Question

Suppose you wanted to build a telescope capable of resolving a planet the size of Earth that is 17 light-years away.

a. Calculate the angular diameter of Earth if it were that far. To see any detail on the surface, we would want to see an angular size about ten times smaller than this.

b. Calculate the diameter of a telescope that would be needed to resolve this angular size if ​observing at 592 nm. Is this feasible?

Answer 1

a. We know, angle = arc/radius

In this case, arc = diameter of the earth = 2*radius of the earth

= (2*6.4*10⁶) m = 12.8*10⁶ m

Radius = 17 light year = (17*9.461*10¹⁵) m = 160.837*10¹⁵ m

So, angular diameter = (12.8*10⁶/160.837*10¹⁵) radian

= 0.0795*10^-9 radian

b. The angular diameter calculated above will be the limit of resolution for the telescope, which is given for a wavelength and diameter d of telescope objective is given by

= 1.22/d

Here, = 592 nm = 592*10^-9 m, = 0.0795*10^-9 radian, thus,

0.0795*10^-9 = 1.22*592*10^-9/d

So, Diameter of telescope objective = (1.22*592/0.0795) m

= 9084.78 m

This size of the diameter of telescope objective is not feasible.

Answer 2

a. Calculate the angular diameter of Earth at a distance of 17 light-years:

To calculate the angular diameter (θ) of Earth, we can use the formula:

θ = (D / D_distance) * (180 / π)

where D is the actual diameter of Earth and D_distance is the distance from the observer to Earth (17 light-years converted to miles).

Given: D = Diameter of Earth ≈ 7,917.5 miles (mean diameter) Distance to Earth (D_distance) = 17 light-years ≈ 100.8 trillion miles (using 1 light-year ≈ 5.88 trillion miles)

Now, calculate θ:

θ = (7,917.5 / 100.8 trillion) * (180 / π)

θ ≈ 2.75 x 10^-9 degrees

b. Calculate the diameter of a telescope needed to resolve this angular size at a wavelength of 592 nm (nanometers):

To calculate the diameter (D_telescope) of the telescope required to resolve the angular size, we can use the formula:

D_telescope = (λ / θ) * 206,265

where λ is the wavelength of light in meters and θ is the angular size in radians.

Given: Wavelength (λ) = 592 nm = 592 x 10^-9 meters (converted to meters) Angular size (θ) ≈ 2.75 x 10^-9 degrees converted to radians ≈ 4.8 x 10^-14 radians (θ = θ° * (π / 180))

Now, calculate D_telescope:

D_telescope = (592 x 10^-9 / 4.8 x 10^-14) * 206,265

D_telescope ≈ 25,513 meters or 25.5 kilometers

Is this feasible? Building a telescope with a diameter of 25.5 kilometers is not currently feasible with our current technology. The largest telescopes on Earth today have diameters in the range of 10 to 20 meters. Even the proposed Extremely Large Telescopes (ELTs) under development are expected to have diameters of up to 39 meters. Constructing a telescope with a diameter of 25.5 kilometers is beyond our current technological capabilities and would present significant engineering and logistical challenges.