When observing distant landmarks from aboard a moving train, it is noticed that those features that are closer to us seem to go past much faster than those that are further away. Indeed, very distant stationary objects traverse our field of view so slowly that they appear to be moving with the train. These otherwise fixed objects can thus appear to someone on the moving train as being in relative motion with one another. This apparent movement effect, caused by variations in proximity, is referred to as parallax.
The effect can be useful inasmuch as it allows for the determination of observer-object distances, as illustrated via Fig. 1, below.
Figure 1: Trigonometric parallax.
Here the train travels in a straight line from A to B. The observer, who is on the train, looks out of the carriage window at A and sees a distant object appear to be located at A'. When the observer reaches B, he peers out of the window again and sees the same distant object (which now seems to be at B' ). If we know the length of the baseline, AB, and the two angles a and b, then the shortest distance, h, between the object (in reality situated at C) and the baseline of the railway track can be worked out as follows:
h = Ax tan a = Bx tan b = (AB - Ax) tan b
h tan a = (AB tan a - h) tan b
h = AB (tan a tan b) / (tan a + tan b)
The tangent function is unstable when applied to an angle approaching 90°, due to division by zero being mathematically undefined, so the method of trigonometric parallax requires a baseline of reasonable length.
A careful examination of photographic plates that have been exposed to the same region of sky, but at times that are a few months apart, will reveal the fact that some stars have shifted their position with respect to the 'background' stars. Such stars are assumed to be closer to us than the (effectively) infinitely far away 'background' stars, and the effect is naturally given the name of stellar parallax. The phenomenon consequently provides the astronomical community with a technique for determining the distance to a 'nearby' star at C. This is because, if the World hurtles around the Sun, then points A and B of Fig. 1 can be regarded as two spatial locations on the World's orbit that are temporally six months apart, in which case the baseline, AB, can be as large as the major axis of the Sun-World system. Even with such a large baseline, the angle ACB turns out to be so small that the claimed closest star, Proxima Centauri, would be 4.3 light-years (l-y) away, if the assumptions of heliocentrism are correct.
Due primarily to the exceptionally detrimental influence of Hollywood on Western societies, the concept of a Star Trek-like universe has been ingrained from infancy into the minds of a very large number of people. One consequence of this is that the stars are generally perceived to be located at phenomenal distances from us. However, such vast distances were originally borne of paradigm-supporting necessity and although they may appear to be confirmed by trigonometric parallax calculations, it should be remembered that this technique is itself based upon the heliocentric assumption that it is used to support.
The Hipparcos Astrometry Satellite
From a NASA website  we read that, "The Hipparcos and Tycho [star] Catalogues are the primary products of the European Space Agency's astrometric mission, Hipparcos. The satellite, which operated for four years, returned high quality scientific data from November 1989 to March 1993.
"Each of the catalogues contains a large quantity of very high quality astrometric and photometric data."
There is considerable on-line ESA documentation  on this mission, and of particular relevance is their "Hipparcos and Tycho Catalogues Volume 1 Introduction and Guide to the Data," , and "Introduction to the Hipparcos and Tycho Catalogues" .
There is also a very useful research tool that enables results from their dataset to be retrieved via a parameter entry table , and we see from the associated 'ReadMe' file that the measured ranges for trigonometric parallax were as follows:
- Hipparcos Catalogue, field H11, -55 mas to 772.33 mas.
- Tycho Catalogue, field T11, -919 mas to 701.5 mas.
where one 'mas' is 0''.001 (for example, 250 mas = 0''.25).
In this discussion we shall refer exclusively to the Tycho Main Catalogue, because this has far more entries than the Hipparcos Catalogue and because the distribution of parallax data in the Tycho Main Catalogue has not been artificially weighted about the zero value by restricting the negative range.
There are 1,058,332 objects in the Tycho Main Catalogue, and these have a median astrometric precision of 7 mas for visual magnitude 9 and below, increasing through 25 mas for visual magnitude 10-11.
Using the ESA's parameter entry table , we selected field three (parallax) and specified a range of -919 (min) to -20 (max) mas, over the entire dataset. This produced 262,100 records of negative parallax objects, or 25% of the total.
Next we selected the positive parallax objects via a minimum value of 20 mas and a maximum of 701.5 mas. This resulted in 310,758 records, or 29% of the total.
The remaining 46% of the Tycho Main Catalogue entries can be assumed to possess zero parallax, within the precision of (0 ± 20) mas.
Section 2.2 Contents of the Tycho Catalogue  makes the following statement regarding Field T11, "The trigonometric parallax, π, is expressed in units of milliarcsec. The estimated parallax is given for every star, even if it appears to be insignificant or negative (which may arise when the true parallax is smaller than its error)."
A further test was conducted, to see if the stars moving across the astrometric instrument slit were directionally different in the northern celestial hemisphere to what they were in the southern celestial hemisphere. In this case, as well as the parallax field, the declination field was also selected. Of the non-zero-parallax stars in the northern celestial hemisphere (0°N ≤ δ ≤ 90°N), 45% of them had a negative parallax, and in the southern celestial hemisphere (0°S ≤ δ ≤ 90°S), 46% of non-zero objects had a negative parallax. So here again is a very symmetrical distribution that would be typical of a naturally occurring phenomenon.
In the geocentric model of the universe, the stars occupy a shell, referred to as the stellatum (see Fig. 2), which rotates diurnally about the World. Walter van der Kamp calculated the distance to the stellatum as being 58 light-days (l-d), which, if correct, points to a universe whose radius is 121.75 billion times smaller than we are taught (60 l-d as opposed to 20 billion l-y), and whose volume is thus at least 1.8 x 10³³ times smaller. Hence, to say that geocentrism does not predict the Star Trek-like monstrosity that mainstream science requires is a bit of an understatement.
Figure 2: Cross-sectional view of the stellatum.
If we assume that stars are positioned within the stellatum such that their density across the thickness of the stellatum follows a normal or Gaussian distribution about the mean distance, μ, from the centre of the World, then the radial density of stars will be represented by a bell-shaped curve (Fig. 3) the precise outline of which will depend upon the standard deviation, σ.
The distribution density, f(x), is given by
for all positive values of σ, where x is the distance from the mean (Kreyszig, pp. 1184 - 1191). As also stated by Kreyszig, many random variables "have a normal distribution or they are approximately normal" and "the normal distribution is a useful approximation of more complicated distributions."
Figure 3: Gaussian density distribution with a mean of zero and a variance ( = σ²) of 0.25, where 46% of the data fall between the two dotted lines (which were calculated using Table A8 in Appendix 5 of Kreyszig, p. A102).
The important thing to keep in mind with the geocentric universe explanation for negative parallax is that it is not the so-called 'background' stars of conventional astronomy that are the furthest away from us, but rather those stars that display negative parallax readings. In other words, the 'background' stars, which are in the majority, are actually displaying parallax with respect to the stars that we associate with negative parallax readings.
In Fig. 3, 46% of all stars are located between the limits indicated by the two dotted lines on either side of the mean (the centre point of the stellatum thickness), and from Fig. 2 we see that this would imply 27% of stellatum stars would be closer to us (and thus display positive parallax) and 27% would be further away than the majority (and thus display negative parallax). I.e., 46% are middle stars (as termed in Fig. 2), 27% are inner stars (c.f. 29% from the Tycho Main Catalogue), and 27% are outer stars (c.f. 25% from the Tycho Main Catalogue).
By definition, any perceived motion within a geostationary universe must be due to the object seen (from the World) to be moving. Just as the Sun travels the full extent of its orbit in one tropical year, so too the stars that exhibit parallax would have to complete their 'orbits' in the same time. The size and cause of this motion is not considered here, rather the emphasis is placed on the capacity of the geocentric model to accommodate negative parallax, whereas negative parallax measurements are totally incompatible with the acentric universe hypothesis and need to be dismissed in that case as simply statistical errors.
It is an indisputable fact that stellar parallax, like the phases of Venus, has been widely cited as 'proof' that the World orbits the Sun. This is unfortunate, since the phenomenon proves no such thing. The only thing it does prove is that either the World is moving with respect to the stars, or that the stars are moving with respect to the World.
At this the geocentrists usually rest their case, claiming that the adoption of a heliocentric philosophy is just as much a matter of faith as the adoption of a geocentric philosophy. However, this invocation of faith is unnecessary and unjustified, for if it were such a simple choice between the World going around the Sun, or some stars moving slightly in order to conveniently give the appearance of the World going around the Sun, then the heliocentrists would have a point of strong probability (as opposed to a point of proof) in their favour, and geocentrism would indeed become more faith than science. Contrariwise it is worthwhile noting that credibility as regards the sizes of the Sun and Moon discs producing the observed solar eclipse effect that we marvel at sits more comfortably with the intelligent design position that geocentrism tends to imply, rather than with the heliocentrists and their claim of coincidence.
The phenomenon of stellar parallax is not what we have been generally led to believe, because in exactly the same way that Eddington 'proved' Einstein's General Theory of Relativity in 1919 by rejecting, omitting or deleting 60% of his measurement data on the bending of starlight, so modern astrophysics maintains the misconception that parallax 'proves' the Kopernikan philosophy of the World hurtling around the Sun, by ignoring and dismissing the entire dataset of negative parallax measurements.
The ESA, unlike Eddington before them, have kept and filed data values which do not fit in with the ruling model of the universe, and should be commended for so doing, but nevertheless they do seem to dismiss a significant proportion of their measurements rather glibly. Of course, they do say that these may arise due to measurement error, but the number and symmetrical distribution of these values would tend to deny this as being anything other than an exception to the rule.
Furthermore, although angular parallax measurements are small (the largest positive value gives an angle ACB, in Fig. 1, on the order of only 0.7 of an arcsecond), the effect is known to be genuine by way of photographic plates taken at various times over a period of twelve months which clearly show the same slight movement of some stars with respect to the background star field. In other words, stellar parallax is an observable phenomenon that is repeatable, rather than being experimental or statistical errors in measurement.
When the full picture is revealed and considered, therefore, it is clearly geocentrism that has the potential to fully and adequately account for the hundreds of thousands of negative parallax observations that have now been recorded, although it is acknowledged that a detailed explanation is not currently available.
Finally, it may be possible to estimate the thickness of the stellatum from the ESA dataset of parallaxes.
(All websites as accessed on 9th February, 2007.)
Kreyszig, E., 1993, "Advanced Engineering Mathematics," 7th ed., John Wiley & Sons, New York.