Poynting–Robertson drag

Poynting–Robertson drag is the tangential component of stellar radiation force caused by aberration in an orbiting dust grain's frame. It removes orbital energy and angular momentum and drives the grain inward.

For grains of fixed diameter $D$ released instantaneously at $r_0$, the drift law $\dot{r}=-\gamma /(rD)$ integrates to the displayed size-radius map. The time to radius $r<r_0$ is $t_r(D)=D(r_0^2-r^2)/(2\gamma )$. Grains therefore arrive in increasing diameter order. This model of Poynting–Robertson drag assumes nearly circular Kepler orbits, fixed positive $\gamma$, and no grain loss or diameter changes before arrival. At a positive time, the map is monotone in $D$ and allows the initial power-law size distribution to be transformed into a spatial distribution or an arrival-time distribution.

For initial dust mass $m_d$ in a ring at $r_0$ with a fixed-diameter power-law size distribution of index $\alpha <4$, burst transport under inverse-size radial drag maps radius to diameter $D_r=2\gamma t/(r_0^2-r^2)$. Set $T_r=t_r(D_{\max })$ and $\eta =D_{\min }/D_{\max }$. The surface density is the azimuthal average: $2\pi r\Sigma dr=(\pi \rho K/6)D_r^{3-\alpha }d{D}_r$. Since $d{D}_r/dr=2r{D}_r/(r_0^2-r^2)$, mass normalization of a power-law size distribution gives the displayed expression. It is supported where $\eta \le t/T_r\le 1$. It applies at $t>0$ before removal at the radius in question; the initial state is a delta-function ring. A local angular density needs additional phase information.

Let the initial dust mass be $m_d$, let its differential number distribution be $K{D}^{-\alpha }$ with $\alpha <4$ between $D_{\min }$ and $D_{\max }$, and put $\eta =D_{\min }/D_{\max }$. Under burst transport under inverse-size radial drag, complete absorption at a fixed radius occurs at $t=TD/D_{\max }$, with $T=t_r(D_{\max })$. The arriving mass is proportional to $D^{3-\alpha }dD$. Substitute $D=D_{\max }t/T$ and $dD/dt=D_{\max }/T$ and use mass normalization of a power-law size distribution to prove the displayed rate for $\eta T\le t\le T$, with zero rate elsewhere. Its integral is exactly $m_d$. Resonant delays, scattering losses, a finite release duration or grain destruction change this ideal pulse.

Consider a circular circumplanetary orbit well inside the Hill sphere, with dust speed $u^2=G{M}_p/a$ and unit plane normal $\hat{\mathbf{k}}$. During one dust orbital period, take the stellar direction $\hat{\mathbf{R}}$ and flux at the planet's stellar radius $a_p$ as constant. The velocity-dependent Poynting–Robertson drag has relative workThe constant terms have zero work over the closed circular orbit. Since $⟨u_i{u}_j⟩=(u^2/2)({\delta }_{ij}-k_i{k}_j)$,The coefficient is three for coplanar stellar and dust orbits; it is two for a dust plane normal to the instantaneous stellar direction. Averaging over a circular planetary year at fixed dust orbital inclination $I$ gives $3-{\sin }^{2}I/2$. Conservative perturbations must remain small enough for this circular-orbit average.

For coplanar circular dust, constant $A=\beta G{M}_{\star }/c$, fixed stellar radius $a_p$ and initial planetocentric radius $a_0>R_p$, integrating $\dot{a}=-3Aa/a_p^2$ givesCompare the inspiral time under Poynting–Robertson drag for a circular circumstellar orbit, $t_{\mathrm{cs}}=(a_p^2-R_{\star }^2)/(4A)$. Planetary arrival is not automatically faster; in the point-star limit it is faster only for $a_0/R_p<e^{3/4}$. A different constant orientation coefficient replaces three.

With $A=\beta G{M}_{\star }/c$ constant, average the drag work $\dot{\epsilon }=-A(2{\dot{r}}^2+v_{\theta }^2)/r^2$ over a bound elliptic orbit. Using $dt=r^{2}df/h$ gives the integral $\pi (2+3e^2){\mu }^2/h^2$, where $\mu$ is the conservative central parameter. Combining with $\dot{a}=2a^2\dot{\epsilon }/\mu$ givesThis is a weak-drag secular average. It also holds when the conservative parameter is $(1-\beta )G{M}_{\star }$, provided a bound orbit exists; the conservative parameter cancels but the drag amplitude remains $A$. For a circular Kepler orbit, $a^2(t)=a_0^2-4At$.

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.

For the Poynting–Robertson drag invariant $C$ and $A=\beta G{M}_{\star }/c$, the orbit-averaged time to the point-star limit isThis tends to $a_0^2/(4A)$ for a circular orbit and to $4Q_0^{1/2}{q}_0^{3/2}/(5A)$ for a highly eccentric orbit. Stellar radius, sublimation, collisions, and failure of orbital averaging can terminate the evolution earlier.

The orbit-averaged Poynting–Robertson drag equations preserve $C=a(1-e^2)e^{-4/5}$ for $0<e<1$. This relates shrinking semi-major axis to decreasing orbital eccentricity; a circular orbit is treated by its separate limiting solution.