geometry of common power-law potentials ii....sep 15, 2015 · geometry of common power-law...
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Geometry of common power-law potentials II. (Ch. 9 of Unit 1)
Review of “Sophomore-physics Earth” field geometry“Outside” Coulomb geometry of -kr
-1-potential and -kr -2-force field
“Inside” Oscillator geometry of kr2/2 potential and -kr1 force field Easy-to-remember geo-solar constants
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
Isotropic Harmonic Oscillator phase dynamics in uniform-body orbits Dual phasor construction of elliptic oscillator orbits
Integrating IHO equations by phasor geometry
Lecture 7 Thur. 9.15.2015
Link → IHO orbital time rates of change Link → IHO Exegesis Plot
Link ⇒ BoxIt simulation of IHO orbits
1Wednesday, September 16, 2015
Review of “Sophomore-physics Earth” field geometry“Outside” Coulomb geometry of -kr
-1-potential and -kr -2-force field
“Inside” Oscillator geometry of kr2/2 potential and -kr1 force field Easy-to-remember geo-solar constants
2Wednesday, September 16, 2015
x=2.0x=0.5 x=1(0,0)
-0.5
-1
-2
-3
-4
--11//xx
--11//xx22
--11//xx
--11//xx22
2.00.5 x=1(0,0)
-1
-2
-3
-4
UU((xx))==--11//xx
--11//xx
--11//xx22
Step1 : Follow line from origin (0,0)
through (x,-1) intercept to (+1,-1/x) intercept.
Transfer laterally to draw (x,-1/x) point.
Step2 : Follow line from origin (0,0)
through (x,-1/x) point to (+1,-1/x2) intercept.
Transfer laterally to draw (x,-1/x2) point.
FF((xx))==--11//xx22
FF((11))
FF((00..55))
FF((00..77))
FF((xx))
11
Δxx
−ΔUU
UU((xx))
Δxx−ΔUUFF((xx))
11==
______
Step3 :(Optional) Display Force vector
using similar triangle constuction based
on the slope of potential curve.
Unit 1Fig. 9.4
Review:Coulomb geometryForce and Potential F(x)=-1/x2 U(x)=-1/x
Outside of r=R⊕
x1
1/x1/x2
x:1=1:(1/x)
1x:(1/x)=1:(1/x2)
3Wednesday, September 16, 2015
4Wednesday, September 16, 2015
Example of contacting lineand contact point
directrixdistance
Directrix
Sub-directrix
focal distance =
2.00.5 x=1(0,0)
-1
-2
UU((xx))==--11//xx
FF((11..00))
FF((xx))==--11//xx22((oouuttssiiddee EEaarrtthh))
FF((xx))==-- xx((iinnssiiddee EEaarrtthh))
FF((11..44))FF((22..00))
FF((22..88))
FF((00..88))
FF((00..44))
FF((--11..00))FF((--11..44))
FF((--22..00))
FF((--00..88))
FF((--00..44))
-0.5-1.5
Directrix
Latusrectum
λ
Focus
UU((xx))==((xx22--33))//22
Parabolic potentialinside Earth
Unit 1Fig. 9.7
The ideal “Sophomore-Physics-Earth” model of geo-gravity
5Wednesday, September 16, 2015
Review of “Sophomore-physics Earth” field geometry“Outside” Coulomb geometry of -kr
-1-potential and -kr -2-force field
“Inside” Oscillator geometry of kr2/2 potential and -kr1 force fieldEasy-to-remember geo-solar constants
6Wednesday, September 16, 2015
d
D
You areHere!
Shell mass element
Shell mass elementM =(solid-angle factor A)D2
Gravity at rdue to shell mass elementsG M - G mD2 d2
D2 - d2
D2 d2( )A = 0
B
Θ
Θ
r
m =(solid-angle factor A) d2
M
m
=(...andweightless!)
You areHere!
OdΩsinΘ
A=
Coulomb force vanishes inside-spherical shell (Gauss-law)
Coulomb force inside-spherical body due to stuff below you, only.
M<
Gravitational force at r< isjust that of planet below r<
r<
M<
F inside(r< ) = GmM <
r<2 = Gm 4π
3
M <
4π
3r<3r< = Gm
4π
3ρ⊕r< = mg
r<R⊕
≡ mg ⋅ x
Note:Hooke’s (linear) force law for r< inside uniform body
Unit 1Fig. 9.6
Cancellation of
7Wednesday, September 16, 2015
d
D
You areHere!
Shell mass element
Shell mass elementM =(solid-angle factor A)D2
Gravity at rdue to shell mass elementsG M - G mD2 d2
D2 - d2
D2 d2( )A = 0
B
Θ
Θ
r
m =(solid-angle factor A) d2
M
m
=(...andweightless!)
You areHere!
OdΩsinΘ
A=
Coulomb force vanishes inside-spherical shell (Gauss-law)
Coulomb force inside-spherical body due to stuff below you, only.
M<
Gravitational force at r< isjust that of planet below r<
r<
M<
F inside(r< ) = GmM <
r<2 = Gm 4π
3
M <
4π
3r<3r< = Gm
4π
3ρ⊕r< = mg
r<R⊕
≡ mg ⋅ x
Note:Hooke’s (linear) force law for r< inside uniform body
Earth surface gravity acceleration: g = G M⊕
R⊕2 = G M⊕
R⊕3 R⊕ = G 4π
3
M⊕
4π
3R⊕
3R⊕ = G 4π
3ρ⊕R⊕ = 9.8m / s
G=6.67384(80)·10-11Nm2/C2 ~(2/3)10-10
Unit 1Fig. 9.6
Cancellation of
8Wednesday, September 16, 2015
d
D
You areHere!
Shell mass element
Shell mass elementM =(solid-angle factor A)D2
Gravity at rdue to shell mass elementsG M - G mD2 d2
D2 - d2
D2 d2( )A = 0
B
Θ
Θ
r
m =(solid-angle factor A) d2
M
m
=(...andweightless!)
You areHere!
OdΩsinΘ
A=
Coulomb force vanishes inside-spherical shell (Gauss-law)
Coulomb force inside-spherical body due to stuff below you, only.
M<
Gravitational force at r< isjust that of planet below r<
r<
M<
F inside(r< ) = GmM <
r<2 = Gm 4π
3
M <
4π
3r<3r< = Gm
4π
3ρ⊕r< = mg
r<R⊕
≡ mg ⋅ x
Note:Hooke’s (linear) force law for r< inside uniform body
Earth surface gravity acceleration: g = G M⊕
R⊕2 = G M⊕
R⊕3 R⊕ = G 4π
3
M⊕
4π
3R⊕
3R⊕ = G 4π
3ρ⊕R⊕ = 9.8m / s
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m
Solarmass :M = 1.9889 × 1030 kg. 2.0 ⋅1030 kg. Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. Solar radius :R = 6.955 × 108m. 7.0 ⋅108m.
G=6.67384(80)·10-11Nm2/C2 ~(2/3)10-10
Unit 1Fig. 9.6
Cancellation of
9Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
10Wednesday, September 16, 2015
A more conventional parabolic geometry...
(a) Parabolic Reflector y=x2
y=1
y=2
y=3
y=4
y=5
(b)Parabolic geometry
λ
λ
Directrix
Latusrectum
reflectsintofocus
Verticalincomingray
DistancetoFocus
Distance =to
directrix
Mλ/2λ/2
Unit 1Fig. 9.3
Better name† for λ : latus radius†Old term latus rectum is exclusive copyright of
X-Treme Roidrage GymsVenice Beach, CA 90017
radius
11Wednesday, September 16, 2015
Review of conventional parabolic geometry...introducing “kites”
p=λ/2
y =-p
0
p
2p
3p
4p
x =0 p 2p 3p 4p
Parabola4p·y =x2=2λ·y
pdirectrix
Latus rectumλ =2p
slope=1tangent slope=-2
Circleofcurvatureradius=λ
(a)
p
p
slope=1/2y =-p
0
p
2p
3p
4p
x =0 p 2p 3p 4p
Parabola4p·y =x2=2λ·y
tangent slope=-5/2
(b)
Unit 1Fig. 9.4
radius
12Wednesday, September 16, 2015
Example of contacting lineand contact point
directrixdistance
Directrix
Sub-directrix
focal distance =
2.00.5 x=1(0,0)
-1
-2
UU((xx))==--11//xx
FF((11..00))
FF((xx))==--11//xx22((oouuttssiiddee EEaarrtthh))
FF((xx))==-- xx((iinnssiiddee EEaarrtthh))
FF((11..44))FF((22..00))
FF((22..88))
FF((00..88))
FF((00..44))
FF((--11..00))FF((--11..44))
FF((--22..00))
FF((--00..88))
FF((--00..44))
-0.5-1.5
Directrix
Latusrectum
λ
Focus
UU((xx))==((xx22--33))//22
Parabolic potentialinside Earth
Unit 1Fig. 9.7
The ideal “Sophomore-Physics-Earth” model of geo-gravity
13Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
14Wednesday, September 16, 2015
surface potential: PE= −G µM⊕
R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
r=R⊕r=0
15Wednesday, September 16, 2015
surface potential: PE= −G µM⊕
R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvescape
2 =G µM⊕
R⊕
r=R⊕r=0
16Wednesday, September 16, 2015
surface potential: PE= −G µM⊕
R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvescape
2 =G µM⊕
R⊕
R⊕-escape-velocity
vescape = 2G M⊕
R⊕
r=R⊕r=0escapefromr=R⊕
17Wednesday, September 16, 2015
surface potential: PE= −G µM⊕
R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvescape
2 =G µM⊕
R⊕
R⊕-escape-velocity
vescape = 2G M⊕
R⊕
r=R⊕r=0escapefromr=R⊕
surface gravity: g = −G M⊕
R⊕2
18Wednesday, September 16, 2015
surface potential: PE= −G µM⊕
R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvescape
2 =G µM⊕
R⊕
R⊕-escape-velocity
vescape = 2G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0escapefromr=R⊕
19Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
20Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
21Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
22Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
Orbit KE=1
2µv
2 =G µM⊕
2R⊕
23Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
Orbit KE=1
2µv
2 =G µM⊕
2R⊕
Orbit E
Total=12µv
2 -GµM⊕
R⊕
=-GµM⊕
2R⊕
24Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
1
2
3
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
Orbit KE=1
2µv
2 =G µM⊕
2R⊕
Orbit E
Total=12µv
2 -GµM⊕
R⊕
=-GµM⊕
2R⊕
(r=R⊕ )-orbit angular frequency:
ω2R⊕=G
M⊕
R⊕2 ⇒ω= G M⊕
R⊕3
25Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
1
2
3
(r=R⊕ )-orbit speed:
v = G M⊕
R⊕
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
Orbit KE=1
2µv
2 =G µM⊕
2R⊕
Orbit E
Total=12µv
2 -GµM⊕
R⊕
=-GµM⊕
2R⊕
(r=R⊕ )-orbit angular frequency:
ω2R⊕=G
M⊕
R⊕2 ⇒ω= G M⊕
R⊕3
26Wednesday, September 16, 2015
Bottom potential: PE= −G 3µM⊕
2R⊕
Dissociation threshold : PE= 0
Sophomore-physics-Earth inside and out: “3-steps out of (or into) Hell”
Geometric(x, y)(Dimensionless)
Scalingrelations
mksvariables(meter-kg-sec)
space coord.: x r = R⊕x x = r /R⊕
PE for x ≥1:
yPE= -1x
PEmks (r)
=GMµR⊕
yPE
PEmks (r) = -GMµr
= -GMµR⊕
1x
Forcefor x ≥1:
yForce= -1x2
Fmks (r)
=GMµR⊕
2 yForce
Fmks (r) = -GMµr2
= -GMµR⊕
21x2
KE = PE relation: 12µvbottom
2 =G 3µM⊕
2R⊕
(r=0)-escape-velocity
vbottom = 3G M⊕
R⊕
surface gravity: g = −G M⊕
R⊕2
escapefromr=0
PE for x <1:
yPE= x2
2− 3
2 PEmks(r)=GMµ
R⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
Forcefor x <1:
yForce= -x Fmks (r) = -GMµR⊕
3 r
r=R⊕r=0
...and surface orbit at r=R⊕
1
2
3
(r=R⊕ )-escape velocity:
vescape = 2G M⊕
R⊕
Centifugal force = surface gravity:
µv2
R⊕
=µg=G µM⊕
R⊕2
Orbit KE=1
2µv
2 =G µM⊕
2R⊕
Orbit E
Total=12µv
2 -GµM⊕
R⊕
=-GµM⊕
2R⊕
(r=R⊕ )-orbit angular frequency:
ω2R⊕=G
M⊕
R⊕2 ⇒ω= G M⊕
R⊕3
27Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
28Wednesday, September 16, 2015
Crushed Earth1/2 radius
8 times as dense1/8 focal distance or λ
1/8 minimum radius of curvature8 times maximum curvature
4 times the surface gravity: g = −G M⊕
R⊕2
2 times the surface potential: PE= −G M⊕
R⊕
2 times surface escape speed: ve= G 2M⊕
R⊕
2x
2 times -orbit energy: E = −G M⊕
2R⊕
2 times -orbit speed: v = G M⊕
R⊕
1
2
3
Orbit at R⊕level : PE= -G M⊕
2R⊕
Suppose Earth radius crushed to 1/2: (R⊕=6.4·106m crushed to R⊕/2=3.2·106m )
Sophomore-physics-Earth inside and out: “3-steps to Hell”
Escape level : PE= 0
(Sit at R⊕ )-level : PE= -G M⊕
R⊕
(Sit at r=0)-level : PE= -G 3M⊕
2R⊕
All formulasidentical toones derivedon p.15 to 27.ImaginereducingR⊕ to R⊕/2
29Wednesday, September 16, 2015
Crushed Earth1/2 radius
8 times as dense1/8 focal distance or λ
1/8 minimum radius of curvature8 times maximum curvature
4 times the surface gravity: g = −G M⊕
R⊕2
2 times the surface potential: PE= −G M⊕
R⊕
2 times surface escape speed: ve= G 2M⊕
R⊕
2x
2 times -orbit energy: E = −G M⊕
2R⊕
2 times -orbit speed: v = G M⊕
R⊕
1
2
3
Orbit at R⊕level : PE= -G M⊕
2R⊕
Suppose Earth radius crushed to 1/2: (R⊕=6.4·106m crushed to R⊕/2=3.2·106m )
Sophomore-physics-Earth inside and out: “3-steps to Hell”
Escape level : PE= 0
(Sit at R⊕ )-level : PE= -G M⊕
R⊕
(Sit at r=0)-level : PE= -G 3M⊕
2R⊕
R⊕ to R⊕/2
8 times (r=R⊕ )-orbit frequency:
ω= G M⊕
R⊕3
All formulasidentical toones derivedon p.15 to 27.ImaginereducingR⊕ to R⊕/2
30Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
31Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3 Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg.
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ =??ρ⊕ = M⊕
(4π /3)R⊕3
32Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3 Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
33Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3 Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
Density of solid Fe=7.9·103kg/m3
Density of liquid Fe=6.9·103kg/m3
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
34Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
35Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
That’s 100·10-27=10-25 kg packed into a volume of 4π/3r3= 4π/3 (3·10-15)3 m3 or about 10-43 m3.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
36π=113~102
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
36Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
That’s 100·10-27=10-25 kg packed into a volume of 4π/3r3= 4π/3 (3·10-15)3 m3 or about 10-43 m3.
Nuclear density is 10-25+43 = 1018kg /m3 or a trillion (1012) kilograms in the size of a fingertip (1cc).
Earth radius crushed by a factor of 0.5·10-5 to would approach neutron-star density.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
Rcrush⊕ 300m
37Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
38Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
That’s 100·10-27=10-25 kg packed into a volume of 4π/3r3= 4π/3 (3·10-15)3 m3 or about 10-43 m3.
Nuclear density is 10-25+43 = 1018kg /m3 or a trillion (1012) kilograms in the size of a fingertip.
Earth radius crushed by a factor of 0.5·10-5 to would approach neutron-star density.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕
Introducing the “Neutron starlet” 1 cm3 of nuclear matter: mass= 1012 kg.
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
Rcrush⊕ 300m
39Wednesday, September 16, 2015
Geometry and algebra of idealized “Sophomore-physics Earth” fields Coulomb field outside Isotropic Harmonic Oscillator (IHO) field insideContact-geometry of potential curve(s) and “kite” geometry“Ordinary-Earth” models: 3 key energy “steps” and 4 key energy “levels”“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”
Earth matter vs nuclear matter: Introducing the “neutron starlet” Fantasizing a “Black-Hole-Earth”
40Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
That’s 100·10-27=10-25 kg packed into a volume of 4π/3r3= 4π/3 (3·10-15)3 m3 or about 10-43 m3.
Nuclear density is 10-25+43 = 1018kg /m3 or a trillion (1012) kilograms in the size of a fingertip.
Earth radius crushed by a factor of 0.5·10-5 to would approach neutron-star density.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg.
Rcrush⊕ 300m
ρ⊕
Introducing the “Neutron starlet” 1 cm3 of nuclear matter: mass= 1012 kg.
Introducing the “Black Hole Earth” Suppose Earth is crushed so that its
surface escape velocity is the speed of light c ≅ 3.0·108m/s. c≡299,792,458m/s (EXACTLY)
G=6.67384(80)·10-11Nm2/C2 ~(2/3)10-10
Vescape =√ (2GM/R⊗)
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
( from p. 15 )
41Wednesday, September 16, 2015
Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3
Nuclear matter Nucleon mass =1.67·10-27kg.~ 2·10-27kg.Say a nucleus of atomic weight 50 has a radius of 3 fm, or 50 nucleons each with a mass 2·10-27kg.
That’s 100·10-27=10-25 kg packed into a volume of 4π/3r3= 4π/3 (3·10-15)3 m3 or about 10-43 m3.
Nuclear density is 10-25+43 = 1018kg /m3 or a trillion (1012) kilograms in the size of a fingertip.
Earth radius crushed by a factor of 0.5·10-5 to would approach neutron-star density.
Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg.
Rcrush⊕ 300m
ρ⊕
Introducing the “Neutron starlet” 1 cm3 of nuclear matter: mass= 1012 kg.
Introducing the “Black Hole Earth” Suppose Earth is crushed so that its
surface escape velocity is the speed of light c ≅ 3.0·108m/s. c≡299,792,458m/s (EXACTLY)
c =√ (2GM/R⊗)
R⊗ = 2GM/c2 = 8.9mm ~1cm (fingertip size!)
G=6.67384(80)·10-11Nm2/C2 ~(2/3)10-10
Vescape =√ (2GM/R⊗)
( from p. 15 )
Earthradius :R⊕ = 6.371⋅106m 6.4 ⋅106m Earthvolume : (4π / 3)R⊕3 1.083⋅1021 ~ 1021m3
ρ⊕ = M⊕
(4π /3)R⊕3
42Wednesday, September 16, 2015
Isotropic Harmonic Oscillator phase dynamics in uniform-body orbits Dual phasor construction of elliptic oscillator orbits
Integrating IHO equations by phasor geometry
43Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
1-D 2-D(Paths are always 2-D ellipses if viewed right!)
Unit 1Fig. 9.10
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
44Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
1-D 2-D(Paths are always 2-D ellipses if viewed right!)
Unit 1Fig. 9.10
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
PEmks(r)=GMmR⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
HO"Spring-constant" (from p. 26) 1
2k=GM
2R⊕3
45Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
1-D 2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
PEmks(r)=GMmR⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
HO"Spring-constant" (from p. 26) 1
2k=GM
2R⊕3
46Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
PEmks(r)=GMmR⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
HO"Spring-constant" (from p. 26) 1
2k=GM
2R⊕3
47Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
48Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ=ω 2E
kcosθ
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3) by (2)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
49Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ=ω 2E
kcosθ ω = dθ
dt= k
m
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3) by (2) divide this by (1)
by def. (3)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
50Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ=ω 2E
kcosθ ω = dθ
dt= k
m
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3) by (2)
by def. (3)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
by integration given constant ω:
θ = ω⋅dt∫ =ω⋅t +α
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
divide this by (1)
51Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ=ω 2E
kcosθ ω = dθ
dt= k
m
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3) by (2)
by def. (3)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
by integration given constant ω:
θ = ω⋅dt∫ =ω⋅t +α
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
divide this by (1)
PEmks(r)=GMmR⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
HO"Spring-constant" (from p. 26) 1
2k=GM
2R⊕3
52Wednesday, September 16, 2015
Fx
v0Fy
F=-r(a) (b)
Total E = KE + PE = 12mv2 +U(x) = 1
2mv2 + 1
2kx2 = const.
1= mv2
2E+ kx
2
2E= v
2E /m⎛
⎝⎜⎞
⎠⎟
2
+ x2E /k
⎛
⎝⎜⎞
⎠⎟
2
1= mv2
2E+ kx
2
2E= cosθ( )2 + sinθ( )2
2Emcosθ = v= dx
dt= dθdt
dxdθ=ω dx
dθ=ω 2E
kcosθ ω = dθ
dt= k
m
Let : v = 2E /m cosθ , and : x = 2E /k sinθ(1) (2)
by (1)by def. (3) by (2)
by def. (3)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
I.H.O. Force lawF =-x (1-Dimension)F =-r (2 or 3-Dimensions)Each dimension x, y, or z obeys the following:
Another example ofthe old “scale-a-circle”trick...
1-D
Unit 1Fig. 9.10
Equations for x-motion[x(t) and vx=v(t)] are given first. They applyas well to dimensions[y(t) and vy=v(t)] and [z(t) and vz=v(t)] in theideal isotropic case.
by integration given constant ω:
θ = ω⋅dt∫ =ω⋅t +α
2-D or 3-D(Paths are always 2-D ellipses if viewed right!)
ω = dθdt
def. (3)
divide this by (1)
PEmks(r)=GMmR⊕
r2
2R⊕2 - 3
2⎛⎝⎜
⎞⎠⎟
HO"Spring-constant" (from p. 26)
12k=GM
2R⊕3 so: ω= G M⊕
R⊕3
53Wednesday, September 16, 2015
Isotropic Harmonic Oscillator phase dynamics in uniform-body orbits Dual phasor construction of elliptic oscillator orbits
Integrating IHO equations by phasor geometry
54Wednesday, September 16, 2015
0 12
3
4
5
6
78-7-6
-5
-4
-3
-2-1
01
2
3 4 5
6
78
-7-6
-5
-4-3
-2-1
(2,4)(3,5)(4,6)
(5,7)
(6,8)
(7,-7)
x-position
x-velocity vx/ω
position
x
velocity vx/ω
(a) 1-D Oscillator Phasor Plot
(b) 2-D Oscillator Phasor Plot
Phasor goes
clockwise
by angle ω t
ω t
(8,-6)(9,-5)
y-velocity
vy/ω
(x-Phase 45°
behind
the y-Phase)y-position
φ counter-clockwise
if y is behind x
clockwise
orbit
if x is behind y
(1,3) Left-
handed
Right-
handed
(0,2)
Fx
v0Fy
F=-r(a) (b)
Isotropic Harmonic Oscillator phase dynamics in uniform-body
Unit 1Fig. 9.10
55Wednesday, September 16, 2015
0 12
3
4
5
6
78-7-6
-5
-4
-3
-2-1
0123 4 5
67
8-7
-6-5-4-3-2-1
(0,4) (1,5)(2,6)(3,7)
(4,8)(5,9)
x-position
x-velocity vx/ω(a) Phasor Plots
for
2-D Oscillator
or
Two 1D Oscillators
(x-Phase 90° behind
the y-Phase)
a
b b
a
a
0 12
3
4
5
6
78-7-6
-5
-4
-3
-2-1
0123 4 5
67
8-7
-6-5-4-3-2-1
(0,0)(1,1)
(5,5)
(2,2)(3,3)
(4,4)
x-position
a
b b
a
a
y-position
y-velocity
vy/ω
x-velocity vx/ω
(b)
x-Phase 0° behind
the y-Phase
(In-phase case)
y-velocity
vy/ω
0 12345
678-7
-6-5-4-3-2-1
(0,4)(1,5)(2,6)(3,7)
b
y-position
b
Fig. 9.11
Unit 1Fig. 9.12
These are more generic exampleswith radius of x-phasor differing from that of the y-phasor.
56Wednesday, September 16, 2015
Isotropic Harmonic Oscillator phase dynamics in uniform-body orbits Dual phasor construction of elliptic oscillator orbits
Integrating IHO equations by phasor geometry
57Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 5.0)
Link ⇒ BoxIt simulation of IHO orbits
58Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
59Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
60Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
61Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
62Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
63Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
64Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
65Wednesday, September 16, 2015
121
2
3
4
56
7
8
9
10
11
121
2
3
4
56
7
8
9
10
11
x
y
x
y
vx/ω
vy/ω
vx/ω
vy/ω
Initial position: r(0)=(7.0 3.0)
Inital velocity: v(0)=(-8.0 6.0)
Arbitrary initial position r(0)=(x(0),y(0)
and initial velocityv(0)=(vx(0),vy(0)
Usually have x and y phasor circles of unequal size
66Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
Initial phase (αx=120°,αy=+135°)
++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 90°:(αx=120°-90°=30°,αy= +135°-90°=45°)
120°
135° r(0°)
vxv y
vx-vyspace
v(0°)
67Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 30°:(αx=120°-30°=90°,αy= +135°-30°=105°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(30°)
++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(30°)
90°
105°
68Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 60°:(αx=120°-60°=60°,αy= +135°-60°=75°)
r(60°)75°
60°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 60°:(αx=120°-60°=60°,αy= +135°-60°=75°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace v(60°)
69Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(90°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 90°:(αx=120°-90°=30°,αy= +135°-90°=45°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(90°)
45°
30°
70Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 120°:(αx=120°-120°=0°,αy= +135°-120°=15°)
r(120°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 120°:(αx=120°-120°=0°,αy= +135°-120°=15°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(120°)
15°
0°
71Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 150°:(αx=120°-150°=-30°,αy= +135°-150°=-15°)
r(150°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 150°:(αx=120°-150°=-30°,αy= +135°-150°=-15°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(150°)-15°
-30°
72Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 180°:(αx=120°-180°=-60°,αy= +135°-180°=-45°)
r(180°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 180°:(αx=120°-180°=-60°,αy= +135°-180°=-45°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(180°)-45°
-60°
73Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(210°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 210°:(αx=120°-210°=-90°,αy= +135°-210°=-75°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(210°)
74Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(240°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 240°:(αx=120°-240°=-120°,αy= +135°-240°=-105°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(240°)
75Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(270°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 270°:(αx=120°-270°=-150°,αy= +135°-270°=-135°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(270°)
76Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(300°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 300°:(αx=120°-300°=-180°,αy= +135°-300°=-165°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(300°)
Link → IHO orbital time rates of change
Link → IHO Exegesis Plot
77Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
121
2
3
4
56
7
8
9
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°° r(330°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 330°:(αx=120°-300°=-210°=+150°,αy= +135°-330°=-195°=+165°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(330°)
78Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(360°)=r(0°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 360°:(αx=120°-360°=-240°=+120°,αy= +135°-360°=-225°=+135°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace v(360°)=v(0°)
79Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
Initial phase (αx=120°,αy=+135°)
++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 90°:(αx=120°-90°=30°,αy= +135°-90°=45°)
120°
135° r(0°)
vxv y
vx-vyspace
v(0°)
80Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 30°:(αx=120°-30°=90°,αy= +135°-30°=105°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(30°)
++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(30°)
90°
105°
81Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 60°:(αx=120°-60°=60°,αy= +135°-60°=75°)
r(60°)75°
60°
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 60°:(αx=120°-60°=60°,αy= +135°-60°=75°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace v(60°)
82Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
r(90°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 90°:(αx=120°-90°=30°,αy= +135°-90°=45°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(90°)
45°
30°
83Wednesday, September 16, 2015
x
y
121
2
3
4
56
7
8
9
10
121
2
3
4
56
7
8
9
10
11
00°°--1155°°--3300°°--4455°°++1155°°++3300°°
++4455°°
--6600°°
++6600°°
--7755°°
++7755°°
--9900°°
++9900°°
--110055°°
++110055°°
--112200°°
++112200°°
--113355°°
++113355°°
--115500°°
++115500°°
--116655°°
++116655°°
118800°°
118800°°++227700°° ++224400°°++330000°°336600°° ++221100°°++333300°°
vx /ω
vy /ω
Δαy-x
xphasor
yphasor
x-yspace
x
y
xvx
y
v y
++334455°°--334455°°
...after time advances by 180°:(αx=120°-180°=-60°,αy= +135°-180°=-45°)
r(180°)
Ellipsometry Contact Plotsvs.
Relative phase Δα=αx-αyΔα=−15°
(Right-polarized anti-clockwise case)
...after time advances by 180°:(αx=120°-180°=-60°,αy= +135°-180°=-45°)
Initial phase (αx=120°,αy=+135°)
vxv y
vx-vyspace
v(180°)-45°
-60°
84Wednesday, September 16, 2015