FORTH, E.N. Economou FIXING by THINKING: FIXING by THINKING: The power of dimensional The power of dimensional analysis analysis E.N. Economou E.N. Economou Dept. of Physics, U of C Dept. of Physics, U of C FORTH FORTH March, March, 200 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
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FORTH, E.N. Economou FIXING by THINKING: The power of dimensional analysis E.N. Economou Dept. of Physics, U of C FORTH March, 2006 In the atomic idea.
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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)
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
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