Von Zeipel theorem

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

${\displaystyle F=-{\frac {L(P)}{4\pi GM_{*}(P)}}g_{\text{eff}},}$

where the luminosity ${\displaystyle L}$ and mass ${\displaystyle M_{*}}$ are evaluated on a surface of constant pressure ${\displaystyle P}$. The effective temperature ${\displaystyle T_{\text{eff}}}$ can then be found at a given colatitude ${\displaystyle \theta }$ from the local effective gravity:[1][2]

${\displaystyle T_{\text{eff}}(\theta )\sim g_{\text{eff}}^{1/4}(\theta ).}$

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 arise in accretion disks.[4]

References

1. Zeipel, Edvard Hugo von (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.