= Solution
In the final <Schwarzschild spacetime>, an outgoing ray with $r>2M_1$ escapes to <future null infinity>, whereas one with $r<2M_1$ cannot do so and ends at the <Schwarzschild singularity>. Their boundary is the outgoing <null geodesic> $r=2M_1$. Since the final region continues indefinitely, the boundary of the <causal past> of <future null infinity> there is exactly that <Schwarzschild event horizon>. Equivalently, integrating the outgoing equation backward from this final boundary selects the complete <event horizon>. Thus
$$
\boxed{r_H(v)=2M_1\qquad(v>v_0).}
$$
The original <Penrose diagram> below shows the right-hand exterior and black-hole interior appropriate to ingoing coordinates. The green band is the ingoing accretion interval; the dashed blue curve is the spherical <apparent horizon> $r=2m(v)$. Outgoing <null geodesics> escaping to <future null infinity> lie to the right of the red <event horizon>; those to its left terminate on the future spacelike singularity. In the final stationary region the two horizons coincide. The red <event horizon> already lies outside $r=2M_0$ before the ingoing matter arrives, because its position is fixed by the future escape criterion.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-54-vaidya-penrose.png]
{title=Penrose diagram of accretion onto a pre-existing Schwarzschild black hole}
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