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Chapter 4: Problem 31

How many pairs of telescopes are there in (a) the VLA with 27 telescopes, (b) the VLBI, with ten telescopes, (c) the proposed Millimeter Array, which may have 40 or 75 telescopes depending on the final design?

Short Answer

Expert verified
There are 351 pairs in VLA, 45 pairs in VLBI, 780 pairs in the 40 telescope Millimeter Array, and 2775 pairs in the 75 telescope Millimeter Array.

Step by step solution

01

- Understanding Combinations

The problem involves finding the number of pairs that can be formed from a set number of telescopes. This can be solved using combinations, denoted as \(_nC_2\), which represents the number of ways to choose 2 elements from a set of n elements.
02

- Formula for Combinations

The formula for combinations is given by: \[ nC_k = \frac{n!}{k!(n-k)!} \] For our case, k=2.
03

- Calculate for VLA (27 telescopes)

Substitute n=27 and k=2 into the formula: \[ 27C_2 = \frac{27!}{2!(27-2)!} = \frac{27 \times 26}{2 \times 1} = 351 \] So, there are 351 pairs of telescopes in the VLA.
04

- Calculate for VLBI (10 telescopes)

Substitute n=10 and k=2 into the formula: \[ 10C_2 = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \] So, there are 45 pairs of telescopes in the VLBI.
05

- Calculate for Millimeter Array (40 telescopes)

Substitute n=40 and k=2 into the formula: \[ 40C_2 = \frac{40!}{2!(40-2)!} = \frac{40 \times 39}{2 \times 1} = 780 \] So, there are 780 pairs of telescopes in the Millimeter Array with 40 telescopes.
06

- Calculate for Millimeter Array (75 telescopes)

Substitute n=75 and k=2 into the formula: \[ 75C_2 = \frac{75!}{2!(75-2)!} = \frac{75 \times 74}{2 \times 1} = 2775 \] So, there are 2775 pairs of telescopes in the Millimeter Array with 75 telescopes.

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Key Concepts

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

Combinatorial Mathematics
To understand how to calculate the number of telescope pairs, we need to delve into combinatorial mathematics. This branch of mathematics deals with counting, arrangement, and combination of objects. When calculating how many ways we can pair a certain number of telescopes, we use the concept of combinations. The formula for combinations is denoted as \( nC_k \), where 'n' is the total number of items, and 'k' is the number of items to choose. Specifically, for our telescope problem, we use \( _nC_2 \).
Radio Astronomy
Radio astronomy is the study of celestial objects that give off radio waves. Unlike optical telescopes that observe light, radio telescopes can observe radio frequencies emitted by stars, galaxies, and other astronomical phenomena. This branch of astronomy allows us to discover and understand the universe's unseen parts, like black holes and dark matter. Telescopes in radio astronomy work together in arrays to increase resolution and gather more detailed data.
Very Large Array (VLA)
The Very Large Array (VLA) is a renowned radio astronomy observatory located in New Mexico, USA. It consists of 27 radio telescopes arranged in a Y-shaped pattern. These telescopes work collectively to detect faint radio emissions from celestial objects. By combining signals from multiple telescopes, the VLA achieves higher resolution images than a single telescope. In our exercise, when we calculate the number of pairs from 27 telescopes in the VLA, we use the combination formula. Plugging in the values, \( 27C_2 = 351 \), we get 351 pairs.
Very Long Baseline Interferometry (VLBI)
Very Long Baseline Interferometry (VLBI) is a technique in radio astronomy where telescopes spread over vast distances (even continents) work together to simulate a massive radio telescope. The distributed nature of VLBI gives it incredibly high resolution. For the VLBI, with 10 telescopes, by using our combination formula \( 10C_2 \), we find there are 45 pairs. The data from these pairs are combined to produce detailed images of astronomical phenomena.
Millimeter Array
A millimeter array is a collection of radio telescopes designed to observe millimeter wavelengths of radio emissions. These wavelengths provide information about the cold universe, including dust clouds and star formation regions. In the proposed millimeter array, with either 40 or 75 telescopes, the combination formula helps determine the number of pairs:
  • For 40 telescopes, \( 40C_2 = 780 \)
  • For 75 telescopes, \( 75C_2 = 2775 \)
These pairs work together to enhance the sensitivity and resolution of the array.

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Most popular questions from this chapter

A diffraction grating has \(N\) lines, a separation \(d\) apart. The spectrum is projected on a screen a distance \(D(\gg d)\) from the grating. Two lines are \(\lambda\) and \(\lambda+\Delta \lambda\) apart. How far apart are they on the screen?

What is the angular resolution (in arc minutes) of (a) a \(100 \mathrm{m}\) diameter telescope operating at \(1 \mathrm{cm}\) wavelength, and (b) a \(30 \mathrm{m}\) telescope operating at \(1 \mathrm{mm}\) wavelength?

Suppose that we use a reflector in the coudé focus, and each of the three mirrors reflects \(95 \%\) of the light. What fraction of the light is lost as a result of these three reflections?

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The sodium D lines in the Sun's spectrum are at wavelengths of 589.594 and \(588.997 \mathrm{nm}\) (a) If a grating has \(10^{4}\) lines/cm, how wide must the grating be to resolve the two lines in first order? (b) Under these conditions what is the angular separation between the two lines? (c) How would the results in (a) and (b) change for second order?
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