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Quadrupole tensor for spherical star (schutz ch9q28))

  1. May 20, 2009 #1
    1. The problem statement, all variables and given/known data

    28. Calculate the quadrupole tensor [itex]I_{jk}[/itex] and its traceless counterpart [itex]\overline{I}_{jk}[/itex] (Eq. (9.78)) for the following mass distributions.
    (a) A spherical star whose density is [itex]\rho(r, t)[/itex]. Take the origin of the coordinates in Eq. (9.73) to be the center of the star.

    2. Relevant equations

    (9.78)
    [tex]I^{jk}=\int T^{00}x^ix^jd^3x[/tex]

    (9.73)
    [tex]\overline{I}_{jk}=I_{jk}-\frac{1}{3}\delta_{jk}I^l_l[/tex]

    3. The attempt at a solution

    Well, first of all [itex]T^{00}=\rho[/itex]
    so 9.78 becomes
    [tex]I^{jk}=\int \rho(r,t)x^ix^jd^3x[/tex]
    and then it should just be a simple case of integrating using spherical polars....right?

    so for [itex]I^{11}[/itex] I should have
    [tex]I^{11}=\int^r_0\int^{2\pi}_0\int^{\pi}_0 \rho(r,t)r^2 r^2sin\theta d\theta d\phi dr[/tex]
    [tex]=\int^r_0\int^{2\pi}_0 2 \rho(r,t)r^4 d\phi dr[/tex]
    [tex]=\int^r_0 4\pi \rho(r,t)r^4dr[/tex]
    [tex]=4\pi \int^r_0 \rho(r,t)r^4 dr[/tex]


    and to me all seems to have gone well, but the answer in the back of the book is
    [tex]=\frac{4\pi}{3} \delta^{ij} \int^r_0 \rho(r,t)r^4 dr[/tex]

    which is close, but not the same. If I look at other components of I they are nothing like this at all! even the components with different j & k don't vanish!

    I'm obviously completely misunderstanding something here....
     
  2. jcsd
  3. May 20, 2009 #2

    dx

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    When you change to spherical coordinates, x1x1 will become r2sin2(θ)cos2(φ), not r2.
     
  4. May 20, 2009 #3

    dx

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    Just to clarify a little bit, x1 = x = r⋅sin(θ)cos(φ), so x1x1 = [r⋅sin(θ)cos(φ)]2.

    The transformation equations from Cartesian to spherical are

    x = r⋅sin(θ)cos(φ)
    y = r⋅sin(θ)sin(φ)
    z = r⋅cos(θ)
     
  5. May 21, 2009 #4
    Oh dear..... yes you are right...
    how silly of me :redface:
    Thanks dx.
     
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