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Pinhole Projector: Can I use a specific solar optical eye piece to focus the Transit of Mercury?

The title pretty much explains this one. Can I use a specific solar optical eye piece, or other optical device, in conjunction with the pinhole projector method to focus the Transit of Mercury? ...
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Linear limb darkening coefficient, u

I was wondering if anyone knew of any resources to understand the linear limb darkening coefficient, $u$. That is to say, how the $\theta$-dependent coefficient $u$ (or sometimes $b_{\nu}$) varies ...