Question

Find the radius of neutron star.

• A. $18.3km$
• B. $9.89km$
• C. $3\times {10}^{3}km$
• D. $13.6\times {10}^{4}km$

Answer 1

The correct option is B $9.89km$

We have $\omega =\frac{2\pi }{T}$ and $K=\frac{1}{2}I{\omega }^2$
or
$K=\frac{2{\pi }^{2}I}{T^2}$
or
$\frac{dK}{dt}=-\frac{4{\pi }^{2}I}{T^2}\frac{dT}{dt}$.
This is the rate of energy loss.
Now moment of inertia
$I=\frac{P{T}^3}{4{\pi }^2}\frac{1}{dT/dt}=\frac{(5\times {10}^{31})(0.0331{)}^3}{4{\pi }^2}\frac{1}{4.22\times {10}^{-13}}=1.09\times {10}^{38}kg{m}^2$
Now radius of neutron star
$R=\sqrt{\frac{5I}{2M}}=\sqrt{\frac{5(1.09\times {10}^{38})}{2\left(1.4\right)(1.99\times {10}^{30})}}=9.9\times {10}^{3}m=9.9km$