Question

the average temperature of the interior of a sun like star is of order 10^8k. estimate the mass of the star in terms of the solar mass if it has a radius of order 10^10cm

Answer 1

Answer on Question 54973, Physics / Astronomy | Astrophysics

Question:

The average temperature of the interior of a sun like star is of 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.

Solution:

We can use temperature, mass relation:

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

We can apply this for both stars. For sun:

$$T_{x}^{4} \approx \frac{M_{x}^{2}}{R_{x}^{4}}$$

For unknown star:

$$T_{x}^{4} \approx \frac{M_{x}^{2}}{R_{x}^{4}}$$ $$\frac{T_{x}^{4}}{T_{x}^{4}} \approx \frac{R_{x}^{4} M_{x}^{2}}{R_{x}^{4} M_{x}^{2}}$$ $$M_{x}^{2} = M_{x}^{2} \frac{T_{x}^{4} R_{x}^{4}}{T_{x}^{4} R_{x}^{4}}$$ $$M_{x} = M_{x} \frac{T_{x}^{2} R_{x}^{2}}{T_{x}^{2} R_{x}^{2}}$$ $$\frac{T_{x}^{2} R_{x}^{2}}{T_{x}^{2} R_{x}^{2}} = \left(\frac{10^{8} K}{1.57 \times 10^{7}} \times \frac{10^{8} m}{7 \times 10^{8} m}\right)^{2} \approx 0.83$$ $$M_{x} = 0.83 M_{x}$$

Answer: $M_{x} = 0.83 M_{x}$

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