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Chapter 17: Problem 5

The Andromeda Galaxy is at a distance of 700 kpc. (a) Suppose we observe the \(\mathrm{CO}(2-1)\) transition at a wavelength of \(1.3 \mathrm{mm},\) with a 30 \(\mathrm{m}\) diameter telescope. What is the linear resolution? (b) Suppose we use a millimeter interferometer with a maximum baseline of \(1 \mathrm{km}\) ? (c) Repeat these calculations for the Whirlpool Galaxy, at a distance of 9.6 Mpc.

Short Answer

Expert verified
For Andromeda: Single Dish: 30.3 pc; Interferometer: 0.92 pc. For Whirlpool: Single Dish: 415.8 pc; Interferometer: 12.55 pc.

Step by step solution

01

Calculate the Angular Resolution for a Single Dish Telescope

The angular resolution of a telescope can be calculated using the formula \theta \approx \frac{\lambda}{D}\. Here, \theta\ is the angular resolution, \lambda\ is the wavelength, and \D\ is the diameter of the telescope. For the CO(2-1) transition at \lambda = 1.3\ mm and \D = 30\ m, \[ \theta \approx \frac{1.3 \times 10^{-3} \ m}{30 \ m} = 4.33 \times 10^{-5} \ radians.\] Converting radians to arcseconds (1 radian = 206265 arcseconds), we get \[ \theta = 4.33 \times 10^{-5} \times 206265 \approx 8.93 \ arcseconds. \]
02

Calculate the Linear Resolution for the Andromeda Galaxy

The linear resolution \Delta l\ can be calculated using the formula \Delta l = d \times \theta\. Here, \d\ is the distance to the object. For the Andromeda Galaxy at \d = 700 \ kpc = 700 \times 10^{3} \ pc\, \[ \Delta l = 700 \times 10^{3} \ pc \times 8.93 \ arcseconds \approx 700 \times 10^{3} \times \left( \frac{8.93}{206265} \right) \approx 30.3 \ pc. \]
03

Calculate the Angular Resolution for Interferometer with Maximum Baseline

For a millimeter interferometer with a maximum baseline, the angular resolution can be estimated similarly to a single dish telescope, replacing diameter \D\ with baseline \B\. For \B = 1\ km and \lambda = 1.3\ mm, \[ \theta \approx \frac{1.3 \times 10^{-3} \ m}{1 \times 10^{3} \ m} = 1.3 \times 10^{-6} \ radians. \] Converting to arcseconds: \[ \theta = 1.3 \times 10^{-6} \times 206265 \approx 0.27 \ arcseconds. \]
04

Calculate the Linear Resolution for the Interferometer for Andromeda Galaxy

Using \theta = 0.27\ arcseconds and \d = 700 \ kpc\, \[ \Delta l = 700 \times 10^{3} \ pc \times 0.27 \ arcseconds \approx 700 \times 10^{3} \times \left( \frac{0.27}{206265} \right) \approx 0.92 \ pc. \]
05

Repeat Calculations for Whirlpool Galaxy

The calculations will be similar for the Whirlpool Galaxy, changing the distance to \d = 9.6 \ Mpc = 9.6 \times 10^{6} \ pc\. \[ \theta_{single dish} = 8.93 \ arcseconds \] Linear resolution for the single dish telescope: \[ \Delta l_{single dish} = 9.6 \times 10^{6} \times \left( \frac{8.93}{206265} \right) \approx 415.8 \ pc. \] For the interferometer: \[ \theta_{interferometer} = 0.27 \ arcseconds \] Linear resolution for the interferometer: \[ \Delta l_{interferometer} = 9.6 \times 10^{6} \times \left( \frac{0.27}{206265} \right) \approx 12.55 \ pc. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Resolution
Angular resolution is a crucial aspect when it comes to telescopes. It indicates how close two objects can be while still being distinguishable by the telescope. Higher angular resolution means finer details can be observed. The formula used to calculate angular resolution is \( \theta \) ≈ \(\frac{\text{λ}}{\text{D}}\), where λ is the wavelength and D is the diameter of the telescope. The result is typically converted from radians to arcseconds since astronomers often use arcseconds to measure angles in the sky.
Linear Resolution
Linear resolution refers to the actual distance between two points that the telescope can distinguish at a given distance. It combines angular resolution with the distance to an object, using the formula \( \Delta l \) = d x \( \theta \), where d is the distance to the object. For example, observing the Andromeda Galaxy, which is 700 kpc away, with an angular resolution of 8.93 arcseconds, results in a linear resolution of approximately 30.3 parsecs.
Single Dish Telescope
A single dish telescope consists of a parabolic radio dish that collects data. Its size, or diameter, is pivotal for determining its angular resolution. Larger dishes provide better resolutions. Using the CO(2-1) transition wavelength of 1.3mm and a 30-meter diameter telescope, the angular resolution achieved is 8.93 arcseconds. Single dish telescopes are common and highly effective but are limited by their size.
Interferometer
Interferometers use multiple telescopes to simulate the effect of a much larger telescope. By adjusting the distance between the telescopes, known as the baseline, they achieve incredibly high angular resolutions. For instance, a millimeter interferometer with a baseline of 1 km and the same wavelength yields an angular resolution of 0.27 arcseconds. This technique allows astronomers to observe finer details that single dish telescopes might miss.
Andromeda Galaxy
The Andromeda Galaxy, located approximately 700 kpc away, is our nearest galactic neighbor and similar in size to our Milky Way. Using a single dish telescope of 30 meters can give a linear resolution of 30.3 parsecs. However, using an interferometer increases the resolution to an impressive 0.92 parsecs. These calculations demonstrate how different equipment can vastly affect our observational capabilities.
Whirlpool Galaxy
Located about 9.6 Mpc away, the Whirlpool Galaxy is another captivating distant galaxy. Following the same calculation steps, with a single dish telescope, the linear resolution achieved is near 415.8 parsecs. When upgraded to an interferometer, the resolution drastically improves to about 12.55 parsecs. Observations of such distant galaxies showcase how advanced equipment significantly enhances our capacity to study the universe in greater detail.

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