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2.b Virial theorem applied to free-fall, pressureless, spherical collapse.

The main result of the previous section is that a spherical, homogeneous cloud contracting under free-fall do it so homologously, with constant density until collapsing in a singalarity in \(t_{ff}\). Using this, we examine how the virial theorem in [1] applies. First, the moment of inertia defined respect to the center of the sphere is \begin{equation}I=\frac{3 M R^2}{5}~~.\end{equation} The double derivative respect to time of this is \begin{equation}\frac{1}{2}\frac{d^2I}{dt^2}=\frac{3 M}{5}\left(\dot{R}^2+R\ddot{R}\right)=\frac{3 M\dot{R}^2}{5}-\frac{3 G M^2}{5R}~~,\label{eq-vir}\end{equation} where we have used that \(\ddot{R}=-GM/R^2\). Clearly, the last term is what we call \(W\) in [1]. We check that the first term correspond to \(2 E_k\): \begin{align} \int_V\rho u^2 dV&=4\pi\rho(t)\int_0^{R(t)}\left(\frac{r\dot{R}(t)}{R(t)}\right)^2r^2dr\quad&\text{Because of homology in the collapsing profile.}\\ &=4\pi\rho(t)\frac{R^3}{5}\dot{R}^2=\frac{3M\dot{R}^2}{5}~~. \end{align} Note that it was not necessary to use the explicit \(R(t)\) solution.

As determined in [1], the complete solution of the spherical collapse is given by \(R(t)=R_0\cos^2\theta(t)\) with \begin{equation}\frac{2 \theta}{\pi}+\frac{\sin(2\theta)}{\pi}=\frac{t}{t_{ff}}~~,\label{eq-tt}\end{equation} where \(t_{ff}=\sqrt{\frac{3\pi}{32 G \rho_0}}\). Note that \(\theta(t)\) grows from \(0\) to \(\pi/2\) with \(t\) going from \(0\) to \(t_{ff}\). Equation \eqref{eq-tt} implies that \begin{align} \dot{R}&=-R_0\sin(\theta)\dot{\theta}\\ &=-\frac{\pi R_0\sin(2\theta)}{2t_{ff}(1+\cos(2\theta))}=-\frac{\pi R_0}{2t_{ff}}\tan(\theta),\\ \label{eq-vel}\end{align} which grows to infinity as \(t\rightarrow t_{ff}\) [2].

The last Equation allows us to deduce energy conservation very explicitly $$ E_k+W=\frac{3M}{5}\left(\frac{\dot{R}^2}{2}-\frac{GM}{R}\right)= \frac{3M}{5}\left(\frac{GM}{R_0}(\tan^2\theta-\sec^2\theta)\right)=-\frac{3GM^2}{5R_0}~~.$$ This derivation also gives us the right hand side of Equation \eqref{eq-vir} \begin{equation}\frac{1}{2}\frac{d^2I}{dt^2}=2 E_k+W=\frac{3GM^2}{5R_0}(\tan^2\theta-1)~~.\end{equation} Therefore, in this collapsing cloud the quantity \(2E_k+W\) goes from zero to infinity.

However, note that the quotient \begin{equation} \alpha_{\rm vir}=\frac{2E_k}{|W|}=\frac{\frac{6GM^2\tan^2\theta}{5R_0}}{\frac{3GM^2\sec^2\theta}{5R_0}}=2\sin^2\theta\label{eq-alfa} \end{equation} is bounded and steadily grows from \(0\) to a maximum of \(2\).

Confusing line profiles: observational properties of the system

What would look the line of a free-falling cloud with no support? What would be the consequences of confusing the line width due to free-fall collapse with that due to to turbulence? Figure 1 shows a \( x {\rm -} y \) cross section of the collapsing sphere of center \(O\). Because of the homology of the velocity field, we see that the surfaces inside the sphere which have the same velocity along the LOS are the parallel circular cross sections perpendicular to the LOS. In Figure 1, one generic of these cross section is represented by a black line crossing point \(J\).

Figure 1: Cross section of the collapsing sphere, showing the surface of equal radial velocity for an homologous velocity field. The three-dimensional model has rotational symmetry around axis \(y\).

Now we may assume that the emissivity in the sphere is given by \(j_\nu=j\phi(\nu-\nu_0)\), where \(\phi_\nu\) is a line profile centered in a frequency \(\nu_0\) and \(j\) is the frequency (or velocity) integrated emmisivity. The intensity in a LOS associated with an impact parameter of \(b\) respect to the center of the sphere, under optically thin conditions, is given by \begin{equation} I_\nu=\int_{-\sqrt{R^2-b^2}}^{\sqrt{R^2-b^2}}j\phi(\nu-\nu_0(s))ds~~,\label{eq-I} \end{equation} where \(\nu_0(s)\) is the central frequency of the line profile, which depends on the location of the cloud because the cloud is moving. That is, the velocity at the position determined by \(b\) and \(s\) determined the Doppler displacement of the central frequency of the line profile. In Equation \eqref{eq-I}, if the velocity field is such that the function \(\nu_0:s\rightarrow\nu_0(s)\) is injective, we can use the change of variable theorem and change the variable from \(s\) to \(\nu_0\), that is, \begin{equation} I_\nu=\int_{\nu_0(-\sqrt{R^2-b^2})}^{\nu_0(\sqrt{R^2-b^2})} j \phi(\nu-\eta)\left|\frac{ds}{d\nu_0}\right|_{\nu_0=\eta}d\eta~~.\label{eq-cambioVarI} \end{equation} In Figure 1, the observer is in \(y=-\infty\), and we assume \(s\) grows with \(y\). Therefore, so for a contracting cloud, \(\nu_0(s)\) grows with \(s\) from redshifted frequencies to blueshifted frequencies and \(ds/d\nu_0>0\). Specifically, for an homologous collapse — as the free-free collapse of a spherical, homogeneous cloud \[\nu_0(s)=\left(1+\frac{s}{R}\frac{u(R)}{c}\right)\bar{\nu}_0~~,\] where \(\bar{\nu}_0\) is the rest frequency and \(u(R)\) is the contraction speed at the border of the cloud. Note that this expression is independent of \(b\), as remarked in Figure 1. Note also that under the non-relativistic approximation we are not considering possible changes in the velocity field during the light crossing time. The previous relation implies that \begin{equation} \frac{ds}{d\nu_0}=\frac{c R}{u(R) \bar{\nu}_0}~~.\label{eq-dnuds} \end{equation}

In the following, we will further assume that \[\phi(\nu)=\delta(\nu-\nu_0)~~,\] that is, the line profile is infinitely narrow. This assumption is sometimes called large velocity gradient approximation (LVG). Under this approximation and using \eqref{eq-dnuds} on Equation \eqref{eq-cambioVarI} we obtain \begin{equation} I_\nu= \begin{cases} \frac{jcR}{u(R)\bar{\nu}_0}&\nu_0(-\sqrt{R^2-b^2})\le\nu\le\nu_0(\sqrt{R^2-b^2})\\ 0&\text{ otherwise.}\\ \end{cases} \end{equation} The condition on \(\nu\) in the previous equation is equivalent to \[b\le R\sqrt{1-\left(\frac{\nu}{\bar{\nu}_0}-1\right)^2\left(\frac{c}{u(R)}\right)^2}\] Therefore, the monochromatic flux density from the cloud — assuming it is at a very large distance \(d\) — is given by \begin{equation} F_\nu=\frac{jcR}{u(R)\bar{\nu}_0}\frac{\pi R^2}{d^2}\left( 1-\left( \frac{\nu}{\bar{\nu}_0}-1\right)^2\left(\frac{c}{u(R)}\right)^2\right)~~,\label{eq-F} \end{equation} for all \(\nu\) for which \(F_\nu\ge0\), otherwise the flux is zero. Note that, consistently, \(\int_0^\infty F_\nu d\nu=(4\pi j)\frac{4\pi R^3/3}{4\pi d^2}\) and that this quantity is independent of \(u(R)\).

It is common in radioastronomy not to express line intensities or flux densities as function of the frequency \(\nu\), but as function of the velocity. That is, relating frequency and velocity using the linear Doppler relation \(\nu=\left(1-v/c\right)\bar{\nu}_0\) or \(v=(\nu/\bar{\nu}_0-1)c\). At the same time, it is common not to express these quantites as distributions over the velocity domain, but to leave them with the units as distributions in the frequency domain. That is, flux density is expressed (or plotted) as a function of the velocity (the \(V_{\rm LSR}\), for example) but in Jy, which is the power per unit area and per unit frequency, not velocity. Following with Equation \eqref{eq-F}, translating to velocity instead of frequency \begin{equation} F_\nu= \Omega_s\frac{j c R}{\bar{\nu}_0 u(R) }\left(1-\left(\frac{v}{u(R)}\right)^2\right)~~,\label{eq-Fv} \end{equation} for all \(|v|\le u(R)\) and where \(\Omega_s=\pi R^2/d^2\). Note that the flux density becomes higher if \(u(R)\) decreases, that is, the line becomes narrower and more "concentrated." While this profile should be recognized as not arising from turbulence, a perturbation from the theory could make the line to resemble something like a Gaussian profile (Figure 2).

Figure 2: In blue the model of line flux density as described by Equation \eqref{eq-Fv}. In black the same line convolved with a Gaussian of width of 10% the width of the original line, and with added Gaussian noise at the 10% level.

Assume a confused astronomer, like ourselves, mistake the line profile of the collapsing cloud with a Gaussian arising from turbulence. The FWHM of the line in Equation \eqref{eq-Fv} is simply \(\sqrt{2}u(R)\). The \(\alpha_{\rm vir}\) defined in here (Eq. [7]) gives then \begin{equation} \alpha_{\rm vir}=\frac{M_{\rm vir}}{M}=\frac{5}{8\ln(2)}\frac{R (\sqrt{2}u(R))^2}{G M} \end{equation} where \(R=R_0\cos^2(\theta)\) and \(|u(R)|=\pi R_0\tan(\theta)/2t_{ff}\) (ref. Equations 5, 6, and 7). Therefore, \begin{equation} \alpha_{\rm vir}=\frac{5}{8\ln(2)}\frac{R_0 \cos^2(\theta) 2 \pi^2 R_0^2\tan^2(\theta)}{4 t_{ff}^2 G M}~~. \end{equation} But \(t^2_{ff}=(3\pi/32G\rho_0)\), hence, \begin{equation} \alpha_{\rm vir}=\frac{5}{16\ln(2)}\frac{\pi^2 R_0^3 \sin^2(\theta)}{t_{ff}^2 G M} = \frac{5}{16\ln(2)}\frac{\pi^2 R_0^3 \sin^2(\theta) 32 G M }{ (4\pi/3) R_0^3 3 \pi G M} =\frac{5}{\ln(4)}\sin^2(\theta)~~, \end{equation} which reaches a maximum of \(\approx3.6\) (instead of \(2\) as in the first section).