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Reimer's Mass-Loss Rate **
Problem: Reimers' (1977) mass-loss rate formula is an empirical formula that estimates the rate of mass loss for stars on the asymptotic giant branch (AGB). Use Reimers' formula to estimate the time that it would take a one-solar-mass red-giant star to be reduced to the mass of its degenerate carbon-oxygen core, approximately 0.6 of its original mass.
Hints:
The mass of the carbon-oxygen core of a one-solar-mass red-giant star is approximate 0.6 of a solar mass.
Use the luminosity formula to calculate the radus of the red giant.
Multiply Reimers' mass-loss formula by M, making the assumption that L, R, and η do not vary significantly with time.
Integrate, and rearrange to solve for M.
Plot M as a function of t (years) for 1 to 600,000.
Calculate the number of years required for the mass to fall to a value of 0.6 solar mass.
Data:
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| (1) |
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| (2) |
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| (3) |
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| (4) |
![M[0] := 1](/view.aspx?SI=154325/24d4d95bc8b87b34b83be08631d51dc6.gif)
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| (6) |
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| (7) |
Useful Equations:
Solution: Solve the luminosity equation for r to obtain:
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| (8) |
and divide by the solar radius to obtain the radius of the red giant in units of the solar radius.
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| (9) |
Multiply Reimers' mass-loss formula by M, making the assumption that L, R, and η do not vary with time. (They do vary with time, but this will give an approximate result.)
M*
![int(M, M = M[0] .. M) = int(-4*10^(-13)*eta*L*rsu, t = 0 .. t)](/view.aspx?SI=154325/f3d3f86b7e8e0fa61bfada544fe0fbf3.gif)
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| (10) |
![M = (M[0]^2-(2*4)*10^(-13)*eta*L*rsu*t)^(1/2)](/view.aspx?SI=154325/65a8173473e4cae3c7f1b6e1aebad014.gif) 
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| (11) |
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| (12) |
Plot the function:
![plot(sqrt(-0.1738700000e-5*t+1), t = 1 .. 600000, title = "Reimers' Mass-Loss Rate", titlefont = ["ARIAL", 15], labels = ["time in years", "fraction of original mass"], labeldirections = ["horizontal", "vertical"])](/view.aspx?SI=154325/0c041704346ad69dbe85720b302eb77a.gif)
To calculate the time required for a 1 solar mass star to be reduced to the mass of the degenerate carbon-oxygen core (0.6 solar masses):
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| (13) |
It would require about 368,000 years.
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Reference
Reimers, D. (1977). On the absolute scale of mass-loss in red giants. A&A, 61, 217-224.
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