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Excerted "Appendix"
A. The gravity deflection for a planet passing behind the Sun
The calculation of the apparent position of a planet involves a relativistic effect, which is the curvature of space by the gravity field of the Sun. This can also be described by a semi-classical algorithm, where the photon travelling from the planet to the observer is deflected in the Newtonian gravity field of the Sun, where the photon has a non-zero mass arising from its energy. To get the correct relativistic result, a correction factor 2.0 must be included in the calculation.
A problem arises when a planet disappears behind the solar disk, as seen from the Earth. Over the whole 6000 year time span of the Swiss Ephemeris, it happens often.
Planet
number of passes behind the Sun
Mercury
1723
Venus
456
Mars
412
Jupiter
793
Saturn
428
Uranus
1376
Neptune
543
Pluto
57
A typical occultation of a planet by the Solar disk, which has a diameter of approx. _ degree, has a duration of about 12 hours. For the outer planets it is mostly the speed of the Earth's movement which determines this duration.
Strictly speaking, there is no apparent position of a planet when it is eclipsed by the Sun. No photon from the planet reaches the observer's eye on Earth. Should one drop gravitational deflection, but keep aberration and light-time correction, or should one switch completely from apparent positions to true positions for occulted planets? In both cases, one would come up with an ephemeris which contains discontinuities, when at the moment of occultation at the Solar limb suddenly an effect is switched off.
Discontinuities in the ephemeris need to be avoided for several reasons. On the level of physics, there cannot be a discontinuity. The planet cannot jump from one position to another. On the level of mathematics, a non-steady function is a nightmare for computing any derived phenomena from this function, e.g. the time and duration of an astrological transit over a natal body, or an aspect of the planet.
Nobody seems to have handled this problem before in astronomical literature. To solve this problem, we have used the following approach: We replace the Sun, which is totally opaque for electromagnetic waves and not transparent for the photons coming from a planet behind it, by a transparent gravity field. This gravity field has the same strength and spatial distribution as the gravity field of the Sun. For photons from occulted planets, we compute their path and deflection in this gravity field, and from this calculation we get reasonable apparent positions also for occulted planets.
The calculation has been carried out with a semi-classical Newtonian model, which can be expected to give the correct relativistic result when it is multiplied with a correction factor 2. The mass of the Sun is mostly concentrated near its center; the outer regions of the Solar sphere have a low mass density. We used the a mass density distribution from the Solar standard model, assuming it to have spherical symmetry (our Sun mass distribution m® is from Michael Stix, The Sun, p. 47). The path of photons through this gravity field was computed by numerical integration. The application of this model in the actual ephemeris could then be greatly simplified by deriving an effective Solar mass which a photon ”sees” when it passes close by or ”through” the Sun. This effective mass depends only from the closest distance to the Solar center which a photon reaches when it travels from the occulted planet to the observer. The dependence of the effective mass from the occulted planet's distance is so small that it can be neglected for our target precision of 0.001 arc seconds.
For a remote planet just at the edge of the Solar disk the gravity deflection is about 1.8”, always pointing away from the center of the Sun. This means that the planet is already slightly behind the Solar disk (with a diameter of 1800”) when it appears to be at the limb, because the light bends around the Sun. When the planet now passes on a central path behind the Solar disk, the virtual gravity deflection we compute increases to 2.57 times the deflection at the limb, and this maximum is reached at _ of the Solar radius. Closer to the Solar center, the deflection drops and reaches zero for photons passing centrally through the Sun's gravity field.
We have discussed our approach with Dr. Myles Standish from JPL and here is his comment (private email to Alois Treindl, 12-Sep-1997):
.. it seems that your approach is
entirely reasonable and can be easily justified as long
as you choose a reasonable model for the density of
the sun. The solution may become more difficult if an
ellipsoidal sun is considered, but certainly that is
an additional refinement which can not be crucial.
link :
http://www.google.com/custom?num=20&hl=en&lr=lang_en&domains=www.astro.com&q=pluto+orbit+%22time+it+takes%22+each+house&btnG=Search&sitesearch=www.astro.com
SWISS EPHEMERIS
The mean apogee (or perigee) moves along the mean lunar orbit which has an ...... This is the time it takes the vernal point to return to the node of the ...www.astro.com/swisseph/swisseph.htm - Cached - Similar
Abril 2004 - Astrodienst
Apr 29, 2004 ... With our ever-increasing ability to observe small, distant objects, we are finding more and more objects similar to Pluto which orbit in the ...www.astro.com/homepage/gb2004_04_p.htm
March 2001 - Astrodienst
Alois adds: It is a recently discovered body outside Pluto's orbit, ..... after 27 1/2 years to the same place (in my case 5 th house - pisces),how you call ...www.astro.com/homepage/gb2001_03_i.htm
