Saha's Equation (helium in a white dwarf) ***
Problem: The atmosphere of a DB white dwarf is pure helium. Use Saha's equation to calculate the ionization ratios N_{II}/N_{I} and N_{III}/N_{II} for temperatures of 5,000 K, 15,000 K, and 25,000 K. Express N_{II}/N_{t }(N_{t }= total number of ions) in terms of these ratios and plot N_{II}/N_{t} for temperatures from 5,000 K to 25,000K. Determine the temperature at which half the helium is ionized.
Hints:
In Saha's equation, the eV value of the Boltzmann constant is used in the exponential term.
Use Saha's equation to find N_{II}/N_{I} at 5,000 K, 15,000 K, and 25,000 K.
Use Saha's equation to find the ratio of N_{III}/N_{II} at 5,000 K, 15,000 K, and 25,000 K.
Put the six results into a table for easy reference.
Express the ratio N_{II}/N_{t} in terms of N_{II}/N_{I} and N_{III}/N_{II}. Note that in this expression, the term N_{III}/N_{II} can be ignored.
Substitute the expression for n_{e} (in "Useful Equations", below) into the alternate version of Saha's equation for N_{II}/N_{I}.
Substitute the N_{II}/N_{I} into the expression for N_{II}/N_{t}, expressed in terms of N_{II}/N_{I} (the N_{III}/N_{II} factor having been dropped).
Multiply both sides of the resulting equation by N_{II}/N_{t}.
Expand and rearrange the equation into the normal form of a quadratic equation.
Let x = N_{II}/N_{t} and solve with the quadratic formula.
Plot the result. The plot should show that half the helium is ionized at approximately 15,000 K.
Data: (All data are in SI units unless otherwise stated.)
N_{I }= total number of unionized helium atoms
N_{II }= total number of ionized helium atoms
N_{t }= total number of helium atoms and ions (N_{I} + N_{II})
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
 (9) 
 (10) 
 (11) 
 (12) 
 (13) 
 (14) 
Useful Equations:
Solution:
Finding the ratio of N[II]/N[I]
At 5,000 K:
 (15) 
 (16) 
At 15,000 K:
 (17) 
 (18) 
At 25,000 K:
 (19) 
 (20) 
Finding the ratio of N[III]/N[II]
At 5,000 K:
 (21) 
 (22) 
At 15,000 K:
 (23) 
 (24) 
At 25,000 K:
 (25) 
 (26) 
Simplifying the ratio N[II]/N[t]
 (27) 
 (28) 
Divide numerator and denominator by N[I].
From the table in (a), above, the last term in the denominator can be ignored for 5000 K to 25,000 K.
0
According to the Saha equation:
where
Substituting into the Saha equation yields:
Substitute this into equation:
to get
Multiply both sides by
Expand and rearrange:
Let x = The equation is a quadratic equation with the solution .
Change back to the original form of the equation, using
 (29) 
 (30) 
 (31) 
Solving for x and plotting the equation, changing x back to N[II]/N[t]:
 (32) 
 (33) 
Half the helium is ionized at approximately 15,000 K.
