Weighing Stars

Useful constants: $G = 6.67 \times 10^{-8}\ \mathrm{cm^3\,g^{-1}\,s^{-2}}$ ; $M_\odot = 1.99 \times 10^{33}\ \mathrm{g}$ ; $L_\odot = 3.828 \times 10^{33}\ \mathrm{erg/s}$ ; $1\ \mathrm{AU} = 1.496 \times 10^{13}\ \mathrm{cm}$ ; $1\ \mathrm{yr} = 3.156 \times 10^7\ \mathrm{s}$ ; $c = 3.0 \times 10^5\ \mathrm{km/s}$ .

Conceptual

1. ⭐ Which star moves more? In a spectroscopic binary, Star A has radial-velocity amplitude $K_A = 30\ \text{km/s}$ and Star B has $K_B = 90\ \text{km/s}$ . (Reminder: the more massive star has the smaller radial-velocity amplitude.)

(a) Which star is more massive? Explain using the center-of-mass condition ( $M_1 a_1 = M_2 a_2$ ).
(b) What is the mass ratio $M_A/M_B$ ?
(c) As a visual binary, which star would trace the larger orbit on the sky?

2. ⭐⭐ The inclination problem. A student says: “We measured $K_1$ and $K_2$ for a spectroscopic binary, so we know both masses exactly.” Explain why this is wrong. Under what circumstances can we determine the true masses? Describe two solutions to the inclination problem.

3. ⭐ Why binaries matter. Explain why binary systems are essential for measuring stellar masses. Why can’t we get the mass of an isolated star from its spectrum or luminosity alone? (Hint: what observable depends on mass for a single star vs. a binary?)

4. ⭐⭐ Binary classification. For each signature, identify the binary type (visual, spectroscopic, eclipsing) and state what physical information it provides:

(a) Two resolvable points of light orbiting each other over decades
(b) Periodic Doppler shifts in the spectral lines of a single star
(c) Periodic brightness dips in an otherwise constant light curve
(d) Two sets of spectral lines shifting in antiphase

Calculation

5. ⭐ Kepler III in solar units. A visual binary has $P = 8.0\ \text{yr}$ and $a = 4.0\ \text{AU}$ .

(a) Using $\frac{M_{\text{total}}}{M_\odot} = \frac{(a/\mathrm{AU})^3}{(P/\mathrm{yr})^2}$ , calculate the total mass in solar masses.
(b) If the stars have equal masses, what is each star’s mass?
(c) Sanity check: is your answer consistent with main-sequence stars?

6. ⭐⭐ Spectroscopic binary masses. An eclipsing, double-lined spectroscopic binary ( $i = 90^\circ$ ) has $P = 10.0\ \text{days} = 8.64 \times 10^5\ \text{s}$ , $K_1 = 40\ \text{km/s} = 4.0 \times 10^6\ \text{cm/s}$ , $K_2 = 120\ \text{km/s} = 1.2 \times 10^7\ \text{cm/s}$ . (Reminder: the more massive star has the smaller $K$ .)

(a) Determine the mass ratio $M_1/M_2$ . Which star is more massive?
(b) Calculate the total separation $a = \frac{(K_1 + K_2)P}{2\pi}$ , in cm and AU.
(c) Calculate the total mass $M_1 + M_2$ via Kepler III in CGS; convert to solar masses.
(d) Solve for the individual masses.
(e) Sanity check: estimate each star’s luminosity from the mass-luminosity relation. What spectral types might they be?

7. ⭐⭐ Mass-luminosity scaling. Using $L/L_\odot \approx (M/M_\odot)^{3.5}$ :

(a) Luminosity of a $3\,M_\odot$ main-sequence star? Show the steps: $3^3 = ?$ , $3^{0.5} = ?$ , so $3^{3.5} = ?$ .
(b) Its main-sequence lifetime relative to the Sun’s ( $t_\odot \approx 10\ \text{Gyr}$ ), using $t_{\text{MS}} \propto M^{-2.5}$ .
(c) A star has $L = 10^4\,L_\odot$ . Estimate its mass using $M/M_\odot = (L/L_\odot)^{1/3.5}$ .

8. ⭐⭐ Lifetime consequences. The most massive main-sequence stars have $M \approx 150\,M_\odot$ ; the least massive, $M \approx 0.08\,M_\odot$ .

(a) Using $L \propto M^{3.5}$ , calculate the luminosity ratio $L_{\text{max}}/L_{\text{min}}$ .
(b) Using $t_{\text{MS}} \propto M^{-2.5}$ , calculate the lifetimes of a $150\,M_\odot$ and a $0.08\,M_\odot$ star, given $t_\odot \approx 10\ \text{Gyr}$ .
(c) The universe is $\sim 13.8\ \text{Gyr}$ old. Which of these stars, if formed at the beginning, would still be on the main sequence today?

Synthesis

9. ⭐⭐ From velocities to fate. A double-lined spectroscopic binary has $P = 2.0\ \text{yr}$ , $K_1 = 15\ \text{km/s}$ , $K_2 = 45\ \text{km/s}$ , $i = 90^\circ$ . (Reminder: the more massive star has the smaller $K$ .)

(a) Determine the mass ratio and total separation in AU. (Hint: $a\sin i = \frac{(K_1 + K_2)P}{2\pi}$ .)
(b) Calculate the total mass in solar masses using the solar-unit Kepler III.
(c) Solve for the individual masses.
(d) Estimate each star’s luminosity from the mass-luminosity relation.
(e) Estimate each star’s main-sequence lifetime. Which evolves off the main sequence first, and by roughly how much sooner?

10. ⭐⭐ Spectroscopic parallax from the mass-luminosity relation. You observe a main-sequence B0 star with apparent magnitude $m = 8.0$ . From the spectral type, $M \approx 15\,M_\odot$ .

(a) Estimate $L/L_\odot$ from the mass-luminosity relation.
(b) Convert to absolute magnitude $M_V = M_{V,\odot} - 2.5\log_{10}(L/L_\odot)$ , with $M_{V,\odot} = +4.83$ .
(c) Use the distance modulus to estimate the distance in parsecs.
(d) What assumptions does this method rely on? Name at least two things that could make the distance wrong.

11. ⭐⭐⭐ The mass-luminosity relation as a diagnostic. A binary has total mass $M_1 + M_2 = 5.0\,M_\odot$ and mass ratio $M_1/M_2 = 4.0$ .

(a) Solve for $M_1$ and $M_2$ .
(b) Predict each star’s luminosity using $L \propto M^{3.5}$ .
(c) What fraction of the system’s light comes from Star 1? (Compute $L_1/(L_1 + L_2)$ .)
(d) In an SB1, only the brighter star’s lines are typically visible. Could this system appear as an SB1? Explain.
(e) A $4\,M_\odot$ main-sequence star has $T_{\text{eff}} \approx 1.4 \times 10^4\ \text{K}$ and a $1\,M_\odot$ star has $T_{\text{eff}} \approx 5{,}800\ \text{K}$ . Use Stefan-Boltzmann in solar units to estimate each radius. Which is larger?