Question

How long will a 5MΘ star burn hydrogen as fuel, given that the Sun will do so for about 10^10
years?

Answer 1

Answer on question #58698, Physics / Astronomy | Astrophysics

How long will a $5\mathrm{M}_{\odot}$ star burn hydrogen as fuel, given that the Sun will do so for about $10^{\wedge}10$ years?

Solution:

Star's lifetime equals to $t = \frac{E_{\text{released}}}{L}$ , where $E_{\text{released}}$ - energy released after all hydrogen fusion into helium, $L$ - star luminosity.

Solar luminosity equals to $3.828 \cdot 10^{33} \, \text{erg/s}$ . For the star with $M = 5M_{\odot}$ the released amount of energy will be proportional to the mass, i.e.

$$5 \cdot E _ {r e l e a s e d} = 5 \cdot 1 0 ^ {1 0} y \cdot 3 1 5 5 6 9 2 6 \frac {s}{y} \cdot 3. 8 2 8 \cdot 1 0 ^ {3 3} \frac {e r g}{s} = 5 \cdot 1. 2 1 \cdot 1 0 ^ {5 1} e r g.$$

The increasing of mass leads to the increase of luminosity which is roughly proportional to $\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^{\alpha}$ , where $a \approx 3.9$

$$\frac {L}{L _ {\odot}} = 5 ^ {3. 9} \approx 5 3 2.$$

Then $L = 532 * L_{\odot} = 532 \cdot 3.828 \cdot 10^{33} \, \text{erg/s}$

The final expression as follows

$$t = \frac {5 \cdot 1 . 2 1 \cdot 1 0 ^ {5 1} e r g}{5 3 2 \cdot 3 . 8 2 8 \cdot 1 0 ^ {3 3} e r g / s} = 9. 4 \cdot 1 0 ^ {7} y$$

Answer: the star will burn its fuel in $9.4 \cdot 10^{7} y$

https://www.AssignmentExpert.com