Place the main-sequence lifetimes of the following stars in order from shortest to longest. a. the Sun: mass \(1 M_{\text {sun }}\), luminosity \(1 L_{\text {sun }}\) b. Capella Aa: mass 3 \(M_{\text {sun }}\), luminosity 76 \(L_{\text {Sun }}\) c. Rigel: mass \(24 M_{\text {sun }}\), luminosity \(85,000 L_{\text {sun }}\) d. Sirius A: mass 2 \(M_{\text {sun }}\), luminosity 25 \(L_{\text {sun }}\) e. Canopus: mass \(8.5 M_{\text {sun }}\), luminosity \(13,600 L_{\text {sun }}\) f. Achernar: mass \(7 M_{\text {sun }}\), luminosity \(3,150 L_{\text {sun }}\)

The main-sequence lifetime of a star can be approximated by the formula: \[ t = \frac{10^{10} \text{ years}}{\frac{M}{L}} \] where \(t\) is the lifetime, \(M\) is the mass of the star in solar masses \(M_{\text{sun}}\), and \(L\) is the luminosity of the star in solar luminosities \(L_{\text{sun}}\).

For each star, use the above formula to calculate the main-sequence lifetime.1. **The Sun**: \[ t_{\text{Sun}} = \frac{10^{10} \text{ years}}{\frac{1}{1}} = 10^{10} \text{ years} \]2. **Capella Aa**: \[ t_{\text{Capella Aa}} = \frac{10^{10} \text{ years}}{\frac{3}{76}} = \frac{10^{10} \text{ years} \times 76}{3} \text{ years} \]\[ t_{\text{Capella Aa}} \approx 2.53 \times 10^9 \text{ years} \]3. **Rigel**: \[ t_{\text{Rigel}} = \frac{10^{10} \text{ years}}{\frac{24}{85,000}} = \frac{10^{10} \text{ years} \times 85,000}{24} \text{ years} \]\[ t_{\text{Rigel}} \approx 3.54 \times 10^5 \text{ years} \]4. **Sirius A**: \[ t_{\text{Sirius A}} = \frac{10^{10} \text{ years}}{\frac{2}{25}} = \frac{10^{10} \text{ years} \times 25}{2} \text{ years} \]\[ t_{\text{Sirius A}} \approx 1.25 \times 10^9 \text{ years} \]5. **Canopus**: \[ t_{\text{Canopus}} = \frac{10^{10} \text{ years}}{\frac{8.5}{13,600}} = \frac{10^{10} \text{ years} \times 13,600}{8.5} \text{ years} \]\[ t_{\text{Canopus}} \approx 1.60 \times 10^8 \text{ years} \]6. **Achernar**: \[ t_{\text{Achernar}} = \frac{10^{10} \text{ years}}{\frac{7}{3,150}} = \frac{10^{10} \text{ years} \times 3,150}{7} \text{ years} \]\[ t_{\text{Achernar}} \approx 4.50 \times 10^8 \text{ years} \]

Arrange the calculated lifetimes in ascending order:1. **Rigel**: \(3.54 \times 10^5\) years2. **Canopus**: \(1.60 \times 10^8\) years3. **Achernar**: \(4.50 \times 10^8\) years4. **Capella Aa**: \(2.53 \times 10^9\) years5. **Sirius A**: \(1.25 \times 10^9\) years6. **The Sun**: \(10^{10}\) years