(a) Mass-radius
(b) Mass-Luminosity for massive stars \(M>M_{Sun}\), assuming that the opacity (cross-section per unit mass is independent of temperature \(\kappa =\) constant.
(c) Mass-Luminosity for low-mass stars \(M<M_{Sun}\), assuming the opacity scales as \(\kappa \sim \rho T^{-3.5}\). This is the Kramer's Law opacity.
(d) Luminosity-effective temperature for the two mass regimes above. The locus of points in the T-L plane is the so-called Hertzsprug-Russell (H-R) diagram. Sketch this as log(L) on the y-axis and log(Teff) running backwards n 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 seller mass. Compare the slope to a real diagram slope.
Let's start with some simplified scaled equations derived in an earlier problem.
\[M \sim r^3\rho\]
\[T_C^4 \sim \frac{L\rho \kappa}{r}\]
\[P \sim \frac{M\rho}{r}\]
\[L\sim r^2 T_{eff}^4\]
Where: \(M\) is mass, \(r\) is radius, \(L\) is luminosity, \(\rho\) is density \(\kappa\) is opacity, \(T_C\) is core temperature, and \(T_eff\) is effective surface temperature.
We will use these to solve for the various relations.
(c) Mass-Luminosity for low-mass stars \(M<M_{Sun}\), assuming the opacity scales as \(\kappa \sim \rho T^{-3.5}\). This is the Kramer's Law opacity.
(d) Luminosity-effective temperature for the two mass regimes above. The locus of points in the T-L plane is the so-called Hertzsprug-Russell (H-R) diagram. Sketch this as log(L) on the y-axis and log(Teff) running backwards n 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 seller mass. Compare the slope to a real diagram slope.
Let's start with some simplified scaled equations derived in an earlier problem.
\[M \sim r^3\rho\]
\[T_C^4 \sim \frac{L\rho \kappa}{r}\]
\[P \sim \frac{M\rho}{r}\]
\[L\sim r^2 T_{eff}^4\]
Where: \(M\) is mass, \(r\) is radius, \(L\) is luminosity, \(\rho\) is density \(\kappa\) is opacity, \(T_C\) is core temperature, and \(T_eff\) is effective surface temperature.
We will use these to solve for the various relations.
(a) Mass-radius
This one is easy. Thanks to ideal gas law, (\(P=\frac{\rho k T\}{m}\)), \(P \sim \rho\) since k, m, and T are all constants.
\[P \sim \frac{M\rho}{r}\]
\[\rho \sim \frac{M\rho}{r}\]
\[\boxed{M \sim r}\]
(b) Mass-Luminosity for massive stars \(M>M_{Sun}\), assuming that the opacity (cross-section per unit mass is independent of temperature) \(\kappa =\) constant.
\[T_C^4 \sim \frac{L\rho}{r}\]
Using the first equation to substitute for \(\rho\) and \(M\) to substitute for \(r\):
Using the first equation to substitute for \(\rho\) and \(M\) to substitute for \(r\):
\[1 \sim \frac{LM^{-2}}{M}\]
\[\boxed{L \sim M^3}\]
(c) Mass-Luminosity for low-mass stars \(M<M_{Sun}\), assuming the opacity scales as \(\kappa \sim \rho T^{-3.5}\). This is the Kramer's Law opacity.
Let's do the same thing again but with a minor adjustment.
\[T_C^4 \sim \frac{L\rho \kappa}{r}\]
\[1 \sim \frac{L M^{-2} M^{-2}}{M}\]
\[\boxed{L \sim M^5}\]
(d) Luminosity-effective temperature for the two mass regimes above. The locus of points in the T-L plane is the so-called Hertzsprug-Russell (H-R) diagram. Sketch this as log(L) on the y-axis and log(Teff) running backwards n 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 seller mass. Compare the slope to a real diagram slope.
\[L\sim r^2 T_{eff}^4\]
\[L\sim M^2 T_{eff}^4\]
\[L\sim \sqrt{L} T_{eff}^4\]
\[\boxed{L\sim T_{eff}^8}\]
If we graph this on log scales, we should take the log of both sides.
\[\boxed{log(L)\sim 8log(T_{eff})}\]
\[\boxed{L\sim T_{eff}^8}\]
If we graph this on log scales, we should take the log of both sides.
\[\boxed{log(L)\sim 8log(T_{eff})}\]
Thus the slope between the two is 8.
Since we graph the x-axis backwards, the slope will be -8.
This will look like this:
An actual diagram look like this:
Our scale are different, but the slopes are the same.
Image from http://skyserver.sdss.org/dr7/sp/astro/stars/images/hr_diagram.gif
I worked with G. Grell and S. Morrison on this problem.
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