[vsnet-chat 1089] Re: eclipse timing

["Fraser Farrell" <fraserf@dove.net.au>]

G'day all,

Gianluca Masi wrote about Heliocentric Correction (in part):
> H.C.= -0.0057755*(R*cos(theta)*cos(alpha)*cos(delta)+
> R*sin(theta)*(sin(obl)*sin(delta)+cos(obl)*cos(delta)*sin(alpha)))

> On my books I have found that the Binnendijk's formula gives an estimate
> for the H.C. (I don't know how precise yet).


If the formula is slightly rearranged to read:

HC = -0.0057755183*R* (cos(theta)*(cos(alpha)*cos(delta))
     +sin(theta)*(sin(obl)*sin(delta)+cos(obl)*cos(delta)*sin(alpha)) )

then all of the trigonometric terms, in effect, are simply giving the
distance of the Earth from a fixed fundamental plane, passing through
the Sun's centre, perpendicular to the Sun-star direction.  The Ecliptic
and the RA/Dec grid are redefined to this fundamental plane for the
moment of interest.

The -0.0057755183*R part of the formula then converts this distance to a
light-time (in days) and corrects for the actual Earth-Sun distance at
the time of interest.

I think the precision of the formula is limited only by the precision of
"R" and "theta".  For these you can use anything from the complete
VSOP87 theory for the Earth down to a simple elliptical orbit.  Note that
some books on astronomical calculation assume the simple elliptical
orbit, or consider only the largest of perturbations.

Jean Meeus' excellent book "Astronomical Algorithms" (Willmann-Bell
1991, ISBN 0-943396-35-2) presents several recipes for calculating "R"
and "theta" including the use of the VSOP87 theory.  To obtain "R" and
"theta" to a precision of 0.000001 (AU and radians, respectively) for
the next millennium or so, the first dozen or so terms of the Earth's
VSOP87 series (listed in Meeus' book) seem to suffice.

This would yield a worst-case HC error of about 0.001 seconds; for a
geocentric observer.  Of course at this level of precision you will need
to worry about other factors such as topocentric correction, clock
errors,....  How precise a HC do we need?


cheers,
Fraser Farrell

http://vsnet.dove.net.au/~fraserf/  email: fraserf@dove.net.au
traditional: PO Box 332, Christies Beach, SA 5165, Australia

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