FORTH, E.N. Economou

FIXING by THINKING:FIXING by THINKING:The power of dimensional The power of dimensional

analysisanalysis

E.N. EconomouE.N. Economou

Dept. of Physics, U of CDept. of Physics, U of C

FORTHFORTH

March, March, 20020066

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

1.1. ATOMIC NATURE OF MATTER (& FORCES)ATOMIC NATURE OF MATTER (& FORCES)

2.2. WAVE/ PARTICLE DUALITYWAVE/ PARTICLE DUALITYQUANTUM QUANTUM MECHANICSMECHANICS

(a) Heisenberg Principle:(a) Heisenberg Principle:

(b) Pauli Principle:(b) Pauli Principle:

(c) “Schrödinger” Principle:(c) “Schrödinger” Principle:

3.3. EQUILIBRIUM STRUCTURES EQUILIBRIUM STRUCTURES MINIMIZATION MINIMIZATION OF (FREE) ENERGY (M.EOF (FREE) ENERGY (M.E))

2

x kin 2/3x p E 4.87

2 mV

2 2/3

kin,t 2 /3

V NV E 2.87N

N / 2 mV

2

1 GROUND 2/3mV

kin 1/ 3

cE 3

V

t

1/ 3

kin,t 1/ 3

cNE 2.32N

V

oror

oror

NEEDED ALSO:Strong force: range 1 – 4 fm, strength 1 –

100 MeV

Weak force: ,

Em force:

Gravity:

AND THE NUMERICAL VALUES OF , , , , ,

en p e ep n e

2 /e r

2 /Gm r

em e c G 0.8pr fm

Mass of proton & neutron is Mass of proton & neutron is the the

kinetic energy of the quarkskinetic energy of the quarks2 2 2

1/3

32 2p u d

cm c m c m c

V

2 2 21/3

32 2n u d

cm c m c m c

V

34

3 pV r

0.8pr fm;

2 917 13 3pm c Exp. 938.27

2 917 14 5nm c Exp. 939.57

Nuclei: A=N+ZNuclei: A=N+Z Strong Interactions: Strong Interactions:

Coulomb Interactions:Coulomb Interactions:

Kinetic Energy:Kinetic Energy:

v nn s s nn s v s nn nn nn1 1 2

A A V A A΄ V ; A A A , A΄ , A 82 2 3

2 2 2/3s 0 sA 4 R / α , V 11 MeV

21 e 3

Z Z 12 R 5

2/3 2/3 2

22/3

Z NZ N 2.87 ΄́

2 2 mV

22 2 2/3nucleus p n 1/3

e / e1

: Zm c Nm c 0 e / 0

0 / 0

3/ 4

3415.5, 16.8, 0.72, 23,

FROM ATOMS TO ASTEROIDS THE FROM ATOMS TO ASTEROIDS THE POTENTIAL ENERGY IS CHARACTERISED BYPOTENTIAL ENERGY IS CHARACTERISED BY

THE KINETIC ENERGY IS CHARACTERISED THE KINETIC ENERGY IS CHARACTERISED BYBY

ANDAND

INSTEAD OF USEINSTEAD OF USE

OTHERS:OTHERS:

e

em

αα

e 0 0 0

m c P Tm ,c,P,T,... , , , ,...

m P T

ee, ,m

2 2

0 0 05 2e Be e

1, P ,

m α km α m α

2

e B 2e

,m ,αm e

αα r f α

AtomsAtoms

αr f α α0.6 f 5d d,

2 2α

αα B

e e

r f α

, α0.5 η 1d d

ffαα increases as we move down the columns increases as we move down the columns of the P.T.E.of the P.T.E.ffαα local minimum for completed p and s/d local minimum for completed p and s/d orbitalsorbitalsffαα local maximum for , local maximum for ,

1ns 1np

Local Maxima for completed p, s/d orbitals Local Maxima for completed p, s/d orbitals

Local Minima for completed p, s/d orbitals plus 1Local Minima for completed p, s/d orbitals plus 1

MoleculesMolecules

α1 α2 α1 α2 Bd r r f f α exception: noble exception: noble gasesgases2

B me

Ed

e e0

α α

m m΄ E

m m 0,8 1,8d d,,

2e e

r r 02α αα1 α2

2 m mE

I m mf f

r0,5 1d d,,

2

0 2e

E 27.2 eVm α

SolidsSolids 3

s4 V

f3 N

3B3

3 ss

m A2.68 g / cm

4 fr3

2

s s s2 2 2e s s s

27.2 eV 625 calE

atom molm r f f

s 1 ,,

211 2 11

s s5 5 5 2e s s s

294 180 NB c c 10 N / m 10

m r f f m

,, sc 0.6

e0 s

e s α s

m 82 Kmα

m r m sf

sα 1.6,,

SolidsSolids

3 2

s 5 5s e

2 10

100 f m α

oD 2

s B

9000K

f A

sB s0.1 β 1d d,,

Comparison with experimental dataComparison with experimental data

4,874,876,486,483,853,853,933,935,285,285,685,684,114,114,634,63 (Κ(Κm/s)m/s)

0,540,540,9980,9981,291,291,371,370,730,730,7220,7221,291,291,681,68B (B ( ))

2,692,694,634,633,823,823,493,493,043,043,393,393,823,824,284,28 (eV/atom(eV/atom))

2,362,362,332,339,019,018,968,962,732,732,792,797,927,927,867,86((gr/cmgr/cm33))

3,183,182,672,672,992,992,672,67

TheoryTheoryExpExp..TheoryTheoryExpExp..TheoTheoryry

Exp.Exp.TheoryTheoryExp.Exp.

SiSiCuCuAlAlFeFe

f

11 210 / mN

Μρ

δΕ

FLUIDSFLUIDS

Sea wavesSea waves2

g,k ,d, ,

3

2 kgkf kd

,, f kd 1 kd 1

kd kd 1, tsunami

f kd tanh kd2

w 0.074 Jm

nn nn

2

1A΄

22 r

wr f α

nn1

0.45 eV / molecule2

nn 8

nn΄ 5

wf = 3.64

2

J0.1

m exp: 0.073 J/m2

WINDWIND INDUCEDINDUCED

TSUNAMTSUNAMII

λ(λ(mm))

1010-3-3

101000

101011

101022

101033

101044

101055

1010-2-2

1010-1-1

1010-1-1

101000

101022

101011

ph

ωυ = m/s

k

0.2320.232

0.84 0.84 km/hkm/h

1.7 cm1.7 cm

σk

ρg

k gd

FLUIDSFLUIDS

Drag forceDrag force , , , 2

α 1F c S ,,2S , LARGE BODIES, HIGH , LARGE BODIES, HIGH

SPEEDSPEEDη 2F c ,, 2c 6 R , SMALL BODIES, LOW , SMALL BODIES, LOW

SPEEDSPEEDαF

ReF /

Reynolds Reynolds numbernumber

watwaterer

0.010.01 0.010.01

airair 0.000180.00018 0.150.15

1 1gcm s 2 1/ cm s

Pr essure time

12

c

16e

2 22 2 2αe Β s w

m 4.13 10 1c c

m 18 1823m α f f

132c 1.72 10 rad / s

931 1

132 2

c 2.44 10 kg c kg2 0.89 10

c ms c ms1.72 10

exp:3 kg

10ms

Black Body RadiationBlack Body Radiation BS, c, , k T

422 2 SB S

SUN s s2 2 3

B B

k Tk TI 4 R 4 R

60 cck T k T

422 E

E s 2 3

k TI 4 R

60 c

Typical Planet: ETypical Planet: EGRAVITATIONALGRAVITATIONAL Ε ΕELECTRICELECTRIC

α u uM = N Am mv

11 333 3

α α

4 4R N r R = N r = N / A r

3 3

v

51 6ΓN 3,59 10 R 6,378 10 m v

2 2

a

2

4

GM eN

R f re

Β

3η , γ 2, f 2, r = fα

5

3322

51v 2 A 55

G

2 1N 2,8 10

f A

12

6

A 55G

3,65R 3,9 10 m

A

Why Earth is round?Why Earth is round? Why are there mountains?Why are there mountains? What is the largest possible height of a mountain in a planet?What is the largest possible height of a mountain in a planet?

When the shear stress exceeds the critical valueWhen the shear stress exceeds the critical value

c

S

B V g 1

V SH3

3 3

3 m

4 f

2g GM / R34

M R3

23

c 5 5e

2 10m f

322 2

B G

RH 0,025f0,3 10

α A

11 2RH 10 m

,

PlanetsPlanets

348

, min 3 3Β

R AN 10 N

f α

v

6minR 10 m (R 10H) t

55 56, max ,min,αστροN 2 10 N 2,3 10v v d

13

8 vmax Β

NR 10 m R fα

A

Jupital:

Pluto:

54 7N 1,14 10 , R 7,15 10 m v

48 6N 8,9 10 , R 1,15 10 m v

55 551.3 10 8.5 10BD ƒ ƒ

STARSSTARS MINIMUM NUMBER OF NUCLEONS:MINIMUM NUMBER OF NUCLEONS:

MAXIMUM NUMBER OF NUCLEONS:MAXIMUM NUMBER OF NUCLEONS:

NUMBER OF NUCLEONS IN OUR SUN:NUMBER OF NUCLEONS IN OUR SUN:

MAXIMUM NUMBER OF NUCLEONS IN A WHITE MAXIMUM NUMBER OF NUCLEONS IN A WHITE DWARF:DWARF:

3/ 4 3/ 257u

,mine G

mN 0.6 0.23 10

m

ν

3/ 259s

,maxG

N 1.3 10

ν ,, s 15

571,2 10

3/ 257

G

11.775 1.71 10

T

11

Z

02R

KIN B1

E2

E

0

2

B3 GM

5 RE

Tign

2 2

KIN e F B3 3

E N E k T5 2

2/322 e

Fe

N3

2m VE

0R R

Tmax

42 44/3e u

max e2eB

0,06 G m m NT N

Nk

4

uign 2

B

e mT 0,1

k

max ignT 2T

STARSSTARS

ΜΜminmin

G kin phP P P

phG kinE1 2 E 1

3 V 3 V 3 V

E

total G ph kinE E E E 0

STARSSTARS

ΜΜmaxmax

For large mass, T becomes large, EFor large mass, T becomes large, Ephph dominates over Edominates over Ekin kin thenthen

s 2

2GMr

c 2

s

1 GMmmc

2 r

1.1.

2.2.

3.3.

4.4.

3

Bs

1 c ck T

4 r 8 GM

2 42 2 4 3 3 B

s 0 3 2

k cdU c dM 4 r T dt M M t

G10 8

3130 0t 2,63 10 M kg years

2 35s P2 3

B P

S A GdU TdS , A 4 r , 1,62 10 m

k 4 c

Black Holes

UniverseUniverse Homogeneous & IsotropicHomogeneous & Isotropic

Expanding according to Hubble’s law:Expanding according to Hubble’s law:

Eucledian geometry Eucledian geometry ((as a result of inflationas a result of inflation))

11stst LAW (for dS LAW (for dS0): dU=d(0): dU=d(εεV)=-pdVV)=-pdV

Three unknown functions:Three unknown functions: R(t), R(t), εε(t), p(t)(t), p(t)

dR= R = H t R

dt

22

κιν G kin G

3 3 GME = MR , E , E E

10 5 R

2

22

R 8πGH = ε

R 3c

(1)(1)

2

R 4πG= - ε + 3p

R 3c

(2)(2)

OBSERVATIONSOBSERVATIONS

WMAP WMAP Wilkinson Microwave Anisotropy ProbeWilkinson Microwave Anisotropy Probe

the equivalent of 6 protons perthe equivalent of 6 protons per m3

This density equals to the critical one with an uncertainty This density equals to the critical one with an uncertainty of 2%of 2%

protons per protons per mm33, i.e. about , i.e. about 4.2%4.2% of of εε

The Rest DARK MATTER

DARK ENERGY

272 3

Kg9.47 10

c m

nucleons2

1

4c

ph2

0c

21.5%

c

v d

21 4%

73 4%

accelerated expansionaccelerated expansion todaytoday

forfor

forfor

nucleons ph dm dee ν

nucleons ph dm de dep p p p p p w ν ,, 1 w 0.73 d

d2 2 2

R 4 G 4 G 4 G3p 3 w 3 0.73 w 1 0

R 3c 3c 3c

R t t , 2.5

270 y t 7Gy

3 d d

inf1

t t 70 y2

d d

phRT const

nucleon ph de3 4

1 1, , const

R R

BB

t

q→B P,n→nuclei

εph→εnucl decoupling

Protostars

Galaxies

R 0 Today10-

4s1s

70 ky

380 ky

180 My

8 Gy

13.7 Gy

500 My