Any recommended observations for amateur astronomers?

If an amateur stargazer wanted to verify the Tychos Model, what observations could they make at home that would convince them of the validity of the model?
Assuming they have a decent telescope and the ability to use it effectively.

I am replying to my own post. Here is one idea sent to me by Matt.

One test would be seeing what star our moon aligns with each and every full moon. According to the Tychos system the moon aligns with the same star every 27.3 days which is not possible in the Heliocentric model.

Indeed - the Moon does pretty much align with the same star every 27.3 days - but keep in mind that there WILL be a (very small) displacement, since the Earth-Moon system moves by 38.428km every day and will thus move (against the starry background) by about 1049 km every 27.3 days:

38.428 km x 27.3 days = 1049.0844 km

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So I am realizing that the moon does not align with the same star every full moon. It aligns every 27.3 days but still goes through its phases.

Here’s one that I have never accomplished; view Mercury.

I don’t think you need a telescope but you might need binoculars, I’m not sure.

I have been trying to see Mercury for a long time and I have never seen it before. It was just last week at it’s best viewing in the morning, prior to sunrise.

Has anyone seen it?