= Solution
\b[<mass> and <energy> signs.] The enclosed <mass> is $M(r)=\int_0^r4\pi s^2\rho(s)\,ds$, so $dM/dr=+4\pi r^2\rho$. The negative sign printed in that equation cannot describe positive enclosed <mass>. Use the physical sign and define <gravitational potential energy> by
$$
\Omega=-\int_0^M\frac{GM_r}{r}\,dM_r<0.
$$
\b[Boundary-pressure <virial theorem>.] Multiply <hydrostatic equilibrium> by $4\pi r^3$ and integrate. The left side is $[4\pi r^3p]_0^{r_0}-3\int_0^{r_0}4\pi r^2p\,dr$, while the right side is $\Omega$. The central boundary contribution vanishes when $r^3p\to0$ and the outer <pressure> is $p(r_0)=p_0$. Thus
$$
\boxed{3\int\frac p\rho\,dM+\Omega=4\pi r_0^3p_0=3p_0V}.
$$
A finite external <pressure> is compatible with a gravitationally self-contained cloud, but is not the zero-surface-force notion of isolation. External ram stress or <mass> flow would require additional terms.
\b[<radiation budget of monatomic cloud collapse>.] For an <ideal gas> with fixed $\gamma$, the specific <internal energy> is $u=p/[\rho(\gamma-1)]$. Therefore the <pressure> <integral> is $3(\gamma-1)U$. Cold monatomic <hydrogen> has $\gamma=5/3$ when only translational degrees of freedom matter. With negligible final surface <pressure>, $2U+\Omega=0$, so $E_f=U+\Omega=\Omega/2$. If the initial cloud is sufficiently extended and cold that $E_i\simeq0$, the emitted <energy> is
$$
\boxed{E_{\rm rad}=E_i-E_f\simeq-\frac\Omega2>0}.
$$
Being initially at rest alone does not set its <binding energy> to zero. If the initial total <energy> is appreciable, retain $E_{\rm rad}=E_i-\Omega/2$; a different <equation of state> also changes the half-energy result.
\b[Inside-out <accretion>.] The combination of $a_T$ and $G$ with dimensions of <mass> per time is
$$
\boxed{\dot M=C\frac{a_T^3}{G}},
$$
where <dimensional analysis> leaves the dimensionless coefficient $C$ undetermined. An isothermal gas keeps $a_T$ constant, giving constant supply in the self-similar idealization. A front at radius $a_Tt$ and an isothermal <mass> scale $M(r)\sim a_T^2r/G$ give the same scaling. The front's swept <mass> is not automatically the accreted central <mass>; the coefficient requires a dynamical similarity solution.<Self-Similar Collapse of Isothermal Spheres and Star Formation by Frank H. Shu>[Inside-out similarity model; only the dimensional scaling is required here]{ref}
\b[<constant-radius accretion energy budget>.] If all available <accretion> power is radiated at the fixed radius, $L_{\rm acc}=GM\dot M/r_0$. Neglecting the seed <mass>,
$$
\int L_{\rm acc}\,dt=\int_0^M\frac{GM'}{r_0}\,dM'
=\frac{GM^2}{2r_0}.
$$
This equals $-\Omega/2$ precisely when
$$
\boxed{\Omega=-\frac{GM^2}{r_0}}.
$$
The positive sign in the final printed <energy> condition is inconsistent with the earlier use of negative <gravitational potential energy>. Equivalently its positive right side must denote the binding-energy magnitude $|\Omega|$. A general structural coefficient $\Omega=-qGM^2/r_0$ leaves a mismatch unless $q=1$ under these other idealizations. Radiation trapped in the object, outflows or release over a different radius also change the budget.
\b[Which <temperature>?] With $M=M_s+\dot M t$ and fixed $r_0$, $L_{\rm acc}\propto M$. If this power emerges thermally from area $4\pi r_0^2$, the <accretion-powered effective-temperature scaling> is
$$
\boxed{T_{\rm eff}=\left(\frac{GM\dot M}{4\pi\sigma_{\rm SB}r_0^3}\right)^{1/4}
\ \propto\ t^{1/4}}
$$
when the seed <mass> is negligible. This is the radiating surface's <effective temperature>, inferred from the preceding accretion-<luminosity> calculation. A homologous ideal-gas interior instead has <virial temperature> $T_c\sim\mu m_HGM/(k_Br_0)\propto t$. Thus \b[the surface-effective and interior-virial scalings are $t^{1/4}$ and $t$, respectively], under their stated assumptions. Inferring a detailed interior <temperature> profile would additionally require a structural and transport model.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2013/paper-1-accretion-energy-temperature.png]
{title=Original constant-radius accretion energy budget and distinct surface-effective and virial temperature scalings}
Back to article page
