‡
4 The instruments used by Hipparchos
by KEITH A.PICKERING
A A revealing gap
More than 2000 years after its compilation, it is now possible to
determine with some confidence the kinds of instruments that were used
to observe the stars of the Ancient Star Catalog, and in which parts of
the sky the various instruments were employed. The fact that there were
multiple instruments, and that the ASC was not,
as stated by Ptolemy, observed with a single ecliptical astrolabe, does
more than provide yet-another proof that the catalog was observed by
Hipparchos (for we have more than enough of those already); it also
allows us a glimpse into the hitherto unknown workings of astronomy and
instrument manufacture as they were practiced at a critical point of
ancient Greek science.
The Almagest divides the ASC into three sections, for the northern,
zodiacal, and southern parts of the sky. Looking at the northern sky,
figure 1 plots the absolute errors in longitude for each northern star
in the ASC, according to its actual longitude at the epoch of the
catalog (which we will take to be -128.0). In looking at
the plot, note particularly that there is an odd gap in the plotted
stars at about 70
° ecliptic longitude. Note also that there is a
similar gap at about 250
° ecliptic longitude, exactly 180
°
away. This
gap can be more easily seen if we overlay the second half of the
longitudes (180-360) on top of the first half, as we have done in
figure 2. Note particularly that the longitude errors increase in
absolute magnitude as we get close to the gap. For purposes of
comparison, figure 3 plots the absolute errors in right ascension by
right ascension: the gap disappears.
This gap is significant because it shows us, first, that the northern
sky was observed primarily with a single instrument; second, that the
instrument was an ecliptic astrolabe; and third, that the astrolabe was
of a somewhat different design from the description given by Ptolemy,
and indeed different from any previously known to have existed.
Figure 1:
Errors in longitude by longitude, for northern
stars in the ASC. Note vertical gaps in the data.
Figure 2:
Errors in longitude by longitude, with longitudes overlain.
Note vertical gap in the data.
Figure 3:
Errors in right ascension by right ascension, for northern stars in the ASC.
Note lack of vertical gaps in the data.
B The astrolabe
The armillary astrolabe used in ancient Greek astronomy is, at first
glance, a bewildering maze of nested rings, fitted closely inside each
other, that rotate in complex ways. Let's look at the way an armillary
astrolabe is contructed, from the inside out. The innermost ring (Ring
1) contains a pair of sighting holes or pinnules, diametrically
opposite each other, through which the star is sighted. Immediately
surrounding Ring 1 is Ring 2, whose inside diameter is fractionally
larger than the outside diameter of Ring 1. Ring 1 is constrained so
that it rotates inside Ring 2, in the same plane, their edges just
touching. Ring 2 has a scale of degrees on its edge, indicating the
rotational position of the pinnules on Ring 1. (See figure 4.) If we
wished, we could mount Ring 2 on the meridian, and then use the Ring
1&2 assembly as a transit instrument. To do this, we would have to
orient Ring 2 so that it points north-south, and so that its
zero-degree points on the scale were horizontal, and 90-degree points were
vertical.
But to make the Ring 1&2 assembly more useful, we will mount it
differently. We construct an outer ring, Ring 3, set vertically
so that the whole Ring 1&2 assembly can pivot within it, around a
vertical axis. We run axle pins from the 90-degree poles of Ring
2 into the inner edge of Ring 3; so now Ring 1&2 can rotate to any
azimuth. To determine the azimuth at which Ring 1&2 is pointing, we
add Ring 4, which is fixed horizontally and at right angles to Ring 3.
Ring 4 carries another scale of degrees, indicating the rotational
position of Ring 2. Rings 3&4 now form a cage, within which Ring 2
rotates freely in azimuth, while Ring 1 rotates freely in altitude
within Ring 2. (See figure 4). The instrument can now be used as a
theodolite, since we can determine the altitude and azimuth of any
star with it. We will call this arrangement the 4-ring instrument.
Figure 4:
The four-ring instrument. Ring 1 (innermost white) carries
pinnules through which the star is sighted. Ring 2 (dark gray) has a scale of
degrees. Ring 3 (outer white) holds the polar axis. Ring 4 (light gray) contains
the second scale of degrees. When Ring 4 is horizontal, the instrument can be
used as a theodolite; when mounted to rotate with the sky, it is an astrolabe.
The 4-ring instrument is capable of pointing to almost any point in
the celestial sphere, making it quite useful. In fact, there is only one
fly in the ointment to this whole arrangement: at certain rotational
positions, Ring 2 becomes so closely aligned with Ring 3 that a star
cannot be seen through the pinnules, because Ring 3 gets in the way.
There are two such rotational positions, exactly 180
° apart. For the
same reason, it is impossible to observe very near to the horizon,
because Ring 4 gets in the
way.
A larger issue with the 4-ring instrument is one of orientation. With
Ring 4 oriented horizontally, it makes a fine theodolite, but
horizon-based coordinates are of limited utility in astronomy, because the sky
moves as the earth rotates. It is much better to mount
the 4-ring instrument so that it rotates too, following the sky.
Recall that the purpose of Ring 3 is entirely structural: it holds the
axis around which Rings 1&2 rotate. So the most obvious
arrangement is to simply extend that axis, and orient the axis toward
the celestial pole. Then the entire 4-ring instrument could be rotated
along with the sky. The astrolabe, if mounted this way, would read
equatorial coordinates directly, because Ring 4 would be permanently
aligned with the celestial equator. All that would be needed would be
a way to align the instrument in right ascension.
Although equatorial coordinates are used extensively today, in ancient
times ecliptical coordinates were more widely used. So in practice,
what was really needed was a way to mount the 4-ring instrument so
that: (a) it could rotate with the sky; and (b) Ring 4 would be
aligned with the ecliptic instead of the celestial equator. And in the
Almagest V.1, Ptolemy describes how this was done: a second axis was
drilled in Ring 3 (this would have been 23
from the first).
Then the 4-ring instrument was mounted so that the second axis was pointed
to the celestial pole. The entire instrument could then rotate (around the
polar axis) to follow the sky; while the coordinate readings from Ring 2 and
Ring 4 are stuck in a different coordinate frame, tilted in exactly the
same manner as the ecliptic is tilted to the equator. And there we
have it: the ecliptic armillary astrobale, nearly the same as
described by Ptolemy in the Almagest.
Except for one big thing. When we drilled the polar axis in Ring 3, at
that moment we permanently fixed the ecliptic longitude of Ring 3
along the 90
°-270
° solstitial colure. This
is the great circle in the
sky through which both the ecliptic poles and celestial poles run, and
now this colure must also run through Ring 3 too, since both
instrumental axes run through Ring 3. Now we know that Ring 3 will get
in the way of some observations, so if we build an astrolabe this way - as
described by Ptolemy - we should expect there to be a gap in observed
stars at 90
° and 270
° ecliptic longitude. As we have
seen, there is a longitudinal gap, but it is not at 90-270; it is at
70-250. This means that the astrolabe which Hipparchos actually used
to observe the ASC was built in a somewhat different manner than the
one described by Ptolemy in the Almagest.
Instead of drilling a second set of axis holes in Ring 3, Hipparchos
(or his instrument maker) must have used separate bearing journals to
hold the polar axis. There would be two such journals clamped or affixed
to opposite sides of Ring 3 at the celestial poles (see fig. 5).
Since the solstitial colure (which defines 90-270 ecliptic longitude) must
contain both axes, the colure no longer contains Ring 3; rather, it is
offset by some amount. In the instrument actually used by Hipparchos,
this amount was about 20 degrees. This arrangement has a structural advantage,
because it avoids putting
another set of holes in Ring 3, which has already been weakened by the
holes for the ecliptic axis.
Figure 5:
The four-ring instrument viewed from above the poles.
Ring 2 (dark gray) pivots around the North Ecliptic Pole (NEP), while the entire
4-ring instrument rotates around the North Celestial Pole (NCP) to follow the sky.
(The NCP axis is affixed to an outer Ring 6, which is not shown in the diagram.)
According to the Almagest, the NCP axis is drilled in Ring 3 (top); in the
astrolabe used by Hipparchos, the NCP axis is carried on a separate bearing
journal (bottom). The 90-270 solsticial colure is the great circle joining the
two axes - along Ring 3 (top) or offset from it (bottom).
If he had used more than one astrolabe for observing the Northern sky,
Hipparchos could have arranged to have the journals on astrolabe #2
mounted on the opposite sides of Ring 3 than the arrangement on
astrolabe #1; so the blind spot of astrolabe #2 would be at 110-290,
and the blind spot of one instrument could be covered by the other.
Therefore it is apparent that large parts of the northern sky were
observed with a single instrument - or nearly so.
You recall that there is another blind spot, along Ring 4. This ring
falls right on the ecliptic, so we might expect to see a gap in the
data here, too, just as we found in the longitudes. In the northern
sky, only one constellation (Ophiuchus) dips all the way down to the
ecliptic; but there is no such gap along the ecliptic in Ophiuchus. In
fact, there is no such gap among the stars of the zodiacal
constellations, either.
So there must have been a different instrument or a different
technique (or both) for observing right at the ecliptic. One
possibility is a second set of
pinnules.
The primary pinnules would be mounted on Ring 1 at diametrically
opposite positions, as already described; while the second set would
be mounted above these, a little more than one ring-width away. Thus,
the sightline through the first set would be exactly parallel to the
sightline through the second set. When the first set was too close to
the ecliptic to observe, the second set would still be able to see
over the top of Ring 4.
I have been unable to find similar gaps in either the Zodiac or the South
sections of the ASC. This implies that the instrument used in the North
was different than the one(s) used in other parts of the sky.
C Gap Characteristics
Are there bright stars in the gap that Hipparchos usually would have taken,
or is the reason for the lack of cataloged stars simply that there are no
bright stars in this region of the sky? In other words, is the gap real?
As it turns out, there are only five stars in the Northern sky brighter
than magnitude 3.9 that Hipparchos left out of the catalog:
χ UMa,
α Lac, 46 LMi, 109 Her, and
α Sct. Two of these five (109
Her and
α Sct) are in the gap. Since the gap represents only about
5% of the sky, this is clearly a significant number.
The gap is caused by the physical presence of Ring 3, which has a constant
physical width. But the longitudinal width of Ring 3 increases toward the ecliptic
pole, because the lines of longitude converge there. We can determine
the relative thickness of Ring 3 by close examination of the edges of the gap.
In figure 6, I have plotted the region near the gap in latitude and
"folded" longitude, along with lines indicating the position that the gap
would have if Ring 3 was centered at 69.5 - 249.5 and had a width of 3.7 degrees.
These parameters fit the actual gap quite well (although smaller widths cannot
be excluded).
Figure 6:
Close view of the gap in cataloged stars. The lines show the
limits of a gap 3.7 degrees wide centered at 69.5 - 249.5 degrees longitude.
Figure 7:
Stars brighter than magnitude 4.5 missing from the northern
part of the ASC. Note that none are missing above latitude 75, possibly indicating
that a different kind of instrument was used in this small region of the sky.
Similarly, in figure 7, I have plotted all stars of magnitude 4.5 or
brighter that are missing from the catalog, with the same gap limits. Note particularly
that there are no missing stars this bright above latitude 75. This is a
good indication of the polar limits of the astrolabe, and shows the region in
which a different instrument was probably used. This also explains why there are
a couple of holdout stars present in the gap: the holdouts are both at very high latitudes.
The edges of the gap are between 4 and 5 degrees apart at the ecliptic. The exact edges
depend on how far north one chooses to assume was observed with this single instrument.
The gap is actually two adjacent gaps: one in which the lower pinnule is blocked by Ring 3,
and one in which the upper pinnule is blocked. Therefore, the 5-degree width of the gap
implies that the physical width of Ring 3 was between 2 and 2
°.5 degrees. The
exact center of the gap is a bit tricky to pin down, but it seems to be very close to
69
°.5 of ecliptic longitude.
Further, between these two adjacent gaps, at the very center, there is
a very narrow "gap within the gap," where a star lying at that precise longitude
should be visible. This is because, when Ring 2 is exactly aligned with
Ring 3, neither pinnule on Ring 1 is blocked by Ring 3: the line of sight passes
along the edge of Ring 3 just as if it were a wide extension of Ring 2. As it
turns out, there is in fact a cataloged star lying almost exactly at the center
of the gap:
ε UMi. But since it also lies at a very high latitude, there
is no guarantee that it was observed through the "gap within the gap," rather than
in the same manner as other stars near the ecliptic pole.
A ring about 2 degrees wide is rather narrow, structurally speaking, which in turn
places limits on the material used to construct the astrolabe. For example, if
Ring 3 was 50 cm in diameter, it could be no more than about 1 cm (perhaps less)
in width. I tend to doubt that a wooden instrument of this narrow aspect ratio could
be stiff enough against the weight it must support to be very accurate; bronze seems
a more likely material.
D Epoch of the Northern Catalog
The single-instrument hypothesis implies that the northern sky
was observed all at once, before the instrument had time to become
worn or damaged; in other words, a matter of months or a few years,
rather than decades. Careful analysis will allow us to determine the
epoch of this northern observational effort.
After subtracting Ptolemy's 2
false precessional constant, we
can reconstruct the actual longitudes of these stars as observed by
Hipparchos. Due to precession, stars advance from west to east
parallel to the ecliptic, maintaining their same ecliptic latitudes,
but increasing their ecliptic longitudes at a rate of about 83' per
century. So, as a first cut, we can simply take these reconstructed
Hipparchan longitudes and assume that they were (on average) correct
as measured, then find the epoch at which such an assumption would be
true. For the northern stars, this works out to -157,
±59 years.
There is a problem with this procedure, however, because the longitudes
observed by Hipparchos were not actually correct, on average. There is a
systematic error which we must account for. The longitude of the stars
is determined ultimately by reference to the Sun. The Sun is observed just
before sunset, on a day just after new Moon. The longitude of the Sun is
known from theory, and the difference between the Sun and Moon gives the
Moon's longitude; then, after sunset of the same day, the difference
between the Moon and a fundamental star is observed, to give the
longitude of the fundamental star; and finally, the longitudes of
individual stars are observed by their difference from the fundamental
star. But each of these steps requires the astrolabe to be briefly
clamped in position while the measurements are being made; and these
successive clampings tend to push the longitudes lower than true,
because the earth rotates during these brief intervals. In other words,
there is a systematic error in rotation of the astrolabe around the
equatorial axis.
Rawlins 1982 has shown that misrotation of the astrolabe with respect
to the real sky will make itself known by the presence of a cosine
error wave in the observed latitudes. Further, the amplitude of this
cosine error wave is proportional to the amount of astrolabe
misrotation. And in fact there is just such an error in the latitudes
of the northern stars. This error wave has an amplitude of
10.6
±1.8 arcmin, implying that
the astrolabe was systematically misrotated by 24.2
±4.2 arcmin. It takes
precession 29.2 years to move a star that far in longitude, meaning
that the actual epoch of observation for the northern stars was
-128
±59 years. This is very nearly the epoch implied by Ptolemy's
precessional constant.
References.
Rawlins, Dennis (1982). An Examination of the Ancient Star
Catalog. PASP 94, 359.
Toomer, G.J. (1998). Ptolemy's Almagest. Princeton University Press.
Footnotes:
Areas near the axis are also blocked by both Ring 3 and Ring 2.
For purposes of simplicity, we have left out Ring 5, which is used
only as an aid to orientation. Ring 6, which is fixed to the earth
as a strutctural support for the whole instrument, holds the second
axis, which points toward the North Celestial Pole.
Ptolemy does not mention a second set of pinnules in the
Almagest, but given the foregoing, he cannot be taken as a wholly
reliable source on the construction of astrolabes.
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