How do we know stellar distances?

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Here is a popular idea among the moderately instructed. Fix stars and objects within them cannot be one light day up, because trigonometry proves alpha Centauri is 4 light years away.

This is probably the reason why I was recommended this video, which arguably is a very good one on trigonometry in a more normal and terrestrial and certain use:

How do trig points actually work?
The Map Reading Company, 27 June 2023
https://www.youtube.com/watch?v=9cc_FtM_WGc

It's just that each triangle actually is measured by two points on a theodolite and one point on a different trig point, which are three bodily objects distant from each other in one and the same moment.

In the case of astronomic trigonometry, Earth is supposed to serve as two different points successively, or alternatingly, as Earth is supposed to move from Summer to Winter position in her supposed circuit around the Sun.

And I think I have pointed out years ago, that the validity or very otherwise of such a move depends very highly on what it is that actually moves.

hglwrites : Geo vs Helio*
https://hglwrites.wordpress.com/2012/05/29/geo-vs-helio/

On the diagram to the right, reflecting Heliocentrism, Earth moves, and the different lines towards the star, in a very exaggerated angle, go out converging to a point outside the picture.

On the diagram to the left, reflecting Geocentrism, the star moves, and the different lines towards the star, same angle, can be cut off basically anywhere. If the star had exactly the same movement as the Sun, yes, the parallax trigonometry would be equal to that in the Heliocentric scenario. I have shown them cut off way closer to Earth, as in star moves considerably less than the Sun.

The Geocentric orbit of Sun around the Zodiac, with the Sun in the middle, is basically the inverse of the Heliocentric orbit of Earth around the Zodiac, with the Sun in the middle.

Aphelion	152,097,597 km	(Apogee)**
Perihelion	147,098,450 km	(Perigee)**

The supposed distance between the two positions of Earth is therefore 299,196,047 km. In Geocentrism, this is the distance between two positions of the Sun, with the same star sign in Zenith. Now, if this were also the distance between two positions of the star — alpha Centauri, it would follow the same mathematics as in Heliocentrism. The star would be 4 light years above us, or more.

Now, what is the real distance travelled by alpha Centauri, if Heliocentrics make it 4 light years up?

Well, if the movement were just the parallax, it would be:

299,196,047 km / (4 * 365) = 204,928.8 km

However, the proper movement executed by the star accounts not just for the parallax, but also for the aberration, c. 20 arc seconds.*** The parallax is just 0.76 arc seconds. In fact, parallax is not measured separately, like by theodolites, in the landscape, they are measured against the background of a much better visible aberration.

204,928.8 km * (20 / 0.76) = 5,392,863.158 km

So, if the overall visible movement (compared to the Zodiac) were taken as parallax, the one mostly taken as aberration, would one take alpha Centauri as more than 20 times further away? No. One would take it as much closer. The smaller the angle, the greater the distance. 0 angle = infinite distance.

So, when, as Geocentric, I cannot know the distance to alpha Centauri from parallax and trigonometry, how do I know it? I first of all know it is so far from Earth as to make Earth virtually a point in relation to the distance. But that's even true of the Sun. Second, I know fix stars are outside of the planets, since the planets come in the way of fix stars when we observe. This means, we can use measures for planetary distances (none of which is as far as one light day) to say stars are even further away. Third, I know the Voyager Space missions have not yet bumped into the stars, and I know they are not fully one light day up.

Hans Georg Lundahl
Paris
XVIIIth Lord's Day After Pentecost
1.X.2023

* This blog of mine was unjustly disconnected. It is still my own material, and I have no qualms linking to it or using it.
** To Heliocentrics, the terms Apogee and Perigee refer to the Lunar such, since only the Moon is supposed to have that, but here we are speaking of Solar Perigee and Apogee.
*** If the Earth moves and light has a finite speed, aberration has to be a physical reality about the light reaching us. However, if the Earth does not move, it's a physical reality about light leaving the star and therefore about where the star is.