TWO NEW D. MATTER PROFILES FOR MILKY WAY HALO GOT FROM ROTATION CURVE M. Abarca

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac TWO NW D.M. DNSITY PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV Autho Mnuel Ac Henndez emil mche1@gmil.com 1. ASTRACT.... INTRODUCTION... 3 3. OSRVATIONAL DATA FROM SOFU. 015 PAPR FOR MILKY WAY... 4 3.1 POWR RGRSSION TO ROTATION CURV... 5 4. DIRCT FORMULA FOR DNSITY ON MILKY WAY HALO GOT FROM ROTATION CURV... 6 4.1 THORTICAL DVLOPPMNT FOR GALACTIC HALOS... 6 4. DIRCT DNSITY FOR MILKY WAY HALO... 6 5. DARK MATTR DNSITY AS POWR OF GRAVITATIONAL FILD... 7 5.1 GRAVITATIONAL FILD THROUGH VIRIAL THORM... 7 5. DARK MATTR DNSITY AS POWR OF GRAVITATIONAL FILD... 7 6. RATIO ARYONIC MASS VRSUS MASS DPNDING ON RADIUS FOR MILKY WAY... 8 7. COMPARISON TWN DIRCT DNSITY AND NFW DARK MATTR DNSITY... 9 8. MASSS IN MILKY WAY... 10 8.1 DYNAMICAL MASS UP TO 44 KPC... 10 8. DYNAMICAL THROUGH DYNAMICAL MASSS... 10 8.3 HALO MASSS THROUGH NFW PROFIL... 11 8.4 DYNAMICAL VRSUS NFW... 1 8.5 DYNAMICAL CORONA HALO COMPARD TO NFW CORONA HALO MASSS... 13 9. D.M. DNSITY AS POWR OF IN MILKY WAY VRSUS D.M. DNSITY AS POWR OF IN M31.. 14 10. HALO RADIUS OF MILKY WAY & M31 ACCORDING D.M. AS POWR OF THORY... 16 10.1 MASSS IN MILKY WAY & M31 AT XTNDD HALOS... 17 11. CONCLUSION... 17 1. ILIOGRAPHYC RFRNCS... 18 1

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 1. ASTRACT In this wok hs een clculted two new density pofiles inside hlo egion of Milky Wy, MW heefte, nd it hs een demonstted tht oth ones e mthemticlly equivlents. The fist is clled diect density ecuse it is got diectly fom velocity s powe egession of dius in hlo ottion cuve. In othe wods velocity of ottion cuve depend on dius s powe function. In fct this function got y powe egession hs coeltion coefficient of 0,85. The second one, density s powe of, is gvittionl field, hs een intoduced y utho in pevious ppes, [8] Ac,M.016, whee it hs een used to study in sevel glxies. It is clled s powe of ecuse density depend on s powe function. Hypothesis which is the sis of theoy is tht is geneted loclly y the own gvittionl field ccoding powe lw. density = A ^ whee A& e coefficients nd is gvittionl intensity of field. To find esons tht utho hs to do so ding sttement, ede cn consult [1] Ac,M.014. Dk mtte model y quntum vcuum. [8] Ac,M.016. Dk mtte density on ig glxies depend on gvittionl field s Univesl lw nd othe ppes quoted in iliogphy. iefly will e explined method followed to develop this ppe. Fistly e pesented ottion cuve nd tle with dt points inside MW hlo. These dt come fom [5] Sofue,Y.015. In ddition it is got powe egession fo ottion cuve points in hlo egion whose function is v = ^ getting coeltion coefficient igge thn 0,85. In fouth chpte it is developed mthemticl method to get new density depending on dius clled diect density ecuse it is got diectly fom powe egession function got in chpte thee. In fifth chpte it hs een demonstted tht powe egession function fo ottion cuve is mthemticlly equivlent tht density depend on gvittionl field, s powe function i.e. density = A ^ whee A& e cleed up depending on & (pmetes of powe egession of ottion cuve). In sixth chpte it hs een got tht fo dius igge thn 44 kpc tio yonic density vesus density is unde % so it is esonle to conside negligile yonic density in ode to develop theoy intoduced in this wok. The seventh is shot chpte whee is comped diect density got with NFW density pofile fitted y Sofue in his ppe. [5] Sofue, Y.015. Reltive diffeences etween oth density pofiles e unde 5% inside min pt of hlo dominion. In ddition it is explined why NFW pofile is igge thn diect thoughout dominion. ighth chpte, clled Msses in Milky Wy, is dedicted to clculte msses though dynmicl method nd NFW pofile. In ddition oth methods e comped t diffeent dius. Results show tht eltive diffeences of msses e unde 0% inside min pt of dius dominion. In ninth chpte is comped density s powe in MW with density s powe in M31, which ws pulished in [ 11] Ac,M.016. Results show tht t specific, oth densities e vey simil. Reltive diffeences e unde 15 % inside min pt of dominion. This fct suppot stongly utho hypothesis out s powe of s Univesl lw. In tenth chpte, it is intoduced new definition fo hlo dius. Accoding theoy geneted y gvittionl field, it is ight to definite hlo s egion whee gvittionl field dominte ove gvittionl field of glctic neighou. So it is clculted tht hlo dius fo MW is 9 kpc nd hlo dius fo M31 is 478 kpc. Also e clculted totl msses which elong these new dius nd it is clculted new tio yonic mss vs totl mss. Such tio is 18% fo MW nd 8 % fo M31.

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac. INTRODUCTION As ede knows Milky Wy is the twin glxy of M31 in Locl Goup of glxies. Its disk dius is ppoximtely 0 kpc nd ccoding [5] Sofue, Y. 015 its yonic mss is M ARYONIC = 1,37 10 11 M SUN In pevious ppe [6] Ac, M.016, utho hs studied inside MW hlo though s powe of. Howeve in such ppe density used it ws NFW pofile povided y [5] Sofue, Y. 015, whees in this ppe density pofile hs een got diectly fom powe egession function t ottion cuve in hlo egion. This new pofile hs een clled diect density ecuse this pofile is fitted diectly fom dt mesues inside hlo egion. In this wok dius dominion egin t 44 kpc ecuse t this distnce yonic density is negligile s it will e shown in chpte six. Theefoe the only one kind of mtte in hlo egion it is supposed to e non yonic dk mtte. It is known tht thee is yonic dk mtte such us gint plnets, cold gs clouds, own dwfs ut this kind of is moe pole to e plced inside glctic disk. Rede cn consult: [ 1] Nieuwenhuizen,T.M. 010. [ 13] Nieuwenhuizen,T.M. 01. [ 14] Nieuwenhuizen,T.M. 010 [ 15] Wyzykowski,L.010. [ 16] M.R.S. Hwkins 015.In fct thee e n impotnt mount of eseches in this wy ecuse yonic nd non yonic e open polems still. Howeve theoy intoduced in [1] Ac, M.014. Dk mtte model y quntum vcuum nd developed in othes ppes quoted in iliogphy efes kind of weid mtte which is geneted y the own gvittionl field. D M ( ) A Theefoe it is needed to conside dius dominion whee yonic mtte would e negligile. In fct ccoding [5] Sofue, Y. 015 dt, in chpte six will e got tht fo dius igge thn 44 kpc yonic mtte density is unde,3 % egding density. This is the eson why dius dominion in this wok is fom 44 kpc up to 190 kpc. In ppe [1] Ac,M.014, it ws postulted tht density depends on gvittionl field. Futhe ppes [] Ac,M.015 nd othes hve studied density s powe of gvittionl field in sevel glxies: M31, NGC3198 nd othes glxies. Results got suppot such hypothesis. D M ( ) A Summising, in this ppe it is got new density clled diect which is clculted diectly fom powe egession function in hlo of ottion cuve of MW As it is supposed tht fo dius igge thn 44 kpc yonic mtte is negligile then diect pofile is pecise function to know non yonic density. In dittion s it is supposed hypothesis powe egession fo ottion cuve in hlo is equivlent lw A& depend on &. D M ( ) A D M ( ) A it is found tht, in hlo s well, whose coefficients As consequence of s powe it is ight to definite hlo dius s egion whee own gvittionl field domintes ove gvittionl field of neighou glxy nd theefoe it is needed to eclculte totl mss ssocited this new hlo. 3

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 3. OSRVATIONAL DATA FROM SOFU. 015 PAPR FOR MILKY WAY Gphic come fom [5] Sofue,Y. 015. dius vel kpc km/s 43,9 1,951+0 50,7 1,754+0 63,85 1,768+0 75,9 1,878+0 91,1 1,790+0 110,3 1,549+0 133,7 1,93+0 158,9 1,059+0 19,5 1,336+0 Dt e lck points. In chpte six will e shown eson why dominion dt egin t 44 kpc in this wok, despite the fct tht it is supposed dius of glctic disk is 0 kpc. 4

Velocity m/s TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 3.1 POWR RGRSSION TO ROTATION CURV It is seen tht expeimentl mesues of ottion cuve hs vey good fitted cuve y powe egession. In pticul coefficients of v e in tle elow. Units e into I.S. Powe egession fo Milky Wy ot. cuve V=*^ 3,4989549+1-3,45408589-01 Coeltion coeff. 0,85 Dt fitted e in gey columns elow. In thid column is shown esults of fitted velocity nd fouth column shows eltive diffeence etween mesues nd fitted esults. Coeltion coefficient is 0.85 which is good coeltion. dius vel vel fitted el diff. dius m m/s m/s % kpc 1,3554+1 1,951+05,01650+05-3,36 43,9 1,56507+1 1,754+05 1,91948+05-9,43 50,7 1,970+1 1,768+05 1,77393+05-0,34 63,85,3405+1 1,878+05 1,67193+05 10,97 75,90,81107+1 1,790+05 1,57059+05 1,6 91,10 3,40353+1 1,549+05 1,47100+05 5,04 110,30 4,1558+1 1,93+05 1,37718+05-6,51 133,70 4,90318+1 1,059+05 1,9808+05 -,58 158,90 5,93997+1 1,336+05 1,1553+05 9,0 19,50 elow is shown gphic with mesues dt nd powe egession function.,500+05 Mesues ottion cuve,000+05 1,500+05 1,000+05 5,000+04 y = 3,4989549+1x -3,45408589-01 R² = 7,166541560-01 0,000+00 0 5+1 1+ Rdius m A coefficient of 0,85 is good coeltion. Theefoe this vlue suppot hypothesis tht ottion cuve of Milky Wy follows lw v whee & e witten ove. 5

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 4. DIRCT FORMULA FOR DNSITY ON MILKY WAY HALO GOT FROM ROTATION CURV 4.1 THORTICAL DVLOPPMNT FOR GALACTIC HALOS Outside disk egion, ottion cuve it is fitted y powe egession with high coeltion coefficient ccoding fomul v velocity esults. As v R M ( ) epesents totl mss enclosed y sphee with dius, y sustitution of G v R M G G 1 If it is consideed outside egion of disk whee yonic mtte is negligile egding dk mtte it is possile to clculte density y simple deivtive. In next chpte will e shown tht fo > 44 kpc yonic mtte is negligile. As density of D.M. is dm D whee dv ( 1) d d m nd dv 4 d so G D ( 1) 4G ( 1) Witing L esults 4G D ( ) L. In cse = -1/ density is ceo which is Keplein ottion. 4. DIRCT DNSITY FOR MILKY WAY HALO Pmetes & fom powe egession of Milky Wy ottion cuve llow clculte esily diect density. Diect density fo Milky Wy hlo 40 < < 300 kpc D ( ) L kg/m^3 L = 4,58164747068706 10 33 - = -,6850817178 eside is such function nd tle. elow is shown esults of density inside its dominion. Clculus e into I.S. Rdius Rdius Diect kpc m kg/m^3 44,00 1,35771+1 8,7388768-4 50,00 1,5485+1 5,87003998-4 70,00,15999+1,37834768-4 90,00,77713+1 1,1119348-4 110,00 3,3947+1 7,0665609-5 130,00 4,01141+1 4,5135967-5 150,00 4,6855+1 3,077804-5 170,00 5,4569+1,19571888-5 190,00 5,8683+1 1,688611-5 10,00 6,47997+1 1,4498977-5 6

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 5. DARK MATTR DNSITY AS POWR OF GRAVITATIONAL FILD As independent vile fo this function is, gvittionl field, peviously will e studied fomul fo in the following pgph. 5.1 GRAVITATIONAL FILD THROUGH VIRIAL THORM As it is known totl gvittionl field my e clculted though Viil theoem, fomul = v /R whose I.S. unit is m/s is well known. Heefte, viil gvittionl field,, got though this fomul will e clled. y sustitution of v in fomul v it is ight to get 1 iefly 1 Rdius Rdius viil m kpc m/s^ 1,35771+1 44,00,9915-11 1,5485+1 50,00,411559-11,15999+1 70,00 1,367918-11,77713+1 90,00 8,956593-1 3,3947+1 110,00 6,386864-1 4,01141+1 130,00 4,819854-1 4,6855+1 150,00 3,78714-1 5,4569+1 170,00 3,066991-1 5,8683+1 190,00,54817-1 6,47997+1 10,00,148186-1 5. DARK MATTR DNSITY AS POWR OF GRAVITATIONAL FILD Accoding hypothesis dk mtte y quntum vcuum D A. Whee A & e pmetes to e clculted. This hypothesis hs een widely studied y utho in pevious ppes. [1] Ac,M. [] Ac,M. [8] Ac,M. [9] Ac,M. [10] Ac,M. ( 1) As it is known diect density D depend on & pmetes which come fom powe 4G egession fomul fo velocity. In pevious pgph hs een shown fomul fo gvittionl field 1 which depend on & s well. Though simple mthemticl tetment it is possile to get A & to find function of density depending on i.e. D A Specificlly fomuls e A 1 ( 1) & 4G. 1 Milky Wy A D A 4,867931919745 10-7 1,593443097574 Accoding pmetes & got in pevious chpte, A& pmetes e: 7

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac elow is tulted density s powe of nd diect density, oth e identicl s it ws expected. 1 D ( ) L D A Rdius Rdius viil Diect pw kpc m m/s^ kg/m^3 kg/m^3 44,00 1,35771+1,9915-11 8,73888-4 8,73888-4 50,00 1,5485+1,41155893-11 5,870040-4 5,870040-4 70,00,15999+1 1,3679184-11,378348-4,378348-4 90,00,77713+1 8,95659316-1 1,11193-4 1,11193-4 110,00 3,3947+1 6,38686417-1 7,06656-5 7,06656-5 130,00 4,01141+1 4,81985393-1 4,51360-5 4,51360-5 150,00 4,6855+1 3,7871444-1 3,07780-5 3,07780-5 170,00 5,4569+1 3,06699080-1,195719-5,195719-5 190,00 5,8683+1,548170-1 1,6886-5 1,6886-5 10,00 6,47997+1,14818606-1 1,44990-5 1,44990-5 As conclusion, in this chpte hs een demonstted tht powe lw fo velocity v is mthemticlly equivlent powe lw fo density depending on. D A 6. RATIO ARYONIC MASS VRSUS MASS DPNDING ON RADIUS FOR MILKY WAY In this pgph will e estimted dius which is needed to conside negligile yonic density egding density in M31 glxy. [5] Accoding Sofue, Y. dt fo M31 disk e Milky Wy yonic Mss t disk d 0 M d 0 d M d = 1,1 10 11 Msun 5,73 kpc 1,134683098 kg/m Whee ) exp( / ) epesents supeficil density t disk. Totl mss disk is given y integtion of ( 0 d supeficil density fom ceo to infinite. M d ( ) d 0 In ode to compe yonic density nd density it is consideed diffeentil yonic mss nd diffeentil msses depending on dius. dm DISK ( ) d whee ) exp( / ) nd dm 4 D ( d whee D ) ( 0 d ( ) ( 1) 4G 0 d 8

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac It is defined tio function s quotient of oth diffeentil quntities dm Rtio dm DISK ( ) D ( ) Rdius Kpc Rdius Rtio () Diect kpc m kg/m^ kg/m^3 30 9,5710+0 1,41011-01 6,0406196-03,3137854-3 3 9,8744+0 1,108914-01 4,608405-03 1,9456453-3 34 1,04914+1 8,663196-0 3,0054470-03 1,6533634-3 36 1,11085+1 6,78553-0,1199366-03 1,418165-3 38 1,1757+1 5,198799-0 1,495387-03 1,64965-3 40 1,348+1 3,998107-0 1,05475-03 1,0686915-3 4 1,9599+1 3,061779-0 7,439851-04 9,3746997-4 44 1,35771+1,335781-0 5,478093-04 8,738877-4 46 1,4194+1 1,77579-0 3,701600-04 7,343009-4 48 1,48114+1 1,345663-0,610995-04 6,5500444-4 50 1,5485+1 1,016775-0 1,841703-04 5,8700400-4 5 1,60456+1 7,661980-03 1,99070-04 5,83985-4 Fo dius 44 kpc tio yonic mtte vesus is only,3 % theefoe is good ppoximtion to conside negligile yonic mss density egding density when dius is igge thn 44 kpc. This is the eson why in this wok dominion fo dius egin t 44 kpc 7. COMPARISON TWN DIRCT DNSITY AND NFW DARK MATTR DNSITY Accoding [5] Sofue,Y., 015. Pmetes of NFW pofile fo Milky Wy e Dk mtte density function pofile NFW Rs = 10,7.9 Kpc Do = 1,318 10-1 kg/m^3 D NFW ( R) D x (1 x) 0 Whee x= dius/ Rs Rs is clled length scle nd Do is density scle. elow e tulted NFW density nd diect density depending on dius oth. Thid column shows eltive diffeences, which oscillte etween 8% nd 15% thoughout dominion. powe NFW Rel diff. Rdius kg/m^3 kg/m^3 % kpc 8,7389-4 1,1461-3 7,815565 44,00 5,87004-4 8,19114-4 8,3366819 50,00,37835-4 3,31014-4 8,1496449 70,00 1,1119-4 1,65345-4 6,7473331 90,00 7,06656-5 9,41639-5 4,9546354 110,00 4,5136-5 5,86354-5 3,0437807 130,00 3,0778-5 3,89555-5 1,1071 150,00,1957-5,71848-5 19,99356 170,00 1,6883-5 1,97171-5 17,390065 190,00 1,4499-5 1,4756-5 15,608596 10,00 It is emkle the fct tht NFW pofile is igge thn diect pofile though the whole dominion. This is n impotnt fct nd in next pgph will e exposed some ides which in utho opinion might explin the fct tht diect pofile gives vlues of density it lowe thn vlues clculted y NFW pofile. As it ws pointed t intoduction, it is known tht thee is yonic dk mtte 9

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac such us gint plnets, cold gs clouds, own dwfs ut this kind of is moe pole to e plced inside glctic disk nd ulge. Rede cn consult these ppes out this open polem: [1] Nieuwenhuizen,T.M. 010. [ 13] Nieuwenhuizen,T.M. 01. [ 14] Nieuwenhuizen,T.M. 010 [ 15] Wyzykowski,L.010. [ 16] M.R.S. Hwkins 015 As it is known, NFW pofile is fitted ove ulge, disk nd glctic hlo nd tking in considetion tht thee is n unknown mount of yonic in ulge nd glctic disk it is ight to conclude tht NFW pofile is fitted though set of dt whose vlues include yonic nd non yonic, especilly in ulge nd disk. Theefoe it is ight to get function fitted, NFW, which poduce high vlues of thoughout its dominion, ulge, disk nd hlo. Howeve diect pofile, s it is got though powe egession function fitted with dt velocity in hlo egion, it is fitted only with non yonic.. 8. MASSS IN MILKY WAY 8.1 DYNAMICAL MASS UP TO 44 KPC It hs een chosen 44 kpc, ecuse t this dius yonic density is negligile egding density in Milky Wy Accoding powe egession lw fo velocity v ottion velocity is v =,0154 10 5 m/s t 44 kpc. And v R M DYNAMICAL ( 44kpc) = 4,15 10 11 Msun i.e. M TOTAL (< 44 kpc) = 4,15 10 11 Msun G In othe wods dynmicl mss is equivlent to totl mss. Accoding Sofue dt Milky Wy yonic mtte = 1,37 0,4 10 11 Msun M DYNAMICAL = Totl mss M31 msses up to 44 kpc yonic mss ulge + disk Dynmicl Dk mtte mss so dynmicl (<44 kpc) =,78 10 11 Msun. It is clled dynmicl ecuse it is got sutcting yonic to totl dynmicl mss. 4,15 10 11 Msun 1,37 10 11 Msun,78 10 11 Msun 8. DYNAMICAL THROUGH DYNAMICAL MASSS Powe egession fo M31 ot. cuve V= 3,4989549+1-3,45408589-01 Coeltion coeff. 0,85 As it ws shown in chpte fou, ottion cuve is fitted with high coeltion y this function elow. v whose pmete e in tle 10

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac Also it is known tht velocity esults v R M ( ) epesents totl mss enclosed y sphee with dius, y sustitution of G 1 v R M ( ). G G So it is possile to clculte enclosed y sphee with R dius y sutction of yonic mtte, tht ccoding Sofue, fo Milky Wy yonic mtte = 1,37 0,4 10 11 Msun whee Msun = 1,99 10 30 kg In tle elow, second column show totl o dynmicl mss inside sphee with dius R nd the thid column show clculted sutcting yonic mss to totl mss. M ARYONIC = 1,37 10 11 Msun. So clculted y this method is clled dynmicl. Velocity Dynmicl Mss M(<R) Dynmicl (<R) Rdius m/s Msun kpc,01544503+05 4,15+11,78+11 44,00 1,9890479+05 4,33+11,953+11 50,00 1,7189105+05 4,806+11 3,436+11 70,00 1,577137393+05 5,0+11 3,83+11 90,00 1,4737047+05 5,541+11 4,171+11 110,00 1,39048300+05 5,841+11 4,471+11 130,00 1,3396737+05 6,110+11 4,740+11 150,00 1,68403837+05 6,355+11 4,985+11 170,00 1,0987508+05 6,58+11 5,1+11 190,00 1,179838177+05 6,793+11 5,43+11 10,00 1,14363951+05 6,990+11 5,60+11 30,00 1,111437399+05 7,176+11 5,806+11 50,00 1,0850188+05 7,35+11 5,98+11 70,00 1,056344344+05 7,519+11 6,149+11 90,00 8.3 HALO MASSS THROUGH NFW PROFIL D0 Fomul of NFW pofile is D NFW ( R) whee x= R / Rs x (1 x) Dk mtte density pofile NFW fo Milky Wy Dt come fom [5] Sofue, Y.015. Rs = 10,7 Kpc Do = 1.318 10-1 kg/m^3 Accoding NFW density pofile, totl enclosed y sphee with R dius is M ( R) 4 R 3 S x x Do ln(1 x) 1 11

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac Clling x f ( x) ln(1 x) Rs x ln(1 / ) nd Z 3 NFW 4R S D0 then 1 Rs NFW ( ) Z f ( x) NFW Accoding dt fo Milky Wy Z NFW =,8 10 11 Msun. Rdius NFW (<R) kpc Msun 44,00,316+11 50,00,554+11 70,00 3,9+11 90,00 3,775+11 110,00 4,33+11 130,00 4,67+11 150,00 4,97+11 170,00 5,80+11 190,00 5,558+11 10,00 5,810+11 30,00 6,04+11 50,00 6,56+11 70,00 6,454+11 90,00 6,640+11 8.4 DYNAMICAL VRSUS NFW eside is tulted totl NFW inside sphee with R dius. y ddition of (< 44 kpc) plus yonic mss it is supposed to hve totl mss. Totl Mss(< 44 kpc) =,31 10 11 + 1,37 10 11 = 3,69 10 11 Msun. At epigph 10.1 ws got tht dynmicl mss(< 44 kpc) = 4,15 10 11 Msun. Theefoe thee is diffeence not negligile which my e cceptle ecuse ccoding Sofue, fo Milky Wy yonic mtte = 1,37 0,4 10 11 Msun.i.e. expeimentl eo fo yonic mtte is simil to mss diffeence. Joining esults got in pevious epigphs it is possile to compe got y these methods. Rdius Dynmicl (<R) NFW (<R) Rel diff. kpc Msun Msun % 44,00,78+11,316+11 16,75 50,00,953+11,554+11 13,5 70,00 3,436+11 3,9+11 6,04 90,00 3,83+11 3,775+11 1,49 110,00 4,171+11 4,33+11-1,48 130,00 4,471+11 4,67+11-3,50 150,00 4,740+11 4,97+11-4,91 170,00 4,985+11 5,80+11-5,91 190,00 5,1+11 5,558+11-6,63 10,00 5,43+11 5,810+11-7,14 30,00 5,60+11 6,04+11-7,50 50,00 5,806+11 6,56+11-7,74 70,00 5,98+11 6,454+11-7,89 90,00 6,149+11 6,640+11-7,98 It is emkle tht t 44 kpc dynmicl is 16 % igge thn NFW, pogessively diffeence decese t 110 kpc NFW ecome igge thn dynmicl nd t 90 kpc NFW is 8 % igge thn dynmicl. 1

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 8.5 DYNAMICAL CORONA HALO COMPARD TO NFW CORONA HALO MASSS As fo dius igge thn 44 kpc it is supposed tht yonic mtte is negligile. This dius is good efeence to clculted enclosed in spheicl coon fom 44 kpc up to R kpc. As ( ) Z f ( x) epesents totl unde dius R, it is ight to get NFW coon hlo mss NFW NFW with this simple fomul: Coon (44 kpc to R ) = Z NFW (f f 1 ). Whee f = f (R ) nd f 1 = f (44 kpc). Similly dynmicl Coon ( 44 kpc to R) = dyn. M(<R) dyn M(<44 kpc) In tle elow in the fouth column is tulted dynmicl coon y sustction of dynmicl mss t R dius minus dynmicl t 44 kpc. In the fifth column is tulted NFW coon y sutction of NFW mss t R dius minus sme kind of mss t 44 kpc. In sixth column is shown eltive diffeence etween oth methods to clculte coon t diffeent dius. Diffeences oscillte etween 8 % nd %. Dynmicl mss (totl) M(<R) NFW (<R) Rdius Dynmicl coon (44<R) NFW Coon (44<R) Rel. Diff. Coons Dynm. Vs NFW Msun Msun kpc Msun Msun % 4,15+11,316+11 44,00 0,000+00 0,000+00 4,33+11,554+11 50,00 1,706+10,373+10 8,1 4,806+11 3,9+11 70,00 6,538+10 9,14+10 8,34 5,0+11 3,775+11 90,00 1,050+11 1,459+11 8,04 5,541+11 4,33+11 110,00 1,389+11 1,917+11 7,53 5,841+11 4,67+11 130,00 1,688+11,311+11 6,93 6,110+11 4,97+11 150,00 1,957+11,656+11 6,31 6,355+11 5,80+11 170,00,03+11,964+11 5,67 6,58+11 5,558+11 190,00,430+11 3,41+11 5,04 6,793+11 5,810+11 10,00,640+11 3,494+11 4,43 6,990+11 6,04+11 30,00,838+11 3,75+11 3,83 7,176+11 6,56+11 50,00 3,04+11 3,939+11 3,4 7,35+11 6,454+11 70,00 3,00+11 4,138+11,67 7,519+11 6,640+11 90,00 3,367+11 4,34+11,1 It is emkle tht NFW coon vlues (tuquoise column) e igge thn Dynmicl coon vlues (gey column). Reson to explin this fct is the sme tht ws exposed in chpte seven egding NFW density nd diect density. 13

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 9. D.M. DNSITY AS POWR OF IN MILKY WAY VRSUS D.M. DNSITY AS POWR OF IN M31 In chpte 5 wee developed theoy of density s powe of. density = A. In this chpte ws demonstted tht mthemticlly it is equivlent lw of powe of. Whee fomul coefficients e: A v 1 ( 1) & 4G fo ottion cuve to lw fo density s. 1 In ddition ws shown fomul fo gvittionl field though viil theoem. 1 Coefficients of Milky Wy e elow. Powe egession fo Milky Wy ot. cuve V=*^ 3,4989549+1-3,45408589-01 Coeltion coeff. 0,85 Milky Wy A D A 4,867931919745 10-7 1,593443097574 In ppe [11] Ac,M.016. Two New Dk Mtte Density Pofiles fo M31 Hlo Got fom Rottion Cuve wee pulished coefficients & fo powe egession velocity depending on dius. Also wee pulished coefficients A& fo density s powe of. elow e couple of dt. Powe egession fo M31 ot. cuve V=*^ 4,15011040+10 -,4755450-01 Coeltion coeff. 0,9554 M31 glxy A D A 3,76651943774 10-6 1,66884753770 Now is going to tulte Viil ccoding Milky Wy dt. Fist nd second columns efes dius egding Milky Wy. At tht dius is clculted gvittionl field, which is used to clculte density fo Milky Wy nd M31 in ode to compe oth esults t the sme. Thid column shown such. Fouth nd fifth column show density s powe of fo Milky Wy nd M31 nd sixth column shows eltive diffeence etween oth densities. Supisingly eltive diffeence fo field elonging to 90 kpc o igge dius, fll elow 1 %. See gey ow. Rdius fo Milky Wy Rdius fo Milky Wy MW Viil powe Milky Wy powe M31 Rel. Diff. MW vs M31 m kpc m/s^ kg/m^3 Kg/m^3 % 1,35771+1 44,9913-11 8,7389-4 1,0976-3 19,65 1,5485+1 50,41156-11 5,87004-4 7,18811-4 18,337,15999+1 70 1,3679-11,37835-4,79050-4 14,770,77713+1 90 8,95659-1 1,1119-4 1,3764-4 1,004 3,3947+1 110 6,38686-1 7,06656-5 7,8839-5 9,73 4,01141+1 130 4,81985-1 4,5136-5 4,89384-5 7,795 4,6855+1 150 3,7871-1 3,0778-5 3,751-5 6,103 14

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 5,4569+1 170 3,06699-1,1957-5,30154-5 4,598 5,8683+1 190,548-1 1,6883-5 1,68337-5 3,40 6,47997+1 10,14819-1 1,4499-5 1,704-5,00 7,09711+1 30 1,8488-1 9,75174-6 9,83657-6 0,86 7,7145+1 50 1,60131-1 7,79560-6 7,78054-6 -0,194 8,33139+1 70 1,40655-1 6,3401-6 6,6639-6 -1,178 8,94853+1 90 1,4698-1 5,339-6 5,1560-6 -,101 9,571+1 300 1,17774-1 4,77795-6 4,6595-6 -,54 Reltive diffeences of density fo MW vs density fo M31 oscillte fom mximum 19,6 % t the iggest field nd deceses up to -,5% t the lowest field. In tle elow hs een clculted Viil though M31 pmetes &. Gvittionl field elong to dius fo M31 tulted in fist nd second columns. Supisingly eltive diffeences of density in M31 nd Milky Wy e minly elow 0 % nd fo field elonging to 150 kpc o igge dius, eltive diffeences fll elow 10 %. Rdius fo M31 Rdius fo M31 M31 Viil powe Milky Wy powe M31 Rel. Diff. MW vs M31 m kpc m/s^ kg/m^3 % 1,348+1 40 5,0368-11 1,89809-3,4570-3,748 1,5485+1 50 3,60799-11 1,11543-3 1,4080-3 0,780,15999+1 70,18166-11 5,0039-4 6,08139-4 17,718,77713+1 90 1,4983-11,74971-4 3,4844-4 15,353 3,3947+1 110 1,10996-11 1,70476-4 1,96891-4 13,416 4,01141+1 130 8,64639-1 1,14504-4 1,9779-4 11,770 4,6855+1 150 6,98098-1 8,1457-5 9,08110-5 10,335 5,4569+1 170 5,78957-1 6,0431-5 6,6453-5 9,061 5,8683+1 190 4,9059-1 4,63641-5 5,03483-5 7,913 6,47997+1 10 4,13-1 3,6584-5 3,93-5 6,868 7,09711+1 30 3,68443-1,94107-5 3,1575-5 5,908 7,7145+1 50 3,558-1,4111-5,53864-5 5,00 8,33139+1 70,89906-1,0078-5,09510-5 4,19 8,94853+1 90,6059-1 1,69306-5 1,7596-5 3,417 9,571+1 300,47653-1 1,56169-5 1,61078-5 3,047 Reltive diffeences of density fo MW vs density fo M31 oscillte fom mximum,7 % t the iggest field nd deceses up to 3 %. density s powe of egding oth glxies e stonishing simil despite the fct tht oth glxies hve vey diffeent ottion cuves coefficients &, thei yonic msses e clely diffeent nd thei glctic disks hve diffeent lws fo supeficil density msses. In ddition it is well known tht expeimentl eos e not negligile despite the fct mesues e mde y pestigious esech tems equipped with high technology tools. Despite these fcts, eltive diffeences fll elow 1% t elonging dius igge 90 kpc in Milky Wy nd fll elow 10 % t elonging dius igge 150 kpc in M31.See ows in gey in tles ove. 15

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac In my opinion this esult suggest stongly tht is geneted y gvittionl field s powe of ccoding Univesl lw. [8] Ac,M.016. D.M. density on ig glxies depend on gvittionl field s Univesl lw. 10. HALO RADIUS OF MILKY WAY & M31 ACCORDING D.M. AS POWR OF THORY Accoding theoy D.M. s powe of it is ight to deduce tht hlo dius of ech glxy extend up to oth glxies hve the sme gvittionl field. Powe egession fo Milky Wy ot. cuve V=*^ 3,4989549+1-3,45408589-01 Coeltion coeff. 0,85 Powe egession fo M31 ot. cuve V=*^ 4,15011040+10 -,4755450-01 Coeltion coeff. 0,9554 In chpte five ws got this fomul of powe egession of ottion cuve in hlo. 1 fo Viil so it is ight clculus of when it is known coefficients In tle elow is clculted fo MW nd M31 t diffeent dius to find dius which give the sme vlue fo in oth glxies. It is supposed tht distnce etween MW & M31 is 770 kpc. Fo exmple t 9 kpc dius of MW elong 478 kpc of dius fo M31. At these dius gvittionl field geneted y oth glxies is lmost identicl s ede cn see t gey ow in tle elow. So ccoding this theoy it is ight to considete 9 kpc hlo dius of Milky Wy nd 478 kpc hlo dius of M31. It is impotnt to see density t these distnces fo oth glxies density MILKY WAY = 5,1376 10-6 nd density M31 = 5,038 10-6. oth vlues diffes only 1,9 %. Astonishing See ow in gey in tle elow. Rdius MW Rdius M31 Rdius MW Rdius M31 MW viil M31 viil M31 -Diect MW-diec kpc kpc m m m/s^ m/s^ kg/m^3 00 570 6,171+1 1,7588+,336-1 9,48587-13 3,474-6 1,4196-5 10 560 6,48+1 1,780+,14819-1 9,7404-13 3,39386-6 1,4499-5 0 550 6,789+1 1,6971+ 1,986-1 1,0006-1 3,54993-6 1,09880-5 30 540 7,097+1 1,6663+ 1,8488-1 1,0845-1 3,7163-6 9,75174-6 40 530 7,406+1 1,6354+ 1,71534-1 1,05760-1 3,89366-6 8,69867-6 50 50 7,714+1 1,6046+ 1,60131-1 1,08815-1 4,08318-6 7,79560-6 60 510 8,03+1 1,5737+ 1,49890-1 1,101-1 4,8588-6 7,01639-6 70 500 8,331+1 1,549+ 1,40655-1 1,15387-1 4,5097-6 6,3401-6 80 490 8,64+1 1,510+ 1,394-1 1,1895-1 4,73577-6 5,75036-6 90 480 8,949+1 1,4811+ 1,4698-1 1,649-1 4,98579-6 5,339-6 300 470 9,57+1 1,4503+ 1,17774-1 1,6571-1 5,5470-6 4,77795-6 91 479 8,979+1 1,4781+ 1,3977-1 1,303-1 5,01180-6 5,18514-6 9 478 9,01+1 1,4750+ 1,36-1 1,3417-1 5,03800-6 5,13760-6 93 477 9,041+1 1,4719+ 1,554-1 1,3804-1 5,06440-6 5,09065-6 16

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac 10.1 MASSS IN MILKY WAY & M31 AT XTNDD HALOS Accoding conclusion pevious epigph it is ight to conside MW hlo dius equl to 9 kpc nd M31 hlo dius equl to 478 kpc. v R As dynmicl mss is M ( ) it is ight to clculte dynmicl msses fo MW & M31 fom powe G egession velocity t hlo dius distnces. Dynmicl Mss Milky Wy (< 9 kpc) = 7,54 10 11 Msun Dynmicl Mss M31 (< 478 kpc) =,0 10 1 Msun Accoding [5] Sofue,Y. 015. yonic Mss MW = 1,37 10 11 Msun nd yonic Mss M31 = 1,61 10 11 Msun So Rtio yonic mss vs Totl mss fo MW = 18, % nd Rtio yonic mss vs Totl mss of M31 = 8 % 11. CONCLUSION This wok is focused in hlo egion of Milky Wy whee yonic density is negligile egding non yonic. Reson is tht the min hypothesis ll my ppes is tht non yonic is geneted loclly y gvittionl field. Theefoe it is needed to study dius dominion whee it is possile to study gvittionl field popgtion without intefeence of yonic mss density o t lest whee this density is negligile. In ode to defend popely conclusion this ppe is impotnt to emphsise esult got in chpte 3 which is tht coeltion coefficient of powe egession t ottion cuve in hlo egion is 0,85. This cceptle vlue of coeltion etween dius nd velocity, suppot tht velocity of Milky Wy ottion cuve follows powe lw egding dius v In chpte fou ws mthemticlly demonstted tht powe lw clled diect in this ppe whose fomul is whose coefficient & wee got in chpte 3. D v ( 1) 4G In chpte five ws demonstted mthemticlly tht powe lw fo velocity mthemticlly equivlent powe lw fo density depending on.. in hlo egion is equivlent density v D A t ottion cuve is 1 eing nd eing A 1 ( 1) 4G &. 1 Theefoe joining chptes 3,4 nd 5 it is concluded tht coeltion coefficient s 0,85 t powe egession lw fo ottion cuve D A v in hlo egion suppot tht density inside hlo egion is powe of gvittionl field whose pmetes A & e witten ove. In chpte seven ws comped diect pofile got in this ppe with NFW fitted y [5] Sofue, Y.015.It ws pointed tht eltive diffeences fluctute etween 15 % nd 8% thoughout dominion in hlo egion nd it ws explined eson why diect pofile is moe pecise thn NFW pofile to descie non yonic density nd why diect density gives lowe vlues thn NFW density. 17

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac Similly in chpte eight ws comped coon msses got though oth methods. Thei eltive diffeence fluctute etween 8 % nd % inside dius dominion eing NFW pofile igge thn diect thoughout dominion. As it ws pointed nd explined in chpte seven, this fct is n impotnt evidence tht suggest stongly out existence of yonic inside ulge nd disk nd existence weid non yonic in hlo egion which is geneted y gvittionl field ccoding hypothesis poposed in this nd othe woks. In chpte nine ws comped density s powe of in Milky Wy with density s powe of in M31 hlo y this fomul D A. To do this compison ws tulted t diffeent dius nd though this set of vlues of wee got density in MW hlo with MW pmetes nd density in M31 hlo with M31 pmetes. Results fo oth sets of vlues e stonishingly simil thoughout hlo dominion since eltive diffeences oscillte etween % nd 3%. The less vlue hs the less eltive diffeence hve oth vlues. This esults suppot stongly utho hypothesis exposed in pevious ppes. [8] Ac,M.016. Dk mtte density on ig glxies depend on gvittionl field s Univesl lw. This hypothesis estte tht this lw D A Univesl, tht is density is simil in diffeent glxies t specific, on condition tht glxies would e simil gint glxies In my opinion these fcts suggest stongly ntue of non yonic. which is vey impotnt ecuse shows wy to develop new quntum gvittion theoy. Finlly in tenth chpte ws intoduced new definition of hlo dius. This new definition ws justified s consequence of theoy of geneted y gvittionl field. It hs een clculted totl mss t these new hlo dius fo MW nd M31nd it hs een estimted tio yonic mss vesus totl mss tht fo MW is 18 % nd fo M31 is 8 %. is 1. ILIOGRAPHYC RFRNCS [1] Ac,M.014,viX:1410.000. Dk mtte model y quntum vcuum [] Ac,M.015,viX:1510.034. Dk mtte density function depending on gvittionl field s Univesl lw [3] Ac,M.015. vix.og/s/151.0309 A new dk mtte density pofile fo NGC 3198 glxy to demonstte tht dk mtte is geneted y gvittionl field. [4] Ac, M.016.viX. 1601.0014 A New Dk Mtte Density Pofile fo M33 Glxy to Demonstte tht Dk Mtte is Geneted y Gvittionl Field [5] Sofue, Y.015. Xiv:1504.05368v1 Dk hlos of M31 nd the Milky Wy. [6] Ac, M.016. vix.160.0047 A new Dk mtte density pofile fo Milky Wy to demonstte tht dk mtte is geneted y gvittionl field. [7] Ac, M.016. vix:1606.0007 A new Dk mtte density pofile fo Milky Wy which depend on gvittionl field. 18

TWO NW D. MATTR PROFILS FOR MILKY WAY HALO GOT FROM ROTATION CURV M. Ac [8] Ac,M.016. vix:1606.079v1 Dk mtte density on ig glxies depend on gvittionl field s Univesl lw. [ 9] Ac,M.016.vix:1607.047 A New Dk Mtte Density Pofile s Powe of Gvittionl Field fo Com Cluste. [ 10] Ac,M.016.vix:1601.04 A New Dk Mtte Density Pofile fo M31 Glxy to Demonstte tht Dk Mtte is Geneted y Gvittionl Field [ 11] Ac,M.016.vix:1609.0035 Two New Dk Mtte Density Pofiles fo M31 Hlo Got fom Rottion Cuve [ 1] Nieuwenhuizen,T.M. 010.Xiv:1011.530v1 Do mico own dwf detections explin the glctic dk mtte? [ 13] Nieuwenhuizen,T.M. 01. Xiv:110.0489v Do the Heschel cold clouds in the Glctic hlo emody its dk mtte? [ 14] Nieuwenhuizen,T.M. 010 Xiv:1003.0453v1 Gvittionl hydodynmics vesus osevtions of voids, jens clustes nd MACHO dk mtte. [ 15] Wyzykowski,L.010.Xiv:101.1154v The OGL View of Micolensing towds the Mgellnic Clouds. III. Ruling out su-sol MACHOs with the OGL-III LMC dt. [ 16] Hwkins M.R.S. 015. Xiv: 1503.01935v1 A new look t micolensing limits on dk mtte in the Glctic hlo. 19