- #1

stevebd1

Gold Member

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## Main Question or Discussion Point

Can anyone shed some light on how to calculate the angular momentum of an oblate spheroid (i.e. a rapidly rotating neutron star). I'm aware the following applies for various shapes-

[tex]J=I\omega[/tex]

where

[tex]\omega=\frac{v}{r}[/tex]

and for a solid sphere-

[tex]I=\frac{2}{5}mr^{2}[/tex]

for a solid disk or cylinder-

[tex]I=\frac{1}{2}mr^{2}[/tex]

for a ring or loop-

[tex]I=mr^{2}[/tex]

where J is the angular momentum, I is the inertia, v is the velocity of the outer edge, m is the mass and r is the radius to the outer edge of the object.

In some cases, I've seen the equation for the solid sphere apply to oblate spheroids but I've also seen ellipticity allowed for. In most cases, this is expressed as the following-

[tex]\epsilon=\frac{a-b}{a}[/tex]

where epsilon is the factor of ellipticity and a is the long radius and b the short radius of the spheroid. I've seen it expressed as the following in equations for oblate spheroids-

[tex]I=\frac{2}{5}mr^{2}(1+\epsilon)[/tex]

in some cases, a and b are expressed as inertias as if the inertia had been calculated for each radius as solid spheres and then plugged into the ellipticity equation in order to get a ratio of some kind.

On a slightly different note, if ellipticity is allowed for based on a and b being radii and the neutron star has a high rotation, the fraction comes out high and pushes the 2/5 (0.4) factor for a solid sphere up to ~0.7 which is higher than the disk or cylinder fraction (0.5) which seems at odds.

Also, would it be safe to assume that the interior is rotating slightly faster than the crust and that also due to different densities, it might be worthwhile calculating the seperate angular momentums for the crust and interior and adding them to get a more realistic figure for J?

regards

Steve

[tex]J=I\omega[/tex]

where

[tex]\omega=\frac{v}{r}[/tex]

and for a solid sphere-

[tex]I=\frac{2}{5}mr^{2}[/tex]

for a solid disk or cylinder-

[tex]I=\frac{1}{2}mr^{2}[/tex]

for a ring or loop-

[tex]I=mr^{2}[/tex]

where J is the angular momentum, I is the inertia, v is the velocity of the outer edge, m is the mass and r is the radius to the outer edge of the object.

In some cases, I've seen the equation for the solid sphere apply to oblate spheroids but I've also seen ellipticity allowed for. In most cases, this is expressed as the following-

[tex]\epsilon=\frac{a-b}{a}[/tex]

where epsilon is the factor of ellipticity and a is the long radius and b the short radius of the spheroid. I've seen it expressed as the following in equations for oblate spheroids-

[tex]I=\frac{2}{5}mr^{2}(1+\epsilon)[/tex]

in some cases, a and b are expressed as inertias as if the inertia had been calculated for each radius as solid spheres and then plugged into the ellipticity equation in order to get a ratio of some kind.

On a slightly different note, if ellipticity is allowed for based on a and b being radii and the neutron star has a high rotation, the fraction comes out high and pushes the 2/5 (0.4) factor for a solid sphere up to ~0.7 which is higher than the disk or cylinder fraction (0.5) which seems at odds.

Also, would it be safe to assume that the interior is rotating slightly faster than the crust and that also due to different densities, it might be worthwhile calculating the seperate angular momentums for the crust and interior and adding them to get a more realistic figure for J?

regards

Steve

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