Using the usual symbols we have:
fdt = dv
therefore fds = vdv
Integrating:
m divided by s = one half times v squared plus c, of which the definite intergral from s1 to s2 is
m divided by s1 minus m divided by s2 = one half times v1 squared minus one half times v2 squared
Hence, since at infinity the velocity is 0, the equation for a fall to a planet's surface from infinity is
m divided by r = one half times v squared
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 p = 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 f or 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 " " " 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 degrees Cent. are as follows: --
Hydrogen 7.4 miles a second Water vapor 2.5 " " " Nitrogen 2.0 " " " Oxygen 1.8 " " " Carbonic dioxide 1.6 " " "
MEANS Polar Diameters 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".011 1° 9.336 Oct (15 to 23 inc) 9.379 0".028 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 July (6 to 22 inc) 9.691 0".11 46°.5 9.672 }9.680 }0".08 Aug (11 to 21 inc) 9.666 0".15 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° 9.514It 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.
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, 1/4 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)
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 above 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,--
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 III
NOTE IV
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.
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