Generalizing MOND to explain the missing mass in galaxy clusters

1.2 MB·

by  Alistair Hodson

Astronomy&Astrophysicsmanuscriptno.29358_Revised_19_Dec_2016 (cid:13)cESO2017
January13,2017
Generalizing MOND to explain the missing mass in galaxy clusters
AlistairO.Hodson1 andHongshengZhao1
1SUPA,SchoolofPhysicsandAstronomy,UniversityofStAndrews,Scotland
e-mail:[email protected]
Received,;accepted,
ABSTRACT
Context. ModifiedNewtonianDynamics(MOND)isagravitationalframeworkdesignedtoexplaintheastronomicalobservations
7
intheUniversewithouttheinclusionofparticledarkmatter.ModifiedNewtonianDynamics,initscurrentform,cannotexplainthe
1
missingmassingalaxyclusterswithouttheinclusionofsomeextramass,beitintheformofneutrinosornon-luminousbaryonic
0
matter. We investigate whether the MOND framework can be generalized to account for the missing mass in galaxy clusters by
2
boostinggravityinhighgravitationalpotentialregions.WeexamineandreviewExtendedMOND(EMOND),whichwasdesignedto

n increasetheMONDscaleaccelerationinhighpotentialregions,therebyboostingthegravityinclusters.
a Aims.WeseektoinvestigategalaxyclustermassprofilesinthecontextofMONDwiththeprimaryaimatexplainingthemissing
J massproblemfullywithouttheneedfordarkmatter.
2 Methods.Usingtheassumptionthattheclustersareinhydrostaticequilibrium,wecancomputethedynamicalmassofeachcluster
1 andcomparetheresulttothepredictedmassoftheEMONDformalism.
Results.WefindthatEMONDhassomesuccessinfittingsomeclusters,butoverallhasissueswhentryingtoexplainthemassdeficit
] fully.Wealsoinvestigateanempiricalrelationtosolvetheclusterproblem,whichisfoundbyanalysingtheclusterdataandisbased
O ontheMONDparadigm.Wediscussthelimitationsinthetext.
C Keywords. Gravitation––Galaxies:clusters:general
.
h
p
-1. Introduction baryonic Tully-Fisher relation, which shows a correlation with
o
total enclosed baryonic mass and galactic outer rotation veloc-
rToexplaindynamicsinastrophysicalsystemfully,asolutionto
t ity (the flat part of the rotation curve) (McGaugh 2005). This
sthe missing mass problem that plagues astrophysics to this day
correlationisexplainednaturallyintheMONDparadigm.Also,
a
mustbeimposed.Commonly,themissingmassinourUniverse
[ MOND is very successful in predicting galaxy rotation curve
is explained away by the existence of some non-baryonic dark profiles;seeforexampleSanders&McGaugh(2002).Afurther

1matter. Although dark matter is thought by most to be the best predictionofMONDwasthenatureofdwarfgalaxies.MOND
vsolutiontothemissingmassproblem,despiteourbesteffortsto
predicts that in the low acceleration regime there should be a
9date,thelackofdirectdetectionhasledsometoexploreotherex-
highmassdeficit(Milgrom1983c).Inotherwords,MONDpre-
6planations.Todothis,itispossibletothinkofNewtoniangravity
dicts that if objects are low surface brightness, they should re-
3
asalimitofamoregeneralgravitylaw.Anexampleofamodi-
quirealotofdarkmatter,whichisexactlywhatisfoundindwarf
3
fiedtheoryofgravity,designedwiththeabsenceofdarkmatter
0 galaxies.Tidaldwarfgalaxies(TDGs)arealsohardtoexplainin
.in mind, is Modified Newtonian Dynamics, hereafter MOND. theΛCDMparadigmastheirformationprocessdoesnotpredict
1TheMONDparadigmworkswiththeprinciplethatNewtonian
the dragging of large amounts of dark matter away from their
0gravitybreaksdowninthelowaccelerationlimit(a<<a where
0 host galaxy (see for example Section 6.5.4 of Famaey & Mc-
7a isanaccelerationconstant≈1.2×10−10ms−2).
1 0 Gaugh (2012) and references therein). The formation of TDGs
: Originally, MOND was proposed and introduced as an em- ismoreeasilyexplainedinMOND(Combes&Tiret2010).For
vpiricallawinthe1983Milgrom1983papers(Milgrom1983a), amoredetailedreviewofthesuccessesofMOND,wereferthe
Xi(Milgrom1983b)and(Milgrom1983c)byanalysinggalaxyro- readertoFamaey&McGaugh(2012).
tationcurves.OneissuewiththeoriginalMONDformulationby DespitethesuccessofMONDwithregardstogalaxydynam-
r
aMilgrom,whichwashighlightedinFelten(1983),wasitsinabil- ics,MONDisnotwithoutitsissues.Firstly,tobeaviablegravity
itytoconservemomentum.ThiswasrectifiedintheBekenstein theorytheremustbearelativisticformulation,analogoustoEin-
andMilgrompaper(Bekenstein&Milgrom1984)byintroduc- stein’s general relativity. There have been a few proposed rela-
ingaLagrangianformulationofMOND,AQUAL(AQUAdratic tivistic theories that produce MOND as a non-relativistic limit
Lagrangian).Recently,Milgromproposedanalternativeformu- including, for example, Tensor-Vector-Scalar gravity (TeVeS)
lationtotheMONDparadigmcalledQuasi-LinearMOND(Mil- (Bekenstein 2004), the Dark Fluid model (Zhao 2007) and Bi-
grom2010),hereafterQUMOND.Thisnewformulationismuch metricMOND(Milgrom2009).However,thesehavetheirlimi-
moredesirableasthequasi-linearwayinwhichtheMONDgrav- tationsanddonotrivalthesuccessofgeneralrelativity.
ityiscalculatedismuchsimplerthaninAQUAL. Another major problem for MOND, and the topic of this
The MOND framework has enjoyed a vast amount of suc- paper, is the limited success when trying to describe the mass
cess,providingreasonabledescriptionsofthedynamicsingalax- deficit in clusters of galaxies; see for example Sanders (1999)
ies.ArguablythemostfamouspredictionofMONDisthatofthe and Sanders (2003). It has been well documented that the in-
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A&Aproofs:manuscriptno.29358_Revised_19_Dec_2016
ferredmassbyMONDingalaxyclustersisnotenoughtokeep namics,perhapsMONDinitscurrentformisnotuniversal,but
the cluster in hydrostatic equilibrium. This result is enough for partofalargergravitationallaw.ThegeneralizationofMOND
some to abandon MOND as a viable theory. It is therefore the shouldnotberuledouthastilygiventheinherentlyempiricalna-
responsibility of those who study MOND to provide a suitable tureofitsformulation.MONDwasdevisedbystudyingrotation
explanationastowhythismissingmassproblemarises.Thesuc- curves,asinformationofgalaxyclusterswaslimited.Anynew
cessfulpredictionsthatweremadeusingMONDhaveledsome adaptationofMONDthatincreasesa inclusterswouldhaveto
0
tobelievethattheinabilitytoreproducethegalaxyclusterdata somehow shield galaxies, where MOND works well, from the
isjustapredictionthatthereexistssomeadditionalmasscompo- effectsofthisvaryingaccelerationscale.Thisideawasformally
nentthatweareyettodetect.OnepossibilityisthatMONDand outlined by Zhao & Famaey (2012) in the form of Extended
dark matter of some kind are not mutually exclusive and there ModifiedNewtonianDynamicsorEMOND.
existbothabreakdownofNewtoniandynamicsinthesmallac- AnotherissueinMONDisthenewlydiscoveredultradiffuse
celeration regime and some kind of elusive particle, increasing galaxies(UDGs)(Mihosetal.2015),(Kodaetal.2015).Ultra-
theavailablemassbudgetinclusters.Thismattercouldbeinthe diffuse galaxies are extremely low density galaxies, residing in
formof2eVneutrinos,whichwassuggestedinSanders(2003). thepotentialwellsofgalaxyclusters,whichrequirealotofdark
However,analysisofasampleofgalaxyclustersbyAngusetal. matter.ThisrequireddarkmattercouldbeexplainedinMOND
(2008)foundthatthe2eVneutrinowasabletoexplainthemass ifitwerenotforthehighexternalfieldexhibitedontheUDGs
deficitintheouterregionsoftheclusterwithmissingmassstill bythegalaxycluster.Theexternalfieldoftheclusterraisesthe
remaining in the central regions. This would warrant the exis- accelerationintheUDG,makingtheMONDpredictioncloserto
tence of a different type of neutrino. An 11 eV sterile neutrino Newtonian.Thiscouldbecounteractedifthevalueofa werein
0
was investigated by Angus (2009), which was found to be suf- factlargerintheseenvironmentsasdeviationsfromNewtonian
ficient to explain the galaxy cluster issue. Further investigation gravitycouldoccurathigheraccelerations.Thefactthatahigher
into solving galaxy clusters in MOND with the aid of neutri- value of a in galaxy clusters could simultaneously explain the
0
noswasconductedinAngus&Diaferio(2011)andAngusetal. massdeficitproblemandtheobservationsoftheseUDGsisin-
(2013). Cosmological simulations showed that it was possible terestingandshouldthusbeexploredinmoredetail.
to form clusters using hot dark matter (sterile neutrinos). How- This paper is outlined as follows. Section 2 outlines the
ever,theresultsshowedanunderabundanceoflowmassclusters MOND formalism and briefly reviews the missing mass prob-
(attributedtotheresolutionofthesimulation)andanoverabun- lem in clusters with MOND. This is followed by a review of
dance of high mass clusters. They go on to explain how the 11 the EMOND formalism. Section 3 discuses the galaxy cluster
eVneutrinowasruledoutasaneutrinomassof30-300eVwas sample that we have used throughout the paper. Section 4 dis-
requiredtoproduceenoughlowmassclusters. cusseshowwellEMONDworksinexplainingthemissingmass
Neutrinos have not been the only attempt to reconcile clus- problem. We find that EMOND has partial success in tackling
ters with MOND. Zhao and Li investigated a family of gravity this problem and thus in Section 5 we try to find an empirical
modelscontrolledbyavectorfield(seeforexampleLi&Zhao relation,basedontheMONDparadigm,whichcanaccountfor
(2009) and Zhao & Li (2010). This family of solutions can re- theclustermassdeficit.Wefindapotentialrelation,withsome
produce different types of behaviour depending on the model successaswell,whichwecomparetotheNavarro-Frenk-White
parameter choices. This is based on the assumption that there (hereaferNFW)fitsdescribedin(Vikhlininetal.2006)inSec-
existssomedarkfluid,describedbythevectorfield,whichisthe tion 6. In Section 7 we discuss the limitations of this empirical
sourceofdeviationfromNewtonianphysics.Anotherattemptto relationandweconcludeinSection8.
solvetheclusterproblemproposedbyKhoury(2015)alsopos-
tulates there exists a dark fluid that permeates space. This idea
ismotivatedbythesuccessofMONDonthegalacticscale,but 2. SummarizingtheMOND-clusterproblem
the success of ΛCDM in cosmological and cluster scales. This
darkfluidformalismaimstorecovergalaxyscaledynamicsviaa 2.1. TheMONDparadigmanditslimitations
MOND-likephononmediatedforceresultingfromthedarkmat-
WebeginbybrieflyreviewingtheMONDparadigmanditsgov-
ter in its superfluid phase. Clusters, in this scenario, would be
erningequations.ThegeneralmotivationforMONDtherecog-
dominated by the normal phase of the superfluid, which would
nition that the Newtonian prediction of gravity only seemed to
behave like particle dark matter. This aims for a best of both
failinlowaccelerationenvironments.Ifthereisnonon-baryonic
worldsscenariowithregardstodarkmatterandMOND.There
dark matter to explain this discrepancy, the gravitational law
has also been a theory proposed by Blanchet (2007), which in-
governing the Universe would have to deviate from Newtonian
cludes a substance called dipolar dark matter. This mechanism
intheselowaccelerationenvironments.
worksontheprinciplethatwedonotactuallyseeabreakdown
InNewtonianphysics,thegravitationalfieldΦisdetermined
ofagravitylawwhenweempiricallyseeMOND,butwhatwe
fromthematterdistributionρviaPoisson’sequation,
are witnessing is an enhancement of gravity due to dipole mo-
ments of this exotic dark matter aligning with the gravitational
field. All these theories have pros and cons, but are not studied
inthiswork. ∇2Φ=4πGρ. (1)
Although the aforementioned solutions may hold the keys
totheclustersuccessinMOND,theneedfordarkmatterofany Determining the gravitational field predicted by MOND re-
kindhasleftsomeunsettled,astheeleganceofMOND,ingalax- quires an alternate Poisson equation (Bekenstein & Milgrom
ies,istheabilitytoaccountforallobservationsfromvisiblemat- 1984),
ter only. Therefore aside from these dark fluid-like theories, it
hasalsobeennotedthatifthevalueofa weretobeincreasedin
clusters, the mass discrepancy could be0alleviated. In the same (cid:34) (cid:32)|∇Φ|(cid:33) (cid:35)
∇· µ ∇Φ =4πGρ, (2)
manner that MOND is a more general form of Newtonian dy- a
0
Articlenumber,page2of17
AlistairO.HodsonandHongshengZhao:GeneralizingMONDtoexplainthemissingmassingalaxyclusters
wherea isauniversalscaleaccelerationandµ(x)istheMOND
0
interpolationfunctionthatbehavessuchthat,
(cid:40)
1 forx(cid:29)1
µ(x)= . (3)
x forx(cid:28)1
The first case of Equation 3 is the Newtonian regime and
thesecondisknownasthedeep-MONDregime.TheNewtonian
Poisson equation is recovered in the x >> 1 case of Equation
3.Oneshouldnoteherethatthecentresofgalaxyclustershave
stronggravity,meaningtheyhaveanx>1MONDvalue.
There are many possible functional forms for µ(x) that sat-
isfy the criteria of Equation 3. We opted to choose the simple Fig.1.Plotshowinghowthedynamical(ortotal)gravitationalacceler-
ationoftheclustercomparestothegravitationalaccelerationpredicted
interpolationfunction(Famaey&Binney2005),
byMONDwitha1:1lineoverplottedforillustrativepurposes.Eachline
isadifferentcluster.Weuse12clustersfromVikhlininetal.(2006).As
x theclusterslieabovethe1:1line(blacksolidlineonplot),therequired
µ(x)= , (4)
1+x gravityismorethanthe‘MONDpredicted’gravity.
forouranalysis.
The MOND paradigm has always struggled with rectifying 6.×10-10
themassdiscrepancyingalaxyclusterswithoutincludinganex- 5.×10-10
tramasscomponentsuchasneutrinosornon-luminousbaryonic
matter. In its usual form MOND is unable to boost the grav- 2-s) 4.×10-10
itational acceleration in clusters sufficiently whilst this frame- m 3.×10-10
(
workisstillabletomatchobservationalconstraintsprovidedby yn
aD2.×10-10
galaxy dynamics. The reason for MOND’s inability to accom-
plishthisisthatthegravitationalaccelerationingalaxyclusters 1.×10-10
isrelativelyhigh,ingeneralx>1,sotheMONDboosttogravity
0
is weak. However, there is an observed mass deficit in galaxy
12.0 12.2 12.4 12.6 12.8 13.0
clustersthatwouldrequiretheclustertobeinthedeep-MOND
regime(x << 1)torectifywithMONDalone.Wedemonstrate Log10(|ΦNFW|)(m2s-2)
this issue briefly by applying MOND to the cluster sample de- Fig.2.Plotshowinghowthedynamicalgravitationalaccelerationvaries
scribedinSection3.Wecanplotthedynamicalacceleration(See withNFWpotentialforeachclusterinoursample.NFWprofileswere
Section 3 for details on how the dynamical acceleration is cal- takenfromVikhlininetal.(2006).Notetheregulartrendbetweenthe
culated)foreachclusteragainstthecorrespondingMONDcom- clusters,whichsuggestsamodifiedgravitylawdependentongravita-
puted acceleration from Equation 2; see (Figure 1). If MOND tionalpotential.
wereauniversallawthatappliedtogalaxyclusters,Fig1should
show1:1correlation(withinobservationalerrors),bywhichwe
meanthatthegravitationalaccelerationpredictedbytheMOND potential of the cluster is equal to the contribution of the dark
formulashouldmatchthegravitationalaccelerationpredictedby matter,describedbyaNFWprofile(Navarroetal.1997),(Zhao
the dynamics of the X-ray gas. In Figure 1, we see that the re- 1996).TheNFWprofileforeachclusterwasgiveninVikhlinin
quired gravitational acceleration, derived from the dynamics of et al. (2006). We made this approximation as the dark matter
the X-ray gas, exceeds the gravitational acceleration predicted NFWprofileisanalyticalandwelldefined,whereascomputing
by the MOND formula. Thus MOND is under-predicting the the gravitational potential from the dynamical data requires us
gravitational acceleration of each cluster. As mentioned previ- to know where the edge of the cluster lies in order to truncate
ously,thisoffsetisoftenreconciledintheMONDparadigmby the mass. Furthermore, we would have to perform an unneces-
the presence of some form of non-luminous baryonic matter or saryadditionalstepofnumericallyintegratingthedynamicalac-
neutrinos,whichhasyettobedetected. celeration to find the potential. This integration would involve
a boundary value that is dependent on the truncation radius of
the cluster. As we are only interested in getting an intuition of
2.2. CanMONDbegeneralizedtoaccountforclusters?
howthepropertiesofthegalaxyclusterscorrelatewithgravita-
AsthesimpleMONDformulationdoesnotholdtrueforgalaxy tionalpotential,theNFWapproximationseemstobereasonable
clusters,effortsweremadetotryandgeneralizeMOND.Inorder comparedtoperformingtheintegrationofthedynamicalaccel-
todothis,onemustidentifywhat,ifany,arethecriticaldiffer- eration.
encesbetweengalaxiesandgalaxyclusters.Onesuchdifference This trend with potential was the main motivation of the
is that the gravitational potential is much larger in galaxy clus- Zhao and Famaey Extended MOND (EMOND) formalism
tersthaningalaxies.Initscurrentform,MONDhasafunctional (Zhao&Famaey2012).InEMOND,thecorrespondingPoisson
dependencesolelyongravitationalacceleration.Perhapsamore equationiswrittenas
universalgravitylawrequiresgravitationalpotentialasamedia-
toroftotalgravity.Again,usingtheclustersampledescribedin
Section 3, we can see how the gravitational acceleration of the
clustervariesasafunctionofgravitationalpotential(seeFigure (cid:34) (cid:32) |∇Φ| (cid:33) (cid:35)
4πGρ=∇· µ ∇Φ −T , (5)
2).InFigure2,wemadetheapproximationthatthegravitational A (Φ) 2
0
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A&Aproofs:manuscriptno.29358_Revised_19_Dec_2016
where the dynamics inside UDGs closer to Newtonian (µ(x) → 1).
However, if this external potential effect was present, the fact
(cid:12) (cid:12) that the UDGs are within the deep potential well of the galaxy
T = 1 (cid:12)(cid:12)(cid:12)d(A0(Φ))2(cid:12)(cid:12)(cid:12)(cid:2)yF(cid:48)(y)−F(y)(cid:3), (6) clusterscouldcauseUDGstoshowMOND-likebehaviourorat
2 8πG (cid:12)(cid:12) dΦ (cid:12)(cid:12) leastexhibitstrongergravitythanthatexpectedfromNewtonian
physics.Theexternalpotentialeffectcouldbeanaturalexplana-
where A (Φ) is a functional alternative to the MOND constant
0 tionastowhyobjectsexistwithinclustersthatbehavesimilarto
a withunitsofacceleration,andµisthesameMONDinterpo-
0 √ UDGs.However,recentestimatesofdarkmatterhaloesinthese
lation function, dF(y)/dy = µ( y) and y = |∇Φ|2/A (Φ)2. It is
0 objectsshowanextremelyhighdarktobaryonicmassratio(van
immediatelyapparentthatEquation5ismuchmorecomplicated
Dokkum et al. 2016). One would need to check that the exter-
than the regular MOND Poisson equation owing to the second nalpotentialeffectrequiredtoreconcilethattheUDGsdoesnot
term.Ithashoweverbeenarguedthatincertainsituations,such
conflictwiththehostgalaxycluster.
asgalaxyclusters,thistermissmallcomparedtothefirst(Zhao
&Famaey2012),givinganapproximateexpression,
3. Analysingagalaxyclustersample
4πGρ≈∇·(cid:34)µ(cid:32) |∇Φ| (cid:33)∇Φ(cid:35), (7) To begin, we first need to select a galaxy cluster sample. We
A (Φ) useaChandragalaxyclustersample(Vikhlininetal.2006)that
0
providesanalyticalfitstothetemperatureanddensityprofilesof
fortheEMONDgravitationalpotential.InSection4weexplic- theX-raygas.Weuse12clustersfromthesample.Theclusters
itlyshowthatwearejustinmakingthisapproximation.Equation arenearby(z<1)andareinthemassrange≈1014−1015M .For
(cid:12)
7, the approximate EMOND Poisson equation, is equivalent to athoroughanalysis,wealsoneedtomodelthebrightestcluster
Equation2withamodifiedscaleacceleration.Theideabehind galaxy (BCG) component. The BCG modelling is outlined in
Equation 7 is to increase the MOND acceleration scale in high Section3.1.1.
potentialenvironments.ThisdecreasesthevalueoftheMOND
interpolation function, µ, thus increasing the total gravitational
accelerationthatispredictedbyMOND(a ≈ a /µ).In 3.1. Clustermodel
MOND Newt
ordertosolvetheEMONDPoissonequation,wemustdefinethe
TomodeltheX-raygasdensityVikhlininetal.(2006)chooses
behaviourofA (Φ),whichwediscussinSection4.Althoughwe
0 an analytical expression to which the data is fit. This model is
determinetheexactbehaviourofA inSection4,weshouldpro-
0 composed of three parts. The first part models the cuspy core
vide some understanding of the general desired behaviour that
usuallyfoundinrelaxedclustersandisachievedbyapowerlaw
wehopetoemulate.Fromourunderstandingofgalaxyrotation
densityprofile.Theythenincludeanadditionaltermthatmodels
curvesinMOND,weknowthatingeneralgalaxiesarewellde-
thesteepeningbrightnessprofileatr>0.3r ,wherer isthe
scribedbyaconstanta ≈ 1.2×10−10 ms−2.Wethereforewant 200 200
0 radius at which the average density of the cluster falls to 200
topreservethisbehaviour.Thereforethestandarda shouldbea
0 times that of the critical density of the Universe. They finally
limitofA inthelowpotentialregimesuchthatA (Φ) → a as
0 0 0 add another power law component to allow the density profile
Φ→0.
to be very general for fitting freedom. Combining these three
componentsleadstotheiroverallemissionprofilefortheX-ray
2.3. Theexternalpotentialeffect gas,
As the modification to the MOND formula we discuss here re-
lies on the gravitational potential, even a constant gravitational (r/r )−α 1 n2
pnoamtenictsia.lTahcirsosfesaatusryestceamnfbreomthoanugehxtteorfnaalssaonurecxetearffneacltpsothteendtiya-l nenp(r)=n20(cid:16)1+(r/rcc)2(cid:17)3β−α/2(1+(r/rs)γ)(cid:15)/γ+(cid:16)1+(r/r0c2)2(cid:17)3β2.
effect (Zhao & Famaey 2012), which is analogous to the ex- (8)
ternal field effect (EFE) that arises in regular MOND; see for
example Derakhshani & Haghi (2014) for work on the EFE in whererc,rc2,andrs areallscaleradiiandα,β,γ,(cid:15),andβ2 are
globularclusters;Blanchet&Novak(2011),whichlooksatthe alldimensionlessparameters.ne andnp areelectronandproton
EFEinthesolarsystem;andHaghietal.(2016),forworkwith numberdensities,andn isaparameterthattakesthemeaningof
0
theEFEanddecliningrotationcurvesinMOND.Thetreatment acentralnumberdensity.Theemissionprofileisconvertedinto
of the external potential effect should be fully examined in fu- themassdensitybyρ (r) ≈ 1.252m (cid:16)n n (cid:17)1/2,wherem isthe
g p p e p
ture work. The original EMOND work assumes that the main
massofaproton.Theparameterfitsforeachgalaxyclusterare
contribution to the external potential of the Milky Way can be
giveninTable1,whichweretakenfromVikhlininetal.(2006).
attributedtotheVirgocluster.Workontheescapevelocityofthe
Wealsorequirethetemperatureprofileofthegasasweare
MilkyWay(Famaeyetal.2007)calculatestheexternalpotential
choosingtorelaxthecommonassumptionthatthegalaxyclus-
for the Milky Way to be approximately Φ ≈ −10−6c2. How-
ext ters are isothermal. Again, we follow the profile provided by
ever,thetypicalgravitationalpotentialsofourgalaxyclustersare
Vikhlininetal.(2006),
muchlargerthanthissoweareabletoexcludethiseffectfrom
ourcalculations.Theimportanceoftheexternalpotentialeffect
ismostprominentingalaxiesandobjectsthatliewithinclusters (r/r +T /T ) (r/r)−a
T(r)=T cool min 0 t , (9)
ornearobjectswithlargegravitationalpotentials.Forexample, 0 (r/rcool)acool +1 (cid:0)(r/rt)b+1(cid:1)c/b
we mentioned earlier that there has recently been a discovery
of many so-called ultra diffuse galaxies (UDGs) inside galaxy which accounts for the cooling of the X-ray gas in the central
clusters. The existence of these objects could pose some threat regionsofthecluster,whichaccordingtoVikhlininetal.(2006)
to MOND as the external field from the clusters should make maybetheresultofradiativecooling.TheparameterT isthe
min
Articlenumber,page4of17
AlistairO.HodsonandHongshengZhao:GeneralizingMONDtoexplainthemissingmassingalaxyclusters
Table1.TableofparametersasgiveninVikhlininetal.(2006)fortheX-raygasemissionprofiledescribedbyEquation8.Weomittedonecluster
from our analysis as Vikhlinin et al. (2006) do not provide NFW fits for dark matter, so we cannot compare our result to that in a consistent
manner.γforeachclusterwaschosentobe3.0.ValuesintablelabelledN/Arefertotheclustersforwhich(Vikhlininetal.2006)didnotprovide
parameters,makingthoseclustersatwo-componentdensitymodelratherthanthree(seeEquation8).
Cluster n r r α β (cid:15) n r β
0 c s 02 c2 2
(10−3cm−3) (kpc) (kpc) (10−1cm−3) (kpc)
A133 4.705 94.6 1239.9 0.916 0.526 4.943 0.247 75.83 3.607
A262 2.278 70.7 365.6 1.712 0.345 1.760 N/A N/A N/A
A478 10.170 155.5 2928.9 1.254 0.704 5.000 0.762 23.84 1.00
A1413 5.239 195.0 2153.7 1.247 0.661 5.000 N/A N/A N/A
A1795 31.175 38.2 682.5 0.195 0.491 2.606 5.695 3.00 1.00
A1991 6.405 59.9 1064.7 1.735 0.515 5.000 0.007 5.00 0.517
A2029 15.721 84.2 908.9 1.164 0.545 1.669 3.510 5.00 1.00
RXJ1159+5531 0.191 591.9 640.7 1.891 0.838 4.869 0.457 11.99 1.00
MKW4 0.196 578.5 595.1 1.895 1.119 1.602 0.108 30.11 1.971
A383 7.226 112.1 408.7 2.013 0.577 0.767 0.002 11.54 1.00
A907 6.252 136.9 1887.1 1.556 0.594 4.998 N/A N/A N/A
A2390 3.605 308.2 1200.0 1.891 0.658 0.563 N/A N/A N/A
central temperature, T0 is a scale temperature, rcool and rt are Table3.r500valuesforeachclusterasgivenin(Vikhlininetal.2006);
r valuescanbecalculatedapproximatelyviar ≈1.5r .
scale radii, and a , a b and c are dimensionless parameters. 200 200 500
cool
Outsidethecentral,coolerregionoftheclusterthetemperature
Cluster r (kpc) r (kpc)
is described by a broken power law. The parameter fits for the 500 min
A133 1007 40
temperaturecanbefoundinTable2,againtakenfromVikhlinin
A262 650 10
etal.(2006).
A478 1337 30
A1413 1299 20
3.1.1. BCGmodel A1795 1235 40
A1991 732 10
Wealsomodelthebrightestclustergalaxy(BCG)foreachclus-
A2029 1362 20
ter, which affects the central parts of the cluster. This is an im- RXJ1159+5331 700 10
portant aspect of the model as it is in the central regions of the
MKW4 634 5
clusterswherethemassdeficitisgreatest.TomodeltheBCG,it
A383 944 25
isreasonabletomaketheapproximationthatitisspherical,with
A907 1096 40
adensityfollowingthewell-knownHernquistprofile,
A2390 1416 80
Mh
ρBCG(r)= 2πr(r+h)3, (10) (2006). For our plots throughout this paper, we choose not to
showanydatabelowthisradius.
Fortheouterradiuswechoosetoextrapolatepastthemaxi-
where M is the BCG mass and h is the BCG scale length. We
mumradiusofeachcluster.Weextrapolatetotheradiusr for
assignaBCGmassproportionaltotheoverallmassoftheclus- 200
tersuchthatMBCG =5.3×1011(cid:16)3×1M051040M(cid:12)(cid:17)0.42M(cid:12),whichfollows (eVacikhhclilnuisntere,twahl.ic2h0c0a6n).bTehcerurd5e0l0yvcaalluceuslaftreodmviVairk2h00lin≈in1.e5tr5a0l0.
theworkofSchmidt&Allen(2007).Here,M500istheenclosed (2006)arealsoshowninTable3.
mass at r , which is the radius at which the average density
500
is500timesthecriticaldensityoftheUniverse.Wealsoassign
eachclusteraHernquistscalelengthof30kpc.Althoughthese 3.2. Dynamicalmassoftheclusters
estimates may be crude, the BCG should only affect the inner
In the following sections we show the results of dynamical
partsoftheclusterandthusshouldnotaffectouroverallconclu-
masses, potentials, and accelerations so we therefore must de-
sionstoomuch.
finethesequantities.Whenwediscuss‘dynamical’quantitiesin
thiswork,werefertothevaluecalculatedbyassumingtheclus-
3.1.2. Radialrangeofthedata tergasisinhydrostaticequilibrium.Thisisacommonassump-
tionindeterminingclustermasses.Theequationforhydrostatic
The radial range of the X-ray data is limited by two factors: equilibriumiswrittenas
an inner boundary or cut-off radius and an upper bound ra-
diuswheretheX-raybrightnessisnotdetected.Vikhlininetal. dΦ GM (r) 1 d (cid:34)ρ kT(r)(cid:35)
(2006) choose an inner radial boundary such that they exclude dyn = dyn =− g , (11)
dr r2 ρ (r)dr wm
thecentraltemperaturebinandanouterradialboundarywhere g p
theX-raybrightnessisnolongerdetectedabove3σorthelimit
of the Chandra field of view was reached. Therefore the model whereΦdyn isthedynamicalpotential,mp isthemassofapro-
isfittoasectionoftheradialextentofthecluster,betweenthese ton (≈ 1.67 × 10−27 kg), and w is the mean molecular weight
two limits, and then extrapolated to lower and higher radii. In (≈0.609).Thevalueρ (r)isthegasdensity,T(r)isthegastem-
g
Table3,weshowtheinnermostradiiasgiveninVikhlininetal. perature, and M (r) is the enclosed dynamical mass at radius
dyn
Articlenumber,page5of17
A&Aproofs:manuscriptno.29358_Revised_19_Dec_2016
Table2.Tableofgastemperatureparametersasgivenin(Vikhlininetal.2006)forEquation9.
Cluster T r a b c T /T r a
0 t min 0 cool cool
kev Mpc kpc
A133 3.61 1.42 0.12 5 10 0.27 57 3.88
A262 2.42 0.35 -0.02 5 1.1 0.64 19 5.25
A478 11.06 0.27 0.02 5 0.4 0.38 129 1.6
A1413 7.58 1.84 0.08 4.68 10 0.23 30 0.75
A1795 9.68 0.55 0 1.63 0.9 0.1 77 1.03
A1991 2.83 0.86 0.04 2.87 4.7 0.48 42 2.12
A2029 16.19 3.04 -0.03 1.57 5.9 0.1 93 0.48
RXJ1159+5531 3.74 0.1 0.09 0.77 0.4 0.13 22 1.68
MKW4 2.26 0.1 -0.07 5.00 0.5 0.85 16 9.62
A383 8.78 3.03 -0.14 1.44 8.0 0.75 81 6.17
A907 10.19 0.24 0.16 5.00 0.4 0.32 208 1.48
A2390 19.34 2.46 -0.10 5.00 10.0 0.12 214 0.08
r.Usingsomemathematicalprowess,Equation11canbetrans- 3.3. Newtonianmassoftheclusters
formedinto
We can find the Newtonian predicted mass from the cluster by
integrating the gas density profile along with the BCG density
kT(r)r (cid:34)dlnρ (r) dlnT(r)(cid:35) profile,
M (r)=− g + . (12)
dyn Gwm dlnr dlnr
p (cid:90) r
InΛCDMlanguage, M wouldbethemassofthebaryons MN(r)= 4πr˜2(ρg(r˜)+ρBCG(r˜))dr˜. (16)
dyn
0
anddarkmatterrequiredtokeeptheclusterinhydrostaticequi-
librium,whichwerefertoasthedynamicalmass.Fromthis,we Weassumeinthisworkthattheonlysignificantstellarmass
cancalculatethedynamicalacceleration, istheBCGmassandthatotherstellarmassissmallcomparedto
thegasmass.TheNewtonianaccelerationandpotentialcanthen
byfoundbyanalogousexpressionofEquation13and14.
GM (r)
a (r)=∇Φ (r)= dyn , (13)
dyn dyn r2
4. ApplyingEMONDtothesampleofclusters
and thus the dynamical potential can be found by integrating
The initial EMOND theory was qualitatively outlined in Zhao
Equation13,
&Famaey(2012)buthasuntilnowundergonenoquantification
with regards to applying the formalism to a sample of galaxy
(cid:90) r clusters.Toaccomplishthis,wefirsttaketheapproximatespher-
Φdyn(r)= adyn(r˜)dr˜. (14) icallysymmetricversionofEquation7,
∞
In Equation 14 the limits are chosen such that the gravita- (cid:32) |∇Φ| (cid:33)
tional potential is negative. Because of the choice of gas den- ∇ΦN ≈µ A (Φ) ∇Φ. (17)
sityandgastemperatureprofileusedtodeterminethedynamical 0
mass,itisverydifficulttodeterminethedynamicalpotentialof WecantheninvertEquation17tofindanexpressionforA (Φ),
0
each cluster from Equation 14. This is because the dynamical
gravitational potential would be sensitive to a choice of cut-off
−|∇Φ ||∇Φ|+|∇Φ|2
radius, i.e. where we truncate the gas density to zero. The rea- A (Φ)= N , (18)
sonforthissensitivityisduetotheintegralfrom∞inEquation 0 |∇Φ |
N
14. We would have to make some assumption about how the
assumingµtakestheformofEquation4.Therefore,wecanem-
clustermassbehavesoutsider .Inordertocircumventthisis-
200 pirically find out if the data favours the EMOND formalism or
sue, we take note of the fact that the NFW dark matter profile
whetherEMONDcannotexplainthemissingmassinclustersby
providesgoodfitstothedynamicalmass.TheNFWprofilehas
determiningifthereisasinglefunctionofA (Φ)thatcanexplain
a well-defined analytical expression for gravitational potential. 0
Therefore we make the assumption that Φ ≈ Φ for each themassdiscrepancyinalltheclustersinoursample.InFigure
dyn NFW 3weplottherequiredA ,givenbyEquation18,asafunctionof
cluster, 0
theNFWgravitationalpotential.Toaccomplishthis,weassume
that the total gravitational acceleration is approximated by the
4πGr3ln(cid:104)r+rs(cid:105)ρ dark matter NFW profile for each cluster. We find that there is
ΦNFW ≈Φdyn =− s r rs s (15) apogteennetiraall,twrehnidchofsetehmesretqouifroeldlowA0ainncerxepasoinnegntwiailthcugrrvaveit(aFtiigounrael
3),forexample,
wherer andρ arethescaleradiusanddensity,respectively.We
s s
should stress at this point that the NFW profile is not an exact
solutionoftheMONDequationsandwethereforeexpectsome (cid:32) Φ (cid:33)
A (Φ)=a exp , (19)
discrepanciesinourourplots.Weusethisassumptionasaguide. 0 0 Φ
0
Articlenumber,page6of17
AlistairO.HodsonandHongshengZhao:GeneralizingMONDtoexplainthemissingmassingalaxyclusters
whereΦ0 isthescalepotentialthatwehaveempiricallychosen 1.×10-8
to be |Φ | ≈ 15000002m2s−2. One can see that in low potential
0
environments, such as galaxies, that A ≈ a and thus MOND 8.×10-9
0 0
dynamicsarepreserved.WeplottedEquation19inFigure3(red )
line).Thisfunctionalformhastheissuethatitrisesveryquickly s2- 6.×10-9
m
icnluhsitgehrsp.oTtheenrteiafolrreegwioentse.sTtehdisacsoeuclodncdafuusnecitsiosune.sinhighermass A(04.×10-9
2.×10-9
A0(Φ)=a0+(A0max−a0)21tanhlog(cid:32)ΦΦ (cid:33)2+ 21, (20) 0 11.8 12.0 12.2 12.4 12.6 12.8 13.0
0
where the acceleration scale, A (Φ), grows like a step func- Log10(ΦNFW)(m2s-2)
0
tionwithminimumvaluea0 inlowpotentialregionsandmaxi- Fig.3.PlotshowingtherequiredvalueofA0accordingtotheEMOND
mumvalue A inhighpotentialregions.Forthisapproach, formalism(Equation18)suchthatpredictedgravitationalacceleration
0max
we are inspired by an alternate modification to gravity that fromEMONDmatchesthedynamicallycalculatedacceleration(Equa-
uses a density dependent modification to gravity to try and tion12)foreachclusterinthesample(thinlines).Weplottedthisre-
explain galaxies and galaxy clusters without the need for quiredA0againsttheNFWgravitationalpotentialasaestimationofthe
dark matter (Matsakos & Diaferio 2016). In this formu- behaviourwithinthecluster.Theshapetakestheformofanexponential
functionoverplottedinredanddescribedbyEquation19.Alsoplotted
lation, gravitational dynamics are described by a modified
inblueisthemorecomplicatedA functiondescribedbyEquation20
Poisson equation, ∇ · ((cid:15)(ρ)∇Φ) = 4πGρ, where (cid:15) is a di- with a parameter choice, Φ = −027000002 m2s−2 and A = 80a .
mensionless function of density. In their work, the func- 0 0max 0
NotetheapparentturnoverofA atthedeepestpotentialisanartefact
0
tional choice of (cid:15)(ρ) is a smooth step function such that, becausethetotalgravitationalpotentialisnotjustthatofanNFWpro-
(cid:15)(ρ) = (cid:15) +(1−(cid:15) )1(cid:104)tanh(cid:104)log(cid:16)ρ(cid:17)q(cid:105)+1(cid:105),where(cid:15) andqare fileasassumedhere.
0 0 2 ρc 0
freeparametersandρ isadensityscale,whichisanalogous
c
toourpotentialscaleΦ .WeoverplotthisfunctioninFigure3
0
(blueline)alongwiththesimpleexponential.
InFigure4weplotthecalculatedA asafunctionofradius
0
fortwoclusters,A133andA2390.WechoosethesetwoasA133
isalessmassiveclusterandA2390isthemostmassivecluster.
WecanseethatinA2390thevalueofA ismostlyatthemaxi-
0
mumvaluethatEquation20allowsasithasalargegravitational
potential, whereas A133 has a much lower A which increases
0
towardsthecentre.
Usingthisrecipe,itisthereforepossibletocalculatetheef-
fectiveenclosedmasspredictedbyEMONDforeachclusterby
whichwemeanNewtonianmassplusphantommass,
r2|∇Φ |
M (r)= EMOND , (21)
EMOND G
where∇Φ isthegravitationalaccelerationgivenbyEqua-
EMOND Fig.4.PlotshowingthecalculatedA for2clusters,A133andA2390,
0
tion 17. To accomplish this, we need to solve a first order dif- using Equation 20. Note that unlike A133, the massive A2390 has a
ferential equation for the potential Φ. This requires a bound- potentialmostlyabovethestepfunction,henceshowslittledependence
ary condition of the potential. For this we picked three values onthechangingoftheboundarypotential(cf.Figure16).
for the boundary, which we set at r = 2r for each clus-
bound 200
ter.ThethreevalueswereΦ(r ) = Φ (r ),Φ(r ) =
0.5Φ (r ), and Φ(r b)ou=nd1.5Φ NFW(r bound). Webdoiudndthis wherey=|∇Φ|2/A0(Φ)2.Therefore,
NFW bound bound NFW bound
fortworeasons.First,wewouldexpectthatthepotentialcanbe
√
approximated by the NFW profile and also we wanted to show 2 y+y √
howmuchthecalculatedmassisaffectedbythechoiceofbound- yF(cid:48)(y)−F(y)= 1+ √y −2log(1+ y). (23)
arypotential.InFigures17to22,whereweshowtheresultsfor
eachcluster usingtheEMONDrecipe, theshadedregion high- AssumingthesimplercasewhereA (Φ)=a exp(Φ/Φ ),
0 0 0
lightsthedependenceoftheboundarypotential.
ThefinalaspectoftheEMONDanalysisthatweshouldpro- d(A (Φ))2 2a2 (cid:32)2Φ(cid:33)
videforcompletenessisprovingthattheEMONDT2 termisin 0 = 0 exp . (24)
fact negligible. To do this, we can solve the right-hand side of dΦ Φ0 Φ0
Equation 5 by again using the NFW profile as an estimation of
We now have all the ingredients to numerically plot the T
thetotalgravityintheclusters.Wecanthencomparethisresult 2
tothesimplifiedPoissonequation,Equation7,toseetheeffect term. To accomplish this, we show, for conciseness, the result
for one cluster, A133, as an example but all clusters show sim-
thattheT termhasontheresult.Assumingthesimpleµfunc-
tion(Equa2tion4),asF(cid:48)(y)=µ(√y),F(y)takestheform, ilar results. In Figure 5 we plot the full right-hand side (RHS)
ofEquation5andtheRHSof7.Thisisessentiallyaplotofthe
predicted Newtonian density profile from the EMOND formu-
√ √
F(y)=−2 y+y+2log(1+ y). (22) lationwiththeT termincludedandneglected.Weseethatthe
2
Articlenumber,page7of17
A&Aproofs:manuscriptno.29358_Revised_19_Dec_2016
0.0
-0.2
-0.4
B
-0.6
-0.8
-1.0
12.0 12.2 12.4 12.6 12.8 13.0 13.2
Log10(ΦNFW)(m2s-2)
Fig.6.PlotshowingthequantityB(Equation6)vs.NFWgravitational
potentialforeachcluster.Byapproximatingthetotalclustergravityas
that of the NFW halo, we illustrate that any correction term B to the
regularMONDformula,whilehavingatrendwiththepotentialofeach
cluster,cannotdescribeallclusterssimultaneously.
Fig.5. Plotshowingthecalculateddensity,predictedbyEMONDfor
clusterA133.Bluedashedlineshowsthedensitywiththeinclusionof
theT term(Equation5)andredlineshowsthedensitycalculatedfrom
2 inoursample,assumingthatthetotalgravitationalacceleration
approximatePoissonequation(Equation7).Thelinesarealmostiden-
isapproximatedbythebest-fitNFWprofileasafunctionoftotal
tical showing the T term was indeed justifiably neglected. Note that
thesmalldifferences2intheouterregionsoftheclusterisanartifactbe- gravitationalpotentialofeachcluster;thisgravitationalpotential
is also approximated by the gravitational potential of the NFW
cause the asymptotic potential of NFW is compatible mathematically
withEMONDonlyifthelatteralsoallowsthedensitydippingintoneg- profile. With this approach, we attempt to determine whether
ativeatlargeradii. there is an additive component to the original MOND formula
thatcanboostthegravityinsideclusters,whilstpreservingreg-
ularMONDingalaxies.WeplotthisresultinFigure6.
plotshowsthattheresultsarealmostidenticalandthuswewere One may notice that although there is not a single function
justifiedinneglectingtheT2 terminourpreviousanalysis.The thatcouldfitalltheseclusters,thereseemstobesomeregularity
values deviate from each other in the outer radii of the cluster. to the plot. To determine whether the result in Figure 6 can be
ThisisbecausetheNFWprofileisnotaperfectsolutionofthe improved,wetriedmodifyingourEquationforB(Equation25)
EMONDequationandthepredictedNewtoniandensityfallsto slightlysuchthat,
zero and eventually negative. The difference in the two lines is
justaresultofthesmalldelaybetweenthetwotermsdrivingthe |∇Φ | (cid:32)|∇Φ| Φ (cid:33)
densitytothiszeropoint. B = N −µ + . (26)
WeconcludefromthisanalysisthatEMONDhashadmixed 2 |∇Φ| a0 Φ0
successasaMONDgeneralizationtoexplainthemissingmass
The result of plotting B against gravitational potential is
2
in galaxy clusters. We therefore look for an alternative in the
shown in Figure 7. This yielded a very tight function of gravi-
followingsection.
tational potential for each cluster. Also, one should notice that
the absolute value of B runs between 0 and 1, which is iden-
2
tical to the behaviour of the MOND interpolating function. Al-
5. Anempiricalalternateformulationexplainingthe
though the trend is very tight, there are some offsets. This may
massdiscrepancy
bebecauseweapproximatedthegravitationalpotentialtobethe
DespitethemixedsuccessofEMONDinansweringthequestion NFWpotential.Thisassumptionmaynothold.
of missing mass in galaxy clusters, the trend of missing mass
versus gravitational potential seems to linger. Also the idea of
5.1. Summarizingandrefiningtheformulae
having a as a non-constant function may not sit comfortably
0
withtheMONDcommunity.Thispromptedustolookforadif- Analysingthedataseemstoshowapossiblealternativeformu-
ferent formalism that could unify galaxies and galaxy clusters lationofMONDthatcouldaccountforthemassdiscrepancyin
withonelaw.Wethoughtitwouldbeagoodideatodetermine this sample of clusters. The modified empirical relation in this
theresidualintheMONDformula,whichwewillcallB,tode- casewouldbe
termineifthereissomecommonthemethroughouttheclusters.
TheMONDresidualisfoundbyrearrangingtheMONDformula |∇Φ | (cid:32)|∇Φ| Φ (cid:33)
suchthat, N −µ + = B (Φ), (27)
|∇Φ| a Φ 2
0 0
|∇Φ | (cid:32)|∇Φ|(cid:33) wherethegravitationalpotential,Φ,andscalepotential,Φ0,
B= N −µ , (25) are negative quantities. As we have seen in Figure 7, B looks
|∇Φ| a0 likeaMONDinterpolationfunction, B (Φ) = µ(Φ/Φ (Φ2)).Im-
2 0
plementing this, our new modified MOND function is written
which is just moving everything in Equation 2 to one side, as-
as
suming spherical symmetry. The hope here was that we could
find a function for B that is 0 in galaxies, thus preserving reg-
ularMOND,butnon-zeroingalaxyclusters.Wecandetermine (cid:34) (cid:32)|∇Φ| Φ (cid:33) (cid:32) Φ (cid:33)(cid:35)
∇Φ = µ + −µ ∇Φ, (28)
whetherthereisatrendifweplotthevalueofBforeachcluster N a Φ Φ
0 0 0
Articlenumber,page8of17
AlistairO.HodsonandHongshengZhao:GeneralizingMONDtoexplainthemissingmassingalaxyclusters
0.0 0.970
0.965
-0.2
0.960
-0.4 μNew
B2 μ 0.955
-0.6 MOND
0.950
-0.8 0.945
0.940
-1.0
12.0 12.2 12.4 12.6 12.8 13.0 13.2 0.01 0.10 1 10 100
Log10(ΦNFW)(m2s-2)
∇Φ/a0
Fig.7.PlotshowingthevalueB (Equation26)asafunctionofNFW
2
gravitationalpotentialforeachcluster(thinlines).Weoverplotathin Fig.8.PlotshowinghowtheMONDinterpolationfunctionandthein-
blue line that shows µ(Φ/Φ ) for Φ = −17000002 m2s−2. Note all terpolationfunctionofourrelation(Equations29and30)comparefor
clustersliefairlyclosetothi0sline, w0ithsomediscrepancyduetoour a low potential object (Φ = Φext) for different values of acceleration.
adoptedNFWprofilenotrepresentingthetotalpotential.This B cor- Wecanseelittlevariation,whichisidealforpreservinggalaxyphysics
2
rectiontoMONDallowstheMONDinterpolationfunctiontorunwith withthisnewrelation.
thepotentialaswellastheacceleration.
becorrect.Ideallyonewouldmakealargerscalesimulationen-
whereΦ isourscalepotentialanalogoustotheMONDacceler- compassingalargeareaofspaceandworkouttheexternalpo-
0
ationscalea . tential(andfield)foreachgalaxy.Thiswasmentionedin(Haghi
0
Equation 28 is a relation based on the MOND paradigm, etal.2016)whousedtheexternalfieldtoexplaindecliningro-
which in the future can hopefully lead to a gravity theory that tation curves. They also mentioned that this was now possible
canexplainawaythemassdiscrepancyingalaxyclusterswhilst thankstotheMONDpatchinRAMSES,PhantomofRAMSES
preservingthedynamicsofgalaxies. (Teyssier2002)and(Lüghausenetal.2015).Thiswouldrequire
We have yet to mention the value for the scale poten- more work to incorporate for example EMOND as the gravity
tial Φ0. For our results, we chose a scale potential of |Φ0| = solverwouldneedtobealteredtoallowvaryinga0values.
17000002m2s−2.WedonotattempttoperformarigorousMonte
Carlo search over the parameter space in this work as we feel
6. ComparingthenewrelationtotheNFWfits
thattodothisproperlywouldrequiresomeindependenttestsof
theequationssuchasgravitationallensingandMilkyWaydata Inthissection,weprovideplotsofthedynamicalmass,themass
toensureourparameterdoesnotcausecontentionwithlocalob- predicted by invoking the new relation and ΛCDM predictions
servations.Thisisbestleftforfurtherwork. for the 12 clusters outlined in Section 3. The dynamical mass
is calculated via Equation 12, the mass predicted from the new
relation,
5.2. ComparisontoMOND
Wedohowevershowaplotshowinghowtheinterpolationfunc-
r2∇Φ
tionofourrelationcompareswiththatofregularMOND(Figure M (r)= New (31)
8).Here, New G
where∇Φ iscalculatedviaEquation28andtheΛCDMmass
New
(cid:34) (cid:32)|∇Φ| Φ (cid:33) (cid:32) Φ (cid:33)(cid:35) from,
µ = µ + −µ (29)
New a Φ Φ
0 0 0
and MΛCDM(r)= Mgas(r)+MBCG+MNFW(r). (32)
The NFW parameters used in Equation 32 are taken from
(cid:32)|∇Φ|(cid:33) (Vikhlininetal.2006)andaredisplayedhereinTable4inwhich
µ =µ . (30)
MOND a theNFWmassisdefinedas
0
thepWoetecnotimalpΦare=EΦquexattaiosndse2fi9neadndin3S0eicntiFoing2u.r3ef8o,radssiffumerienngtvthaal-t M (r)=4πρ r3(cid:34)ln(cid:32)1+ r (cid:33)−(cid:18)1+ rs(cid:19)−1(cid:35), (33)
uesofaccelerationarangingfrom0.01a0−100a0.Wealsoshow NFW s s rs r
rotationcurvesfortwogalaxies,M33andNGC4157,usingthe
where ρ and r are the scale potential, and radius of the NFW
new relation and EMOND (Figures 9 and 10). These rotation s s
profile r is defined as r = r /c , where the values for r
curves have been used in Sanders (1996) for M33 and Sanders s s 500 500 500
andc (theNFWconcentrationparameter)canbefoundinTa-
&Verheijen(1998)forNGC4157.WeoverplottheMONDpre- 500
bles 3 and 4, respectively. The value ρ is calculated from the
dictionforcomparison.ThereislittledeviationfromMONDfor s
equation,
thesetwoexamples.Theremaybesomedegeneracybetweenfit-
tingthescalepotentialforagalaxyandthestellarmasstolight.
We do not discuss this point here, but mention it as a possible 500(cid:32)r (cid:33)3 ρ
avenue for future work. We have used a fixed background po- ρ = 500 crit , (34)
tential for both galaxy cases whereas in practice, this may not s 3 rs log(cid:104)1+ r500(cid:105)−(cid:104)1+ rs (cid:105)−1
rs r500
Articlenumber,page9of17
A&Aproofs:manuscriptno.29358_Revised_19_Dec_2016
M33 NGC4157
120
200
100
180
1-) 1-)
s s
m 80 m 160
k k
( (
city 60 city 140
o o
el el 120
V V
40
100
20 80
1-s) 1.5 1-s) 4
m m
k k
( (
e e
dul 1. dul 3
si si
e e
R R
city 0.5 city 2.
o o
el el
V V
0 1
0 2 4 6 8 5 10 15 20 25
Radius(kpc) Radius(kpc)
Fig.9.TopplotshowingtherotationcurveofM33.Bluepointsarethe Fig.10.SameasFigure9forgalaxyNGC4157.Weuseastellarmass
datawitherrorbars,theblackdashedlineistheregularMONDfit,the tolightof1.69foreachcase.
redlineisthenewrelation,andthebluelineisEMOND.Inthebottom
plotwealsoshowtheresidualbetweenMOND,thenewrelation,and
EMOND.Notethedifferencebetweentheserotationcurvesareactually tionalenvironments(seeSection7).Perhapsarefinementofthe
tiny(∼1km/s).Hereweadoptedabackgroundpotentialof10−6c2given formalisminthesehighpotentialregionsmaysolvetheproblem
inZhao&Famaey(2012)),andastellarmasstolightof0.722. of A2390. We also acknowledge that a rigorous error analysis
has not been performed and thus it is hard to gauge how well
Table 4. NFW concentration parameters at radius r . NFW profiles thisnewrelationhasfairedinpredictingclustermasses.
500
canbedeterminedbycombiningthesewithther500parametersgivenin Also, we do not truncate the gas density in our calculation.
Table3,thevaluesforrsandthevalueforρsgiveninEquation34. Itisunrealisticthatthegasdensityextendstor200.Aswedonot
havethenecessarydatatoconstrainthis,wealsoleavethisasan
Cluster c
500 openissue.
A133 3.18
A262 3.54
A478 3.57 7. Limitationsofthenewrelationandrequired
A1413 2.93 testingoftheformalism
A1795 3.21
A1991 4.32 Althoughthenewrelationcanrecoverthemassoftheclusterto
A2029 4.04 reasonableprecision,thereisasevereflawintheempiricallaw.
RXJ1159 1.7 Both the gravitational potential and acceleration are large near
MKW4 2.54 stars and black holes. Therefore we have |∇Φ| + Φ >> 1 and
A383 4.32 Φ >>1.Thisresultsin a0 Φ0
Φ
A907 3.48 0
A2390 1.66 (cid:34) (cid:32)|∇Φ| Φ (cid:33) (cid:32) Φ (cid:33)(cid:35)
µ + −µ →1−1→0. (35)
a Φ Φ
0 0 0
whereρ isthecriticaldensity.
crit
andthusEquation28becomes,
We can see from the dynamical mass plots (Figures 11-16)
for each cluster that on average the new relation can provide a
reasonableboost,oratleastareasonablematchtotheNFWpro- ∇Φ=∇Φ /0→∞. (36)
N
file, for the sample. The most noticeable outlier from our new
relationisA2390,whichhasaverypoorfit.A2390isthelargest Therefore, although this new formula can cover cluster and
clusterandthushasthelargestgravitationalpotential.Thisnew galaxy scales, it inevitably fails on the stellar scale as the pre-
relation,inthiscurrentfrom,hasaprobleminveryhighgravita- dicted gravity tends towards infinity. This problem is inherent
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