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AN ALTERNATIVE MODEL FOR THE ROTATION OF
SPIRAL GALAXIES: THE CONNECTIONS AMONG
SHAPE, KINEMATICS AND EVOLUTION
Mario Everaldo de Souza Universidade Federal de Sergipe, Departamento de Física e-mail: [email protected] Av. Marechal Rondon, s/n, Campus Universitário, Jardim Rosa Elze 49100-000, São Cristovão, Sergipe, Brazil
Abstract
It is proposed that the arms of spiral galaxies are formed by the continuous outflow of matter from their centers. It is then shown that the ratio between the radial and tangential velocities of the outflow is the parameter responsible for the logarithmic spiral structure of spiral galaxies. The fitting of some spiral galaxies to the model allows the calculation of the radial velocities of matter in these galaxies and such values completely agree with the observational data. An approximate general equation is proposed for the description of the arms of spiral galaxies with or without bars. Some important consequences are discussed with respect to dark matter, galactic evolution and cosmology. It is concluded that a quantitative representation of the dynamics of the spiral galaxies can be entirely represented with data on their formation without any need to use the far-reaching conjecture of dark matter. We finally indicate that this conclusion is fully in agreement with the absence of dark matter inferred from Santilli's IsoRedShift within the inhomogeneous and anisotropic inner galactic medium that light has to traverse before reaching intergalactic spaces, as well as with the recent demise of symmetries and other recent advances.
Keywords: Rotation models of spiral galaxies; Rotation of spiral galaxies; Spiral galaxies; Spiral structure; Galactic evolution; Santilli IsoRedShift; Dark matter.
Introduction The current status quo for the rotation of spiral galaxies is still based on the density wave theory which states that the matter of the disk becomes distributed in spiral arms due to the action of a wave-like perturbation in the form of quasi-steady global modes of the disk ([1],[2],[3]). These three references are just examples of a long list of proposals within the same general framework. Other approaches defend that the spiral structure is a short-lived, transient phenomenon triggered by gravitational instabilities. Two of many references in this line are Goldreich & Lynden-Bell [4], and Julian & Toomre [5]. A quite simple argumentation against the density wave theory is that according to this theory we would have to have many spirals with 3, 4, and even 5 arms. Observations, however, have shown that almost all spirals have only two arms. A very recent study by Foyle et al. [6] based on observations of 12 spiral galaxies discard the density wave theory in its simplest form as being an “important aspect of explaining spirals in large disk galaxies”, but the authors wrongly conclude that the spiral structure is not a long-lived phenomenon. We present an alternative model for the description of spiral galaxies based on the outflow of matter from their cores and show that the spiral structure is inherent to the existence of the galaxy. Outflows of matter from the centers of galaxies have been reported since a long time ago. Let us present some examples. Very recent data [7] of NGC 6240, which is considered a typical protogalaxy show that “approximately 70% of the total radio power at 20cm originates from the nuclear region ( 1.5 kpc ), of
which half is emitted by two unresolved ( R 30 pc ) cores and half by a diffuse component. Nearly all of the other 30% of the total radio power comes from an arm- like region extending westward from the nuclear region”. NGC 2992 presents a jet- like structure and a circum-nuclear ring [8]. Falcke and Biermann [9] report that there is a large scale emission-like jet going outward from the core of NGC 4258 with a mass of about 4 × 10^35 kg and with a kinetic power of approximately 10^42 ergs/s and expansion velocity of about 2000km/s. Brunthaler et al. [10] report the first superluminal jet with a velocity of about 1.25c in the Seyfert spiral galaxy III Zw 2. For superluminal as well as subluminal speeds within physical media verifying causality laws in view of the universal Lorentz-Poincaré-Santilli (LPS) isosymmetry for interior dynamical problems, one can see Ref. [11], and the comments in the Concluding Remarks. Balmaverde and Capetti [12] have reported in 2006 that “Considering the radio structure, several objects of our CoreG sample have a radio-morphology with well developed jets and lobes: UGC 7360, UGC 7494 and UGC 7654 are FR I radio- galaxies part of the 3C sample (3C 270, 3C 272.1 and 3C 274), while in the Southern sample we have the well studied radio-galaxies NGC 1316 (Fornax A), a FR II source, NGC 5128 (Cen A) and IC 4296. A literature search shows that at least another 11
of spirals, clearly showing that their disks are younger than their bulges and that the hydrogen has its origin in the centers of galaxies. Taking a closer look at the morphologies of some galaxies we can clearly see jets/arms coming out or their nuclei. It is the case of some galaxies classified as peculiar galaxies. All pictures of UBVR images (UGC ´s and VV 114) shown below are credited to Hibbard, Liu & Armus [21]. A very enlightening image is that of UGC 04264 (Fig 1) which shows two young arms in the lower spiral galaxy. The lower arm is not affected by tidal forces.
Fig.1: UBVR image of UGC 04264*. The arms in the spiral galaxy are young and the lower arm is not affected by tidal forces from the interacting galaxies.
Still another clear example is the image of UGC 06748* (Fig. 2) where we can observe young arms in the spiral galaxies being formed. Observe that the top arm of the left galaxy and the right arm of the middle galaxy do not suffer tidal forces.
Fig. 2. UBVR image of UGC 06748*. On the left we see a pair of interacting spiral galaxies with young arms.
The remarkable image below (Fig. 3) of UGC 08929* reveals the formation of a very young arm in the spiral galaxy. The arm/jet is just beginning to curve and on the other side we already see some protuberance being formed.
Fig. 3. UBV image of a very young arm just beginning to curve. Observe the protuberance on the opposite side.
2 The model Taking into account what was shown and discussed above this work proposes that the spiral arms are formed by the shedding of matter from the nuclei of spiral galaxies. This is actually an old idea, proposed in 1964 by Oki et al. [23]. So, let us consider that a certain extended mass of gas m is ejected from the bulge of the galaxy with a radial velocity vr as is shown in Fig. 5. In
the bulge the mass m was rotating with an angular velocity . When it leaves the bulge at a later time v is not affected by the radial driving forces that cause the shedding of matter, and as it is shown below the mass keeps its angular momentum maintaining the tangential velocity approximately constant. Let us recall again that m is not pointlike. The Milky Way and other galaxies show that the mass m frequently has the form of an arc of matter which gets approximately distributed along a spiral so that we have an equation of the form for the angular momentum of m
i i i i
mRv m rv (2)
where R is the radius of the bulge, (^) v i are the velocities of the different parts
of the extended mass m which are located at ri , just after having left the bulge.
We notice that after having left the bulge the mass m can continue with an average v given by
i i
i i i i i i
m r v m rv v mR m R
^ (3)
in which, since mi / m 1 and ri / R 1 , v i can thus have values around v .
We obtain more detail on this if we analyze the behavior of the kinetic energy. Just before leaving the bulge the mass m (in the form or an arc, for example) has the kinetic energy
K m vr v (4)
and just after having left the bulge the mass m has the kinetic energy
2 i^ i i r i
K m v v (5)
Disregarding the small variation of the gravitational potential, K is conserved and thus we have
i i^ i i^0
r i r i
dK dv^ dv m v v dt dt dt
^ (6)
whose solutions are
ri ri^ i i^0
dv dv v v dt dt
(^) (7)
with i 0
dv dt
^ , dvri 0 dt
, and
i i
v r const v const
^
In the case of (7) ri
dv dt
and i
dv dt
have opposite signs. The analysis of Shetty et
al. [24] on streaming of matter in M51 clearly shows that vri and v i vary
together about their mean average values. However, observations have shown that Nature prefers the solution given by (8), and our fittings below show that this is indeed the case. The reason for this lies in the fact that the driving forces that shed matter outward are radial forces and, hence, do no work in the direction of v i. The simple arguments above
show that the constancy of the tangential velocity of spiral galaxies is directly
R r
From (10) we also obtain
r
R v d dt dt dr r rv
^ (12)
where we have made use of the fact that vr dr dt
The first approximation is to consider the radial velocity vr approximately
constant. In this case the integral of (12) is just
r Re ^ (13)
where r v v
. (13) is Danvar equation, but now we see that is a very
important parameter, directly related to the kinematics of the galaxy. It was deduced by de Souza quite some time ago [25,26]. Substituting (13) into (11) we obtain
e ^ (14)
and thus the arms lag the bulge exponentially with respect to . Since d dt
^ ,
1 e ^ 1
This relation is important because knowing the maximum for a certain spiral arm we can find the maximum value of and find out how much the bulge
rotated since the beginning of the formation of the spiral arms. Of course, this is very important for studying galactic evolution. We can also define the lagging angle (Fig. 6).
It is important to mention that the beginnings of the formations of the two arms may occur at different times, and the arms can have different radial velocities. That is why there are asymmetric spirals with an arm much more developed than the other one. There are even spiral galaxies with a single arm such as NGC 4725.
Fig. 6. The lagging angle which is measured with respect to the initial stream of matter, across a diameter of the bulge by an observer corotating with the bulge.
In the data of Ganda et. al. [27] of 18 galaxies we find galaxies where vr
increases slightly with (^) r , although in most of them vr decreases slightly with
r. Therefore, our approximation above is quite justified. If we consider the variation in vr with the gravitational field we should also consider other effects
such as interaction between consecutive arms, and this is a very complex effect and is hard to be taken into account.
3 Application of the model to some spirals We begin the application of (13) to the grand design galaxies M51 and M and to the barred spiral NGC 1300. In all galaxies below the data points were visually captured with the use of the software plot digitalizer. Since we are dealing with the logarithm of a fraction in the fitting of , the error bar is quite small because the logarithm smoothes out errors. Thus we estimate for the fitted values of an error bar of about 1-2%.
Table 1. The calculated values of for spiral galaxy M51.
r / R 30 o 1.23 0. 60 o 1.45 0. 90 o 1.82 0. 120 o^ 2.09^ 0. 150 o 2.49 0. 180 o 2.86 0. 210 o 4.45 0. 240 o^ 4.82^ 0.
3.2 M
In this case we chose the spiral arm shown in Fig. 8 and considered points corresponding to bright stars around the middle of the arm. We took the lagging angle as in the case of M51.
Fig. 8. The fitted spiral of NGC 628 considered for the calculation. The spiral is along the bright patch of the arm (NASA/HST photo).
Table 2. The calculated values of for spiral galaxy M74.
r / R 60 o 1. 38 0.3 1 90 o 1. 75 0.3 6 120 o 2. 25 0.3 9 150 o^ 2.^63 0.3^7 180 o 3.13 0.3 6 210 o 3.63 0. 240 o 3.88 0.3 2 270 o^ 4.25^ 0. 300 o 4.63 0.
According to Kamphuis & Briggs [28] v for NGC 628 (M74) is about 200
km/s and according to Ganda et al. [27] its vr is about 70 km/s, and thus, 70
200 which is quite close to our calculated values above (Table 2)
whose average is 0.34.
3.3 NGC 1300 The considered value for R was half of the diameter between the ends of the dust lanes in the bright nucleus. Then this line was displaced to the beginning of the spiral arms for the calculation of (Fig. 9). As in the previous examples, intervals of 30 o were used along the dust lane of the arm.
Table 3. The calculated values of for the barred spiral galaxy NGC 1300.
r / R 60 o^ 1.^31 0. 90 o 1. 74 0. 53 120 o 2.34 0. 54 150 o 3. 00 0. 42
Table 4. The calculated values of for the spiral galaxy NGC 4030. rb means the distance to the center of the nucleus across the baseline.
r^ /^ rb 30 o 1. 38 0. 40 60 o 1. 46 0.3 6 90 o^ 1.77^ 0.3^6 120 o 2. 15 0.3 7
Ganda et al [27] report a value vr 90 km/s and Mathewson & Ford [31] say
that v is about 236 km/s, and, thus, 0.38which is quite close to the above
calculated average value.
Fig. 10. The fitted arm of the spiral NGC 4030 (Penryn, California photo).
3.5 NGC 1042
We performed the fitting in the longer arm, as shown below in Fig. 11 and measured the ratios (^) r / R in intervals of 30o. The calculated values for (^) are listed in Table 5. Their average value is about 0.69.
Table 5. The calculated values of (^) for the spiral galaxy NGC 1042.
r / R 60 o 2.0 0. 66 90 o 3.0 0. 70 120 o^ 4.6^ 0.^73 150 o 5.6 0. 66
Fig. 11. The arm of NGC 1042 considered in the fitting (NASA/HST photo).
Table 6. The fitted values for arm A of NGC 4254.
30 o 0. 60 o 0. 50 90 o^ 0.^53 105 o 0. 51
Table 7. The fitted values for arm B of NGC 4254.
60 o^ 0.3^2 90 o 0.3 2 120 o 0.3 3 150 o 0. 180 o^ 0. 210 o 0. 33 240 o 0.3 3
Table 8. The fitted values for arm C of NGC 4254.
30 o 0.3 1 60 o 0.3 2 90 o 0.3 7
And now we can understand why this galaxy is so asymmetric: arm A has a larger radial velocity than arms B and C, and has values of arms of barred spirals. With the above average values of ( B 0.33, C 0.33) for arms B
and C we find that their radial velocities are approximately equal to 49.5 kms- which is much smaller than the radial velocity of arm A.
4. Discussion of results 4.1 Connections among shape, kinematics and evolution We clearly see that the results are consistent and the parameter (^) b of the Danvar equation is the ratio vr v and is, thus, directly connected to the kinematics of the galaxy. This means that the shape of a spiral galaxy is directly connected to its kinematics and evolution. Young spiral galaxies are small and old spiral galaxies are large (unfolded). A spiral galaxy unfolds itself from the inside out throughout time up to the exhaustion of the mass of its nucleus. And it does not get tightly wound as a consequence of the unfolding and winding because of the radial velocity vr. Of course, the bulge
should diminish slowly with time since its mass is shed outward. The Milky Way is still shedding matter outwards and there is a lot of mass in its center yet, and so it will keep on going during quite a while, probably a couple of billion years. The calculation of the parameter for a galaxy from the shape of its arms provides important information on its kinematics and will be very useful for the study of spiral galaxies. The above calculations and results mean that if v varied too much with r
spiral galaxies would not exist at all. We immediately observe that most galaxies have not rotated much because of the following argumentation. Considering the ends of the arms of a spiral and taking the angular difference between them we obtain a certain (^) , and so we have the approximate relation
and, thus, ~2-1. For M51A, for example, ~ 2 × 0.37-1^ rad = 5.4 rad =
1.72 π rad. Taking a look at its photo we observe that it has barely completed a
full turn. The same holds for M74, for which~ 2 × 0.31-1^ rad = 6.45 rad =
2.05 π rad. For NGC 1300 we obtain the value of~ 2 × 0.5-1^ rad = 4 rad =
1.27 π rad, which is very consistent.
4.2 Distinction between barred and non-barred spirals We observe that with respect to the tangential velocity, barred spirals have larger radial velocities (expansion velocities) than non–barred spirals and that is why their spiral arms are more open, that is, less tightly wound. Analyzing
