respect. Astronomers were not content during the recent transit to use
Halley’s and Delisle’s two time methods (which may be conveniently
called the duration method and the epoch method), but endeavoured to
determine the position of Venus on the sun’s face directly, both by
observation and by photography. The heliometer was the instrument
specially used for the former purpose; and as, in one of the new
methods to be presently described, this is the most effective of all
available instruments, a few words as to its construction will not be
out of place.
The heliometer, then, is a telescope whose object-glass (that is,
the large glass at the end towards the object observed) is divided
into two halves along a diameter. When these two halves are exactly
together—that is, in the position they had before the glass was
divided—of course they show any object to which they may be directed
precisely as they would have done before the glass was cut. But if,
without separating the straight edges of the two semicircular glasses,
one be made to slide along the other, the images formed by the two
no longer coincide.[9] Thus, if we are looking at the sun we see
two overlapping discs, and by continuing to turn the screw or other
mechanism which carries our half-circular glass past the other, the
disc-images of the sun may be brought entirely clear of each other.
Then we have two suns in the same field of view, seemingly in contact,
or nearly so. Now, if we have some means of determining how far the
movable half-glass has been carried past the other to bring the two
discs into apparently exact contact, we have, in point of fact, a
measure of the sun’s apparent diameter. We can improve this estimate by
carrying back the movable glass till the images coincide again, then
further back till they separate the other way and finally are brought
into contact on that side. The entire range, from contact on one side
to contact on the other side, gives twice the entire angular span of
the sun’s diameter; and the half of this is more likely to be the true
measure of the diameter, than the range from coincident images to
contact either way, simply because instrumental errors are likely to be
more evenly distributed over the double motion than over the movement
on either side of the central position. The heliometer derived its
name—which signifies sun-measurer—from this particular application of
the instrument.
It is easily seen how the heliometer was made available in determining
the position of Venus at any instant during transit. The observer
could note what displacement of the two half-glasses was necessary to
bring the black disc of Venus on one image of the sun to the edge of
the other image, first touching on the inside and then on the outside.
Then, reversing the motion, he could carry her disc to the opposite
edge of the other image of the sun, first touching on the inside and
then on the outside. Lord Lindsay’s private expedition—one of the
most munificent and also one of the most laborious contributions to
astronomy ever made—was the only English expedition which employed
the heliometer, none of our public observatories possessing such an
instrument, and official astronomers being unwilling to ask Government
to provide instruments so costly. The Germans, however, and the
Russians employed the heliometer very effectively.
Next in order of proximity, for the employment of the direct surveying
method, is the planet Mars when he comes into opposition (or on the
same line as the earth and sun) in the order
Sun____________________________Earth__________Mars,
at a favourable part of his considerably eccentric orbit. His distance
then may be as small as 34½ millions of miles; and we have in his case
to make no reduction for the displacement of the background on which
his place is to be determined. That background is the star sphere,
his place being measured from that of stars near which his apparent
path on the heavens carries him; and the stars are so remote that
the displacement due to a distance of six or seven thousand miles
between two observers on the earth is to all intents and purposes
nothing. The entire span of the earth’s orbit round the sun, though
amounting to 184 millions of miles, is a mere point as seen from all
save ten or twelve stars; how utterly evanescent, then, the span of
the earth’s globe—less than the 23,000th part of her orbital range!
Thus the entire displacement of Mars due to the distance separating
the terrestrial observers comes into effect. So that, in comparing the
observation of Mars in a favourable opposition with that of Venus in
transit, we may fairly say that, so far as surveying considerations are
concerned, the two planets are equally well suited for the astronomer’s
purpose. Venus’s less distance of 25 millions of miles is effectively
increased to 34⅓ millions by the displacement of the solar background
on which we see her when in transit; while Mars’s distance of about
34½ millions of miles remains effectively the same when we measure his
displacement from neighbouring fixed stars.
But in many respects Mars is superior to Venus for the purpose of
determining the sun’s distance. Venus can only be observed at her
nearest when in transit, and transit lasts but a few hours. Mars can
be observed night after night for a fortnight or so, during which his
distance still remains near enough to the least or opposition distance.
Again, Venus being observed on the sun, all the disturbing influences
due to the sun’s heat are at work in rendering the observation
difficult. The air between us and the sun at such a time is disturbed
by undulations due in no small degree to the sun’s action. It is true
that we have not, in the case of Mars, any means of substituting time
measures or time determinations for measures of position, as we have
in Venus’s case, both with Halley’s and Delisle’s methods. But to say
the truth, the advantage of substituting these time observations has
not proved so great as was expected. Venus’s unfortunate deformity of
figure when crossing the sun’s edge renders the determination of the
exact moments of her entry on the sun’s face and of her departure from
it by no means so trustworthy as astronomers could wish. On the whole,
Mars would probably have the advantage even without that point in his
favour which has now to be indicated.
Two methods of observing Mars for determining the sun’s distance are
available, both of which, as they can be employed in applying one of
the new methods, may conveniently be described at this point.
An observer far to the north of the earth’s equator sees Mars at
midnight, when the planet is in opposition, displaced somewhat to
the south of his true position—that is, of the position he would
have as supposed to be seen from the centre of the earth. On the
other hand, an observer far to the south of the equator sees Mars
displaced somewhat to the north of his true position. The difference
may be compared to different views of a distant steeple (projected,
let us suppose, against a much more remote hill), from the uppermost
and lowermost windows of a house corresponding to the northerly and
southerly stations on the earth, and from a window on the middle
story corresponding to a view of Mars from the earth’s centre. By
ascertaining the displacement of the two views of Mars obtained from
a station far to the north and another station far to the south,
the astronomer can infer the distance of the planet, and thence the
dimensions of the solar system. The displacement is determinable by
noticing Mars’s position with respect to stars which chance to be close
to him. For this purpose the heliometer is specially suitable, because,
having first a view of Mars and some companion stars as they actually
are placed, the observer can, by suitably displacing the movable
half-glass, bring the star into apparent contact with the planet, first
on one side of its disc, and then on the other side—the mean of the
two resulting measures giving, of course, the distance between the star
and the centre of the disc.
This method requires that there shall be two observers, one at a
northern station, as Greenwich, or Paris, or Washington, the other at
a southern station, as Cape Town, Cordoba, or Melbourne. The base-line
is practically a north-and-south line; for though the two stations may
not lie in the same, or nearly the same, longitude, the displacement
determined is in reality that due to their difference of latitude only,
a correction being made for their difference of longitude.
The other method depends, not on displacement of two observers north
and south, or difference of latitude, but on displacement east and
west. Moreover, it does not require that there shall be two observers
at stations far apart, but uses the observations made at one and the
same stations at different times. The earth, by turning on her axis,
carries the observer from the west to the east of an imaginary line
joining the earth’s centre and the centre of Mars. When on the west
of that line, or in the early evening, he sees Mars displaced towards
the east of the planet’s true position. After nine or ten hours the
observer is carried as far to the east of that line, and sees Mars
displaced towards the west of his true position. Of course Mars has
moved in the interval. He is, in fact, in the midst of his retrograde
career. But the astronomer knows perfectly well how to take that
motion into account. Thus, by observing the two displacements, or the
total displacement of Mars from east to west on account of the earth’s
rotation, one and the same observer can, in the course of a single
favourable night, determine the sun’s distance. And in passing, it may
be remarked that this is the only general method of which so much can
be said. By some of the others an astronomer can, indeed, estimate the
sun’s distance without leaving his observatory—at least, theoretically
he can do so. But many years of observation would be required before he
would have materials for achieving this result. On the other hand, one
good pair of observations of Mars, in the evening and in the morning,
from a station near the equator, would give a very fair measure of
the sun’s distance. The reason why the station should be near the
equator will be manifest, if we consider that at the poles there
would be no displacement due to rotation; at the equator the observer
would be carried round a circle some twenty-five thousand miles in
circumference; and the nearer his place to the equator the larger
the circle in which he would be carried, and (_cæteris paribus_) the
greater the evening and morning displacement of the planet.
Both these methods have been successfully applied to the problem of
determining the sun’s distance, and both have recently been applied
afresh under circumstances affording exceptionally good prospects of
success, though as yet the results are not known.
It is, however, when we leave the direct surveying method to which
both the observations of Venus in transit and Mars in opposition
belong (in all their varieties), that the most remarkable, and, one
may say, unexpected methods of determining the sun’s distance present
themselves. Were not my subject a wide one, I would willingly descant
at length on the marvellous ingenuity with which astronomers have
availed themselves of every point of vantage whence they might measure
the solar system. But, as matters actually stand, I must be content
to sketch these other methods very roughly, only indicating their
characteristic features.
One of them is in some sense related to the method by actual survey,
only it takes advantage, not of the earth’s dimensions, but of the
dimensions of her orbit round the common centre of gravity of herself
and the moon. This orbit has a diameter of about six thousand miles;
and as the earth travels round it, speeding swiftly onwards all the
time in her path round the sun, the effect is the same as though the
sun, in his apparent circuit round the earth, were constantly circling
once in a lunar month around a small subordinate orbit of precisely the
same size and shape as that small orbit in which the earth circuits
round the moon’s centre of gravity. He appears then sometimes displaced
about 3000 miles on one side, sometimes about 3000 miles on the other
side of the place which he would have if our earth were not thus
perturbed by the moon. But astronomers can note each day where he is,
and thus learn by how much he seems displaced from his mean position.
Knowing that his greatest displacement corresponds to so many miles
exactly, and noting what it seems to be, they learn, in fact, how large
a span of so many miles (about 3000) looks at the sun’s distance.
Thus they learn the sun’s distance precisely as a rifleman learns the
distance of a line of soldiers when he has ascertained their apparent
size—for only at a certain distance can an object of known size have a
certain apparent size.
The moon comes in, in another way, to determine the sun’s distance
for us. We know how far away she is from the earth, and how much,
therefore, she approaches the sun when new, and recedes from him when
full. Calling this distance, roughly, a 390th part of the sun’s, her
distance from him when new, her mean distance, and her distance from
him when full, are as the numbers 389, 390, 391. Now, these numbers
do not quite form a continued proportion, though they do so very
nearly (for 389 is to 390 as 390 to 391-1/400). If they were in exact
proportion, the sun’s disturbing influence on the moon when she is at
her nearest would be exactly equal to his disturbing influence on the
moon when at her furthest from him—or generally, the moon would be
exactly as much disturbed (on the average) in that half of her path
which lies nearer to the sun as in that half which lies further from
him. As matters are, there is a slight difference. Astronomers can
measure this difference; and measuring it, they can ascertain what the
actual numbers are for which I have roughly given the numbers 389, 390,
and 391; in other words, they can ascertain in what degree the sun’s
distance exceeds the moon’s. This is equivalent to determining the
sun’s distance, since the moon’s is already known.
Another way of measuring the sun’s distance has been “favoured”
by Jupiter and his family of satellites. Few would have thought,
when Römer first explained the delay which occurs in the eclipse of
these moons while Jupiter is further from us than his mean distance,
that that explanation would lead up to a determination of the sun’s
distance. But so it happened. Römer showed that the delay is not in the
recurrence of the eclipses, but in the arrival of the news of these
events. From the observed time required by light to traverse the extra
distance when Jupiter is nearly at his furthest from us, the time in
which light crosses the distance separating us from the sun is deduced;
whence, if that distance has been rightly determined, the velocity of
light can be inferred. If this velocity is directly measured in any
way, and found not to be what had been deduced from the adopted measure
of the sun’s distance, the inference is that the sun’s distance has
been incorrectly determined. Or, to put the matter in another way, we
know exactly how many minutes and seconds light takes in travelling to
us from the sun; if, therefore, we can find out how fast light travels
we know how far away the sun is.
But who could hope to measure a velocity approaching 200,000 miles in a
second? At a first view the task seems hopeless. Wheatstone, however,
showed how it might be accomplished, measuring by his method the yet
greater velocity of freely conducted electricity. Foucault and Fizeau
measured the velocity of light; and recently Cornu has made more exact
measurements. Knowing, then, how many miles light travels in a second,
and in how many seconds it comes to us from the sun, we know the sun’s
distance.
The first of the methods which I here describe as new methods must next
be considered. It is one which Leverrier regards as the method of the
future. In fact, so highly does he esteem it, that, on its account, he
may almost be said to have refused to sanction in any way the French
expeditions for observing the transit of Venus in 1874.
The members of the sun’s family perturb each other’s motions in a
degree corresponding with their relative mass, compared with each
other and with the sun. Now, it can be shown (the proof would be
unsuitable to these pages,[10] but I have given it in my treatise
on “The Sun”) that no change in our estimate of the sun’s distance
affects our estimate of his mean density as compared with the earth’s.
His substance has a mean density equal to one-fourth of the earth’s,
whether he be 90 millions or 95 millions of miles from us, or indeed
whether he were ten millions or a million million miles from us
(supposing for a moment our measures did not indicate his real distance
more closely). We should still deduce from calculation the same
unvarying estimate of his mean density. It follows that the nearer
any estimate of his distance places him, and therefore the smaller it
makes his estimated volume, the smaller also it makes his estimated
mass, and in precisely the same degree. The same is true of the planets
also. We determine Jupiter’s mass, for example (at least, this is the
simplest way), by noting how he swerves his moons at their respective
(estimated) distances. If we diminish our estimate of their distances,
we diminish at the same time our estimate of Jupiter’s attractive
power, and in such degree, it may be shown (see note), as precisely to
correspond with our changed estimate of his size, leaving our estimate
of his mean density unaltered. And the same is true for all methods of
determining Jupiter’s mass. Suppose, then, that, adopting a certain
estimate of the scale of the solar system, we find that the resulting
estimate of the masses of the planets and of the sun, _as compared with
the earth’s mass_, from their observed attractive influences on bodies
circling around them or passing near them, accords with their estimated
perturbing action as compared with the earth’s,—then we should infer
that our estimate of the sun’s distance or of the scale of the solar
system was correct. But suppose it appeared, on the contrary, that
the earth took a larger or a smaller part in perturbing the planetary
system than, according to our estimate of her relative mass, she
should do,—then we should infer that the masses of the other members
of the system had been overrated or underrated; or, in other words,
that the scale of the solar system had been overrated or underrated
respectively. Thus we should be able to introduce a correction into our
estimate of the sun’s distance.
Such is the principle of the method by which Leverrier showed that in
the astronomy of the future the scale of the solar system may be very
exactly determined. Of course, the problem is a most delicate one. The
earth plays, in truth, but a small part in perturbing the planetary
system, and her influence can only be distinguished satisfactorily
(at present, at any rate) in the case of the nearer members of the
solar family. Yet the method is one which, unlike others, will have
an accumulative accuracy, the discrepancies which are to test the
result growing larger as time proceeds. The method has already been
to some extent successful. It was, in fact, by observing that the
motions of Mercury are not such as can be satisfactorily explained by
the perturbations of the earth and Venus according to the estimate of
relative masses deducible from the lately discarded value of the sun’s
distance, that Leverrier first set astronomers on the track of the
error affecting that value. He was certainly justified in entertaining
a strong hope that hereafter this method will be exceedingly effective.
We come next to a method which promises to be more quickly if not more
effectively available.
Venus and Mars approach the orbit of our earth more closely than any
other planets, Venus being our nearest neighbour on the one side, and
Mars on the other. Looking beyond Venus, we find only Mercury (and the
mythical Vulcan), and Mercury can give no useful information respecting
the sun’s distance. He could scarcely do so even if we could measure
his position among the stars when he is at his nearest, as we can
that of Mars; but as he can only then be fairly seen when he transits
the sun’s face, and as the sun is nearly as much displaced as Mercury
by change in the observer’s station, the difference between the two
displacements is utterly insufficient for accurate measurement. But,
when we look beyond the orbit of Mars, we find certain bodies which are
well worth considering in connection with the problem of determining
the sun’s distance. I refer to the asteroids, the ring of small planets
travelling between the paths of Mars and Jupiter, but nearer (on the
whole[11]) to the path of Mars than to that of Jupiter.
The asteroids present several important advantages over even Mars and
Venus.
Of course, none of the asteroids approach so near to the earth as Mars
at his nearest. His least distance from the sun being about 127 million
miles, and the earth’s mean distance about 92 millions, with a range of
about a million and a half on either side, owing to the eccentricity of
her orbit, it follows that he _may_ be as near as some 35 million miles
(rather less in reality) from the earth when the sun, earth, and Mars
are nearly in a straight line and in that order. The least distance of
any asteroid from the sun amounts to about 167 million miles, so that
their least distance from the earth cannot at any time be less than
about 73,500,000 miles, even if the earth’s greatest distance from
the sun corresponded with the least distance of one of these closely
approaching asteroids. This, by the way, is not very far from being the
case with the asteroid Ariadne, which comes within about 169 million
miles of the sun at her nearest, her place of nearest approach being
almost exactly in the same direction from the sun as the earth’s place
of greatest recession, reached about the end of June. So that, whenever
it so chances that Ariadne comes into opposition in June, or that the
sun, earth, and Ariadne are thus placed—
Sun________Earth________Ariadne,
Ariadne will be but about 75,500,000 miles from the earth. Probably no
asteroid will ever be discovered which approaches the earth much more
nearly than this; and this approach, be it noticed, is not one which
can occur in the case of Ariadne except at very long intervals.