# Astronomy: Local Thermodynamic Equilibrium, Gaunt Factor

• Rokon117
This is due to the higher frequency of photons at the Balmer limit, resulting in a larger absorption cross-section. In summary, the problem involves showing that a hydrogen gas cloud at 10000K is twice as optically thick at the Balmer limit compared with the Paschen limit, using the equations an(f)=[64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n) and hf1/k=1.578*10^5K. This is due to the higher frequency of photons at the Balmer limit, resulting in a larger absorption cross-section.
Rokon117
1. The problem statement, all variables and given/known

## Homework Equations

show that a hydrogen gas cloud at 10000K is about twice as optically thick at the Balmer limit compared with the Paschen limit. Assume that the H atoms are in Local Thermodynamic Equilibrium, the Gaunt factor is unity at all energy levels, and include absorption by H to levels up to n=10. Note that hf1/k=1.578*10^5K for lyman limit photon of frequency f1.

## The Attempt at a Solution

an(f)=[64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)
gbf(n)=1
sorry just don't know how to type those symbols

.a0=0.0529nmfn=1.5809*10^15 Hz at Balmer limitfp=7.9579*10^14 Hz at Paschen limitn=1 to 102. Relevant equationsan(f)=[64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)3. The attempt at a solutionan(f) at Balmer limit = [64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)= [64/(3*3^0.5) A] (pi 0.0529nm^2)n(1.5809*10^15 Hz/1.578*10^5K)^3 1= 8.106*10^-19 m^2an(f) at Paschen limit = [64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)= [64/(3*3^0.5) A] (pi 0.0529nm^2)n(7.9579*10^14 Hz/1.578*10^5K)^3 1= 4.049*10^-19 m^2Therefore, the hydrogen gas cloud at 10000K is about twice as optically thick at the Balmer limit compared with the Paschen limit.

## 1. What is Local Thermodynamic Equilibrium (LTE) in astronomy?

Local Thermodynamic Equilibrium (LTE) in astronomy refers to a state in which the temperature and density of a gas are uniform throughout a volume, and the gas is in thermal equilibrium with its surroundings. This is an important concept in understanding the physical processes that occur in stars and other astronomical objects.

## 2. How does LTE affect the formation of spectral lines?

In LTE, the gas particles are in constant collisions with each other, causing them to reach a balance between the rates of emission and absorption of radiation. This results in the formation of spectral lines with specific intensities and wavelengths, which can be used to study the composition and physical properties of celestial objects.

## 3. What is the Gaunt factor in astrophysics?

The Gaunt factor is a dimensionless factor that accounts for the deviation of spectral line intensities from their expected values in LTE. It takes into account the effects of quantum mechanics and relativity on the emission and absorption of radiation, and is often used in calculations of spectral line intensities in astrophysics.

## 4. How is the Gaunt factor calculated?

The Gaunt factor is calculated using a formula that takes into account the temperature, density, and electron number density of the gas, as well as the energy levels and transition probabilities of the spectral line being studied. It is also affected by the ionization state and magnetic field of the gas.

## 5. What role does the Gaunt factor play in astrophysical research?

The Gaunt factor is an important factor in understanding the physical processes that occur in stars and other astronomical objects. It is used in calculations of spectral line intensities, which can provide valuable information about the composition, temperature, density, and other properties of celestial objects. It also helps scientists to better understand the behavior of matter in extreme environments, such as the cores of stars and the interstellar medium.

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