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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 17-solar-mass star, and (ii) a 0.34-solar-mass 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 pp-chain 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)

: p-p 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 1010 years.

(B.i) Finding the Main Sequence lifetime of a 17-solar-mass 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 17-solar-mass star:

 (26)

(B.ii) Finding the Main Sequence lifetime of a 0.34-solar-mass 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.

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Reference

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