TheMoonasaTestBodyforGeneralRelativity

AWhitePapertothePlanetaryScienceDecadalSurvey

CorrespondingAuthor

StephenM.Merkowitz,NASAGoddardSpaceFlightCenter

Stephen.M.Merkowitz@nasa.gov,(301)286­9412

Co­Authors

EdwardAaron,ITEInc.NeilAshby,UniversityofColorado

DavidCarrier,LunarGeotechnicalInstituteDouglasCurrie,UniversityofMaryland,CollegePark

JohnJ.Degnan,SigmaSpaceCorporationSimoneDell’Agnello,INFNLaboratoriNazionalidiFrascati

GiovanniDelleMonache,INFNLaboratoriNazionalidiFrascatiJanMcGarry,NASAGoddardSpaceFlightCenter

ThomasW.Murphy,Jr.,UniversityofCalifornia,SanDiegoKennethNordtvedt,NorthwestAnalysis

RobertD.Reasenberg,Harvard‐SmithsonianCenterforAstrophysicsIrwinI.Shapiro,Harvard‐SmithsonianCenterforAstrophysics

SlavaG.Turyshev,JetPropulsionLaboratoryJamesG.Williams,JetPropulsionLaboratory

ThomasZagwodzki,NASAGoddardSpaceFlightCenter

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Summary

Gravityistheforcethatholdstheuniversetogether, yet a theory that unifies it withother areas of physics still eludes us.Testing the very foundation ofgravitational theories, like Einstein’stheory of general relativity, is critical inunderstanding the nature of gravity andhow it relates to the rest of the physicalworld.

Lunar laser ranging (LLR) has been aworkhorse for testing general relativityover the past 40 years. The threeretroreflector arrays put on themoon bytheApolloastronautsandoneoftheSovietLuna arrays continueto be useful targets,and have providedthe most stringenttests of the StrongEquivalence Principleandthetimevariationof Newton’s gravita‐tional constant.However, groundstation technologyhas now reached a point where furtheradvances in range precision, andconsequently in the tests of generalrelativity, will be limited by errorsassociatedwiththelunararrays.

Significantadvancesinlunarlaserrangingwill require placing modernretroreflectorsand/oractivelaserrangingsystems at new locations on the lunarsurface.Rangingtonewlocationswillalsoenable better measurements of the lunarlibrations, aiding in our understanding ofthe interior structure of themoon. Moreaccurate range measurements will allowustostudyeffectsthataretoosmalltobeobserved by the current capabilities aswell as enabling more stringent andcrucialprobesofgravity.

More advanced retroreflectors are nowavailable. They have the potential toreducesomeoftheerrorsassociatedwithusingtheexistingarrays,resultinginmoreaccurate range measurements. Retro‐reflectors are extremely robust, do notrequirepower,andwilllastfordecadesastheApolloarrayshavedemonstrated.Thislongevity is important for studying long‐term effects such as a possible timevariation in the gravitational constant.Active laser ranging systems, such asasynchronouslasertransponders,arealsopotential options. They additionallyhaveapplicationsforrangingtoMarsandotherinterplanetary bodies with science goals

similar to those oflunarranging.The principalscientific products ofLLR fall into twocategories: gravita‐tional science andlunar science. In thiswhite paper, wediscuss how the next

generation of LLR addresses four keygravitational science questions. Inaddition, we discuss the current state ofretroreflector technology and describeways in which further advances can bemade in both retroreflector andtransponder technologies that wouldenable lighter and more accurate LLRinstruments. Lunar science associatedwith LLR is discussed in another whitepaper[1].

KeyScienceQuestions:

• IstheEquivalencePrincipleexact?• Doesthestrengthofgravityvary

withspaceandtime?• Doextradimensionsorothernew

physicsaltertheinversesquarelaw?• Whatisthenatureofspacetime?

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Introduction

Overthepast40years,lunarlaserranging(LLR) from a variety of observatories toretroreflector arrays placed on the lunarsurface by the Apollo astronauts and theSoviet Luna missions have dramaticallyand continually increased ourunderstanding of gravitational physicsalong with Earth and Moon geophysics,geodesy, and dynamics. We propose tobuild upon this legacy by starting a newactivitytoputadditionalLLRinstrumentson the moon for advanced gravitationaland lunar interior studies. At present,further technological improvements inthe ground­based segment of LLR arerendered futile by the 40­year­oldarraysonthemoon.

Installation of retroreflectors was a keypartoftheApollomissions,soitisnaturalto ask if future lunar missions shouldinclude them as well. We seek to placehardware on the lunar surface that willsustainLLRprogress fordecadestocomeandsupportfutureupgradestothegroundstations. A distributed array of state‐of‐the art solid retroreflectors is baselined,but we also discuss the possibility ofdeploying large hollow retroreflectors,laser transponders, and lasercommunication terminals. The proposedactive LLR instruments have the addedbenefitthattheycanbeadaptedforuseonMarsorotherplanetarybodiesbeyondthemoon.ProgressinLLRenabledscienceislimitedby both the properties of the currentretroreflector arrays and by theirdistribution on the lunar surface. Theavailable retroreflectors all lie within 26degrees latitude of the equator, and themost useful (Apollo) ones within 24degrees longitude of the sub‐earthmeridian as shown in Figure 1. This

clustering is sub‐optimal and weakenstheir geometrical strength. New retro‐reflectors placed at locations other thanthe Apollo sites would enable bettermeasurements of the lunar orientation,impacting the overall value of all rangedata. In addition, more advancedretroreflectorsarenowavailable thatwillreduce some of the measurement errorsassociatedwithusingtheApolloarrays[2].

Figure 1: Location of the lunarretroreflector arrays. The three Apolloarrays are labeled AP and the two Lunaarrays are labeled LUN. LUN1 isunavailable. ORI and SHK show thepotential location of two additional sitesthat would strengthen the geometriccoverage and increase the sensitivity tolunarorientationbyasmuchasafactoroffour[2].

Table 1 provides a comparative frame‐work for currentand futuresciencegoalsstemming from LLR. In the followingsectionswesummarize thecurrentstatusofansweringfourkeygravitationalsciencequestions and provide some theoreticalmotivation.

IstheEquivalencePrincipleexact?

The Equivalence Principle (EP), which isbasedon theequalityofgravitationalandinertial mass, is a cornerstone of generalrelativity, putting the theory on ageometric footing. The assumption that

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the EP holds in all its forms is built intogeneralrelativity,yeteffortstoformulateaquantumdescriptionofgravitygenericallyintroducenew scalar or vector fields thatviolate the EP [3,4]. Therefore, it isimperativethatwesubjectthisapparentlyexact empirical result to the greatestpossiblescrutiny.Two flavors of the EP may be tested viaLLR: the weak EP (WEP), and the strongEP (SEP) [5]. The WEP pertains to non‐gravitational contributions to mass:namely, Standard Model contributions ofnuclear and electromagnetic energy, plusquark masses and their kinetic energies.Nucleons of differing fractional electro‐weak andnuclear binding energiesmightexhibitdifferentcouplingstogravityinthecaseofaWEPviolation. TheSEPextendsthe WEP to include gravitational self‐energyofabody,addressingthequestionof how gravity pulls on itself and,therefore, accessing the non‐linear aspectof gravity. Laboratory masses lackmeasurable gravitational self‐energy, sobodies of astronomical sizemust be usedtoprovidesensitivitytoSEPviolation.LLR

providesthebestavailabletestoftheSEPtodate.

AviolationoftheEPwouldcausetheEarthandMoonto fallatdifferentrates towardthe Sun, resulting in a polarization of thelunarorbit.ThispolarizationshowsupinLLRasadisplacementalongtheEarth‐Sunline with a 29.53 day synodic period.Recent solutions using LLR data yield a WEP test numerically comparable with present laboratory limits, at a part in 1013 [6,7]. Combining the laboratory and LLR results together, the possibility of a conspiratorial cancellation is excluded, leaving a rigorous test of the SEP: Δ(MG/MI)SEP=(‐2.0±2.0)x10‐13[6]. Because Earth’s self-energy contributes 4.5×10−10 of its total mass, this translates to a SEP test of ~0.04%. Millimeter precision ranges to the moon should deliver order-of-magnitude improvements in EP tests [8].

Doesthestrengthofgravityvarywithspaceandtime?

Modern attempts to provide a quantum‐mechanical framework for gravity ofteninvoke extra dimensions that may becompactifiedatthePlanckscaleof10−35m.

Science Timescale Current(cm) 1mm 0.1mm

WeakEquivalencePrinciple Fewyears |Δa/a|<1.3×10−13 10‐14 10‐15

StrongEquivalencePrinciple Fewyears |η|<4.4×10−4 3×10‐5 3×10‐6

Time variation of G ~10years 9×10−13yr‐1 5×10‐14 5×10‐15

InverseSquareLaw ~10years |α|<3×10‐11 10‐12 10‐13

PPNβ Fewyears |β−1|<1.1×10‐4 10‐5 10‐6

Table 1: Current and future science deliverables from LLR. Aside from theWEP, LLR iscurrently the best test of each item. The timescale indicates the approximate datacampaign length for achieving the particular science goal. The 1 mm goals arestraightforwardtoassess, thoughthe0.1mmgoalsare lessrigorouslyestimatedandwilldepend on significant model development. The estimate for the Parameterized Post­Newtonian (PPN) parameter β derives from the SEPη = 4β−γ−3, with PPN parameter γ determinationprovidedbyothermeans.

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New scalar fields—for instancerepresentedbythedilatonandmodulithatregulate these dimensions—typicallyevolve over cosmological time scales andproduce secular variations in thefundamental “constants.” The same maybe said of a residual field left over frominflation, which might be responsible fortheapparentaccelerationoftheexpansionof the universe seen today. Thus thegravitationalconstant,G,thefine‐structureconstant,andtheelectron‐to‐protonmassratio may evolve with time or varyspatially.

LLR is sensitive to variations in G. AchangingGaffectsboththemonthlylunarorbit and the annual Earth‐Moon orbit.For the lunar orbit, changingG and tidal‐friction both change the semimajor axisand the orbital period, but with differentproportions. Solar perturbations on thelunarorbitarelarge.Secularchangeintheannual orbital period from changing Gaccumulates as an orbital longitudeperturbation evolving quadratically withtime, t. The t2 effect on the phase of thesolarperturbationsprovidesastronglimitwhen measured over decades. Currently,LLR provides the best limits:

/year [6], whichtranslates to less than 1% variation overthe 13.7 billion year age of the universe.Anticipated improvements in LLR dataquality, volume, and longevity willaccelerate our search for new physicsobserved via temporal variations in thefundamentalconstantsofnature.

Doextradimensionsorothernewphysicsaltertheinversesquarelaw?

Theinversesquarelaw(ISL)ofgravityhasbeen meaningfully tested over lengthscales spanning 20 orders of magnitude,eliminating Yukawa‐like couplings

competitive with the strength of gravityfrom10−4to1016meterlengthscales.Thedeepestprobeof the ISL is fromLLRatascaleof~108meters,whereanynewforcemustbeweakerthangravitybymorethanten orders‐of‐magnitude [9]. Short‐rangetests of the ISL have recently beenprompted by the energy scale of thecosmological acceleration, suggesting anew‐physicslengthscaleat~0.1mm[10].On cosmological scales, brane‐inspiredmodifications to gravity (such as DGPgravity) attempt to account for theapparentaccelerationoftheuniverse,andin so doing produce influences on thelunar orbitwithin a factor of about 10 ofcurrent measurements [11,12]. Finally,covariant versions of the “modifiedNewtonian dynamics” (MOND) paradigm,suchasTeVeS[13],maybesubjecttoclearfalsification by precision rangingmeasurements within the solar system.Current LLR capabilities fall short oftesting these ideas, but notinsurmountably. Improvedequipmentonboth theMoonandEarthcanconceivablycoveranothertwoordersofmagnitudeinthe coming decade opening up thepossibilityforamajordiscovery.

Whatisthenatureofspacetime?

The recent and unexpectedmeasurementof the accelerating expansion of theuniversehasprovidednewmotivationforexploringthenatureofspacetime.Modelsthatpredictmodificationofgravityatlargedistances, such as brane‐world models,have recently become of interest [14]. These theories exhibit a strong coupling phenomenon that makes the gravitationalforce source dependent. These theoriesbecome testable at shorter distanceswhere the coupling sets in for lightersources[15].

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TheEarth‐Moonsystemprovidesatestbedforinvestigatingthenatureofspacetimeatsolar‐system scales. For example, generalrelativitypredictsthatagyroscopemovingthrough curved spacetime will precesswith respect to a rest frame. This isreferred to as geodetic or de Sitterprecession. The Earth‐Moon systembehaves as a gyroscope with a predictedgeodetic precession of 19.2 msec/year.This is observed using LLR bymeasuringthelunarperigeeprecession. Thecurrentlimit on the deviation of the geodeticprocession from the general relativitypredictionis:Kgp=(‐1.9±6.4)x10‐3[6].Thismeasurement can also be used to set alimitonapossible cosmological constant:Λ<10‐26km‐2[16],whichhasimplicationsforourunderstandingofdarkenergy.

It is also useful to look at violations ofgeneral relativity in the context ofmetrictheories of gravity. Parameterized Post‐Newtonian (PPN) formalism provides aconvenient way to describe a class ofdeviations from general relativity. ThemostoftenconsideredPPNparametersareγ andβ: γ indicates howmuch spacetimecurvatureisproducedperunitmass,whileβ indicateshownonlineargravity is (self‐interaction).γandβareidenticallyoneingeneral relativity. Limits on γ can be setfrom geodetic precession measurements[17], but the best limits presently comefrom measurements of the gravitationaltime delay of light, i.e. the Shapiro effect.Doppler measurements to the Cassinispacecraftsetthecurrentlimitonγ:(γ–1)=(2.1±2.3)×10‐5[18].ThiscombinedwithLLRSEPresultsprovidesthebestlimitonβ:(β–1)=(1.2±1.1)×10‐4[6].Scalartensortheories with “attractors” for the cosmicbackground scalar‐field dynamics predicta residual γ‐1 and perhaps β‐1 of order10‐7‐10‐5 today [3], within reach of

advanced LLR and spacecraft time‐delaymeasurements.

NextGenerationLLR

Five retroreflector arrayswere placed onthe Moon in the period 1969–1973. USastronautsplaced threeduring theApollomissions(11,14,and15),andtwoFrencharrays were sent on Russian Lunokhodrovers.AlltheApolloarraysandthearrayon Lunokhod 2 are still viable targetstoday.

Figure 2: The Apollo 15 retroreflectorarray was made from 300 fused silicacubes. The physical size of the array isnowlimitingrangingmeasurements.

The first LLR measurements had aprecision of about 20 cm. Since 1969,several stations have successfully rangedto the lunar retroreflectors and haveincreasedtherangeaccuracybyafactorof10tothelevelofafewcentimeters. Poordetection rates have historically limitedLLR (not every laser pulse sent to theMoonresultsinadetectedreturnphoton).However, the relatively new APOLLOsystemusesthelargecollectingareaoftheApache Point telescope and has veryefficientavalanchephotodiodearrayssuchthat thousandsofdetectionsarerecorded(even multiple detections per pulse)leadingtoastatisticaluncertaintyofabout1 mm for timescales of less than 10minutes.

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The dominant random uncertainty perphoton received by modern LLR stationsstems from the size and changingorientation of the reflector array due tothe lunar librations and the associatedspreadofpulsereturntimes.Additionally,at the millimeter level of precision,systematic errors associated with lunararrays (such as the thermal expansion ofthe array support structure andunderlying regolith) start to becomesignificant. Progressingbeyond this levelof precision will require new lunarretroreflectors or laser transpondersdesigned to be thermally stable and toreduceoreliminateorientation‐dependentpulsespreading.

Figure 3: Improvements in the groundstation technologyover thepast 40 yearshave increased the range accuracy by 2orders of magnitude. Errors associatedwith the existing retroreflectors are nowbecomingalimitation.

Large single cube corners (>10cm) canpotentially be made to provide similarreturnratesas theApolloarrays,withoutsignificant pulse spreading. To furtherincrease the total response, several cubescould be deployed around a landing sitewith enough separation that responsesseenbytheEarthstationsdonotoverlap.

Solidcubecornerretroreflectors(upto11cm) have flown on over a hundredmissions, for both satellite and lunararrays. The proposed large cube pushes

thestateoftheart,butrecenttestsofa10cm cube have demonstrated it meetsrelevant requirements for the lunarenvironment. Designsforthehousingarestillindevelopmentanditsqualificationismakinggoodprogress[19].

Figure 4: A 10 cm solid cube cornerreflector was recently qualified for thelunar environment. Also shown forcomparisonisa3.8cmApolloengineeringmodelcubecorner.

Hollowcubecornersarealsoapromisingalternative to the traditional solid cubes.They potentiallyweigh less, have smallerthermal distortions, do not suffer fromthermal changes in their index ofrefraction,anddonotintroducesignificantpolarization effects. Thus, they can bemadelargerwithoutsacrificingasmuchinoptical performance. Hollow cubes haveflownonafewmissions,butaregenerallynot used on satellites for laser rangingbecause of a lack of test data and someindications of instabilities at hightemperatures. Advancesinadhesivesandothertechniquesforbondinghollowcubesmakes it worthwhile to further developthistechnologyforlunarapplications.

IsolationfromgroundmotionandthermalchangesarealsokeyforgoingbeyondtheApollo array capabilities. Each reflectorshould be rigidly grounded to directlysense lunar body motion and be locatedfar enough away from normal human

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activity to avoid vibration andcontamination (dust) from affecting thecubes. Advanced retroreflectors wouldalsobenefit frombeingthermallycoupledto the ground below the surface layer.Thiswouldrequiredrillingaholeaboutameter deep and inserting a thermallystable rod (high conductivity, lowcoefficient of thermal expansion). Theretroreflector would then be mounted tothe exposed end of the rod. A thermalblanket positioned over the lunar surfacearound and below the retroreflectorwouldalsobeofbenefit.

Activelasertranspondersareapromisingalternative topassiveretroreflectors [20].Active transpondersaredevices thatbothsend and receive predictable signals andcanbeusedforrangingandtimetransfer.Laser transponders have approximately aR2 linkadvantageoverdirectranging lossof 1/R4, essentially because the signal ispropagating in only one direction beforebeing regenerated. With thedevelopment and inclusion of lasercommunications for spaceflightmissions, it is logical to include anopticaltransponderthatusesthesameopto­mechanical infrastructure withminimal impact on the missionresources. These instruments could beused to support the proposed science inaddition to providing communicationssupport to the astronauts and/or otherscientific instruments. These lunarinstruments would also provide apathfinder for applications on Mars andotherplanetarybodies.While both the retroreflector andtransponder options would benefit fromfurther development, existing technologyis sufficient to support near‐termopportunities such as the InternationalLunarNetwork(ILN)[21].Puttingasinglelarge cube corner on the ILN would

immediately provide better geometriccoverage, thus increasing the overallmeasurement sensitivity to the lunarmotion.Thisopportunitycouldserveasapathfinder for more sophisticatedemplacement by the astronauts, enablingunprecedented tests of our fundamentalunderstandingofgravity.

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