DARK MATTER AND THE DYNAMICS OF GALAXIES: A NEWTONIAN APPROACH 1. INTRODUCTION

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1 DARK MATTER AND THE DYNAMICS OF GALAXIES: A NEWTONIAN APPROACH Mugu B. RĂUŢ Coesponding autho: Mugu RĂUŢ, m_b_aut@yahoo.om Abstat In this pape I popose a oetion to the well-known Newtonian gavitational potential, a oetion whih explains the fom of the adial veloities as a funtion of the disoid galaxies adii. The main sope of this wok is to find a oetion to the Newtonian gavitational potential whih has to fulfil two majo onditions: a) to take into aount the entie amount of the expeimental data; b) the esulting potential to be a onsequene of ondition a) fom a physial pespetive. As a esult, the oeted fom of the Newtonian gavitational potential was found to belong to a physial ause and this ause an be the existene of the dak matte, evenly distibuted within galaxies. This distibution makes dak matte to at as a binde fo odinay matte, so that the disoid galaxies not otate as a fluid (as standad Newtonian theoy states), but as some igid fames (as the obsevational data state). Key wods: modified Newtonian dynamis; dak matte; MOND theoy; adial veloities uve.. INTRODUCTION Dynamis of galaxies is uently one of the majo poblems of the theoy of gavity, until its disovey, the fist half of -th entuy, []. Most of the galaxies exeute aound thei entes of mass a otating movement that has some featues that distinguish them. If the matte fom whih the galaxies ae onsisting would be subjet only the law of Newtonian gavity, then galaxies would otate like some ideal fluids, ineasingly faste towads the ente and deeasingly slowe towads the edges. But, in eality, the otation of galaxies takes plae as if they ae some igid bodies. At a distane fom thei entes, alled itial adius, the adial veloity beomes patially independent of the adius. Ove the time, thee have been seveal attempts to explain the behavio of the otating galaxies. Fistly, we talk about the existene of dak matte, []. This exoti matte ats like a bend fo odinay matte and the esulting dynamis is the obseved one. Fom an expeimentally point of view, the empiial Tully-Fishe elation must be valid. Consequently, v M, whee β=/3-/4, v is the adial veloity and M the galaxy mass. The exponent β epesents an inteval, not the exteme values of an inteval. This exponent takes diffeent values when the obsevations ae made in diffeent wavelengths, [3]. The dak matte theoy explains, in piniple, the dynamis of galaxies, but not fom Tully-Fishe elation pespetive. Anothe theoy whih ty to explain this dynamis is based on geneal elativity, [4]. The dak matte is no moe onsideed, but this theoy is vey geneal and it has no expeimental patiulaities. The Weyl-Dia based on theoy, [5], explains the dynamis of galaxies but, one again, geneally and with poo efeenes to expeimental data.

2 Fom all these theoies only the Modified Newtonian Dynamis, MOND, takes into aount, with some degee of auay, the expeimental data. MOND onsides only β=/4 as epesentative value fo wavelengths whih haateize a lage vaiety of galaxy masses, [6]. Exepting this fat, MOND exhibits some majo disadvantages. Fist of all it is an effetive theoy; it bings no ausal justifiations fom physial ode to eluidate the behavio of otating galaxies. MOND only postulates the modified dynamis laws and these postulates have not a physial basis. The aim of this pape is to oet, somehow, these disadvantages. Fist of all, we intend to onside the entie inteval, β= [/3;/4], in ou theoetial evaluations. Then, we intend to give ou theoy a physial signifiation.. ATTRACTIVE FORCES AND THE DYNAMICS OF THE GALAXIES In the following we suppose that one an obtain a theoy to explain the otation uves of disoid galaxies, a theoy based on dak matte. Theefoe we have a gavitational potential of the fom: GM A () whih is the esult of solving Poisson's equation: 4G( ue) () The seond tem on the ight side of equation () is an empiial tem attahed to the Newtonian gavitational potential, a tem that desibes the ation of an attative gavitational potential, due to an unknown fom of enegy. We have theefoe A>. With this potential () we ty to show that this unknown enegy may ause the adial movement of the disoid galaxies obseved in eality. Fom () we find the expession of the fist deivative of this equation, the aeleation: GM g A (3) whih have a geat impotane in the development of easoning in the following. The gavitational aeleation (3) is entiely attative and it balanes with the entifugal aeleation of galaxy, whih is epulsive. Suppose that neithe of them tip the balane one way o anothe, so at equilibium we must have: v GM A (4) whih leads afte a obvious a multipliation with to a simple expession: GM v A (5) Fom obsevations made on the motion of galaxies and the Tully-Fishe empiial expession it esults v L M, whee L epesent the luminosity and. 3 4 So, now we an empiially detemine the pope α in two ases oesponding to the two expessions of the potential () whih povide the shape of the otation uves of disoid galaxies aoding to obsevations. Theefoe we have: 3 v M (6) and the ase in whih we have a patiula inteest: 4 v M (7) The itial adius fom whih the veloity v( ) onst. ome fom the ondition:

3 Thus, expession (5) beomes: and the itial adius is: v( ) GM ( ) A GM ( ) A (8) Then this is the adius fom whih the veloity is independent fom it: v A 4 4 / ( GM) [( ) ( ) ] To be in aodane with the obsevations, onditions (6) and (7), we must have: 4 with γ=/3 and γ=/4. Unde these estitions we find two values fo α, onsequently ½ and. The ase oesponding to α= leads to: v 4 4GMA () a value independent fom adius. Unde these onditions equation () is valid without the need to onside MOND theoy. If we admit that a have not the same meaning as in the MOND theoy but totally due to othe auses, having no onnetion with the expansion of the univese but only with intenal dynamis of galaxies, is a onstant speifi to eah galaxy in pat, then all we have talked so fa is valid. Othewise the plae of a may be taken by, the geneal value A whih an be detemined fom expeimental uves. Amazingly, if we do this we get to the esult 4A a (whih was also detemined fom expeimental uves, as a mean value, fo disoid galaxies). In this ase a annot be oneived as in MOND theoy, [6], but as a galati haateisti without any onnetion with the expansion of the univese. The diffeene fom the MOND theoy appeas to be the double value of the itial adius: GM () a But the fom () an be avoided if we take A a. It leads us unexpetedly lose to the MOND theoy, fom itial adius pespetive, but adial veloity is a little bit bigge than the MOND-like veloity. Indeed, if we onside A a we obtain fom () exatly: 4 v 4GMa () whih is the well-known expession obtained in the MOND theoy fo the adial veloities independent fom the galaxies adius, multiplied with fou, atually less impotant. What is impotant hee is that we got these esults in the appoximation: and not in the appoximation: g g N a g ( g a ) / N like in MOND theoy, [6]. The diffeene lies in the fat that ou theoy has not woked out with an expession of a modified inetia like MOND theoy does, Newton's Seond Law emaining unhanged. So, with () and (7), some MOND theoy esults an be obtained without the need to hange the law of inetia. We just need to hange the definition of onstant a as an atifat of galaxies, supposed due to an unknown enegy and its distibution into eah galaxy in pat. If this aeleation is dietly onneted with dak matte, as a popety of dak matte, than a should be a univesal onstant. It is independent of quantity of dak (9)

4 matte existent in a galaxy. We ould think, the small deviations fom this value an be assigned to the fom of eah galaxy in pat. How dak matte is distibuted in galaxy diffe fom anothe galaxy, so the onstant A (o a ) ould be diffeent. This aeleation ould explain the anomaly of Pionee spaeaft, also. Thee is an attative onstant foe in ou galaxy, supplementay to the Newtonian one, whih is dietly due to ompession tend of the galaxy aused by the amount of dak matte unifomly spead in it. Equation (4) is valid beause dak matte is opposing to the tend of the dispesion aused by the otation of the galaxy. But all these omments ae valid only if dak matte is the oigin of the potential () fo α=. 3. POSSON S EQUATION VERIFICATION The seond ase we disuss now is oesponding to α=/. Following the same steps as in pevious ase, fom (8) we find: fo itial adius, and fom (9): / 5 GM (3) 3A 9 3 / 5 / 3 / v A ( GM) [(3/ ) (/3) ] (4) fo adial veloity, esults that should fit the obsevational data. The onstant A is, this time, no moe aeleation. The aeleation indued by the unknown enegy is now ineasing with adius. Fo the same magnitude of onstant A as in pevious ase, it esults ineased values fo itial adius and adial veloity. The question now is whih one of the tow ases is oet fom Poisson s equation point of view. The potential () fo α= has no soue, exept the nomal matte. Even if it veifies the Poisson s equation (), so it is a valid gavitational potential, it an be eated fom nomal matte only. Ionially we find appoximately the same esults in MOND theoy and we have to onlude that the Modified Newtonian Dynamis is oet. The absene of a physial ause and validity of ase (7) make this possible. Consequently, thee is an equivalene between MOND theoy and the theoy of modified Newtonian potential () fo α= we peviously pesent. That s why the MOND theoy is an effetive theoy and not a physial theoy, the potential () fo α= has soue nomal matte only. Theefoe, if we want to take into aount the effets of an unknown enegy to galati nomal matte we must onside the seond ase, the potential () fo α=/. It ould have dak matte as soue; othewise the Poisson s equation () would have no sense. This is the eason why a Newtonian theoy based on a modified Newtonian potential that an desibe the motion of galaxies due to dak matte is valid only in this ase, fo whih: / A 4G dm (5) The onstant A is deeasing/ineasing popotionally to the density of dak matte, hene is a featue of eah galaxy in pat. And we have finally a Newtonian theoy whih desibes lose to eality the effets of dak matte in tems of dynamis of galaxies. But the sign minus fom equation (5) seems to ontadit this affimation. The only physial justifiation of it is that the dak matte density omes fom a negative pessue: p dm v (6) as a esult of self inteation between dak matte s patiles in motion into a homogenous ompessible fluid, with negative ompessibility. The expession (6) is an intuitive one, beause the gavity nomal foe should be a esult of odinay matte motion though the dak matte fluid: p v (7) This is the eason why the veloity in equations (6) and (7) ould not be the same.

5 4. REPULSIVE FORCES AND THE DYNAMICS OF THE GALAXIES If the natue is moe supising than we expet and odinay matte somehow sueeds to geneate a gavitational potential in fom () fo α=, then it will podue the obsevational data whih ae wongly intepeted as dak matte effets. Fat is, the obsevational data we have, in both foms (6) and (7), an so easily give ise to misintepetations. Aoding to this model only the obsevational data in fom (6) an be attibuted to dak matte. The est of it is due to othe auses. Whih ae these auses, we don t know. But in the following onsideations we will show that the ause fo all obsevational data ould be the dak matte. Assume this time that odinay matte sueeds to desibe the dynamis of disoid galaxies in absene of dak matte, though a epulsive gavitational potential. This potential is equivalent with: GM B (8) fo some values α whih will be alulated to aomplish the obsevational equiements. The equation (8) is the esult of solving Poisson s equation: G( ) 4 ue The tem ue is geneally alled it unknown enegy. Conening the potential (8) the balane of aeleations is: v GM B whih is diffeent fom (3). In ode to wite oetly the balane of all aeleations involved in dynamis of a galaxy in this ase, we must onside, again, the ontibution of dak matte. If we oneive a galaxy full of some sot of matte that eat with an opposite aeleation to the expansion tend of the galaxy, than this matte ould be the dak matte. Theefoe we have: GM v a B B (9) At a=, afte we eahed a itial adius value: GM v B B () we an find the equilibium onditions fom whih we an detemine the itial adius expession and the adial veloity expession. Hene, fom: GM b () the expession of this itial adius will be: GM () b On eahing the itial adius, the equilibium of the galaxy will not be omplete unless we onside: v b esult whih leads, by taking into aount (), to: 4 4 ( ) v b GM (3) The expeimental estitions (6) and (7), in equation (3), lead to the same ases, α= and α=/. If b B, the equations () and (3) have no sense. The geneal onstant b R indiates an additional epulsive foe exept the assumed epulsive foe B. Hene the ase b=b is the only valid. Whene, afte we eplae the onstant B with a and α=, we an detemine the adial veloity of the galaxy as:

6 4 v GMa whih is the well-known expession obtained in the MOND theoy fo the adial veloities independent fom the galaxies adii, and the itial adius fom whih the small aeleations appoximation ous in the MOND theoy: GM a The ase oesponding to α=/ is not a valid one beause it is expessing, though (8), a epulsive ation whih annot be attibuted to dak matte. 5. DISCUSSIONS The esults obtained in the pevious setion ae speifi to MOND theoy. So we must onlude that ou goal, to find a Newtonian theoy fo desibe the dynamis of the galaxies due to dak matte, is eahed if we onside both potentials, () and (8) simultaneously, () fo α=/ and (8) fo α= Moe than that, the obsevational data we have ae in fom of an inteval, β=/3-/4, i.e. [,/ ], R. The above disussed ases efe to the limits of this inteval, β=/3 and β=/4. In ode to povide an auate desiption of the dak matte ation we must to establish what ae the limits fo whih the ases oesponding to () fo α=/ and (8) fo α= ae valid. To do this we must solve the Poisson s equation () with the potential (). It esults: A G dm Fom this equation we obseve that the potential () has soues fo (,/ ]. Only fo α= the potential () has no soues and the potential (8) is moe pope to desibe the dak matte ation in this ase. Theefoe, the dak matte effets ae desibed ompletely by the potential: GM A, fo (,/ ], GM B, fo α= (4) It is an empiial potential, beause thee is theoy eated only fo obsevational data fitting. It is obvious that the gavitational potential whih desibes popely the ation of dak matte is the fom (4), without the value oesponding to α=. A gavitational potential due to some physial auses annot geneate simultaneously an ation and a eation. The theoy pesented in [5] dedues only a potential like (4) fo α=. It is podued by odinay matte, like in ou theoy. The effets ould not be those pesented above, beause its effets ae muh smalle than a potential, geneated by the same odinay matte. If we appoximate the Yukawa gavitational potential, [7]: / Ce (5) in tems of powe seies with espet to the galaxy adius than we find a gavitational potential like (4), fo α=. The equation (5) epesents the potential pe mass unit, theefoe we meet seious obstales to popely alibate the onstants in ode to make (4) looks like (5). If we take the onstant C in the fom GM than the onstant B must be fouteen times bigge than the onstant a (fo ou galaxy). Taking into aount the above onsideations we must admit that the fom (4) annot momentay be epodued by a physially gounded mathematially oet theoy. Unde these onditions equation (4),

7 without the fom oesponding to α=, is valid without MOND theoy s onfimation. In this final ase the geneal onstant A an be detemined fom expeimental uves. Ou theoy is only an effetive theoy, not bette than MOND theoy, and nothing moe. But it involves dak matte in galaxies dynamis. Instead, it has the same disadvantages as MOND theoy: it doesn t solve the mass disepanies poblem. The onstant A fom (4) depends on the galaxies adii, theefoe in the ase of some small spheial galaxies and some big galati lustes, an anomalous mass disepany will ou: too lage fo small spheial galaxies and modeate fo big galati lustes, [6]. Pehaps, moe auate measuements of mass/luminosity onvesion fato M/L, will laify this poblem. 6. CONCLUSIONS In this pape we popose an altenative Newtonian theoy fo MOND, a theoy whih desibes the effets of dak matte in the dynamis of galaxies. Unde the hypothesis that fo the shape of the adial veloity uves of galaxies ae esponsible an attative and a supising epulsive fom of enegy, this influene is found to be expessed by a supplementay potential whih it must be added to the Newtonian gavitational potential. Somewhat supising, we found the esults speifi to MOND theoy, but this time with a modified Newtonian potential. 7. REFERENCES. ZWICKY F. (937), On the masses of Nebulae and of Clustes of Nebulae, Astophysial Jounal, 86, 7. RUBIN V. C., FORD W. K. (97), Rotation of the Andomeda Nebula fom a Spetosopi Suvey of Emission Regions, Astophysial Jounal, 59, TULLY R. B., FISHER J. R. (977), A New Method of Detemining Distanes to Galaxies, Astonomy and Astophysis, 54, COOPERSTOCK E. I., TIEU S. (7), Galati Dynamis via Geneal Relativity: A Compilation and New Developments, Intenational Jounal of Moden Physis, A, MIRABOTALEBI S., JALATZADEH S., MOHAVED S., SEPANGI H. R. (8), Weyl-Dia Theoy Peditions on Galati Sales, Monthly Noties of Royal Astonomy Soiety, 385, MILGROM M. (983), A Modifiation of the Newtonian Dynamis as a Possible Altenative to the Hidden Mass Hypothesis, Astophysial Jounal, 7, SNEDDON I. N., THORNHILL C. K. (949), A Popety of the Yukawa Potential, Poeedings of the Cambidge Philosophial Soiety, 45, 38-3.