The three principal <black-hole seed> channels occupy different mass ranges.

* A <Population III remnant black-hole seed> forms after a massive metal-free star exhausts its fuel and collapses. Weak line-driven winds preserve more mass than at high metallicity, although pair-instability can leave gaps in the remnant distribution. Typical light seeds are $10$--$10^2M_\odot$.
* A <runaway stellar-collision black-hole seed> forms in a dense young cluster whose core-collapse time is shorter than the lifetimes of its massive stars. Mass segregation and repeated stellar collisions build a very massive star, which collapses to a seed of roughly $10^2$--$10^4M_\odot$.
* A <direct-collapse black-hole seed> forms in a rapidly inflowing atomic-cooling halo where molecular cooling and fragmentation are suppressed. Gas builds a supermassive star or quasistar and leaves a heavy seed of roughly $10^4$--$10^6M_\odot$.

For radiative efficiency $\eta=0.1$, growth at <Eddington ratio> $f_{\rm Edd}$ obeys
$$
M(t)=M_{\rm seed}\exp\!\left[
\frac{1-\eta}{\eta}
\frac{f_{\rm Edd}\Delta t}{t_E}
\right],
$$
where $t_E=0.45\,\mathrm{Gyr}$. Between $z=25$ and GN-z11 at $z=10.6$,
$$
\Delta t=0.430-0.128=0.302\,\mathrm{Gyr},
$$
so the exponent is $6.04f_{\rm Edd}$. Equivalently, the <Salpeter time> is about $0.05\,\mathrm{Gyr}/f_{\rm Edd}$.

Reaching $10^6M_\odot$ requires
$$
f_{\rm Edd}=\frac{\log(10^6M_\odot/M_{\rm seed})}{6.04}.
$$
Representative values are
$$
\begin{array}{c|c|c}
M_{\rm seed}/M_\odot&\text{channel}&f_{\rm Edd}\text{ required}\\ \hline
10^2&\text{Population III remnant}&1.52\\
10^3&\text{runaway cluster}&1.14\\
10^4&\text{runaway or direct collapse}&0.76\\
10^5&\text{direct collapse}&0.38
\end{array}
$$
Continuous Eddington-limited growth supplies only $e^{6.04}\simeq420$, so a $100M_\odot$ seed reaches about $4\times10^4M_\odot$, whereas a seed above roughly $2.4\times10^3M_\odot$ can reach the target at a unit duty cycle. Light seeds require sustained mildly super-Eddington accretion, an earlier start, mergers, or lower effective efficiency. Heavy direct-collapse seeds need only a moderate time-averaged Eddington ratio and are therefore easier to reconcile with the short available time. No channel is ruled out by the mass alone because seed masses, obscuration, duty cycles, super-Eddington episodes, mergers, and the observational mass estimate are uncertain.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 347 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-347/3/a/solution