Costantino Sigismondi's Proposal to Determine the Solar Diameter
from Video/CCD Observations of the Partial Solar Eclipse of 2000
December 25
Updated: 2000 Dec. 19 U.T., 1h U.T.
Measuring the Solar Diameter during the Eclipse
of Dec. 25, 2000
by Costantino Sigismondi
e-mail: sigismondi@icra.it
Yale University, Dept. of Astronomy,
Astronomical Observatory of Rome and ICRA,
International Center for Relativistic Astrophysics,
Department of Physics, University of Rome
"La Sapienza",
P.le A.Moro 2 00185
Rome, Italy
[Costantino is now in Italy, and will be returning to Yale
University in February. I have corrected some errors in
this modified version of Costantino's paper. He also
proposed techniques and accuracies that are beyond the
capabilities of nearly all observers, which have not been
corrected yet in his original paper, but which he agreed
in e-mail messages were unrealistic, and every case agreed
to new requirements that I proposed that can be met by
many observers with video equipment. His original
proposal, without these corrections, is in a TeX file that
can be obtained here. It is
also available at his anonymous ftp site at Yale.
Besides the TeX file, it also contains a figure that is also
here. The ftp site also contains
large PostScript versions of the same paper and figure.
David Dunham, Dec. 18]
Abstract
On Christmas Day, 2000, the partial solar eclipse will present
suitable conditions to allow a determination of the solar diameter
within a few tens of milliarcseconds of accuracy.
[Below, "^" means "raised to the power of" and "~" means
"approximately" or "approximately equal to". DWD]
Introduction
Such a determination can be compared to the
determination of the polar diameter of the Sun coming
from the analysis of historical total solar eclispes
in order to recover secular variations in the solar
diameter.
With respect to other determinations of the solar
diameter based upon the observations of meridian
transits of the Sun or of the transits of Mercury on
the photosphere, the observations of the instant of
the totality are not affected by the atmospheric
seeing.
An error of 10 m on the determination of the edges of
the totality band gives about 0.1 arcsec of
uncertainty in the solar diameter [actually, it should
be 100 m for 0.1 arcsec uncertainty in the solar
diameter - DWD]. Another source of uncertainty arises
from the knowledge of the Moon's limb of about the
same amount 0.1 arcsec [Watts 1963] [actually, around
0.2 to 0.3 arcsec, but by using contacts in the same
lunar valleys for two separate eclipses, this error
can be largely eliminated - DWD]. The best
determinations with total solar eclipses can not reach
an uncertainty better than 200 milliarcsec (mas)
[actually, 5 to 10 times more accurate if the same
lunar valleys can be used; even without the same
valleys at similar librations, an accuracy of better
than 100 milliarcsec is possible by analyzing dozens
of contacts. A more serious error source might be
definition of the edge of the Sun. - DWD].
The determination of the instant of the first
contact during a partial eclipse, until the
possibility to use CCD camera or Active Optics, was
also affected by atmospheric seeing.
The transit of Mercury of Nov 15, 1999 was a `grazing'
transit [0], not really useful for an accurate measure of
the solar diameter. The previous one occurred in 1985 well
before the Active Optics Techniques and the large
diffusion of CCD cameras.
Moreover the solar diameter changes with an hourly
rate up to 10 mas/hr due to the orbital motion of
the Earth; this effect is strongly reduced around the
apsides on July 4th and January 4th, less than 2
mas/hr and that is the favourable case of the next
partial eclipse.
Method and expected accuracy
The procedure for measuring the ellipsoidal shape of
the Sun (solar oblateness) with an array of small
telescopes during that eclipse is presented with the
corresponding expected uncertainties of the measures.
In order to achieve such goal are needed accuracy on
the determination of the solar diameters (polar and
equatorial) of the order of Delta D/D about 10^-6. The
future applications of such measures are also
outlined.
The technology of CCD cameras allows to overcome the
distortion of the image done by integrating in time
the seeing's effects with the naked eye or
photography. Quick exposures with Delta t less than or
equal about 0.01s [or the 0.0167s half-frame rate of
standard video] are well below seeing's timescale
and meanwhile can be gathered a lot of Sun's photons
even with a semi-professional telescope of diameter d
of 0.2 m or more with a bandpass filter.
A CCD frame can fix the instantaneous wavefront path.
In night time observations a guiding star's
instantaneous spread function allows to reconstruct
the unperturbed wavefront according to the Active
Optic Techniques (AOT). No guiding star is available
during day time, but the assumption of perfectly
spherical Sun is valid up to one part in 10^6 [1]. So
every larger deviation of this shape is due to the
seeing. Choosing the waveband for the filter, the same
for all the instruments of the array, say 6300 +/-800
Anstroms, in the optical range the chromospheric
features can not affect the smoothness of the profile
of the Sun. [See the more extensive discussion of the
wavelength range and filters in the message that David
Dunham distributed early on December 18. Costantino
also provided the following information about filters:
Regarding the LPR filters,
I found general web sites
http://www.4w.com/pac/filters.htm
http://sciastro.net/portia/advice/filters.htm
and any seller (MEADE e.g.) knows such filters.
The aim of using LPR is to select a waveband in order
to make a comparison with SDS data (630+/-80 nm).
In fact the Sun can have different diameters at
different wavelenghts.]
Framing for 100 s the first and the last contact of
the lunar limb on to the solar disk, the irregular
profile of the Moon can be recognized by AOT with a
precision of theta ~1.22 x lambda/50 x d= 20 mas,
50 times better than the limit imposed by the
diffraction [2]. The relative velocity of the Moon's
limb over the Sun's photosphere is about 0.5 arcsec
per second, and in 100s quite an arcminute of the
Moon's limb becomes visible.
In this way we have to examine only a small part of
the lunar limb and not all the profile, as in the
total solar eclipse method. So the Watts tables can be
substitued by the most recent data of the Clementine
spacecraft [8], avoiding the 100 mas of error
arising from the bad knowledge of the Moon's limb
profile. [In general, Clementine data won't help much
with the lunar profile since their data arcs are separated
by almost 3 deg. of longitude, and they cover land
that often won't contribute to the lunar profile (only
the highest ridges and mountains do). Probably
much better (than Watts) profile data can be obtained
only after the Japanese Selene mission a few years
from now. In the meantime, Watts' profile data, with
systematic (local area) corrections applied where
possible from Clementine and lunar occultation video
observations, can provide a somewhat better reduction
than Watts' data alone. By using a few dozen profile
points (the observed arc will contain such numbers of
them), the Watts error can be statistically reduced
considerably. DWD]
The instants t0 and t1 of the contacts can be
determined with an accuracy Delta t less than 0.01 s.
[but the times to 0.0167s of the half-frame of NTSC video
observations will be all right.] In fact t0 and t1 can
be deduced interpolating the motion of the Moon's profile
of which the accuracy becomes better as the eclipse goes
on.
In this way each telescope (3 are needed) can fix two
points on the Moon's limbs A and B and two instants
for the contacts. A total of six points describe 3
chords on to the Sun.
The correction for the orbital motion of the system
Earth-Moon around the Sun, for which the solar
diameter increases, on December 25th, of a very
small amount because we are close to the perihelion,
is negligible. That effect can become as large as
~10 mas per hour in the months of April and
October. Being the motion of the Moon known with an
accuracy up to 2 mas per year [3], [4] and the
Astronomical Unit better than one part in 10^8, we
can neglect those factors as source of error.
By the parallax effect, observing the eclipse from
different locations separated by 10s of degrees of
latitude northwards in North America, the
corresponding points A, B move northwards on the
Moon's limbs determining a maximum eclipse magnitude
ranging from 0.66 at 51o of latitude, to 0.56 at 41o
and 0.41 at 31o [9]. [of course, this is longitude-
dependent. There are NO observers where the eclipse
magnitude is 0.66, but it would be great to get some
observations from Montreal, Quebec City, or other parts
of southern Quebec; northern Maine; northwestern New
Brunswick; or Sudbury, Ont. where the eclipse mag. is
about 0.62. But we hope we can at least get observations
at eclipse mag. 0.58, including Halifax, Buffalo,
Toronto, Duluth, and Winnipeg, or from mag. 0.57 for
Boston and Detroit, and many other places with a slightly
smaller value. The chances for observations at smaller
magnitudes increases as one moves towards the southern
and especially southwestern U.S.A. and Mexico.]
At those latitudes even many amateur observers can obtain
images as precise as requested with CCD technique and a
precise timer Delta t less than 10^-6s. [Again, the
0.0167s limiting accuracy of NTSC video is all right.]
The position of their telescope is also to be transmitted
within 5 cm of tolerance, corresponding about to a 1 mas
in latitude. [This precision in location is virtually
impossible to obtain. But it will be good enough to
obtain a position accurate to 10 m or better, possible by
averaging GPS observations for 15 minutes or more if the
view is relatively unobstructed above 15 deg. altitude in
all directions. 10 m corresponds to 5 mas at the Moon's
distance, far below other error sources. DWD].
The observers can fix six points A, B, A', B', A'',
B'' on the Moon's limb and six time determinations.
The time correction for the orbital motion of the
Earth that is reflected on the Sun as a motion
eastwards of ~1/12 degree in two hours, as the
eclipse lasts, and the correction for the parallax,
allow to get finally six points on the celestial
sphere on which the solar edge is projected at the
instant at t0.
The accuracy on the position of such points is Delta
x/x ~20 mas/1000as ~2 times 10^-5. That corresponds
to an accuracy on $\Delta D/D ~50 times 10^-5, with
6 points matching about 1/4 of the whole
circumference [9]. That accuracy is enough for
detecting the oblateness of the Sun. More than three
telescopes can allow to improve the detection of the
shape of the Sun minimizing the residuals of the
best fitting ellipse.
This method can be used for obtaining data useful
for the absolute calibration of measures of
instruments balloon-borne (SDS [6]) or satellite-
borne (Piccard [7]) in the filter's waveband of 6300
+/-800 Angstroms with a precision of Delta D less
than 40 mas.
References
[0] Alpo web site
[1] Sofia, Lydon... on the oblateness of the Sun with SDS
[2] HST resolution power...
[3] Bertotti et al PRL (1987)
[4] Shapiro et al PRL (1988)
[5] Watts, 1963
[6] something on SDS
[7] something on Piccard
[8] something on Clementine
[9] Emapwin computer program per eclipses
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A criticism of the project by Isao Sato in Japan is here.