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Main Sequence Lifetime **

``

restart

 

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:

 

X := .707

.707

(1)

M[sun] := 1.989*10^33*Unit('g')

0.1989000000e34*Units:-Unit('g')

(2)

``

bolmag[sun] := 4.74

4.74

(3)

``

M[17] := -3.3

-3.3

(4)

``

bolcor[17] := -3.34

-3.34

(5)

M[34] := 12

12

(6)

bolcor[34] := -1.52

-1.52

(7)

M[sc] := .15

.15

(8)

m[p] := 1.673*10^(-24)*Unit('g')

0.1673000000e-23*Units:-Unit('g')

(9)

`Δm` := 4.4*10^(-26)*Unit('g'): p-p chain reaction

0.4400000000e-25*Units:-Unit('g')

(10)

c := 2.998*10^10*Unit('cm')/Unit('s')

(11)

L[sun] := 3.845*10^33*Unit('erg')/Unit('s')

0.3845000000e34*Units:-Unit('erg')/Units:-Unit('s')

(12)

je := 10^7*Unit('erg')/Unit('J')

10000000*Units:-Unit('erg')/Units:-Unit('J')

(13)

sy := 3.156*10^7*Unit('s')/Unit('yr')

31560000.00*Units:-Unit('s')/Units:-Unit('yr')

(14)

 

Useful Equations:

 

t = E/L and E/L = .15*X*M*Unit('g')*`Δm`*c^2/(4*m[p]*L*Unit('g')*Unit('cm')^2/Unit('s')^2*(1/Unit('s')))

 

bolmag = M[v]+BC

 

M[a] := M[sc]*X*M[sun]

 

N[p] := M[a]/m[p]

 

N[r] := (1/4)*N[p]

 

E := N[r]*`Δm`*c^2 

 

 

Solution:

 

(A) Calculating the Main Sequence lifetime of the Sun:

``

The available mass of hydrogen:

 

M[a] := M[sc]*X*M[sun]

0.2109334500e33*Units:-Unit('g')

(15)

 

The number of available protons:

 

N[p] := M[a]/m[p]

0.1260809623e57

(16)

Number of reactions equals the number of protons per reaction:

 

N[r] := (1/4)*N[p]

0.3152024058e56

(17)

``

Total energy equals the number of reactions times the energy per reaction:

 

E := simplify(N[r]*`Δm`*c^2)

0.1246537813e45*Units:-Unit('J')

(18)

``

Convert to ergs:

``

E[erg] := 1.246537813*10^44*Unit('J')*je

0.1246537813e52*Units:-Unit('erg')

(19)

``

Solar lifetime in seconds for constant luminosity:

 

ts := E[erg]/L[sun]

0.3241970905e18*Units:-Unit('s')

(20)

In years:

 

ts/sy

0.1027240464e11*Units:-Unit('yr')

(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:

 

unassign('E')

````

t = E/L

``

t = M[sc]*X*M*Unit('g')*`Δm`*c^2/(4*m[p]*L*Unit('g')*Unit('cm')^2/Unit('s')^2*(1/Unit('s')))

t = 0.6267158434e18*M*Units:-Unit('s')/L

(22)

``

Convert to solar units and years:

``

6.267158440*10^17*M*Unit('s')*M[sun]/(L*L[sun]*sy)

0.1027240465e11*M*Units:-Unit('s')*Units:-Unit('g')*Units:-Unit('yr')/(L*Units:-Unit('erg'))

(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,

 

bolmag = M[v]+BC

 

bolmag[17] := M[17]+bolcor[17]

-6.64

(24)

From Table 1, the absolute bolometric magnitude of the Sun is 4.81 - 0.07 = 4.74

 

Using Equation 10 in the text:

 

M-M[bol] = -2.5*log[10](L/L[s])

 

solve(bolmag[17]-bolmag[sun] = -2.5*log[10](x), x)

35645.11334

(25)

NULL

Using Equation 22, above:

NULL

NULL

1.03*10^10*M*Unit('yr')/L

NULL

For a 17-solar-mass star:

``

1.03*10^10*(17/35645)

4912329.920

(26)

or about 4.9 million years.

 

 

(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,

 

bolmag = M[v]+BC

 

bolmag = M[34]+bolcor[34]

bolmag = 10.48

(27)

 

Using Equation 10 in the text:

 

M-M[bol] = -2.5*log[10](L/L[s])

 

solve(10.48-4.74 = -2.5*log[10](x), x)

0.5058246620e-2

(28)

NULL

Using Equation 22, above:

``

``

1.03*10^10*M*Unit('yr')/L

``

1.03*10^10*(.34/(0.506e-2))

0.6920948617e12

(29)

NULL

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.

 

 

NULL