It is estimated that the Coma cluster (see Figure 24-21) contains about \(10^{13} \mathrm{M}_{\odot}\) of intracluster gas. (a) Assuming that this gas is made of hydrogen atoms, calculate the total number of intracluster gas atoms in the Coma cluster. (b) The Coma cluster is roughly spherical in shape, with a radius of about \(3 \mathrm{Mpc}\). Calculate the number of intracluster gas atoms per cubic centimeter in the Coma cluster. Assume that the gas fills the cluster uniformly. (c) Compare the intracluster gas in the Coma cluster with the gas in our atmosphere \(\left(3 \times 10^{19}\right.\) molecules per cubic centimeter, temperature \(300 \mathrm{~K}\) ); a typical gas cloud within our own Galaxy (a few hundred molecules per cubic centimeter, temperature \(50 \mathrm{~K}\) or less); and the corona of the Sun \(\left(10^{5}\right.\) atoms per cubic centimeter, temperature \(10^{6} \mathrm{~K}\) ).

Step by step solution

01

Calculating Total Number of Gas Atoms

Given that the Coma Cluster contains about \(10^{13} M_{\odot}\) of intracluster gas. To convert \(M_{\odot}\) to kilograms, use the solar mass constant, \(1 M_{\odot} = 1.989 \times 10^{30}\) kg. Therefore, the mass of the intracluster gas in kilograms is \(10^{13} M_{\odot} = 10^{13} \times (1.989 \times 10^{30})\) kg. To compute the number of hydrogen atoms, divide this mass by the mass of a hydrogen atom (\(1.67 \times 10^{-27}\) kg): \(\frac{10^{13} \times (1.989 \times 10^{30})} {1.67 \times 10^{-27}}\).

02

Calculating Gas Atoms per Cubic Centimeter

Given that the radius of the Coma Cluster is 3 Mpc (1 parsec (\(pc\)) = \(3.09 \times 10^{16}\) m = \(3.09 \times 10^{18}\) cm), the volume of the cluster, treated as a sphere, can be calculated by the formula \(\frac{4}{3}\pi r^{3}\). After calculating the volume in cubic centimeters, divide the total number of gas atoms (calculated in step 1) by the volume to find the number of gas atoms per cubic centimeter.

03

Comparing Intracluster Gas Data with Other Celestial Bodies

To make the comparison, simply list down the calculated value from step 2 side-by-side with the given densities of gas in our atmosphere, gas cloud within our galaxy and the corona of the sun. Don't forget to take note of the temperatures that come with each entity for a more rigorous comparison.

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Key Concepts

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

Coma Cluster

The Coma Cluster is an immense collection of over a thousand galaxies located approximately 300 million light-years from Earth. As a crucial part of our understanding of galactic structures, the Coma Cluster serves as an excellent example of a large-scale cosmic system. Within the cluster, there is a wealth of intracluster gas, which is a thin plasma that fills the space between the galaxies.

This gas is predominantly made up of hydrogen atoms—the simplest and most abundant element in the universe. The mass of the cluster's intracluster gas is estimated to be about \(10^{13} \mathrm{M}_{\odot}\), where \(\mathrm{M}_{\odot}\) signifies the solar mass constant. Understanding this key aspect of the Coma Cluster aids us not only in grasping the structure of one of the nearest large galaxy clusters but also illuminates the behavior and composition of similar galactic structures throughout the cosmos.

Hydrogen Atom Mass

The mass of a hydrogen atom plays a pivotal role in astronomical calculations. A hydrogen atom has an approximate mass of \(1.67 \times 10^{-27}\) kilograms, a value that might seem inconsequential on the human scale but becomes immensely significant when dealing with the stupendous quantities of matter in space.

Key Role in Astronomical Calculations

In calculating the total number of hydrogen atoms in an entity such as the Coma Cluster's intracluster gas, the mass of a single hydrogen atom is necessary for converting from total mass to number of atoms.

This conversion, from the solar mass constant to the mass of individual hydrogen atoms, allows astronomers to determine the extensive number of hydrogen atoms present in vast structures like the Coma Cluster.

Solar Mass Constant

Astronomical calculations often involve the solar mass constant, denoted as \(\mathrm{M}_{\odot}\), which represents the mass of the Sun. This constant is approximately \(1.989 \times 10^{30}\) kilograms. It provides a baseline for measuring masses on an astronomical scale, like those of stars, galaxies, and clusters.

Use in Textbook Exercise

The solar mass constant was utilized in the textbook exercise to convert the mass of the Coma Cluster's intracluster gas, stated in solar masses, to kilograms. This conversion facilitates further calculations, as different astronomical objects can be compared and analyzed in a consistent unit of mass. The exercise illustrates how crucial and common the use of solar masses is to understand and describe the myriad celestial entities.

Astronomical Calculations

Astronomical calculations involve quantifying the incomprehensibly large distances, masses, and other physical properties of celestial bodies. These calculations often require a fundamental understanding of physics and apply constants such as the hydrogen atom mass and the solar mass.

Volume and Density in the Coma Cluster

Specifically, when calculating the density of intracluster gas in the Coma Cluster, these constants have allowed us to derive the number of hydrogen atoms per cubic centimeter.

This involves not only converting mass to the number of atoms but also determining the vast spatial volume of the spherical cluster. By combining distance measurements, such as megaparsecs (Mpc), with volume formulas for spheres, the problem underlines the importance of mathematical application in astronomical studies.

Galactic Structures

Galactic structures such as the Coma Cluster are composed of not only stars and planets but also of interstellar and intracluster material. This includes gas, dust, and dark matter.

Majestic Cosmic Scale

The sheer scope of these structures is difficult to imagine, but it becomes more tangible through the lens of scientific computation. The textbook exercise that computes the density of intracluster gas within the spherical confines of the Coma Cluster provides insight into the mass distribution within such vast cosmic conglomerations.

Comparing the density of intracluster gas with other types of celestial bodies, as seen in the exercise, allows us to appreciate the different environments that exist in our universe, from the dense atmosphere of Earth to the rarified coronas of stars and the vast stretches of space that constitute galactic structures.