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1. At the lower end of the main sequence,
M  0.085M Sun
Te  2.74 103 K
L  5.05 104 LSun
Such low-mass stars are entirely convective,
so all the hydrogen (70% by mass) is available
for fusion. What is the lifetime of such a
star?
E  m c 2
L  5.05 10 4 LSun  1.93 10 23W
 0.0070.7  0.085M Sun c 2
t
 7.42  10 J
43
E 7.42 1043

 3.8 1020 s  1.2 1013 years
L 1.93 1023
At the upper end of the main sequence,
M  90M Sun
Te  5.27 104 K
L  1.1106 LSun
Only the central ~10% of the mass is
available for hydrogen fusion, because the
star is not fully convective. What is the
lifetime of such a star?
E  m c 2
L  1.1106 LSun  4.20 1032W
 0.0070.1 90M Sun c 2
 1.12 10 46 J
t
E 1.12 1046

 2.7 1013 s  8.4 105 years
32
L 4.2 10
2. Estimate the temperature of a dust
grain that is located d=100 AU from a
newly formed F0 main sequence star.
The rate of energy absorbed by the
grain is:
 rg2 
dEab

 L* 
 4d 2 
dt


The rate of energy emitted by the
grain at temperature Tg is:
Setting the rates equal:
 rg2 
  4rg2Tg4
L* 
 4d 2 


L  1 
Tg4  *  2 
4  4d 
R*2T*4

4d 2
For an F0 star, R~1.6 RSun and T~7200 K
Tg 
dEem
 4rg2Tg4
dt
R*
T*  43.9 K
2d
3. Relate the extinction in magnitudes to the
optical depth, .
In the case of pure absorption, the intensity of light

depends on the optical depth: I   I  ,0e  . Thus the
change in magnitude due to this extinction is just:
m  2.5 log 10 e    1.086 
So we have an expression for the extinction in terms of
optical depth:
a  1.086 
The Mie approximation:
The optical depth can also be written in terms of the
s
number density of scattering particles, n,
    n( s)  ds
and the cross-section :
0
In the Mie approximation,
size of a dust grain.
Q 

a 2
, where a is the typical