Gravitational structures!

Transkript

1 Gravitational structures! Macroscopic Schrödinger equation! -Observational tests-! Laurent Nottale CNRS LUTH. Paris-Meudon Observatory 1

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

3 Solar system: masses of planets Log(planet mass/earth mass) Log(planet mass / earth mass) J Outer Solar System S U N IS P distance (A.U.) V E Inner Solar System m M C H distance (A.U.) density 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 = distance to Sun (astronomical units) Ref: Nottale, Schumacher & Gay 97 Uranus Hierarchical model of Solar System formation 3 Neptune

4 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

5 Intramercurial solar system: transient IR dust rings Base: 216 km/s, n=2 144 km/s, 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

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

7 Outer solar system: Kuiper belt (SKBOs) Rank n SKBO 10 8 Earth (inner system) Jupiter Saturn Uranus Neptune Pluton + KBO Number Semi-major axis (A.U.) Ref: Da Rocha Nottale 03 7

8 Outer Solar System: Kuiper belt (SKBOs) Validation of predicted probability peak at 55 AU Rank n SKBO UB 313 («Eris»): validation 8 of second peak Earth (inner system) Jupiter Saturn Uranus Neptune Pluton + KBO Number Semi-major axis (A.U.) Ref: Da Rocha Nottale 03 8

9 Orbit of Sedna 9

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

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

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

13 2002 GB 32: a= CR 105: a= FP 185: a= VB 12: a= OO 67: a= SQ372: a=915 Sedna + distant SKBOs n ex =7 Number SKBOs 2001 FP 185 Sedna 2003 VB SQ372 Confirmation: 2 new objects in n=2, 1 new object in n=3 n dss =( a / 57 UA ) 1/2 Predicted, AU (57) Observed

14 Sedna + distant SKBOs, 2008! 14

15 Solar system: obliquities and inclinations of planets and satellites Number obliquity / inclination (degree) Schrödinger equation for rotational motion of solids: Probability peaks for: Solar system: n = 7 15 Ref: LN 97, 98

16 Solar System: space debris around Earth 16

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

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

19 Hierarchical structure of the Solar System 19

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

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

22 Extra-solarplanetary systems Velocities Earth Mars Venus Mercury 22

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

24 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 : first detection HARPS (Mayor et al) 24 OGLE-TR-132:confirmed exoplanet (Bouchy et al)

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

26 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 =

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

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

29 Exoplanets 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

30 Exoplanets 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

31 Exoplanets 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 ) k = e n 31

32 Exoplanets 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 ) Number k = e n 32

33 Exoplanets 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

34 Exoplanets

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

36 Exoplanets Semimajor axes 36

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

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

39 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

40 Planets around the pulsar PSR B ! 40

41 PSR B ! *Discovered by Wolszczan (1990): P rot = (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: M T, P=25.3 j) *Confirmation 3rd planet (Wolszczan et al 2000) 41

42 PSR B planets: " orbital elements! Orbital Elements Projected semimajor axis A B C (2) (2) (2) Eccentricity (3) (3) Orbital Period (d) (100) (3) (5) Wolszczan et al ApJ 528,

43 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 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) = ; n B (B,C) = (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

44 PSR : Comparison predictions-observations! C a (observed, AU) A 4 5 B x a (predicted, AU) Nottale L., 1996, A&A Lett. 315, L9 1998, CSF, 9, 1043 (

45 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 ) = while ( /2) / ( /2) = and (5/8) = 0.62 (T B 1/3 /T C 1/3 ) = while ( /2) / ( /2) = and (7/8) = Even though the agreement with the 'peak' formula could already be considered as excellent (relative differences of resp and 0.004), the agreement with our more precise expectation, the 'mean' formula, is more than ten times better (resp and ). 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 = ± 0.6 km/s, and taking k = 3 strictly, we obtain: M PSR = 1.48 ± 0.02 M sol. 45

46 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 ± and ± 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 x = *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 Even with the too high and not adopted) choice n m = 20, one would still get the significant result P = 3 x

47 Extrasolar planetary system:" PSR B ! Period (days) A B C 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

48 Prediction of most probable periods of other possible bodies around PSR B ! (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/ ? ? (1) (98.22)

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

50 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

51 Analysis of residuals! Days PSR , 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

52 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/ ) Gravitational structure formation in scale-relativity.! * Da Rocha D. & Nottale, L., 2003, arxiv : astro-ph/ On the morphogenesis of stellar flows. Application to planetary nebulae.! 52

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

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

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

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

59 Galactic structures :" double stars! Number km/s v (km/s) <V>=289.4 ± 3.0= 2 x 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

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

61 Galactic structures :" proper motion near the galactic center! Number Velocity (km/s) Data: Eckart & Genzel 96. Ref.: Da Rocha & Nottale 03 61

62 Extragalactic structures:" rotation curves of spiral galaxies! km/s Number 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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65 Extragalactic structures:" groups of galaxies, formation! n=0 n=1 n=2 n=2 (2,0,0) (1,1,0) 65

66 Extragalactic structures:" isolated galaxy pairs! 10 8 Galaxy pairs 144 km/s 72 km/s 24 km/s Number exp (-v / 100) r Tricottet & Nottale 2001, Da Rocha & Nottale

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

68 Extragalactic structures:" clusters of galaxies! 14 N A Radial velocity N A N Coma Radial velocity Radial velocity 68

69 Extragalactic structures:" local supercluster! Radial Velocity (km/s) km/s 30 Power km/s Frequency Number radial velocity difference (km/s) Analysis of data of Guthrie-Napier 96. Ref. Da Rocha & Nottale 03 69