Luminosity, Radius, and Effective Temperature of a Low-Mass Star Ascending from the Main Sequence to the Red Giant Branch

Question:
Once a low-mass star leaves the main sequence and ascends the red giant branch, its luminosity is provided by hydrogen burning in a thin shell surrounding a degenerate helium core. As hydrogen is burned in the shell, the mass of the helium core grows and the star continues to ascend the giant branch. Both the luminosity and radius of red giants can be written as functions of their helium core mass (nearly independent of any other quantity). These expressions are represented (empirically) by:

where mc is the helium core mass, and L and R are the luminosity and radius of the (entire) red giant, respectively. Use these relations to compute the evolution of a star (L and R as functions of time) as it ascends the red giant branch. Take the initial core mass value to be and the final value (at the tip of the giant branch) to be . Note that,

where ∆M/∆t represents the rate that hydrogen is burned in the shell around the core, and is proportional to the rate of growth of the degenerate helium core. Plot the radius, luminosity, and effective temperature of the star as a function of time on the giant branch. Also, plot the track of the evolving giant on an H-R diagram (i.e., log L versus log Te). Assume a standard composition of X = 0.70, Y = 0.28, Z = 0.02.

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