Solutions to Astrophysics Problems (Pt. 2)

1) For a  hydrogen plasma find:

(P _e N _{i + 1} /N _i)  =  P _e N _HII /N _HI    

at a temperature of 5040 K, given the hydrogen partition functions are:

Z _{i + 1}  =  Z ₂    =   1   and:  Z _i =    Z ₂    =   2      with  c _i =   13.6 eV

Solution:

By the Saha equation:   log(P _e N _{i + 1} /N _i)  = 

-0.48  + log (2 Z _{i + 1} / Z _i ) + 2.5 log T – 5040 c _i/ T

=   -0.48  +  log (1) + 2.5 log (5040)  – 5040 (13.6)/ 5040

=   -0.48 + 0  +   9.25 – 13.6  =  -4.83

Antilog (-4.83) =   1.5 x 10 ^-5    Ans.

2)     For the temperature and conditions of problem (1) of the previous set find the ratio of the probability that the system will be found in any of the eight degenerate states of energy level E2 to the probability the system will be in any of the two degenerate states of energy level E1.

Solution:

We have:

P(E2) / P(E1)   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

With  g ₁ = 2(1)² = 2

And:   g ₂ = 2(2)² = 8

For that problem also:  N₂ =  N₁     and:

E2 =  - 13.6 eV and E1 = -3.4 eV, therefore:

[g₂ / g₁ ] exp (- E2 – E1) / kT

=  4 exp [- 13.6eV – (-3.4 eV) ]/ kT

N₂ /  N₁ = 1 = 4 exp (-10.2 eV)/ kT

Taking natural logs:

ln (4)  =    (10.2 eV)/ kT

Then:    

P(E2) / P(E1)   =     ln (4)    = 1.38

Thus, the probability that the system will be found in any of the eight degenerate states of energy level E2 is 1.38 times that of the probability the system will be in any of the two degenerate states of energy level E1.

Or:   P(E2) =     1.38 P(E1)

3)     An H-alpha line undergoes triplet splitting in the vicinity of a sunspot. The undisturbed line is measured at   l_o  =   6.62 x 10  ^-5    cm.  The line shift on either side is: 0.0025 A. Use this information to find the strength of the magnetic field in: a) gauss and b)Tesla

We have:  D  l  =     (l_o)²  e H/ 4 π  m_e c²  

Solve for H:

H =   4 π  (D  l)  m_e c²   /  (l_o)²  e

Where: c =  3 x 10  ¹⁰ cm/s

m_e  =    9.1 x 10  ^-28 g

(D  l) =    2 (0.0025 A) = 0.005 A

= 5.0 x 10  ^-11    cm

e = 4.8  x 10  ^-10 e.s.u.

H = 

 4 π(5.0 x 10  ^-11    cm)  (9.1 x 10  ^-28 g)( 3 x 10  ¹⁰ cm/s) ² / (6.62 x 10  ^-5  cm) ² e

=  2.44  x 10  ² G

But 1 T = 10,000 G, so:

H =    2.44  x 10  ² G/  (10  ⁴ G/ T) =   0.024 T