Worksheet 12.1, Problem 5: Stellar Scaling Relationships on the Main Sequence

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?

Before we begin, it is important to know four equations of stellar structure: hydrostatic equilibrium (\(dP/dr\)), radiative diffusion (\(dT/dr\)), mass conservation (\(dM/dr\)), and the equation of state (for matter inside a star, not accounting for radiation pressure).

\(\dfrac{dP(r)}{dr}=-\dfrac{GM(r)\rho (r)}{r^2}\)

\(\dfrac{dT(r)}{dr}=-\dfrac{L(r)\kappa \rho (r)}{\pi r^2 acT(r)^3}\)

\(\dfrac{dM(r)}{dr}=4\pi r^2\rho (r)\)

\(P_g=\dfrac{\rho (r) k_B T(r)}{\overline{m}}\)

We can express the differentials above as being similar to non-differential fractions:

\(\dfrac{dP(r)}{dr}\sim \dfrac{P(r)}{r}\)

\(\dfrac{dT(r)}{dr}\sim \dfrac{T(r)}{r}\)

\(\dfrac{dM(r)}{dr}\sim \dfrac{M(r)}{r}\)

Then, we can use the following boundary conditions for each relationship from the center of the star to the surface of the star to solve for scaling relationships.

\(P(r=0)=P_C\)  and  \(P(r=R_\star)=0\)

\(T(r=0)=T_C\)  and  \(T(r=R_\star)=T_{eff}\ll T_C\)

\(M(r=0)=0\)  and  \(M(r=R_\star)=M_\star\)

Also, since density is equal to mass divided by volume, we can say that:

\(\rho (r=R_\star )\sim \dfrac{M_\star}{R_\star^3}\)

Using this and our boundary conditions, we can solve for the scaling relationships:

\(P_C\sim \dfrac{M_\star^2}{R_\star^4}\)

\(T_C^4\sim \dfrac{L_\star M_\star\kappa}{R_\star^4}\)

Now we have the information we need to solve this problem

a) Mass-radius (i.e. assume \(R\sim M^\alpha\) and find \(\alpha\)). 

From the equation of state, we can find another relation for \(P_C\):

\(P_C\sim \dfrac{\rho (r) k_B T(r)}{\overline{m}}\sim \dfrac{M_\star}{R_\star^3}T_C\sim \dfrac{M_\star}{R_\star^3}\)

Since we are only looking for a proportionality, we can drop the constants \(k_B\), \(\overline{m}\), and \(T_C\). Setting this relationship equal to the relationship we determined previously, we get:

\(P_C\sim \dfrac{M_\star}{R_\star^3}\sim \dfrac{M_\star^2}{R_\star^4}\)

\(M_\star \sim R_\star\)

So we have found that \(\alpha=1\), and the mass and radius have a linear relationship. 

b) Mass-luminosity (\(L\sim M^\alpha\)) for massive stars \(M>1 M_\odot\), assuming the opcaity (cross-section per unit mass) is independent of temperature \(\kappa=\text{ constant}\).

From the scaling relationship we found for temperature:

\(T_C^4\sim \dfrac{L_\star M_\star\kappa}{R_\star^4}\)

\(T_C\) is just a constant, so we can drop the constants, which gives us the relationship:

\(R_\star^4\sim L_\star M_\star\)

From part (a), we determined that the radius is linearly related to the mass, so we can rewrite the expression:

\(M_\star^4\sim L_\star M_\star\)

Which gives us the relationship:

\(L_\star\sim M_\star^3\)

So here we get \(\alpha=3\).

c) Mass-luminosity for low-mass stars \(M\le 1M_\odot\), assuming the opacity (cross-section per unit mass) scales as \(\kappa \sim \rho T^{-3.5}\). This is the so-called Kramer's Law opacity.

Here, we can no longer treat \(\kappa\) as a constant in our equation for radiative diffusion. Instead, we have \(\kappa\sim \frac{M_\star T_C^{-3.5}}{R_\star^3}\). So we get the new relation:

\(T_C^4\sim \dfrac{L_\star M_\star\kappa}{R_\star^4}\sim \dfrac{L_\star M_\star^2 T_C^{-3.5}}{R_\star^7}\)

Again, dropping\(T_C\) as a constant, and using the proportionality \(R_\star \sim M_\star\), we get the relation:

\(L_\star \sim M_\star^5\)

So we have found that \(\alpha =5\) in the mass-luminosity scaling relationship.

d) Luminosity-effective temperature (\(L\sim T_{eff}^\alpha\) for the two mass regimes above. This locus of points in the T-L plane is the so-called Hertzsprung-Russel (H-R) diagram. Sketch this as \(\log(L)\) on the y-axis and \(\log(T_{eff})\) 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 \(0.1<M<10M_\odot\). Also find a sample H-R diagram showing real data using Google Images. How does the slope of the observed H-R diagram compare to yours?

From part (b) and (c), we found the mass-luminosity scaling relationships for massive and low-mass main sequence stars. An average of these two solutions gives us the relation \(L_\star \sim M_\star ^4\). We also know, from our linear relationship \(R_\star \sim M_\star\), that \(L_\star \sim R_\star^4\).

Then, using the Stefan-Boltzmann Law, \(L_\star =4\pi R_\star ^2\sigma T_{eff}^4\), we get the relation:

\(L_\star \sim R_\star^2 T_{eff}^4\)

Now, we can use our relationship between the radius and the luminosity to get an expression between luminosity and effective temperature:

\(L_\star \sim L_\star^{1/2} T_{eff}^4\)

\(L_\star \sim T_{eff}^8\)

The plot of this relationship on an H-R diagram looks similar to the following image:

The y-axis is labeled for both mass and luminosity, and the x-axis is the logarithm
 of the effective temperature. The slope of the line \(L\sim T_{eff}^8\) is 8.

If you look up an H-R diagram online that is using real data, they look like the diagram below. 

H-R Diagram
Image Credit: https://www.le.ac.uk/ph/faulkes/web/images/hrcolour.jpg

This graph of real data for main sequence stars is not perfectly linear and has some fluctuations, but it is very similar. If we had done our calculations with more precision, we may have found a result more like the one shown in this graph.

I would like to thank Charles Law and Daniel Chen for their help in solving this problem.