Orbit of a Test Particle and Rotation Curves of Galaxies in an Expanding Universe Hossein Shenavar∗ Department of Physics, Ferdowsi University of Mashhad, P.O. Box 1436, Mashhad, Iran (Dated: May 14, 2014) A new equation of motion, which is derived in an accompanying article by considering spacetime measurement via geometrodynamic clocks [1], is surveyed. It is shown that the new term in the equation of motion suggest a small correction in orbits of outer planets; thus it is compatible with the solar system data. Then a typical system of particles is investigated to have a better understanding of galactic structures and the general form of the force law is introduced. As the first example, rotation curve and mass discrepancy functions of an axisymmetric disk of stars are derived. It is shown that the general form of rotation curve could be justified. In addition, it is suggested that the mass discrepancy as a function of centripetal acceleration becomes significant near a0 . An important feature of this model is the prediction of a constant acceleration in outer parts of galaxies. The critical surface density, σ0 = a0 /G, has a significant role in rotation curve and mass discrepancy plots. The specific form of NFW mass density profile at small radii, ρ ∝ 1/r, is explained too. PACS numbers:
I.
INTRODUCTION
Observations have shown that the dynamics of galactic and extragalactic systems do not agree with the observed mass content of them [2], [3]. Rotation curves of spiral galaxies demonstrate this problem very clearly [4] ,[5]. The point is that in the weak-field and low velocity limit of outer parts of spiral galaxies, which one expects a Newtonian regime, and so a Keplerian decline in rotation velocity of orbiting objects, observations have shown flat rotation curves (rotation curve: rotation velocity as a function of radius). Thus, there should be more mass in the system to make the flat rotation curve possible. Another problem is the stability of galactic systems [6], [7]. Numerical simulations have shown severe instabilities in the cold, self-gravitating disks of spiral galaxies. So, in order to explain the existence of such systems, one needs to embed the disk in a halo of ”dark matter”. Historically, it was Zwicky that first pointed to such ” mass discrepancy”. He realized that radial velocities of individual galaxies in the Coma cluster are too high for the amount of visible mass of the cluster [3]. Now, by developing radio telescopes, we have a better understanding of the baryonic content of galactic and extragalactic systems. As a result, the amount of missing mass that we are expecting is smaller ( specially compared to Zwicky’s estimation)[3]; however, we still need to explain rotation curve and stability of such systems. Nevertheless, by developing particle physics, physicists begun to consider new particles as possible candidates to solve the dark matter problem. Now, it seems that the famous cold dark matter (CDM) model could provide a good explanation for these apparent mass discrepancies. These particle should be massive, nonbaryonic with no electromagnetic interactions [2]. However, there are some other observations that relate the dynamics of galaxies and their baryonic content. For example, we could mention the Tully-Fisher law, that originally was found as an empirical relation between galaxy’s luminosity and its HI line-width [8], [9], [10]. Because luminosity and line-width are related to the mass and rotation velocity respectively, one could interpret Tully-Fisher law as a mass-rotation velocity relation. However, further investigations have shown a direct relation between the amount of baryonic mass of galaxies and its rotation velocity [2]. In addition, it seems that surface density factor of spiral galaxies plays an important role in galactic phenomena. Moreover, investigations about mass discrepancy of galaxies have revealed some relations with centripetal and Newtonian acceleration [12], [2]. Also, an interesting evidence in galactic scales appears as close link between luminous and dark component: ”For any feature in the luminosity profile there is a corresponding feature in the rotation curve (Renzo’s rule)” [11], [12]. Nevertheless, there have been some attempts to justify the mentioned discrepancy, not by adding more mass to systems, but by changing governing theories of gravity; i.e. Newtonian mechanics and general theory of relativity. For example, Milgrom considered the possibility that the inertia term of Newton’s second law is not proportional to
∗ Electronic
address:
[email protected]
2 the acceleration of the object, but rather is a more general function of it: mµ(
a )a = f a0
where the function µ plays a significant role at small accelerations (for example in galactic scales) [13], [14], [15]. It is possible to interpret the last equation as a modification of inertia or a modification of gravity. Anyway, modified Newtonian dynamics (MOND) could simply explain asymptotic flatness of rotation curves [4], [19], [20], [21]. In addition, it seems that mass discrepancy is related to acceleration in units of a0 [12]. Ga0 which is the zero-point of the baryonic TullyFisher relation could naturally be derived from MOND equation of motion. The same is true for the zero-point of the Faber-Jackson relation in isothermal systems like elliptical galaxies [16]. There is also another quantity, a0 /G, which plays a significant role in galactic scales. Disks with mean surface density less than a0 /G have added stability [17]. Also a0 /G defines a transition central surface density and a0 /2πG is very close to the central surface density of dark halos [18]. Moreover, this model simply explains that features in the rotation curve should follow the baryonic distribution; so the Renzo’s rule. The main point is that, after so many data fittings of experts working on MOND model, the role of a constant acceleration, a0 , in galactic scales seems irrefutable now [2]. On the other hand, alternative theories of gravity have some proposals to solve the mass discrepancy problem too [22]. It is difficult to even mention all of these theories here; however we should briefly consider fourth-order conformal theory of gravity that is related to our special model here [23], [24], [25], [26], because this theory predicts a constant acceleration too. The Field equation is derived from the following action: Z √ Iw = −α d4 x −gCλµνκ C λµνκ where Cλµνκ is the conformal Weyl tensor and α is a purely dimensionless coefficient. This action leads to a fourthorder field equation which has an exact vacuum solution of the form ds2 = B(r)dt2 −
dr2 − r2 dΩ B(r)
where B(r) = 1 − ar + b + cr + λr2 . As it is clear, the fourth term of the rhs of B(r) establishes a constant acceleration. It has been shown that by using this added linear potential, one could capture the general trend of the rotation curve data to a good degree without needing any dark matter. In addition the obtained mass to light ratios, M/L, are close to the values of the local solar neighbourhood [27] [28]. Conformal gravity claims that mass discrepancy in galactic scales is due to a global cosmological effect on local galactic motions. In additions, the cosmological constant problem could be suppressed if we include the amount by which it gravitates [29]. However, Youngsub Yoon has argued that Mannheim’s conformal gravity does not reduce to Newtonian gravity at small scales unless one assumes undesirable singularities of the mass density of the proton [30]. The reason is that the gravitational force in this theory depends not only on the total mass, but also on the mass distribution which is problematic, because Cavendish-type experiments have shown no dependency to the density distribution. In addition, it is proved that the linear term in the potential, c above, is negative; however we need extra attraction, and thus a positive linear gravitational potential, to fit the galaxy rotation curve [30]. Hitherto, Flanagan has claimed that the conformal invariance is not a fundamental property of this theory but it is an artifact of the choice of variables [31]. Then it is proved that in the weak field and small velocity limit of solar system, the theory does not agree with the predictions of general relativity. In galactic scales we usually assume that space is nearly flat and so one could use Newtonian dynamics to describe galactic phenomena. We also assume that although all galaxies are sitting in an accelerated expanding universe, this evolution of the large scale has no effect on galactic structures. In an accompanying paper I have questioned the latter assumption and then I proposed that one should use the following equation of motion a = g − cH(t)
(1)
˙ in weak field limits [1]. In the last equation H(t) = R(t)/R(t). The key point to derive this equation is to use geometrodynamic clocks as the method of measurement. This procedure of measurement is based on free particles and light signals and one can prove that If gµν is the metric as measured with the geometrodynamic clock, then the motion of particles in free fall is along the geodesics of gµν [32]. In addition, since in a nonstatic flat universe we have d2 x R(t) dx dt = c for all null geodesics, then it is easy to see that R(t) dt2 + cH(t) = 0. It is easy to check that although this modification is not important near very massive objects, it has significant consequences in galactic scales. It has to be mentioned that the equation (1) has been proposed before by assuming a five-dimensional brane world, i.e. a space-time-velocity Riemannian manifold, to describe an expanding universe. This is Carmeli’s special and
3 general theory of relativity that claims to be able to explain the Tully-Fisher relation and rotation curves of spiral galaxies [33], [34], [35]. However, as we illustrate here [1], there is no need for a new theory of gravity but one has to just recognize the role of measuring spacetime intervals in an expanding universe. In this paper, I start to derive some basic results of this new equation of motion. In the next section we study the motion of a single particle. Then I will show that the new equation of motion is compatible with the established results of the solar system. In the third section the motion of a system of particles is investigated which is crucial for our future studies. In the next two sections I will derive the rotation curve and mass discrepancy of disk galaxies. At last it is shown that the constant acceleration a0 = cH0 could be replaced by an equivalent hypothetical mass distribution - the dark matter distribution- to justify the rotation curve of disk galaxies.
II.
THE MOTION OF A SINGLE PARTICLE
We first consider motion of a particle m in a gravitational field. As we discussed before, the modified equation of motion of this particle in a centrally directed gravitational field is as follows: d2 r + a0 eˆr = g(r)eˆr dt2
(2)
where r = rˆ e is the position of the particle, a0 = cH0 is a constant acceleration and g(r) is the Newtonian field. Here we could treat the new term a0 = cH0 as a coordinate transformation or a new force when we transfer a0 to the rhs of Eq. (2). These two approaches have same consequences. For example one could show that both of these scenarios result in the following conservation of energy: 1/2mv2 + ma0 r + V (r) = E
(3)
Rr
where V (r) = − r0 mdx.g(x) is the gravitational potential energy. One could consider F = m(g(r) − a0 )eˆr as a new force and use the definition of work done against this force in moving a particle from r0 to r1 : Z r W (r) = dx.F (4) r0
and then derive the conservation of energy (3). Another way is to directly integrate Eq. (2) to derive conservation of energy Eq. (3). As we mentioned, these two approaches are equivalent; however we prefer to use the concepts of force and potential. The reason is that the central force problem, which we need here, could be find in any standard textbook of theoretical mechanics. Therefore it is customary to use follow this approach. d From Eq. (2) one could prove that dt (r × dr dt ) = 0 which is expected because the problem is spherically symmetric. Therefore the particle moves in a plane, i.e. the orbital plane. By using plane polar coordinate (r, ψ) and define the Lagrangian of the motion as follows: L=
1 ˙ 2) − Φ m(r˙ 2 + (rψ) 2
(5)
where Φ = ΦN + ma0 r is the total potential. Then one could drive equations of motion from Euler-Lagrange equation [36]: dΦ m¨ r − mrψ˙ 2 + =0 dr
(6)
d ˙ =0 (mr2 ψ) dt
(7)
The quantity l = mr2 ψ˙ is the angular momentum which according to (7) is another constant of the motion. The equation of motion in terms of r, if we express ψ˙ in terms of l, contains only r and its derivatives. Therefore, it is equivalent with a one dimensional problem in which a particle is subjected to an effective potential as: Ψ = ΦN + ma0 r +
l2 2mr2
(8)
4 where the third term is due to the familiar centrifugal force. From Eq. (3) one could rewrite the appropriate energy conservation theorem as: 1 E = Ψ + mr˙ 2 2
(9)
Now we will consider this one dimensional problem for the specific case of an attractive inverse-square law of force; i.e. the Kepler problem ΦN = − kr , where for positive k describes a force toward the center. First consider the escape velocity of a particle which is defined as the minimum velocity required to escape from the gravitational field. According to the discussion we had in the first paper [1], if the particle reaches the radius R0 ≈ aΛ0 , which Λ is the cosmological constant, then it is free. The reason is that in this radius the repulsive force due to cosmological constant, F1 ≈ mΛr, overcomes the total attractive force F = m(g(r) − a0 ). Although for most physical cases, one could neglect g(r) at a distance like R0 because g(R0 ) a0 . Thus from Eq. (9) the escape velocity is approximately v02 ≈ a0 R0 and every particle with an energy larger than v02 could come from infinity, strike the repulsive centrifugal barrier, be repelled and then travel back to infinity. On the other hand all particles with smaller velocities than v0 are bounded to the central object. For example consider a particle with energy E1 ; See Fig. 1. For this particle there are two turning points, r1 and r2 , also known as apsidal distances. According to Bertrand’s theorem, which states that the only central forces that result in closed orbits are the inverse-square law and Hooke’s law [36], orbits of Eqs. (6)and (7) are not closed. However if energy of the particle coincides with the minimum of the effective potential, then r1 = r2 and the orbit is a circle. Another important feature of Eq. (2) is that for this new equation of motion the famous Laplace-Runge-Lens vector is not a constant of the motion anymore. In the Kepler problem one could show that beside four independent constants of the motion, i.e. the angular momentum vector and the energy, LRL vector indicates another constant which always points in the same direction [36]. The cross product of Eq. (2) with the constant angular momentum l, could result (after a little manipulation): d r (p × l − mk ) = m2 a0 (rr˙ − rr) ˙ dt r
(10)
where p is momentum of the particle, k depends on the masses of the particle and the source of the gravity and A = p × l − mk rr is the LRL vector. Although A is not constant anymore, a natural questions is whether there is any other constant of the motion. The answer is related to the nonclosed nature of orbits. See [36] page 105. Orbitals of Eq. (2) are nonclosed, therefore as ψ goes around, the particle never retraces its footsteps on any previous orbit. Thus r is an infinite-valued function of ψ and so the additional conserved quantity of the motion should involve an infinite-valued function of ψ too. So there is no more simple-defined constant of the motion. Before we turn to the problem of a system of particles, which is needed to describe galactic phenomena, we shall consider the perturbation effect of a0 -term in Eq. (2) on the outer objects of the solar system. To do so, I propose η = a0 /gN as a measure of the strength of a0 relative to Newtonian gravitational field gN . In the solar system one has, for near the surface of the Sun η ≈ 10−13 , the Earth η ≈ 10−8 , Neptune η ≈ 10−5 and Eris (one of the last dwarf planets) η ≈ 10−4 . Therefore the new term is very small compared to the Newtonian gravity term in the solar system. One could use canonical perturbation theory to evaluate precession rate averaged over a period of unperturbed orbit τ as [36]: ¯˙ = ∂∆H ω ∂l
(11)
Y 10
E1
5
r 2
4
6
8
10
-5
font=scriptsize
FIG. 1: The ferent values of
-10
equivalent one-dimensional a0 . In this graph
effective potential we have chosen
for k
the =
potential m =
Ψ l
and =
dif1
5 where ∆H = ma0 r is the perturbation Hamiltonian and ma0 ∆H = τ
Z
τ
rdt
(12)
0
is the time average of the perturbation. Then by using the conservation of angular momentum, i.e. ldt = mr2 dψ, the last integral converts to: Z m2 a0 l2 3 2π dψ ∆H = (13) ( ) lτ mk [1 + e cos(ψ − ψ 0 )]3 0 in which e is the eccentricity and ψ 0 is one of the turning angles of the orbit. One could evaluate the integral of the last equation for any given eccentricity; the result, say C, is most likely of the order of 10. See Table I. Then, from Eq. (11) the averaged precession rate is as follows: 2 ¯˙ = 4C a a0 (1 − e2 )2 ω τ GM
(14)
where M is the mass of Sun and we have used the relation l2 = mka(1 − e2 ) which is correct for an unperturbed orbit with a semimajor axis a. You may also find the perihelion shift of a test particle in fourth-order conformal gravity by tracking timelike geodesics here [37], which is compatible with the present result. Eq. (14) is independent from the mass of the planet as one expects because gravity is a geometrical theory. Also from Eq. (14) it is clear that orbits with larger semimajor axis have larger precession rate. Therefore this effect is important in outer parts of the solar system. See Table I for perihelion precession per revolution for two planets -Uranus and Neptune- and two dwarf plants - Pluto and Eris. In this table we have assumed that a0 = 10−10 m/s2 . TABLE I: Precession of perihelion per revolution for two planets and two dwarf plants in arcseconds. Data from [38]. Object
e
Uranus Neptune Pluto Eris
0.047 0.009 0.249 0.433
a (AU)
C
¯˙ per revolution ω
19.18 6.32 30.06 6.28 39.48 7.60 68.04 11.54
31.9800 78.3900 143.9400 487.3100
As we know general relativity predicts another correction to Newtonian motion that can be described by an 1/r3 potential. This term comes from Schwarzschild solution of Einstein field equation in strong field limit around the Sun [32]. By using the same method as we applied here, one can evaluate precession rate averaged over a period for this 6π GM 00 ¯ case as ω¯˙1 = τ (1−e per revolutuion 2 ) ( c2 a ). See [36] for full derivation. For the planet Mercury we have ω˙1 ≈ 0.10 00 or 42 per century. As we see this last correction is smaller than correction due to Eq. (14); however in a period of a century the angular advance of the perihelion of the Mercury is larger than angular advance of the planets of Table I. The reason is that the sidereal orbit period of the Mercury, τ , is much smaller than outer planets and dwarf planets. Table I demonstrates our model’s prediction about outer planets. In the case of inner planets like Mercury and the Earth, for the reason that will be explained in the next section, the effect of the Carmelian term a0 decreases by a factor of r/R where r is the average scale of the orbit of the inner planet and R is a typical scale for the solar 1AU system, say the semimajor axis of the Neptune. Therefore, for example for the Earth and Mercury we put 30AU a0 0.387AU and 30AU a0 respectively, instead of a0 in Eq. (14) to obtain the precession rate. I have derived precession rate of inner planets in Table II. We find 0.0700 and 0.2900 per century for Mercury and the Earth respectively. TABLE II: Precession of perihelion per revolution for inner plants in arcseconds. Data from [38]. Object Mercury Venus Earth Mars
e
a (AU) C
r/R
¯˙ per revolution ω
0.205 0.387 7.14 0.387/30 (1.7449 × 10−4 )00 0.009 0.723 6.28 0.723/30 0.001100 0.0167 1.00 6.28 1/30 0.002900 0.093 1.38 6.45 1.38/30 0.007700
6 For an orbit with a large eccentricity, the value of C grows rapidly and therefore we expect a larger precession rate. For example, for e = 0.8 and e = 0.9 we have C = 106.65 and C = 561.01 respectively. This is usually the case for comets and it could be a helpful clue in our future investigations. Consider for now the Halley’s Comet with eccentricity of e = 0.967 and semimajor axis of 17.94AU [38]. It makes a perfect case for the reason that it has a ¯˙ = 160.82. The factor (1 − e2 )2 is significant here too very high eccentricity and one can derive C = 8589 and ω because it decreases the precession for large eccentricities. On the other hand, consider for example Chiron comet ¯˙ = 18.91. Thus the precession with a = 13.7AU and e = 0.383 [38]. For this comet one could find C = 10.02 and ω rate in the orbit of Halley’s comet is larger by an order of magnitude than the precession rate in the orbit of Chiron. The question is whether this new term could explain the motion of comets in the solar system. However this matter needs a thorough survey and it is beyond the scope of this article, the point here is that if Eq. (2) supposed to be correct, one should be able to detect its corrections at all scales in the solar system. There are some other independent evidence too. For example, tracking data of Pioneer 10 and 11 spacecraft have shown a systematic unmodeled acceleration of (8.74 ± 1.33) × 10−10 m/s2 directed toward the Sun [39], [40], [41]. If there is indeed such added acceleration at this scale, it must have some effects on small objects of the solar system too. In 2007, Wallin, Dixon and Page [42] considered this idea and selected a well-observed sample of transNeptunian objects with orbits between 20 and 100 AU from the Sun, and placed tight bound on the magnitude of the added acceleration. According to this research, the deviation from inverse square law of gravity should be a0 = (8.7 × 10−11 ± 1.6 × 10−10 ) m/s2 . Even in larger lengths like the Oort cloud, Iorio [43] has shown that a constant acceleration toward the Sun could make bound trajectories and these trajectories radically differ from the Newtonian ones. We see that this acceleration scale appears in independent observations in solar system and therefore, it is now safe to test Eq. (2) in larger scales like galaxies and clusters of galaxies. To do so, we need to investigate the problem of a system of particles all interacting according to (2). III.
SYSTEM OF PARTICLES
Suppose Fei be the external force on the ith particle, with mass mi , and let FN ij be the Newtonian force exerted on the ith particle by the jth particle and finally Faij0 be the term due to acceleration transformation. Clearly one has a0 N FiiN = Fiia0 = 0 and also FN ij = −Fji . On the other hand because Fij is a force due to coordinate transformation we have: Fija0 mi (15) a0 = Fji mj The equation of motion for the ith particle is as follows: m¨ri = Fei +
X
a0 (FN ij + Fij )
(16)
j
where, if the last equation is summed for all other particles, it takes the form: X X X a0 mi ¨ri = Fei + (FN ij + Fij ) i
i
(17)
ij
P P dri P P dri ri and Π = Now it is possible to introduce four useful quantities R = 1/M mi ri , P = mi dt , % = dt that stand for center of mass or weighted average of the radii vectors of particles, total linear momentum of the particles, summation of radii and velocity vectors respectively. Then it is possible to prove the following important theorem: Theorem 1 a) If the total external force is zero and Fa0 is negligible, then the total linear momentum P is conserved, b) If the total external force is zero and the Newtonian gravitational force FN is negligible, then the summation of velocity Π is conserved and, c) If the total external force is equal to zero, but neither Newtonian gravitational force nor the force due to coordinate transformation Fa0 is negligible, then neither P nor Π is conserved. Proof. Part a). Under the conditions of part (a) which refer to strong field limit, the rhs of (17) is equal to zero since N the law of action and reaction states that each pair FN ij + Fji is zero. Thus the total linear momentum is conserved. Part b). In this case, which refers to weak field limit, it is easier to start from (2) and sum over all particles. Then it follows: X X Fe X FN ij i ¨ri = + ( + a0ij ) (18) m m i i i i ij
7 Acceleration transformation between every two particles are equal and opposite; so a0ij + a0ji = 0. If the total external force is zero and the Newtonian gravitational force FN is negligible, then it is clear that the rhs of Eq. (18) is equal to zero and so Π is a constant of the motion. Part c). This case is clear from Eqs. (17) and (18). However there is an important exception here: If all particles in the system have the same mass then both P and Π are conserved. As an example, let us consider the important and interesting case of a two-body system. Suppose that the total external force is negligible, then by adding equations of motions of these two particles, it is possible to find equation of motion of the center-of-mass: ¨ = (m2 − m1 )a0 MX
(19)
where X = 1/M (m1 x1 + m2 x2 ) is the center of mass position and M is the total mass of the system. Therefore the conservation of total linear momentum of the system is violated unless the two particles have same masses m2 = m1 . To derive the relative motion of m2 and m1 one should multiply the equation of motion of the former by m2 /M and the latter by m1 /M and then subtract to achieve: µ¨ x = F12 + 2µa0
(20)
1 m2 where x = x1 − x2 is the relative position of two mass and µ = mM is called the reduced mass. As one expects, for the case that two particles have same masses we attain m¨ x = F12 + ma0 , which is exactly of the form of equation of motion of single particle in gravitational field F12 . To obtain the net force on a particle mi at position x from a system of particles, one should simply add the small contribution from each small particle. However, for a typical galaxy by about 1011 stars this method is not practicable. A good solution is to assume a smooth mass distribution that is everywhere proportional to the local star density. According to Eq. (2), the force of a small element d3 x0 from mass distribution ρ(x0 ) Q on point particle mi is as follows:
δF(x) = Gmi
x0 − x x0 − x 0 3 0 ρ(x )d x + m a i 0 |x0 − x|3 |x0 − x|
Then the total gravitational force of all distribution Q on point mass mi becomes: Z Z x0 − x a0 x0 − x 0 3 0 ρ(x )d x + ρ(x0 )d3 x0 ) F(x) = mi (G 0 3 |x − x| M |x0 − x|
(21)
(22)
where M is the total mass of the distribution Q. To obtain the second term of rhs of (22) we have used this fact that the acceleration due to coordinate transformation a0 between mi and mass distribution Q should be the same. This term could also be interpreted as the average of the acceleration a0 weighted in proportion to the mass of the particles of Q. An interesting and important example to apply Eq. (22) is the force between a spherical shell of matter of mass M and a particle of mass m inside or outside of the shell. For the first integral on the rhs of Eq. (22) this problem is the famous shell theorems: A uniform shell exerts no gravitational force on a particle inside the shell; the shell attracts an external particle as if all the mass of the shell were concentrated at its center . For the second integral on the rhs of Eq. (22) it is straightforward to show that the force is ma0 (1 − R2 /3r2 ) for an external particle and 2ma0 r/3R for an internal one [52]. R is the radius of the shell and r is the distance of the particle from shell’ s center. Deriving the force for other symmetrical mass distributions shows that this behaviour is general. This is the reason that in the previous section we demanded to use Rr a0 instead of a0 in the case of inner planets like the earth. See [52] for detailed derivation of force around a spherical shell, a homogeneous sphere, an exponential sphere and an exponential disk. It is easy to check that the new force is always toward the center of the shell. From the symmetry, it is clear that there is no net force on a particle located at r = 0. We will use these results to have a better understanding of the solutions and their asymptotic behaviour. As usual, we prefer to work with scalar quantities; thus we define the total potential as: Z Z a0 ρ(x0 )d3 x0 + ρ(x0 )d3 x0 |x0 − x| (23) Φ = −G |x0 − x| M where the first term is the Newtonian gravitational potential and the second one is the new potential due to coordinate transformation. This new term is positive and so it provides extra attracting force as we need in galactic scales. It is clear that the Newtonian gravitational potential fulfils the Poisson equation; though the total potential Φ can not fulfil
8 any second-order Poisson equation because of the presence of the Carmelian potential Φa0 . It is possible, however, to show that the new potential Φ satisfies a new fourth order Poisson equation: ∇4 Φ = 4πG∇2 ρb +
8πa0 ρb M
(24)
We leave this matter to another paper in which we discuss the distribution of mass in galaxies [52]. IV.
ROTATION CURVE OF SPIRAL GALAXIES
According to Newton equation of motion, it is expected that around a massive object, rotation velocity as a function of radius falls as v 2 ∼ 1r . This behavior is known as Keplerian fall-off of velocity and it is highly supported by solar system data. The question is whether the observed mass distribution of galaxies is compatible with their rotation curve without need for extra matter. Because hydrogen gas is distributed in galaxies to much further distances than their stars, the HI gas observation provides a powerful tool to test the Keplerian behavior of rotation curve. See [3] for an excellent history. It is now clear that there is a mass discrepancy at least for galaxies which have extended rotation curves beyond their optical radius. According to Casertano and van Gorkom [45], the rotation curves data of galaxies could be divided into three categories. The rotation curve of low luminosity dwarf galaxies, with maximum velocity smaller than 100km/s, are typically rising. Those of of intermediate to high luminosity galaxies, with 100 < vmax < 180 km/s and vmax > 180 km/s respectively, are found to be flat. The rotation curves of the very highest luminosity galaxies are generally declining from 15% for NGC2903 to 30% for NGC2683. Galaxies have very diverse mass distributions; however, for the simplicity we consider here the special case of spiral galaxies. The disk of spiral galaxies contains stars, gas and dust. There is also an spiral structure with various shapes and lengths. We usually measure distribution of stars in a galactic disk by observing total stellar luminosity emitted by unit area of the disk, i.e the surface brightness. Observations suggest that the surface brightness is approximately an exponential function of radius: I(r) = I exp(−r/Rd ); where R is the radius and Rd is the disk scale length. −1 Scale lengths range from 1 h−1 7 kpc to more than 10 h7 kpc [46]. It is believed that most spiral galaxies contain a bulge, which is a centrally concentrated stellar system. Galactic disks are thin because mass density falls off much faster perpendicular to the equatorial plane than in the radial direction. Here we seek a thin disk rotation curve; the extension to a separable thick disk or a disk with a spherical bulge is straightforward. v vR 4 4.0 3 2.0 2
1.0 0.5
1
0.2 0.1
font=scriptsize
0
2
4
6
8
10
y = rH2Rd L
FIG. 2: Rotation velocity in units of vR as a function of radius for different values of σ0 /Σ0 . Low luminosity galaxies have a clear rising rotation curve; those of intermediate- to high-luminosity galaxies have a relatively flat rotation curve, and those of the highest luminosity galaxies have a Keplerian decline in their rotation curves. It is important to note that the actual rotation velocity in outer parts of galaxies would be smaller, because this velocity is dependent to gas distributions which typically extend far beyond optical disks. Therefore, gas distributions have larger scale lengths Rd and their rotation velocity increase more slowly. If we suppose that the disk surface mass density is also exponential; Σ(R) = Σ0 exp(−r/Rd )
(25)
then the total mass of the disk is M = 2πΣ0 Rd2 . Following an approach which Casertano has developed [44], by differentiating Eq. (23) with respect to r one could derive the circular velocity of the exponential disk: v 2 (r) = r
∂Φ = 4πΣ0 GRd y 2 [I0 (y)K0 (y) − I1 (y)K1 (y)] + 4a0 Rd y 2 I1 (y)K1 (y) ∂r
9 in which I and K are modified Bessel functions and y = r/2Rd . Complete derivation of the last equation is presented here [25], [29], [52]. One could also find the potential of a separable thick disk and a spherical bulge in the latter reference. If the structure is composed of two or more elements, for example a cold exponential disk plus some HI gas distribution with a specific profile, then one has to evaluate rotation curve for each element and add them to derive the total rotation curve. For example, O’Brien and Mannheim have approximated the gas profile as single exponential disk with scale length equal to four times those of the corresponding optical disk [28]. This is a very successful approximation as fitted rotation curves simply show here [28]. a 4
2.0
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0
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y = rH2Rd L
FIG. 3: Centripetal acceleration a in units of a0 as a function of radius for different values of Σ0 /σ0 . Regardless of the mass, size and shape of galaxies the centripetal acceleration approaches to a constant value a0 . Newtonian theory of gravity predicts a decreasing centripetal acceleration as 1/r2 . See [27] [28] for two sample of galaxies which clearly suggest a constant acceleration at their last data point. All departure from the standard Newtonian mechanics are embodied in the a0 -dependent term of the Eq. (26). This term is competitive with Newtonian one in outer parts of galaxies and it can be shown that the more luminous the galaxy, the less important a0 term is; or the less hypothetical dark matter is needed. To see this, let us divide the last equation by the maximum velocity of an equivalent spherical distribution GM/Rd : v 2 (r) 2 σ0 2 = 2y 2 [I0 (y)K0 (y) − I1 (y)K1 (y)] + y I1 (y)K1 (y) GM/Rd π Σ0
(26)
where σ0 = a0 /G is a critical mass density. Therefore for galaxies with larger mass density, Σ0 , the second term is less important. The main point is that the total mass of galaxies are not an important factor here. See Fig. 2 for the rotation curve of a cold disk of stars without any additional element. Although one should note that in practice the rotation velocity in outer parts of galaxies is smaller than what is shown in this figure. As we explained before, to achieve this smaller value, Mannheim and his colleagues assume that the gas profile around any galaxy as a single exponential disk with scale length larger than those of the corresponding optical disk. This is a reasonable approximation and a successful one. However, one might expect that a thorough description of the gas component include the collisional nature of that too. Indeed, this idea might be successful because collisions naturally decrease the radial velocity and increase random motions. To deal with this problem, we should build a statistical theory of gravitating systems and in fact, the final result for the rotation curve of a collisional system need a simulation. This is beyond the scope of this paper. Detailed rotation curve data fitting will be presented elsewhere; here we seek theoretical clues of Eq. (2) for future investigations. Another important feature of this equation is that it suggests a constant acceleration at large radii from the center of the disk. See Fig. 3. Outside of any disk the acceleration could be approximated by GM/r2 + a0 − 1.5a0 Rd2 /r2 which is in agreement with the conclusion we had for the shell in the last section. Use asymptotic behaviour of modified Bessel functions [47] to derive the last result. More massive galaxies reach a higher peak in their acceleration plot however all data should have an asymptotic acceleration equal to a0 eventually. This behaviour is strongly supported by Mannheim and O’Brien investigations [27] [28]. They have studied two different galaxy samples consisting of high surface brightness (HSB), low surface brightness (LSB) and dwarf galaxies with different masses, scales and rotation velocities. In the first sample they have considered 111 HSB, LSB and dwarf spiral galaxies. The second sample is consisted of 27 objects which most of them are dwarf galaxies. The data are such that the value of the quantity (v 2 /r)last ≈ 3 × 10−11 m/s2 as measured at the last data point is near universal. This quantity is also very close to the constant acceleration of the conformal Weyl gravity which is numerically extracted from the data fitting of rotation curves. MOND theory [13] and Moffat’s metric skew-tensor theory, which have succeeded to explain galactic rotation curve, posses some universal parameters too. For Mond theory one has a0 ≈ 10−10 m/s2 and for MSTG theory G0 M0 /r02 c2 ≈ 7.67 × 10−29 cm−1 [29]. Therefore it seems that this quantity should be regarded as an important empirical clue for galactic systems.
10 V.
MASS DISCREPANCY
In any alternative theory of gravity it is important to find new quantities to illustrate differences between new 0 theory and the Newtonian one. For example, the mass discrepancy is defined as ( vvN )2 = ΦΦ0 , where v is the observed N velocity and vN is the velocity attributable to Newtonian gravity of visible baryonic matter [12], [2]. It has been shown that this quantity is well correlated with acceleration, and increases systematically with decreasing acceleration below a0 = 1.2 × 10−10 m/s2 [12]. It is easy to derive the mass discrepancy from equation (26) as a function of radius r, 2 centripetal acceleration a = vr , or Newtonian acceleration gN as follows: (
v 2 σ0 I1 (y)K1 (y) ) =1+ vN Σ0 I0 (y)K0 (y) − I1 (y)K1 (y)
(27)
(
v 2 a ) = vN a − 2a0 yI1 (y)K1 (y)
(28)
(
2a0 v 2 ) =1+ yI1 (y)K1 (y) vN gN
(29)
Equation (27) shows that in galaxies whose surface mass densities are much smaller than σ0 , mass discrepancy exists everywhere. On the other hand, galaxies with a high surface density show a larger mass discrepancy in their outer parts. Thus, mass discrepancy as a function of radius depends on the ratio σ0 /Σ0 for any single galaxy. However, the plot for many different galaxies with different σ0 /Σ0 would be a scatter one. You may find such data plot here: [12]. On the other hand if there is a bulge at the center of the galaxy then one should expect a higher mass discrepancy at small radii. However mass discrepancy outside of the galaxy will not change significantly. HvvN L2 10 1.5
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8 6 4
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FIG. 4: Mass discrepancy as a function of radius for different values of σ0 /Σ0 . For larger values of σ0 /Σ0 (LSB galaxies) mass discrepancy exists everywhere; for smaller values (HSB galaxies ) mass discrepancy exists in outer parts ( r > Rd ). Although one should note that the mass discrepancy is generally smaller than the prediction of this graph. The reason, as we mentioned for the case of rotation curve is that the gas extends far beyond optical disk. McGaugh have shown that, regardless of the choice of stellar mass-to-light ratio, acceleration is the physical scale with which the mass discrepancy correlate best [12], See also [2]. On the one hand, equation (28) illustrates that for any galaxy when centripetal acceleration approaches to a0 , we see a large mass discrepancy (see figure 5). In addition, the special form of (28) seems generic for spiral galaxies; i.e. if equation (2) is correct, all of spiral galaxies should have the same relation between mass discrepancy and centripetal acceleration. This is an important point because the previous results, viz. Eqs. (26) and (27), seems to vary from galaxy to galaxy because they depend on the central surface density Σ0 ; however equation (28) shows a unique character for all spiral galaxies. On the other hand, in solar system scales which centripetal acceleration is much larger than a0 , mass discrepancy is very small. In this limit the mass discrepancy approaches to one. See the horizontal asymptote of figure (5). In addition, for any radius there is a vertical asymptote at a . a0 . From equation (29) we see a large mass discrepancy when gN approaches to zero. However when gN is much larger than a0 , mass discrepancy is very small again. To conclude, we should say that if equation (2) is the correct description of object’s acceleration in galactic scales, equations (27), (28) and (29) should be applicable in any such scales, except around centers of galaxies which we should use general theory of relativity.
11
One last point: Comparing the modified velocity v and Newtonian velocity vN could be very useful. From Eq. (26) it is easy to see that: 2 v 2 − vN = 2a0 yI1 (y)K1 (y) r
(30)
where v and r are observable quantities and vN could be derived [12]. Data plot of this last equation could directly disprove Eq. (2) if we find inconsistent results. On the other hand it is possible to define at any radius r, a dynamical 2 mass which is attributable to the rotation velocity: MD = rv G . Then one could obtain the difference between 2 dynamical and Newtonian mass as follows: MD − MN = 8σ0 Rd yI1 (y)K1 (y); where MN is the amount of Newtonian or baryonic - mass inside radius r. Now we interpret the quantity MDM = MD − Mb as the amount of ”missing mass” or the dark matter mass, where clearly is related to the critical surface density σ0 but not to the surface density Σ0 of galaxies. Therefore the dark matter density distribution varies as (ρDM = dMdVDM ) [47]: ρDM =
σ0 {I1 (y)K1 (y) + y(I0 (y)K1 (y) − I1 (y)K0 (y))} 4πRd
(31)
which is a very interesting result . The first point is the asymptotic expansion of the last equation: ρDM ≈ ρ0 (σ0 /2π)[r−1 +O(r−1 )]. This behavior is very close to NFW profile for galaxy dark matter halos, ρDM = (r/rs )(1+(r/r 3 s )) , and explains the efficiency of this mass profile. Navarro, Frenk and White (NFW) used high-resolution N-body simulations to study density profiles of dark matter halos [48], [49] . They found that all such profiles have the same shape, specially independent of the halo mass. Their specific mass profile changes gradually from 1/r to 1/r3 beyond a certain critical distance from the center, rs . This special form of mass distribution reproduces galactic phenomena; however it provides an infinite mass for the dark matter halo [46]. The second point about Eq. (31) is that it predicts σ0 a constant central surface density for dark halos 2π ; i.e. the central surface density is independent of the total mass of galaxies. Nevertheless, this is a famous observational result that was first noticed by Donato et al. [50]. See also [2]. Then Milgrom proved that MOND predicts such central surface density for the dark halos too [18]. Eqs. (26) and (31) could explain another important feature of our model. From these two equations it is clear that all departure from pure Newtonian theory are included in the a0 -dependent terms. Therefore any deviation in rotation curve data and dark matter profile should occur simultaneously. This is known as the Renzo’s rule; i.e. ”For any feature in the luminosity profile there is a corresponding feature in the rotation curve” [11], [12]. In fact any pure baryonic theory could explain this observational rule. It is important to note that all these results are achieved without assuming the dark matter hypothesis.
VI.
CONCLUSION
In this paper I started to investigate a new equation of motion in weak field limit which is motivated by considering spacetime measurement in an expanding universe. The new term in Eq. (2) is an intermediate effect between mere curved spacetime - which we use to describe strong fields around massive objects - and a pure FRW spacetime that HvvN L2 4
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aa0 0
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FIG. 5: Mass discrepancy as a function of centripetal acceleration for different values of y = r/Rd . The horizontal line,( vvb )2 = 1 , indicates where there is no mass discrepancy; i.e. baryonic mass content is enough to explain the observed motion. It is clear that the mass discrepancy becomes important near a0 which is expected according to our model.
12 contains these objects. It is proved that the new equation is compatible with the previous established results in solar system scales, because the perturbed term is small compared to the Newtonian results at the mentioned scales. A system of particles is investigated and then rotation curve and mass discrepancy for a disk galaxies are derived. Same results could be achieved for other types of galaxies. It seems that one could justify galactic phenomena by using this simple model. An important feature of this model is the prediction of a constant acceleration in outer parts of galaxies. This is a key point of our new model in galactic scales. In addition, I proved that the mass discrepancy is related to acceleration in units of a0 . Moreover I showed that the critical surface density, σ0 = a0 /G, has an important role in the rotation curve and mass discrepancy plots. Furthermore, a mass density profile with an asymptotic behavior of ρDM ∼ [r−1 + O(r−1 )] and a central surface density of σ0 /2π is derived too. This is related to the NFW mass profile. However, from a theoretical point of view, there still remains much work to do. In fact, compared with the earlier obstacles of this topic, the problem in galactic scales has some other features than the rotation curve conspiracy of galaxies and high velocity objects in clusters. We mentioned some of these problems in the first section of this paper. Maybe the most important problem is that, according to Newtonian theory of gravity, galaxies and clusters are unstable. This is in contrast with our repeated observations of old galaxies and clusters that seem to be quite stable. Another important conspiracy is the mass estimation of galactic structures. For example, it is expected that for a cold disk of stars the total mass and virial velocity be related through the following equation: v 3 ∼ M . Instead, according to successive observations, these two parameters are related through the famous Tully-Fisher law: v 4 ∼ M . In a forthcoming paper I will show that Eq. (2) could explain the stability of galactic structures. However, numerical simulations have already proved that a linear potential is capable of providing stable disks without needing further matter or dark matter [51] . Furthermore, the Tully-fisher relation could naturally be derived from the virial theorem plus the stability condition for cold disks. In addition, Ga0 which is the zero point of baryonic Tully-Fisher relation, will be justified. Moreover the mass distribution in cold and hot galaxies will be studied [52]. ACKNOWLEDGMENTS I would like to thank Stacy McGaugh for his helpful criticism.
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