Average distance between a star and planet in an eccentric orbit Available to Purchase

See for example W. K. Hartmann and C. Impey, Astronomy: The Cosmic Journey (Brooks/Cole, 2002), 6th ed., p. 59;

J. Bennett, M. Donahue, N. Schneider, and M. Voit, The Cosmic Perspective (Addison–Wesley, Reading, MA, 2002), 2nd ed., p. 136;

M. A. Seeds, Horizons: Exploring the Universe (Brooks/Cole, 2002), 7th ed., p. 54. Perhaps the best example of the “average-distance” statement is that made by the late astronomer Carl Sagan in Episode 3 of his popular COSMOS television series which aired in the Fall of 1980 on PBS.

M. Zeilik, S. A. Gregory, and E. v. P. Smith, Introductory Astronomy and Astrophysics (Saunders College Publishing, Philadelphia, 1992), 3rd ed., p. 8. Here, the authors state that “(t)he mean distance from the Sun to a planet in (an) elliptical orbit is just the semi-major axis $a.$ We prove this fact by noting that for each point … on an ellipse there is a symmetrical point (about the minor axis) from the focus … (and) the average of these distances is $a.$” Unfortunately, the authors do not say that a circumferential averaging of the planet–sun separation is the only way to obtain this result analytically.

For a review, see

D. M.

Williams

and

D.

Pollard

, “,” (), – ().

Actually, the problem is more widespread. See for example the advanced texts by B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics (Addison–Wesley, Readings, MA, 1996), p. 27, and J. M. A. Danby, Fundamentals of Celestial Mechanics (Willmann-Bell, 1988), 2nd ed., p. 5.