Lecture 16 Solutions: The H-R Diagram
Practice Problem Solutions
Student note: These are model solutions written to show setup, units, and reasoning clearly. Your own work can be shorter as long as the physics, units, and logic are sound.
Use these after you try the problems yourself. The goal is to compare your reasoning to a clean worked example, not to skip the thinking step.
Core Problems
Problem 1: Spectral Type and Temperature
Restatement: Rigel is B8 and Betelgeuse is M1. We want to identify which is hotter and estimate the temperature difference.
Key insight: Spectral type is a temperature sequence. B stars are very hot; M stars are cool.
Answer: Rigel is hotter. A B8 star is typically about \(1.1 \times 10^4\) to \(1.2 \times 10^4\ \text{K}\), while an M1 supergiant is about \(3.5 \times 10^3\ \text{K}\).
So the temperature difference is roughly
\[ 12{,}000 - 3{,}500 \approx 8{,}500\ \text{K}. \]
That means Rigel is hotter by about \(8{,}000\) to \(9{,}000\ \text{K}\).
Common misconception: Students sometimes read spectral letters alphabetically and assume M comes “later” because it is hotter. The sequence is historical, not alphabetical.
Problem 2: Reading the H-R Diagram
Restatement: A star has spectral type A5 and luminosity \(10\,L_\odot\). We want to decide whether that location lies on the main sequence.
Key insight: On the H-R diagram, a star belongs on the main sequence only if its temperature and luminosity are consistent with the main-sequence band.
Answer: Yes, this point is consistent with the main sequence. An A5 star is hotter than the Sun, and a luminosity of about \(10\,L_\odot\) is reasonable for a mid-A main-sequence star. On the diagram it would lie above and to the left of the Sun, but still within the main-sequence band.
Common misconception: Students sometimes think “above the Sun” automatically means giant. On an H-R diagram, you have to consider both temperature and luminosity together.
Problem 3: Stefan-Boltzmann Reasoning
Restatement: Two main-sequence stars have the same luminosity, but Star A is hotter. We want to know which star has the larger radius.
Key insight: Luminosity depends on both temperature and radius. If two stars have the same luminosity, the hotter one must compensate with a smaller surface area.
Answer: Start with the Stefan-Boltzmann law:
\[ L = 4\pi R^2 \sigma T^4 \]
If \(L\) is fixed, then
\[ R^2 \propto \frac{1}{T^4}. \]
So as temperature increases, radius must decrease to keep the luminosity the same.
Therefore Star A is smaller.
Common misconception: “Hotter” does not automatically mean “larger.” Radius and temperature both matter.
Problem 4: Giants vs. Dwarfs
Restatement: We want to explain how a cool red giant can outshine a hot white dwarf.
Key insight: Temperature matters, but total luminosity depends on surface area too. A huge cool star can emit more total energy than a tiny hot one.
Answer: Use the Stefan-Boltzmann relation:
\[ L \propto R^2 T^4. \]
A red giant has a very large radius, so its emitting area is enormous. A white dwarf is much hotter, but it is tiny, roughly Earth-sized. The giant’s huge \(R^2\) term can outweigh the white dwarf’s larger \(T^4\) term.
So a cool red giant can still be much more luminous than a hot white dwarf because it has far more surface area.
Common misconception: “Hotter means brighter” is only true if size stays the same. Here size changes enormously.
Problem 5: The Main Sequence as Mass
Restatement: If you move up and to the left along the main sequence, what happens to stellar mass?
Key insight: Along the main sequence, hotter and more luminous stars are generally more massive.
Answer: You are moving toward more massive stars. On the main sequence, the upper-left region contains stars that are hotter, brighter, and more massive than stars lower down and to the right.
Common misconception: The H-R diagram is not a map of physical position in space. “Up” and “left” describe temperature and luminosity, not location.
Challenge Problems
Challenge 1: Radius from the H-R Diagram
Given: - Sirius A: \(L = 26\,L_\odot\), \(T = 9940\ \text{K}\) - Sun: \(L_\odot\), \(T_\odot = 5778\ \text{K}\)
Find: - (a) \(R/R_\odot\) - (b) Sirius A’s location relative to the Sun on the H-R diagram
Equation: \[ \frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4 \]
Solve for radius:
\[ \frac{R}{R_\odot} = \frac{\sqrt{L/L_\odot}}{(T/T_\odot)^2} \]
Solution:
Substitute the given values:
\[ \frac{R}{R_\odot} = \frac{\sqrt{26}}{(9940/5778)^2} \]
Now evaluate the pieces:
\[ \sqrt{26} \approx 5.10, \qquad \frac{9940}{5778} \approx 1.72 \]
\[ \left(\frac{9940}{5778}\right)^2 \approx (1.72)^2 \approx 2.96 \]
So
\[ \frac{R}{R_\odot} \approx \frac{5.10}{2.96} \approx 1.7 \]
For the H-R diagram position: Sirius A is hotter than the Sun, so it sits to the left. It is also more luminous, so it sits above the Sun.
Unit check: The ratio form uses only dimensionless quantities, so \(R/R_\odot\) is dimensionless as it should be.
Sanity check: Sirius A is hotter and much more luminous than the Sun, so a radius modestly larger than the Sun’s is reasonable.
Answer: Sirius A has radius about \(1.7\,R_\odot\). On an H-R diagram it lies left of the Sun and above the Sun.
Challenge 2: A Mysterious Star
Restatement: A star has temperature around \(6000\ \text{K}\) but luminosity only \(10^{-4}L_\odot\). We want to identify what kind of star it is and estimate its radius.
Given: - \(T \approx 6000\ \text{K}\) - \(L = 10^{-4}L_\odot\) - \(T_\odot = 5778\ \text{K}\)
Find: - (a) whether it is on the main sequence - (b) what region of the H-R diagram it occupies - (c) its radius - (d) a likely stellar type
Equation: \[ \frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4 \]
so
\[ \frac{R}{R_\odot} = \frac{\sqrt{L/L_\odot}}{(T/T_\odot)^2} \]
Solution:
At about \(6000\ \text{K}\), a main-sequence star would have luminosity of order \(1\,L_\odot\), not \(10^{-4}L_\odot\). So this object is not on the main sequence.
A star that is relatively hot but extremely dim belongs in the white dwarf region of the H-R diagram.
Now estimate the radius:
\[ \frac{R}{R_\odot} = \frac{\sqrt{10^{-4}}}{(6000/5778)^2} \]
\[ \sqrt{10^{-4}} = 0.01, \qquad \frac{6000}{5778} \approx 1.04 \]
\[ \left(\frac{6000}{5778}\right)^2 \approx 1.08 \]
So
\[ \frac{R}{R_\odot} \approx \frac{0.01}{1.08} \approx 9 \times 10^{-3} \]
This is about \(0.01\,R_\odot\), which is roughly Earth-sized.
Unit check: Again, the ratio form leaves us with a dimensionless radius ratio.
Sanity check: White dwarfs are supposed to be hot, faint, and very small, so this result fits that picture well.
Answer: This star is not on the main sequence. It lies in the white dwarf region, has radius about \(0.01\,R_\odot\), and is most likely a white dwarf.
Challenge 3: The H-R Diagram and Stellar Lifetime
Restatement: A \(10\,M_\odot\) star is about \(10{,}000\) times more luminous than the Sun. We want its main-sequence lifetime and the physical reason massive stars are rare.
Given: - \(M = 10\,M_\odot\) - \(L = 10{,}000\,L_\odot\) - \(\tau_\odot \approx 10^{10}\ \text{yr}\)
Find: - (a) the main-sequence lifetime - (b) why O and B stars are rare compared with M dwarfs
Equation: \[ \tau \propto \frac{M}{L} \]
So relative to the Sun,
\[ \frac{\tau}{\tau_\odot} = \frac{M/M_\odot}{L/L_\odot} \]
Solution:
Substitute the given values:
\[ \frac{\tau}{\tau_\odot} = \frac{10}{10{,}000} = 10^{-3} \]
Now multiply by the Sun’s lifetime:
\[ \tau = 10^{-3} \times 10^{10}\ \text{yr} = 10^7\ \text{yr} \]
So the star lives about ten million years on the main sequence.
For part (b): O and B stars are rare because they burn through their fuel extremely quickly. Even though they are bright and easy to notice, they do not stay around long. M dwarfs are much less luminous but live for enormous spans of time, so they accumulate in the Galaxy.
Unit check: The ratio \(M/L\) is being compared to the Sun’s value, so the units cancel and the final lifetime stays in years.
Sanity check: Massive stars are known to live much shorter lives than the Sun, so a lifetime millions of years long rather than billions is reasonable.
Answer: A \(10\,M_\odot\) star lives about \(10^7\) years, or 10 million years, on the main sequence. O and B stars are rare because they live fast and die young, while M dwarfs survive for far longer.