The VLBA uses an array of radio telescopes ranging across 8,000 km of Earth's surface from the V

The VLBA uses an array of radio telescopes ranging across...

The VLBA uses an array of radio telescopes ranging across $8,000 \mathrm{km}$ of Earth's surface from the Virgin Islands to Hawaii. a. Calculate the angular resolution of the array when radio astronomers are observing interstellar water molecules at a microwave wavelength of $1.35 \mathrm{cm}$. b. How does this resolution compare with the angular resolution of two large optical telescopes separated by 100 meters and operating as an interferometer at a visible wavelength of $550 \mathrm{nm} ?$

The VLBA uses an array of radio telescopes ranging across $8,000 \mathrm{km}$ of Earth's surface from the Virgin Islands to Hawaii.
a. Calculate the angular resolution of the array when radio astronomers are observing interstellar water molecules at a microwave wavelength of $1.35 \mathrm{cm}$.
b. How does this resolution compare with the angular resolution of two large optical telescopes separated by 100 meters and operating as an interferometer at a visible wavelength of $550 \mathrm{nm} ?$

Key Concepts

Angular Resolution

Angular resolution refers to the minimum angle between two objects that an instrument can distinguish as separate. In the context of telescopes and interferometers, it is often approximated by the ratio of the wavelength of the observed radiation to the effective aperture or baseline of the array. This concept is critical in comparing the performance of different observational systems in astronomy.

Interferometry

Interferometry is a technique that combines the signals received at two or more telescopes to produce interference patterns. These patterns are then used to reconstruct high-resolution images of astronomical objects. The method allows for the effective resolution of a telescope array to be equivalent to that of a single telescope whose diameter is equal to the maximum separation (baseline) between the individual telescopes.

Baseline

The baseline is the separation distance between the telescopes in an interferometric array. It directly influences the angular resolution achievable by the array, with larger baselines providing finer resolution. The concept of the baseline is fundamental in both radio and optical interferometry, as it determines the scale of the details that can be observed in the target objects.

Wavelength Dependence

The wavelength of the observed radiation plays a crucial role in determining the angular resolution of a telescope system. Since the resolution is proportional to the wavelength (when divided by the baseline or aperture), shorter wavelengths (such as visible light) can yield significantly higher angular resolution compared to longer wavelengths (such as microwaves), assuming similar baseline lengths. This concept is essential when comparing the performance of optical and radio interferometers.

Transcript

00:01 In problem number 40, we're asked to calculate the angular resolution of the very long baseline array, or vlba, as it observes in the microwave form of the spectrum at 1 .35 centimeters.

00:18 Now, to calculate a telescope's angular resolution, we use the formula theta, where theta is our angular resolution.

00:30 Theta is equal to 2 .06 times 10 to the 5th lambda over d or seconds.

00:49 The lambda is the wavelength at which we're observing, and d is the diameter of the aperture and the telescope we're using.

00:57 For this problem, we're given the distance and the wavelength.

01:02 The distance is 8 ,000 kilometers, a very long distance, but that's why they call it the very long baseline array.

01:13 And we're also given wavelength.

01:17 Our wavelength is in the microwave part of the spectrum.

01:20 It is 1 .35 centimeters.

01:25 Now, to keep our units consistent, we're going to be dividing these two numbers.

01:34 We want to convert both of them to the same units.

01:38 Let's convert them both to meters.

01:41 So 1 .35 centimeters.

01:46 In one meter, there are 100 centimeters.

01:53 So those units cancel.

01:55 And what we're left with is our wavelength is now 0 .135 meters.

02:06 And now for the bigger distance, the kilometers, in one kilometer there are a thousand meters.

02:18 And that gives our distance as 8 million, or as i'm going to write it in scientific notation, 8 times 10 to the 6th meters.

02:30 All right.

02:32 To find our angular resolution, we just plug all these numbers in for our formula and solve for theta.

02:39 So we get theta is equal to 2 .06 times 10 to the 5th times 0 .0135 meters...

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