## Sunday, June 16, 2013

### Astrophysics Solutions (1)

1.  We know:   P = (N o + Ne) k T

Where N o  is the number of atoms of all kinds, both neutral (unionized) and ionized and  Ne  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 + Ne k T

Then:

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

1/ m  =   H Ne /  r  =  (1 + X)/ 2

m =  2/ (1 + X)