The HR Diagram

Useful constants: $\sigma = 5.67 \times 10^{-5}\ \mathrm{erg\,cm^{-2}\,s^{-1}\,K^{-4}}$ ; $L_\odot = 3.828 \times 10^{33}\ \mathrm{erg/s}$ ; $T_\odot = 5{,}800\ \mathrm{K}$ ; $R_\odot = 6.96 \times 10^{10}\ \mathrm{cm}$ ; $M_\odot = 1.99 \times 10^{33}\ \mathrm{g}$ ; $c = 3.0 \times 10^5\ \mathrm{km/s}$ ; $1\ \mathrm{pc} = 3.086 \times 10^{18}\ \mathrm{cm}$ .

Conceptual

1. ⭐ The backward scale. Rank from brightest to faintest in apparent magnitude: the Sun ( $m = -26.74$ ), Sirius ( $m = -1.46$ ), Polaris ( $m = 1.98$ ), the faintest naked-eye star ( $m \approx 6$ ). Then explain why “brightest” corresponds to the smallest (most negative) number.

2. ⭐⭐ Reading the HR diagram. Three stars occupy different regions: Star X (upper left, hot and luminous), Star Y (lower right, cool and faint), Star Z (upper right, cool and luminous).

(a) Which is most likely a main-sequence O star? A main-sequence M dwarf? A red giant?
(b) Which must have the largest radius? Explain using Stefan-Boltzmann reasoning.
(c) Which is most massive, assuming the two main-sequence stars follow $L \propto M^{3.5}$ ?

3. ⭐⭐ Observer vs. theorist. Explain the difference between the observer’s HR diagram ( $M_V$ vs. spectral type) and the theorist’s ( $\log L$ vs. $\log T_\text{eff}$ ). Why do the same patterns appear on both? What insight does the theorist’s version add?

4. ⭐ Why 10 parsecs? Absolute magnitude is the apparent magnitude a star would have at $d = 10~\text{pc}$ . Why is a standard distance necessary for comparing stars? What would happen to the HR diagram if you plotted apparent magnitude on the vertical axis?

Calculation

5. ⭐ Magnitude-flux conversions. Two stars have $m_A = 2.0$ and $m_B = 7.0$ .

(a) What is $\Delta m$ ?
(b) What is the flux ratio $F_A/F_B$ ? (Use $F_A/F_B = 10^{0.4 \times \Delta m}$ .)
(c) If Star A is at $d_A = 10~\text{pc}$ and Star B at $d_B = 100~\text{pc}$ with the same luminosity, verify the magnitude difference is consistent with the inverse-square law.

6. ⭐⭐ Distance modulus practice. A cluster has a main-sequence A0 star with known $M_V = +0.6$ ; you measure $m = 10.6$ .

(a) Calculate the distance modulus $m - M$ .
(b) Solve for the distance in parsecs.
(c) Express in kiloparsecs. Is this star in the Milky Way’s disk (scale ${\sim}10~\text{kpc}$ )?
(d) Sanity check: how many times fainter does it appear than the same star at $10~\text{pc}$ ?

7. ⭐⭐ Radius from the HR diagram. Use $L/L_\odot = (R/R_\odot)^2 (T/T_\odot)^4$ to find each radius:

(a) A red giant with $L = 200\,L_\odot$ , $T_\text{eff} = 4{,}600~\text{K}$ . Show $(R/R_\odot)^2$ first, then the square root.
(b) A white dwarf with $L = 0.005\,L_\odot$ , $T_\text{eff} = 1.5 \times 10^4~\text{K}$ .
(c) Compare both to the Sun. Is the giant larger or smaller than Earth’s orbit ( $1~\text{AU} \approx 215\,R_\odot$ )?

8. ⭐⭐ Lines of constant radius. A line of constant radius satisfies $\log(L/L_\odot) = 2\log(R/R_\odot) + 4\log(T/T_\odot)$ .

(a) For $R = 10\,R_\odot$ , calculate the luminosity at $T_\text{eff} = 3{,}000~\text{K}$ and at $T_\text{eff} = 3 \times 10^4~\text{K}$ .
(b) For $R = 0.01\,R_\odot$ (Earth-sized), calculate the luminosity at $T_\text{eff} = 10^4~\text{K}$ .
(c) Does the $R = 0.01\,R_\odot$ line pass through the main sequence, the giant branch, or the white dwarf region?

Synthesis

9. ⭐⭐ Placing stars on the HR diagram. For each, calculate the missing quantity, identify the region (main sequence, giant, white dwarf), and justify:

(a) $T_\text{eff} = 3{,}500~\text{K}$ , $L = 2{,}000\,L_\odot$ . Find $R/R_\odot$ .
(b) $T_\text{eff} = 2.5 \times 10^4~\text{K}$ , $R = 0.01\,R_\odot$ . Find $L/L_\odot$ .
(c) $L = 30\,L_\odot$ , $T_\text{eff} = 9{,}500~\text{K}$ . Find $R/R_\odot$ .
(d) For star (c), use the mass-luminosity relation to estimate its mass. Consistent with an A-type main-sequence star?

10. ⭐⭐ The main sequence as a mass sequence. Using the reading’s reference table:

(a) Verify $L \propto M^{3.5}$ approximately holds for an O5 star ( $M = 40\,M_\odot$ , $L = 5 \times 10^5\,L_\odot$ ) and an M5 dwarf ( $M = 0.1\,M_\odot$ , $L = 0.001\,L_\odot$ ). How close is $3.5$ to the actual exponent for each?
(b) Using $t_\text{MS} \propto M^{-2.5}$ , verify the listed lifetimes for the O5 and M5 stars ( $t_\odot = 10~\text{Gyr}$ ).
(c) The universe is $13.8~\text{Gyr}$ old. What is the approximate mass of the most massive star that could still be on the main sequence in a $13.8~\text{Gyr}$ -old cluster? (Set $t_\text{MS} = 13.8~\text{Gyr}$ and solve for $M$ .)

11. ⭐⭐⭐ Spectroscopic parallax: a complete inference chain. You observe a star: spectral type G2 V, apparent magnitude $m = 9.83$ .

(a) From the spectral type, assign $M_V$ using the reference table.
(b) Calculate the distance modulus and solve for the distance in parsecs.
(c) Its Doppler shift gives $v_r = 25~\text{km/s}$ (receding). Calculate $\Delta\lambda/\lambda_0$ .
(d) What assumptions have you made? List at least three, and for each, describe how a violation would affect your distance estimate.

12. ⭐⭐⭐ Why giants are giant. A star begins as a $2\,M_\odot$ main-sequence star and later becomes a red giant with $L = 100\,L_\odot$ , $T_\text{eff} = 4{,}500~\text{K}$ .

(a) Estimate the main-sequence luminosity and temperature of a $2\,M_\odot$ star (reference table) and its main-sequence radius in $R_\odot$ .
(b) Calculate the red giant’s radius in $R_\odot$ .
(c) By what factor did the radius increase?
(d) The mass barely changed. Explain qualitatively why luminosity increased even though surface temperature decreased. (Which factor in $L = 4\pi R^2 \sigma T^4$ changed more?)
(e) Sketch (qualitatively) the path from main sequence to giant branch. Which direction did it move first? Why does the HR diagram look like an evolution diagram?