5. Assuming the core temperature, \(T_c\), 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?
(a) Mass-radius (i.e. assume R „ Mα and find α).
To begin, we can consider the equation of state, which is shown below: \[P = \frac{ \rho k T_c}{ \bar{m}} \] Since we are primarily concerned with ratios in this problem, we can ignore the constants above. This gives us the following relationship \[ P \sim \rho \] Since we know that density is simply mass over volume, we can expand this to: \[ P \sim \rho \sim \frac{M}{R^3}\] Next, we can consider the relationship of pressure within a star to mass and density. This relationship is presented as follows: \[P \sim \frac{M \rho}{R} \sim \frac{M^2}{R^4} \] Setting our Pressure relation proportions equal to each other, we get: \[ \frac{M}{R^3} \sim \frac{M^2}{R^4} \] Reducing this, we get our final Mass-Radius relationship: \[M \sim R\]
(b) Mass-luminosity (L Mα) for massive stars M ą 1 Md, assuming the opacity (cross-section per unit mass) is independent of temperature κ “ const.
Utilizing the equation of state and pressure relationship established in part A, we know that \[P \sim \frac{M^2}{R^4} \sim \rho T\] Additionally, we know that \[\rho \sim \frac{M}{R^3}\] By combining these, we achieve the relationship that \[T \sim \frac{M}{R} \] Next, we can use the Luminosity proportion of \[L \sim \frac{T^4R^4}{M}\] Inserting \(T \sim \frac{M}{R} \), we get \[L \sim \frac{M^4}{M} \sim M^3\] This gives us our ultimate luminosity-mass equation of \[ L \sim M^3 \]
(c) Mass-luminosity for low-mass stars M <= 1 Md, assuming the opacity (cross-section per unit mass) scales as k ~ ρT^3.5 . This is the so-called Kramer’s Law opacity.
Including constant values in our luminosity proportion, we get the following: \[L \sim \frac{{T_c}^4R}{k \rho}\] Plugging in Kramer's law for opacity, and assuming T as a constant, we get: \[L \sim \frac{{T_c}^4R}{k \rho} \sim \frac{R}{ {\rho}^2}\] Since \(\rho \sim \frac{M}{R^3} \) and \( M \sim R \), our L relationship can be re-written as \[ L \sim \frac{M}{ {\rho}^2}\] Additionally, density can be re-written as \[ \rho \sim \frac{1}{M^2}\] Plugging this density value into our L proportion, we get: \[ L \sim \frac{M}{ \frac{1}{M^4}} \sim M^5\] Therefore, for low-mass stars, the Mass-Luminosity Relationship is \[ L \sim M^5\]
(d) Luminosity-effective temperature (L „ T α eff) 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 ă M ă 10 Md). 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?
We can begin with the Luminosity formula: \[L = 4 \pi R^2 \sigma T^4 \] \[ L \sim R^2 {T_{eff}}^4 \] Applying our proportions of \(L \sim M^3 \) and \( M \sim R \), we get a new relationship of \[ L \sim M^2 {T_{eff}}^4 \sim L^{0.5}{T_{eff}}^4\] Isolating L, we get: \[L^{0.5} \sim {T_{eff}}^4\] \[L \sim {T_{eff}}^8 \]
On a log scale, this produces a slope of 8. Since this would be inverted using the astronomers' unorthodox graphing practice, it would actually show a slope of -8. Comparing this to the H-R plot below, we can see that the relationship holds reasonably well.
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| http://casswww.ucsd.edu/archive/public/tutorial/HR.html |
Acknowledgements: Team EE
Good job.
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