forth, e.n. economou fixing by thinking: the power of dimensional analysis e.n. economou dept. of...
Post on 21-Dec-2015
217 views
TRANSCRIPT
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