Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 19, Problem 42Q

(a)

To determine

The average separation between the two stars of the overcontact binary system W Ursae in kilometers. It is given that the two stars have mass 0.99 M and 0.62 M, repectively.

(a)

Expert Solution
Check Mark

Answer to Problem 42Q

Solution:

1.65×106km

Explanation of Solution

Given data:

The mass of two stars are 0.99 M and 0.62 M.

Formula used:

Write the expression for Newton’s form of Kepler’s third law:

(m1+m2)p2=a3

Here, m1 is the mass of the first star, m2 is the mass of the second star, p is the period of revolution of one star around the other in years, and a is the semi-major axis of the orbit of one star around the other. In the above formula, if the two masses are in terms of the solar mass (M) and the period is in years, then the semi-major axis is in astronomical unit (au).

In case of binary stars, a becomes the distance between the two stars.

Explanation:

It is known that the period of the binary star system W Ursae is of 8 hours. Convert this time period in years.

8hours=13days=13×365years=11095years

Now, recall the expression for Newton’s form of Kepler’s third law.

(m1+m2)p2=a3

Substitute 0.99M for m1, 0.62M for m2, and 11095years for p and calculate a.

a3=(0.99M+0.62M)(11095)2=1.34×106(au)3a=1.10×102au

Convert a from au to km using the following conversion,

1au=1.496×108km

So, the distance between the two binary stars is,

a=(1.10×102)(1.496×108)=1.65×106km

Conclusion:

Hence, the value of average separation between the stars is 1.65×106km.

(b)

To determine

To check: That W Ursae is an overcontact binary star system, using the radii of its two stars given as 1.14R and 0.83R and the result obtained in part (a).

(b)

Expert Solution
Check Mark

Answer to Problem 42Q

Solution:

W Ursae is an overcontact binary star system since the distance between the two binary stars in is almost comparable to the sum of their individual radii.

Explanation of Solution

Introduction:

An overcontact binary star system is one where the two stars are in contact with each other.

The radii of the two stars of W Ursae are 1.14R and 0.83R, respectively. For this system to be an overcontact binary, the sum of these radii should be equal to the distance between the stars.

Explanation:

Caluclate the sum of the two given radii as ‘Rt’.

Rt=1.14R+0.83R=1.97R

Convert unit of Rt to km.

Rt=(1.97)(6.96×105)=1.37×106km

Now, compare the value of Rt with the value calculated in part (a), and observe that they are close to each other. This proves that the system is an overcontact binary.

Conclusion:

Hence, it is found that the W Ursae is an overcontact binary star system as the sum of the radii of the two stars is almost the same as the distance between them.

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