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Chapter 26: Problem 43

What would be the critical density of matter in the universe \(\left(\rho_{c}\right)\) if the value of the Hubble constant were (a) \(50 \mathrm{~km} / \mathrm{s} / \mathrm{Mpc}\) ? (b) \(100 \mathrm{~km} / \mathrm{s} / \mathrm{Mpc}\) ?

Short Answer

Expert verified
The critical density of the universe for (a) 50 \(\mathrm{km/s/Mpc}\) is \(2.96 \times 10^{-26} kg m^{-3}\) and for (b) 100 \(\mathrm{km/s/Mpc}\) is \(1.186 \times 10^{-25} kg m^{-3}\).

Step by step solution

01

Understanding the concept and formula

The critical density is the measure of minimum required density of matter in the universe to halt the expansion and make the universe flat. It is calculated using the formula \(\rho_{c} = \frac{3H^2}{8\pi G}\), where \(H\) is the Hubble's constant and \(G\) is the Gravitational constant of \(6.67 \times 10^{-11} m^3 kg^{-1} s^{-2}\).
02

Conversion of units

The given Hubble constant needs to be converted into the same base units as gravitational constant for the correct calculations. 1 Mpc is approximately equal to \(3.086 \times 10^{19}\) km. Hence, (a) 50 km/s/Mpc = \(50/(3.086 \times 10^{19}) = 1.62 \times 10^{-18} s^{-1}\) and (b) 100 km/s/Mpc = \(100/(3.086 \times 10^{19}) = 3.24 \times 10^{-18} s^{-1}\).
03

Calculating Critical Density

Substitute the values of \(H\) and \(G\) in the given formula to calculate \(\rho_c\). For (a) \(\rho_{c} = \frac{(3)(1.62 \times 10^{-18})^{2}}{8\pi(6.67 \times 10^{-11})}= 2.96 \times 10^{-26} kg m^{-3}\) and for (b) \(\rho_{c} = \frac{(3)(3.24 \times 10^{-18})^{2}}{8\pi(6.67 \times 10^{-11})}= 1.186 \times 10^{-25} kg m^{-3}\)

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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 as \(H\), has been a pivotal element in cosmology since its discovery. It represents the rate at which the universe is expanding by relating the velocity at which a galaxy is moving away from us to its distance.

Specifically, it provides us with a simple yet profound equation: \( v = H \times d \), where \(v\) is the velocity of the galaxy, \(H\) is the Hubble constant, and \(d\) is the distance of the galaxy from us. Thus, a higher value of \(H\) suggests a faster expansion rate. This knowledge helps us estimate the age of the universe, the distances to faraway galaxies, and subsequently, their movements over time.
Gravitational Constant
The gravitational constant, denoted by \(G\), is a cornerstone of Newton's law of universal gravitation. It is a key component in the equation \(F = G\frac{m_1m_2}{r^2}\), which allows us to calculate the force of gravity between two masses, \(m_1\) and \(m_2\), separated by a distance \(r\).

This fundamental constant is also crucial for understanding the critical density of the universe. Given that \(G\) has a fixed value of approximately \(6.67 \times 10^{-11} m^3 kg^{-1} s^{-2}\), it provides an invariant measure in the scaling and calculation of celestial mechanics, including the orbits of planets and the structure of galaxies.
Expansion of the Universe
The universe is in a state of continuous expansion, a notion first introduced by Edwin Hubble, giving rise to the term 'Hubble expansion'. Observations across astrophysics, including the redshift of distant galaxies, support this groundbreaking concept.

Here, the idea is that the farther away a galaxy is from us, the faster it appears to be moving away. This apparent velocity is not due to the galaxy's own motion, but rather the expansion of the space itself. Recognizing this expansion is fundamental to determining various cosmological parameters, including the critical density that delineates whether the universe will expand forever, stabilize, or collapse.
Cosmology
Cosmology is a branch of astrophysics focused on understanding the origins, evolution, structure, and eventual fate of the universe. It encompasses a wide range of studies from the cosmic microwave background radiation to the distribution of galaxies, dark matter, and dark energy.

Cosmologists employ theoretical models, grounded in Einstein's general theory of relativity, as well as observational data to describe the universe on its grandest scales. Calculating critical density is one aspect of cosmology that ties into models predicting the ultimate future of the cosmos, whether it be in continued expansion, a big freeze, a big rip, or a big crunch.
Astrophysics
Astrophysics is the scientific discipline that employs the principles of physics to explain the phenomena observed in outer space. Here, we explore not just stars and planets, but also more exotic constituents such as black holes, neutron stars, and cosmic strings.

From the cosmic scale down to subatomic particles, astrophysics provides a framework for understanding the physical processes that govern the universe. Critical density ties into astrophysics by way of determining the balance of gravitational forces and the expansive pressure exerted by dark energy, these such subtleties reveal much about the future dynamics of the cosmos.

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Most popular questions from this chapter

Temperatures in the Early Universe. Access the Active Integrated Media Module "Blackbody Curves" in Chapter 26 of the Universe Web site or eBook. (a) Use the module to determine by trial and error the temperature at which a blackbody spectrum has its peak at a wavelength of \(1 \mu \mathrm{m}\). (b) At the time when the temperature of the cosmic background radiation was equal to the value you found in (a), was the universe matter-dominated or radiation- dominated? Explain your answer.

Why did Isaac Newton conclude that the universe was static? Was he correct?

Can you see the cosmic background radiation with the naked eye? With a visible-light telescope? Explain why or why not.

Use Starry Night Enthusiast \({ }^{\mathrm{TM}}\) to compare the distances of objects in the Tully Database with the radius of the Cosmic Light Horizon, the limit of our observable universe. As you will find, the most distant galaxies in this database are a long way away from the Earth and yet these distances are only a small fraction of the distances from which we can see light in our universe. Select Favourites \(>\) Deep Space \(>\) Tully Database to display this collection of galaxies in their correct 3-dimensional positions in space around our position. Stop Time and click on View \(>\) Feet to remove the image of the astronaut's suit from the view. Select Preferences from the File menu (Windows) or the Starry Night Enthusiast menu (Macintosh). In the Preferences dialog, select Cursor Tracking (HUD) in the drop-down box and ensure that Distance from observer, Name and Object type are selected. The view shows the boundaries of the Tully database as a cube. Use the location scroller (hold down the Shift key and mouse button while moving the mouse) to rotate the cube to allow you to choose galaxies on the outer fringes of this space. Use the Hand Tool to examine a selection of the furthest objects from the Earth, which is centered in the view, and write a list of \(10-20\) objects, noting the \(\mathbf{O b}\) ject type and Distance from observer. (a) In your sample, is there a predominance of any one kind of galaxy? If so, what type of galaxy appears to be most common at these distances? (b) Select the furthest of these galaxies and compare their distances with the radius of the cosmic light horizon. What fraction of the radius of the observable universe is covered by the Tully database?

Would it be correct to say that due to the expansion of the universe, the Earth is larger today than it was \(4.56\) billion years ago? Why or why not?
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