a permanent asymmetric_dust_cloud_around_the_moon
TRANSCRIPT
LETTERdoi:10.1038/nature14479
A permanent, asymmetric dust cloud aroundthe MoonM. Horanyi1,2,3, J. R. Szalay1,2,3, S. Kempf1,2,3, J. Schmidt4, E. Grun1,3,5, R. Srama6 & Z. Sternovsky1,3,7
Interplanetary dust particles hit the surfaces of airless bodies in theSolar System, generating charged1 and neutral2 gas clouds, as wellas secondary ejecta dust particles3. Gravitationally bound ejectaclouds that form dust exospheres were recognized by in situ dustinstruments around the icy moons of Jupiter4 and Saturn5, buthave hitherto not been observed near bodies with refractory rego-lith surfaces. High-altitude Apollo 15 and 17 observations of a‘horizon glow’ indicated a putative population of high-densitysmall dust particles near the lunar terminators6,7, although laterorbital observations8,9 yielded upper limits on the abundance ofsuch particles that were a factor of about 104 lower than that neces-sary to produce the Apollo results. Here we report observations of apermanent, asymmetric dust cloud around the Moon, caused byimpacts of high-speed cometary dust particles on eccentric orbits,as opposed to particles of asteroidal origin following near-circularpaths striking the Moon at lower speeds. The density of the lunarejecta cloud increases during the annual meteor showers, especiallythe Geminids, because the lunar surface is exposed to the samestream of interplanetary dust particles. We expect all airless plan-etary objects to be immersed in similar tenuous clouds of dust.
The Lunar Atmosphere and Dust Environment Explorer (LADEE)mission was launched on 7 September 2013. After reaching the Moonin about 30 days, it continued with an instrument checkout period ofabout 40 days at an altitude of 220–260 km. LADEE began its approxi-mately 150 days of science observations at a typical altitude of 20–100km, following a near-equatorial retrograde orbit, with a characteristicorbital speed of 1.6 km s21 (ref. 10). The Lunar Dust Experiment(LDEX) began its measurements on 16 October 2013 and detected atotal of approximately 140,000 dust hits during about 80 days ofcumulative observation time out of 184 total days by the end of themission on 18 April 2014. LDEX was designed to explore the ejectacloud generated by sporadic interplanetary dust impacts, includingpossible intermittent density enhancements during meteoroidshowers, and to search for the putative regions with high densities of0.1-mm-scale dust particles above the terminators. The previousattempt to observe the lunar ejecta cloud by the Munich DustCounter on board the HITEN satellite orbiting the Moon (15February 1992 to 10 April 1993) did not succeed, owing to its distantorbit and low sensitivity11.
LDEX is an impact ionization dust detector (Methods subsection‘The LDEX instrument’). When pointed in the direction of motion ofthe spacecraft, LDEX recorded average impact rates of about 1 andabout 0.1 hits per minute of particles with impact charges of q $ 0.3and q $ 4 fC, corresponding to particles with radii of a> 0.3 mm anda> 0.7 mm, respectively (Fig. 1). Approximately once a week, LDEXobserved bursts of 10 to 50 particles in a single minute. Particlesdetected in a burst are most likely to originate from the same well-timed and well-positioned impact event that happened just minutesbefore their detection on the ground-track of LADEE. Several of the
yearly meteoroid showers generated sustained elevated levels of LDEXimpact rates, especially those where the majority of the incomingmeteoroids hit the lunar surface near the equatorial plane, greatlyenhancing the probability of LADEE crossing their ejecta plumes.The Geminids generated the strongest enhancement in impact ratesfor 61.5 days centred around 14 December 2013.
The distribution of the detected impact charges remained largelyindependent of altitude, and throughout the entire mission it closelyfollowed a power law: pq(q) / q2(1 1 a) (Fig. 2). This alone indicatesthat the initial mass distribution of the ejecta particles is, to a goodapproximation, independent of their initial speed and angular distri-butions (Methods subsection ‘Dust ejecta clouds’), and that the num-ber of ejecta particles generated on the surface per unit time withmass greater than m follows a power law: N1(.m) / m2a. TheLDEX measurements indicate a < 0.9, surprisingly close to the valueaG 5 0.8 suggested by the Galileo mission at the icy moons of Jupiter12
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Figure 1 | Impact rates throughout the mission. The daily running average ofimpacts per minute of particles that generated an impact charge of q $ 0.3 fC(radius a> 0.3 mm) and q $ 4 fC (radius a> 0.7 mm) recorded by LDEX. Theinitial systematic increase until 20 November 2013 is due to transitions from thehigh-altitude checkout to the subsequent science orbits. Four of the severalannual meteoroid showers generated elevated impact rates lasting several days.The labelled annual meteor showers are: the Northern Taurids (NTa); theGeminids (Gem); the Quadrantids (Qua); and the Omicron Centaurids (oCe).The observed enhancement peaking on 25 March 2014 (grey vertical line) doesnot coincide with any of the prominent showers. During the Leonids meteorshower around 17 November 2013, the instrument remained off due tooperational constraints. From counting statistics, we determine that the dailyaverage impact rate of particles generating a charge of at least 0.3 fC is 1.25 hitsper minute and, hence, the 1s relative error is about 2%, while for particlesgenerating an impact charge . 4 fC the average rate is 0.08 hits per minute and,hence, the 1s relative error is about 10%.
1Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80303, USA. 2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA. 3Institutefor Modeling Plasma, Atmospheres, and Cosmic Dust (IMPACT), University of Colorado, Boulder, Colorado 80303, USA. 4Astronomy and Space Physics, University of Oulu, FI-90014 Oulu, Finland.5Max-Planck-Institut fur Kernphysik,D-69117Heidelberg, Germany. 6Institut fur Raumfahrtsysteme,Universitat Stuttgart, Raumfahrtzentrum Baden Wurttemberg, 70569Stuttgart, Germany. 7AerospaceEngineering Sciences, University of Colorado, Boulder, Colorado 80309, USA.
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and to laboratory experimental results of ejecta production fromimpacts13. The derived ejecta size distribution also represents the sizedistribution of the smallest lunar fines (very small particles) on thesurface because most ejecta particles return to the Moon and comprisethe regolith itself, unless these small particles efficiently conglomerateon the lunar surface into larger particles.
The characteristic velocities of dust particles in the cloud are of theorder of hundreds of metres per second, which is small compared totypical spacecraft speeds of 1.6 km s21. Hence, with the knowledge ofthe spacecraft orbit and attitude, impact rates can be converted directlyinto particle densities as functions of time and position. This approachis expected to result in a relative error ,20%, on the basis of a com-plete ejecta cloud model12 (Extended Data Fig. 2). Both the derivedaverage number density as a function of height, and the initial speeddistribution match expectations only for altitudes h> 50 km(Extended Data Fig. 3). This indicates that, for the lunar surface, the
customary assumption of a simple power-law speed distribution with asingle sharp cut-off minimum speed u0 needs revision for speeds belowabout 400 m s21. At higher values the speed distribution follows asimple power law (Extended Data Table 1), as predicted by existingmodels12. An ejecta plume opening cone angle of y0 < 30u is consist-ent with our measurements, including those taken during the observedbursts of impacts. The average total mass of the dust ejecta cloud isestimated to be about 120 kg, approximately 0.5% of the neural gasatmosphere14.
We found that the density distribution is not spherically symmetricaround the Moon (Fig. 3), exhibiting a strong enhancement near themorning terminator between 5 and 7 h local time (LT), slightly cantedtowards the Sun from the direction of the motion of the Earth–Moonsystem about the Sun (6 LT). The observed anisotropy reflects thespatial and velocity distributions of the bombarding interplanetarydust particles (Extended Data Fig. 4) responsible for the generationof the ejecta clouds (Methods subsection ‘Dust production fromimpacts’). This observed anisotropy is in contrast to the roughly iso-tropic ejecta clouds engulfing the Galilean satellites, where the vastgravitational influence of Jupiter is efficiently randomizing the orbitalelements of the bombarding interplanetary dust particles15. The aniso-tropic ejecta production is consistent with existing models of the inter-planetary dust distributions near the Earth that combine in situ dustmeasurements, visible and infrared observations of the zodiacal cloud,as well as ground-based visual and radar observations of meteors in theatmosphere16,17. The dust production on the lunar surface is domi-nated by particles of cometary origin, as opposed to slower asteroidaldust particles, which follow near-circular orbits as they migratetowards the Sun, owing to Poynting–Robertson drag18. Meteoroidsthat are of asteroidal origin would be able to sustain only a muchweaker, more azimuthally symmetric ejecta cloud, contrary to LDEXobservations.
In addition to bombardment by interplanetary dust, the exposure ofairless surfaces to ultraviolet radiation and solar wind plasma flowcould result in the lofting of small dust particles, owing to electrostaticcharging and subsequent mobilization19. The charging processes areexpected to be most efficient over the terminators, where strong loca-lized electric fields could exist over the boundaries of lit and darkregions. High-altitude horizon glow observations near the lunar ter-minator suggested a population of grains, characteristically of radius0.1 mm, with a density of n < 104 m23 at an altitude of h 5 10 km,
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Figure 3 | Lunar dust density distribution. a, The top-down view of the dustdensity n(a> 0.3mm) projected onto the lunar equatorial plane. While pointednear the direction of the motion of the spacecraft, LDEX did not makemeasurements between 12 and 18 LT. White colouring indicates regions whereLADEE did not visit or was not set up for normal operations. b, The density as a
function of LT at three different altitude bins showing a persistent enhancementcanted towards the Sun away from the direction of the orbital motion of theEarth–Moon system. Error bars were calculated by propagating the
ffiffiffiffiNp
errorthrough the density calculation, where N is the number of detected dustimpacts.
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Figure 2 | Slope of the charge and mass distributions. The exponent ofthe power-law distributions of the impact charges pq(q) / q2(1 1 a) fitted toLDEX measurements as functions of altitude (15 km bins) and time (10 daybins). The colour indicates the value of the charge distribution index 2(1 1 a),and the size of a circle is inversely proportional to its absolute uncertainty(largest circle: 60.1; smallest circle 60.5). The inset shows the impact chargedistribution for all heights for the entire mission, resulting in a x2 minimizingfit21 of a 5 0.910 6 0.003.
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increasing towards the surface to n 5 5 3 105 m23. Follow-upobservations8,9 indicated a drastically lower upper-limit of the lofteddust densities. At an altitude of 10 km, our dust current measurementsshow an upper limit for the density of particles of radius 0.1 mm that isapproximately two orders of magnitude below the Apollo estimates20.However, the LDEX dust current measurements (Methods subsection‘The LDEX instrument’) of Jdust < 105 electrons per second (Fig. 4)remained independent of altitude and, hence, gave no indication ofthe relatively dense cloud of 0.1-mm-sized dust that was inferred fromthe Apollo observations over the lunar terminators.
Online Content Methods, along with any additional Extended Data display itemsandSourceData, are available in the online version of the paper; references uniqueto these sections appear only in the online paper.
Received 7 October 2014; accepted 15 April 2015.
1. Auer, S. & Sitte, K. Detection technique for micrometeoroids using impactionization. Earth Planet. Sci. Lett. 4, 178–183 (1968).
2. Collette, A., Sternovsky, Z. & Horanyi, M. Production of neutral gas bymicrometeoroid impacts. Icarus 227, 89–93 (2014).
3. Hartmann, W. K. Impact experiments: 1. Ejecta velocity distributions and relatedresults from regolith targets. Icarus 63, 69–98 (1985).
4. Kruger,H., Krivov,A.,Hamilton,D.&Grun, E.Detectionofan impact-generateddustcloud around ganymede. Nature 399, 558–560 (1999).
5. Spahn, F. et al. Cassini dust measurements at Enceladus and implications for theorigin of the E ring. Science 311, 1416–1418 (2006).
6. McCoy, J. E. Photometric studies of light scattering above the lunar terminatorfrom Apollo solar corona photography. Proc. Lunar Sci. Conf. 7, 1087–1112(1976).
7. Zook, H. A. & McCoy, J. E. Large scale lunar horizon glow and a high altitude lunardust exosphere. Geophys. Res. Lett. 18, 2117–2120 (1991).
8. Glenar, D. A., Stubbs, T. J., Hahn, J. M. & Wang, Y. Search for a high-altitude lunardust exosphere using Clementine navigational star tracker measurements. J.Geophys. Res. Planets 119, 2548–2567 (2014).
9. Feldman, P. D. et al. Upper limits for a lunar dust exosphere from far-ultravioletspectroscopy by LRO/LAMP. Icarus 233, 106–113 (2014).
10. Elphic, R. C. et al. The Lunar Atmosphere and Dust Environment Explorer mission.Space Sci. Rev. 185, 3–25 (2014).
11. Iglseder, H., Uesugi, K. & Svedhem, H. Cosmic dust measurements in lunar orbit.Adv. Space Res. 17, 177–182 (1996).
12. Krivov, A. V., Sremcevic, M., Spahn, F., Dikarev, V. V. & Kholshevnikov, K. V. Impact-generated dust clouds around planetary satellites: spherically symmetric case.Planet. Space Sci. 51, 251–269 (2003).
13. Buhl, E., Sommer, F., Poelchau, M. H., Dresen, G. & Kenkmann, T. Ejecta fromexperimental impact craters: particle size distribution and fragmentation energy.Icarus 237, 131–142 (2014).
14. Stern, S. A. The lunar atmosphere: history, status, current problems, and context.Rev. Geophys. 37, 453–492 (1999).
15. Sremcevic, M., Krivov, A. V., Kruger, H. & Spahn, F. Impact-generated dust cloudsaround planetary satellites: model versus Galileo data. Planet. Space Sci. 53,625–641 (2005).
16. McNamara, H. et al. Meteoroid Engineering Model (MEM): a meteoroid model forthe inner Solar System. Earth Moon Planets 95, 123–139 (2004).
17. Dikarev, V. et al. The new ESA meteoroid model. Adv. Space Res. 35, 1282–1289(2005).
18. Nesvorny, D. et al. Dynamical model for the zodiacal cloud and sporadic meteors.Astrophys. J. 743, 129 (2011).
19. Rennilson, J. J. & Criswell, D. R. Surveyor observations of lunar horizon-glow. Moon10, 121–142 (1974).
20. Glenar, D. A., Stubbs, T. J., McCoy, J. E. & Vondrak, R. R. A reanalysis of the Apollolight scattering observations, and implications for lunar exospheric dust. Planet.Space Sci. 59, 1695–1707 (2011).
21. Markwardt, C. B. in Astronomical Data Analysis Software and Systems XVIII (edsBohlender, D. A., Durand, D. & Dowler, P.) ASP Conf. Ser. 411, 251 (2009); preprintat http://arxiv.org/abs/0902.2850.
Acknowledgements The LADEE/LDEX project was supported by NASA. Tests andcalibrations were done at the dust accelerator facility of the University of Colorado,supported by NASA’s Solar System Exploration Research Virtual Institute (SSERVI). Weare grateful for engineering and technical support fromthe Laboratory for Atmosphericand Space Physics (LASP), especially from M. Lankton (project manager), S. Gagnardand D. Gathright (mission operations), and D. James (calibration).
Author Contributions M.H. was the instrument principal investigator, directed the dataanalysis, and was primarily responsible for writing this paper. J.R.S. developed the dataanalysis software. S.K. was responsible for the calibration of the instrument andcontributed to the data analysis. J.S. led the modelling effort. E.G. and R.S. contributedto the analysis and interpretation of the data. Z.S. designed the instrument andcontributed to the data analysis.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to M.H. ([email protected]).
Lunar sunrise terminator
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rren
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lectr
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nd
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Apollo (1976, ref. 6)
Apollo (2011, ref. 20)
Clementine (2014, ref. 8)
LRO (2014, ref. 9)
LDEX
Figure 4 | LDEX current measurements. The accumulated charge collectedby LDEX in dt 5 0.1 s intervals (Jdust), averaged over the sunrise terminatorbetween 5:30 and 6:30 LT. The coloured lines show the predicted value of Jdust
based on the impacts of small particles alone using the upper limits of the dustdensities estimated by remote sensing visible6,8,20 and ultraviolet9 observations.Error bars show 1s on logJdust as the current measurements are log-normallydistributed. Because Jdust < 105 electrons per second is two orders of magnitudelower than the Apollo estimates near an altitude of 10 km and exhibits noaltitude dependence, LDEX measurements show no evidence for the existenceof the suggested relatively dense clouds of 0.1-mm-sized dust particles. LRO isthe Lunar Reconnaissance Orbiter.
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METHODSThe LDEX instrument. The Lunar Dust Experiment (LDEX) is an impact ion-ization dust detector, which measures both the positive and negative charges of theplasma cloud generated when a dust particle strikes its target22. The amplitude andshape of the waveforms (signal versus time) recorded from each impact are used toestimate the mass of the dust particles. The instrument has a total sensitive area of0.01 m2, which gradually decreases to zero for particles arriving from outside itsdust field-of-view of 668u off from the normal direction. LDEX is sensitive toultraviolet; hence, its operations, in general, excluded the Sun in its optical field-of-view of 690u.
Particles below the single impact detection threshold generate a plasma cloud inthe same way as larger impacts, but without triggering a full waveform capture.Their collective signal is integrated independently of the impacts of large particles.In addition to small dust particles, the integrated ion signal is also sensitive to anumber of possible background contributions, most importantly ultravioletphotons scattering into the ion detector, generating photoelectrons. To identifythe background contributions in the collective signal, the acceleration potentialbetween the target and ion collector is intermittently switched from its nominal2200 V to 130 V, making the instrument ‘blind’ to dust. The contributors to thecurrent in nominal mode (JN) are ions from dust impacts, photoelectrons andlow- and high-energy ions. In switched mode (JS) the contributors are photo-electrons and high-energy ions. High energy in this case indicates .30 eV, asthese ions can reach the microchannel plate even in switched mode. Hence,the difference JN 2 JS represents the collective signal of small dust particlesand low-energy ions only. Each dust particle with a radius of 0.1 mm and impactspeed of 1.6 km s21 is expected to generate23 Qi < 100e. Their collective current isJdust < AvnQi, where A is the detector sensitive area, v is the speed of dust particlerelative to the spacecraft, and n is the density of the small particles. Hence, attrib-uting JN 2 JS to small dust particles alone allows us to set an upper limit for n.Alternatively, estimates for n from independent observations can be used topredict Jdust, which we can then compare to our measurements. The low-energyion contribution may be due to back-scattered solar wind protons24, energeticneutral atoms25,26 and the lunar ionosphere27. Any contribution of low-energy ionsto JN 2 JS would further reduce our estimate of n.Dust ejecta clouds. We compare the steady state, spherically symmetric model ofa dust cloud12 to the LT-averaged LDEX observations. The phase space density ofdust above the surface based on a model for an impact-generated ejecta cloud canbe written as12,28
n(v,h,w; r)~Nz
8p2Rrfu(u(v))fy(y(v,h))
vu(v)2 sin h cos y(v,h)ð1Þ
Here, the variables v, h, w denote the velocity vector of dust grains at a radialdistance r from the Moon (lunar radius R 5 1,737 km). The distance r is regardedas a parameter of the distribution, not a variable. Further, h is the angle between thevelocity vector and the radial direction and w is the velocity azimuth angle (anti-clockwise around the radius vector). The distribution does not depend explicitlyon w for a spherically symmetric cloud. We retain the azimuthal dependence toperform averages over quantities that do depend on w. N1 is the total rate ofgrains produced on the surface. We denote by u the starting speed of ejecta onthe surface and by y the ejection cone angle measured from the surface normal.Using the conservation of energy and angular momentum of the two-body prob-lem we have y 5 y(v, h) and u 5 u(v). For the distribution of starting velocities, fu,we use a power law with exponent m (equivalent to c 1 1 in other customarynotation12), normalized to unity in the range u[½u0,?�:
fu(u)~m{1
u1{m0
u{m
For the ejecta cone angles, we use a uniform distribution, fy, normalized to unity inthe range y[½0,y0�:
fy(y)~sin y
1{ cos y0
The mass distribution of the grains is uncorrelated with the velocity and isdescribed as a power law that is normalized to the total rate of mass productionin the range m[½mmin,mmax� (refs 12 and 18). The generalized version of equation(1) becomes
n(m,v,h,w; r)~Mz
8p2Rr1{a
m1{amax {m1{a
minm{(1za) fu(u(v))fy(y(v,h))
vu(v)2 sin h cos y(v,h)ð2Þ
where M1 denotes the total mass production rate that is related to N1 as
Nz~Mz 1{a
a
m{amin{m{a
max
m1{amax {m1{a
min
The exponent is expected12,30 to be in the range 0.5 # a # 1. Equation (2) gives thenumber of particles found at distance r from the centre of the moon in the phasespace volume element d3vd3rdm.
When a grain of mass m hits the dust detector at a velocity v it produces animpact charge29
q~cmvb ð3ÞThe combination of equations (2) and (3) can be used to estimate the distributionof charges to be recorded by LDEX in the following manner.
Typically the LADEE spacecraft followed a nearly circular orbit around themoon with speed vsc relative to the surface. The boresight of LDEX pointed inthe direction of spacecraft motion (apex) so that the detector encountered dustgrains of velocity v at a relative velocity vsc 2 v. The number of grains DN withvelocity in d3v and mass in dm, that can reach the detector during time Dt, is givenby the number of such grains found in a cylindrical volume spanned by thedetector surface A and the relative velocity vector (Extended Data Fig. 1)
DN~Dtd3vdmA(v) cos vHH( cos v)n(m,v,h,w; r) v{vscj j ð4Þwhere v is the angle between the boresight and the relative velocity vector. TheHeaviside function, HH(cosv), guarantees that we count only grains that can enterthe detector. With A(v) we account for the fact that the effective detector area ofLDEX depends on the angle v. The effective area is maximal for v 5 0, dropping tozero for v 5 68 degrees (ref. 22). We evaluate cosv in terms of the spacecraft anddust velocities, as well as the angles w, h by noting that for a circular spacecraft orbit
cos v~vsc{v sin h cos w
v{vscj j ð5Þ
Dividing (4) by Dt we obtain the differential rate dc of particles that impact thedetector
dc~dvv2dhdw sin hdmA(v)(vsc{v sin h cos w)HH(vsc{v sin h cos w)
n(m,v,h,w; r)ð6Þ
where A(v) is expressed with equation (5) as a function of v, h, w. We re-arrangethe right-hand side of (6) into a mass and a velocity distribution as
dc~dvdhdwdmpm(m)p(v,h,w; r,y0,m,u0)
where
pm(m)~Mz 1{a
m1{amax {m1{a
minm{(1za)HH(m{mmin)HH(mmax{m) ð7Þ
and
p(v,h,w; r,y0,m,u0)~A(v)
8p2Rr(vsc{v sin h cos w)
HH(vsc{v sin h cos w)vfu(u(v))fy(y(v,h))
u(v)2 cos y(v,h)
We define
pv(v)~
ðp
0
dh
ð2p
0
dwp(v,h,w; r,y0,m,u0)
Using equation (3) we can then p p express the model prediction for the distri-bution of charges detected by LDEX as
pq(q)~
ðdcd(q{mcvb)~
ð?
0
dmpm(m)
ð?
0
dvpv(v)d(q{mcvb)
~
ð?
0
dvpv(v)
cvbpm
qcvb
� �
Inserting equation (7) and sorting terms gives
pq(q)~Mz
mmax
1{a
1{ mminmmax
� �1{a
1q
cmmax
q
� �a ðvmax(q)
vmin(q)
dvpv(v)vab ð8Þ
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where the boundaries
vmin(q)~q
cmmax
� �1=b
, vmax~q
cmmin
� �1=b
follow from the normalization of the mass distribution, equation (7).Evaluating pv(v) shows that the integral in equation (8) depends only weakly on
the charge q via the boundaries. Quantitatively,
0:13v
Ðvmax(q)
vmin(q)
dvpv(v)vab
Ð?0
dvpv(v)vab
v0:23
when changing q from 0.1 fC to 1,000 fC. Therefore, the distribution pq(q) isdominated by far by the power law, so that we expect to see
pq(q)!q{(1za)
in the data. Hence, measuring the exponent of the impact charge distributionyields the exponent of the mass distribution of the particles recorded by LDEX.Dust production from impacts. Extended Data Fig. 3 and Extended Data Table 1were generated assuming that the LT-averaged ejecta cloud is spherically symmetricto determine the bulk properties of the cloud and allow for direct comparison withprevious studies12,30. Here we address the anisotropic nature of the dust influx to thelunar surface. To this end we replace the single global dust mass production M1 withan LT- and time (t)-dependent function of mass production per unit surface areaM1(LT, t).
The mass flux of bombarding interplanetary dust particles with mass m andvelocity v is a function of both the position on the lunar surface and time: F(m, v,LT, t). A single dust particle striking a pure silica surface generates a large number ofejecta particles with a total mass m1 5 mY(m, v), where the yield, Y , m0.2v2.5, isdetermined on the basis of laboratory experiments31. The mass flux of impactors isdominated by particles with characteristic mass m0 < 1028 kg (about 100 mm inradius)32. Our detected impacts are dominated by ejecta particles generated along theground track of the spacecraft that followed a nearly equatorial orbit. Hence, it isconvenient to track the position on the lunar surface in LT, with LT 5 0, 6, 12, 18marking midnight, the dawn terminator, the sub-solar point and the dusk termin-ator, respectively. The mass production rate per surface area as function of LT isfound by integrating the product of the interplanetary dust flux F and the yield Yaround the lunar equator
Mz(LT,t)~ðð
F(m,v,LT,t)Y(m,v)dmdv ð9Þ
We evaluated equation (9) using both NASA’s MEM16 and ESA’s IMEM17 models,which agree well near one astronomical unit. The flux F and the predicted massproduction rates M1(LT, t) are shown in Extended Data Fig. 4, and are consistentwith the asymmetric dust ejecta cloud observed by LDEX. We note that themeteoroid population still remains one of the most uncertain space environmentcomponents33.Data availability. All LDEX data are available through NASA’s Planetary DataSystem (http://sbn.psi.edu/pds/resource/ldex.html).Code availability. The code used for calculating the flux of interplanetary dustparticles reaching the lunar surfaces is available upon request from NASA (http://www.nasa.gov/offices/meo/software).
22. Horanyi, M. et al. The Lunar Dust Experiment (LDEX) onboard the LunarAtmosphere and Dust EnvironmentExplorer (LADEE) mission. SpaceSci. Rev.185,93–113 (2014).
23. Dietzel, H. et al. The HEOS 2 and HELIOS micrometeoroid experiments. J. Phys. E 6,209–217 (1973).
24. Saito, Y. et al. Solar wind proton reflection at the lunar surface: low energy ionmeasurement by MAP-PACE onboard SELENE (KAGUYA). Geophys. Res. Lett. 35,L24205 (2008).
25. Saul, L. et al. Solar wind reflection from the lunar surface: the view from far andnear. Planet. Space Sci. 84, 1–4 (2013).
26. Allegrini, F. et al. Lunar energetic neutral atom (ENA) spectra measured by theinterstellar boundary explorer (IBEX). Planet. Space Sci. 85, 232–242 (2013).
27. Poppe, A. R., Halekas, J. S., Szalay, J. R., Horanyi, M. & Delory, G. T. Model-datacomparisons of LADEE/LDEX observations of low-energy lunar dayside ions. LunarPlanet. Sci. Conf. Abstr. 45, 1393 (2014).
28. Sremcevic, M., Krivov, A. V. & Spahn, F. Impact-generated dust clouds aroundplanetary satellites: asymmetry effects. Planet. Space Sci. 51, 455–471 (2003).
29. Auer, S. in Interplanetary Dust (eds Grun, E., Gustafson, B., Dermott, S. & Fechtig., H.)387–438 (Springer, 2001).
30. Kruger, H., Krivov, A. V. & Grun, E. A dust cloud of Ganymede maintained byhypervelocity impacts of interplanetary micrometeoroids. Planet. Space Sci. 48,1457–1471 (2000).
31. Koschny, D. & Grun, E. Impacts into ice-silicate mixtures: ejecta mass and sizedistributions. Icarus 154, 402–411 (2001).
32. Grun, E., Zook, H. A., Fechtig, H. & Giese, R. H. Collisional balance of the meteoriticcomplex. Icarus 62, 244–272 (1985).
33. Drolshagen, G. Comparison of Meteoroid Models. IADC Report No. 24.1 (Inter-Agency Space Debris Coordination Committee, 2009).
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Extended Data Figure 1 | Detection geometry. A particle of velocity v isrecorded by a detector of sensitive area A. The surface normal of the detectorarea points along the velocity vector of the spacecraft vsc. The particle enters theinstrument with an angle v measured between the instrument boresight andthe relative velocity vector of the particle vsc 2 v.
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Extended Data Figure 2 | Systematic approximation error and itsdependence on ejection parameters. a, The calculated density for a standardset of parameters listed in Extended Data Table 1 for the model ejecta cloud12 asfunction of altitude (black line) normalized to the production rate N1. Thedensity is recalculated using n 5 c/(Avsc) (red line), the approach taken in thispaper to infer the dust density from the measured impact rates c, indicating an
underestimate of ,20% for altitudes below 100 km. b, Contour plot of theratio of the ‘true’ model density over the recalculated density at the altitudeh 5 50 km, as a function of the opening cone angle of the ejecta plume y0 andthe exponent of the power-law initial-speed distribution m, appropriatelysetting the minimum speed u0, while keeping the maximum speed constant at2vescape, maintaining a constant total kinetic energy of the ejecta particles.
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Extended Data Figure 3 | Comparison of observed and modelled cloudproperties. a, The dust density n(h) of the lunar ejecta cloud as function ofaltitude and size (colour scale). The continuous black line shows the modelprediction12 using the best-fit parameters listed in Extended Data Table 1.b, The cumulative dust mass in the lunar exosphere. The continuous blue lineshows the ejecta model prediction (Extended Data Table 1). c, The initial
normalized vertical velocity distribution f(u) calculated from n(h) using energyconservation. The continuous line shows f(u) / u23.4 6 0.1 matched to the dataat u $ 400 m s21 (altitude h < 50 km). Error bars were calculated bypropagating the
ffiffiffiffiNp
error through the various calculations, where N is thenumber of detected dust impacts.
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Extended Data Figure 4 | Modelled flux and mass production in the lunarequatorial plane. a, The calculated flux of interplanetary dust particles Fimp
reaching the lunar equatorial region as a function of LT and t (colour coded for
monthly averages). b, The mass production rate, equation (9), calculated usinga model for the spatial and velocity distributions of interplanetary dust particlesnear the Earth16, consistent with the observed asymmetric dust cloud.
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Extended Data Table 1 | Parameters of the theoretical ejecta cloud model12 for the Moon
These parameters form a consistent set, and are not independent of each other30.
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