associate research professor / senior research scientist
TRANSCRIPT
1
Statistical Calibration Relative to the Meter and the Triple Point of Water
David B. Pollock
Associate Research Professor / Senior Research Scientist
Electrical Computer Engineering / Center for Applied Optics
University of Alabama Huntsville
301 Sparkman Drive
OB 444
Huntsville, AL 35899
Tuesday, August 20, 2015
Purpose: To answer the questions: “Is Radiometry more accurate than Thermometry when Noise
Equivalent Power is the Relative Standard Uncertainty Estimate, specifically when an energy fraction is
kinetic and a fraction is momentum. Is radiometry or thermometry relatively more accurate?”
Overview: Noise-Equivalent-Power is a linear, mean square Relative Uncertainty Scale. Planck
Temperature, TP / Boltzmann Constant, σ ratio, expressed as a Fraction is 1 / 6 ppm. While the
Luminance scale is the first radiation constant c1 L / h-bar, expressed as a Fraction is 0.01 ppm. The
second radiation constant scale is c2 / k / hc, expressed as a Fraction is 0.1 ppm. The velocity-of-light /
impedance scale c0 / ΩZ0 ratio ( 2.99792E-8 / 376.730315 )-1 expressed as a Fraction 12566370614 1/3.
And, the inverse velocity of light expressed as a Fraction is 33356409 1/2 exact.
For example, compare two atomic-clock transition rates, 1-Hertz relative time N(1,2) periods. The
difference of Clock rate #1 vs: Clock rate #2 produces an analytical comparison process similar to the one
used to compare two units of energy propagating whether electrons, photons or pick your favorite
particles. The periodic amplitude difference estimation process reveals whether Clock #1 rate or Clock #2
rate deviated from linear time, ∆n t / N. Here the quantity N is greater than one period for a rate n / N e-nt.
The current standard clock rate, a specific hyperfine-transition rate between two states of a Cesium atom
isotope, 55 Cs, (Eh) atomic mass unit energy. The specific isotope mean state change rate is a whole
positive number of periods relative to a mean time square rate, a ratio, n / N (t – t0)δt.
The Planck equation predicts the bound-state of periodic energy as an energy / energy-specific density
ratio which is a periodic function of time, Temperature and length (distance). Further, as the Bureau
International des Poids et Mesures, BIPM, state “All other SI units can be derived from these, (SI units)
by multiplying together different powers of the base units.” i A periodic ratio rate is c / λ, period c ~ 3 E-
8, m-s-1, specific density 1 c / λ2, spatial energy period 8πhc, ~ 5E-40J.
We attempt to follow the National Institute of Standards and Technology (NIST) definitions. “The
standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y.” A “relative
standard uncertainty ur(y) of a measurement result y is defined by ur(y) = u(y) / |y|, where y is not equal to
0.” Further, the NIST definitions specify Using concise notation as “If, for example, y = 1 234.567 89 U
and u(y) = 0.000 11 U, where U is the unit of y, then Y = (1 234.567 89 ± 0.000 11) U. A more concise
form of this expression, and one that is in common use, is Y = 1 234.567 89(11) U, where it is understood
that the number in parentheses is the numerical value of the standard uncertainty referred to the
corresponding last digits of the quoted result.” 1
Introduction – Order of presentation is: Invariant Velocity-of-Light, Accuracy, Energy-Temperature-
Substance, Convergence, The Planck Function, Boltzmann - Stefan-Boltzmann - Planck, Bernoulli, Color,
1 Current (2014), Standard unit values are accessible at http://physics.nist.gov/cuu/index.html
2
Radiant Power, One-period-Wavelength-Temperature Product, Homogenous, In-homogeneous Energy
Density, Bound and Constrained Periodic Information, Temperature and Power Relative to the Triple
Point of Water, Bessel Function, a Calibration Recommendation.
Invariant statistical velocity-of-light, c – Has a value is continuous, a monotonic scale periodic length
(m / m-square), periodic time (s / s-square). However, Temperature is continuous (K / K4 ). We note that
one period of length is 1.000 000(00), m; one period of time is 1.000 000(00), s; one period of
temperature is 1.000 273(16), K. We conclude a specific periodic specific density scale is 3.333 333(16)
correlated periodic length, time and temperature.
The total derivative of periodic velocity of light, a time dependent frequency ν and frequency rate c / λ, is
shown as Equation(1.1). The reference rate value is exact, 2.997 92+E8 m-sec-1, NIST (Dr. Taylor, Barry
N.; Dr. Mohr, Peter J.; Douma, Michael, 2010).
2
2
1
c
cd dc d
cd c dc d
c dd c dc
, 1 / s. (1.1)
Accuracy – An accurate 3-dimension, periodic exponential function
N
2/3 2/3 2/3
n 0
df (X,Y, Z) x y z
is illustrated by Figure 1 X, Y, Z, singular values of x, y, z and
Figure 2 Singular values x, y, z. The point being singular exponential functions converge to first order
values 3,0; 198,0.
3
Figure 1 X, Y, Z, singular values of x, y, z
Figure 2 Singular values x, y, z
-200, 40000 200, 40000
-0.5, -1.25992105
0.5, 1.25992105
-250.00 -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 250.00
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.00
5,000.00
10,000.00
15,000.00
20,000.00
25,000.00
30,000.00
35,000.00
40,000.00
45,000.00
-250 -200 -150 -100 -50 0 50 100 150 200 250
x^( - 1 / 3 ) Value, #
Exp
on
enti
al A
mp
litu
de
x^(
-1
/ 3
),
Val
ue
Am
pli
tud
e x^2
Val
ue,
#
x^( 2 ) Value, #
Accurate Exponential Calculationsx^(2) = x^( - 1 / 3 ) =
-0.5, -1.25992105
0.5, 1.25992105
-200, 34.19951893
3, 0 198, 0
402, 34.19951893
-300 -200 -100 0 100 200 300 400 500
-5
0
5
10
15
20
25
30
35
40
-200 -150 -100 -50 0 50 100 150 200
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
+ / - x ^ ( - 2 / 3 )
Exp
on
enti
al A
mp
litu
de,
+ 2
/ 3
x ^ ( - 1 / 3 ), #
Exp
on
enti
al A
mp
litu
de
x^(-
1 /
3 )
Val
ue,
#
Accurate Exponential Numbers
x^( - 1 / 3 ) = - x^( 2 / 3 ) = + x^( 2 / 3 ) =
4
Energy-Temperature-Substance – Given substance A(Eh) and B(Eh), specific density ρa, ρb and atomic
mass unit u(amu) independent. Substance A at (Eh) Temperature T(a) and substance B(Eh) at Temperature
T(b) respectively continuously radiate electromagnetic energy whose field lines do not cross. Thermal
balance is achieved when the respective radiant emission ε(a,b) and absorption α(a,b) have a respective
steady-state value ∆T/T K. When photon kinetic energy specific density is sufficiently great (i.e. ionizing
radiation) thermal mechanical damage occurs. We restrict our discussion here to thermal momentum
transfer, a capacitance change rate dC / di δ(t-t0) an electric current. A solid-state sensor with some
detective quantum efficiency creates this electrical current proportional to the capacitive coupling as
incident energy frequency interacts with such a sensor. This type sensor samples frequency, a clock time
dependent sample rate (also a frequency).
An absolute Temperature scale relative to the Triple-Point of Water is exactly 273.160(07)K ppm.
Substance A and B each continuously radiate a small amount of spectral energy, ε = c / λ (εσT4) until an
equilibrium state A, B exists. The periodic rate ab is dK(t / λ)δ(t) (hc / λσK). The Radiant power is
8 , Jhc s, Concurrently A and B absorb and radiate energy until the equilibrium state is realized, a
radiative exchange a singular, mean Temperature value T A T B . With the accepted value for T0
273.160(06) as an absolute linear temperature scale and the Stefan-Boltzmann Constant σ = 5.670
373(21) 10-8, it would seem that using delta-temperature, Δ(2T(0) – σT), unit energy per estimate, as a
long-term peak signal-to-rms-noise, a ratio to estimate stability is appropriate.
4
3 4 1
2 3 2
3
4
'' 12
''
( )
( 4 )
(' )24 12
T
T T dT T
T
T
dT
d T
d T
T dT T
T T dT T
(2.1)
Optical Frequency, electromagnetic-energy, is constrained as the energy propagates along a single E-field
line a Value of + / - 1 Hz radiated by a specific Substance at a specific Temperature. We perceive only the
envelope of the frequency, either the eye or an optical sensor. Normal usage spectral emittance is ε, an
emission rate, and spectral absorbance is α, unit energy half-life. Transmittance plus Emittance minus the
product α exp(-αt) is equal to One.
The perception, eye or sensor, is relative to the Velocity-of-Light, Unit 2.99792E-8 m / s, exact. This
Value is less than the prime number Value 3. The 3 is the commonly used approximation of the rate at
which energy propagates. Unit Value for a Kelvin, Planck Temperature T(P) is 1.41683E-32K,
approximated as 1.5E-32K. The International reference Temperature is 273.16K, the Triple-point-of-
Water with an uncertainty of 6.00E-6.
To use a 3-color, monochromatic, 2.66 GHz computer to calculate Temperature specific density relative
to 300K we create a 3-dimensional matrix of specific red-green-blue points. Analytical examples (plots)
which follow use the computer Relative Absolute Temperature Color scale for R, G, B; for 1-dimension
Figure 3; for 2-dimensions Figure 4; for 3-dimensions Figure 5; for correlated periodicity Figure 6; and,
for constant rate Figure 7.
5
Figure 3 Absolute 3-color, 1-Dimension, 23 periods of the average absolute Temperature density scale relative to the
Triple-Point-of-Water.
Figure 4 Absolute 3-color, 2-Dimension, 23 periods, time-dependent, average, absolute Temperature density scales
relative to the Triple-Point-of Water.
6
Figure 5 Absolute 300 Temperature-Periods scale of 100K uniform temperature, Relative to SI Unit Values.
Figure 6 Correlated W - s, x = N c / λ, s-1
-6.00E+00
-4.00E+00
-2.00E+00
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
1 5 9
13
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
10
1
10
5
10
9
11
3
11
7
12
1
12
5
12
9
13
3
13
7
14
1
14
5
14
9
15
3
15
7
16
1
16
5
Correlated, W - s = ( 5 x^4 Exp( -x ) ) - ( x^5 Exp( - x ) ), x = N (1 … 10^3) c / λ
Constant Error Bar, +2
7
Figure 7 A mean quantum energy propagation rate per 100 periods.
Convergence – Two binary, exponential series that converge to a singular value 1 as estimate number
n , are Equations (2.2), and (2.3).
2 3
0
11 1
1 1
xx x x x nx
x xn
ee e e e e
e e
(2.2)
2 3
0
11
1
xx x x x nx
x xn
ee e e e e
e e
(2.3)
We note the exponent of the Planck Function is Unit less. It however does have Unit values and Inverse
Unit values Figure 8. Shown is correlated energy and energy, each at an energy rate for 100 periods.
Three 6-degrees of freedom polynomials are shown. The Sum ( x / ( ex – 1 ) ), is shown on the computer
RGB scale: Red ( 187,000,000 ); Green ( 000,224,000 ); and Blue ( 000,000,227 ). Note the periodic rates
converge to +1 at zero and never cross.
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
1 5 91
31
72
12
5
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
101
Am
pli
tud
e
8-bit byte
Mean Quantum Rate = [ ( n - σ )^2 - 1 ] / 2
8
Figure 8 A unit-less exponential function that converges periodically at integer 10.
The Planck function – A Chi-square, exponential series, density distribution with specific Constant
values, Equations (2.2), (2.3) constant value products, energy-meter and meter-temperature. The energy-
distance-scale for Temperature is exponential with specific characteristic density function values C1 and
C2. The quantity C1 a constant scalar energy-distance dependency. The quantity C2 a constant scalar,
distance-temperature dependency. These two constants represent a “capacity” to radiate electromagnetic
energy as well as the inverse of C2 as a “capacity” to transform incident radiation into multiple Unit
Values of temperature, time, length, energy (kinetic, potential). The International System of Units
maintained by (Bureau International des Poids et Mesures, 8th edition, 2006; updated in 2014 ).
Eight power series converge with a limited number of terms, Equations (3.1) through (3.11). Respective,
specific, periodic unit values relative to the velocity of light in Vacuum, c, m / s, 299 792 458 (exact) and
temperature, K, 273.160(06) 1ppm, and distance, m, c, inverse seconds.
24
1 1
2
2 2
16 32 31 1
0
8 4.992482 532 38 7.6 ,10 J m
1/ 1/ 1.438 7770(13) 13 , 10 m K
1/T 1
, / , 299 792
.416833(85)6 ,10 7.0579910 , 1 K
458
P
C c hc ppm
hcC
c m s exact
c ppmk
ppm E64
(3.1)
The Planck equation for spectral radiant intensity, Luminance, is Equation (3.2).
y = 2E-10x6 - 5E-08x5 + 7E-06x4 - 0.0004x3 + 1.014x2 + 1.7961x + 1
R² = 1
y = 2E-10x6 - 7E-08x5 + 8E-06x4 - 0.0005x3 + 0.016x2 + 0.7776x + 1
R² = 1
y = 2E-10x6 - 7E-08x5 + 8E-06x4 - 0.0005x3 + 0.016x2 - 0.2224x + 1
R² = -0.234
0 20 40 60 80 100 120
- 2/10
- 1/10
0
1/10
2/10
3/10
4/10
5/10
6/10
7/10
8/10
9/10
1.000E+00
1.000E+03
1.000E+06
1.000E+09
1.000E+12
1.000E+15
1.000E+18
1.000E+21
1.000E+24
1.000E+27
1.000E+30
1.000E+33
1.000E+36
1.000E+39
1.000E+42
0 20 40 60 80 100 120
x, Value
x,
Fra
ctio
n (
3 /
10
)
Lo
g x
, V
alu
e
x, Value
The Unitless Planck Function Exponent, x ≥1; 6-degrees of freedom
x Sum e-nx Sum ( x / ( ex -1 ) ) ( ex -1 ) / x x / (1 - e-x )
x / ( ex -1 ) Poly. (x Sum e-nx) Poly. (x / (1 - e-x )) Poly. (x / ( ex -1 ))
9
1/2 4 1
5
1, , ,Hz m K
1
, m - Hz
Planck Constant, #
Veocity of Light, m / s
Distance, m
Stefan-Boltzmann Constant, #
Temperature, K
hc
kT
hcL T t
kTe
c
h
c
k
T
(3.2)
Change variables:
2 2 2
2 2 2
2 2 2
2 2 2
, #
2 , m
cos cos , m cos2
cos , m cos2
, m-Hz2
hcx
kT
x y z
x y z
x y z
x y z
x y z
x y z
hhc
(3.3)
Now differentiate with respect to ν and T Equation (3.4) and then integrate. Expand the result as a power
series to obtain specific in-band energy, U(2)-U(1), and in-band energy specific density ρ, Equations (3.5)
, (3.6), (3.7), (3.8), (3.9), (3.10), and (3.11). 2
4
0 0
23
3
0 0 2
1, , 4
2 1
Integrate and T over the range from 0 :
4 1, , ,Hz K m
21
x
kT d TL T
e d dT
k T d TL T
d dTe
(3.4)
5
2 5
12
2 nx
n
M Thc x e
C
m – K5 (3.5)
4
2
412
2 6 6 3nx
n
T eM hc nx nx nx
C n
J – s-1 – m2 (3.6)
10
4
4
12
2q nx
n
M Tc x e
C
K – s-1 – m-1 (3.7)
2 2
24
2 2
2x x
M kc x
Te e
J – s-1 – m2 (3.8)
4
2
412
224 24 12 4
nx
n
M hc T enx nx nx nx
T T C n
K7 – s-1 – J-3 – m-2 (3.9)
3
312
26 6 3
nxq
n
M c T enx nx nx
T T C n
K7 – s-1 – J-4 – m-3 (3.10)
1
0
, J – m – K1
q
q nx
Peakxn
Peak Peak
MM
M xM x eT T e
T T
(3.11)
The Planck equation is a statistical, exponential peak at a specific periodic distance (commonly expressed
as a frequency) relative to the velocity of light, a specific rate expressed as meters / second.
Using Monochromatic (zero-harmonics) computer color (op. cit.) relationships to display results: Figure 9
One energetic quanta propagating in free-space.; and Figure 10. This is usually written as δ(t-t0) for an
event begun at t0.We note there is a continuous, one-quarter period lag for electromagnetic radiation,
(causality), the blue area of Figure 6 Correlated W - s, x = N c / λ, s-1.
11
Figure 9 One energetic quanta propagating in free-space.
Velocity of Light – Expressed as a Fraction, ¼ periods, Log3 Joule per meter-second, specific Spectral
Radiant Power Density values relative to Fraction, ¼ periods, Log2 Wavelength m2, are Figure 10, and
Figure 11. They illustrate 0-to-101 Unit Energy Values for the Equivalent Source values dν / ν, λ / dλ, and
the mean energy ν (t) with specific log scales noted by the Axes Titles.
1, 1.43878E-02
2, 2.87755E-02
3, 4.31633E-02
4, 5.75511E-02
5, 7.19389E-02
6, 8.63266E-02
7, 1.00714E-01
8, 1.15102E-01
9, 1.29490E-01
10, 1.43878E-01
0 20 40 60 80 100 120 140 160 180 200
-6.00E+00
-4.00E+00
-2.00E+00
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
0.0
0.1
1.0
10.0
100.0
1.E+00 1.E+01
Correlated Quanta, W - s
Qu
an
ta R
ate
, J
/ s
Log
Pea
k T
emp
eratu
re,
K
Log Quanta, N ( c / λ ), s-1 - K-1
Planck Correlated Energy
N ( c / λ ) Correlated, W - s
12
Figure 10 Unit Temperature, Frequency, and Distance
Figure 11 Average, Mean Square Energy, and Frequency Values with Constant Temperature T, Kelvin
13
Boltzmann, Stefan-Boltzmann, Planck – Relative to the Triple-point-of-water, tpw, per unit wavelength
per unit frequency plotted as discrete points is illustrated by three plots: Figure 12 “Point” source
Boltzmann Statistics, Constant value is 5.670 373(21) E-8 21 ppm, W-m-2- K4. Figure 13 Stefan-
Boltzmann Statistics plotted relative to tpw = 273.160(11) 1ppm. Figure 14 Planck Statistics plotted
relative to tpw = 273.160(11) 1ppm K (#). We call attention to the convergence to four, independent time
periodic values, photons “mingle”, relative to the velocity of light. The Figure 13 frequency peaks are
negative, source radiant energy reduction.
Figure 12 “Point” source Boltzmann Statistics, Constant value is 5.670 373(21) E-8 21 ppm, W-m-2- K4.
5.67037E-08,
1.00274E-07
5.67037E-08,
1.10302E-07
5.67037E-08,
1.20329E-07
5.67037E-08,
1.30356E-07
0.00000E+00
2.00000E-08
4.00000E-08
6.00000E-08
8.00000E-08
1.00000E-07
1.20000E-07
1.40000E-07
Wavelength, 1 E-6
Sca
led
Bo
ltzm
ann C
onst
ant
Boltzmann Statistics, Peak A = σ / 2 π λ = 5.6703 / ( 2 π 10.1 ) E-2
ε = 1 ε = 1.1 ε = 1.2 ε = 1.3
14
Figure 13 Stefan-Boltzmann Statistics plotted relative to tpw = 273.160(11) 1ppm.
Figure 14 Planck Statistics plotted relative to tpw = 273.160(11) 1ppm K (#)
2, -2.29E+00
18, 2.00E-01
32, 8.53E-01
2, -3.82E+00
18, -1.72E-01
32, 7.84E-01
2, -5.82E+00
18, -6.59E-01
2, -8.40E+00
18, -1.29E+00
32, 5.79E-01
2, 1.08289E-07
18, 4.45188E-07
32, 2.41845E-06
0
0.0000005
0.000001
0.0000015
0.000002
0.0000025
0.000003
-1.00E+01
-8.00E+00
-6.00E+00
-4.00E+00
-2.00E+00
0.00E+00
2.00E+00
0 5 10 15 20 25 30 35
Sp
ectr
al T
emp
erat
ure
, W
/ μ
m /
sr
W / m3
Stefan-Boltzmann, εW / m3 / K4
εK = 1 εK = 1.1 εK = 1.2 εK = 1.3 σ = 5.67000E-08 σ / λ / 2 / π
0.01, 0.00E+00
11.00, 6.21E-30
25.00, 1.69E-30
73.00, 5.43E-32 100.00, 1.71786E-32
0.01, 0.00E+00
11.00, 6.21E-30
12.00, 6.01E-30
56.00, 1.38E-31
73.00, 5.43E-32
99.00, 1.78299E-32
0 1/81 1/27 1/9 1/3 1 3 9 27 81 243
-1.00E-30
0.00E+00
1.00E-30
2.00E-30
3.00E-30
4.00E-30
5.00E-30
6.00E-30
7.00E-30
0.00E+00
1.00E-30
2.00E-30
3.00E-30
4.00E-30
5.00E-30
6.00E-30
7.00E-30
0 1/32 1/8 1/2 2 8 32 128
Cut-off Wavelength { Fraction 21 / 25 }, Log3 λc 273.15K
Po
wer
( 2
73
.15
K )
, W
/ c
m2
/ μ
m /
K /
sr
Po
wer
, W
/ c
m2
/ μ
m /
K /
sr
Cut-off Wavelength Fraction , 21 / 25 Log2 { λc ,μm ( 273.17K ), λc μm ( 273.16K ) }
Bound, Blackbody, Normal Spectral Radiance re:tpw K, 0.01-100, μm
273 1/6 273 1/6 273 1/7
15
Bernoulli Statistics apply to real physical properties for which a bound population N = 1 or greater and
has a fixed ratio, N / N2n / n! Repetitive increment n estimates converge with a finite number of terms
(estimates n) to a mean, μ = 0.500, and mean square value, σ2 = 0.250. The linear incremental, real scale
is +1.000(000) to +25.00(000), ppm. Correlated estimates n = 0.50(11) – 4.50(11) / 9.00(11).
1
1
2
( ) 0,1
0.7
( )
x x
ixt x x
it
f x p q x
t e p q
q pe
p
pq
(4.1)
With a population of 100x fixed value numbers, μ = 50.000, σ = 29.15475947. The mean square Value
and the mean square variance Value, 2 2
/ 2 1( ) 675 , respectively fixed values. Random
computer clock cycle start time is shown Figure 15 Bernoulli numbers, random computer clock cycle
values for R, G, B Scale Values, x, y. The two dimension population average = 15 random computer
clock time numbers, dependent upon the specific clock cycle start for a calculation. Two number sets, red
and green, randomly return order ( 0, 0.# ) while blue returns ( 0, 0 ). RG&B return 100. Computer Clock
calculation start time is chosen randomly when the ctl-Save command is given. The delta time is one
clock period which is ambiguous to a computer as a 2-dimensional sub- matrix, (1, nan) or (nan, 1), nan
being not-a-number. A typical 2-D, Red, Green, Blue map is Figure 15, and the Red, Green, Blue
Histograms, a bar-chart, is Figure 16.
16
Figure 15 Bernoulli numbers, random computer clock cycle values for R, G, B Scale Values, x, y.
Color – We note Unit CIE tri-stimulus luminosity scale, (Technology), R, G, B, is commonly used with
3-vector unit values R = (187, 000, 000), G =(000, 224, 000), B = (000, 000, 227) to create an average
unit value false, spectral-intensity, color scale relative to 0 units, Figure 16 is 20-singluar monochromatic
periods. The contrast scale counts up 0 to 1 and down 1 to 0 by tenths, (0.1). These are real, specific, 3-
dimension Red, Green, Blue values R(187), G(224), B(227), decimal ∑(638) relative to 1 Color Space
f(x,y,z) = 1–x–y–z. These 3-color (actually singular, periodic, real) values are used throughout this paper
to illustrate individual real, physical single-period properties. These three values are from the
International Committee of Illumination (CIE) are appropriate for a singular Temperature u300K,
Illuminant “C”. (Technology)These three singular values are fixed for time and 3-space. Computer
computation start time is a “real” integer value. Random file save commands start on a random computer
clock cycle when the calculations are updated prior to a save command execution. For Figure 15 the Red,
Green and Blue change their respective zero order values randomly 0, n concurrent with their 2-D
position on the plot. The variability is random relative to the random clock cycle start value between 0
and 99 periods. For Figure 16 the colored areas are used by the computer as monochromatic energy,
increment 0.1. The 1-0-1 eight-bit byte exponential value scale is 25 + 1 = 1, 21 and 25 – 1 = 1, 0 or 0, 1.
The latter is ambiguous. These specific values lead to ambiguous results relative to computer clock values
(1, nan) and (nan, 1) when real data is coded counting up and decoded counting down. The bit shift
between counting up and counting down results in a correct average computer-clock-time-dependent
value, but an incorrect (negative random computer-clock-time-dependent value) there is a distribution of
clock values. Radiant Power, One-period, Wavelength-Temperature Product – The value of the
normal, specific frequency at specific integer 100K temperature is shown by Figure 17 Normal Energy,
Specific Density, J – s.
0, 29 100, 39.50, 15
100, 7.5
0, 0
100, 732.5042662
0.00E+00
1.00E+02
2.00E+02
3.00E+02
4.00E+02
5.00E+02
6.00E+02
7.00E+02
8.00E+02
0 20 40 60 80 100 120
SI
Un
it V
alu
e, #
Estimate n, #
Time correlated True =15
T + 2 σ / 4 Random 0-99 / 4 n ( σ / 4 )
17
Figure 16 False Intensity, respective RGB average color for 20-singular monochromatic periods.
Figure 17 Normal Energy, Specific Density, J – s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 19 37 56 75 94 112 131 150 168 187 168 150 131 112 94 75 56 37 19 0022
4567
90112
134157
179202
224202
179157
134112
9067
4522
00
23
45
68
91
114
136
159
182
204
227
204
182
159
136
114
91
68
45
23
00
100
200
300
400
500
600
700
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
ME
AN
CO
LO
R,
%
AVERAGE COLOR INTENSITY NUMBER, #
RGB Color Bit#
Mean I, % R 187 G 224 B 227
100, 1.548607
-99, 1.548690
0, 0.000000
100, 0.451393
-99, 0.451310
0.000000, 2.000000
0.836080, 1.548607
0.000000
0.500000
1.000000
1.500000
2.000000
2.500000
0.000000
0.500000
1.000000
1.500000
2.000000
2.500000
-150 -100 -50 0 50 100 150
Sp
atia
l F
req
uen
cy,
Hz
/ r
Estimation Frequency, 1 / s
Normal Spatial Visibility, Average Specific Density, T = 100 K
I (ω) max , ( 1 / s ) ( 1 / K ) I (ω) min , ( 1 / s ) ( 1 / K ) Period = Sin x / N
18
Homogenous, in-Homogenous Energy Density – Specific substances, A(Eh) B(Eh) are with specific
quantum energy states assigned their respective Atomic Number density of states. However, Temperature
is continuous. Two mathematical representations are applied to describe homogenous continuous
periodicity and in-homogenous discrete periodicity, common origin, Equation (5.1). The two expressions
respectively are the sphere, S, and the complete elliptical integral, E, illustrated by Figure 18 Constrained
Periodic Information and Figure 19 The respective projected surface areas are a continuous circle and a
continuous parabola, each respective radius of curvature, circle c, parabola p. value Rc (c, p ) is constant
with time. The parabolic curvature is a product of the major-minor axes, ab. The circle curvature is r.
Periodic frequency is further illustrated by Figure 21 Periodic, single valued information frequency,
Figure 22 Space-time bandwidth, periodic energy quanta and Figure 23 Bound, periodic energy
frequency. These Figures further illustrate periodic energy frequency. Note the absence of information, 4th
quadrant, Figure 20 Stationary parabolic periods and 1st Half-period, Figure 21 Periodic, single valued
information frequency.
Bound and Constrained Periodic Information – The complete Elliptical Integral has a projected area A
given by the projected area, π ab when the semi-axes are 4 a E exactly, Equation (5.1). A sequence of
Figures follow to illustrate the consequences when energy is bound and constrained.
19
2 2
2 2
0
2 2x
20
2
Given an Ellipse, E, with semi-axes a,b.
a bC(circumference ) 2
2
The Elliptic integral, E, second kind is:
x sin
E ,k 1 k sin d
1 k zE ,k dx
1 z
The complete Elliptic integral, K, is:
dK
1 k s
/2
20
/22 2
0
2 2
2 2
F k,2in
E 1 k sin d E k,2
F k, E k, 2k,2 2 4
F k, E k, 02 2
When semi-axes a, b are 4aE exactly then:
a b2k
2a
a b2k
2b
and the circumscribed e
lliptical Area, A, is exactly:
A ab
(5.1)
20
Figure 18 Constrained Periodic Information
Figure 19 Bound Periodic Energy
3.33E-01, 4.36E-03
4.08E-17, -2.50E-01
-3.33E-01, 4.36E-03
-8.17E-17, 2.50E-01
1.53E-17, 3.33E-01
-2.50E-01, 4.08E-17
-4.59E-17, -3.33E-01
2.50E-01, -8.17E-17
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
n C
os
( x )
/ 4
, n (
Sin
x )
/ 3
n ( Sin x ) / 3 , n ( Cos x ) / 4
3-Dimensionsal Surfaces, Constrained Constant Periodicity
n ( Cos x ) / 4 n ( Sin x ) / 3
3.33E-01, 1.53E-17
-3.33E-01, -4.59E-17
-4.36E-03, 3.33E-01
-2.50E-01, 4.08E-17
-4.59E-17, -3.33E-
01
2.50E-01, -8.17E-17
-40.00% -20.00% 0.00% 20.00% 40.00%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
-40.00% -20.00% 0.00% 20.00% 40.00%
-30%
-20%
-10%
0%
10%
20%
30%
n ( Cos x ) / 4
n (
Sin
x )
/ 3
n ( Sin x ) / 3
n (
Co
s x )
/ 4
2-Dimensional Closed Integral Surfaces Bound ( x, y )( 0 → 400 radians )
n ( Cos x ) / 4 n ( Sin x ) / 3
21
Figure 20 Stationary parabolic periods
Figure 21 Periodic, single valued information frequency
1.50000, 0.00000
0.00000, -4.50000
-1.50000, 0.00000
-8
-6
-4
-2
0
2
4
6
8
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Rad
ians
Radians
Staionary Parabolic Periods0.253125 Cos x 0.51625 Cos x 1.125 Cos x 2.25 Cos x
4.5 Cos x 6 Sin x 5 Sin x 4 Sin x
-3E+136
-2E+136
-1E+136
0
1E+136
2E+136
3E+136
1
21
41
61
81
10
1
12
1
14
1
16
1
18
1
20
1
22
1
24
1
26
1
28
1
30
1
30
9
28
9
26
9
24
9
22
9
20
9
18
9
16
9
14
9
12
9
10
9
89
69
49
29 9
Am
pli
tud
e, #
Period Δn ( c / λ ), # m - s
Periodic Energy Frequency, x ( c / λ )x Exp -( x + 1 ) / n -x Exp ( x - 1 ) / n
22
Figure 22 Space-time bandwidth, periodic energy quanta
Figure 23 Bound, periodic energy frequency
-6
-4
-2
0
2
4
6
-7
-5
-3
-1
1
3
5
7
1
12
23
34
45
56
67
78
89
10
0
11
1
12
2
13
3
14
4
15
5
16
6
17
7
18
8
19
9
21
0
22
1
23
2
24
3
25
4
26
5
27
6
28
7
29
8
30
9
31
0
29
9
28
8
27
7
26
6
25
5
24
4
23
3
n S
in x
n C
os
xSin x 1.5 Sin x 2 Sin x 3 Sin x 4 Sin x 5 Sin x 6 Sin x
Cos x 2 Cos x 2.25 Cos x 4.5 Cos x 4 Cos x 5 Cos x 15 Cos x
-50
-40
-30
-20
-10
0
10
20
30
40
50
1
13
25
37
49
61
73
85
97
109
121
133
145
157
169
181
193
205
217
229
241
253
265
277
289
301
313
305
293
281
269
257
245
233
Am
pli
tud
e, L
og
2
Period, Radians
Constant Bandwidth, Amplitude n Period, Cos x
15 Cos x 5 Cos x 4.5 Cos x 4 Cos x 3 Cos x 2.25 Cos x 2 Cos x 1.5 Cos x
23
Temperature and Power Relative to the Triple Point of Water – One-hundred units of Spectral
Radiant Power specific density relative to the Triple-Point of Water 273.16K are shown by Figure 24 and
Figure 25. The temperature scale, K, is a continuous source of electromagnetic energy emission. The
specific radiant substance A(Eh), H2O, specific state 273.15K. These figures illustrate power specific
density of Eh(H2O), Z18 as the specific illumination frequency received from another substance B(Eh).
Figure 24 Relative Spectral Radiance.
1, 6.99E+03
1, 2.02E+03
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Rel
ativ
e In
ten
sity
Fra
ctio
n (
3 /
1
0 )
, W
-cm
2 -
μm
-1 -
sr-1
Fraction ( 3 / 10 ) μm - K
Third-order Relative Intensity, (1 … 6991)μm - K
= Max, μm = Std Dev Array
24
Figure 25 Energy specific density relataive to the Triple-Point of Water.
Visibility – Statistical Optics/Coherence theory recognizes Visibility as an energy-density-ratio, an
ensemble of time-space variant correlated energy written as 1 1 2 2, . Electromagnetic radiation
is linear shift-invariant and complex auto-correlated by an optical system, Figure 26. A computer false
color Red-Blue scale is Figure 27 (bit wise monochromatic). These two Figures are 8-bit bitmaps. Note
Information is correlated and Zero Order field lines do not cross.
2.02E+03, 6.99E+03 2.02E+03, 6.99E+03
6.99E+03, 2.02E+03 6.99E+03, 2.02E+03
1 3 9 27 81
0
500
1000
1500
2000
2500
0
1000
2000
3000
4000
5000
6000
7000
8000
1 10 100
Log3 { Peak Temperature Fraction ( 3 / 10 ) }, K / m
{ S
td D
ev F
ract
ion (
3 /
10
) }
, K
{ P
eak T
emp
erat
ure
Fra
ctio
n (
3 /
10
) }
, K
Log { Peak Temperature Fraction ( 3 /10 ) }, K / m
Third-order Statistics, (1 … 6991)μm - K
= Max, μm = Std Dev Array
25
Figure 26 Correlated Information.
Figure 27 Real 8-bit values for RB on the RGB Illuminant Scale
y = 2E-09x6 - 6E-09x5 + 0.25x4 - 3E-09x3 + 5E-09x2 - 1.25x + 1
R² = 1
y = 1E-08x6 - 3E-08x5 + 0.5x4 - 2E-08x3 + 6E-09x2 - 2.5x + 2
R² = 1
0.00 0.20 0.40 0.60 0.80 1.00 1.20
-0.50
0.00
0.50
1.00
1.50
2.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Reverse Order, Argument, x
Co
rrel
ated
Rad
iant
Ener
gy
Co
rrel
atio
n B
and
wid
th,
m -
Hz
Argument, x
Periodic Correlated Electromagnetic Radiationf ( x ), Correlation 2 f ( x / 2 ), Correlated Poly. (2 f ( x / 2 ), Correlated )
1.50, 0.20
1.50, -0.20
y = -4.9352x6 + 31.471x5 - 77.167x4 + 90.819x3 - 51.923x2 + 12.216x
R² = 0.4117
y = 4.9352x6 - 31.471x5 + 77.167x4 - 90.819x3 + 51.923x2 - 12.216x
R² = 0.4117
0.000.501.001.502.002.503.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Blue periodicity = 227 ( 8-bits )
Blu
e b
it x
-val
ue
Red
bit
x-v
alu
e
Red peiodicity = 187 ( 7-bits )
Real 8-bit, R, B Values
r1 (ν), r2 (ν) Period = x / 2 r1 (ν), r2 (ν) Period = - x / 2
Poly. (r1 (ν), r2 (ν) Period = x / 2) Poly. (r1 (ν), r2 (ν) Period = - x / 2)
26
Figure 28 Periodic, correlated frequency, ν.
Bessel Functions – The first kind are symmetric in frequency illustrated by Figure 29 Frequency density
plus and minus 100 periods of frequency. and, by Figure 30 Potential electromagnetic energy, eV, concise
electron mass equivalent energy, 8.187 105 06(36) x 10-14 J. Note the frequency scale offsets, 100 periods
and 15 periods respectively. Note also the spatial and temporal scales over-shoot, an under-damped
periodic system of values, causality at A and B are random in time, respectively complex-auto-correlated
units of energy emitted or sensed. It is important to recall the Bessel function describes a frequency
envelope Value centered at 1 and period + / - 3 Units of Frequency.
0.00, 0.00
1.00, -1.00
0.00, 1.00
1.00, 0.00
y = -2E-09x6 + 5E-09x5 - 7E-09x4 + 3E-09x3 + 3E-09x2 - 1x
R² = 1
y = 2E-09x6 - 6E-09x5 + 0.25x4 - 3E-09x3 + 5E-09x2 - 1.25x + 1
R² = 1
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Per
iod
icit
y (
+ /
-x )
, #
Period x, #
Correlated Periodicity
r1 (ν), r2 (ν) Period = - x f (x) = ( 5 (1 - x ) -1+ x^4 ) / 4
Poly. (r1 (ν), r2 (ν) Period = - x) Poly. (f (x) = ( 5 (1 - x ) -1+ x^4 ) / 4)
27
Figure 29 Frequency density plus and minus 100 periods of frequency.
Figure 30 Periodic Energy Density
0.20, 0.50
2.00, 1.33 2.00, 3.32E-01 -2.00, 3.32E-01
-100.00-50.000.0050.00100.00150.00200.00
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
2.50E-01
3.00E-01
3.50E-01
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-100.00 -50.00 0.00 50.00 100.00 150.00 200.00
Reverse Order { Log2 [ 2 π ω ( t - t0 ) }
Pow
er,
J /
s
En
ergy,
J /
s
Log2 [ 2 π ω ( t - t0) ]
Lambetian Source Energy Distribution, + π sr
Phase Period = J0 ( x ) / 2 Energy E = [ 2 J1 ( x ) ]^2 , J / s / sr Power = [1 - 2 Cos x ]^2 / 4 , J / s / sr
12, 0.17210, 0.17
12, 1.16 210, 1.16
12, 1.33 210, 1.330446016
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1 6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
126
131
136
141
146
151
156
161
166
171
176
181
186
191
196
201
206
211
Inte
nsi
ty A
mp
litu
de,
N /
2
Period, N
Half-Energy Density, 0 - 101 N / 2
Half-Energy E / 2 = [ J1 ( x ) ]^2 / 2 , J / s / sr Δ Intensity = ( E - E / 2 ) Energy E = [ 2 J1 ( x ) ]^2 , J / s / sr
28
Figure 31 Relative Energy Intensity
Figure 32 Spectral Intensity Envelope
-2, 0.6652 2, 0.6652
0, 1.0000
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-100 -80 -60 -40 -20 0 20 40 60 80 100
Inte
nsi
ty A
mp
litu
de,
I (
t )
Period ( x ), m - s
Relative Intensity[ Bessel J1 (x) ]^2 [ Bessel J0 (x) ]^2
0.0000
0.5000
1.0000
1.5000
2.0000
2.5000
-100 -80 -60 -40 -20 0 20 40 60 80 100
Sca
lar
Inte
nsi
ty
Periodic Scale, (Frequency x )
Relative Spectral Intensity
[ Bessel J1 (x) ]^2 + 2*[ Bessel J0 (x) ]^2 2 [ Bessel J1 ( x ) ]^2
ABS { [ Bessel J1 (x) ]^2 - 2*[ Bessel J0 (x) ]^2 }
29
Recommended Calibration
Calibrate an Optical Sensor relative to the Triple-Point of Water!
It is suggested that a power spectral density is an adequate estimate of either a passive or active remote
sensor, periodic energy, eV 0.5110 041 (16) MeV 3.1 ppm, and, 931.4812 MeV, 5.5ppm. The peak-
power, a frequency of period ν, easily saturates a sensor without causing damage. Long-half lives and
accordingly persistent energy with time result in a single photon undergoing multiple reflections and
transmission periods before its energy depletion is less than the continuum, a sensitivity reduction.
Momentum exchange between an energetic photon and a molecular mass ensemble of substance is
significantly weak, 1-Joule is equivalent to 1.602 176 565(35) E(-19)eV.
The suggestion is made that a Performance ratio, 1-minus a ratio of a mean signal to mean-square signal
is appropriate to periodically monitor a sensor stability (Marathay, et al., 2010). Amplitude is easily
saturated and as Figure 29, Figure 30, and, Figure 31 show there is an intensity pole, a “hole” on the
sensor optical axis. At 3200x digital image magnification a white square embedded on a black
background appears at the center of a data frame. The conclusion is an array-average to array-mean-
square value, a ratio, / 2Expectation 2 , is an effective performance estimator relative to the velocity
of light and temperature (the triple point of water).
The impact on a calibration instrument or chamber with 4-π sr temperature inhomogeneity greater than
0.01K is significant for a calibration 0.01ppm. As a practical matter this isn’t realistic. However, when the
magnitude of an array average is greater than 90 relative to a mean-square-average noise less than 10 for a
single data frame, a valid inhomogeneity estimate is made. The average value greater than 90 and mean-
square value less than 10 form a statistical power / per data frame ratio relative to the velocity-of-light in
a vacuum. An array average value is an intrinsic, dc electrical current noise off-set numerically added to a
peak energy frequency estimate. Average energy Peaks greater than 90, steady state oscillating field,
periodically capacity sampled at some impedance Z0 is a steady state current, dc. Subtraction corrects
(removes) the intrinsic steady state dc term pixel by pixel when an array average signal is less than 10.
Residuals greater than 7-σ are either dead, inhomogeneous, or non-linear pixels. Dynamic contrast bounds
signals greater than 90 when kT noise is less than 10. One Joule meter mol-1, NA hc, is concisely 0.119
626 565 779(84)i . The ratio (differential) of the inverse Planck temperature value, (TP)-1 and (TP) is
1E+64. While the ratio (differential) of ( h c / k )-1 and ( h c / k ) is 100 000 000.9.
Typical analyzed data illustrating the points made herein (Statistical Calibration, Relative to the Triple
Point of Water, 2014), is available from the CALCON 2014 Proceedings, see specifically the analyzed
data shown on page 96. The experiment configuration and details appear on slide 67.
30
References Bureau International des Poids et Mesures. 8th edition, 2006; updated in 2014 . BIPM - SI base units.
[Online] 8th edition, 2006; updated in 2014 . http://www.bipm.org/en/measurement-units/base-units.html.
Dr. Taylor, Barry N.; Dr. Mohr, Peter J.; Douma, Michael. 2010. The NISTReference on Constant,
Units, and Uncertainty. Physical Measurement Laboratory of NIST, National Institute of Standards and
Technology. Gaithersburg : Department of Commerce, 2010. CODATA analysis of 2014.
Marathay, Arvind S., McCalmont, John E. and Pollock, David B. 2010. Chapter 7; Radiometry, Wave
Optics, and Spatial Coherence. [book auth.] Markus Testorf, Bryan Hennelly and Jorge Ojeda-Castaeda.
[ed.] Taisuke Soda. Phase Space Optics. s.l. : McGraw-Hill, 2010, pp. 217 - 236.
Statistical Calibration, Relative to the Triple Point of Water. Pollock, D. B. 2014. Logan, UT : Space
Dynamics Laboratory, 2014. CALCON 2014.
Technology, Rochester Institue of. http://www.cs.rit.edu/~ncs/color/t_chroma.html. [Online] [Cited:
July 28, 2015.]
31
i See URL - http://www.bipm.org/en/measurement-units/base-units.html
And URL - http://physics.nist.gov/cuu/index.html
CO-Data 2014
Normal Value Standard Uncertainty Relative Std Unc,
Sci Concise Reciprocal Std Unc, Fraction
J m mol-1, NA hc = 0.119626566 0.000 000 000 084 1.00E-10 0.119 626 565 779(84) 8 1/3
m s-1, c, c0 = 2.99792E-08 exact exact 2.99792E-08 33356409 1/2
Ω, Z 0 = 376.7303135 exact exact 376.7303135 0
1 / 6, TP / σ = 2.49866E-25 1.67E+00 4002146336230170000000000
c0 / Ω Z0 = 7.95775E-11 Exact 12566370614 1/3
W m-2 K-4, σ = 5.67037E-08 0.000 021 x 10E(-8) 3.60E-06 5.670 373(21) x 10(-8) 17635524 1/7
K, TP = 1.41683E-32 0.000 085 x 10E(32) 6.00E-06 1.416 833(85) x 10(32) 70579948377825800000000000000000
m K , c2 = 0.01438777 0.000 0013 x 10E(-2) 9.10E-07 1.438 7770(13) x 10(-2) 69 1/2
m-1 k -1, k / h c = 69.503476 0.000 063 9.10E-07 69.503 476(63) 0
Hz K-1, k / h = 20836618000 0.000 0019 x 10E(10) 9.10E-07 2.083 6618(19) x 10E(10) 0
J K-1, k = 1.38065E-23 0.000 0013 x 10E(-23) 9.10E-07 1.380 6488(13) x 10(-23) 72429715652525100000000
J s, h = 6.62607E-34 0.000 000 29 x 10(-34) 4.40E-08 6.626 069 57(29) x 10E(-34) 1509190311746150000000000000000000
W m2 sr-1, c1L = 1.19104E-16 0.000 000 053 x 10E(-16) 4.40E-08 1.191 042 869(53) x 10(-16) 8396003418748480
J - s, h-bar = 1.05457E-34 0.000 000 047 x 10E-(34) 4.40E-08 1.054 571 726(47) x 10(-34) 9482522386533240000000000000000000
J, eV = 1.60218E-19 0.000 000 035 x 10E(-19) 2.20E-08 1.602 176 565(35) x 10(-19) 6241509343260180000
A J-1, e / h = 2.41799E+14 0.000 000 053 x 10(14) 2.20E-08 2.417 989 348(53) x 10(14) 0
32
Other Ratios
m k , hc / k = 0.01438777 1.59E+04 1.10E+06 1.438776948E-10 69 1/2
K-1, TP-1 = 7.05799E+31 5.26316E-05 1.67E+05 7.05799E-33 0
J-1 s-1 , h-1 = 1.50919E+33 3.44828E-28 2.27E+07 6.62607E-34 0
J, 8 π hc = 4.99248E-40 4.992 482 532 (25)E-40 π 2.00301E+39 2003011514892740000000000000000000000000
m2 s, c / λ2 = 2.99792E-10 1.00E+01 3335640952 3335640952
J sr-1 s-1, 16 hc / 2π = 5.05844E-41 1 / π 1.97689E+40 19768931262818000000000000000000000000000
3 m k , hc / k = 0.043163309 3.00E+00 23.16782533 23 1/6
1, h c / λ k TP = 2.77391E+32 1.00E+00 3.60502E-33 0
4 J m Hz-1, hc = 7.94578E-41 4.00E+01 1.25853E+40 12585292520485600000000000000000000000000
2 T(0) - σ, K-8 = 546.3181178 2.00E+00 0.001830435 0
2 T(0), K = 546.320002 2.00E+00 0.001830429 0
Δ K, = 0.001884198 Δ.00E+00 530.7297013 530 3/4
http://us.wow.com/wiki/Triple_point?s_chn=11&s_pt=source2&type=content&v_t=content
http://www.bipm.org/en/measurement-units/base-units.html
The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
Statistical Calibration
Relative to the Meter and the Triple
Point of WaterDavid B. Pollock
Associate Research Professor / Senior Research Scientist
Electrical Computer Engineering / Center for Applied Optics
University of Alabama, 301 Sparkman Drive, OB444
Huntsville, AL 35899 (256) 824-2514 Email: [email protected]
08 - 24 - 2015 1
Accurate
Radiometry or Thermometry?• Radiometry – Mean Square Spectral Energy Intensity incident a
Substance A from a Substance B. Inversely B from A.
• Thermometry – Mean Quadratic Thermal Energy Density Differential dTa (A) / dTb ( B )
• Time Standard – Cesium Isotope, 1 / s
• Temperature Standard – Triple-Point-of-Water, 1 / 273.16K
• Length Standard – c m-s exact.
• Standard Frequency – c m-s-1 exact.
• Relative Standard Uncertainty, Random Periodic Difference
– Time c d(t-t0), 1 / s
– dT / T, 1 / K
– dL / L, 1/ m
• Accuracy – RSS { Time2 + Temperature2 + Length2 }
08 - 24 - 2015 2
08 - 24 - 2015 3
Frequency Velocity of Light c,
Nth Order Bound Quantum Force
2
2
0
2
0 2
1
2 ( )
2 ( )
c
cd dc d
cd c dc d
d f t t t
dc cf t t t c d
2 3 4 5 60
-1
1
1 1 2 6 24 61
1 =
-1 ;
!0 & = ;
positive integer, 0n
n axn
n
Information
a a a a a a
nx e dx
an
na
a
n a
1 2q q;
2
0
2
Force F = F = m a4 r
Energy E = mc
08 - 24 - 2015 4
0.90, 1.68E+55
1.00, 5.00E-01 1.10, 5.00E-01
y = 5E+53x6 - 6E+54x5 + 3E+55x4 - 5E+55x3 + 4E+55x2 - 5E+54x + 0.1
R² = 0.2236
-4.00E+54
-2.00E+54
0.00E+00
2.00E+54
4.00E+54
6.00E+54
8.00E+54
1.00E+55
1.20E+55
1.40E+55
1.60E+55
1.80E+55
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50
Po
wer
Sp
ectr
al S
pec
ific
Den
sity
, W
/ c
m2
/ μ
m /
sr
@ 2
73
.16
K
Wavelength, ( 0.09 … 3.90 )μm
Energy Specific Density Distribution @ TPW, 273.160(04)K
0.00E+00 273.16 Poly. (0.00E+00 273.16)
08 - 24 - 2015 5
1, 1.000000
2, 1.452623
3, 0.640985
3, 0.576725
3, -0.576725
1, 1.000000
-100-50050100
-60.000000%
-40.000000%
-20.000000%
0.000000%
20.000000%
40.000000%
60.000000%
80.000000%
100.000000%
120.000000%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
140.00%
160.00%
-100 -50 0 50 100
Specific Value Unit ( x ), u + / - 100.00(1,0) u
Sp
ecif
ic V
alu
e, u
, %
Sp
ecif
ic D
ensi
ty, u
, %
Specific Density Unit ( x ), u - / + 100.00(0,1) u
Bound Information x, + / - 100u
∆^2 = { J1 ( x,1 ) / x } { J1 ( x,0 ) / x }
08 - 24 - 2015 6
1, 2
1, 3
0, -1
1, 511
0 1/5 2/5 3/5 4/5 1 1 1/5
-1.0E+00
9.9E+01
2.0E+02
3.0E+02
4.0E+02
5.0E+02
6.0E+02
0.0E+00
1.0E+00
2.0E+00
3.0E+00
4.0E+00
5.0E+00
6.0E+00
0 1/5 2/5 3/5 4/5 1 1 1/5
Fraction
8-b
it C
ounts
, +
/ -
1
Pea
k-t
o-P
eak I
nfo
rmat
ion
, 8
-bit
Byte
Fraction
E-field Information
[ Peak - ( Counts - 1 ) ] / 2 Peak-to-Peak ∆ | Counts | ∆Counts - 2
Electron Mass, 1 Exp-8 eV = 0.5110041(16) MeV 3.1 ppm
08 - 24 - 2015 7
y = 2E-10x6 - 5E-08x5 + 7E-06x4 - 0.0004x3 + 1.014x2 + 1.7961x + 1
R² = 1
y = 2E-10x6 - 7E-08x5 + 8E-06x4 - 0.0005x3 + 0.016x2 + 0.7776x + 1
R² = 1
y = 2E-10x6 - 7E-08x5 + 8E-06x4 - 0.0005x3 + 0.016x2 - 0.2224x + 1
R² = -0.234
0 20 40 60 80 100 120
- 2/10
- 1/10
0
1/10
2/10
3/10
4/10
5/10
6/10
7/10
8/10
9/10
1.000E+001.000E+021.000E+041.000E+061.000E+081.000E+101.000E+121.000E+141.000E+161.000E+181.000E+201.000E+221.000E+241.000E+261.000E+281.000E+301.000E+321.000E+341.000E+361.000E+381.000E+401.000E+421.000E+44
0 20 40 60 80 100 120
x, Value
x, F
ract
ion (
3 /
10
)
Lo
g x
, V
alu
e
x, Value
The Planck Function Exponent, x ≥1u
x Sum e-nx Sum ( x / ( ex -1 ) ) ( ex -1 ) / x x / (1 - e-x )
x / ( ex -1 ) Poly. (x Sum e-nx) Poly. (x / (1 - e-x )) Poly. (x / ( ex -1 ))
08 - 24 - 2015 8
0, 0.001
1, 1.001
2, 2.001
3, 3.001
4, 4.0015, 5.0016, 6.001
7, 7.001
0, 0.00 1, 1.00 2, 2.00 3, 3.00 4, 4.00
5, 5.00 6, 6.00 7, 7.00
0, 0.00E+00
1, 2.73E+00
2, 1.48E+01
3, 6.04E+01
4, 2.19E+02
5, 7.44E+02
6, 2.43E+03
7, 7.70E+03
y = x + 0.001
R² = 1
0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00 8.00E+00
-1.00E+03
0.00E+00
1.00E+03
2.00E+03
3.00E+03
4.00E+03
5.00E+03
6.00E+03
7.00E+03
8.00E+03
9.00E+03 0
1/729
1/243
1/81
1/27
1/9
1/3
1
3
9
012345678
Scientific #
Inte
ger
Sci
enti
fic
Val
ue
Lo
g3
{ I
nte
ger
Fra
ctio
n (
31
2 /
94
3 )
}, #
Integer #
Smoothed, Scientific, Linear Binary Analysis, w / 8-bit ByteA + 0.001 A - 0.001 A exp (A + 0.003) Linear (A + 0.001)
Red = 187, 000, 000
Green = 000, 224, 000
Blue = 000, 000, 227
08 - 24 - 2015 9
3.33E-01, 1.53E-17
-3.33E-01, -4.59E-17
-4.36E-03, 3.33E-01
-2.50E-01, 4.08E-17
-4.59E-17, -3.33E-01
2.50E-01, -8.17E-17
-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
-30%
-20%
-10%
0%
10%
20%
30%
n ( Cos x ) / 4
n (
Sin
x )
/ 3
n ( Sin x ) / 3
n (
Cos
x )
/ 4
2-Dimension, Closed Integral, Bound Surfaces, ( x, y )( 0 → 400 radians )
n ( Cos x ) / 4 n ( Sin x ) / 3
08 - 24 - 2015 10
3.33E-01, 4.36E-03
4.08E-17, -2.50E-01
-3.33E-01, 4.36E-03
-8.17E-17, 2.50E-01
1.53E-17, 3.33E-01
-2.50E-01, 4.08E-17
-4.59E-17, -3.33E-01
2.50E-01, -8.17E-17
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
n C
os
( x
) /
4, n
( S
in x
) /
3
n ( Sin x ) / 3 , n ( Cos x ) / 4
4-Dimension, Constrained Surfaces 4π sr, Constant Periodicity
n ( Cos x ) / 4 n ( Sin x ) / 3
08 - 24 - 2015 11
1.50000, 0.00000
0.00000, -4.50000
-1.50000, 0.00000
-8
-6
-4
-2
0
2
4
6
8
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Rad
ian
s
Radians
3-Periods, Stationary Parabolic Information
0.253125 Cos x 0.51625 Cos x 1.125 Cos x 2.25 Cos x 4.5 Cos x
6 Sin x 5 Sin x 4 Sin x 3 Sin x 1.5 Sin x
08 - 24 - 2015 12
399, 159400.3752100, 33340000.00
0 50 100 150 200 250 300 350 400
0.00
5000000.00
10000000.00
15000000.00
20000000.00
25000000.00
30000000.00
35000000.00
40000000.00
0.00
20,000.00
40,000.00
60,000.00
80,000.00
100,000.00
120,000.00
140,000.00
160,000.00
180,000.00
0 50 100 150 200 250 300 350 400
h / 2 = { | r1 | + | r2 | } / ( 3 π ), m
Vo
lum
e (
r 1-
r 2 )
/ (
π2
h )
, m
2/
m
No
rmal
Pro
ject
ed S
urf
ace
Are
a, h
(r 1
,r2 ) /
π,
m2
h = ( r1 + r2 ) / 2, m
2-Period Stationary Hyper-sine Information
Projected Surface Area Normal ( h, r1 , r2 ) = ( r1 + r2 ) ^ [ h^2 + ( r1 - r2 )^2 ) ^ 0.5, sr
Normal Volume ( h, r1 , r2 ) = { ( r1 )2 + ( r1 r2 )2 + ( r2 )2 } / 3, m^3
08 - 24 - 2015 13
-0.5, -1.25992105
0.5, 1.25992105
-200, 34.19951893
3, 0 198, 0
402, 34.19951893
-300 -200 -100 0 100 200 300 400 500
-5
0
5
10
15
20
25
30
35
40
-200 -150 -100 -50 0 50 100 150 200
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
+ / - x ^ ( - 2 / 3 )
Ex
po
nen
tial
Am
pli
tud
e, +
2 /
3
x ^ ( - 1 / 3 ), #
Val
ue,
#
Accurate Calculations with Exponential Numbers
x^( - 1 / 3 ) = - x^( 2 / 3 ) = + x^( 2 / 3 ) =
08 - 24 - 2015 14
-200, 40000 200, 40000
-0.5, -1.25992105
0.5, 1.25992105
-250.00 -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 250.00
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.00
5,000.00
10,000.00
15,000.00
20,000.00
25,000.00
30,000.00
35,000.00
40,000.00
45,000.00
-250 -200 -150 -100 -50 0 50 100 150 200 250
x^( - 1 / 3 ) Value, #
Ex
po
nen
tial
Am
pli
tud
e x
, V
alu
e, #
Am
pli
tud
e x
^2
Val
ue,
#
x^( 2 ) Value, #
Accurate Calculations with Exponential, Periods
x^(2) = x^( - 1 / 3 ) =
08 - 24 - 2015 15
-3, 2.55E+101 3, 2.55E+101
-11 -9 -7 -5 -3 -1 1 3 5
0.00E+00
5.00E+100
1.00E+101
1.50E+101
2.00E+101
2.50E+101
3.00E+101
1.0000E-01
1.0000E+04
1.0000E+09
1.0000E+14
1.0000E+19
1.0000E+24
1.0000E+29
1.0000E+34
1.0000E+39
1.0000E+44
1.0000E+49
1.0000E+54
1.0000E+59
1.0000E+64
1.0000E+69
1.0000E+74
1.0000E+79
1.0000E+84
-10 -5 0 5 10
2N + N2
Mea
n F
lux
= 1
, V
aria
nce
= 1
.41
42
1
Lo
g {
Gau
ssia
n F
lux
, P
h /
s }
Estimate Variance Number, #N
Statistical Distribution Photon Flux, N
Variance = 1 Variance = 2 Variance = 3 Variance = 4Variance = 5 Variance = 6 Variance = 7 Variance = 8
08 - 24 - 2015 16
4.80E+01, 2.55E+101 1.68E+02, 2.55E+101
1.00E+00 3.00E+00 9.00E+00 2.70E+01 8.10E+01 2.43E+02 7.29E+02
0.00E+00
5.00E+100
1.00E+101
1.50E+101
2.00E+101
2.50E+101
3.00E+101
2.38419E-074.76837E-079.53674E-071.90735E-063.81470E-067.62939E-061.52588E-053.05176E-056.10352E-051.22070E-042.44141E-044.88281E-049.76563E-041.95313E-033.90625E-037.81250E-031.56250E-023.12500E-026.25000E-021.25000E-012.50000E-015.00000E-011.00000E+002.00000E+004.00000E+008.00000E+001.60000E+013.20000E+016.40000E+011.28000E+022.56000E+02
3.0 9.0 27.0 81.0 243.0
Log 3 { 2 N + N 2 }, #
Lo
g 3
{ μ
= 1
, σ
= 2
0.5
}
Lo
g 2
{ F
lux
Den
sity
}#
-m
2
Log3 { 2 N + N2 }, #
Flux Density, n - m2
μ = σ = 0 μ = σ = 1 μ = σ = 2 μ = σ = 3 μ = σ = 4 μ = σ = 5
μ = σ = 6 μ = σ = 7 μ = σ = 8 μ = σ = 9 μ = σ = 10 μ, σ = 1, 2^0.5
Integral xn e-ax dx = Г (n +1 ) / a n+1
n > -1, a > 0 & = n! / a n+1
n positive integer, a > 0
08 - 24 - 2015 17
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0019
3756
7594
112131
150168
187168
150131
11294
7556
3719
00
22
45
67
90
112
134
157
179
202
224
202
179
157
134
112
90
67
45
22
00
23
45
68
91
114
136
159
182
204
227
204
182
159
136
114
91
68
45
23
00
100
200
300
400
500
600
700
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
ME
AN
CO
LO
R,
%
AVERAGE COLOR INTENSITY NUMBER, #
RGB Color Bit#
Mean I, % R 187 G 224 B 227
08 - 24 - 2015 18
1 2 3 4 5
= Population Average 5.05E+01 3.57E+01 5.15E+01 3.52E+03 9.56E+03
= Population Stadard Deviation 2.93E+01 2.07E+01 2.93E+01 3.11E+03 8.46E+03
= Avg / Std 1.72E+00 1.72E+00 1.76E+00 1.13E+00 1.13E+00
1, 1.72E+00 2, 1.72E+003, 1.76E+00
4, 1.13E+00 5, 1.13E+00
0.00E+00
5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
-4.00E+03
-2.00E+03
0.00E+00
2.00E+03
4.00E+03
6.00E+03
8.00E+03
1.00E+04
1.20E+04
1.40E+04
AV
ER
AG
E J
/ K
PE
RIO
DIC
ITY
OF
L
IGH
T,
M /
S
AXIS TITLE
Energy Specfic Density Distribution Relative to
The Velocity of Light SI
( Population 0 - 101 ), J / m / s / K
= Population Average = Population Stadard Deviation = Avg / Std
08 - 24 - 2015 19
1 - Cos N / x
0.000000
0.100000
0.200000
0.300000
0.400000
0.500000
0.600000
0.700000
0.800000
0.900000
1.000000
1
10
19
28
37
46
55
64
73
82
91
10
0
10
9
11
8
12
7
13
6
14
5
15
4
16
3
17
2
18
1
19
0
19
9
20
8
21
7
22
6
23
5
24
4
25
3
26
2
27
1
28
0
28
9
29
8
AM
PL
ITU
DE
, 1
-N
/ X
DIS
TA
NC
E,
M /
S
Visibility
∆I( ω )min , 100.00(10)
0.000000-0.100000 0.100000-0.200000 0.200000-0.300000 0.300000-0.400000 0.400000-0.500000
0.500000-0.600000 0.600000-0.700000 0.700000-0.800000 0.800000-0.900000 0.900000-1.000000
08 - 24 - 2015 20
-6.00E+00
-4.00E+00
-2.00E+00
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
1 6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
10
1
10
6
11
1
11
6
12
1
12
6
13
1
13
6
14
1
14
6
15
1
15
6
16
1
Correlated, W - s = ( 5 x^4 Exp( -x ) ) - ( x^5 Exp( - x ) ), x = N (1 … 10^3) c / λ
Constant Error Bar, +2
08 - 24 - 2015 21
-1.000000
-0.500000
0.000000
0.500000
1.000000
1.500000
2.000000
1 8
15
22
29
36
43
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
155
162
169
176
183
190
197
204
211
218
225
232
239
246
253
260
267
274
281
288
295
Per
iod N
Period, x / N
300 Temperature Periods, Frequency Constant, Uniform Temperature, 100K
Maximum Energy Frequency = 1 + Cos x / N Period = Sin x / N Minimum Energy Frequency = 1 - Cos N / x ∆ν, m / s = 1
f { x } = ( h c / λ k T )
Concluding Remarks
• Accuracy is Statistical Relative Intensity, root 2 / 2
• One binary count is the mean square uncertainty of information content (energy).
• Contrast 0-to-One on a binary scale, one-bit of an 8-bit Byte, 1:210 , 1:1024
• Frequency-period, substance, time, space, energy.
• Substance, hartree, molecular radiation/absorption, energy (mass, qm),
• Avogadro’s Number N = 6.022169(40) 1023 mole-1
08 - 24 - 2015 22
Statistical Calibration
Relative to the Meter and the Triple
Point of Water
Supplement
08 - 24 - 2015 23
Units SI and derived
08 - 24 - 2015 24
C0-Data 2014
Normal Value Standard Uncertainty Relative Std Unc, Sci Concise Reciprocal Std Unc, Fraction
J m mol-1, NA hc = 0.119626566 0.000 000 000 084 1.00E-10 0.119 626 565 779(84) 8 1/3
m s-1, c, c0 = 2.99792E-08 exact exact 2.99792E-08 33356409 1/2
Ω, Z 0 = 376.7303135 exact exact 376.7303135 0
1 / 6, TP / σ = 2.49866E-25 1.67E+00 4002146336230170000000000
c0 / Ω Z0 = 7.95775E-11 Exact 12566370614 1/3
W m-2 K-4, σ = 5.67037E-08 0.000 021 x 10E(-8) 3.60E-06 5.670 373(21) x 10(-8) 17635524 1/7
K, TP = 1.41683E-32 0.000 085 x 10E(32) 6.00E-06 1.416 833(85) x 10(32) 70579948377825800000000000000000
m K , c2 = 0.01438777 0.000 0013 x 10E(-2) 9.10E-07 1.438 7770(13) x 10(-2) 69 1/2
m-1 k -1, k / h c = 69.503476 0.000 063 9.10E-07 69.503 476(63) 0
Hz K-1, k / h = 20836618000 0.000 0019 x 10E(10) 9.10E-07 2.083 6618(19) x 10E(10) 0
J K-1, k = 1.38065E-23 0.000 0013 x 10E(-23) 9.10E-07 1.380 6488(13) x 10(-23) 72429715652525100000000
J s, h = 6.62607E-34 0.000 000 29 x 10(-34)4.40E-08
6.626 069 57(29) x 10E(-34) 1509190311746150000000000000000000
W m2 sr-1, c1L = 1.19104E-16 0.000 000 053 x 10E(-16)4.40E-08
1.191 042 869(53) x 10(-16) 8396003418748480
J - s, h-bar = 1.05457E-34 0.000 000 047 x 10E-(34)4.40E-08 1.054 571 726(47) x 10(-34)
9482522386533240000000000000000000
J, eV = 1.60218E-19
0.000 000 035 x 10E(-19)2.20E-08 1.602 176 565(35) x 10(-19)
6241509343260180000
A J-1, e / h = 2.41799E+14 0.000 000 053 x 10(14) 2.20E-08 2.417 989 348(53) x 10(14) 0
Other Ratios
m k , hc / k = 0.01438777 1.59E+04 1.10E+06 1.438776948E-10 69 1/2
K-1, TP-1 = 7.05799E+31 5.26316E-05 1.67E+05 7.05799E-33 0
J-1 s-1 , h-1 = 1.50919E+33 3.44828E-28 2.27E+07 6.62607E-34 0
J, 8 π hc = 4.99248E-40 4.992 482 532 (25)E-40 π 2.00301E+39 2003011514892740000000000000000000000000
m2 s, c / λ2 = 2.99792E-10 1.00E+01 3335640952 3335640952
J sr-1 s-1, 16 hc / 2π = 5.05844E-41 1 / π 1.97689E+40 19768931262818000000000000000000000000000
3 m k , hc / k = 0.043163309 3.00E+00 23.16782533 23 1/6
1, h c / λ k TP = 2.77391E+32 1.00E+00 3.60502E-33 0
4 J m Hz-1, hc = 7.94578E-41 4.00E+01 1.25853E+40 12585292520485600000000000000000000000000
2 T(0) - σ, K-8 = 546.3181178 2.00E+00 0.001830435 0
2 T(0), K = 546.320002 2.00E+00 0.001830429 0
Δ K, = 0.001884198 Δ.00E+00 530.7297013 530 3/4
http://us.wow.com/wiki/Triple_point?s_chn=11&s_pt=source2&type=content&v_t=content
http://www.bipm.org/en/measurement-units/base-units.html
The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
08 - 24 - 2015 25
-100
0
100
200
300
400
500
600
123456789101112131415161718192021
RE
LA
TIV
E I
NT
EN
SIT
Y,
C /
2 Λ
COLOR X, Y, Z,
Relative Intensity, CIE Color
∆Counts - 2 [ Peak - ( Counts - 1 ) ] / 2
08 - 24 - 2015 26
0.0
0.4
0.8
1.2
1.6
2.0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Qu
antu
um
rat
e, n
q(t
)
Bndwidth x, #
Complex auto-correlated Contrast
2 [ Bessel J0 ( x / 2 ) ]^2 ( Bandwidth, x / 2 ) / 2 ( Bandwidth, x )
( Bandwidth, 2 x ) / 2 ( Bandwidth 4 x ) ( Bandwidth 3 x )
08 - 24 - 2015 27
-1, -5060532.623 1, -5060532.623
-1, 5060533.6225 1, 5060533.6225
-30 -20 -10 0 10 20 30
-1,000,000.0
0.0
1,000,000.0
2,000,000.0
3,000,000.0
4,000,000.0
5,000,000.0
6,000,000.0
-6,000,000
-5,000,000
-4,000,000
-3,000,000
-2,000,000
-1,000,000
0
1,000,000
-30 -20 -10 0 10 20 30
Space-Time Estimate, #
Co
mp
lex
Co
rrel
ated
Sp
ectr
al R
adia
nce
, J
-s
-m
Co
mp
lex
Co
rela
ted I
nte
nsi
ty, m
-s
/ sr
Peak Intensity, ν( t ) = c / λ
Periodic Information, N
Spatial Period Spatial -Temporal Period 2 [ Bessel J0 (x) ]^ 2 Temporal Period
08 - 24 - 2015 28
-2, 0.6652
0, 0.0000
2, 0.6652
0, 1.0000
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-60 -40 -20 0 20 40 60
Inte
nsi
ty A
mp
litu
de,
I (
t )
Period ( x ), m - s
Relative Intensity
[ Bessel J1 (x) ]^2 [ Bessel J0 (x) ]^2
08 - 24 - 2015 29
0, 0.00E+00
2, -0.1535-2, -0.1535
-50 -30 -10 10 30 50
-2.0E+89
0.0E+00
2.0E+89
4.0E+89
6.0E+89
8.0E+89
1.0E+90
1.2E+90
1.4E+90
1.6E+90
1.8E+90
2.0E+90-2.0E+00
-1.5E+00
-1.0E+00
-5.0E-01
0.0E+00
5.0E-01
1.0E+00
1.5E+00
2.0E+00
2.5E+00
3.0E+00
-50 -30 -10 10 30 50
Period, 1 / N
Fre
qu
ency
, 1
/ N
Fre
qu
ency
, N
Period, N
Periodic Rate of Closure, N x m / s
Position error Two-body Separation Cos( J1( x ) ) - 2 J1 ( x )
Two-body Separation Abs ( Cos( J1 ( x ) ) + 2 J1 ( x ) Two-body Separation + ( x / 2 + Cos( x ) )
Two-body Period
08 - 24 - 2015 30
0.09, 3.45E-24
0.10, 2.26E-24
0.20, 1.41E-25
0.30, 2.79E-26
0.09, 3.45E-24
0.10, 2.26E-24
0.20, 1.41E-25
1/10 1/10 3/10 5/10124
-5.00E-25
0.00E+00
5.00E-25
1.00E-24
1.50E-24
2.00E-24
2.50E-24
3.00E-24
3.50E-24
4.00E-24
-5.E-25
0.E+00
5.E-25
1.E-24
2.E-24
2.E-24
3.E-24
3.E-24
4.E-24
4.E-24
1/10 1/10 3/10 5/10 1 2 4
Cut-off Wavelength Fraction ( 3 / 10 }, λc μm
Po
wer
, W
-cm
2-
um
-sr
Po
wer
, W
-cm
2-
μm
-sr
Cut-off Wavelength Fraction ( 3 / 10 }, λc μm
Blackbody, Spatial-Spectral Radiance
Specific Density re: tpw + / - 0.01K, μm
tpw ( 0 ) 0.00E+00 273.16 K 1.00E-04 273.17 K -1.00E-04 273.15 K
08 - 24 - 2015 31
-3E+136
-2E+136
-1E+136
0
1E+136
2E+136
3E+136
1
18
35
52
69
86
10
3
12
0
13
7
15
4
17
1
18
8
20
5
22
2
23
9
25
6
27
3
29
0
30
7
30
6
28
9
27
2
25
5
23
8
22
1
20
4
18
7
17
0
15
3
13
6
11
9
10
2
85
68
51
34
17
Am
pli
tud
e, #
Period Δn ( c / λ ), # m - s
Periodic Energy Frequency, x ( c / λ )x Exp -( x + 1 ) / n -x Exp ( x - 1 ) / n
08 - 24 - 2015 32
-0.600000
-0.400000
-0.200000
0.000000
0.200000
0.400000
0.600000
0.800000
1.000000
1.200000
1
12
23
34
45
56
67
78
89
10
0
11
1
12
2
13
3
14
4
15
5
16
6
17
7
18
8
19
9
21
0
22
1
23
2
24
3
25
4
26
5
27
6
28
7
29
8
Dif
fere
nti
al J
( x
) /
x
0 d
ba,
ev
Radar Harmonic ReturnsSpectral Intensity, W / sr
-0.600000--0.400000 -0.400000--0.200000 -0.200000-0.000000
0.000000-0.200000 0.200000-0.400000 0.400000-0.600000
0.600000-0.800000 0.800000-1.000000 1.000000-1.200000
08 - 24 - 2015 33
-5.00000
-4.00000
-3.00000
-2.00000
-1.00000
0.00000
1.00000
2.00000
3.00000
4.00000
5.00000
123
45
67
89
11
113
315
517
719
922
124
326
528
730
933
135
337
539
738
336
133
931
729
527
325
122
920
718
516
314
111
997
75
53
31 9
Am
pli
tud
e, D
egre
es
Period, Radians
4.5 Cos x, Degrees
4.5 Cos x, Degrees
08 - 24 - 2015 34
-6.00000
-4.00000
-2.00000
0.00000
2.00000
4.00000
6.00000
-7
-5
-3
-1
1
3
5
71
13
25
37
49
61
73
85
97
10
912
113
314
515
716
918
119
320
521
722
924
125
326
527
728
930
131
330
529
328
126
925
724
523
3
n S
in x
n C
os
x
Number, n
Space-Time Bandwidth, Constant n (hc/λkT = 0) Sin x 1.5 Sin x 2 Sin x 3 Sin x 4 Sin x 5 Sin x 6 Sin x
Cos x 2 Cos x 2.25 Cos x 4.5 Cos x 4 Cos x 5 Cos x 15 Cos x
08 - 24 - 2015 35
-5
-4
-3
-2
-1
0
1
2
3
4
5
0.0
5236
0.5
2094
0.9
7670
1.4
0841
1.8
0545
2.1
5802
2.4
5746
2.6
9638
2.8
6891
2.9
7080
2.9
9954
2.9
5442
2.8
3656
2.6
4884
2.3
9591
2.0
8398
1.7
2073
1.3
1511
0.8
7712
0.4
1752
-0.0
5236
-0.5
20
94
-0.9
7670
-1.4
0841
-1.8
0545
-2.1
5802
-2.4
5746
-2.6
9638
-2.8
6891
-2.9
7080
-2.9
9954
-2.9
5442
-2.8
3656
-2.6
4884
-2.3
9591
-2.1
5802
-2.4
5746
-2.6
9638
-2.8
6891
-2.9
7080
-2.9
9954
-2.9
5442
-2.8
3656
-2.6
4884
-2.3
9591
Rad
ian
s
Radians
2-Dimension ( x, y ) Ellipsoid Amplitude and Periodicity
4.5 Cos x 1.5 Sin x
08 - 24 - 2015 36
n Cos ( x ) / 4
-100
-80
-60
-40
-20
0
20
40
60
0
1.0
48
328
214
3.9
34
584
331
8.3
56
238
926
13
.84
301
551
19
.79
348
492
25
.52
230
677
30
.31
485
509
33
.48
542
864
34
.43
495
612
32
.70
408
181
28
.01
776
369
20
.31
801
868
9.7
82
180
675
-3.1
750
480
12
-17
.916
428
13
-33
.613
467
-49
.292
621
34
-63
.893
859
45
-76
.338
137
36
-85
.599
656
18
-90
.778
328
57
-91
.167
720
07
-86
.313
865
24
-76
.060
784
97
-83
.732
794
83
-90
.246
107
89
-91
.434
751
68
-87
.665
133
43
-79
.571
728
53
-68
.001
939
19
Dep
th, 1
Am
pli
tude,
A (
x, y,
z )
Incrmental Period, n Sin ( x ) / 3
3-Dimensionsal ( x, y, z ) Surfaces of Constant Period ( Projected Area )
-100--80 -80--60 -60--40 -40--20 -20-0 0-20 20-40 40-60
08 - 24 - 2015 37
-50
-40
-30
-20
-10
0
10
20
30
40
50
111
21
31
41
51
61
71
81
91
10
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
130
929
928
927
926
925
924
923
9
Am
pli
tud
e, L
og
2
Period, Radians
Constant Bandwidth, n Cos x
Amplitude n, Period Cos x
15 Cos x 5 Cos x 4.5 Cos x 4 Cos x 3 Cos x 2.25 Cos x 2 Cos x 1.5 Cos x
08 - 24 - 2015 38
0.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000010.0000000070.0000000600.0000004770.0000038150.0000305180.0002441410.0019531250.0156250000.1250000001.0000000008.000000000
64.000000000512.000000000
4,096.00000000032,768.000000000
262,144.0000000002,097,152.000000000
16,777,216.000000000134,217,728.000000000
1,073,741,824.0000000008,589,934,592.000000000
68,719,476,736.000000000549,755,813,888.000000000
4,398,046,511,104.00000000035,184,372,088,832.000000000
281,474,976,710,656.0000000002,251,799,813,685,250.000000000
18,014,398,509,482,000.000000000
1
13
25
37
49
61
73
85
97
109
121
133
145
157
169
181
193
205
217
229
241
253
265
277
289
301
313
305
293
281
269
257
245
233
Tan
gen
t, L
og
2 (
n T
an x
)
Angle x, n radians
Periodic Information Stacked Area, radians
Tan x Sin x Cos x
08 - 24 - 2015 39
y = 2E-15x6 - 3E-12x5 + 2E-09x4 - 6E-07x3 + 9E-05x2 - 0.0052x + 0.4
R² = -1.25
-1.20E+00
-1.00E+00
-8.00E-01
-6.00E-01
-4.00E-01
-2.00E-01
0.00E+00
2.00E-01
4.00E-01
6.00E-01
-99.000 1.000 101.000 201.000 301.000 401.000 501.000
Co
ntr
ast
dT
Temperature, K
Incremental Contrast, Delta 500K
Contrast, Delta T Poly. (Contrast, Delta T)
CIE Luminance, Binary 3-Vector
Red 187, 000, 000
Green 000, 224, 000
Blue 000, 000, 227
08 - 24 - 2015 40
λ
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
1
14
27
40
53
66
79
92
10
5
11
8
13
1
14
4
15
7
17
0
18
3
19
6
20
9
22
2
23
5
24
8
26
1
27
4
28
7
30
0
31
3
32
6
33
9
35
2
36
5
37
8
39
1
Sp
ectr
al R
adia
nt
Inte
nsi
ty, W
-m
2-μ
m -
sr-1
+ / - 2 / 3 ( J1 TPW )2, μm-1
Polychromatic Surface, 1E-8, 0.001μm, Δλ / λ
0.00-5.00 5.00-10.00 10.00-15.00 15.00-20.00
20.00-25.00 25.00-30.00 30.00-35.00 35.00-40.00
08 - 24 - 2015 41
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
177
153
229
305
381
457
533
609
685
761
837
913
989
1065
1141
1217
1293
1369
1445
1521
1597
1673
1749
1825
1901
1977
2053
2129
2205
2281
2357
No
rmal
-S
pec
tral
Rad
ian
t In
ten
sity
, W
-m
2-
μm
-sr
Period ( 0.9 … 240.0 ) λ, μm
Normal Stefan-Boltzmann, 240 Periods
0.00E+00-5.00E+01 5.00E+01-1.00E+02 1.00E+02-1.50E+02
1.50E+02-2.00E+02 2.00E+02-2.50E+02
08 - 24 - 2015 42
0 20 40 60 80 100
0.000
50.000
100.000
150.000
200.000
250.000-30000.00
-25000.00
-20000.00
-15000.00
-10000.00
-5000.00
0.00
0 20 40 60 80 100
Number ( N π / 2 ), 1
Am
pli
tud
e A
, #
Rel
ati
ve
T,
K
Relative Temperature, Period NR ( K ) = - ( 1 + N T ( Sin^2 Nx + Cos^2 Nx ) )
CIE Luminance, Binary 3-Vector
Red 187, 000, 000
Green 000, 224, 000
Blue 000, 000, 227
08 - 24 - 2015 43
0 20 40 60 80 100 120 140 160
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 20 40 60 80 100 120 140 160
N S
in N
x, ra
d
Co
s 2
Nx
, ra
d
Spatial Period, N x = N ( π / 2 ), rad
Uniform Temperature Distribution, 4 π sr
Peak Temperature ( K ) = Cos 4 N x Spatial Temperature T = N Sin N x
CIE Luminance, Binary 3-Vector
Red 187, 000, 000
Green 000, 224, 000
Blue 000, 000, 227
08 - 24 - 2015 44
y = -6E-28x6 + 9E-23x5 - 5E-18x4 + 8E-14x3 + 7E-11x2 - 1E-05x + 1
R² = 0.0021
y = 2E-27x6 - 5E-22x5 + 4E-17x4 - 2E-12x3 + 3E-08x2 - 0.0002x + 0.5
R² = -0.06
0 50 100 150 200 250
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0 50 100 150 200 250
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
2 x / N π, Degrees
Min
imu
m, T
= T
( S
in π
x /
2 N
),K
2 x / N π, Degrees
Tem
per
atu
re A
mp
litu
de,
T =
T (
Co
s π
x /
2 N
), K
3-Color, Continuous Contrast, ΔTPeak Temperature Contrast = T [ 1+ ( Cos^2 - Sin^2 ) / ( Cos^2 + Sin^2 ) ] Minimum Temperature = T Sin x / 2 N
Mean Temperature = T Cos x / N Poly. (Peak Temperature Contrast = T [ 1+ ( Cos^2 - Sin^2 ) / ( Cos^2 + Sin^2 ) ])
Poly. (Mean Temperature = T Cos x / N)
CIE Luminance, Binary 3-Vector
Red 187, 000, 000
Green 000, 224, 000
Blue 000, 000, 227
08 - 24 - 2015 45
050100150200250300
0.000
100.000
200.000
300.000
400.000
500.000
600.000-100.000
0.000
100.000
200.000
300.000
400.000
500.000
600.000
0 50 100 150 200 250 300
Peak Temperature Specific Density, K
Tem
per
atu
re S
pec
ific
, K
Max
imu
m S
pec
ific
Tem
per
atu
re D
ensi
ty, K
Minimum Temperature Specific Density, K
Correlated Specific Temperature Density, 300K
Minimum Temperature Specific Density = 2 A Sin^2 ( N π A / 2 ) Contrast, Delta T Peak Temperature Specific Density = 1 + 2 Cos^2 ( N π A / 4 )
08 - 24 - 2015 46
0
50
100
150
200
250
300
350
400
450
500
-1.8E+05
-1.6E+05
-1.4E+05
-1.2E+05
-1.0E+05
-8.0E+04
-6.0E+04
-4.0E+04
-2.0E+04
0.0E+00
2.0E+04
0 50 100 150 200 250
Pea
k T
emp
erat
ure
Sp
ecif
ic D
ensi
ty E
stim
ate
N, K
/ r
Sam
ple
Per
iod
, m
/ s
Area Sample Surface Normal / sr , N π / 2, m2
Constant Hemisphere Temperature Surface, N π / 2, r
Probability T > T (Min) R ( K ) = - ( 1 + N T ( Sin^2 Nx + Cos^2 Nx ) ) Minimum Temperature Specific Density = 2 A Sin^2 ( N π A / 2 )
Peak Temperature Specific Density = 1 + 2 Cos^2 ( N π A / 4 )
CIE Luminance, Binary 3-Vector
Red 187, 000, 000
Green 000, 224, 000
Blue 000, 000, 227
08 - 24 - 2015 47
-1.00E+01
-8.00E+00
-6.00E+00
-4.00E+00
-2.00E+00
0.00E+00
2.00E+00
1 3 9 27 81
Sp
ectr
al T
emp
erat
ure
, W
–μ
m-1
sr-1
Log3 W - m-3 Fraction ( 3 / 10 )
Stefan-Boltzmann, εW - m-3 – K-4
σ K^4 / π λ K = 1 σ K^4 / π λ K = 1.1 σ K^4 / π λ K = 1.2 σ K^4 / π / λ, K = 1.3
08 - 24 - 2015 48
1.11416E-06, 1.11416E-06
1.80E+06, 1.80E+06
1.1, 1.97E+06
1.2, 2.15E+06
1.3, 2.33E+06
0
500000
1000000
1500000
2000000
2500000
0.0
00
00E
+0
0
2.0
00
00E
+0
5
4.0
00
00E
+0
5
6.0
00
00E
+0
5
8.0
00
00E
+0
5
1.0
00
00E
+0
6
1.2
00
00E
+0
6
1.4
00
00E
+0
6
1.6
00
00E
+0
6
1.8
00
00E
+0
6
2.0
00
00E
+0
6
Rel
ativ
e F
ract
ion
, (
4 /
8 )
, W
/ m
/ K
/ s
/ s
r /
4
Wavelength 0.09 μm, K / sr
Normal Boltzmann Statistics Specific Spatial Density, 0.09μm, T(peak)
5.67037E-08 = σ / ( 2 π λ^2 ) 1 = 2 ε / σ - 1 1.1 = 2 ε / σ - 1 1.2 = 2 ε / σ - 1 1.3 = 2 ε / σ - 1
08 - 24 - 2015 49
5.67000E-08,
5.67000E-08
4.81E-01, 4.81E-01
1.1, 2.39E-01
1.2, -7.71E-02
1.3, -4.84E-01
-6.00000E-01
-4.00000E-01
-2.00000E-01
0.00000E+00
2.00000E-01
4.00000E-01
6.00000E-01
W /
m2
/ K
/ s
/ s
r
Temperature, K / sr
Boltzmann Statistics (peak)
σ = 5.67000E-08 σ / λ / 2 / π σ K^4 / π λ K = 1 σ K^4 / π λ K = 1.1 σ K^4 / π λ K = 1.2 σ K^4 / π / λ, K = 1.3
08 - 24 - 2015 50
1, -1
315, -7.3583E+138
1, 0
315, -2.336E+136
0 50 100 150 200 250 300 350
-2.50E+136
-2.00E+136
-1.50E+136
-1.00E+136
-5.00E+135
0.00E+00-8.00E+138
-7.00E+138
-6.00E+138
-5.00E+138
-4.00E+138
-3.00E+138
-2.00E+138
-1.00E+138
0.00E+00
0 50 100 150 200 250 300 350
- x ( c / λ / n ), Integer #
-x E
xp
( x
-1
), V
alu
e #
-x E
xp
-(
x +
1 )
, V
alu
e #
- x ( c / λ ), Integer #
Exponential Information, Period x ( c / λ )
-x Exp ( x - 1 ) -x Exp ( x - 1 ) / n
08 - 24 - 2015 51
1.00, 0.35
0.71
1.411.77
2.83
5.66
11.31
22.63
99.00, 35.00
1.00, 1.09E+01
10.00, 3.29E+02
101.00, 2.77E+04
1.00, 1.00
100.00, 100.00
1.0E+00 1.0E+01 1.0E+02 1.0E+03
1
3
9
27
81
243
729
2,187
6,561
19,683
59,049
1/4
1/2
1
2
4
8
16
32
64
1 2 4 8 16 32 64 128
Log10 Energy, J ( m - s )
Lo
g3
Mea
n E
ner
gy D
ensi
ty, J
/ (
m -
s )
Lo
g2
MS
En
erg
y, F
ract
ion
1 /
4 (
J /
s )
x 2
Log2, Fraction 1 / 4 ( J / s )
Lambertian Source ( T ), KMean Energy 0-to-101, Joules / m
dν / ν = J / s, r m s λ / (dλ) = ν exp (1 - c / λ^2), J / ( m - s ) Mean Energy, ν(t) J / s
08 - 24 - 2015 52
4, 1.09E+01
10201, 2.77E+04
2
6
18
54
162
486
1458
4374
13122
39366
1 4 16 64 256 1024 4096 16384Lo
g3
En
erg
y D
ensi
ty R
elat
ive
to t
he
Kel
vin
,, F
ract
ion 1
/4 J
/ (
m -
s )
Log 2 λ, Fraction 1/4 ( m2 )
Spectral Radiant Power Density1 K, 1 Hz, 1 m ( Bandwidth )
λ / (dλ) = ν exp (1 - c / λ^2), J / ( m - s )
Conclusion
100 % Energy Estimate is Uncertain!
An Accuracy Statement Answers
THE Question?
What is the Relative Fraction Energy
Reflected, Transmitted, Absorbed
R + T – α E-αt / 2 = 2(u)
when propagated a length c, m – s.
08 - 24 - 2015 53