HUGHES: HORROCKS’S LAW
Horrocks’s bogus law
David W Hughes wonders why 17th-century astronomers, notably Jeremiah Horrocks, claimed a link
between planetary size and distance from the Sun.
ABSTRACT
In the early 17th century it was generally
accepted that the size of a planet was
related to its distance from the Sun. The
Lancashire astronomer Jeremiah
Horrocks (1619–42) took this a stage
further. For the planets known at the
time he suggested that the planet–Sun
distance was about 15000 times the
planetary radius. This was equivalent to
saying that all the planetary spheres
subtended an angle of 28arcsec at the
Sun. This so called “Horrocks’s Law”
was confirmed by his Sunday 24
November 1639 (old style) measurement
of the angular diameter of the transiting
Venus. It leads to a solar parallax of
14arcsec, a value considered rather
small at the time, but refreshingly close
to the modern value of about
8.798arcsec.
O
ne strange consequence of the recent
transit of Venus (8 June 2004) was that
it briefly returned “Horrocks’s Law”
from its rightful resting place in forgotten obscurity. Following Aughton (2004), where this law
is stated, and first (to my knowledge) given its
name, this can be quoted as: “All planets (with
the exception of Mars) are the same angular size
when seen from the Sun, this size being 28 seconds of arc.” Without this law (figure 1), Horrocks could not have used his 76 arcsec transit
observation of the angular size of Venus to confirm his 15 000 RE (94 000 000 km) estimate for
the astronomical unit. Here RE is the radius of
the Earth, which in the early 17th century was
taken to be somewhere between 5800 and
6900 km (Horrocks does not quote the value
that he used for the size of the Earth but according to Chapman [1990], he might have followed
Richard Norwood [1637] who had computed
that a meridianal degree of latitude was 69.54
statute miles in length). In this short article I
investigate how this bogus law originated, and
where the figure of 28 arcsec came from.
The publication of Copernicus’s De Revolutionibus Orbium Coelestrium in 1543 initiated a
drastic change in the perceived spacing of the
planets. Since the 12th century the old-fashioned
1.14
Sun
Mercury Venus
Earth
Mars
Jupiter
Saturn
1 (above): A simplistic (not to scale) representation
of Horrocks’s Law: “All planets (with the exception
of Mars) are the same angular size when seen from
the Sun, this size being 28arcsec.” This is equivalent
to saying that the planet–Sun distance is 15000
times greater than the planetary radius.
2 (right): Johannes Kepler’s geometrical “vision”
of the construction of the inner solar system.
(See Tabula III in Mysterium Cosmographicum
[Mystery of the Cosmos] 1597 Tübingen)
geocentric cosmos had the distances between the
Earth and Moon, Mercury, Venus, Sun, Mars,
Jupiter and Saturn as about 64, 170, 1100, 1200,
8500, 14 000 and 20 000 RE respectively (see,
for example, Van Helden 1976). The original
Copernicus heliocentric system had the sizes of
the orbits of Mercury, Venus, Earth, Mars,
Jupiter and Saturn in the ratio of 0.3763:0.7193:
1.000:1.5198:5.2192:9.1743 (see, for example,
Stephenson 1994). About 60 years later these
ratio values were recalculated by Kepler, from
the observations of Tycho Brahe. He obtained
0.389:0.724:1.000:1.523:5.200 and 9.510, values that are considerably closer to the modern
figures (see Van Helden 1985).
Mystical mind
Kepler was convinced that these numbers were
not random. His mystical mind imbued them
with deep teleological significance, echoing an
underlying divine harmony and purpose. The
work of God, the ultimate mathematical creator,
was beautifully exemplified by Kepler’s wellknown geometrical construction of the inner
solar system (figure 2) with its nested planetcontaining spheres separated by the five regular
solids, the cube, tetrahedron, dodecahedron,
icosohedron and octahedron, published in
Mysterium Cosmographicum (Tübingen 1597).
It was, however, not just the planetary distance
ratios that were non-random. Kepler was also
convinced that their sizes were non-random too:
“Nothing is more in concord with nature
than that the order of magnitudes should be
the same as the order of the spheres, so that
among the six primary planets, Mercury
should have the least body, because it is
inmost, and should obtain the most narrow
sphere; that next to Mercury should be
Venus, which is larger, but still smaller than
the Earth’s…” (see Kepler 1618).
Unfortunately Kepler was unclear as to
whether it was the diameter, surface area or volume of the planetary body that was proportional to the average heliocentric distance. Like
all good theoretical astronomers, Kepler turned
to observations. His friend and correspondent
Johannes Remus Quietanus had noted, in 1618,
that Jupiter at opposition was about 50″ across
and Saturn about 30″ and Mars appeared larger
than Jupiter but not by much (see Van Helden
op cit. p85). Planetary volume proportional to
heliocentric distance seemed to fit best.
When Jeremiah Horrocks wrote Venus in Sole
Visa in around 1640 he made it clear that he
regarded the most important result coming from
A&G • February 2005 • Vol. 46
Predictions
Using the data in his newly completed Rudolphine Tables, Kepler produced an ephemerides
for the years 1629 to 1639. This was published
in 1630. It contained an “admonition” encouraging astronomers to look out for the two transits predicted to occur in 1631, Mercury
crossing the Sun on 7 November and Venus on 6
December. This notice was also widely distributed in the form of a small tract (Bartsch 1630).
Kepler stressed that the transits would provide
an excellent opportunity to obtain an accurate
measurement of the size of the two inferior
planets. Irradiance problems and the lack of
accurate micrometers meant that direct measurements of planetary diameters were
extremely difficult at the time. Observations of a
bright planet against a dark sky were much less
easy to interpret than observations of a dark
planet against the background of the solar disc.
A&G • February 2005 • Vol. 46
3(a): The planetary apparent
magnitude is plotted at 10-day
intervals, the zero corresponding to 6 January 2004. In this
three-year period Mars had a
brilliant opposition on 28
August 2003.
(b): The apparent magnitude of
Mercury is plotted at 10-day
intervals, the zero
corresponding to 6 January
2004. The cyan diamonds
represent Mercury’s evening
apparitions and the purple
squares the morning
apparitions. The Earth–Mercury
distance typically varies from
about 0.7AU to 1.3AU and the
visible phase of the planet from
about 0.10 to 0.94. Both these
have a considerable effect on
the brightness as seen from
Earth, and it is clear that the
interpretation of this rapid
hundredfold variation in
brightness would have been
very daunting to the
astronomers of antiquity.
3
Mars
2
1
–600
–400
–200
Saturn
200
–1
400
Jupiter
–2
–3
Venus
–4
–5
day number
4
apparent magnitude of Mercury
his Venus transit observations was the measurement of the apparent angular diameter of Venus
at inferior conjunction. For this he obtained a
value of about 76″, his friend William Crabtree
in nearby Broughton, Manchester, obtaining
63″. From previous measurements Horrocks
was expecting a value of about 1 arcmin, this
being a great deal less than the 3′ suggested by
Tycho Brahe, the 7′ by Kepler and the 11′ by
Lansberg (see Chapman 1990).
In fact, ever since the 12th and 13th-century
translations of the works of Ptolemy and the
Moslem astronomers had circulated throughout
Europe, the accepted angular diameters of the
Moon, Mercury, Venus, Sun, Mars, Jupiter and
Saturn, as seen from Earth, had been taken to
be 30′, 2′, 3′, 30′ 1.5′ 2.5′ and 1.67′ respectively
(see Van Helden 1976). These values apply to
planets at their mean distance from Earth. Converting to the heliocentric system, Kepler, in his
Admonitio (see below) had predicted that Venus
would have an angular diameter of almost a
quarter that of the Sun, at the 1631 transit.
In calculating the mean angle that Venus subtended at the Sun, Horrocks did not ignore the
small eccentricities (0.0068 and 0.0167) of the
orbits of Venus and Earth. The zero-eccentricity
approximation gives Venus a heliocentric distance of 0.723 AU and a geocentric distance of
0.277 AU. So the 76″ angle that Venus subtended at Earth corresponds to a
76 × 0.277 / 0.7323 = 29.1″ angle subtended at
the Sun. Horrocks showed a great interest in
contemporary planetary tables. He used a more
exact method. The Sky astronomy software
package indicates that the heliocentric distances
of Venus and Earth at the time of the Horrocks
observation were 0.720367 and 0.984557 AU
respectively, so the angle subtended by Venus at
the Sun was really 27.873″. Horrocks used
0.72000 and 0.98409 AU (see Horrocks 1662
p123, Davis 1967 p88) giving 27.876″.
apparent magnitude of planet
HUGHES: HORROCKS’S LAW
3
2
1
–600
Van Helden (1976) notes that Pierre Gassendi
was the only observer to produce wellpublicized, reliable contemporary observations
of the 1631 Mercury transit. When he first saw
Mercury on the morning of 7 November,
Gassendi was extremely surprised as to how
small it appeared. In fact he was only convinced
that he was not seeing a sunspot after he had
recorded the speed with which Mercury was
moving across the disc. Gassendi obtained a
rather inexact angular diameter of “the third
part of a minute”, i.e. 20 arcsec, and this result
was published in 1632. Horrocks would have
known of this result. At the time of transit The
Sky indicates that the heliocentric distances of
Mercury and Earth were 0.312865 and
0.989368 respectively, so Mercury’s 20 arcsec
diameter, as seen from Earth, corresponds to a
43 arcsec diameter as seen from the Sun. (Van
Helden [1985 p97] notes that two other observations have subsequently come to light [see
also Humbert 1936]. Johann Baptist Cysat, at
Innsbruck, obtained a value of 25 arcsec, and
Johannes Remus Quietanus, at Rouffach, about
18 arcsec. Neither of these would have been
known to Horrocks.) Horrocks would, however, have used Kepler’s value for the contemporary Mercury–Earth/Mercury–Sun distances.
This gave Mercury an angular diameter of about
28 arcsec as seen from the Sun.
So Horrocks had observed that Venus subtended an angle of about 28″ at the Sun, and that
calculations using Gassendi’s data and Kepler’s
orbit indicated that Mercury did likewise. This
was too good a coincidence to overlook. Horrocks jumped to the conclusion that all the planets, including Earth, had diameters that
–400
–200
200
400
–1
–2
day number
subtended the same angle at the Sun, and that
this angle was around 28″. In his Astronomia
Kepleriana Defensa & Promota (1673 London),
a posthumous work pieced together by John
Wallis (1616–1703) Savilian Professor of Geometry at Oxford, Horrocks concluded that the
solar parallax was 14 arcsec (see Wallis 1672).
Sometime after the death of Horrocks, the Belgian astronomer Gottfried Wendelin (1580–
1667) added his weight to the “all the planetary
spheres subtended an angle of about 30 to
28 arcsec at the Sun” hypothesis. In 1647 Wendelin returned to the lunar dichotomy method
of measuring the astronomical unit. After much
careful observation he found that the dichotomy
occurred when the Sun–Earth–Moon angle was
89° 45′. Using the established Earth–Moon
distance, this leads to the astronomical unit
being 13 740 RE. So the solar parallax is
tan–1 (1 / 13740) = 4.17 ×10–3 degrees = 15.01″.
This gives the angle mentioned above as 30arcsec.
Kepler’s three laws of planetary motion relied
on accurate estimations of the orbital periods of
the planets. But astronomers could also estimate
planetary brightnesses, which clearly interested
Horrocks. In Venus in Sole Visa he wrote:
“I have often compared Jupiter with Venus,
which may be done with certainty, as they
shine so equally. On the morning of the 25th
February 1640, I thought him rather less; on
the 2nd March, I thought him equal or perhaps rather larger; on the 6th, I thought him
evidently larger” (see Whatton 1859 p210).
Maybe Horrocks’s rather crude brightness
estimates were used to bolster his faith in
“Horrocks’s Law”. The brightness, b, of a
planet (ignoring phase effects) is given by
1.15
log relative brightness
HUGHES: HORROCKS’S LAW
Table 1: Typical mean apparent magnitudes and mean
Earth–planet distances for the three years 2002.00–2004.99
2
1.5 Venus
1
Jupiter
Mean distance each year (AU)
2002
2003
2004
0.5
0
–0.5
–1
0.2
0.4 0.6
Mars
0.8
Saturn
1
1.2
log mean distance from Earth
4: The log of the mean relative brightness of
Venus, Mars, Jupiter and Saturn (the means
brightness of Saturn being taken as unity) is
plotted as a function of the log of their mean
distance from the Earth (in AU). If Horrocks’s
Law is valid there should be a linear
relationship between these two quantities of
gradient –2.0. This is shown as a straight line.
ALR2
b = 8πr2∆2
(1)
where A is the albedo of the planet, L is the luminosity of the Sun, r is the planet–Sun distance
and ∆ is the planet–Earth distance (see, for example, Cole and Woolfson 2002). Horrocks’s Law
intimated that R ∝ r, so if we assume that all
planets have similar albedos equation (1) gives
1
b ∝ 2
(2)
∆
Figures 3a and b show a typical three-year run
of recent planetary apparent magnitudes. A similar variability would have been experienced by
observers throughout history. It can be seen that
Venus, the brightest planet, has a typical apparent magnitude of −4.0, followed by Jupiter at
–2.1 and Saturn at −0.05. Discounting the brilliant opposition in 2003, Mars has a more typical magnitude of about +1.5. Mercury’s apparent
magnitude would, historically, be very difficult
both to assess and explain (see figure 3b).
The typical mean planet–Earth distances and
mean apparent magnitudes over the three-year
test period are given in table 1. Following Horrocks’s Law and equation 2, a plot of the logarithm of the average planetary brightness as a
function of the logarithm of the mean planet–
Earth distance (figure 4), should be a straight
line of gradient −2.0. Figure 4 indicates that this
supposition is not too fanciful.
The uncritical acceptance by Horrocks of
what now appears to be a very strange and scientifically unjustifiable “law” might have something to do with his veneration of Kepler. On the
one hand Horrocks has been pictured as a “new
man”. Jeremy Shakerley (1626–1655?) referred
to him as a “Noble Genius”, and praised him as
the English Galileo. Here we sense the underlying revolutionary implications of a challenger
of the old order of astronomy. Horrocks was an
observer at heart. Accuracy was important. He
strongly believed that any hypothesis not in
agreement with observations was, of necessity,
false. On the other hand we must note that Horrocks’s planetary orbital work and ephemeris
production was firmly anchored in the past.
1.16
Venus
Mars
Jupiter
Saturn
1.006
2.204
5.362
9.096
1.409
0.716
5.475
9.057
0.884
2.052
5.497
9.306
Chapman (op cit 1990 p345) stresses the fact
that at the top of Horrocks’s personal list of scientific heroes was the mystical, religious “John
Kepler, the prince of astronomers”.
Kepler’s geometrical masterpiece (figure 2)
might seem strange and contrived to us now. But
just think how strange, inexplicable and “magical” Kepler’s third (harmonic) law would be to
Horrocks and the astronomers of Horrocks’s
time (see Wallis 1637 p295). In those preNewtonian days astronomers had no idea of the
motive force behind the motion of the planets.
Most contemporary notions assumed that the
Sun was magnetized and that the spinning Sun
somehow wafted the planets along. The inexplicable fact that the cube of the planet–Sun distance was proportional to the square of the
planetary orbital period would have made contemporary astronomers receptive to the suggestion that the diameter of the planetary globe was
proportional to its distance from the Sun. At the
time there was no obvious underlying scientific
reason for either statement. They could both
easily be just part of God’s underlying plan
behind the construction of the solar system.
Red herring
Horrocks’s Law, his suggestion that the radius
of a planet was proportional to its distance from
the Sun, might at the time be seen as a vital clue
to the forces behind Kepler’s Harmonic Law.
At least Horrocks also sowed the seeds for his
law’s destruction. Like all good astronomers
Horrocks was cautious: “I do not put forward
this conjecture as a certain demonstration, but
as a probability” (see Venus in Sole Visa p143).
Horrocks was an assiduous observer of the
heavens. For five years he used an astronomical
radius and a series of occulting strings and wires
in pioneering investigations of planetary angular
diameters. These investigations were to continue
over the following centuries and it soon became
abundantly clear that the law was false.
The Horrocks value of 28″ for the Earth’s
mean angular diameter as seen from the Sun is
fairly close to today’s value of 17.8″. Prior to
Horrocks, the authoritarian status of Ptolemy’s
writings and the stature of the opinions of Tycho
Brahe added weight to the acceptance of an
angle of 6 arcmin for this mean diameter. This
acceptance was prevalent even though Kepler
had re-examined the solar parallax problem and
Mean app.
magnitude
Overall mean
distance (AU)
–4.0
+1.5
–2.1
–0.05
1.10
2.13
4.45
9.15
suggested that the angle was not greater than
2 arcmin and was likely to be much below this
suggested upper limit.
The importance of Horrocks’s 28 arcsec is that
it firmly places him as the first astronomer to
have a reasonably correct appreciation of the
immensity of the solar system, the true size of
the Sun, and the vastness of astronomical distances in general. ●
David W Hughes, Dept of Physics & Astronomy,
The University, Sheffield S3 7RH, UK;
[email protected]
Acknowledgments: I would like to thank Dr Adam
Hart-Davis for initiating this enquiry and Dr Allan
Chapman for his encouragement.
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A&G • February 2005 • Vol. 46