Worksheet 12.1: Problem 5: Squiggles Abound

The problem: Assuming the core temperature, TC, of a Sun-like star is pretty much constant (nuclear fusion is a threshold process with a steep temperature dependence on the reaction rate), what are the following relationships?

Part A: Mass-radius

We know the equation of state and can simplify from there: $$P_c = \frac{ \rho \kappa T }{\bar{m}}$$ Taking out the constants gives us $$P \sim \rho T$$   $$\rho T \sim \frac{M^2}{R^4}$$  We know $$\rho = \frac{M}{R^3}$$ so $$\frac{M^2}{R^4}  \sim \frac{M}{R^3}$$ and finally $$M \sim R$$

Part B: Mass-luminosity (L = Mα) for massive stars M > Mo, assuming the opacity (cross-section per unit mass) is independent of temperature κ = const.

From question 3 we know $$L \sim \frac{T_c^4  R}{\kappa \rho}$$ When we take out the constants, this gives, $$L \sim \frac{ R}{ \rho}$$ Once again using $\rho = \frac{M}{R^3}$ and $M \sim R$ from above lets us simplify to $$L \sim \frac{ R}{ \frac{M}{R^3}}$$ $$L \sim M^3$$

Part C: Mass-luminosity for low-mass stars M < 1 Mo, assuming the opacity (cross-section per unit mass) scales as κ = ρT^3.5 . This is the so-called Kramer’s Law opacity.

Once again we can use $L \sim \frac{T_c^4  R}{\kappa \rho}$ but this time $\kappa$ is not constant so $$L \sim \frac{T_c^{7.5} R}{\rho^2}$$ And one more time using $\rho = \frac{M}{R^3}$ and $M \sim R$ we can simplify to $$L \sim \frac{R}{\rho^2} \sim \frac{R}{\frac{M^2}{R^6}} \sim \frac{R^7}{M^2}$$ $$L \sim M^5$$  

Part D: Luminosity-effective temperature $T_{eff}^4  \sim L^{\alpha}$ for the two mass regimes above. This locus of points in the T-L plane is the so-called Hertzsprung-Russell (H–R) diagram. Sketch this as log L on the y-axis, and log Teff running backwards on the x-axis. It runs backwards because this diagram used to be luminosity vs. B-V color, and astronomers don’t like to change anything. Include numbers on each axis over a range of two orders of magnitude in stellar mass (0.1 < M < 10 Mo). For your blog post, look up a sample H-R diagram showing real data using Google Images. How does the slope of the observed H-R diagram compare to yours?

For both cases we know $$T_{eff} = \frac{L}{4 \pi R_{\star}^2}$$ which scales to $$T_{eff}^4 \sim \frac{L}{R_{\star}^2}$$ which is really $$T_{eff}^4  \sim \frac{L}{M_{\star}^2}$$  

For high mass stars, we know $L \sim M^3$ so $$T_{eff}^4  \sim \frac{L}{L^{\frac{2}{3}}}$$ $$T_{eff}^4  \sim L^{\frac{1}{3}}$$  

For low mass stars, we know $L \sim M^5$ so: $$T_{eff}^4  \sim \frac{L}{L^{\frac{2}{5}}}$$ $$T_{eff}^4  \sim L^{\frac{3}{5}}$$

For simplicity, in this class we will use the average $L \sim M^4$ and $L \sim T^8$. Plotting this against two log scales gives:

Which is roughly in line with an actual HR diagram:

Source

I worked with Sean, Barra, and April on this problem.