36. Comparative Altitudes of Planetary Atmospheres
The total energy of equal masses of the gases oxygen, nitrogen, and aqueous vapour, as estimated by the method above, are respectively in the ratios
1 : 1·06 : 6·8
Referring back once more to the phenomena described with reference to the gravitational equilibrium of a gas, let it be assumed that the gaseous substance liberated on the surface of the planetary body is oxygen, and that the planetary body itself is of approximately the same constitution and dimensions as the earth. The oxygen gas thus liberated will expand against gravity, and envelop the planet in the manner already described (§ [34]). Now the total energy of a mass of one pound of oxygen has been estimated under certain assumptions (§ [35]) to be 164,000 ft. lbs. The value of the gravitative attraction of the planet on this mass is the same as under ordinary terrestrial conditions, so that if the entire energy of one pound of the gas were utilised in raising itself against gravity, the height through which this mass would be raised, and at which the material would attain the level of absolute zero of temperature, assuming gravity constant with increasing altitude, would be simply 164,000 ft. or approximately 31 miles. The whole energy would not, of course, be expended in the expansive movement; only the outermost surface material of the planetary gaseous envelope attains to absolute zero of temperature. In estimating the altitude of this surface, however, the precise mass of gaseous substance assumed for the purpose of calculation is of little or no importance. Whatever may be the value of the mass assumed, its total energy and the gravitative attraction of the planetary body on it are both alike entirely and directly dependent on that mass value. It is therefore clear that no matter how the mass under consideration be diminished, the height at which its energy would be completely worked down, and at which its temperature would be absolute zero, is the same, namely 31 miles. At the planet's surface, the total energy of an infinitesimally small portion of the gaseous mass is proportional to that mass. This amount of energy is, however, all that is available for transformation against gravitation in the ascent. But at the same time, the gravitative force on the particle, that force which resists its upward movement, is proportionately small corresponding to the small mass, so that the particle will in reality require to rise to the same altitude of 31 miles in order to completely transform its energy and attain absolute zero of temperature. When the expansive process is completed, the outer surface of the spherical gaseous envelope surrounding the planet is then formed of matter in this condition of absolute zero; this height of 31 miles is then the altitude or depth of the statical atmospheric column at a point on the planetary surface where the temperature is 50° F.
It is to be particularly noted that this height is entirely dependent on the gravitation, temperature, and energy conditions assumed.
With respect, also, to the assumption made above, of constant gravitation with increasing altitude, the variation in the value of gravity within the height limits in which the gas operates is so slight, that the energy of the expanding substance is completely worked down long before the variation appreciably affects the estimated altitude of absolute zero. In any case, bearing in mind the approximate nature of the estimate of the energy of the gases themselves, the variation of gravity is evidently a factor of little moment in our scheme of comparison.
Knowing the maximum height to be 31 miles, a uniform temperature gradient from the planetary surface to the outermost surface of the atmospheric material may be readily calculated. In the case of oxygen, the decrease of temperature with altitude will be at the rate of 16° F. per mile, or 1° F. per 330 ft.
If the planetary atmosphere were composed of nitrogen instead of oxygen, the height of the statical atmospheric column under the given conditions would then be approximately
31 × 1·06 = 33 miles,
and the gradient of temperature 15·5° F. per mile.