= Solution
Assume negligible fragment release speed relative to a circular parent <planetesimal>. Its initial speed is $v_b=\sqrt{GM_\star/r_b}$, but <radiation pressure> reduces the grain's gravitational parameter to $(1-\beta)GM_\star$. Its <specific orbital energy> and <specific angular momentum> are
$$
\varepsilon_0=\frac{GM_\star}{r_b}\left(\beta-\frac12\right),\qquad h_0^2=GM_\star r_b.
$$
For $0\leq\beta<1/2$, the resulting <dust orbit released from a circular parent ring> has
$$
\boxed{a_0=\frac{(1-\beta)r_b}{1-2\beta},\qquad e_0=\frac\beta{1-\beta},\qquad q_0=r_b,\qquad Q_0=\frac{r_b}{1-2\beta}.}
$$
Release is at <periapsis>: the inherited tangential speed exceeds the new circular speed. Low-$\beta$ grains remain near the <dust birth ring>; higher bound $\beta$ gives increasingly eccentric orbits extending well outside it. Uniform release longitude produces an approximately axisymmetric <dust halo>, with long residence times near <apoapsis>. At $\beta=1/2$ the conservative release orbit is parabolic; above this <radiation-pressure blowout threshold> it is initially unbound. For $\beta\geq1$, zero or repulsive effective gravity also prevents a bound <ellipse>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-316-dust-release.png]
{title=Dust release orbits and <inspiral time under Poynting–Robertson drag>}
{description=Left: representative orbits released at one point on the birth ring, including the parabolic release limit. Right: the formal averaged lifetime of bound grains as a function of the radiation-pressure coefficient.}
The left panel shows one release longitude; rotating it around the ring fills the halo. At longer times <Poynting–Robertson drag> also carries bound grains inward, so an initially outward-spanning release distribution does not imply an empty inner region in a continuously supplied, collision-free population.
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