Some consequences of a Universal Tension arising from Dark Energy for structures from Atomic Nuclei to Galaxy Clusters

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1 unning Head: Universal Tension fro DE Article Type: Original esearch Soe consequences of a Universal Tension arising fro Dark Energy for structures fro Atoic Nuclei to Galaxy Clusters C Sivara Indian Institute of Astrophysics, Banore, 560 0, India Telephone: ; Fax: e-ail: sivara@iiap.res.in Kenath Arun Christ Junior College, Banore, , India Telephone: ; Fax: e-ail: kenath.arun@cjc.christcollege.edu Kiren O V Christ Junior College, Banore, , India Telephone: ; Fax: e-ail: kiren.ov@cjc.christcollege.edu Abstract: In recent work, a new cosological paradig iplied a ass-radius relation, suggesting a universal tension related to the background dark energy (cosological constant), leading to an energy per unit area that holds for structures fro atoic nuclei to clusters of axies. Here we explore soe of the consequences that arise fro such a universal tension.

2 . Introduction In recent papers (Sivara, 99a; 99b; 008; Sivara & Arun, 0a; 0) a new kind of cosological paradig was invoked wherein the requireent that for a hierarchy of large scale structures, like axies, axy clusters, super-clusters, etc. their gravitational (binding) self energy density ust at least equal or exceed the background repulsive dark energy density (a cosological constant as current observations strongly suggests) iplies a ass-radius relation of the type: c... () G G c c c (i.e. gives rise to a universal tension, T ) 8 8G G This is the background curvature x superstring tension (Sivara, 99a; 005). Or this can also be inferred as the local ass x local curvature. (Superstring tension is cosological dark energy) c ~, is the G This paradig focuses on the universe s fundaental structures and syetries and ephasises a new universal paraeter underlying systes fro the sallest (atoic nuclei) to the largest (clusters of axies), encopassing nearly 80 orders of agnitude in ass and nearly 0 orders of agnitude in size. (Oldershaw, 987; Sivara, 99; 00; 005). Nuclear Tension c The energy per unit area (surface tension) given by above equations, i.e. T c, has G the sae nuerical value as that in nuclear physics, (Sivara, 005; 008) like the surface tension in the nuclear liquid drop odel of nucleus and nuclear atter. ~ 0 ergs / c. This has consequences for the In the nucleus this nuclear surface tension balances the Coulob repulsion: Z e T... ()

3 Where 0 A, 0 ~.50 c For T ~ 0 ergs / c, this sets a liit of: Z A 0 which agrees with the usual Bohr-Wheeler criterion. (Bohr & Wheeler, 99) Considering also the rotation (spin) of the nucleus we have:... () Z e T... () 5 Where the ters on the left are repulsive in nature and that on the right is attractive. This gives the radius of the size of the nucleus as (where P A, P is the proton ass): Z e 5 T A P... (5) The angular frequency dependence on nuclear size is given in figure (): Figure : Dependence of Angular Frequency with Nuclear size

4 For the liiting case as T 5 : P A... (6) This sets a liit on the frequency as: 0T... (7) A P For A = we have: 0 s... (8) And the corresponding tie period of: ~ 0 s... (9) This corresponds to the nuclear tie scale. Figure () gives the variation of angular frequency with the nuber of nucleons. Figure : Variation of angular frequency with A

5 This liit on the frequency will also put a constraint on the rotational energy levels of nucleus: n... (0) 5 7 n 0A... () For A = 0, we have n 00 These states correspond to the yrast states, which are the lowest excited level at high angular oentu ~ 70 as suggested in the following reference (Grover, 967). Later observations indicate high angular oentu ~ 00. Figure () gives the alost linear dependence of the order (n) with the ass nuber. Figure : Variation of n with ass nuber. Gravitationally bound structure, Angular oentu and Dark Energy In the case of large, gravitationally bound structures such as axies, axy clusters, etc. the requireent is that gravitational self energy density should be coparable to the background cosic vacuu energy density for the object to be an autonoous structure. That is: G c... () 8 8G 5

6 This would also give the sae result as equation (), i.e.: Where is the cosological constant with an observed value of 0 G c 56 c. This equation holds for a whole range of large scale structures, including the Hubble volue. (Sivara, 008 and references there in) The relation is suggestive of a surface tension which has the sae universal value for all the large scale cosic structures fro globular clusters, large olecular clouds, all the way to the Hubble universe (Sivara, 99a; 008). A kind of universal surface tension, suggesting the holographic picture! (Sivara & Arun, 0) (It also holds for the electron!) The universality of this surface tension again constrains the size of a neutron star. For a neutron star coposed on N neutrons: G NS NST N... () NS For a solar ass neutron star, detected neutron star. (Crawford, et a 006) Considering also the rotation of the neutron stars we have: G NS T N NS NS NS NS 57 N 50, which atches the observations for the heaviest... () This sets a liit on the rotational frequency and the corresponding tie period of the neutron star as: 0 s, 0. 5s... (5) This is consistent with the observations of the illisecond pulsar having the fastest rotational period detected so far, which is ~.s. (Hessels, et a 006) In the case of axies, this surface tension balancing the rotational energy can possibly explain the flat rotation curve of the axies. That is, for axies, their rotation balances this surface tension. This gives: T... (6) 6

7 Where the rotation frequency and the corresponding tie periods are given as: T 0 s, 0 s... (7) Since is a constant even for a axy, the relation given by equation (6) leads to: constant... (8) Therefore, the velocity, which is given by: v T... (9) will also be a constant as expected fro the axy rotation curves! This suggests a velocity independent of radial distance (flat rotation curve) without invoking dark atter. It is interesting to note that this dependence of rotational frequency going as inverse of the size hold true even right down to the atoic nucleus, as indicated by figure. Figure : Variation of rotational frequency with size 7

8 In an earlier work (Sivara & Arun, 0b) a priordial cosic rotation was suggested which can give rise to the observed rotation angular oenta of axies, axy clusters, stellar planetary systes, etc., the origin of which is otherwise not clearly understood. The angular oentu of the axy given by have (again since is a constant): J, is conserved. Therefore we J v is a constant... (0) This also leads to constant and therefore a velocity independent of radial distance.. Dynaics of evolving structures The requireent that the gravitational self energy density ust at least equal or exceed the background repulsive dark energy density iplied a ass-radius relation as given by equation (), for a hierarchy of large scale structures, like axies, axy clusters, super-clusters, etc. This ass-radius relation holds good for nebulae too. Any perturbation to it will lead to its collapse and eventual foration of the star (and possibly planetary syste). For a typical star of ass, the order of star 0 g, the condition that iplies that the initial size of the nebula be of Neb 6 ~ 0 c. It is also of interest to note that the sae value for the tension (arising as we have seen, fro the cosic dark energy ( ter)) which we have used for axies, axy clusters, atoic nuclei, etc. also sees to be relevant for the diensions of planets and stars. For exaple, for a typical planetary ass of gravitational self energy, i.e. 8 ~ 0 g, balancing surface energy and G T... () we get the radius, which is given by: 8

9 G T... () For 8 ~ 0 g, we get 5000k (the earth radius). The above equation also gives a Jupiter radius of ~ 0 5 k for the corresponding ass. For a typical stellar ass of ~ 0 g, the above equation iplies ~ 0 c. So the range of stellar and planetary sizes is also given by the sae value of T! This suggests a deep underlying connection between the background dark energy (-ter, which gives the background curvature) and all the structures ebedded in this background. For the large structures we had balance of gravitational energy densities with the background dark energy density. For the planetary and stellar objects, the balance was with surface energies and gravitational self energies. 5. Densities of various structures As noted above, we had a universa ratio, i.e. a ubiquitous surface tension of c G 0 ergs / c know that nuclear density is, underlying all entities fro nuclei to axy superclusters! But we ~ 0 understand this diversity in densities? 5 g / cc, superclusters have a density of ~ 0 g / cc. How to It is just that the average density is ~, so that if we have the universal T, i.e. c G g / T ~ c! (As T, the densities of the various structures considered would scale as, so is just T ) As T is a universal constant the density siply scales as tension, this is the Laplace pressure for a droplet).. (In connection with surface 9

10 Thus for a nucleus ~ feri, we have ~ 0 0 g / cc. For a axy ~ 0 c, we have 5 5 ~ 0 g / cc. For a super-cluster ~ 0 c, ~ 0 g / cc. And for the Hubble volue, ~ 9 ~ ~ 0 g / cc, just what is observed! H So we have another universal result: constant Holding fro nuclei to the universe!... () Figure 5: Variation of density with size 6. Nuclear Vibrational and otational Energy levels In connection with the energy levels of the nucleus, including both vibrational and rotational levels, we can invoke the liquid drop odel of nuclei. In the drop odel there is equilibriu between surface tension and Coulob repulsion. Sall perturbations of the drop surface of radius by r gives changes in surface energy (surface given by Fr,, constant), which can be expanded in spherical haronics (like in fluid echanics of incopressible liquid spheres). 0

11 Thus: C l Y... (), l Y l L Yl, l,... (5) The surface energy is perturbed as: T * ES l l C C... (6) While the electrostatic (repulsive) Coulob energy is perturbed as: Z e l * EC C C... (7) l Finally we can write the Hailtonian including also the kinetic energy: 5 C l, C l *... (8) The lowest ode being l we have the energy levels of a five-diensional haronic oscillator as: E n 5 l... (9) This for l gives the ground state energy level as: E Z e T 0... (0) An For a nuclei of Z 0, A 0, the above equation gives a ground state energy of: E 0 5 ergs 0eV 0 The higher levels will be in ultiples of 0eV.... () For l 0, stability is given by: Z e T, A 0 The higher vibrational excited states are given by n,,..., etc.... ()

12 We can include the rotational energy levels (like in atoic spectroscopy). Thus rotational levels are n rot. The liiting values of rot for various A have been given above. Energy levels of rotation are: E rot ll I... () Where the oent of inertia of the nuclei is: The rotational energy is then given as: I 5 E rot 0. ev... () And the total energy is: E total E E... (5) vib rot For various n, etc. Siilar relations as those above hold also for (nuclei of) priordial axies, provided we replace the Coulob energy ter with the gravitational energy. This would also have a negative sign as it is binding energy. In other words the replaceent is perturbed as: Ze by G would give the result. That is, the surface energy T * ES l l C C... (6) And the gravitational energy is perturbed as: (Lab, 95) 6 G l * EG C C... (7) l The tension (T) ter would be the sae. Scaling relations are as before and equation (0) will not apply to axies! The frequency of oscillation due to the perturbation for the axies is given as: 8 T G... (8)

13 For a typical axy of 0 g, 0 g / cc, the frequency is 0 s. These oscillations will eit gravitational waves, where the quadrupole gravitational power is given by: (Sivara & Arun, 0) G 6 P GW... (9) 5 c And for a typical axy of 0 g, 0 c, this gives: P GW ~ 0 5 ergs / s... (0) GEGW The corresponding strain produce on a detector, which is given as h ~ 0 c r within the liits of proposed space based gravitational wave observatories like LISA. 0, which is 6. Conclusion In this paper, we have extended our earlier work (which gave rise to a ass-radius relation) with a universal value of a surface tension ~ 0 ergs / c arising fro the requireent that the binding energy density of gravitationally bound objects be at least equal or exceed the background repulsive dark energy density. This universal tension arising fro dark energy doinating three-fourths of the universe, leads to various consequences for a hierarchy of objects, fro atoic nuclei to axy clusters. This can for instance set a liit on the rotational energy levels of a nucleus; set the diensions of planets and stars; to even explain the flat rotation curve of axies without invoking dark atter and liit the size of axy clusters. In short, we have a new paradig encopassing features of structures ranging over eighty orders in ass and forty orders in length scale. eference: Bohr, N & Wheeler, J. H. 99, Phys. ev., 56, 6 Crawford, F. et a 006, Astrophysical J., 65, 99 Grover, J.. 967, Phys. ev., 57, 8 Hessels, J. et a 006, Science,, 90

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