# Mars (Lowell)/Appendix

Note I
The critical velocity at the surfaces of the planets is found as follows:—
Using the usual symbols we have:
$fdt=dv\,$
$\therefore fds=vdv$.
And as $f={\frac {-m}{s^{2}}}$, since the force tends to decrease the coördinates, this becomes $-{\frac {mds}{s^{2}}}=vdv$ Integrating:
${\frac {m}{s}}={\frac {1}{2}}v^{2}+c$, of which the definite integral from s$_1$ to s$_2$ is
${\frac {m}{s_{1}}}-{\frac {m}{s_{2}}}={\frac {1}{2}}v_{1}^{2}-{\frac {1}{2}}v_{2}^{2}$
Hence, since at infinity the velocity is 0, the equation for a fall to a planet's surface from infinity is
${\frac {m}{r}}={\frac {1}{2}}v^{2}$,
r being the radius of the planet and v the velocity acquired at its surface from a fall from infinity, which is the same as the velocity needed for projection from its surface to infinity.
To find m we have in the case of the Earth g = 32 ft. a second at its surface; this gives us m in terms of g, that is, f. For the other planets we need only to introduce their masses and radii in terms of those of the Earth and then multiply the value for the Earth by the square root of the ratio.
The result is that we find the critical velocity for the several planets and for the Sun to be as follows:—
| Mercury | 2.2 | miles | a | second | (probable | value). |
| Venus | 6.6 | " | " | " | " | " |
| Earth | 6.9 | " | " | " | | |
| Moon | 1.5 | " | " | " | | |
| Mars | 3.1 | miles | a | second. |
| Jupiter | 37. | " | " | " | (mean | value). |
| Saturn | 22. | " | " | " | " | " |
| Uranus | 13. | " | " | " | " | " |
| Neptune | 14. | " | " | " | " | " |
| Sun | 382. | " | " | " |
While the probable maximum speed of the molecules of some of the common gases at 0° Cent. are as follows:—
| Hydrogen | 7.4 | miles | a | second. |
| Water vapor | 2.5 | " | " | " |
| Nitrogen | 2.0 | " | " | " |
| Oxygen | 1.8 | " | " | " |
| Carbonic dioxide | 1.6 | " | " | " |
Note II
The change in the apparent size of the equatorial diameter as compared with the polar one as the phase increased, suggesting the unconscious measurement of a twilight upon the planet, becomes still more striking when, in addition to the October-November measures mentioned in the text, the measures from July to October are considered in connection with them. Tabulated chronologically, the whole are as follows:—
MEANS Polar Diameters :
| | Cor. for ref. irr. tilt and phase Irr. 0".10 | Prob. error | Angle of phase | Cor. for ref. irr. tilt and phase Irr. 0".15 |
| July (6 to 22 inc.) | 9.976 | 0".13 | 0° | 9.933 |
| Aug. (11 to 21 inc.) | 9.362 | 0".04 | 0° | 9.325 |
| Sept. (20 to Oct. 5 inc.) | 9.401 | 0".012 | 0° | 9.355 |
| Oct. (12 & 24 to 30 inc.) | 9.375 | 0".028 | 1° | 9.336 |
| Oct. (15 to 23 inc.) | 9.379 | 0".011 | 2°.5 | 9.339 |
| Oct. (12 & 24 to 30 inc.) | 9.375 | 0".028 | 1° | 9.336 |
| Nov. (2 to 21 inc.) | 9.390 | 0".012 | 4° | 9.350 |
Equatorial Diameters :
| July (6 to 22 inc.) | 9.691 | 9.680 | 0".11 | 0".08 | 46°.5 | 9.672 |
| Aug. (11 to 21 inc.) | 9.666 | 9.680 | 0".15 | 0".08 | 41° | 9.645 |
| Sept.(20 to Oct. 5) | 9.523 | | 0".010 | | 20°.5 | 9.490 |
| Oct. (12 & 24 to 30 inc.) | 9.457 | | 0".016 | | 7° | 9.417 |
| Oct. (15 to 23 inc.) | 9.429 | | 0".010 | | 1° | 9.385 |
| Oct. (12 & 24 to 30 inc.) | 9.457 | | 0".016 | | 7° | 9.417 |
| Nov. (2 to 21 inc.) | 9.545 | | 0".015 | | 19° | 9.514 |
It will be seen that, except for the July value, the size of the polar diameter comes out essentially the same throughout. Now, during July the polar cap was very large, and covered the southern part of the disk at the point where the polar diameter was measured. As it was much brighter than the rest of the disk, its irradiation must have been correspondingly great, and this would have had the effect of increasing the apparent length of the polar diameter beyond its true value.
The equatorial measures, on the other hand, show a systematic increase as the phase increased; and they do this on both sides of opposition. The increase, it will be noticed, is much greater than the probable errors of observation.
Note III
As the statement has been widely circulated that recent spectroscopic observations negative an atmosphere on Mars, it may be well to mention in a note that the observations in question neither affirm nor deny its presence, as their self-disclosed measure of precision, ¼ of an atmosphere, proves them incapable of it. They simply concur in showing that atmosphere to be thin. As a matter of fact, if spectroscopic observations did deny the existence of an atmosphere on Mars, such assertion would be fatal, not to the atmosphere, but to the observer or his instrument, as the existence of an atmosphere is demonstrated by the fundamental laws of ​physics, inasmuch as no change could take place on the planet's surface without it, and that changes do take place is undeniable. (See page 31 et seq.)
Note IV
Mars has two satellites, discovered by Hall in 1877, and known as Deimos (Dread) and Phobos (Fear), named in keeping with the God of War.
Deimos, at a distance of 14,600 miles from the planet's centre, makes his circuit in 30 hours and 18 minutes; Phobos, at a distance of 5,800, in 7 hours and 39 minutes. As Mars himself rotates in 24 hours and 39 minutes, Phobos goes round the planet faster than the planet turns upon itself, and, in consequence, would appear to any observers on the planet's surface to break the otherwise universal conformity of stellar motions by rising in the west and setting in the east. Deimos, too, is just as unconventional in its way, for it remains for two days at a time about the horizon. Furthermore, with each, owing to its nearness to the planet, its distance from any place on the surface varies at different times, and with its distance varies its apparent size in a somewhat startling manner.
As for themselves, they are very minute bodies, though not so difficult to see as is commonly stated. In the clear air of Arizona, both were conspicuous objects. They appear as stars of about the 12th and 10th magnitudes respectively; Phobos being much larger, relatively to Deimos, than its hitherto accepted value would indicate. Observations at Flagstaff by both Mr. Douglass and by me agree in making its relative brilliancy such as to give it a diameter about 3.6 times that of Deimos. It is not usually so conspicuous as Deimos, in spite of its size, because of its proximity to the planet, and the consequent much greater illumination of the field upon which it is seen. Considering their most probable albedoes as somewhat less than that of our moon, ​we find from their stellar magnitudes, taking the stellar magnitude found for Deimos by Pickering in 1877 as basis, their diameters to be,—
             Deimos, about 10 miles;
             Phobos, about 36 miles.
Phobos would thus, at its closest approach to the surface of the planet, that is, when it was in the zenith, just show a disk like the Moon. Otherwise both satellites would appear as stars.
Neither satellite shares the red tint of the planet.
Note V.
As the means employed in any astronomical observation are of interest, I may add that the telescope used in these researches was an 18-inch refractor, made by Brashear, of Alleghany, Pa., the largest he has yet made. The powers used varied from 320 to 1305 diameters, the usual ones being, for visual purposes, 440 and 617, and, for micrometric measurements, 862. There is, not unnaturally, much misconception prevalent as to the magnification possible in a telescope. The highest powers of a glass can never be used on planetary detail, as the tremors of the air blur the image. Thus we come back again to the question of atmosphere, which is indeed the crux observationis. With regard to work on the planets, the important point about an observatory is not so much what is its lens as what is its location.
Plate XXIV
MAP OF MARS
on Mercator's Projection
Lowell Observatory
Flagstaff, A. T. 1895