Conformal dynamical equivalence and applications

Transcription

1 Journal of Physs: Conferene Seres Conformal dynamal equvalene and applatons To te ths artle: N K Spyrou 11 J. Phys.: Conf. Ser Related ontent - A onventonal form of dark energy K Kleds and N K Spyrou - Geodes motons versus hydrodynam flows K Kleds and N K Spyrou - Observatonal osmology M S Longar Vew the artle onlne for updates and enhanements. Ths ontent was downloaded from IP address on 8/1/18 at 3:48

2 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 Conformal dynamal equvalene and applatons 1 N K Spyrou Astronomy Department, Arstoteleon Unversty of Thessalonk, Thessalonk, Maedona, Hellas (Greee) Emal: spyrou@astro.auth.gr Abstrat. The Conformal Dynamal Equvalene (CDE) approah s brefly revewed, and some of ts applatons, at varous astrophysal levels (Sun, Solar System, Stars, Galaxes, Clusters of Galaxes, Unverse as a whole), are presented. Aordng to the CDE approah, n both the Newtonan and general-relatvst theores of gravty, the sentrop hydrodynam flows n the nteror of a bounded gravtatng perfet-flud soure are dynamally equvalent to geodes motons n a vrtual, fully defned flud soure. Equvalently, the equatons of hydrodynam moton n the former soure are funtonally smlar to those of the geodes motons n the latter, physally, fully defned soure. The CDE approah s followed for the dynamal desrpton of the motons n the flud soure. After an observatonal ntroduton, takng nto aount all the nternal physal haratersts of the orrespondng perfet-flud soure, and based on the property of the sentrop hydrodynam flows (qute reasonable for an solated physal system), we examne a number of ssues, namely, () the lassal Newtonan explanaton of the elebrated Poneer-Anomaly effet n the Solar System, () the possblty of both the attratve gravty and the repulsve gravty n a non-quantum Newtonan framework, () the evaluaton of the masses - theoretal, dynamal, and mssng - and of the lnear dmensons of non-magnetzed and magnetzed large-sale osmologal strutures, (v) the explanaton of the flat-rotaton urves of ds galaxes, (v) possble formaton mehansms of wnds and jets, and (v) a bref presentaton of a onventonal approah - toy model to the dynams of the Unverse, haraterzed by the domnant ollsonal dark matter (wth ts subdomnant lumnous baryon ontamnaton ), orretly nterpretng the osmologal observatonal data wthout the need of the notons dark energy, osmologal onstant, and unversal aeleratng expanson. 1. Introduton and Motvaton Aordng to many urrent observatonal data, the realst pture and morphology of an astrophysal-osmologal struture dffers greatly from ts orrespondng optal pture. The Solar 1 Ths s an extended verson of the talk, under the same ttle, gven by the author durng the Conferene Reent Developments n Gravty (NEB) XIV (Ioannna, Hellas, 8-11 June 1), n whh the man results of the researh work, durng approxmately the last ten or so years, on the general subjet Conformal Dynamal Equvalene, were presented. The members of the Study Group, partpatng n the above researh, are the followng: M. Plons (1 and ), C. G. Tsagas (1), K. Kleds (1 and 3), S. Baslakos (4), K. Zagkours (1 and 5) and N. K. Spyrou (1). 1: Astronomy Department, Arstoteleon Unversty of Thessalonk, Thessalonk, Hellas, : Insttute of Astronomy and Astrophyss, Natonal Observatory of Athens, Athens, Hellas, 3: Tehnologal Eduaton Insttute of Serres, Serres, Hellas, 4: Researh Center on Astronomy and Appled Mathemats, Natonal Aademy of Athens, Athens, Hellas, 5: Oxford Unversty, Oxford, Unted Kngdom. Publshed under lene by Ltd 1

3 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 System, for example, vewed as a star aompaned by a number of nearby small objets and planets, dffers from the one observed embedded n the solar wnd, the outer boundares of whh extend up to the orrespondng outer boundares of the wnds from nearby stars. Also, far beyond the Kuper Belt, extendng out of the orbts of the outer planets Neptune and Pluto, there s an enormous spheral array of y worlds orbtng the Sun, alled the Oort Comet Cloud, whose lnear dmensons are of the order of 1 5 AU. Ths means that the Solar System extends up to half the dstane of the alpha-proxma Centaur, the star nearest to the Sun. Furthermore, aordng to far-ultravolet spetrosop observatons of the propertes of the hghly-onzed oxygen, of the hot gas, and of the hgh-veloty louds n the halo of the Mlky Way galaxy and the Loal Group [1] (see also [] and referenes theren), there exsts an extended, hot (T > 1 6 K), and suffently dffuse Galat Corona, prevously undeteted by other means (e.g., X-rays), far beyond the nearby halo deteted prevously. Ths mples that the Galat Corona, and hene the Mlky Way galaxy tself, possbly extends out to the Magellan Clouds. So, the expeted lnear dmensons of the Mlky Way galaxy are at least kp, almost ten tmes larger than ts optal lnear dmensons (~3 kp), and even larger on the bass of at least some of the hgh-veloty louds at estmated dstanes as large as 85 kp [3]. If suh a result ould be onsdered as typal, then the mutual (mnmal) dstane of the outer boundares of the Mlky Way and the nearby Andromeda Galaxy, at a dstane of approxmately 6 kp from t, s of the order of ther lnear dmensons, approxmately kp. Suh a result for neghborng galaxes s generally beleved to be vald for typal lusters of galaxes, n whh the mutual dstanes of neghborng members are of the order of a few hundred kp. Under suh ondtons, the lnear dmensons of galaxes wth haloes and oronas (n the range.1 kp 1 Mp, dependng on the type of the galaxes, e.g., ellptal, spral, rregular) s not neessarly muh smaller than ther mutual dstanes. As a onsequene, the dynamal desrpton of the observed motons n a typal osmologal struture s not that of a system of gravtatng pont masses (namely, no tdal nteratons), but rather of the form of hydrodynam flows n a more or less ontnuous gravtatng soure. Moreover, the struture of the nterstellar medum tself n ellptal galaxes and of the ntraluster medum n luster of galaxes present speal nterest. Thus, on the one hand, the nterstellar medum onssts prmarly of hot (T > 1 6 K) plasma; several moleular louds wth dmensons -5 p and masses ~1 6 m n t, are partly assoated wth HII regons and embedded n a lower-densty nterstellar medum; the masses (~ m ) of the entral dark objets n ellptal galaxes are only a few perent of the moleular-gas mass n the sub-klo parse regon, as mpled by the observatonal study of water-maserng soures [4] - [7]. On the other hand, a luster of galaxes an be treated [8] as an approxmately sothermal sphere of hot onzed hydrogen (of dmensons a few Mp, temperature, whh an be as hgh as ~1 8 K, number densty of eletrons ~1-4 m -3, and number-densty of moleular louds ~1-4 to 1-3 m -3 ); the ntraluster gas ontans X-rays, flls the spae between the galaxes, oupes muh of the luster s volume, and the X-ray lumnostes fall n the range 1 43 to 1 45 erg. se -1 ; the mass of the ntraluster medum exeeds the total mass of the lumnous parts of the luster s galaxes by a fator of several. (However, one must not forget that the surveys for the largest osmologal features so far over only.1% of the observable unverse!!!). In ths sprt, t s nterestng that, as a onsequene of the mprovement of the observng tehnques, urrently we wtness n varous ways (see, e.g., [9] [1]; or even ontnuous tatons-reports, pror the offal publaton) ontnuous observatonal detetons of many new galaxes (e.g., small ones eah orbtng another larger and brghter galaxy, or numerous, tdally-dsrupted and densely-populated galaxes (ultra-ompat dwarf; UCD galaxes) nhabtng at the entral regons of huge lusters of galaxes, lke the Fornax and the Vrgo galaxy lusters), of ntergalat globular lusters (beyond the ~15 ones surroundng our Mlky Way galaxy) and even stars [13], and also of hot flamentary networks onnetng and surroundng the galaxes of a luster of galaxes, seen n the ultravolet but unseen n the optal, nfrared and rado wavelengths [], [3]. Aordng to all the above, the morphology of galaxes s very dfferent from the smple pture of spral, ellptal, or, even rregular galaxes, n the sense that the galaxes (as well as the

4 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 lusters of galaxes, and, possbly, the seond-order lusters - super lusters - of galaxes) are almost spherally symmetr, very omplex, pratally ontnuous, and of muh larger lnear dmensons osmologal strutures than prevously assumed. Ths stuaton beomes even more nterestng, onsderng, n partular, the shapes of galat dark-matter haloes derved from gravtatonallensng studes, N-body smulatons, stellar traers (e.g., globular lusters), HI studes and polar rngs, studes of satelltes (e.g., varous tals), as well as from X-ray, and absorpton lnes studes [14] (also, n the same volume, see, partularly, the ontrbutons by Fllp and Sepulveda, Ryden and Tnker, Zepf, Sparke, Johnston, Buote, Bowen). The observable Unverse dffers very muh from the smple pture of a olleton of galaxes (or hgher-order osmologal strutures), n whh the mutual dstanes of the neghborng members are muh larger than ther lnear dmensons. Consequently, the onsttuents elements of the Unverse and the Unverse as a whole, an qute satsfatorly be treated dynamally as ontnuous gravtatonal systems and, more spefally, bounded, gravtatng perfetflud soures, the physal-dynamal desrpton of whh s very well establshed n both the Newtonan and the general-relatvst levels. So, we arrve at the very rual result, that the motons of and n these onsttuents should be onsdered as hydrodynam flows rather than geodes motons. It s exatly ths result, that enables us, through the dynamal-equvalene approah, to reast the geodes motons, now takng nto aount the ontrbuton to the observatonally determned mass of the flud soure of all of ts nternal physal haratersts, not only ts mass densty, as soures of the observed motons. Also, we reall [15], [16] that, aordng to the observatons of the osm mrowave bakground by the Wlknson Mrowave Ansotropy Probe (WMAP), the Cosmos s beleved to be omposed of heavy elements (.3 %), ghostly neutrnos (.3 %), stars (.5 %), free hydrogen and helum (4%) (and so baryon matter ~5%), dark matter ( %), and dark energy (73 %). Fnally, we pont out that the osmologal flud s mostly onsdered as ollsonless, and that, n the ontext of a osmology domnated by a osmologal onstant, t s not urrently known what defntely ths unknown s, and whether t s strtly onstant, nreasng, or dereasng wth tme. In vew of the above, we have appled the model of a bounded, ontnuous, gravtatng perfet-flud soure of lassal hydrodynams or/and magneto hydrodynams, n order to desrbe the sentrop motons n the large-sale osmologal strutures, all ther soures, and ther onsequenes. Ths s n aordane wth the approah of the Conformal Dynamal Equvalene (CDE) between the hydrodynam flows and the geodes motons, proposed and further developed and elaborated by the author and ollaborators (for detals and applatons, see [17] [38]). Exatly, the results so far of the study group on ths subjet wll be brefly presented here. The struture of ths artle s as follows: After an observatonal ntroduton n the begnnng of the present Seton 1, we reall n the next Seton that, n general, the onservaton of rest mass (baryon-number onservaton) and the onservaton of entropy are not mutually ndependent requrements of general relatvty, a very mportant result frequently overlooked. Then, n Seton 3, we frst outlne the dynamal equvalene n the ase of the Newtonan theory of gravty. We prove that the sentrop flows n a bounded, gravtatng perfet-flud soure are dynamally equvalent to geodes motons n a generalzed salar potental (V). Ths generalzed potental s expltly expressed n terms of the standard gravtatonal potental (U, satsfyng the standard gravtatonal Posson equaton), mass densty, and, addtonally, the nternal physal haratersts of the gravtatng soure, namely, mass densty, sotrop pressure and nternal thermodynam energy densty. Ths enables us to defne, through a Posson-type equaton for the generalzed potental, the generalzed mass-energy densty, ρ V (and the orrespondng generalzed mass, m V ) produng the generalzed potental. As an astrophysal byprodut-applaton, we prove n Seton 3, that, n the Newtonan theory of gravty for a stat onfguraton, the mass-densty and pressure funtonal laws annot be hosen arbtrarly from eah other, but ther hoe depends on eah other s one. Ths an onsderably lmt the form of the (wdely used n the lterature, Plummer-type) mass-densty law of an sentrop and sothermal equaton of state. 3

5 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 In Seton 4, t s proved that the generalzed mass-energy densty, ρ V, an be postve/vanshng/negatve, and that the generalzed mass-energy, m V, an, smlarly, be postve/vanshng/negatve. As a onsequene, based on the form of the generalzed Euler-Newton equatons of moton, we prove that, n the Newtonan theory of gravty, a postve/vanshng/negatve generalzed mass-energy densty, ρ V, mples a spatally dereasng/non-hangng/nreasng aeleraton, and, hene, domnane of the nwards aeleraton (attratve gravty)/non hangng aeleraton/outwards aeleraton (repulsve gravty), all of them stemmng from the relatve mportane of the mass densty and of the other nternal physal haratersts of the flud soure. In Seton 5, based on the results of Seton 4, we explore the dsrmnaton of the regons of prevalene of the attratve gravty and repulsve gravty, leadng to the determnaton of the so-alled nverson dstane, the latter s mportane n determnng-explanng the onventonal lnear dmensons of the soure onsdered, as well as ts temperature dependene. In Seton 6, we apply the results of the dynamal equvalene of Seton 3 to a typal nhomogeneous (wth a Plummer-type mass-densty onfguraton), and sothermal and sentrop perfet-flud-osmologal struture. We explore the relatve mportane of the soure s mass, m (theoretally evaluated through the Plummer-type densty law) and ts observatonally determned mass, m V (dynamal mass) and the latter s dependene on the mass densty, ρ, (and mass, m) and on the flud s nternal physal haratersts. Ths relaton enables us to prove that, under ondtons generally met n the physal Unverse, the mass m of the osmologal struture generally dffers from, and atually t s muh larger than the observatonally determned mass m V (dynamal mass). In ths way, we propose a possble, lassal (namely, not quantum-mehanally orented) partal explanaton of the mssng-mass problem. In Seton 7, the dstnton between the theoretal mass, m, and the observatonally determned (dynamal) mass, m V, (under the assumpton and use of planar, equatoral, rular geodess n ds galaxes), allow us to propose a lassal explanaton of the flat-rotaton urves problem. Addtonally, the dependene of the rotatonal veloty on the struture s (spatally onstant) temperature (beyond ts gravtatonal feld) s rtally examned. In Seton 8, we outlne the use of the CDE approah n presentng a lassal explanaton of the elebrated Poneer Anomaly Effet n the Solar System and, also, we provde the frst analytal determnaton of the true lnear dmensons of the Solar System (t extends up to almost half the dstane to our losest star α-proxma Centaury), and the determnaton of the nternal thermodynam energy at the near and far regons of the Solar System. In Seton 9, based on the notons of the repulsve gravty and nverson dstane, we outlne the use of the CDE approah towards provdng a possble theoretal physal explanaton of the mehansm of formaton of wnds and jets, wth possble applatons to the phenomena of the expanson of a red-gant star and a supernova. In Seton 1, we outlne the use of the CDE approah n the determnaton of the (osmologally-dynamally mportant) masses of the globular lusters of galaxes, omposed of both dark matter and baryon matter, by relatng theory and observatonal results. In Seton 11, we outlne the generalzaton of the CDE approah n the ase of a magnetzed gravtatng perfet-flud soure and the mportane of the new magnet ontrbuton. In Seton 1, we outlne the general-relatvst osmologal applatons of the CDE, n the ase of a ollsonal-dark-matter model. We prove that the extra energy, needed to ompromse for the urrently avalable osmologal data, an be ompensated by the energy of the nternal motons of the ollsonal osmologal flud, that the post-reombnaton Unverse remans ever-deeleratng, and, also that the explt form of the equaton of state for the osmologal flud an be determned. In other words, n the ollsonal-dark-matter model, no dark energy and no aeleraton of the unversal expanson are neessary. Also, we ndependently verfy the partularly nterestng fats, that, n the ontext of the standard, although, physally, less probable, ollsonless-dark-matter model, the extra dark-energy omponent (to ompromse the osm-mrowave-radaton-bakground data) s, n fat, needed, the dstant lght soures appear to be dmmer than expeted, and the expanson of the Unverse 4

6 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 keeps balanng between aeleraton and deeleraton, dependng on the (fully and expltly determned approprate) value of the osmologal red-shft parameter. Therefore, these results of the ollsonless-dark-matter model seem to be a msnterpretaton of the osmologal observatonal data by an observer, who (although lvng n a Unverse flled, manly, wth, ollsonal dark matter) nssts n adoptng the tradtonal (ollsonless-dark matter) and less physal approah. Fnally, we onlude n Seton 13.. Adabatty and Isentropty versus Baryon Conservaton n General Relatvty It s known that, n general relatvty, the feld equatons for a gravtatng perfet-flud soure 8 G k 4 T k (.1) are suppled by the equatons of moton k T (.) ; k In Eqs. (.1) and (.), G s the gravtatonal onstant and the veloty of lght n vauum and, n the standard notaton, k s the Ensten tensor, T ( p) u u pg (.3) k k k s the flud s energy-momentum tensor, where g κ are the ovarant omponents of the metr tensor, u the ovarant omponents of the four-veloty, a sem-olon denotes ovarant dervatve, and the mass-energy densty ε s assumed (see, e.g., [39], pp , 9-94) to splt as (.4) wth the proper quanttes ρ, p and ρπ beng, respetvely, the rest-mass densty, proper sotrop pressure, and nternal spef-energy densty of the flud soure, the latter beng the thermodynam energy densty that hanges durng the expansons and/or ontratons of the flud. A dret onsequene of Eqs. (.) and (.4) s [4] u ( p) u (.5a), ; or, equvalently, 1 p u, p u ;, (.5b) where a omma denotes usual partal dervatve wth respet to the spae-tme oordnates. Therefore, n general relatvty, the onservaton of rest-mass (baryon-number onservaton) requred by the equaton s ompatble wth the equatons of moton f, and only f, ( u ) (.6) ; 1, p u (.7), Ths last equaton s obvously satsfed under the assumpton that the flud s hydrodynam flows are adabat, n other words, the net hange, dq, of the thermal ontent of a flud s volume element, defned through the frst lassal axom of thermodynams 1 dq d pd (.8) s always zero, namely, 5

7 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 1 dq Q, u d pd (.9) whene Eq. (.7) s dentally satsfed. Proeedng further to sentrop flows, t mght be helpful to reall here that n reahng the general result (.9) we an also use (as, e.g., n [17]) the equlbrum hydrodynams hypothess, n the form of the onstany of the entropy, S, along the flow lnes [41], [4], namely, n vew of the seond lassal thermodynam axom, dq TdS (.1) T beng the flud dstrbuton s absolute temperature, ds S u, dq d pd T T (.11) 1 1, p u T, In fat, n relatvst astrophyss and osmology, we usually assume S to be a group nvarant [43], [44], whene Su,, for every u and T (.1) or, equvalently, 1, p (.13), dret onsequene of whh s the adabatty ondton (.9) (or (.7)). Aordng to all the above, the onservaton of rest-mass (Eq..6) and the onservaton of entropy (Eq..13) are not ndependent requrements n the framework of general relatvty. The 1 reason of ther ndependene n the Newtonan lmt " ", s that Eq. (.5b) redues smply to ontnuty equaton ( ) (.14) t We emphasze that sentropty (and, hene, also adabatty) of the hydrodynam flows ould be physally neessary and useful. Atually, then, beyond the onstany of the matter ontent (the number of baryons) of a fnte flud volume element (Eqs. (.6) or (.14)), also ts thermodynam ontent remans unhanged along the flow lne, n aordane wth suh a usual assumpton for an solated physal system. As emphaszed by Chandrasekhar [4], the onservaton of rest-mass (.6), under the assumpton of the absene of any dsspatve mehansms n the flud soure, should be onsdered as a fundamental physal law supplementng the feld equatons (.1). If, aordng to [4], we furthermore assume that the only baryons present are protons and neutrons of pratally equally rest masses, the onservaton law (.6) s equvalent to the equally fundamental law of the onservaton of the baryon-number (pratally Eq. (.6), wth ρ replaed by the baryon-number densty). Consequently, f we assume that the rest-mass of a baryon remans onstant, we onlude that the onservaton of rest-mass (.6) s atually expressng the onservaton of the baryon mass. All these physal results onernng the adabatty or/and sentropty of the hydrodynam flows and ts dependene on the onservaton of rest-mass has long ago been emphaszed by Chandrasekhar [4], but, n our opnon, sne then, they, unfortunately, have been 6

8 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 hghly overlooked, espeally n astrophysal applatons. Here, we shall rely heavly on the adabatty of the hydrodynam flows and, hene, on ther dynamal equvalene to geodes motons (see, e.g., [17]). 3. Newtonan Dynamal Equvalene of Hydrodynam Flows and Geodess It has been suggested [17] that, n both the Newtonan and the general-relatvst theores of gravty, t s possble to gve to the equatons of the hydrodynam flow motons n the nteror of a bounded gravtatng perfet-flud soure the form of the equatons of geodes motons. In ths Seton 3, we lmt ourselves to the Newtonan ase (for the relatvst hydrodynam ase, see [17], [33], [37], [38] and for the magneto-hydrodynam ase, see [34], [35]; see also Setons and 1 herewth). More presely, n the ase of the nteror of a non-magnetzed flud, the Newtonan equatons of the moton of a test partle (geodes moton n the nteror of the flud) are d U (3.1) dt where the Newtonan gravtatonal potental U obeys the Newtonan feld equaton (Posson s equaton) U 4G (3.) wth ρ beng the gravtatonal soure s mass-densty funton. On the other hand, the Euler s equatons for the hydrodynam flow moton of a flud volume element (n the nteror of the non-magnetzed flud), are d 1 U p (3.3) dt suppled by the ontnuty equaton (.14). These two knds of moton n the nteror of the flud soure dffer from eah other due to exstene of the pressure-gradent term n Eq. (3.3). We remark that the ontnuty equaton (.14), as an expresson of the onservaton of the flud s total mass, orresponds, physally, to the assumed onstany of the test-partle s mass. Now we assume that the flud-volume element s hydrodynam flow s adabat and sentrop, whene, there s no exhange of heat quanttes (Q) between the flud-volume element and the rest of the flud soure. Then, aordng to Eq. (.9), dq d d 1 p dt dt dt or, equvalently, 1 1 p p (3.4) t t In analogy to Eq. (.1), for dq to be dentally zero, Eq. (3.4) should be vewed as an dentty for every value of the three veloty,, of the volume element, so that the followng two equatons 1 1 p, p (3.5a,b) t t suffe to hold smultaneously. Note that, for a tme-ndependent soure ( =), Eqs. (3.4) and t (3.5a) are equvalent to eah other. In vew of Eq. (3.5a), the Euler s equatons of moton (3.3) are wrtten n the form d V (3.6) dt 7

9 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 where the generalzed (salar) potental V s defned as p V U. (3.7) Eqs. (3.6) are of the same funtonal form as the geodes equatons of moton (3.1), namely, the aeleraton three-vetor s equal to the gradent of a salar potental. Therefore, the adabat perfet-flud hydrodynam flows are dynamally equvalent to the geodes motons n the generalzed potental V. Interestngly enough, p and Π, beyond ρ, appear now as soures of geodes moton, n ontrast to the non-adabat-flud ase, n whh only ρ (through U, Eq. (3.)) produes geodes motons. Next we mght ask: What s the mass-energy densty ρ v, produng the generalzed potental V? Obvously, n vew of the orrespondng Newtonan geodes equatons (3.1) and (3.), the densty ρ v an be defned through the Posson-type feld equaton V 4G v (3.8) The generalzed feld equaton (3.8), and, hene, the ntroduton of ρ v s a natural onsequene of the generalzed equatons of moton (3.6) (a property shared by the general-relatvst equatons (.1) and (.), but not of lassal Newtonan gravty). In ontrast to ths, the standard Posson equaton (3.), s not a onsequene of the Euler s equatons (3.3), but, ndependently of the latter, t, smply omplements them. It s straghtforward to prove, wth the ad of Eq. (3.7) along wth Eqs. (3.) and (3.5a), that v (3.9) where 1 p 1 1 p (3.1) 4 G 4 G wll be alled the nternal-mass densty, so that Eq. (3.8) s wrtten n the form V 4 G( ) (3.11) It s obvous that the mass m v, whh orresponds to the densty ρ v produng the generalzed potental V, s 3 m d x (3.1) where v s the three-dmensonal volume of the soure onsdered. In vew of Eqs. (3.9) and (3.1), m m m (3.13) namely, the mass m v dffers from the mass by the nternal mass m m v v v v 3 d x (3.14) v (3.15) v 3 d x Hene, the extra ngredent ρ to the generalzed mass densty, ρ v, stemmng from the soure s nternal physal haratersts, results n an extra mass, ndependent of the mass m of the soure (namely, the volume ntegral of ρ), the so-alled nternal mass (namely, the volume ntegral of ρ ), beng also a part of the observatonally determned mass (the dynamal mass; see Seton 6, below). 8

10 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 We note that, n general, the mass m v s not onstant. If we would wsh to postulate the onservaton of m v ths would be equvalent to the generalzed ontnuty equaton dmv m m o ns t. (3.16) dt v t v (3.17) whh, n turn, generalzes the lassal ontnuty equaton (.14). The onstany of m + m permts m and m to hange n tme. For both to be onstant n tme, the orrespondng two ontnuty equatons must hold smultaneously and ndependently of eah other at every moment. In any ase, the possbly assumed neessty of Eqs. (3.16) and (3.17) would lmt the harater and propertes of the physal soure onsdered, probably, n a qute desve way. The astrophysal and osmologal sgnfane of the above results ( ) les n the fat that (see Setons 6 and 7, below) the mass determned at any moment wth the ad of the geodes motons n the generalzed potental V s not smply the mass m, determned wth the ad of the geodess n the feld of the gravtatonal potental U, as t s generally beleved, but the mass m v, whh s dfferent from m. Moreover, t s obvous that V s not the standard gravtatonal potental U modfed by hand ; nstead t s a natural generalzaton of U resultng smply from the assumpton of the sentropty of the hydrodynam flows, and, so, we are not dealng here wth a modfed-gravty approah. Also, n the generalzed potental V, the volume element moves as a pont mass, but, now, arryng along all the nternal physal haratersts of the flud soure (not only ts mass densty). We emphasze that both the above two remarks are of speal mportane n the ontext of both the Newtonan and the general-relatvst theores of gravty (see also Seton 1, below). d In the speal ase of a stat dstrbuton,, from Eqs. (3.6) and (3.8) we dt obtan V, V, v (3.18) a stuaton smlar to the moton n «vauum», and also 1 p 4G (3.19) The physal sgnfane of Eq. (3.19) s best seen n the ase of a spherally-symmetr flud dstrbuton wth on equaton of state of the form p f( ) (3.) Then, Eq. (3.19) s wrtten as 1 df 1 df d 1 df ( ) 4 G (3.1) d r d d d where a prme denotes total dervatve wth respet to the radal dstane r. Eq. (3.1) an be seen as ether a onssteny ondton between Eqs. (3.) and (3.), for a gven densty dstrbuton () r, or, as a dfferental equaton to be solved for ρ(r). Therefore, for stat and spherally-symmetr gravtatng perfet-flud soures, under adabat ondtons, the funtonal form of an arbtrarly hosen equaton of state f(ρ) depends on the funtonal form of the mass-densty dstrbuton law ρ(r). In other words, the laws of the two parameters, mass densty and pressure, annot be hosen arbtrarly from eah other, but ther hoe depends on eah other s one. 9

11 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 As an applaton we onsder the adabat and sothermal (adabat ndex γ = 1) equaton of state of the form (6.1) below and a Plummer-type rest-mass densty of the form (6.) below. In ths ase, Eq. (3.1) redues to or, furthermore, 4 G r n 1 (3.) r r 3n r 4 G r n (3.3) r r r r r Applyng Eq. (3.3) for r ~, r = r and r = R (the radus of the dstrbuton), and omparng the values of κ n these three ases, we onlude that neessarly ln 3 n (3.4α) ln and, sne n s, by assumpton, an nteger, fnally n 3 (only!!!) (3.5β) Suh a relaton between the adabat ndex γ (here 1) and n has already been derved ndependently and more generally by Kleds and Spyrou [17], namely, 1 1 (3.6) n and so n n Eq. (6.) below s, pratally, the adabat ndex γ ( see also Seton 5 of [17]). 4. Attratve Gravty and Repulsve Gravty n Newtonan Gravty Now we remark that, ontrary to the ase of the equatons of moton (3.1), the equatons of moton (3.6) do not neessarly mply an nwards aeleraton sοlely (namely, only attratve gravty), beause d ( V ) V 4G v (4.1) dt Therefore d dt v (4.) or, equvalently, Spatally dereasng d mples v ( V 4 Gv ) (4.3a) dt Spatally non-hangng d mples v ( V ) (4.3b) dt Spatally nreasng d mples v ( V 4 Gv ) dt (4.3) and ve versa. Aordng to the above, v mples domnane of nwards aeleraton (4.4a) (attratng gravty) mples non hangng (spatally) aeleraton (4.4b) v v mples domnane of outwards aeleraton (antgravty, repulsve gravty) (4.4) 1

12 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 The sgnfane of the above results (4.1) (4.4) s that, n ontrast to the general belef, they reveal the possblty of repulsve gravty n the framework of the Newtonan gravty. Obvously, ths possblty of repulsve gravty s a onsequene of the CDE approah, aordng to whh the soure s nternal physal haratersts appear to at a soures of gravtatonal geodes motons. Furthermore, t s rather straghtforward from the defnton (3.1) for ρ, that, n realst ases, ρ annot be a onstant or vanshng (see Seton 6, below) and, hene, ρ v an, n prnple, have any sgn,, as stated above. v 5. The Inverson Dstane and the CDE Approah We reall that the generalzed mass-densty (3.9) produng the geodes motons (3.6) and (3.7) an be ether postve, or negatve, or even vansh. Ths mples the possblty of a spatally-nreasng, or spatally-dereasng, or even spatally-unhangng aeleraton, dependng on the dstane from the entre of the soure as ompared to the so-alled nverson dstane. Next, usng Eqs. (3., for n=3), (3.9) and (3.1), we fnd z (5.1) z where 1 r z 1, ( r r ) (5.) r and 3kT 1 mh 4 G r (5.3) Then, frst we note that, by standard theoretally-derved assumpton, the radus of the nnermost stable rular orbt (dentfed wth r ) around a entral Shwarzshld blak hole of mass M s three tmes the blak hole s gravtatonal radus, R, If we approxmate [8] M as r 6GM 3Rs s we fnd whene M r ( r r ) r (5.4a) G r, (5.4b) ( M / M ) gr. m (5.4) and 3 kt (5.5) mh Although the hoe (5.4a) seems arbtrary, the ondtons (5.4a,b,) an be ompatble wth the assumpton of a blak hole as a fttous homogeneous dark objet of mass M, radus r and mean mass densty equal to ρ(r=r ). Atually, n the ase of a super-massve blak hole of mass M ~

13 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 M ʘ, we fnd r ~ 1 15 m, whene the ondton (5.4b) mples ρ ~1-14 gr.m -3, namely, about ten orders of magntude larger that the mass densty of the Mlky Way n the solar neghborhood (~1-4 gr.m -3 ). Then, under the ondtons assumed up to now, we note that the ondton s equvalent to The dsrmnant of the bnomal φ(z) satsfes 1 (equvalently, v) (5.6) ( z) z z (5.7) (5.8) Δ, when TT lm (5.9) wth (see, however, also Eqs. (5.3) - (5.5)) Tlm mh ~ K 6k (5.1) where the threshold temperature T tt s defned va Eq. (7.4). Therefore (see also Eqs. (4.1)), T T ( ), ( z), (5.11) For lm For T T ( ) lm s aeptable, namely,, from the two solutons of the Eq. (5.7) only one, the largest z, 1 r nv r rnv; 8 r 4 1 (5.1) suh that a) If r rnv, then (nwards aeleraton or attratve gravty domnates) (5.13a) b) If r rnv, then (spatally unhangng aeleraton) (5.13b) ) If r rnv, then (outwards aeleraton or repulsve gravty domnates) (5.13) In the speal ase, namely, T Tlm, whh s beleved to apply usually n almost all of the osmologal large-sale strutures, Eqs. (5.1) and (5.) redue to, respetvely, rnv 1 mh r 3 kt (5.14) wth rnv znv ~ ( xnv, for r r ) (5.15) r 1

14 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 ASTRONOMICAL SYSTEM r Inverson Dstane r nv Numeral Values SOLAR SYSTEM R (6.96X1 1 m).391 T / () 8 AU AU, for T 3 ~ 1 K GALAXY 3R s (Central Blak Hole M ~1 6 M ~8.86x1 11 m) 1.48 T / p (6) T Kp, for ~ 1 K (Conventonal Outskrts of the Mlky Way) CLUSTER OF GALAXIES 3R s (Central Galaxy Blak Hole M ~1 1 M ~8.86x1 17 m) T / Kp (8) T Mp, for ~ 1 K (Conventonal Dmensons of a Cluster of Galaxes) SUPERCLUSTER OF GALAXIES 3R s (Central Galaxy Blak Hole M ~1 1 M ~8.86x1 17 m) 1.48 T / Kp (9) T Mp, for ~ 1 K (Conventonal Dmensons of a Super-luster of Galaxes) Table 1: The nverson dstane for typal osmologal strutures Representatve values of the nverson dstane, r nv, based on Eqs. (5.14) and (5.15), are shown n Table 1, and are seen to be omparable to the urrently aepted lnear dmensons of the orrespondng struture. Ths means that the real dmensons of the struture are muh larger, than thought up to now. We onlude that, for nreasng r, smaller than the nverson dstane r nv, the gravtatonal repulson of the negatve nternal mass s smaller than the gravtatonal attraton of the postve restmass. The matter aelerates nwards and eventually reahes an equlbrum stuaton due to the aton of some knd of pressure (here, thermal pressure). At the nverson dstane, r nv, the gravtatonal repulson and gravtatonal attraton anel eah other. Beyond the nverson dstane an nverson ours, namely, the gravtatonal repulson domnates over the gravtatonal attraton, and matter aelerates outwards. Therefore, we onlude that, n general, mass should exst also beyond the onventonal lmts of the struture. 13

15 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/ A Smplfed Classal Treatment of the Problem of the Mssng Mass Today, the general osmologal belef s that the (domnant) dark matter oexsts wth ts (subdomnant, small) baryon ontamnaton. The latter s tghtly bounded to the former, and the two onsttute the osmologal gravtatng (perfet) flud, n the sense that on top of a volume element of dark matter sts the baryon (lumnous) matter. So, n the sprt of CDE approah, the osmologal strutures, e.g., the galaxes, should be treated as fnte volume elements of the osmologal gravtatng flud n the ontext of a ontnuous Unverse (n ontrast to a olleton of gravtatng pont masses (galaxes) at mutual dstanes large ompared to ther lnear dmensons). In usng the results of Seton 3 (as n Setons 1 and 1 below), we note that, generally, ρ s the total mass-energy densty, nludng both the dark and the baryon matter, and the equaton of state p(ρ) = κρ, wth κ beng a postve onstant, refers to the total pressure, namely, due to both the dark and baryon masses. However, n ths Seton and for reasons of larfyng the physal mportane of the results of Setons 3-5, we shall treat the soure s matter densty and pressure as smply of baryon orgn, and examne the relatve mportane of the (smply baryon) mass-densty, ρ, and the nternalmass densty, ρ, n astronomal and osmologal strutures, and ts onsequenes. Generally, all the physal parameters desrbng an astrophysal or osmologal soure an be funtons of both spae and tme. Here we shall lmt ourselves to the ase of physal parameters beng expltly funtons of only the spatal oordnates. So, we assume that a gravtatng flud soure (of absolute temperature T and mean moleular weght μ) s spherally-symmetr of radus R, and s haraterzed by an adabat and sothermal (adabat ndex γ = 1) equaton of state, suh that kt p m (6.1) H where k and m H are, respetvely, the Boltzmann onstant and rest-mass of the atom hydrogen, obvously ompatble wth the sentropty ondtons (3.5a,b). Furthermore, n hoosng the baryon-mass-densty dstrbuton law, we shall rely on the well-known observaton-based unversal profle of Navarro, Frenk and Whte [45], [46] and the Hernqust [47] profle (see also [48] [5]) for the densty dstrbuton law of lusters of galaxes. So we shall adopt a speal ase of the above profles, namely, the Plummer-type densty desrbed by whene where n r ( r) 1 ( r,, n: a postve nteger) (6.) r Then usng, addtonally, the Gauss- Stokes theorem, from Eqs. (3.13) - (3.15) we fnd 3 nkt R m (6.3) GmH R r 1 r nkt m ( R r ) R Gm (6.4) H Then, for n=3 and for every upper lmt x max, we fnd m T 6 xmax (7) (6.5) m ln x max r R x, x, r 3R.87 1 M p (6.6) r 5 max s (8) r R s, T (7) and M (8) beng, respetvely, the Shwarzshld radus of the entral dark objet, the regon s (onstant) temperature n unts 1 7 K and the mass-energy, M, of the entral dark objet n unts of 14

16 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/ M. Therefore, n ths ase, the nternal mass s defntely negatve and, absolutely, t dverges wth the soure s lnear dmensons. Under suh ondtons, Eq. (3.13) takes on the form m mv ( m) mv (6.7) Therefore, the baryon mass m, determned wth the ad of the geodes motons n the gravtatonal potental U, s larger than the mass m v, determned wth the ad of the geodes motons n the generalzed gravtatonal potental V (for a proof of the above theorem n the relatvst ase, see [33]). Obvously, t s the geodes motons n the potental V that must be used rather than those n the gravtatonal potental U, beause n the former (V) all the nternal physal haratersts of the gravtatng soure are taken nto aount as soures of the observed motons. In other words, for smply physal reasons, as mass determned on the bass of geodes motons, the mass m v must be used, not the mass m. But then the baryon mass, m, of the (thermal) soure s larger than the mass m v determned observatonally. We shall apply the above results to galat and osmologal levels (for detals see [17], [] [31], [37], [38]). Thus n the ase of maserng galaxes (e.g., NGC 458 and NGC 168) and non-maserng galaxes (e.g., NGC 461) we fnd that, dependng on the lnear dmensons of the rumnulear regon onsdered (rangng from the sub-parse up to the kloparse), -m s not always neglgble ompared to the mass M of the entral dark objet (1-5 < -m / M < 1 - ). On the other hand, -m an be omparable to the total rest-mass, m, of the rumnulear regon (1-3 < -m / m <.6; the upper bound.6 orrespondng to the mass of the blak hole beleved to lurk n the Mlky Way s nulear regon). Spefally, n the ase of the Mlky Way, for the lnear dmensons (R) of the regon onsdered beng n the range.1 p and 1 kp, the rato -m /m falls n the range (1 to 1 3 ) T (7) /μ. Ths means that, e.g., n the nnermost regon of dmensons.1 p, ths rato an be larger than unty, provded that the temperature there s at least 1 8 K, and also that onsderng the broader regon of dmensons 1 kp, the same rato an be at least 1 3 for temperature of the order 1 K. It s very nterestng that the above ondtons an generally be met n the physal Unverse. Thus, as mentoned n the Seton 1, the nterstellar medum n ellptal galaxes onssts prmarly of hot ( T > 1 6 K ) plasma; several moleular louds wth dmensons -5 p and masses ~1 6 solar masses (m ) are partly assoated wth HII regons and embedded n a lower-densty nterstellar medum; the masses (~1 8 m ) of the entral dark objets n galaxes are only a few perent of the moleular-gas mass n the sub one kloparse regon. All the above data are n aordane wth the modern aspet of a galaxy, dfferent from that dedued on the bass of smply the galaxy s optal vew. Analogously, n the ase of typal lusters of galaxes wth dmensons [11] of a few Mp, the rato m /m falls n the range (1 3 to 1 5 ) T (7) /μ and so t an be larger than unty, provded that T/μ > 1 4 to1 5 K. Agan ths s generally true, beause, as mentoned n Seton 1, a luster of galaxes an be treated as an approxmately sothermal sphere of hot onzed hydrogen (of dmensons a few Mp, temperature ~1 8 K, number densty of eletrons ~1-4 m -3, and number densty of moleular louds ~1-4 to 1-3 m -3 ); the ntraluster gas ontans X-rays, flls the spae between the galaxes, oupes muh of the luster s volume, and the X-ray lumnostes fall n the range 1 43 to 1 45 erg. se -1 ; the mass of the ntraluster medum exeeds the total mass of the lumnous parts of the luster s galaxes by a fator of several. Fnally, n the ase of a seond-order luster (luster of lusters or super-luster) of galaxes, the rato m /m s of the order of 1 3 T (7) /μ, and so t an exeed unty, for T/μ > 1 3 K, a ondton that agan s met (note, however, that urrent observatons do not seem to support the dea of a spherally-symmetr super-luster of galaxes; on ths, see also, espeally, [3], [8]). 15

17 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 r max -m /m -m /m 1.1 p.53 T (7) /μ T(7) T , K 1 p.449 T (7) /μ T(7) T 7.3,.3 1 K 1 p T (7) /μ T(7) T 5.931,.93 1 K kp T (7) /μ T(7) 3 T , K 15 kp x 1 3 T T (7) /μ T(7) 4 T 3.481,.48 1 K 3 kp 8.1 x 1 3 T (7) /μ T(7) 4 T , K 1 kp.549 x 1 4 T (7) /μ T(7) 5 T 3.91, K kp x 1 4 T (7) /μ T(7) 5 T.11,.1 1 K Table : Relatve mportane m / m, of the negatve nternal mass, m, and the postve rest mass m T 6 xmax (7) m, for the Mlky Way: , M (8) =1-7, r.87 1 p m ln x From all the above we onlude that, usng the geodes motons (nsde and outsde of) the osmologal strutures for the observatonal determnaton of masses, n onjunton wth the CDE approah, the extra (negatve) mass m (whh s meanngless n the ontext of the standard geodes moton (3.1) of a test partle) appears n the observatonally determned dynamal mass and, n many ases, t an largely exeed the mass of the gas of the (orrespondng regon of the) large-sale osmologal struture under onsderaton. We onsder ths as an nterestng ontrbuton towards larfyng the onept of the mssng mass. The above results, for dfferent values of M (and r ), are shown analytally n Tables and 3. max 16

18 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 A typal luster of galaxes: m x T m ln x 6 max (7), max r.87 1 M p 5 (8) Μ (8) r max -m /m -m /m Mp.79 x 1 3 T (7) /μ T T 3.581, (7) 4 3 K 1 Mp T (7) /μ T T.931, (7) 5 K A typal nd order luster of galaxes : m x T m ln x 6 max (7), max r.87 1 M p 5 (8) M (8) r max -m /m -m /m 1 1 Mp.7917 x 1 3 T (7) /μ T T 3.581, K 1 (7) 4 3 Table 3: Relatve mportane m / m, of the negatve nternal mass, m, and the postve rest mass m, for: () A typal luster of galaxes and () a typal nd order luster of galaxes. Fnally, we stress that our method has been based on the assumpton that only the baryon densty s onsdered, whle the densty of dark matter s gnored. However, our method an be generalzed, n the sprt of Setons 1 and 1 below, so as to take nto aount both the above denstes. Ths subjet s urrently under srutny. 7. Dynamal Masses and Flat Rotaton Curves of Ds Galaxes In ths Seton, we shall demonstrate that the flat rotatonal urves of the dsk galaxes an naturally be explaned, based on the generalzed Euler s equatons of moton (3.6), n whh ρ s agan smply the baryon densty. Thus, the ase of a spherally-symmetr perfet-flud soure of generalzed densty ρ v (r), the veloty, υ f (r), of an equatoral, rular geodes orbtal moton at a dstane r from the enter s Gmv () r G f ( r) m( r) m( r) r r (7.1a) replang the standard one (on the bass of the equatons of moton (3.1)) Gm() r () r, (7.1b) r 17

19 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 where, we emphasze, that, by defnton, the masses m(r) and mv () r are the volume ntegrals of the orrespondng denstes ρ(r) and ρ v (r), manfestng themselves n the orrespondng Posson equatons, (3.) and (3.8). We reall that, for a homogeneous soure of densty ρ, Eq.(7.1b) redues to 1 4G () r r (7.) 3 Eq. (7.), wth the exepton of the rumnulear regon, s n ontradton wth observatons. On the other hand, for ρ v to be dfferent than ρ, an sothermal gravtatng perfet-flud annot be homogeneous. Moreover, as dedued from the 1 m radaton, n dsk galaxes the HI louds, at dfferent dstanes from the enter of the galaxy, all orbt at more or less the same speed. Ths an be explaned, n the present ontext, however, f we assume a non-homogeneous, spherally-symmetr soure and use Eq. (6.), whene r 3 1 r r r m( r) 4r ln 1 ln 1 1 (7.3a) r r r 1 r r 4 r ln for r r (7.3b) 3 r If, addtonally, we assume the equaton of state, Eq. (6.1), we fnd, n analogy to Eqs. (7.3) (for n=3), 3 3kT r m () r (7.4) GmH r r 1 r Insertng Eqs. (7.3) and (7.1a) nto Eq. (7.8a) and usng also the ondton (5.4a,b) we fnd f ( x) 1 ln 1 x 1) x (7.5) 1 1 x ln 1 x 1 x x 1 where we have put r 3kT 13 T x,.75 1 (7.6 ) r m H The nterest n Eq. (7.5) s, among others, n the explt dependene of ( x) on the dstrbuton s temperature. A more prese expresson, however, for the veloty ( x), to be used for omparson wth observatonal data, should take nto aount the mass, M, of the entral dark objet, beyond of the densty ρ v. The veloty BH () r, due to the mass M, of an equatoral, rular geodes orbt at the dstane, from the enter s f f 18

20 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 GM GM GM GM BH r r x 3R GM sx 3 x 6x So, the total rular veloty, V, s V f BH 1/ 1 ln 1 x 1 x x x 1 ln(1 ) x x 1 6x 1/ (7.7) (7.8a) or 1/ 1 ln 1 x 1 x km V s 1 1 x 1 ln 1 x 1 x x 1 6x or, equvalently, (7.8b) kT B x 1 V.9891 ln 1 x 1 ln 1 km s x x 1 x μmh x 1 6x (7.8) In the form (7.8a) or (7.8b) or (7.8), the veloty V s, generally, preferable than n the form (7.5), espeally lose to the entral dark objet. In Fgure 1, the form of the veloty V s plotted as a funton of x, for Τ/μ~1. The expeted flat harater of the rotaton urve s promnent, resemblng the ase of the Mlky Way (V km s ). Smlar results are presented n Fgure (the group of urves n the mddle), for several values of the temperature, namely, T/μ~1.8 x 1 6 (γ~5.1x1-7 ), Τ/μ~1.45x1 6 (γ~3.99x1-7 ), Τ/μ~1 5 (γ~.75x1-8 ), T/μ ~1 4 (γ~.75x1-9 ) and, fnally, Τ/μ~1 3 (γ~.75x1-1 ). The fourth ase, n partular, s n good agreement wth, e.g., the typal velotyurve for a S spral galaxy [9]. The man dfferene, beyond the dfferent sale, between the theoretally-derved veloty urve and the observed one, refers to the ponts lose to the enter, whh are not onsdered n the theoretal treatment. 1 19

21 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 It s remarkable that the flat part of the rotaton urve orrespond to the same range of 1 1 values of x (>1 1) and, generally speakng, for suh large values of x (>1 1), only a slght (but not vanshng) dependene of V on the temperature s noted. Also, n our ase, the 1 theoretally derved flat rotaton urve ontnues up to the dstane of at least kp ( x 1 ). In the ase of the Mlky Way galaxy, suh a dstane (beyond the Magellan Clouds!!!) defnes the galaxy s real dmensons, as ontrasted to ts onventonal radus ~15 kp [1]. (For an ndependent method of determnaton of the true lnear dmensons, appled n the ase of the Solar System and based on the vanshng of the generalzed aeleraton V (Eq. (3.6)), see Seton 8 below; and also Table 1 for the nverson dstane). We onsder the results of Setons 6 and 7 as a partal lassal soluton to the dark-matter problem. Obvously, as wth Seton 6, our method an be generalzed, n the sprt of Setons 1 and 1 below, so as to nlude both the dark-matter and the baryon denstes. Ths subjet s urrently under srutny.

22 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 Fgure : Flat rotaton urves for several values of the temperature, n large-sale strutures. 8. A Classal Treatment of the Poneer-Anomaly Effet n the Solar System (wth K. Zagkours) The Poneer 1 and 11 spae probes have been launhed n 197s and are urrently travellng outwards our Solar system. There has been a measured devaton n ther (and other spae probes n the Solar System) speeds, whh shows a small extra nward aeleraton towards the Sun, amountng n, approxmately, the above spae probe (and not only them), durng ther lfetme up to now, loosng a dstane approxmately equal to the dstane of the Moon from the Earth. Ths extra aeleraton has been gven the name the Poneer Anomaly Effet. Currently, no unversally aepted explanaton exsts for ths anomaly. In order to solve ths mysterous aeleraton, many theores have been proposed, most of whh onentrate n observatonal errors, reordng errors, new gravty soures n the Solar system and, fnally, even new physs. We studed the Poneer Anomaly Effet n the framework of the CDE approah. In our study we use a model of the Solar System wth the spherally-symmetr (or, equvalently, pont-mass) Sun of mass M at ts entre and an extra mass densty (generally, both baryon and dark matter) 1

23 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 dspersed around t. The mass-densty profle s desrbed by a Plummer-type equaton (see Eq. (6.)). In the numeral applatons, just for smplty, we treat the extra, dspersed mass as a baryon sothermal perfet flud soure (nevertheless, ths may not be neessary, beause the same steps an be followed for any mxture of dark matter and baryon matter). For ths system, the generalzed Euler s hydrodynam equatons for the spherally-symmetr total aeleraton of the sentop moton are wrtten n the form γ G r M mv r r γ γ ln r GM r r r 4πGρ r 3k r r r r the two parts on the r.h.s. beng due to, respetvely, the entral spherally-symmetr Sun and the extra mass (flud) dspersed around t. Furthermore, reallng that n the CDE approah the flud volume element moves as a test partle, n aordane wth the generalzed laws (3.6) and (3.7), namely, but, now, arryng along all the physal haratersts of the flud soure, we dentfy the (Poneer or other) spae probe wth the above volume element-test partle, movng as above. From our analyss, appled to a baryon and sothermal mass dstrbuton, dspersed around the entral Sun, we end up we some nterestng results two of whh are worth mentonng here. The frst result s that we an atually evaluate an extra aeleraton at the regon of the Poneer spae probe and the numeral results are n agreement wth the observed data, namely, the extra 8 aeleraton evaluated n our analyss s equal to γ = m s, whle the measured extra 8 aeleraton s γ = m s. We onsder the above result as a qute aeptable lassal explanaton of the elebrated Poneer Anomaly Effet n the Solar System. (8.1) Fgure 3: The extra aeleraton as a funton of the dstane from the Sun.

24 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 The behavour, wth the dstane from the Sun, of the evaluated extra aeleraton s shown n Fgures 3 and 4, and the orrespondng result for the total aeleraton s shown n Fgure 5. The extra aeleraton s an nward aeleraton up to ertan dstane, after whh t beomes an outward aeleraton. In ontrast, the total aeleraton s always an nward aeleraton vanshng at some far dstane, R. Ths last result permts the frst theoretal analytal determnaton of the true lnear dmensons of the Solar System, R, as the soluton of the algebra equaton (for r) γ =, for r = R. Fgure 4: The extra aeleraton as a funton of the dstane from the Sun. In fat, for T/μ = 1, the Solar System s found to extend up to, approxmately, the dstane 1 kau, whh, remarkably enough, s the radus of the Oort Cloud, and amounts to almost half the dstane to our losest star α-proxma Centaur. Ths result shows that the Solar system s not as small as we thought up to now, but t may atually be a ontnuous entty that almost touhes the nearest neghbourng stellar system. Fnally, we note that our model permts also the evaluaton of thermodynam parameters, lke the nternal thermodynam energy (per unt mass), Π, whose values at the surfae of the Sun s 9-8 r m s and at the Solar System s boundary s R m s. Of ourse, our model permts the analogous treatment of other spae probes n the Solar System (e.g., New Horzons). For further detals and results on the applatons of the CDE approah to the Solar System, see [36]. 3

25 Journal of Physs: Conferene Seres 83 (11) 135 do:1.188/ /83/1/135 Fgure 5: The total aeleraton as a funton of the dstane from the Sun. 9. A possble theoretal explanaton of the formaton mehansm of wnds, jets, supergant stars, and supernovae As desrbed n Setons 4 and 5, the generalzed gravty an be repulsve, provded that the generalzed densty v s negatve. In ths way, usng the noton of the nverson dstane, we an desrbe and justfy the outwards aeleraton neessary for the formaton of wnds and jets. Also, possble applatons to the phenomena of the expanson of a red-gant star and a supernova are onsdered. Ths subjet s, urrently, under srutny. 1. Determnaton of the masses of lusters of galaxes (wth M. Plons and S. Baslakos) Of obvous speal osmologal mportane and nterest s the determnaton of the masses of the globular lusters of galaxes. Usually, the total gravtatng mass (dark matter and baryons) of a typal luster of galaxes s lassally beng treated usng the hydrodynam propertes of the nter-luster gas. More spefally (Plons, prvate ommunaton), for a luster n hydrostat equlbrum, from the ombned use of the Posson equaton for the total gravtatonal potental (due to the total mass densty, namely, dark matter and baryons), an equaton of state for the gas (namely, the total pressure due only to the densty of the gas.e., ts baryon mass), and an assumed Plummer-type gas-densty profle (wth, e.g., n = 3), all plugged nto the Euler equaton of hydrostat motons, the hydrodynam mass and the total mass-densty are evaluated. Smlar results are obtaned for the same perfet-flud luster, usng the dea of the CDE approah. The latter method an be generalzed for a non-stat luster of galaxes, so as to nlude the total mass-densty and the total pressure of the non-stat soure, namely, onsderng the ontrbutons to them of both the dark matter and the baryon matter, and, addtonally, usng for the non-stat soure the notons of the spetral shft and the generalzed dynamal mass. More presely, assumng an equaton of state n terms of the total densty and total pressure, p wρ, w (1.1) we evaluate the generalzed densty ρ n terms of the mass densty of the baryon mass w ρ 3 ρ ρ, m ρ d x V 4πG ρ (1.a, b) V V 4