Chapter 13, Problem 38QAP

(a)

To determine

The distance of the distant star.

Answer to Problem 38QAP

The distance of the distant star is $30303.77\text{parsec}$.

Explanation of Solution

Write expression for distance of delta star.

    $d=\frac{1}{p}$        (1)

Here, $d$ is distance of delta star and $p$ is parallax.

Write expression for luminosity of delta star.

    $L_{Delta}=335\left(P\right){\text{L}}_{\text{Sun}}$        (2)

Here, $L_{Delta}$ is luminosity of delta star and $P$ is period.

Write expression for luminosity of distant star.

    $L_D=335\left(P_D\right){\text{L}}_{\text{Sun}}$        (3)

Here, $L_D$ is luminosity of distant star and $P_D$ is period of distant star.

Write expression for apparent brightness of delta star.

    $B_{Delta}=\frac{L_{Delta}}{4\pi {\left(d\right)}^2}$        (4)

Here, $B_{Delta}$ is apparent brightness of star.

Write expression for apparent brightness of distant star.

    $B_D=\frac{L_D}{4\pi {\left(d_D\right)}^2}$

Here, $B_D$ is apparent brightness of distant star and $d_D$ is the distance of distant star.

Rearrange above expression for $d_D$.

    $d_D=\sqrt{\frac{L_D}{4\pi B_D}}$        (5)

Conclusion:

Substitute $0.0033\mathrm{arcsec}$ for $p$ in equation (1).

    $\begin{array}{l}d=\frac{1}{0.0033\mathrm{arcsec}}\\ d=303.03\text{parsec}\end{array}$

Substitute $5.4\text{days}$ for $P$ in equation (2).

    $\begin{array}{c}{L}_{Delta}=335\left(5.4\text{days}\right)\\ =1809{\text{L}}_{\text{Sun}}\end{array}$

Substitute $54\text{days}$ for $P_D$ in equation (3).

    $\begin{array}{c}{L}_D=335\left(54\text{days}\right){\text{L}}_{\text{Sun}}\\ =18090{\text{L}}_{\text{Sun}}\end{array}$

Substitute $1809{\text{L}}_{\text{Sun}}$ for $L_{Delta}$ and $303.03\text{parsec}$ for $d$ in equation (4).

    $\begin{array}{c}{B}_{Delta}=\frac{1809{\text{L}}_{\text{Sun}}}{4\pi {\left(303.03\text{parsec}\right)}^2}\\ =1.5656\times {10}^{-3}{\text{L}}_{\text{Sun}}{\text{arcsec}}^{\text{-2}}\end{array}$

Substitute $18090{\text{L}}_{\text{Sun}}$ for $L_D$ and $0.001B_{Delta}$ for $B_D$.

    $d_D=\sqrt{\frac{18090{\text{L}}_{\text{Sun}}}{4\pi \left(0.001B_{Delta}\right)}}$

Substitute $1.5656\times {10}^{-3}{\text{L}}_{\text{Sun}}{\text{arcsec}}^{\text{-2}}$ for $B_{Delta}$ in above expression.

    $\begin{array}{l}{d}_D=\sqrt{\frac{18090{\text{L}}_{\text{Sun}}}{4\pi \left(0.001\left(1.5656\times {10}^{-3}{\text{L}}_{\text{Sun}}{\text{arcsec}}^{\text{-2}}\right)\right)}}\\ d_D=30303.77\text{parsec}\end{array}$

Thus, the distance of the distant star is $30303.77\text{parsec}$.

(b)

To determine

The distance of more distant Cepheid variable is measurable by parallax.

Answer to Problem 38QAP

The distance of more distant star cannot be measured by parallax method.

Explanation of Solution

To measure the distance of any astronomical object requires a very large baseline and highly precise angular measurements. Due to revolution of Earth around Sun, the direction of a nearby stars changes from Earth. Nearby stars appear to move back and forth relative to more distant stars and the motion known as stellar parallax.

Cepheids have very high luminosities, they are variable stars in which the period of variability is directly related to the luminosity. As they have very high luminosity they are easy to detect and only period of variability is required to find the true luminosities. From the apparent brightness of the Cepheid, the distance can easily be calculated.

The distance of distant star is more than $1000\text{parsec}$. The parallax method is not applicable to measure distance of such distant stars. Thus, the distance of more distant star cannot be measured by parallax method.

Conclusion:

Thus, the distance of more distant star cannot be measured by parallax method.