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 \[L \sim \frac{ R}{ \frac{M}{R^3}}\] \[ L \sim M^3 \]
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 \[ L \sim \frac{R}{\rho^2} \sim \frac{R}{\frac{M^2}{R^6}} \sim \frac{R^7}{M^2}\] \[L \sim M^5\]
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 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:
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:
I worked with Sean, Barra, and April on this problem.
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