geometry of common power-law potentials ii....sep 15, 2015 · geometry of common power-law...

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

http://www.uark.edu/ua/modphys/markup/RelaWavityWeb.html?plotType=1%7C0&semiMajor=1.0&semiMinor=0.125





http://www.uark.edu/ua/modphys/markup/BoxItWeb.html




“Inside” Oscillator geometry of kr2/2 potential and -kr1 force field Easy-to-remember geo-solar constants


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)


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




“Inside” Oscillator geometry of kr2/2 potential and -kr1 force fieldEasy-to-remember geo-solar constants


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


d

D

You areHere!

Shell mass element



D2 - d2

D2 d2( )A = 0

B

Θ

Θ

r


M

m


You areHere!

OdΩsinΘ

A=



M<


r<

M<


r<2 = Gm 4π

3

M <

4π

3r<3r< = Gm

4π

3ρ⊕r< = mg

r<R⊕

≡ mg ⋅ x


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


d

D

You areHere!

Shell mass element



D2 - d2

D2 d2( )A = 0

B

Θ

Θ

r


M

m


You areHere!

OdΩsinΘ

A=



M<


r<

M<


r<2 = Gm 4π

3

M <

4π

3r<3r< = Gm

4π

3ρ⊕r< = mg

r<R⊕

≡ mg ⋅ x


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


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


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


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


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



R⊕




Scalingrelations



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



R⊕




Scalingrelations



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


2 =G µM⊕

R⊕

R⊕-escape-velocity

vescape = 2G M⊕

R⊕

r=R⊕r=0escapefromr=R⊕



R⊕




Scalingrelations



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


2 =G µM⊕

R⊕


vescape = 2G M⊕

R⊕


surface gravity: g = −G M⊕

R⊕2



R⊕




Scalingrelations



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


2 =G µM⊕

R⊕


vescape = 2G M⊕

R⊕


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



Bottom potential: PE= −G 3µM⊕

2R⊕




Scalingrelations



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⊕


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:


3 r

r=R⊕r=0



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0

...and surface orbit at r=R⊕



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0


Centifugal force = surface gravity:

µv2

R⊕

=µg=G µM⊕

R⊕2



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0



µv2

R⊕

=µg=G µM⊕

R⊕2

Orbit KE=1

2µv

2 =G µM⊕

2R⊕



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0



µv2

R⊕

=µg=G µM⊕

R⊕2

Orbit KE=1

2µv

2 =G µM⊕

2R⊕

Orbit E

Total=12µv

2 -GµM⊕

R⊕

=-GµM⊕

2R⊕



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0


1

2

3


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



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0


1

2

3

(r=R⊕ )-orbit speed:

v = G M⊕

R⊕


µv2

R⊕

=µg=G µM⊕

R⊕2

Orbit KE=1

2µv

2 =G µM⊕

2R⊕

Orbit E

Total=12µv

2 -GµM⊕

R⊕

=-GµM⊕

2R⊕


ω2R⊕=G

M⊕

R⊕2 ⇒ω= G M⊕

R⊕3



2R⊕




Scalingrelations



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


2 =G 3µM⊕

2R⊕


vbottom = 3G M⊕

R⊕


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:


3 r

r=R⊕r=0


1

2

3

(r=R⊕ )-escape velocity:

vescape = 2G M⊕

R⊕


µv2

R⊕

=µg=G µM⊕

R⊕2

Orbit KE=1

2µv

2 =G µM⊕

2R⊕

Orbit E

Total=12µv

2 -GµM⊕

R⊕

=-GµM⊕

2R⊕


ω2R⊕=G

M⊕

R⊕2 ⇒ω= G M⊕

R⊕3


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


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


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


Examples of “crushed” matterEarth matter Density ~ 6.0·1024-21 ~6·103kg/m3 Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg. ρ⊕


ρ⊕ = M⊕

(4π /3)R⊕3


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


ρ⊕ = M⊕

(4π /3)R⊕3


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⊕


ρ⊕ = M⊕

(4π /3)R⊕3




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.


36π=113~102


ρ⊕ = M⊕

(4π /3)R⊕3





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.



ρ⊕ = M⊕

(4π /3)R⊕3

Rcrush⊕ 300m





Nuclear density is 10-25+43 = 1018kg /m3 or a trillion (1012) kilograms in the size of a fingertip.



Introducing the “Neutron starlet” 1 cm3 of nuclear matter: mass= 1012 kg.


ρ⊕ = M⊕

(4π /3)R⊕3

Rcrush⊕ 300m







Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg.

Rcrush⊕ 300m

ρ⊕


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⊗)


ρ⊕ = M⊕

(4π /3)R⊕3

( from p. 15 )







Earthmass :M⊕ = 5.9722 × 1024 kg. 6.0 ⋅1024 kg.

Rcrush⊕ 300m

ρ⊕


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 )


ρ⊕ = M⊕

(4π /3)R⊕3


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.


Fx

v0Fy

F=-r(a) (b)



1-D 2-D(Paths are always 2-D ellipses if viewed right!)

Unit 1Fig. 9.10


2mv2 + 1

2kx2 = const.

PEmks(r)=GMmR⊕

r2

2R⊕2 - 3

2⎛⎝⎜

⎞⎠⎟

HO"Spring-constant" (from p. 26) 1

2k=GM

2R⊕3


Fx

v0Fy

F=-r(a) (b)



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.


2mv2 + 1

2kx2 = const.

PEmks(r)=GMmR⊕

r2

2R⊕2 - 3

2⎛⎝⎜

⎞⎠⎟


2k=GM

2R⊕3


Fx

v0Fy

F=-r(a) (b)


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)



Another example ofthe old “scale-a-circle”trick...

1-D

Unit 1Fig. 9.10


2-D or 3-D(Paths are always 2-D ellipses if viewed right!)

PEmks(r)=GMmR⊕

r2

2R⊕2 - 3

2⎛⎝⎜

⎞⎠⎟


2k=GM

2R⊕3


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3)




1-D

Unit 1Fig. 9.10



ω = dθdt

def. (3)


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3) by (2)




1-D

Unit 1Fig. 9.10



ω = dθdt

def. (3)


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3) by (2) divide this by (1)

by def. (3)




1-D

Unit 1Fig. 9.10



ω = dθdt

def. (3)


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3) by (2)

by def. (3)




1-D

Unit 1Fig. 9.10


by integration given constant ω:

θ = ω⋅dt∫ =ω⋅t +α


ω = dθdt

def. (3)

divide this by (1)


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3) by (2)

by def. (3)




1-D

Unit 1Fig. 9.10





ω = dθdt

def. (3)

divide this by (1)

PEmks(r)=GMmR⊕

r2

2R⊕2 - 3

2⎛⎝⎜

⎞⎠⎟


2k=GM

2R⊕3


Fx

v0Fy

F=-r(a) (b)


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


by (1)by def. (3) by (2)

by def. (3)




1-D

Unit 1Fig. 9.10





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


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)


Unit 1Fig. 9.10


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.


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




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



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


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







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°)


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°°






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°


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(60°)75°

60°






vxv y

vx-vyspace v(60°)


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°)






vxv y

vx-vyspace

v(90°)

45°

30°


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(120°)






vxv y

vx-vyspace

v(120°)

15°

0°


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°)






vxv y

vx-vyspace

v(150°)-15°

-30°


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(180°)






vxv y

vx-vyspace

v(180°)-45°

-60°


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°)






vxv y

vx-vyspace

v(210°)


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°)






vxv y

vx-vyspace

v(240°)


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°)






vxv y

vx-vyspace

v(270°)


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°)






vxv y

vx-vyspace

v(300°)

Link → IHO orbital time rates of change

Link → IHO Exegesis Plot






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°)




...after time advances by 330°:(αx=120°-300°=-210°=+150°,αy= +135°-330°=-195°=+165°)


vxv y

vx-vyspace

v(330°)


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°)




...after time advances by 360°:(αx=120°-360°=-240°=+120°,αy= +135°-360°=-225°=+135°)


vxv y

vx-vyspace v(360°)=v(0°)


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°°


120°

135° r(0°)

vxv y

vx-vyspace

v(0°)


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°°






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°


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(60°)75°

60°






vxv y

vx-vyspace v(60°)


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°)






vxv y

vx-vyspace

v(90°)

45°

30°


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(180°)






vxv y

vx-vyspace

v(180°)-45°

-60°


geometry of common power-law potentials ii....sep 15, 2015 · geometry of common power-law...

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