Chapter 33, Problem 45GP

To determine

Doppler shift.

Answer to Problem 45GP

Doppler shift observed from the galactic center is $2\times {10}^{-3}$

Explanation of Solution

Given:

At the galactic center, super massive black hole carries mass of about 2 billion solar masses. The radius from the galactic center to the gas clouds is 60 light-years.

Formula used:

Doppler shift is given as

  $\lambda ={\lambda }_0\sqrt{\frac{1+v/c}{1-v/c}}$

Calculation:

Since gravity causes centripetal force. Thus, speed of the rotating gas would be

  $\begin{array}{c}{F}_{gravity}=F_{centripetal}\\ G\frac{m_{gas}{m}_{blackhole}}{r^2}=\frac{m_{gas}{v}_{gas}^2}{r}\\ v_{gas}=\sqrt{\frac{G{m}_{blackhole}}{r}}\\ v_{gas}=\sqrt{\frac{(6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^2{\text{/kg}}^2)(2\times {10}^9)(1.99\times {10}^{30}\text{ kg})}{(60\text{ ly})\left(\frac{9.46\times {10}^{15}\text{ m}}{1\text{ ly}}\right)}}\\ =6.84\times {10}^5\text{ m}/s\end{array}$

Using red-shift equation,

  $\begin{array}{c}z=\frac{\Delta \lambda }{{\lambda }_0}\\ =\frac{v}{c}\\ =\frac{6.84\times {10}^5\text{ m/s}}{3\times {10}^8\text{ m/s}}\\ \approx 2\times {10}^{-3}\end{array}$

Conclusion:

Thus, Doppler shift observed from the galactic center is $2\times {10}^{-3}$