ArticlePDF Available

Abstract and Figures

The non-spherical shape of the Sun has been invoked to explain the anomalous precession 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. Comment: 12 pages, 5 figures. To be published in Nuovo Cimento B
Content may be subject to copyright.
arXiv:physics/0501083v1 [physics.hist-ph] 17 Jan 2005
Solar Oblateness from Archimedes to Dicke
Costantino Sigismondi(
) 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 [1] 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
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.
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
55”. With
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)[4]].
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[15]).
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
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)[13] 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 [14]).
Recently Sanchez-Cuberes et al. (2000)[15] 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
simulation of the eclipse geometry, having included the lunar limb topography as given
by the Watts’ profiles (Watts, 1963[7]).
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
= 0
, 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[19]; Fia la et al. 1994[2]) and to other
determinations of the solar diameter based upon the observations of meridian transits
of the Sun (see e. g. Winlock, 1853[3] and Ribes et al., 1988[16] 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[5] a nd Maunder and Moore, 2000[6] 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 )[7],
as determined fro m pairs of photoelectrically timed occultations (Van Flander n, 1970[17];
Morrison a nd Appleby, 1981[18]).
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[19]).
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[22]) 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.
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[8]), 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
and t
of the external contacts can be determined with an accuracy
better than the frame rate t 10
s. In fact t
and 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.
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;[10]
1996[10]) or satellite-borne (Dam´e et al., 1999[12]) 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
25 10
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[9], 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 [19] 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 [20] 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 [21]. Sofia, Fiala et al. [22] 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. [20] 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 [22] 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 [23] 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. [22] 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
selection criteria and on different solar and lunar ephemerides adopted, as it is shown in
[2] 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 [24] 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[25] 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) [26]. 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 =
T ac
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 =
T ac
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 [27] 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
pole R
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 [28], II.
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 [28], 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
[30]. They considered a complication raised in Hill’s work [31] 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 [32]. 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
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 [33]. 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 [34] wher e Brown and Chr istensen-Dalsgaard adopted adjustments to
the modified IAU value for the astronomical unit (value of 1.4959787066 · 10
Mm, US
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
[35] 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
[36]. 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[37], 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[11]. 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.
[1] Archimedes, Psammites, The Sand Reckoner, Italian edition in Classici della Scienza 19
UTET Torino (1974) p. 443-470.
[2] Fiala, A. D.; Dunham, D. W.; Sofia, S., Sol ar Physics, 152 (1994) 97
[3] Winlock, J., Astronomical Journal, 3 (1853) 97-103
[4] Howe, H. A., Popular Astronomy, 33 (1925) 280
[5] Parkinson, J. H.; M orrison, L. V.; Stephenson, F. R.,Nature, 288 (1988) 548-551 The
constancy of the Solar diameter over the past 250 years
[6] Maunder, M. and P. Moore,Transit when a planet crosses the Sun Springer-Verlag
[7] Watts,C. B., The Marginal Zone of the Moon 1963 Astronomical Papers prepared for the
use of the American Ephemeris and Nautical Almanac XVII (United States Government
Printing Office, Washington)
[8] Westfall, J. E., 1999˜rhill/alpo/transitstuff/merc11
[9] Hirata , N. et al., General overview of the lunar imager/spectrometer in ”New Views of the
Moon, Europe, Future Lunar Exploration, Science Obj ectives, and Integration of Datasets”,
David Heather editor, (Berlin, Germany) 2002
[10] Sofia, S., W. Heaps, and L. Twigg, Astrophys. J., 427 (1994) 1048 The Solar Diameter
and Oblateness Measured by the Solar Disk Sextant on the 1992 September 30 Balloon Flight
[11] Sofia, S. Lydon T. J., Physical Review Letters, 76 (1996) 177-179 A measurement of the
shape of the solar disk: The solar quadrupole moment, the solar octopole moment, and the
advance of perihelion of the planet Mercury
[12] Dam
e, L. et al, Advances in Space Research, 24 (1999) 205-214 PICARD: simultaneous
measurements of the solar diameter, differential rotation, solar constant and their variations
[13] L¨o fdhal, M. G., et al., Phase-diversity Restoration of two Simultaneous 70-minute
Photospheric Sequences, Bulletin of the American Astronomical Society, 29, Volume 29,
Number 2 AAS 190th Meeting, Winston-Salem, NC, June 1997 1997
[14] Criscuoli, S., J.A. Bonet, F. Berrilli, D. Del Moro and A. Egidi, Phase diversity
procedure in F95 for future Themis application, Meeting - THEMIS and the New Frontiers
of Solar Atmosphere Dynamics Roma 19-21 March, 2001 2001
[15] S
anchez Cuberes, M., Bonet, J. A., V
azquez, M . , Wittmann, A. D., Astrophys.
J., 538 (2000) 940-959 Center-to-Limb Variation of Solar Granulation from Partial Eclipse
[16] Ribes, E., et al., Nature, 332 (1988) 689 Size of the Sun in the Seventeenth Century
[17] Van Flandern, T., Astronomical Journal, 75 (1970) 744 Some Notes on the Use of the
Watts Limb-Correction Charts
[18] Morrison L. V. and G. M. Appleby,MNRAS, 196 (1981) 1005 Analysis of Lunar
[19] D.W. Dunham and J.B. Dunham,Moon, 8 (546) 1973
[20] D.W. Dunham, S. Sofia, A.D. Fiala et al.,Science, 210 (1980) 1243-1245
[21] Brown, E. W.,Astron. J., 37 (1926) 9-19
[22] Sofia, Dunham & Dunham and Fiala, Nature, 304 (1983) 522-526
[23] J.A. Eddy and A.A. Boornazian,Bull. Am. Astron. Soc., 11 (1979) 437
[24] Shapiro, I. I.,Bull. Am. Astron. Soc., 208 (1980) 51
[25] Brans, C. and Dicke, R.H.,Phys. Rev., 124 (1961) 925
[26] Dicke, H.R.,Nature, 202 (anno?) 432
[27] Dicke, H.R. and Golenberg, H.M.,Phys. Rev. Letters, 18 (1967) 313
[28] Dicke, H.R. and Golenberg, H.M.,Astrophys. J. supplement series, 241 (1974) 27:131-
[29] Dicke, H.R.,Astrophys. J., 159 (anno?) 1
[30] Hill, H.A. and Stebbins, R .T.,Astrophys. J., 200 (1975) 471-483
[31] Hill, al.,Phys. Rev. Letters, 33 (1974) 1497
[32] Hill, H.A., Stebbins, R.T. and Oleson, J.R., Astrophys. J., 200 (1975) 484-498
[33] T.M. Brown, D.F. Elmore, L. Lacey and H. Hull, Applied Optics, 21 (1982) 19
[34] Brown, T.M . and Christensen-Dalsgaard, J., Astrophys. J., 500 (1998) L195-L198
[35] Toulmonde, M., Astron. & Astrophysics, 325 (1997) 1174-1178
[36] Sofia, S., Maier, E. and Twigg, L., Adv. Space Res., 11 (1991) (4)123-(4)132
[37] Sigismondi, C., M easuring the angular solar diameter using two pinholes, Am. J. of
Physics, 70 (2002) 1157
... Buildings with a facade facing south intercepted the Sun's rays and stored heat energy in Roman baths, and this simple technology is still used today. There is also a legend related to the Greek scientist Archimedes, who was said to have set fire to enemy ships of the Roman Empire before they could reach the shore, using the Sun's rays reflected from special shields made of bronze (Sigismondi and Oliva 2005). This would be a prototype solar laser. ...
... The Northern and the Southern limit measurements together yield a direct measurement of the polar diameter of the Sun while center line measurements are referred to the equatorial diameter. With the present values of accuracy for Watts' profiles, because the center line Baily Beads span much fewer degrees in position angles than at polar limits, the goal of 0.01 arcsec of accuracy on both diameters corresponds to one part over 2 × 10 5 , enough to detect a relativistically significant difference between the two diameters [11]. ...
Full-text available
The occasion of a central solar eclipse can be exploited to measure the solar diameter up to 0.01 arcsec of accuracy, individuating the umbral or the annularity path limits within a few meters. We discuss an experimental design with six observers. The comparison with other eclipses allows us to ascertain the solar variations. Measurements from the northern and the southern limits of the eclipse's path give the polar solar diameter while observations near the center line give the equatorial diameter. The accuracy is good enough to detect a quadrupole momentum J2 ≥ 10-5.
... If solar radius correction is ∆R=0 at its maximum it means that the Sun is shrinking during the last decades. The effect of limb darkening (Rogerson, 1959) combined with atmospheric and optical filtering produce an artificial reduction of the observed photospheric radius (Sigismondi and Oliva, 2005b) but that effect seems to be negligible at this level of accuracy. ...
Full-text available
Data on Baily beads observed in total eclipse of March 29, 2006 (Egypt) and those of annular eclipses of September 22, 2006 (French Guyana) and October 3, 2005 (Spain) are used to evaluate the variations of solar radius with respect to its standard value during a whole draconitic year. A portable observatory has to be set on the shadow limit of central eclipses, where lunar limb is grazing to the solar one and the number of beads is large. The observation of solar corona during Egyptian eclipse for several minutes during maximum eclipse on shadow's limits is studied in parallel with the eclipse observed by Clavius in 1567. From fall 2005 to fall 2006 the solar radius does not show significant changes (0.00 to −0.01 arcsecs ) with respect to its standard value of 959.63 arcsec within errorbars of 0.17 arcsecs. This is its value at minimum of cycle 23 of solar activity.
... The asphericity is the observed variability of the radius with the solar latitude . The asphericity of the Sun, and in particular the oblateness: f = (R eq − R pol )/R eq , has great implications on the motion of the bodies around the Sun through the quadrupole moment (Sigismondi and Oliva, 2005; Sigismondi, 2011). It can also provide important informations on solar physics as the other global parameters. ...
Full-text available
The Total Solar Irradiance varies over a solar cycle of 11 years and maybe over cycles with longer period. Is the solar diameter variable over time too? We introduce a new method to perform high resolution astrometry of the solar diameter from the ground, through the observations of eclipses by reconsidering the definition of the solar edge. A discussion of the solar diameter and its variations must be linked to the Limb Darkening Function (LDF) using the luminosity evolution of a Baily's Bead and the profile of the lunar limb available from satellite data. This approach unifies the definition of solar edge with LDF inflection point for eclipses and drift-scan or heliometric methods. The method proposed is applied for the videos of the eclipse in 15 January 2010 recorded in Uganda and in India. The result shows light at least 0.85 arcsec beyond the inflection point, and this suggests to reconsider the evaluations of the historical eclipses made with naked eye.
A discussion of the solar diameter and its variations must be linked to the limb darkening function (LDF). We introduce a new method to perform high resolution astrometry of the solar diameter from the ground, through the observations of eclipses, using the luminosity evolution of Baily's Bead and the profile of the lunar edge available from satellite data. The method proposed is applied for the videos of the eclipse on 15 January 2010 recorded in Uganda and in India. We obtained a detailed profile constraining the inflection point position. The result suggests reconsidering the evaluations of the historical eclipses observed with a naked eye.
Observing Solar eclipses at the totality path limits allows to measure solar diameter variations and lunar ephemerides corrections with an accuracy of a few hundredths of arcsecond. After a review of secular variations of solar diameter, presenting the eclipses of Rome 1567, London 1715, and New York City 1925, we present the current research on this topic and the phenomena occurring on the edges of the totality path. For the forthcoming eclipses, the location of observers must be chosen after calculations of limits and simulations of bead appearances and timings. We present our example in Valoria La Buena (Spain) for the 3 October, 2005 eclipse. Small unfiltered refracting telescopes, equipped with projection screen are recommended for standard observations, in order to avoid artificial reduction in the observed solar diameter due to filtering effects. Preliminary data reductions on eclipses of 1995 India and 1998 Caribbean, and on 2004 Venus Transit are also presented.
Total and annular eclipses allow us to measure the angular solar diameter at unit distance up to an accuracy of some hundredths of arcsecond. Data of lunar limb features from Japanese mission Kaguya will be useful to detect also the solar oblateness signal, relevant from a General Relativistic point of view. Useful eclipse data are available for 1567, 1715, 1869, 1925 and from 1966 to 2009 with uneven sampling: only these data can allow a study of solar diameter evolution with significant resolution on secular basis. KeywordsSolar physics–Astrometry–Eclipses–Solar activity
Full-text available
Solar diameter measurements have been made nearly continuously through different techniques for more than three centuries. They were obtained mainly with ground-based instruments except for some recent estimates deduced from space observations. One of the main problems in such space data analysis is that, up to now, it has been difficult to obtain an absolute value owing to the absence of an internally calibrated system. Eclipse observations provide a unique opportunity to give an absolute angular scale to the measurements, leading to an absolute value of the solar diameter. However, the problem is complicated by the Moon limb, which presents asphericity because of the mountains. We present a determination of the solar diameter derived from the total solar eclipse observation in Turkey and Egypt on 29 March 2006. We found that the solar radius carried back to 1 AU was 959.22±0.04 arcsec at the time of the observations. The inspection of the compiled 19 modern eclipses data, with solar activity, shows that the radius changes are nonhomologous, an effect that may explain the discrepancies found in ground-based measurements and implies the role of the shallow subsurface layers (leptocline) of the Sun.
Full-text available
In the annular or total eclipses of 3 October 2005, 29 March 2006, 22 September 2006, and 1 August 2008, observational campaigns were organized to record the phenomenon of Baily’s beads. These campaigns were internationally coordinated through the International Occultation Timing Association (IOTA) at both its American and European sections. From the stations in the northern and southern zones of grazing eclipse, the eclipses have been recorded on video. Afterward, as many beads as possible have been identified by analyzing the video data of each observing station. The atlas presented in this paper includes 598 data points, obtained by 23 observers operating at 28 different observing stations. The atlas lists the geographic positions of the observing stations and the observed time instants of disappearance or reappearance of beads, identified by an angle measured relative to the Moon’s axis of rotation. The atlas will serve as a basis for determining the solar diameter.
Full-text available
The measurement of the solar diameter is introduced in the wider framework of solar variability and of the influences of the Sun upon the Earth's climate. Ancient eclipses and planetary transits would permit to extend the knowledge of the solar irradiance back to three centuries, through the parameter W=dLogR/dLogL. The method of Baily's beads timing during eclipses is discussed, and a significant improvement with respect to the last 40 years has been obtained by reconstructing the Limb Darkening Function's inflexion point from their light curve and the corresponding lunar valleys' profiles. The case of the Jan 15, 2010 annular eclipse has been studied in detail, as well as the last two transits of Venus. The atlas of Baily's beads, realized with worldwide contributions by IOTA members is presented along with the solar diameter during the eclipse of 2006. The transition between the photographic atlas of the lunar limb (Watts, 1963) and the laser-altimeter map made by the Kaguya lunar probe in 2009 has been followed. The other method for the accurate measurement of the solar diameter alternative to the PICARD / PICARD-sol mission is the drift-scan method used either by the solar astrolabes either by larger telescopes. The observatories of Locarno and Paris have started an observational program of the Sun with this method with encouraging results. For the first time an image motion of the whole Sun has been detected at frequencies of 1/100 Hz. This may start explain the puzzling results of the observational campaigns made in Greenwich and Rome from 1850 to 1955. The meridian line of Santa Maria degli Angeli in Rome is a giant pinhole telescope and it permits to introduce didactically almost all the arguments of classical astrometry here presented. The support to the PICARD mission continues with the analyses of the transit of Venus and the total eclipse of 2012.
Full-text available
This paper analyzes the behavior of a family of finite exact perturbations of Friedmann-Robertson-Walker cosmologies constructed with the inverse scattering technique of Belinskii and Zakharov (1978). The application of this method in a five-dimensional spacetime with a massless scalar field and a subsequent Kaluza-Klein dimensional reduction makes it possible to construct models with perfect fluid material content in four dimensions. The behavior of the energy momentum and Weyl tensors are studied as characterizations of the gravitational and material behaviors. The important case of solitonic perturbations on de Sitter backgrounds is also treated.
Full-text available
A technique for locating the edge of the sun is proposed. The technique uses a finite Fourier transform of the observed limb-darkening function to achieve reduced sensitivity to atmospheric and instrumental effects and heightened sensitivity to the shape of the intrinsic limb-darkening function. A theory is developed that predicts these sensitivities. In order to facilitate wider application, general relations are calculated. A testing program which complements the theory is also reported. The location of the edge is shown to be influenced only by solar phenomena down to the milli-arcsecond range.
Although transits of planets across the Sun are rare (only Mercury and Venus orbit the Sun closer than us, and so can transit the Sun's disc) amateur astronomers can observe, record and image other kinds of transit, which are very much more frequent. Transit is in two parts, the first telling the fascinating story of the early scientific expeditions to observe transits. The second part is for practical observers, and explains how to observe transits of all sorts - even transits of aircraft as they fly between the observer and the Sun!
New observations show a small difference between the sun's polar and equatorial limb darkening functions. This excess equatorial brightness varies in time and can be of sufficient magnitude to account for the solar oblateness inferred by Dicke and Goldenberg from their measurements. This removes the serious consequence of their work for Einstein's general theory of relativity. The problems of a solar-edge definition and the derivation of a solar mass quadrupole moment are discussed.
Nature is the international weekly journal of science: a magazine style journal that publishes full-length research papers in all disciplines of science, as well as News and Views, reviews, news, features, commentaries, web focuses and more, covering all branches of science and how science impacts upon all aspects of society and life.
A decrease in the solar radius is determined using the technique of Dunham and Dunham (1973), in which timed observations are made just inside the path edges. When the method is applied to the solar eclipses of 1715, 1976, and 1979, the solar radius for 1715 is 0.34 + or - 0.2 arc second larger than the recent values, with no significant change between 1976 and 1979. The duration of totality is examined as a function of distance from the edges of the path. Corrections to the radius of the sun derived from observations of the 1976 and 1979 eclipses by the International Occultation Timing Association are also presented.
This paper suggests that there has been no detectable change in the sun's diameter since at least AD 1700. Previous reports of contraction of the solar diameter are attributed to misinterpretations of the meridian circle observations. Observations of long term changes in solar diameter taken with the transit of Mercury across the sun are compared with the horizontal meridian circle observations of Eddy and Boornazian (1979). Linear regression analysis of the Mercury transits produces a secular trend of -0.14 + or - 0.08 arc sec per century in the solar semi-diameter. Data on total solar eclipses are also considered. Using the Mathers (1966) film of an eclipse, a correction of -0.22 + or - 0.20 arc sec has been deduced to the adopted semi-diameter of the sun.
Timings of 66000 occultations of stars by the Moon are analysed for systematic corrections to the limb-profile heights taken from Watts' charts of the marginal zone of the Moon. The radius, shape and location of the centre of the datum implicit in the charts are found to vary with libration and these variations produce systematic errors in the limb-profile heights which attain 0.4 arcsec in some position angles.