Collisionless"Boltzmann"Eq"(="Vlasov"eq)""""""

Transcription

1 CollisionlessBoltzmannEq(=Vlasoveq) S+Gsec3.4(notcovering3.4.2) Whenconsideringthestructureofgalaxies,onecannotfolloweach individualstar(10 11 ofthem!), ConsiderinsteadstellardensityandvelocitydistribuLons.However,afluid modelisnotreallyappropriatesinceafluidelementhasasinglevelocity, whichismaintainedbyparlclepparlclecollisionsonascalemuchsmaller thantheelement. Forstarsinthegalaxy,thisisnot$true$Pstellarcollisionsareveryrare,and evenencounterswherethegravitalonalfieldofanindividualstaris importantindeterminingthemolonofanotherareveryinfrequent Sotakingthistoitslimit,treateachparLcleasbeingcollisionless,moving undertheinfluenceofthemeanpotenlalgeneratedbyalltheother parlclesinthesystemφ(x,t) 1 CollisionlessBoltzmannEqseeMBWsec5.4.2S&G 3.4 ThedistribuLonfuncLonisdefinedsuchthatf(r,v,t)d 3 xd 3 v$specifies thenumberofstarsinsidethevolumeofphasespaced 3 xd 3 v$ centeredon(x,v)atlmet AlternaLvelyf(r,v,t)istheprobabilitythatarandomlychosenstarhas thespecifiedsetofcoordinatespfisascalar AtLmet,afulldescripLonofthestateofthissystemisgivenby specifyingf(x,v,t)d 3 xd 3 v f(x,$v,t)iscalledthe distribulonfunclon (or phasespacenumber density )in6dimensions(xandv)ofthesystem. f 0sincenonegaLvestardensiLes SincethepotenLalissmooth,nearbyparLclesinphasespacemove 2 togetherppfluidapprox.

2 CollisionlessBoltzmannEqseeMBWsec5.4.2S&G3.4 Foracollisionlessstellarsystemindynamicequilibrium,the gravitalonalpotenlal,φ,relatestothephasepspacedistribulonof stellartracersf(x,$v,t),viathecollisionlessboltzmannequalon numberdensityofparlcles:n(x,t)= f(x,v,t)d 3 v$ averagevelocity:$$ <v(x,t)>= f(x,v,t)v$d 3 v/ f(x,v,t)d 3 v=$ (1/n(x,t)) f(x,v,t)vd 3 v$ $ bold$variables$are$vectors$ $ 3 SeeS&Gsec3.4 ThecollisionlessBoltzmannequaLon(CBE)islikethe equalonofconlnuity, dn/dt= n/ t+ (nv) x=0(s&g3.83)e.gparlcles areconserved butitallowsforchangesinvelocityandrelatesthechanges inf(x,v,t)totheforcesaclngonindividualstars Inonedimension,theCBEis(DerivaLonpg142ofS&G) f/ t+v f/ xp[ φ(x,t)/ x] f/ v=0(3.86) holdsifstarsareneithercreatednordestroyed,andif theychangetheirposilonsandvelocilessmoothly. thecbeimpliesthatthephasepspacedensityarounda givenparlcleremainsconstant 4

3 AnalogywithGasPconLnuityeqseeMBWsec4.1.4 ρ/ t+ (ρv)=0whichisequivto ρ/ t+v ρ=0 Intheabsenceofencountersf salsfiestheconlnuityeq,flowis smooth,starsdonojumpdisconlnuouslyinphasespace ConLnuityequaLon: definew=(x,v)pair(generalizeto3pd) dw/dt=(v,p φ) 6Pdimensionalspace df/dt=0 f/ t+ 6 (f dw/dt)=0 5 AnalogyofStellarSystemstoGases PDiscussionduetoMarkWhiile Similari7es$:$$ comprisemany,interaclngobjectswhichactaspoints(separalon>>size) canbedescribedbydistribulonsinspaceandvelocityegmaxwellianvelocity distribulons;uniformdensity;sphericallyconcentratedetc. StarsoratomsareneithercreatednordestroyedPPtheybothobeyconLnuity equalonspnotreallytrue,galaxiesaregrowingsystems! AllinteracLonsaswellasthesystemasawholeobeysconservaLonlaws(egenergy, momentum)if$isolated But$$: TherelaLveimportanceofshortandlongrangeforcesis!radically!different: atomsinteractonlywiththeirneighbors starsinteractconlnuouslywiththeenlreensembleviathelongrange airaclveforceofgravity eguniformmedium:f~g(ρ dv)/r 2, ;dv~r 2 dr;f ~ρdr $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$~$equal$force$from$all$distances$$ 6

4 AnalogyofStellarSystemstoGases PDiscussionduetoMarkWhiile TherelaLvefrequencyofstrongencountersisradicallydifferent: PPforatoms,encountersarefrequentandallarestrong(ieδV~V) PPforstars,pairwiseencountersareveryrare,andthestarsmoveinthesmooth globalpotenlal(e.g.s+g3.2) Someparallelsbetweengas(fluid)dynamicsandstellardynamics:manyofthe sameequalonscanbeusedaswellas: PPP>conceptssuchasTemperatureandPressurecanbeappliedtostellarsystems PPP>weuseanalogstotheequaLonsoffluiddynamicsandhydrostaLcs therearealsosomeintereslngdifferences PPP>pressuresinstellarsystemscanbeanisotropic PPP>selfPgravitaLngstellarsystemshavenegaLvespecificheat 2K+U=0!E=K+U=PK=P3NkT/2!C=dE/dT=P3Nk/2<0 andevolveawayfromuniformtemperature. 7 FullUpEquaLonsofMoLonPStarsasanIdealFluid (SS+Gpgs140P144,MBWpg163) ConLnuityequaLon(parLclesnotcreatedordestroyed) dρ/dt+ρ.v=0;dρ/dt+d(ρv)/dr=0 Eq'sofmoLon(Eulerseq) dv/dt=p P/ρ Φ& & Poissonseq 2 Φ(r)=P4πGρ(r)(examplepotenLal) 8

5 CollisionlessBoltzmannEq Thisresultsin(S+Gpg143) theflowofstellarphasepointsthroughphasespaceis incompressible thephasepspacedensityofpointsaround agivenstarisalwaysthesame ThedistribuLonfuncLonf isafunclonofsevenvariables (t,x,v),sosolvingthecollisionlessboltzmannequalonin generalishard.soneedeithersimplifyingassumplons (usuallysymmetry),ortrytogetinsightsbytaking momentsoftheequalon. TakemomentsofaneqPPmulLpyingf bypowersofv 9 CollisionlessBoltzmannEqPMoments(pg143) n(x,t)asthenumberdensityofstarsatposilonx TheaveragevalueofaquanLtyQ!intheneighborhoodofx$ isq(x,!t)!1/n(x,!t) Q!f!d 3 v,!!n(x,!t) f!d 3 v$=ρ(x,!t)/m! SeungQ=1wegetthezerothmoment n/ t+ (nv)/ x=0;thesameeqasconlnuityequalonofa fluid SeungQ=v firstmoment: n v/dt+nv v/dx=pn φ/ x / x(nσ 2 ) 10

6 CollisionlessBoltzmannEqPMoments(pg143) SimilartoEuler'seqforgas σ is the velocity dispersion <v 2 (x,!t)>=<v(x,!t)> 2 +σ 2 ; where v is the 'coherent, streaming, velocites and σ is the 'random' velocity But unlike fluids, we do not have thermodynamics to help out... & Ingeneral,theJeansequaLonshavenineunknowns (threestreamingmoloncomponentsv!! andsixindependentcomponentsofσ) 11 JeansEquaLonsMBWsec5.4.3 SincefisafuncLonof7variables,obtainingasoluLonischallenging Takemoments(e.g.integrateoverallv) letnbethespacedensityof'stars' n/ t+ (n<v i > )/ x i =0;conLnuityeq.zerothmoment firstmoment(mullplybyvandintegrateoverallvelociles) (n<v j > / t)+ (n<v i v j >)/ x i +n φ/ x j =0 equivalently n (<v j > / t)+n<v i > <v j >/ x i =Pn φ/ x j P (nσ 2 ij )/ x i where nistheintegralovervelocityoff;n= fd 3 v <v i > isthemeanvelocityinthei th direclon=(1/n) fv i d 3 v σ 2 ij=<(v i P<v i >)(v j P<v j >)> stresstensor =<v i v j >P<v i ><v j > 12

7 CollisionlessBoltzmannEq astronomicalstructuralandkinemalcobservalonsprovide informalononlyabouttheprojec7ons$of$phase$space$distribu7ons$ along$lines$of$sight, limilngknowledgeaboutfandhencealsoaboutφ. ThereforealleffortstotranslateexisLngdatasetsintoconstraintson CBEinvolvesimplifyingassumpLons. dynamicequilibrium, symmetryassumplons parlcularfunclonalformsforthedistribulonfunclonand/orthe gravitalonalpotenlal. 13 JeansEq n (<v j > / t)+n<v i > <v j >/ x i =Pn φ/ x j P (nσ 2 ij )/ x i Sowhataretheseterms?? Gasanalogy:Euler seqofmolon ρ v/ t+ρ (v. )v =P P ρ Φ n φ/ x j :gravitalonalpressuregradient nσ 2 stresstensor islikeapressure,butmaybe ij anisotropic,allowingfordifferentpressuresin differentdireclonspimportantinelliplcalgalaxies andbulges'pressuresupported'systems(withabit ofcoordinatetransformonecanmakethis symmetrice.g.σ 2 14 ij=σ 2 ji)

8 JeansEqCont n v/dt+nv v/dx=pn φ/ xp / x(nσ 2 ) SimplificaLons:assumeisotropy,steadystate,nonProtaLng!termsonthele vanish JeanEqbecomes:Pn!φ =!(nσ 2 )& using Poisson eq:! 2 φ = 4πGρ Generally, solve for ρ (massdensity)& 15 SphericalsystemsPEllipLcalGalaxiesandGlobular Clusters ForasteadyPstatenonProtaLngsphericalsystem,theJeans equalonssimplifiesto (1/n)d/dr(n<v 2 r >)+ 2β<v2 r >/r=pgm(r)/r2 wheredφ/dr=gm tot (r)/r 2 andn(r),<v 2 r >andβ(r)describes the3pdimensionaldensity,radialvelocitydispersionand$ orbital$anisotropyofthetracercomponent(stars) β(r)=1p<v 2 θ >/<v2 r >;β=0isisotropic,β=1isradial 16

9 SphericalsystemsPEllipLcalGalaxiesandGlobular Clusters Wecanthendescribethemassprofileintermsof observablesas GM(r)=Pr<v 2 r>[(dlnn/dlnr)+(dln<v 2 r>/dlnr)+2β] wherev 2 ristheradialvelocityprofile,nisthedensityand β=1p<v θ > 2 /<v r > 2 hastobemodeled ThiscanbealternaLvelywriienas M(r)=[V 2 r/g]+(σ r2 r/g)[p(dlnn/dlnr)p(dlnσ r2 /dlnr)p(1pσ θ 2/ σ r2 )P(1Pσ φ 2/ σ r2 )] wherethesubsriptsr,θ andφ refertosphericalcoordinates Thiswillbecomeveryimportant forelliplcalgalaxies 17 whileapparentlysimplewehave3setsof unknowns<v 2 r >,β(r),n(r) and2setsofobservablesi(r)psurfacebrightnessof radialon(insomewavelengthband)andthelines ofsightprojectedvelocityfield(firstmomentis velocitydispersion) Itturnsoutthatonehasto'forwardfit'Pe.g. proposeaparlcularformfortheunknownsandfit forthem.

10 JeansEquaLons:AnotherFormulaLon JeansequaLonsfollowfromthe collisionlessboltzmannequalon; Binney&Tremaine(1987),MBW S+Gsec3.4. cylindricalcoordinatesandassuming anaxipsymmetricandsteadypstate system,theacceleralonsinthe radial(r)andverlcal(z)direclons canbeexpressedintermsof observablequanlles: thestellarnumberdensitydistribulon ν * And5velocitycomponents ParotaLonalvelocityv φ P4componentsofrandomvelociLes (velocitydispersioncomponents) σ φφ,σ RR,σ ZZ,σ RZ wherea Z,a R areacceleralonsinthe appropriatedireclonsp giventhesevalues(whichare thegradientofthegravitalonalpotenlal), thedarkmaiercontribuloncanbe eslmateda eraccounlngforthe contribulonfromvisiblemaier 19 UseofJeansEqs:SurfacemassdensitynearSun Selectatracer!populaLonofstarsandmeasureitsdensityn(z) atheightz!abovethedisk smidplane Measuren(z)andv z Lookinghighabovetheplane,v z n(z) 0; Using the moment equation v/ t+$v$ v/ z= φ/ x 1/n / x[nσ 2 (z,!t)]!the!terms!in!blue!vanish! andassumingthingsdonotchangewithlme n(z) φ/ z 1/ / z[nσ 2 (z] Using Poisson's eq 4πGρ(R,!z)= 2 φ(r,!z)andgoingtocynlidrical coordinates 4πGρ(R,!z)= φ 2/ z 2 +1/R!( φ/ R![R φ/ R])!eq!3.92! 20

11 UseofJeansEqs:SurfacemassdensitynearSunSec 3.4.1inS&G Now use Jeans eq: nf z =P (nσ 2 z)/ z+(1/r) / R(Rnσ 2 zr); ifr+zareseparable, e.gφ(r,z) =φ(r)+φ(z) then σ 2 zr ~0 and voila! (eq 3.94 in S+G) Σ(z) =-(1/2πGn) (nσ 2 z)/ z; Σ(<!z)isthesurfacemassdensityΣ(<!z)needtoobservethe numberdensitydistribulonofsometracerofthepotenlal abovetheplane[goesasexp(pz/z 0 )] anditsvelocitydispersiondistribulonperpendiculartothe planetogetthemassdensity. 21 MoLonPerpendiculartothePlanePAlternateAnalysisP (S+Gpgs140P144,MBWpg163) thenthe1pdpoisson'seq4πgρ tot (z,r)=d 2 φ(z,r)/dz 2 whereρ tot isthetotalmassdensitypputitalltogethertoget 4πGρ tot (z,r)=pdk(z)/dz(s+g3.93) d/dz[n * (z)σ z (z) 2 ]=n * (z)k(z) togetthedatatosolvethis,wehavetodeterminen * (z)andσ z (z)for thetracerpopulalons(s) 22

12 UseofJeansEqForGalacLcDynamics AcceleraLonsinthezdirecLonfromtheSloan digitalskysurveyfor 1)allmaier(toppanel) 2) known'baryonsonly(middlepanel) 3)raLoofthe2(boiompanel) BasedonfullPupnumericalsimulaLonfrom cosmologicalcondilonsofamwlikegalaxypthis 'predicts'whata Z shouldbenearthesun (Loebmanetal2012) ComparewithresultsfromJeanseq(ν isdensityof tracers,v φ istheazimuthalvelocity(rotalon)) 23 WhatDoesOneExpectTheDataToLookLike NowusingJeanseq 1kpcx1kpcbins;acceleraLonunitsof2.9x10 P13 km/sec 2 No7ce$that$it$is$not$ smooth$or$monotonic$ $$$$$and$that$the$simula7on$is$ neither$perfectly$ rota7onally$symmetric$nor$ steady$state.. errorsareontheorderof 20P30%Pfigureshows comparisonoftrueradial andzacceleralons comparedtojeansmodel fits 24

13 Jeans(ConLnued) Usingdynamicaldataandvelocitydata,getesLmateof surfacemassdensityinmw Σ total ~70+/P6M /pc 2 Σ disk ~48+/P9M /pc 2 Σ star ~35M /pc 2 Σ gas ~13M /pc 2 weknowthatthereisveryliilelightinthehalosodirect evidencefordarkmaier 25 NowontotheLocal group!! End