Chapter 5, Problem 81GP

To determine

To find: The mass of the core.

To compare: The obtained mass with the mass of Sun.

Answer to Problem 81GP

Mass $=5.2\times {10}^{39}$ kg and almost ${10}^9$ times heavier than Sun.

Explanation of Solution

Given:

  $v=780$ km/s

  $r=5.7\times {10}^{17}$ m

Formula Used:

The centripetal force is

  $F_c=\frac{m{v}^2}{r}$

Gravitational force is

  $F=\frac{GMm}{r^2}$

Calculation:

The centripetal force balances the gravitational force.

  $\frac{GMm}{r^2}=\frac{m{v}^2}{r}$

Substituting the values

  $M=\frac{r{v}^2}{G}=\frac{5.7\times {10}^{17}\times {\left(78\right)}^2\times {10}^8}{6.67\times {10}^{-11}}$

That is

  $M=5.2\times {10}^{39}$ kg

The mass of the Sun is $2.0\times {10}^{30}\text{kg}$

Therefore, the number of solar masses $=\frac{M_{\text{galay}}}{M_{\text{sin}}}$

  $\begin{array}{l}=\frac{5.2\times {10}^{39}\text{kg}}{2\times {10}^{30}\text{kg}}\\ =\frac{2.6\times {10}^9\text{solar masses}}{}\end{array}$

Conclusion:

Therefore the mass of core is $5.2\times {10}^{39}$ kg which is almost ${10}^9$ times heavier than Sun.