Aps/MaclaurinSpheroidFreeFall

Free-Fall Collapse of an Homogeneous Spheroid

Free-Fall Collapse of an Homogeneous Spheroid

"In the course of researches on the formation of galaxies one meets the following idealized problem. What is the form of the collapse under gravitational forces of a uniformly rotating spheroidal gas cloud? In the special case where initially the gas is absolutely cold and of uniform density within the spheroid, we show that the collapse proceeds through a series of uniform, uniformly rotating spheroids until a disk is formed."

— Drawn from 📚 D. Lynden-Bell (1962, Math. Proc. Cambridge Phil. Soc., Vol. 58, Issue 4, pp. 709 - 711)

Simplified Governing Relations

When studying the dynamical evolution of strictly axisymmetric configurations, it proves useful to write the spatial operators in our overarching set of principal governing equations in terms of cylindrical coordinates and to simplify the individual equations as described in our accompanying discussion. The resulting set of simplified governing relations is as follows:

Equation of Continuity

$\frac{d\rho }{dt}+\frac{\rho }{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \dot{\varpi }\right]+\rho \frac{\partial }{\partial z}\left[\rho \dot{z}\right]=0$

Euler Equation

${\hat{e}}_{\varpi }[\frac{d\dot{\varpi }}{dt}-\varpi {\dot{\phi }}^2]+{\hat{e}}_{\phi }[\frac{d(\varpi \dot{\phi })}{dt}+\dot{\varpi }\dot{\phi }]+{\hat{e}}_z\left[\frac{d\dot{z}}{dt}\right]=-{\hat{e}}_{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \varpi }+\frac{\partial \mathrm{\Phi }}{\partial \varpi }]-{\hat{e}}_z[\frac{1}{\rho }\frac{\partial P}{\partial z}+\frac{\partial \mathrm{\Phi }}{\partial z}]$

Adiabatic Form of the
First Law of Thermodynamics

$\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)=0$

Poisson Equation

$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \frac{\partial \mathrm{\Phi }}{\partial \varpi }\right]+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}=4\pi G\rho .$

Key References

• 📚 D. Lynden-Bell (1962, Math. Proc. Cambridge Phil. Soc., Vol. 58, Issue 4, pp. 709 - 711): On the gravitational collapse of a cold rotating gas cloud
  NOTE … according to the new ADS listing, the authors associated with this paper should be, D. Lynden-Bell & C. T. C. Wall (Charles Terence Clegg "Terry" Wall); however, the archived article, itself, lists Lynden-Bell as the sole author while indicating that the paper was simply being communicated by Wall.

Comment by J. E. Tohline: In §II of this "1964" article, Lynden-Bell references his 1962 article with an incorrect year (Lynden-Bell 1963); within his list of REFERENCES, the year (1962) is correct, but the journal volume is incorrectly identified as 50 (it should be vol. 58).

 

• D. Lynden-Bell (1964), ApJ, 139, 1195 - 1216: On Large-Scale Instabilities during Gravitational Collapse and the Evolution of Shrinking Maclaurin Spheroids
• Classic paper by C. C. Lin, Leon Mestel, and Frank Shu (1965, ApJ, 142, 1431 - 1446) titled, "The Gravitational Collapse of a Uniform Spheroid."

See Also