Chapter 17, Problem 68Q

(a)

To determine

The distance to the visual binary 70 Ophiuchi from the Earth in parsecs. It is given that the observed length of the semi major axis through a telescope is 4.5 arcsecs and the parallax angle of 70 Ophiuchi is 0.2 arcsecs.

Answer to Problem 68Q

Solution:

The distance of 70 Ophiuchi from the the Earth is 5 pc.

Explanation of Solution

Given data:

The observed length of the semi major axis through a telescope is 4.5 arcsecs and the parallax angle of 70 Ophiuchi is 0.2 arcsecs.

Formula used:

The expression for distance of a star in pc when its parallax angle is known is given blow.

$d=\frac{1}{p}$

Here, $d$ is the distance of the star from the observer (Earth) in parsecs and $p$ is its parallax angle in arc second.

Explanation:

The parallax method is very useful to find the distance of far off objects in the universe. By knowing the parallax angle of the object one can find out its distance from the point of observation.

Refer to the expression for distance of a star in pc when its parallax angle is known.

$d=\frac{1}{p}$

Substitute $0.2\text{arsec}$ for $p$.

$\begin{array}{c}d=\frac{1}{0.2}\\ =5\text{pc}\end{array}$

Conclusion:

Hence, the distance of the visual binary 70 Ophiuchi from the Earth is 5 pc.

(b)

To determine

The length of the semi-major axis of the visual binary 70 Ophiuchi in astronomical unit. It is given that the observed length of the semi major axis through a telescope is 4.5 arcsecs and the parallax angle of 70 Ophiuchi is 0.2 arcsecs.

Answer to Problem 68Q

Solution:

The actual length of the semi-major axis is 22.5 au.

Explanation of Solution

Given data:

The apparent length of the semi-major axis when observed using a telescope is 4.5 arcsecs and the parallax angle of 70 Ophiuchi is 0.2 arcsecs.

Formula used:

Write the expression for distance of a star in pc when its parallax angle is known.

$d=\frac{1}{p}$

Here, $d\text{and}p$ are the distance of the star from the observer (Earth) in parsecs and its parallax angle in arcsecs, respectively.

Write the expression of small-angle formula (when distance between the star and the observer is measured in pc and the apparent length is measured in arcsecs).

$a=\alpha d$

Here, $a\text{,}\alpha \text{and}d$ are the actual length of the semi-major axis (in au), apparent length of the semi-major axis (in arcsecs) and the distance between the binary star system and the observer (in pc), respectively.

Explanation:

Use the distance between the observer and 70 Ophiuchi calculated using parallax method in the previous part to calculate the length of the semi-major axis.

Refer to the expression for actual length of semi-major axis.

$a=\alpha \left(\mathrm{arcsec}\right)\times d\left(\text{pc}\right)$

Substitute $5\text{ parsec}$ for $d$ [Refer to part (a)] and $4.5\text{ arec}$ for $\alpha$

$\begin{array}{c}a=\left(4.5\text{ arsec}\right)\times \left(5\text{ parsec}\right)\\ =22.5\text{ au}\end{array}$

Conclusion:

Hence, the actual length of the semi-major axis of 70 Ophiuchi is 22.5 au.

(c)

To determine

The sum of the masses of the two stars in the visual binary 70 Ophiuchi in terms of the solar mass. It is given that the time period, the apparent length of the semi-major axis and the parallax angle of 70 Ophiuchi are 87.7 years, 4.5 arcsecs and 0.2 arcsecs, respectively.

Answer to Problem 68Q

Solution:

$1.5{\text{ M}}_{\odot }$

Explanation of Solution

Given data:

The time period, the apparent length of the semi-major axis and the parallax angle of 70 Ophiuchi are 87.7 years, 4.5 arcsecs and 0.2 arcsecs, respectively.

Formula used:

Write the expression for the distance of a star in pc when its parallax angle is known.

$d=\frac{1}{p}$

Here, $d\text{and}p$ are the distance of the star from the observer (Earth) in parsecs and its parallax angle in arcsecs, respectively.

Write the small-angle formula (when distance between the star and the observer is measured in pc and the apparent length is measured in arcsecs).

$a=\alpha d$

Here, $a\text{,}\alpha \text{and}d$ are the actual length of the semi-major axis (in au), apparent length of the semi-major axis (in arcsecs) and the distance between the binary star system and the observer (in pc) respectively.

Write the expression of Kepler’s third law for a binary star system.

$M_1+M_2=\frac{a^3}{P^2}$

Here, $M_1\text{and}{M}_2$ are the masses of the two stars in 70 Ophiuchi, $a$ is the acutal length of the semi-major axis of the orbit of the binary system and $P$ is the period of the binary system.

Explanation:

Use the value of the actual length of the semi-major axis calculated in part (b).

Refer to the expression of Kepler’s third law and substitute this value of $a$ and time period as 87.7 years to calculate the sum of the masses of the two stars.

$M_1+M_2=\frac{a^3}{P^2}$

Substitute $22.5\text{ au}$ for $a$ [Refer to part (b)] and $87.7\text{ year}$ for $P$.

$\begin{array}{c}{M}_1+M_2=\frac{{\left(22.5\text{ au}\right)}^3}{{\left(87.7\text{ year}\right)}^2}\\ =1.5{\text{ M}}_{\odot }\end{array}$

Conclusion:

Hence, the sum of the masses of the two stars in binary visual 70 Ophiuchi is $1.5{\text{ M}}_{\odot }$.