# My estimate for the distance of the farthest Sun-size star that could be focused as a single-whole-star, by a 0.001''-precision telescope, is 30.53 light years. What is your estimate? Same, or different?

Oct 7, 2016

If $\theta$ is in radian measure, a circular arc, subtending an

$\angle \theta$ at its center, is of length $\left(r a \mathrm{di} u s\right) X \theta$

This is an approximation to its chord length

$= 2 \left(r a \mathrm{di} u s\right) \left(\frac{\theta}{2} + O \left({\left(\frac{\theta}{2}\right)}^{3}\right)\right)$, when $\theta$ is quite small.

For the distance of a star approximated to a few significant (sd)

digits only in large distance units like light year or parsec, the

approximation (radius) X theta is OK.

So, the limit asked for is given by

( star distance ) $X \left(\frac{.001}{3600}\right) \left(\frac{\pi}{180}\right)$ = size of the star

So, star distance d = (star size)/ $\left(\frac{.001}{3600}\right) \left(\frac{\pi}{180}\right)$

=(diameter of the Sun)/$\left(4.85 X {10}^{- 9}\right)$, for a sun-size star

$= \left(\frac{1392684}{4.85}\right) k m$

$2.67 X {10}^{14} k m$

$= \left(\frac{2.67}{1} , 50\right) X {10}^{6} A U$

$= 1.92 X {10}^{6} A U$

$= \frac{1.92 X {10}^{6}}{6.29 X {10}^{4}} l i g h t y e a r s \left(l y\right)$

$= 30.5 l y .$