Abstract

In a spherically symmetric inhomogeneous system, certain relations between refractive index and distance from the center of symmetry give closed solutions in elementary functions of the refraction integral. A careful formulation of the fundamental equations reveals that all such solutions reduce to special cases of one of eight standard forms, and that the theory of plane-parallel systems is mathematically identical. One special case gives a better representation of the Earth’s atmosphere than the traditionally used hypotheses of Cassini and Simpson.

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  1. Equation (1) is rigorously derived by Kerr (Ref. 5).
  2. Some authors have used the term differently, as meaning the actual displacement in arc of the image from the true direction of the body. The definitions coincide for a body infinitely distant from the center of symmetry, i. e., for astronomical refraction.
  3. Note that in order to secure these elegant symmetrical relationships of δ12 and θ12, I have set the limits of integration in Eq. (2) so that δ12 is negative if At decreases continuously outwards.
  4. The quantities δ12, etc., should be read "delta one, two," etc.
  5. Propagation of Short Radio Waves, edited by D. E. Kerr (Radiation Laboratory Series) (McGraw-Hill, New York, 1951).
  6. A. I. Mahan, Appl. Opt. 1, 497 (1962).
  7. The equivalent homogeneous atmsophere is that which has the same density throughout its depth as has the real atmosphere at the Earth's surface, and the same total mass.
  8. Refraction Tables in The Nautical Almanac (any recent edition) (U.S. Government Printing Office, Washington, D.C. 20402, and H. M. Stationery Office, London).
  9. B. Garfinkel, Astron. J. 72, 235 (1967).
  10. I have used the data: height of equivalent homogeneous atmosphere = 7.99 km, radius of Earth = 6370.9km, constant of refraction = 58. 294 arcseconds.
  11. A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).
  12. H. C. Plummer, Mon. Not. R. Astron. Soc. 92, 25 (1931).
  13. R. S. Heath, Geometrical Optics (Cambridge U. P., Cambridge, England, 1897).

Fletcher, A.

A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).

Garfinkel, B.

B. Garfinkel, Astron. J. 72, 235 (1967).

Heath, R. S.

R. S. Heath, Geometrical Optics (Cambridge U. P., Cambridge, England, 1897).

Mahan, A. I.

A. I. Mahan, Appl. Opt. 1, 497 (1962).

Plummer, H. C.

H. C. Plummer, Mon. Not. R. Astron. Soc. 92, 25 (1931).

Other (13)

Equation (1) is rigorously derived by Kerr (Ref. 5).

Some authors have used the term differently, as meaning the actual displacement in arc of the image from the true direction of the body. The definitions coincide for a body infinitely distant from the center of symmetry, i. e., for astronomical refraction.

Note that in order to secure these elegant symmetrical relationships of δ12 and θ12, I have set the limits of integration in Eq. (2) so that δ12 is negative if At decreases continuously outwards.

The quantities δ12, etc., should be read "delta one, two," etc.

Propagation of Short Radio Waves, edited by D. E. Kerr (Radiation Laboratory Series) (McGraw-Hill, New York, 1951).

A. I. Mahan, Appl. Opt. 1, 497 (1962).

The equivalent homogeneous atmsophere is that which has the same density throughout its depth as has the real atmosphere at the Earth's surface, and the same total mass.

Refraction Tables in The Nautical Almanac (any recent edition) (U.S. Government Printing Office, Washington, D.C. 20402, and H. M. Stationery Office, London).

B. Garfinkel, Astron. J. 72, 235 (1967).

I have used the data: height of equivalent homogeneous atmosphere = 7.99 km, radius of Earth = 6370.9km, constant of refraction = 58. 294 arcseconds.

A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).

H. C. Plummer, Mon. Not. R. Astron. Soc. 92, 25 (1931).

R. S. Heath, Geometrical Optics (Cambridge U. P., Cambridge, England, 1897).

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