Astrophysics Solutions (1)

1.  We know:   P = (N _o + N_e) k T

Where N _o  is the number of atoms of all kinds, both neutral (unionized) and ionized and  N_e  is the number of free electrons (the greater this number, the higher the degree of ionization). The total gas pressure therefore is the sum of the two pressures:

 

P = N _o k T + N_e k T

Then:

N_e k T =  (1.45 x 10 ¹⁹)( 1.38 x 10^-23 J/K) (5650 K) = 1.13 Pa

But P, the total gas pressure is 8.3 x 10³ Pa, so:

P = 8.3 x 10³ Pa  =  N _o k T + N_e k T = N _o k T + 1.13 Pa

Then: N _o k T = 8.3 x 10³ Pa   -  1.13 Pa » 8.3 x 10³ Pa 

N o = (8.3 x 10³  Pa ) / k T =

(8.3 x 10³  Pa ) / (1.38 x 10-23 J/K) (5650 K)

N _o = 1.06 x 10 ²³

Where N _o    is the total number of neutral and ionized atoms.

2) We assume negligible amounts of heavier elements so:

m        = 4/ (6X + Y + 2)

Where X = 0.9 and Y = 0.1.  Substituting those values:

m        = 4/ (6(0.9) +   0.10 + 2) 

= 4/ (0.54 + 0.10 + 2) = 4/ 2.64

m        = 1.51

(Note that this molecular weight is in proton masses)

3)  We have: 

1/ m  =  (2X + 3Y/4 + ½Z)

But the no. of electrons is:

N_e  =   r /H [X +  (Y + Z)/ 2]  »  r [1 + X]/ 2H

1/ m  =   H N_e /  r  =  (1 + X)/ 2

m =  2/ (1 + X)