{{Short description|Astrophysics concept}} In astrophysics, the '''von Zeipel theorem''' states that the radiative flux <math>F</math> in a uniformly rotating star is proportional to the local effective gravity <math>g_\text{eff}</math>. The theorem is named after Swedish astronomer Edvard Hugo von Zeipel.
The theorem is:
: <math>F = -\frac{L(P)}{4\pi G M_*(P)} g_\text{eff},</math>
where the luminosity <math>L</math> and mass <math>M_*</math> are evaluated on a surface of constant pressure <math>P</math>. The effective temperature <math>T_\text{eff}</math> can then be found at a given colatitude <math>\theta</math> from the local effective gravity:<ref>{{cite journal | last = Zeipel | first = Edvard Hugo von | title = The radiative equilibrium of a rotating system of gaseous masses | journal = Monthly Notices of the Royal Astronomical Society | volume = 84 | pages = 665–719 | date = 1924 | issue = 9 |bibcode = 1924MNRAS..84..665V | doi = 10.1093/mnras/84.9.665 | doi-access = free }}</ref><ref>{{cite journal | last = Maeder | first = André | title = Stellar evolution with rotation IV: von Zeipel's theorem and anistropic losses of mass and angular momentum | journal = Astronomy and Astrophysics | volume = 347 | pages = 185–193 | date = 1999 |bibcode = 1999A&A...347..185M }}</ref>
: <math>T_\text{eff}(\theta) \sim g_\text{eff}^{1/4}(\theta).</math>
This relation ignores the effect of convection in the envelope, so it primarily applies to early-type stars.<ref>{{cite journal | title=Gravity-Darkening for Stars with Convective Envelopes | last=Lucy | first=L. B. | journal=Zeitschrift für Astrophysik | volume=65 | page=89 | date=1967 | bibcode=1967ZA.....65...89L }}</ref>
According to the theory of rotating stars,<ref>{{cite book |last1=Tassoul |first1=J.-L. |title=Theory of Rotating Stars |date=1978 |publisher=Princeton: Princeton Univ. Press}}</ref> 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.<ref>{{cite journal |last1=Kley |first1=W. |last2=Lin |first2=D. N. C. |title=Two-Dimensional Viscous Accretion Disk Models. I. On Meridional Circulations In Radiative Regions |journal=The Astrophysical Journal |date=1998 |volume=397 |pages=600–612 |doi=10.1086/171818 |bibcode = 1992ApJ...397..600K |doi-access=free }}</ref> <!--The von Zeipel theorem has been used for the better part of a century to predict the difference in surface gravity, brightness and temperature between a rapidly rotating star's poles and its equator. In 2011, using a technique called interferometry, University of Michigan researchers essentially zoomed in to take close-up pictures and measurements of the winter star Regulus. It is the brightest star in the constellation Leonis, and if it were spinning just a few percent faster, it would fly apart. The astronomers found that the actual difference in temperature between its equator and poles is much less than the old theory predicts. "Our model fitting of interferometry data shows that while the law correctly describes the trend of surface temperature variation, it deviates quantitively," said Xiao Che, a doctoral student in the Department of Astronomy who is first author of a paper on the findings to be published in ''Astrophysical Journal'' on April 20. "It is surprising to me that von Zeipel's law has been adopted in astronomy for such a long time with so little solid observational evidence." It's important to get this number right, says John Monnier, an associate professor in the U-M Department of Astronomy. "In some cases, we found a 5,000-degree Fahrenheit [2800 kelvins] difference between what the theory predicts and what our actual measurements show," Monnier said. "That has a big effect on total luminosity. If we don't take this into account, we get the star's mass and age and total energy output wrong."-->
==References== {{reflist}}
Category:Stellar astronomy Category:Equations of astronomy
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