=================================Originally Posted by tusenfem
Densemass equations
Densemass equations
Sirs,
The following paper is about non-requirement of “MISSING MASS” in galaxies. This discusses requirement of a (1.) huge central mass depending on the size of the Galaxy, (2.) Gravitation force of external Galaxies, and (3.) Star masses in the Galaxy. All these three are required simultaneously. Then we will get the Star rotation curves as observed in various Galaxies. Other cases were discussed in the paper and results were shown as graphs.
The equations given in the paper can be used. You can substitute the values for all the three above. Plot the circular velocities against distance from the center, you will get the Observed circular velocity curves. Here we don’t use Special Relativity and General relativity effects and we use only Newtonian Gravitation Attraction forces. We also observe that the bodies do not collapse into the center. They are compensated by the Dynamic Multi body Tautness (Centrifugal) forces.
You are free to ask any questions….
Warm regards to you all…
=snp
Sirs , It seems this gave only text, no pictures, tables how to show them here?????? should be less than 15000 chars????How to show a doc file in total????
=snp
==================================
Missing Mass In Galaxies In Dynamic Universe
Model Of
Cosmology Using Regression Analysis
SNP. Gupta
Bhilai Steel Plant, 1b / 57 / sector 8 , Bhilai,490008, India, snp.gupta@gmail.com; snp.gupta@indiatimes.com
Abstract. In this present work SITA simulations were used to find out Theoretical star circular velocity curves in a Galaxy (star circular velocity verses star distance from the center of galaxy), depends on various initial conditions and are never half bell shaped curves as predicted by Bigbang cosmologies. Here we are presenting four main cases. In the first case A Galaxy with a huge central mass with star like masses in presence of external galaxies were taken. Theoretical predictions of circular velocities were matching with the observed velocities. In the later cases either Huge central mass was absent or external galaxies were absent or both were absent, the theoretical circular velocities did not match the observations. Hence the question of missing mass does not arise.
BASIS: DYNAMIC UNIVERSE MODEL OF COSMOLOGY
In SITA Non uniform mass distribution and 3-D positioning done and Newtonian attraction forces and Special Relativistic effects were used. No long distance repulsive (λ) General relativistic effects. No space-time continuum. SITA results are encouraging. The masses do not collapse into lump but they are orbiting each other. Large-scale structures of universe could not be explained by Big bang based theories without using additional repulsive forces like “Einstein’s l“, as it requires isotropy and homogeneity. Our universe is neither isotropic nor homogeneous. It is LUMPY. And it is not empty. Special theory of Relativity is sufficient. Things can be explained by Newtonian gravitation. This proves Galaxy disk formation require some external forces other than self-gravitation of Galaxy it self. Here the there is universal gravitational effect at that position and time are calculated due to ALL the bodies present in the universe. This forms a repulsive force. This force varies with time, position, structure, masses, their distances, their dynamic movement etc. Our universe is not a Newtonian type static universe. Theoretical star circular velocities in a Galaxy, in Bigbang cosmologies are predicted as shown in the left in the Picture. The observed rotation curves are shown on the right side. Is this means that the mass of the Galaxy increases with increasing distance from the center? The observed rotation curves are shown on the right side of Pic 1 Is this means that the mass of the Galaxy increases with increasing distance from the center? {Ref 56} Here we have predicted using Dynamic universe model cosmology in four different cases and present. Here we have predicted using Dynamic universe model cosmology in SIX different cases and present.
MATHEMATICAL FORMULATION:
Lets assume a non-homogeneous and anisotropic set of N particles moving under mutual gravitation as a system, and these particles are also under the gravitational influence of other systems with a different number of particles in different systems. Also lets assume no uniformity in the particle distribution with respect to size, mass or internal distances. Lets call this set of all the systems of particles as an Ensemble. Lets further assume that there are many Ensembles each consisting of a different number of systems with different number of particles. And similarly lets further call a group of Ensembles as Aggregate. Lets further define a Conglomeration as a set of Aggregates. And let a further higher System may have a number of conglomerations and so on and so forth. Lets start with. Ensemble Equations (Ensemble consists of many systems)
d2 Iγjk
------- = Wγjk + 2 Kγjk - 2Fγjk (18-E)
dt2
Here γ denotes Ensemble.
This Fγjk is the external field produced at system level. And for system
d2 Ijk
------- = Wjk + 2 Kjk - 2Fjk (13)
dt2
Assume ensemble in a isolated place. Gravitational potential fext(a)produced at system level is produced by Ensemble and fγ ext(a) = 0 as ensemble is in a isolated place.
Nγ G mγβ
fγtot(a) = fγext - Σ ----------------- (19)
α≠β ˝xγβ - xγα˝
β=1
There fore
Nγ G mγβ
fext(a) = fγtot = - Σ ----------------- (20)
α≠β ˝xγβ - xγα˝
β=1
And 2Fjk =
d2 Ijk
= - ------- + Wjk + 2 Kjk
dt2
N
= Σ [ Ń fext ] mα xαj
α=1
N
+ Σ [ Ń fext ] mα xαk (21)
α=1
AGGREGATE Equations (Aggregate consists of many Ensembles )
d2 Iδγjk
------- = Wδγjk + 2 Kδγjk - 2Fδγjk (18-A)
dt2
Here δ denotes Aggregate.
This Fδγjk is the external field produced at Ensemble level. And for Ensemble
d2 Iγjk
------- = Wγjk + 2 Kγjk - 2Fγjk (18-E)
dt2
Assume Aggregate in an isolated place. Gravitational potential fext (a) produced at Ensemble level is produced by Aggregate and f δγ ext(a) = 0 as Aggregate is in a isolated place.
Nδγ G m δγβ
f δγtot(a) = f δγext - Σ ----------------- (22)
α≠β ˝x δγβ - x δγα˝
β=1
There fore
Nδγ G m δγβ
fδγtot a = fγext (a) = - Σ ----------------- (23)
α≠β ˝x δγβ - x δγα˝
(Aggregate) (Ensemble) β=1
And 2Fγjk =
Nγ
= Σ [ Ń fδext ] mα xδαj
α=1
N
+ Σ [ Ń fδext ] mα xδαk (24)
α=1
Total AGGREGATE Equations (Aggregate consists of many Ensembles and systems)
Assuming these forces are conservative, we can find the resultant force by adding separate forces vectorially from equations (20) and (23).
Nγ G m γβ
f ext(a) = - Σ -----------------
α≠β ˝x γβ - x γα˝
β=1
Nδγ G m δγβ
- Σ ----------------- (25)
α≠β ˝x δγβ - x δγα˝
β=1
This concept can be extended to still higher levels in a similar way.
How missing mass problem arouse?
There is a usual conceptual mistake. Newtonian Gravitation or Einstein’s General theory of Relativity treated the Multi-body dynamical problem as a single body static problem. The usual situation is, there are many Galaxies present in the universe. In each Galaxy there is a Galaxy center of huge mass and many stars rotating about it. How to tackle this problem? It is usual consider the Galaxy is a solid rotating child’s top about a central axis. It may be of disk shape or sphere shape. No external Galaxies, no heavy-duty center, but with uniform density. Hence the rotation curves are drooping curves. But the observations star circular velocities in a Galaxy, gives us a different picture. People went for speculating invisible missing mass. There is a search but in vain.
Here I have shown five cases as explained below. In two cases I have not taken the bulky Galaxy center at all, but some stars with or without external Galaxies. In another two cases, I have taken the Galaxy with huge center and stars with or without external Galaxies. It is clearly seen that the predicted theoretical star circular velocities in a Galaxy are same as the observations made by the astronomers, in only one case when both the Galaxy center and External Galaxies are present.
Resulting Graphs
In all these cases, I have used SITA simulations for Multi-body dynamic systems, with same initial conditions. I took some stars at star distances and external Galaxies at Galactic distances. I made 100 iterations for uniformity.
Case 1 : From starting positions to positions after 100 iterations showing disk formation and velocities achieved graph. This is with a Huge central mass at the center of galaxy, sun like stars and external galaxies xy, zx position graphs. This Graph shows the theoretical star circular velocity curves in a Galaxy (star circular velocity verses star distance from the center of galaxy)
Case 2 : Similar as above, showing disk formation and velocities achieved graph. This is without a Huge central mass at the center of galaxy, sun like stars and external galaxies xy, position graphs
Case3. Similar ascase1, showing disk formation and velocities achieved graph. This is wih a Huge central mass at the center of galaxy, sun like stars and no external galaxies xy, position graphs.
Case 4. Similar as case 1, showing no disk formation and velocities achieved graph. This is wihout a Huge central mass at the center of galaxy, sun like stars and no external galaxies xy, position graphs.
Case 5. Theoretical star circular velocity curves in a Galaxy (star circular velocity verses star distance from the center of galaxy) in gravitationally stabilized system of masses after forming a galaxy disk. I did it’s stability analysis tests, by giving perturbations and Jeans swindle test.
Table : Theoretical Galaxy Circular Vel vs radius Graphs in different cases with start end of 100 iterations positions
Case Starting positions End of 100 iterations Velocity vs Gal Radius
1 Case 1 : From starting positions to positions after 100 iterations showing disk formation and velocities achieved graph. This is with a Huge central mass at the center of galaxy, sun like stars and external galaxies xy, zx position graphs.
2 Case 2 : From starting positions to positions after 100 iterations showing disk formation and velocities achieved graph. This is without a Huge central mass at the center of galaxy, sun like stars and external galaxies xy, zx position graphs.
3 Case 3 : From starting positions to positions after 100 iterations showing disk formation and velocities achieved graph. This is wih a Huge central mass at the center of galaxy, sun like stars and no external galaxies xy, zx position graphs.
4 Case 4 : From starting positions to positions after 100 iterations showing no disk formation and velocities achieved graph. This is wihout a Huge central mass at the center of galaxy, sun like stars and no external galaxies xy, zx position graphs.
5 Case 5: Theoretical star circular velocity curves in a Galaxy (star circular velocity verses star distance from the center of galaxy) in gravitationally stabilized system of masses after forming a galaxy disk when it’s stability analysis was done by giving perturbations and jeans swindle test
Cases 1,2,3 & 4 show cases with and without central mass and / or external galaxies. We can see clearly ext Galaxies and Central mass in Galaxy is required as dist velocity curves are near to actual observational results. These N-body calculations and results are showing theoretical star circular velocity curves. Do the Galaxies to be assumed to have some missing mass? Is that required?
