In a hillingar mirage, the Earth’s surface appears flat, because nearly horizontal light rays have the same curvature as the Earth. A linear temperature profile is traditionally inferred; its gradient is calculated to give this curvature to the exact horizontal ray. To see an image, however, a bundle of rays is required. To ensure that each ray in the bundle have the same curvature, the temperature profile must contain a small positive quadratic term, the coefficient of which is derived.
REFERENCES
1.
Several good introductory books on atmospheric optical phenomena are the following: R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., Cambridge, 1980);
M. G. J. Minnaert, Light and Color in the Outdoors (Springer, New York, 1993);
and D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge U.P., Cambridge, 1995).
2.
R. Cleasby and G. Vigfusson, An Icelandic/English Dictionary (Clarendon, Oxford, 1957), 2nd ed.
The antiquity of the word hillingar is not discussed, but it is used in the discussion of Norse exploration in G. Jones, The Norse Atlantic Saga (Oxford U.P., London, 1964), p. 6.
3.
John Ross, A Voyage of Discovery (John Murray, London, 1819). This book includes a beautiful engraving of the image seen by Ross. The image contains none of the usual features such as distortion, anomalous magnification, or inversion that characterize most mirages.
4.
W. H.
Hobbs
, “Conditions of Exceptional Visibility within High Latitudes, Particularly as a Result of Superior Mirage
,” Ann. Assoc. Am. Geographers
27
, 229
–240
(1937
).5.
W. H.
Lehn
, “The Novaya Zemlya effect: An arctic mirage
,” J. Opt. Soc. Am.
69
, 776
–781
(1979
).6.
Magnification is defined as the size of an observed image with respect to a reference image that would be seen with exactly straight light rays.
7.
J. M. Pernter and F. Exner, Meteorologische Optik (Braumüller, Vienna, 1922), 2nd ed., p. 63.
8.
W. G.
Rees
, “Mirages with linear image diagrams
,” J. Opt. Soc. Am. A
7
, 1351
–1354
(1990
). The image diagram (transfer characteristic) is a plot of ray elevation angle at the eye versus the actual elevation at which the same ray intersects the object being viewed.9.
If the refractive index is a linear function of elevation, then the corresponding temperature profile does contain a small quadratic term. But this term is an order of magnitude smaller than the one derived here, and moreover it has the opposite sign.
10.
R. G. Fleagle and J. A. Businger, An Introduction to Atmospheric Physics (Academic, New York, 1980), 2nd ed., Sec. 7.13.
11.
M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1986), 6th ed., Sec. 3.2.1.
This content is only available via PDF.
© 2001 American Association of Physics Teachers.
2001
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
