from which we may find that, with the units here employed, the earth's mass as the unit of mass, the mean solar day as the unit of time, and the mile as the unit of distance—
k = 1830 × 1010.
If we introduce this value of k into the corresponding equation, which represents the motion of the earth around the sun, we shall have—
a3/(365.25)2 = 1830 × 1010 (333,000 + 1),
where the large number in the parenthesis represents the number of times the mass of the sun is greater than the mass of the earth. We shall find by solving this equation that a, the mean distance of the sun from the earth, is very approximately 93,000,000 miles.
113. Another method of determining the sun's distance.—This will be best appreciated by a reference to [Fig. 17]. It appears here that the earth makes its nearest approach to the orbit of Mars in the month of August, and if in any August Mars happens to be in opposition, its distance from the earth will be very much less than the distance of the sun from the earth, and may be measured by methods not unlike those which served for the moon. If now the orbits of Mars and the earth were circles having their centers at the sun this distance between them, which we may represent by D, would be the difference of the radii of these orbits—
D = a'' - a',
where the accents '', ' represent Mars and the earth respectively. Kepler's Third Law furnishes the relation—
(a'')3/(T'')2 = (a')3/(T')2;
and since the periodic times of the earth and Mars, T', T'', are known to a high degree of accuracy, these two equations are sufficient to determine the two unknown quantities, a', a''—i. e., the distance of the sun from Mars as well as from the earth. The first of these equations is, of course, not strictly true, on account of the elliptical shape of the orbits, but this can be allowed for easily enough.