Detection and Characterization of Micrometeoroids with LISA Pathfinder
aa r X i v : . [ a s t r o - ph . E P ] O c t Noname manuscript No. (will be inserted by the editor)
Detection and Measurement of Micrometeoroids withLISA Pathfinder
J. I. Thorpe · C. Parvini · J. M. Trigo-Rodr´ıguez
May 18, 2018
Abstract
The Solar System contains a population ofdust and small particles originating from asteroids, comets,and other bodies. These particles have been studied us-ing a number of techniques ranging from in-situ satellitedetectors to analysis of lunar microcraters to ground-based observations of zodiacal light. In this paper, wedescribe an approach for using the LISA Pathfinder(LPF) mission as an instrument to detect and char-acterize the dynamics of dust particles in the vicin-ity of Earth-Sun L1. Launching in late 2015, LPF isa dedicated technology demonstrator mission that willvalidate several key technologies for a future space-based gravitational-wave observatory. The primary sci-ence instrument aboard LPF is a precision accelerom-eter which we show will be capable of sensing discretemomentum impulses as small as 4 × − N · s. We thenestimate the rate of such impulses resulting from im-pacts of micrometeoroids based on standard modelsof the micrometeoroid environment in the inner solarsystem. We find that LPF may detect dozens to hun-dreds of individual events corresponding to impacts ofparticles with masses > − g during LPF’s roughlysix-month science operations phase in a 5 × km by8 × km Lissajous orbit around L1. In addition, we J. I. Thorpe · C. ParviniGravitational Astrophysics Laboratory, NASA GoddardSpace Flight Center, Greenbelt, MD 20771, USAE-mail:[email protected]. ParviniDepartment of Aerospace Engineering, The George Washing-ton University, Washington, DC 20052, USAJ. M. Trigo-Rodr´ıguezMeteorites, Minor Bodies and Planetary Sciences Group, In-stitute of Space Sciences (CSIC-IEEC), Campus UAB Bel-laterra, Carrer de Can Magrans, s/n 08193 Cerdanyola delVall´es (Barcelona), Spain. estimate the ability of LPF to characterize individualimpacts by measuring quantities such as total momen-tum transferred, direction of impact, and location ofimpact on the spacecraft. Information on flux and di-rection provided by LPF may provide insight as to thenature and origin of the individual impact and helpconstrain models of the interplanetary dust complex ingeneral. Additionally, this direct in-situ measurement ofmicrometeoroid impacts will be valuable to designers offuture spacecraft targeting the environment around L1.
Keywords
Meteoroids, micrometeoroids · dust PACS · · Our current understanding of the interplanetary dustcomplex is informed by a number of measurement tech-niques including photographic and visual meteors [8,9,18], radio meteors, atmospheric collections [6], ob-servation of zodiacal light in thermal [13] and visual[14] wavelengths, in-situ penetration and ionization de-tectors [19,20], and analysis of micro-craters in lunarsamples [1]. Additionally, NASA’s Stardust mission [10]and ESA’s Rosetta mission [16] have made in-situ mea-surements of the dust environments near the cometsWild 2 and 67/Churyumov-Gerasimenko respectively.The combined picture from all of these techniques, eachwith varying ability to detect and characterize varioussub-populations of the dust complex, is used to con-strain theoretical models that account for the sources,sinks, and dynamics of the dust complex.The LISA Pathfinder (LPF) mission [2], is a tech-nology demonstration mission dedicated to validatingseveral key technologies for a future space-based ob-servatory [3,12] of astrophysical gravitational waves [5,
J. I. Thorpe et al. × km by 8 × km Lissajous orbitaround the first Sun-Earth Lagrange Point (L1). LPFis currently planed to launch in December of 2015 withscience operations beginning in early 2016 and lastingfor approximately six months with a possible extensionof an additional six months. In this paper, we outlinean approach for using LPF as an instrument to detectand characterize interplanetary dust in the vicinity ofL1. This technique does not require any modificationto the hardware or operations of LPF and may providean important new source of information to constrainmodels of the dust complex.The outline of the paper is as follows. In section 2 wedescribe the working principle of the LPF instrumentand its application as a dust detection instrument. Insection 3 we make a simple estimate for the detectionthreshold of LPF as a function of transferred momen-tum using a simplified 1-D model. In section 4 we esti-mate the rates of events above this threshold and thetotal number of likely events during the LPF mission. Insection 5 we estimate the precision with which LPF willmeasure parameters of individual impacts such as mo-mentum transferred and direction of impact. In section6, we summarize our results and outline likely avenuesfor refining the estimates made here. Inside the LPF are two inertial sensor units each con-taining a test mass comprised of a 46 mm cube of Au-Ptalloy with a mass of 1 .
96 kg. During launch and cruise,these test masses are mechanically supported by a lockmechanism, which is retracted prior to beginning sci-ence operations leaving the test masses freely-fallinginside the inertial sensors with no mechanical contacts.Each inertial sensor unit measures the position and at-titude of its respective test mass relative to the LPFspacecraft in all six rigid-body kinematic degrees-of-freedom (DoFs) using a capacitive sensing system. Theinertial sensor unit can apply forces and torques to thetest mass via electrostatic actuation. Additional infor-mation is provided by a laser interferometer that mea-sures a reduced set of DoFs with higher precision anda star tracker which measures the three angular DoFsof the spacecraft relative to a background inertial refer-ence frame. A control system converts the position andattitude measurements from the two inertial sensors,the interferometer, and the star tracker into force andtorque commands that are applied to the test masses and the spacecraft, the latter being actuated using amicrothruster system.The main purpose of the control system is to re-duce external force disturbances on the test masses;low-disturbance reference masses are a key technologyfor LISA-like instruments. A byproduct of this effort isan exquisite measurement of external force disturbanceson the LPF spacecraft either through the residual mo-tion between the spacecraft and the test masses or inthe applied control forces and torques. One such sourceof external force disturbance is an impulse generated byan impact of a micrometeoroid or dust particle. In thefollowing sections, we estimate the detection thresholdand rate for such impact events.
The momentum transferred to the LPF spacecraft byan individual micrometeoroid impact will combine withstochastic disturbances on the spacecraft from a num-ber of sources such as Solar radiation pressure, out-gassing, and noise from the microthruster system. The measured momentum transfer will also be effected bynoise in the position sensing system and force distur-bances on the test masses. To simplify our estimate ofthe threshold at which LPF can detect and individualimpact, we consider a simplified one-dimensional modelin a linear DoF. We adopt the maximum-likelihood for-malism, where the signal-to-noise ratio (SNR) of anevent is defined using a noise-weighted inner-product: ρ = h F | F i , (1)where ρ is the SNR, F is the ‘waveform’ of the force onthe spacecraft resulting from the impact and h . . . | . . . i denotes the noise-weighted inner product defined as: h a | b i ≡ R Z + ∞ ˜ a ( f )˜ b ∗ ( f ) S n ( f ) df, (2)where tildes denote Fourier-domain signals and S n ( f )is the one-sided power-spectral density of the equiva-lent force noise on the spacecraft. Given a waveform F and an estimate of the noise characterized by S n ( f ),the SNR can be calculated using (2) and (1). An SNRthreshold for detection, typically set around 5-10, canbe used to determine whether an individual event islikely to be observed.While the error budget developed for LPF containsdozens of individual effects that contribute equivalentforce noise to the test masses and spacecraft, there aretwo effects that dominate force noise on the spacecraftin the measurement band 0 . ≤ f ≤
100 mHz.The first is the noise of the microthruster system itself, etection and Measurement of Micrometeoroids with LISA Pathfinder 3 which is characterized by a flat power spectral densitywith a level of S th ≈ − N / Hz , (3)The thruster noise dominates at low frequencies butis eclipsed above a few mHz by the equivalent forcenoise of the inertial sensor. The inertial sensor noise ischaracterized by a flat displacement amplitude spectraldensity at a level of ∼ / Hz / . This can be con-verted to an equivalent force noise on the spacecraftby multiplying by the spacecraft mass ( M = 422 kg)and taking two time derivatives. The resulting powerspectral density of this equivalent spacecraft force noisefrom the inertial sensor is given by S is ≈ h (2 × − m / Hz / ) · M · (2 πf ) i = 1 . × − · f N / Hz . (4)The individual and combined components of thespacecraft equivalent force noise are shown in Figure1. Fourier Frequency [ Hz] -4 -3 -2 -1 E q u i v a l e n t F o r ce N o i s e k g m s H z -18 -16 -14 -12 -10 ThrusterInertial SensorTotal
Fig. 1
Power spectral density of equivalent spacecraft forcenoise in a simplified one-dimensional model of LPF.
An impact by a micrometeoroid is modeled as animpulsive force occurring at time τ and imparting atotal momentum P over a characteristic timescale ∆ . This can be written in the time and frequency domainsrespectively as: F ( t ) = P∆ [ Θ ( t − τ ) − Θ ( t − τ − ∆ )] , (5)˜ F ( f ) = P sinc ( f ∆ ) e − πif ( τ + ∆/ . (6)where Θ ( t ) is the Heaviside step function, and sinc ( x ) ≡ sin ( πx ) /πx is the normalized sine cardinal function.One would generally expect that the characteristic im-pulse times for micrometeoroids would be small com-pared to the standard sampling rate for LPF data of1 Hz. In this limit,lim ∆ → ˜ F ( f ) ≈ P e − πifτ . (7)This provides us with a waveform parametrized bytwo parameters, total momentum transferred P andimpact time τ . Using the expression for ˜ F ( f ) in (7)and a generic total equivalent force noise of the form S n ( f ) ≡ S + S f , the formula for SNR in (1) can beused to determine the SNR as a function of P : ρ = P/P c ,P c ≡ √ π (cid:0) S S (cid:1) / (8)where P c is the characteristic threshold momentum. Forthe case described above of S = 10 − N / Hz and S =1 . × − N / Hz , the characteristic momentum P c ≈ . × − N · s. For an event to be detected with ρ ≥ P ≥ P c ≈ . × − N-s.
The population of dust and micrometeoroids in the in-ner solar system is derived primarily from the collisionalprocessing of asteroids and comets. The particles mak-ing up this population vary in mass, size, composition,and orbit and combine to form a dust complex witha complex morphology [17]. The most commonly-usedmodel of the population is the model of Gr¨un, et al.[7], which estimates the cumulative flux of micromete-oroids in the inner solar system as a superposition ofthree distinct populations: Φ ( m ) = (cid:0) . × m . + 15 . (cid:1) − . , − g < m < g ,Φ ( m ) = 1 . × − (cid:0) m + 10 m + 10 m (cid:1) − . , − g < m < − g ,Φ ( m ) = 1 . × − (cid:0) m + 10 m (cid:1) − . , − g < m < − g ,Φ ( m ) = 3 . × [ Φ ( m ) + Φ ( m ) + Φ ( m )] . (9) J. I. Thorpe et al.
The flux Φ ( m ) represents the total number of parti-cles with mass greater than m grams impacting a unitarea from a single hemisphere (2 π steradians) in a sin-gle year. Figure 2 contains a plot of this model. Muchof the literature on micrometeoroid flux addresses is-sues related to suppression or enhancement of variousregions in this power-law due to effects of the Earth andMoon, but for LPF’s orbit around Sun-Earth L1, theunmodified Gr¨un model is most appropriate. Particle Mass, m [g] -18 -14 -10 -6 -2 F l u x f o r m ≥ m y r · m · ( π s t r ) -8 -4 Φ Φ Φ Φ Fig. 2
Flux of micrometeoroids with mass m ≥ m in theinner Solar system from the Gr¨un, et al. [7] model. The dashedlines show the individual components and the solid line showsthe total flux as described in (9). For detection with LPF, the mass distribution of themicrometeoroid population is just part of the requiredinformation. The transferred momentum additionallydepends on the relative velocity between the microme-teoroid and the spacecraft. As a zeroth-order estimate,we assume that the characteristic velocity of impactsis equivalent to the orbital velocity of LPF around theSun, or roughly 30 km / s. Under the admittedly crudeassumption that all impacts occur with this velocity, wecan compute the rate of LPF detections as R ≈ AΦ (cid:18) P c ρ ¯ v (cid:19) , (10)where Φ ( m ) is the Gr¨un model flux from (9), P c isthe characteristic momentum from (8), ρ is the SNRthreshold for detection, ¯ v is the characteristic velocity,and A is the cross-sectional area of the spacecraft. For ρ = 8 and A = 3 m , (10) gives R ≈ . × yr − orabout 60 events over the 180-day baseline LPF sciencemission.To improve upon this rate estimate, we employedthe NASA Meteoroid Engineering Model (MEM) [15],which was developed to help spacecraft designers andmission planners assess micrometeoroid risk from im-pacts. To quantify such risk, it is crucial to know the total number of impacts, their mean velocity, and thesize of the biggest meteoroid that the spacecraft is likelyto encounter. The MEM takes as input a state vectordescribing the orbit of a spacecraft and returns the to-tal flux and velocity distribution of impacts for particleshaving a mass m ≥ − g. Because the MEM concen-trates on these higher-mass events, which are more rare,it is not by itself useful in determining the number of theevents likely to be detected by LPF. However, we canuse the MEM to provide an improved estimate of thedistribution of impact velocities under the assumptionthat this distribution is roughly constant as we extendto lower masses. Using a representative state vector forthe LPF science orbit provided by ESA, the MEM wasused to produce the velocity distribution in Figure 3.The MEM results are reasonably well fit by a normaldistribution with a mean velocity of ¯ v = 21 . / s anda standard deviation of σ v = 9 . / s. Velocity [ kms ] f r a c t i o n o f E v e n t s [ % ] MEM ResultsGaussian Fit
Fig. 3
Probability distribution of impact velocities for LPFmicrometeoroid collisions during science operations. His-togram in blue are estimates from the NASA Meteoroid En-gineering Model and a representative ephemeris for LPF. Thefit in red is a best-fit normal distribution.
Combining the mass flux from the Gr¨un model andthe velocity distribution from Figure 3 produces theimpact flux as a function of particle momentum in Fig-ure 4. Note that for a threshold momentum of 8 P c ≈ . × − N · s, the predicted event rate assuming a 3 m area is approximately 3 × events per year, about 2 . etection and Measurement of Micrometeoroids with LISA Pathfinder 5 Impact Momentum, P [N · s] -9 -8 -7 -6 -5 -4 -3 F l u x f o r P ≥ P y r · m · ( π s t r ) -2 -1 Fig. 4
Flux of LPF particle impacts with transferred mo-menta P ≥ P estimated using the Gr¨un model mass flux inFigure 2 and the fit to the MEM-derived velocity distributionin Figure 3. or cometary meteoroid showers that intersect LPF’s or-bit. A key feature of LPF as a micrometeoroid instrument isthe ability to characterize in addition to simply detect-ing micrometeoroid impacts. For example, each LPFtest mass will measure the transferred linear momen-tum from the impact in three orthogonal directions.The error of in the estimate of each of these momentumcomponents will be ∼ ρ − where ρ is the SNR definedin (8). Under the assumption that the error in eachmomentum component is independent, the two anglesdescribing the impact direction will be measured witherrors of roughly σ θ ∼ σ φ ∼ √ ρ − . (11)For example, an event with ρ = 10 will have typicalerrors in the impact angles of 0 .
17 rad ≈ ◦ . Knowl-edge of the direction of the impact combined with anephemeris for LPF will allow reconstruction of the im-pactor’s orbit, a key piece of information for distin-guishing different populations of micrometeoroids. Forexample, it may be possible to measure an excess ofimpacts coming from a particular direction that couldbe associated with a known comet. This would allow thevelocity of the impact to be inferred from the ephemeridesof LPF and the comet which would in turn allow an es-timate of the mass of each impacting particle from themeasured momentum transfer.LPF will also measure three components of the an-gular momentum transferred to the spacecraft. To es-timate the noise floor for angular degrees of freedom, we follow a similar procedure to the analysis in section3. The angular sensing noise of the LPF inertial sensorunits is characterized by a flat spectrum with an ampli-tude spectral density of ∼
200 nrad / Hz / . This can beconverted into an equivalent torque noise by multiply-ing by the spacecraft moment of inertia ∼
200 kg · m and taking two time derivatives. The resulting powerspectral density of the equivalent torque noise is S Nis ≈ h (2 × − rad / Hz / ) · I · (2 πf ) i = 2 . × − · f N · m / Hz . (12)The equivalent torque noise of the micropropulsionsystem can be estimated by multiplying the force noiseby a characteristic ‘lever-arm’ corresponding to the per-pendicular distance between the spacecraft center ofmass and a line along the thrust vector passing throughthe thruster mounting point. A rough estimate of thisvalue is 0 . S Nth ≈ . × − N · m / Hz . Figure 5 shows thepower spectral density of the total equivalent torquenoise for this simplified model. Fourier Frequency [ Hz] -4 -4 -3 -1 E q u i v a l e n t T o r q u e N o i s e k g m s H z -18 -16 -14 -12 -10 -8 -6 ThrusterInertial SensorTotal
Fig. 5
Power spectral density of equivalent spacecraft torquenoise in a simplified model of LPF.
The characteristic angular momentum for LPF canthen be computed using (8) with S = 2 . × − N · m / Hz and S = 2 . × − · f N · m / Hz . The resultis L c = 5 . × − N · m · s. The typical errors on impactlocation can then be estimated as σ r = ρ − L c P c ≈ ρ − . ρ = 10 event would have typical localization er-rors of ∼
16 cm. A more detailed reconstruction of theimpact location would include a mechanical model of
J. I. Thorpe et al. the spacecraft and might improve the impact localiza-tion somewhat. While the impact location is not asuseful a measurement for studying the micrometeoroidpopulation as the total momentum or impact direction,it may be of interest to LPF’s operations team.
The next logical step in this research effort is to refinethe estimates made here, taking advantage of the de-tailed information available about the LPF spacecraftand additional nuances of the known or modeled mi-crometeoroid environment. Examples of improvementsin the LPF model might include proper modeling ofthe noise and noise correlations in the inertial sensorand thruster subsystems, including the second inertialsensor as a quasi-independent sensor, and an improvedrate estimate including a geometrical model of the LPFspacecraft and its orientation during the science oper-ations.Modifications to the micrometeoroid flux model mightinclude recent radar data ( [11] and references therein)that indicate a bimodal distribution of velocities in whichthe fraction of high-velocity meteoroids peaking at 50 −
60 km/s. If such a high-velocity population exists in thevicinity of L1, it would extend LPF’s detection capabil-ity to lower mass particles and further boost the detec-tion rate. Similarly, micrometeoroids coming from high-eccentricity cometary orbits collide at higher velocitythan the asteroidal one, so the momentum transferredwill be higher and the threshold for mass detection willbe lower. We think that such capability to computethe flux from the cometary source and its possible fine-structure associated with the disintegration of periodiccomets may also be of interest.In addition to improving estimates of detection thresh-olds, event rates, and parameter estimation capabilities,an analysis pipeline could be developed that would de-tect and characterize events from a stream of LPF sci-ence data. Techniques that have already been developedfor the analysis of data from ground- and space-basedgravitational wave detectors would likely be applicableto this problem.
We have outlined a concept for using the LPF space-craft as an instrument for studying the micrometeoroidenvironment in the vicinity of Earth-Sun L1 and maderough estimates of the detection threshold, event rates,and errors with which physical parameters are likely tobe measured.These estimates suggest that LPF should be capable of detecting dozens to hundreds of individ-ual events during its operational lifetime and will mea-sure their individual momenta, impact directions, andimpact locations with interesting precision.LPF represents a unique and novel opportunity tomake in-situ measurements of the micrometeoroid en-vironment in the vicinity of L1, a region for which noin-situ measurements exist. Such measurements can beused to constrain models of the micrometeoroid pop-ulation in the inner solar system and will complementexisting measurements using other techniques. Our pro-posed method requires no modification to LPF’s hard-ware or operations, it simply requires a dedicated anal-ysis of the data that will be collected as part of normalscience operations.
Acknowledgements
The authors would like to acknowledge Ian Harrison forproviding the representative LPF ephemeris file usedto estimate the distribution of impact velocities in sec-tion 3. JMTR’s research was supported by the SpanishMinistry of Science and Innovation (project: AYA2011-26522). CP’s research was supported by the 2015 NASAGoddard Space Flight Center Summer Internship Pro-gram. Copyright (c) 2015 United States Governmentas represented by the Administrator of the NationalAeronautics and Space Administration. No copyrightis claimed in the United States under Title 17, U.S.Code. All other rights reserved.
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