Chapter 5, Problem 5.1P

(a)

To determine

The radial velocity of Barnard’s star.

Answer to Problem 5.1P

The radial velocity of Barnard’s star is $113\text{km}/\text{s}$.

Explanation of Solution

Write the expression for radial velocity of Barnard’s star.

    $v_r=c\frac{\left({\lambda }_{\text{obs}}-{\lambda }_{\text{rest}}\right)}{{\lambda }_{\text{rest}}}$        (1)

Here, $v_r$ is the radial velocity, $c$ is the speed of light, ${\lambda }_{\text{obs}}$ is the observed wavelength, ${\lambda }_{\text{rest}}$ is the rest wavelength.

Conclusion:

Substitute $3\times {10}^8\text{m}/\text{s}$ for $c$, $656.034\text{nm}$ for ${\lambda }_{\text{obs}}$, $656.281\text{nm}$ for ${\lambda }_{\text{rest}}$ in the equation (1).

    $\begin{array}{c}{v}_r=\left(3\times {10}^8\text{m}/\text{s}\right)\frac{\left(656.034\text{nm}-656.281\text{nm}\right)}{656.281\text{nm}}\\ =\left(3\times {10}^8\frac{\left(1\text{m}\times \frac{1\text{km}}{1000\text{m}}\right)}{\text{s}}\right)\frac{\left(656.034\text{nm}-656.281\text{nm}\right)}{656.281\text{nm}}\\ =113\text{km}/\text{s}\end{array}$

Thus, the radial velocity of Barnard’s star is $113\text{km}/\text{s}$.

(b)

To determine

The transverse velocity of Barnard’s star.

Answer to Problem 5.1P

The transverse velocity of Barnard’s star is $89.4\text{km}/\text{s}$.

Explanation of Solution

Write the expression for distance between star and earth.

    $r=\frac{1}{p^{″}}\text{pc}$        (2)

Here, $p^{″}$ is the parallax angle.

Write the expression for transverse velocity of Barnard’s star.

    $v_{\theta }=r\mu$        (3)

Here, $r$ is the distance between star and earth, $\mu$ is the proper motion of the star.

Conclusion:

Substitute $0.54901$ for $p^{″}$ in equation (2).

    $\begin{array}{c}r=\frac{1}{0.54901}\text{pc}\\ =\text{1}\text{.82}\text{pc}\end{array}$

Substitute $\text{1}\text{.82}\text{pc}$ for $r$, $10.3577^{″}{\text{yr}}^{-1}$ for $\mu$ in equation (3).

    $\begin{array}{c}{v}_{\theta }=\left(\text{1}\text{.82}\text{pc}\right)\left(10.3577^{″}{\text{yr}}^{-1}\right)\\ =89.4\text{km}/\text{s}\end{array}$

Thus, the transverse velocity of Barnard’s star is $89.4\text{km}/\text{s}$.

(c)

To determine

The speed of Barnard’s star through space.

Answer to Problem 5.1P

The speed of Barnard’s star through space is $144.09\text{km}/\text{s}$.

Explanation of Solution

Write the expression for speed of Barnard’s star.

    $v=\sqrt{v_r^2+v_{\theta }^2}$        (4)

Here, $v_r$ is the radial velocity Barnard’s star, $v_{\theta }$ is transverse velocity of Barnard’s star.

Conclusion:

Substitute $113\text{km}/\text{s}$ for $v_r$ and $89.4\text{km}/\text{s}$ for $v_{\theta }$ in the equation (4).

    $\begin{array}{c}v=\sqrt{{\left(113\text{km}/\text{s}\right)}^2+{\left(89.4\text{km}/\text{s}\right)}^2}\\ =\sqrt{20761.3600{\text{km}}^2/{\text{s}}^2}\\ =144.09\text{km}/\text{s}\end{array}$

Thus, the speed of Barnard’s star through space is $144.09\text{km}/\text{s}$.