Abstract

The ir transmission of the terrestrial atmosphere is calculated at four altitudes of interest: Mauna Kea at 4.2 km (2–1000 μm), aircraft at 14 km (5–1000 μm), and balloon at 28 km and 41 km (10–1000 μm). We show both high resolution spectra (0.05 cm−1) and broadband averages. The results are intended to serve both as a detailed guide to the interference that is expected from the atmosphere for astronomical spectroscopy and also as an indicator of the relative change in absorption and emission that can be expected at various observing altitudes. One salient result for the spectral region around 100 μm is that the absorption (and emissivity) of the atmosphere drops by a factor of 10 for each increase in altitude of 15 km throughout the aircraft and balloon range; thus balloon-borne astronomical photometry and spectroscopy should both enjoy a considerable advantage over aircraft observations in the 30–300-μm region.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.
  2. I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
    [CrossRef]
  3. H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
    [CrossRef]
  4. U.S. Standard Atmosphere, and Supplements, NASA, USAF, USWB (U.S. Government Printing Office, Washington, D.C., 1962 and 1966).
  5. C. W. Allen, Astrophysical Quantities (Athlone, London, 1964).
  6. C. B. Farmer, Can. J. Chem. 52, 1544 (1974).
    [CrossRef]
  7. R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Vol. 1.
  8. D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
    [CrossRef]
  9. J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
    [CrossRef]
  10. A. Goldman, D. G. Murcray, F. H. Murcray, W. J. Williams, J. N. Brooks, Appl. Opt. 12, 1045 (1973).
    [CrossRef] [PubMed]
  11. C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
    [CrossRef] [PubMed]
  12. S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).
  13. N. P. Carleton, W. A. Traub, Science 177, 988 (1972).
    [CrossRef] [PubMed]
  14. J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).
  15. W. A. Traub, N. P. Carleton, in Exploration of the Planetary System, Proc. IAU Symposium 65, A. Wosczcyk, C. Iwaniszewska, Eds. (ReidelDordrecht, 1974), p. 223.
    [CrossRef]
  16. J. H. Van Vleck, V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).
    [CrossRef]
  17. G. Rybicki, Center for Astrophysics; private communication (1967);a listing of this Voigt profile subroutine is given in the Smithsonian Astrophysical Observatory Special Report 303 by E. H. Avrett, R. Loeser (1969).
  18. W. A. Traub, M. T. Stier, Center for Astrophysics Preprint369 (1975);copies are available from the authors on request. The total spectral range covered in this report is 2–5000 μm at 4.2 km, 5–1000 μm at 14 km, and 10–1000 μm at both 28 km and 41 km.
  19. C. B. Farmer, P. J. Key, Appl. Opt. 4, 1051 (1965).
    [CrossRef]
  20. J. E. Vernazza, E. H. Avrett, R. Loeser, to be published in Astrophys. J. Suppl., Jan.1976.
  21. J. E. Harries, Nature 241, 515 (1973).
    [CrossRef]
  22. J. P. Baluteau, E. Bussoletti, Nature Phys. Sci. 241, 113 (1973).
    [CrossRef]
  23. J. Kauppinen, Appl. Opt. 14, 1987 (1975).
    [CrossRef] [PubMed]

1975

W. A. Traub, M. T. Stier, Center for Astrophysics Preprint369 (1975);copies are available from the authors on request. The total spectral range covered in this report is 2–5000 μm at 4.2 km, 5–1000 μm at 14 km, and 10–1000 μm at both 28 km and 41 km.

J. Kauppinen, Appl. Opt. 14, 1987 (1975).
[CrossRef] [PubMed]

1974

C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
[CrossRef] [PubMed]

C. B. Farmer, Can. J. Chem. 52, 1544 (1974).
[CrossRef]

1973

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

J. E. Harries, Nature 241, 515 (1973).
[CrossRef]

J. P. Baluteau, E. Bussoletti, Nature Phys. Sci. 241, 113 (1973).
[CrossRef]

A. Goldman, D. G. Murcray, F. H. Murcray, W. J. Williams, J. N. Brooks, Appl. Opt. 12, 1045 (1973).
[CrossRef] [PubMed]

1972

N. P. Carleton, W. A. Traub, Science 177, 988 (1972).
[CrossRef] [PubMed]

1971

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
[CrossRef]

1965

1945

J. H. Van Vleck, V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).
[CrossRef]

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Athlone, London, 1964).

Avrett, E. H.

J. E. Vernazza, E. H. Avrett, R. Loeser, to be published in Astrophys. J. Suppl., Jan.1976.

Baluteau, J. P.

J. P. Baluteau, E. Bussoletti, Nature Phys. Sci. 241, 113 (1973).
[CrossRef]

Benedict, W. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Bohlander, R. A.

H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
[CrossRef]

Brooks, J. N.

Burch, D. E.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Burkhardt, E. G.

C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
[CrossRef] [PubMed]

Bussoletti, E.

J. P. Baluteau, E. Bussoletti, Nature Phys. Sci. 241, 113 (1973).
[CrossRef]

Calfee, R. F.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Carleton, N. P.

N. P. Carleton, W. A. Traub, Science 177, 988 (1972).
[CrossRef] [PubMed]

J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).

W. A. Traub, N. P. Carleton, in Exploration of the Planetary System, Proc. IAU Symposium 65, A. Wosczcyk, C. Iwaniszewska, Eds. (ReidelDordrecht, 1974), p. 223.
[CrossRef]

Carruthers, G. P.

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

Clough, S. A.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Connes, P.

J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).

Cruikshank, D. P.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

Farmer, C. B.

Fox, K.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Garing, J. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Gebbie, H. A.

H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
[CrossRef]

Goldman, A.

Goody, R. M.

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Vol. 1.

Harries, J. E.

J. E. Harries, Nature 241, 515 (1973).
[CrossRef]

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

Kauppinen, J.

Key, P. J.

Lambert, C. A.

C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
[CrossRef] [PubMed]

Loeser, R.

J. E. Vernazza, E. H. Avrett, R. Loeser, to be published in Astrophys. J. Suppl., Jan.1976.

Martin, T. Z.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

McClatchey, R. A.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Morrison, D.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

Murcray, D. G.

Murcray, F. H.

Murphy, R. E.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

Nolt, I. G.

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

Noxon, J.

J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).

Pardoe, G. W. F.

H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
[CrossRef]

Patel, C. K. N.

C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
[CrossRef] [PubMed]

Robinson, G. A.

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

Rothman, L. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

Rybicki, G.

G. Rybicki, Center for Astrophysics; private communication (1967);a listing of this Voigt profile subroutine is given in the Smithsonian Astrophysical Observatory Special Report 303 by E. H. Avrett, R. Loeser (1969).

Sinton, W. M.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

Stier, M. T.

W. A. Traub, M. T. Stier, Center for Astrophysics Preprint369 (1975);copies are available from the authors on request. The total spectral range covered in this report is 2–5000 μm at 4.2 km, 5–1000 μm at 14 km, and 10–1000 μm at both 28 km and 41 km.

Swann, N. R. W.

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

Traub, W. A.

W. A. Traub, M. T. Stier, Center for Astrophysics Preprint369 (1975);copies are available from the authors on request. The total spectral range covered in this report is 2–5000 μm at 4.2 km, 5–1000 μm at 14 km, and 10–1000 μm at both 28 km and 41 km.

N. P. Carleton, W. A. Traub, Science 177, 988 (1972).
[CrossRef] [PubMed]

J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).

W. A. Traub, N. P. Carleton, in Exploration of the Planetary System, Proc. IAU Symposium 65, A. Wosczcyk, C. Iwaniszewska, Eds. (ReidelDordrecht, 1974), p. 223.
[CrossRef]

Van Vleck, J. H.

J. H. Van Vleck, V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).
[CrossRef]

Vernazza, J. E.

J. E. Vernazza, E. H. Avrett, R. Loeser, to be published in Astrophys. J. Suppl., Jan.1976.

Weisskopf, V. F.

J. H. Van Vleck, V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).
[CrossRef]

Williams, W. J.

Wood, C. W.

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

Appl. Opt.

Can. J. Chem.

C. B. Farmer, Can. J. Chem. 52, 1544 (1974).
[CrossRef]

Center for Astrophysics Preprint

W. A. Traub, M. T. Stier, Center for Astrophysics Preprint369 (1975);copies are available from the authors on request. The total spectral range covered in this report is 2–5000 μm at 4.2 km, 5–1000 μm at 14 km, and 10–1000 μm at both 28 km and 41 km.

Infrared Phys.

J. E. Harries, N. R. W. Swann, G. P. Carruthers, G. A. Robinson, Infrared Phys. 13, 149 (1973).
[CrossRef]

J. Atmos. Sci.

I. G. Nolt, T. Z. Martin, C. W. Wood, W. M. Sinton, J. Atmos. Sci. 28, 238 (1971).
[CrossRef]

Nature

H. A. Gebbie, R. A. Bohlander, G. W. F. Pardoe, Nature 230, 521 (1971).
[CrossRef]

J. E. Harries, Nature 241, 515 (1973).
[CrossRef]

Nature Phys. Sci.

J. P. Baluteau, E. Bussoletti, Nature Phys. Sci. 241, 113 (1973).
[CrossRef]

Publ. Astron. Soc. Pac.

D. Morrison, R. E. Murphy, D. P. Cruikshank, W. M. Sinton, T. Z. Martin, Publ. Astron. Soc. Pac. 85, 255 (1973).
[CrossRef]

Rev. Mod. Phys.

J. H. Van Vleck, V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945).
[CrossRef]

Science

C. K. N. Patel, E. G. Burkhardt, C. A. Lambert, Science 184, 1173 (1974).
[CrossRef] [PubMed]

N. P. Carleton, W. A. Traub, Science 177, 988 (1972).
[CrossRef] [PubMed]

Other

J. Noxon, W. A. Traub, N. P. Carleton, P. Connes, to be published (1975).

W. A. Traub, N. P. Carleton, in Exploration of the Planetary System, Proc. IAU Symposium 65, A. Wosczcyk, C. Iwaniszewska, Eds. (ReidelDordrecht, 1974), p. 223.
[CrossRef]

S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

G. Rybicki, Center for Astrophysics; private communication (1967);a listing of this Voigt profile subroutine is given in the Smithsonian Astrophysical Observatory Special Report 303 by E. H. Avrett, R. Loeser (1969).

J. E. Vernazza, E. H. Avrett, R. Loeser, to be published in Astrophys. J. Suppl., Jan.1976.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, Air Force Cambridge Research Laboratory Atmospheric Absorption Line Parameters Compilation, AFCRL-TR-73-0096 (1973).Copies of this compilation are available on magnetic tape from the U.S. Department of Commerce, National Climatic Center, Federal Building, Asheville, No. Carolina, 28801. Dr. McClatchey has recently informed us that the currently available tape is a revised version of that used for the present paper; in particular, the tape now includes the O2and CO lines mentioned in section II.

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Vol. 1.

U.S. Standard Atmosphere, and Supplements, NASA, USAF, USWB (U.S. Government Printing Office, Washington, D.C., 1962 and 1966).

C. W. Allen, Astrophysical Quantities (Athlone, London, 1964).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Atmospheric transmission at four altitudes with an air mass of 2.0 and a rectangular bandpass of 0.05-cm−1 width.

Fig. 2
Fig. 2

Atmospheric transmission at four altitudes with an air mass of 2.0 and a rectangular bandpass of 0.05-cm−1 width.

Fig. 3
Fig. 3

Atmospheric transmission at four altitudes with an air mass of 2.0 and a rectangular bandpass of 0.05-cm−1 width.

Fig. 4
Fig. 4

Atmospheric transmission at four altitudes with an air mass of 2.0 and a rectangular bandpass of 0.05-cm−1 width.

Fig. 5
Fig. 5

Atmospheric transmission at four altitudes with an air mass of 2.0 and a rectangular bandpass of 0.05-cm−1 width.

Fig. 6
Fig. 6

Integrated transmission and emissivity at four altitudes with an air mass of 2.0. The bandpass varies from 5 cm−1 to 100 cm−1 in order to keep the resolution roughly constant. Note that in the neighborhood of 100 μm that the atmospheric emissivity decreases by about a factor of 10 for each altitude step of 15 km above Mauna Kea; this is also about the same rate at which the pressure falls. The mid-ir windows between 2 μm and 28 μm are clearly present, as are the poorer windows at λ > 350 μm and λ ≈ 32 μm.

Tables (2)

Tables Icon

Table I Molecular Abundances, Effective Pressures, and Temperatures Used in the Curtis-Godson Approximationa

Tables Icon

Table II Atmospheric Emissivity () and Radiated Flux (B) at Four Altitudes and for the Wavelength Range from 1000 μm to 10 μm, with Extensions down to 2 μm at Lower Altitudesa

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

f υ = n / n TOTAL = N / N TOTAL
N TOTAL ( cm 2 ) = p / m ¯ g = 2.119 × 10 22 p ( m bar ) ,
dI / I = η kdh = d τ ,
k = SF ( σ ) n ,
F ( σ ) d σ = 1 ,
I / I 0 = exp ( τ ) ,
τ = η kdh = η SF ( σ ) N .
F L ( σ ) = α / π α 2 + ( σ σ 0 ) 2 ,
α = α 0 ( p / p 0 ) ( T 0 / T ) 1 / 2 .
n ( h ) / n ( 0 ) = p ( h ) / p ( 0 ) = exp ( h / H ) ,
τ ( σ ) = η S n 1 H 2 π α 1 ln [ 1 + ( α 1 σ σ 0 ) 2 ] .
τ ( σ ) = η SnH α π 1 α 2 + ( σ σ 0 ) 2 .
F D ( σ ) = β 1 π 1 / 2 exp [ ( σ σ 0 ) / β ] 2 ,
β = 4.30 × 10 7 σ 0 ( T / M ) 1 / 2 ,
F V ( x , A ) = F L ( α ) * F D ( β ) ,
τ ¯ 0 = 2 SN η π Δ σ tan 1 ( Δ σ 2 α ) ,
E = 1 exp ( h σ c / kT ) ;
= 1 T ,
B = P ,

Metrics