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

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

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 continuity equation and the Poisson equation remain unchanged; the equation formulated by the first law of thermodynamics becomes irrelevant; and the two components of the Euler equation become,

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

Gravitational Potential

Here, our specific interest is in modeling the free-fall collapse of a uniform-density spheroid. Ignoring, for the moment, the time-dependent nature of this problem, we appreciate from a separate, detailed derivation that the gravitational potential inside (or on the surface) of an homogeneous, triaxial ellipsoid with semi-axes $(x,y,z)=(a_1,a_2,a_3)$ is given, to within an arbitrary additive constant, by the expression,

$\mathrm{\Phi }(\overrightarrow{x})=\pi G\rho [A_1{x}^2+A_2{y}^2+A_3{z}^2],$

where the three, spatially independent coefficients, $A_1,A_2,$ and $A_3$ are functions of the chosen lengths of the three semi-axes. When deriving mathematical expressions for each of the three $A_i$ coefficients, in our accompanying discussion we have found it useful to initially attach a subscript, $(\ell ,m,\mathrm{o}\mathrm{r}s)$ — indicating whether the coefficient is associated with the (largest, medium-length, or smallest) semi-axis — before specifying how, for a given problem, $(\ell ,m,s)$ are appropriately associated with the three $(x,y,z)$ coordinate axes.

Oblate Spheroids

For example, for an oblate-spheroidal mass distribution, by definition the "largest" and "medium-length" semi-axes are equal to one another. Hence, $a_{\ell }=a_m$ and, according to our associated derivation,

$A_{\ell }=A_m$

$=$

$-\frac{(1-e^2)}{e^2}+\frac{(1-e^2{)}^{1/2}}{e^3}\left[{\sin }^{-1}e\right],$

where, $e\equiv (1-a_s^2/a_{\ell }^2{)}^{1/2}$; and,

$A_s$

$=$

$\frac{2}{e^2}-\frac{2(1-e^2{)}^{1/2}}{e^3}\left[{\sin }^{-1}e\right].$

Conventionally, the $z$-axis is aligned with the symmetry (in this case, shortest) axis of the mass distribution, so we set $A_1=A_2=A_{\ell }$, and $A_3=A_s$. Therefore — see also our parallel discussion — we appreciate that, for oblate-spheroidal mass distributions,

$\mathrm{\Phi }(\overrightarrow{x})=\pi G\rho [A_{\ell }{\varpi }^2+A_s{z}^2].$

These same coefficient expressions may also be found in, for example:  Chapter 3, Eq. (36) of [EFE]; §4.5, Eqs. (48) & (49) of [T78]; and the first column of Table 2-1 (p. 57) of [BT87].

Prolate Spheroids

For a prolate-spheroidal mass distribution, by definition the "smallest" and "medium-length" semi-axes are equal to one another. Hence, $a_m=a_s$ and, according to our associated derivation,

$A_{\ell }$

$=$

$\frac{(1-e^2)}{e^3}\cdot \ln \left[\frac{1+e}{1-e}\right]-\frac{2(1-e^2)}{e^2},$

where, as above, $e\equiv (1-a_s^2/a_{\ell }^2{)}^{1/2}$; and,

$A_s=A_m$

$=$

$\frac{1}{e^2}-\frac{(1-e^2)}{2e^3}\cdot \ln \left(\frac{1+e}{1-e}\right).$

If the symmetry (in this case, longest) axis of this prolate-spheroidal mass distribution is aligned with the $z$-axis of the coordinate system, then we should set $A_1=A_2=A_s$, and $A_3=A_{\ell }$. This means that the expression for the gravitational potential is,

$\mathrm{\Phi }(\overrightarrow{x})=\pi G\rho [A_s{\varpi }^2+A_{\ell }{z}^2].$

These same coefficient expressions may also be found in, for example, the second column of Table 2-1 (p. 57) of [BT87].

NOTE:  If, following [EFE], we instead align the longest (and, in this case, symmetry) axis of the prolate mass distribution with the $x$-axis, then $A_2=A_3=A_s$ and $A_1=A_{\ell }$. This matches the coefficient expressions presented in our parallel discussion of the potential inside and on the surface of a prolate-spheroidal mass distribution.

Consider a Time-Varying Eccentricity

If the eccentricity of an homogeneous spheroid varies with time — that is, if $e\to e(t)$ — while it remains homogeneous, the result will be a potential of the form,

$\mathrm{\Phi }(\overrightarrow{x},t)=A(t){\varpi }^2+C(t)z^2,$

whether the spheroid is oblate or the spheroid is prolate.

Lynden-Bell's (1962) Insight

Motion of a Single Particle

Consider a particle that, at time $t=0$, is at position $({\varpi }_0,{\phi }_0,z_0)$ and is moving about the $z$-axis with velocity, ${\varpi }_0\mathrm{\Omega }\Rightarrow j_0={\varpi }_0^2\mathrm{\Omega }$. Consider furthermore that its acceleration is subject to the force arising from an axisymmetric gravitational potential of the form,

$\mathrm{\Phi }(\overrightarrow{x},t)=A(t){\varpi }^2+C(t)z^2.$

(This is the gravitational potential adopted by 📚 Lynden-Bell (1962) — see his equation (1) — except he employed an opposite sign convention. He would therefore have also employed an opposite sign convention to the gradient of the potential that appears on the RHS of the Euler equation.) In this case, the two components of the Euler equation that govern the particle's motion are,

${\widehat{\mathbf{𝐞}}}_{\varpi }$:	$\ddot{\varpi }-\frac{j_0^2}{{\varpi }^3}$	$=$	$-2A\varpi ,$
${\widehat{\mathbf{𝐞}}}_z$:	$\ddot{z}$	$=$	$-2Cz,$

where we have adopted the familiar shorthand notation, $d\dot{\varpi }/dt\to \ddot{\varpi }$ and $d\dot{z}/dt\to \ddot{z}$. If we divide the first of these relations by ${\varpi }_0$ and the second by $z_0$, then adopt the dimensionless variables, $R\equiv \varpi /{\varpi }_0$ and $Z\equiv z/z_0$, we can write,

$\frac{\ddot{\varpi }}{{\varpi }_0}-\frac{j_0^2}{{\varpi }_0{\varpi }^3}$	$=$	$-2A\left(\frac{\varpi }{{\varpi }_0}\right)$
$\Rightarrow \ddot{R}-\frac{{\mathrm{\Omega }}^2}{R^3}$	$=$	$-2AR,$

and,

$\frac{\ddot{z}}{z_0}$	$=$	$-2C\left(\frac{z}{z_0}\right)$
$\Rightarrow \ddot{Z}$	$=$	$-2CZ.$

In addition, conservation of angular momentum implies that, at any moment in time, the particle's frequency of rotation about the symmetry axis will be,

$\dot{\phi }$

$=$

$\frac{\mathrm{\Omega }}{R^2}.$

These last three expressions appear, respectively, as equations (10), (11), and (9) of 📚 Lynden-Bell (1962).

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