Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2

With constant isothermal sound speed $a_T$, the equation of state is $p=a_T^2\rho$. Spherical hydrostatic equilibrium and mass continuity requireTry $\rho =A/r^2$. With no central point mass, $M(r)=4\pi Ar$, and hydrostatic balance becomes $2a_T^{2}A=4\pi G{A}^2$. The positive-density branch is thereforeThe constant external pressure fixes the truncation radius $r_s$ through $p(r_s)=p_0$, givingThis is the pressure-truncated singular isothermal gas sphere, one hydrostatic solution, not a claim that every isothermal cloud has a singular centre. Its central density diverges, but its enclosed mass tends to zero as $r\to 0$.

For inside-out collapse of a singular isothermal sphere, the expansion front is at $r_f=a_{T}t$. The original mass swept by this front isThus the only time-independent accretion scale is $a_T^3/G$. The front-sweeping estimate has coefficient two, but gas takes time to fall from the front to the object; it is not an exact central accretion coefficient. Writefor fixed mean particle mass $\mu m_H$. Determining the dimensionless coefficient $m_0$ requires the collapse solution; it is not fixed by front propagation alone and is not needed for the requested temperature scaling.^[ref]

Well inside the front, where the point mass dominates and infall is approximately pressureless and locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. Conserving the radial mass flux gives the free-fall density profile around a growing point mass:The inner slope is $-3/2$, versus $-2$ in the undisturbed outer cloud. For constant ${\dot{M}}_{\star }$ and negligible initial object mass, $M_{\star }\propto t$ and the inner normalization varies as $t^{-1/2}$. These are inner asymptotes, not an assertion of an abrupt exact power-law join at the front.

At distance $r$, one grain intercepts power through its projected area and emits through its full surface:The blackbody dust sublimation radius follows by setting $T_d=T_{\mathrm{sub}}$:Here the supplied luminosity is $3.8\times 10^{37}\mathrm{W}$. The grain radius cancels because both cross-sections scale as $a^2$. The temperature exterior to this boundary scales as $T_d\propto r^{-1/2}$. This estimate assumes isotropic illumination, no shielding, and unit absorption/emission efficiency as specified; real grain emissivities can change it. The numerical evaluation uses the external conversion values ${\sigma }_{\mathrm{SB}}\simeq 5.6704\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ and $1\mathrm{pc}\simeq 3.0857\times 10^{16}\mathrm{m}$.^[ref]

The dust-cooled equilibrium is thermally stable under the stated constant irradiation, with fixed dust properties and dust-to-gas ratio. At a fixed position, absorbed power per grain is independent of its temperature, while emission is $4\pi a^2{\sigma }_{\mathrm{SB}}{T}^4$. A positive temperature perturbation increases cooling, causing a restoring energy loss; a negative perturbation reduces cooling, allowing reheating.

For heat capacity $C>0$ of a grain together with its thermally coupled gas,so the perturbation decays. The thermal stability of externally heated blackbody dust relies on remaining below sublimation with the grain population intact; destruction, variable illumination or finite dust-gas equilibration can change the response. An isobaric parcel with a fixed grain-to-gas ratio has the same restoring sign when heating and cooling are compared per particle.

First specify the perturbation: take fixed gas density, fixed ionic abundance and the same temperature-independent heating, so thermal stability requires $d\mathcal{C}/dT>0$ at an equilibrium. Let $\theta =\Delta E/k_B$, $n_e$ be the collider density and $n_A=n_1+n_2$ the atomic density. A two-level optically thin balance isThe critical density of an atomic transition is approximately $A_{21}/q_{21}$. At low density, each rare collisional excitation is followed by an escaping photon, soThe isochoric thermal stability of a collisionally excited line therefore givesAt $T=2\theta$ the linear restoring coefficient vanishes; the cooling maximum is marginal to first order and is not robust two-sided stability.

At high density, collisional de-excitation competes successfully with radiative decay, and collisions establish local thermodynamic equilibrium level populations. Detailed balance gives $n_2/n_1=g{e}^{-\theta /T}$, where $g=g_2/g_1$. The cooling is thenThus the high-density fixed-density branch is stable for finite $T$, becoming weakly restoring as the upper fraction saturates at high temperature. It is incorrect to retain the low-density factor $T^{-1/2}$ in the radiative cooling law once populations are thermalized.^[ref]

The isobaric thermal stability of a two-level coolant is a distinct question. If the intended heating is a fixed rate per particle, then $n\propto T^{-1}$ and stability tests the cooling per particle. The low-density boundary becomes $T<2\theta /3$; the high-density cooling per particle remains proportional to $f_u$, so remains stable. If instead heating is constant per volume, the isobaric low-density boundary is $T<2\theta /5$, and the high-density stability condition is $\theta (1-f_u)/T>1$. The source gives no pressure constraint or explicit heating normalization: the boxed $2\theta$ result is the fixed-density answer, with these conditional alternatives stated to avoid asserting a universal boundary.