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Chapter 1: Problem 24

Hubble's constant: Astronomers believe that the universe is expanding and that stellar objects are moving away from us at a radial velocity \(V\) proportional to the distance \(D\) from Earth to the object. a. Write \(V\) as a function of \(D\) using \(H\) as the constant of proportionality. b. The equation in part a was first discovered by Edwin Hubble in 1929 and is known as Hubble's law. The constant of proportionality \(\mathrm{H}\) is known as Hubble's constant. The currently accepted value of Hubble's constant is 70 kilometers per second per megaparsec. (One mega- \(\rightarrow\) parsec is about \(3.085 \times 10^{19}\) kilometers.) With these units for \(H\), the distance \(D\) is measured in megaparsecs, and the velocity \(V\) is measured in kilometers per second. The galaxy G2237 \(+305\) is about \(122.7\) megaparsecs from Earth. How fast is G2237 + 305 receding from Earth? c. One important feature of Hubble's constant is that scientists use it to estimate the age of the universe. The approximate relation is $$ y=\frac{10^{12}}{H} $$ where \(y\) is time in years. Hubble's constant is extremely difficult to measure, and Edwin Hubble's best estimate in 1929 was about 530 kilometers per second per megaparsec. What is the approximate age of the universe when this value of \(H\) is used? d. The calculation in part c would give scientists some concern since Earth is thought to be about \(4.6\) billion years old. What estimate of the age of the universe does the more modern value of 70 kilometers per second per megaparsec give?

Short Answer

Expert verified
a. \( V = H \cdot D \); b. 8590 km/s; c. 1.89 billion years old; d. 14.3 billion years old.

Step by step solution

01

Write Velocity as a Function of Distance

To express the velocity \( V \) as a function of the distance \( D \), we use the proportionality given in Hubble's law. This can be written as: \( V = H \cdot D \). Here, \( H \) is Hubble's constant, which indicates how fast galaxies are receding per unit distance.
02

Calculate Velocity for Galaxy G2237 +305

Given that the distance \( D \) to galaxy G2237 +305 is 122.7 megaparsecs and \( H = 70 \) km/s/Mpc, we can use the formula \( V = H \cdot D \) to find the velocity: \[ V = 70 \text{ km/s/Mpc} \times 122.7 \text{ Mpc} = 8590 \text{ km/s}. \] The galaxy G2237 +305 is receding from Earth at a velocity of 8590 kilometers per second.
03

Estimate Age of Universe with Hubble's 1929 Value

Using the approximate relation \( y = \frac{10^{12}}{H} \), where \( H = 530 \) km/s/Mpc, we calculate the universe's age: \[ y = \frac{10^{12}}{530} \approx 1.89 \times 10^9 \text{ years}. \] This estimate suggests the universe is approximately 1.89 billion years old.
04

Estimate Age of Universe with Modern Hubble Constant

With the modern value \( H = 70 \) km/s/Mpc, we recalculate the age: \[ y = \frac{10^{12}}{70} \approx 1.43 \times 10^{10} \text{ years}. \] This results in the estimated age of the universe being approximately 14.3 billion years, which is consistent with current scientific consensus and older than Earth.

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Key Concepts

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

Expanding Universe
The concept of an expanding universe suggests that the universe has been increasing in size since its formation. This means that galaxies and other stellar objects are receding from one another over time. Imagine blowing up a balloon; as the balloon expands, the dots on the surface (representing galaxies) move apart. This explanation helps us understand why we observe galaxies moving away from each other. The notion of an expanding universe was first introduced by Edwin Hubble in 1929. His observations showed that galaxies are moving away from us, contributing to the conclusion that the universe itself is expanding. This discovery provided a significant turning point in cosmology, shaping our understanding of the universe's large-scale structure.
Hubble's Constant
Hubble's Constant, represented by the letter \( H \), is a crucial component in understanding the expansion of the universe. It quantifies the rate at which the universe is expanding by relating the velocity at which galaxies recede from us to their distance.In mathematical terms, this relationship is expressed as \( V = H \cdot D \), where \( V \) stands for the radial velocity at which a galaxy is moving away, \( D \) is the distance from Earth to the galaxy, and \( H \) is Hubble's Constant. The accepted modern value for Hubble's Constant is 70 kilometers per second per megaparsec. This means that for every megaparsec (roughly 3.085 x 10\(^{19}\) kilometers) of distance, a galaxy appears to move away at a velocity of 70 kilometers per second.Understanding Hubble's Constant helps astronomers estimate not just how fast the universe is expanding, but it also becomes instrumental in calculating other vital cosmological parameters.
Age of the Universe
Estimating the age of the universe relies significantly on Hubble's Constant. Once we understand the rate at which the universe is expanding, we can work backward to deduce when the universe would have theoretically been a single point.The formula used to estimate the age of the universe is \( y = \frac{10^{12}}{H} \), where \( y \) represents the age in years and \( H \) is the Hubble's Constant. For example, if we use the historically overestimated value of \( H = 530 \) km/s/Mpc, we would calculate the age of the universe to be approximately 1.89 billion years. However, with the more accurate modern measurement of \( H = 70 \) km/s/Mpc, we arrive at a more credible age of about 14.3 billion years.The age estimation is fundamental as it aligns with other scientific observations and measurements, confirming that the universe is much older than the Earth itself.
Radial Velocity and Distance Relationship
The relationship between radial velocity and distance is a direct outcome of Hubble's Law, showing how velocity relates to the expansion of the universe. According to this law, the farther away a galaxy is, the faster it is moving away from Earth. This relationship can be observed in the formula \( V = H \cdot D \),
  • \( V \): Radial velocity of the galaxy.
  • \( H \): Hubble's Constant, the rate of expansion.
  • \( D \): Distance from the galaxy to Earth.
For instance, using this formula, we can determine how fast specific galaxies, like galaxy G2237 +305 at 122.7 megaparsecs away, are receding. Substituting into the formula, we find it moves away at 8590 km/s.Understanding this relationship reinforces the concept that not just space, but all distant galaxies are a part of this ongoing cosmic expansion, painting a picture of an ever-changing universe.

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Most popular questions from this chapter

Reynolds number: The Reynolds number is very important in such fields as fluid flow and aerodynamics. In the case of a fluid flowing through a pipe, the Reynolds number \(R\) is given by $$ R=\frac{v d D}{\mu} $$ Here \(v\) is the velocity of the fluid in meters per second, \(d\) is the diameter of the pipe in meters, \(D\) is the density of the fluid in kilograms per cubic meter, and \(\mu\) is the viscosity of the fluid measured in newton- seconds per square meter. Generally, when the Reynolds number is above 2000 , the flow becomes turbulent, and rapid mixing occurs. \({ }^{10}\) When the Reynolds number is less than 2000 , the flow is streamline. Consider a fluid flowing through a pipe of diameter \(0.05\) meter at a velocity of \(0.2\) meter per second. a. If the fluid in the pipe is toluene, its viscosity is \(0.00059\) newton- seconds per square meter, and its density is 867 kilograms per cubic meter. Is the flow turbulent or streamline? b. If the toluene is replaced by glycerol, then the viscosity is \(1.49\) newton-seconds per square meter, and the density is \(1216.3\) kilograms per cubic meter. Is the glycerol flow turbulent or streamline?

A car that gets \(m\) miles per gallon: The cost of operating a car depends on the gas mileage \(m\) that your car gets, the cost \(g\) per gallon of gasoline, and the distance \(d\) that you drive. a. How much does it cost to drive 100 miles if your car gets 25 miles per gallon and gasoline costs 289 cents per gallon? b. Find a formula that gives the cost \(C\) as a function of \(m, g\), and \(d\). Be sure to state the units of each variable. c. Use functional notation to show the cost of driving a car that gets 28 miles per gallon a distance of 138 miles if gasoline costs \(\$ 2.99\) per gallon. Use the formula from part b to calculate the cost.

A rental: A rental car agency charges \(\$ 49.00\) per day and 25 cents per mile. a. Calculate the rental charge if you rent a car for 2 days and drive 100 miles. b. Use a formula to express the cost of renting a car as a function of the number of days you keep it and the number of miles you drive. Identify the function and each variable you use, and state the units. c. It is about 250 miles from Dallas to Austin. Use functional notation to express the cost to rent a car in Dallas, drive it to Austin, and return it in Dallas 1 week later. Use the formula from part b to calculate the cost.

Sales income: The following table shows the net monthly income \(N\) for a real estate agency as a function of the monthly real estate sales \(s\), both measured in dollars. $$ \begin{array}{|c|c|} \hline s=\text { Sales } & N=\text { Net income } \\ \hline 450,000 & 4000 \\ \hline 500,000 & 5500 \\ \hline 550,000 & 7000 \\ \hline 600,000 & 8500 \\ \hline \end{array} $$ a. Make a table showing, for each of the intervals in the table above, the average rate of change in \(N\). What pattern do you see? b. Use the average rate of change to estimate the net monthly income for monthly real estate sales of \(\$ 520,000\). In light of your answer to part a, how confident are you that your estimate is an accurate representation of the actual income? c. Would you expect \(N\) to have a limiting value? Be sure to explain your reasoning.

Radioactive substances change form over time. For example, carbon 14, which is important for radiocarbon dating, changes through radiation into nitrogen. If we start with 5 grams of carbon 14 , then the amount \(C=C(t)\) of carbon 14 remaining after \(t\) years is given by $$ C=5 \times 0.5^{t / 5730} $$ a. Express the amount of carbon 14 left after 800 years in functional notation, and then calculate its value. b. How long will it take before half of the carbon 14 is gone? Explain how you got your answer. (Hint: You might use trial and error to solve this, or you might solve it by looking carefully at the exponent.)
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