FIXING by THINKING: The power of dimensional analysis

1. FIXING by THINKING:The power of dimensional analysis E.N. Economou Dept. of Physics, U of C FORTH March, 2006 In the atomic idea “there is an enormous amount of information about the world, if just a little imagination and thinking are applied”. R.P.Feynman FORTH, E.N. Economou

2. ATOMIC NATURE OF MATTER (& FORCES) • WAVE/ PARTICLE DUALITYQUANTUM MECHANICS (a) Heisenberg Principle: (b) Pauli Principle: (c) “Schrödinger” Principle: • EQUILIBRIUM STRUCTURES  MINIMIZATION OF (FREE) ENERGY (M.E) or or

3. NEEDED ALSO: Strong force: range 1 – 4 fm, strength 1 – 100 MeV Weak force: , Em force: Gravity: AND THE NUMERICAL VALUES OF , , , , ,

4. Mass of proton & neutron is the kinetic energy of the quarks ; Exp. 938.27 Exp. 939.57

5. Nuclei: A=N+Z • Strong Interactions: • Coulomb Interactions: • Kinetic Energy:

6. FROM ATOMS TO ASTEROIDS THE POTENTIAL ENERGY IS CHARACTERISED BY • THE KINETIC ENERGY IS CHARACTERISED BY AND • INSTEAD OF USE • OTHERS:

7. Atoms , , • fα increases as we move down the columns of the P.T.E. • fαlocal minimum for completed p and s/d orbitals • fα local maximum for ,

8. Local Maxima for completed p, s/d orbitals • Local Minima for completed p, s/d orbitals plus 1

9. Molecules exception: noble gases , ,

10. Solids , , ,

11. Solids ,

12. Comparison with experimental data Fe Al Cu Si Exp. Theory Exp. Theory Exp. Theory Exp. Theory 2,67 2,99 2,67 3,18 (gr/cm3) 7,86 7,92 2,79 2,73 8,96 9,01 2,33 2,36 (eV/atom) 4,28 3,82 3,39 3,04 3,49 3,82 4,63 2,69 B () 1,68 1,29 0,722 0,73 1,37 1,29 0,998 0,54 (Κm/s) 4,63 4,11 5,68 5,28 3,93 3,85 6,48 4,87

13. FLUIDS • Sea waves ,

14. exp: 0.073 J/m2

15. 102 101 100 0.232 10-1 0.84 km/h 10-3 10-2 10-1 100 101 102 103 104 105 TSUNAMI WINDINDUCED λ(m) 1.7 cm

16. FLUIDS • Drag force , , LARGE BODIES, HIGH SPEED , , SMALL BODIES, LOW SPEED Reynolds number

17. exp:

18. Black Body Radiation

19. Typical Planet: EGRAVITATIONAL ΕELECTRIC

20. Why Earth is round? • Why are there mountains? • What is the largest possible height of a mountain in a planet? When the shear stress exceeds the critical value ,

21. Planets Jupital: Pluto:

22. STARS • MINIMUM NUMBER OF NUCLEONS: • MAXIMUM NUMBER OF NUCLEONS: • NUMBER OF NUCLEONS IN OUR SUN: • MAXIMUM NUMBER OF NUCLEONS IN A WHITE DWARF: ,

23. STARS Μmin T Tmax Tign 0 R

24. STARS Μmax For large mass, T becomes large, Eph dominates over Ekin then

25. Black Holes 1. 2. 3. 4. years

26. Universe • Homogeneous & Isotropic • Expanding according to Hubble’s law: • Eucledian geometry (as a result of inflation) • 1st LAW (for dS0): dU=d(εV)=-pdV • Three unknown functions:R(t), ε(t), p(t) (1) (2)

27. OBSERVATIONS WMAP Wilkinson Microwave Anisotropy Probe

30. the equivalent of 6 protons per m3 • This density equals to the critical one with an uncertainty of 2% • protons per m3, i.e. about 4.2% of ε The Rest • DARK MATTER • DARK ENERGY

31. , • accelerated expansion • today for for

32. BB q→B P,n→nuclei εph→εnucl decoupling Protostars Galaxies Today 10-4s 1s 70 ky 380 ky 180 My 500 My 8 Gy 13.7 Gy t