Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 8, Problem 52Q

(a)

To determine

The angular distance in arcseconds between the star 2M1207 and its planet, as seen from Earth, considering this star is 170 light years from Earth.

Universe, Chapter 8, Problem 52Q

(a)

Expert Solution
Check Mark

Answer to Problem 52Q

Solution:

1.1 arcsec.

Explanation of Solution

Given data:

The star 2M1207 is 170 light years from Earth.

Formula used:

Write the expression for the small angle formula.

α=D(206265)d

Here, α is the small angle, D is the orbital distance, and d is the distance from the observer.

Explanation:

Recall the expression for the small angle formula.

α=D(206265 arcseconds)d

Substitute 55 au for D, 170 ly for d.

α=(55 au)(206265 arcsec)(170 ly)(63,240 au1 ly)=1.1 arcsec

Conclusion:

Hence, the angular distance in arcseconds between the star 2M1207 and its planet is 1.1 arcsec.

(b)

To determine

The orbital period of the orbiting star 2M1207, whose mass is 0.025 times that of the Sun, by considering that the distance between the star and its planet is the semi-major axis of the orbit.

(b)

Expert Solution
Check Mark

Answer to Problem 52Q

Solution:

2580 yrs.

Explanation of Solution

Given data:

The mass of star 2M1207 is 0.025 times that of the Sun.

Formula used:

Write the formula for the relation between orbital period and orbital distance according to Kepler’s third law.

P2=(4π2GM)a3

Here, P is the period, G is the gravitational constant, M is the mass of planet or star, and a is the orbital distance.

Explanation:

The formula for the relation between orbital period and orbital distance for Sun, according to Kepler’s third law is written as,

PSun2=(4π2GMSun)aSun3 …… (1)

Here, subscript ‘Sun’ is used for the respective quantities of the Sun.

The formula for the relation between orbital period and orbital distance for star 2M1207, according to Kepler’s third law is written as,

P2M12072=(4π2GM2M1207)a2M12073 …… (2)

Here, subscript ‘2M1207’ is used for the respective quantities of the star 2M1207.

Divide equation (2) by equation (1).

P2M12072PSun2=(4π2GM2M1207)a2M12073(4π2GMSun)aSun3P2M12072=(MSun)(a2M12073)(M2M1207)(aSun3)(PSun2)

Substitute 1 yr for PSun, 0.025MSun for M2M1207, 1 au for aSun and 55 au for a2M1207.

P2M1207=(MSun)(55 au)3(0.025MSun)(1 au)3(1 yr)2=2580 yrs

Conclusion:

Hence, the orbital period for star 2M1207 is 2580 yrs.

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