= Solution
For a spherical grain of density $\rho$, the <radiation-pressure coefficient> is
$$
\beta=\frac{3L_\star\langle Q_{\rm pr}\rangle}{8\pi cGM_\star\rho D}.
$$
The $r^{-2}$ dependence of <radiation pressure> cancels that of <stellar gravity>. Here <radiation-pressure efficiency> includes absorption and the momentum-transfer part of scattering, averaged over the stellar spectrum. In the <geometrical optics> regime, $D$ exceeds the important stellar wavelengths, $Q_{\rm pr}$ is of order unity, and \b[$\beta\propto D^{-1}$]: cross-sectional area grows as $D^2$, whereas mass grows as $D^3$. Around the stellar wavelength, <Mie scattering> can produce a broad maximum and material-dependent structure. In the <Rayleigh scattering> regime, absorption gives $Q_{\rm abs}\propto D$, while scattering gives $Q_{\rm sca}\propto D^4$. Thus \b[small absorbing grains approach an approximately constant $\beta$, whereas nearly transparent grains have $\beta\propto D^3$]. A universal fall to zero at small $D$ is therefore inappropriate without specifying the optical properties. The sketch shows both possible small-grain limits; its vertical normalization is illustrative.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-316-dust.png]
{title=Radiation-to-gravity ratios for absorbing and scattering grains, and the signed angular lag over five dust orbits for beta equal to 0.01 and 0.04}
Assume release with negligible velocity relative to the <planet>, negligible planetary gravity after release, and a constant <radiation-pressure coefficient>. Put $\mu=GM_\star$ and $\mu_d=(1-\beta)\mu$. The inherited speed and <specific angular momentum> are $v_p=\sqrt{\mu/a_p}$ and $h=a_pv_p$. The new <specific orbital energy> is
$$
\varepsilon_d=\frac{v_p^2}{2}-\frac{\mu_d}{a_p}=\frac{\mu}{a_p}\left(\beta-\frac12\right).
$$
For $0\leq\beta<1/2$, the <dust orbit released from a circular parent ring> is an ellipse. Using $\varepsilon_d=-\mu_d/(2a_d)$ and $h^2=\mu_da_d(1-e_d^2)$ gives
$$
\boxed{\frac{a_d}{a_p}=\frac{1-\beta}{1-2\beta},\qquad e_d=\frac{\beta}{1-\beta}}.
$$
For $\beta>0$, release is at <periapsis>, since $a_d(1-e_d)=a_p$. At the <radiation-pressure blowout threshold>, $\beta=1/2$, the orbit is parabolic. For $1/2<\beta<1$ it is hyperbolic: the signed <semimajor axis> is negative and $e_d>1$. For $\beta\geq1$ the central force ceases to be attractive, so the elliptical interpretation and the positive-eccentricity formula cannot be extended unchanged.
With $\theta_d$ measured from release, it is the <true anomaly>. The <polar equation of a Kepler orbit> and conservation of <specific angular momentum> give
$$
r=\frac{h^2/\mu_d}{1+e_d\cos\theta_d}=\frac{a_p}{1-\beta+\beta\cos\theta_d},\qquad
\boxed{\dot\theta_d=\frac h{r^2}=n_p(1-\beta+\beta\cos\theta_d)^2}.
$$
Define the signed <dust-tail angular lag> by $\Delta\theta=\theta_d-\theta_p$, with $\theta_p=n_pt$ and both angles unwrapped. For fixed $\theta_d$, expansion to first order in $\beta$ gives
$$
n_pt=\int_0^{\theta_d}\frac{d\psi}{(1-\beta+\beta\cos\psi)^2}
=\theta_d+2\beta(\theta_d-\sin\theta_d)+O(\beta^2\theta_d),
$$
so \b[$\Delta\theta=2\beta(\sin\theta_d-\theta_d)+O(\beta^2\theta_d)$]. The dust trails the <planet>, hence the signed lag is negative. Its slope is $-4\beta\sin^2(\theta_d/2)$; it decreases monotonically, has horizontal tangents at successive release-direction passages, and has a sinusoidal ripple about $-2\beta\theta_d$. At the end of orbit $j$, $\theta_d=2\pi j$ and $\Delta\theta\simeq-4\pi\beta j$. At five orbits this is \b[$-0.2\pi$ for $\beta=0.01$ and $-0.8\pi$ for $\beta=0.04$]. The upper-$\beta$ sketch remains a first-order approximation; its accumulated error is $O(\beta^2\theta_d)$.
For the subsequent positive-distance statements, set $s=-\Delta\theta=\theta_p-\theta_d\geq0$. The <mean motion> of the grain is
$$
\frac{n_d}{n_p}=\sqrt{\frac{1-\beta}{(a_d/a_p)^3}}=\frac{(1-2\beta)^{3/2}}{1-\beta}=1-2\beta+O(\beta^2).
$$
A full relative wrap, $s=2\pi$, therefore takes $t_{\rm wrap}\simeq\pi/(\beta n_p)$ after averaging over the orbital ripple. During this time <Poynting–Robertson drag> changes the <semimajor axis> by
$$
\frac{|a_d(t_{\rm wrap})-a_d(0)|}{a_d(0)}
\simeq\frac{2\beta\mu}{ca_p^2}\frac{\pi}{\beta n_p}
=\boxed{\frac{2\pi v_p}{c}}.
$$
The signed change is negative. This estimate treats $a_d\simeq a_p$ as fixed in the rate and computes the drag accumulated over the radiation-pressure wrap time. Since $v_p/c\ll1$, drag causes little migration over that time. For an arbitrarily tiny $\beta$, however, this small migration can itself alter the relative phase appreciably: neglecting that feedback on the wrap time additionally requires $v_p/c\ll\beta$.
Rapid <dust sublimation> is a plausible sink near a hot <planet>: a grain can evaporate long before it completes a relative wrap. Destructive <collisions> can also remove visible grains. <Poynting–Robertson drag> alone, with the estimate above, does not explain immediate removal from a short tail. If the destruction time is $t_{\rm loss}$, a short tail requires roughly $2\beta n_pt_{\rm loss}\ll2\pi$.
For a <steady dust-tail continuity equation>, assume a constant injection rate $\dot N$, grains with the same fixed $\beta$, negligible release-speed dispersion, and negligible destruction within the segment being calculated. A finite destruction lifetime can terminate that segment, or multiply its density by a survival probability. To first order,
$$
\dot s=n_p-\dot\theta_d=4\beta n_p\sin^2(\theta_d/2)+O(\beta^2),\qquad
n(s)=\frac{\dot N}{\dot s}.
$$
To obtain the explicit trigonometric profile, make the additional <secular phase approximation for a dust tail>, $s\simeq2\beta\theta_d$, discarding the bounded $-2\beta\sin\theta_d$ term in the angle-to-age map while retaining the periodic instantaneous drift speed. Then
$$
\boxed{n(s)\simeq\frac{\dot N}{4\beta n_p}\frac1{\sin^2(s/(4\beta))}}.
$$
Since the square is even, the same written profile applies to the signed $\Delta\theta$ if number intervals are interpreted with positive orientation. The result is an approximate phase-substitution profile, not a uniformly valid consequence of $\beta\ll1$. The consistent first-order description is instead parametric:
$$
\boxed{s=2\beta(\theta_d-\sin\theta_d),\qquad
n(s)=\frac{\dot N}{4\beta n_p\sin^2(\theta_d/2)}}.
$$
In particular, immediately after release, $s\sim\beta\theta_d^3/3$ and $n\propto s^{-2/3}$, whereas the phase-substitution expression would give $s^{-2}$. Nor does $s\ll2\pi$ by itself justify neglecting the orbital ripple. The formal enhancements where $\dot s=0$ are <dust-tail caustics>; a spread of <radiation-pressure coefficients>, finite release velocities, and destruction smooth them. With constant lifetime $\tau$, the parametric profile acquires $\exp[-t(\theta_d)/\tau]$. These qualifications state precisely the extra assumptions behind the explicit profile.
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