Main Sequence Lifetime **
Problem: (A) Calculate the Main Sequence lifetime of the Sun. (B) Devise a formula in solar units relating the Main Sequence lifetime to stellar mass, and use it to calculate the Main sequence lifetimes of (i) a 17solarmass star, and (ii) a 0.34solarmass star. (Information is from Irwin (2007).)
Hints:
Use cgs units.
15% of mass is used in core fusion.
Luminosity is constant over time.
For the Sun: Find the available mass of hydrogen; find the number of available protons; find the number of ppchain reactions at one reaction per four protons; find the total energy released; convert this to ergs; divide by luminosity to find the time.
For Main Sequence stars in general: find an equation relating mass to luminosity in solar units; use absolute mass and bolometric correction to find the absolute bolometric magnitude of the star; use the absolute bolometric magnitude to find the luminosity of the star; use the magnitude and luminosity equation that you devised to find the lifetime of the star.
Data:
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
 (9) 
: pp chain reaction
 (10) 
 (11) 
 (12) 
 (13) 
 (14) 
Useful Equations:
Solution:
(A) Calculating the Main Sequence lifetime of the Sun:
The available mass of hydrogen:
 (15) 
The number of available protons:
 (16) 
Number of reactions equals the number of protons per reaction:
 (17) 
Total energy equals the number of reactions times the energy per reaction:
 (18) 
Convert to ergs:
 (19) 
Solar lifetime in seconds for constant luminosity:
 (20) 
In years:
 (21) 
or approximate 10 billion years.
(B) Main Sequence Stars in General
Using the procedure above, derive an equation for the lifespan of a Main Sequence star:
 (22) 
Convert to solar units and years:
 (23) 
To find the lifetime in years of a Main Sequence star, multiply the ratio of the star's mass and luminosity in solar units times approximately 10^{10} years.
(B.i) Finding the Main Sequence lifetime of a 17solarmass star:
From Table 1 in the text, a star of 17 solar masses has an absolute visual magnitude of 3.3 and a bolometric correction of 3.34. By Equation 9 in the text,
 (24) 
From Table 1, the absolute bolometric magnitude of the Sun is 4.81  0.07 = 4.74
Using Equation 10 in the text:
 (25) 
Using Equation 22, above:
For a 17solarmass star:
 (26) 
or about 4.9 million years.
(B.ii) Finding the Main Sequence lifetime of a 0.34solarmass star:
From Table 1 in the text, a star of 0.34 solar masses has an absolute visual magnitude of 12.0 and a bolometric correction of 1.52. By Equation 9 in the text,
 (27) 
Using Equation 10 in the text:
 (28) 
Using Equation 22, above:
 (29) 
or about 692 billion years, which is many times the age of the Universe.

Reference
Irwin, J. (2007). Astrophysics: Decoding the Cosmos. Chicester: John Wiley and Sons.
