Calculating the Amount of Hydrogen on the Sun's surface ***

Problem: (A) Use two lines of the Paschen series of hydrogen, along with the general curve of growth of the Sun, to calculate the number of hydrogen atoms with electrons in the n = 3 orbital above each square metre of the Sun's surface. The Paschen series results from an electron making an upward transition from the n = 3 orbital of the hydrogen atom. (B) Then use the Bolztmann and Saha equations to calculate the total number of hydrogen atoms above each square metre of the Sun's surface.

Hints:

Use the two Paschen lines given in the table and calulate log₁₀(W/λ).

Find the value of log₁₀(f (λ/500 nm) for each of the two lines.

Find log₁₀  (where  = the number of atoms) for each of the two lines.

Find the average log₁₀  and solve for .

Use the Boltzmann equation to find the ratios of the numbers of atoms in the three levels.

From this result, calculate the total number of hydrogen atoms per square metre at the solar surface.

Use the Saha equation to check the ionization state of the atoms to determine whether or not the Boltzmann result can be accepted.    

Data:

The following table, from Aller (1971), gives the wavelength (λ), line width (W), and oscillator strength (f) of the Paschen gamma and Paschen delta lines of hydrogen in the Sun.

The following is the curve of growth of the Sun, from Aller (1971).

Energies for the n = 3, n = 2, and n = 1 states, respectively:

Partition numbers for levels I and II:

Useful Equations:

log₁₀(f (λ/500 nm) = log₁₀(f  * λ/500)

log₁₀ N_a =  log₁₀[f N_a(λ/500 nm)] -  log₁₀(f (λ/500 nm)  

where N refers to the number of atoms in a given state, g = the number of degenerate states (g=2n² for hydrogen), E is the energy, k = the Bolzmann constant, and T is the temperature in kelvins.

where the pressure (P_e) = 1 N m^-2, Z_I = 2, Z_II = 1, and χ_i = 13.6 eV.

Solution (A):

For both lines, calculate log₁₀(W/λ).

for Paγ, and

for Paδ.

Trace these figures on the solar curve of growth to find corresponding values for log₁₀[f N_a(λ/500 nm)]. The results are 19.21 for the 1093.8 nm line and 18.96 for the 1004.9 nm line. Next, the value of log₁₀(f (λ/500 nm) is calculated for each line.

for Paγ, and

for Paδ.

Find log₁₀ na for both lines:

for Paγ, and

for Paδ.

The average of these values is

Therefore,

According to the solar curve of growth, there are approximately 1.51 * 10²⁰ atoms of hydrogen with electrons in the n = 3 orbital above every square metre of the Sun's surface.

Solution (B) To find the total number of hydrogen atoms above every square metre of the Sun's surface, the Boltzmann and Saha equations can be used. Using Boltzmann's equation, the ratio of the number of hydrogen atoms in the third state to that in the second state is

and the ratio of the number of hydrogen atoms in the third state to that in the first state is

Therefore, for every neutral hydrogen atom in n = 3, there are

in the n = 2 state and

in the n = 1 state. Using the solar curve of growth, this would give a total of

hydrogen atoms per square metre of the Sun's surface.

The ratio of ionized to neutral atoms according to the Saha equation is

This shows that almost all of the hydrogen atoms are neutral. Therefore, the result of the Boltzmann equation can be accepted.

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Reference

Aller, L. (1971). Atoms, Stars, and Nebulae. (Rev. Ed.). Cambridge, MA: Harvard University Press.