Galactic mass distribution without dark matter or

1astro-ph/0309762v2,revised1Mar2007GalacticmassdistributionwithoutdarkmatterormodifiedNewtonianmechanicsKennethFNicholson,retiredengineerkeywords:galaxies,rotation,massdistribution,darkmatterAbstractGiventhedimensions(includingthickness)ofagalaxyanditsrotationprofile,amethodisshownthatfindsthemassanddensitydistributioninthedefinedenvelopethatwillcausethatrotationprofilewithnear-exactspeedmatches.Newton'slawisunchanged.Surface-lightintensityanddarkmatterarenotneeded.Resultsarepresentedindimensionlessplotsallowingeasycomparisonsofgalaxies.Ascomparedwiththepreviousversionofthispaperthemethodsarethesame,butsomedataarepresentedinbetterdimensionlessparameters.Alsothepartonthicknessrepresentationissimplifiedandextended,thecontentsarerearrangedtohavealogicalbuildupintheproblemdevelopment,andmoreexamplesareadded.1.IntroductionMethodsusedforfindingmassdistributionofagalaxyfromrotationspeedshavebeenmostlythoseusingdark-mattersphericalshellstomakeupthemassneededbeyondtheassumedloadingused(asdonebyvanAlbadaetal,1985),orthosethatmodifyNewton'slawinawaytoapproximatelymatchmeasureddata(forexampleMOND,Milgrom,1983).Howevertheuseofthedark-matterspheresisnotacorrectapplicationofNewton'slaw,and(excludingrelativisticeffects)modificationstoNewton'slawhavenotbeenjustified.Intheapplicationsofdark-mattersphericalshellsitisassumedthatallgalaxiesareintwoparts,athindiskwithanexponentialSMDloading(acorrectsolutionforrotationspeedsbyFreeman,1970)andaseriesofsphericalshellscenteredonthegalaxycenter.Thetwoconstantsneededfortheexponentialdiskloadingaresupposedtobeobtainedfromsurfacelightintensityandareasonablemass/lightratio.Howeverfewgalaxiesifanyhavediskswithanexponentialloadinglikethat,andthesphericalshellshadtobeaddedtohaveenoughmassintheouterradiitocausesufficientspeedstomatchexperiment.Sotheassumptionthatallgalaxydisksmusthavethattypeofexponentialloadingisfalse,andsofartherehasbeennoevidenceofthesphericalshells(calleddarkmatter)exceptthattheycanbeusedmathematicallytogetanapproximatematchforthemeasuredrotationspeeds.Thestatementoftenmadethattheshellsarenecessarytosatisfythegravitationaleffectsforgalaxiesisalsofalse,aswillbeshownhere.Furtherproblemsareassociatedwiththeuseoftheaddedsphericalshells.Multiplesolutionsarepossibleforagivenrotationprofile,evenallowingsolutionswithallspheresandnodisk,buttheremustbeonlyonemassdistributioninagivengalaxygeometrythatcancauseagivenrotationprofile.Alsotheattractionsonashellofthediskpartsthatareoutside,andofdiskpartsandspheresinside,havenotbeenincluded.Theseeffectswouldcausetheshellstobepulledintothedisk,orrequireorbitingcomponentsoftheshells.Ifthereweresuchorbitingcomponents,theireffectswhenpassingthroughthediskwouldbeeasilyseennearoursolarsystem.ModificationstoNewton'slawhaverequiredanincreaseinstrengthofthegravityfieldasitgetsweakwithdistance,aneffectthatisnotshownbyexperimentforanythingthatnormallydecreaseswiththesquareofthedistance.IfitcanbeshownthatNewton'slawcanbeusedcorrectly,tofind,withonlyonepossiblesolution,themassdistributioninsideagalaxy'senvelopethatcausesameasuredrotationprofile,2therewouldbenobasisfortheuseofextramatteroutsideagalaxy'senvelope,orforthemodificationofNewton'slaw.Itcouldthenbesaidwithconfidence(atleastforgalaxies)thatdarkmatterdoesn'texist,andthatNewton'slaw(excludingrelativisticeffects)stillworksfine.InthispaperNewton'slawisunchanged,witheachparticleattractingeveryother,andallmatterinagalaxyremainsinsideanenvelopedefinedbyitsdimensions.Likemanyotherpapersonthissubject,itisassumedthatthegalaxiesareaxisymmetricandsymmetricinz(normaltothedisk),therotationspeedsaremeasuredatthecenterplane,andthereisadefinitegalaxyrimwherethelastspeedreadingisobtained.Alsoitisassumedthatthereissolidrotationinsidethemostinnerspeedreading.Actually,thespeedsareprobablymeasuredatthesurface,thegalaxiesarenotoftencloselysymmetric,andtheremaybesomemorematteroutsidethelastreading,buttheseassumptionsallowareasonablesolution.Onlygravityeffectsareconsidered,withnogaspressure,shearforce,electromagnetic,orrelativisticeffects.2.MethodAgalaxyisrepresentedbyaseriesofrings,eachsplitinto720constant-density(equivalent)segmentsrepresentedbyrodswithaxesnormaltothegalaxyplane.Usinganestimateddensitydistributioninthezdirectionnormaltothediskplane,anequivalentthicknessisfoundthathasaconstantdensityinz.Thisconstant-densityequivalentthicknesswillcausearingtohavethesameSMDandnearlythesamegravitationaleffectsastheonewiththeestimateddensitydistribution.Withitsequivalentthicknessandaunitdensity,eachringiwillcauseanaccelerationonatestmassatradiusjofamounta(i,j).ThetotalaccelerationA(j)atradiusj,isthenfoundbyaddingtheeffectsofallringswiththeirappropriatedensities.Thisistheforwardproblem,findingtheaccelerationprofile(andthusrotationspeeds)causedbyagivenmassdistribution.Thereverseproblem,massdistributionfromtherotationprofile,canbedoneinasimilarway,sinceallnecessarydataareknown.Thematrixequationsare:Theseequationsshowthatforagiventhicknessanddensitydistributiontherecanbeonlyonerotationprofile.Alsoforagiventhicknessandrotatio