EBK AN INTRODUCTION TO MODERN ASTROPHYS
EBK AN INTRODUCTION TO MODERN ASTROPHYS
2nd Edition
ISBN: 9781108390248
Author: Carroll
Publisher: YUZU
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Chapter 7, Problem 7.1P
To determine

The proof that the semi major axis of the orbit of the reduced mass is given by a=a1+a2.

Expert Solution & Answer
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Answer to Problem 7.1P

The semi major axis of the orbit of the reduced mass is a1+a2.

Explanation of Solution

Write the expression for general relation between the distance and semi major axis of the star orbit.

    r=a(1e2)1+ecosθ        (1)

Here, r is the distance, e is the eccentricity of its orbit and a is the semi major axis.

Write the expression for distance when two stars are in different direction.

    r=r1(r2)=r1+r2        (2)

Here, r1 is the position vector of first star and r2 is the position vector of second star.

Write the expression for relation between distance of the stars and their semi major axis.

    r2r1=a2a1r2=a2a1r1        (3)

Here, a1 is the semi major axis of first star and a2 is semi major axis of second star.

Write the expression for semi major axis of reduced mass.

    a=rp+ra2        (4)

Here, rp is the perihelion distance and ra is the aphelion distance.

Conclusion:

Substitute a2a1r1 for r2 equation (2).

    r=r1+a2a1r1=r1(a1+a2a1)=(a1+a2)(r1a1)        (5)

Substitute (1e2)1+ecosθ for r1a1 in equation (4).

    r=(a1+a2)(1e2)1+ecosθ        (6)

Substitute 0° for θ in equation (6).

    rp=(a1+a2)(1e2)1+ecos(0°)=(a1+a2)(1e2)1+e=(a1+a2)(1e)

Substitute 180° for θ in Equation (6).

    ra=(a1+a2)(1e2)1+ecos(180°)=(a1+a2)(1e2)1e=(a1+a2)(1+e)

Substitute (a1+a2)(1e) for rp and (a1+a2)(1+e) for ra in Equation (4).

    a=(a1+a2)(1e)+(a1+a2)(1+e)2=2(a1+a2)2=a1+a2

Thus, the semi major axis of the orbit of the reduced mass is a1+a2.

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