Concept explainers
(a)
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.
(a)
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.
Here,
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.
Substitute
Conclusion:
Hence, the distance of the visual binary 70 Ophiuchi from the Earth is 5 pc.
(b)
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.
(b)
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.
Here,
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).
Here,
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.
Substitute
Conclusion:
Hence, the actual length of the semi-major axis of 70 Ophiuchi is 22.5 au.
(c)
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.
(c)
Answer to Problem 68Q
Solution:
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.
Here,
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).
Here,
Write the expression of Kepler’s third law for a binary star system.
Here,
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
Substitute
Conclusion:
Hence, the sum of the masses of the two stars in binary visual 70 Ophiuchi is
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Chapter 17 Solutions
Universe
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