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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Celestial data for the planet Mercury
From: Paul Hirose
Date: 2016 Feb 24, 23:36 -0800

```On 2016-02-23 23:18, Herbert Prinz wrote:
> Paul,
> I agree that any significant discrepancy indicates an error. Now I am
curious: How do you arrive at 95 mas difference? I get a little over 101 mas.
What is it that one of the two of us is doing wrong (or differently from the
other) in a simple subtraction and multiplication?

The coordinates posted by Dave Walden are:

************************

2016 Feb 01 11:00:00.0     19 14 15.421     - 20 38 03.89
************************
http://ssd.jpl.nasa.gov/horizons.cgi#results
2016-Feb-01 11:00     19 14 15.4172 -20 38 03.896
***************
14 15.42394
-20 38 3.8918

The last two differ by 95 mas. To get that figure, the first step is to
convert from spherical format (assume radius = 1) to rectangular.

+.297 943 774  -.887 153 335  -.352 403 843  HORIZONS
+.297 944 211  -.897 153 195  -.352 403 824  IMCCE

The dot product of those vectors is 1.000 000 000. That's the cosine of
the separation angle.

The magnitude of the cross product of the vectors is 4.59e-7. That's the
sine of the separation angle. In this case we simply take the arc sine
and obtain 2.63e-5 degrees, or .0947 seconds.

A more general solution is to treat the cosine and sine as x and y
respectively, and do a rectangular to polar conversion. That yields a
precise result whether the angle is tiny or large. In addition, the
result is correct even if the vectors have different magnitudes.

In practice, the computation was simpler than what I showed above. My HP
49G calculator allows vector input in rectangular, spherical, or
cylindrical format. Also, it has functions to compute dot and cross
products without any explicit conversion to rectangular format. The
hardest part is entering the digits without a mistake.

Regarding the reason for the 95 mas discrepancy, in a private
communication someone suggested the IMCCE may still be using the IAU
1976 precession and 1980 nutation models. Let's test that with the DE430
ephemeris, IAU 1976/80 precession and nutation, with and without frame
bias. Geocentric apparent place at 2016 Feb 1 11:00:00 UTC:

19 14 15.42394  -20 38 03.8918  IMCCE
19 14 15.4244   -20 38 03.892   IAU 76/80 (with bias)
19 14 15.4239   -20°38'03.893   IAU 76/80 (no bias)

The first and second lines agree within 6 mas. The first and third are
only 1 mas different. So my guess is that IMCCE does not apply the frame
bias correction. That's wrong.

Frame bias is the difference between the mean equator and equinox
coordinate system at J2000.0 (= 2000 Jan 1 0 hours Terrestrial time) and
the ICRS (International Celestial Reference System). When the latter was
created, it was intended to be as close as practical to the former.

We now know there's a small offset. It's negligible if you're only
working to a tenth of an arc second. But at higher accuracy there's a
disconnect. Modern star catalogs and solar system ephemerides are based
on the ICRS. On the other hand, the IAU precession models transform
coordinates from the J2000.0 system to the mean equator and equinox
system of date. Thus in precise work you're supposed to get the ICRS
coordinates of the body (including effects such as aberration and light
time), apply frame bias to obtain its J2000.0 coordinates, then apply
precession, then nutation.

It's ironic that the IMCCE coordinates have the most decimal places, but
throw away much of their precision due to an oversight. When precise
computations of a body's position differ by nearly a tenth of an arc
second, it usually indicates a mistake somewhere. With reasonably modern
tools and data you shouldn't be that far off.

```
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