# 1911 Encyclopædia Britannica/Parallax

**PARALLAX** (Gr. παραλλάξ, alternately), in astronomy, the apparent change in the direction of a heavenly body when viewed from two different points. Geocentric parallax is the angle between the direction of the body as seen from the surface of the earth and the direction in which it appears from the centre of the earth. Annual parallax is the angle between the direction in which a star appears from the earth and the direction in which it appears from the centre of the sun. For stellar parallaxes see Star; the solar parallax is discussed below.

*Solar Parallax*.—The problem of the distance of the sun
has always been regarded as the fundamental one of celestial
measurement. The difficulties in the way of solving it are very
great, and up to the present time the best authorities are not
agreed as to the result, the effect of half a century of research
having been merely to reduce the uncertainty within continually
narrower limits. The mutations of opinion on the subject
during the last fifty years have been remarkable. Up to about
the middle of the 19th century it was supposed that transits of
Venus across the disk of the sun afforded the most trustworthy
method of making the determination in question; and when
Encke in 1824 published his classic discussion of the transits
of 1761 and 1769, it was supposed that we must wait until the
transits of 1874 and 1882 had been observed and discussed
before any further light would be thrown on the subject. The
parallax 8.5776″ found by Encke was therefore accepted without
question, and was employed in the *Nautical Almanac* from 1834
to 1869. Doubt was first thrown on the accuracy of this number
by an announcement from Hansen in 1862 that the observed
parallactic inequality of the moon was irreconcilable with the
accepted value of the solar parallax, and indicated the much
larger value 8.97″. This result was soon apparently confirmed
by several other researches founded both on theory and observation,
and so strong did the evidence appear to be that the
value 8.95″ was used in the Nautical Almanac from 1870 to 1881.
The most remarkable feature of the discussion since 1862 is that
the successive examinations of the subject have led to a continually
diminishing value, so that at the present time it seems
possible that the actual parallax of the sun is almost as near to
the old value of Encke as to that which first replaced it. The
value of 8.848″, determined by S. Newcomb, was used from 1882
to 1900; and since then the value 8.80″ has been employed,
having been adopted at a Paris conference in 1896.^{[1]}

Five fundamentally different methods of determining the distance of the sun have been worked out and appHed. They are as follows:—

I. That of direct measurement.—From the measures of the
parallax of either Venus or Mars the parallax of the sun can
be immediately derived, because the ratios of distances in
the solar system are known with the last degree ofMethods of

Determina-

tion.
precision. Transits of Venus and observations of various
kinds on Mars are all to be included in this class.

II. The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the earth. The velocity is now well determined; the difficulty is to determine the time of passage.

III. The third method is through the determination of the mass of the earth relative to that of the sun. In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity. When we know the mass of the earth in gravitational measure, its product by the denominator of the fraction just mentioned gives the mass of the sun in gravitational measure. From this the distance of the sun can be at once determined by a fundamental equation of planetary motion.

IV. The fourth method is through the parallactic inequality in the moon's motion. For the relation of this inequality to the solar parallax see Moon.

V. The fifth method consists in observing the displacement in the direction of the sun, or of one of the nearer planets, due to the motion of the earth round the common centre of gravity of the earth and moon. It requires a precise knowledge of the moon's mass. The uncertainty of this mass impairs the accuracy of the method.

I. To begin with the results of the first method. The transits
of Venus observed in 1874 and 1882 might be expected to hold
a leading place in the discussion. No purelyTransits of

Venus.
astronomical enterprise was ever carried out on so
large a scale or at so great an expenditure of money
and labour as was devoted to the observations of these transits,
and for several years before their occurrence the astronomers of
every leading nation were busy in discussing methods of observation
and working out the multifarious details necessary to
their successful application. In the preceding century rehance
was placed entirely on the observed moments at which Venus
entered upon or left the limb of the sun, but in 1874 it was
possible to determine the relative positions of Venus and the
sun during the whole course of the transit. Two methods
were devised. One was to use a heliometer to measure the
distance between the hmbs of Venus and the sun during the
whole time that the planet was seen projected on the solar disk,
and the other was to take photographs of the sun during the
period of the transit and subsequently measure the negatives.
The Germans laid the greatest stress on measures with the
hehometer; the Americans, English, and French on the photographic
method. These four nations sent out well-equipped
expeditions to various quarters of the globe, both in 1874 and
1882, to make the required observations; but when the results
were discussed they were found to be extremely unsatisfactory.
It had been supposed that, with the greatly improved telescopes
of modern times, contact observations could be made with much
greater precision than in 1761 and 1769, yet, for some reason
which it is not easy to explain completely, the modern observations
were but little better than the older ones. Discrepancies
difficult to account for were found among the estimates of even
the best observers. The photographs led to no more definite
result than the observations of contacts, except perhaps those
taken by the Americans, who had adopted a more complete
system than the Europeans; but even these were by no means
satisfactory. Nor did the measures made by the Germans with
heliometers come out any better. By the American photographs
the distances between the centres of Venus and the sun, and the
angles between the line adjoining the centres and the meridian,
could be separately measured and a separate result for the
parallax derived from each. The results were:—

Transit of 1874: Distances; par.=8.888″.

Pos. angles; par.=8.873″.

Transit of 1882: Distances; par.=8.873″.

Pos. angles; par.=8.772″.

The German measures with the heliometer gave apparently
concordant results, as follows:—

Transit of 1874: par. =8.876".

Transit of 1882: par. =8.879".

The combined result from both these methods is 8.857", while the combination of all the contact observations made by all the parties gave the much smaller result, 8.794". Had the internal contacts alone been used, which many astronomers would have considered the proper course, the result would have been 8.776".

In 1877 Sir David Gill organized an expedition to the island of
Ascension to observe the parallax of Mars with the heliometer.
By measurements giving the position of Mars among
neighbouring stars in the morning and evening, the effect of parallax could be obtained as well as
by observing from two different stations; in fact the rotation
of the earth carried the observer himself round a parallel of
latitude, so that the comparison of his own morning and
evening observations could be used as if they had been made at
different stations. The result was 8.78". The failure of the
method based on transits of Venus led to an international
effort carried out on the initiative of Sir David Gill to measure
the parallax by observations on those minor planets which
approach nearest the earth. The scheme of observations was
organized on an extended scale. The three bodies chosen
for observation were: Victoria (June 10 to Aug. 26, 1889);
Iris (Oct. 12 to Dec. 10, 1888); and Sappho (Sept. 18 to Oct. 25,
1888). The distances of these bodies at the times of opposition
were somewhat less than unity, though more than twice as great
as that of Mars in 1877. The drawback of greater distance
was, however, in Gill's opinion, more than compensated by the
accuracy with which the observations could be made. The
instruments used were heliometers, the construction and use of
which had been greatly improved, largely through the efforts of
Gill himself. The planets in question appeared in the telescope
as star-like objects which could be compared with the stars with
much greater accuracy than a planetary disk like that of Mars,
the apparent form of which was changed by its varying phase,
due to the different directions of the sun's illumination. These
observations. were worked up and discussed by Gill with great
elaboration in the Annals of the Cape Observatory, vols. vi. and
vii. The results were for the solar parallax π:—

From Victoria, π = 8.801"±0.006".

From Sappho, π = 8.798"±0.011".

From Iris, π = 8.812"±0.009".

The general mean result was 8.802". From the meridian observations of the same planets made for the purpose of controlling the elements of motion of the planets Auwers found π=8.806".

In 1898 the remarkable minor planet Eros was discovered,
which, on those rare occasions when in opposition near perihelion,
would approach the earth to a distance of 0.16. On these
occasions the actual parallax would be six times greater than that
of the sun, and could therefore be measured with much greater
precision than in the case of any other planet. Such an approach
had occurred in 1892, but the planet was not then discovered.
At the opposition of 1900–1901 the minimum distance was
0.32, much less than that of any other planet. Advantage
was taken of the occasion to make photographic measures for
parallax at various points of the earth on a very large scale.
Owing to the difficulties inherent in determining the position
of so faint an object among a great number of stars, the results
have taken about ten years to work out. The photographic
right ascensions gave the values 8-80" + 0.007" ± 0.0027"
(Hinks) and 8.80" + 0.0067" + 0.0025" (Perrine); the
micro metric observations gave the value 8.8o6"+0.004 (Hinks).^{[2]}

II. The velocity of light (*q.v.*) has been measured with all
the precision necessary for the purpose. The latest result is
299,860 kilometres per second, with a probable error of perhaps
30 kilometres—that is, about the ten-thousandth part of the
quantity itself. This degree of precision is far beyond any we
can hope to reach in the solar parallax. The other element
which enters into consideration is the time required for light to
pass from the sun to the earth. Here no such precision can
be attained. Both direct and indirect methods are available.
The direct method consists in observing the times of some
momentary or rapidly varying celestial phenomenon, as it
appears when seen from opposite points of the earth's orbit.
The only phenomena of the sort available are eclipses of Jupiter's
satellites, especially of the first. Unfortunately these eclipses
are not sudden but slowly changing phenomena, so that they
cannot be observed without an error of at least several seconds,
and not infrequently important fractions of a minute. As the
entire time required for light to pass over the radius of the earth's
orbit is only about 500 seconds, this error is fatal to the method.
The indirect method is based upon the observed constant of
aberration or the displacement of the stars due to the earth's
motion. The minuteness of this displacement, about 20.50",
makes its precise determination an extremely difficult matter.
The most careful determinations are affected by systematic
errors arising from those diurnal and annual changes of temperature,
the effect of which cannot be wholly eliminated in astronomical
observation; and the recently discovered variation of
latitude has introduced a new element of uncertainty into the
determination. In consequence of it, the values formerly
found were systematically too small by an amount which even
now it is difficult to estimate with precision. Struve's classic
number, universally accepted during the second half of the 19th
century, was 20.445". Serious doubt was first cast upon its
accuracy by the observations of Nyren with the same instrument
during the years 1880–1882, but on a much larger number of stars.
His result, from his observations alone, was 20.52"; and taking
into account the other Pulkowa results, he concluded the most
probable value to be 20.492". In 1895 Chandler, from a general
discussion of all the observations, derived the value of 20.50".
Since then, two elaborate series of observations made with
the zenith telescope for the purpose of determining the variation
of latitude and the constant of aberration have been carried
on by Professor C. L. Doolittle at the Flower Observatory near
Philadelphia, and Professor J. K. Rees and his assistants at the
observatory of Columbia University, New York. Each of these
works is self-consistent and seemingly trustworthy, but there
is a difference between the two which it is difficult to account
for. Rees's result is 20.47"; Doolittle's, from 20.46" to 20.56".
This last value agrees very closely with a determination made by
Gill at the Cape of Good Hope, and most other recent determinations
give values exceeding 20.50". On the whole it is probable
that the value exceeds 20.50"; and so far as the results of direct
observation are concerned may, for the present, be fixed at
20.52". The corresponding value of the solar parallax is 8.782".
In addition to the doubt thrown on this result by the discrepancy
between various determinations of the constant of aberration,
it is sometimes doubted whether the latter constant necessarily
expresses with entire precision the ratio of the velocity of the
earth to the velocity of hght. While the theory that it does
seems highly probable, it cannot be regarded as absolutely
certain.

III. The combined mass of the earth and moon admits of being
determined by its effect in changing the position of the plane
of the orbit of Venus. The motion of the node ofMass of the

Earth.
this plane is found with great exactness from observations
of the transits of Venus. So exact is the latter
determination that, were there no weak point in the subsequent
parts of the process, this method would give far the most certain
result for the solar parallax. Its weak point is that the apparent
motion of the node depends partly upon the motion of the
ecliptic, which cannot be determined with equal precision. The
derivation of the distance of the sun by it is of such interest
from its simplicity that we shall show the computation.

From the observed motion of the node of Venus, as shown by the
four transits of 1761, 1769, 1874 and 1882, is found

Mass of (earth+moon) = Mass of sun332600 In gravitational units of mass, based on the metre and second as units of length and time,

Log. earth's mass = 14.60052

Log. moon's mass = 12.6895.

The sum of the corresponding numbers multiplied by 332600 gives

Log. sun's mass = 20.12773.

Putting *a* for the mean distance of the earth from the sun, and *n* for its mean motion in one second, we use the fundamental equation

*a*^{2}*n*^{2} = M_{0}M′,

M_{0} being the sun's mass, and M′ the combined masses of the earth and moon, which are, however, too small to affect the result. For the mean motion of the earth in one second in circular measure, we have

*n* = π31558149; log. *n* = 7.29907

the denominator of the fraction being the number of seconds in the sidereal year. Then, from the formula

*a*^{2} = M_{0}*n*^{2} = [20.12773]—15.59814

we find

Log. *a* in metres = 11.17653

Log. equat. rad. ⨁ 6.80470

Sine ☉'s eq. hor. par. 5.62817

Sun's eq. hor. par. 8.762″.

IV. The determination of the solar parallax through the parallactic inequality of the moon's motion also involves two elements—one of observation, the other of purely
Motion of

Moon. mathematical theory. The inequality in question has its greatest negative value near the time of the moon's first quarter, and the greatest positive value near the third quarter. Meridian observations of the moon have been heretofore made by observing the transit of its illuminated limb. At first quarter its first limb is illuminated; at third quarter, its second limb. In each case the results of the observations may be systematically in error, not only from the uncertain diameter of the moon, but in a still greater degree from the varying effect of irradiation and the personal eqtiation of the observers. The theoretical element is the ratio of the parallactic inequality to the solar parallax. The determination of this ratio is one of the most difficult problems in the lunar theory. Accepting the definitive result of the researches of E. W. Brown the value of the solar parallax derived by this method is about 8.773″.

V. The fifth method is, as we have said, the most uncertain
Motion of

Earth. of all; it will therefore suffice to quote the result, which is

= 8.818".

The following may be taken as the most probable values of the solar parallax, as derived independently by the five methods we have described:—

From measures of parallax . 8.802″

From velocity of light . . 8.781″

From mass of the earth . . 8.762″

From par. ineq. of moon . . 8.773″

From lunar equation . . 8.818″

The question of the possible or probable error of these results is one on which there is a marked divergence of opinion among investigators. Probably no general agreement could now be reached on a statement more definite than this; the last result may be left out of consideration, and the value of the solar parallax is probably contained between the limits 8.77″ and 8.80.″ The most likely distance of the sun may be stated in round numbers as 93,000,000 miles. (S. N.)