Die normierte Planckfunktion
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Die spektrale Energieverteilung innerhalb eines 'schwarzen'
Hohlraumes, dessen Wände die Temperatur T haben, wird durch das
Plancksche Strahlungsgesetz beschrieben. Die Strahlungsleistung
dP, die pro Flächenelement dA in das Raumwinkelelement dΩ
(orthogonal zu dA) im Wellenlängenintervall dl als Funktion der
Vakuumwellenlänge l abgestrahlt wird, ist nach Planck:
┌ ┐
2∙h∙c² 1 │ W │
dP = ──────∙──────────── , Dimension: │────────│ , (1)
l^5 ┌ c2┐ │m^3∙rad²│
exp│───│ - 1 └ ┘
└T∙l┘
mit der zur Abkürzung eingeführten Größe:
h∙c
c2 := ─── = 1.43877485e-2 [m∙K] . (2)
k
Die in (1) bzw. (2) vorkommenden Größen sind im Einzelnen:
h := 6.6260693e-34 [J∙s] Plancksches Wirkungsquantum,
c := 2.99792458e+8 [m/s] Vakuum-Lichtgeschwindigkeit,
k := 1.3806505e-23 [J/K] Boltzmann-Konstante,
T := [K] absolute Temperatur,
l := [m] Wellenlänge .
Gefragt ist nun die über alle Wellenlängen integrierte Leistung.
Wie sich zeigen wird, ist (1) geschlossen über l integrierbar.
Dazu benutzen wir die Integraldarstellung der Riemannschen
Zeta-Funktion Z(s):
∞
1 ⌠ u^(s-1)
Z(s) := ────∙ │ ────────────∙du für Re(s) > 1. (3)
Γ(s) ⌡ exp(u) - 1
u=0
Insbesondere ist:
π^4
Z(4) = ──── , so daß
90
∞
⌠ u^3 π^4 π^4
│ ────────────∙du = Z(4)∙Γ(4) = ────∙3! = ──── . (4)
⌡ exp(u) - 1 90 15
u=0
Nun soll die Funktion (1) so normiert werden, daß das Integral
über alle Wellenlängen [0,∞] den Wert 1 ergibt. Das Integral von
(1) über alle Wellenlängen l ist:
∞
⌠
│ ┌ ┐
│ dl │ W │
dP = 2∙h∙c²∙│ ────────────────── , │───────│ , (5)
│ ┌ ┌ c2┐ ┐ │m²∙rad²│
│ l^5∙│exp│───│ - 1│ └ ┘
⌡ └ └T∙l┘ ┘
l=0
und mittels der Substitution:
c2 c2 -c2
─── = u bzw. l = ─── und dl = ────∙du (6)
T∙l T∙u T∙u²
geht es über in:
┌ ┐5 0
│ T│ ⌠ -u^5 c2
dP = 2∙h∙c²∙│──│ ∙ │ ───────────∙────∙du ,
│c2│ ⌡ exp(u) - 1 T∙u²
└ ┘ u=∞
┌ ┐4 ∞
│ T│ ⌠ u^3
dP = 2∙h∙c²∙│──│ ∙ │ ───────────∙du ,
│c2│ ⌡ exp(u) - 1
└ ┘ u=0
und wenn man (4) einsetzt:
┌ ┐4 ┌ ┐
2∙h∙c²∙π^4 │ T│ │ W │
dP = ──────────∙│──│ , │───────│ . (7)
15 │c2│ │m²∙rad²│
└ ┘ └ ┘
Um die Leistung dP zu erhalten, die vom Flächenelement dA
einseitig in den Halbraum (Raumwinkel 2∙π) abgestrahlt wird, muß
man noch über dΩ integrieren, das ergibt den Faktor π, so daß man
erhält:
┌ ┐4 ┌ ┐
2∙h∙c²∙π^5 │ T│ │ W │
dP = ──────────∙│──│ , │────│ , (8)
15 │c2│ │ m² │
└ ┘ └ ┘
und ersetzt man in (8) c2 nach (2), so folgt:
┌ ┐
2∙h∙c²∙π^5∙k^4 2∙π^5∙k^4 │ W │
dP = ──────────────∙T^4 = ─────────∙T^4 = σ∙T^4 , │────│. (9)
15∙h^4∙c^4 15∙h^3∙c² │ m² │
└ ┘
Das ist das Stefan-Boltzmann-Gesetz mit der sogenannten
Stefan-Boltzmann-Kontanten σ:
2∙π^5∙k^4
σ = ───────── = 5.67040124e-8 [W/(m²∙K^4)] . (10)
15∙h^3∙c²
Kommen wir nach diesen Vorbemerkungen nun zur Bestimmung der
'normierten' Plankfunktion.
Dividiert man die Funktion (1) durch die Größe (7), so erhalten
wir eine 'normierte' Funktion dPn, deren Integral über l von 0 bis
∞ den Wert 1 ergibt:
┌ ┐4 ┌ ┐
│ c2│ 1 │ 1 │
dPn = 15∙│───│ ∙────────────────── , │───│ . (11)
│π∙T│ ┌ ┌ c2┐ ┐ │ m │
└ ┘ l^5∙│exp│───│ - 1│ └ ┘
└ └T∙l┘ ┘
Hat man z. B. Meßwerte yi der Dimension [W/(m²∙m)] für
verschiedene Wellenlängen li vorliegen, so würde man zum Bestimmen
der Funktion durch Ausgleichen ansetzen:
┌ ┐4
│ c2 │ 1
yi = A1∙15∙│────│ ∙───────────────────── , (12)
│π∙A2│ ┌ ┌ c2 ┐ ┐
└ ┘ li^5∙│exp│─────│ - 1│
└ └A2∙li┘ ┘
mit den zu bestimmenden Parametern A1 und A2. A1 stellt dann die
(integrierte) Gesamtleistung pro Fläche über alle Wellenlängen
dar, A2 ist die Temperatur.
Ein andere Ansatz berücksichtigt zusätzlich einen (durch
fehlerhafte Meßtechnik begründeten) Untergrund:
┌ ┐4
│ c2 │ 1
yi = A1 + A2∙15∙│────│ ∙───────────────────── , (13)
│π∙A3│ ┌ ┌ c2 ┐ ┐
└ ┘ li^5∙│exp│─────│ - 1│
└ └A3∙li┘ ┘
mit A1: Untergrund,
A2: Gesamtleistung pro Fläche
A3: Temperatur.
Anhang
▀▀▀▀▀▀
Die partiellen Ableitungen nach den zu bestimmenden Parametern A1
und A2, Fall 1, bzw. A1,A2,A3, Fall 2, sollen hier notiert werden.
Fall 1, (12):
┌ ┐4
dyi │ c2 │ 1
─── = 15∙│────│ ∙───────────────────── ,
dA1 │π∙A2│ ┌ ┌ c2 ┐ ┐
└ ┘ li^5∙│exp│─────│ - 1│
└ └A2∙li┘ ┘
┌ ┐
│ │
dyi A1 dyi │ c2 ┌ c2 ┐ 1 │
─── = ──∙───∙│ -4 + ─────∙exp│─────│∙──────────────│ .
dA2 A2 dA1 │ A2∙li └A2∙li┘ ┌ c2 ┐ │
│ exp│─────│ - 1│
└ └A2∙li┘ ┘
Fall 2, (13):
dyi
─── = 1 ,
dA1
┌ ┐4
dyi │ c2 │ 1
─── = 15∙│────│ ∙───────────────────── ,
dA2 │π∙A3│ ┌ ┌ c2 ┐ ┐
└ ┘ li^5∙│exp│─────│ - 1│
└ └A3∙li┘ ┘
┌ ┐
│ │
dyi A2 dyi │ c2 ┌ c2 ┐ 1 │
─── = ──∙───∙│ -4 + ─────∙exp│─────│∙──────────────│ .
dA3 A3 dA1 │ A3∙li └A3∙li┘ ┌ c2 ┐ │
│ exp│─────│ - 1│
└ └A3∙li┘ ┘
* * *
Numerisches Beispiel mit zwei Parametern für die Sonnenstrahlung
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Daten von RReDC:
http://rredc.nrel.gov/solar/spectra/am0/ASTM2000.html.
------------------------------------------------------------------
Equations N = 880, Unknowns M = 2
Data reading: 0.00sec
Iteration AM0(1) AM0(2) delta
0 +1.4000e+003 +5.7000e+003 +0.0000e+000
1 +1.4470e+003 +5.4124e+003 +8.5644e-002
2 +1.4433e+003 +5.4517e+003 +9.7929e-003
3 +1.4448e+003 +5.4465e+003 +2.0128e-003
4 +1.4447e+003 +5.4472e+003 +2.4896e-004
5 +1.4447e+003 +5.4471e+003 +3.2517e-005
6 +1.4447e+003 +5.4471e+003 +4.2039e-006
Observations Positions Weights Mean obs. Mean errors Residuals
Y [W/m^3] X [m] W YM [W/m^3] FYM [W/m^3] V [W/m^3]
+5.0000e+6 +1.9950e-7 +1.0 +6.0934e+7 +1.5910e+6 -5.5934e+7
+7.0000e+6 +2.0050e-7 +1.0 +6.3486e+7 +1.6446e+6 -5.6486e+7
+7.0000e+6 +2.0150e-7 +1.0 +6.6111e+7 +1.6993e+6 -5.9111e+7
+8.0000e+6 +2.0250e-7 +1.0 +6.8807e+7 +1.7548e+6 -6.0807e+7
+9.0000e+6 +2.0350e-7 +1.0 +7.1577e+7 +1.8113e+6 -6.2577e+7
+9.0000e+6 +2.0450e-7 +1.0 +7.4421e+7 +1.8687e+6 -6.5421e+7
+1.0000e+7 +2.0550e-7 +1.0 +7.7339e+7 +1.9269e+6 -6.7339e+7
+1.0000e+7 +2.0650e-7 +1.0 +8.0332e+7 +1.9861e+6 -7.0332e+7
+1.1000e+7 +2.0750e-7 +1.0 +8.3401e+7 +2.0461e+6 -7.2401e+7
+1.5000e+7 +2.0850e-7 +1.0 +8.6546e+7 +2.1069e+6 -7.1546e+7
+2.4000e+7 +2.0950e-7 +1.0 +8.9767e+7 +2.1686e+6 -6.5767e+7
+2.8000e+7 +2.1050e-7 +1.0 +9.3066e+7 +2.2311e+6 -6.5066e+7
+3.4000e+7 +2.1150e-7 +1.0 +9.6441e+7 +2.2944e+6 -6.2441e+7
+3.0000e+7 +2.1250e-7 +1.0 +9.9895e+7 +2.3585e+6 -6.9895e+7
+3.2000e+7 +2.1350e-7 +1.0 +1.0343e+8 +2.4233e+6 -7.1427e+7
+4.1000e+7 +2.1450e-7 +1.0 +1.0704e+8 +2.4889e+6 -6.6038e+7
+3.7000e+7 +2.1550e-7 +1.0 +1.1073e+8 +2.5552e+6 -7.3727e+7
+3.4000e+7 +2.1650e-7 +1.0 +1.1450e+8 +2.6222e+6 -8.0495e+7
+3.6000e+7 +2.1750e-7 +1.0 +1.1834e+8 +2.6899e+6 -8.2343e+7
+4.5000e+7 +2.1850e-7 +1.0 +1.2227e+8 +2.7582e+6 -7.7269e+7
+4.8000e+7 +2.1950e-7 +1.0 +1.2628e+8 +2.8272e+6 -7.8276e+7
+4.8000e+7 +2.2050e-7 +1.0 +1.3036e+8 +2.8968e+6 -8.2362e+7
+3.9000e+7 +2.2150e-7 +1.0 +1.3453e+8 +2.9671e+6 -9.5528e+7
+5.1000e+7 +2.2250e-7 +1.0 +1.3877e+8 +3.0378e+6 -8.7774e+7
+6.6000e+7 +2.2350e-7 +1.0 +1.4310e+8 +3.1092e+6 -7.7099e+7
+5.8000e+7 +2.2450e-7 +1.0 +1.4750e+8 +3.1811e+6 -8.9505e+7
+5.4000e+7 +2.2550e-7 +1.0 +1.5199e+8 +3.2535e+6 -9.7990e+7
+4.1000e+7 +2.2650e-7 +1.0 +1.5655e+8 +3.3263e+6 -1.1555e+8
+4.1000e+7 +2.2750e-7 +1.0 +1.6120e+8 +3.3997e+6 -1.2020e+8
+5.4000e+7 +2.2850e-7 +1.0 +1.6592e+8 +3.4734e+6 -1.1192e+8
+4.8000e+7 +2.2950e-7 +1.0 +1.7073e+8 +3.5476e+6 -1.2273e+8
+5.6000e+7 +2.3050e-7 +1.0 +1.7561e+8 +3.6222e+6 -1.1961e+8
+5.0000e+7 +2.3150e-7 +1.0 +1.8057e+8 +3.6971e+6 -1.3057e+8
+5.5000e+7 +2.3250e-7 +1.0 +1.8561e+8 +3.7724e+6 -1.3061e+8
+4.6000e+7 +2.3350e-7 +1.0 +1.9072e+8 +3.8480e+6 -1.4472e+8
+3.9000e+7 +2.3450e-7 +1.0 +1.9592e+8 +3.9239e+6 -1.5692e+8
+5.7000e+7 +2.3550e-7 +1.0 +2.0119e+8 +4.0001e+6 -1.4419e+8
+4.9000e+7 +2.3650e-7 +1.0 +2.0654e+8 +4.0765e+6 -1.5754e+8
+5.3000e+7 +2.3750e-7 +1.0 +2.1196e+8 +4.1531e+6 -1.5896e+8
+4.2000e+7 +2.3850e-7 +1.0 +2.1746e+8 +4.2300e+6 -1.7546e+8
+4.6000e+7 +2.3950e-7 +1.0 +2.2304e+8 +4.3069e+6 -1.7704e+8
+4.3000e+7 +2.4050e-7 +1.0 +2.2869e+8 +4.3841e+6 -1.8569e+8
+5.2000e+7 +2.4150e-7 +1.0 +2.3442e+8 +4.4613e+6 -1.8242e+8
+7.2000e+7 +2.4250e-7 +1.0 +2.4022e+8 +4.5387e+6 -1.6822e+8
+6.5000e+7 +2.4350e-7 +1.0 +2.4609e+8 +4.6161e+6 -1.8109e+8
+6.2000e+7 +2.4450e-7 +1.0 +2.5204e+8 +4.6936e+6 -1.9004e+8
+5.1000e+7 +2.4550e-7 +1.0 +2.5806e+8 +4.7711e+6 -2.0706e+8
+5.1000e+7 +2.4650e-7 +1.0 +2.6415e+8 +4.8486e+6 -2.1315e+8
+5.7000e+7 +2.4750e-7 +1.0 +2.7031e+8 +4.9261e+6 -2.1331e+8
+4.5000e+7 +2.4850e-7 +1.0 +2.7654e+8 +5.0035e+6 -2.3154e+8
+5.8000e+7 +2.4950e-7 +1.0 +2.8284e+8 +5.0809e+6 -2.2484e+8
+5.9000e+7 +2.5050e-7 +1.0 +2.8920e+8 +5.1581e+6 -2.3020e+8
+4.7000e+7 +2.5150e-7 +1.0 +2.9564e+8 +5.2353e+6 -2.4864e+8
+4.4000e+7 +2.5250e-7 +1.0 +3.0214e+8 +5.3123e+6 -2.5814e+8
+5.5000e+7 +2.5350e-7 +1.0 +3.0871e+8 +5.3891e+6 -2.5371e+8
+6.1000e+7 +2.5450e-7 +1.0 +3.1534e+8 +5.4657e+6 -2.5434e+8
+8.9000e+7 +2.5550e-7 +1.0 +3.2204e+8 +5.5422e+6 -2.3304e+8
+1.0700e+8 +2.5650e-7 +1.0 +3.2880e+8 +5.6184e+6 -2.2180e+8
+1.2900e+8 +2.5750e-7 +1.0 +3.3562e+8 +5.6944e+6 -2.0662e+8
+1.3400e+8 +2.5850e-7 +1.0 +3.4250e+8 +5.7700e+6 -2.0850e+8
+1.0800e+8 +2.5950e-7 +1.0 +3.4945e+8 +5.8454e+6 -2.4145e+8
+1.0200e+8 +2.6050e-7 +1.0 +3.5645e+8 +5.9205e+6 -2.5445e+8
+1.0300e+8 +2.6150e-7 +1.0 +3.6351e+8 +5.9952e+6 -2.6051e+8
+1.2100e+8 +2.6250e-7 +1.0 +3.7063e+8 +6.0696e+6 -2.4963e+8
+1.7500e+8 +2.6350e-7 +1.0 +3.7781e+8 +6.1436e+6 -2.0281e+8
+2.7400e+8 +2.6450e-7 +1.0 +3.8504e+8 +6.2173e+6 -1.1104e+8
+2.8000e+8 +2.6550e-7 +1.0 +3.9233e+8 +6.2905e+6 -1.1233e+8
+2.6000e+8 +2.6650e-7 +1.0 +3.9967e+8 +6.3632e+6 -1.3967e+8
+2.7000e+8 +2.6750e-7 +1.0 +4.0706e+8 +6.4355e+6 -1.3706e+8
+2.6000e+8 +2.6850e-7 +1.0 +4.1450e+8 +6.5074e+6 -1.5450e+8
+2.5200e+8 +2.6950e-7 +1.0 +4.2200e+8 +6.5787e+6 -1.7000e+8
+2.9300e+8 +2.7050e-7 +1.0 +4.2954e+8 +6.6496e+6 -1.3654e+8
+2.3200e+8 +2.7150e-7 +1.0 +4.3713e+8 +6.7199e+6 -2.0513e+8
+2.1500e+8 +2.7250e-7 +1.0 +4.4477e+8 +6.7897e+6 -2.2977e+8
+2.0400e+8 +2.7350e-7 +1.0 +4.5245e+8 +6.8590e+6 -2.4845e+8
+1.3700e+8 +2.7450e-7 +1.0 +4.6018e+8 +6.9276e+6 -3.2318e+8
+2.0000e+8 +2.7550e-7 +1.0 +4.6795e+8 +6.9957e+6 -2.6795e+8
+2.5800e+8 +2.7650e-7 +1.0 +4.7577e+8 +7.0632e+6 -2.1777e+8
+2.4000e+8 +2.7750e-7 +1.0 +4.8362e+8 +7.1301e+6 -2.4362e+8
+1.6600e+8 +2.7850e-7 +1.0 +4.9152e+8 +7.1963e+6 -3.2552e+8
+8.9000e+7 +2.7950e-7 +1.0 +4.9945e+8 +7.2619e+6 -4.1045e+8
+1.1200e+8 +2.8050e-7 +1.0 +5.0742e+8 +7.3269e+6 -3.9542e+8
+2.3100e+8 +2.8150e-7 +1.0 +5.1543e+8 +7.3912e+6 -2.8443e+8
+3.0700e+8 +2.8250e-7 +1.0 +5.2348e+8 +7.4548e+6 -2.1648e+8
+3.3000e+8 +2.8350e-7 +1.0 +5.3156e+8 +7.5177e+6 -2.0156e+8
+2.4400e+8 +2.8450e-7 +1.0 +5.3967e+8 +7.5799e+6 -2.9567e+8
+1.4100e+8 +2.8550e-7 +1.0 +5.4781e+8 +7.6413e+6 -4.0681e+8
+3.2000e+8 +2.8650e-7 +1.0 +5.5599e+8 +7.7021e+6 -2.3599e+8
+3.7100e+8 +2.8750e-7 +1.0 +5.6419e+8 +7.7621e+6 -1.9319e+8
+3.0700e+8 +2.8850e-7 +1.0 +5.7243e+8 +7.8214e+6 -2.6543e+8
+4.5600e+8 +2.8950e-7 +1.0 +5.8069e+8 +7.8799e+6 -1.2469e+8
+6.2300e+8 +2.9050e-7 +1.0 +5.8897e+8 +7.9376e+6 +3.4028e+7
+6.0000e+8 +2.9150e-7 +1.0 +5.9728e+8 +7.9946e+6 +2.7163e+6
+5.4500e+8 +2.9250e-7 +1.0 +6.0562e+8 +8.0508e+6 -6.0619e+7
+5.4500e+8 +2.9350e-7 +1.0 +6.1398e+8 +8.1062e+6 -6.8977e+7
+5.0900e+8 +2.9450e-7 +1.0 +6.2236e+8 +8.1608e+6 -1.1336e+8
+5.4800e+8 +2.9550e-7 +1.0 +6.3076e+8 +8.2146e+6 -8.2756e+7
+4.9200e+8 +2.9650e-7 +1.0 +6.3918e+8 +8.2675e+6 -1.4718e+8
+5.3100e+8 +2.9750e-7 +1.0 +6.4761e+8 +8.3197e+6 -1.1661e+8
+4.1300e+8 +2.9850e-7 +1.0 +6.5607e+8 +8.3710e+6 -2.4307e+8
+4.8500e+8 +2.9950e-7 +1.0 +6.6454e+8 +8.4215e+6 -1.7954e+8
+4.0300e+8 +3.0050e-7 +1.0 +6.7302e+8 +8.4712e+6 -2.7002e+8
+4.4500e+8 +3.0150e-7 +1.0 +6.8152e+8 +8.5200e+6 -2.3652e+8
+4.8400e+8 +3.0250e-7 +1.0 +6.9003e+8 +8.5679e+6 -2.0603e+8
+6.3100e+8 +3.0350e-7 +1.0 +6.9855e+8 +8.6151e+6 -6.7548e+7
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Unknowns AM,
Std. deviation FAM of AM,
Std. deviation FY of single measurement Y[i]:
+1.4447e+003 +5.4471e+003
±8.0543e+000 ±1.7099e+001
±1.5305e+008
Correlations of AM (diagonal = sample variances):
+6.4871e+001
-5.8951e-001 +2.9238e+002
Calculation: 0.11sec
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Daten von RReDC.
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