Thermal Radiation from a Hemispherical Source in Contact with an Infinite Lambert Plane

J. E. Kester; J. E. Minardi

doi:10.1364/AO.2.000283

Applied Optics
Vol. 2,
Issue 3,
pp. 283-290
(1963)
•https://doi.org/10.1364/AO.2.000283

Thermal Radiation from a Hemispherical Source in Contact with an Infinite Lambert Plane

J. E. Kester and J. E. Minardi

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J. E. Kester and J. E. Minardi, "Thermal Radiation from a Hemispherical Source in Contact with an Infinite Lambert Plane," Appl. Opt. 2, 283-290 (1963)

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Abstract

Expressions are derived for the radiant energy emanating directly from a hemispherical source, the energy per unit area reflected from an infinite Lambert plane, and the energy received by a surface due to reflections from the plane. Numerically derived values for these energies are presented as fractions of E/2πR² for the direct energy and aE/2πR² for the reflected energies, where E is total emitted energy, R is the distance between receiver and center of hemisphere, and a is the albedo of the plane. Self-shadowing by the hemisphere is shown to be a significant factor even at very long ranges.

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	Scaled Range ℛ

Angle α	2	3	5	7.5	10	15	25	35	50	100	∞
0	0.096	0.198	0.307	0.369	0.401	0.434	0.460	0.471	0.480	0.490	0.500
10	0.101	0.199	0.300	0.357	0.387	0.418	0.443	0.454	0.462	0.472	0.482
20	0.113	0.198	0.282	0.328	0.353	0.378	0.399	0.408	0.415	0.423	0.431
30	0.129	0.196	0.256	0.288	0.305	0.322	0.336	0.342	0.346	0.352	0.357
40	0.144	0.190	0.225	0.241	0.249	0.256	0.262	0.264	0.266	0.268	0.271
50	0.157	0.180	0.188	0.188	0.188	0.187	0.186	0.186	0.185	0.185	0.184
60	0.164	0.161	0.146	0.135	0.129	0.122	0.116	0.114	0.112	0.110	0.108
70	0.160	0.131	0.101	0.085	0.076	0.067	0.060	0.057	0.054	0.052	0.049
80	0.138	0.096	0.063	0.046	0.038	0.029	0.022	0.020	0.017	0.015	0.012
90	0.115	0.073	0.043	0.028	0.021	0.014	0.008	0.006	0.004	0.002	0

	Scaled range ℛ

Angle α	2	3	5	7.5	10	15	25	35	50	100	00
0	0.096	0.198	0.307	0.369	0.401	0.434	0.460	0.471	0.480	0.490	0.500
10	0.098	0.198	0.300	0.361	0.392	0.424	0.450	0.461	0.469	0.479	0.489
20	0.105	0.197	0.289	0.342	0.370	0.401	0.423	0.433	0.441	0.449	0.459
30	0.114	0.194	0.273	0.317	0.340	0.363	0.383	0.392	0.398	0.405	0.412
40	0.122	0.191	0.253	0.285	0.302	0.319	0.333	0.339	0.343	0.348	0.353
50	0.132	0.187	0.229	0.249	0.258	0.268	0.276	0.279	0.281	0.283	0.287
60	0.141	0.179	0.198	0.205	0.209	0.211	0.213	0.214	0.215	0.215	0.215
70	0.146	0.159	0.158	0.155	0.152	0.149	0.146	0.144	0.143	0.142	0.142
80	0.132	0.119	0.103	0.093	0.087	0.079	0.075	0.072	0.071	0.070	0.070
90	0.058	0.037	0.021	0.014	0.011	0.007	0.004	0.003	0.002	0.002	0

	Scaled Range ℛ

Angle α	2	3	5	7.5	10	15	25	35	50	100	∞
0	0.096	0.198	0.307	0.369	0.401	0.434	0.460	0.471	0.480	0.490	0.500
10	0.101	0.199	0.300	0.357	0.387	0.418	0.443	0.454	0.462	0.472	0.482
20	0.113	0.198	0.282	0.328	0.353	0.378	0.399	0.408	0.415	0.423	0.431
30	0.129	0.196	0.256	0.288	0.305	0.322	0.336	0.342	0.346	0.352	0.357
40	0.144	0.190	0.225	0.241	0.249	0.256	0.262	0.264	0.266	0.268	0.271
50	0.157	0.180	0.188	0.188	0.188	0.187	0.186	0.186	0.185	0.185	0.184
60	0.164	0.161	0.146	0.135	0.129	0.122	0.116	0.114	0.112	0.110	0.108
70	0.160	0.131	0.101	0.085	0.076	0.067	0.060	0.057	0.054	0.052	0.049
80	0.138	0.096	0.063	0.046	0.038	0.029	0.022	0.020	0.017	0.015	0.012
90	0.115	0.073	0.043	0.028	0.021	0.014	0.008	0.006	0.004	0.002	0

	Scaled range ℛ

Angle α	2	3	5	7.5	10	15	25	35	50	100	00
0	0.096	0.198	0.307	0.369	0.401	0.434	0.460	0.471	0.480	0.490	0.500
10	0.098	0.198	0.300	0.361	0.392	0.424	0.450	0.461	0.469	0.479	0.489
20	0.105	0.197	0.289	0.342	0.370	0.401	0.423	0.433	0.441	0.449	0.459
30	0.114	0.194	0.273	0.317	0.340	0.363	0.383	0.392	0.398	0.405	0.412
40	0.122	0.191	0.253	0.285	0.302	0.319	0.333	0.339	0.343	0.348	0.353
50	0.132	0.187	0.229	0.249	0.258	0.268	0.276	0.279	0.281	0.283	0.287
60	0.141	0.179	0.198	0.205	0.209	0.211	0.213	0.214	0.215	0.215	0.215
70	0.146	0.159	0.158	0.155	0.152	0.149	0.146	0.144	0.143	0.142	0.142
80	0.132	0.119	0.103	0.093	0.087	0.079	0.075	0.072	0.071	0.070	0.070
90	0.058	0.037	0.021	0.014	0.011	0.007	0.004	0.003	0.002	0.002	0

Abstract

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Figures (10)

Tables (2)

Equations (49)

Applied Optics