What angular resolution would you need to see the Sun and Jupiter as distinct points of light if you were looking at it from 15 light years away?

Answer 1

Great question to chew on ! And I think I can work it out, at least

to an order-of-magnitude approximation.

Radius of Jupiter's orbit is around 5 AU = 5 x 93 million = 465,000,000 miles.

15 light years = 15 x 5.8787 x 1012 = 8.818 x 1013 miles

(Tangent)-1 of (5 AU / 15 LY) = 0.000302 degree = roughly like 1.09 arc-second.

There's a rule for the resolution you need in order to completely resolve two sources

that subtend a given angle, and I don,t remember what it is. But the order of magnitude

for this particular case is about 1 arc-second ... with one bright object and a dim one.

That calculation looks right and the separation can theoretically be resolved by a telescope with a 4½" aperture. But there is a big difference in brightness because the Sun would be a star of magnitude 3.2 while Jupiter's magnitude would be about 21. So a considerably bigger telescope would be needed because to see something that dim needs a big aperture, round about 100".

Answer 2

10 light years = approx 5.8786 x 1013 miles

Sun/Earth distance = approx 9.3 x 107 miles

Forget about the size/diameter of the sun or Earth. Let's just see what kind of angle

93 million miles looks like from 10 LY away.

tangent ( Θ ) = approx 9.3 x 107 / 5.8786 x 1013 = 1.582 x 10-6

For angles like that, tan( Θ ) = approx ( Θ )

Θ = 1.582 x 10-6 radian = approx 9.064 x 10-5 degree = 0.326 second of arc.

Use the ratio 1.582 x 10-6 to create a situation that we can relate to:

A US dime (10¢) is 17.9 mm in diameter.

When you look at a dime that's 7.031 miles from your face, it has the same apparent size

as the space between the sun and Earth as seen by an observer 10 light years away from

our solar system.