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Chapter 19: Problem 44

The Hubble time \(\left(1 / H_{0}\right)\) represents the age of a universe that has been expanding at a constant rate since the Big Bang. Assuming an \(H_{0}\) value of \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\) and a constant rate of expansion, calculate the age of the universe in years. How is the age different if \(H_{0}=75 \mathrm{km} / \mathrm{s} / \mathrm{Mpc} ?\left(\text { Note: } 1 \text { year }=3.16 \times 10^{7}\right.\) seconds and \(1 \mathrm{Mpc}=3.09 \times 10^{19} \mathrm{km} .\)

Short Answer

Expert verified
For \(H_0=70 \ \text{km/s/Mpc}, the age of the universe is 1.40 \times 10^{10} \ \text{years}. For \(H_0=75 \ \text{km/s/Mpc}, it is 1.30 \times 10^{10} \ \text{years}.

Step by step solution

01

Understand the Hubble Constant

The Hubble Constant, denoted by \(H_0\), describes the rate at which the universe is expanding. It is given in units of \(\text{km} / \text{s} / \text{Mpc}\).
02

Convert Hubble Constant to Compatible Units

Convert \(H_0\) from \(70 \ \text{km} / \ \text{s} / \ \text{Mpc}\) to \(\text{s}^{-1}\): Use the conversion \(1 \text{Mpc} = 3.09 \times 10^{19} \ \text{km}\) to find \[ H_0 = \frac{70 \ \text{km}}{\text{s} \ \times 3.09 \times 10^{19} \ \text{km}} = 2.27 \times 10^{-18} \ \text{s}^{-1} \] Repeat for \(H_0 = 75 \ \text{km} / \ \text{s} / \ \text{Mpc}\) to find \[ H_0 = \frac{75 \ \text{km}}{\text{s} \ \times 3.09 \times 10^{19} \ \text{km}} = 2.43 \times 10^{-18} \ \text{s}^{-1} \]
03

Calculate the Hubble Time

Use the formula for Hubble Time: \( t_H = \frac{1}{H_0} \). For \(H_0 = 2.27 \times 10^{-18} \ \text{s}^{-1}\): \[ t_H = \frac{1}{2.27 \times 10^{-18}} = 4.41 \times 10^{17} \ \text{s} \] For \(H_0 = 2.43 \times 10^{-18} \ \text{s}^{-1}\): \[ t_H = \frac{1}{2.43 \times 10^{-18}} = 4.12 \times 10^{17} \ \text{s} \]
04

Convert Hubble Time to Years

To convert seconds to years, use the conversion \(1 \ \text{year} = 3.16 \times 10^7 \ \text{s}\). For \(H_0 = 70 \ \text{km} / \ \text{s} / \ \text{Mpc}\), \[ t_H = \frac{4.41 \times 10^{17}}{3.16 \times 10^7} = 1.40 \times 10^{10} \ \text{years} \] For \(H_0 = 75 \ \text{km} / \ \text{s} / \ \text{Mpc}\), \[ t_H = \frac{4.12 \times 10^{17}}{3.16 \times 10^7} = 1.30 \times 10^{10} \ \text{years} \]

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Key Concepts

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

Hubble Constant
The Hubble Constant, denoted by \(H_0\), is a crucial value in cosmology. It indicates the rate of expansion of the universe. This means it tells us how quickly galaxies are moving away from each other. The standard unit for \(H_0\) is \(\text{km/s/Mpc}\). This stands for kilometers per second per megaparsec. Here, 1 megaparsec (Mpc) is a distance unit equal to about 3.09 x 10^{19} km. Most recent estimates of \(H_0\) range between 70 and 75 \(\text{km/s/Mpc}\), depending on different methods of measurement. Understanding \(H_0\) helps us measure the age and size of the universe.
Universe Expansion
The universe is expanding, a concept first proposed by Edwin Hubble in the 1920s. This expansion means that galaxies are moving away from each other. The Hubble Constant \(H_0\) directly measures this expansion rate. More precisely, it connects the velocity at which a galaxy moves away to its distance from us. The farther away a galaxy is, the faster it appears to move. This discovery has profound implications, showing the universe has been expanding since the Big Bang. This expansion helps answer fundamental questions about the past and future of our cosmos.
Age of the Universe
Calculating the age of the universe involves understanding the Hubble time \( \left(1 / H_{0} \right) \). If we assume the universe has been expanding at a constant rate since the Big Bang, this can give us an estimate of its age. For instance, using \(H_{0} = 70 \text{km/s/Mpc}\), we convert it to compatible units (like seconds) to get the Hubble time. The age comes out to be around 1.40 x 10^{10} years. If \(H_{0}\) is 75 \text{km/s/Mpc}, the age is slightly lower, around 1.30 x 10^{10} years. These calculations provide vital clues in cosmology.
Cosmology
Cosmology is the scientific study of the universe as a whole, its structure, origin, and evolution. Key concepts in cosmology include the Big Bang, cosmic inflation, dark matter, and dark energy. The Hubble Constant \(H_0\) plays a central role in cosmology, offering a means to estimate the universe's size and age. Current cosmological models suggest the universe is around 13.8 billion years old. Maps of cosmic microwave background radiation and galaxy surveys help refine our cosmological models. Cosmology bridges astronomy, particle physics, and general relativity to explain the universe's vast phenomena.
Unit Conversion
Unit conversion is essential in calculating cosmological values. For instance, converting the Hubble Constant from \(\text{km/s/Mpc}\) to \(\text{s}^{-1}\) involves specific steps. First, recognize that 1 Mpc equals 3.09 x 10^{19} km. Given \(H_0 = 70 \text{km/s/Mpc}\), we convert it by dividing 70 km/s by 3.09 x 10^{19} km, resulting in \(H_0 \approx 2.27 x 10^{-18} \text{s}^{-1}\). This conversion helps in calculating the Hubble time. Similarly, converting time from seconds to years uses the relation \(1 \text{year} = 3.16 x 10^7 \text{s}\). Such conversions allow accurate determinations of the universe's properties.

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Most popular questions from this chapter

In astronomy, isotropy means that the universe is the same a. in all locations. b. in all directions. c. at all times. d. at all size scales.

Assume that the most distant galaxies have a redshift \(z=10\) The average density of normal matter in the universe today is \(4 \times 10^{-28} \mathrm{kg} / \mathrm{m}^{3} .\) What was its density when light was leaving those distant galaxies? (Hint: Keep in mind that volume is proportional to the cube of the scale factor.)

The temperature of the CMB is \(2.73 \mathrm{K}\). What is the peak wavelength of its Planck blackbody spectrum expressed both in microns and in millimeters?

Why is it not possible to use the measured radial velocities of nearby galaxies, such as the Andromeda Galaxy, to evaluate the Hubble constant \(\left(H_{0}\right) ?\)

When they look into the universe, astronomers observe that nearly all galaxies are moving away from the Milky Way. This observation suggests that a. the Milky Way is at the center of the universe. b. the Milky Way must be at the center of the expansion. c. the Big Bang occurred at the location of the Milky Way. d. an observer in a distant galaxy would make the same observation.
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