Answer to Question #169878 in Astronomy | Astrophysics for Lanna

Question

In the late 19th century, Percival Lowell claimed he could see canals on the surface of Mars; long, dark, narrow features.  This question looks at the reasonableness of his claim.

a) What is Mars's angular diameter (in arcseconds, or 1/3600 of a degree) when it is at opposition (i.e., closest approach to Earth)? (Look up the distance Mars is from earth on average at opposition.)

b) The smallest angle that Lowell's telescope could resolve (meaning the smallest angle an observer could make out) was 1 arcsecond.  This means that Lowell's canals would have had to be at least this wide to have been detected.  What is the narrowest width (km) Lowell's putative canals could be and would these be anything like canals on Earth?

Answer 1

a) The eccentricity of the orbit of Mars is quite large, so the distance between the Earth and Mars in opposition may be quite different, from $a_m(1-e) - a_e$ to $a_m(1+e) - a_e$.

As an average distance we may take $l=a_m - a_e = 1.52 - 1 = 0.52\,\mathrm{AU}.$

The angular diameter is (in radians) $d = \dfrac{D_m}{l} = \dfrac{2\cdot3390\,\mathrm{km}}{0.52\cdot1.5\cdot10^8\,\mathrm{km}} = 8.7\cdot10^{-5}\,\mathrm{rad} = 17.9''.$

b) If we assume the width of the canal to be equal or greater than 1'', than we may write a proportion

$\dfrac{x\,\mathrm{km}}{1''} = \dfrac{2\cdot3390\,\mathrm{km}}{17.9''}\,,$ $x = 379\,\mathrm{km}.$

This width is extremely large, and the length of canals should be the same order of magnitude as the diameter of Mars. We have no such kind of canals on the Earth (nowadays, maybe).