Von Zeipel theorem

In astrophysics, the von Zeipel theorem states that the radiative flux $F$ in a uniformly rotating star is proportional to the local effective gravity $g_\textrm{eff}$. Specifically,

$F=-\frac{L(P)}{4\pi G M_{*}(P)}g_\textrm{eff}$

where the luminosity $L$ and mass $M_{*}$ are evaluated on a surface of constant pressure $P$. The effective temperature $T_\textrm{eff}$ can then be found at a given colatitude $\theta$ from the local effective gravity

$T_{\textrm{eff}}(\theta)\sim g_{\textrm{eff}}^{1/4}(\theta)$.^[1][2]

The theorem is named after Swedish astronomer Edvard Hugo von Zeipel.

According to the theory of rotating stars,^[3] if the rotational velocity of a star depends only on the radius, it cannot simultaneously be in thermal and hydrostatic equilibrium. This is called the von Zeipel paradox. The paradox is resolved, however, if the rotational velocity also depends on height or there is a meridional circulation. A similar situation may be arisen in accretion disks.^[4]

References

1. von Zeipel, Edvard Hugo (1924). "The radiative equilibrium of a rotating system of gaseous masses". Monthly Notices of the Royal Astronomical Society 84: 665–719. Bibcode:1924MNRAS..84..665V. doi:10.1093/mnras/84.9.665.
2. Maeder, André (1999). "Stellar evolution with rotation IV: von Zeipel's theorem and anistropic losses of mass and angular momentum". Astronomy and Astrophysics 347: 185–193. Bibcode:1999A&A...347..185M.
3. Tassoul, J.-L. (1978). Theory of Rotating Stars. Princeton: Princeton Univ. Press.
4. Kley, W.; Lin, D. N. C. (1998). "Two Dimensional Viscous Accretion Disk Models. I. On Meridional Circulations In Radiative Regions". The Astrophysical Journal 397: 600–612. Bibcode:1992ApJ...397..600K. doi:10.1086/171818.