What did Copernicus and Tycho Brahe actually contend about stellar parallax? [duplicate]

It's often stated that Tycho Brahe objected to Copernicus' heliocentric model on the basis that, if Earth were revolving around the Sun, stellar parallax would be observable, due to the changing position of the Earth.

In describing what model Copernicus and Brahe used for stellar parallax, most things I've seen give an explanation of the more common modern model, in which stars believed to be in the 'foreground' relative to Earth are observed against stars contended to be much further away (For the purposes of the formula, unmoving stars).

This couldn't be the the model meant by Copernicus and Brahe, however; Both believed the stars to be set upon an outer sphere (Although Brahe believed this sphere revolved around Earth and Copernicus believed it to be unmoving), and as such neither would have suggested the version of the stellar parallax model which sets stars contended to be in the 'foreground' against 'background' stars.

My question is, what was actually being contended by Brahe and by Copernicus, respectively? What form of stellar parallax model were they holding to with regards to this?

• Stars in the "background" are just a visual convenience. As already Aristotle and Aristarchus knew, there has to be a shift in the angular direction to a fixed star if the Earth moved. And it has to be there regardless of whether there is anything in the "background" or not. Commented Oct 3, 2021 at 3:08
• @Conifold What does a 'shift in the angular direction' consist of? In terms of parallax itself, I understood the common modern definition, at least, to necessarily be the observance of a closer body moving against a distant, 'fixed' body. For example, observing a nearby tree moving against the backdrop of a far-off mountain as you pass by it. Commented Oct 3, 2021 at 3:54
• The angle between the line connecting your eyes and the line connecting the midpoint between them to the tree changes when you move around, even if there is no far-off mountain in the backdrop or no backdrop at all. Of course it is easier to spot when there is a distant reference object, but such an object is not necessary. Commented Oct 3, 2021 at 3:57
• Do you mean the line connecting the midpoint between to possible positions to the tree? But I fail to see how this form of trigonometric parallax would demonstrate the movement of the Earth to Tycho Brahe without further points of reference. Commented Oct 3, 2021 at 4:25
• Funnily enough, that question is actually one of the reasons I posted this. Going back, though, I think I understand what's meant by angular distance now. I hadn't considered that the angle between stars as perceived would change if the Earth itself were to move. Calling this 'parallax' seems like a misnomer, though, at least insofar as that most sources refuse to differentiate between it and the common modern model. Thanks. Commented Oct 3, 2021 at 6:00