Chapter 17, Problem 10Q

(a)

To determine

The radial velocity of a star which has a wavelength of $H_{\alpha }$ Balmer line is equal to $656.15\text{nm}$. It is given that the laboratory value of the wavelength of $H_{\alpha }$ Balmer line is $656.28\text{nm}$.

Answer to Problem 10Q

Solution:

$-59.42\text{km/s}$

Explanation of Solution

Given data:

The wavelength of $H_{\alpha }$ Balmer line in the spectrum of the star is $656.15\text{nm}$ and the laboratory value of the wavelength of $H_{\alpha }$ Balmer line is $656.28\text{nm}$.

Formula used:

Write the expression for radial velocity of the star.

$v_r=\frac{\Delta \lambda }{{\lambda }_o}c$

Here, $v_r$, $\Delta \lambda$, ${\lambda }_o$ and c are the radial velocity of the star, difference observed in the wavelength of $H_{\alpha }$ Balmer line, the laboratory value of the wavelength of $H_{\alpha }$ Balmer line when the star is not moving and the speed of light, respectively.

Write the expression for the difference in the value of wavelength of $H_{\alpha }$ Balmer line ($\Delta \lambda$).

$\Delta \lambda =\lambda -{\lambda }_o$

Here, $\lambda$ is the observed value of the wavelength of $H_{\alpha }$ Balmer line of the star.

Explanation:

The change in the observed wavelength of the spectrum of a star moving away from or moving closer to an observer is a perfect example of redshift observed due to the Doppler’s effect.

Write the expression for the difference in the values of wavelength of $H_{\alpha }$ Balmer line ($\Delta \lambda$).

$\Delta \lambda =\lambda -{\lambda }_o$

Substitute the values of $656.15\text{ nm}$ for $\lambda$ and $656.28\text{ nm}$ for ${\lambda }_o$.

$\begin{array}{c}\Delta \lambda =656.15\text{ nm}-656.28\text{ nm}\\ =-0.13\text{nm}\end{array}$

Recall the mathematical expression for the radial velocity of a star.

$v_r=\frac{\Delta \lambda }{{\lambda }_o}c$

Substitute the values of $-0.13\text{nm}$ for $\Delta \lambda$, $656.28\text{ nm}$ for ${\lambda }_o$ and $3\times {10}^8\text{m/s}$ for $c$ in the expression.

$\begin{array}{c}{v}_r=\frac{-0.13\text{nm}}{656.28\text{ nm}}\left(3\times {10}^8\text{m/s}\right)\\ \text{=}-59.42\text{km/s}\end{array}$

Conclusion:

Hence, the radial velocity of the star is $-59.42\text{km/s}$.

(b)

To determine

If a star is moving away or moving closer to an observer. It is given that the oberved value of the wavelength of $H_{\alpha }$ Balmer line of the star is $656.15\text{nm}$ and the laboratory value of the wavelength of $H_{\alpha }$ Balmer line is $656.28\text{nm}$.

Answer to Problem 10Q

Solution:

The star is moving towards the observer.

Explanation of Solution

Given data:

The oberved value of the wavelength of $H_{\alpha }$ Balmer line of the star is $656.15\text{nm}$ and the laboratory value of the wavelength of $H_{\alpha }$ Balmer line is $656.28\text{nm}$.

Explanation:

The value of the radial velocity of the star is calculated in the part (a) as $-59.42\text{km/s}$. The velocity of the star has a negative value. It implies that the star is moving towards the Earth.

Conclusion:

Hence, the star is coming closer to the observer.

(c)

To determine

The observed value of the wavelength of the $H_{\alpha }$ Balmer line of a star. It is given that the laboratory value of the wavelength of $H_{\alpha }$ Balmer line is 486.13 nm.

Answer to Problem 10Q

Solution:

$486.03\text{nm}$

Explanation of Solution

Given data:

The laboratory value of the wavelength of $H_{\alpha }$ Balmer line is $\text{486}\text{.13}\text{nm}$.

Formula used:

Write the expression for the radial velocity of a star.

$v_r=\frac{\Delta \lambda }{{\lambda }_o}c$

Here, $v_r$, $\Delta \lambda$, ${\lambda }_o$ and c are the radial velocity of a star, difference observed in the wavelength of the $H_{\alpha }$ Balmer line, the laboratory value of the wavelength of $H_{\alpha }$ Balmer line when the star is not moving and the speed of light, respectively.

Write the expression for the difference in the values of the wavelength of $H_{\alpha }$ Balmer line $\Delta \lambda$.

$\Delta \lambda =\lambda -{\lambda }_o$

Here, $\lambda$ is the observed value of wavelength of $H_{\alpha }$ Balmer line of the star.

Explanation:

Use the radial velocity of the star calculated in part (a) and use the mathematical expression to calculate the value of redshift to calculate the change in the wavelength of $H_{\alpha }$ Balmer line.

Rewrite the mathematical expression for the radial velocity of a star.

$v_r=\frac{\Delta \lambda }{{\lambda }_o}c$

Rearrange the above expression for $\Delta \lambda$.

$\Delta \lambda =\frac{v_r{\lambda }_o}{c}$

Substitute the known values of $-59.42\text{km/s}$ for $v_r$, $486.13\text{ nm}$ for ${\lambda }_o$ and $3\times {10}^5\text{ km/s}$ for c.

$\begin{array}{c}\Delta \lambda =\frac{-59.42\text{ km/s}\left(486.13\text{ nm}\right)}{3\times {10}^5\text{ km/s}}\\ =-0.096\text{nm}\\ \approx -0.10\text{nm}\end{array}$

Recall the mathematical expression for the difference in the value of the wavelength of $H_{\alpha }$ Balmer line $\Delta \lambda$.

$\Delta \lambda =\lambda -{\lambda }_o$

Rearrange the above expression for $\lambda$.

$\lambda =\Delta \lambda +{\lambda }_o$

Substitute the known values of $-0.01\text{nm}$ for $\Delta \lambda$ and $486.13\text{ nm}$ for ${\lambda }_o$.

$\begin{array}{c}\lambda =-0.10\text{nm}+486.13\text{ nm}\\ =486.03\text{nm}\end{array}$

Conclusion:

Hence, the observed value of the wavelength of $H_{\alpha }$ Balmer line of the star is $486.03\text{nm}$.

(d)

To determine

If the caluclated value of the wavelength of $H_{\alpha }$ Balmer line of a star depends on the distance of the star from the Sun.

Answer to Problem 10Q

Solution:

The calculated value of the wavelength of $H_{\alpha }$ Balmer line of a star does not depend on its distance from the Sun.

Explanation of Solution

Given data:

The laboratory value of $H_{\alpha }$ Balmer line is 486.13 nm.

Explanation:

The observed spectrum of a star depends only on the relative motion of the observer and the star. The distance between these two does not play any role in the Doppler Effect.

Conclusion:

Hence, the value of the observed wavelength calculated does not depend on the distance between the star and the Sun.