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A Derivation of Modified Newtonian Dynamics

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A Derivation of Modified Newtonian Dynamics Journal of The Korean Astronomical Society http://dx.doi.org/10.5303/JKAS.2013.46.2.93 46: 93 ∼ 96, 2013 April ISSN:1225-4614 c 2013 The Korean Astronomical Society. All Rights Reserved. http://jkas.kas.org A DERIVATION OF MODIFIED NEWTONIAN DYNAMICS Sascha Trippe Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea E-mail : [email protected] (Received February 14, 2013; Revised March 20, 2013; Accepted March 29, 2013) ABSTRACT Modified Newtonian Dynamics (MOND) is a possible solution for the missing mass problem in galac- tic dynamics; its predictions are in good agreement with observations in the limit of weak accelerations. However, MOND does not derive from a physical mechanism and does not make predictions on the transitional regime from Newtonian to modified dynamics; rather, empirical transition functions have to be constructed from the boundary conditions and comparisons with observations. I compare the formalism of classical MOND to the scaling law derived from a toy model of gravity based on virtual massive gravitons (the “graviton picture”) which I proposed recently. I conclude that MOND naturally derives from the “graviton picture” at least for the case of non-relativistic, highly symmetric dynamical systems. This suggests that – to first order – the “graviton picture” indeed provides a valid candidate for the physical mechanism behind MOND and gravity on galactic scales in general. Key words : Gravitation — Galaxies: kinematics and dynamics −10 −2 1. INTRODUCTION where x = ac/am, am ≈ 10 ms is Milgrom’s con- stant, and µ(x)isa transition function with the asymp- “ It is worth remembering that all of the discus- totic behavior µ(x) → 1 for x ≫ 1 and µ(x) → x for sion [on dark matter] so far has been based on x ≪ 1. The first limiting case corresponds to stan- the premise that Newtonian gravity and general dard Newtonian dynamics. The second limiting case relativity are correct on large scales. In fact, 4 ≈ there is little or no direct evidence that conven- leads to vc GM0 am = const.; this explains the tional theories of gravity are correct on scales asymptotic flattening of galactic rotation curves, the Tully-Fisher/Faber-Jackson relations, and (via division much larger than a parsec or so. ” 2 — Binney & Tremaine (1987), Ch. 10.4, p. 635 by r ) the surface brightness–acceleration relation of galaxies (see, e.g., Famaey & McGaugh 2012 for a re- Since the seminal works by Zwicky (1933) and Ru- cent review). bin et al. (1980), it has become evident (e.g., Binney Despite its success, MOND is obviously incomplete. & Tremaine 1987; Sanders 1990) that the dynamical Firstly, it does not derive from a physical mechanism masses of galaxies and galaxy clusters exceed their lu- a priori. Second, even though it provides the correct minous (baryonic) masses by up to one order of mag- limiting cases by construction, MOND does not provide nitude – the well-known missing mass problem. A pos- µ(x) itself and makes no prediction on the transitional sible solution is provided by Modified Newtonian Dy- regime from Newtonian to modified dynamics. This is namics (MOND) which postulates a modification of especially unfortunate given the fact that the transi- the classical Newtonian laws of inertia and/or gravity tional regime has been explored by observations: the in the limit of weak accelerations (Milgrom 1983a,b,c; empirical mass discrepancy–acceleration (MDA) rela- Bekenstein & Milgrom 1984; Sanders & McGaugh 2002; tion (McGaugh 2004) shows that the ratio Mtot/M0 Bekenstein 2006; Ferreira & Starkman 2009; Famaey & (the mass discrepancy) is a characteristic function of McGaugh 2012). Assuming a test particle on a circu- the accelerations ac and gn; here Mtot is the total dy- 2 lar orbit around a baryonic mass M0 at distance r with namical mass given by ac = GMtot/r . If a prediction ∗ circular speed vc, MOND relates the centripetal ac- 2 for µ(x) was available, it could be tested by compari- celeration ac = vc /r and the Newtonian acceleration 2 son to the empirical MDA relation in a straightforward gn = GM0/r , with G being Newton’s constant, as manner. gn Recently, I proposed a scheme for gravitational in- = µ(x) (1) teraction on galactic scales (the “graviton picture”) ac which is based on the ad-hoc assumption that grav- ity is mediated by virtual massive gravitons that obey ∗For simplicity, I only regard absolute values of velocities and certain reasonable rules of interaction (Trippe 2013). accelerations; the orientations are evident from the assumed geometry. The “graviton picture” predicts a theoretical MDA re- – 93 – 94 S. TRIPPE lation which is in good agreement with observations; it where a0 is a constant of the dimension of an accel- comprises expressions for limiting cases that agree with eration. The limiting case ac ≫ 8πa0 corresponds those of MOND (and likewise agree with observations). to the usual Newtonian dynamics. The limiting case 4 The present work follows up on, and amends, Trippe ac ≪ 8πa0 leads to vc ≈ 8π GM0 a0 = const.; from (2013). I realized only after publication of Trippe comparison to the corresponding MOND result we find (2013) an additional consequence of my toy model of am ≡ 8πa0. gravity introduced there: A comparison of classical In contrast to MOND, the “graviton picture” com- MOND and “graviton picture” shows that MOND nat- prises a scaling law for the transitional regime from urally derives from the “graviton picture” at least for Newtonian to modified dynamics (Eq. 5) a priori. In the case of non-relativistic, highly symmetric dynam- analogy to Eq. 1 we can define a transition function ical systems. This comparison is the subject of the present work. −1 gn M0 1 ξ(x)= = = 1+ (6) ac Mtot x 2. ANALYSIS with x = ac/am as before. This results in 2.1 Transition Functions in MOND x Eq. 1 provides the defining properties of MOND; the ξ(x)= = µ1(x) . (7) transition function µ(x) is constrained by (1) limiting 1+ x cases that have to be consistent with observations (2) As we see, the transition function ξ(x) provided by the the condition that xµ(x) increases monotonically with “graviton picture” is identical to the MOND transition x (Famaey & McGaugh 2012). These conditions are function µn(x) for n = 1. fulfilled (e.g., Milgrom 1983a; Famaey & Binney 2005; Furthermore, we can define an inverse transition Famaey & McGaugh 2012) by the set of functions function in analogy to Eq. 3 as x µn(x)= 1 ; n =1, 2, 3, ... (2) Mtot ac gn x (1 + xn) /n ζ = = −→ y = = . (8) M0 gn am ζ Alternatively, one may re-write Eq. 1 as Via Eq. 5 this leads to the quadratic equation ac = ν(y) (3) 2 1 gn ζ − ζ − =0 . (9) y −1 2 where y = gn/am; ν(y) → 1 for y ≫ 1 and ν(y) → y / for y ≪ 1. For reasons analogous to those for the case Solving this expression for ζ and ignoring the unphys- of µ(x), a set of valid transition functions is given by ical negative root, we find (Famaey & McGaugh 2012) 1 2 1 4 / 1 ζ(y)= 1+ 1+ = ν1(y) . (10) 1/2 /n 2 y 1 4 " # ν (y)= 1+ 1+ . (4) n 2 yn ( " #) In this case, the transition function ζ(y) provided by the “graviton picture” is identical to the MOND transi- Comparison to the empirical MDA relation suggests tion function νn(y) for n = 1. I illustrate the functions n = 1 or n = 2 (Kroupa 2012). Nevertheless, none −1 −1 of these scaling relations follows from first principles: ξ (x) = µ1 (x) and ζ(y) = ν1(y) in Fig. 1; the di- technically, arbitrary alternative transition functions agrams should be compared to Fig. 10 of Famaey & can be constructed from comparison to the data. McGaugh (2012) and Fig. 11 of Kroupa (2012). 2.2 The “Graviton Picture” 3. DISCUSSION The “graviton picture” (Trippe 2013) employs the “ It is principally the elegance of general relativ- ad-hoc assumption that gravity is mediated by virtual ity theory and its success in solar system tests gravitons with non-zero mass that obey certain reason- that lead us to the bold extrapolation that the gravitational acceleration has the form GM/r2 able rules of interaction. This leads to the formation of 21 26 a “graviton halo” with a mass density profile ρ ∝ r−2 on scales 10 − 10 cm that are relevant for the around a baryonic source mass M0. The total dynam- solar neighborhood, galaxies, clusters of galax- ” ical mass Mtot scales with centripetal acceleration ac, ies, and superclusters. providing a theoretical MDA relation — Binney & Tremaine (1987), Ch. 10.4, p. 635 Historically, the missing mass problem has usu- Mtot a0 =1+8π (5) ally been approached by postulating non-luminous and M0 ac DERIVATION OF MOND 95 Fig. 1.— The transition functions derived from comparison of MOND and “graviton picture”, assuming am = 1.1 × −10 −2 10 m s . a. The inverse of the function µ1(x) vs. ac. b. The function ν1(y) vs. gn. Please note the logarithmic–linear axis scales. These diagrams should be compared to Fig. 10 of Famaey & McGaugh (2012) and Fig. 11 of Kroupa (2012). non-baryonic dark matter (Ostriker & Peebles 1973; The “graviton picture” of gravitation (Trippe 2013) Einasto et al. 1974), eventually evolving into the starts off from a physical mechanism: the (ad-hoc) as- ΛCDM standard model of cosmology (e.g., Bahcall et sumption that gravity is mediated by virtual massive al. 1999). In recent years, it has become clear that this gravitons that obey certain rules of interaction. From approach is incomplete.
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