Question

1. d) The average temperature of the interior of a sun-like star is of the order 10^8 K. Estimate
the mass of the star in terms of the solar mass if it has a radius of order 10^10 cm.

Answer 1

Answer on Question #50909, Physics, Astronomy | Astrophysics

d) The average temperature of the interior of a sun-like star is of the order $10^{\wedge}8$ K. Estimate

the mass of the star in terms of the solar mass if it has a radius of order $10^{\wedge}10$ cm.

Answer:

Will use temperature, mass relation from wiki:

http://en.wikipedia.org/wiki/Mass%E2%80%93luminosity Relation

$$T ^ {4} \sim \frac {M ^ {2}}{R ^ {4}}$$

Will apply it for both stars

For sun:

$$T _ {s} ^ {4} \sim \frac {M _ {s} ^ {2}}{R _ {s} ^ {4}}$$

And given star:

$$T _ {x} ^ {4} \sim \frac {M _ {x} ^ {2}}{R _ {x} ^ {4}}$$ $$\frac {T _ {s} ^ {4}}{T _ {x} ^ {4}} = \frac {R _ {x} ^ {4}}{R _ {s} ^ {4}} \frac {M _ {s} ^ {2}}{M _ {x} ^ {2}}$$ $$M _ {x} ^ {2} = M _ {s} ^ {2} \frac {T _ {x} ^ {4}}{T _ {s} ^ {4}} \frac {R _ {x} ^ {4}}{R _ {s} ^ {4}}$$ $$M _ {x} = M _ {s} \frac {T _ {x} ^ {2}}{T _ {s} ^ {2}} \frac {R _ {x} ^ {2}}{R _ {s} ^ {2}}$$ $$\frac {T _ {x} ^ {2}}{T _ {s} ^ {2}} \frac {R _ {x} ^ {2}}{R _ {s} ^ {2}} = \left(\frac {1 0 ^ {8} K}{1 . 5 7 \cdot 1 0 ^ {7}} \frac {1 0 ^ {8} m}{7 \cdot 1 0 ^ {8} m}\right) ^ {2} \approx 0. 8 3$$ $$M _ {x} = 0. 8 3 \cdot M _ {s}$$

So given star have the 0.83 sun mass.

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