Problem 7 - Astronomy Today

Question

For a Hubble constant of \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\), the critical density is \(9 \times 10^{-23} \mathrm{kg} / \mathrm{m}^{3} .\) (a) How much mass does this correspond to within a volume of 1 cubic astronomical unit? (b) How large a cube would be required to enclose 1 Earth mass of material?

Step-by-Step Solution

Answer

(a) About \(3.02 \times 10^{11} \mathrm{kg}\) of mass in 1 AU cube. (b) Cube side length is \(4.02 \times 10^{15} \mathrm{m}\) for Earth mass.

Step 1: Understanding the Value of 1 AU in SI Units

An Astronomical Unit (AU) is the average distance from the Earth to the Sun. In meters, this is approximately\(1 \text{ AU} = 1.496 \times 10^{11} \text{ m}\).

Step 2: Determining Volume of 1 AU Cubed

The volume of a cube with side length of 1 AU is found by cubing the side length: \[V = (1.496 \times 10^{11})^3 \, \text{m}^3\]. Calculating this gives \(V \approx 3.35 \times 10^{33} \, \text{m}^3\).

Step 3: Calculating Mass Inside 1 AU Cubic Volume

Using the volume calculated and the critical density, the mass can be determined using the formula \(M = \text{density} \times \text{volume}\). Substituting the given values gives \(M = 9 \times 10^{-23} \, \text{kg} / \text{m}^3 \times 3.35 \times 10^{33} \, \text{m}^3\), resulting in \(M \approx 3.02 \times 10^{11} \, \text{kg}\).

Step 4: Understanding Earth's Mass in Kilograms

Earth's mass is approximately \(5.972 \times 10^{24} \, \text{kg}\).

Step 5: Finding Volume Required for Earth's Mass

The required volume to enclose a mass equal to Earth's uses \(V = \frac{\text{Earth's mass}}{\text{density}}\). Substitute the known values: \(V = \frac{5.972 \times 10^{24} \, \text{kg}}{9 \times 10^{-23} \, \text{kg} / \text{m}^3}\), leading to \(V = 6.63 \times 10^{46} \, \text{m}^3\).

Step 6: Calculating Side Length of the Cube

To find the side length of the cube with this volume, calculate \(L = V^{1/3}\). Performing the calculation gives \(L = (6.63 \times 10^{46})^{1/3} \, \text{m} \approx 4.02 \times 10^{15} \, \text{m}\).

Key Concepts

Hubble Constant

The Hubble Constant is a key concept in cosmology, defining the rate at which the universe is expanding. It is denoted by the symbol \( H_0 \) and typically measured in kilometers per second per megaparsec (\( ext{km/s/Mpc} \)). This essentially means how fast galaxies are moving away from each other, per unit distance. For example, a Hubble Constant of \( 70 \text{ km/s/Mpc} \) indicates that for every megaparsec (approximately 3.26 million light-years) between two points, the space is expanding by 70 kilometers per second. - It helps us understand the universe's age and size. - It provides insight into the universe's ultimate fate, whether it will keep expanding, slow, or eventually collapse. Cosmologists refine the exact value of the Hubble Constant with precise measurements from various methods, like observing supernovae or the cosmic microwave background radiation.

Astronomical Unit

An Astronomical Unit (AU) is a measure of distance that's crucial for understanding the vast scales within our solar system. It stands for the average distance between the Earth and the Sun, approximately \( 1.496 imes 10^{11} \) meters. This unit provides a convenient way of expressing distances without excessive calculations and is widely used for:- Planetary distances in our solar system.- A basis for scaling measurements beyond the solar system. One light year is about \( 63,240 \) AU, and though many cosmic distances are measured in light-years, AU gives a grounded scale for understanding solar system proximities.

Earth Mass

The mass of Earth is a fundamental constant used in many scientific calculations. Approximately equal to \( 5.972 imes 10^{24} \) kilograms, it provides a basis for calculating gravity, orbits, and other planetary conditions. This mass helps us explore:- The gravitational interaction with the Moon and other celestial bodies.- Calculating the gravitational pull it has, which keeps our atmosphere in place and allows objects to orbit Earth. In cosmology, Earth's mass becomes a reference point when examining larger cosmic structures, including calculations of density and accumulation of mass over spaces, like in the step-by-step problem discussed.

Cosmology Calculations

Cosmology calculations are a set of mathematical operations that help understand the universe's large-scale structure, dynamics, and history. These calculations often involve: - Determining the universe's critical density, due to its significance in understanding whether the universe will expand forever or collapse back. - Using formulas that relate density, volume, and mass to understand various celestial phenomena. For example, to calculate the mass within a unit of space, we often multiply the space's volume by its density, like the critical density calculation in our exercise. Such cosmology calculations can empower scientists to predict scenarios and validate theoretical models about the universe's future, making them foundational to scientific inquiry in cosmology.