Journal of Modern Physics, 2017, 8, 2179-2188 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196
Galactic Structure and Holography
T. R. Mongan
84 Marin Avenue, Sausalito, CA, USA
How to cite this paper: Mongan, T.R. Abstract (2017) Galactic Structure and Holography. Journal of Modern Physics, 8, 2179-2188. Galactic structure, involving bulk motion of matter conserving mass and an- https://doi.org/10.4236/jmp.2017.814133 gular momentum, is analyzed in a holographic large scale structure model. In isolated galaxies, baryonic matter inhabits dark matter halos with holographic Received: November 18, 2017 Accepted: December 17, 2017 radii determined by total galactic mass. The existence of bulgeless giant spiral Published: December 20, 2017 galaxies challenges previous models of galaxy formation. In sharp contrast, holographic analysis is consistent with the finding that 15% of a sample of Copyright © 2017 by author and 15,000 edge-on disk galaxies in the sixth SDSS data release are bulgeless. Ho- Scientific Research Publishing Inc. This work is licensed under the Creative lographic analysis also indicates that E7 ellipticals are less than 2% of elliptical Commons Attribution International galaxies. License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Keywords Open Access Galactic Structure, Holography
1. Introduction
This paper does not address the wide range of processes, mechanisms, and mer- gers generating galactic structures and leading to the myriad galaxy types ob- served by astronomers. Instead, it identifies the range of isolated galactic struc- tures consistent with conservation of mass and angular momentum in a holo- graphic large scale structure (HLSS) model [1]. Current observations indicate that our universe can be approximated as a closed vacuum-dominated Friedmann universe. Futhermore, “the universe is dominated by an exotic nonbaryonic form of matter largely draped around the galaxies. It approximates an initially low pressure gas of particles that interact only with gravity…” [2]. The holographic principle [3] applied in a vacuum-dominated universe led to an approximate large scale structure model [1] where isolated
structures with total mass M g inhabit spherical holographic screens with M radius S = g if the Hubble constant H =67.8 km ⋅⋅ sec−−11 Mpc . 0.183 g cm2 0
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M The HLSS model considers galactic matter density distributions ρ (r) = g , 4πSr 2 where r is distance from the galactic center. Then, mass within radius R is
(RS) Mg , and tangential velocity of stars or star clusters in circular orbits within
M g the 1 r 2 density distribution around the galactic center is VG= , where S G =6.67 × 10−831 cm ⋅⋅ g −− sec 2. This analysis treats galaxies as spherical halos of dark matter with radius S and
(1−α ) M g density distribution ρ (r) = surrounding total baryonic mass 4πSr 2
MMbary= α g . Cosmological data indicate matter fraction of total energy density of the universe is ≈ 0.31 and baryonic fraction of total energy desity is ≈ 0.05 , so baryonic fraction of matter is α ≈=0.05 0.31 0.16 and dark matter fraction is α −=1 0.84 . The analysis uses α = 0.16 , but a distribution of values centered around α = 0.16 could be used to assess galactic systems formed in parts of the universe with slightly higher or lower ratio of baryons to dark matter. This paper considers galactic structures at redshift z = 0 , but the analysis is readily extended to z > 0 using results in Ref. [1].
2. Black Holes in the Galactic Core M There is no singularity in galactic matter density distribution ρ (r) = g 4πSr 2 because mass inside a core radius C at the galactic center is concentrated in a C central black hole with mass MM= , where C is the holographic radius CBH S g of star clusters that can inhabit circular orbits just outside the core without being
disrupted and drawn into the central black hole [1]. The mass M St comprising
the dark matter halo with mass M H plus baryonic matter with mass M bary is
MSt=−=+= MM g CBH M H M bary 0.84 MSt + 0.16 MSt . Total angular momentum of a galaxy is estimated using the moment of inertia 2 I= MS2 of a rotating sphere with galactic holographic radius S and density gg9 M ρ (r) = g . From the holographic relation MS= 0.183 2 , total angular 4πSr 2 g 2 2 M J = g ω ω momentum of the galaxy is gg, where g is galactic angular 9 0.183
velocity. Galactic angular velocity ωg is estimated by considering a mass m fixed on the rotating holographic screen at radius S. Radial acceleration GmM aS= −ω 2 of that mass results from gravitational force F = − g attracting g S 2 it to the centroid of the structure. Then, (correcting a misprint in Ref. [1]) GM 3 0.5 22g G 2 2 G 2 ωg = = (0.183g cm ) and JMgg= . In the HLSS S 3 M 9 0.25 (0.183M g )
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model [1], average galactic mass at z = 0 is 8.3× 1043 g and average galactic −16 2 angular momentum is JMgg=9 × 10 , 15% higher than Paul Wesson’s [4] −16 2 empirical value JMgg=8 × 10 .
Central black hole angular momentum JCBH max cannot exceed maximum 2 G22 GC Kerr black hole angular momentum JMCBH max =±=±KCBH Mg , c cS
2 J GC 2 G0.5 CBH = and ≤± 0.25 . In the HLSS model at z 0 , J cS 9 g (0.183M g )
39 45 C galactic mass ranges from ≈×4 10 g to ≈10 g and ≤1. So S −4 −3 JJCBHmax g ranges from ≈×2 10 to ≈×5 10 , and central black hole angular momentum is negligible. Core radius C equals the holographic radius of star cluster sub-elements of the galaxy that can inhabit circular orbits around the galactic center without being disrupted by the central black hole. Maximum 39 HLSS star cluster mass is 4.64× 10 g , so Cmax = 0.052 kpc , and minimum star 34 cluster mass is 2.11× 10 g , so Cmin = 0.00011 kpc .
3. Spiral Galaxies
Spiral galaxies predominate in underdense regions of the universe. This analysis treats baryonic matter in isolated spiral galaxies as disk structures with constant
M g tangential velocity VG= surrounding a central sphere with constant S density.
4. Conservation of Mass and Angular Momentum
Baryonic matter outside core radius C around the central black hole consists of a
bulge, with mass M B in a hollow sphere with inner radius C and outer radius
B, and a disk with mass M DS . So Mbary =0.16( MMg −=+ CBH) MM B DS . Total mass within B and outside the core in the 1 r 2 density distribution around the M galactic center is (BC− ) g and tangential velocity at B is S
M M g V= GB = VG = . If M , the baryonic part of total mass within B, is B BSB in a hollow sphere with inner radius C and outer radius B, M M=0.16( BC − ) g (1) B S Conservation of mass then requires a disk mass
M g B MDS =0.16 Mg − M CBH −−( BC) =0.16 1 − Mg (2) SS
Angular momentum conservation in spiral galaxies requires
JJgCBHHBDS= + J ++ JJ, where terms on the right of the equation are
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respectively the angular moment of the central black hole ( JCBH ), the dark
matter halo ( J H ), the central bulge ( J B ), and the disk ( J DS ). Since −3 JJCBH ≤×5 10 g ,
JJg≈ H ++ JJ B DS (3) β M A hollow sphere with density ρ (r) = g , inner radius C, and outer radius 4πSr 2 2 RC33− R has moment of inertia β M g , so angular momentum Jbary of 9 RC− galactic baryonic matter comprising the bulge and disk with mass
Mbary= MM B + DS is 2 SC33− Jbary=+=− JJ B DS J g (0.84) Mg ωg (4) 9 SC−
5. Bulge Radius and Disk Thickness
If the hollow sphere of baryonic matter constituting the bulge has constant M 4 ρ =−−g 33 density B 0.16(BC) π(BC) , tangential velocity of baryonic S 3 C matter in the bulge rises from VV= at core radius C to V at B. The hollow C B
sphere of the bulge with constant density ρB has moment of inertia 55 2 (BC− ) M B . To approximate observed velocity curves, angular velocity of 5 (BC33− ) the constant density bulge is found by setting tangential velocity at the edge of V the bulge VBBB= ω equal to disk tangential velocity V, so ωB = and B
55 2 V (BC− ) JMBB= . (5) 5 B (BC33− )
From Equation (4), disk angular momentum is 22SC33−− V BC55 V =−− JJDS g (0.84) Mg MB 33. (6) 95SC− S BC− B
Maximum bulge radius Bmax occurs when there is no disk and all baryonic JJ−= J SC BC> mass is bulge mass. From gH Bmax , using and max , 22 5 (0.16) M SV≈ (0.16) M B V and BSS≈=0.56 . 95ggmax max 9 Next, consider a disk extending from RB= to RS= with density (RB− ) − 2 z K ρρ(Rz,) = 4B sech e , approximating an isothermal disk with constant z0
scale height z0 . Setting scale height zH0 = 8 , the integral H zH 22 4sech2 dzz= 8 tanh= H tanh( 4) = 0.9993H. With constant disk ∫0 0 zz00 2
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height H encompassing 99.9% of disk matter, disk mass is approximately
(RB−−) (SB) S −− M=2π HρρeKK RR d2=π H K( K +−+ B) ( K S)e DS B ∫B B
and disk angular momentum is approximately
(RB− ) S − J= 2π Hρ VedK RR2 DS B ∫B (SB− ) − 2 22 2K =2πHρB KV(22 K+ KB +− B) ( 22 K + KS + S ) e
For B= fSB , Sf(1− ) − B K (K+ fSB ) −+( K S)e VM DS = Sf(1− B ) J Ds − 2 22 2 2 K (22K+ KfBB S + f S) −( 22 K ++ KS S ) e
Setting radial scale length K= xS , the following equation can be solved for x by trial and error
(1− f ) − B x ( xf+B ) −+( x1e) SVM DS = (1− fB ) J Ds − 2 22 x (22xxff+BB +) −( 221e xx ++)
Given x, HB2 is found from H (MB2 ) = DS (1− f ) 2B − B 2 x 2πρBBSx( x+ f) −+( x 1e)
If sufficient data are available, the model can be extended to consider situations where different stellar populations are in sub-disks with different radial scale lengths and scale heights This is relevant because analysis of the Milky Way [5] found different star populations in sub-disks with scale height inversely related to sub-disk radial scale lengths.
6. Percentage of Bulgeless Spiral Galaxies
Kormendy et al. [6] note that bulgeless galaxies with V >150 km sec 44 (M g >×2 10 g) “challenge our picture of galaxy formation by hierarchical
clustering.” In this analysis, spiral galaxies with mass M g have bulge radii * between C and Bmax . If bulge radius B approximates the demarcation BC* − between spirals with and without a bulge, is the bulgeless fraction of BCmax − spiral galaxy configurations consistent with mass and angular momentum H conservation. Estimating B* from =1, 6% to 10% of galaxies with mass 2B*
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ranging from 1× 1044 g (with V =169 km sec ) to 1× 1045 g (with
V = 300 km sec ) are bulgeless. With scale height zH0 = 8 , the integral H zH 24 4sech2 dzz= 8 tanh= H tanh( 2) = 0.964H, so a central subdisk ∫0 0 zz00 4 of height H/2 contains 96.4% of disk mass. Estimating B* from H/2 = 2B*, 19% to 23% of galaxies with masses between 1× 1044 g and 1× 1045 g are bulgeless. These estimates bracket the 15% estimate of bulgeless galaxies found among 15,127 edge-on disk galaxies in the sixth release of SDSS data [7]. So, holographic analysis has no difficulty accounting for bulgeless giant spiral galaxies.
7. MOND Is Unnecessary
Dark matter can account for flat velocity curves in spiral galaxies, and observations of the colliding “bullet cluster” galaxies 1E0657-558 provide further evidence for dark matter. To avoid using dark matter to account for flat velocity curves, Modified Newtonian Dynamics (MOND) assumes the law of gravity is different at large distances. Ref. [8] cites the baryonic Tully-Fisher relation and
mass discrepancy-acceleration (or VVobserved Newtonian ) relation as “challenges for the ΛCDM model,” and accounts for those relations using MOND with an −82 acceleration threshold a0 ≈×1.2 10 cm sec . However, the HLSS model [1] within the ΛCDM paradigm readily accounts for the MOND acceleration threshold, baryonic Tully-Fisher relation, and mass discrepancy-acceleration
(VVobserved Newtonian ) relation. First, radial acceleration at holographic radius S is GM GS(0.183 2 ) aG=g = =0.183 = 1.2 × 10−82 cm sec , equal to MOND S SS22 acceleration. Second, radial acceleration at radius R due to dark matter is GR aMDM = DM and at R sufficiently distant from the galactic center that R2 S
total baryonic mass of the galaxy MMBg= 0.16 can be treated as concentrated at the galactic center, Newtonian radial acceleration from baryonic matter is
GM B a = . Radius Rγ where aa= is found from B R2 DM B
G Rγ GM = B = = 22M DM . Since MMDM 0.84 g , RSγ 0.19 , and at that radius RRγγS −82 aaDM= B =5.4 × 10 cm sec , near the MOND estimate. Third, tangential 2 velocity V at R relates to radial acceleration ar by V= Rar . So, 2 V Ra( + a) V a RM observed ≈ DM B , and observed =+=+11DM DM . When VNewtonian RaB VNewtonian aBB SM 2 Vobserved a0 M DM R R 0.19M B a0 RS 0.19 , ≈1. Using = , = VNewtonian aBB0.19 MS SMDM a B
Vaobserved RM DM 0.19M DM 0 a0 and =+=+11 . When aaB 0 , 1 VNewtonian SM B MBB a aB
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1 4 4 Vobserved aao 0 Vaobserved 0 and =+≈1 0.998 . So, when aaB 0 , = VNewtonian aaBB VaNewtonian B GM GM and using a = B and V = B , B R2 Newtonian R2 2 GM R2 4 = B = Vobserved 2 a0 GMB a0 , known as the baryonic Tully-Fisher R GM B −−11 relation. Fourth, if H0 =67.8 km ⋅⋅ sec Mpc , the cosmological constant −−56 2 −82 Λ=1.12 × 10 cm , and accelerations cH0 =6.6 × 10 cm sec and Λ c2=5.5 × 10− 82 cm sec are consistent with the acceleration 3 −82 aaDM= B =5.4 × 10 cm sec . So, the MOND hypothesis is unnecessary.
8. Elliptical Galaxies
Elliptical galaxies predominate in overdense regions of the universe, and may result from mergers and collisions of progenitor galaxies. Baryonic matter interacts more strongly than dark matter, so collisions between galaxies can produce torques transferring angular momentum from baryonic matter to the composite halo, yielding baryonic ellipsoids with angular velocity less than the angular velocity of the halo and the galaxy as a whole. This analysis treats elliptical galaxies as constant density ellipsoids within a surrounding spherical isothermal dark matter halo. Dynamical instability of thin galaxies (the firehose instability) limits long axes of ellipsoids to about three times the length of short axes. So elliptical galaxies in this model range from oblate spheroids with radius A along the spin axis equal to one third the radius
R⊥ perpendicular to the spin axis to prolate spheroids with radius A along the
spin axis equal to three times the radius R⊥ perpendicular to the spin axis. In the holographic approach, galaxies are arrangements of bits of information identifying the location of matter in the galaxy. There is no reason for one arrangement of information satisfying conservation of mass and angular momentum to be preferred over another. So, different configurations satisfying conservation of mass and angular momentum with the same total mass should be equally likely. Baryonic ellipsoids outside the core radius C surrounding the central black
hole have baryonic mass ME =0.16( MMg − CBH ) and, since SC and
2 2 RC , the moment of inertia around the spin axis I≈ MR⊥ . Prolate and ⊥ EE5 oblate spheroidal ellipsoids are analyzed as deformed versions of spheres with S radius R = , so the spheroids fit within the holographic screen. Prolate and o 3
oblate spheroids with radius ARo along the spin axis have the same mass and
angular momentum as spheres with radius RR= o . For prolate spheroids capable of avoiding firehose instability, radius along the spin axis ranges from
Ro to 3Ro , ( A =13 → ) and for corresponding oblate spheroids it ranges
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1 from Ro 3 to Ro , A = →1 . Ellipsoid angular velocity depends on how 3 much baryonic angular momentum was transferred to the halo during elliptical galaxy formation. Hubble class of prolate spheroidal elliptical galaxies with major axes A varying B from Ro to 3Ro and minor axis BR= o is E =10 1 − rounded to the A nearest integer, so limits imposed by firehose instability only allow prolate spheroidal elliptical galaxies of Hubble class E0 to E7. We only see projections of ellipsoids on the plane perpendicular to our line of sight, so actual Hubble class AEi of an elliptical galaxy and its projected (observed) Hubble class Ei are not the same. Prolate elliptical galaxies rotate around their major axis and oblate elliptical galaxies rotate around their minor axis, so results hold for oblate galaxies when 1 A is replaced by A, and the projection effect can be
understood by considering only prolate spheroids. Table 1 shows ARio major axis lengths marking transitions between Hubble classes for prolate spheroids,
where Ai is A projected on the plane perpendicular to the line of sight. When the angle between rotation axis of prolate elliptical galaxies and the line of sight decreases from 90˚ to 0˚, the projection PA(θθ) = sin on the plane perpendicular to the line of sight of the actual A along the axis of rotation gets smaller, so projected shape of the galaxy gets rounder and higher actual Hubble classes appear as lower projected Hubble classes as the axis of rotation gets closer to the line of sight. Hubble class distribution of prolate spheroid shapes projected on the plane perpendicular to the line of sight is found by integrating π from A =1 to A = 3 and from θ = 0 to θ = . As above, this analysis 2 assumes information (matter), in prolate elliptical galaxies capable of avoiding firehose instability with a given mass and angular momentum and corresponding
radius Ro perpendicular to the rotation axis, is evenly distributed among
prolate spheroids with radius a= ARo along the spin axis ranging from Ro to
3Ro ( A =13 → ) . Total occupancy of prolate elliptical galaxies with given mass
1 A8 =3 and angular momentum is ∫ dA =1. The interval AE0 from A0 to A1 2 A0 =1
1 A1 contributes ∫ dA = 0.026 to projected Hubble class E0. The interval AE1 2 A0=1
Table 1. Major axis lengths Ai marking transitions between Hubble classes for prolate spheroids.
Hubble class E0-E1 E1-E2 E2-E3 E3-E4 E4-E5 E5-E6 E6-E7 change
1 10 1− 0.5 1.5 2.5 3.5 4.5 5.5 6.5 Ai
Transition A1 = 1.05 A2 = 1.18 A3 = 1.33 A4 = 1.54 A5 = 1.82 A6 = 2.22 A7 = 2.86 Ai
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Table 2. Projected Hubble class Ei of elliptical galaxies versus actual Hubble class AEi of elliptical galaxies.
E0 E1 E2 E3 E4 E5 E6 E7 Total
AE7 0.005 0.001 0.002 0.003 0.005 0.009 0.032 0.015 0.071
AE6 0.029 0.008 0.011 0.018 0.032 0.076 0.143 0 0.317
AE5 0.030 0.008 0.013 0.021 0.048 0.082 0 0 0.202 AE4 0.031 0.009 0.015 0.032 0.052 0 0 0 0.140 AE3 0.033 0.011 0.023 0.036 0 0 0 0 0.103 AE2 0.036 0.017 0.026 0 0 0 0 0 0.078 AE1 0.043 0.019 0 0 0 0 0 0 0.062 AE0 0.026 0 0 0 0 0 0 0 0.026 Projected 0.233 0.074 0.089 0.110 0.137 0.168 0.174 0.015
1 A2 = from A1 to A2 contributes ∫ dA 0.062 to projected Hubble classes E0 2 A1 1 AA21θ ( ) π 2 and E1, broken down as ∫∫dA sinθθ d+ ∫ sin θθ d where 2 AA110 θ ( )
−1 Ai θi ( A) = sin , the first term in the integral contributes to projected Hubble A class E0, and the second term contributes to projected Hubble class E1.
Similarly, the interval AEi from Ai to A11+ contributes to all projected Hubble classes from E0 to Ei, broken down as
1 AAij++11θθ( ) i−1 1( A) π 2 θθ++θθ θθ ∫∫dA sin d ∑ ji= ∫sin d ∫ sin d where the first term 2 Ai0 θθji( AA) ( ) contributes to projected Hubble class E0, the last term contributes to projected Hubble class Ei, and intervening terms contribute to projected Hubble classes
θ j+1( A) from E1 to Ei( −1) . Use sinθθ d=( cos θjj( AA) − cos θ+1 ( )) and ∫θJ ( A)
2 − AAjj cos sin 1 =1 − to get AA
θ j+1( A) 1 θθ=22 −− 22 − ∫ d sin AAjj AA+1 , and θ j ( A) A ( ) 1 22 22− 1 ξ dAA −=ξ A −+ ξξtan . This results in Table 2, ∫ 22 A A −ξ listing fractional distribution of projected Hubble class of elliptical galaxies. The result suggests Hubble class E7 galaxies comprise < 2% of elliptical galaxies.
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https://doi.org/10.1103/RevModPhys.74.825 [4] Wesson, P. (1981) Physical Review D, 23, 1730. https://doi.org/10.1103/PhysRevD.23.1730 [5] Bovy, J., et al. (2012) The Astrophysical Journal, 753, 148. [6] Kormendy, J., et al. (2010) The Astrophysical Journal, 723, 54. [7] Kautsch, S.J. (2009) Astronomische Nachrichten, 330, 100-106. [8] Famaey, B. and Mc Gaugh, S. (2012) Living Reviews in Relativity, 15, 10.
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