A thermodynamic degree of freedom solution to the galaxy cluster problem of MOND. Abstract

Transcription

1 A themodynamic degee of feedom solution to the galaxy cluste poblem of MOND E.P.J. de Haas (Paul) Nijmegen, The Nethelands (Dated: Octobe 23, 2015) Abstact In this pape I discus the degee of feedom paamete of the emegent Dak Matte foce. I show how this degee of feedom paamete N esults in a possible diffeence between obseved mass and appaent mass, in cases whee N is lage than one. This might solve the galaxy cluste mass discepancy of MOND. In my model the degee of feedom of galaxies in clustes influences thei numbe of micostates inside the cluste and thus the entopy. And then it also influences the appaent bayonic mass in the emegent Dak Matte foce that will appea a facto in between two o thee bigge than can be diectly obseved. CONTENTS I. The entopic Dak Matte foce as deived fom the fist law of themodynamics 2 II. The paametes of the entopic model 3 III. The galaxy cluste poblem of MOND as caused by the degee of feedom paamete 4 Refeences 5 haas2u@gmail.com 1

2 I. THE ENTROPIC DARK MATTER FORCE AS DERIVED FROM THE FIRST LAW OF THERMODYNAMICS In my elementay paticle Dak Matte halo model, see [1] fo futhe infomation and efeences, I stat with the gavitational souce mass and I get gavitational potential at as m g = m 0 + m dm = m 0 + dm m 0 = m 0 φ = GM 0 ( 1 + ) dm (1) GM 0 dm = φ 0 + φ dm. (2) Fo the esulting foce of gavity on a classical chage mass m we get the unchanged Newtonian esult F = m φ = m φ 0 + m φ dm = m φ 0 = GM 0m 2 ˆ. (3) The gavitational enegy is howeve affected by the Dak Matte potential as U g = mφ = mφ 0 + mφ dm = GM 0m GM 0m dm. (4) I assume that the viial theoem is still valid. Using 2U k = U g I get v 2 = φ fo obiting satellites and v 2 = φ = GM 0 + GM 0 dm. (5) At all times in the galactic disk, the centipetal foce F c must match with the viial theoem, so F g = F c. The diffeence between the needed F c and the Newtonian foce of gavity F N must be deliveed by the emegent Dak Matte foce F dm. Assuming the obits to be cicula, we can inset Eqn.(5) in the fomula fo F c to get F c = m 0v 2 and so the entopic Dak Matte foce must esult in = GM 0m GM 0m 0 dm = F N F dm (6) F dm = GM 0m 0 dm (7) This foce cannot be deived fom the divegence of the potential enegy. To esolve this poblem I go to the fist law of themodynamics du = T ds F, witten as du ds F g = + T = F N + F dm (8) S dm 2

3 Newtonian gavity is deived fom dun F N = S and I will deive the entopic Dak Matte foce fom. (10) I define the Dak Matte entopy on the oute flat otation cuve pats of the galactic disks as Udm S = k B ln W = k B ln k B T m Fom the entopy we can deive the entopic DM foce using = T d Udm k k B T B ln m = Udm U d ln m (9) (11) = = GM 0m 0 dm. (12) Because the Dak Matte halo enegy is negative, the entopy deceases outwads, ceating an entopic foce inwads. II. THE PARAMETERS OF THE ENTROPIC MODEL The numbe of micostates W of an ideal gas paticle m with a micostate adius m in a volume with macostate adius is taditionally given by the times one can fit this small o micoscopic volume in the lage o macoscopic volume, so by the themodynamic gas in a bottle numbe of micostates ln W = ln V V m = 3 ln m. (13) But if we have such a paticle moving with only one degee of feedom on a cicula tajectoy with adius and this paticle can be consideed a quantum paticle with a de Boglie wavelength λ = h/p, then the numbe of micostates can be defined as the numbe of micoscopic wavelengths that fit onto the macoscopic cicumfeence. This gives the numbe of micostates as ln W = ln 2π λ with S as the phase space of the paticle. p = ln h = ln S h (14) In the case of galactic neutal hyogen gas paticles, this phase space is huge. This appoach also gives us a way to look at a elativistic 3

4 genealization of the model. And it connects the Dak Matte halo of elementay paticles to a de Boglie subquantum themodynamics. This intepetation tuns the model into Quantum Gavity. If we look at the entopic foce, deived fom a system with N degees of feedom d ln W = d N ln = N d m ln N d ln m = N d ln = N (15) then it is clea that m is a fee paamete of ou theoy fom the pespective of the deived F dm. This means that both a themodynamic and a quantum intepetation of m ae possible, in pinciple. Pactical consideations should detemine the choice of model fo m. The quantum intepetation has the advantage to look like a move towads a quantum theoy of gavity, but as long as it emains a fee paamete, this has no specific use. On the othe hand, a sta obiting a galaxy at a lage distance has an incedible small de Boglie wavelength but clealy a much smalle numbe of micostates elative to the length of its obit, as compaed to a neutal hyogen atom. So fo lage objects the volume of classical mass adius intepetation seems to make the most sense. The othe fee paamete of ou theoy is the tempeatue T, because its intepetation doesn t effect the esulting F dm eithe, as can be seen in = T d Udm k k B T B ln m = T T d ln = m. (16) The tempeatue as a fee paamete of ou theoy only woks as fa the tempeatue of the obiting objects is independent fom the adius at which they obit. The key non-fee paametes of this entopic foce deivation ae the Dak Matte enegy, the adius and the degee of feedom N. III. THE GALAXY CLUSTER PROBLEM OF MOND AS CAUSED BY THE DE- GREE OF FREEDOM PARAMETER In my model, objects on a disk have one degee of feedom, objects moving feely on a sphee have two degee s of feedom and objects that behave as in a mono-atomic gas have thee degees of feedom. Fo the entopic Dak Matte foce this degee of feedom 4

5 paamete N with value between 1 and 3 can be inseted to give ( ds d = ln m ) N = N = G(NM 0)m 0 dm. (17) Without this numbe of micostates degee of feedom elated facto N, in cetain situations the needed bayonic mass might be oveestimated by a facto between 2 and 3. The paamete N will neve be exactly thee because such systems behave as a fee gas and do not display gavitational attaction phenomena. In the case of galaxy clustes, the degee of feedom cannot be 2 o smalle because then the cluste should have been shaped like a disk o a ecognizable sphee. Neithe can it be 3 because then the cluste would dispese like a fee gas. So its degee of feedom should be somewhee in between 2 and 3, giving it an appaent bayonic mass 2M b < M a < 3M b. The fact that MOND has this poblem, indication a distinct diffeence between MOND and my model. In my model this would not be a poblem but a chance to measue the degee of feedom facto in galaxy clustes. This paamete should contain infomation egading the pocess of fomation of such clustes. [1] E. P. J. de Haas, The Dak Matte Entopic Foce and Newtons Enegetic Foce as a Complete Fist Law of Themodynamics Set of Gavitational Foces, vixa: (2015). 5