signatures that the kinetic models can provide.

II. KINET-X MISSION AND INSTRUMENTATION OVERVIEW

A. Mission design

KiNET-X mission's sounding rocket launched at 00:44:0.1122 UT on May 17, 2021 from Wallops Flight Facility, Virginia. On the flight's upleg, two canisters containing barium thermite were deployed from the rocket at $\sim$ 151.94 and $\sim$ 152.94 s after launch, respectively; they moved away from the main payload for several hundred seconds until arriving at the detonation configuration illustrated in Fig. 1. The PIP-Bob 14 and PIP-Bob 16 were ejected from the main platform at 186 and 318 s into the flight, respectively, and are also shown. The first barium canister detonated at T + 593.1778 s and the second barium canister detonated at T + 627.1578 s, corresponding to altitudes of about 400 and 350 km, respectively (Delamere , 2024). At first detonation time, the main payload was 3.825 km up the local magnetic field line, with a 55 m perp-B offset (Tibbetts, 2024), from release 1's detonation point as shown in the left plot of Fig. 1. At the second canister detonation time, the main payload was 2.624 km up the magnetic field line, with a 26 m perp-B offset, from the release's detonation point as shown in the right plot of Fig. 1.

FIG. 1.

View large Download slide

Positions of PIP-carrying platforms (i.e., main and ejected subpayloads) relative to the detonation point, which defines the origin of each plot, at (left) release 1's detonation time and (right) release 2's detonation time. East, north, and up geographic coordinates are used here. The local geomagnetic field line connecting the main payload and detonation point are indicated by the black arrow on each plot. Note that at T + 600 s, the entire array is moving at about 1.6 km/s east, −0.5 km/s north and −1.4 km/s up.

B. Main payload and ground-based instruments of interest

1. Petite-Ion-Probes

The main payload instruments whose data are the primary focus of this paper are the Petite-Ion-Probes (PIPs). The PIPs are small retarding potential analyzers that measure PIP anode current (I) induced by incident thermal ion flux with energies greater than the PIP screen bias voltage (V), swept from 5 to 0 V in 28 steps, with a measurement cadence of 45 sweeps/second. These IV curves contain interlocked information about the plasma's ion temperature and density as well as the spacecraft potential. The PIP data analysis procedure, called “fitting,” involves forward-modeling IV curves to each measured curve and will be discussed in Sec. III. Note that the measured anode current is converted to a voltage by the PIP's pre-amp for digitization by the parent Arduino-Shield processor (called a BobShield). The PIP divides the current by a gain (⁠

G_{PIP}

⁠) value, determined by the PIP pre-amp's resistor, and adds a 1 V offset to the PIP's measured voltage. Note that there can also be some slight, low voltage noise introduced into this 1 V offset from other onboard electronics. Post-flight, each PIP's measured currents are retrieved using the formula

I_{PIP} = (V_{PIP} - V_{Off}) / G_{PIP},

(1)

where

G_{PIP}

is in units of volts/ampere and

V_{Off}

is either 1 V or set to an average of

V_{PIP}

values from before launch (usually within 1 V

\leq V_{Off} \leq

1.1 V). However, the stored voltage (⁠

V_{PIP}

⁠) must be in the range 0–5 V. Hence, a PIP's gain sets an upper limit on its recorded anode current, thereby selecting the observable ranges of a population. For KiNET-X, PIPs 18-0, 18-1, 21-0, and 21-1 had low gains of 40 mV/nA and PIPs 19-0, 19-1, 20-0, 20-1 had high gains of 320 mV/nA. The low gain PIPs have a better view of the thermal core, and the high gain, of the event tails. The

G_{PIP}

⁠,

V_{Off}

⁠, and upper limits of anode current for all KiNET-X main payload PIPs are given in Appendix A and Table V.

The KiNET-X main payload contained eight PIPs, grouped in pairs, on the main payload's deck as shown in Fig. 2. Each pair's PIPs had the same gains and were positioned with their look-directions at 45° to each other. Each pair was located 90° from its neighboring pair. All PIPs were mounted so that their look-direction was elevated 14.5° from the deck, which is in the plane perpendicular to the local geomagnetic field during the events. Thus, the main payload PIPs covered all spin angles on the main payload at an elevation from the deck optimized (accounting for payload ram velocity) for the event times. As we will discuss in Sec. III, the PIPs' data analysis yields more accurate results in complex environments when its forward-modeling procedure is constrained using other instruments' data. As the measured plasma for KiNET-X included two species of ions (ambient $O^{+}$ and injected $B a^{+}$ ⁠), other instruments' data were essential for eliminating the number of unknown values in our forward-model, as discussed below in Sec. III. For this study, we used data from the onboard electron retarding potential analyzer (ERPA) and Goddard Space Flight Center's (GSFC's) electric field probes as well as the ground-based Millstone Hill (MLH) incoherent scatter radar (ISR) and cameras located in Bermuda and on a NASA aircraft.

FIG. 2.

View large Download slide

Diagram of PIPs' arrangement on the main payload deckplate as seen looking toward the nosecone. The spin axis is oriented to be aligned to the magnetic field during the two release events. Purple rectangles represent high gain (320 mV/nA) PIPs and blue rectangles represent low gain (40 mV/nA) PIPs. The light semi-circles extending from each rectangle indicate the rough angular field-of-view of each PIP. The dashed lines/arrows and angle labels indicate the rocket's y and z axes and the deck angles of the main payload platform.

2. Additional in situ instruments

The main payload platform carried the University of New Hampshire (UNH) ERPA (Frederick-Frost , 2007). The ERPA measured the ambient ionospheric electron temperature over the flight. As Sec. III A will discuss, measured electron temperature is linearly related to the spacecraft potential, which affects the PIPs' response. Hence, our forward-modeling procedure can use the ERPA data for validation of fitting results and/or as a constraint on the fitting model.

GSFC's electric field probes onboard the main payload provided electric field data in the directions parallel to and perpendicular to the local magnetic field (Pfaff, 1996; Pfaff and Marionni, 1998). These electric field measurements reveal which electromagnetic and/or plasma waves the main payload passed through. The duration and magnitudes of any observed waves can be used to estimate the expected change in the background ionospheric species' temperatures caused by wave-particle interactions for comparison with the PIP data-derived oxygen ion (⁠ $O^{+}$ ⁠) temperatures. Additionally, the GSFC's direct current electric field (DCE) probes' measurements yielded the $\vec{E} \times \vec{B}$ velocities needed for the PIP data analysis (see Sec. III).

3. Ground-based instruments

At T + 39.8878 s, the MLH ISR's 46 m steerable antenna (MISA) collected a slant-range, fixed direction profile of a region within KiNET-X's flight path (Erickson, 2021). This fixed direction profile included ion temperature and electron density measurements. We assume that the ISR's measured electron density is equal to the ambient (⁠ $O^{+}$ ⁠) ion density. The measured points along the fixed direction are not co-located in latitude, longitude, and altitude with the rocket; thus, we must also assume that the ambient ionospheric density vs altitude profile does not vary greatly over a few degrees latitude and longitude. Over the altitude region of interest, the MLH beam goes from 200 km north of the rocket trajectory's plane to 200 km south of it. Using the known altitude of the rocket over time, we found an approximate trajectory-time-dependent ambient oxygen ion density (⁠ $n_{O^{+}}$ ⁠) profile for the main payload from this ISR slant-range electron density altitude profile. Comparison of the ISR-derived $n_{O^{+}}$ profile along the flight path with the PIP-derived $n_{O^{+}}$ prior to the barium releases was used to check the accuracy of our forward-modeling procedure for the pre- and post-release portions of the flight. Additionally, as we will discuss in Sec. III B, the ISR-derived $n_{O^{+}}$ profile can be used to constrain the ambient oxygen ion population's density for the multi-species fit analysis of the PIPs' data inside the barium release cloud.

In order to observe the injected barium's evolution, the University of Alaska Fairbanks (UAF) fielded optical instruments at a site in Bermuda and on a NASA aircraft, flying north of Bermuda. These optical instruments included two time-lapse cameras with blue filters for 455.4 nm $B a^{+}$ resonance emission observation and green filters for 553.5 nm for observing the neutral barium (Delamere , 2024; Barnes , 2024). The filtered time-lapse cameras were chosen to measure the expansion velocity of each release's neutral barium cloud, each release's barium ionization rate, and the net barium ion yield for each release. These data are essential for the interpretation of the PIPs' barium ion density observations as we will detail in Sec. IV B.

III. PIP DATA ANALYSIS METHOD: THERMAL ION FORWARD MODELING

A. PIP data analysis overview

We obtain plasma parameters from PIP data by fitting modeled IV curves, for an identically oriented PIP in a parameterized modeled plasma, to the measured IV curves. Roberts (2017b) and Fraunberger (2020) describe the forward modeling of a PIP moving at subsonic relative velocities to a single-species Maxwellian plasma of one temperature. In that case, the collected anode current of a PIP, with a collector area of A and a combined grid transparency of

η

⁠, due to a particle flux (F) onto the collector is

I = η eAF,

(2)

where e is the unit charge of the particle. The particle flux reaching a PIP's anode depends on the plasma distribution function, the velocity and attitude of the PIP and, when flown on subsonic platforms such as sounding rockets, the energy-dependent PIP's collector area Roberts (2017b).

The particle flux for ionized particles with thermal velocities between

{\vec{v}}_{1} = v_{x 1} \hat{x} + v_{y 1} \hat{y} + v_{z 1} \hat{z}

and

{\vec{v}}_{2} = v_{x 2} \hat{x} + v_{y 2} \hat{y} + v_{z 2} \hat{z}

in the PIP frame is given by

F = \int_{v_{x 1}}^{v_{x 2}} \int_{v_{y 1}}^{v_{y 2}} \int_{v_{z 1}}^{v_{z 2}} v_{x} f (\vec{v} - {\vec{v}}_{D}) d v_{z} d v_{y} d v_{x},

(3)

where

f (\vec{v} - {\vec{v}}_{D})

is the plasma distribution function,

\vec{v}

is the ion thermal velocity, and

{\vec{v}}_{D}

is the bulk plasma motion into the PIP. Note that the PIP frame, specific to each individual PIP, is defined as follows:

\hat{x}

(the PIP's “look-direction”) is perpendicular to the anode's plane,

\hat{y}

is along the short side of the PIP's anode and

\hat{z}

is along the long side of the PIP's anode. The constraint on the velocity component perpendicular to the anode plane (⁠

v_{x}

⁠) is the net potential of the PIP's screen voltage (⁠

V_{b}

⁠) and the time-dependent sheath voltage (⁠

V_{s}

⁠) from charging of the payload during flight. The sheath effects cannot be ignored in this case where

V_{b}

is larger than the ram energy. Thus, the minimum value of

v_{x}

for an ion of mass m in order to reach the anode of a PIP when the screen is at bias

V_{b}

and the payload is charged to a voltage of

V_{s}

is

v_{c} = \sqrt{2 e (V_{b} + V_{s}) / m} .

(4)

The remaining two velocity components parallel to the anode plane (⁠

v_{y}

and

v_{z}

⁠) are limited by both

v_{x}

and the geometry of the PIP at subsonic sounding rocket platform velocities as shown in (Roberts , 2017b). Thus, the expression for the modeled collected PIP current at screen bias voltage (⁠

V_{b}

⁠) is

I (V_{b}) = η e \int_{0}^{w} \int_{0}^{h} (\int_{\sqrt{\frac{2 e (V_{b} + V_{s})}{m}}}^{\infty} \int_{- y \frac{v_{x}}{L}}^{(w - y) \frac{v_{x}}{L}} \int_{- z \frac{v_{x}}{L}}^{(h - z) \frac{v_{x}}{L}} v_{x} f (\vec{v} - {\vec{v}}_{D}) d v_{z} d v_{y} d v_{x}) d z d y,

(5)

where h, w, and L are the width of the PIP's short side, the length of the PIP's long side and the depth of the PIP's anode from the entrance aperture, respectively (Roberts , 2017b).

The expression for the bulk flow of the plasma (⁠

{\vec{v}}_{D}

⁠) into a specific PIP, in the PIP-frame, is

\begin{matrix} {\vec{v}}_{D} = \hat{x} (- {\hat{x}}_{PIP, ENU} \cdot ({\vec{v}}_{flow, ENU} - {\vec{v}}_{ram, ENU})) \\ + \hat{y} (- {\hat{y}}_{PIP, ENU} \cdot ({\vec{v}}_{flow, ENU} - {\vec{v}}_{ram, ENU})) \\ + \hat{z} (- {\hat{z}}_{PIP, ENU} \cdot ({\vec{v}}_{flow, ENU} - {\vec{v}}_{ram, ENU})), \end{matrix}

(6)

where

{\vec{v}}_{flow}

is the geophysical plasma flow (geoflow) velocity (i.e.,

\vec{E} \times \vec{B}

velocity),

{\vec{v}}_{ram}

is the ram velocity of the platform and the “ENU” subscript indicates that the vector is in East-North-Up (ENU) coordinates. Note that since

{\vec{v}}_{ram}

is the motion of the payload, the relative velocity to the PIP is its negative, as shown in Eq. . The main payload platform's Global Positioning System (GPS) data provides the time-dependent

{\vec{v}}_{ram, ENU}

⁠. The time-dependent

{\vec{v}}_{flow}

determination method can vary depending on the mission. For KiNET-X, GSFC provided the time-varying

\vec{E} \times \vec{B}

velocity using the known magnetic field and the measured electric field during flight. Hence, the geoflow velocity (⁠

{\vec{v}}_{flow, ENU}

⁠) in this paper's scalar fits is assumed to be equal to GSFC's

\vec{E} \times \vec{B}

velocity, linearly interpolated onto each PIP's time-base. Finally, note that a PIP's axes vectors in the ENU frame (i.e.,

{\hat{x}}_{PIP, ENU}

⁠,

{\hat{y}}_{PIP, ENU}

⁠,

{\hat{z}}_{PIP, ENU}

⁠) are sensor-specific and time-dependent given the payload's spin and cone motion, and must therefore be determined for each individual PIP using the NASA Sounding Rocket Operations Contract (NSROC) payload attitude solution's Direction-Cosine-Matrices (DCMs) along with knowledge of each PIP's orientation on the payload. A default limitation on PIP data used is that

arccos (- \hat{x} • \frac{{\vec{v}}_{D}}{| {\vec{v}}_{D} |}) \leq 30 °

⁠; for some cases a more restrictive

15 °

mask is used.

Previously, our scalar fitting method described in Roberts (2017b) and Fraunberger (2020). used the following assumptions:

the singly ionized, all oxygen plasma follows a Maxwellian distribution
any current offset in the tail of the IV curve is due to noise or high-energy (above 5 eV) auroral electron/particle flux and should be subtracted before fitting
“Goodness of Fit” is quantified by the mean of the residual.

Thus, the modeled current due to a single species (i) collected by a PIP at a screen voltage bias of

V_{b}

with these assumptions is

I_{fit} [V_{b}] = η e n_{i} {(\frac{m}{2 π k_{B} T_{i}})}^{\frac{3}{2}} \int_{0}^{w} \int_{0}^{h} (\int_{\sqrt{\frac{2 e}{m} (V_{b} + V_{s})}}^{\infty} \int_{v_{y 1}}^{v_{y 2}} \int_{v_{z 1}}^{v_{z 2}} v_{x} exp (- \frac{m}{2 k_{B} T_{i}} | \vec{v} - {\vec{v}}_{D} |^{2}) d v_{z} d v_{y} d v_{x}) d z d y,

(7)

where

v_{y 2}

⁠,

v_{y 1}

⁠,

v_{z 2}

⁠, and

v_{z 1}

are the limits of integration for

d v_{y}

and

d v_{z}

⁠, respectively, given in Eq. . In order to determine what values of scalar plasma parameters (e.g.,

n_{i}

⁠,

T_{i}

⁠, and/or

V_{s}

⁠) produce the closest match to the observed anode current, we constrain the solution search with known values of

{\vec{v}}_{D}

and

{\vec{v}}_{ram}

as discussed, and then use the minimizer function in the LMFit python library (Newville , 2021) to modify the requested scalar values and determine the best fit that minimizes the residual array,

| \frac{I_{true} [V_{b, min} : V_{b, max}] - I_{fit} [V_{b, min} : V_{b, max}]}{I_{error} N_{b}} |^{2},

(8)

for select points in the sweep (i.e., all points at bias voltages (⁠

V_{b}

⁠) from

V_{b, min}

to

V_{b, max} \leq 5 V

⁠). Here,

N_{b}

is the number of select points,

I_{true}

is an array of measured currents for each screen bias voltage and

I_{error}

is the measurement error (associated with the physical PIP system). The selected points at which the above comparison is made are those at sweep voltages

\geq

some minimum voltage (⁠

V_{b, min} \geq V_{s}

⁠).

The results of a single-species (⁠

O^{+}

⁠) Maxwellian scalar fit to determine

n_{i}

⁠,

T_{i}

⁠, and

V_{s}

for KiNET-X are shown in Fig. 3, indicating that this fit is good match to other observations at times outside of the barium release events. These single-species scalar fit results are compared to values obtained from other instruments' data. The payload potential (⁠

V_{s}

⁠) can be determined from the electron temperature (⁠

T_{e}

⁠) data collected by the onboard ERPA using the following formula:

V_{s} = - 5 k_{B} T_{e} + Φ,

(9)

where

Φ

is a time-dependent work function determined from the slope of an initial scalar fit values for

V_{s}

over a quiet period of the flight and

k_{B}

is Boltzmann's constant (Kelley, 1989; Siddiqui , 2011). The MLH ISR measured an ion density-vs-altitude profile at 0:44:40 UT, just after launch. Using the rocket's altitude-vs-time from the attitude solution, the MLH ISR

n_{i}

-vs-altitude profile can be mapped to a

n_{i}

-vs-time profile for use with the scalar fit. Figure 3 shows that, outside the events, the single species fit works well for the PIPs, matching the external diagnostics, and other instruments' ambient

O^{+}

ion density data and

V_{s}

data are good approximations. However, the release events' time period of interest has a multi-species plasma as the injected barium is ionized; additionally, some of the ambient

O^{+}

plasma population is heated. These deviations from an ideal single-species Maxwellian plasma appear in the major deviations of Fig. 3's fitted data from the ISR and ERPA measurements around the times of the releases. Thus, a scalar fit method that allows for a multi-species plasma must be used during these event periods.

FIG. 3.

View large Download slide

Data from a scalar fit of low gain (light blue) PIP and high gain (brown) PIP data for parameters $n_{O^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $V_{s}$ assuming a single-species $O^{+}$ Maxwellian plasma. Data from other instruments overplotted for comparison. The release times are indicated by the dashed green lines. Note that these single Maxwellian fits in the multiple-species plasma of the event times (i.e., from T + 592–650 s) are not expected to be physically realistic.

B. PIP data analysis for a multi-species plasma

Scalar fits for more complex environments can be done by adding constraints to the fitting process, which allows flexibility beyond a single species/temperature Maxwellian distribution. The forward-modeling fitting code can be constrained for the given mission using any relevant data provided by other onboard instruments or ground-based remote sensing data. For KiNET-X, the following constraints, justified in Sec. III A, apply:

Time varying values of $V_{s}$ are set using ERPA data, allowing the payload potential estimate to be determined by the measured $T_{e}$ ⁠.
Ambient, background $O^{+}$ ion population density over time and altitude is set using a MLH ISR $n_{i}$ profile at launch time.
Geoflow velocity data is derived from GSFC's DC electric field probe data.
Payload ram velocity and attitude are taken from NSROC and GPS data.

In order to model the PIP anode current in a multi-species plasma, the current from each species must be calculated and then added together to obtain the full current as shown below:

I_{fit} [V_{b}] = \sum_{i} η e n_{i} {(\frac{m_{i}}{2 π k_{B} T_{i}})}^{\frac{3}{2}} \int_{0}^{w} \int_{0}^{h} (\int_{\sqrt{\frac{2 e}{m_{i}} (V_{b} + V_{s})}}^{\infty} \int_{v_{y 1}}^{v_{y 2}} \int_{v_{z 1}}^{v_{z 2}} v_{x} exp (- \frac{m_{i}}{2 k_{B} T_{i}} | \vec{v} - {\vec{v}}_{D, i} |^{2}) d v_{z} d v_{y} d v_{x}) d z d y,

(10)

where i =

O^{+}

⁠,

B a^{+}

for KiNET-X. Here we make the approximation that the

{\vec{v}}_{D, i}

for the barium is the same as that of the ambient oxygen, that is, that the barium is entrained with the ionosphere. Using the constraints described above, we can limit the number of unknowns in our fit by setting some of the parameters to time-varying values derived from other measurements. Thus, the values to be determined in the multi-species scalar fit are

n_{B a^{+}}

⁠,

T_{B a^{+}}

⁠,

T_{O^{+}}

⁠. Note that this is still a three-parameter fit as now

V_{s}

and

n_{O^{+}}

are set to values derived from other instruments.

Another adjustment from previous analyses involves the tail end (i.e., high sweep bias voltage portion) of the IV curves. As stated in Sec. III A, for a single species $O^{+}$ plasma, the offset of the IV curves' tails from zero are normally subtracted from the entire IV curve, as shown in the “ $O^{+}$ Maxwellian Fit” subplots of Fig. 4, and the maximum value of $V_{b}$ is usually set at a value that excludes most of each IV curve's tail from the residual array calculation so as not to bias the fit. This is done in auroral studies because high-energy electron precipitation causes signatures on the anode. However, as shown in the “ $O^{+}$ and $B a^{+}$ Maxwellians Fit” subplots of Fig. 4, the modeled $B a^{+}$ IV curve (dark pink dashed lines) raises the modeled net current and introduces a slope in the net modeled IV curve's tail. Therefore, we cannot force the tail of an IV curve measured in plasma containing $B a^{+}$ to zero and we must include more tail $V_{b}$ values in the residual array calculation.

FIG. 4.

View large Download slide

Comparisons for two scalar fit models at (a) a time prior to the first release and (b) a time following barium release 2. Each sub-figure (e.g., a and b) compares (left) an $O^{+}$ Maxwellian plasma fit for $n_{O^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $V_{s}$ ⁠, and (right) a two-species' (⁠ $O^{+}$ ⁠, $B a^{+}$ ⁠) Maxwellian fit for $n_{B a^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $T_{B a^{+}}$ ⁠. The top plots show the full measured and modeled IV curves, whereas the bottom plots are zoomed to view the tails of the curves. Note that the “Raw Flight Data” curves in all plots are the initial IV curves extracted from that PIP's voltage data with Eq. (1), whereas “Flight Data” and “Flight Data Fit To” curves in the “ $O^{+}$ Maxwellian Fit” plots have had their tails subtracted off in preparation for the fit as described in Sec. III A.

For a mission such as KiNET-X where a second ion species is introduced by a chemical release, two-species fitting is only activated near the first release time, as prior to then the second species is not present. In the case of KiNET-X, we activate our two-species fit at 590 s after launch, which is almost 3 s before the first release is detonated. Then, for each PIP, the goodness-of-fit for potential scalar values of $n_{B a^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $T_{B a^{+}}$ is evaluated using Eq. (8) with a modeled IV curve (⁠ $I_{fit}$ ⁠) determined from Eq. (10) for the potential scalar values, and with $n_{O^{+}}$ ⁠, $V_{s}$ ⁠, and ${\vec{v}}_{D}$ constrained by data from the other instruments as previously discussed.

C. Results of multi-species analysis method

Figure 5 presents the results of our multi-species fit to the KiNET-X main payload PIPs' data as well as the two profiles from other instruments used to constrain this fit. Note that Fig. 5 shows only fit results for PIP data at times when a given PIP's look-direction was within 30° of ram. The top subplot shows the fitted $B a^{+}$ density profile with the scaled $O^{+}$ density profile from the ISR overlaid. The second subplot presents the fitted $O^{+}$ temperature profile over time. The third panel plots the final fitted parameter of the $B a^{+}$ temperature. The final panel shows the spacecraft potential over the flight, derived from the ERPA $T_{e}$ data, used in the fit. Note that there is some noise in the fits' output $n_{B a^{+}}$ before the first barium release time because of small current offsets in the tail due to electrical noise in the environment. As in Fig. 3, fit results for high gain and low gain PIPs are differentiated by color in these plots. This reveals several differences between the high and low gain PIPs that will be discussed in the following paragraphs.

FIG. 5.

View large Download slide

Data from the multi-species scalar fit of low gain (light blue) PIPs' and high gain (brown) PIPs' data for parameters $n_{B a^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $T_{B a^{+}}$ ⁠, assuming a co-located $B a^{+}$ Maxwellian plasma and $O^{+}$ Maxwellian plasma. Time-dependent $V_{s}$ and $n_{O^{+}}$ data, from other instruments, used in the multi-species scalar fit are also plotted. Note that the plotted $n_{O^{+}}$ data were scaled by a factor of 10 to fit on the same plot as the $n_{B a^{+}}$ data. The release times are indicated by the dashed green lines.

There are significant differences between the fitted $T_{O^{+}}$ values for low gain PIPs and those for high gain PIPs seen in Fig. 5. Within a short measurement time period such as one main payload spin cycle (⁠ $≲$ 2 s), after the initial “shock” of a barium release, there is a greater spread in the fitted $T_{O^{+}}$ values for high gain PIPs than in those found for low gain PIPs. A modeled PIP IV curve's slope in the region about the inflection point is very sensitive to the temperature of the modeled ion species. In our multi-species plasma, the $O^{+}$ fit strongly controls the resulting net IV curve's shape at low to mid sweep voltages between approximately 1.5–3.25 V as shown in the top plots of Figs. 4(b) and 6. If the measured current reaches the given PIP's saturation value at sweep voltages within this region, then much of the portion of an IV curve that is most sensitive to ion temperature is obscured, as illustrated in the right plots of Fig. 6. Thus, the ion temperature values from fits to measured saturated IV curves (i.e., curves that reached the PIP's saturation value) are more prone to errors than those from fits to unsaturated IV curves. The saturation current for low gain PIPs is just under 100 nA, about eight times higher than the high gain PIPs' saturation current of 12.5 nA. As shown in Fig. 6, inside the barium ion clouds the induced anode currents exceed the high gain PIPs' saturation values. Thus, the high gain PIPs' fitted $T_{O^{+}}$ values are more susceptible to error and not self-consistent as saturation removed a large portion of the measured data that the model needed to fit to inside the portion of the curve most sensitive to $T_{O^{+}}$ changes. However, the low gain PIPs' IV curves do not reach their saturation value during the mission. Therefore, for this mission we will only use the low gain PIPs' temperature values in our investigation.

FIG. 6.

View large Download slide

Comparison of a two-species' (⁠ $O^{+}$ ⁠, $B a^{+}$ ⁠) Maxwellian fits for $n_{B a^{+}}$ ⁠, $T_{O^{+}}$ ⁠, and $T_{B a^{+}}$ to (left) a low gain PIP's data and (right) a high gain PIP's data following barium release 2. The top plots show the full measured and modeled IV curves, whereas the bottom plots are zoomed in the y-axis to view the tails of the curves. The dashed curves on the plots show the calculated single-species IV curves for $O^{+}$ (light pink) and $B a^{+}$ (dark pink), whose sum is the net IV curve for a single PIP (blue).

Although the high gain PIPs' saturation obscures the $O^{+}$ population's temperature data, it does not obscure the portion of each IV curve that is sensitive to the $B a^{+}$ population's temperature and density values. As shown by the dark pink curve in the left plots of Fig. 4(b) and all plots of Fig. 6, the $B a^{+}$ population's effects on the modeled net IV curve are an added offset in current to the entire IV curve (e.g., raising the curve's tail up from 0 A) and a slope in the tail of the IV curve at higher sweep voltages (greater than 3.25 V). However, in the first panel of Fig. 5, there is more spread in the low gain PIPs' fitted $n_{O^{+}}$ values over one spin cycle (⁠ $≲$ 2 s) than in the high gain PIPs' fitted $n_{O^{+}}$ values. The source of this difference is the effect PIP saturation has on the weighting of an IV curve's tail in the “goodness-of-fit” determination. The saturation of a high gain PIP's IV curve at lower sweep voltages leads to a fit that is heavily weighted by the curve's tail values at high sweep voltages, where the signature of the $B a^{+}$ population is most clearly differentiated from that of the $O^{+}$ population. Hence, there is less spread in the high gain PIPs' fitted $n_{B a^{+}}$ than in the low gain PIPs' $n_{B a^{+}}$ as the “goodness-of-fit” only depends on the tail values and does not include a comparison of the net IV curve's shape at lower sweep voltages. However, as expected, the low gain PIPs' fitted $n_{B a^{+}}$ values follow the general trend of the high gain PIPs' fitted $n_{B a^{+}}$ values. Therefore, we use the fitted $n_{B a^{+}}$ values from both low and high gain PIPs in our study for a better time cadence.

Finally, consider the fitted $T_{B a^{+}}$ values shown in the third panel of Fig. 5. There is a large spread in $T_{B a^{+}}$ values from fits to both high gain PIPs and to low gain PIPs. Following each release, there is a very rough trend in the highly variable $T_{B a^{+}}$ data of a rise in $T_{B a^{+}}$ followed by a decline. However, there is so much spread in $T_{B a^{+}}$ values over both short and long measurement periods that such trends are difficult to highlight. The spread in $T_{B a^{+}}$ stems from the limitations of our assumption that both ion species follow Maxwellian distributions. There is no reason to expect the $B a^{+}$ population to be Maxwellian. Although the fit solution for the density moment of the distribution appears robust, higher moments (e.g., $T_{B a^{+}}$ ⁠) are more sensitive to assumptions about the shape of the plasma distribution.

As our study is focused on the injected $B a^{+}$ spatial evolution and the transfer of energy to the ambient population, the main PIP data products of interest to the study are the oxygen ions' temperatures over time provided by the low gain PIPs and the barium ions' density over time provided by both the low and high gain PIPs. These primary PIP fitted datasets are shown in Fig. 7. Note that the large variation over each spin cycle in $T_{O^{+}}$ from PIP 18-1 is caused by fitting limitations, which are discussed in Sec. 1 of Appendix B. A key feature of the $T_{O^{+}}$ data are the increase in the ambient, $O^{+}$ population's temperature following each releases' canister detonation, more clearly in release 2. The $n_{B a^{+}}$ data show that the PIPs observed the injected $B a^{+}$ population following each release, with the second release observed sooner after detonation than the first release was observed. Also, a higher peak $n_{B a^{+}}$ was observed following the second release than following the first release. Finally, although not seen in this overview figure, there is a sharp, precursor enhancement in $B a^{+}$ density following the second release but not the first one. These features of interest will be discussed in Sec. IV.

FIG. 7.

View large Download slide

Summary of key datasets across both events: PIP-measured $O^{+}$ temperature data obtained by a multi-species scalar fit to main payload low gain PIPs' data (top) and PIP-measured $B a^{+}$ density data obtained by the same scalar fit to all main payload PIPs' data along with MLH ISR $O^{+}$ density (bottom). Times of the releases' canister detonations are indicated by dashed lines. The plotted PIP data here are limited to data collected when each PIP is within 15° of ram. Note that important $n_{B a^{+}}$ signatures at T + 629 s, discussed in Sec. IV B, are compressed in this figure's two-event view of these data.

IV. OBSERVATIONS OF BARIUM RELEASES AND INTERPRETATION

A. PIP observations

An illustration of aggregated main payload PIPs' combined raw, unfitted observations, over the entire payload's field-of-view (FOV) is shown in Fig. 8 for each event. The panels show each PIP's anode current for a screen bias voltage (⁠ $V_{b}$ ⁠) of 3 V (i.e., mostly the $B a^{+}$ tail) as a function of the PIP's angle-of-attack (AoA), defined as the angle between the PIP's look-direction (⁠ ${\hat{x}}_{PIP}$ ⁠) and the relative plasma velocity from the payload's ram motion and geoflow, for each measured time. These plots are referred to as “PIPograms.” These plots provide a spatial and temporal representation of the main payload PIPs' unprocessed, unfitted data. A comparison of the two subplots reveals two features of note. First, the first release produced a much weaker signature in the PIPs' data than the second release did. Second, the PIPogram for the second event shows that the PIPs within +/-50° of AoA observed a brief, sharp precursor enhancement in anode current about 1 s after the canister detonation, which quickly decreased before the primary current enhancement feature in these data (from about 1.5 to 8 s after canister detonation). This precursor enhancement appears as a spike in barium ion density in the fitted data for event 2, as we will show below, which is coincident with the observed field-aligned electron beam from the onboard Electron PLASma Instrument (EPLAS) (Delamere , 2024).

FIG. 8.

Plots of main payload PIPs' aggregated anode current (at a screen bias voltage ( Vb) of ∼ 3 V) vs these PIPs' angles-of-attack (i.e., cos−1(x̂PIP•(v̂flow−v̂ram))) over time for periods around the first release (top) and the second release (bottom). Times of each release's canister detonation are indicated by vertical dashed lines. Horizontal dashed lines indicate −30° and 30°, which are typical limits placed on PIP data during fitting. The precursor to event 2 is seen just after T + 628 s.

View large Download slide

Plots of main payload PIPs' aggregated anode current (at a screen bias voltage (⁠ $V_{b}$ ⁠) of $\sim$ 3 V) vs these PIPs' angles-of-attack (i.e., $c o s^{- 1} ({\hat{x}}_{PIP} • ({\hat{v}}_{flow} - {\hat{v}}_{ram}))$ ⁠) over time for periods around the first release (top) and the second release (bottom). Times of each release's canister detonation are indicated by vertical dashed lines. Horizontal dashed lines indicate −30° and 30°, which are typical limits placed on PIP data during fitting. The precursor to event 2 is seen just after T + 628 s.

B. Interpretation of spatial structuring of injected ions

Figure 9 compares the two events' $B a^{+}$ density evolution by plotting them on separate subplots with a shared x-axis of time relative to each event's release detonation time. There are four key differences in the observed barium ion density between release 1 and release 2 as illustrated in Figs. 7–9. The first difference is that the maximum observed barium density following release 2 was nearly six times as much as that observed following release 1. Second, the time between canister detonation and main payload PIPs' observation of barium ions was much shorter for the second release than it was for the first. The difference in maximum density and the time difference are expected qualitatively as the second release's canister was closer to the main payload at its detonation time than the first release's canister was to the main payload at its detonation time. However, as we will discuss later, the large amplitude difference is caused by the different neutral yields of the two releases. Third, the shape of release 2's density profile differs from that of release 1 in that: the release 2 profile's growth period is shorter than that of release 1, and release 1's profile plateaus between 4 and 5 s whereas release 2's profile almost immediately decays after reaching its peak density value. Fourth, there is a brief, sharp precursor enhancement of barium ion density about 1 s after the second release's canister detonation, during the second event's growth period, that was not observed during the growth period of the first release. This precursor enhancement is not an artifact of the LMFit fitting as it also appears in the PIPs' raw data, shown in Fig. 8, as a brief enhancement in measured PIP anode current at T + 628.25 s that precedes the main current enhancement feature.

FIG. 9.

View large Download slide

Comparison of the PIP-measured $B a^{+}$ density data before and after the first barium release (top) and the second barium release (bottom). The x-axes of the two subplots are the same, but the y-axes are different in order to show features. The detonation times are indicated by dashed lines. The precursor enhancement of the second release, noted in Fig. 8, is highlighted by the blue ellipse.

1. Particle trace model overview

In order to better understand the physics behind these ion signatures, we developed an idealized particle tracing model of a barium release following simple kinematics of charged particles in a magnetized environment. It evolves a time-sequence of multiple barium ion sub-clouds (henceforth referred to as “clouds”), each ionized at a different time (in steps of $Δ t_{clouds} = 0.1 s$ ⁠) after the onset of radial expansion of the neutral barium. The model calculates the trajectories of the newborn ions moving under the Lorentz force in each cloud. By adding these ionized clouds together, we can obtain the net spatial distribution of the released ions as a function of time and position. In the model, the canister detonation time is t = 0 s and the coordinate system is defined with the canister detonation point as the origin and the z-axis along the local geomagnetic field direction. The model's reference frame is moving along the magnetic field at a velocity equal to the field-aligned component of the main payload GPS ram velocity. The model assumes that the barium neutrals radially expand from a point source moving at the GPS ram velocity. The ionization time (⁠ $t_{0}$ ⁠) of the first cloud is $t_{0}$ = 0.1 s and the ionization times for subsequent clouds increase in intervals of $Δ t_{clouds} = 0.1$ s.

Each one of these clouds, ionized at time

t_{0}

⁠, consists of a number of particles,

N_{cloud} (t_{0}) = N_{cloud} (0.1) \frac{N_{0}}{N_{i} (0.1)} (1 - e^{- t / τ}),

(11)

where

N_{cloud} (0.1)

is the user-selected number of particles in the first modeled cloud (⁠

t_{0}

= 0.1 s) and the quantities

N_{i} (0.1)

⁠,

N_{0}

⁠, and

τ

are determined from optical data of each release, such as those shown in Sec. IV B 2, below. The cloud's particles have initial velocities following a Gaussian distribution of standard deviation

v_{exp} / \sqrt{3}

about the cloud's central position, where

v_{exp}

is the observed mean radial expansion velocity of the neutrals. This cloud is initially centered on some point

(x_{center}, y_{center}, z_{center})

⁠, determined from the constant drift velocity (i.e., GPS ram velocity) of the constituent neutrals up to time

t = t_{0}

⁠. The initial position of each particle “i” in the cloud at the cloud ionization time (⁠

t_{0}

⁠) is

\begin{matrix} x_{i, 0} = x_{center} + v_{i, x 0} t_{0}, \\ y_{i, 0} = y_{center} + v_{i, y 0} t_{0}, \\ z_{i, 0} = z_{center} + v_{i, z 0} t_{0}, \end{matrix}

(12)

where

\begin{matrix} v_{i, x 0} = v_{i, g x} + v_{RAM, x 0}, \\ v_{i, y 0} = v_{i, g y} + v_{RAM, y 0}, \\ v_{i, z 0} = v_{i, g z}, \end{matrix}

(13)

is the particle's total initial velocity and

{\vec{v}}_{i, g} = \hat{x} v_{i, x g} + \hat{y} v_{i, y g} + \hat{z} v_{i, z g}

is the particle-specific initial velocity component of

{\vec{v}}_{i, 0}

selected from the aforementioned Gaussian distribution. The position and velocity of each particle “i” in the cloud is then calculated for subsequent times,

\begin{matrix} x_{i} (t) = - \frac{v_{i, y 0}}{ω_{c}} cos (ω_{c} (t - t_{i, 0})) + \frac{v_{i, x 0}}{ω_{c}} sin (ω_{c} (t - t_{i, 0})) + \frac{v_{i, y 0}}{ω_{c}} + x_{i, 0}, \\ y_{i} (t) = \frac{v_{i, x 0}}{ω_{c}} cos (ω_{c} (t - t_{i, 0})) + \frac{v_{i, y 0}}{ω_{c}} sin (ω_{c} (t - t_{i, 0})) - \frac{v_{i, x 0}}{ω_{c}} + y_{i, 0}, \\ z_{i} (t) = v_{i, z 0} (t - t_{i, 0}) + z_{i, 0}, \end{matrix}

(14)

and

\begin{matrix} v_{i, x} (t) = v_{i, y 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, x 0} cos (ω_{c} (t - t_{i, 0})), \\ v_{i, y} (t) = - v_{i, x 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, y 0} cos (ω_{c} (t - t_{i, 0})), \\ v_{i, z} (t) = v_{i, z 0}, \end{matrix}

(15)

where

t_{i, 0}

is a particle-specific ionization time. The particle-specific ionization time is a modification of

t_{0}

to represent the slight, random variations in the ionization times of each ion in the cloud and in our model takes the form

t_{i, 0} = t_{0} + Δ t_{clouds} * (ξ_{i} - 1 / 2),

(16)

where

ξ_{i}

is a random decimal in the range of 0–1 (inclusive). Figure 10 (in a coordinate system aligned to

\vec{B}

⁠, unlike the geographic coordinate system of Fig. 1) illustrates the model initialization of a cloud ionized at 0.1 s and its evolution over the following 0.4 s. Note that we only model the bottom “hemisphere” of each cloud because the observation points (i.e., the PIPs) remain in this region throughout the time periods of interest.

FIG. 10.

View large Download slide

(Left) Cloud ionized at 0.1 s plotted at 0.1 s (blue) and 0.5 s (green). (Right) Select particles' trajectories within this cloud. Both of these plots are in a geomagnetic frame, which is moving along the magnetic field at a velocity equal to the field-aligned-component of the GPS ram velocity (compare to the geographic representations of Fig. 1). The plots' origin is defined by the release's detonation location (green star), with the subpayloads' relative positions indicated by the large points. Note that negative z is increasing altitude as the local magnetic field direction (⁠ $\hat{z}$ ⁠) points downward and northward in this region (black arrow).

Once we have combined all of the time-dependent ionized clouds following the imagery-determined ionization profile for a given release, the density observed at a measurement point can be calculated. First, the time-dependent East-North-Up (ENU) coordinates of a measurement point, referenced to the release detonation location, are rotated into the model's frame using a Rodrigues rotation matrix (Belongie, 2003). Then, for each time, the number of modeled particles within a set radius of 100 m from the measurement point's position is found. Figure 11 illustrates the combined clouds at 1.5 s after release 1, with the region where the main payload density is being queried shown by the pink box (although actually the query region is spherical). The queried particles then undergo an additional filtering for each PIP on the main payload to remove any particles whose velocities would not be within the field-of-view (FOV) of that PIP.

FIG. 11.

View large Download slide

Combined model barium ion clouds with 1000 particles/cloud (out of roughly a million) randomly selected for the plot (green) along with main payload density query region (pink box) and each main payload PIP's look-direction (i.e., ${\hat{x}}_{PIP}$ rotated into the model's reference frame) at one instant (pink arrows), all plotted in the frame of $\hat{z}$ aligned with the local magnetic field and moving along the magnetic field at a velocity equal to the field-aligned-component of the GPS ram velocity. Note that the actual query region of the model is spherical, but a wireframe box is used here to illustrate sampling a query region.

It should be noted that our model does not take into account collisions. Using a hard sphere approximation we find that the mean free time for collisions between the barium atoms and the background oxygen atoms to be 21 s at 350 km, and 72 s at 400 km, using the atomic radii of Slater (1964), and the empirical densities from the MSIS 2.0 model (Emmert , 2021). Thus, we conclude that collisions with the background atmosphere are a minor perturbation in the release dynamics.

2. Model results without skidding or delayed ionization

When modeling the ion cloud of event 1, we considered the ionization profile and neutral cloud expansion velocity determined from ground-based camera observations of the clouds following release 1, shown in Fig. 12. For event 1, the mean expansion velocity was 1.5 km/s and, for the first few seconds, the observed ionization profile is approximated by

N_{i} (t) = N_{0} (1 - e^{- t / τ}),

(17)

with

N_{0}

= 8.75 × 10²³

m^{- 3}

and

τ

of 4–8 s, as justified by Fig. 12. However, as shown in Sec. 3 of Appendix B, these values of

τ

did not give results resembling the observed density data. Instead, after testing different

τ

values, we arrived at

τ

= 28 s for event 1, which is closer to the theoretical value of

τ

for barium (Barnes , 2024). Hence, the ionization profile used for our model was

N_{i} (t) = N_{0} (1 - e^{- t / 28}),

(18)

for release 1, where

N_{0}

= 8.75× 10²³

m^{- 3}

⁠. The modeled release 1 density expected at the query point is plotted along with the measured release 1 density in the top plot of Fig. 14. The growth phase of the barium cloud as seen by the main payload following release 1 is very well represented by the model. Also, the decay of the barium density is mostly captured by this simple model. For a quantitative comparison of different modeled profiles to the PIPs' data for event 1, see Sec. 2 of Appendix B and Table VI in Sec. 3 of Appendix B. Therefore, we conclude that the physics of the evolution and motion of the ion cloud formed from the first barium release is well represented by our model without additional processes.

FIG. 12.

View large Download slide

Number of barium ions over time as derived from camera images of release 1.

We do not have a similar optical measurement of the ionization profile and neutral cloud expansion velocity for release 2 as we did for release 1. However, we do have observations that illustrate the relative amounts of neutral barium present over time for both releases, shown in Fig. 13 (Barnes , 2024). The neutral barium count of Fig. 13 indicated that release 2's detonation yielded about three times as much neutral barium than did release 1's detonation. Thus, for event 2, we are using the ionization profile approximated by Eq. (18), except with $N_{0} = 3 * (8.75 \times 10^{23} m^{- 3}) = 2.625 \times 10^{24} m^{- 3}$ ⁠. As shown in Sec. 4 of Appendix B and in Table I, comparisons of density profiles modeled with different values of $τ$ and different expansion velocities led us to select the same $τ$ = 28 s and a slightly slower neutral expansion velocity of 1.2 km/s for release 2.

FIG. 13.

View large Download slide

Intensity of light from optical data that was filtered for neutral barium emissions (Barnes , 2024). This is a proxy for the neutral barium yield over time for the two releases, whose detonation times are indicated by the dashed black lines.

TABLE I.

Comparisons of effect each parameter has on key features of release 2's modeled profile and on the associated Pearson correlation coefficient. Note that unless otherwise specified $τ$ = 28 s, $v_{exp}$ = 1.2 km/s, $t_{skid}$ = 0 s, and $t_{delay}$ = 0 s except for the $τ$ comparisons where $v_{exp}$ = 1.5 km/s. See Sec. 2 of Appendix B for details on how each value is calculated. The bold text indicates the preferred choices (see text).

Value	Pearson coefficient	$t_{PIP, onset} - t_{trace, onset}$	$t_{PIP, peak} - t_{trace, peak}$	$\frac{n_{B a^{+}, trace} (t_{trace, peak})}{n_{B a^{+}, PIP} (t_{PIP, peak})}$
$τ$ = 4 s	0.56	0.24 s	1.10 s	4.65
$τ$ = 6 s	0.61	0.24 s	1.06 s	3.59
$τ$ = 8 s	0.64	0.22 s	1.16 s	2.82
$τ$ = 20 s	0.69	0.14 s	1.14 s	1.31
$τ$ = 24 s	0.68	0.18 s	1.06 s	1.08
$τ$ = 28s	0.69	0.08 s	1.12 s	0.93
$v_{exp}$ = 1.0 km/s	0.94	−0.30 s	0.14 s	0.93
$v_{exp}$ = 1.2 km/s	0.88	−0.12 s	0.62 s	0.95
$v_{exp}$ = 1.5 km/s	0.69	0.08 s	1.12 s	0.93
$v_{exp}$ = 2.5 km/s	0.15	0.40 s	1.82 s	0.92
$t_{skid}$ = 0.0 s	0.88	−0.12 s	0.62 s	0.95
$t_{skid}$ = 0.3 s	0.90	−0.12 s	0.58 s	1.09
$t_{skid}$ = 0.5 s	0.90	−0.18 s	0.70 s	1.18
$t_{skid}$ = 0.6 s	0.91	−0.18 s	0.64 s	1.26
$t_{skid}$ = 0.7 s	0.91	−0.18 s	0.52 s	1.27
$t_{skid}$ = 0.8 s	0.90	−0.18 s	0.58 s	1.32
$t_{skid}$ = 0.9 s	0.92	−0.20 s	0.38 s	1.39
$t_{skid}$ = 1.0 s	0.93	−0.22 s	0.52 s	1.34
$t_{delay}$ = 0.0 s	0.88	−0.12 s	0.62 s	0.95
$t_{delay}$ = 0.3 s	0.90	−0.10 s	0.56 s	0.95
$t_{delay}$ = 0.5 s	0.89	−0.08 s	0.68 s	0.99
$t_{delay}$ = 0.6 s	0.89	−0.10 s	0.72 s	0.98
$t_{delay}$ = 0.7 s	0.90	−0.06 s	0.60 s	0.97

Value	Pearson coefficient	$t_{PIP, onset} - t_{trace, onset}$	$t_{PIP, peak} - t_{trace, peak}$	$\frac{n_{B a^{+}, trace} (t_{trace, peak})}{n_{B a^{+}, PIP} (t_{PIP, peak})}$
$τ$ = 4 s	0.56	0.24 s	1.10 s	4.65
$τ$ = 6 s	0.61	0.24 s	1.06 s	3.59
$τ$ = 8 s	0.64	0.22 s	1.16 s	2.82
$τ$ = 20 s	0.69	0.14 s	1.14 s	1.31
$τ$ = 24 s	0.68	0.18 s	1.06 s	1.08
$τ$ = 28s	0.69	0.08 s	1.12 s	0.93
$v_{exp}$ = 1.0 km/s	0.94	−0.30 s	0.14 s	0.93
$v_{exp}$ = 1.2 km/s	0.88	−0.12 s	0.62 s	0.95
$v_{exp}$ = 1.5 km/s	0.69	0.08 s	1.12 s	0.93
$v_{exp}$ = 2.5 km/s	0.15	0.40 s	1.82 s	0.92
$t_{skid}$ = 0.0 s	0.88	−0.12 s	0.62 s	0.95
$t_{skid}$ = 0.3 s	0.90	−0.12 s	0.58 s	1.09
$t_{skid}$ = 0.5 s	0.90	−0.18 s	0.70 s	1.18
$t_{skid}$ = 0.6 s	0.91	−0.18 s	0.64 s	1.26
$t_{skid}$ = 0.7 s	0.91	−0.18 s	0.52 s	1.27
$t_{skid}$ = 0.8 s	0.90	−0.18 s	0.58 s	1.32
$t_{skid}$ = 0.9 s	0.92	−0.20 s	0.38 s	1.39
$t_{skid}$ = 1.0 s	0.93	−0.22 s	0.52 s	1.34
$t_{delay}$ = 0.0 s	0.88	−0.12 s	0.62 s	0.95
$t_{delay}$ = 0.3 s	0.90	−0.10 s	0.56 s	0.95
$t_{delay}$ = 0.5 s	0.89	−0.08 s	0.68 s	0.99
$t_{delay}$ = 0.6 s	0.89	−0.10 s	0.72 s	0.98
$t_{delay}$ = 0.7 s	0.90	−0.06 s	0.60 s	0.97

The resulting modeled barium density following the second release, assuming no skidding and no delayed ionization, is shown in the lower plot of Fig. 14. The results of our quantitative analysis of various model parameters, described in Sec. 2 of Appendix B, for event 2 are presented in Table I. Note that there is a slight mismatch of both the maximum $n_{B a^{+}}$ value and the time it occurred between the modeled and measured data as well as a difference between the modeled and measured $B a^{+}$ profiles' growth and decay times. Examining the early growth period, release 2's modeled barium density is offset in time from the measured density data with the modeled profile's onset lagging about 0.12 s behind. However, the model has a shorter growth period with the peak in the model density occurring over half a second before the peak in the measured profile. Additionally, the modeled density is much lower than that measured throughout most of the decay period (about 3–9 s after release 2). Finally, the observed spike in barium density around 1 s after the second release's detonation is not reproduced in the simple model of the barium density. Thus, other processes, such as skidding or delayed ionization, are needed to resolve these inconsistencies between the measured and modeled data for release 2.

FIG. 14.

View large Download slide

Comparison of fitted PIP data to initial model result (no skidding or delayed ionization). Plot of main payload PIPs' measured densities of barium ions (grey) with modeled barium ion densities from modeled clouds without skidding (purple).

3. Modifications to model for special cases: skidding and delayed ionization

Given the shortcomings of the model results for release 2, we consider optional modifications to the particles' motions and ionization times in order to test if our results from release 2 are influenced by processes such as skidding and/or delayed ionization. In order to incorporate skidding into our model, we consider the form of the motion during the transition from the neutral particle motion to gyro-motion (i.e., the skidding period). As posited in Sec. I, during the skidding period the injected ions move as if they were not fully coupled to the ambient plasma, for instance because of the presence of an electric field such that

\vec{E} \times \vec{B} / B^{2}

is equivalent to the perpendicular component of the neutral radial velocity. Delamere (2000) modeled skidding as a momentum transfer from the injected Ba⁺ plasma to the ambient plasma. Thus, Delamere (2000) arrived at the expression for the recently-ionized particles' radial velocity (⁠

{\vec{v}}_{radial, ⊥}

⁠) over time in the direction perpendicular to

\vec{B}

as

{\vec{v}}_{radial, ⊥} = {\vec{v}}_{0, ⊥} e^{\frac{- (t - t_{0})}{t_{skid}}},

(19)

where

t_{skid}

is the skid timescale and

{\vec{v}}_{0, ⊥}

is the particle's initial velocity in the plane perpendicular to

\vec{B}

(i.e., the perp-B plane). Hence, we modify the model's perpendicular velocity components of Eq. to incorporate the exponential transfer of the ionized particle's initial (radially expanding) neutral motion to gyro-motion. Thus, the velocity of a skidding particle is calculated as follows:

\begin{matrix} v_{i, x} (t) = (v_{i, y 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, x 0} cos (ω_{c} (t - t_{i, 0}))) (1 - e^{\frac{- (t - t_{i, 0})}{t_{skid}}}) + v_{i, x 0} e^{\frac{- (t - t_{i, 0})}{t_{skid}}}, \\ v_{i, y} (t) = (- v_{i, x 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, y 0} cos (ω_{c} (t - t_{i, 0}))) (1 - e^{\frac{- (t - t_{i, 0})}{t_{skid}}}) + v_{i, y 0} e^{\frac{- (t - t_{i, 0})}{t_{skid}}}, \\ v_{i, z} (t) = v_{i, z 0}, \end{matrix}

(20)

for

0 < t_{skid}

⁠, where

t_{i, 0}

is that given by Eq. . Integration of the above velocity components with respect to time, using the SymPy package (Meurer , 2017), yields the following expressions for the time-dependent position of a skidding particle:

\begin{matrix} x_{i} (t) = - \frac{v_{i, y 0}}{ω_{c}} cos (ω_{c} (t - t_{i, 0})) + \frac{v_{i, x 0}}{ω_{c}} sin (ω_{c} (t - t_{i, 0})) + \frac{v_{i, y 0}}{ω_{c}} + x_{i, 0} + x_{i, skid} (t), \\ y_{i} (t) = \frac{v_{i, x 0}}{ω_{c}} cos (ω_{c} (t - t_{i, 0})) + \frac{v_{i, y 0}}{ω_{c}} sin (ω_{c} (t - t_{i, 0})) - \frac{v_{i, x 0}}{ω_{c}} + y_{i, 0} + y_{i, skid} (t), \\ z_{i} (t) = v_{i, z 0} (t - t_{i, 0}) + z_{i, 0}, \end{matrix}

(21)

where

\begin{matrix} x_{i, skid} (t) = v_{i, x 0} t_{skid} (1 - e^{\frac{- (t - t_{i, 0})}{t_{skid}}}) - \frac{1}{ω_{c}^{2} t_{skid}^{2} + 1} (ω_{c} v_{i, y 0} t_{skid}^{2} + v_{i, x 0} t_{skid} \\ + ω_{c} t_{skid}^{2} (- v_{i, y 0} cos (ω_{c} (t - t_{i, 0})) + v_{i, x 0} sin (ω_{c} (t - t_{i, 0}))) e^{\frac{- (t - t_{i, 0})}{t_{skid}}} - t_{skid} (v_{i, y 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, x 0} cos (ω_{c} (t - t_{i, 0}))) e^{\frac{- (t - t_{i, 0})}{t_{skid}}}), \\ y_{i, skid} (t) = v_{i, y 0} t_{skid} (1 - e^{\frac{- (t - t_{i, 0})}{t_{skid}}}) + \frac{1}{ω_{c}^{2} t_{skid}^{2} + 1} (ω_{c} v_{i, x 0} t_{skid}^{2} - v_{i, y 0} t_{skid} \\ - ω_{c} t_{skid}^{2} (v_{i, x 0} cos (ω_{c} (t - t_{i, 0})) + v_{i, y 0} sin (ω_{c} (t - t_{i, 0}))) e^{\frac{- (t - t_{i, 0})}{t_{skid}}} \\ + t_{skid} (- v_{i, x 0} sin (ω_{c} (t - t_{i, 0})) + v_{i, y 0} cos (ω_{c} (t - t_{i, 0}))) e^{\frac{- (t - t_{i, 0})}{t_{skid}}}) . \end{matrix}

(22)

Thus, our model, when skidding is enabled, represents skidding as the exponential transition/transfer of the ion's perp-B motion from the initial (radially expanding) neutral motion to gyro-motion about a magnetic field line. This is illustrated in the second panel of Fig. 15. Note that, following the modified trace, the model calculates the density in the region of the main payload in the same manner described previously.

FIG. 15.

View large Download slide

Modeled particles' trajectories in the unmodified model (left), the model with skidding (center) and the model with delayed ionization (right). For each model, the plotted particles are selected from the model's cloud initialized at $t_{0}$ = 0.3 s. Each modeled particle's plotted trajectory begins with their initial position at detonation, then follows their positions from 0.3 to 1 s after detonation in steps of 20 ms with their ionization point indicated by the star marker (outlined in black) and their positions at each time step indicated by the colored circles.

It is also of interest to consider the possibility of delays in the ionization process of the injected barium neutrals. A short delay in the ionization of the neutral barium within some time

t_{delay}

after detonation can be represented in our model by a continuation of neutral motion (i.e., radial expansion) with velocity

{\vec{v}}_{i, 0}

for times

t \leq t_{delay}

for particles in clouds with

t_{0} \leq t_{delay}

⁠. For times

t > t_{delay}

⁠, these particles' motions follow Eqs. and with modified initial positions of

\begin{matrix} x_{i, 0} = x_{center} + v_{i, x 0} t_{delay}, \\ y_{i, 0} = y_{center} + v_{i, y 0} t_{delay}, \\ z_{i, 0} = z_{center} + v_{i, z 0} t_{delay}, \end{matrix}

(23)

and a modified

t_{i, 0}

of

t_{i, 0} = t_{0} + Δ t_{clouds} * (ξ_{i} - 1 / 2) + (t_{delay} - t_{0}),

(24)

where

t_{delay}

is in seconds since detonation. Note that the density calculation portion of the model's code will not treat these particles as ions until after

t_{delay}

⁠. This is achieved by placing a condition on the density calculation that for all times

t \leq t_{delay}

the density is zero, where t is in seconds since release detonation. The trajectories of particles modeled with delayed ionization are shown in the third panel of Fig. 15.

4. Release 2 modeled with skidding

As discussed in Sec. I, electric fields parallel to the local magnetic field at the edges of an ion cloud are theorized to be important in initiating skidding (Delamere , 2000). Following the second release the onboard energetic electron spectrometer (i.e., the Electron Plasma Instrument) onboard the main payload measured a field-aligned electron beam approximately 1 s after detonation; the electric field instrument also measured associated strong parallel electric fields (Delamere , 2024). These observations support the hypothesis that skidding occurred following the second release. However, given the localized nature of these two observations, it could not be conclusively determined whether or not similar electric fields and electron beams formed following the first release but were not coincident with the instruments (Delamere , 2024). The PIPs' observations of the second release showed features specific to this event, as did the other instruments. The electron beam was coincident with the PIPs' observed spike in ion density approximately 1 s after release 2, highlighted by the blue circle in Fig. 9's bottom plot. These event 2-specific observations across multiple instruments following the second release and the failure of initial model runs without skidding to reproduce the observed $B a^{+}$ density spike justified a study of the effects of ion skidding for various lengths of time. Note that, unlike in (Delamere , 2000), our model does not attempt to model the mechanisms that produce skidding. Instead, we examine the impact of the skidding motion, described by Eqs. (20)–(22) as an exponential transfer from expansion motion to gyro-motion, on our model results and attempt to determine if skidding occurred by comparison of these model results with our measured data.

The panels of Fig. 16 show the resulting densities from models with different skidding durations. Note that only the $t_{skid}$ = 0 s calculation uses the original velocity and position equations [Eqs. (14) and (15)]. All $t_{skid} >$ 0 runs use the skid-modified velocity and position equations [Eqs. (20)–(22)]. As the input $t_{skid}$ increases from 0 to 0.3 s, the time and magnitude of the model's peak density moves closer to that observed, as quantified in Table I. Also, the modeled profile's decay period grows closer in magnitude and time to that observed, seen qualitatively in Fig. 16 and quantitatively by the increased Pearson correlation coefficient value in Table I. As $t_{skid}$ increases to 0.7 s, the decay period of the modeled profile grows closer in magnitude and time to that observed. However, the models' onset times grow later and peak density values increase past that observed by the PIPs. Strangely, the time of peak density initially increases with $t_{skid}$ = 0.5 s, but then decreases as $t_{skid}$ increases to 0.7 s. As the skidding duration is increased past 0.7 s, the modeled profiles' onset times become later and peak densities become higher. Also, the time of peak density again increases and then decreases with increasing $t_{skid}$ values. Thus, past $t_{skid}$ = 0.7 s the modeled profiles' deviate more from the observed profile during the growth phase even while the decay portion of the modeled profiles remains mostly unchanged. The $t_{skid}$ value that produces the modeled profile that best captures the onset, peak and the decay period of the observed profile is 0.3 s. As shown in quantitatively in Table I, $t_{skid}$ = 0.3 s has a better correlation coefficient than $t_{skid}$ = 0 s, along with the best pair of peak density times and values out of all of the profiles with skidding. However, none of these models reproduce either the initial density spike nor the onset time of the observed profile. Hence, these results indicate that skidding could have occurred and show how such skidding would likely impact our observations. However, it cannot explain all of the differences between the profiles of our model and that observed.

FIG. 16.

View large Download slide

Plots of PIP data vs model data for models with different skidding durations.

5. Release 2 modeled with ionization delay

The inclusion of skidding in the model did not reproduce the observed density precursor or resolve the offset between the measured and observed onset times. Also, as discussed in Barnes (2024), the optical instruments' observed shape of release 2's $B a^{+}$ cloud could have been produced by either a delay in the onset of ionization or skidding. In order to verify that delayed ionization produces a distinguishable effect on our modeled results compared to those of skidding, as well as investigate if delayed ionization could be responsible for the observed onset time, we ran the no-skidding model with delayed ionization, as described in Sec. IV B 3, for various ionization delays (⁠ $t_{delay}$ ⁠). Intuitively, one might expect that delayed ionization might shorten the travel time to the observation point, as the radial neutral particle velocity path is more direct. The results of these models are plotted alongside the observed data in Fig. 17. Comparison of the model results shown in Fig. 16 to those of Fig. 17 reveal that delayed ionization does not reproduce the same profiles as the skidding model. Additionally, delayed ionization does not produce the observed precursor in the model's output density. Longer delays in ionization do push the onset time back slightly, but not enough for the modeled profile to match the onset time observed by the PIPs. Unlike the skidding case, delayed ionization does not significantly improve the modeled profiles' decay periods, as shown qualitatively in Fig. 17 and quantitatively by the various $t_{delay}$ profiles' correlation coefficient values that never exceed that of $t_{skid}$ = 0.3 s. Note that the correlation coefficients' values for different $t_{delay}$ cases only improve, compared to the undelayed case, due to the decreased offset between the modeled and measured profiles' onset and peak value times, but it does not improve the decay phase. Thus, we can conclude that impacts of delayed ionization on our model are distinct from those of skidding. Based on these results, delayed ionization would only affect a density profile's early growth period, and would not influence the density evolution past the peak.

FIG. 17.

View large Download slide

Plots of PIP data vs model data for models where ionization is delayed for different lengths of time following release 2's canister detonation.

6. Explanation of model onset time offset from data

Thus far, our model throughout its three permutations has not reproduced the observed precursor signature of event 2, highlighted by the blue circle in Fig. 9. Note that our model assumed that release 2 produced a single $B a^{+}$ population whose ions' velocities followed a Gaussian spread about a mean expansion velocity of $v_{exp}$ = 1.2 km/s. However, it may be possible that a small population of the barium neutrals had $v_{exp} >$ 1.2 km/s. This subset might affect only the observed density near the onset time but not affect later measurements. A comparison of the effects of different $v_{exp}$ values on our simple model, without skidding or delayed ionization, is shown in Sec. 4 of Appendix B and Fig. 26. In that figure, $v_{exp}$ = 1.5 km/s gives the correct onset time. In order to investigate whether a smaller population with a mean expansion velocity of 1.5 km/s or higher could qualitatively reproduce the spike, we ran our model for a small subset of particles that followed the ionization profile of Eq. (18) except with $N_{0}$ equal to half that used for release 2's ionization profile (i.e., $N_{0} = N_{0, sub} = (2.625 \times 10^{24} m^{- 3}) / 2 = 1.3125 \times 10^{24} m^{- 3}$ ⁠). A qualitative comparison of different $v_{exp}$ values on this simple model, without skidding or delayed ionization, is shown in Fig. 18.

FIG. 18.

View large Download slide

Plots of event 2's PIP data vs model data for models with different $v_{exp}$ values. Release 2's detonation time is indicated by the green dashed line.

The value of $v_{exp}$ that produces an onset time closest to that observed is $v_{exp}$ = 1.5 km/s. However, the density spike's magnitude is best reproduced by the model with $v_{exp}$ = 2.5 km/s. Note that the ionization profile for this fast $B a^{+}$ population may differ from that of the main $B a^{+}$ profile or that used for this section's models in $τ$ and/or $N_{0}$ ⁠. However, these results support the hypothesis that there was a population of $B a^{+}$ which had a faster expansion velocity than the main population, and this faster population is responsible for the early $B a^{+}$ precursor observations.

C. Ion temperature wave-particle heating comparison

Now we turn from analysis of the releases' injected barium ions and consider the releases' impact on the ambient ionospheric (⁠ $O^{+}$ ⁠) plasma. A major part of KiNET-X's mission was to investigate wave-particle interactions, especially energy coupling/transfer. As shown in Figs. 7 and 19, and the top plots of Fig. 20, the main payload PIPs observed an increase in the background ambient $O^{+}$ plasma of about 150–200 K following the first release and about 500–600 K following the second release. (Note that the GSFC fields' data, and plots in this Sec. IV C, use a full-second flight time T0 convention of T0 = 00:44:00 UT rather than 00:44:0.1122 UT used elsewhere in this paper). The electric field probes onboard the main payload observed waves in the lower hybrid (LH) frequency (⁠ $\sim$ 6 kHz) and cyclotron frequency (⁠ $\sim$ 38 Hz) ranges coincident with the PIP-observed increases in the ambient (⁠ $O^{+}$ ⁠) ion population's temperature following each release as shown in Fig. 20. One or more of these wave modes could have contributed to the heating of the ambient plasma.

FIG. 19.

View large Download slide

Plot of PIPs' observed oxygen ion temperatures around the (top) first release and (bottom) second release close to the events. Unlike previous figures showing $n_{B a^{+}}$ ⁠, the plotted PIP data here are limited to data collected when each PIP is within 15° of AoA. Release detonation times are indicated by the dashed green lines.

FIG. 20.

View large Download slide

(a) For a time period around the first release, the following are shown: a lineplot of main payload PIPs' measured ambient $O^{+}$ plasma temperature (with zoomed y-axis to show feature of interest) (top), a log-y spectrogram of the perpendicular electric field at low frequencies on the same time axis (middle) and a log-y spectrogram of measured electric field in the perp-B direction at high frequencies (bottom). (b) The same style plots as (a) for the time period around the second release. Note that all plots in a column (i.e., sub-figure) share the same x axis. The spectrograms corresponding to the first release show low frequency interference corresponding to twice the rocket spin rate.

Figure 20 shows electric field spectrograms from a double probe detector in the spin plane which was oriented perpendicular to the geomagnetic field. Here, each spectrogram's colorbar gives the measured values of

p = 10 l o g_{10} (\frac{E^{2}}{Δ f_{B W}}),

(25)

where

E^{2}

is the electric field magnitude squared in units of

{(mV / m)}^{2}

and

Δ f_{B W}

(Hz) is the frequency width of the feature of interest on the plot. Thus, solving Eq. for

E^{2}

gives

E^{2} = Δ f_{B W} 10^{p / 10} {(\frac{m V}{m})}^{2} / H z = Δ f_{B W} 10^{p / 10} {(10^{- 3} \frac{V}{m})}^{2} / H z .

(26)

The $E^{2}$ values observed for each wave mode are essential in the following two sub-sections' calculations of the change in ambient $O^{+}$ temperatures that each of the observed wave modes could produce.

1. Lower hybrid wave heating

The calculation of the expected change in temperature based on the observed lower hybrid waves is adapted from the calculations detailed in the appendix of Bell (1991). For a single-species ion population, the change in energy (⁠

ξ_{i}

⁠) of species i due to a LH wave acting on the ion population for a period of time (⁠

τ

⁠) can be approximated by

Δ ξ_{i} \approx {(τ π q_{i}^{2} \frac{E^{2}}{Δ k} v_{i, ⊥ 0})}^{1 / 2},

(27)

where

q_{i}

is the ion charge, E is the electric field of the wave,

Δ k

is the (perpendicular-to-B) wavenumber spread, and

v_{i, ⊥ 0}

is the ions' initial velocity perpendicular to the magnetic field (prior to encountering the LH wave). Note that in Bell (1991),

ξ

is defined as

ξ_{i} = \frac{1}{2} m_{i} v_{i ⊥}^{2},

(28)

where

m_{i}

is the ion mass (Bell , 1991). Now, recall that the equation for the thermal energy of a particle with

γ

degrees of freedom is

U_{i} = \frac{γ}{2} k_{B} T_{i},

(29)

where

k_{B}

is the Boltzmann constant. Now, assuming that

Δ ξ_{i}

is equal to the change in thermal energy, the change in temperature

Δ T_{i}

is obtained by equating Eqs. and , with

T_{i}

replaced by

Δ T_{i}

⁠. Therefore, the change in temperature due to a LH wave interacting with ion species i for a duration of time (⁠

τ

⁠) is

Δ T_{i} \approx \frac{2}{γ k_{B}} {(τ π q_{i}^{2} \frac{E^{2}}{Δ k} v_{i, ⊥ 0})}^{1 / 2} .

(30)

We still need expressions for

v_{i, ⊥ 0}

and

Δ k

⁠. The initial thermal velocity of the ions can be calculated from the prerelease PIP-measured ambient ion temperature (⁠

T_{i, 0}

⁠) by equating Eqs. and to obtain

v_{i 0} \approx \sqrt{\frac{γ}{m_{i}} k_{B} T_{i 0}} .

(31)

Assuming the ambient plasma follows an isotropic Maxwellian distribution prior to release, there are two degrees of freedom perpendicular to the magnetic field (i.e.,

γ

= 2), and thus

v_{i, ⊥ 0} \approx \sqrt{\frac{2}{m_{i}} k_{B} T_{i 0}} .

(32)

Now, following the assumption

v_{i, ⊥ f} ≫ v_{i, ⊥ 0}

from Bell (1991), the wavenumber spread is approximated by

Δ k = \frac{ω_{L H, i}}{v_{i, ⊥ 0}} - \frac{ω_{L H, i}}{v_{i, ⊥ f}} \approx \frac{ω_{L H, i}}{v_{i, ⊥ 0}},

(33)

where

ω_{L H, i}

is the lower hybrid wave frequency for species i

ω_{L H, i} = \sqrt{\frac{ω_{c e} ω_{c i}}{1 + ω_{c e}^{2} / ω_{p e}^{2}}} = \sqrt{\frac{(e B / m_{i}) (e B / m_{e})}{1 + {(e B / m_{e})}^{2} / [ε_{0} m_{e} / (n_{e} e^{2})]}} .

(34)

Finally, we can approximate

Δ T_{i}

from Eq. with values for

Δ E

⁠,

v_{i, ⊥ 0}

⁠, and

Δ k

calculated from our measured values using Eqs. , , and , respectively. The relevant values and resulting

Δ T_{O^{+}}

from LH heating are given for the two events in Tables II and III.

TABLE II.

Release 1 $Δ T_{O^{+}}$ calculation values and result.

Quantity	Lower hybrid	Cyclotron
f (Hz)	$f_{L H} \sim$ 6 kHz	$f_{c} \sim$ 38 Hz
$ω$ (radians/s)	$ω_{L H, O^{+}} \sim$ $4 x 10^{4}$	$ω_{c, O^{+}} \sim$ 239.5
$Δ f_{B W}$ (Hz)	1 kHz	10 Hz
p $({(mV / m)}^{2} / Hz)$	−52 $\leq$ p $\leq$ −45	−50 $\leq$ p $\leq$ −45
$T_{O^{+}, 0}$ (K)	1100 K	1100 K
Time period (s)	595.5–597.125 s	594.5–600 s
$τ$ (s)	1.625 s	5.5 s
$Δ T_{O^{+}}$ (K)	11–25 K	105–332 K
$Δ T_{O^{+}, PIPs}$ (K)	$\sim$ 100–150 K	$\sim$ 100–150 K

Quantity	Lower hybrid	Cyclotron
f (Hz)	$f_{L H} \sim$ 6 kHz	$f_{c} \sim$ 38 Hz
$ω$ (radians/s)	$ω_{L H, O^{+}} \sim$ $4 x 10^{4}$	$ω_{c, O^{+}} \sim$ 239.5
$Δ f_{B W}$ (Hz)	1 kHz	10 Hz
p $({(mV / m)}^{2} / Hz)$	−52 $\leq$ p $\leq$ −45	−50 $\leq$ p $\leq$ −45
$T_{O^{+}, 0}$ (K)	1100 K	1100 K
Time period (s)	595.5–597.125 s	594.5–600 s
$τ$ (s)	1.625 s	5.5 s
$Δ T_{O^{+}}$ (K)	11–25 K	105–332 K
$Δ T_{O^{+}, PIPs}$ (K)	$\sim$ 100–150 K	$\sim$ 100–150 K

TABLE III.

Release 2 $Δ T_{O^{+}}$ calculation values and result.

Quantity	Lower hybrid	Cyclotron
f (Hz)	$f_{L H} \sim$ 6 kHz	$f_{c} \sim$ 38 Hz
$ω$ (radians/s)	$ω_{L H, O^{+}} \sim$ $4 \times 10^{4}$	$ω_{c, O^{+}} \sim$ 239.5
$Δ f_{B W}$ (Hz)	2 kHz	10 Hz
p $({(mV / m)}^{2} / Hz)$	−52 $\leq$ p $\leq$ −45	−40 $\leq$ p $\leq$ −35
$T_{O^{+}, 0}$ (K)	1125 K	1125 K
Time period (s)	628.75–631.5 s	628.125–631.5 s
$τ$ (s)	2.75 s	3.375 s
$Δ T_{O^{+}}$ (K)	24–54 K	263–834 K
$Δ T_{O^{+}, PIPs}$ (K)	$\sim$ 400–500 K	$\sim$ 400–500 K

Quantity	Lower hybrid	Cyclotron
f (Hz)	$f_{L H} \sim$ 6 kHz	$f_{c} \sim$ 38 Hz
$ω$ (radians/s)	$ω_{L H, O^{+}} \sim$ $4 \times 10^{4}$	$ω_{c, O^{+}} \sim$ 239.5
$Δ f_{B W}$ (Hz)	2 kHz	10 Hz
p $({(mV / m)}^{2} / Hz)$	−52 $\leq$ p $\leq$ −45	−40 $\leq$ p $\leq$ −35
$T_{O^{+}, 0}$ (K)	1125 K	1125 K
Time period (s)	628.75–631.5 s	628.125–631.5 s
$τ$ (s)	2.75 s	3.375 s
$Δ T_{O^{+}}$ (K)	24–54 K	263–834 K
$Δ T_{O^{+}, PIPs}$ (K)	$\sim$ 400–500 K	$\sim$ 400–500 K

2. Ion cyclotron oscillation heating

The calculation of the ion cyclotron heating presented here is based on that presented in Chang (1986). For a population of gyrotropically distributed ions, the net increase in these ions' energy in the direction perpendicular to B due to ion cyclotron oscillations over a time period

τ

is

Δ W_{⊥, res} = \frac{1}{2} \frac{q_{i}^{2} E_{⊥}^{2}}{m_{i}} τ^{2},

(35)

which was discussed in Chang (1986). Thus, the net change in an ion population's energy (⁠

Δ ξ_{i}

⁠) is assumed to be equal to

Δ W_{⊥, res}

⁠. As in the LH heating calculation, the change in ion temperature (⁠

Δ T_{i}

⁠) associated with this change in energy can be determined by equating Eq. to the thermal energy equation [Eq. ] for 2 degrees of freedom and solving for

Δ T_{i}

⁠. Thus, the change in temperature due to ion cyclotron heating over a period of time

τ

can be calculated by

Δ T_{i} = \frac{1}{2 k_{B}} \frac{q^{2} E^{2}}{m_{i}} τ^{2} .

(36)

3. Calculation results

The values used in the calculations of $Δ T_{O^{+}}$ from each heating method and the results for release 1 and release 2 are shown in Tables II and III, respectively. In both cases, the PIP-observed temperature changes fall within range of $Δ T_{O^{+}}$ produced by cyclotron heating. The calculated $Δ T_{O^{+}}$ due to LH heating in both cases is much lower than that observed. Therefore, we conclude that the ion cyclotron oscillations are the primary coupling mechanism between the electric field and the ambient oxygen ions.

V. CONCLUSIONS

Modifications were made to the PIP data analysis' forward-modeling tool, presented in Fraunberger I. (2020) and Roberts (2017b), in order to investigate the multi-species plasma of the KiNET-X experiment. Using this multi-species analysis technique, the main payload PIPs' observations indicate increases in the temperature of the background, ambient ionospheric $O^{+}$ plasma following each release. Lower hybrid waves and ion cyclotron oscillations were observed by the main payload's electric field probe during the same time periods as the $O^{+}$ temperature increases following each release. By estimating the magnitude of the ion temperature change that the observed lower hybrid waves and the observed ion cyclotron oscillations would produce in the ambient plasma and comparing each to the observed temperature increases, we determined that the ion cyclotron oscillations were the primary heating mechanism after both releases.

The multi-species PIP data analysis technique was used to investigate the evolution of the density of the barium ion (⁠ $B a^{+}$ ⁠) clouds resulting from each release. Following both releases, the main payload PIPs observed the increase and decrease in the barium ions' density. Three key features of these data were: a higher maximum $B a^{+}$ density following the second release than the first one, the different shapes of the first and second releases' $n_{B a^{+}}$ profiles, and a precursor spike in $B a^{+}$ density that was also seen in the PIPs' raw current data following the second release that was not observed following the first release. In order to understand these features, we developed a simple model of each release's $B a^{+}$ cloud by modeling the trajectories of the cloud's $B a^{+}$ ions moving under the Lorentz force. Then the model's $B a^{+}$ density in the region around the main payload could be estimated for each release. The modeled first release's $B a^{+}$ density closely resembled the PIP-observed $B a^{+}$ density data during the growth and decay periods, with a slight mismatch in the total $n_{B a^{+}}$ between the model and data. Thus, for the first release, the expansion and ionization of the neutral barium created an ion cloud with a shape similar to that of our model in the plane perpendicular to $\vec{B}$ ⁠. As the barium ionized over the first few seconds after the first release, the ions were trapped and their motion constrained to gyration about their magnetic field line.

In contrast, for the second event, the model's maximum $n_{B a^{+}}$ value occurred slightly earlier than observed. Additionally, modeled $B a^{+}$ density profiles with different conditions for the second release did not reproduce the observed $B a^{+}$ precursor density spike nor did they follow the observed $n_{B a^{+}}$ growth and decay rates. As the ionization rate could have been different for release 2, we tested different ionization profile time constants as shown in Table I and concluded that $τ$ = 28 s was valid for the second release as well as the first. As release 2 did not have an observed value for $v_{exp}$ ⁠, comparisons of density profiles modeled with different expansion velocities led us to select an expansion velocity of 1.2 km/s whose profile best captured both the onset time and the later growth phase's slope.

The remaining differences in the modeled profile from the observed profile led us to consider the impacts of different phenomena on the model's output density profile. One possibility was that the ionization of the barium was delayed for several hundred milliseconds within the first second following the second release (during the ionization of the full barium cloud) leading to the particles expanding further than they otherwise would before becoming trapped in the cyclotron motion around magnetic field lines in the plane perpendicular to $\vec{B}$ ⁠. As shown in Table I, delayed ionization moved the modeled density profiles' onset times closer to that of the observed profile. However, as seen qualitatively in Fig. 17 and quantitatively in these profiles' Pearson correlation coefficients, the delayed ionization did not bring the decay period of the modeled profiles closer to that observed. Another possibility was that instead of instantly transitioning from the (neutrals') radial expansion motion to cyclotron motion upon ionization, the newly ionized barium particles experienced a transition from their previous radial expansion motion to cyclotron motion over some period of time. This skidding hypothesis is based on observations from other sounding rocket studies of some newly-ionized particles briefly continuing their previous, neutral motion through a magnetic field before becoming trapped (Haerendel, 2019; Delamere , 2000). This phenomenon is sometimes referred to as “skidding” (Haerendel, 2019). This model variant represents some features of our event 2 observations and illustrates how skidding would manifest generally in thermal ion data.

In order to understand why skidding may have been observed during the second release only, consider “the timescale of momentum transfer from injected plasma to ambient plasma via Alfvén waves” as in (Delamere , 2000; Delamere and Bagenal, 2013). As shown in Delamere and Bagenal (2013), if the rate of ambient plasma mass becoming coupled to an injected plasma (⁠

\frac{d M_{O^{+}}}{d t}

⁠) via an Alfvén wave is close to the rate at which the injected plasma's mass is growing (⁠

\frac{d M_{B a^{+}}}{d t}

⁠), then momentum transfer occurs quickly and skidding is minimized. As shown in Delamere and Bagenal (2013),

\frac{d M_{O^{+}}}{d t} = 2 n_{O^{+}} m_{O^{+}} v_{A} A,

(37)

where

n_{O^{+}}

is the ambient

O^{+}

density,

m_{O^{+}}

is the mass of

O^{+}

⁠,

v_{A}

is the ambient Alfvén speed, and A is the cross-sectional area of the boundary between the ambient plasma and injected plasma cloud. Within the first second after each release, the cloud is roughly spherical with a time-dependent radius of

R = v_{exp} t

⁠; thus

\frac{d M_{O^{+}}}{d t} \approx 2 n_{O^{+}} m_{O^{+}} v_{A} (π {(v_{exp} t)}^{2}),

(38)

where t is the number of seconds after the release and

v_{exp}

is the cloud's expansion velocity. This approximation is justified as Delamere (2024) reported that Alfvén waves only persisted after each release for a period on the order of tenths of a second. The value of

\frac{d M_{B a^{+}}}{d t}

can be determined from the expression for the barium ion population's total mass in time,

M_{B a^{+}} = m_{B a^{+}} N_{B a^{+}} (t) = m_{B a^{+}} N_{0} (1 - e^{- t / τ}),

(39)

where

N_{0}

is the release's neutral yield. Differentiating with respect to time gives

\frac{d M_{B a^{+}}}{d t} = \frac{m_{B a^{+}} N_{0}}{τ} e^{- t / τ},

(40)

where

τ

= 28 s as in Eq. . As Delamere (2024) stated that the Alfvén waves had velocities on the order of a few 100 km/s and only persisted for several tenths of a second after the releases, we will consider the values of

\frac{d M_{O^{+}}}{d t}

and

\frac{d M_{B a^{+}}}{d t}

at t = 0.75 s for

v_{A}

= 200 km/s. Our calculations' results, summarized in Table IV, show that the rate of ambient plasma mass coupling to injected plasma for release 1 was very close to the rate of injected plasma mass growth. Hence, the

B a^{+}

plasma injected by release 1 quickly coupled to the ambient ionosphere, and hence no significant skidding period would precede

B a^{+}

gyro-motion onset. In contrast, for release 2, the estimated

B a^{+}

population's

B a^{+}

mass growth was much greater than the estimated ambient plasma's mass coupling rate. Thus, it took longer for the injected plasma to couple to the ambient ionosphere and skidding can be expected to be observable.

TABLE IV.

Comparisons of $\frac{d M}{d t}$ values of each population for each release.

Event	$v_{exp} (km / s)$	$N_{0}$	$n_{O^{+}} (m^{- 3})$	$\frac{d M_{O^{+}}}{dt}$ (kg/s)	$\frac{d M_{B a^{+}}}{dt}$ (kg/s)
1	1.5	8.75 × 10²³	1.5 × 10¹¹	0.006	0.007
2	1.2	2.625 × 10²⁴	2.25 × 10¹¹	0.006	0.021

Event	$v_{exp} (km / s)$	$N_{0}$	$n_{O^{+}} (m^{- 3})$	$\frac{d M_{O^{+}}}{dt}$ (kg/s)	$\frac{d M_{B a^{+}}}{dt}$ (kg/s)
1	1.5	8.75 × 10²³	1.5 × 10¹¹	0.006	0.007
2	1.2	2.625 × 10²⁴	2.25 × 10¹¹	0.006	0.021

We introduced skidding to our model of release 2 by modifying the equations for the components of each ion's velocity in the plane perpendicular to $\vec{B}$ such that the initial velocity of each ion would transition from radial motion to cyclotron motion following an exponential function with a time constant $t_{skid}$ ⁠. The main impacts of skidding on the modeled profile were a change in the time of peak modeled density and a decay period signature closer in amplitude to that observed, which impacted the calculated Pearson correlation coefficient. The $B a^{+}$ density profile produced by the model with $t_{skid}$ = 0.3 s had a peak close in magnitude and time to that observed by the main payload PIPs and a high correlation coefficient. This model's profile also closely followed the observed profile through the decay period. However, the model failed to reproduce the precursor in density as well as the onset time of the observed profile. Also, the addition of skidding led to deviation in the slopes of the modeled and measured profiles during the later portion of the growth period. Thus, skidding with a time constant of 0.3 s is one explanation for some features of the second release cloud's particular shape, but it does not explain the precursor spike. If some small fraction of the released barium had a $v_{exp}$ of 1.5 km/s, the onset time of the precursor spike can be replicated in the model.

Each barium cloud's spatial structure as well as evolution can be further investigated by incorporating the multi-point PIP measurements from the ejected subpayloads. The multi-point PIP data analysis techniques and results for KiNET-X will be discussed in an upcoming paper.

ACKNOWLEDGMENTS

NASA (Grant Nos. 80NSSC18K0797 and 80NSSC21K2009) supported this paper. The KiNET-X science team thanks Matthew Blandin and Kylee Branning for their support during the launch, and for the dedicated work by the NASA Wallops Sounding Rockets Program Office (SRPO) and NSROC teams. Magdalina Moses was also supported by UNH NASA Space Grant (Grant No. 80NSSC20M0051) and by NH Epscor (No. 80NSSC22M0047). Also, the science team thanks Ralph Gibson for support in the PIPs' assembly. Finally, the colorblind simulator tool at https://davidmathlogic.com/colorblind/ was used to check that as many figures as possible were colorblind-friendly.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

M. L. Moses: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). K. Lynch: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). P. A. Delamere: Conceptualization (lead); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (lead); Writing – review & editing (equal). M. Lessard: Data curation (equal); Investigation (equal); Resources (equal). R. Pfaff: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Visualization (equal); Writing – review & editing (equal). M. Larsen: Investigation (equal); Resources (equal). D. L. Hampton: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Visualization (equal). M. Conde: Resources (equal). N. P. Barnes: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Visualization (equal); Writing – review & editing (equal). P. A. Damiano: Resources (equal). A. Otto: Resources (equal). C. Moser-Gauthier: Data curation (equal); Resources (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: PIP DATA ANALYSIS

1. PIP gains and offsets

As discussed in Sec. II B 1, the PIPs onboard KiNET-X's main payload had different gains. Also, in the data retrieval different voltage offsets, determined from the pre-launch baseline, were used for different PIPs. These values as well as the saturation currents of each PIP are shown in Table V.

TABLE V.

KiNET-X main payload PIPs' gain, voltage offset and saturation values. The bold text indicates the preferred choices (see text).

PIP IDa	Gain (⁠ $G_{PIP}$ ⁠) (mV/nA)	Offset voltage (⁠ $V_{Off}$ ⁠) (V)	Saturation current (nA)
18-0	40	1.005 2084	$\sim$ 99.9
18-1	40	1.007 5067	$\sim$ 99.8
19-0	320	1.003 9404	$\sim$ 12.5
19-1	320	1.007 9592	$\sim$ 12.5
20-0	320	1.010 1768	$\sim$ 12.5
20-1	320	1.011 2245	$\sim$ 12.5
21-0	40	1.015 6335	$\sim$ 99.6
21-1	40	1.010 9512	$\sim$ 99.7

PIP IDa	Gain (⁠ $G_{PIP}$ ⁠) (mV/nA)	Offset voltage (⁠ $V_{Off}$ ⁠) (V)	Saturation current (nA)
18-0	40	1.005 2084	$\sim$ 99.9
18-1	40	1.007 5067	$\sim$ 99.8
19-0	320	1.003 9404	$\sim$ 12.5
19-1	320	1.007 9592	$\sim$ 12.5
20-0	320	1.010 1768	$\sim$ 12.5
20-1	320	1.011 2245	$\sim$ 12.5
21-0	40	1.015 6335	$\sim$ 99.6
21-1	40	1.010 9512	$\sim$ 99.7

^a

In this column, the number before the dash indicates which shield that PIP is connected to.

APPENDIX B: SENSITIVITY STUDIES

1. Source of spread in $O^{+}$ temperature from low gain PIPs

As shown in the top plot of Fig. 21, there is a significant instrumental variation, particularly for PIP 18-1's fitted $T_{O^{+}}$ data. Also, PIP 18-1's fitted $T_{O^{+}}$ often differs greatly from that of PIP 18-0. This comes from the best-fit values chosen by LMFit. As shown in the IV curves in the subplots of Fig. 22, for the times highlighted in the top plot of Fig. 21, the choices LMFit makes for $T_{O^{+}}$ and the other fit parameters can lead to slightly different results for similar curves. However, for the most part, the LMFit results for PIPs 18-0, 21-0, and 21-1 agree with each other. Thus, the mean of these three PIPs' fitted $T_{O^{+}}$ can be trusted.

FIG. 21.

View large Download slide

Low gain PIPs' scalar fit $O^{+}$ temperatures (top), $B a^{+}$ and $O^{+}$ densities (middle) and ERPA-derived payload potentials (bottom) during the two releases at times that each PIP was looking within 15° of ram. The release detonation times are indicated by the dashed green lines. The times shown in the subplots of Fig. 22 are indicated by the arrows.

FIG. 22.

View large Download slide

Four subplots of PIPs' IV curves from around the times indicated in Fig. 21. Each subplot shows IV curves from two PIPs that made measurements closest to that time. The top plots of each subplot show the full IV curves and the bottom plots show a zoom of the IV curve's tail.

2. Methods for evaluating correlation between model results and PIP data

In this study, beyond qualitative assessments, several quantitative metrics are used to evaluate how well a given modeled density profile matches that observed by the PIPs. In order to calculate these metrics, the observed PIP data needs to be “matched” in time to the modeled density profiles' uniform time axis. This consists of finding the timestamp (if any) of the measured PIP data that is closest to the modeled PIP's time (within no more than 0.011 s) for every modeled time. Then, we can use these two matched datasets, plotted in blue on Fig. 23, in our correlation analysis. The Pearson correlation coefficient provides a measure of the strength of a linear relationship between the observed and the modeled $B a^{+}$ densities (Milton and Arnold, 2003). For this study, we used the SciPy's statistics package (Virtanen , 2020) to determine the Pearson correlation coefficient for each modeled result. However, the (longer) decay period will dominate the (shorter) growth periods in the calculation of the Pearson correlation coefficient for the entire period of the event. Thus, we also consider the difference (⁠ $t_{PIP, onset} - t_{trace, onset}$ ⁠) between the observed profile's onset time and that of a modeled profile, determined by taking a moving average of each profile and defining the start time as the time where the density profile exceeds a set value. Also, we consider the difference (⁠ $t_{PIP, peak} - t_{trace, peak}$ ⁠) between the times of the PIP-observed and the modeled peak density, determined by taking a moving average of the derivative of the smoothed density profile and finding where this smoothed derivative is zero. Finally, we consider the ratio (⁠ $\frac{n_{B a^{+}, trace} (t_{trace, peak})}{n_{B a^{+}, PIP} (t_{PIP, peak})}$ ⁠) in the peak density value of the modeled profile to that of the measured profile. Figure 23 illustrates the smoothed moving window curve, which is used to calculate these diagnostics from the matched data.

FIG. 23.

View large Download slide

Plots of PIP (top) and model (bottom) data which has undergone time matching (blue), smoothed density data (brown), and smoothed derivative of these smoothed density data (green).

3. Selection of ionization profile for trace models

As illustrated in Fig. 12, the observed ionization profile can be reasonably approximated by several values of

τ

in the model equation

N_{i} (t) = N_{0} (1 - e^{t / τ}),

(B1)

where

N_{0}

is the maximum ionization yield, t is time since detonation, and

τ

is the ionization time constant. However, none of these

τ

values produce a modeled

B a^{+}

density profile of release 1 that resembles the observed profile, as shown in the top three plots of Fig. 24. Additionally, observations from previous barium release experiments gave an expected time constant for barium ionization on the order of 28 s (Barnes , 2024). Thus, we perform several comparisons of our model's results for different

τ

values with the measured data, in the absence of skidding and delayed ionization.

FIG. 24.

View large Download slide

Comparisons of event 1 PIP data to particle tracer model's results with different values of $τ$ ⁠.

First, consider the particle trace model results for release 1 compared to the PIPs' data shown in Fig. 24. As $τ$ increases from 4 to 8 s, the modeled $n_{B a^{+}}$ maximum value and profile shape move closer to those observed by the PIPs, but the modeled density values are higher than were observed. Also, the modeled profile's values move closer to each other, decreasing the spread in the modeled density. As $τ$ is increased past 8 s, the spread in modeled density decreases, and the modeled density values approach the observed values. Examining the parameters for each $τ$ in Table VI, we see that $τ$ of 24 and 28 s produce models with the highest correlation coefficients. Additionally, $τ$ = 28 s gives the closest modeled peak density magnitude and time to that observed. Therefore, for event 1, $τ$ = 28 s results in the modeled density profile closest to that observed.

TABLE VI.

Comparisons of effects $τ$ and $t_{skid}$ have on key features of release 1's modeled profile and on the associated Pearson correlation coefficient. Note that, unless otherwise specified, $v_{exp}$ = 1.5 km/s, $τ$ = 28 s, $t_{skid}$ = 0 s, and $t_{delay}$ = 0 s. The bold text indicates the preferred choices (see text).

Value	Pearson coefficient	$t_{PIP, onset} - t_{trace, onset}$	$t_{PIP, peak} - t_{trace, peak}$	$\frac{n_{B a^{+}, trace} (t_{trace, peak})}{n_{B a^{+}, PIP} (t_{PIP, peak})}$
$τ$ = 4 s	0.32	0.40 s	1.82 s	4.57
$τ$ = 6 s	0.34	0.34 s	1.72 s	3.91
$τ$ = 8 s	0.39	0.18 s	1.88 s	3.09
$τ$ = 20 s	0.43	0.10 s	1.74 s	1.57
$τ$ = 24 s	0.49	0.00 s	1.48 s	1.23
$τ$ = 28 s	0.48	0.00 s	1.38 s	1.16
$t_{skid}$ = 0.0 s	0.48	0.00 s	1.38 s	1.16
$t_{skid}$ = 0.3 s	0.50	0.00 s	1.44 s	1.22

Value	Pearson coefficient	$t_{PIP, onset} - t_{trace, onset}$	$t_{PIP, peak} - t_{trace, peak}$	$\frac{n_{B a^{+}, trace} (t_{trace, peak})}{n_{B a^{+}, PIP} (t_{PIP, peak})}$
$τ$ = 4 s	0.32	0.40 s	1.82 s	4.57
$τ$ = 6 s	0.34	0.34 s	1.72 s	3.91
$τ$ = 8 s	0.39	0.18 s	1.88 s	3.09
$τ$ = 20 s	0.43	0.10 s	1.74 s	1.57
$τ$ = 24 s	0.49	0.00 s	1.48 s	1.23
$τ$ = 28 s	0.48	0.00 s	1.38 s	1.16
$t_{skid}$ = 0.0 s	0.48	0.00 s	1.38 s	1.16
$t_{skid}$ = 0.3 s	0.50	0.00 s	1.44 s	1.22

For completeness, we perform the same comparison of $τ$ 's effect on the event 2 modeled density profile with the same expansion velocity, without any skidding or delayed ionization. Note that we know from the neutral observations that release 2 had a larger barium yield than release 1, leading to our scaling of $N_{0}$ by three for release 2's ionization profile. As shown in Fig. 25 and in Table I, for all values of $τ$ ⁠, the modeled profiles' reach their maximum density values earlier than the observed profile does. However, as $τ$ is increased, the modeled profiles' peak density values and onset times grow closer to those of the observed profile. The choice of $τ$ = 28 s for event 2 is justified as it has one of the highest correlation coefficients in addition to the best onset time and closest peak density value compared to the other choices of $τ$ ⁠.

FIG. 25.

View large Download slide

Comparisons of event 2 PIP data to particle tracer model's results with different values of $τ$ ⁠.

4. Selection of mean expansion velocity for release 2

As there was no optical measurement of release 2's mean neutral expansion velocity, we must test the impact of the expansion velocity on the modeled density profiles. Note that all of the following model runs used $τ$ = 28 s and an ion yield three times that of release 1, justified by Fig. 13. The modeled profiles for different expansion velocities are shown in Fig. 26. As shown in these plots and in Table I, $v_{exp}$ of 1.0 km/s models the later period of the growth phase and the decay period the best; however, its onset is much later than that observed. In contrast, $v_{exp}$ = 1.5 km/s captures the onset time, but not the time of peak density nor the decay period. The profile with $v_{exp}$ = 1.2 km/s has an onset time closer to that observed than $v_{exp}$ = 1 km/s and a closer peak time to that observed than $v_{exp}$ = 1.5 km/s. As shown in Table I, $v_{exp}$ = 1.2 km/s has the closest peak density value to that observed out of all the $v_{exp}$ choices and a decent correlation coefficient. Thus, we used $v_{exp}$ = 1.2 km/s in the main body of this paper.

FIG. 26.

View large Download slide

Comparisons of event 2 PIP data to particle tracer model's results with different values of $v_{exp}$ ⁠.

Remaining parameter sensitivity discussions are contained in Sec. IV B of the main text.

REFERENCES

1.

Achilleos

,

N.

,

André

,

N.

,

Blanco-Cano

,

X.

,

Brandt

,

P. C.

,

Delamere

,

P. A.

, and

Winglee

,

R.

, “

1. Transport of mass, momentum and energy in planetary magnetodisc regions

,”

Space Sci. Rev.

187

,

229

–

299

(

2015

).

https://doi.org/10.1007/s11214-014-0086-y

Google Scholar

Crossref

2.

Barnes

,

N. P.

,

Delamere

,

P. A.

,

Hampton

,

D. L.

,

Lynch

,

K. A.

,

Moses

,

M.

,

Lessard

,

M.

,

Moser-Gauthier

,

C.

,

Pfaff

,

R.

,

Larsen

,

M.

, and

Otto

,

A.

, “

Kinetic-scale energy and momentum transport experiment (KiNET-X) contextualized with simulations

,” Phys. Plasmas (submitted).

3.

Bell

,

T. F.

,

Helliwell

,

R. A.

, and

Hudson

,

M. K.

, “

Lower hybrid waves excited through linear mode coupling and the heating of ions in the auroral and subauroral magnetosphere

,”

J. Geophys. Res.

96

,

11379

, https://doi.org/10.1029/91JA00568 (

1991

).

Google Scholar

Crossref

4.

Belongie

,

S.

, see https://mathworld.wolfram.com/RodriguesRotationFormula.html for “

Rodrigues' rotation formula. From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein

” (

2003

).

Google Scholar

5.

Chang

,

T.

,

Crew

,

G. B.

,

Hershkowitz

,

N.

,

Jasperse

,

J. R.

,

Retterer

,

J. M.

, and

Winningham

,

J. D.

, “

Transverse acceleration of oxygen ions by electromagnetic ion cyclotron resonance with broad band left-hand polarized waves

,”

Geophys. Res. Lett.

13

,

636

–

639

, https://doi.org/10.1029/GL013i007p00636 (

1986

).

Google Scholar

Crossref

6.

Delamere

,

P. A.

and

Bagenal

,

F.

, “

Magnetotail structure of the giant magnetospheres: Implications of the viscous interaction with the solar wind

,”

J. Geophys. Res. Space Phys.

118

,

7045

–

7053

, https://doi.org/10.1002/2013JA019179 (

2013

).

Google Scholar

Crossref

7.

Delamere

,

P. A.

,

Bagenal

,

F.

,

Paranicas

,

C.

,

Masters

,

A.

,

Radioti

,

A.

,

Bonfond

,

B.

,

Ray

,

L.

,

Jia

,

X.

,

Nichols

,

J.

, and

Arridge

,

C.

, “

Solar wind and internally driven dynamics: Influences on magnetodiscs and auroral responses

,”

Space Sci. Rev.

187

,

51

–

97

(

2015

).

https://doi.org/10.1007/s11214-014-0075-1

Google Scholar

Crossref

8.

Delamere

,

P. A.

,

Lynch

,

K.

,

Lessard

,

M.

,

Pfaff

,

R.

,

Larsen

,

M.

,

Hampton

,

D. L.

,

Conde

,

M.

,

Barnes

,

N. P.

,

Damiano

,

P. A.

,

Otto

,

A.

,

Moses

,

M.

, and

Moser-Gauthier

,

C.

, “

Alfvén wave generation and electron energization in the KiNET-X sounding rocket mission

,”

Phys. Plasmas

31

,

112108

(

2024

).

https://doi.org/10.1063/5.0228435

Google Scholar

Crossref

9.

Delamere

,

P. A.

,

Stenbaek-Nielsen

,

H. C.

, and

Otto

,

A.

, “

Reduction of momentum transfer rates by parallel electric fields: A two-fluid demonstration

,”

Phys. Plasmas

9

,

3130

–

3137

(

2002

).

https://doi.org/10.1063/1.1485772

Google Scholar

Crossref

10.

Delamere

,

P. A.

,

Stenbaek-Nielsen

,

H. C.

,

Swift

,

D. W.

, and

Otto

,

A.

, “

Momentum transfer in the combined release and radiation effects satellite plasma injection experiments: The role of parallel electric fields

,”

Phys. Plasmas

7

,

3771

–

3780

(

2000

).

https://doi.org/10.1063/1.1287741

Google Scholar

Crossref

11.

Emmert

,

J. T.

,

Drob

,

D. P.

,

Picone

,

J. M.

,

Siskind

,

D. E.

,

Jones

,

M.

,

Mlynczak

,

M. G.

,

Bernath

,

P. F.

,

Chu

,

X.

,

Doornbos

,

E.

,

Funke

,

B.

,

Goncharenko

,

L. P.

,

Hervig

,

M. E.

,

Schwartz

,

M. J.

,

Sheese

,

P. E.

,

Vargas

,

F.

,

Williams

,

B. P.

, and

Yuan

,

T.

, “

NRLMSIS 2.0: A whole‐atmosphere empirical model of temperature and neutral species densities

,”

Earth Space Sci.

8

,

e2020EA001321

(

2021

).

https://doi.org/10.1029/2020EA001321

Google Scholar

Crossref

12.

Erickson

,

P.

, see https://w3id.org/cedar?experiment_list=experiments/2021/mlh/16may21&file_list=mlh210516k.002.hdf5 for “

Data from the CEDAR madrigal database

” (

2021

).

Google Scholar

13.

Fraunberger

,

M.

,

Lynch

,

K. A.

,

Clayton

,

R.

,

Roberts

,

T. M.

,

Hysell

,

D.

,

Lessard

,

M.

,

Reimer

,

A.

, and

Varney

,

R.

, “

Auroral ionospheric plasma flow extraction using subsonic retarding potential analyzers

,”

Rev. Sci. Instrum.

91

,

094503

(

2020

).

https://doi.org/10.1063/1.5144498

Google Scholar

Crossref

PubMed

14.

Frederick‐Frost

,

K. M.

,

Lynch

,

K. A.

,

Kintner

,

P. M.

,

Klatt

,

E.

,

Lorentzen

,

D.

,

Moen

,

J.

,

Ogawa

,

Y.

, and

Widholm

,

M.

, “

SERSIO: Svalbard EISCAT rocket study of ion outflows

,”

J. Geophys. Res. Space Phys.

112

, A08307, https://doi.org/10.1029/2006JA011942 (

2007

).

Google Scholar

15.

Haerendel

,

G.

, “

Experiments with plasmas artificially injected into near-earth space

,”

Front. Astron. Space Sci.

6

,

29

(

2019

).

https://doi.org/10.3389/fspas.2019.00029

Google Scholar

Crossref

16.

Kelley

,

M. C.

,

The Earth's Ionosphere: Plasma Physics and Electrodynamics

(

Academic Press, Inc

.,

San Diego, CA

,

1989

).

Google Scholar

17.

Meurer

,

A.

,

Smith

,

C. P.

,

Paprocki

,

M.

,

Čertík

,

O.

,

Kirpichev

,

S. B.

,

Rocklin

,

M.

,

Kumar

,

A.

,

Ivanov

,

S.

,

Moore

,

J. K.

,

Singh

,

S.

,

Rathnayake

,

T.

,

Vig

,

S.

,

Granger

,

B. E.

,

Muller

,

R. P.

,

Bonazzi

,

F.

,

Gupta

,

H.

,

Vats

,

S.

,

Johansson

,

F.

,

Pedregosa

,

F.

,

Curry

,

M. J.

,

Terrel

,

A. R.

,

Roučka

,

v.

,

Saboo

,

A.

,

Fernando

,

I.

,

Kulal

,

S.

,

Cimrman

,

R.

, and

Scopatz

,

A.

, “

SymPy: Symbolic computing in python

,”

PeerJ Comput. Sci.

3

,

e103

(

2017

).

https://doi.org/10.7717/peerj-cs.103

Google Scholar

Crossref

18.

Milton

,

J. S.

and

Arnold

,

J. C.

,

Introduction to Probability and Statistics

, 4th ed. (

McGraw Hill

,

New York, NY

,

2003

).

Google Scholar

19.

Newville

,

M.

,

Otten

,

R.

,

Nelson

,

A.

,

Ingargiola

,

A.

,

Stensitzki

,

T.

,

Allan

,

D.

,

Fox

,

A.

,

Carter

,

F.

,

Michał

,

Osborn

,

R.

,

Pustakhod

,

D.

,

Lneuhaus

,

Weigand

,

S.

,

Glenn

,

Deil

,

C.

,

Mark

,

Hansen

,

A. L. R.

,

Pasquevich

,

G.

,

Foks

,

L.

,

Zobrist

,

N.

,

Frost

,

O.

,

Beelen

,

A.

,

Stuermer

,

Azelcer

,

Hannum

,

A.

,

Polloreno

,

A.

,

Nielsen

,

J. H.

,

Caldwell

,

S.

,

Almarza

,

A.

, and

Persaud

,

A.

, “

lmfit/lmfit-py: 1.0.3

,” Zenodo (

2021

). https://doi.org/10.5281/zenodo.5570790

Google Scholar

20.

Pfaff

,

R. F.

, “

In-situ measurement technique for ionospheric research

,” in

Modern Ionospheric Science: A Collection of Articles Published on the Occasion of the Anniversary 50 Years of Ionospheric Research in Lindau

, edited by

H.

Kohl

,

R. R.

Rüster

, and

K.

Schlegel

(

European Geophysical Society

,

Katlenburg-Lindau

,

1996

), pp.

459

–

551

.

Google Scholar

21.

Pfaff

,

R. F.

and

Marionni

,

P. A.

,

Multiple‐Baseline Spaced Receivers

, Geophysical Monograph Series (

American Geophysical Union

,

Washington, DC

,

1998

), Vol.

103

, pp.

161

–

167

.

Google Scholar

22.

Roberts

,

T. M.

,

Lynch

,

K. A.

,

Clayton

,

R. E.

,

Disbrow

,

M. E.

, and

Hansen

,

C. J.

, “

Magnetometer-based attitude determination for deployed spin-stabilized spacecraft

,”

J. Guid., Control, Dyn.

40

,

2941

–

2947

(

2017a

).

https://doi.org/10.2514/1.G002591

Google Scholar

Crossref

23.

Roberts

,

T. M.

,

Lynch

,

K. A.

,

Clayton

,

R. E.

,

Weiss

,

J.

, and

Hampton

,

D. L.

, “

A small spacecraft for multipoint measurement of ionospheric plasma

,”

Rev. Sci. Instrum.

88

,

073507

(

2017b

).

https://doi.org/10.1063/1.4992022

Google Scholar

Crossref

PubMed

24.

Siddiqui

,

M. U.

,

Gayetsky

,

L. E.

,

Mella

,

M. R.

,

Lynch

,

K. A.

, and

Lessard

,

M. R.

, “

A laboratory experiment to examine the effect of auroral beams on spacecraft charging in the ionosphere

,”

Phys. Plasmas

18

,

092905

(

2011

).

https://doi.org/10.1063/1.3640512

Google Scholar

Crossref

25.

Slater

,

J. C.

, “

Atomic radii in crystals

,”

J. Chem. Phys.

41

,

3199

–

3204

(

1964

).

https://doi.org/10.1063/1.1725697

Google Scholar

Crossref

26.

Tibbetts

,

B.

, private communication (May 4,

2024

).

27.

Virtanen

,

P.

,

Gommers

,

R.

,

Oliphant

,

T. E.

,

Haberland

,

M.

,

Reddy

,

T.

,

Cournapeau

,

D.

,

Burovski

,

E.

,

Peterson

,

P.

,

Weckesser

,

W.

,

Bright

,

J.

,

van der Walt

,

S. J.

,

Brett

,

M.

,

Wilson

,

J.

,

Millman

,

K. J.

,

Mayorov

,

N.

,

Nelson

,

A. R. J.

,

Jones

,

E.

,

Kern

,

R.

,

Larson

,

E.

,

Carey

,

C. J.

,

Polat

,

İ.

,

Feng

,

Y.

,

Moore

,

E. W.

,

VanderPlas

,

J.

,

Laxalde

,

D.

,

Perktold

,

J.

,

Cimrman

,

R.

,

Henriksen

,

I.

,

Quintero

,

E. A.

,

Harris

,

C. R.

,

Archibald

,

A. M.

,

Ribeiro

,

A. H.

,

Pedregosa

,

F.

,

van Mulbregt

,

P.

, and

SciPy 1.0 Contributors

, “

SciPy 1.0: Fundamental algorithms for scientific computing in python

,”

Nat. Methods

17

,

261

–

272

(

2020

).

https://doi.org/10.1038/s41592-019-0686-2

Google Scholar

Crossref

PubMed

© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License .

Single-point in situ measurements of thermal ions during the KiNET-X ionospheric sounding rocket mission Open Access

I. INTRODUCTION

II. KINET-X MISSION AND INSTRUMENTATION OVERVIEW

A. Mission design

B. Main payload and ground-based instruments of interest

1. Petite-Ion-Probes

2. Additional in situ instruments

3. Ground-based instruments

III. PIP DATA ANALYSIS METHOD: THERMAL ION FORWARD MODELING

A. PIP data analysis overview

B. PIP data analysis for a multi-species plasma

C. Results of multi-species analysis method

IV. OBSERVATIONS OF BARIUM RELEASES AND INTERPRETATION

A. PIP observations

B. Interpretation of spatial structuring of injected ions

1. Particle trace model overview

2. Model results without skidding or delayed ionization

3. Modifications to model for special cases: skidding and delayed ionization

4. Release 2 modeled with skidding

5. Release 2 modeled with ionization delay

6. Explanation of model onset time offset from data

C. Ion temperature wave-particle heating comparison

1. Lower hybrid wave heating

2. Ion cyclotron oscillation heating

3. Calculation results

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: PIP DATA ANALYSIS

1. PIP gains and offsets

APPENDIX B: SENSITIVITY STUDIES

1. Source of spread in O+ temperature from low gain PIPs

2. Methods for evaluating correlation between model results and PIP data

3. Selection of ionization profile for trace models

4. Selection of mean expansion velocity for release 2

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

Single-point in situ measurements of thermal ions during the KiNET-X ionospheric sounding rocket mission

1. Source of spread in $O^{+}$ temperature from low gain PIPs