We need to first mention that the smallest size which the human eye can
easily resolve an object is approximately one arc-minute, or one sixtieth
of a degree. So, the Sun will appear as a ‘dot’ when its angular diameter
equals or is less than one arc-minute. At what distance will the Sun’s
angular diameter equals one arc-minute?

We’ll determine this distance by using the formula

α=2arctan(D/2r)

where α = is the object’s angular diameter; D = object’s diameter; r =
distance from object.

one arc-minute = 1/60th of a degree = 0.0167
the sun’s true diameter = 865,370 miles

Assuming that our calculation is correct, we determine:

The distance at which the Sun’s angular diameter of one arc-minute is
approximately 2.969 billion miles, or about at Pluto’s perihelion, or point
of least separation distance from the Sun.*

However, at this distance, the Sun will still appear about 185 times
brighter than the full moon. (One could still read a book by Sunlight at
this distance.)

How far would one have to travel in order for the Sun to appear as bright
as Sirius, the brightest star in the night sky?

At approximately 1.46 light years from the Sun, our parent star and Sirius
will appear equally bright.**
This assumes that one is traveling toward Sirius. This equals the distance
of the remotest Oort Cloud shell, at which the Sun’s gravitational
influence becomes quite tenuous.

As a person’s separation distance from the Sun increases, its apparent
magnitude (brightness) decreases. The Sun will actually vanish from our
view at a distance of about 56 light years.**

*Pluto’s distance varies between perihelion (about 2.7 billion miles) to
aphelion (4.58 billion miles)

**d⋅10M−4.855/10^(M−4.855/5) +1 = d☉

The formula for calculating the distance at which the Sun and a star with a
given absolute magnitude appear equally bright.

*** m- M = 5logd - 5

m = apparent magnitude; M = absolute magnitude; d = distance in parsecs
We assume that the dimmest star’s apparent magnitude is 6.0. The Sun’s
absolute magnitude equals 4.83.