Quadrupole tensor for spherical star (schutz ch9q28))

In summary, the conversation discusses the calculation of the quadrupole tensor and its traceless counterpart for a spherical mass distribution, with the use of transformation equations from Cartesian to spherical coordinates. The attempt at a solution includes integration using spherical polars, but the correct answer is found by transforming the components of the tensor.
  • #1
Mmmm
63
0

Homework Statement



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.

Homework 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]

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...
 
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  • #2
When you change to spherical coordinates, x1x1 will become r2sin2(θ)cos2(φ), not r2.
 
  • #3
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)
 
  • #4
Oh dear... yes you are right...
how silly of me :redface:
Thanks dx.
 

1. What is a quadrupole tensor for a spherical star?

A quadrupole tensor is a mathematical representation of the mass distribution within a spherical star. It describes the shape and orientation of the star's mass relative to its center of mass.

2. How is the quadrupole tensor calculated?

The quadrupole tensor can be calculated using the moments of inertia of the star, which are related to the distribution of mass within the star. These moments of inertia are calculated by integrating over the volume of the star.

3. What is the significance of the quadrupole tensor in astrophysics?

The quadrupole tensor is used to study the internal structure and dynamics of stars. It can provide information about the star's rotation, stability, and the presence of internal density variations. It is also important in understanding gravitational waves and their effects on stars.

4. Can the quadrupole tensor be used to determine a star's shape?

No, the quadrupole tensor only describes the mass distribution within a star, not its actual physical shape. A star's shape can be affected by factors such as rotation and tidal forces, which are not captured by the quadrupole tensor.

5. How is the quadrupole tensor related to the moment of inertia tensor?

The quadrupole tensor is a sub-component of the moment of inertia tensor, which also includes the mass and angular momentum of the star. The quadrupole tensor is specifically related to the second-order moments of inertia, while the moment of inertia tensor includes all orders of moments.

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