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Chapter 19: Problem 32

Suppose the number density of galaxies in the universe is, on average, \(3 \times 10^{-68}\) galaxy \(/ \mathrm{m}^{3} .\) If astronomers could observe all galaxies out to a distance of \(10^{10} \mathrm{pc}\), how many galaxies would they find?

Short Answer

Expert verified
About 1.2 galaxies.

Step by step solution

01

Understand the Given Values

The number density of galaxies is given as \(3 \times 10^{-68}\) galaxies per cubic meter \(\mathrm{m}^3\) and the distance out to which we can observe is \(10^{10}\) parsecs (pc).
02

Convert Parsecs to Meters

We need to convert the distance from parsecs to meters. The conversion factor is: 1 parsec \( = 3.086 \times 10^{16}\) meters. Thus, \(10^{10}\) parsecs is \[ 10^{10} \times 3.086 \times 10^{16} \mathrm{m} \].
03

Calculate Volume of the Observational Sphere

We need to calculate the volume of the sphere with radius \(10^{10} \times 3.086 \times 10^{16} \mathrm{m}\). The volume V of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). Use the radius from the previous step.
04

Use the Number Density to Find Number of Galaxies

Multiply the volume from Step 3 by the number density \( 3 \times 10^{-68}\) galaxies per cubic meter. This gives the total number of galaxies.
05

Perform the Calculation

Substitute all values into the formulas and calculate step-by-step, making sure to keep track of the exponents carefully.

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Key Concepts

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

Galaxy Density
Understanding galaxy density is crucial for calculating the number of galaxies in a given volume. Galaxy density refers to the number of galaxies per unit volume. In this exercise, it's given as \(3 \times 10^{-68} \text{ galaxies/m}^3\). This means that, on average, there are 3 galaxies in each cubic meter of the universe, albeit spread incredibly thin due to the vast distances involved. Knowing this density helps astronomers and physicists estimate the number of galaxies in different sections of space.
Parsec to Meter Conversion
A parsec is a common unit of distance used in astronomy, equivalent to about 3.086 \(\times \) 10\(^{16}\) meters. Converting parsecs to meters ensures that all measurements are in compatible units. In this exercise, the observational radius is given as \(10^{10}\) parsecs. To convert this distance to meters:
  • Multiply the number of parsecs by the conversion factor:
    \( 10^{10} \times 3.086 \times 10^{16} \text{ meters} \)
  • This equals \(3.086 \times 10^{26} \text{ meters} \).
So, astronomers can observe distances up to \(3.086 \times 10^{26} meters\). Converting these distances is vital for subsequent calculations.
Volume of a Sphere
To determine the number of galaxies within a certain observational distance, you need to find the volume of space contained within that distance. This involves calculating the volume of a sphere. The formula for the volume of a sphere is: \( V = \frac{4}{3} \pi r^3 \). Here,
  • \(r\) is the radius of the sphere— in this case, \(3.086 \times 10^{26} \text{ meters} \)
  • Plugging the radius into the formula:
    \( V = \frac{4}{3} \pi (3.086 \times 10^{26})^3 \)
Remember to correctly handle the exponent arithmetic in subsequent steps.
Exponent Arithmetic
Exponent arithmetic is essential for accurately handling astronomical distances and densities. When you multiply exponential terms, remember to add the exponents: \((a \times 10^m) \times (b \times 10^n) = ab \times 10^{m+n}\). For instance, squaring \(3.086 \times 10^{26}\) in the volume calculation involves raising each part of the term:
  • For \((3.086)^3\) and \((10^{26})^3\),
    • \((10^{26})^3 = 10^{78}\)
  • Then, multiply by the other components in the volume formula.
  • Finally, multiply the resulting volume by the galaxy density \(3 \times 10^{-68} \text{ galaxies/m}^3\).
Careful exponent management ensures that your calculations remain accurate and meaningful in the universe's vast scales.

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Most popular questions from this chapter

The Hubble constant is found from the a. slope of the line fit to the data in Hubble's law. b. \(y\) -intercept of the line fit to the data in Hubble's law. c. spread in the data in Hubble's law. d. inverse of the slope of the line fit to the data in Hubble's law.

Go to the Astronomy Picture of the Day app or website (http://apod.nasa.gov/apod) and look at some recent pictures of galaxies. In each case, consider the following questions: Was the picture taken from a large or small telescope; from the ground or from space? Are galaxies in the image face on, edge on, or at an angle? What wavelengths were used for making the image? Are any of the colors "false colors"? If the picture is a combination of images from several telescopes, what do the different colors indicate?

a. Go to the website for the Fermi Gamma-ray Space Telescope (http://fermi.gsfc.nasa.gov). Scroll down to click on “Full News Archive" and look for a story about dark matter. What has this telescope discovered about dark matter? b. Go to the website for the Alpha Magnetic Spectrometer (http://ams02.org), a particle physics detector located on the International Space Station to search for dark matter, including WIMPs. What are the latest results?

Astronomers determine the radius of an AGN by measuring a. how much light comes from it. b. how hard it pulls on stars nearby. c. how quickly its light varies. d. how quickly it rotates.

The spectrum of a distant galaxy shows the H \(\alpha\) line of hydro\(\operatorname{gen}\left(\lambda_{\mathrm{rest}}=656.28 \mathrm{nm}\right)\) at a wavelength of \(750 \mathrm{nm} .\) Assume that \(H_{0}=70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). a. What is the redshift \((z)\) of this galaxy? b. What is its recession velocity \(\left(v_{r}\right)\) in kilometers per second? c. What is the distance of the galaxy in megaparsecs?
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