Chapter 1, Problem 1.9P

(a)

To determine

The coordinate of Proxima Centauri to epoch $\text{J2010}\text{.0}$.

Answer to Problem 1.9P

The processed coordinates are $\alpha ={14}^h{30}^m{29.4}^s$ and $\delta =-62°42^{\prime }25.26^{″}$.

Explanation of Solution

Write the expression for the changes in the coordinates

    $\Delta \alpha =M+N\sin \alpha \sin \delta$        (1)

Write the expression for the changes in Coordinate.

    $\Delta \delta =N\cos \alpha$        (2)

Here, $\Delta \alpha$ and $\Delta \delta$ are the change in coordinate.

Conclusion:

Substitute $217.2619°$ for $\alpha$ and $-62.6794°$ for $\delta$ $0.128127°$ for $M$ and $0.055674°$ for $N$ in equation (1)

    $\begin{array}{c}\Delta \alpha =\left(0.128127°\right)+\left(0.055674°\right)\sin \left(217.2619°\right)\sin \left(-62.6794°\right)\\ =0.193628°\end{array}$

Substitute $217.2619°$ for $\alpha$ and $M$ and $0.055674°$ equation (2).

    $\begin{array}{c}\Delta \delta =\left(0.055674°\right)\cos \left(217.2619°\right)\\ =-0.044211°\end{array}$

Thus, the processed coordinates are $\alpha ={14}^h{30}^m{29.4}^s$ and $\delta =-62°42^{\prime }25.26^{″}$.

(b)

To determine

The change in $\alpha$ and $\delta$ due to proper motion between $2000$ and $2010$.

Answer to Problem 1.9P

The change in $\alpha$ is $-{5.46}^{\text{s}}$ and the change in $\delta$ is ${7.984}^{\text{s}}$.

Explanation of Solution

Write the expression for the change $\alpha$.

    $\Delta \alpha =\left(\mu t\right)\frac{\sin \varphi }{\cos \delta }$        (3)

Here, $\Delta \alpha$ is change in $\alpha$, $\mu$ is the proxima centauri and $t$ is the difference in years.

Write the expression for change in $\delta$.

    $\Delta \delta =\left(\mu t\right)\cos \varphi$        (4)

Here, $\Delta \delta$ is the change in $\delta$.

Conclusion:

Substitute $3.84^{″}{\text{yr}}^{-1}$ for $\mu$, $10\text{yr}$ for $t$, $-62.6794°$ for $\delta$ and ${282}^{\text{o}}$ for $\varphi$ in equation (3)

    $\begin{array}{c}\Delta \alpha =\left(3.84^{″}\left(\frac{1^{\text{s}}}{15^{″}}\right)\times 10\text{yr}\right)\frac{\sin 282°}{\cos \left(-62.6794°\right)}\\ =-{5.46}^{\text{o}}\end{array}$

Substitute $3.84^{″}{\text{yr}}^{-1}$ for $\mu$, $10\text{yr}$ for $t$, ${282}^{\text{o}}$ for $\varphi$ in equation (4).

    $\begin{array}{c}\Delta \delta =\left(3.84^{″}\left(\frac{1^{\text{s}}}{15^{″}}\right)\times 10\text{yr}\right)\cos \left(-62.6794°\right)\\ =7.984^{″}\end{array}$

Thus, the change in $\alpha$ is $-{5.46}^{\text{s}}$ and the change in $\delta$ is ${7.984}^{\text{s}}$.

(c)

To determine

The factor which has largest contribution for changes in coordinates.

Answer to Problem 1.9P

The Precession has largest contribution for changes in coordinates.

Explanation of Solution

Proxima Centauri is red star which has the mass eight times greater than the Sun with average density 33 times greater than the Sun. the orbital period of the Proxima Centauri is 550000 years.

Since the change in value of $\alpha$ is higher than the change in value of $\delta$ ( the changes in $\delta$ is in $7.984^{″}$ and the change in $\alpha$ is $5.46°$).

Conclusion:

Therefore, the Precession has the largest contribution for changes in the coordinates of Proxima Centauri.