Ephemeris Comparison

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NOTE, on 13 August 2013: Due to a computer problem, I have lost the data upon which the first version of this web page was based, and I also lost my related notes. Only what had already been published on Internet remains. Unfortunately, after re-gathering and re-processing the data, I haven't found exactly the same results as before ― apparently I haven't been able to recreate the exact same configuration of some of the data sources as before, and the precise details of what that configuration was has been lost.

Additionally, my method for measuring the time shift between planet positions from different sources turns out not to be as good as I had hoped, so I have withdrawn my discussion of that until I've found a better method.

There are various computer programs, program libraries and web pages that can produce tables of positions of planets. Here I compare the following sources:

- The "HORIZONS" system of the Jet Propulsion Laboratory that belongs to the NASA: http://ssd.jpl.nasa.gov/?horizons. You can use that web page to create tables of the position of various celestial bodies in the Solar System. I extracted Cartesian heliocentric coordinates, with the HORIZONS "ephemeris type" equal to "VECTOR".
- The Swiss Ephemeris http://www.astro.com/swisseph/. This
offers very many different options for calculating coordinates. I
calculated Cartesian heliocentric coordinates, using
`swetest -eswe -fX -p3 -hel -true -j2000 -icrs -head -j662198.5 -s90 -n23536`

. - The iauPlan94 function from the SOFA program library of the IAU, available at http://www.iausofa.org/.
- The VSOP87A theory of the Bureau des Longitudes in Paris, available at http://vizier.cfa.harvard.edu/viz-bin/ftp-index?/ftp/cats/VI/81. The scientific paper describing the model can be found at http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?1988A%26A...202..309B.
- the SOLEX program from http://chemistry.unina.it/~alvitagl/solex/. This offers very many different options for the calculation of coordinates.

If there are differences between the results from various sources, then here are some possible causes for those differences:

- The sources use coordinate systems with different orientations ― for example, the one uses ecliptical coordinates and the other one uses equatorial coordinates.
- The sources use coordinate systems with different equinoxes ― for example, the one uses J2000.0 and the other one uses B1950.0.
- The sources use coordinate systems with different zero points ― for example, the one uses the Solar System barycenter as zero point and the other one uses the center of the Sun as zero point.
- The sources use different precession models.
- The sources use different time scales and that hasn't been taken into account ― for example, the one uses CT (UTC) and the other one uses TT, and for both the same clock time was entered (for example 11:43:12). Because the time scales aren't the same, the same clock time does not indicate the same instant in the different time scales.
- The sources don't take the same coordinate effects into account. For example, the one corrects for aberration or light deflection by the force of gravity, but the other one does not.
- The sources are based on different physical models of the Solar System and/or of gravity.

Distances between celestial bodies do not depend on the orientation of the coordinate system, so those are suitable for detecting differences in time scales and in the underlying models.

We compare these sources using heliocentric positions of Venus between the years −2900 and 2900 at intervals of 90 days.

Figure 1 compares the heliocentric geometrical distance of Venus as calculated using the iauPlan94 function from the SOFA library of the IAU and using the VSOP87A model. Along the horizontal axis is displayed the heliocentric geometrical distance of Venus according to the VSOP model, measured in AU. Along the vertical axis is displayed the difference between the distance according to the IAU model and the distance according to the VSOP model, also measured in AU.

To show the variation in time somewhat, the period of nearly 6000 years has been divided into three parts of about 2000 years each. The data points for the first period (the years −2900 through about −1000) are shown in blue, the points for the second period (roughly −1000 to 1000) in black, and the points for the third period (roughly 1000 to 2900) in red. The points are drawn in that order, so a black point may hide a blue point, and a red point may hide a black and/or blue point.

The distribution of points along the horizontal axis shows that the distance from Venus to the Sun varies between about 0.717 and 0.730 AU, but that that variation decreases with time: the horizontal range declines from blue via black to red.

In the vertical direction we see that the difference (in the
heliocentric geometric distance of Venus) between IAU and VSOP during
the "red" period is not greater than about 5 × 10^{−6} AU = 750 km, and is
about the same for all distances for which there are red points. In
the "blue" period the difference is often greater, up to about 8 × 10^{−5} AU
= 12000 km, and depends on the distance: for smaller distances the
difference is often negative, and for larger distances it is often
positive. This indicates that the IAU model has a slightly greater
distance variation in the blue period than the VSOP model has.

If that were the only difference, then we would not find any blue points clearly to the upper left or lower right of the midpoint of all points, but we do find such points. That indicates that there is a phase difference in the blue period between the IAU model and the VSOP model, so that the perihelia of Venus are on average not at the same instants according to the IAU model as according to the VSOP model.

To see if these conclusions are warranted, we take a look at some appropriate other graphs.

Fig. 2: Ephemeris Comparison: Venus / IAU - VSOP (detail)

Figure 2 shows the heliocentric distance of Venus for two periods of 50 days in the year −2899, according to the VSOP model and the IAU model. The upper graph shows an aphelion, and the lower graph a perihelion. In both graphs, the curved lines show the distances according to the scale along the left-hand side. The tenfold difference between the IAU distances and the VSOP distances is also indicated by the difference between the far less curved sloping line and the horizontal line in the graphs. For example, in the upper graph the sloping IAU line is near 0.7286 on the left-hand side, and the horizontal VSOP line is at 0.728 there, so the difference between the IAU distance and the VSOP distance is one tenth the difference between those two values, i.e. 0.00006 AU.

The upper graph shows that the aphelion distance according to the IAU model is greater than according to the VSOP model, and the lower graph shows that the perihelion distance according to the IAU model is less than according to the VSOP model. The difference between the aphelion distance and the perihelion distance is therefore greater according to the IAU model than according to the VSOP model, which fits with what we concluded earlier from Figure 1. Apparently the IAU model assumes a slightly greater eccentricity for the orbit of Venus for that period than the VSOP model does.

Moreover, we see that the aphelion and perihelion occur slightly earlier according to the IAU model than according to the VSOP model, so there is a phase difference between them, as we deduced already from Figure 1.

Fig. 3: Ephemeris Comparison: Venus / IAU - VSOP (Phase Difference)

The temporal variation of the phase difference (IAU versus VSOP) during the nearly 6000 year period is shown in Figure 3. We see that the phase difference is about 8 hours around the year −2900 and then gradually declines.

Our deductions from Figure 1 were accurate, so such figures are useful.

Fig. 4: Ephemeris Comparison: Venus

Figure 4 shows diagrams like Figure 1 for
all non-trivial combinations of the five investigated sources. The
columns and rows have the same order: JPL, SWE, VSOP, IAU, SOLEX. The
difference shown along the vertical axis is that between the source of
the row and the source of the column. Figure 1 is
included (reduced in size) in column 3, row 4. In the lower left half
of the figure, all vertical axes have the same range, from −10^{−4} to
+10^{−4} AU. In the upper right half, the vertical axis of each graph
has a range that fits that graph ― usually that range is a lot less
than in the lower left half. Those ranges (in AU) are listed in the
following table.

SWE | VSOP | IAU | SOLEX | |

JPL | 5 × 10^{−9} | 10^{−6} | 8 × 10^{−5} | 2 × 10^{−8} |

SWE | 10^{−6} | 8 × 10^{−5} | 2 × 10^{−8} | |

VSOP | 8 × 10^{−5} | 10^{−6} | ||

IAU | 8 × 10^{−5} |

*http://aa.quae.nl/en/reken/efemeridenvergelijking.html;
Last updated: 2016−02−07*