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Supposing heliocentrism to be true, and supposing a space mission had a linear trajectory throughout the voyage, Earth at the position it started from would be in origo, and Earth at other annual positions would not be.

Earth would be moving in and out of the origo of the trajectory.

The space mission would in such a case be going in an apparent zig zag as seen from Earth. Or Earth as seen from the space mission would be moving. Note, we are not talking of a manned mission stationed on the Moon, where the daily "movement of Earth" could be parallactically connected to the Moon's real daily movement around Earth. We are talking about unmanned missions in trajectories that are supposed to be linear according to the laws of physics.

One can account for their apparent non-spiralling as compared to surrounding space by assuming a daily movement around Earth with the aether, with the surrounding space, thus a real spiral around Earth.

But if there was a zig zag, how can a geocentric account for it?

And if there was none, how can a heliocentric account for it?

However, this question about what is moving each year is not one where one can easily document an observed answer.

Hans Georg Lundahl wrote to diverse space or science sites and asked about the zig zag. One affirmed, then referred to an article where the answer was not apparent and then the correspondents for that article did not respond to questions. This correspondence is copied onto his blog.

Despite lack of direct answer to this person, an indirect answer may be hinted at in the article by Nadia Drake: Guided By Starlight, New Horizons Speeds Toward Pluto:

http://phenomena.nationalgeographic.com/2015/07/12/guided-by-starlight-new-horizons-speeds-toward-pluto/

Relevant quote:

Riding aboard New Horizons are two star cameras, each aimed in a different direction.

“They’re used for determining which direction the spacecraft is pointed in, or where all the cameras and instruments are oriented,” Rogers says.

Ten times each second, the cameras snap images of their starfields. They compare those images with an onboard map of more than 10,000 stars. Based on that information, the spacecraft can figure out if it’s tipped up or down, or slightly swiveled.

“All you need is about three or four stars,” Rogers says. If something is amiss, the spacecraft will automatically adjust its pointing.

In other words, there is no camera pointing to Earth. It is not watching our purported in and out of its origo.

Nor can it be watched optically from Earth, it is too small, but we can watch what its cameras are watching because it is transmitted.

And this transmission does not provide us with the wanted zig zag proof for Heliocentrism, as far as the blogger Lundahl could get at by his correspondence questions.

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