Aps/MaclaurinSpheroidFreeFall

Free-Fall Collapse of an Homogeneous Spheroid

Free-Fall Collapse of an Homogeneous Spheroid

"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, $(\varpi ,\phi ,z)$, and to simplify the individual equations as described in our accompanying discussion. The resulting set of simplified governing relations is …

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

${\hat{e}}_{\varpi }[\frac{d\dot{\varpi }}{dt}-\frac{j^2}{{\varpi }^3}]+{\hat{e}}_z\left[\frac{d\dot{z}}{dt}\right]=-{\hat{e}}_{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \varpi }+\frac{\partial \mathrm{\Phi }}{\partial \varpi }]-{\hat{e}}_z[\frac{1}{\rho }\frac{\partial P}{\partial z}+\frac{\partial \mathrm{\Phi }}{\partial z}]$
where, the specific angular momentum,     $j(\varpi ,z)\equiv {\varpi }^2\dot{\phi }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}(\mathrm{i}\mathrm{.}\mathrm{e}\mathrm{.},\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e})$

Adiabatic Form of the
First Law of Thermodynamics

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

Poisson Equation

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

Here, our specific interest is in modeling the free-fall collapse of a uniform-density spheroid. This study is closely tied to our separate discussion of the free-fall collapse of uniform-density spheres. For example, by definition, an element of fluid is in "free fall" if its motion in a gravitational field is unimpeded by pressure gradients. The most straightforward way to illustrate how such a system evolves is to set $P=0$ in all of the governing equations. In doing this, the equation formulated by the first law of thermodynamics becomes irrelevant; and the two components of the Euler equation become,

${\widehat{\mathbf{𝐞}}}_{\varpi }$:	$\frac{d\dot{\varpi }}{dt}-\frac{j^2}{{\varpi }^3}$	$=$	$-\frac{\partial \mathrm{\Phi }}{\partial \varpi },$
${\widehat{\mathbf{𝐞}}}_z$:	$\frac{d\dot{z}}{dt}$	$=$	$-\frac{\partial \mathrm{\Phi }}{\partial z}.$

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