Gravitational structures!

Gravitational structures! Macroscopic Schrödinger equation! -Observational tests-! Laurent Nottale CNRS LUTH. Paris-Meudon Observatory http://www.luth.obspm.fr/~luthier/nottale/ 1

Solar System : inner and outer systems 7 49 55 UA 6 P 36 a (obs.) 5 4 3 2 1 SE New predictions (at that time) 0.043 UA/M sol 0.17 UA/M sol SI N J S m U V N T M Hun 1 2 3 4 5 6 7 8 9 rank n Ref: LN 1993, Fractal space-time and microphysics (World Scientific) Chap. 7.2 C H 10 Hil 25 16 9 4 1 a (AU) 2

Solar system: masses of planets Log(planet mass/earth mass) Log(planet mass / earth mass) 3 2 1 0-1 -2-3 1 0 J Outer Solar System S U N IS P 10 20 30 40 distance (A.U.) V E Inner Solar System m M C H 0.5 1 1.5 2 2.5 3 3.5 distance (A.U.) density 0.5 0.4 0.3 0.2 0.1 Inner Solar System n = 1 1 m V T M n = 1 0 Outer Solar System n = 2 1 Jupiter n = 3 1 Saturn n = 4 1 n = 5 1 5 10 15 20 25 30 distance to Sun (astronomical units) Ref: Nottale, Schumacher & Gay 97 Uranus Hierarchical model of Solar System formation 3 Neptune

Solar System: Sun, solar cycle Wave function: Fundamental period: On the surface of the Sun: (equator) If the Sun had kept its initial rotation: τ would then be the Kepler period, (Pecker Schatzman) But, like all stars of solar-type, the Sun has been subjected to an important loss of angular momentum since its formation (cf. Schatzman & Praderie, The Stars, Springer) Result: Observed period:11 ans Ref: LN, Proceedings of CASYS 03, AIP Conf. Proc. 718, 68 (2004) 4

Intramercurial solar system: transient IR dust rings Base: 216 km/s, n=2 144 km/s, 0.043 UA, n=3 51 Peg-like exoplanets Observations: Peterson 67, MacQueen 68, Koutchmy 72, Lena et al. 74, Mizutani 84 Ref: Da Rocha Nottale 03 5

Intramercurial solar system: «sungrazer» comets Ref: Schumacher & Nottale 02 6

Outer solar system: Kuiper belt (SKBOs) 1 2 3 4 5 6 Rank n 7 8 9 10 SKBO 10 8 Earth (inner system) Jupiter Saturn Uranus Neptune Pluton + KBO 6 4 2 Number 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Semi-major axis (A.U.) Ref: Da Rocha Nottale 03 7

Outer Solar System: Kuiper belt (SKBOs) Validation of predicted probability peak at 55 AU 1 2 3 4 5 6 Rank n 7 8 9 10 SKBO 10 2003 UB 313 («Eris»): validation 8 of second peak Earth (inner system) Jupiter Saturn Uranus Neptune Pluton + KBO 6 4 2 Number 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Semi-major axis (A.U.) Ref: Da Rocha Nottale 03 8

Orbit of Sedna http://www.spitzer.caltech.edu/ 9

Comparison of Kuiper belt dwarf planets and their moons to Earth (2006) 10

Distant outer solar system: SEDNA scattered Kuiper belt objects (SKBOs) Number 2001 FP185 SEDNA 2003 VB12 SKBOs (a / 55 AU) 1/2 11

New planet:sedna n ex =7 Number SKBOs 2001 FP 185 Sedna 2003 VB 12 ( a / 57 UA ) 1/2 Predicted, AU (57) 228 513 912 1425 2052 Observed 57 227 509 12

2002 GB 32: a=219 2000 CR 105: a=222 2001 FP 185: a=228 2003 VB 12: a=495 2000 OO 67: a=517 2006 SQ372: a=915 Sedna + distant SKBOs n ex =7 Number SKBOs 2001 FP 185 Sedna 2003 VB 12 2006 SQ372 Confirmation: 2 new objects in n=2, 1 new object in n=3 n dss =( a / 57 UA ) 1/2 Predicted, AU (57) 228 513 912 1425 2052 Observed 57 227 509 915 13

Sedna + distant SKBOs, 2008! 14

Solar system: obliquities and inclinations of planets and satellites 7 0 1 2 3 4 5 6 7 Number 6 5 4 3 2 1 25 50 75 100 125 150 175 obliquity / inclination (degree) Schrödinger equation for rotational motion of solids: Probability peaks for: Solar system: n = 7 15 Ref: LN 97, 98

Solar System: space debris around Earth 16

Solar system: space debris around Earth (mean) Base: 144 km/s 17 Ref: Da Rocha & Nottale 2003

Solar system: space debris around Earth. 2004 data. Data 2-Jan-2004 (US Space Surveillance Network Two Line Element, 13th edition of breakup Book) 18

Hierarchical structure of the Solar System 19

Extrasolar planetary systems: semi-major axes. Data 2003. over star mass 89 planets such that σ n < 0.25 20 Refs: LN 93, 96, LN Schumacher & Lefevre 00, LN &TranMinh 02, Da Rocha LN 03, LN & Galopeau 03, 04

Extrasolar planetary systems: semi-major axes. Data 2003. 89 planets such that σ n < 0.25: 61/89 in the interval [-0.25,+0.25] Probability: P = 2.8 10-4 21

Extra-solarplanetary systems Velocities Earth Mars Venus Mercury 22

Exoplanets: «intramercurial» sub-system OGLE candidates Ref: Nottale, Galopeau, Ceccolini, Da Rocha, Schumacher, Tran-Minh 2003 23

Exoplanets at 0.02 AU/M sol Validation of a theoretical prediction of a new sub-structure with base w 0 = 432 km/s (Keplerian velocity at radius of solar-type stars) Ref: Nottale, Schumacher et Lefèvre, 2000, A&A 361, 379 n 432 = 2 a/m = 0.02 UA/M sol OGLE-TR-56 Candidats OGLE HD 330075: first detection HARPS (Mayor et al) 24 OGLE-TR-132:confirmed exoplanet (Bouchy et al)

Extra-solar planets: eccentricities 83 planets such that σ k < 0.5: Probability: P = 3 10-5 Ref: LN, Galopeau, Ceccolini, Da Rocha, Schumacher, Tran-Minh 2003 25

a and e Base 144 km/s 78 planets such that σ n < 0.25 and σ k < 0.50 : 42 on 78 in the central square [-0.25,+0.25]. Probability: P = 4.8 10-8 26

Extrasolar systems: semi-major axes, history Nombre Mercury Venus Earth Mars Flora Ceres Cybele Animation, all exoplanets (18-02-2003) 27

Extra-solar planetary systems: semi-major axes 28 Animation, exoplanets such that σ n < 0.25, chronologic order of discovery ( up to 18-02-2003)

Exoplanets 27-10-04 Semi-major axes: Number w 0 =145 km/s 108 planets such that σ n < 0.25 (P 10-4 ) Eccentricities: 100 planets such that σ k < 0.50 Nombre (P 10-7 ) k = e n 29

Exoplanets 27-10-04 Semimajor axes and eccentricities. Combined analysis. Histogram: delta k delta n w 0 =145 km/s 93 planets such that σ n < 0.25 et σ k < 0.50 (P 10-9 ) 30

Exoplanets 17-02-05 Semimajor axes: Number w 0 =145 km/s 126 planets such that σ n < 0.25 (P = 10-4 ) Eccentricities: 122 planets such that σ k < 0.50 Number (P 5 10-7 ) k = e n 31

Exoplanets 17-02-05 Mercure Venus Terre Mars Ceres Semimajor axes: Number w 0 =145 km/s 126 planets such that σ n < 0.25 (P = 10-4 ) Eccentricities: 122 planets such that σ k < 0.50 (P 5 10-7 ) Number k = e n 32

Exoplanets 17-02-05 Semimajor axes and eccentricities. Combined analysis. Histogram: delta k delta n w 0 =144 km/s 113 planets such that σ n < 0.25 et σ k < 0.50 (P 10-6 ) 33

Exoplanets 15-06-05 34

Exoplanets 15-06-05 Semimajor axes w 0 =144 km/s: 127 planets such that σ n < 0.25 (P = 10-4 ) 35

Exoplanets 15-06-05 Semimajor axes 36

Exoplanets (data 2008, N=301) (main peak cut) Predicted probability peaks Number Proba = 5 x10-7 (P / M * )^(1/3) 37

Exoplanets (data 2008, N=301) N Mercury Venus Earth Mars Ceres Hygeia 1! 3! 5! 7! 9! Predicted (1993) fundamental level, 0.043 AU/ M sol 38

Exoplanets: power spectrum analysis 276 exoplanets between n = 0.7 and n = 8.7 (2008 data) Peak at k=8 --> period Δn = 1, p=16, Proba = 1.13x10-7 :>5 sigmas! 39

Planets around the pulsar PSR B1257+12! 40

PSR B1257+12! *Discovered by Wolszczan (1990): P rot =6.218 531 938 028 3(2) ms *Firts extrasolar planets (Wolszczan & Frail 1992) (B: 3.4 M T, P=66.5 j; C: 2.8 M T, P=98.2 j) *Confirmation: mutual gravitational perturbation of planets B and C (Wolszczan 1994) *Third planet (Wolszczan 1994), ~moon mass (A: 0.015 M T, P=25.3 j) *Confirmation 3rd planet (Wolszczan et al 2000) 41

PSR B1257+12 planets: " orbital elements! Orbital Elements Projected semimajor axis A B C 0.0000033(2) 0.0013106(2) 0.0014134(2) Eccentricity 0 0.0183(3) 0.0264(3) Orbital Period (d) 25.3144(100) 66.5352(3) 98.2228(5) Wolszczan et al. 2000 ApJ 528, 907 42

Analyse.1.! (i) Since, from Kepler's third law, a 1/2 = M 1/6 T 1/3 a 0 1/2 (n + 1/4) (in Solar System units, resp. AU, M sol and yrs), we expect the T 1/3 differences between the three planets to be quantized in terms of integers (up to very small differences). This first prediction is very well verified, since (T B 1/3 T A 1/3 )/(T C 1/3 T B 1/3 ) = 1.984, (12) that differs from Δn = 2 by only 0.016. This implies that the three planets must rank as n A = n B 2, n B, n C = n B + 1. We have assumed for simplicity n C n B = 1, but any other choice would be equivalent (it would multiply k by an integer). (ii) We thus obtain the two following equations: T A 2/3 / T C 2/3 = [ n B 2 (7/2) n B + 3 ] / [ n B 2 + (5/2) n B + (3/2) ] (13A) T B 2/3 / T C 2/3 = [ n B 2 + (1/2) n B ] / [ n B 2 + (5/2) n B + (3/2) ] (13B) Considering period ratios allows us to eliminate the unknown pulsar mass. We find, respectively from Eqs. 13A and 13B: n B (A,C) = 7.016 ; n B (B,C) = 6.971. (14) The quantization is once again confirmed with a remarkable precision, yielding n A = 5, n B = 7 and n C = 8 (see Figure). 43

PSR1257+12: Comparison predictions-observations! 0.6 9 0.5 C.472.474.474.473 8.472. a (observed, AU) 0.4 0.3 0.2 0.1 A 4 5 B 6 7.366.365.364.364.366.192.191.190.190.192 x 50 3 1 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 a (predicted, AU) Nottale L., 1996, A&A Lett. 315, L9 1998, CSF, 9, 1043 (http://wwwusr.obspm.fr/~nottale/arpsr.pdf) 0.7 44

Analyse.2.! (iii) We can go even farther: The precision reached by this system is so good that we can expect to be able to make the difference between the 'mean' and 'peak' formulae (this means to test for second order terms in the theory). We find: (T A 1/3 /T C 1/3 ) = 0.63666 while (5 2 + 5/2) / (8 2 + 8/2) = 0.63593 and (5/8) = 0.62 (T B 1/3 /T C 1/3 ) =0.87827 while (7 2 + 7/2) / (8 2 + 8/2) = 0.87866 and (7/8) = 0.875. Even though the agreement with the 'peak' formula could already be considered as excellent (relative differences of resp. 0.018 and 0.004), the agreement with our more precise expectation, the 'mean' formula, is more than ten times better (resp. 0.0011 and 0.0004). The pulsar mass is given by: M = (w 13 /2πG) (T n /(n 2 + n/2) 3/2 ). (16) Assuming a standard neutron star mass of 1.4 ± 0.1 M sol, we obtain a confirmation of the value of the structuration constant: k = (w 1 /144) = 2.96 ± 0.07, (17) within 0.04 of the quantized value k = 3. We can now compute the pulsar mass. From w 0 = 144.7 ± 0.6 km/s, and taking k = 3 strictly, we obtain: M PSR = 1.48 ± 0.02 M sol. 45

Analyse.3.! Estimate of the probability to get such an observed configuration by chance: *The observed ratios (T A /T C ) 1/3 and (T B /T C ) 1/3 fall within ±7.3 10 4 and ±3.9 10 4 of two of the predicted quantized ratios. *The number of possible configurations is (C nm3 ) = n m (n m 1)(n m 2) / 6, where n m is the maximal reasonable value for the quantum number n. *Taking n m = 10 yields a probability P =( C 103 ) x 1.46 10 3 x 7.8 10 4 = 1.4 10 4. *Also accounting for the fact that k falls within 0.1 of an integer using standard neutron star masses, we find a highly significant total probability P = 3 x 10 5. Even with the too high and not adopted) choice n m = 20, one would still get the significant result P = 3 x 10 4. 46

Extrasolar planetary system:" PSR B1257+12! Period (days) 0 10 20 30 40 50 60 70 80 90 100 110 1 2 3 4 5 6 7 8 A B C 24 25 26 66 67 97 98 99 days days days Data: Wolszczan 94, 00 M psr =1.4 ± 0.1 M sol --> w = (2.96 ± 0.07) x 144 km/s, i.e. 432 km/s = Keplerian velocity for R sol Proba < 10-5 of obtaining such an agreement by chance Prediction of other orbits: P 1 =0.322 j, P 2 =1.958 j, P 3 =5.96 j Residuals in Wolszczan s data 00: P = 2.2 j (2.7 σ) Refs: Nottale 96, 98, Da Rocha & Nottale 03 47

Prediction of most probable periods of other possible bodies around PSR B1257+12! (mean, gravity center conservation ) Rank n (P n /P 8 ) pred P n (d) pred P n (d) obs P 1/3 pred = a 1/2 P 1/3 obs = a 1/2 1 0.0033 0.322 0.1485 2 0.0199 1.958 2.2? 0.2712 0.28? 3 0.0607 5.960 0.3930 4 0.1362 13.38 0.5145 5 0.2572 25.26 25.34 0.6359 0.6366 6 0.4343 42.66 0.7573 7 0.6784 66.63 66.54 0.8787 0.8783 8 (1) (98.22) 98.22 1 1 9 1.4099 138.48 1.1213 10 1.9188 188.46 1.2426 48

Research of possible other planets! n=6 n=5 n=4 Wolszczan et al, preprint (2000) Observations 430 MHz 1990-1994 (residuals after fit of B & C) 49

Research of fourth planet (close to neutron star)! Wolszczan et al 2000: dayly observations during one month (Arecibo, 1999) 430 MHz (whites dots), 1130 MHz (black dots) Residuals before and after fit of planet A 50

Analysis of residuals! 4 2 0-2 -4 0 1 2 3 4 5 Days PSR 1257+12, donnees Wolszczan 1999 Analyse residus post correction de A,B,C Meilleur fit: Periode: 2.22 jours Amplitude 1.1 microseconde Signification statistique: 2.7 sigma Periode predite: n=2: 1.96 jours Marginal: to be confirmed by new observations 51

Stars:" morphology of planetary nebulae! Spherical harmonics: angular distribution of probability density * Da Rocha D. & Nottale, L., 2003, Chaos, Solitons and Fractals, 16, 565 (arxiv : astroph/0310036) Gravitational structure formation in scale-relativity.! * Da Rocha D. & Nottale, L., 2003, arxiv : astro-ph/0310031 On the morphogenesis of stellar flows. Application to planetary nebulae.! 52

Stars:" morphology of planetary nebulae! Predicted morphologies: angles of maximal probability 53

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Stars: morphology of planetary nebulae! Comparison with observational data Da Rocha 2000, Da Rocha & Nottale 2003 56

Stars:" ejection and accretion! SN 1987A, deprojected angle : 41.2 ± 1.0 d predicted angle (l=4, m=2): 40.89 d Da Rocha D. & Nottale, L., 2003, arxiv : astro-ph/0310031 On the morphogenesis of stellar flows. Application to planetary nebulae. 57

Galactic structures" star systems, formation! n=0 n=1 n=2 n=2 (2,0,0) (1,1,0) 58

Galactic structures :" double stars! Number 140 120 100 80 60 40 20 0 289.4 km/s 100 200 300 400 500 600 700 v (km/s) <V>=289.4 ± 3.0= 2 x 144.7 km/s Catalogue of eclipsing binaries: Brancewicz et Dvorak 80 Fit by Laguerre polynomial (Kepler potential) a/m = (289.4/V) 2 Ref: Nottale & Schumacher 98, Da Rocha & Nottale 03 59

Galactic structures:" high velocity clouds! From Pietz et al. 96. Ref: Da Rocha & Nottale 03 60

Galactic structures :" proper motion near the galactic center! 20 15 Number 10 5 72 144 216 288 360 432 504 Velocity (km/s) Data: Eckart & Genzel 96. Ref.: Da Rocha & Nottale 03 61

Extragalactic structures:" rotation curves of spiral galaxies! 120 144 km/s 100 80 Number 60 40 20 0 0 50 100 150 200 250 300 350 Rotation Velocity (km/s) Fit by Hermite polynomials (harmonic oscillator potential, constant density): Peak velocity = 142 ± 2 km/s 967 spiral galaxies, catalogue of Persic & Salucci 95. Ref: Da Rocha & Nottale 03 62

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Extragalactic structures:" groups of galaxies, formation! n=0 n=1 n=2 n=2 (2,0,0) (1,1,0) 65

Extragalactic structures:" isolated galaxy pairs! 10 8 Galaxy pairs 144 km/s 72 km/s 24 km/s Number 6 4 2 0 0.2 0.4 0.6 0.8 1 exp (-v / 100) r Tricottet & Nottale 2001, Da Rocha & Nottale 2003 66

Extragalactic structures: local group of galaxies Position Velocities Da Rocha 2001 67

Extragalactic structures:" clusters of galaxies! 14 N A 576 12 10 8 6 4 2 10000 11000 12000 13000 14000 15000 Radial velocity N A 2634 12 10 N Coma Radial velocity 8 6 4 2 0 5500 6000 6500 7000 7500 8000 8500 9000 Radial velocity 68

Extragalactic structures:" local supercluster! Radial Velocity (km/s) 20 200 100 65 50 40 36 km/s 30 Power 15 10 432 km/s 5 0 20 40 60 80 100 Frequency Number 6 5 4 3 2 1 50 100 150 radial velocity difference (km/s) Analysis of data of Guthrie-Napier 96. Ref. Da Rocha & Nottale 03 69