Problem 40 - Astronomy: At Play in the Cosmos

Question

The temperature in a region of coronal gas is \(10^{6} \mathrm{K},\) while a nearby H II region has a temperature of \(10,000 \mathrm{K}\). Assuming the pressure of the two regions is equal, what is the ratio of the density of the coronal gas to that of the H II region?

Step-by-Step Solution

Answer

The density ratio \( \frac{n_1}{n_2} \) is 0.01.

Step 1: Identify the Formula for Pressure

Pressure in a gas can be calculated using the ideal gas law, given by the formula \( P = nkT \), where \( P \) is the pressure, \( n \) is the number density of particles, \( k \) is Boltzmann's constant, and \( T \) is the temperature in Kelvin.

Step 2: Set the Pressures Equal

Since the problem states that the pressures of the two regions are equal, we can set their expressions equal to each other: \( n_1 k T_1 = n_2 k T_2 \). Here, \( n_1 \) and \( T_1 \) are the density and temperature of the coronal gas, while \( n_2 \) and \( T_2 \) are for the H II region.

Step 3: Simplify the Equation

The constant \( k \) (Boltzmann's constant) appears on both sides of the equation and thus cancels out, simplifying it to \( n_1 T_1 = n_2 T_2 \).

Step 4: Solve for the Density Ratio

Rearrange the equation \( n_1 T_1 = n_2 T_2 \) to find the ratio of densities: \( \frac{n_1}{n_2} = \frac{T_2}{T_1} \).

Step 5: Substitute Known Values

Substitute the given temperatures into the equation: \( \frac{n_1}{n_2} = \frac{10,000}{10^{6}} \).

Step 6: Calculate the Density Ratio

Perform the calculation: \( \frac{n_1}{n_2} = \frac{10,000}{1,000,000} = 0.01 \).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation used to relate different properties of gases, namely pressure, volume, and temperature. It is represented by the formula \( PV = nRT \), or in terms of pressure per molecule, \( P = nkT \). Here:

• \( P \) stands for pressure, the force exerted by the gas molecules within a defined space.
• \( n \) is the number density of particles, essentially the number of molecules or atoms per unit volume.
• \( k \) is Boltzmann's constant, crucial for converting temperature into energy units.
• \( T \) is the temperature of the gas, expressed in Kelvin.

Using this law, we can determine how changes in temperature affect pressure or density, providing key insight into gaseous behavior in various environments.
In scenarios where temperature or particle density change, this law helps maintain or evaluate physical balance.

Coronal Gas

Coronal gas refers to extremely hot, ionized gas found in the outer atmosphere of stars, including the Sun's corona. It typically exists at temperatures around \(10^{6} \) Kelvin or higher.
At these temperatures:

• Atoms in the gas are fully ionized, meaning electrons are stripped away from their nuclei, creating a plasma of ions and electrons.
• This gas emits X-rays, a property that astronomers use to study its characteristics.

The Sun's corona is a well-studied example, where its high temperature and low density cause fascinating phenomena such as solar flares and coronal mass ejections.
Understanding coronal gas helps scientists learn more about solar activity and its impact on space weather, which affects Earth's magnetosphere.

H II Region

An H II region is a cloud of ionized hydrogen found around young, hot stars. The ionizing radiation from these stars strips electrons from hydrogen atoms, thus creating a region filled with free protons and electrons. These regions have

• Temperatures around \(10,000 \) Kelvin.
• Bright emissions visible across various wavelengths due to recombination of electrons and protons.

They are indicators of recent star formation and can often be found in spiral arms of galaxies.
Studying H II regions provides valuable insight into the lifecycle of stars and the processes involved in cosmic evolution.

Pressure Equilibrium

Pressure equilibrium is the state where the pressure in one area is equal to the pressure in another, causing no net movement of matter between the two regions. In astrophysical scenarios:

• It is crucial for maintaining stability across different environments, such as across different layers of a star's atmosphere.
• In our exercise, the coronal gas and the H II region have equal pressure, meaning their physical causes balance each other's outward and inward forces.

Using the ideal gas law helps astronomers assess and compare these pressures by analyzing the relation between temperature and density, as in calculating the density ratio between two gases.
This understanding aids in studying large-scale structures and behavior of gases in astronomical bodies and interstellar space.