EBK AN INTRODUCTION TO MODERN ASTROPHYS
EBK AN INTRODUCTION TO MODERN ASTROPHYS
2nd Edition
ISBN: 9781108390248
Author: Carroll
Publisher: YUZU
bartleby

Concept explainers

Question
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
Check Mark

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.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Knowledge Booster
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Recommended textbooks for you
  • College Physics
    Physics
    ISBN:9781305952300
    Author:Raymond A. Serway, Chris Vuille
    Publisher:Cengage Learning
    University Physics (14th Edition)
    Physics
    ISBN:9780133969290
    Author:Hugh D. Young, Roger A. Freedman
    Publisher:PEARSON
    Introduction To Quantum Mechanics
    Physics
    ISBN:9781107189638
    Author:Griffiths, David J., Schroeter, Darrell F.
    Publisher:Cambridge University Press
  • Physics for Scientists and Engineers
    Physics
    ISBN:9781337553278
    Author:Raymond A. Serway, John W. Jewett
    Publisher:Cengage Learning
    Lecture- Tutorials for Introductory Astronomy
    Physics
    ISBN:9780321820464
    Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
    Publisher:Addison-Wesley
    College Physics: A Strategic Approach (4th Editio...
    Physics
    ISBN:9780134609034
    Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
    Publisher:PEARSON
  • College Physics
    Physics
    ISBN:9781305952300
    Author:Raymond A. Serway, Chris Vuille
    Publisher:Cengage Learning
    University Physics (14th Edition)
    Physics
    ISBN:9780133969290
    Author:Hugh D. Young, Roger A. Freedman
    Publisher:PEARSON
    Introduction To Quantum Mechanics
    Physics
    ISBN:9781107189638
    Author:Griffiths, David J., Schroeter, Darrell F.
    Publisher:Cambridge University Press
    Physics for Scientists and Engineers
    Physics
    ISBN:9781337553278
    Author:Raymond A. Serway, John W. Jewett
    Publisher:Cengage Learning
    Lecture- Tutorials for Introductory Astronomy
    Physics
    ISBN:9780321820464
    Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
    Publisher:Addison-Wesley
    College Physics: A Strategic Approach (4th Editio...
    Physics
    ISBN:9780134609034
    Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
    Publisher:PEARSON