Answer to Question #105844 in Astronomy | Astrophysics for hitendra

Question #105844
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.
1
Expert's answer
2020-03-19T11:30:55-0400

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:


LstarLSun=(rstarrSun)2(TstarTSun)4.\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:

TSun=5778 K,rSun=6.96108 m,mSun=1.98911030 kg.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=mstarmSun=210311.98911030=10.05.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<202<m<20, we have


LstarLSun=m3.5, Tstar=TSunm3.54rSunrstar=20961 K.\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}.

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