To Explain the World

The Discovery of Modern Science

Steven Weinberg, Bvtn 509 Wei, 2015 Harper Collins

Steven Weinberg is a theoretician who does physics with mathematics, not measurement. This book is about the development of mathematical explanations for observations, not how those observations are made or validated, nor how experimental apparatus is constructed and calibrated.

I only skimmed the book. It whetted my interest in how Tycho and Kepler performed their measurements and especially how they estimated the errors in their imprecise pre-telescope pre-photography pre-standards astronomical measurements. A companion book should be written by an experimentalist instead of a pure theoretician.

A footnote on page 168 may be result from narrow theoretical focus:

• The main effect of the ellipticity of planetary orbits is not so much the ellipticity itself as the fact that the Sun is at a focus rather than the center of the ellipse. To be precise, the distance between either focus and the center of an ellipse is proportional to the eccentricity, while the variation in the distance of points on the ellipse from either focus is proportional to the square of the eccentricity, which for a small eccentricity makes it much smaller. For instance, for an eccentricity of 0.1 (similar to that of the orbit of Mars) the smallest distance of the planet from the Sun is only ½ percent smaller than the largest distance. On the other hand, the distance of the Sun from the center of this orbit is 10 percent of the average radius of the orbit.

The underlined text above is incorrect. The radius r of a Keplerian orbit (with a semimajor axis of a and an eccentricity of e ) varies with angle \theta by the formula:

\Large r ~ = ~ \LARGE { { a ( 1 - e )^2 } \over { 1 + e \cos( \theta ) } }

This equation can be derived from the conservation of energy and angular momentum in an inverse-square gravity field, and is well known to astronomers and space engineers. For Mars, the semimajor axis a is 2.279392e11 meters or 1.523679 AU and the eccentricity e is 0.0934. At perihelion (closest to the Sun), \theta = 0, so the perihelion radius is:

\Large r_{p} ~ = ~ \LARGE { { a ( 1 - e )^2 } \over { 1 + e \cos( 0 ) } } \Large ~ = ~ \LARGE { { a ( 1 - e )^2 } \over { 1 + e } } ~ = ~ a ( 1 - e ) ~ \approx ~ 2.067e11 m or 1.3814 AU

At aphelion (farthest from the Sun), \theta = \pi, so the aphelion radius is:

\Large r_{a} ~ = ~ \LARGE { { a ( 1 - e )^2 } \over { 1 + e \cos( \pi ) } } \Large ~ = ~ \LARGE { { a ( 1 - e )^2 } \over { 1 - e } } ~ = ~ a ( 1 + e ) ~ \approx ~ 2.492e11 m or 1.6660 AU

The ratio of these distances is ( 1 + e ) / ( 1 - e ) ~ \approx ~ 1.206. About 20%, not 1/2 %.