The smallest detail visible with ground-based solar telescopes is about 1 arc second. How large a region (in km) does this represent on the Sun? Hint: Use the small-angle formula: angular diameter (in arc seconds) 2.06 ✕ 105 = linear diameter distance. (Note: The average distance to the Sun can be found in Celestial profile: The Sun.)

x = 727.5 km

Explanation:

With the conditions given using trigonometry, we can find the tangent

tan θ = CO / CA

With CO the opposite leg and CE is the adjacent leg which is the distance from the Tierral to Sun

D = 150 10⁶ km (1000m / 1 km)

D = 150 10⁹ m.

We must take the given angle to radians.

1º = 3600 arc s

π rad = 180º

θ = 1 arc s (1º / 3600 s arc) (pi rad / 180º) =

θ = 4.85 10⁻⁶ rad

That angle is extremely small, so we can approximate the tangent to the angle

θ = x / D

x = θ D

x = 4.85 10-6 150 109

x = 727.5 103 m

x = 727.5 km