STRENGTH OF THE ETHER
To show that the ether cannot be the slight and rarefied substance
which at one time, and indeed until quite lately, it was thought to
be, it is useful to remember that not only has it to be the vehicle of
light and the medium of all electric and magnetic influence, but also
that it has to transmit the tremendous forces of gravitation.
Among small bodies gravitational forces are slight, and are altogether
exceeded by magnetic and electric or chemical forces. Indeed
gravitational attraction between bodies of a certain smallness can be
more than counterbalanced even by the pressure which their mutual
radiation exerts--almost infinitesimal though that is;--so that as a
matter of fact, small enough bodies of any warmth will repel each
other unless they are in an enclosure of constant temperature, i.e.
unless the radiation pressure upon them is uniform all round.
The size at which radiation repulsion over-balances gravitational
attraction, for equal spheres, depends on the temperature of the
spheres and on their density; but at the ordinary temperature to
which we are accustomed, say 60° Fahrenheit or thereabouts, equality
between the two forces will obtain for two wooden spheres in space if
each is about a foot in diameter; according to Professor Poynting's
data (_Philosophical Transactions_, Vol. 202, p. 541). For smaller or
hotter bodies, radiation repulsion overpowers mutual gravitation; and
it increases with the fourth power of their absolute temperature. The
gravitational attractive force between particles is exceedingly small;
and that between two atoms or two electrons is negligibly small, even
though they be within molecular distance of each other.
For instance, two atoms of, say, gold, at molecular distance, attract
each other gravitationally with a force of the order
γ (10⁻²² x 10⁻²²) / (10⁻⁸)² =
10⁻⁴⁴ / 10⁻¹⁶ x 10⁻⁷ = 10⁻³⁵ dyne;
which would cause no perceptible acceleration at all.
The gravitational attraction of two electrons at the same distance is
the forty-thousand-millionth part of this, and so one would think must
be entirely negligible. And yet it is to the aggregate attraction of
myriads of such bodies that the resultant force of attraction is
due;--a force which is felt over millions of miles. The force is not
only felt indeed, but must be reckoned as one of prodigious magnitude.
When dealing with bodies of astronomical size, the force of
gravitation overpowers all other forces; and all electric and magnetic
attractions sink by comparison into insignificance.
These immense forces must be transmitted by the ether, and it is
instructive to consider their amount.
SOME ASTRONOMICAL FORCES WHICH THE ETHER HAS TO TRANSMIT.
_Arithmetical Calculation of the Pull of the Earth on the Moon._
The mass of the earth is 6000 trillion (6 × 10²¹) tons. The mass of
the moon is 1/80th that of the earth. Terrestrial gravity at the
moon's distance (which is 60 earth radii) must be reduced in the ratio
1:60²; that is, it must be 1/3600th of what it is here.
Consequently the pull of the earth on the moon is
6 × 10²¹ / 80 × 3600 tons weight.
A pillar of steel which could transmit this force, provided it could
sustain a tension of 40 tons to the square inch, would have a diameter
of about 400 miles; as stated in the text, page 102.
If this force were to be transmitted by a forest of weightless pillars
each a square foot in cross-section, with a tension of 30 tons to the
square inch throughout, there would have to be 5 million million of
them.
_Arithmetical Calculation of the Pull of the Sun on the Earth._
The mass of the earth is 6 × 10²¹ tons. The intensity of solar
gravity at the sun's surface is 25 times ordinary terrestrial gravity.
At the earth's distance, which is nearly 200 solar radii, solar
gravity will be reduced in the ratio of 1:200 squared.
Hence the force exerted by the sun on the earth is
(25 × 6 × 10²¹)/(200)² tons weight.
That is to say, it is approximately equal to the weight of 37 ×
10¹⁷ ordinary tons upon the earth's surface.
Now steel may readily be found which can stand a load of 37 tons to
every square inch of cross-section. The cross-section of a bar of such
steel, competent to transmit the sun's pull to the earth, would
therefore have to be
10¹⁷ square inches,
or say 700 × 10¹² square feet.
And this is equivalent to a million million round rods or pillars each
30 feet in diameter.
Hence the statement in the text (page 26) is well within the mark.
_The Pull of the Earth on the Sun._
The pull of the earth on the sun is, of course, equal and opposite to
the pull of the sun on the earth, which has just been calculated; but
it furnishes another mode of arriving at the result, and may be
regarded as involving simpler data--i.e. data more generally known.
All we need say is the following:--
The mass of the Sun is 316,000 times that of the Earth.
The mean distance of the sun is, say, 23,000 earth radii.
Hence the weight or pull of the sun by the earth is
316000/(23000)² × 6 × 10²¹ tons weight.
In other words, it is approximately equal to the ordinary commercial
weight of 36 × 10¹⁷ tons, as already calculated.
_The Centripetal Force acting on the Earth._
Yet another method of calculating the sun's pull is to express it in
terms of the centrifugal force of the earth; namely, its mass,
multiplied by the square of its angular velocity, multiplied by the
radius of its orbit;--that is to say,
F = M (2φ/T)² r
where T is the length of a year.
The process of evaluating this is instructive, owing to the
manipulation of units which it involves:--
F = 6 x 10²¹ tons x (4φ² x 92 x 10⁶ miles)/(365¼ days)²
which of course is a mass multiplied by an acceleration. The
acceleration is--
(40 x 92 x 10⁶)/133300 x (24)² miles per hour per hour
= (3680 x 10⁶ x 5280)/133300 x 576 x (3600)² feet per sec. per sec.
= (115 x 5280)/133300 x 576 x 12·96 feet per sec. per sec.
= g/1640
Hence the Force of attraction is that which, applied to the earth's
mass, produces in it an acceleration equal to the 1/1640th part of
what ordinary terrestrial gravity can produce in falling bodies; or
F = 6 × 10²¹ tons × g/1640
= 6/1640 x 10²¹ tons weight;
which is the ordinary weight of 37 × 10¹⁷ tons, as before.
The slight numerical discrepancy between the above results is of
course due to the approximate character of the data selected, which
are taken in round numbers as quite sufficient for purposes of
illustration.
If we imagine the force applied to the earth by a forest of round
rods, one for every square foot of the earth's surface--i.e. of the
projected earth's hemisphere or area of equatorial plane,--the force
transmitted by each would have to be 2700 tons; and therefore, if of
30-ton steel, they would each have to be eleven inches in diameter, or
nearly in contact, all over the earth.
_Pull of a Planet on the Earth._
While we are on the subject, it seems interesting to record the fact
that the pull of any planet on the earth, even Neptune, distant though
it is, is still a gigantic force. The pull of Neptune is 1/20,000th of
the sun's pull: i.e. it is 18 billion tons weight.
_Pull of a Star on the Earth._
On the other hand, the pull of a fixed star, like Sirius--say a star,
for example, which is 20 times the mass of the sun and 24 light years
distant--is comparatively very small.
It is easily found by dividing 20 times the sun's pull by the squared
ratio of 24 years to 8 minutes; and it comes out as 30 million tons
weight.
Such a force is able to produce no perceptible effect. The
acceleration it causes in the earth and the whole solar system, at its
present speed through space, is only able to curve the path with a
radius of curvature of length thirty thousand times the distance of
the star.
_Force required to hold together the Components of some Double Stars._
But it is not to be supposed that the transmission of any of these
forces gives the ether the slightest trouble, or strains it to
anywhere near the limits of its capacity. Such forces must be
transmitted with perfect ease, for there are plenty of cases where the
force of gravitation is vastly greater than that. In the case of
double stars, for instance, two suns are whirling round each other;
and some of them are whirling remarkably fast. In such cases the force
holding the components together must be enormous.
Perhaps the most striking case, for which we have substantially
accurate data, is the star ββ Aurigæ; which, during the general
spectroscopic survey of the heavens undertaken by Professor Pickering
of Harvard, in connexion with the Draper Memorial, was discovered to
show a spectrum with the lines some days double and alternate days
single. Clearly it must consist of a pair of luminous objects
revolving in a plane approximately containing the line of vision; the
revolution being completed every four days. For the lines will then be
optically displaced by the motion, during part of the orbit--those of
the advancing body to the right, those of the receding body to the
left,--while in that part of the orbit which lies athwart the
direction of vision, the spectrum lines will return to their proper
places,--opening out again to a maximum, in the opposite direction, at
the next quadrant.
The amount of displacement can be roughly estimated, enabling us to
calculate the speed with which the sources of light were moving.
Professor Pickering, in a brief statement in _Nature_, Vol. XLI, page
403, 1889, says that the velocity amounts to about 150 miles per
second, and that it is roughly the same for both components.
Taking these data:--
Equality and uniformity of speeds,
150 miles per second each,
Period 4 days,
we have all the data necessary to determine the masses; and likewise
the gravitative pull between them. For the star must consist of two
equal bodies, revolving about a common centre of gravity midway
between them, in nearly circular orbits.
The speed and period together easily give the radius of the circular
orbit as about 8 million miles.
Equating centrifugal and centripetal forces
mv² / r = γ m² / (2r)²
and comparing the value of 4r³ / T² so obtained with the r³ /
T² of the earth, we find the mass of each body must be about 30,000
times that of the earth, or about 1/10th that of the sun.
* * * * *
(This is treating them as spheres, though they must really be pulled
into decidedly prolate shapes. Indeed it may seem surprising that the
further portions can keep up with the nearer portions as they revolve.
If they are of something like solar density their diameter will be
comparable to half a million miles, and the natural periods of their
near and far portions will differ in the ratio (17/16)^{3/2} = 1·1
approximately. Tenacity could not hold the parts together, but
gravitational coherence would.)
* * * * *
This, however, is a digression. Let us continue the calculation of the
gravitative pull.
We have masses of 3 × 10⁴ × 6 × 10²¹ tons, revolving with
angular velocity 2π ÷ 4 days, in a circle of radius 8 × 10⁶ miles.
Consequently the centripetal acceleration is 4 π² × 8 × 10⁶ / 16
miles per day per day; which comes out 32 / 2·2 ft. per sec. per sec.,
or nearly half ordinary terrestrial gravity.
Consequently the pull between the two components of the double star ß
Aurigæ is
_g_ / 2.2 × 18 × 10²⁵ tons,
or equal to the weight of
80 × 10²⁴ tons on the earth,
which is more than twenty million times as great as is the pull
between the earth and our sun.
* * * * *
Simple calculations such as these could have been made at any time;
there is nothing novel about them, as there is about the estimate of
the ether's density and vast intrinsic energy, in Chapters VI and VII.
But then there is nothing hypothetical or uncertain about them either;
they are certain and definite: whereas it may be thought there is
something doubtful about the newer contentions which involve
consideration of the mass and size of electrons and of the uniform and
incompressible character of etherial constitution. Even the idea of
"massiveness" as applied to the ether involves an element of
uncertainty, or of figurativeness; because until we know more about
ether's peculiar nature (if it is peculiar), we have to deal with it
in accordance with material analogies, and must specify its
massiveness as that which would have to be possessed by it if it
fulfilled its functions and yet were anything like ordinary matter. It
cannot really _be_ ordinary matter, because ordinary matter is
definitely differentiated from it, and is presumably composed of it;
but the inertia of ordinary matter, however it be electrically or
magnetically explained, must in the last resort depend on something
parentally akin to inertia in the fundamental substance which fills
space. And this it is which we have attempted in Chapters VI and VII
to evaluate and to express in the soberest terms possible.