For constant $v_m$, $M_r$ obeys the <linear transport equation>
$$
\partial_tM_r+v_m\partial_rM_r=0.
$$
The <method of characteristics> therefore gives
$$
\boxed{M_r(r,t)=M_r(r-v_mt,0)}.
$$
The localized profile translates inward without changing shape: material initially concentrated near $r_0$ reaches the central star after
$$
\boxed{t_{\rm acc}\simeq\frac{r_0}{|v_m|}}.
$$

Each annulus loses <specific angular momentum> as its characteristic moves to smaller $r$. The negative <magnetic torque> transmits that angular momentum along the large-scale field to the anchored interstellar medium. Thus the disk's angular momentum decreases even before mass crosses $r=0$; angular momentum is conserved only after the field and its anchoring medium are included.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-321/1/b/ii/solution