Chapter 23, Problem 30Q

To determine

(a)

The recessional velocity of the RD1 galaxy.

Answer to Problem 30Q

$2.79\times {10}^5{\text{kms}}^{\text{-1}}$ and it is $0.93c$

Explanation of Solution

Given:

The redshift $\left(z\right)$ of the galaxy is $\mathrm{5.34.}$

Formula Used:

$z=\frac{v}{c}$

Where,

$\begin{array}{c}z=\text{red shift of the object}\\ v=\text{recessional velocity of a galaxy}\\ c=\text{speed of light}\end{array}$

Calculation:

$\begin{array}{l}z=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}\\ \Rightarrow z^2=\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\\ \Rightarrow z^2\times \left(1-\frac{v}{c}\right)=1+\frac{v}{c}\\ \Rightarrow z^2-\frac{z^{2}v}{c}=1+\frac{v}{c}\\ \Rightarrow z^2-1=\frac{v}{c}+\frac{z^{2}v}{c}=\frac{v}{c}\left(1+z^2\right)\\ \Rightarrow \frac{v}{c}=\frac{z^2-1}{z^2+1}=\frac{{\left(5.34\right)}^2-1}{{\left(5.34\right)}^2+1}=0.93\end{array}$

$\begin{array}{c}v=\text{0}{\text{.93×3×10}}^{\text{8}}{\text{ms}}^{\text{-1}}\\ =2.8\times {10}^{\text{8}}{\text{ms}}^{-1}\\ =2.8\times {10}^5\text{km/s}\end{array}$

Conclusion:

Therefore, the recessional velocity of the RD1 galaxy is $2.8\times {10}^{\text{5}}{\text{kms}}^{\text{-1}}$ and it is $0.93$ as a fraction of the speed of light.

To determine

(b)

The recessional velocity of the RD1 galaxy, if erroneously used the low-speed formula for the calculation and whether it is a large error or not.

Answer to Problem 30Q

The recessional speed obtained of the galaxy using the low-speed formula is $1.6\times {10}^{6}km/s$ and it is a large error using this formula to calculate the recessional speed.

Explanation of Solution

Given data:

The recessional speed of the galaxy is $1.6\times {10}^{6}km/s.$

Formula used:

The recessional velocity of a galaxy can be found by using the formula (however valid for low speeds only),

$z=\frac{v}{c}$

Where,

$\begin{array}{c}z=\text{red shift of the object}\\ v=\text{recessional velocity of a galaxy}\\ c=\text{speed of light}\end{array}$

Calculation:

$\begin{array}{c}z=\frac{v}{c}\\ v=5.34\times 3\times {10}^{8}m{s}^{-1}\\ =1.6\times {10}^{9}m{s}^{-1}\\ =1.6\times {10}^{6}km/s\end{array}$

From the calculation, the recessional velocity can be obtained as $1.6\times {10}^{6}km/s$ which exceeds the ultimate speed limit. Whenever redshift is larger than $1.5$, that means the larger expansion in universe and that effect cannot be explained using the Doppler shift. Although this answer isn’t violating the Einstein’s laws, it is a large error using this formula to calculate the recessional speed of the galaxy.

Conclusion:

The recessional speed obtained of the galaxy using the low-speed formula is $1.6\times {10}^{6}km/s$ and it is a large error using this formula to calculate the recessional speed of the galaxy since the formula is valid only for redshifts lower than $\mathrm{1.5.}$

To determine

(c)

The distance from the Earth to the galaxy RD1 according to the Hubble law.

Answer to Problem 30Q

The distance from the Earth to the galaxy RD1 according to the Hubble law is $3,821.9Mpc\text{ or }1.24\times {10}^{10}\text{light years}\text{.}$

Explanation of Solution

Given data:

Hubble constant $H_0=73km/s/Mpc.$

$1Mpc=3.26\times {10}^6\text{light years}$

Formula used:

Hubble law can be interpreted as

$v=H_{0}d$

Where,

$\begin{array}{c}v=\text{recessional velocity of a galaxy}\\ H_0=\text{Hubble constant}\\ d=\text{diatance to the galaxy}\end{array}$

Calculation:

Rearranging the formula,

$\begin{array}{c}d=\frac{v}{H_0}\\ =\frac{2.79\times {10}^{5}km{s}^{-1}}{73km/s/Mpc}\\ =3,821.9Mpc\\ \\ d=3,821.9\times 3.26\times {10}^6\text{light years}\\ \text{=1}\text{.24}\times {\text{10}}^{10}\text{light years}\end{array}$

Conclusion:

Therefore, the distance from the Earth to the galaxy RD1 according to the Hubble law is $3,821.9Mpc\text{ or }1.24\times {10}^{10}\text{light years}\text{.}$