Question

The world's largest observatory, ALMA, is comprised of telescopes which serve as interferometric elements. The telescopes are placed up to 16km apart. They are designed to detect light of wavelength in the range from 0.3 to 3.6mm. a) What is the highest achievable angular resolution of ALMA? b) Can ALMA resolve the size of Earth's orbit in a planetary system such as our own, seen at 100pc away from us? A parsec (pc) corresponds to 3.086 x 1016 m. Earth's orbit, known as astronomical unit (au), is 1.496 x 1011 m c) Suppose a 10-km wide asteroid (comparable in size to the one that killed the dinosaurs) were approaching the Earth at some speed v. Suppose we used ALMA to try and resolve such objects to see if they are a single solid object, and also suppose that it would take one year to prepare a response to deflect such an asteroid. How fast could the asteroid be going for this warning system to work?

Answer 1

a] Highest achievable angular resolution can be found using Rayleigh's diffraction limit criteria.

$dsin\theta = 1.22\lambda$

for small angles,

$sin\theta \approx \theta$

so, $\theta = 1.22\frac{\lambda}{d}$

the highest achievable resolution will have a very small value of this angle which will happen for the smallest wavelength (0.3mm)

therefore, $\theta = 1.22\frac{0.3\times 10^{-3}}{16000} = 2.287\times 10^{-8}rad$

b]

For small angles, $\theta \approx tan\theta = \frac{x}{L}$

L = 100 pc = 3.086 x 10¹⁸ m

so, x = 3.086 x 10¹⁸ x 2.287 x 10^-8 = 7.06 x 10¹⁰ m

this is smaller than the size of Earth's orbit of 1.496 x 10¹¹ m and so ALMA will be able to resolve Earth's orbit.

c] L = x/(2.287 x 10^-8) = (10000 / 2.287 x 10^-8) = 4.372 x 10¹¹ m

this is how far the asteroid is from Earth.

Assuming that it takes 1 year to cover this distance, its speed will be:

v = L/t = 4.372 x 10¹¹/(365 x 24 x 3600) = 13865.23 m/s = 13.865 km/s

this is how fast the asteroid will be moving if the system is going to work.