Answer to Question #105844 in Astronomy | Astrophysics for hitendra

Question

A main sequence star has mass 2×1031 kg and radius 3×109
m. Obtain an estimate of the
average temperature throughout the star. Examine if Newton’s theory would be adequate
for the study of this star.

Answer 1

For a main sequence star with mass 2×10³¹ kg and radius 3×10⁹ m the average temperature can be found relative to the Sun according to the following expression:

$$\frac{L_{\text{star}}}{L_{\text{Sun}}}=\bigg(\frac{r_{\text{star}}}{r_{\text{Sun}}}\bigg)^2\bigg(\frac{T_{\text{star}}}{T_{\text{Sun}}}\bigg)^4.$$

Here we have:

$T_{\text{Sun}}=5778\text{ K},\\ r_{\text{Sun}}=6.96\cdot10^8\text{ m},\\ m_{\text{Sun}}=1.9891\cdot10^{30}\text{ kg}.$

Find the mass ratio relative to the Sun:

$$m=\frac{m_{\text{star}}}{m_\text{{Sun}}}=\frac{2\cdot10^{31}}{1.9891\cdot10^{30}}=10.05.$$

For such a ratio, since $2<m<20$, we have

$$\frac{L_{\text{star}}}{L_{\text{Sun}}}=m^{3.5},\\ \space\\ T_{\text{star}}=T_{\text{Sun}}m^{\frac{3.5}{4}}\sqrt{\frac{r_{\text{Sun}}}{r_{\text{star}}}}=20961\text{ K}.$$