Question

Use the concepts in this chapter to obtain an estimate for the number of atoms in the universe. Make the following assumptions: (a) All of the atoms in the universe are hydrogen atoms in stars. ( This is not a ridiculous assumption because over three-fourths of the atoms in the universe are in fact hydrogen. Gas and dust between the stars represent only about 15\(\%\) of the visible matter of our galaxy, and planets compose a far smaller fraction.) (b) The sun is a typical star composed of pure hydrogen with a density of 1.4 \(\mathrm{g} / \mathrm{cm}^{3}\) and a radius of \(7 \times 10^{8} \mathrm{m}\). (c) Each of the roughly 100 billion stars in the Milky Way galaxy contains the same number of atoms as our sun.(d) Each of the 10 billion galaxies in the visible universe contains the same number of atoms as our Milky Way galaxy.

Short answer

The estimated number of atoms in the universe is the result of the product: number of atoms in the sun \( \times \) number of stars in the Milky Way galaxy \( \times \) number of galaxies in the visible universe.

Step-by-step solution

Calculate the volume of the sun

The volume of the sun can be found using the formula for the volume of a sphere, which is \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius of the sphere. For the sun, use the radius given \( r = 7 \times 10^8 \) meters.

Calculate the mass of the sun

To find the mass of the sun, you multiply the volume calculated in Step 1 by the density of hydrogen (\( \rho = 1.4 \mathrm{g} / \mathrm{cm}^{3} \) or \( 1.4 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3} \) since there are \( 10^3 \mathrm{cm}^{3} \) in \( 1 \mathrm{m}^{3} \) and \( 10^3 \mathrm{g} \) in \( 1 \mathrm{kg} \) ). The mass of the sun (\( M \) ) can be calculated by \( M = \rho \times V \) .

Calculate the number of atoms in the sun

The number of atoms in the sun is determined by dividing the mass of the sun by the mass of one hydrogen atom. The mass of one hydrogen atom is approximately \( 1.67 \times 10^{-27} \mathrm{kg} \) . Thus, the number of hydrogen atoms \( N \) in the sun is \( N = \frac{M}{m_{\text{hydrogen}}} \) .

Calculate the number of atoms in the Milky Way galaxy

Assuming each star has the same number of atoms as our sun, we multiply the number of atoms in the sun by the number of stars in the Milky Way galaxy, approximately 100 billion (\( 10^{11} \)).

Calculate the number of atoms in the visible universe

Finally, to estimate the number of atoms in the universe, multiply the number of atoms in the Milky Way galaxy by the number of galaxies in the visible universe (10 billion or \( 10^{10} \) ).

Key concepts

Volume of a Sphere

Beyond its geometric elegance, understanding the volume of a sphere is crucial in multiple scientific areas, including estimating stellar magnitudes like the sun. The formula to calculate the volume of a sphere is \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius, and \( \pi \) is a constant (approximately 3.14159). For giant celestial bodies like the Sun, with a radius of about \( 7 \times 10^8 \), meters calculating the volume allows us to progress further in determining the number of atoms contained within such a massive space.

When we apply the formula to the sun's radius, we obtain a solar volume immense to fathom, but essential to estimate the sun's constitution and, by extension, that of other spheres in the universe.

Mass of the Sun

The mass of the sun plays a pivotal role in the structure and dynamics of our solar system. To calculate the sun's mass, we use the volume derived from the volume formula, and then apply the formula \( M = \rho \times V \), wherein \( \rho \) represents the density. The sun's density, primarily being that of hydrogen (\( 1.4 \mathrm{g} / \mathrm{cm}^{3} \) or \( 1.4 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3} \)), allows us to evaluate the total mass.

Knowing this mass is essential not just for our solar system's physics, but for comparing our sun to other stars within the Milky Way galaxy, and consequently assisting in our understanding of the stellar mass distribution across the galaxy.

Number of Hydrogen Atoms

Hydrogen, being the simplest and most abundant element in the universe, forms the core of astronomical mass calculations. One hydrogen atom has a mass of approximately \( 1.67 \times 10^{-27} \mathrm{kg} \). By dividing the mass of the sun by the mass of a hydrogen atom, we can determine the astounding quantity of hydrogen atoms present in a star.

Comprehending the quantity of hydrogen atoms within a star has deep implications in astrophysics, shedding light on the processes of nuclear fusion and the lifecycle of stars. This knowledge also scales up, informing us on the chemical composition of galaxies themselves.

Milky Way Galaxy Stars

Estimating the constituents of the Milky Way galaxy allows astronomers to grasp the vastness and composition of the cosmos. Our galaxy alone has an estimated 100 billion stars, each thought to hold as many hydrogen atoms as our sun. By determining the number of atoms in our sun and multiplying it by the number of stars, scientists approximate the total number of atoms in the Milky Way.

This calculation helps in understanding not just our galaxy's matter distribution but also offers a template for approximating other galaxies' mass in the observable universe—a key step in measuring the cosmic scale.

Atoms in the Visible Universe

The atoms in the visible universe amount to a figure beyond human comprehension, yet can be estimated by combining our knowledge of the volume of a sphere, the mass of the sun, and the number of hydrogen atoms. By extrapolating the data from our Milky Way to the estimated 10 billion galaxies in the visible universe, astronomers can make informed estimates about the quantity of matter that constitutes the cosmos.

The endeavor to quantify the atoms in the universe encapsulates not only the scientific urge to understand the cosmos but also the mathematical beauty of using fundamental principles to reach for the stars, literally.