arXiv:physics/0501083v1 [physics.hist-ph] 17 Jan 2005
IL NUOVO CIMENTO Vol. ?, N. ? ?
Solar Oblateness from Archimedes to Dicke
) and Pietro Oliva(
) University of Rome La Sapienza
) ICRA, International Center for Relativistic Astrophysics, and University of Rome La Sapienza,
Piazzale Aldo Moro 5 00185 Roma, Italy.
The non-spherical shape of the Sun has been invoked to explain the anomalous pre-
cession of Mercury. A brief history of some methods for measuring solar diameter is
presented. Archimedes was the ﬁrst to give upper and lower values for solar diameter
in third century before Christ; we also show the method of total eclipses, used after
Halley’s observative campaign of 1715 eclipse; the variant of partial eclipses useful
to measure diﬀerent chords of the solar disk; the method of Dicke which correlates
oblateness with luminous excess in the equatorial zone.
PACS 95.10.Gi – Eclipses, transits, and occultations -.95.30.Sf - Relativity and grav-
itation -95.55.Ev - Solar instruments
1. – Archimedes
Archimedes of Syracuse  gave out an evaluation of the angle subtended by the
Sun with the vertex on the observer’s eye. He knew that a perfect determination is not
possible due to the observer bias and systematic errors, so he proposed to ﬁnd out the
Fig. 1. – Geometry of angular solar diameter measurement in c ase of point-like eye. Gray angle
is the angular lower diameter, black angle is the upper limit. The ruler is the black horizontal
Societ`a Italiana di Fisica 1
2 COSTANTINO SIGISMONDI AND PIE TRO OLIVA
Fig. 2. – Geometry of angular solar diameter measurement in case of real eye. A real eye in th e
same position of th e point-like eye sees a major portion of solar surface. In order to see only
the limb it is required to approach more the eye to the cylinder.
upper and lower angles allowed by his method: in this way we have a range within solar
angular diameter lies. To do this we must have a big ruler (a strip of wood more likely)
mounted on a basement in a pla c e such that is possible to see the dawn (Syracuse in
Sicily has an eastern free sea horizon). When the Sun rises up and it is still near the
horizon, so that is possible to aim at it, we point the r uler toward Sun; suddenly we put
on the ruler a rounded cylinder such that, if we look from one edge of the ruler pointed at
the Sun, the Sun is hidden behind that. Afterwards we need to shift the cylinder on the
ruler till we see a very little Sun limb from the e dges of the cylinder. Now we block the
cylinder in this p osition. Let’s ﬁnd now, the lower evaluation angle. Archimedes argued:
if the observer’s eye was point-like (ﬁg.1), then the angle identiﬁed by the lines drawn
from eye’s site such to be tangent to cylinder’s edges (the gray angle in ﬁg.1), would
be smaller than the angle made by the lines from the eye to the edges of the Sun (the
black a ngle in ﬁg.1). This is because we set the cylinder such that to see a portion of
the Sun. Indeed the observer’s eye is not point-like but extended: we are overestimating
the lower limit angle (see captions in ﬁg. 2 and 3). This lea d to the situation described
in ﬁg. 2 where we put, instead of the eye, a rounded surface with similar dimensions
to those of the real eye. It is possible to ﬁnd an appropriate rounded s urface by taking
two thin cylinders c oloured black and w hite. Then, we need to put the black one clo se
to the eye while the white one is placed more far in the line of sight. The right size of
the surface is the one such that the black shape perfectly hides the white one and vice
versa. In this way we’re sure to take a rounded surface not smaller than the eye. Now
we need to ﬁnd the upper limit angle: this is easier, we just need to shift the cylinder on
the ruler until the Sun completely disappears and we block it. The angle identiﬁed by
Fig. 3. – Real eye geometry: the true solar angular diameter is obtained leading tangents from the
eye border to the cylinder and up to the solar limb. Evaluating th e solar angular d iameter from
the center of the eye yields an overestimate of it, as Archimedes pointed out in the Arenarius
or Sand Reckoner.
SOLAR OBLATENESS FROM ARCHIMEDES TO DICKE 3
Table I. – Last Contact timings of Total Eclipse of January 24, 1925
Aperture (inches) Magnif ication power T ime
2”.75 33 h : 7 46
3”.4 80 h : 7 47
5”.0 55 h : 7 46
6”.0 60 h : 7 46
20”.0 175 h : 7 46
The ﬁrst three of these times were noted by the correspondent observers with stop watches;
the last two chronographically.
the line from the eye’s border tangent to the cylinder is surely bigger (or equal at least
but not smaller) than the true one because now we can’t see the Sun no more (ﬁg.3).
Quantitatively, Archimedes measured the lower limit 1/200 times the squared angle and
that the upper limit 1/164 times the squared angle, i.e. 27
such accuracy no oblateness was detectable; unless the apparent one due to atmospher ic
refraction, q uantitatively studied by Tycho Brahe in sixteenth century.
2. – Measuring solar diameter with eclipses
The method of measuring the solar diameter w ith eclipses explo its the same principle
of that one of the transits of Mercury: to recover the solar disk from more than three
points. Those points correspond to the external (ﬁrst and fourth) or internal (second
and third, not present in a partial eclipse) contacts of the Moon or Mercury with the
solar disk, as seen by diﬀerent observers at diﬀerent locations on the Earth (see ﬁgure
The observations of the instants of totality (second and third contact) are not aﬀected
by atmospheric seeing, be cause of the sudden change of the overall luminosity. Conversely
the determination of the instants of the external contacts during a partial eclipse (and all
contacts during a transit), until the possibility of using CCD cameras or ada ptive-optics
techniques, was heavily aﬀected by the seeing and the resolution of the telescope.
1. Historical data. – The eﬀect due to the telescope resolution matching with the
seeing conditions, is evident from the data retrieved in the 1925 total ec lips e at the
Chamberlin Observatory in Denver, Colorado [Table 1 from Howe (1925)].
From this dataset the eclipse has lasted more when observed with larger instruments,
exception made for the second data.
2. New perspectives. – We propose to monitor the external contacts of a partial
eclipse with CCD cameras whose frame rate ∆t ∼ 0.01 s is b e low the timescale of at-
mospheric seeing. Many solar photons ca n be gathered even with a semi-professional
telescope of diameter d ≥ 0.2 m with a bandpass ﬁlter. CCD fr ames can ﬁx the instanta-
neous wavefront pa th. The presence of the lunar limb helps to evaluate the instantaneous
point spread function for reconstructing the unperturbed wavefront according to current
image-restoration techniques (Sanchez-Cuberes et al., 2000).
The identiﬁcation of the lunar limb features and the solar limb near the contact event,
in each image , and the absolute timing of each frame with WWV radio stations, will allow
4 COSTANTINO SIGISMONDI AND PIE TRO OLIVA
Fig. 4. – In a partial eclipse we have only external contacts between solar and lunar limbs: (A)
ﬁrst and (B) last. Total and annular eclipses have also internal contacts which delimitate the
phases of totality or annularity of the eclipse.
to k now precisely the lunar fea tur e and the time of the contact’s event.
If our partial eclipse method can be made to succe de in pra c tice, it will open up more
possibilities for measuring the sola r diameter, espec ially because partial eclipses can be
seen fr om large observatories relatively often.
3. Image-restoration techniques. – The combination between ﬁnite resolution of
the telescop e s, atmosphere’s turbulence and stray lights from other regions of the solar
disk (both due to scattering in the Earth atmosphere and by optical surfaces) can be
quantitatively studied during a par tial eclipse. In fact the degradation eﬀects made by
the imaging system (atmosphere +telescope) are to be cons idered at the exact instant of
the exposure, and when the Moon’s limb crosses the solar disk, it serves as a reference
object to estimate the amplitudes o f the instantaneous optical transfer function.
Once known that function, there are se veral methods developed to compensate for
the atmos pheric eﬀects (Sanchez-Cuberes et al., 2000).
For example, with the 50 cm Swedish Vacuum Solar Telescope SVST at La Palma
L¨ofdhal et al. (1997) used phase-diversity speckle restoration technique to study the
evolution of bright points (0.2 arcsec of apparent dimension). The application of phase
diversity restoration technique allows to reach the limits imposed by the diﬀraction to
the instrument and can help adaptive optics to improve them (Criscuoli et al. 2001 ).
Recently Sanchez-Cuberes et al. (2000) have studied a t high resolution (0.53
arcsec at the solar limb, and better in other regions) s olar granulation features from
CCD images of 13 ms of e xposure taken with the SVST during the eclipse of May 10,
1994. Their idea was to match the lunar limb present in each frame to the numerical
SOLAR OBLATENESS FROM ARCHIMEDES TO DICKE 5
simulation of the eclipse geometry, having included the lunar limb topography as given
by the Watts’ proﬁles (Watts, 1963).
Although past eﬀorts to determine the s olar diameter using observatio ns of partial
solar eclipses have failed due to atmospheric seeing, the possibility to restore video CCD
images can succeed in the goa l of determining with a great accuracy, for each observer
whose position is known within 10 m of accuracy:
• the features of the lunar limb which ﬁrstly ‘hits‘ the solar limb, and that one which
is the last; with an accuracy of 0.2 arcseconds.
• the instant of the external co ntacts of the actual lunar limb with the solar limb,
with an accuracy of 0.01 s.
3. – Solar diameter measurements using total eclipses and transits
The method we present here is to be compared to the determination o f the North-
South diameter of the Sun (which is the polar one only when P
, at apsides) from the
analysis of total solar eclispes obse rved at the edges in order to recover secular variations
in the solar diameter (Dunham and Dunham, 1 973; Fia la et al. 1994) and to other
determinations of the solar diameter based upon the observations of meridian transits
of the Sun (see e. g. Winlock, 1853 and Ribes et al., 1988 ac c ounting on the
observations made by Picard in seventeenth century) or of the transits of Mercury across
the photosphere (see Parkinson et al., 1980 a nd Maunder and Moore, 2000 for a
complete histo rical review).
1. Total eclipses from centerline. – Totality occurs when the solar limb disappears
behind the last valley of the eastern lunar limb and ends when the Sun reappears from
another depression of the western lunar limb.
A source of error in the evaluation of sola r diameter arises from the knowledge of
Moon’s limb features. There are about 0.2 arcsec of uncertainty in Watts’ tables (1963 ),
as determined fro m pairs of photoelectrically timed occultations (Van Flander n, 1970;
Morrison a nd Appleby, 1981).
Therefore if one relys on Watts’ proﬁle the best determinations with total solar eclipses
can not reach an accuracy better than 0.2 arcsec. But the accuracy on the evaluation
of the solar diameter can be considerably improved by measuring the times of doz e ns of
Baily’s beads phenomena, involving a similar number of Watts’ points, thereby decreasing
the error statistically.
2. Total eclipses from edges. – Even better is to make measurements relative to the
same polar lunar valley bottoms at similar latitude librations, possible since all solar
eclipses occur on the ecliptic with negligible latitude librations. It means to observe the
total e clipse near the edges (Dunham and Dunham, 1973).
Moreover, it is possible to ex ploit also situations of sa me longitudinal libration angle.
The eclipses of 1925 and 1979 (after three complete Saros cycles, an ‘Exeligmos
years and 34 days) where also exactly at the same longitudinal libration angle: their
comparison (Soﬁa e t al., 1983) removes the uncertainty on the measured variations
of the solar diameter due to the Watts’ errors almost entirely.
With current lunar proﬁle knowledge, then total and annular eclipses are better for
determining the solar diameter, because they can produce polar Baily’s beads when
observed at the edges of their totaly (annularity) path.
6 COSTANTINO SIGISMONDI AND PIE TRO OLIVA
An error of 10 m in the determination of the edg es of the band of totality gives about
0.006 arcsec of uncertainty in the evaluation of solar diameter.
Regarding the timing of the beads events, the solar intensity goes to almost zero very
quickly, then atmospheric seeing errors are more dire c tly eliminated.
3. Transits. – The transits of Mercury of November 15, 1999, was a ‘grazing’ transit
(Westfall, 1999), not useful for an accurate measure o f the solar diameter, because
it did not allow to sample points of the solar disk enough spaced between them. The
previous tr ansit occurred in 1985 well before adaptive optics techniques and the large
diﬀusion of CCD cameras. The transits of Mercury of May, 7 2003, and Venus (June 8,
2004 a nd 2012) have to be considered also for this pourpose.
4. – Expected accuracy with partial eclipses evaluations
1. Positions of the observers. – An error of 10 m in the determination of the edges
of the band of totality gives about 0.006 arc sec of uncertainty in the evaluation of so lar
An accuracy of 10 m in geogra phical position of the obse rver can b e achieved with
about 10 minutes of averaging GPS.
2. Bandpass ﬁlter . – The o bs e rva tions have to be done with a ﬁlter with waveband
of 6300 ± 800
A, in order to have data always comparable between them, and in the same
waveband of Solar Disk Sextant (SDS, see last paragraph).
3. Duration of the imaging of the external contacts. – The eclipse magnitude m is
the frac tion of the Sun’s diameter obscured by the Moon.
The relative velocity of the Moon’s limb over the Sun’s photosphere is about v = 0.5
arcsec p e r second, along the centerline of a total eclipse. For a partial eclipse the velocity
of penetration of the dar k ﬁgure of the Moon (perpendicularly to the solar radius) is
v ∼ 0.5 · (1 − m) arcsec/s, then for having about 1 arcminute of Moon already in the
solar disk it is necessary to continue to take images for ∆ t = 120/(1 − m) s after the
ﬁrst c ontact and before the last contact.
The instants t
of the external contacts can be determined with an accuracy
better than the frame rate ∆t ≤ 10
s. In fact t
can be deduced by interpolating
the motion of the rigid Moon’s proﬁle, which becomes better deﬁned as the eclipse
In this way each observer (2 at least are needed) can ﬁx two points on the Moon’s
limbs and two instants for the contacts.
4. Expected accuracy in the solar diameter measurements. – The accuracy of the
determination of the lunar features producing the externa l contacts for a given observer
is therefore limited by the Watts’ proﬁles er rors (0.2 a rcsec).
Two observers enough distant (500 to 1000 Km in latitude for a East-West path of
the eclipse) allow to have 4 po ints and 4 times for recovering the apparent dimensions of
the solar disk at the moment of the eclipse within few hudredth of arcsecond of accuracy.
The accur acy becomes worse as the points sample a smaller pa rt of the s olar circle.
The following table shows how the error on the determination of the solar diameter
changes from having three po ints within 6 0 degrees to 240 degrees.
SOLAR OBLATENESS FROM ARCHIMEDES TO DICKE 7
That accuracy can allow the detection of the oblateness of the Sun. Therefore more
than three observers can allow improvement o f the detection of the shape of the Sun by
minimizing the residuals of the best ﬁtting ellipse.
5. – Perspectives on eclipse methods
We have pr oposed an accurate measurement of the solar diameter during partial solar
eclipses. This method is the natural extension of the method of measuring the solar
diameter during total eclipses. It exploits modern techniques of image processing and
fast CCD video records to overcome the problems arising from atmospheric turbulence.
With this method professional and semi-pro fessional obs e rvato ries can be involved in
such a measure ments, much mo re often than in total eclipses.
Moreover this method can be used for obtaining data useful for the absolute cali-
bration of measurements by instruments that are balloon-borne (Soﬁa et al., 1994;
1996) or satellite-borne (Dam´e et al., 1999) with a precision of ∆D ≤ 40 10
It is a lso to note that from the ﬁrst to the fourth contact of eclipses there a re about
two hours. The apparent solar dia meter changes with a maximum hourly rate up to
arcsec/hr due to the orbital motion of the Earth; this eﬀect is strongly r e -
duced around the apsides on July 4
and January 4
, ≤ 2 10
arcsec/hr and this
is a favourable case for eclipses in December-January or June-July.
In the future Watts’ tables can be substitued by the upcoming (2004) data o f the Se-
lene Japanese spacecraft, and the systematic errors arising from them will be avoided.
6. – Secular variations of solar diameter
It was the 3rd of May 1715 when solar eclipse was observed in England from both
edges of the paths of totality. Following Dunham a nd Dunham method  it is pos sible
to extract solar radius informa tio n by determining the edges of the path of the totality.
Unluckily there are elements of uncertainty on the eﬀective positions of the observers on
the edg e s  and this causes a remarkable error on the radius determination using 1 715’s
eclipse data. Another eclipse on January 24 , 192 5 was very accurately obs e rved by more
than 1 00 employees of the Aﬃliated Electric Companies of NY City and many other
adva nce d a mateurs in response to the campaign led by E .W. Brown and a detailed study
was made after the observation . Soﬁa, Fiala et al.  used Brown’s data and found
a correction of (0.21±0.08) arcsec. for the standard solar radius value of 959.63 arc sec at
a distance of 1UA, for 1925 eclipse. Analyses of the eclipse in Australia in 1976 and of the
eclipse in North America in 1979, were made by Soﬁa, Fiala et al.  but no appreciable
changes in the solar radius were found between those two eclipses. However, the solar
radius deter mined for 1715 was found to be (0.34±0.2) arcsec larger than 1979 value. On
the other hand, Soﬁa, Fiala, Dunham and Dunham  found that between the 1925 and
the 1979 eclipses, the solar ra dius decreas e d by 0 .5 arcsec but the solar size between 1925
and 1715 did not signiﬁcantly changed. Ther e fo re they concluded that the solar radius
changes are not secular. Eddy and Boornazian  in the same year reported results
over observations made between 1836 and 1953 at the Royal Greenwich Obser vatory.
They found a secular decrease trend in the horizontal solar diameter amounting to more
than 2 arc sec/century while the solar vertical diameter seemed to change with about
half of this rate. With the same data Soﬁa e t al.  had found out that any secular
changes in the solar diameter in the past century, could not have exceeded 0.25 arc sec.
The disag reement between the r e sults of diﬀerent groups depends on the diﬀerent data
8 COSTANTINO SIGISMONDI AND PIE TRO OLIVA
selection criteria and on diﬀerent solar and lunar ephemerides adopted, as it is shown in
 for the analyses of the annular eclipse of May 30, 1984 . Another measurement of the
solar radius, independent on lunar ephemerides, was made by Shapiro  who analyzed
data from 23 transits of Mercury between 1736 and 1973. His conclusion was that any
secular solar radius decrease was below 0.15 arc sec/century. This method has been
criticized for the black drop eﬀect which aﬀects the exact determination of the instans
of internal contacts, ﬁrst pointed out by Captain Cook during Venus’ transit of 1769.
7. – The method of Dicke for measuring solar oblateness
Around 1961, R. H. Dicke and others tried to point out the possible eﬀects due
to existence of a scalar ﬁeld in the framework of Eins tein’s General Relativity. The
presence o f such a scalar ﬁeld would have important cosmological eﬀects. T he gravi-
tational deﬂection of light and the relativistic advancement of planetary perihelia are
two eﬀects that could have b een inﬂuenced by a scalar ﬁeld: w ith respect to classical
General Relativity both eﬀects were expe cted to be about 10% less in the case the scalar
ﬁeld would be present. For this reason Dicke showed that the advancement o f the line
of apsides of Mercur y was not to be considered as a good test for General Relativity,
which was believed before, because of the entanglement of its cause s (scalar ﬁeld and
classical Genera l Relativity) . A small solar oblateness (∆R/R ∼ 5 · 10
) caus e d
by internal rotation in the Sun would cause the 1 0% eﬀect of perihelion a dvancement
without invoking any relativistic eﬀect. It was clear that until such oblateness could be
excluded or conﬁr med from observational data, the interpretation of the advancement of
Mercury’s line of apsides would was a mbiguous. The Einstein relativistic motion of the
longitude of the perihelion is
(1) ˙π =
where a is the planetary semimajor ax is, e is the eccentricity and T is the period; on
the other hand we have the rotation of the perihelion due to an oblate Sun which is
(2) ˙π =
where ∆ is the ratio between (solar e quatorial radius - polar radius) and (mean
radius). The scalar-tensor theory of gravitation could have been brought in agreement
with observational data, if the Sun possessed a small oblateness and a mas s quadrupole
In 196 6, Dicke and Goldenberg  measured the diﬀerence in ﬂux between the equa-
tor and pola r limb of the Sun. The idea was simple: using a chopper with apertures
made to show only a small section of the solar limb (see ﬁg. 5), they measured the ﬂux
at the poles and at the equato r of the Sun. We must consider two ipothesis:
• If the temperature at the pole is equal to the one at the equator, ﬁnding a ﬂux
diﬀerence can only mean that there is an oblateness such that the radius at the
and the radius a t the equator R
, diﬀers from a quantity ∆R. Then the
ﬂux diﬀerence ∆F sho uld be constant if we change the expose d limb by changing
the chopper’s aperture.
• On the contrary, if there is no oblateness but we still have a ﬂux diﬀerence ∆F ,
means that there is a temperatur e gradient. Then ∆F should be proportional to
the amount of exposed limb.
Dicke and Goldenber g found that ∆F remained about constant so that the Sun should
have a small oblateness. The theory of the measurement is well explained in , II.
SOLAR OBLATENESS FROM ARCHIMEDES TO DICKE 9
Fig. 5. – Measurement of the ﬂux F within th e chopper mask at the solar poles. By changing
the fraction f
of exposed limb it is possible to detect if there is any ∆F between polar and
equatorial diameters, and if it changes with f
. Only if ∆F 6= 0 and it is constant with f
is a consequence of th e solar oblateness, otherwise it would be a consequence of temperature
The instrument use d a nd the measurement procedure is also explained in , IV, V,
VI and VII. Mainly both the instrument and the measuring procedure were designed to
eliminate systematic errors. Dicke found for ∆R to be ∆R = 43.3 ± 3.3 10
The oblateness of ∆R/R = (4.51 ± 0.34) · 10
implies a quadrupole moment of
J = (2.47 ± 0.23) · 10
At the end of his analysis, Dicke found that new independent measurements of the
solar oblateness were needed, to make c omparison between data taken with diﬀerent
faculae activity o n the Sun. In 1975, new observations were made by Hill and Stebbins
. They considered a complication raised in Hill’s work  of a time vary ing excess
of eq uatorial brightness due to sunspots and faculae. It is clear that to measure the
diﬀerence between the polar radius and the equatorial radius, we must ﬁrst be sure on
which point to take as equatorial solar edge. The point is to give out a consistent def-
inition of the solar limb. This can be done by using a proper limb darkening function.
Hill et al. demonstrated that the excess brightness can be easily monitored by using a
proper analytic deﬁnition of the s olar edge, using the FFTD . It was pointed out
that the main problem in this kind of measurements is identifying some point on the
limb darkening curve as the so lar edge . It is clear that if more points on the darkening
curve can be taken as solar edge, many diﬀere nt deﬁnition of solar radii c an be gave and
many diﬀerent measures of solar oblateness done. The diﬀerences be tween these values
will co ntain information about the shapes of the limb proﬁles. The value obtained from
Hill for the intrinsic visual oblateness is (18.4 ± 12.5) · 10
arcsec which is obviously
10 COSTANTINO SIGISMONDI AND PIE TRO OLIVA
in conﬂict with the va lue of Dicke-Goldenbe rg. In this confused situation another g roup
decided to construct an instrument to measure long term changes: the Sola r Diameter
Monitor (SDM) at The High Altitude Observatory . Their purpose was to determine
which kind of solar diameter variation was taking place, if any, within a re asonable period
of time (3-5 years). The SDM beg an operation in Aug. 1981. An accurate discussion
on the measured duration of so lar meridian transit during six years between 1981 and
1987 is made in  wher e Brown and Chr istensen-Dalsgaard adopted adjustments to
the modiﬁed IAU value for the astronomical unit (value of 1.4959787066 · 10
Nava l Observatory, 1997) to take into account for the mean displacements between the
telescope’s noontime location and the Earth’s centre. T hey also corrected for the dis-
placements of the Sun’s centre relative to the barycentre of the Ear th-Sun system. They
found the s olar radius to be R
= (695.508 ± 0.026) Mm which is about 0.5 Mm smaller
than the Allen (19 73) value of 695.99 Mm. Moreover, Brown and Christensen- Da lsgaard
found no signiﬁcant variations in the solar diameter during their observational period:
their annual averages for the years 1981-1987 all agree within ±0.037 Mm. Toulmonde
 discuss e d about 71000 measurements regarding almost 3 00 years of data: he did not
ﬁnd evidence of any secular variation in his data.
8. – Solar Disk Sextant measurements
Further attempts to measure the solar oblateness have been ma de with the Solar
Disk Sex tant (SDS) which is an instrument made to monitor the size and shape of the
Sun. The principle of the instrument is well described in Soﬁa, Maier and Twigg work
. Basically a pr ism whit an opening angle very stable along the years is posed in
front of the objective of a telescope, and it produces tweo images of the Sun at focal
plane. The distance between the center of those images is dep e nding on the focal lenght
of the telescope, while the gap between the two limbs depends on the angular diameter
of the Sun. The same idea is exploited in using two pinholes instead of one and has
been proposed for simpler proto types of SDS, whose images are unaﬀected by optical
distortions. Considered that the so lar radius changes until now repo rted are to b e of the
order of 1 arc se c per century, the SDS instrumental accuracy was as ked to keep calibrated
on 0.01 arc sec/year and a stability of 0.003 arc sec/year was reached. The really good
feature of the SDS consists in the fact that the instrument accuracy requirements are fo r
relative rather than absolute va lues of the radius which led to a solar edge point detection
accurate to 1/10 pixel on the instrument. With statistical methods one can have a further
reduction of a 10 factor. The SDS e arly version was developed to be carried into space
during Space Shuttle ﬂights, but unlikely the Challenger accident took place. This led to
the needs to change strategy avoiding important delays. SDS was mounted on a system
for ground based observations but it was soon clear that no valuable s c ientiﬁc data could
be obtained from ground because the atmosphere’s inﬂuence. So the SDS was mounted
on a stratospheric balloon and it measured solar oblateness. A complete analysis of
his 4 ﬂights data (1992 , 1994, 19 95 and 1996) is still in progress.
∗ ∗ ∗
Costantino Sigis mondi thanks Drs. Terry Girard, David Dunham and Elliot Horch,
who encouraged him, during his scholarship at Yale University (2000-2002), to pursue
the idea of partial eclipse measurements of solar diameter.
SOLAR OBLATENESS FROM ARCHIMEDES TO DICKE 11
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